DUKE UNIVERSITY LIBRARY DURHAM, N. C. Rec’d Digitized by the Internet Archive in 2016 with funding from Duke University Libraries https://archive.org/details/comprehensiveari01 luma A Comprehensive Arithmetic v FOR GRAMMAR, HIGH and COMMERCIAL SCHOOLS REVISED AND EDITED BY JOHN A. LUMAN, A. M. Vice-Principal of Peirce School FOURTH EDITION PHILADELPHIA : Published by PEIRCE SCHOOL Nos. 9 1 7-9 1 9 Chestnut Street I 908 COPYRIGHT, 1 908 BY PEIRCE SCHOOL T. C. DAVIS & SONS. PRINTERS. PH1LA \ p-|q \s =- AT. “CH- J5D 5 II X^I PREFACE I N the preparation of this book, great care and attention have been given to the logical relation of the science, and its adaptation to practical affairs ; to definiteness and exactness in principles and rules ; to brevity and clearness in the presentation of the subject matter ; and to modern progressive methods. The aim has been to present the truths and principles of a practical science in a practical, business-like way, and to develop them in a natural, simple, effective manner, with a variety of well-selected problems of a character suited to students of grammar, high and commercial schools. The features of this book are : i — The logical development of number. ( a ) The properties of number. (<£) The relation of a quantity to a fixed unit of measure. ( 228 Exchange 234 Foreign Exchange 237 Taxes 240 Duties ... 243 Ratio and Proportion 245 Ratio 245 Proportion 247 Compound Proportion 250 Alligation 253 General Average 257 Equation of Accounts 261 Product Method 263 223632 6 CONTENTS PAGE Interest Method 264 Cash Balance 269 Accounts Bearing Interest 278 Savings Fund Accounts 280 Bankruptcy 282 Partnership Settlements 283 Involution and Evolution . 298 Involution 298 Evolution 299 Square Root 300 Similar Figures 301 Cube Root 302 Similar Solids 306 Arithmetical Progression 307 Geometrical Progression 309 Mensuration 311 Right-angled Triangles 311 Surfaces 313 Solids . . 318 Latitude and Longitude 324 Longitude and Time 325 Review Mental Problems 328 General Review Problems . 335 Metric System ... 361 ARITHMETIC DEFINITIONS AND PRINCIPLES 1. Quantity is a limited portion of any natural object; as of time, space, weight, etc., or of any solid or fluid substance. 2. All mathematical operations deal with measures of quantity , and quantity can be measured only by comparing it with some known quantity of the same kind taken as a standard. 3. Quantity can be changed, mathematically, only by increase or diminution. 4. A quantity may be increased in either of two ways: (a) By combining with it one or more quantities of the same kind, greater or less than itself ; this is called Addition, (b) By combining with it any given number of quantities each exactly equal to itself ; this is called Multiplication. 5. A quantity may be diminished in either of two ways : (a) By taking from it a quantity not greater than itself ; this is called Subtraction, (b) By continuing to take from it, as many times as possible, a given quantity less than itself ; this is called Division (because the quantity is separated into a number of equal parts, with or without a remainder). 6 . The two mathematical principles, increase and diminution , thus give rise to the four fundamental operations of arithmetic — Addition, Subtraction, Multi- plication and Division. 7. Arithmetic, as a science, is the science of numbers. As an art, it embraces all known methods of computation by means of figures, all of which are but processes or combinations of adding, subtracting, multiplying and dividing. 8. The Signs of the four cardinal operations are as follows : Of Addition, + plus; of Subtraction, — minus ; of Multiplication, X multiplied by ; of Divi- sion, -r- divided by. 9. The Result in addition is called the sum; in subtraction, the difference or remainder ; in multiplication, the product] and in division, the quotient. 10. The Sign of Equality is =, read equals or equal to. 11. A Unit, represented by the figure 1, signifies one thing of any kind ; as, 1 dollar, 1 yard, 1 day. When not applied to any thing, it represents the abstract idea of unity. 12. A Number is a unit or a group of units, expressed by one or more figures, and considered as one quantity. Numbers may be abstract, as 1, 5, 12 ; or concrete, as 6 feet, 25 barrels. 7 NOTATION AND NUMERATION 13. Notation is the art of writing numbers. For purposes of calculation the Arabic nofation, expressing numbers in figures, is used exclusively; but the Roman notation, a system of representing numbers by combinations of letters, is still used for numbering chapters, dates of imprint and inscriptions, dials, etc. 14. Numeration is the art of reading or naming numbers. 15. Arabic Notation expresses numbers by the following ten characters : 1, one; 2, two; 3, three; 4, four; 5, five; 6, six; 7, seven; 8, eight; 9, nine; 0, naught, cipher or zero. They are written as follows : Any one of these ten characters or figures is called a digit, and a whole number represented by any one of them an integer. The number of units expressed by any one of these figures when standing alone is shown by its name ; but when used in combination with other figures the number of units expressed is indicated by its place, as shown in the table which follows. A figure occupying any one of these places expresses as many times the number of units it names as is indicated by the name of the place it occupies, being in each case ten times as many as it would represent in the next place to the right. Thus the figure 8, with one figure at the right of it — as in 80, 82, or 86 — represents 8 tens, because it occupies the second, or tens place ; with two figures following to the right— as in 800, 805, or 837 — it represents 8 hundreds, because it occupies the third, or hundreds place, etc. The cipher, by itself, represents nothing, and is used merely to fill vacant places. 16. NUMERATION TABLE NAMES OF PLACES FLACES, OR ORDERS P o HQ o a SbO 1 XU Oi ZZ t- 3- = £.2 : c3 > 3 1C* > .33 c : sm o ;hh ; c ==: ; a; .— ; h q p S S2 £ Q p g 2 c3 t 3 3 y y £ S y 3 S 3 3 S 3 S 3 y S 2 y S y / y £ 2 S 2 2 3 / S y 223 2 f 7 y y y 3 £ y £ S £ 27 y S 7 y y 3 2 3 7 S £ y s 2 y y / 7 3 y 2 *2 £ 3 / 2 / 3 y 3 s 7 y 3 y 3 / 2 y y 2 3 y 2 y y / S y 7 2 £ 3 3 y s S y y y 3 £22 £ 3 3 y 3 3~ y £ £ 3 y 3323 / y y y / y £ s' 33 y y 3 3 3 y 3 2 s 3 £ S 3 £ y y / s 2 2 3 y £ S 3 7 s £7773 23 £ 7 y s y z Note. — The above arrangement is recommended for sums of six columns in ten rows. These are known as “six by tens,” and should be dictated as follows (taking the first table as an example) : “Forty-five, sixty-one, sixty-two, twenty-one, forty-nine, forty-five,’’ etc. For beginners, the dictation should occupy about one minute, at the end of which say, “Add.’’ Allow sixty seconds for the adding of a table, and begin the dictation of the next promptly, continuing until the last one has been added, wben results may be called for or read off by the teacher and checked by the student. At first very few students will be able to obtain the correct results within the time limit, but as they become more expert many will obtain them in from twenty to thirty seconds. In adding columns insist upon the student’s naming only the results of combinations, omitting the unnecessary words “and are,” or “and is.” ADDITION 15 35. Tables of “ eight by fifteen,” as shown below, may be given to more advanced students in the same manner as the “ six by ten,” having the time limit of ninety seconds or less, depending on the accuracy and skill of students in handling figures. Teachers should prepare many similar examples for daily drill. 57836489 91445S27 68563394 26497764 37195183 25377836 79213152 57427435 88239176 98446712 77492682 3S768699 82389724 53668795 26582358 59344193 47547362 48993286 62275846 71323494 94563324 53974829 72465137 25784159 84951873 28959271 29438196 28926784 43852971 65267962 35425723 74884675 54799321 36616389 64269762 58517261 89366339 98523173 22458165 71951831 83649177 44277686 55858487 68665653 84467125 62785921 79939831 96439565 29926997 65823589 49269482 38243299 24944828 71S14554 13234947 34412847 54714523 32382649 82992241 49518736 56339768 53666987 73935986 76647862 52679624 92678429 85439665 44793198 21252791 63742615 83764791 44762786 57885837 36528939 97643284 36. Addition of More than One Column in One Operation 3 2. / 3 2 S ' S / / S S Process 24 + 50= 74+1= 75 75+20= 95+5=100 100+10=110 + 3=113 113 + 30=143+2=145 This method proceeds by adding the tens and then the units. 3 S 2 / Z / Process S y 2- 372+200+30+^+100+20+1+^0+40+2 = 1069. / O The student should be encouraged to use above methods. They give varia- tion of method and train to quickness in mental extension. 16 ADDITION 37 . A common practise among bookkeepers and bankers, especially when frequent interruptions occur, is to indicate at the right the sum (including carrying figure) of each column, the result being the addition of the last column and the last figure or cipher of each preceding sum in regular order. By the civil service method, the sums of the columns are set to the right in manner indicated and re-added. Bookkeepers’ Method Civil Service Method 23 y b r y 3 0 3 o V b b cT 3 2 *3 / j r y 4 i / & y 3 S 3 / 2 3 b 7 44 3 f 3 b 4 2 / /. r / 2 r 2 S' <3 7 / S 2 2 1/ 2 2 2 4/ f f S'/ 0 2 t/ f JS / o CROSS ADDITION AND TABULAR WORK 38 . The ability to add numbers horizontally, quickly and accurately, is almost as valuable as the ability to add vertically. Frequent drills should be given, and care should be taken to add only figures of the same order. WRITTEN PROBLEMS 39 . 1. Find the total number of yards in ten pieces containing the follow- ing numbers of yards : 25, 42, 37, 29, 35, 51, 47, 39, 31, 43. 2. Cash Sales Monday Tuesday j Wednesday 1 Thursday Friday Saturday 1 Total $123.45 $272.68 j $89.07 $431.22 $345.16 $650.05 3. Coal Shipments — Pounds Egg Stove Pea Buckwheat Totals Monday 327050 472150 72140 27040 Tuesday 438160 384060 48120 3S160 Wednesday 358100 538010 38100 35100 Thursday 376080 368070 68170 360S0 Friday 420120 240180 40080 18040 Saturday 110060 125100 25010 10020 Totals Total pounds of egg coal shipped during week ? Stove ? Pea ? Buckwheat ? Total coal shipped each day ? Total of all kinds for the week '? ADDITION 17 A Sums disbursed from Peabody Fund, 1876-1880, inclusive. 1876 1877 1878 1879 1880 Totals Virginia $17800 $18250 $15350 $9850 $6800 North Carolina 8050 4900 4500 6700 3050 South Carolina 4150 4300 3600 4250 2700 Georgia 3700 4000 6000 6500 5800 Tennessee 10100 15850 14600 12000 10900 Totals 5 . Five salesmen made the following returns of sales during a week : On Monday— No. 1, $307.50; No. 2, $416.70; No. 3, $178.25; No. 4, $281.05; No. 5, $198.67. On Tuesday — No. 1, $209.10 ; No. 2, $258.40; No. 3, $227.72; No. 4, $203.50; No. 5, $268.80. On Wednesday— No. 1, $201.48; No. 2, $165.10; No. 3, $245; No. 4, $176.50; No. 5, $198.79. On Thursday— No. 1, $146.60; No. 2, $160.36; No. 3, $236.45; No. 4, $245.15; No. 5, $35L70. On Friday— No. 1, $185.50; No. 2, $240.30; No. 3, $149.30; No. 4, $194.82; No. 5, $282.70. On Saturday— No. 1, $301.70; No. 2, $337.40; No. 3, $341.80; No. 4, $380.78; No. 5, $417.80. Arrange in tabular form and answer the following questions: Lpon what day was the greatest amount of sales made? Which salesman sold the greatest amount of goods? What was the total amount of sales for the week? The following are the daily receipts of the treasurer of a borough for fourteen consecutive weeks : Weeks Monday Tuesday Wednesday Thursday Friday Saturday Totals First 486 24 360 50 48 21 516 18 233 40 10s 96 Second 132 25 897 56 437 52 213 78 478 09 980 43 Third 731 64 930 4S 376 83 481 63 704 45 173 26 F ourtli 897 26 270 45 103 48 862 24 238 27 527 28 Fifth 188 25 390 31 976 82 932 21 752 34 438 21 Sixth 297 11 434 56 542 26 276 43 177 30 222 43 Seventh 729 55 721 14 297 31 274 90 249 32 576 87 Eighth 253 54 490 78 864 ! 44 831 82 757 42 543 85 Ninth 238 80 809 66 198 23 582 64 523 45 350 67 Tenth 487 56 789 40 169 45 346 24 434 56 456 78 Eleventh 913 27 841 54 786 32 123 45 876 43 784 27 Twelfth 1963 29 1187 95 864 21 487 91 798 63 1428 95 Thirteenth 1048 63 968 42 1298 74 689 40 1698 75 1078 64 Fourteenth 857 93 1236 97 1187 42 1693 75 1574 83 1870 71 Totals --s> 1 V r> ■1\S\ Find the total receipts for each day ; total receipts for each week ; total receipts for the fourteen weeks. SUBTRACTION 40. Subtraction is the process of finding the difference between two numbers. 41. The Difference between two numbers is a number which, added to the less, will make it equal to the greater. 42. The Subtrahend is the number to be subtracted. 43. The Minuend is the number from which the subtrahend is taken. 44. The Sign of Subtraction is — , read minus, and denotes that the num- ber following it is to betaken from the number preceding it. 45. The Principles of Subtraction are : 1. Only like numbers can be subtracted. 2. Units of the same order only can be directly subtracted. 3. The difference is a number of the same kind as the minuend and sub- trahend. f. If the minuend and subtrahend be equally increased or diminished, the difference remains the same. 46. To find the difference between two numbers. Examples. — (1) Subtract 476 from 847 ; (2) from 10000 take 1346. ^ After writing the subtrahend under the minuend so that / units are under units, tens under tens, etc., we begin with — ‘T- 7 fio units and say, “Six from seven leaves one,” and write 1 / 3 *-/■ 4 % y ^ under the units column. In the tens column we cannot ^ ' take 7 from 4, so we borrow 1 (hundred) from the 8, which equals 10 tens, and add it to the 4 and say, “ Seven from fourteen leaves seven,’’ and write 7 under the tens. Having decreased 8 by borrowing from it we now say, “Four from seven leaves three,” and write 3 under the hundreds, which gives for a result, 371 — the difference. Note. — A nother method of subtracting which has its advantages in such cases as Example 2. is as follows : Instead of borrowing 1 from the highest order and regarding it as a succession of 9’s in each of the other orders excepting the units, where it must be considered as 10, the operation is per- formed by saying, “Six from ten leaves four ; five from ten leaves five ; four from ten leaves six ; two from ten leaves eight” — increasing the subtrahend figure by 1 at each step after the first, and regard- ing the minuend each time as 10. 47. R ule. — Write the subtrahend under the minuend so that units of the same order stand in the same column. Beginning with the units, subtract each figure from the one above it, uniting the difference under the figure subtracted. If any figure in the minuend is less than the figure to be subtracted, increase it by ten, by taking 1 from the next order above, then subtract. 18 SUBTRACTION 19 48 . Proof. — Add the difference to the subtrahend ; if correct, the sum will equal the minuend. Another proof of subtraction is to cast out the 9’s in the minuend and sub- trahend, then subtract the excess figure of subtrahend from excess figure of min- uend. If the latter is the smaller, add 9. The difference must equal excess figure of the remainder. 49 . The following exercises are merely suggestive. When a figure in the minuend is the smaller, borrow a unit from the next higher order which is equal to ten of th.e lower order. Add ten and then subtract. In oral drills, avoid such expressions as “ from ” and “ leaves.'’ Name results only. MENTAL EXERCISE 7 9 8 6 5 7 8 6 9 i J 8 6 8 9 3 4 5 2 3 4 2 4 3 5 2 o O 3 6 6 27 36 45 18 19 37 39 35 28 38 48 12 14 13 13 12 11 15 14 17 18 23 49 56 45 48 54 47 65 44 36 37 67 33 ■ 32 21 27 30 35 43 23 23 22 25 57 76 84 96 79 84 59 46 86 75 66 23 32 31 43 67 52 24 21 53 21 22 43 32 34 37 51 34 41 36 50 42 35 19 25 18 19 16 15 19 17 21 27 19 46 53 49 40 52 34 47 64 45 52 32 18 26 23 22 23 ORAL 25 19 EXERCISE 27 28 23 19 50 . 1- From 28 take 17. take 19. From 63 take 36. From 52 lake 25. From 35 5 from 9 leaves what? from 56 ? 34 from 43 ? 3 from 8 ? ( 15 from 35? 37 from 73? IS 3. 28 — 19= = ? 34 -23= ? 37 — 28= 9 75 — 59= ? 37 —18= _ 9 If.. What is the difference between 34 and 57? 48 and 29? 85 and 57? 91 and 19? 57 and 75? 49 and 94? 5. What is the difference between $17 and $65? $83 and $29? $67 and $83 ? $.53 and $.27 ? $.67 and $1 ? $.75 and $2 ? 6. Two boys had respectively $1.10 and $.85; if each spends $.35, how much money will they have left? 7. A boy had 73 cents and his father gave him a half dollar ; if he spends 48 cents, how much has he remaining? 20 SUBTRACTION 8. Two newsboys together bought newspapers to the value of 75 cents; one sold 55 cents worth, the other 65 cents worth. What did the\ r gain? 9. A man has two fields, one of 32 acres, the other of 57 acres; he puts 48 acres in wheat and the rest in corn; how many acres in corn? 10. A man sold two horses for $135 and $110, respectively; what did lie gain or lose, if he had paid $250 for the pair? 11. What is the value of 13+15—9? Of 23 + 33—17? Of 83—50+23? Of 24+48— 13? Of 55— 16— 7 + 6? IS. Add 46 to 33 and take away 27. Take away 29 from 77 and add 15. 13. Add 7, 9, 8, 6 and 5 and from the result take the sum of 6, 4, 7. 7+ I sell a piano for $225, and receive cash $165 and a note to balance account. What is the face of the note? 15. A merchant buys a bill of merchandise for $383, giving a note for $175 and cash for balance. How much cash does he pay? CASHIERS’ CALCULATIONS 51 . 1. A salesman handed in a sales check for $3.73 and a $5 bill. Hand him the change. Ans. — 2 pennies, a quarter and a dollar. S. Required the change, in the fewest denominations, for a sale of $4.19 to be taken out of a $5 bill. Ans. — 1 penny, 1 nickel, 1 quarter, 1 half. 3. Amount of sale. Money tendered. $ 3.19 $10 bill .75 5 bill 3.79 20 bill .18 2 bill 1.17 1 bill and $J silver 15.68 Two 10 bills 4.49 10 bill 3.33 5 bill 8.90 5 bill and two $2 bills .65 2 bill 14.91 20 bill 1.93 5 bill 1.67 1 bill and $4 and SJ silver 13.23 10 bill and $5 bill 3.09 2 bill and $1 and $^ silver 11.21 50 bill 83.37 100 bill 68.12 Four 20 bills 16.71 10 bill and $5 bill and 82 bill SUBTRACTION 21 f A salesman’s check shows items 35 cents, 70 cents and 18 cents; a $20 bill is tendered. What change is handed back ? 5. Salesman’s items. Money tendered. $.50, .20, .18 $2 bill .39, .15, .25, .10 5 bill .75, .20, .30 10 bill .36, .13, 1.20 2 bill .87, .33, .37 20 bill 52 . A good mental drill in subtraction is to prepare lists of subtrahends to be taken from 50, 100, 500, 1000, etc. Start with 100 as a fixed minuend, and call off in rapid succession all subtrahends from 10 to 20, in irregular order; as “ fifteen,” “ twelve,” “ eighteen,” etc., and have students write down the remainders ; as 85, 88, 82, etc. Finish with 20, and then call off in irregular order the num- bers between 20 and 30, finishing with 30. Proceed in this manner by increasing subtrahends, the students putting down the decreasing remainders. From time to time call off the remainders and have students check their results. In using 50 for a fixed minuend, begin with 25 as your subtrahend and work alternately up and down from this point. After some proficiency has been attained in these, 500 or 1000 may be used for the minuend in the same manner. This exercise may be used for oral drill also ; but for class drill the written work has the advantage of being quieter, and also of benefiting in a greater degree the slower students who are generally disposed to remain silent in concert oral drills. WRITTEN PROBLEMS Balancing Accounts Example. — Find the balance of the following account : Dr. Cr. 2 3 7-7^ r / 2 O f V 3 ■ 6 7 r 2 . f 6 / 3 3.BB / 3 0 f'f * r 17 o 0.0 <2 600.00 r o . 0 0 Bala n ee, 2. A B . f? 2. " / 3 0 f . f A 53 . Rule. — Find the sum of the larger side , set the total beneath : add the columns of the other side, setting down in the place for the balance the figure necessary to make the required figure in the total : verify by adding and totaling all the items of the smaller side. 54 . 1. Find the balance of the following account: Debits $243.75; $18.94; $309.90; $475.60. Credits 385.20; 10.50; 930.27; 50.40. Note. — A rrange amounts in columns and find balance as in model. 22 SUBTRACTION 2 . The debit items of a trial balance are $644.50, $454.70, $27, $44.80. $104; and the credit items, excepting capital account, are $212.50, $62.50. What is the amount of capital ? 3 . Cash items received during a day were $1000, $28.50, $63, $75 ; and the cash items paid out were $300, $50, $100, $72. What is the balance on hand? J. The amount of bills receivable received during a month were $400, $100, $200, $150, and the amount paid on them was $150. How much remains unpaid ? 5 . Merchandise items bought in a month were respectively: $2440, $74.50, and $901.25. Sales made $286.25, $400, $116.25, $314, $441.50, $572, $240, $309, $227.50, and $755.25. If all sold, what amount of gain or loss does the balance show ? 6 . The debit items of a loss and gain account are as follows: $250, $225, $153.60, $150, $254.76, $398.83, $12, $8.17, $71.27, $17.50. The credit items are $14.53, $150, $310. What is the balance of the account, and does it show a gain or a loss ? 7. During the month of March a merchant allowed discounts to his customers as follows : $2.20, $5, $1.25, $1.53, $1.83, $14.58. He was allowed dis- counts by others $1.75, $25.39, $4.20, $9.58. Does his discount account show a gain or loss, and how much? 8 . A merchant deposited in bank during a month $8692.60, $11833.93, $7933.93, and checked out $1600, $2800, $2670.60, $5870.60, $5792.60. What is his balance in bank ? 9 . A farmer made an inventory of his effects and found that he had land worth $12600, buildings $3450, live stock $2365, wheat $218.75. farming imple- ments $680, household furniture $1475, J. Wright’s note $300. He owed a mort- gage of $6000, notes held by others $1575, and outstanding accounts $2367.80. What was his net worth ? 10 . A business has cash on hand $11780, notes on hand $650. It owes $3475. What is the net worth of the business ? 11 . A business man makes the following bank deposits : S1578.25. $614. $342.50, $595.69, $734, $110.10 ; and draws the following checks : $224.73, $219.20, $163.57, $163.75, $591.32, $176.67. What is his balance in bank? 12 . Gash balance on hand, $9462.50 ; receipts for the day, $14165.75: cash paid out, $18328.62. Find cash balance at close of day. 13 . To the merchandise credits, $17342.33, add the value of inventory, $4555.70, and take away the merchandise debits, $21876.66. What was gained? 11 /.. Jones’s account shows sundry debits as follows: $345.50, $427.15. $S7.93. $1503.34, and the following credits : $250, $300.50, $475, $290.90, $75.8100. $175, $120.25. What is the balance of his account ? 15 . The gross weight of a car and its contents is 82525 pounds; if the car weighs 38500 pounds, what is the weight of the contents ? 16 . The resources of a business are: Cash $2351.23, Merchandise 357S1.16. Real Estate $8200, Bills Receivable $1250, Personal Accounts due 81462.50. The liabilities are: Bills Payable $2550, Personal Accounts owing $714.69. The capital is the excess of the resources above the liabilities. What is its amount ? SUBTRACTION 23 17. An erroneous trial balance is as follows: Dr. Cr. T. W. Baylor $1600.00 $15000.00 Cash 6C000.00 55000.00 Real Estate 4000.00 5000 00 Merchandise 21000.00 27500.00 Expense 4500.50 Bank Stock 800.00 1050.50 R. R. Stock 1000.00 Interest 350.75 100.75 Commission 95.00 Personal Accounts 13910.00 Personal Accounts 2499.00 How much is it out ? 18. Check Stub. Add deposits. Subtract checks drawn. Balance $1500.00 Deposits 465.50 Drawings 332.25 Balance Deposits 825.10 Drawings 342.90 Balance Deposits 119.62 Drawings 485.00 Balance Deposits 526.30 Drawings 310.80 Balance Deposits 48.60 Drawings 1200.00 Balance 19. Arrange items and perform operation as in problem No. 18. William Bray had, on Monday, a bank balance of $685.61 ; he deposited $412.80 and drew checks for $122.59, $145.80, and $157.10. On Tuesday he drew checks for $158.99, $100.67, $149.27, and $120. What is the balance to his credit on Tuesday night ? SO. John Cake had, on Monday, a bank balance of $22.60 and deposited $768.25. On Tuesday he drew checks for $156.28, $191.75, $156.28, and $201. What is the balance to his credit on Tuesday night ? MULTIPLICATION 55. Multiplication is the process of finding the product of two numbers. 56. The Product of two numbers is the sum obtained by adding one of the numbers to itself as many times as there are units in the other number. 57. The Multiplicand is the number to be multiplied. 58. The Multiplier is the number which shows how many times the multi- plicand is to be repeated. 59. The Sign of Multiplication is X, read multiplied by, and denotes that the number preceding it is to be multiplied by the number following it. 60. The Principles of Multiplication are: 1. The product is a number of the same kind as the multiplicand. 2. The multiplier is art abstract number, because it denotes the number of times the multiplicand is repeated to produce the product. 3. The product of two abstract numbers is the same, whichever is made the multiplier. Ip. If the multiplier be separated into parts, and the multiplicand be multi- plied by each part separately, the sum of the partial products will be the entire product ; as, fifteen times a number is equal to ten times the number plus five times the number. 61. To find the product of two numbers. Example. — Multiply 436 by 2S7. Write the multiplier beneath the multiplicand, and dravXa line. Seven times 6 units are 42 units — write 2 directly under the 7 (so that it will be in units column ) and carry 4 tens. Seven times three tens are 21 tens (4 tens- carried) are 25 tens — write 5 in the tens column and carry 2 hundreds. Seven times 4 hundreds (2 hundreds carried) are 30 hundreds — write 30. Eight times 6 are 48 — write 8 under the tens figure (5) of the first par- tial product, and carry 4. Eight times 3 (4 carried) are 28 hundreds — write 8 under hundreds column, and carry 2. Eight times 4 (2 carried) are 34 — write 34. Two times 6 are 12— write 2 under hundreds column, and carry 1. Two times 3 (1 carried) are 7 — -write 7 under thousands column. Two times 4 are 8 — write 8 under ten-thousands column. U 3 4 f 7 3 0 S3 3 r f 72 / 2- / 3 2l Draw a line and add the three partial products, which gives 125132, or 287 times 436. 24 MULTIPLICATION 25 62 . Rule. — Write the multiplier under the multiplicand , so that units of the same order shall stand in the same column. Beginning with the units, multiply the multiplicand by each figure of the multi- plier, separately, writing the first figure of each partial product under that figure of the multiplier which produced it Add the partial products. 63 . Proof. — Multiply the multiplier by the multiplicand ; if the same result is obtained, it is almost sure to be correct. A shorter method is to cast out the 9’s of the multiplicand apd multiplier and multiply the excess figures together. If correct, the excess figures of the two products will agree. 64 . The multiplication of any two numbers, however large, is but a succes- sion of multiplications of a single figure by a single figure. Any multiplication whatever may be performed by the student who has mastered the followingjtable, which contains all the products of any two of the figures 2, 3, 4, 5, 6, 7, 8, 9 : MULTIPLICATION TABLE 2 X 2 = = 4 O O X 3 = = 9 4 X 4 = = 16 5 X 5 = = 25 2 X 3 = = 6 3 X 4 = = 12 4 X 5 = = 20 5 X 6 = = 30 2 X 4 = = 8 3 X 5 = = 15 4 X 6 = = 24 5 X 7 = = 35 2 X 5 = = 10 3 X 6 = = 18 4 X 7 = = 28 5 X 8 = = 40 2 X 6 = = 12 3 X 7 = = 21 4 X 8 = = 32 5 X 9 = = 45 2 X 7 = = 14 O O X 8 = = 24 4 X 9 = = 36 6 X 6 = = 36 2 X 8 = = 16 3 X 9 = = 27 7 X 7 = = 49 6 X 7 = = 42 2 X 9 = = 18 8 X 8 = = 64 7 X 8 = = 56 6 X 8 = = 48 9 X 9 = = 81 8 X 9 = = 72 7 X 9 = = 63 6 X 9 = = 54 65 . While the foregoing table is all that is absolutely necessary, the student, in order to become a rapid calculator, should, from time to time, increase his store of memorized products by careful study and drill upon the following extended tables, until he can name instantly the product of any two numbers up to 25 times 25. This will enable him to perform with great rapidity many commercial calculations, especially in making extensions on invoices, etc. 26 MULTIPLICATION Multiplication Table Extended to 12X12 1 2 9 O 4 5 6 7 8 9 10 11 12 2 4 6 8 10 12 14 16 18 20 22 24 3 6 9 12 15 18 21 24 27 30 33 36 4 8 12 16 20 24 28 32 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 70 77 84 8 16 24 32 40 48 56 64 72 80 88 96 9 18 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 80 90 100 110 120 11 22 33 44 55 66 77 88 99 110 121 132 12 24 36 48 60 72 84 96 10S 120 132 144 Note. — The numbers in bold-faced type — products of two equal factors, or a number multi- plied by itself — are called squares ; as, 16 is the square of 4. ORAL MULTIPLICATION DRILL 66 . A very useful drill to enable the student to master readily the extended tables is as follows: Name all the products of one factor in regular order from the lowest to the highest, by mentally adding at each step, as in addition, and then repeat them again from highest to lowest, mentally subtracting at each step, as in subtraction. It is, of course, necessary to name the factors as well as the products — other- wise it would be merely an addition and subtraction drill. “Twice 13 are 26, three times 13 are 39, four times 13 are 52, five times 13 are 65, six times 13 are 78,” etc., to “ twenty-five times 13 are 325.” Then reverse — “Twenty-four times 13 are 312, twenty-three times 13 are 299,” etc. MULTIPLICATION 27 Multiplication Table Extended to 20 X 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 13 26 39 52 65 78 91 104 117 130 143 156 169 182 195 208 221 234 247 260 14 28 42 56 70 84 98 112 126 140 154 168 182 196 . 210 224 238 252 266 280 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 16 32 48 64 80 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320 17 34 51 68 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340 18 36 54 72 90 108 126 144 162 18 ' 198 216 234 252 270 288 306 324 342 360 19 38 57 76 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 Multiplication Table Extended to 25 X 25 l o 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 21 42 63 84 105 126 147 168 189 210 231 252 273 294 315 336 357 378 399 420 441 462 483 504 525 22 44 66 88 110 132 154 176 198 220 242 264 286 308 330 352 374 396 418 440 462 484 506 528 550 23 46 69 92 115 138 161 184 207 230 253 276 299 322 345 368 391 414 437 460 483 506 529 552 575 24 48 72 96 120 144 168 192 216 240 264 288 312 336 360 384 408 432 456 480 504 528 552 576 600 25 50 75 100 125 150 175 200 225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 | 625 1 ORAL EXERCISE 67 . Ex. —If a train runs 15 miles an hour, how far will it run in 8 hours? Ans. — It will run 8 times 15 miles, which is 120 miles. 68 . 1 . W hat will 15 roses cost at 5 cents each? 2 . At 9 cents a yard, what will 18 yards of ribbon cost? 3 . A clerk earns $14 a w r eek ; how much will he earn in 9 weeks? A steamer goes 14 miles an hour; how many miles will it go in 11 hours ? 5 . What will be the cost of 6 oranges at 5 cents each and 12 lemons at 3 cents each ? 6 . What is the value of 7 cows at $25 each ? 7. What will be the cost of 7 yards muslin at 12 cents a yard, 3 spools thread at 2 cents each, and a paper of needles 5 cents? 28 MULTIPLICATION 8. If three men can do a piece of work in 14 days, how long would it require one man to do it? 9. B having 35 sheep sold 18 of them and then bought 3 times as many as he had left ; how many did he then have ? 10. An engine travels 30 miles an hour and a trolley car 22 miles an hour: how much farther will the engine travel in 8 hours than the trolley car? SALESMEN’S AND CASHIERS’ CALCULATIONS 69 . Required the amount of change in the fewest denominations for each of the following sales : Items of Sale Money Tendered 1. 6 lbs. Coffee @ $.38 $2 bill 2. 3 “ Tea U .60 Two silver dollars 3. 25 “ Sugar (6 .05 $1 bill and $4 silver 1 24 cans Corn a .06 5 bill 5. 6 doz. Oranges u .30 2 bill 6. 12 cans Tomatoes u .08 2 bill 7. 5 lbs. Tea u .75 10 bill 8. 11 “ Beef u .12 5 bill 9. 6 “ Butter u .27 2 bill 10. 8 “ Prunes u .14 5 bill 11. 10 “ Soap (< .07 2 bill 12. 5 “ Starch .08 1 bill 13. 6 “ Rice u .07 4 silver u: 3 yds. Silk u .65 5 bill 15. 12 “ Ribbon a .15 5 bill 16. 12 spools Cotton u .03 2 bill 17. 12 yds. Chintz il .18 5 bill 18. 25 “ Muslin u .08 10 bill 19. 6 pairs Hose u .25 5 bill 20. 12 yds. Flannel a .18 2 bill and $1 silver WRITTEN EXERCISE 70 . 1. Multiply $475.25 by 25. 2. Multiply 28375 by 153. 3. Multiply 62137 by 328. Multiply 1756 by 488. 5. Multiply 3991 by 998. 6. Multiply $486.65 by 117. 7. Multiply 3937 by 4256. 8. Multiply 5236 by 973. 9. Multiply 215042 by 9364, 10. Multiply 231 by 9997. 11. Multiply 28317 by 472. 12. Multiply 6452 by 324. 13. Multiply 28375 by 756. U. Multiply 25293 by 4938. 15. Multiply 311035 by 426. 16. Multiply 16387 by 729. 17. Multiply $442.37 by 674. 18. Multiply 5029 by 943. 19. Multiply $316.75 by 749. 20. Multiply 91442 by 3664. SHORT METHODS 29 SPECIAL RULES AND SHORT METHODS 71. To multiply by 10, 100, 1000, etc. 72. Rule. — At the right of the multiplicand place as many ciphers as are found in the multiplier. 73. To find the product of two numbers when either or both end in ciphers. Example. — Multiply 2590 by 38000. 74. Rule. — Find the product of the significant figures of the two factors, and to the right annex as many ciphers as are found at the right of both. i y / 3 ri 0 O O O 2 0/2 7 7 7 f M ^ 2- O O O 0 Note. — The judgment of the teacher must he exercised as to the introduction of short rules. Some may be used to advantage with beginning grades ; others with more advanced students. 75. To multiply by 25. Example.— Multiply 31416 by 25. „ . By annexing two ciphers the number is multiplied by 100, 7 v m t-A OO or 4 times the value of 25. I/ ) 3 / O O 76. Rule. — Multiply by 100 (by annexing 2 ciphers ) and divide by f EXERCISE 77. 1. Multiply 3048 by 25. 6. 2. Multiply 9144 by 25. 7. 3. Multiply 5029 by 25. 8. f Multiply 16093 by 25. 9. 5. Multiply 6452 by 25. 10. Multiply 92903 by 25. Multiply 8361 by 25. Multiply 25293 by 25. Multiply 4047 by 25. Multiply 16387 by 25. 78. To multiply by 250, 125, etc., is but an extension of the above rule. Multiply by 1000 (by annexing three ciphers) and divide by 4 and 8 respec- tively, 1000 being 4 times the value of 250, or 8 times the value of 125. Use Exercise 77 for drill. 79. To multiply by a number a little less or more than 100, 1000, etc. Example. — Multiply 3937 by 998. 3/3/000 f f' Y i/- (3937x2) Subtract. 3 / 2 - 7 / 2- 3 80. Rule. — Annex to the multiplicand 2 ciphers if the multiplier is 99 or 101 ; 3 ciphers if 999 or 1001, etc. To this result add the multiplicand when the multi- plier is 1 more, or subtract when the multiplier is 1 less ; if 2 more, add twice the multiplicand ; if 2 less, subtract twice the multiplicand, etc. 30 MULTIPLICATION WRITTEN EXERCISE 81. 1. Multiply $1875.24 by 97. 2. Multiply $234.70 by 99. 3. Multiply $558.96 by 998. If. Multiply $437.10 by 95. 5. Multiply $442.37 by 997. 6. Multiply $120.92 by 9997. 7. Multiply $316.75 by 96. 8 . Multiply $242.71 by 999. 9. Multiply $716.32 by 94. 10. Multiply $176.19 by 9999. 82. To multiply when one part of the multiplier is a factor of the other part. Examples. — ( 1 ) Multiply $325.50 by 357 ; (2) 7854 by 428. / 3 2 3~. SO y p ^ 7 22. r 2 2 7 f cT O Product by 7 3 / ^3 / Product by 4. / / 3 3 2 S' 3 “ “ 5X7. 2/33/2 “ “ 7X4. /// 2- O 3 . S O 3 3 / 2 83. Rule. — First multiply by one factor ; then multiply the partial product by the other factor. Note. — T he factors of a number are the numbers which when multiplied will produce it ; as 7 and 4 are the factors of 28. WRITTEN EXERCISE 84. 1. Multiply 28317 by 284. 2. Multiply 7645 by 328. 3. Multiply 3624 by 324. F Multiply 9463 by 637. 5. Multiply 3785 by 426. 6. Multiply 3524 by 735. 7. Multiply 2835 by 636. 8. Multiply 9072 by 954. 9. Multiply 648 by 432. 10. Multiply 311035 by 864. 85. To multiply by 11, 111, etc. Example. — Multiply 193 by 11, the result by 111, and that result by 1111. 86. Rule. — To multiply by 11, set down the right-hand figure of the multiplicand, then the sum of the first and second figures, the sum of the second and third, and so on, carrying as usual, and writing the / / / / left-hand figure of the multiplicand (plus any carrying 2 / / F / 0 aP F' 3 figure) as the last figure or figures of the product. To multiply by 111, set doivn the right-hand figure of the multiplicand as before, then the sum of the first and second, then the sum of the first, second and third, of the second, third and fourth, and so on, carrying as usual. For 1111, use each figure four times. / f 3 / / 2- / IF 3 / / / 2- 3 3~ & 3~ 3 SHORT METHODS 31 WRITTEN EXERCISE 1 . 3732 X 11, 111. 7. 22046 X 11, 111. 2. 3937 X 11, 111. 8. 15432 X 11, 111. 3. 10936 X 11, 111. 9. 3215 X 11, 111. 4- 26417 X 11, 111. 10. 2679 X 11, 111. 5. 28375 X 11, 111. 11. 2150 X 11, 111. 6. 3527 X 11, 111. 12. 1728 X 11, 111. CROSS MULTIPLICATION 88. Cross Multiplication consists in multiplying one number by another without writing partial products, setting down the result only, performing all of the calculations mentally. It is especially useful in bill work. By 'practise, operations can be performed readily with multipliers of three or more figures. 89. To multiply two figures by two figures, writing product only. Example. — Multiply 46 by 32. l/ 6, 3 2 / // 7 2 “ 2 times 6 are 12 ” — write 2, and carry 1. “ 3 times 6 are 18 and 1 (carried) makes 19 ; 2 times 4 are 8, plus 19 makes 27 ” — write 7, and carry 2. “ 3 times 4 are 12, and 2 (carried) makes 14.” 90. Rule. — Multiply units by units and set down right-hand figure. Multiply units of multiplicand by tens of multiplier, add the carried figure, and to the sum add product of tens of multiplicand by units of multiplier, setting down right-hand figure. Multiply tens by tens, add the carried figure, and write both figures of result. WRITTEN EXERCISE 91. 1. 27 X 27 = ? 6. 39 X 74 = 2. 49 X 26 = ? 7. 48 X 34 = 3. 33 X 63 = ? 8. 93 X 89 = If.. 57 X 75 = ? 9. 72 X 67 = 5. 37 X 86 = ? 10. 64 X 53 = ? 11. 28 X 35 = ? 16. 37 X 43 = ? ? 12. 52 X 29 = ? 17. 79 X 38 = ? ? 13. 45 X 73 - ? 18. 54 X 83 = ? ? U. 68 X 56 = ? 19. 87 X 94 = ? ? 15. 47 X 69 = ? 20. 95 X 58 = ? 92. To multiply three figures by two figures, writing product only. Example. — Multiply 436 by 78. 8X6=48 — write 8. 436X78=34008 7X6=42, plus 4, 46 pins (8X3) 24=70 — write 0. 7X3=21, plus 7, 28 plus (8X4) 32=60 — write 0. 7X4=28, plus 6=34 — write 34. 93. R ule. — 1 . Multiply the units {of the multiplicand) by units {of the multi- plier), write right-hand figure. 2. Multiply units by tens, add carried figure, then tens by units, add, write right-hand figure. 3. Multiply tens by tens, add carried figure, then hundreds by units, add, write right-hand figure. If.. Multiply hundreds by tens, add carried figure, write result. 32 MULTIPLICATION WRITTEN EXERCISE 94. 2. 234 X 46 = ? 6. 613 X 28 = 0 27. 845 X 64 = ? 16. 282 X 48 = 9 2. 368 X 32 = ? 7. 438 3 421 X 26 = ? 8 347 £ 732 X 34 = ? 9. 764 5. 525 X 72 = 9 10. 956 47 = ? 12. 529 X 52 63 = ? 13. 672 X 74 78 = ? U- 438 X 65 59 = •? 15. 124 X 57 ? 17. 375 X 37 = ? ? 18. 913 X 82 = ? ? 19. 742 X 94 = ? ? 20. 412 X 76 = 9 95. To multiply three figures by three figures, writing product only. Example.— Multiply 521 by 346. 1X6 = 6. (lX4) + 0 carried +(2X6) =16. 521 X 346=1 80266 dX3)+i “ +(2X4)+(5X6)=42. (2X3)+4 “ +(5X4) =30. (5X3) +3 “ =18. 96. Rule. — 1. Units X units. 2. (Units X tens ) + (tens X units.) 3. (Units X hundreds ) +- (tens X tens) + (hundreds X units.) 4.. (Tens X hundreds) + (hundreds X tens.) 5. Hundreds by hundreds. WRITTEN EXERCISE 97. 1 . 329 X 642 = ? 2. 473 X 567 = ? 3. 234 X 319 = ? I 748 X 325 = ? 5. 456 X 538 = ? 6. 918 x 475 = ? 7. 237 X 646 = 9 <5. 786 X 372 = ? 9. 589 X 716 = ? 10. 745 X 824 = ? 98. To square a number of two figures. 11. 678 X 343 = ? 12. 834 X 937 = ? 13. 712 X 846 = ? 14- 576 X 672 = ? 15. 483 X 519 = ? Examples. — (2) Find the square of 42 ; (2) of 64. 42 (2 X 2) 64 (4 X 4) 42 (4 + 4) X 2 64 (6 + 6) X 4 + 1 1764 (4 X 4) + 1 4096 (6 X 6) + 4 99. Rule. — (7) Multiply together the rigth-hand figures ; (2). multiply the sum of the left-hand figures by the right-hand figure of the multiplier ; ( 3 ) multiply together the left-hand figures ; carry at each step, as in ordinary multiplication. 100. To find the product of two numbers whose units add to 10 and whose tens are alike. Examples. — (2) Multiply 66 X 64; 66 6 X 4 — write 24 .64 '6 + 1 X 6 — write 42 4224 (2) 88 X 82. 88 8 X 2 — write 16 82 S + 1 X S — write 72 7216 101. Rule. — Multiply together the units and ivrite the result ; add 1 to the tens of the multiplicand, multiply by the tens of the multiplier, and prefix the result to the product of the units. SHORT METHODS 33 WRITTEN EXERCISE 1. 45 X 45 6. 36 X 36 0. 95 X 95 /V / . 56 X 54 3. 27 X 27 8. 38 X 32 s 44 X 44 9. 67 X 63 5. 64 X 64 10. 85 X 85 SLIDING METHOD 103 . The figures of the multiplier are written on a separate piece of paper and moved from right to left or left to right as indicated in examples. The figures in either the multiplicand or the multiplier must be written in reverse order. If figures in the former are reversed, the multiplication will proceed from left to right ; if the latter, from right to left. At the first step only one figure is multiplied (units X units) then paper is moved one place to the right ; at the second, two figures are multiplied (tens X units and units X tens) ; at the third, three figures, (hundreds X units, tens X tens and units X hundreds) ; and so on, moving the paper a place each time, multiplying together at each step the vertical figures of the multiplicand and multiplier, and adding mentally the several products, carrying as usual. When the left-hand figure of the multiplier is under the right-hand figure of the multiplicand (or vice versa), the last operation is performed and both figures of the result written down. Example 1. — Multiply 456 by 144. (Figures of multiplicand reversed.) First Step Second Step Third Step Fourth Step Fifth Step b S' 44 b A - 44 6> .jT 44 b S' 44 b S' 44 / // // / 44 // / 44 // / 44 44 / o4y4 44 b 44 b b 44 S' b b 44 b S' b b S 6X4=24. (5X4)+2 + (6X4). (4>l)+4+(5X4) + (lX6). (4X4)+4 + (5Xl). (4Xl)+2. Write 4 Write 6 Write 6 Write 5 Write 6 Carry 2 Carry 4 Carry 4 Carry 2 Result 65664 Note. — A ny of the foregoing exercises may be used for practise. Example 2 — Multiply 456 by 144. (Figures of multiplier reversed.) Fifth Step 44 S' b u 44/ b S~ b b 44 Result 65664 Fourth Step i/bS b 44 44 / S' b b 44 2 Carried Third Step 44 S' b 44 44 / b b 44 4 Carried Second Step 44^ ^ 44 44 / b 44 4 Carried First Step 44 S b ^44/ 44 2 Carried 34 MULTIPLICATION WRITTEN PROBLEMS 104 . 1 . What is the cost of 3456 sheep at $6 per head? 2 . A man bought a farm containing 495 acres at $28 an acre. What was the cost of the farm? 3 . A farmer sold 96 bushels of wheat at 57 cents a bushel. How much money did he receive? J. At 63 cents a bushel, what will 325 bushels of potatoes cost? 5 . Find the value of 16 crates of eggs, each crate containing 32 dozen at 22 cents a dozen. 6 . A cotton factory contains 245 looms; if each loom makes 37 yards of cloth daily, how much cloth will the factory make in 70 days? 7. There are four fields, each containing 585 hills of potatoes, and every hill averages 12 potatoes. How many potatoes in the four fields? 8 . If a saw-mill saws 5768 feet of boards in a day, how many feet will it turn out in 47 weeks of 6 days each ? 9 . If ninety-five men can do a piece of w r ork in 45 hours at a cost of IS cents per hour, what will the work cost? 10 . A house requires 4765 shingles on each of two sides of the roof. How many shingles are necessary for an operation of 80 houses? 11 . A coal dealer bought 45 cars of coal containing 25 tons each. What was the cost of it all at $4.50 a ton ? 12 . A farmer harvested 42 bushels of grain per acre from a field of 22 acres. What did he receive for it at 95 cents a bushel? 13 . Mr. H. Humphreys bought 6 bureaus at $S.75 each ; 3 easy-chairs at $15.32 each ; 12 dining-room chairs at $4.67 ; 15 mattresses at $13.75 : 2 extension tables at $17.85 ; 4 mirrors at $9.79. What was the total amount of the bill? 11 /.. James White bought of John Black 26 yards of cloth at $3.25 per yard ^ 147 yards of sheeting at 26 cents a yard; 28 yards of table linen at $1.75 per yard; 23 dozen towels at $3.25 per dozen. He sold him 27 lambs at $7.15 each ; 15 yearlings at $15 each ; 47 chickens at 57 cents each. Which owes the other and how much? 15 . A railway 1326 miles in length was constructed at an average cost of $63,275 per mile. What was the total cost ? 16 . A carpenter works every day in the year but Sundays. He receives $5 per day. How much does he save in one year if his expenses amount to $12 per week ? DIVISION 105. Division is the process of finding how many times one number is contained in another. 106. The Dividend is the number to be divided. 107. The Divisor is the number by which the dividend is divided. 108. The Quotient is the number which shows how many times the divisor is contained in the dividend. 109. The Remainder is that part of the dividend left after dividing, when the dividend does not contain the divisor an exact number of times. 110. The Sign of Division is -5- , read divided by, and denotes that the number preceding it is to be divided by the number following it. 111. The Principles of Division are : 1. The divisor must be a number of the same kind as the dividend ; for division is a continued subtraction of the divisor from the dividend (the quotient showing how many times it is taken), and it is evident that nothing could be taken from the dividend but a part of itself. 2. The quotient is an abstract number, because it shows the number of times the dividend contains the divisor. 3. If both dividend and divisor be multiplied or divided by the same number , the quotient of the results obtained ivill be the same as the quotient of the original numbers. SHORT DIVISION 112. The term “ short division ” is used when the divisor is small enough to perform the operation mentally and write down only the quotient. 113. To divide by short division. Examples. — (1) Divide 32741 by 6 ; (2) 265085 by 12. Rule. — Write the divisor on the f, j 3 2 7 4^ / left of the dividend, with a curved IT S line between them, and draw a horizontal line under the dividend. / 2 )2 6 2 2 0^ of/2 35 3G DIVISION Begin with the first figure at the left of the dividend (or, if that figure is less than the divisor, use the first two figures), divide and write the quotient under the figure divided Thus in Ex. 1, since 6 is not contained in 3, we say, “Six in 32 five times, and 2 over,” and write 5 under the 2. When the division is not even or exact, prefix the remainder ( mentally ) to the next figure of the dividend before dividing it. Thus in Ex. 1, prefixing the remainder 2 to the 7, we say, “Six in 27, four times, and 3 over,” and write 4 under 7 ; then proceeding, as before, we say, “Six in 34 five times, and 4 over, six in 41 six times and 5 over,” writing the 4 and 6 in their proper positions and placing the last remainder 5 over the divisor 6 (with a line between them) at the right of the quotient. Whenever a partial dividend (except the first ) is less than the divisor, write a cipher (0) in the quotient and proceed with the division, including the next figure in the partial dividend. Thus in Ex. 2, the second division leaves a remainder of 1, which prefixed to the cipher in the dividend makes 10, which is less than 12, so we write 0 in the quotient, and, prefixing the 10 to the 8, we say, “ 12 in 108 nine times ” — no remainder. Then, 12 being less than 5, we place another cipher in the quotient and write the remainder 5 after it, with the divisor under it in the form of a fraction, T 5 3 . Note. — A “ partial dividend ” is that part of the dividend used at each of the successive steps in the operation. ORAL EXERCISE 114 . 1. At 3 cents each, how many oranges can be bought for 36 cents? 2. At 8 cents a yard, how many yards of ribbon cau be bought for 64 cents? 3. Gave $60 for sheep at the rate of $5 a head, how many did I buy? 1^. If a wheelman rides 12 miles an hour, how long will it take him to travel 96 miles? 5. How many kegs of 10 gallons each can be tilled from barrels containing 140 gallons? 6. How many days will it take a man to earn $96 if he receives 86 a day? 7. How many 7 bushels of wheat can a man buy for $84 if he pays 86 for 5 bushels? 8. A man gave a group of boys 56 cents ; if each boy received 14 cents, how many in the group? 9. If coal is $5 a ton, how many tons can be bought for $95 ? 10. If 12 hams weigh 180 lbs., what is the average weight of a ham? DIVISION 11. Plow many are 36 plus 14, divided by 5? 25 plus 15, divided by 8 ? 32 plus 18, less 2, divided by 8 ? IS. How man} r are 100 minus 12, divided by 11 ? 80 plus 2, minus 5, divided by 7? 144 less 12, plus 8, divided by 14? 13. 3 times a number, plus 5 times a number, minus the number, plus 4 times the number, equals how many times the number? 9 times a number, divided by 3, multiplied by 8, divided by 6, multiplied by 5, divided by 4, equals how many times the number? 11^. Think of a number, multiply it by 8, divide by 4, divide by 2, add 16, subtract the number thought of, divide by 8, and the quotient is wdiat? 15. Think of a number, multiply it by 8, divide it by 4, multiply by 4, divide by 8, add 20, subtract the number thought of, divide by 10, and the quotient is what? 16. If 8 tons of hay cost $200, what is the value of 1 ton? 17. If a train runs 210 miles in six hours, what is its average rate of speed ? 18. If a man buys $555 worth of flour at $3 a barrel, how many barrels should he receive? 19. Enough linoleum to cover a floor of 320 square feet is sold for $9.60 ; what is the price per square foot ? SO. How many bushels of oats at 50 cents a bushel can be bought for $18'. WRITTEN EXERCISE 115 . 1. Divide 79171923 by 2. 2. Divide 87682835 by 3. 3. Divide 96303847 by 4. 4- Divide 15274652 by 5. 5. Divide 84205561 by 6. 6. Divide 65425622 by 7. 7. Divide 82562828 by 8. 8. Divide 12412730 by 9. 9. Divide 37142897 by 11. 10. Divide 31S12719 by 12. 11. Divide 3453488739 by 4. IS. Divide 4568392825 by 6. 13. Divide 7643912848 by 7. 11 Divide 5324769214 by 8. 15. Divide 6843179157 by 9. 16. Divide 7289374673 by 11. 17. Divide 8924321866 by 12. 18. Divide 382841 1382 by 7. 19. Divide 1431287191 by 11. SO. Divide 6897205400 by 12. 38 DIVISION LONG DIVISION 116. Long Division is applied when the divisor is too large to be handled as a purely mental operation — that is, when the figures of the successive multi- plications and subtractions must be written down. How to divide a number by long division. Example. — Divide 13279 by 47. 117. Rule . — Write the divisor to the left of the dividend with a curved line between them, and another curved line to the right of the dividend to separate the quotient from it. Determine, by inspection, how many times the divisor is contained in the fewest left- hand figures of the dividend, and write the number to the right of the dividend. Thus, in the accompanying Example, since 47 is not contained in 1 or in 13, we take 132 as the first partial dividend. By comparing them we see that the left-hand figure of the divisor (4) would he contained in 13 three times, hut by inspection, we see that multiplying by 3 would give 2 to carry from the 3 times 7, which added to 3 times 4 would make 14. Hence, we conclude that the first quo- tient figure should be 2, which we place in the quotient to the right of the dividend. 44 yj / 3 z 7 *? ( Z f z W 44 Jf/ 3 7 6 / / f f 44 2 S Result, 282-ff. Multiply the divisor by the quotient figure just found, and ivrite the product under the partial dividend. Draw a line, subtract and write the remainder beneath. Multiplying 47 by 2 gives us 94, which, subtracted from the partial dividend, leaves 38. Bring down the next figure of the dividend and place it at the right of the remain- der, making the next partial dividend. Bringing down the 7 gives us 387 for a partial dividend, which, by inspection as before, we can see will contain 47 eight times ; write 8 in the quotient. Proceed as with first partial dividend, and continue until all the dividend figures have been brought down. Multiplying 47 by 8 gives 376, which, subtracted from 387, leaves 11, to which we annex the last figure (9) of the dividend and proceed as before. Write the remainder, if any, at the right of the quotient with the divisor below it. 118. Proof. — Multiply the quotient by the divisor, and add the remainder to the product. The sum should equal the dividend; or, in an even division, after casting out 9’s in the dividend and quotient, multiply excess figures of divisor and quotient. The excess figure of product must then equal excess figure of dividend. If there is a remainder, its excess figure is subtracted from the excess figure of the dividend or that figure +9 (if smaller than remainder excess), which gives the excess figure of dividend. SHORT METHODS 39 WRITTEN EXERCISE 119 . 1. Divide 63247 by 53. 11. 2. Divide 24876 by 89. 12. 3. Divide 31845 by 67. 13. S Divide 27328 by 73. If. 5. Divide 486329 by 213. 15. 6. Divide 675471 by 379. 16. 7. Divide 39754 by 437 17. 8. Divide 298546 by 913. 18. 9. Divide 3875213 by 634. 19. 10. Divide 4783976 by 719. 20. 5254361 by 2483. 3989768 by 7356. 42864374 by 8429. 56893573 by 3067. 67654826 by 5246. 32120487 by 4983. 74S973254 by 35826. 656438602 by 48375. 392876348 by 25764. 487326454 by 384213. 3 2 7*0= < 0 = SPECIAL RULES IN DIVISION 120. To divide by 10, 100, 1000, etc. Example.— 34837-^100. J ^ J = .3/ 121. Rule. — Cut off at the right of the dividend as many figures as there are ciphers in the divisor. The figures remaining to the left will be the quotient, and the figures cut off from the right will be the remainder. 122. To divide by any number ending in ciphers. Example. — Divide 4456328 by 32700. Result, 136JLLUL 123. Rule. — Cut off the ciphers at the right of the divisor and an equal number of figures at the right of the dividend. Find the quotient by dividing the re- maining figures of the dividend, by the significant figures of the divisor, writing the quotient figure at each step directly above the right-hand figure of the partial dividend. To the remainder, after this operation, annex at the right the figures cut off from the dividend, which gives the entire remainder. 124. To divide by 25, etc. / 3 6 ) z/ z/ s 6 3 3 2 7 2 r / / r 6 f F / 2 2 S' 3 / f 4 2 f / 2 t Examples. — (1) Divide 62438 by 25; {2) 45659-^-250. 52:4-4=13. Remainder. Result, 2497^. 2 v '3 r 2^77 S 2 44 s s 7 / 7 2 3 & 125. Rule. — Multiply the dividend by If, and divide the product by 100. ( This can be extended, as in multiplication, to 250; 2500, 125, 1250, etc.) 40 DIVISION WRITTEN EXERCISE 126. 1. Divide 3248 by 25. 2. Divide 4199 by 25. 3. Divide 6327 by 25. If. Divide 14832 by 25. -5. Divide 71248 by 25. 6. Divide 34862 by 25. 7. Divide 41780 by 25. 8. Divide 63745 by 25. 9. Divide 42150 by 25. 10. Divide 37895 by 25. 127. To divide by a composite number. Example. — Divide 3479 by 28. 7)3479 4)497 124i 28=7X4- Dividing by one factor, 7, gives 497, which, divided by the other factor, 4, gives the quotient. Example. — Divide 4265 by 64. 8)4265 8)533 + 1 Rem. 66+5 Rem. 5x8=40, plus 1=41 True Rem. Result, 66|^. 64=8X8 Divide as above. The second remainder is a part of 533 times 8 — a number of eighths , and must be multiplied by the first divisor before adding to the first remainder, to get the whole remainder. 128. Rule — Separate the divisor into two factors, and divide the dividend by one factor ; then divide the quotient by the other factor. If the first division leaves no remainder, write the second remainder over the second divisor, and place at the right of quotient. If both divisions leave remainders, multiply the second remainder by the first divisor, add to first remainder, and write sum over the entire divisor, placing at right of quotient. WRITTEN EXERCISE 16. Divide 1312432 by 48 (8 X 6) 129. 1. Divide 34826 by 14 (7 X 2). 2. Divide 47563 by 15 (5 X 3). 3. Divide 57648 by 16 (4 X 4). If. Divide 37849 by 18 (6 X 3) 5. Divide 96876 by 21 (7 X 3). 6. Divide 134762 by 22 (11 X 2). 7. Divide 312843 by 24 (6 X 4). 8. Divide 525301 by 27 (9 X 3). 9. Divide 626884 by 28 (7 X 4). 10. Divide 317783 by 32 (8 X 4). 11. Divide 498632 by 33 (11 X 3). 12. Divide 653728 by 35 (7 X 5). 13. Divide 329849 by 36 (6 X 6). If. Divide 483764 by 42 (7 X 6). 15. Divide 959388 by 45 (9 X 5). 17. Divide 3298467 by 49 (7 X 7). 18. Divide 1128321 by 54 (9 X 6). 19. Divide 2557637 by 56 (8 X 7). 20. Divide 4887572 bv 63 (9 X 7). 21. Divide 5112863 by 64 (8 X S). 22. Divide 6768459 by 72 (8 X 9). 23. Divide 8384536 by 77 (11 X 7. 2 If. Divide 9394281 by 81 (9 X 9). 25. Divide 3838633 by 84 (12 X 7). 26. Divide 5637344 by 88 (11 X 8). 27. Divide 8565787 by 108 (12 X 9). 28. Divide 3254063 by 121 (11 X 11). 29. Divide 4080362 by 132 (12 xU). 30. Divide 6302011 by 144 (12 X 12). DIVISION 41 WRITTEN PROBLEMS 130 . 1 . At $125 each, how many horses can be bought for $4250? 2. A bin contains 128560 lbs. How many bushels of 56 lbs. each does it contain ? 3. If potatoes yield 46 bushels per acre, how many acres will be required to yield 3312 bushels? I,.. A clerk saves $57 a month. How many months will it require to save $1311? 5. A fruit grower received $1755 for 195 barrels of cranberries. What was the price per barrel? 6. How many lots at $321 each can be bought for $772326? 7. One share of bank stock is worth $98. How many shares can be bought for $22050? 8. A public library has a yearly circulation of 5696600 books. How many books are taken daily, if the library is open 313 days in the year? 9. A company of 547 men took equal shares in a mine valued at $705083. How much money did each man invest? 10. If a saw-mill turns out 1678152 feet of boards in 294 days, how many feet are sawed daily ? 11. Two cities 294 miles apart are connected by railroad at a cost of $6845630 ; what is the cost of one mile of railroad ? 12. A manufacturer produced 8643759 yards of goods at a cost of 129656385 cents. What was the cost of manufacturing one yard? 13. The dividend is 72987 and the divisor 45; required the quotient and remainder. Ilf. The divisor is 587 and the quotient 8723; what is the dividend? 15. If 28 horses cost $3864, what is the cost of one horse ? 16. A man left $2535 to each of his four children, but, one of them dying, the money was divided among the three living. How much did each receive? REVIEW PROBLEMS Addition, Subtraction, Multiplication and Division. 131 . 1. A merchant’s bank balance on Saturday was $564.32. On Monday, he deposited $365.45 and checked out $260.17 ; on Wednesday, deposited $85.60 ; on Thursday, deposited $125 and checked out $468.57; on Friday, deposited $93.75. What is his bank balance after these transactions? 2. A clerk who earns $65 a month, pays $25 a month board, $150 a year for clothes, incidental expenses $138, and invests the remainder in books at $4 a volume. How many volumes can he buy yearly? 3. A clerk’s present worth is $832. If his salary is $100 a month, and his expenses $14 a week, how many years must he work to be worth $4000? 42 DIVISION If.. A is worth $525 ; B is worth 3 times as much less $80 : C is worth 8 times as much as A and B. How many horses valued at $180 each can they buy with their total money ? 5. Jones owes $864.25, then pays $376.15 ; he then buys for $1514.27 and pays $935.63 ; he then buys for $564.28 and pays $438.20 ; he now pays $1068.75. How much does he still owe? 6. Bought 544 bushels of clover seed at $3.85 a bushel and paid for it with linen worth 17 cents a yard. How many yards did it require? 7. Bought a farm of 135 acres. Paid $5130 cash and $145 monthly for 4 years and 6 months. How much did I pay per acre ? 8. A manufacturer’s weekly sheet shows total product as follows: 1st week, 5684 yards; 2d week, 8647 yards; 3d week, 7863 yards; 4th w 7 eek, 6435 yards ; 5th week, 4639 yards ; sold 24951 yards at 88 cents a yard, and the remainder at 84 cents a yard. What w 7 as the average receipt for each yard produced ? 9. A lady owns a farm valued at $6300, and 3 stores valued at $15600. She gives her daughter of the farm and J of the value of the stores ; the remainder is shared equally among 5 sons. How much does the daughter receive more than each son ? 10. Owning 45 acres of land, I sell 15 acres for $2250 ; the remainder I sell at $135 an acre. I receive $6000 cash and the balance in pigs at $5 each. How many pigs did I receive ? 11. Bought 35 barrels of pork at $26 per barrel ; 3 barrels were damaged and sold for $5 per barrel less than cost ; the remainder were sold at a profit of 83 per barrel. What was the gain ? 12. A farmer sold a grocer 50 bushels of potatoes at 68 cents a bushel; 12 barrels of vinegar at $6 a barrel ; 75 bushels of apples at 72 cents a bushel ; 52 pounds of butter at 42 cents a pound, and 84 dozen eggs at 15 cents a dozen. The farmer bought 3 barrels of sugar, 386 pounds each, at 5 cents a pound ; 2 barrels of fish at $18 a barrel ; 5 boxes raisins, 40 pounds each, at 12 cents a pound ; 2 cases prunes, 80 pounds each, at 9 cents a pound ; 3 barrels of coal oil, 45 gallons each, at 15 cents a gallon ; 18 bushels of salt at 27 cents per bushel. Which owes the other, and how much ? 13. A farmer buys a farm of 110 acres at $75 an acre; he pays $2200 down and the remainder in 5 yearly instalments. What was paid each year? Ilf. A builder finds the cost of an operation of 65 houses to be as follows : land $325,000, material $185,687, labor $205,947. At what price must he sell each house to gain $475 a piece? 15. In constructing a railroad 25 miles long, the grading cost $586,742; 10 bridges were built at an average cost of $45S67 ; a tunnel was dug at a cost of $168,738. What was the average cost per mile ? 16. In a business exchange 125 horses valued at $31200 were given for 25 pianos. What was gained by selling each piano for $1500? PROPERTIES OF NUMBERS 132. A unit is one ; as, one book, one year. 133. A number is a collection of units ; as, three, four boxes, five cents. 134. Numbers are divided into integral and fractional, concrete and abstract, prime and composite, odd and even, similar and dissimilar. 135. An integral number expresses whole things ; as, six, three years. 136. A fractional number expresses parts of things : as, one half, one fourth of an inch. 137. A concrete number is a number applied to particular objects ; as, two desks, seven apples. 138. An abstract number is a number considered apart from objects ; as, three, four, five. 139. A prime number has no exact divisor, except itself and one; as, two, three, five, seven. 140. A composite number is the product of factors other than itself and one, and, consequently, is exactly divisible by them ; as, 6, 8, 9, 15. 141. An odd number is one that leaves a remainder of one when divided by two ; as, 5, 7, 9. 142. An even number is one exactly divisible by two ; as, 8, 10, 12. 143. Similar numbers are groups of the same kind of units ; as, 2 feet, 6 feet, 8 feet. 144. Dissimilar numbers are groups of different kinds of units; as, two ■oranges, five apples. 145. The prime factors of a number are the numbers which when multiplied together will produce it. A prime factor is a factor that is a prime number. The factors of 12 are 2, 2 and 3 ; of 15, 3 and 5. 146. The result of taking the same number two or more times as a factor is called a power; as 5x5—25, the second power (called the square) of five; 3 X3 X3=27, the third power (called the cube) of three. The power is usually indicated by a small figure, called the exponent, which is written above and at tlie right of the factor ; as, 3 2 means 3x3 ; 4 3 means 4X4X4. The small figure indicates the degree of the power; as 2 s , the fifth power of two ; 5 4 , the fourth power of five. 43 FACTORING 147. Factoring is the process by which a composite number is reduced to its factors. This is done by division, the number which we divide being the multiple , and the divisors, the factors. 148. To factor a number is to find the prime factors. Example. — Find the prime factors of 63. 3)63 3)21 7 3X3X7=63. 149. Rule. — Divide the number to be factored by any prime factor that will exactly measure it; in like manner divide the quotient thus obtained and the successive quotients until one is obtained that is prime. The several divisors and the prime quo- tient are the prime factors sought. WRITTEN EXERCISE 150. Find the prime factors of 1. 27. 6. 1825. 11. 3025. 16. 9020. 2. 81. 7. 1890. 12. 819. 17. 17017 3. 132. 8. 5346. 13. 1785. 18. 1009. 4. 210. 9. 3465. 14- 8729. 19. 971. 5. 1732. 10. 7007. 15. 1287. 20. 1063. DIVISIBILITY OF NUMBERS 151. The following properties of numbers will be found useful in reducing fractions to their lowest terms, in resolving numbers into their prime factors, etc., as they serve to abridge the labor of calculation. 1. Even numbers are divisible by 2. 2. When the sum of the figures of a number is divisible by three, the num- ber is divisible by 3. 3. When 4 divides the number expressed by the two right-hand figures of a given number, the given number is divisible by 4- 4- When the right-hand figure of a number is 5 or 0, the number is divisible by 5. 5. When an even number is divisible by 3, it is divisible by 6. 6. When 8 divides the number expressed by the three right-hand figures of a given number, the given number is divisible by 8. 7. When the sum of the figures of a number is divisible by 9, the number is divisible by 9. 8. 1001 and its multiples, which are usually easily recognized, are divisible by 7, 11, and 13. 9. When the sum of the odd figures of a number equals the sum of the even figures, or when the difference between these sums is 11 or a multiple of 11, the number is divisible by 11. 44 GREATEST COMMON DIVISOR 152. A common divisor of several numbers is a number that will exactly divide each of them. 153. The greatest common divisor of two or more numbers is the greatest number that will exactly divide each of them. 154. A number is only an exact divisor of such other numbers as contain all its prime factors. 155. To find the greatest common divisor of two or more numbers by factoring. Example. — Required the greatest common divisor of 12, 18 and 36. ( 12=2X2X3 The factors of < 18=2X3X3 [ 36=2X2X3X3 The factors common to all are 2 and 3 ; therefore, their product, 2X3, or 6, is the greatest common divisor. 2 )/z —/ r — a & < 3 ) 6 — f — / r ' 2 - 3 — T Explanation. — Since 2 is an exact divisor of each number, 2 is a factor of the greatest common divisor. The quotients being divisible by 3, 3 is a factor of the greatest common divisor. The only factors common to all are 2 and 3 ; their product is 6, the greatest common divisor. 156. Rule. — Divide the numbers by any prime number that exactly divides all of them ; proceed in the same manner with the quotients ufitil no number can be found that will divide all of the last quotients. The product of all the divisors used will be the greatest common divisor. 157. To find the greatest common divisor when the numbers are large and not easily factored. Example 1. — Required the greatest common divisor of 437 and 897. ^23 2 0 7 2 0 7 45 46 GREATEST COMMON DIVISOR Example 2.-Required the greatest common divisor of 9388. 16429 and 18776. Abbreviated Method 9388 16429 1 G. C. D. 2347 18776 8 9388 18776 1 7041 7041 3 2347 7041 Note. — This arrangement simply avoids the repetition of figures and thus saves time. 158 . Rule. — Divide the larger number by the smaller, and each preceding divisor by the remainder until there is no remainder. The last divisor will be the greatest common divisor. WRITTEN EXERCISE 159 . Find the greatest common divisor of 1. 24, 32 and 48. 9. 81, 135 and 216. 2. 53, 63 and 81. 10. 390, 910 and 1365. 3. 72, 108 and 120. 11. 96, 108, 132 and 156. 4. 39, 52 and 65. 12. 36, 84, 108 and 144. 5. 81, 105 and 135. 13. 365, 511 and 803. 6. 210, 315 and 420. n. 102, 153 and 255. 7. 104, 182 and 351. 15. 140, 250, 360 and 270. 8. 180, 216 and 276. 16. 144, 256, 192 and 204. 17. A farmer has three strips of timber, 80, 96 and 102 feet, respectiv What is the length of the longest pieces, all of the same length, that may be cut from them ? 18. Three schools containing 120, 210 and 360 students, respectively, are divided into classes, each containing the same number of students. What is the greatest number of students each class can contain, and how many classes of this size are there in each school ? 19. A man distributed $240, $336 and $480 among the employees of 3 mills in equal sums, the sums being as large as possible. Required the amount of the equal sums and number of employees. 20. A triangular lot, the sides of which are respectively 168, 192 and 180 feet, is to be inclosed with a tight board fence, 7 feet high, the boards to run lengthwise. How many boards will it take if the}'' are 6 inches wide and of the greatest possible length that will require no cutting? LEAST COMMON MULTIPLE 160. A multiple of a number is a number that is exactly divisible by it ; as, 10 is a multiple of 5 ; 12 of 4 ; 9 of 3. 161. A common multiple of two or more numbers is a number exactly divisible by each of them. Thus, 18 is a common multiple of 6, 3 and 2. 162. The least common multiple of two or more numbers is the least number that is exactly divisible by each of them. Thus, 12 is the least common multiple of 3 and 4. 163. A common multiple of two or more numbers contains all the prime factors of each of those numbers. 164. To find the least common multiple of two or more numbers. Example. — Find the least common multiple of 16, 18, 24 and 60. D, ) / & - / ? -24 - 6 O 2 )f - 4 - / 2 -3 O iw - y - 6 - / 1 3)2 - V - 3 -/^T y 2 - 3 - / - S' 2 X 2 X 2 X 2 X 3 X 3 X or- 2*x3 x x>r = y20 165. Rule. — IT rite the numbers in a row ; draw a line underneath, and, select- ing for a divisor the smallest prime number that will exactly divide two or more of them, divide as many as you can, bringing down below the line those numbers which cannot be divided, together with the quotients. Repeat this process with the same divisor until it will no longer divide two of the numbers beloiv the line ; then take the next smallest prime factor that will divide two or more, and proceed as before. Con- tinue thus until the numbers below the line are prime to each other. Finally , multiply together the numbers beloiv the last line and all the divisors, and the product will be the least common multiple. WRITTEN EXERCISE 166. Find the least common multiple of 1. 20 and 36. 2. 21 and 42. 3. 12, 18 and 24. f. 15, 30 and 45. 5. 16, 24 and 30. 6. 42, 84 and 105. 7. 45, 72 and 120. 8. 54, 90 and 270. 9. 40, 80 and 120. 10. 63, 84 and 98. 11. 45, 60 and 75. 12. 18, 36, 72 and 120. 13. 8, 10, 40 and 60. If. 27, 36, 45 and 60. 15. 28, 35, 49 and 70. 16. 96, 120, 160 and 132. 17. 36, 60, 120 and 204. 18. 150, 180.320 and 480. 19. What is the shortest piece of goods that can be cut into pieces of 12 yds., 18 yds. and 24 yds. and nothing remain? 20. What is the smallest tract of land that can be exactly laid out in fields of either 8, 12, or 16 acres? How many fields of each kind can be made ? 47 CANCELATION 167. Cancelation is employed to shorten the work of division, and consists in casting out the same factor from both dividend and divisor. 168. To cast out a factor of any product is simply to divide that product by the factor eliminated, and to cast out the same or equal factors from both divi- dend and divisor is equivalent to dividing both dividend and divisor by the same number ; it does not change the quotient. Thus, dividing 30 by 10, or three times 30 by three times 10, or one-half of 30 by one-half of 10, all produce the same quotient. (30-^10=3; 90-r-30=3; 15 -h5=3.) 169. To find a quotient by cancelation. Example. — Divide 8X9X12 by 3X6X10. The factor 3 is rejected from both dividend and divisor, by canceling the 3 below the line and dividing the 9 above the line by 3, cancel- ing the 9 and writing the quotient 3 above it. The factor 3 is again rejected, canceling the 3 above the 9 and dividing the 6 below the line by 3, canceling the 6 and writing 2 below it. The factor 2 is then rejected, canceling the 2 below the 6 and dividing the 8 above the line by 2, cancel- ing the 8 and writing 4 above it. The factor 2 is again rejected, canceling the 4 above the 8 and writing 2 above it, and canceling the 10 below the line and writing 5 below it. This leaves 2X12, or 24, above the line, and 5 below it ; dividing, we have the result, 4|. 170. Rule. — Write the factors of the dividend above a line, and the factors of the divisor below it. Cancel all the factors which are found in both dividend and divisor, and divide the product of the remaining factors in the dividend by the product of the remaining factors in the divisor. WRITTEN EXERCISE 171. 1. Divide 12X15X18 by 4X6X9. 2. Divide 15 X 14X22 by 3 X 11X7. 3. Divide 18X26X28 by 14x9x13. If. Divide 20x32x48 by 12x10X15. 5. Divide 72x36x24 by 18X48X16. 6. Divide 56X77X84 by 63x42x33. 7. Divide 180X108X270 by 120X18X45. 5. Divide 121x132x84 by 144x66x22. 9. Divide 75X320X160 by 20X180X150. 10. Divide 240X120x350 by 180X75X80. 11. Divide 360X146X145 by 365X100X87. 12. Divide 1728X195X363 by 144X65X36. 2 V J? X Jp X oT 7 48 CANCELATION 49 WRITTEN PROBLEMS 172 . 1 . If 10 apples cost 35 cents, what will 14 apples cost? 2 . If 7 barrels of flour cost $84, what will 10 barrels cost? 3 . A grocer bought 324 bushels of potatoes at 40 cents per bushel, and paid in molasses at 60 cents per gallon. How many gallons did it require? 4 - How many sheep worth $2.25 per head must be given for 12 tons of hay at $15 per ton ? 5 . What must be paid for shipping 896 pounds of iron at $6 per 2240 lbs.? 6 . How many bushels of wheat at 90 cents per bushel must be exchanged for 120 bushels of oats at 45 cents per bushel? 7. How many times can a cask, holding 3 gallons, be filled from 27 bottles each holding two pints ? 8 . How many cords of wood at $4.50 per cord should be given for 18 barrels of flour at $5 per barrel ? 9 . A sold 15 barrels of pork, each containing 180 pounds, at 18 cents per pound, and received in payment a number of hogsheads of molasses, each con- taining 90 gallons, at 60 cents per gallon. How many hogsheads did he receive ? 10 . A dairyman sold 36 firkins of butter, each containing 56 pounds, at 25 cents per pound, and with the money bought 30 pieces of calico at 8 cents per yard. How many yards in each piece ? 11 . How many bushels of apples worth 45 cents per bushel must be given in exchange for 60 pieces of muslin, each containing 48 yards at 9 cents per yard ? 12 . A milkman has 20 cows, which give daily 6 quarts each, which he sells at five cents per quart. How many hams, each weighing 15 pounds, at 16 cents per pound, can he buy with 4 days’ milk ? 13 . How many barrels of apples, each containing three bushels, at 35 cents per bushel, must be given for 16 barrels of sugar, each containing 240 pounds, at 7 cents a pound ? 14 . If a laborer receives 15 cents per hour, how many days of 10 hours each will it require him to earn the value of 6 barrels of flour at $4.50 per barrel ? 15 . How many crates of eggs, each containing 30 dozen, at IS cents per dozen, should be given in exchange for 120 rolls of paper, each containing 15 yards, at 32 cents per yard ? 16 . A dealer bought 625 pounds of cheese at 12. cents per pound and exchanged it for five loads of corn, each load containing 18 bags, and each bag 3 bushels. What was the price per bushel ? 17 . A cotton dealer exchanged 24 bales of cotton, averaging 410 pounds per bale, worth 12 cents per pound, for 48 loads of apples, each load containing 15 bags of 2 bushels each. What was the price per bushel of the apples ? 18 . A coal merchant exchanged 5 car loads of coal, each containing 15 tons, at $4.80 per ton, for 5 cases of children’s shoes at $3. How many pairs in each case? COMMON FRACTIONS 173. A fraction is one or more of the equal parts into which a unit is divided ; as, one half (f), two thirds (-§•). 174. A common fraction is one which has a written denominator, and is distinguished from a decimal fraction, which has its denominator indicated by the decimal point. 175. Common fractions are divided into 'proper, improper, simple, compound , and complex. 176. A proper fraction is one whose numerator is less than its denominator ; as, three fourths (f). 177. A n improper fraction is one whose numerator is equal to or greater than its denominator ; as, five fifths (J-), four thirds (-f). It is simply a fractional form of a whole or mixed number. 178. A simple fraction consists of a single fraction ; as, five sixths (f). 179. A compound fraction is a fraction of a fraction ; as, -f of f of f ; 1 of i 8 U1 5 - 180. A complex fraction is one whose numerator or denominator, or both, 31 3 2 contain fractions; as, ~ * ■ 4 % x 181. The terms of a fraction are the denominator and the numerator ; the denominator is the part written below the line and indicates the number of parts into which the unit is divided ; the numerator is the part written above the line and indicates how many parts are taken. 182. A mixed number consists of a whole number and a fraction ; as, 21 WRITTEN EXERCISE 183. Express in figures : 1. One half. 2. One third. 3. Three fourths. 4- Five sixths. 5. Seven eighths. 184. What do the following 1 -2- 1. 3 . 9 3 4 • 5. £ T £ 6. Eight ninths. 7. Nine tenths 8. Ten twelfths. 9. Eleven fifteenths. 10. Fifteen sixteenths. expressions mean ? X _9_ 71 4 o. 10 . /. 17 . 6 11 £ 15 . U. 12 . o. 33. Q 3i 3 5 * io- if 50 FRACTIONS 51 REDUCTION OF FRACTIONS 185. Reduction of fractions means changing their form without changing their value. 186. There are five different steps or operations in the reduction of frac- tions : ( 1 ) reducing improper fractions to whole or mixed numbers ; ( 2 ) reducing mixed numbers to improper fractions ; (3) reducing fractions to higher terms ; (4) reducing fractions to their lowest term ; (5) reducing fractions to their least common denominator. 187. To reduce improper fractions to whole or mixed numbers. Example. — Change to a mixed number. £ 2 ) 4 / <6 7 y 2 2 / 4 y / 2 f / y Result 14-|-f 188. Rule. — Divide the numerator by the denominator. WRITTEN EXERCISE 189. Reduce to whole or mixed numbers : i. n 6. 41. 11 411 16. 141 2. V- 7 . 41 . 12. 4iL. -/O' 12 0 5 ■ 6 1 3 ¥■ X 34. O. 3 . 13. 4&1. IQ 3967 10 • 215- 4- ¥• 9. 404. n. w- IQ 4867 ±v. 219 . 5. 10. 424 15. -W. ran 6 7 8 9 1 ZU. 3 8 9 - 190. To reduce mixed numbers to improper fractions. Example. — Change Bd^- to an improper fraction. 3 .2 3 / 0 2 & r s_ y p y Result 191. Rule. — Multiply the whole number by the denominator , add the numerator to the product, and write the sum over the denominator. WRITTEN EXERCISE 192. Change to improper fractions : 1. 81 5. 7ii. p. 73if- 2. Ill 6. 9fi 10. 89ff-. 3. 13f. 7. 2311. u. IO 8 J 3 . l 16 t V 8. 45fi. 12. 13711. 13. 227if. H. 524f|. 15. 8971-fl. 16. 967114. 52 FRACTIONS 193. To reduce to higher or lower terms. 194. Multiplying or dividing both numerator and denominator of a fraction by the same number changes the form but not the value of the fraction. 195. To reduce fractions to higher terms. Example. — Change f to thirty-sixths. 7 _ f 3 6 196. Rule. — Multiply both terms of the fraction by such number as will raise its denominator to the required denominator. 197. Change the following by inspection : 1. | to 9 tbs. 11. f and f to 21sts. 2. f to 16ths. 12. f and f- to 18ths. 3. f to 15ths. 13. f and f to 28ths. 1 f to 18ths. U- J and f to 12ths. 5. f to 28ths. 15. f and t 9 t to 55ths. 6. f to 24ths. 16. f-, f and Y 2 to 36ths. 7. f to 27ths. 17. f, f and to 70ths. 8. ff to 30ths. 18. f, i> A an( i if to 48ths. 9 ff to 44th s. 19. ts> if and if t0 84ths - 10. fi to 36th s. 20. §> T 9 > if an( ^ ri to 156ths. 198. To reduce fractions to lowest terms. Example. — Reduce f-f to lowest terms. J_L <5 if 6 3 Dividing both numerator and denominator successively by 9 and 2, we obtain the result §. 199. Rule. — Divide both terms by their greatest common divisor. Or divide both terms by such numbers as will exactly measure them until the numerator and denominator are prime to each other. ORAL EXERCISE 200. Change by inspection to lowest terms: 1 -S- 1 • 12 - 6. if. n. M- 16. 1 9 o 7 * O 8 2 T* 'y 18 / . 2 4 • 12. ff. 17. 1 1 TT 2 - Q 9 36- 8 AS. ° ' 2 4' 13- if- 18. 132 1 44' / I 2 4- T 6 ’ 0 14 3~5 ’ U. ff. 19. 9 9 1 0 S' X 6 o. 9 . 10. if. 15- fi- 20. 7 9 1* WRITTEN EXERCISE 201. Change to lowest terms : 1 6 6 16 5- '2 7 6 315' 11 A3 - 2 11 - 648' 16. 720 115 5' a 35 /S - TITS'' 7 19 2 ' ' 2 16' 16) 8 64 9 3 6 ' 17. 1009 2 0 0 2 ' 0 7 4 5 7 6 ' O 7 6 8 9 6 0' 1® 7 8 0 1092' 18. 1134 1 2 8 7' 1 1 1 9 4- 13 3' Q 7 9 2 // 1512 19. 12 8 7 1 0 0 8 ' 35 2 8 - 1 7 2 S ' c 8 0 140' 10 392 5 0 4' ir 8 19 10 ■ 3 0 25' 20. 3025 17 0 17' FRACTIONS 53 ADDITION OF FRACTIONS 202. The denominator determines the fractional unit, and since only like units can be added or subtracted, there must be a common denominator. 203. The least common denominator of fractions is the least number into which all the denominators can be evenly divided. This may often be deter- mined by inspection, as 12 is the least common denominator of ^ 204. When the common denominator is not seen readily, select the largest denominator, observe whether the next largest will divide it evenly; if not, select the smallest number into which each of these two will divide evenly ; with this number take the third largest denominator and proceed in the same manner until you have a number which all denominators will divide evenly This number is the least common denominator. 205. When the least common denominator cannot be determined by inspection, it can be found by getting the least common multiple of the denom- inators, as indicated in the example. This may be abbreviated by excluding all denominators that will divide evenly into a larger denominator. 206. To find the least common denominator. Example. — f, ¥ V Regular Method Abbreviated Method 2) / b 3 b 2j i/ — y — / r f — 9 / 2 — J UJ / — 3 x x 3 7 7 1 1 5 > 8 > 1 0 ’ 12 - / 2 . 1 _1 5_ v- 3i 6i in 12- denominator of the following fractions by 5. 6 . 7. 8 . 3 7 3 1 1 O’ 8) 10’ 1 2 * .3 3 _ 5 . 11 4 > 7 ’ 8 ’ 14 - 1 3 5 5 1 1 2 > 4 ’ 7 ’ 14 ' 16 - 2 3 5 11 _ 9 _ 3 ’ 4 ) 8 ) 12 ’ 16 - 54 FRACTIONS 208. To add fractions. Example. — Add ■§, f and §. 105 L. C. D. 6 3 0 -?&- / J3 / 1. TT and H- 2. 3 5> 9 ailU 1 8* 8. TT> ¥’ 1 and T2- 3. 1 2’ T. T aild $> 9. 7 5 8 o n r 5 3 8’ T’ 9 d ' llu 1 1- 4~ 2 3' f and -sV 10. 11 2 17 a n r] 23 1 8 ’ ¥> 24 t,iiu 36- 5. 4 7’ td A aud A- 11. 65 43 11 5 r mr 12 84’ 48’ 2D 14 1:111 u T- 6. 4 7 8 Q-prl 7 15’ 8> 9 “ 1JU 54' 12. 2 3 8 9 10 ot,^ 11 3’ 8’ 9’ 10’ 11 dlla 1 2 212. To add mixed numbers. Example. — Add 4J, 3§ , 5£ and 4|. 12 L. C. D. ^ 3# = r S% = /0 - 3 / _ (7 Least common denominator of the fractions, 12 ; fractions added equal 2§ ; adding 2 in with whole numbers gives the result 18|. 2 33 = 2% / rfi / 2 Note. — A dd the fractions separately, then combine the result with the whole numbers. ADDITION OF FRACTIONS 55 WRITTEN EXERCISE 213 . Add the following: 1. 4f, 3J and 7^-. 2. 8 §, 7f and 9J. 3. 34 6 f and 8 ^. f- 9f, 5f and 7 X V- A 6 i 7|, 4i and 84. 6. 3f, 6 f, 8 i and Ilf 7. Hi, 131 gi and 12*. A 154, 7i 3i and 33f. 9. 23i, 48| and 67f. 10. 87|, 1264 and 168& 11. 554, 67f and 83f. 93-3-, 6 Sf and 75-f. IS. Find the sum of 357f bu., 189f bu., 697-f bu. and 378^- bu. 11 Add 897f ft., 10S9J ft., 8784 ft., 39541- f t . an( j 689f ft, 15. A merchant had carpets in quantities respectively as follows : 27Sf yds., 348J yds., 195f yds., 482f yds. and 378J yds. How many yards in all? WRITTEN PROBLEMS 214 . 1. A merchant sold from a piece of muslin 16f yds., 27f yds., 47f yds. and there still remained unsold 25jf yds. How many yards were in the piece? 2. A farmer sold ten loads of hay weighing as follows : 37^- cwt., 454 cwt., 39fi cwt., 38^ cwt., 47f cwt., 314 cwt., 34f£ cwt., 41fg cwt., 33|-f cwt., 50f| cwt. Find total weight. 3. A bought f of a yard of cloth for $f, f of a yard for $41 of a yard for $f, and -| of a yard for How much money did he spend? f. A family burned in December 3f tons of pea coal and 14 tons of nut coal ; in January, 2f tons of pea coal and 1 jf- tons of nut coal ; in February, 3f tons of pea coal and 2| tons of nut coal. How much of each kind was burned during the three months? 5. A boy worked 5f days for S4f, 7J days for $5f, 3f days for $3f, 6 § days for $5^-, and Ilf days for $12f. How many days did he work, and how much money did he receive ? 6. I purchased 5 hams, net weight of each, being respectively, 17f lbs., 23f lbs., 31f lbs., 2241 lbs., 19f lbs. Find total weight. 7. Find the weight of 5 cars of coal of 13f tons, lly^- tons, 12ff tons, 14^- tons, and 9f tons. 8. Four cars of iron weighed respectively, 28^- tons, 30-^- tons, 26325 - tons and 294 tons. What was the total weight? 9. A merchant received money in several sums, as follows : £102J, £79J, £83^-, £26-2^-, £5 and £62f. What was the sum of these receipts ? 10. Four wheelmen rode in a day respectively, 125f miles, 107f miles, 89f miies and 95§ miles. What was their total mileage for the day ? 56 FRACTIONS SUBTRACTION OF FRACTIONS 215. To be subtracted, fractions must have a common denominator. 216. To find the difference between two fractions. Example. — Find the value of -§ — f. 63L.C.D. Y&3 217. Rule. — Reduce to a common denominator, and write the difference of the numerators over the common denominator. ORAL EXERCISE 218. By inspection find the value of 1 . 1 2 1 3* 5 . 9 1 0 2 5- 9 . 2 . 3 4 2 3* 6 . 7 8 5 1 2- 10 . 3 . 5 _ 6 1 2 ’ 7 . 7 8 G 7 ■ 11 . 1 7 8 3 4* 8 . 1 7 1 8 2 3‘ 12 . 2 4 3 15 - 1 1 5 1 2 1 6 ' 1 7 3 2 0 4 ’ 8 5 9 6 - WRITTEN EXERCISE 219. Perform the following subtractions : 1. From pf take f-. 2. From if take f . 3. From ff take f. If. From ff- take f-. 5. From -ff take T 5 6 7 8 2 -. 6. From ff take -f. 7. From f§ take -ff. 8. From f-f take 9. From 1 f 5 z g - take f. 10. From ^ take ff. 11. From -fff take iff. 12. From ff take ff. 13. From ff take ff. U- From fff take 15. From f§f take f. 16. From ff take ff. 220. To find the difference between two mixed numbers. Example.— Find the difference between 14f and 3f. /7ffY7 = = / //y 7 =y 2 , 3'A = 3 7 A' = 3 In practise it is sufficient 3 '/j 74 to write : 7 / ^ / 0*/*' 221. Rule. — Subtract the fractions and the whole numbers separately , then com- bine the results. Note. — I f the fraction of the minuend is smaller than the fraction of the subtrahend, borrow one unit from the whole number, reduce it to an improper fraction and add it to the fraction of the minuend. WRITTEN EXERCISE 222. Find the difference between 1. 6-f- and 2^. 2. If and f. 3. 3f and If. If. 17 and If. 5. 13f and 2f. 6. 15-f and 7-f. 7. 237ff and 159f . 8. 587ff and 258-f. 9. 67Sif and 399ff 10. 1037-^ and 674yf. 11. 367^ and 287ff. 12. 973f- and 5S7^%. SUBTRACTION OF FRACTIONS 0/ WRITTEN PROBLEMS 223 . 1 . What is the value of (f+f — 1)+§? S. Find the value of 3f+5f-+34 — 2-f. 3 Find the value of (16§-f-124) — (5^+6^-) What is the value of 564 — 38§ + 19^ — 84? 5 . From the sum of 144 and Ilf take the difference between 35f and 264- 6. From take To 3 l T T and add 14. 7. From a barrel containing 37f gallons, 29| gallons leaked out. How much remains in the barrel ? 8. A farmer sold 560^- bu. from a bin containing 927-f bu. How many bushels still in the bin? 9 . A lady bought a shawd for $13£, a hat for $12f, and a dress for $25f, and in payment gave six ten-dollar bills. How much change should she receive? 10 . A man’s family expenses for the month of January are: rent, $25f, clothes, $17f, coal, $4f, groceries, $15^4. If he receives 8874 a month, how much can he save ? 11. I bought 9J tons of coal and had it delivered in 5 loads, the first con- taining If tons, the second If tons, the third 14 tons, and the fourth 1J tons. What did the fifth load contain? IS. A had 375 lbs. of wire and sold 25f lbs. to B, 75§ lbs. to C, and 37^- lbs. to D. How much had he left ? 13 A merchant sold from a piece of cloth 37§ yards to A, 23f to B, and 12§ to C, and there remain 27/g- yards unsold. Find number of yards in the piece. Ilf. From a piece of muslin of 74 yds., there were sold to different persons 64 yds., 84 yds., 124 yds., 15f yds. and 27f yds.; how many yards remain unsold? 15 . From a tract of land of 472|- acres, there were sold three farms respec- tively, 112| acres, 220f acres and 88f- acres. How many acres were retained ? 16 . In locating a park, tracts of land were bought as follows : 33|- acres, 178§ acres, 82-f- acres, 13f acres and 3f acres. Roads and building sites occupied 18| acres, and woodlands, 87f acres; the remainder was in grass. How many acres in grass? 17 . From 25000 bushels of wheat, there were taken four carloads of 8374 bushels, 936J bushels, 1006f bushels and 8834 bushels. How many bushels were left? 18 . Atrip of 3230 miles was made as follows: 4784 miles the first day, 518f miles the second day, 490| miles the third day, 537f miles the fourth day, 487| miles the fifth day and 427| miles the sixth day. How many miles were traveled on the seventh day? 19 . From a hogshead of molasses containing 72f gallons, were drawn 9§ gallons, 15f gallons, 13| gallons and 27f gallons. How many gallons remain? SO. A wheelman traveled the first day 1264 miles, the second day 137| miles, the third day 117f miles, the fourth day 27f miles, the fifth day 141 f miles. How many miles must he ride to complete 752^ miles? 58 FRACTIONS MULTIPLICATION OF FRACTIONS 224. Multiplication of fractions is the process of finding the product of two factors when one or both of them are fractions. Note. — W hen any number is multiplied by a number greater than 1, whether a whole or mixed number, the product is greater than the multiplicand ; but when a number is multiplied by a fraction less than 1, the product is less than the multiplicand. 225. To multiply a fraction by a fraction. Example. — What is the cost of T 7 y of a yard of cloth at per yard? -4- y-Z- / o / X & 3 = f2_L / XO 4 0 02 n 3 / o 40 u 226. Rule. — Multiply the numerators for a new numerator and the denominators for a new denominator. Note. — T he work may often be abbreviated by cancelation. WRITTEN EXERCISE 227. Find the product of 1. 2 v 3 3 y\ 4 . l 2 4 v 7 4 9 v 9- 7. 9 V 6 8 17-^84- 10. 2 V 7 V 3 X A 2 A -5 . 2. 4 V 9 5 16- 5. 1 1 V 4 8 T6 X '5ir- 8 6 v 1 19 v 2 4' 11. 4 v 3 V 2 8 9 A 7 A 33 . 3. 7 v 1 2 8 A 15- 6. 3 V 1 1 11 A 7 5- 9. 3 v 5 V 6 4 A G A 7 . 12. 5 V 3 V 9 6 A 8 A x 5 . Note. — C ompound fractions are connected by the word of and are simplified by the operation of multiplication. Find the product of 13. f offXlf. U. f of * off 15. fofHoftfXf 16 *X«X*of*. J'y 4 v 7 8 v 13 2 \ / 1 7 1 ' • 3 3 A 9 9 A 5 5 5 /N 3 2 * 18. -f of 4 of ut j^-. 19 - tfXtfXtt. Of) 4 , 6 S V 1 1 / 1 o /SU - jAj x 9 - A T3 A T9’ 228. To multiply a whole number by a fraction. Example. — Multiply 57 by f. ^7/ 2 FbjT <3 / /? 02 • /9 £ 3/% 3 229. Rule. — Multiply the whole number by the numerator, and divide the product by the denominator. 230. Multiply 1. 4 by 21. 2 f by 16. 3. 4 by 17. f I by 26. WRITTEN EXERCISE 5. by 38. 6. by 68. 7. by 25. 8. |4 by 75. 9 ff by 165. 10. by 85. 11. 66 by f. 12. 14 by Jy. MULTIPLICATION OF FRACTIONS 59 231. To multiply a mixed number by a whole number. Example. — Multiply 229f by 7. 5 2 / & o / 7 times |=^=4f ; write f- and carry 4. 232. Rule. — Multiply the fraction by the whole number , adding the result to the product of the whole members. WRITTEN EXERCISE 233. Multiply 1. 337f by 9. 5. 397f by 38. 2. 426f by 15. 6. 439!! by 45. 3. 568f by 21. 7. 897f by 87 1^. 287-f- by 35. 8. 989f by 237. 234. To multiply a mixed number by a mixed number Example. — Multiply 37J by 354. 9. 1378|f by 10. 3789fi by 11. 4288|| by 12. 506314 by 235. Rule. — Multiply , separately, the whole 37X:T numbers, each fraction by the opposite whole number, (3 ) ^35 and the fraction by the fraction ; add these products. (4) £X£ 423. 629. 325. 724. 3 / r 3 / / / / 3 3 / 14 - // 4 / 236. Multiply 1. 56f by 271. 2. 37f by 25|. 3. 87f by 19f. I 437f by 18f. WRITTEN EXERCISE 5. 178f by 15f. 6 267* by 37|. 7 468fby35|. 8. 276H by 127|. 9. 468-f by 2391. 10. 5671 by 256£. 11. 6081 by 19711 12 . 763* by 230*. WRITTEN PROBLEMS 237. Find the cost of 1. 385! pounds of paper at 2-f cents per pound. 2. 378f pounds of beef at 7f cents per pound. 3. 878f yards of muslin at 9! cents per yard. Jf. 676£ bushels of timothy seed at $5.62! per bushel. 5. 173f dozen eggs at 18£ cents per dozen. 6. 375|- gallons of wine at $2.37! per gallon. 7. 739f quarts of berries at 5! cents per quart. 8. 378 T 3 T acres of land at $127-f per acre. 9. 39|- peeks of apples at 37f cents per peck. 10. 837f pounds of coffee at 33f cents per pound. 60 FRACTIONS DIVISION OF FRACTIONS 238. Division of fractions is the process of finding the quotient when the dividend or divisor, or both, are fractions Note. — T he quotient is larger than the dividend if the divisor is less than 1, and smaller if the divisor is greater than 1. 239. The reciprocal of a number is 1 divided by that number; as, the reciprocal of 5 is i ; of 9 is f . 240. The reciprocal of a fraction is 1 divided by that fraction, and simply inverts the fraction ; as, the reciprocal of f is f ; of % is f 241. To divide a whole number by a fraction. Example. — Divide 2 by f. It will be observed by the following steps in reasoning that the divisor naturally inverts itself. To divide by f is to divide by J of 2. If we divide 2 by 2. we have as a quotient 1. A divisor I as great would produce a quotient 3 times as great, that is, 3 times 1 or 3 ; or ^ is contained 3 times as often as 1, or 3X2 or 6 times, and f is contained 2 as often as 4 or £ of 6 times or 3. One may be expressed under the whole number, and the division performed similar to a fraction by a fraction. 242. Rule. — Multiply the dividend by the divisor inverted. (In other words, multiply the dividend by the reciprocal of the divisor.) WRITTEN EXERCISE 243. Divide 1. 1 by f. I 15 by if 7. 39 by if. 2. 9 by f. 5. 20 by ff. 8. 45 by qf. 3 16 by f 6. 25 by ff 9. 168 by ff. 10. 224 by ff. 11. 375 by ff. 12. 670 by ff. 2 _x_ / 3 _ 2 _ 3 ^*4 OZ‘ X X A. 2 3 _ 3 2 b — 3 2 244. To divide a fraction by a whole number. Example 1. — Divide f by 2. x _;_ 2=f or Dividing the numerator divides the fraction. Example 2 —Divide f by 3. f- -:- 3=Yy Or f-t-f =f Xf . Multiplying the denominator divides the fraction. c t \C 245. Rule. — Divide the numerator by the whole number, or 'multiply the denom- inator by the whole number. DIVISION OF FRACTIONS 61 WRITTEN EXERCISE 246. Divide 1. | by 2. 4- \ by 4. 7. ff by 5. 10. if by 10. 2. f by 3. 5. m by 9. 8. 44 by 7. 11. 44 by 15. 8. 1 by 2. 6. ff by 8. 9. & by 9. 12. ff by 25. 247. To divide a fraction by a fraction. Example. — Divide f by f. To divide by a fraction, the principle is the same, | Xf=f =lf. whether the dividend be a whole number or a fraction. 248. Rule. — Multiply the dividend by the divisor inverted. WRITTEN EXERCISE 249. Divide 1. f by f. 4- f by f. 7. ff by f. 10. If by f. 8 i by f. 5. if by 8. ff by f 11. 2f by H. 3 yV by f. d. by T 9 T . P. ff by yf. 12. 3£ by If. 250. To divide a mixed number by a whole number. Example. — Divide 37 J by 2. //z = 37 divided by 2 gives 18, with a remainder of 1. Adding this remainder to the fraction of the divi- dend, and reducing to an improper fraction, we have f. Dividing § by the divisor we have f, which completes the quotient. 251. Rule. — First divide the integral part of the mixed number ; if there be a remainder, add it to the fractional part ; then divide the fractional part. 252. Divide 1. 75J by 4. 2. 137f by 9. 3. 148-f by 19. 4. 267f by 23. WRITTEN EXERCISE 5. 3971 f by 27. 9. 6. 4781f by 33. 10 7. 5678f by 35. 11. 8. 6438f by 39. 12. 87378f by 129. 98375fi by 137. 876341 by 147. 93675 t 4 x by 165. 253. To divide a whole number by a mixed number. Example. s. — (1) Divide 243 by 4f ; (2) 7854 by 30J. tffej 2 y S tf _4_ 3 O f / ¥ = /~ 3 ) 7 2 f 7 ^ IH m 7 / 4 77 -y f s-i/ y l / _ z y s f _2 s f % 7 7 / / / /~ " 254. Rule. — Reduce both dividend and divisor to improper fractions having a common denominator (by multiplying both by the denominator of the fraction in the divisor), then divide the numerator of the dividend by the numerator of the divisor. Or, give the whole number a denominator of 1, invert the divisor and proceed as in multiplication. 62 FRACTIONS 255. Divide 1. 46 by 64. 2. 32 by 3§. 79 by 7$. I 122 by 9$. WRITTEN EXERCISE 5. 364 by Ilf 6. 892 by 13f. 7. 478 by 24$f 8. 572 by 3114. 9. 1219 by 56fl. 10. 3924 by 42if 11. 4685 by 74 ¥ 3 T . 12. 62830 by 126$£ 256. To divide a mixed number by a mixed number. Example. — Divide 358$ by 18f. / rfij 3 j r'A / 7 / 2 . 2 2 S) U 3 0 0 ^ 2 2 S 2 0 S 0 2 0 2 S 2 S 2 2 ) / S' 7 0 O . 3 / / 4 % Z / S' / 6 % EXPLANATION. (3) 164=4 of 100. (4) 84=* of 100. 7 41 a 3 i 0 0 3_ & 3 3 f f f c y 3 270. Rule . — Add or subtract the fractional part greater or less than 10, 100, etc. 271. Multiply 1. 574 by 1124. 2. 872 by 874- 3. 196 by 116$. b 572 by lllf WRITTEN EXERCISE 5. 472 by 834. 6. 784 by 75. 7. 847 by 114$. 5. 686 by 85f 9 384 by 1334. 10. 784 by 66$. 11. 1210 by 109JL. 12. 750 by 125. 272. To divide a whole number by a mixed number that is %, y%, of 10, 100, etc. Examples. — (1) Divide 960 by 124; (2) 385 by 334- f 6, 0 3 ? s r 3 7 ^ r / / s s CL / O o // z o 273. Rule. — Multiply the dividend by the denominator of the equivalent frac- tion. Mark off one right hand figure if the divisor is a fractional part of 10 ; two, if a fractional part of 100, etc. The remainder will be tenths if a fractional part of 10, and hundredths if a fractional part of 100, etc. 274. WRITTEN EXERCISE 1. Divide 432 by 124- 2 Divide 276 by 84. 3. Divide 595 by 16f. If. Divide 348 by 14$. 5. Divide 384 by 334- 6. Divide 189 by 124. 7. Divide 297 by 94 t- 8. Divide 1462 by 1334. 66 RELATION OF NUMBERS RELATION OF NUMBERS 275. Finding the relation which one number bears to another is a process of division, and the result is expressed in times either integrally or fractionally. Ex. — What is the relation of 10 to 5 ? Ans. — 10 is 2 times, or twice 5. Ex. — What is the relation of 5 to 10 ? Ans. — 5 is one-half 10. Note. — O nly numbers of the same denomination can be thus compared MENTAL PROBLEMS 276. 7. What is the relation of 8 to 4? Of 12 to 6 ? Of 36 to 12 ? Of 12 to 84 ? Of 72 to 6 ? 2. What is the relation of 96 to 8? Of 11 to 121 ? Of 132 to 12 ? Of 132 toll? Of 132 to 6? Of 8 to 144? 3. What part of 8 apples is 6 apples ? f What part of 16 bushels is 10 bushels ? 5. What part of 35 gallons is 20 gallons ? 6. What part of 42 yards is 30 yards? 7. What part of 75 acres is 35 acres? 8. What part of 44 quarts is 32 quarts ? 9. What part of 10 is 24 ? 3J? 5? If? If ? If? 10. What part of 100 is 12f? 25? 16-|? 14f? SJ? 9^? 33f? 20? 6f? 111? 3i? 24 ? 374? 624? 11. What part of 1000 is 250? 333i? 125? 166f? 83f? 200? 12. What part more than 10 is 12|? 13f ? 15? What part less than 10 is 7i? 6| ? 8f? 13. What part more than 100 is 125 ? 133f? 1124? 116f? 120? 108f? 150 ? 1111 ? 109 ^? 1141 ? n. What part less than 100 is 75 ? 87| ? 66f ? 83J ? 90 ? 85f ? 8Sf ? 90if ? 80? 75. 6f is what part of 13i ? Of 20? Of33f? Of66§? Of 8? 16. If 4 yards of silk cost $12, what will 8 yards cost? Solution. — 8 yards will cost twice as much as 4 yards, or $24. 17. If 10 pounds of sugar cost 50 cents, what will 100 pounds cost? 18. If 7 yards of chintz are worth 91 cents, what is the value of 21 yards? 19. If 8 pairs of shoes cost $15, what are 32 pairs worth? 20. If 12 oranges cost 35 cents, what is the value of 36 oranges? 21. If 15 horses cost $800, what are 75 horses worth? 22. If 9 men can build 17 rods of wall in a given time, how many rods can 54 men build in the same time? RELATION OF NUMBERS 67 23. A man bought 11 pigs for $9, how many could he have bought for $45 ? 2Jf.. 4 men plowed 28 acres in 6 days; how many acres could they plow in 18 days? 25. How much will 5 books cost if 25 books cost $15 ? Suggestion. — 5 books are 1 of 25 books. 26. What cost 7 tons of hay if 28 tons cost $168? 27. If a wheelman rode 510 miles in 6 days, how far could he ride in 2 days ? 28. If 84 bushels of wheat are worth $77, what is the value of 12 bushels? 29. If 63 gallons of molasses are worth $27, what will 7 gallons cost ? 30. A man had $30 and lost $25, what part of the original amount has he left ? 31. A man had $35 and received $70 additional; how does his present amount compare with the first amount ? 32. A man has $45. He spends $9. What part of the original amount has he spent ? What part remains ? S3. In a bin of 60 bushels of wheat, 20 bushels are damaged. What part of the wheat is sound? WRITTEN PROBLEMS 277 . 1 . A contributed $4500 to a charitable institution, B $7500, and C $3000. What part of the total did each contribute? 2. A grocer bought 560 pounds of Rio coffee and 440 pounds of Java. What part of the whole amount is each ? 3. A merchant bought 3 rolls of carpet. The first contained 360 yards, the second 560 yards, and the third 840 yards. What part of the total in each roll ? J. Jones has a capital of $2250 and gains $750; what part of his present worth is his gain ? 5. A merchant’s business expenses outside of rent for a year amount to $3500 and his rent is $700; what part of his total expenses is his rent? 6. A shipper sends goods over one railroad 600 miles and over another 400 miles ; if his freight charges are $30, how much should each road receive? 7. In a shipment of grain there are 6500 bushels wheat and 3500 bushels oats ; what part of the shipment is wheat? What part is oats? 8. The receipts from certain agricultural products amount to $3000 from hay, $2500 from potatoes, and $2000 from straw ; what part of the whole amount was received from each ? 9. In a partnership A invested $4000, B $3000, and C $2000 ; what part of the gain should each receive? 68 RELATION OF NUMBERS 10. A grocer mixed 40 pounds of Mocha coffee with 120 pounds of Java coffee ; what part of the mixture is Java? What part is Mocha? 11. At a partnership closing, D received $2000 as his share of gain, E received $3000, and F $1000; what part of the gain did each receive as his share ? 12. Two kinds of coffee are blended at the rate of 2 pounds of one to 3 pounds of the other; how many pounds of each in 180 pounds of the mixture? 13. The relation of $2500 is to a second sum as 5 to 9 ; what is the second sum ? Ilf.. A merchant’s assets, $15200, are to his liabilities as 5 to 2 ; what are his liabilities ? 15. In a shorthand school of 220, the ladies are to the gentlemen as 7 to 4 : how many of each sex in the school? 16. A business man owes three creditors $2600 in the ratio of 6, 4 and 3; what is due each creditor? 17. A business man withdrew from bank $1250, which was f- of the amount in bank. How much had he in bank at first? 18. I pay $156 for a horse, which is- f of the amount paid for a carriage ; what did both cost me ? 19. A cycler wheeled 135 miles, which was ■§■ of the distance he traveled by steamer; what was the length of his trip ? 50. If f of a vessel cost $15000, what is the value of -f of it? 51. If T 7 g- of an estate is worth $101500, what would be the portion of an heir to f of it? SS. Bought f of a mill for $15250 ; what is the value of 4 of it at the same price ? S3. If ^ of a factory is valued at $14014, what is -f- of it worth ? Slf. If I spend -f of $15, the remainder is what part of my friend’s money, who has $25 ? 55. A has $45 and B has f as much plus $22; what part of A's money equals B’s? 56. What is the relation of 124 to 374? Of 374 to 124? What is the relation of 25 to 874? 57. A man having of a ton of coal sold § of it ; what part of a ton remained ? 58. A merchant owned § of the stock of a store and sold f- of his interest, what interest in the entire stock did he retain ? 59. A and B each had of a ton of hay ; B sold A f of his hay r ; what part of A’s ecptals B’s after the sale? REVIEW PROBLEMS IN FRACTIONS 69 REVIEW PROBLEMS IN FRACTIONS 278. 1 . Simplify -}-■ f • f of 6 2. Add 27f, 671 38f and 3. What is the value of of ff-f 7-|X3-^ of yf- ? A The product of two numbers is 405 and one of them is 27§. What is the other ? 5. There are two numbers whose difference is 49|- and one number is of the other; what are the numbers? 6. If a man travels 4 miles in -f- of an hour, how far would he travel in 2f hours at the same rate ? 7. At $f- a yard, how many yards of silk can be bought for $10f? 8. If 48 is f of some number, what is § of the same number ? 9. A gentleman owning T 7 g- of a vessel sold ^ of his share; how much of the vessel does he now own ? 10. If a person contracts to finish a piece of work in 27 days, what part of it should he have done in 12 days ? 11. A merchant lost f of his capital, after which he gained $700 and had $2500 ; what amount did he lose ? 12. A and B own a steamboat ; A owns yk- and B the remainder of the interest. If $7000 has been gained during a season, how much should each have ? 13. Divide $4500 between two persons so that one shall have j- as much as the other. Ilf.. Suppose the cargo of a vessel to be worth 2-f times the vessel ; what is the value of the vessel if the cargo is worth $17500? 15. If 8|- tons of hay cost $100, how many tons can be bought for $125 at the same rate ? 16. If of 10 bushels of wheat cost $7|-, how much will of 8 bushels cost at the same rate ? 17. A cistern being full of water sprung a leak, and before the leakage was stopped ■§ of the water had run out, but f as much as had run out ran in. What part of the cistern had been emptied when the leak was stopped ? 18. How many suits of clothes requiring 5-| yards each can be cut from 121f yards of cloth ? 19. A tank has two outlet pipes ; one can empty it in 5 hours, the other in 8 hours. In what time will the tank be emptied if both be opened ? 20. A can do a job of work in 6 hours and B can do it in 5 hours ; in what time can both working together do it? 21. A and B together have $1640. If B’s money is equal to |- of A’s, how much has each ? 22. A farmer sold 6 bales of hay, weighing respectively, 112f lbs., 120f lbs., 98f lbs., 137f lbs., 118f lbs., 108-| lbs. ; find cost at If cents per pound. 70 FRACTIONS 23. If 17J bushels of potatoes are equally divided among 7 families of 3 persons each, how many should be given to each family and how many to each person ? 2 If.. A man paid $35 for a colt, f of which cost was of the cost of a horse. What did the horse cost? 25. A grocer bought 107-g- bushels of potatoes and sold 36f- bushels. He afterwards purchased 54f bushels of one man, 37f bushels of another and then sold 47| bushels. How many bushels had he then remaining? 26. What is the total value of 307 1 yards of prints at 6 2 cents per yard ; 407 3 yards cloth at $2.06 1 ; 325 2 yards cassimere at $4.37 2 ? 27. Bought a watch and chain for $140, the chain costing i as much as the watch. Find cost of each. 28. A speculator has 4 of his money in city lots, \ in farm land, and the remainder, $17,230, in cash. Find amount invested in city lots and farm land. 29. How many bushels of apples worth 624 cents per bushel must a farmer exchange for 125f yards of carpet at $1.12 per yard and 33^ yards of cloth at $2| per yard ? 30. From a barrel containing 37J gallons of syrup, a grocer sold 25f gallons. What part of the barrel is unsold? 31. A man’s income is $10f per week, and his expenses are $8|. Flow many weeks will it require him to save $137J? 32. An importer bought a number of bales of silk, each containing 157 yards, at $lf per yard, and sold it at $2f, gaining $623. How many bales did he buy ? S3. A. B and C have a certain sum of money. A has f of the sum, B 4 and C the remainder. Find the sum each has, if A has $42. 3 If. A jeweler engaging in business lost -f- of the sum invested, after which he sold the remaining stock, gaining $632, and received $3162. Find loss. 35. A can do a piece of work in 5 days and B in 6 days. In what time can they do it working together? 36. A and B can do a piece of work in 12 days; B can do it alone in 20 days. In what time can A do it? 37. A Liverpool merchant bought 56 bales of cotton, each containing 3674 lbs. at 16}f cents per lb., and sold it for 19^f cents per lb. Find his gain if the freight amounted to $97.50. 38. How many yards of cloth can a merchant buy for $244, if $f is paid for f of a yard ? 39. A bushel of wheat produces 40 pounds of flour. How many bushels must be taken to the mill to produce 300 pounds of flour, if the miller takes for toll ? IfO. What is the value of 9 pieces of goods containing, respectively, 47. 49 3 . 45 2 , 37 1 , 56 3 , 48 1 , 43 2 , 41 3 , and 46 1 yards, at 62J cents per yard ? Ifl. A man bought f of 321 1 acres of land at $674 per acre, and sold i- of 4 of the land at $764 per acre, and the remainder at $75f per acre. Find his gain. GENERAL REVIEW PROBLEMS 71 42 . How many lemons would have to be sold to gain 20 cents, if bought at the rate of 2 for 1 cent and sold at the rate of 5 for 3 cents? 43. A man bequeathed 4 of his estate to his wife, § of the remainder to his son, and what was still left to his daughter. If the son’s share was $46754 more than the wife’s, what was the value of the estate and the share of each? 44- A man purchased a horse and buggy for $1267.50, paying 4 more for the horse than for the buggy. Find the cost of each. 45. A and B have $1265. If A has f- less money than B, how much has each ? 46. A father divided $1260 between his son and daughter, giving the daughter -f- as much as his son. How much did each receive? 47. A lady spent 4 of her monthly earnings for board, of the remainder for clothes, and still has left $17J. What did she receive per month? 48. In how many days will a boy earn $130 by sawing wood at $f per cord, provided he saws 2J cords per day? 49. How many bushels of potatoes, bought at $f per bushel and sold at 624 cents, will realize a gain of $25? 50. I own \ of a mill worth $8043 and sell f of my share. How much do I still own and what is its value? 51. If I own -f- of a property, and f- of my share is worth $3955, what is the entire property worth ? 52. A, B and C own a factory. A owns B owns 4 more than A. What part does C own? 53. Divide $3000 into two parts so that one shall be f as much as the other. 54 ■ A can do a piece of work in 8 days, B in 12 days, and C in 15 days. After A and B work together 3 days, how long will it require B and C to finish the work ? 55. A grocer buys sugar at the rate of 16 pounds for $1 and sells it at the rate of 12 pounds for $1. How much will he have to sell to gain $1 ? 56. A, B and C engage in trade. A puts in $4000, B $6000 and C $5000. At the end of the year they find they have gained $4500. How much should each receive ? 57. A grocer bought f of a barrel of molasses and sold § of it for the entire cost and the remainder at cost, and gained by the transaction $24. Find price per barrel. 58. It requires 8 minutes for the larger of two pipes, and 12 minutes for the smaller, to fill a cistern. Find time required for both together. 59. 4, of the difference between two numbers is 882. If -f of the larger is equal to the smaller, what are the numbers? 60. A young man agreed to work for a year for $600 and a suit of clothes. His employer discharged him at the end of nine months, giving him $422.50 and the suit. What was the value of the suit? 61. A lady spent $224 for a dress, $9f for a hat and $5f for shoes. She then found that she had left f of the sum of money she originally had. How much had she at first ? 72 FRACTIONS 62. If t 7 ¥ of a property is worth $762 1 .30, what is the value of jj - at the same 7 rate ? 63. B has in bank $237J. His income is $6f per week and his expenses $9J. How many weeks will his reserve fund hold out? 6If. A broker invested $37056 in land, and sold it for times its cost, gain- ing $4 per acre. How many acres did he buy ? 65. If § of —tpof 14 times a man’s money is $2626, what is 4 of - ^ of 3^ times 4 I2 his money ? 66. A can mow a field in 6 days, and A and B together in 2J days. How- many days will it require B alone to mow- the field ? 67. A, B and C can do a piece of work in 10, 12 and 15 hours, respectively. How long will it take all of them, working together, to do the work? If they receive $45, how should it be divided ? 68. A father divided $2151 between his two sons, so that f of what the younger received equaled -f of what the older received. Find amount received by each. 69. A man owning f of a vessel sold -f of his share for $21242. Find value of the vessel and present value of his share. 70. How long will it require to fill a cistern with a capacity of 337f gallons, if there is a pipe discharging into it 23f gallons per hour and there is a leak by which it loses 24 gallons per hour ? 71. If a boy buys peaches at the rate of 6 for 4 cents and sells them at the rate of 4 for 3 cents, how many must he buy and sell to gain 25 cents? 72. I spent -f- of my money for an overcoat and f- of the remainder for a dress suit. If the overcoat cost $111 less than the suit of clothes, what was the cost of each ? 73. A boy earned $5621 in two years, and § of what he earned the first year and $50 equals what he earned the second year. Find sum earned each year. 74- B lost | of his money, and after earning f as much as he had lost he had $237J. How much had he at first? 75. A boy and a man receive for a year’s work $1276. If the boy can do f as much work as the man, how should the money be divided ? 76. A grocer bought a tub of butter containing 84 pounds at ISf cents per pound, and sold -f of it for $14 and the remainder at a profit of 41 cents per pound. Find gain. 77. A miller bought 72J bushels of wheat at S7J cents per bushel and sold 1 of it at 921 cents per bushel, and the remainder at $1,121 P er bushel. Find his gain. 78. Bought 768 pounds of coffee, and sold sufficient to amount to $87.50 at 27J cents per pound. How much is unsold ? 79. A lady withdrew ^ of her money from bank and spent § of what she withdrew for a dress. If the dress cost $24, how much money has she still in bank ? GENERAL REVIEW PROBLEMS 73 80. A purchased 325 head of horses and sold them so that § of what he received for them equaled the cost. If his gain was $9875, what did they cost him per head ? 81. If £ of a farm is worth $450 more than -f of it, how much is the whole farm worth ? 83. £ of the sum paid for a house is the sum paid for a lot. What is the cost of each if together they cost $14014.21 ? 83. f of B’s money is equal to £ of A’s and the sum of their money is $19008. Find share of each. 84- A speculator bought a tract of land at $67£ per acre. He sold £ of it at $874 per acre, -f of the remainder at $76£ per acre, and 105f acres, or what still remained, at $80£. Find his profit. 85. A dealer expended $5460 for grain. £ for wheat at $1.37 2 per bushel, £ for corn at $.60§ per bushel, and the remainder for rye at $.89 per bushel. How many bushels of each kind did he purchase? 86. A and B each had f- of a ton of hay ; B sold A § of his hay. What part of A’s equals B’s after the sale ? 87. A’s farm consisting of 325f acres, cost him $624 an acre. He sold £ of it at $67 an acre, £ of it at $70 an acre, and the remainder at cost. Find his gain. 88. I bought 6 hams, weighing respectively, Ilf lbs., 12£ lbs., 7j4> lbs., 9£ lbs., 15| lbs., and 13££ lbs., at 11£ cents per pound. I sold the first 3 at 15f cents per pound and the last 3 at 16£ cents per pound. Find my gain. 89. A dealer bought 5 cars of coal of 13f tons, lly^ tons, 12££ tons, 14 T 9 ¥ tons and 9^2^- tons. Find the total cost at $6.75 a ton. 90. From a chain 546£ yards long traces £ of a yard long were cut. How many pairs were cut, and what part of a yard remained ? 91. A farm consisting of 253f acres cost $62f an acre ; £ of it was sold at $67£ an acre, £ of the remainder at $72£ an acre and the remainder at cost. Find the gain. 92. How many lots containing 2^- acres can be sold from a field containing 45f acres, and what will the remainder be worth at $158 an acre? 93. A dealer bought 5 turkeys weighing respectively, 17§ lbs., 23 T 9 F lbs., 21f lbs.,22££ lbs., and 19£ lbs. at 13f cents per pound, and sold them at 16£ cents per pound. Find his gain. 94 ■ Sold £ of the difference between 758£ and 948£ tons of coal for $1328£. What was the price per cwt. ? 95. A farmer sold to a merchant 3 hams weighing lOf lbs., 11| lbs., and I2££ lbs. at 12 £ cents a pound ; 12£ dozen eggs at 254 cents a dozen. He took £ the value of these commodities in cash (business result), 4 of the exact remainder in sugar at 6£ cents a pound, and the balance of exact result in muslin at 124 cents a yard. How much cash did he receive? How many pounds of sugar? How many yards of muslin? DECIMAL FRACTIONS 279. A decimal fraction, or a decimal, is a fraction whose denominator is 10 or a power of 10 ; as, T V, jfa, ToVo- 280. In decimal fractions, a unit is divided into 10 equal parts, named tenths; the tenths are divided into 10 equal parts named hundredths. In descending, each order is y 1 -^ part of the preceding order. 281. In a decimal the denominator is indicated by a point (.) called the decimal point. Sufficient ciphers must be prefixed when necessary to make the places equal the ciphers in the denominator. Thus, ^ is written decimally ; lino -03 ; toVo> -003. 282. The decimal point is also called the separatrix when it separates whole numbers and decimals. In United States money it separates dollars and cents. 283. A pure decimal consists of decimal figures only; as, .7, .45, .375. 284. A mixed decimal consists of a whole number and a decimal ; as, 6 9, 24.75, 3.1416. 285. A complex decimal is a decimal which has at its right a common fraction; as, .16§, .18f, .083J. 286. The decimal notation applies to integers and decimals, and the pro- cesses are the same whether the scale is ascending or descending. As we ascend from units, each place in the integral scale is increased tenfold and in the decimal scale decreased in the same ratio, the first place above units being tens and the first place below units being tenths, etc. DECIMAL NUMERATION 287. The similarity and relation of the periods and orders in a decimal to those in a whole number are shown by the following: NUMERATION TABLE CO 9876543210 . 123456789 Integral Periods and Orders Decimal Periods and Orders 74 DECIMALS 75 WHOLE NUMBERS AND DECIMALS COMPARED 288. General Principles : 1. Both whole numbers and decimals increase from right to left and decrease from left to right in a tenfold Sa. The farther an integral fig- ure is from the units order the greater its value. 3a. Placing a cipher after a whole number multiplies it by 10 ; two ciphers by 100, etc. Note. — C iphers are not properly placed be- fore whole numbers. ratio. Sb. The farther a decimal figure is from the decimal point the less its value. 3b. Placing ciphers after a deci- mal does not change its value, while placing a cipher between the figures and the decimal point divides it by 10 ; two, by 100, etc NUMERATION OF DECIMALS 289. How to read a pure decimal. Example. — .1375 is read, “One thousand three hundred seventy-five ten- thousandths or, “Thirteen hundred seventy-five ten-thousandths.” 290. Rule. — First : Begin at the point and, numerate the decimal figures to determine the name of the right-hand order. The first place, being one-tenth of units, is called tenths ; the second, hundredths ; the third, thousandths ; the fourth, ten-thousandths , etc. Observe carefully the written expression in decimals and integers. Second : Read the decimal as if it were a whole number, adding the name of the right-hand order. 291. How to read a mixed decimal. Examples. — (1) 24.75 is read, “ Twenty-four and seventy-five hundredths;” (2) 354.10865 is read, “Three hundred fifty-four and ten thousand eight hundred sixty-five hundred-thousandths.” 292. Rule. — After numerating the decimal figures, read the whole number, fol- lowing it with “ and” at the separatrix ; then read the decimal after the manner of a pure decimal. Note. — Do not read “ and ” except at the decimal point. Thus, in the last example do not say, “Three hundred and fifty-four and ten thousand eight hundred and sixty-five,” etc. 293. How to read a complex decimal. Examples. — (1) Read ,83-J- ; (2) 83.166f . Note. — The common fraction at the end of a complex decimal is not to be regarded as occupy- ing a decimal place, but as being a part of 1 in the last place before it. Hence, the above examples should be read, (1) “ Eighty-three one-third hundredths ; ” (2) “Eighty-three and one hundred sixty- six two-thirds thousandths.” 76 DECIMALS WRITTEN EXERCISE 294. Write in words or read orally the following numbers : 1 . .5 8. 2.5 15. 70.06 2. .05 9. 7.055 16. 349.52433 3. .005 10. .055 17. 2000.0025 V .1 11. .1003 18. .2025 5. .15 12. ,002o5 19. 100.1875 6. .025 13. .000109 20. 100.18 7. .77 n- 32.10005 21. 9.003 NOTATION OF DECIMALS 295. A decimal should contain as many places as there are ciphers in the denominator of the equivalent common fraction. In the common fraction, it will be observed that the denominator is usually written and may be any number; in the decimal, the denominator is simply indicated and mud be 10 or some power of 10. 296. How to write a decimal. Examples — (1) Write forty-five hundredths; (2) fifty-six thousandths; (3) fourteen hundred twenty-five ten-thousandths; (4) three hundred seventy-five hundred-thousandths; (5) nineteen thousand sixty-four millionths. (1) .45; (2) .056 ; (3) .1425; (4) .00375: (5) ,olo064. Note. — It is important to distinguish between the number part and the name part- of a decimal expressed in words. When such decimals as those in the examples above are recognized as 45 hun- dredths, 56 thousandths, 1425 teD-thousandths, 375 hundred-thousandths, etc., the correct writing of them will be an easy matter by simply observing the following : 297. Rule — First write the figures of the decimal as if they comprised a whole number, then place the point to the left of as many figures ( counting from, the right- hand) as there would be ciphers in the denominator if written as a common fraction. Verify the notation by numerating the decimal from left to right, beginning at tenths. WRITTEN EXERCISE 298. Write the following : 1. 9 tenths. 2. 27 hundredths. 3. 83 hundredths. 125 thousandths. 5. 65 thousandths. 6. 1375 ten-thousandths. 7. Fourteen hundredths. 8. Twenty-five thousandths. 9. Sixty-six ten-thousandths. 10. Twelve hundred-thousandths. 11. Eighty-eight millionths. 12. Ninety-five tenths. DECIMALS 77 13. 72 ten-thousandths. 17. Nine and five-tenths lit.. 15 hundred-thousandths. 18. Ninety-nine ten-millionths. 15. 95 hundredths. 19. One hundred-millionths. 16. 4175 thousandths. SO. Eleven hundred eleven hundiedths. SI. One hundred one thousandths. SS. One hundred and one thousandth. S3. Fifteen hundred fifteen millionths. Sj. Fifteen hundred and fifteen millionths. 25. Twenty-five thousand and twenty-five thousandths. 26. Three and fourteen hundred sixteen ten-thousandths 27. Four and eiglity-six hundred sixty-five ten-thousandths. 28. Two thousand thirty-seven and two thousand thirty -seven ten- thousandths. 29. Four hundred five and five hundred four hundred-thousandths. 30. Ninety-nine million nine hundred ninety thousand ninety-nine and nine hundred ninety million ninety-nine thousand nine hundred ninety-nine billionths. REDUCTION OF DECIMALS 299. To reduce a common fraction to a decimal. Examples — (1) Reduce f r/3. 0 0 0 3 7 cT f/J .dfr^T ft-) f/c = ./T/5T fe) = . oj-37-r to a decimal ; (2) / ^7,^ h. 0 0 0 0 / b / 4 0 / i r / 2 0 / / 2 r o r o 3 2 0 2 3 7 or h. 0 ' 2 s / a z t, r ? / s / 2 i/ r 7 s o 70 ' by 3.18 ; (3) $4.8665 by (3) a) u.ir & & / 2 / 2 S / 2 / 312. Rule. — Multiply as in whole numbers and point off as many decimal places in the product as there are decimal places in both multiplicand and multiplier. Note. — The small figures to the right in Examples 1 and 2 above are proof-figures, the first being proved by casting out the 9’s, and the second by casting out the ll’s. WRITTEN EXERCISE 313. 1 . Multiply 278.05 by 231. 2. Multiply 2150.42 by 868. 3. Multiply 28.575 by 3.1416. Ip. Multiply 33.5 by 33.5 by .7854. 5 Find the product of 3. 5X3.5 X. 5236. 6. Multiply 347.176 meters by 39.37. 7. Multiply 875.65 francs by .193. 8. Multiply £68.125 by 4.8665. 9. Multiply 475.5 liters by 1.0567. 10. Multiply 250.05 hectoliters by 2.8375. 11. Multiply 478.5 kilometers by .62137. 12. Multiply 6.5 meters by 5.4 by 2.5 by 1.375. 13. Multiply 175.6 kilos by .18. Iff The diameter of a circle is 14.7 feet ; to find its circumference multiply the diameter by 3.1416. What is the circumference? 15. Multiply 5.5 by 5.5 and that product by .7854. 16. Multiply $1575.25 by .06 and that product by 287.17. 17. Multiply 5050.65 by .01125. 18. 10.89X108.9X1.089=? 19 33 X 33 X 33 X .5236= ? 20. 25.5 X 25.5 X. 7954= ? 21. What is the product of 375f and 101.29? 22. What is the value of 312.2 meters of silk at 5.18 francs a meter? 23. Multiply 15.25 liters by 7.55 and that product by .193. 2ff A ship having a cargo valued at $22500 being disabled at sea, .22} ot its cargo was thrown overboard ; what was the loss to a shipper who owned .18 of the cargo ? 82 DECIMALS DIVISION OF DECIMALS 314. To divide by a whole number, a pure decimal or a mixed decimal. 315. The following method will be found easy and safe for beginners, as it avoids errors in pointing off the decimal places. Examples. — (1) $198.72=9 ; (2) (3) .409652-^-47 ; (4) 231.3937=78,54. .75=. 625 ; J / # r. 7 7 2 2 p r 3 0 ./o 2 f . y S' 2 3 / . 3 f 7 3 7 0 0"' / vT ,7 0 r 7 + 3 / 3 7 <2 7 r If 3 7 2 7 7 3 / V 7 + r 3 / 0 2 / 7 / 2 2 / / 2+ 7 7 3 7 7 v+ <7 7 0 0 o o f y / &> *£ <7 y 7 cT .2 3 7 & 3 3 & 3 2 f 7 f - 4-7 - 2 f 2 2. r z Result .008716. Result 2.9461 -(-. Proof Divisor Rem. 6. Quotient Rem. 4 ; 6X4=24=6. Dividend Rem. 1. Remainder, 4. (1+9)— 4=6. 316. Rule. — Write the divisor on the left of the dividend with a curved line between them, as in the division of whole numbers. Draw a horizontal line below the dividend if the division is to be “ short,” or above it if the question requires long division. When the divisor contains more decimal places than the dividend, equalize them by annexing ciphers to the dividend. ( See Ex. 2) Draw a vertical line through the dividend (at right angles to the horizontal line) as many places to the right of the decimal point as there are decimal places in the divisor — through the decimal point when the divisor is a whole number. Proceed with the division, being careful to write the first figure of the quotient directly beneath (in short division ) or directly above (in long division) the right-hand figure of the first partial dividend. All that part of the quotient to the left of the deci- mal line will be a whole number, and that to the right of it a decimal. DECIMALS 83 WRITTEN EXERCISE 1 317 . Obtain exact results for the following : 1 . Divide .625 by 2.5. 2 . Divide 15.25 by .05. 3 . Divide .0156 by .003. 4 - Divide 23 1 by .07. 5 . Divide 2.31 by .7. 6 . Divide 345.15 by .75. 7 . Divide 7 5 by .015 8 . Divide 37.06 by .017. 9 . Divide 142.74 by .61. 10 . Divide 110 by .44. 11 . Divide 24.75 by 2.25. 12 . Divide 3.65 by 7.3. 13 . Divide 2425.5 by .32. 14 - Divide 56.43 by .064. 15 . Divide 79.64 by 5.5. 16 . Divide 30.25 by 16 17 . Divide 2150.4 by 67.2. 18 . Divide 5280 by 16 5. 19 . Divide 43560 by .375. 20 . Divide .7854 by .0034. 21 . Divide 9 5 byl9. 22 . Divide 36 5 by 73. 23 . Divide 1750 by .875. 24 . Divide 461.975 by 5.435. 25 . Divide 3.6 by .18. 26 . Divide 343 68 by 53.7. 27 . Divide 44.44 by 1.1 . 28 . Divide 128.625 by .08. 29 . Divide 169.39 by 13. 30 . Divide 162.54 by 27. 31 . Divide 5236 by .008. 32 . Divide 7276 5 by 31.5. 33 . Divide .07958 by .16. 34 . Divide 93.75 by .0625. 35 . Divide 272.25 by 25. 36 . Divide 39.37 by .32. 37 . Divide 30.25 by 6.4. 38 . Divide 165 by .055. 39 . Divide 183.15 by 55. 40 . Divide 105.67 by 3.2. WRITTEN EXERCISE 2 318 . Extend the following to at least four decimal places when necessary : 1 . Divide .175 by 2. 2 . Divide 35.104 by 5. 3 . Divide .3562 by 5. 4 - Divide 360 by .06. 5 . Divide 14.2672 by .08. 6 . Divide 360 by .04J. 7. Divide 568.6 by 30.25. 8 . Divide 765.28 by 2.3J. 9 . Divide 360 by .05. 10 . Divide 4.8665 by .01}. 11 . Divide 2150.4 by 8. 12 . Divide 48.4 by .16. 13 . Divide 24 255 by .11. 14 .. Divide 384 by 1.2. 15 . Divide 3.45 by .005. 16 Divide 30.4 by 19. 17 . Divide 16.9 by .13. 18 . Divide 37.5 by .00}. 19 . Divide 7276.5 by 31.5. 20 . Divide 864.65 by 272}. 21 . Divide 375 bj 1 2 3 * 5 6 7 8 9 10 11 12 .19}. 22 . Divide 56} bv .13}. 23 . Divide 272} by .375. 24 - Divide 34.33} by .03}. 25 . Divide 17.16§ by 83}. Note. — A repeating decimal may be extended to the required number of places in the dividend, but should be handled as a mixed number in the divisor. 84 DECIMALS MISCELLANEOUS EXERCISE 3 319 . Ca rry results to four decimal places when necessary: 1. Divide 4378.75 by 320. 2. Divide 9654.5 by 24.75. 3. Divide 138.48 by 30±. 4- Divide 7500 by 24f. 5. Divide 2747.7 by 57.75. 6 Divide 39.37 by 36. 7. Divide 5.18 by .193. 8. Divide 4.8665 by 240. 9. Divide .7854 by 231. 10. Divide 329.84 by 5.19. 11. Divide 82432 by .37. IS. Divide 12.4328 by .61. 13. Divide $245.50 by .23. 14- Divide $545.25 by 12.3. 15. Divide $1250.60 by $1 25. 16. Divide 34.5685 by .97. 17. Divide 112.3456 by 1.37. 18. Divide 3.39924 by8.716. 19. Divide .00540625 by .00865 50. Divide 21.984375 by 78.65. 51. Divide 21.984375 by .375. SS. Divide $86.73 by 1239. S3. Divide $939.12 by 13416. #4- Divide $13878.72 by $48.19. S5. Divide $9615.36 by $50 08. 26. Divide .409652 by .047. 27. Divide 3 1.833 J by 18-f-. 28. Divide 96.8 by .238. 29. Divide 175.25 by .000f. 30. Divide 62.8 bv .14. 31. Divide 653.21056 by 1.8 32. Divide 94.265 by 32. 33. Divide 242.55 by 75.796J. 54- Divide 741.85f by 124. APPLICATION OF DECIMALS TO INVENTORIES 320 . It is usually sufficient in ordinary commercial operations to carry a decimal division to four places, but in estimating the cost of manufactured prod- ucts to get a unit of value on millions, it is necessary to carry the result to eight or nine decimal places. This is true also in figuring bankruptcy and part- nership settlements by the percentage method, and in constructing compound interest tables. WRITTEN PROBLEMS 321 . 1 . If the cost of producing 7500000 barrels of a certain grade of flour was $31829437.95, what should be the company’s inventory value of 2345560 barrels ? 2. The cost of producing 5000000 tons of pig iron is $47953274.87, what should be the mill inventory value of 2897450 tons? 3. If the cost of manufacturing 7586000 rolls of building paper was $4492673.65, what should be the manufacturer’s inventory of 3815967 rolls? 4- The cost of producing 9847600 rolls of wall paper was $102S922.45, what should be the manufacturer’s inventory value of 31472569 rolls? 5. If the cost of making 44562975 pounds of a certain commodity was $2943647.50, what should be the manufacturer’s inventory value of 29675875 Dounds ? x 6. If the cost of refining 24756800 gallons of oil was $12S5675.70, what should be the company’s inventory value of 6938450 gallons? 7. The cost of making 25000000 tons of steel was $487755692, what was the company’s inventory value of 9398670 tons? DECIMALS 85 SHORT METHODS IN DECIMALS Multiplication 322. To multiply by simply moving the decimal point. 1. Multiply 24.75 by 10. f. Multiply .5236 by 100. 2. Multiply 31.5 by 100. 5. Multiply 2150.4 by 10000. 3. Multiply 3.1416 by 1000. 6. Multiply 172.8 by 10. 323. Rule. — Moving the decimal point one place to the right multiplies by 10 ; two places , by 100 ; three places , by 1000, etc. 324. To multiply by a decimal, the equivalent of which is etc., of 1. 1 2 ) 1. Multiply 43560 by .5. 2. Multiply 5280 by .25. 3. Multiply 2150.4 by .125. 4- .Multiply 2747.7 by .33J. 5. Multiply 24.75 by .16|. 6. Multiply 31.5 by .081 7. Multiply 4.8665 by 12J. 8. Multiply 231 by .14-f-. 9. Multiply 160 by .2J. 10. Multiply 78.54 by .3334- 325. Rule. — Divide by the denominator of the equivalent common fraction. 326. To multiply by a decimal, the equivalent fraction of which is l, etc., greater or less than 1. 1. Multiply 2150.4 bv .75. 2. Multiply 2747.7 by .66 1. 3: Multiply 31.5 by .83^-. 4- Multiply 5280 by 874. 5. Multiply 4.86 by .99. 6. Multiply 43560 by .91f. 7. Multiply 2150 4 by 1.25. 8. Multiply 2747.7 by 1.334. 9. Multiply 31.5 by 1.16f. 10. Multiply 5280 by 1.125. 11. Multiply 4.8665 by 1.10. 12. Multiply 43560 by I.O84. 327. Rule. — From the multiplicand subtract such part of itself as the multiplier is less than l, or add such part as it is greater than 1. MISCELLANEOUS EXERCISE 328. Multiply 1. 30.25x2.5. 2. 3.1416X25. 3. 31.5X12.5. I 24.75 X 33J-. 5. 4.8665x500. 6. 30.25x74- 7. 3.1416X750. 8. 24.75X874. 9. 2150.4 X6f. 10. 31.5X66§. 11. 30.25X125. 12. 3.1416X1250. 13. .238X112.5. If. 2150.41 X 1334. 15. .193X1500. Division 329. To divide by moving the decimal point. 1. Divide 24.75 by 10. 5. Divide 37.85 by 100. 2. Divide 2150.4 by 100. 6. Divide 4.8665 by 1000. 3. Divide 31.5 by 1000. 7. Divide 31.416 by 10. f. Divide 1728 by 100. 8. Divide 144 by 100. 330. Rule. — Moving the decimal point one place to the left divides by 10 ; two places, by 100 ; three places, by 1000, etc. 86 DECIMALS 331. To divide by a decimal of which the equivalent fraction is J, etc. 1. Divide 3.1416 by .5. 2. Divide .7854 by .25. 3. Divide 31.5 by .334. I/.. Divide 24.75 by .16f. 5. Divide 16 5 by .12J. 6. Divide 144 by .08J. 7. Divide 1728 by ,14-f-. 8. Divide 2150.4 by .125. 9. Divide 4.8665 by ,2J. 10. Divide 30.25 by .3334 332. Rule — Multiply by the denominator of the equivalent fraction. 333. To divide by a decimal whose equivalent value is 4, 4> etc., greater or less than 1. Example— 36 a-. 75=36 a-|=36X 3=48. Suggestion. — 36X4 is equivalent to adding 4 of the dividend to itself. 1. Divide 24 75 by .75. 2. Divide 2150.4 by .874. 3. Divide 2747.7 by .66|. f. Divide 144 by .834. 5. Divide 1728 b} r ,85-f-. 6. Divide 31.5 by .74 ; by ,6f. 7. Divide 24.75 by 1.25. 8. Divide 2150.4 by 1.334 9. Divide 2747.7 by 1.50. 10. Divide 144 by 1.124. 11. Divide 1728 by 1.25. 12. Divide 31.5 by 1.16f. 334. Rule. — Divide by the numerator of the equivalent common fraction and add the result when the divisor is less than 2, or subtract when it is greater than 1. 335. To divide by a whole number ending in ciphers. 1. Divide 435.6 by 300. 2. Divide 528.85 by 700. 3. Divide 75864.15 by 6000. f Divide 4576.75 by 4000. 5. Divide 21504 by 320. 6. Divide 9178.39 by 1300. 7. Divide 37477.88 by 11000. 8. Divide 4866 5 by 1600. 336. R ule. — Drop the ciphers on the right of the divisor , move the decimal point in the dividend an equal number of places to the left , and divide. REVIEW EXERCISE 337. In simplifying expressions connected by the signs of the fundamental operations ( + , — , X, and a-) observe the following order : 1. Multiplication. 2. Division. 3. Addition and Subtraction as they may occur. Expressions contained in marks of aggregation, (parenthesis, brackets, braces) must be simplified according to the above order, and the result used for the whole expression in the marks; then simplify the original expression as directed above. DECIMALS Example.— 4X8+14=8+ [16— (25=5) +7]=29. Operations : (Restatement). 1. 8+14=22. a. 4X22=88. 2. (25=5)=5. b. 88=8=11. 3. (16— 5+7)=18. c. 11 + 18=29. + 4x22=8+18 338 . 1 - 12.5 XI 2.5 X. 7854=144=? 2. 26.728 X $1.05= ? 3. 13.450 X $18.25=? + 43.75X$11 + 132.80X$7= ? 5. 184.80 X $.625=? 6. 1 2.456 X $8.75=? 7. 28165X$.08=$28.75= ? 8. £765 5 X $4.8665= ? 9. £12345 6 X $4.8665=$. 193=? 10. 687.5X39.37=36=? 11. 785.125X36=39.37=? 12. 7912X25000X640=? 13. 791 2 X 791 2 X 791 2 X. 5236=? U 24X20X8X4 x$025=s? 15. 60X24X30 9 144 16. 60x49xl44=(14x20)= ? 17. 17X22X62.5=? 18. (28 X 6600=9) X $.92= ? 19. ($900=.05)X 1.1675= ? 20. ($9.21=. 03) X. 65= ? 0)1 75x5280x48 *- O0 _ 2L 43660 + *' 2U - 22. 60X3.1416X8X1728=231=? 23. (1 ,5 X 5280 X 75=272.25) X $1 .10 21 187X187 X. 7854=272.25=? 25. 2240X7=8X4X168=? 26. 14X3.1416X38=9=? 27. 59X5280=60=5.5=? 28. $76000 X. 75X1.04=? $950 X 93 X. 06 9 360 ~ • Yl $6500 X 61 X .05 9 "365 ~~ ' $1896 X (91 — 8) X .06 9 360 — ' 32. £124.5025 X $4.87+ $1.50= ? 33. Divide 7.5 by 62.5. 3£. $1890=.045 X 1.05= ? 35. $1940.50=4.875=? 36. 640 X $6.25 X. 02= ? 37. ($7200=.08)X 1.1625=? 38. £37.804 X $4.8665= ? 39. 4775 X $.82— (4175 X $-82 X. 04)= 4.0. (2X40 + 2 X60)X9X 3.5 X $.30= 41. $972.50=(162X 4=9 X $3.10)= 42. 15290 bu.=92 bu.=102= ? 43. 165+3. u75 — .60863x40.07=. 1309= ? 44- (2X16X9)+(2X18X9)+(16X18)=? 4.5. 102 X 102 X 102 X .5236 X 1728=231= ? 46. (22X22 X .7854)= 144 X 24 X 450 X $16.50= ? 47. 2240 X 12=(26 X 26 X .7854=144 X 1 66)= ? 48. 144 X 144 X. 7854 X8Xl728=(31. 5X9X24)= ? 49. $31 7829.32+ $61378.12 — ($46312.85 +$301449.72)= ? 50. (385 X 385 X .7854=43560) X $1.59= ? 88 DECIMALS REVIEW PROBLEMS 339 . 1. $197.87* + $67.37f-K$27.47ix3|)— .75^=? 9. $37i+$1.87f+$g ; 4 3. Find the cost of 10 loads of hay at $16J per ton, containing respectively 2| tons, tons, 1.375 tons, 2\ tons, 2.8 tons, 1-^ tons, If tons, 2.15 tons, If tons and 24 tons. f. If 9.3 barrels of flour cost $48,825. what will 15.25 barrels cost at the same rate ? 5. A farmer sowed three fields in wheat, containing respectively 1171- acres, 125§ acres, and 325.561 acres. He harvested from the first an average of 27J bushels per acre, from the second 30.25 bushels per acre, and from the third 29f bushels per acre. Find the value of his wheat at $1.12J per bushel. 6. A speculator bought 325.45 acres of land at $15f per acre and sold it in plots of 1.15 acres each at $112.50 a plot. Find his gain or loss. 7. Two trains are 587.25 miles apart, and are running toward each other. One train has a speed of 45f miles per hour and the other 53J miles per hour. After they have been running for 4.6 hours, how far are they apart? 8. A grain dealer bought 1275f- bushels of wheat at $.874 per bushel. He sold f of it at $.90f per bushel, .25 of the remainder at $.94f per bushel, and what still remained at cost. What did he gain? 9. If a bushel contains 2150.4 cubic inches, how many bushels in 37560 cubic inches? What is the value of this quantity at 624 cents per bushel? 10. During a year a man spends .37J of his income for clothes, .274 for board, .16 for incidentals, and deposits the rest in bank. If his bank deposit is $570, what is his income? 11. A merchant deposited in a bank at one time $575.25, and at another $487.30. If he checked out $387.50, how many yards of cloth can he buy, at $1.25 per yard, with the money he still has in bank? 13. If .625 of a ton of coal cost $3.25, what will 7f tons cost ? A£. Divide $9 between two boys so that one shall receive .25 more than the other. 15. If a load of hay weighing 1665 pounds cost $8.75, how much should, at that rate, be paid for a long ton (2240 pounds)? 16. How many barrels of flour at $5,124 must be given for 127.374 bushels of com at 30 cents per bushel and 127J bushels of wheat at $.874 per bushel ? 17. If the freight on 75.8 barrels of apples is $9.47^, what should be paid on 103f barrels ? 18. If there are 63360 inches in a mile, and 39.37 inches in a meter, what will it cost to pave a street one mile long at $1.12J per meter? 19. The circumference of a body is 3.1416 times the diameter. What is the diameter of the earth if the circumference is 24899.024 miles ? 34 12. Change to a decimal fraction. 3 DECIMALS 89 20 . The value of a franc in United States money is $.193. How many francs in $458.37 J, and how many yards of silk could be purchased with this sum at 9.5 francs per yard ? 21 . If .37 of a boy’s wages for a week is $3.14J, what should he receive for 7f weeks ? 22 . If a certain kind of iron ore contains .37J foreign substance, how much pure iron in 968.9 tons of the ore ? 23 . A tailor bought 32 pieces of cloth containing 23J yards each at $3.60J per yard and sold it so as to gain $372, after allowing $14.88 for freight. How much did he receive per yard? 24 .. A man in disposing of his property gave J to his younger son, J of the remainder to his elder son, .125 to a church, and the remainder, $32151.35, to his wife. Find value of his property and amount each son received. 25 . A clerk spent .37J of his money and still had remaining $237.25. How much did he spend? 26 . A house and lot together cost $7805.38. If the house cost 1.75 times as much as the lot, what was the cost of each ? 27 . The product of three numbers is 331.81071, and two of the numbers are 3.79 and 23.1. What is the third number? 28 . A grocer invested $526.50 in Rio and Java coffee. If .625 of the amount invested in Rio equals the amount invested in Java, what sum was invested in each, and how many pounds of each could he buy at 22|- cents per pound ? 29 . At the end of the first year a merchant’s inventory is $37325. 62J, which is equal to .375 of his sales for the year. If during the year he gained $12725. 37J, what was the amount of his purchases ? 30 . Two men commenced business with equal capital. At the end of the year one had gained $3725.40, the other had lost $2376.80, and together they had $7897.30. How much had each at first ? 31 . A merchant sold 450 yards of cloth from a piece containing 875 yards. What decimal part of the piece remained unsold ? 32 . A, B and C own a mill. A owns .265 of it, B .378 of it, and C the remainder. If A’s share is $7234.50, what is the value of B’s and C’s shares? 33 . If .375 of a ton of coal cost $4.95, what will .0875 of a ton cost? 34 - A invested .375 of his money in a factory, .24 of the remainder in a city property, and deposited the balance, $17432.50, in a bank. Find amount of his money, and amount invested in the factory. 35 . The private secretary of the governor receives $2000, of which he spends $525 for rent, $675 for living expenses, $297.50 for clothes. What decimal part of his salary does he save? 36 . If 25.5 barrels of flour cost $85, how many can you get for $40.25 ? 37 . A man traveled 697^ miles by rail and 011 bicycle. If he traveled .55 as far on bicycle as by rail, how far did he travel by rail and how far on bicycle? 38 . If the value of an English sovereign is $4.8665, how many sovereigns are worth $100 ? What is the value of 156 sovereigns ? QUANTITY, PRICE AND COST 340. The relation of these three elements is that of multiplicand, multi- plier and product. Quantity is the number of units. Price is the cost of one unit. Cost is the value of a quantity. Formula : 1. Quantity X Price=Cost. 2. CostH-Price=Quantity. 3. Cost-uQuantity=Price. 341. An aliquot is a number or quantity (either integral or fractional) that is contained in another number or quantity an exact number of times. Thus, 6 is an aliquot of 18; 25 is an aliquot of 100 ; J is an aliquot of 4, and of f. 342. An aliquot part of a number or quantity is one of its even parts, which is given a fractional expression by placing it over such number or quantity and reducing it to its lowest terms. 343. A simple aliquot is one that can be reduced to a fractional expression having a numerator of 1 ; as i, 4, etc. 344. A compound aliquot is comprised of two or more simple aliquots ; as, t=f+(i ofi). 345. An aliquot unit is the number or quantity of which the aliquot is an even part. 346. A commercial unit is a fixed unit of measure upon which the price of a commodity is based ; as, a yard, a bushel, a gallon, a pound, a ton, etc. 347. Simple Aliquot Parts of $1.00. 50c. 1 2 16fc. 1 6 10c. — i 1 0 6|c — i i r> 33g-c. — 1 3 14fc. 1 7 9QrC. — i i i 5c. - i 2 0 25c. 1 4 124c. — 1 8 Sic. - i 1 2 3|c — i • — To 20c. — 1 5 H|c. 1 9 6fc. - 1 1 5 24c, - i ■ 40 348. If the price given should be $1,334, or 4 more than $1.00; or 66§ c i less than $1.00, the result can be readily obtained by adding or subtracting the fractional part, as the case may be. This can be applied to every simple aliquot with any easy unit as a base, whether 25c., 50c., $1.00, $100 or 100 per cent.., the object being to simplify multiplication and division in business computations. Require students to make up tables of the aliquot parts of 50c., 25c., 10c., similar to table given above. 90 QUANTITY, PRICE AND COST 91 APPLICATION OF ALIQUOT PARTS 349. General Rule. — Divide the quantity by the number of commercial units that $ 1 will buy. Explanation — At 121c., $1 will buy 8 units ; at 33Jc., 3 units ; at 50c., 2 units; at 66|c., f units, or 11; at 75c., f or 1£ units; at $1.25, 4 of a unit ; at $1.33J, | of a unit ; at $1.50, § of a unit. Note. — -When the quantity contains a common fraction, it should he reduced to a decimal before dividing. If the fraction will not reduce to a simple decimal, extend it to 4 decimal places. 350. To find the cost when the price is a simple aliquot of $1. Find the cost of Exercise 1 1. 175 yds. @ 50c. 2. 84f yds. @ 33Jc. 3. 127 gal. @ 25c. If. 75J gal. @ 20c. 5. 156 lbs. @ 16fc. 6. 99tL yds. @ 12Jc. 7. 350 yds. @ 14fc. 8. 894 lbs. @ 10c. 9. 810 qts. @ 114c. 10. 495J qts. @ 9^0. 11. 480 gal. @ 8Jc. 12. 230 yds. @ 6jC. 13. 2791 yds. @ 331c. i.£. 236f lbs. @ 25c. 15. 7S9f- yds. @ 16fc. 351. To find the cost when the price is a simple aliquot less than $1 or more than $1. Exercise 2 1. 248 gal. @ 871c. 2 3561 gal. @ 90c. 3. 189 yds. @ 75c. If. 456 yds. @ 66§c. 5. 360f yds. @ 80c. 6. 35 6 gal. @ 95c. 7. 750 yds. @ 871c. 8. 632 yds. @ $1.50. 9. 156| yds. @ $1.33^. 10. 256 doz. @ $1.25. 11. 1691 doz. @ $1.16f. 12. 573| yds. @ 75c. 352. Rule. — Add to the quantity or subtract from it such aliquot part of itself as the price is greater or less than $1. 353. To find the cost when the price is a simple aliquot more or less than $2, $3, $4, $5, etc. Exercise 3 1. 960 bbls. @ $2,121. 2. 480 bbls. @ $2.16|. 3. 560 bbls. @ $3.14|-. If.. 728 bbls. @ $3.25. 5. 360 bbls. @ $4,331. 6. 244 yds. @ $1.87f. 7. 342 yds. @ $1.83f 8. 572 yds. (cy $1.75. 9. 456f yds. @ $2,331 10. 196 yds. @ $2.75. 11. 128 tons @ $5.12J. 12. 96f tons @ $4.66f. 13. 240 tons @ $3.87J. If 85J tons @ $6.25. 15. 375 tons @ $2.66f. 354. R ule. — Multiply the quantity by the nearest dollar price and add to the result, or subtract from it, such aliquot part of the quantity as the real price is above or below the assumed price. 92 QUANTITY, PRICE AND COST 355. To find the cost using 50c., 25c., or 10c. as a base. Find the cost of 1. 164 yds. @ 55c. 2. 132 gal. @ 45c. 3. 284 gal. (aj 374c. Ip. 716 gal. @ 56|c. Exercise 4 5. 76J yds. @ 224c. 6. 120 yds. @ 18fc. 7. 92J yds. @ 35c. 8. 132 yds. @ 15c. 9. 48f yds. @ 134c, 10. 152 lbs. @ 7Jc. 11. 69f lbs. @ 6fc. 12. 175 lbs. @ 3Jc. Note.— T hese exercises are to test the student’s power of invention. 356. To find the cost when the price is an aliquot part of $10, $100, $1000, or an aliquot part more or less than $10, etc. 1. 318f yds. @ $2.50. 2. 2744 yds. @ $3,334- 3. 157f yds. @ $5. Ip. 894 tons @ $334- Exercise 5 5. 534 doz. @ $15. 6. 28 bbls. @ $7.50. 7. 67J tons @ $25. 8. 39f tons @ $33J. P. 160 acres @ $125. 10. 714 acres @ $250. 11. 27f acres @ $500. 7^. 145| acres @ $150. 357. To find the cost when the quantity is 10, 100, 1000, etc., or when it may be factored upon some power of 10 taken as an aliquot unit. 1. 10 yds. @ $1,354. 2. 100 yds. @ 73fc. 3. 1000 yds. @ 9|c. Ip. 500 yds. @ 86-fc. Exercise 6 5. 250 gals. @ $1.15|. 6. 125 gals. @ $1,194. 7. 75 lbs. @ 231-e. 8. 2500 yds. @ 64fc. 9. 1500 gals. @ 894c. 10. 2000 lbs. @ 494c. 11. 500 tons @ $3,784-. 12. 750 bbls. @ $2.96f 358. To find the cost when the numerator of the aliquot is greater than 1. 359. Rule. — Resolve the aliquot into two or more simple aliquots. Illustration : 374c.=$f=4 + (i °f $)• Exercise 7 1. 144 yds. @ 374c. 2. 256 gals. @ 624c. 3. 1964 g ro - @ 85c. Ip. 288 yds. @ $1,374- 5. 512 yds. @ $1,624 6. 640 bbl. @ $1,724. 7. 84 bbls. @ $2,374. 8. 56 bbls. @ $3,624. 9. 68 bbls. @ $2,224. 10. 96 tons @ $5,374. QUANTITY, PRICE AND COST 93 360. To find the quantity when the cost is given and the unit price is a simple aliquot of $1. Exercise 8 Cost. Unit Price. Cost. Unit Price. Cost. Unit Price. 1. $52.50 25c. 6. $14.85 3ic. 11. $19.45 20c. 2. 368.25 12ic. 7. 8.75 2ic. 12. 73.25 8^c. 3. 97.75 16fc. 8. 4.721 5c. 13. 16.50 7ic. f- 142.50 33JC. 9. 184.50 14f( ■» If. 24.75 6ic. 5. 295.75 50c. 10. 59.10 11* ■> 15. 36.20 66§c. 16. $78.75 (a ) 25c. a yard. 21. $175. .50 @ $1.25 a yard. 17. $156.25 ( @ 33^-c. a yard. 22. $264. 80 @ $1.33J a yard. 18. $24.75 (a } 16fc. a pound. 23. $363. 75 @ $1.50 per barrel. 19. 147.25 (a ) 6Jc. a gallon. 2f. $1975 @ $12.50 a ton. 20. $385 @ 87£c. yard. 25. $26.58 @ 75c. a yard. 361. Rui ,e. — Multiply the cost by the number of commercial units that $ 1 will buy. 362. To find the price of a commercial unit. Example. — What is the unit price when 100 yds. cost $43,625 ? Solution. — M oving the point in the cost two places to the left gives $.43625, or 43f cents, as the price of a yard. Exercise 9 363. Find the unit price if 1. 100 yds. cost $15,874. 2. 1000 lbs. cost $66.25. 3. 250 yds. cost $188.75. f. 125 yds. cost $177.50. 5. 3000 tons cost $116.25. 6. 333J yds. cost $575. 7. 2000 lbs. cost $127.50. 8. 150 yds. cost $78.75. 9. 1500 gals, cost $97.12J. 10. 750 yds. cost $2986.50. 364. Find the cost of 1. 389 yds. @ 25c. 2. 246f yds. @ 33Jc. 3 369 lbs. @ 16fc. f. 166f yds. @ 12Jc. S. 750 gals. @ 8Jc. Review Exercise 10 6. 156 lbs. @ $1.33j-. 7. 562J yds. @ $2.50. 8. 116 gals. @ $1.12J. 9. 3126 lbs. @ 2Jc. 10. 964 ft. @ 15c. 11. 1190 bbls. @ $4.1 2J. 1 2. 41 8f yds. @ $3.16f. 13. 273J yds. @ $3.87|-. If. 168 tons @ $6.25. 15. 178J yds. @ 62Jc. 94 QUANTITY, PRICK AND COST SPECIAL RULES 365. To find the cost when merchandise is sold by the hundred or the thousand. Many kinds of merchandise, such as lumber, cigars, meal, grain, etc., are sold by the hundred or thousand. Note.— C is the abbreviation for 100 ; M for 1000. A cental of grain is 100 pounds. A cwt. is 100 pounds avoirdupois. A net ton is 2000 pounds ; a long ton, 2240 pounds. Example. — Find the value of 2604 lbs. of wheat at $1.12 per cental. 366. Rule. — Point off two places in the quantity if the price is per hundred, or three places if per thousand ; multiply by the price and point off as in the multiplcation of decimals. Note. — F or net tons, point off three places and divide by 2. 2 C >.0 4 /./ 2 . S 2 o r -2 r 6> 44 f i f . / £ 4 r WRITTEN PROBLEMS 367. 1- Find the cost of 4500 feet of pine boards at $22.50 per M. 2. Find the cost of 75 boxes cigars, each containing 50, at $15.50 per M. 3. What will 1112 pounds of hay cost at $1.40 per cwt,? 4- Find the cost of a car load of bran, weighing 27350 lbs., at 65 cents a cwt. 5. Find the cost of insuring a property valued at $7250 at 87J cents per $100. 6. Find the cost of 40 boxes envelopes, each containing 500, at $2.16§ per M. 7. What are 17610 feet of oak lumber worth at $27.25 per M ? 8. Find the cost of the following merchandise: 2175 lbs. Middlings at $.65 a cwt. 875 “ Cracked Corn “ .50 200 “ Oat Meal “ 1.25 3125 “ Bran “ .38 9. Find the cost of the following invoice of lumber : 26281 ft. Hemlock at $12.25 per M. 4278 “ Scantling “ 12.50 “ 7500 Shingles “ 17.50 “ 10. What is the cost of a car load of hay, 18750 lbs., at $13.50 per ton ? 11. Find the freight on hay in preceding problem at 8 cents a C. 12. 13560 pounds of wheat, purchased at $1.37 a cental, were sold at $1 a bushel of 60 lbs. What amount was gained or lost by the transaction? 13. Find the cost of 1812 pounds of hay at $14.50 a ton. 11/.. What is the value of 65 pounds of wheat at $1.25 a cental ? 15. 1 sold a bill of hemlock lumber, 585622 feet, and gained $1.30 a thous- and. What was my entire gain ? QUANTITY, PRICE AND COST 95 368. To find cost when merchandise is sold by the ton. Example. — What is the value of 5 tons 6 cvvt. of coal at $5.75 per ton ? 5 tons 6 cwt. =5.30 tons. 369. Rule. — Multiply the hundredweights by .05 and annex to the tons; multiply by the price per ton and point off required places. Note. — The above rule applies to both the “short ” or standard ton of 2000 pounds, and the “ long ” ton of 2240 pounds, the latter being accompanied by the long hundred of 112 pounds, which equals or .05 of a ton. /s-. 7 s S3 / 7 2. s i r 7 .r /3 O. 4 7 s PROBLEMS 370. 1. Find the cost of 11 tons 18 cwt. of coal at $3 25. 2. Find the cost of 9 tons 2 cwt. of coal of $5.50. 3. What is the value of 2 cars of coal, one containing 11 tons 3 cwt., the other 13 tons 15 cwt., at $2.85 per ton? Make the extensions in the following : 1 car Pea Coal, 11 tons 6 cwt. at $3.25 1 “ Egg Coal, 12 tons 19 cwt. “ 4.75 1 “ Stove Coal, 10 tons 13 cwt. “ 4 50 1 “ “ “ 13 tons “ 4.50 Total 5. What is the value of 18 tons 17 cwt. of iron ore at $18.75 per ton ? 6. What is the freight on 43 tons 7 cwt. of iron at $1.35 a ton ? 7. A man bought his year’s supply of coal, 12 tons 6 cwt., at $5.75 ; what was the amount of the bill ? 8. Three cars of iron weighed respectively 25 tons 7 cwt., 23 tons 18 cwt., and 27 tons 2 cwt. ; what was the value of the iron at $14.80 a ton? 9. What would be the freight on the iron in preceding problem at 35 cents a ton ? 10. How many tons of 2000 pounds in 25 tons 9 cwt. long tons? 371. To find the cost of commodities sold by the standard or “ short ” ton of 2000 pounds, when the weight is given in pounds. Examples. — (1) Find the cost of 2840 pounds of hay at $18 a ton ; (2) of 3500 pounds of bran at $25 a tou ; (3) of 1960 pounds of fertilizer at $32.50 a ton. 2r s/o 2 P 4 /S s/ 0. ^ ' fo) 7 r. 3S\ r 0 O - VML- 7S- s /f / 0 f ro 2 s/S f3/ ss fff 2 O . / O. , frj 2.5 0 // 2.5 0 // Since 1 cent a pound gives $20 a ton, and since most ton prices can be easily figured upon that basis, use the following 372. Rule. — Point off two decimal places in the number of pounds, which gives $20 a ton ; then add to or subtract from this result such part of it as the price is more or less than $20. 96 QUANTITY, PRICE AND COST ORAL EXERCISE 373. Analyze the following prices upon a basis of $20 a ton and apply them to such quantities as may he given by the teacher : 1. $10 a ton. 2. $30 “ “ 3. $16 “ “ ff $24 “ “ 5. $22 “ “ 6. $15 “ “ 7. $12.50 a ton. 8. $17.50 “ “ 9. $22.50 “ “ 10. $27.50 “ “ 11. $40 “ “ 12. $35 “ “ 13. $45 a ton. 74. $50 “ “ 15. $47.50“ “ 16. $37,50 “ “ 17. $21.50 “ “ 18. $65 “ “ 374. To find the cost of commodities sold by the bushel of so many pounds. Note. — I t is evident that 1 cent a pound for any commodity sold by the bushel, is as many cents a bushel as there are pounds in a bushel of the commodity in question. Thus, 1 cent a pound equals 60 cents a bushel for wheat, clover seed, potatoes, beans, etc. (these commodities each weighing 60 pounds to the bushel) ; it also gives 32 cents a bushel for oats, 45 cents for timothy seed, 56 cents for shelled corn. Examples. — (1) Find the cost of 14680 lbs. wheat at 90c. a bu.; (2) of 4368 lbs. oats at 40c. a bu. / iff 6 7 3 2 0 ro = Ho_ = h 0 3 Of v. " ... ... A 2 0 = fOff it 3 / O ?2 fSff (0 0 = 32 f - rt o 375. Rule. — Point off two decimal places in the quantity and increase, or dimmish this value (as the price may require ) by the use of aliquot parts. 376. To find cost without changing to bushels. WRITTEN EXERCISE 1. 24370 lbs. wheat at $0.75 a bu. 11. 29640 lbs. oats at 0 -f 0 c/it 2. 54940 a tt U .90 “ 12. 13664 << U t< .45 3. 94580 tt tt u .971 “ 13. 9900 a u tt .50 4 . 73690 u u u 1.05 “ Iff 97980 “ shelled corn tt .70 5. 66460 it tt t( 1.03 “ 15. 57428 U it tt it .60 6. 9720 “ beans u 1.50 “ 16. 7245 “ timothy seed tt 1.05 7. 17230 tt It u 2.70 “ 17. 6440 “ barley tt .78 8. 8850 “ peas u 1.95 “ 18. 5280 “ millet tt 1.35 9. 3190 “ clover seed u 6.90 “ 19. 12240 “ rye it .84 10. 2775 u u u tt 7.50 “ 20. 99750 “ corn in ear tt .42 QUANTITY, PRICE AND COST 97 REVIEW PROBLEMS 377 . 1. A farmer hauled to market 25 loads of wheat averaging 2796 lbs. to the load. One-half the wheat was graded as “No. 1,” for which he received $1.05 a bushel; 4 as “No. 2,” for which he was paid $1 a bushel; and the remainder “ No. 3,” worth 90 cents a bushel. How much did he get for his wheat? Five carloads of wheat, weighing respectively 52475, 54980, 46325, 49590 and 51650 pounds, were shipped from Kansas to Philadelphia. If the dealer who shipped the wheat paid 90 cents a bushel for it, 274 cents per cwt. for freight, .and $75 for other expenses, and then sold it for $1.20 a bushel, what was his profit? 3. Ten loads of hay weighing respectively 2450, 2380, 2590, 2725, 2675, 2295, 2885, 2565, 3020 and 2745 pounds, were sold at $22.50 a ton. How much did he receive for the hay ? A A stock dealer bought 125 head of hogs, weighing them in five lots; the first lot weighed 10500 lbs., the second lot, 9975 lbs., the third, 11150 lbs., the fourth, 8995 lbs. and the fifth, 9780 lbs. What was the average weight of the hogs? If he paid 6J cents a pound for them, and sold them at cents a pound after paying out $105 for freight and other expenses, what was his profit? 5. A Pittsburg grain merchant bought 245760 lbs. of shelled corn at 42c. a bu., 385600 lbs. of corn in the ear at 40c. a bu., 296440 lbs. of oats at 36c. a bu., and 582380 lbs. of wheat at 93c. a bu. What was the entire cost? 6. If the merchant mentioned in the preceding problem had the corn in the ear shelled, losing thereby 4 of its weight in the cobs, but disposing of the cobs at $5 a ton, then selling all the corn at 49c. a bu., the oats at 45c. a bu., and the wheat at $1.03 a bu., what was his net gain on the whole, his expenses amounting to $593.45 ? 7. How much better would the merchant in the preceding problem have done by having the wheat converted into flour, at an expense of 334c. per barrel for the milling and 20c. apiece for barrels, and then selling the flour at $5.25 a bbl. (196 lbs.), the bran and middlings at $22 a ton, estimating a bushel of wheat to make 42 lbs. of flour and allowing any fraction of a barrel of flour for waste? 8. A New Jersey farmer marketed 50 loads of potatoes of 175 baskets each. If the average net weight of the potatoes was 374 lbs. to the basket and he received 75c. a bushel for them, what did he realize from the crop? ANALYSIS 378 . Many arithmetical problems may be solved readily by analysis, pro- ceeding from the known to the unknown, according to the conditions of the problem, using the unit as the basis of comparison. MENTAL, PROBLEMS Example. — What will be the cost of 5 oranges, if 3 oranges cost 12 cents? Solution. — 1 orange will cost 4 cents and 5 oranges will cost 20 cents. Note. — It is deemed unnecessary that the student go through a long formula of words ; let him use sufficient words to show that the mental trend is connected and logical, and require no more. 379 . 1. What will 6 lemons cost at the rate of 8 lemons for 16 cents? 2. What is the value of 7 pair of shoes, if 4 pair are worth $12? 3. If 7 men can mow 14 acres of grass in a day, how many acres can 12 men mow in the same time? 1/.. What are 8 oranges worth, if 9 oranges cost 18 cents? 5. If 9 pounds of beef cost 54 cents, what will 7 pounds cost ? 6. At the rate of 6 barrels of flour for $30, what will 11 barrels cost? 7. If a man travels 64 miles in 8 days, how far can he travel in 14 days ? 8. If a drover feeds hay at the rate of 22 tons in 11 weeks, how many tons will he feed in 7 weeks? 9. How many apples can be bought for 42 cents at the rate of 14 for 7 cents ? 10. A wheelman rode at the rate of 84 miles in 7 hours; how far would he ride in 9 hours? 11. If 9 men can mow 18 acres of grass in a day, how much can 11 men mow in the same time? 1 2. If 8 men can dig 24 rods of ditch in a day, how much can 9 men dig in the same time? 13. If 7 boys can do a piece of work in 15 days, how long will it take 21 boys to do it? 11/.. How many men will be required to build ashed in 3 days, if 5 men can do it in 12 days ? 15. How many men can mow as much grass in 5 days as 4 men can mow in 40 days ? 16. If it requires 12 men 6 days to build a wall, how many men will be required to build it in 12 days ? 17. If 8 yards of cloth cost $24, what will f- of 20 yards cost ? 18. If a farmer gave 9 bushels of wheat for 2 barrels of flour, what was the wheat worth a bushel if 8 barrels of flour cost $72 ? 19. A farmer exchanged apples worth $2 a barrel for 4 } T ards of cloth which cost at the rate of $21 for 7 yards ; how many barrels of apples did he give ? 20. If f of a yard of muslin cost 9 cents, what will 2 yards cost? Solution. — 1 of a yard will cost 3 cents, 1 yard will cost 12 cents and 2 yards will cost 24 cents. 98 ANALYSIS 99 21 . If of a box of tea cost $10, what will 1 box cost? 22 . What will 3 boxes of soap cost, if f of a box cost $6 ? 23 . If | of a yard of cloth cost $8, what will 3 yards cost? 24 - If f of a barrel of flour cost $4, what is the the value of 12 barrels? 25 . If | of a yard of cloth cost $7, what will 8 yards cost? 26 . If A can walk 24 miles in -f- of a day, how far can he walk in 6 days ? 27 . If f of a box of raisins cost $5, what will 7 boxes cost? 28 . How much will 8 barrels of apples cost if of a barrel cost 70 cents ? 29 . How much will 7 tons of hay cost if $20 will buy f of a ton ? 30 . A watch cost $42 and f of its cost was twice the cost of a chain ; what was the cost of the chain? 31 . A horse cost $100 and 4 of its cost was the cost of a carriage ; what did both cost ? 32 . If 11 yards of cloth cost $5J, what will 5 yards cost? Solution. — §5£ are $V-, 1 yard will cost and 5 yards will cost $4 or $24 33 . If 5 oranges are worth 7J cents, what is one dozen worth ? 34 -. What cost 18 bananas at the rate of 1 \ cents for 5 bananas? 35 . If 11 chickens cost $4f , what will 15 chickens cost ? 36 . If J of 16 yards of cloth cost $34, w T hat will 4 of 18 yards cost? 37 . If -g- of 42 yards of muslin cost 49 cents, what will § of 36 yards cost ? 38 . If 3 apples cost f of a cent, what will 10 apples cost? 39 . If 8 pair of shoes cost $-2A, what will 10 pair cost ? 40 . If 3 shaddocks cost f of 1 dollar, what will 8 shaddocks cost? 41 . How much are 12 lamps ivorth, if 7 lamps are worth $^g 4 -? 42 . What will f- of 15 yards of cloth cost, if 4 of 24 yards cost 4 of $36 ? 43 . What is the value of 6 mirrors, if 9 mirrors are worth $^-? 44 ■ A boy is 12 years old, which is J his father’s age ; what is the father’s age? Solution. — If 12 years is 4 the father’s age, the father is 4 times 12 years, or 48 years old. 45 . A sheep cost $8, which w r as f the cost of a cow ; what would 3 cows cost? 46 . James has 20 pennies, which is § of John’s number ; how many have both ? 47 . Martha’s hat cost $7, which was 4 of the cost of her dress ; what was the cost of both ? 48 . A clerk spent $20, which is 4 of the sum he has left ; how much had he at first ? 49 . A man earned $30, which is f of what he has in a savings bank ; how much has he in bank ? 50 . A watch was bought for $25, which is 4 of 5 times what the chain cost \ what was the cost of both ? 51 . The head of a fish is 5 inches long, which is 4 of twice the length of the body ; what is the length of the body ? 100 ANALYSIS 52. Wheeler paid $5 for a saddle, which is 4 of J of the cost of his wheel ; what did his wheel cost? 53. The head of a fish is 4 inches long and the tail 5 inches long, which together is f of J of the length of the body ; how long is the fish? 54 . Thomas, who is 15 years old, is •§- of Harry’s age ; how old is Harry ? Solution. — £ of Harry’s age is \ of 15 years or 3 years ; Harry is 6 times 3 years old, or 18 years old. 55. A woman found $12, which was f of what money she then had; how much had she at first? 56. A farmer sold a cow for $32, which was f of the cost of the cow ; what did the cow cost? 57. A drover sold a horse for $120, thereby gaining 4 of its cost ; w T hat was its cost ? 58. Leary sold a book for 80 cents, which was 4 of -f of its cost when new ; what was its cost? 59. Harkness sold a horse for $140, which was § of 4 of its value; what was the horse worth ? 60. 18 feet of a pole is in the water, which is f of -f of the length in the air ; what is the length of the pole? 61. A pole stands 40 feet in the air, which is J-of f of the length of the pole; what is the length of the pole? 62. A horse cost $150, and § of the cost of the horse is twice the cost of the harness; what was the cost of both? 63. A vest cost $5, which is f of of the cost of a coat and J of f the cost of trousers; what did the suit cost? 64- A cow cost $35, which was 4 of twice the cost of a horse ; what did both cost ? 65. A boy’s pony cost $44, which was -f of f- of the cost of his wagon ; what was the cost of his wagon ? 66. If 5 boys can earn $6f in a week, how much can 6 boys earn in the same time ? 67. How far will a man walk in 3 hours if in 7 hours he walks 194 miles? 68. If a man earns $54 in 3 days, how much can he earn in 4 days? 69. If 4J tons of coal cost $27, how much will 7 tons cost at the same rate? 70. How much will a man earn in a week at the rate of $24 a day ? 71. If 3 loads of hay cost $154, what will 6 loads cost? 72. If a vessel can sail 23 miles in 4J- hours, how far can it sail in 9 hours ? 73. If a man rides 14 miles in 2J hours, how far will he ride in 4f hours? 74 ■ If 4 cows eat 2 tons of hay in 8 weeks, how long will the same amount last 5 cows? 75. If 2 men can do a piece of work in 16 days, how long will it take 8 men to do the same work ? ANALYSIS 101 76. A watch cost $50, which is f of 3 times what the chain cost. Required the cost of both. 77. A man having § of a barrel of flour bought f of a barrel ; how much had he then ? 78. A man’s money increased by its f equals $90 ; how much money had he? 79. A man owned f of a boat and sold l of the boat ; what part of the boat did he still own ? 80. The difference between J- of my money and of my money is $10 ; what is the amount of my money ? 81. A boy having 36 pennies, lost f of them and then found -§ as many as he had at first ; how many had he then ? 82. Margaret has $12 and her brother has f- as much ; how much have both ? 83. f of is jig- of the cost of a chain ; how much did the chain cost? 8 If.. A hat cost f of $10, which is J- of the cost of a coat. Required the cost of both. 85. If 5 pints of milk cost 13 cents, how many pints of milk can you buy for 26 cents ? 86. If 12 sheep cost $25, what will 15 sheep cost at the same rate? 87. A chain cost $12 and J of its cost is f of the cost of a watch ; what is the cost of both ? 88. If a yard of cloth costs § of $1, how many yards can be bought for $10 ? 89. How many yards of cloth can be bought for $6, if f of $1 buys 2 yards? 90. If 6 men can do a piece of work in 6-f days, how long will it take 5 men to do it ? 91. A man being asked his age said, that if to his age its 4 and its § were added, he would be 52 years old ; what was his age? 92. A man having lost f of his money had $24 remaining; how much did he lose ? 93. f of the length of a pole is in the air, J in the water, and the remainder, 18 feet, in the earth ; what is the length of the pole ? 91f. What number is that which being increased by its J, and that sum diminished by its §, the remainder is 30? 95. What is a man’s money if twice his money diminished by $12 equals $88 ?' 96. A man’s money increased by $80 equals three times his money; how much money has he? 97. If a tree’s height be increased by § its height and 20 feet, the sum will equal twice its height; what is its height? 98. A man and a boy earned $24 in a week ; what did each earn if the man earned twice as much as the boy? 99. A watch and chain cost $70 ; what was the cost of each if of the cost of the watch equals the cost of the chain ? 100. A pole 40 feet long was broken in two unequal parts so that f of the longer equals the shorter ; what is the length of each part ? 102 ANALYSIS WRITTEN PROBLEMS 380 . Example. — If 5 men reap ten acres of wheat in a day, how many acres will 7 men reap in the same time? Solution. — If five men reap ten acres of wheat in a day, one man will reap one-fifth as much or two acres, and seven men will reap fourteen acres. Example — If 10 men dig a ditch in 4 days of 10 hours each, in how many days will 12 men dig a similar ditch, working 8 hours a day ? Solution. — If 10 men dig it in 4 days of 10 hours each, it would require one man 40 days of 10 hours, or 400 hours ; 12 men would do it in -jj of 400 hours, or 33J hours ; and working eight hours a day would require days. Example. — A can do a piece of work in 10 days, B in 12 days; in what time can both working together do it? Solution. — A can do r ' ff of it in one day ; B can do in one day ; both working together can do X V and x \, or H in one day. They can do f$, the whole piece of work, in 5 t 5 j days. 381 . 1 ■ If a post 5 feet high casts a shadow 12 feet long, how high is a telegraph pole which at the same time casts a shadow 65 feet long? A pipe will fill a cistern in 10 hours, another pipe in 6 hours ; if both be opened at the same time, how long will be required to fill the cistern ? 3 . A man can do a piece of wor k in 6 days, and a boy can do it in 10 days. In what time can they do it by working together? J. An inlet pipe will fill a tank in 6 hours, and an outlet pipe will empty it, when full, in 10 hours. If both be left open when the tank is empty, how long will be required to fill the tank ? 5 . Jones can dig a ditch in 12 days, Smith in 15 days, and Brown in 18 days. How long will be required for them to dig a ditch twice as deep, if they all work together? 6 . B can do ^ as much work in a day as A. How long would it require both working together to do that which B alone can do in 20 days? 7. Green can unload a car of corn in 6 hours; Black can unload it in 5 hours. How many hours would both require, working together, to unload 3 similar cars ? 8 . A can do a piece of work in 16 days. B in 20 days. If both together begin the work, and A leaves when it is f done, how long will B require to complete it ? 9 . If I buy goods at 4 yards for $5 and sell at the rate of 5 yards for §7, how many yards must be sold to gain $70? 10 . If 10 men can do a piece of work in 24 days, and 4 men retire when the}' have worked 10 days, in what time will the remaining men finish the work ? 11 . If 5 men or 9 boys can do a piece of work in 12 days, in how many days should 9 men and 5 boys do the same work? ANALYSIS 103 12. 10 men agreed to do a piece of work, but 2 of them did not report, on account of which the work took 3 days’ more time. In what time could the 10 men have done the work? 13. A cau carry a ton of coal to the fourth story of a building in \ of a day, B in f of a day, and C in •§■ of a day. How long will it require A, B and C, together, to carry 1 ton ? Ilf.. A, B and C can do a piece of work in 8, 10 and 15 days respectively. After the 3 have worked 2 days, how long will it require B and C together to finish it? 15. A does a piece of work in J day, B in § of a day, C in 2 days, and D in If days. How long will it require to perform the work if they all work together ? 16. If a loaf of bread weighs 8 ounces when flour is $3.75 a barrel, what should it weigh wdien flour is $4.50 a barrel? 17. If I gain $1250 by selling $20000 worth of goods, how much should I sell to gain $6250? i/ 18. If 9| pounds of tea cost $6.75, what would 174 pounds be worth ? i/ 19. If $7.60 will buy 144 pounds of soap, how many pounds can be bought for $19.75 ? 20. If 114 acres of land will pasture 33 cows, how many acres will be required for 327 cows ? 21. If 6 men can do a piece of work in 17 days, how many men would be required to do the work in 11 days? 22. If a cycler can ride 225 miles in 2 4 days of 8 hours each, how many days of ten hours each would be required to ride 625 miles ? 23 . If 6 men can mow a field in 10 hours, how many men must be added to mow it in 6 hours ? 21f. If a man can do 4 of a piece of work in 6 days of 10 hours each, in what time can he do the remainder, working 8 hours a day? 25. A, B and C own a vessel, A owning § and each of the others half the remainder ; B sells f of his share for $3000 ; what is the value of A’s and C’s share at the same rate ? 26. If 8 horses or 7 cows eat 6f tons of hay in 23 days, how long will it take 7 horses and 8 cows to eat the same quantity ? 27. If 15 men can do a piece of work in 10 days, how long will be required to do it if 10 men quit when the work is half done ? 28. One pump can fill a cistern in 4 hours and another in 5 hours, and an outlet pipe can empty it in 1 hour ; if the tank be empty and the pumps both working, in what time can it be filled? 29. If the tank in the preceding problem be full with both pumps working and the outlet pipe open, in what time will the tank be emptied ? 104 ANALYSIS 30. A can mow an acre in 5J hours, and A and B together can mow twice as much in 4^ hours ; in what time can B working alone mow 2J acres? 31. Two men start in business with the same amount of capital ; the first gains \ of his capital and the second gains \ of his capital, when the first has $100 more than the second. What was the capital of each ? 32. C can reap a field of grain in 6 days and D in 8 days. In how many days can they do it reaping together? 33. A cistern has two pipes, lyythe first of which it may be filled in 12 hours, and by the second in 8 hours; how long will both be in filling it? 3 If.. D and E working together can make a china closet in 8 days ; D work- ing alone could do it in 12 days; how long would it take E to make it? 35. A owns J of a tract of land and B owns y 1 ^. A’s share is worth $250 more than B’s. Find the value of the tract. 36. A and B agree to do a piece of work, A to receive $2 a day and B $3 a day. A works twice as many days as B, and they together receive $70. How many days does each labor ? 37. I sold a book and lost £ of its cost. Had it cost $1 less, I would have- gained | of its cost. What was its cost? 38. A man sold a book, making £ of the cost. Had the book cost $1 less and sold for the same profit, he would have gained three times the cost. Find the cost. 39. A and B invest equal sums of money ; A gains £ of his investment and B loses $20; B’s money is then equal to £ of A’s. How much does each invest?" IfO. A can do a piece of work in 5 days ; B can do it in 6 days. In how many days can they do it working together? Ifl. G can reap a field of grain in 6§ days and D in 8f days. In how many days can they do it reaping together? If.2. A tank has two pipes, by the first of which it may 7 be filled in 12 hours, and by the second in 15 hours; how long will both be in filling it? If3. E and F working together can make a china closet in 5 days; E work- ing alone could do it in 9 days; how long would it take F to make it? Iflf. A boy hired to a mechanic for twenty weeks on condition that he should receive $20 and a coat. At the end of twelve weeks the boy ceased work, when it was found that he was entitled to $9 and the coat. What was the value of the coat ? MEASURES Measures of Value 382. U. S. Money 10 Mills = 1 Cent E $ d ct. m 1 = 10 = 100 = 1000 = 10000 10 Cents = 1 Dime 1 = 10 = 100 = 1000 10 Dimes = 1 Dollar 1 = 10 = 100 10 Dollars = 1 Eagle 1 = 10 383. English Money 4 Farthings = 1 Penny £ s. d. f. 1 = 20 = 240 = 960 12 Pence = 1 Shilling 1 = 12 = 48 20 Shillings = 1 Pound 1 = 4 £ 1 = $4.8665* (Gold) 384. French Money 100 Centimes = 1 Franc Fr. Cent. 1 = 100 1 Franc = $0,193* (Gold). 385. German Money 100 Pfennigs = 1 Mark M. Pf. 1 = 100 1 Mark = $0,238* (Gold). Other Foreign Moneys Notes. — The Latin Union countries — France, Belgium, Switzerland, Greece and Italy have the same monetary unit, called the “ franc” in the first three, the “drachma” in Greece, and the“lira ” in Italy. The standard gold and silver coins circulate freely in the different countries. Finland, Spain and Venezuela have monetary units of the same value as the franc of France ($0,193), called respectively “mark,” “peseta’’ and “bolivar.” Denmark, Norway and Sweden have a like monetary unit called the “crown”; value $0,268. Mexico, Guatemala, Honduras, Nicaragua, Salvador, Chile, Uruguay, Argentine Republic and the Philippine Islands have each a monetary unit known as the “ peso,” varying in value according to the pure silver or gold contents of the coins of those countries, the lowest being tliat of Chile, worth $0,365, and the highest, that of Uruguay, worth $1,034. (The silver coins fluctuate in value, while the gold coins do not. ) Besides the United States, the following six countries or colonies have the “dollar ” as a mone- tary unit, the value of which is appended in each case : British Honduras ($1.00), Canada ($1.00), Columbia ($1.00), Hong-Kong ($0,463), Liberia ($1.00), Newfoundland ($1,014). These coins are all of gold, except that of the British colony of Hong-Kong, which is of silver valued as above on April 1, 1908. * These monetary unit values are determined by the Director of the United States mint, and proclaimed in a circular issued every three months by the Secretary of the Treasury. They repre- sent the intrinsic value of the precious metal in the coins and are used in estimating the value of imported merchandise upon which duties are collected at the various custom houses. They have noth- ing to do with the commercial rates at which bills of exchange are bought and sold, the latter fluctua- ting with the supply and demand in the foreign exchange market. The commercial rates may, there- fore, be higher or lower than these intrinsic, or par value rates. 105 106 MEASURES OF EXTENSION AND SURFACE Measures of Extension and Surface 386 . Long 12 Inches = 1 Foot 3 Feet = 1 Yard 54 Yards = 1 Rod 320 Rods = 1 Mile M EASURE Mi. rd. yd. ft. in. 1 = 320 = 1760 = 5280 = 63360 1= 5i= 16J= 198 1 = 3 = 36 1 = 12 39.37 Inches = 1 Meter A knot is equal to 6087 feet. The hand, used in measuring the height of horses, equals 4 inches. A pace is equal to three feet, and 5J paces approximate a rod. A rod is sometimes called a perch or pole. A statute mile in the United States and England is 5280 feet, and is distinguished from the geo- graphic mile which is 6087 feet. The yard is the standard unit of extension of the United States and also of Great Britain. 387 . Surveyors’ Long Measure 7.92 Inches 1 Link 25 Links 1 Rod 100 Links (4 rods) 1 Chain 80 Chains 1 Mile The unit of surveyors’ long measure is the Gunter’ s chain, which is 4 rods, or 66 feet in length. Since the chain has 100 links, links may be written as hundredths of a chain. Thus 18 chains, 22 links = 18.22 chains. 388 . Square 144 Square Inches = 1 Square Foot 9 Square Feet =1 Square Yard 30| Square Yards = 1 Square Rod 160 Square Rods = 1 Acre 640 Acres = 1 Square Mile Measure A. sq. rd. sq. yd. sq. ft. sq. in. 1 = 160 = 4840 = 43560 = 6212640 1 = 30i = 2721 _ 39182 1 = 9 = 1296 1 = 144 1.196 Square Yards = 1 Square Meter With the exception of the acre, the units of square measure are derived from the corresponding units of long measure. 389 . Surveyors’ Square Measure 625 Square Links = 1 Square Rod 16 Square Rods = 1 Square Chain 10 Square Chains, or 160 Square Rods = 1 Acre 640 Acres = 1 Square Mile 36 Square Miles = 1 Township The unit of land measure is the acre. In surveying public lands, a square mile, 640 acres, is called a “ section.” MEASURES OF VOLUME OR CAPACITY 107 Measures of Volume or Capacity 390 . Solid Measure 1728 Cubic Inches = 1 Cubic Foot 27 Cubic Feet = 1 Cubic Yard 128 Cubic Feet = 1 Cord 40 Cubic Feet = 1 Ton (for freight) 24f Cubic Feet = 1 Perch of Stone. (16* X 1* X l = 24f.) cu. yd. cu. ft. cu. in. 1 = 27 = 46656 1 - 1728 1.308 Cubic Yards = 1 Cubic Meter 1 Cubic Meter or Stere is the metric unit of Wood Measure. The unit of Lumber Measure is a Board Foot, which is T \ of a Cubio Foot. 391 . Liquid Measure 4 Gills = 1 Pint 2 Pints = 1 Quart 4 Quarts = 1 Gallon gal. qt. pt. gi. 1 = 4 = 8 = 32 1 = 2 = 8 1 = 4 The Wine Gallon contains 231 cubic inches. For estimating, a barrel contains 31* gallons, and a hogshead 2 barrels. Commercially, the exact contents of barrels are found by gauging. 1.05668 Liquid Quarts = 1 Liter (used for milk and wine) 392 . Dry Measure 2 Pints = 1 Quart 8 Quarts = 1 Peck 4 Pecks = 1 Bushel bu. pk. qt. pt. 1 = 4 = 32 = 64 1 = 8 =16 1 = 2 A bushel, stricken measure (the Winchester bushel) contains, approximately, 2150.4 cubic inches, and is used in measuring grain, such as wheat, corn (shelled), beans ; also, small fruits, chestnuts, etc. A heaped bushel contains, approximately, 2747.7 cubic inches, and is used in measuring large fruits (apples, pears, etc.), vegetables, corn in the ear, coal, coke, lime, etc. The British Imperial bushel contains, approximately, 2218.2 cubic inches. The dry gallon or half peck, contains 268.8 cubic inches, or * of 2150.4. .9081 Dry Quarts = 1 Liter 100 Liters = 1 Hectoliter 2.8375 Bushels = 1 Hectoliter 108 MEASURES OF WEIGHT Measures of Weight 393 . Troy Weight 24 Grains = 1 Pennyweight 20 Pennyweights = 1 Ounce 12 Ounces = 1 Pound lb. oz. pwt. gr. 1 = 12 = 240 = 5760 1 = 20 = 480 1 = 24 The Troy pound, 5760 grains, is the unit of weight. Troy weight is used in weighing gold, silver and precious stones. The carat, for weighing diamonds, is 3.2 Troy grains. The carat, indicating fineness of gold, means A part. Gold 20 carats fine has 20 parts gold and 4 parts alloy. 394 . Apothecaries’ Weight it. § 3 9 gr. 20 Grains = 1 Scruple 1 = 12 = 96 = 288 = 5760 3 Scruples — 1 Dram 1 = 8 = 24 = 480 8 Drams - 1 Ounce 1 = 3 = 60 12 Ounces = 1 Pound 1 = 20 This weight is used in weighing small quantities of drugs and medicines. For large quantities avoirdupois weight is used. 395 . • Avoirdupois Weight T. cwt. lb. oz. 16 Ounces = 1 Pound 1 = 20 = 2000 = 32000 100 Pounds = 1 Hundredweight 1 = 100 = 1600 20 Hundredweights = 1 Ton 1 = 16 The gross or long ton used at U. S. Custom Houses, and by wholesale dealers in coal and iron, is 2240 pounds, the hundredweight 112 pounds, the quarter 28 pounds. 1 Pound Avoirdupois = 7000 Troy Grains 15.432354 Troy Grains 2.2046 Pounds Av. (approximately 2204.6 Pounds “ 1000 Grams 1000 Kilos ■ 1 Gram. = 1 Kilogram. = 1 Ton or Tonneau. = 1 Kilogram = 1 Ton or Tonneau. 396 . W eights of Grains Article Measure Pounds Exceptions Article Measure Pounds Exceptions Barley bushel 48 Pa. 47 Hemp bushel 44 Beans bushel 60 Millet bushel 50 Buckwheat Clover Seed bushel bushel 50 60 Pa. 48 N. J. 64 Oats bushel 32 ( N. J. 30 \ Md. 26 Corn in ear bushel 70 Peas bushel 60 Corn, shelled bushel 56 Rye bushel 56 Flaxseed bushel 56 N. J. 55 Timothy Seed bushel 45 Wheat bushel 60 The above table and the one following contain the weights of the common grains, fruits, and vegetables, as used by a majority of the States. Exceptions are given in Pa., Md. and N. J. MEASURES 109 397 . Weights of Fruits, Vegetables, Etc. Article Measure Pounds Exceptions Article Measure Pounds Apples bushel 50 N. J. 48 Beef barrel 200 Peaches, dried bushel 33 Pork barrel 200 Potatoes bushel 60 Pa. 56 Flour barrel 196 Sweet Potatoes bushel 54 Fish quintal 100 Peanuts bushel 22 Grain cental 100 Corn Meal bushel 50 Nails keg 100 Salt, coarse bushel 70 Pa. 85 Lime, unslacked bushel 80 Onions bushel 57 Pa. 50 398 . Time Measure 60 Seconds 60 Minutes 24 Hours 7 Days 365 Days 12 Calendar Months = 1 Minute = 1 Hour = 1 Day = 1 Week = 1 Year Add one day for Leap Years. All years exactly divisible by 4, excepting Centennial Years, are Leap Years. Centennial Years exactly divisible by 400 are Leap Years. 399 . Circular Measure 60 Seconds = 1 Minute 60 Minutes = 1 Degree 360 Degrees = 1 Circle This table is used for measuring angles and arcs, and for latitude and longitude. The signs are degree (°), minute ('), second ("). Longitude and Time 400 . As the circle of the earth, 360°, passes under the sun in 24 hours, in one hour ^4 of the circle, or 15°, would pass; hence, 15° of longitude make a difference of 1 hour in time; hence, to find difference in longitude, multiply difference in time by 15; and to find difference in time, divide difference in longitude by 15. 401 . Miscellaneous 20 Units = 1 Score 12 Units = 1 Dozen 12 Dozen = 1 Gross 12 Gross = 1 Great Gross 24 Sheets = 1 Quire 20 Quires = 1 Ream 6 Feet =1 Fathom 1|- Miles = 1 Knot 3 Knots = 1 League 60 Knots or Geographical Miles 69-jt Statute Miles 1 Degree DENOMINATE NUMBERS 402. Denominate numbers are concrete numbers expressing divisions of money, weight, measure, etc. They may be reduced, added, subtracted, multi- plied and divided. 403. To reduce to lower denominations. Example. — Reduce £13 9s. lOd to pence. 20 12 £ s. d. 13 9 10 £13 9 S . I0d. = 3238d. 269 12 3238 404. Rule. — Multiply the number of higher denomination by the number of the next lower denomination in a unit of the higher, adding the number of the lower denomination to the product. Proceed until the required denomination is reached. WRITTEN EXERCISE 405 . 1 Reduce 18 lb. 11 oz. 3 pwt. 16 gr. to grains. 2. Reduce 12 lb. 5s 63 19 14 gr. to grains. 3. Reduce 77 bu. 1 pk. 1 pt. to pints. 4- Reduce 12 mi. 30 rd. 5 yd. 2 ft. 10 in. to inches. 5. Reduce 365 da. 5 hr. 48 min. 49 sec. to seconds. 6. Reduce 77° 26' 45" to seconds. 7. Change 9 tons 12 cwt. 60 lb. 12 oz. to ounces. 8. Reduce 145 mi. 124 rd. 4 yd. to feet. 9. Change 215 gal. 3 qt. 1 pt. to pints. 10. How many pence in £472 10s.? 406. To reduce or change to higher denomination. Example. — Reduce 7534 pt. to bushels. 2)7534 8)3767 4) 470+ i qt. 7534 pt. =117 bu. 2 pk. 7 qt. 117 + 2 pk. 407. Rule. — Divide the lower denomination by the number in a unit of the next higher denomination, the remainder, if any, being put down as a part of the result. Proceed until the required higher denomination is obtained. no DENOMINATE NUMBERS 111 WRITTEN EXERCISE 408. 1 ■ Change 1213 pt. to higher denominations. 2. Change 478260 gr., apothecaries’ weight, to higher denominations. 3. Change 95346 gr., Troy weight, to higher denominations. If. Change 288692 oz., avoirdupois weight to higher denominations. 5. Change 4841 far. to higher denominations. 6. Change 1456 gi. to gallons. 7. Change 436614 in. to miles. 8. Change 18500 sq. ft. to higher denominations. 9. Change 108000 seconds of time to higher denominations. 10. Change 160000 seconds of circular measure to higher denominations. 409. To reduce denominate fractions and decimals. Example 1. — Reduce \ mi. to feet. 80 m ii l x As = 880 ft. f 1 2 2 Example 2. — Reduce | ton to lower denominations. 5 50 Zx 2 ° = - = 174 ix^ = 50 8 2 2 1 2 Result, 17 cwt. 50 lb. Example 3. — Reduce .255 bu. to lower denominations. .255 4 1.020 8 .16 2 .32 Result, 1 pk., 0.32 pt. Example 4. — Reduce f ft. to fraction of a rod. I jL xl=' rd. 11 22 112 DENOMINATE NUMBERS Example 5. — Change .3759 pt. to decimal of a gallon. 2) .3759 4 ) .18795 .0469875 gal. . — Reduce 8 oz. 7 pwt. 20 24 oz. pwt. gr. 8 7 12 20 167 24 680 1 lb. =5760 gr. 4020 _ 201 _ 67 lb 5760 288 96 334 ^ 4020 gr. g7 Example 7. — The same expressed decimally: ~g = -6979| WRITTEN EXERCISE 410 . 1 . Change of a mile to the decimal of a rod. 2. Reduce - 1 1 7 3 4 1 0 of a ton to the decimal of a pound. 3. Change .007 of an acre to the decimal of a square rod. If.. Change ^ of a gross to the decimal of a dozen. 5. Reduce of a gallon to lower denominations. 6. Reduce of a square mile to lower denominations. 7. Change yjVo °f a pound to the decimal of a scruple. 8. What decimal of a ton is .00625 of an ounce ? 9. Change .025 bu. to lower denominations. 10. Change f- of a foot to the decimal of a rod. 11. Change § of a pound to the decimal of a ton. 12. What decimal of a cubic yard is 430 cu. in.? 13. What decimal of a pound sterling is 5 d.? Ilf.. Change } of a pint to the decimal of a bushel. 15. What decimal of an entire circumference is 40"? 16. What decimal of a degree is 52J'? 17. What part of a square rod is 4J sq. yd.? 18. What decimal of an acre is 6^ sq. ft.? 19. What part of a mile is an inch? 20. Change yt of a mile to lower denominations. 21. What is the value of ^ of a bu. in pints? 22. Reduce T 9 T of a square mile to lower denominations. DENOMINATE NUMBERS 113 23. If § of a pound of gold is \yorth $174.25. what is the value of § of a pennyweight ? Sit.. A man had lOf tons of coal after buying 2.5 tons ; how many pounds had he at first? 25. From f of a pound sterling take .05 of a shilling. 26. What part of 1.25 miles is 150 rd. 2 yd. 6 in. ? 27. Reduce 65 65 29 to the fraction of a pound. 28. Reduce 4 oz. 10 pwt. 20 gr. to the decimal of a pound. 29. Reduce 14 hr. 50 min. 30 sec. to the decimal of a day. 30. Change 10s. 6d. to the fraction of a pound sterling. 31. What part of 3 pk. 6 qt. is 4 qt. 1 pt. ? 32. What part of 6 score is 14 dozen ? 33. What part of a cubic yard is 2 cu. ft. 280 cu. in. ? 34-. What part of 8^ bu. is 2 pk. 64 qt. ? 35. If a man earns 80 cents in 2 hr. 40 min., what will he earn in 6 days if he works 8 hours a day ? 411. To add denominate numbers. Example. — Find the sum of 26 bu. 3 pk. 4 bu. 5 qt. 4 8 bu. pk. qt. 26 3 4 18 1 3 17 0 5 62 1 4 qt. ; 18 bu. 1 pk. 3 qt.; and 17 Result, 62 bu. 1 pk. 4 qts. 412. Rule. — Find the sum of the lowest denomination, reduce this sum to next higher, setting down remainder, if any ; carry the number of next higher denomina- tion, find sum, and reduce in like manner ; so continue until addition is completed. 413. To subtract denominate numbers. Example. — From 31 gal. 2 qt. 1 pt. take 15 gal. 3 qt. gal. qt. pt. 31 2 1 2 - g q Result, 15 gals. 3 qts. 1 pt. 15 3 1 414. Rule. — Find the difference of lowest denomination ; if the subtrahend num- ber is greater than the minuend number, borrow one of the next higher denomination, adding to the minuend number as many of the lower denomination as make one of the higher, then find difference. After borrowing, diminish the minuend number of the next higher denomination by one, then find difference. So continue until the subtrac- tion is completed. 114 DENOMINATE NUMBERS 415 . To multiply denominate numbers. Example. — Multiply 3 yd. 2 ft. 5 in. by 120. 3 12 5 X 120 = 600 in. = 50 ft, y d - ln - 2 X 120 = 240 + 50 = 290 ft,= 3 2 5 96 yd. 2 ft. 120 3 X 120 = 360 + 96 = 456 yd. Result 456 yd. 2 ft. 456 2 0 416 . Rule. — Multiply as in whole numbers and reduce to next higher, setting down remainder, if any, and carrying the number of higher to product of next higher. Proceed in like manner until full product is obtained. 417 . To divide denominate numbers. Example — Divide 176° 42' 12" by 12. 522 372 12)176° 42'^ 12" 14° 43' 31" 418 . Rule .—Divide the number of highest denomination and set down the quo- tient ; reduce the remainder, if any, to next lower denomination, add to result the number in the dividend of same denomination, set down quotient and proceed in like manner until division is completed. If the lowest denomination divided has a remainder, it may be expressed fractionally. 419 . The methods of adding, subtracting, multiplying and dividing here exemplified are not much used in commercial operations. Custom usually des- ignates some denomination as the one to be used, and quantities are named in it and fractional parts of it. WRITTEN PROBLEMS 420 . 1. If a book cost 6d, how many may be bought for 3£ 5s. 6d. ? 2. Change 192000 oz. to tons. 3. A haystack contains 23000 lb.; what is its value at $15 a ton ? f. How many grains Troy in 12 lb. 4 oz. 12 pwt.? 5. How many 2 drachm powders of magnesia can be put up from 13 lb. 5 oz. ? 6. What is the cost of three bushels of tomatoes at 12 cents a quart? 7. Reduce 15 gal. 1 qt. 1 pt. to gills. 8. How many acres of land in a tract containing 2S774S0 sq. ft. ? 9. How many half-acre lots in a piece of ground containing S200 sq. rd. 30 sq. yd.? State the surplus, if any, in square feet. 10. What is the value of a diamond weighing of a carat at $75 a carat? 11. What is the value of the pure gold in an ornament weighing 4 pwt., 18 carats fine, at $20 an ounce for pure gold? 12. Reduce 80 pounds Avoirdupois to denominations of Troy weight, 13. Change 224 oz. Troy to denominations of Avoirdupois weight. If. What is the value of 20000 pounds of oats at 35 cents a bushel ? DENOMINATE NUMBERS 115 15. How many yards of cloth at 7s. 3 d a yard can be bought for £8 10s.? 16. What is the value of £11 5s. 4 d at $4.8665 a pound sterling? Notk. — A business rule for changing shil- lings and pence is to multiply the shillings by 5, and point off 2 places ; the pence by 4| , point oil 3 places. Suggestion. 1 s. = £ 2 ir or .05. 1 d. = £370 or -004^. 17. What is the value in francs of £25 7s. 3d ? 18. What is the value in United States money of 220 yards broadcloth at 8s. 5 d a yard ? 19. What is the value in English money of $1200? 20. Bought 250 meters silk at 5.10 francs a meter and sold it at $1.25 a yard. How much was gained ? 21. A ship sails 7512 kilometers in 60 days. How many knots does it average per hour ? 22. What would be the cost of excavating a cellar 12 meters long, 6 meters wide, 2.5 meters deep, at 36 cents a cubic yard ? 23. How many cords of wood in a pile 30 feet long, 6 feet high, the sticks being 4 feet long ? 24- A cask of wine contains 160 liters worth a franc a liter. How much will be gained or lost by selling it at a dollar a gallon ? 25. The five-cent nickel piece weighs 5 grams; how many could be coined from a kilo of the metal ? 26. Change f-J of a mile to lower denominations. 27. Change £.2685 to lower denominations. 28. Reduce £11 4s. 2 Id to pounds sterling. 29. What decimal of an acre is 112 sq. rd. ? 30. How many slates covering 2f sq. ft. each will cover a square of 100 sq. ft. ? 31. How many reams of paper will be needed to print an edition of 20000 copies of a book containing 216 pages, each sheet folding into 8 leaves? 32. How many degrees in f of a circle? How many minutes? How many seconds? 33. What does 1 of the circumference of a circle measure in degrees, minutes and seconds ? ^ // dr 4- 'A -2qr o / 6>% o / / / . A 4 4 u.r 6, 6 o r / 3 3 4 y ^ 4 4 7 s f 6 <70/2? 4 '-S O 6 4 3-2. V V 3 z f 23 3 3 Res. $54.83. 116 DENOMINATE NUMBERS 34 - What part of a circumference is 16° 18' 25"? 35 . Find the time by compound subtraction from the date of the Declaration of Independence until to-day. Reduce to minutes, counting 30 days to the month. 36 . If the equator is 25000 miles long, what is the length of 10 degrees? 37 . What is the length of a quadrant, if a degree measures 4 inches? 38 . Give the length of a degree in a sextant, when the circumference meas- ures 12 ft. 6 in. 39 . How many scores in 160 dozen? 4 . 0 . How many quires in 500 reams ? 41 . Buy pens at $1.25 a gross and sell them at a cent apiece. What will be the gain in a sale of a great gross? 4 %. How long has a sum of money been on interest, that was placed July 4, 1876? (Time by compound subtraction.) 43 . How many cubic feet in a cistern that will hold 135 barrels? 44 - How many cubic feet in a bin that will hold 200 bushels of wheat? 45 . What is the difference in time between a place in longitude 69° 50' west and a place 120° 25' west? 46 . Change -^ 111 0 of a degree to the decimal of a minute. 47 . What decimal of a square rod is | of a square foot? 48 . If f of a pound of gold is worth $172.50, what is the value of .375 pwt. ? 49 . What would a man gain by selling by the short ton at the same price that he had paid in purchasing by the long ton ? 50 . Find what fractional part of a bushel 3 pk. 6 qt. is, and change the result to a decimal. 51 . Bought 172.5 meters silk at 5.18 francs per meter and sold it at $1.93 a yard. How much was gained? 52 . Buy wheat at 12.35 francs a hectoliter and sell it at a dollar a bushel. How much is gained on a hectoliter? 53 . What would be gained by buying wine at 1 franc a liter and selling it for 20 cents a quart? 54 - Bought 100 kilos French candy at 5.18 francs a kilo and sold it for half a dollar a pound. What w T as my gain or loss on a pound ? 55 . What part of a tonneau would he gained by buying coal by the long ton and selling by the tonneau at the same price? 56 . What part of a ton would be lost by buying coal by the tonneau and selling it by the long ton at the same price? 57 . How many square rods in a piece of land containing 7 square chains? 58 . If the circumference of the earth is 24899 miles, how many kilometers is it? 59 . If the circumference of earth is 25000 miles, what is the length of a degree? PRACTICAL MEASUREMENTS 421. A line has one dimension only — length. 422. A surface has two dimensions — length and width. 423. A square unit is a square in which ihe units of length and width are alike. 424. A solid has three dimensions — length, width and thickness. 425. A cubic unit is a cube in which the units of length, width and thick- ness are alike. SURFACE MEASUREMENTS Note. — U nder this head come such practical measure- ments as carpeting, plastering, papering, painting, roofing, flooring, paving, land measurements, etc. 426. To find the area of rectangular sur- faces. RECTANGLE Example. — How many square feet in a tight board fence 6 feet high and 40 rods long? 40 rods = 660 ft. 660 ft. X 6 = 3960 sq. ft. Note. — W hen the dimensions are expressed in different denominations, they should be reduced to units of the same denomination before multiplying; as, inches to feet , or feet to inches. 427. Rule. — Multiply the length by the width ; the product will be the area. WRITTEN PROBLEMS 428. 1. What is the value of a lot 32 feet front, 120 feet deep, at 25 cents a square foot ? 2. How many square yards of linoleum will be required to cover a floor 22J feet long, 14J feet wide? 3. What will be the cost of painting a wainscoting 40 feet in length, 3 feet 6 inches high, at 25 cents a square yard ? 117 118 PRACTICAL MEASUREMENTS f A railroad 60 feet wide passes half a mile through a tract of land. If the damage is assessed at $150 an acre, what must the company pay? 5. What will it cost, at $2.25 a square yard, to pave with asphalt a street 50 feet wide for a length of 1600 feet ? What must a property owner pay who has a frontage of 40 feet ? 6. How many square feet of tin will be required to cover a flat roof 40 feet long and 20 feet wide ? What will be the cost at $5.25 per square of 100 square feet? 7. At $67.50 per thousand square feet, what will it cost to floor and ceil a room 60 feet long and 25 feet wide ? 8. A ceiling 18 feet 10 inches long, 14 feet 7 inches wide may be of hard wood, costing 12Jc. a square foot, or of corrugated iron at $1.35 a square yard. Which is the cheaper and how much ? 9. The floor of a church 80 feet long, 45 feet wide, is covered with matting. Find the cost of cleaning it at 3 cents a square yard. 10. A bill-board 78 feet 6 inches long and 8 feet high was made at a cost of 6Jc. a sq. ft. What was the cost ? ROOFING AND PAVING 429. To find the number of bricks, shingles, slates, etc., required to cover a given surface. Examples. — (1) Find the number of bricks 4x8 inches required to lay a walk 5 feet wide and 40 feet long ; (2) Find the number of pieces each 2 inches square required to inlay a vestibule floor, 6 feet 8 inches by 5 feet 6 inches. /o /t S X ^Q-X - 430. R ule. — Divide the given surface by the surface of a unit to find the required number. Note. — T he work may he greatly shortened by cancelation, as shown in examples above. PRACTICAL MEASUREMENTS 119 WRITTEN PROBLEMS 431. 1 - How many shingles, averaging 4 inches in width, laid five inches to the weather, will be required to cover two sides of a roof 40 feet long, each side 18 feet wide ? What will the shingles cost at $12 per M ? Note. — The ordinary shingles are 16 in. by 4 in. and are put up in bundles of 250 each. Noth- ing less than an entire bundle should be considered if purchased by the bundle. 2 . At $12.50 per M, what will the shingles cost for a roof 60 ft. long, 18ft. from eaves to ridge on one side, and 24 ft. on the other, the shingles being 4 inches wide, laid 4J inches to the weather ? At $3 per bundle? 3 . How many slates each 10 inches wide, laid 5J inches to the weather will be required for a roof 50 ft. long and 20 ft. from eaves to ridge ? 4- How many bricks, 8X4, laid flat, will be required to pave a 6-foot walk 800 feet long ? What will the bricks cost at $14 a thousand ? What will the laying cost at 60 cents a square yard ? 5 . What will it cost to pave a section of street 40 feet wide, 600 feet long, wiih vitrified brick laid on edge, 9X24 in., at $11 per thousand and 72 cents a square yard for laying ? 6 . How many wooden blocks each 2J in. thick, 6 in. wide and 10 in. long, placed on end, will it take to pave a street 50 ft. wide, for a distance of 2J miles. What will the paving cost at $4.75 a sq. yd. for material and labor? PLASTERING, PAPERING AND PAINTING 432. To find entire surfaces of rooms, boxes, etc. Formula : (Lengthq-Width)X2=Perimeter. P. XHeight=Surface of Walls. L.XW.=Surface of top or bottom. Note. — In problems on plastering, etc., the surface of the walls+the surface of the ceiling, divided by 9 gives the number of square yards. If any allowance is to be made for the space, occupied by doors and windows, deduct such allowance before reducing to square yards. yj r el i v — 1 FT.— J— 1 FT. -J— 1 ft.— 3 F E E T 1 SQUARE YARD Example. — H ow many square yards of plastering in the walls and ceiling of a room 24 feet long, 16 feet wide and 12 feet 6 inches high above the base- board, no allowance for doors and windows? 120 PRACTICAL MEASUREMENTS 433 . Rule — Multiply the perimeter ( distance around ) by the height, and to this resu lt add twice the product of the length by the width. /3 r # < ?j / 3 r# WRITTEN PROBLEMS 434 . 1. How many square feet of surface has a box that is 5 ft. 6 in. long, 3 ft. 6 in. wide and 4 ft. high ? 2. How many square feet of surface has a room that is 18 feet long, 14 feet wide, 9J feet high ? What would be the cost of plastering the walls and ceiling at 35 cents a square yard, no allowance for doors and windows ? 3. At 42 cents a square yard, what will it cost to plaster the walls and ceil- ing of a hallway 80 ft. long, 9 ft. wide and 15 ft. high, allowing for half the space occupied by 12 doors, each 5 ft. 4 in. wide and 9 ft. high? 4- Find the cost of lathing, plastering and flooring a hall 30 ft. by 50 ft. and 20 ft. high at 85 cents a sq. yd. for the lathing, 35 cents a sq. yd. for plaster- ing, and $4.50 per 100 sq. ft. for the flooring ; allowance to be made in the lathing for the full space occupied by 12 windows, each 5 ft. wide and 8 ft. 6 in. high, and 2 doors 6 ft. 6 in. by 12 ft., and § of the door and window space to be allowed in the plastering. 5. How many rolls of paper 8 yards long, 18 inches wide, would be required for the sides, ends and ceiling of a room 15 ft. 6 in. long, 14 ft. 3 in. wide and 9 ft. high ? Note. — S ingle rolls of wall paper are 8 yards long, double rolls, 16 yards long. Paper hangers estimate the number of rolls required by measuring the unbroken wall space ; it is found that the irregular spaces above doors and windows can be covered from the remnants of the rolls. 6. What will be the cost of papering the sides and ends of a room 18 feet long, 15 feet wide, 10 feet high, deducting 10 feet from the perimeter for doors and windows, if the paper costs 50 cents a double roll ? 7. Estimate the cost of papering the walls and ceiling of 5 rooms, each 24 ft. long, 16 ft. wide and 12 ft. high, the baseboard being 12 in. high and the border 18 in. wide. Six inches are allowed to each strip for matching, and 4 single rolls are to be deducted for doors and windows in each room. The paper costs 90 cents a double roll, and the border 12J cents a yard. The work can be done by 2 men in lj days, at $2.75 each per day. PRACTICAL MEASUREMENTS 121 CARPETING 435. Carpet is sold by the yard, a yard of carpet being 3 feet long, regard- less of its width. Linoleum and oil cloth are generally sold by the square yard. In estimating the number of yards of carpet for a room it is always neces- sary to find the number of strips required, and to do this we must know whether they are to run lengthwise or crosswise. They are generally laid lengthwise, but if the room is nearly square they may run either way, and sometimes there is a considerable saving in laying them crosswise. 436. To find the number of yards of carpet required for a room. 437. Rule. — If the strips are to be laid lengthwise, divide the width of the room in inches by the width of the carpet in inches ; the result will be the number of strips, counting any fraction as a whole strip. If the strips are to be laid crosswise, divide the length of the room in inches by the width of the carpet in inches, and count any fraction left over as a whole strip. Multiply the length of the strips in feet, plus any allowance for waste in matching, by the number of strips, and divide by 3. Example. — Which will be the cheaper, and how much, to lay the strips lengthwise or crosswise on a floor 18 ft. by 20 ft., carpet 27 in. wide, costing $1.50 a yard, if there is no waste in matching either way ? Carpet laid crosswise : / 20Fee ir8% T = 24 Strips IV/OE O/nchl 27/ncnt f l r— % \ t ** Sf hi s y> < 0, <5> 18X12= 8 strips, each 20 ft. 6f yds. long. 27 6f yds.X8=53j yds. Or, 8 strips each 20 ft.=160 ft. 160 ft. -=-3=53 J yds. Note. — The heavy lines in the diagrams indicate the direction of the strips, and the dotted lines the number of yards in each strip. >0X12=^: w 9 : 8| strips, or 9 strips, each 18 ft., or 6yds. long=54 yds. 54 yds. — 53 } yds.=f yds. less length- wise. f- yds. at $1.50=$1.00 the result. 122 PRACTICAL MEASUREMENTS WRITTEN EXERCISE 438 . Find the number of yards and th e cost of each of the following: Length Width Width Strips Waste Price of of of to in of Room. Room. Carpet. Run. Matching. Carpet. 1 . 25 ft. 19 ft. 27 in. Lengthwise None $1.50 2. 40 ft. 31 ft. 27 in. Lengthwise 6 in. 3.25 3. 28 ft. 27 ft. 27 in. Lengthwise 1 ft. 2.50 If. 20 ft. 18 ft. 27 in. Crosswise None 1.25 5. 35 ft. 22 ft. 36 in. Lengthwise 9 in. .95 6. 19 ft. 6 in. 15 ft. 9 in. 27 in. Cheaper way None 1.10 7. 72 ft. 9 in. 40 ft. 6 in. 36 in. Lengthwise 1 ft. 3 in. 1.12i 8. 50 ft. 40 ft. 27 in. Cheaper w T ay 1 ft. 1.50 9. 13 ft. 6 in. 12 ft. 27 in. Crosswise 10 in. 1.75 10. 16 ft. 14 ft. 31J in. Cheaper way 3 in. 1.33J 11. 30 ft. 8 in. 21 ft. 11 in. 36 in. Lengthwise 4 in. 1.25 12. 22 ft. 6 in. 20 ft. 10 in. 27 in. Cheaper way None 1.16f 13. 24 ft. 18 ft. 31| j u . Length wflse 1 ft. 1.35 Ilf. 27 ft. 24 ft. 6 in. 27 in. Cheaper way None 1.50 15. 11 ft. 3 in. 10 ft. 9 in. 36 in. Crosswise 6 in. 2.25 WRITTEN PROBLEMS 439 . 1. How many yards of carpet £- yd. wide (27 in.) will be required for a room 15 ft. wide and 22 ft. long, if the strips are laid lengthwise and there is a waste of 1 foot on each strip in matching? 2. What will it cost at 87J cents a yard to carpet a room 17 ft. by 22 ft. 6 in., the carpet being 1 yd. wide, and laid the more economical way, no allow- ance for waste in matching? 3. Find the cost at $1.12J a yard to carpet a room 17 ft. 6 in. wide and 30 ft. 9 in. long with carpet 27 in. wide, the strips to be laid lengthwise and 9 in. to be allowed on each strip for waste in matching. If. At 97 J cents a yard for the carpet and 5 cents a square yard for lining, what would it cost to cover a floor 30 ft. wide and 50 ft. long with carpet 314 in. wide, no allowance for waste in matching? 5. At $1.50 a yard, what will it cost to make a rug, the body of which is to be 3 yards wide and 6 yards long, made of 27-inch Axminster, and the border surrounding it to be 24 inches wide, with no waste except that caused by the diagonal folds at the corners of the border? 6. A lady desiring to cover the floor of her parlor, 20 feet long and 17 feet wide, has the choice of three kinds of carpet, viz : 27 inches wide, with an allowance of 6 inches for matching, at 624c. a yard ; ^ of a yard wide, with S inches allowance for matching, at 80 cents a yard ; or 1 yard in width, with 10 inches allowance for matching, at $1 a yard. Laying the strips lengthwise, which is the most economical ? PRACTICAL MEASUREMENTS 123 THE CIRCLE 440. A circle is a plane figure bounded by a curved line every point of which is equally distant from a point within called the center. 441. The circumference of a circle is its boundary line. 442. The diameter of a circle is any line passing through the center, ending in the circumference. 443. The radius of a circle is one-half the diameter, or any line drawn from the center to the circumference. 444. Concentric circles are cir- cles having the same center. 445. To find the circumference of a circle. 446. Rule. — Multiply the diameter by 3.14-16. 447. To find the diameter of a circle. 448. Rule. — Divide the circumference by 3.1416. 449. To find the area of a circle. 450. Rule. — Multiply the circumference by \ of the diameter, or the diameter by 4 of the circumference. WRITTEN PROBLEMS 451. 1. What is the circumference of a wheel whose diameter is 60 inches? 3. What is the diameter of a circle whose circumference is 628.32 rods? 3. A carriage wheel whose diameter is 3f feet made 1400 revolutions in going a certain distance. Find the distance. 4 . What is the area of a fish-pond whose diameter is 25 ft.? 5. Find the cost of building a wall around a circular garden containing 250000 sq. ft., at $5.50 per linear yard. 6. The side of a square is 22 ft. Find the diameter of a circle equal in area to the area of the square. 7. What will it cost, at 18 cents a square yard, to gravel a walk 6 ft. wide outside a circular plot of ground whose diameter is 80 yards? 8. At $2 a square (100 sq. ft.) what will it cost to paint the surface of a smoke-stack 31 ft. 6 in. in circumference and 120 ft. high ? 9. The radius of the smaller of two concentric circles is 10 feet, and that of the larger circle is 15 feet; what is the area of the ring between the two circles? 124 PRACTICAL MEASUREMENTS SOLID MEASUREMENTS 452. Under this head come such practical measurements as masonry, brick- work, excavations, lumber, contents of heavy timbers, wood measure, capacities of tanks and bins. 453. A volume has length, breadth, and thickness or height. 454. A cube is a rectangular solid, the six- faces of which are equal. 455. Students should note carefully the fol- lowing facts : 1. 27 cu. ft.=l cu. yd. or load ; 24f cu. ft.=a perch of stone. 2. In estimating the amount of material required for a wall, deductions are made for all open- ings and, in a building, for the corners, the latter being four times the thickness of the wall. 3. In estimating the value of masons’ labor on walls, no allowance is made for corners and only- part allowance for the openings. J. Because of the fact that the number of cubic feet in a perch varies so much in different locali- ties, the practise of estimating by the perch is being discontinued, and all kinds of heavy masonry, whether brick, stone or concrete, is now quite generally estimated by the cubic yard. 5. Since 2*, the difference between 27 cubic feet and 24f cubic feet, is T J T of the latter or T q of the former, cubic yards may be changed to perches by adding yU, or perches may be reduced to cubic yards by subtracting T \. 6. In southeastern Pennsylvania and adjacent parts of New Jersey, it is customary, in the absence of a special contract, to allow only 22 cubic feet to the perch in measuring walls to estimate either labor or material 456. To find the contents of walls or other solids in cubic feet, cubic yards, or perches. Example. — How many cubic feet in a stone wall 81 ft. long, 22 in. thick and 8 ft. high ? How unury cubic yards? How many perches? Note. — O nly like dimen- sions should be multiplied, hence, write 22 in. as \\ ft. Suggestion. — 24| = - 9 ^ which, being the divisor, is in- verted in the solution, and the work shortened by cancelation. 457 . Rule. — Multiply together the three dimensions— length, width (or thickness) and height (or depth) ; the result will be the contents in cubic units of the given denomination. r/ x x fr / 2 / tr ir f / X 2 2 X r setts' /2 x xy ' X / X 22 X X X £/ fX /2 X f f PRACTICAL MEASUREMENTS 125 WRITTEN PROBLEMS 458 . 1 - How many perches of stone in a wall 57 ft. long, 7 ft. high, 2J ft. thick ? What will be the cost of laying at 75 cents a perch ? 3 . How many perches of stone in a cellar wall 18 in. thick, measuring 36 ft. by 24 ft. outside, 8 ft. deep ? 3 . What will it cost to lay a stone wall 150 yards long, 4 ft. high, 20 in. thick, at 95 cents a perch ? If. What will it cost to excavate a reservoir 50 meters square, 2 meters deep, at 65 cents a cubic meter? 5 . What will it cost to cover the bottom of the reservoir in preceding problem with concrete 1 inch thick, at 45 cents a square yard ? 6. How many cubic feet of earth must be removed in digging a ditch 120 rods long, 30 inches wide, 3J feet deep ? What will be the cost of the work at 55 cents a cubic yard ? 7. Find the cost of digging a cellar 24 feet wide, 54 feet long and 6 feet deep at 35c. a cubic yard for f of it, and 45c. a cubic yard for the remainder. 8. Find the cost of the material for the walls of a cellar 25 ft. 6 in. long, 18 ft. 9 in. wide, 6 ft. high and 1J ft. thick at $3.75 a perch. 9 . What will it cost to construct a retaining wall 90 feet long, 28 in. thick, and of an average height of 7 ft., at $2.70 a perch for material and 60 cents a perch for labor ? At $3.60 a cubic yard for both ? 10 . At $3.25 a perch for the material and $1.30 a perch for the labor, what will it cost to build a stone house 60 ft. long, 42 ft. 6 in. wide, 28 ft. high, the walls being 16 inches thick, allowing 524 cu. ft. for doors and windows? 11. A stone wall is 65 ft. long, 2 ft. thick and 16J ft. high. How many perches does it contain ? How many cubic yards ? 13 . The foundation of a house is to be 40 ft. long and 24 ft. wide. If built 20 inches thick and ft. high, what will it cost at $3 a perch for the material (net measure) and $2.50 a perch for the mason work (outside measure), no allow- ance being made for openings? 13 . Find the cost of digging a cellar 108 ft. long, 46 ft. wide and 10 ft. deep at 44c. a cubic yard. Ilf. What will it cost to build a foundation wall in the cellar of the pre- ceding problem, the wall to be 2J ft. thick and 12 ft. high, at $3.75 a perch for the material, and $2.75 a perch for the labor, if in estimating the material full allowance is made for 2 openings each 6 ft. X8 ft. and 10 openings each 3 ft. X6 ft., and f of these openings is deducted in estimating the masonry ? 126 PRACTICAL MEASUREMENTS 15. A foundation wall 2 ft. thick and 8 ft. high for a building 60 ft. long and 28 ft. 6 in. wide contains 2 openings for doors, each 7 ft. by 4 ft., and 10 windows, each 6 ft. by 3| ft. Find the cost of the material at $3.20 a perch, allowing for all the openings; also find the cost of the labor at $3.50 a perch, allowing for half the openings. 16. How many bricks 8 in. X 4 in. X 2 in. will be required for a wall 60 feet long, 24 feet high and three bricks thick? Note. — The num- ber of bricks required for a wall is usually found by calculating the number needed for a course and multiply- ing by the number of courses. An allowance is made for mortar, usually from | of an inch to £ of an inch, to each dimension. This allowance need not be considered in working these prob- lems. 17. What would be the cost of bricks required in preceding problem at $8.50 per M ? 18. In a brick-yard there is a pile of common bricks, 72 feet 8 inches long, 32 feet 4 inches wide and 12 feet high. What are the bricks worth at $8.50 per M ? 19. How many common bricks (8 in. X 4 in. X 2 in.) are required for the walls of a house 48 ft. long, 30 feet 6 inches wide, 25 feet high and 2 feet thick, deducting 132 cubic feet for windows and doors? 20. How many bricks will be required to build a bouse, the walls of which are 50 ft. 8 in. long, 25 ft. 6 inches wide, 42 ft. high and 2 ft. thick, allowing 21 bricks to a cu. ft.? Allow for 4 doors 8 ft. high, 3 ft. 6 inches wide, and 10 windows 7 ft. long, 3 ft. wide. What will the bricks cost at $8.50 per M? 21. A brick house measures on the outside 40 ft. 9 in. long, 34 feet wide and 32 ft. 6 in. high, the walls are 24 inches thick. Making an allowance for 12 windows 6 ft. 3 in. long, 3 ft. wide, and 6 doors 8 ft. long, 4 ft. wide, what would be the cost of the bricks at $8 per M, allowing 21 bricks to the cu. ft.? 22. How many perches of stone in three bridge piers, each 24 ft. long, 40 ft. high, 12 ft. thick at the base and 5 ft. thick at the top? How many cubic yards? PRACTICAL MEASUREMENTS 127 LUMBER 459. An amount of lumber is usually given in board feet, except in the case of a few rare hardwoods, such as black ebony, lignum vitae, German black walnut and vermilion, which are sold by the pound ; and mouldings, which are sold by the running foot at so much per C. 460. A board foot is a square foot of board one inch thick or less. Note. — That is, in lumber dealers’ usage, a foot of lumber is (or is the equivalent of) a board twelve inches long, twelve inches wide, and one inch thick or less — surface alone being considered ; but if the board be more than one inch thick, a square foot of it makes proportionately more than a board foot. Thus if 11, 11, or 2 inches thick, a square foot of board equals one and one-fourth, one and one- half, or two board feet ; and so on with greater thicknesses. A mouIding=foot is 1 inch wide, 1 inch thick and 1 foot long. Mouldings are estimated by the size of the pieces from which they are cut, as shown by the thick edge and back ; that is, if the moulding is made from 2 in. X 3 in. lumber, every foot in length is counted 6 moulding-feet. 461. To find the amount of lumber. Example. — How many feet of lumber (board feet) in 26 pieces, 3 inches thick, 4 inches wide, 12 feet long ? Note. — In a lumber bill this is written 26 pc., 3X4 — 12. / 2 X 4 X^3 X 2 & = 3 / 2 Formula (when width is in inches): Length X Width X Thickness X Pieces _ , „ — 2 — Board Feet. Note. — T he operation may frequently be shortened by cancelation. WRITTEN EXERCISE 462. 1 ■ How many board feet in 200 inch boards 10 in. wide, 16 feet long? How many board feet in 150 2-inch planks, 9 inches wide, 18 feet long? 3. What amount of lumber in 80 joists 3X7 — 20? A How many feet in 140 scantlings 3X4 — 18 ? 5. How much lumber in 36 sills 6X8 — 30 ? 6 . What amount of lumber in 36 plates 4X6 — 30? 7. Find the amount of lumber in 180 pieces 2x4 — 16. 8 . Find the amount of lumber in 225 planks 2x9 — 14. ,9. How many feet in 75 rafters 3X5 — 22? 10 . How many feet in 24 girders 3X10 — 28? 128 PRACTICAL MEASUREMENTS 11 . Find amount of lumber in following order : Solution 18 pc. 3X7—20 18 pc. 3X7—20 630 36 “ 3X4—18 36 “ 3X4—18 648 50 “ 2X10—12 50 “ 2X10—12 1000 2278 12 . Find the amount of lumber in the following bill: 40 pc. 3X4—20 12 “ 4X6—30 8 “ 4X6—20 60 “ 2X4—20 140 “ 2X10—12 13 . Find board feet in the following order: 50 pc. 10X12—36 120 “ 6X 8—30 170 “ 4X 6—20 2500 ft. 2 in. plank 14 - What amount of lumber in the following bill? 65 pc. 10X12—44 150 “ 6X 8—36 224 “ 4X 6—24 880 “ 3X 7—36 880 “ 2X 6—36 15 . What is the value of 280 pc. 3x4 — 16 @ $14 per M? Solution 280X^X^X16^480 11. n 1 4.480 14 17.920 44.80 Result $62,720 16 . What is the cost of the following bill of hemlock at $27.50 per M? 60 pc. 4 X 6 — 24 75 “ 3 X 4—18 120 “ 2X10—12 17 . What is the value of the following pine lumber at $55.50 per M ? 20 pc. 14x20 — 45 62 “ 12X16—40 18 . What is the cost of the following bill of oak at $56 per M? 50 pc. 4X6 — 20 50 “ 3X4—10 100 “ 2X6—10 1200 ft. boards PRACTICAL MEASUREMENTS 129 WOOD MEASURE 463. The cord is the unit of measure for rough wood. 464. A cord of wood is a pile 8 feet long, 4 feet high and each stick 4 feet long. It contains 128 cubic feet. 465. A cord foot is a portion of the cord 1 foot long. 466. Rule. — Multiply together the length , height , and width ( each expressed in feet ) and divide by 128 to get cords. WRITTEN PROBLEMS 467. 1. A pile of wood is 84 ft. long, 12 ft. high, and the sticks are 6 ft. long. What is it worth at $3.25 a cord ? 2. A train of 18 cars, each 44 ft. long, 74 ft. wide, is piled with wood to a height of 7 ft. Of how many cords does the load consist? 3. A shed 32 ft. long and 14 ft. wide is high enough to hold 35 cords. How much will it cost to paint the outside at 4J cents a square foot? 4~ A roof projecting 18 in. on all sides, covers a pile of wood 27 ft. wide. The boards for the roof cost $47,564 at $22.65 per M. What is the value of the wood piled under it to a height of 15 ft. 8 in., at $3.78 a cord? 5. A pile of wood 25 ft. wide, 64 ft. long and 15 ft. high, contains how many cords ? 6. What is the value of a pile of wood 30 ft. long, 8 ft. wide and 7J ft. high, at $4.50 a cord ? 7. From a pile of pulp-wood 84 ft. long, 8 ft. wide and 12 ft. high, 48 cords were sold. What was the length of the pile remaining? 130 PRACTICAL MEASUREMENTS CAPACITIES 468. To find the capacity of boxes, bins, or tanks, in gallons or bushels. 469. R ule. — Find contents in cubic inches and divide by number of cubic inches in a gallon ( Liquid 231) to get gallons ; by 2150.1/. cubic inches to get stricken bushels, or by 271/7.7 cubic inches to get heaped bushels. WRITTEN PROBLEMS 470. 1. A bin 7 ft. 6 in. long, 4 ft. 9 in. wide and 5 ft. 6 in. deep is filled with wheat. What is it worth at 974 cents a bushel? 2. A rectangular tank is 8 ft long, 5 ft. wide and 3 ft. 4 in. deep. How many gallons of water will it hold? How many barrels? 3. If the tank in the preceding problem were converted into a bin, how many bushels of wheat would it hold? How many bushels of potatoes? 1/. How many bushels of wheat will a bin 6 feet square and 4 feet deep contain ? 5. What is the capacity, in barrels, of a tank 8 feet square, 10 feet deep? 6. A farmer has a bin 24 ft. long, 8 ft. 4 in. wide and 9 ft. high. How many bushels of corn in the ear will it hold? How many bushels of wheat in it when it is filled to within IS inches of the top ? 7. A bin 15 ft. long, S ft. 4 in. wide and 6 ft. 3 in. deep was filled with apples, from which enough cider was pressed to fill a tank 9 ft, 4 in. long, 4 ft. 6 in. wide and 3 ft. deep. If the apples were bought at 374 cents a bushel and the cider was sold at 33J cents a gallon, how much was gained, if the charge for making the cider was 75 cents a barrel ? PRACTICAL MEASUREMENTS 131 GAUGING 471. Gauging is a process of finding the capacity or volume of tanks, casks, barrels, etc. 472. To find the capacity, in gallons, of a round tank of uniform diameter. 473. Rule. — Multiply the square of the diameter in inches by the height or depth of the tank in inches, and this product by .0034- WRITTEN PROBLEMS 474. 1. What is the capacity of a tank 8 feet 4 inches in diameter and 12 feet 6 inches deep ? 2. Find the capacity of a stand-pipe 10 feet in diameter and 180 feet high. 3. An oil tank 5 ft. 8 in. in diameter and 24 ft. 2 in. long (inside measure- ment) is filled with oil. What is it worth at 6J cents a gallon? 4 ■ A well 3 ft. 4 in. in diameter has 12 feet of water in it. How many barrels of water are there in the well? 475. To find the capacity of a cask or a barrel in gallons. A cask is equivalent to a cylinder, having the same length and a diameter equal to the mean diam- eter of the cask. 476. Rule. — To find the mean diameter of a barrel or cask, add to the head diameter § of the difference between the head and the bung diameters, expressed in inches ; or, if the staves are but little curved, -f. Multiply the square of the mean diameter by the length ( both expressed in inches), and this product by .0034 >' the result will be the capacity in gallons. WRITTEN PROBLEMS 477. 1. How many gallons in a cask whose head diameter is 24 inches, bung diameter 30 inches and its length 34 inches ? 2. What are the contents of a cask 3 feet 6 in. long, the head diameter being 26 inches and the bung diameter 30 inches? 3. A merchant received ten casks of New Orleans molasses, having the following dimensions : head diameter, 30 inches ; bung diameter, 38 inches and length 42 inches, at an average cost of 38 cents per gallon. He retailed the same at 12 cents per quart. What was his gain on the entire lot? 4 . How much did a farmer realize from the sale of 7 barrels of vinegar at 12f cents a gallon, if the staves were slightly curved, the head diameter being 2 ft., the bung diameter 2 ft. 3 in. and the length 2 ft. 10 in.? 5. The bung diameter of a hogshead is 45 inches, its head diameter 40 inches and its length 55 inches ; what is its capacity ? 6. How many gallons in a cask of slight curvature, 3 ft. 6 in. long, the head diameter being 26 in. and the bung diameter 31 in.? 132 PRACTICAL MEASUREMENTS GENERAL REVIEW PROBLEMS 478 . 1 . How many yards will it take to make a rug 4 yds. wide and 9 yds. long, including a border 18 inches wide, if the strips in the body of the rug are made of 27-inch Brussels and 9 in. are lost on each strip in matching ? 2. What weight of water is contained in a tank 14 in. X 9| in. X 8J in., a b-inch cube of water weighing 125 oz.? 3. Find the cost of slating a board 23 ft. 6 in. long and 2 ft. 8 in. wide at 9 •cents per square foot. 4 Divide J of 25.08J miles by % and change to proper denominations. 5. Find the value in American money of express money orders amounting to £17 9s. 6 d. (Exchange $4.8065.) 6. What is the total cost of 24 tons (2240 lb.) of wheat at 90 cents per bushel and 140 centals of wheat at 87J cents per bushel? 7. What is the depth of a tank 7 feet long and 4 feet wide that will hold 42 barrels? 8. What is the weight of shelled corn that will exactly fill a bin 8 ft. long, ■5 ft. wide, 6 ft. deep ? 9. How many Philadelphia bricks 8J X 4| X 2§ in a pile 80 feet X 6 feet X 12 feet? What is their value at $8 per M ? 10. A tank 14 feet long, 3 feet 8 inches wide and 11 feet deep is filled with petroleum. What is its value at 15 cents per gallon? 11. If 48 gallons of wine and 12 gallons of water are mixed, how much wine is there in 4 qt. 1 pt. of the mixture? 12. If a miller takes § for toll, and a bushel of wheat produces 40 pounds of flour, how many bushels must be taken to mill to obtain 10 barrels of flour 9 13. What will a tract of land 270 rods long and 495 yards wide cost at $624 per acre ? 111.. It requires 24 men 30 days to do a piece of work After they have worked 8 days, 6 men leave. In what time can the entire work be completed? 15. The sum of two numbers is 367 and their difference is 11. What are the numbers ? 16. What will the following lumber cost at $56.25 per M? 75 pc. 5 X 7 — 28 75 “ 3 X 7 — 14 150 “ 2 X 8 — 14 300 “ 1 X 4 — 14 17. A man wishing to build a tank in his barn found that he could not safely place there a weight of more than 24 tons of water. How long may he make a tank 4 feet wide and 4 feet deep, water weighing 624 pounds per cubic foot ? 18. What will it cost to pump the water out of a flooded cellar, 30 feet by 18 feet, the water being 4J feet deep, at 8 cents a hogshead? 19. How many bricks 8 in. X 4 in. X 2 in. will be required to pave a street 14 miles long and 30 feet wide, if the bricks are laid edgewise? PRACTICAL MEASUREMENTS 133 20. If f of A’s money equals f of B’s, and § of B’s equals f of C’s, and together they have $23300, how much money has each? SI. What is the cost of 550 inch boards, each 18 feet long and 4 inches wide, at $18J per M ? SS. A farmer sold 45 bushels of potatoes and 55 bushels of rye for $55 50, receiving 10 cents a bushel more for the rye than for the potatoes. What was the price of each per bushel? S3. Find the cost of carpeting two rooms, each 21 feet by 18 feet, with Brus- sels carpet f yard wide at $1.12J per yard ; carpet to be laid lengthwise and 1 foot allowed to each strip for waste. Sj . £. Divide .06§ acres by 1.66§, and multiply the quotient by 9.3§. 55. A grain elevator is 40 feet long, 18 feet wide and 25 feet deep; it is § full of wheat. How many bushels are in it? What is the weight of the wheat in it 9 56. If a bin would hold 1000 bushels of grain, how much water would it contain ? 27. What is gained or lost by buying 15 bushels of chestnuts at $5 per bushel, dry measure, and selling them at 25 cents per quart, liquid measure? 28. Bought in Paris, France, 325.5 meters of silk at 15.60 francs per meter. Find cost in United States money. 29. If the driving -wheels of a locomotive are 10 feet 6 inches in circumfer- ence, and average 8 revolutions per second, how long will it require to go 76 mi. 108 rd. 3 yd.? 30. A rectangular field 140 rods long produces 1764 bushels of wheat at the rate of 32 bushels per acre. Find its width. 31. At 25 cents per square foot, find the cost of a walk 3 feet wide around a grass plat 81 feet long and 60 feet wide; also the cost of a drive 10 feet wide around the walk at the same price per square foot. 32. Bought 36500 tons of iron at £14 10s. 6 d. per ton. What is its value in United States money ? How should it be sold per ton in United States money to gain $2J per ton, and allow $2J per ton duty ? 33. A bought 320 long tons of hay at $15 per ton and sold it at $16.25 per short ton. Find his gain. 31t- A rectangular iron tank, including the top, is made of 2-inch plates. If the inside dimensions of the tank are 8 feet long, 5 feet wide and 10 feet deep, how many cubic ft. of iron in the material ? 35. Find the weight of the tank in preceding problem, and its contents when filled with water, if iron is 7 times heavier than water. 36. How many feet of siding would be required for a house 37 feet 6 inches long, 25 feet wide and 18 feet high, with two gables each 12 feet high, adding ^ for lap and waste ? 37. How many perches of stone in a wall 396 feet long, 44 feet wide and 6.3 feet deep? What will it cost to build the wall at $3.25 per perch? 38. When wheat is quoted at $1.25 per cental, what are 250 bushels worth ? 134 PRACTICAL MEASUREMENTS 39 . 45 men agreed to do a piece of work, but 25 of them did not come and the work was prolonged 5 days. In what time could the 45 have done the work ? 40 . What should be paid for a piece of cloth 35 J yards long and 1J yards wide, if $52.50 is paid for a piece of the same quality 15f yards long and yards wide ? 41 . Find the cost of painting the walls, floor and ceiling of a room 32 feet by 21 feet and 15 feet 6 inches high at 19J cents per square yard. 42 . Find the cost of the following, hemlock being worth $40.50 per M, and pine $24.50 per M : 80 pc. Hemlock 6 X 8 — 32 160 “ “ 4X6—24 240 “ “ 2X10—24 12000 ft. Pine boards 4-5. What will 18 two-inch planks cost at $13J per M, each 14 feet long, 10 inches wide at one end, 8 inches at the other? 44- A grocer purchased eggs at 20 cents per dozen and sold them at the rate of 6 for 19 cents. How many must he sell to gain $1.20? 45. A pharmacist bought 210 pounds of drugs at 72 cents a pound, avoir- dupois, and sold them at 72 cents a pound, apothecaries’ weight. Find bis gain. 46 . Find the cost of a lot 176 feet front and 88 feet deep, at $375.50 per acre. 47. A bin 15 feet long, 6 feet wide and 9 feet deep, is f full of wheat. What is the value at 90 cents per bushel ? 4 . 8 . How many board feet in a board 26 feet long, 15 inches wide and 4 inch thick ? 49 . What will be the cost of 9.37125 miles of insulated wire at 3J cents a foot ? 50 . If a contractor receives $7425 for grading a roadbed 1| miles long, what should he get for 325.9 feet? 51 . A man bought 12 acres of land at $325 per acre and laid it out in town lots 1 chain 50 links long by 1J chains deep. He sold the lots at S825 each. What was his gain ? 52 . Certain city lots are 24 feet by 100 feet. How many such lots in an acre ? 53 . At 2J cents a square inch, what will it cost to bronze a cube the edge of which is 3J feet ? 54 • A farmer sowed 3 bu. 1 pk. 1 qt. of seed and harvested from it 87 bu. 2 pk. and 3 qt. How many bushels did he harvest from a bushel of seed ? 55 . What would it cost, at 16 cents per square yard, to paint the outside of a house 56 feet by 24 feet by 23 feet 6 inches in height ? 56 . A can plow -f of a field in 3 days ; B f of it in 6 days. How long will it require A and B working together to plow it? 57 . How many meters in a mile? How many yards in 565.25 meters? 58 . If a five-cent piece is of a meter in diameter, bow many of them will extend a yard ? PRACTICAL MEASUREMENTS 135 59. How many rods of fence will be required to fence a railway f of a mile' long? How many posts, each 6 feet apart? How many feet of wire, if the fence is 5 wires high ? 60. Reduce § of an ounce avoirdupois to the decimal of a ton. 61. If a lady weighs 125 pounds avoirdupois, what would she weigh Troy? 63. The height of a flight of stairs is 18 feet. How many steps, each 8 inches high ? How many yards of carpet would be required if the tread is ten inches, allowing 1 yard for waste? 63. Divide .0974 of a mile by 2.5, and multiply the quotient by .03J. Change the results to proper denominations. 64.. How many boards, each 18 feet long and 6 inches wide, will be required to make a cubical box 6 feet long ? 65. Find the cost of the masonry of a building at $7.25 per perch, the dimensions of which are 31 feet 6 inches by 23 feet 6 inches and 7 feet high, 18 inches thick. 66. What will be the thickness of a piece of timber 50 feet long, 10 inches wide, to contain 240 board feet ? 67. If f of $ 1 7 g - will buy oh dozen oranges, how many dozens can you buy for*®? b 68. What will it cost, at $16 per M, to erect a tight board fence 6 feet high around a lot 60 yards long and 76 feet wide ? 69. Multiply 193.875 miles by .083, and divide the result by .055. 70. Find the cost at $90 per acre of a field 15 chains 75 links long and 9 chains 25 links wide. 71. A paper manufacturer paid $357 at $8.50 a cord for a pile of pulp-wood 32 feet long, 5J feet wide. How high was it? 73. If 8 men or 12 boys can do a piece of work in 10 days, how long will it require 8 men and 15 boys to do the work ? 73. If a blackboard 21 feet long contains 9 sq. yd. 6 sq. ft., how wide is it ? 74- Add .0| sq. yd., .03J sq. ft. and .18 sq. in. and multiply the sum by 12.31. 75. Which will be the cheaper and how much, to lay carpet 27 inches wide lengthwise or crosswise in a room 26 feet long and 21 feet wide, at $1.25 per yard, allowing one foot to each strip for waste? 76. A bin 9 feet long, 5 feet wide, contains 540 bushels. Allowing 1 \ cu. ft. 4o a bushel, bow deep is it? 77. Allowing 21 bricks to a cubic foot, how many bricks would be required for the main walls of a building 40 feet long, 28 feet 9 inches wide, 24 feet 6 inches high and 24 inches thick ? What would be the cost of the bricks at $10.50 per M ? 78. Divide 155 inches by .0625 and change to higher denominations. 79. If 8 barrels of potatoes and 9 barrels of apples are worth $30, and 16 barrels of potatoes and 12 barrels of apples are worth $48, what is the price of one barrel of each? 136 PRACTICAL MEASUREMENTS 80. How many bundles of laths will be required for the walls and ceiling of a room 28 feet long, 15 feet 6 inches wide and 10 feet high, each bundle being estimated to cover 5 sq. yd.? 81. A case of cranberries containing 3 T 3 g- bushels was bought at $3 per bushel and retailed at 10 cents per quart, liquid measure. How much was gained thereby ? 2 3 82. If 3 of a property cost $700, what should be given for t of a property 4 % at the same rate? 83. Find the cost of the following bill of lumber : 300 inch boards, 14 ft. long, 8 in. wide, at $32.50 per M. 50 joists 12 ft. long, 4 by 5 in. “ 28 00 “ “ 60 planks, 20 ft. long, 10 by 2J in. “ 9.50 “ C 84- A grain dealer paid $118 for 22 barrels of flour, giving $6 for the first grade and $4 for the second grade. How many barrels did he purchase of each ? 85. At $.56 per load, what will the excavation of a cellar cost, the dimen- sions of which are 29 feet 6 inches by 22 feet 8 inches and 6.3 feet deep? 86. How many centals of grain will a rectangular box 9 feet long, 4 feet 2 inches wide and 18 inches deep hold, allowing 60 pounds to a bushel ? How many gallons of water? 87. From £17 8.s. 6d. deduct .05 of itself. 88. How many sheets of tin 20 inches by 14 inches will cover a roof 60 feet long and 23 feet from eaves to ridge? 89. A railroad passes through 1^ miles of A’s farm. If the road is 60 feet wide, what is the cost of the right of way at $66 per acre? 90. If 45 T. 15 cwt. of coal are worth $222.25, what is the value of 23 T. 12 cwt.? 91. The floor of a pavilion 87 j r ards long and 26 feet wide is 2J inch oak. What is it worth at $37J per M? 92. Find weight in long tons of the ice 64 inches thick on a pond, the dimensions of which are 180 yards by 36 feet, if water expands jig- of its volume in freezing and a cubic foot of water weighs 1000 ounces. 93. If 8 barrels of flour cost £15 8s. 6d., what will 5 barrels cost at the same rate? (Result in English and United States money.) 94- Find the cost, at 8 cents a pound, of lining with zinc a cistern (including the top) 9 feet 6 inches by 4 feet 3 inches and 9 feet deep, if for every 5 sq. ft. 12 pounds are required. 95. How many half acre lots can be laid out in a tract of land containing 105 A. 30 sq. rd ? 96. At $15 an acre, what is the cost of a tract of land 34 miles square? 97. How many feet of inch boards will be required to build a fence around a lot 35 rods long and 22 rods wide, if the fence is made 6 feet high? What will be the cost of the boards at $12.50 per M ? PRACTICAL MEASUREMENTS 137 APPROXIMATE . MEASUREMENTS 479. For practical purposes the following rules approximate accuracy. They will be found convenient for use by farmers and others in cases in which only approximately accurate results are required. GRAIN 480. To find the number of stricken bushels of grain in a bin or box. Note. — The relation of 2150.4 eu. in. to 1728 cn. in. is, approximately, as 5 to 4 ; orljcu. ft. are nearly a bushel ; so in practical work the number of eu. ft. diminished by 1 will give an equivalent in bushels and the bushels increased by 1 will give an equivalent in cu. ft. 481. Rule. — Multiply together the three dimensions expressed in feet, and their product by 4 , or .8. Note 1. — For greater accuracy add 1 bushel for every 100 cu. ft. Note 2. — If the bin or box is only partly filled use the depth of the grain for the third dimension. 482. To find the number of heaped bushels in a box or bin. 483. Rule. — Find the contents in cubic feet and multiply by .63. 484. To find the number of bushels of well-seasoned corn in the ear, in a bin or crib. 485. Rule. — Multiply the cubic feet by and divide by 9, (if the corn be of good quality); if of inferior quality, simply multiply the cubic feet by .Jf. Note 1. — The above rule is based on the fact that 2J cu. ft. of good corn in the ear, or 2J cu. ft. of inferior corn, will yield 1 bu. of shelled corn — 56 lbs. Note 2.— If the crib be a flaring one — wider at the top than at the bottom — take for the third dimension, the width at half the height to which it is filled with corn, the corn being leveled before measuring. HAY 486 . Owing to the variety of conditions affecting the kind, it is difficult to make an accurate estimate of the quantity of hay in a given space or bulk. Under ordinary conditions, however, the following facts will give a fair approxi- mate : A ton of hay unpressed in a load or loft is 540 cu. ft.; in a common covered hay barn or in a low stack, 405 cu. ft.; if timothy hay, in mow com- pressed with grain or in butts of large stacks, 324 cu.ft.; in well-settled large stack or mow, 450 cu. ft.; if clover hay, 550 cu. ft. 487 . Rule. — Find contents in cu. ft. and divide by the number of cubic feet in kind of hay required. COAL 488 . Lehigh white ash coal, egg size, 34J cubic feet in a ton; Schuylkill white ash coal, 35 cubic feet ; red ash stove coal, 36 cubic feet. 489 . Rule. — Find contents in cubic feet and divide by the number of cubic feet for a ton. 138 PRACTICAL MEASUREMENTS LIQUIDS 490. To find approximately the capacity in gallons or barrels of a rectangular tank or cistern. 491. Rule. — Multiply the contents in cubic feet by 7\ for gallons, or divide the cubic feet by 4-f {4--®) for barrels. Note. — For greater accuracy in gallons, deduct 1 gallon for every 50 cubic feet. 492. To find the approximate capacity of a round tank, well, cistern, etc., the dimensions of which are given in feet. 493. R ule. — Multiply the square of the diameter in feet by the depth in feet and this product by 5%. The result will be the capacity in gallons. Note — In a flaring or sloping tank (one smaller at the top than at the bottom), the average or mean diameter is, approximately, l the sum of the top and bottom diameters. BRICKS AND SHINGLES, OR SLATES 494. To find the approximate number of bricks required for a wall. Note. — 22 common bricks laid in mortar equal 1 cu. ft. 495. Rule. — Multiply the number of cubic feet in the wall by 22. Note 1.— The dimensions of common bricks are 8 in. X 4 in. X 2 in.; of Philadelphia and Baltimore bricks, 81 X 4J X 2§ in.; of North River bricks, 8 X 34 X 21 in. ; of Milwaukee bricks, 8J X 41 X 2| in. ; and of Maine bricks, 71 X 3f X 2f in. Note 2. — To find the number of fancy bricks for outside courses of walls, multiply the square feet of surface by the number of bricks required for 1 sq. ft. of surface, which number is obtained by dividing 144 by the exposed surface of 1 brick, making proper allowance for mortar. The number of fancy bricks multiplied by the number of bricks in thickness back of them will give the approximate number of common bricks required for the rest of the wall. 496. To find the approximate number of shingles or slates for a roof. Note 1. — If laid 5 inches to the weather it requires 71 shingles to make a square foot of roof, or 720 to the square of 100 sq. ft. ; if laid 4 inches to the weather, it requires 8 to the square foot, or 800 to the square. To make proper allowance for waste, builders usually estimate 800 to 1000 shingles (or about 4 bundles) to every 100 square feet of roof. Note 2. — Roofing-slate is sold by the square (100 sq. ft. ), the price depending on the color and the quality, as well as on the number of pieces required. Sizes vary from 6" X 12 // to 14 // X 24" — the length always being in even inches, but the width being in either even or odd inches. The number of pieces to the square depends, of course, on the exposure, which dealers usually reckon as 4 of the remainder after deducting 3 from the length of the slate. Thus the exposure of a slate 18 in. long is PRACTICAL MEASUREMENTS 139 LOOS 497 . In estimating the amount of lumber in square-edged inch-boards that can be sawed from a round log, lumbermen and others make use of the follow- ing rules : 498 . Doyle’s Rule. — From the diameter in inches, subtract ^ ; the square of the remainder will be the number of board feet yielded by a log 16 feet long. Or, from the diameter in inches, subtract f, square f of the remainder and multiply the product by the length in feet. 499 . Ropp’s Rule. — From the square of the diameter in inches, subtract 60, multiply the remainder by J the length in feet, and point off the right-hand figure of the product. 500 . Two -thirds’ Rule. — From the diameter deduct } for saw kerf (cut) and slab ; square the remainder, multiply by the length of the log, and divide this product by 12 501 . Three-fourths’ Rule. — Same as preceding rule, except that J is deducted for saw kerf and slab. Example. — How many square feet of inch boards can be cut from a log 24 in. in diameter and 20 ft. long ? SOLUTIONS Doyle’s Rule 24 — 4 = 20 i of 20 = 5 5 X 5 X 20 = 500 Result 500 ft. Ropp’s Rule 24 X 24 = 576 576 — 60 — 516 516 X 10 = 516.0 Result 516 ft. Two-thirds’ Rule 24 — 8 =16 16 X 16 = 256 256 X 20 = 5120 5120 = 12 = 4261- Result 426 ft. Three-fourths’ Rule 24 — 6 =18 18 X 18 = 324 324 X 20 = 6480 6480 12 = 540 Result 540 ft. Note 1. — The term “ diameter ” is understood to meau the diameter inside the bark at the top or smaller end of the log. Note 2. — No rule can be relied upon for absolute accuracy under all conditions. Doyle’s rule, though in general use, is too favorable to the buyer of small logs and to the seller of large ones. “The Woodman’s Handbook,” issued by the Bureau of Forestry, Department of Agriculture, Wash- ington, D. C., gives a complete list of rules. WRITTEN PROBLEMS 502 . 1. How many bushels of wheat or other small grain can be stored in a granary 24 ft. 6 in. long, 7 ft. wide and 6 feet high? 2. How many bushels of apples, potatoes, or turnips can be put into a bin 10 ft. 6 in. long, 4 ft. wide, filled to a depth of 3 feet. 9 in.? 3. A box car 28 ft. 6 in. long and 8 ft. wide (inside measurement) is filled with lime to a depth of 4 ft. 3 in. How many bushels does it contain? Jf. A corn crib 30 ft. long, 9 ft. high, 5 ft. wide at the bottom and 8 ft. wide at the top is filled with corn of good quality. How many bushels, approximately, are there in the crib, and what should be its weight when shelled? 140 PRACTICAL MEASUREMENTS 5. How many tons, approximately, of clover hay are there in a stack 20 ft. long, 15 ft. wide and 12 ft. high, and what is the hay worth at $15 per ton? 6. How many tons of Lehigh stove coal can be put into a bin 14 ft. long, 10 ft. wide and 8 ft. deep? How many tons of Schuylkill white ash stove coal? How many tons of red ash stove coal ? 7. A dealer has a bin filled with Schuylkill white ash stove coal that cost him $405, at $4.50 per ton. The bin is 25 ft. long and 15 ft. wide. What is the depth of the coal ? 8. A farmer sold the timothy hay in a mow 30 ft. long, 18 ft. wide and 11 ft. high, at $10.25 a ton of 324 cu. ft. How much did he receive for it? 9. A contractor paid $150 for a stack of hay. It was 30 ft. long, 10 ft. wide and 12 ft. high. What was the price of the hay per ton, if 450 cubic feet were taken as the equivalent of a ton ? 10. A rectangular tank 8 ft. long, 3J ft. wide and 2J ft. deep will contain how many gallons when filled? 11. A round tank 4 ft. 8 in. in diameter and 9 ft. long will hold how many gallons ? How many barrels ? 13. A well 3 ft. 6 in. in diameter and 30 ft. deep is filled with water to within 14 ft. of the top. How many hogsheads of water in it, approximately? 13. Find the capacity in barrels of a tank 9 feet, 4 inches in diameter at the top, 11 feet 8 inches at the bottom and 18 feet 4 inches deep. Ilf.. Find the contents in board feet of two logs each 18 in. in diameter at the top and 24 ft. long. (Results by different rules.) 15. How many bricks will it take to build a wall 7 ft. high, 17 inches (or 4 bricks) thick, on three sides of a garden 50 ft. square, allowing for a gate 6 feet wide ? 16. How many fancy bricks, 8 in. X 2 in. on the face, will it take for the outside course of a building, 45 ft. long and 28 ft. wide, the walls being 21 ft. 6 in. high from the foundation to the cornice, deducting 4 doorways, each 5 ft. by 10 ft. and 16 windows, each 4J ft. by 8 ft.? What will they cost at $27.50 per M ? 17. If the outside walls in the preceding problem are 5 bricks thick, and 18 in. all around the top of them is made entirely of common bricks, besides a total length of 150 ft. of 3-brick partition walls in the building, averaging 20 ft. high, and 80 ft. of 2-brick partitions 10 ft. high, how many common bricks will be required? What will they cost at $12.50 per M? 18. A farmer hauled to a sawmill 20 straight, smooth logs of the following dimensions: 2 logs 48 in. in diameter, 12 ft. long; 2 logs 36 in. diameter, 14 ft. long ; 2 logs 30 in. in diameter, 16 ft. long ; 3 logs 24 in. in diameter, IS ft. long ; 3 logs, 18 in. in diameter, 20 ft. long ; 2 logs, 15 in. in diameter, 22 ft. long ; 2 logs, 12 in. in diameter, 24 ft. long ; 4 logs 10 in. in diameter, 28 ft. long. What did it cost him to have these logs sawed into lumber at $6 per thousand board feet ? (Results by different rules.) PERCENTAGE 503. Percentage is a method of computation by hundredths. 504. Per cent, is an abbreviation of the Latin phrase per centum, and means by the hundred. The sign for per cent, is %. Note. — 10 % means 10 of every hundred, or ten hundredths (bfo)- 505. There are three principal elements considered in percentage. The Base, the Rate and the Percentage. 506. The Base is the number of which a number of hundredths are taken. 507. The Rate is the number of hundredths. 508. The Percentage is the result obtained by taking a certain number of hundredths of the base. 509. In computing percentage, it is frequently more convenient to use the equivalent common fraction than the decimal. The following equivalents should be memorized by the student: 50% 1 2 * 14*% i 7‘ 10% i — io- 22*% 9 40- 33* % x — 3- 25% 1 4- 15% 3 — 2 0- 20% i 5- 35% 7 2 0- 66* % 2 3- 12 *% 1 8* 7J% 3 40- 30% 3 — io- 45% 9 2 0- 16-1% 1 6’ 75% 3 4* 2i% 1 4 0- 40% 2. 5 ‘ 1 14- 83*% 5 6- 37i% 3 8* 13*% 2 15- 60% 3. 5 ‘ 22*% 2 9- 11*% 1 9 • 62i% 5 8* 3*% 1 30- 70% 7 — io- 44*% 4 9- 9*% 1 1 1 87i% = i* 18f% 3 16- 80% 4 5 " 28*% 2 7 ‘ 8*% 1 12 5% 1 — 2 0* 17*% 7 40- 90% 9 — io- 6f% _U_ 1 5 ‘ 6* % 1 — TS General Percentage Formula ^ 0^ 0 j C|aAK/vv *uxaA- -vwbu XKbsJ, av cG/cAjmaAo l Explanation. — If A’s farm is worth 35% more than B’s, then 100% or B’s + 35% of B’s or 135% of B’s=$4050 or A’s farm. If 135% of B’s is $4050, 1- 35) 4050.00 P. 100%, or B’s, is $3000. 30 00 B. 516. Rule. — To find the base, divide the percentage by the rate expressed decimally. Note. — It is frequently convenient to change the rate to its equivalent common fraction and solve by analysis. MENTAL PROBLEMS 517. 1 . A jeweler sold a watch for $30 and thereby gained 25 per cent.; what was the cost of the watch? Solution. — T he gain equals 1 of the cost, which, added to the cost, is f of the cost, or $30 ; 1 of the cost equals 1 of $30 or $6, and the whole cost equals 4 times $6 or $24. 2. A merchant sold a shawl for $14, which was at a gain of 40 per cent.; what did the shawl cost him ? 3. A farmer sold a cow for $32, and thereby gained 33^ per cent, of the cost; what was the cost of the cow? It.. An agent sold a library for $120, thereby losing 33^ per cent, of its value ; what was its value? 5. A merchant sold cloth at $6 per } r ard, thereby gaining 20 per cent.; how should he have sold it to gain 40 per cent.? 6. If by selling land at $75 per acre, 25 per cent, was gained, what was its cost ? 7. On muslin sold at 9 cents a yard, 50 per cent, was gained ; how much did it cost a yard ? 8. A boat was sold for $70, which was at a loss of 16f per cent.; at what price should it have been sold to gain 16f per cent.? 9. A wagon was sold for $90, which was 10 per cent, below its value ; what would have been gained by selling it for $125? 10. A dealer sold 2 wheels for $30 each. On the one he gained 25 per cent, and on the other he lost 25 per cent. ; how much did he gain or lose by the transaction ? 11. A man gained 25 per cent, by selling a watch for $25 more than it cost ; required the cost. 12. A hat was sold for 25 cents more than its cost, which was a gain of 10 per cent.; what was its cost ? 13. A watch was sold at a loss of 33^ per cent. If the price received was $60, what was its cost? 146 PERCENTAGE 14 - A man gained $20 by selling a boat for 25 per cent, more than it cost; what would he have gained by selling it for $90? 15 . A piano was sold for $50 less than its value, which was a loss of 12| per cent. ; what would have been the gain if it had been sold for $440 ? 16 . A certain kind of merchandise yields 5 per cent, profit ; how much should I buy to gain $90 ? 17 . I received $300 or 16§ per cent, of an amount owed to me ; what was the amount of the debt? 18 . A merchant marks goods at an advance of 20 per cent.; what is the cost of goods marked $360 ? 19 . Carpets sold at $125 make a gain of 66§ per cent; what was the cost of the carpets ? 20 . A man offers to pay $500 for a 25 per cent, interest in a business ; what is the valuation of the business ? WRITTEN EXERCISE 518 . 1 ■ $425 is 5 % of what? 2 . 720 lbs. is 9 % of what ? 3 . 740 is 7J% less than what number ? 4 . $817.32 is 6J% of what amount ? 5 . What number increased by 12J% of itself equals 896? 6 . $1252.16 is 8% more than what amount? 7. $32.87 is %% of what sum ? 8 . What sum diminished by 3% of itself equals $1763.24? 9 . $9378.25 is 87f % of how much ? 10 . 12 yds. is \ % of what distance ? 11 . 5 bu. 2 pk. is 11% of what quantity? 12 . 12J% of an amount of money is $250; what is the amount? 13 . If 12J% of the distance from Philadelphia to New York is 11-J miles, what is the whole distance? 14 - 19 bus. are 66f% of what number of bushels? 15 . $10 is 2J% more than what sum ? 16 . 666f is 66f% less than what number? 17 . The percentage is 11 and the rate is 18f % ; what is the base? 18 . 990 miles is 16f% of how many miles? 19 . £20 4s. is 30% of how many pounds sterling? 20 . $36 is 10% more than 15% of what number? 21 . 4 mi. 18 rd. 12 ft. is 18f % of how many miles? 22 . lS-g-% of a man’s salary is $300; what is his salary? 23 . What number will leave 740 after deducting 7J% of it? 24 • Through a reduction of 28f% from list price, I gain $40 on a purchase. W1 lat is the list price? PERCENTAGE 147 WRITTEN PROBLEMS 519 . 1 ■ If 24% of a contract can be executed in 108 days, in what time can the whole contract be executed ? 2 . A man owing a bill pays 35% of it by paying $83.65. How much does he still owe? 3 . A man received a cash discount of $60 on a bill at the rate of 2% ; what was the amount of the bill? J. A man bought a table at a discount of 124% from the marked price and saved $4; what was the price of the table? 5 . A man obtained $660 for goods sold, which was at a gain of 20% to him ; what had he paid for the goods? 6 . A farm was sold for $18000, which was 15% less than the price asked for it ; what was asked for it ? 7. Jones and Smith together own $24500, and Smith owns 16§ % less than Jones; how much does each own? 8 . If I sell $30000 worth of property, what is left will be worth 85% of the value of the whole property ; what is the value of the property I have left? 9 . $1980 is 20% of the cost of B’s farm ; and the cost of B’s farm is 90% of the cost of A’s farm. How much more did A’s farm cost than B’s? 10 . Brown has his property insured for 80% of its value. If his property is damaged 25% by fire, and the insurance company pays him $2340, being 25% of his policy, how much does he lose by the fire? 11 . A broker’s commission, at |%, amounted to $62.50; what was the amount of the transaction? 12 . By cutting down his expenses 3^%, a merchant saves $482.18. What are his expenses? 13 . A man owning f of a ship, sells 274% of his share for $680. What is the value of the whole vessel at that rate? 7J. A sold his horse for $7840, which was 164% more than it cost. What did it cost? 15 . 174% °f Jones’s stock of goods was destroyed by fire, and he received $2162.18 insurance, which was f of the loss. What was his whole stock worth ? 16 . After deducting 3% for prompt payment, a merchant receives $1284.92 for a bill of goods. What was the whole amount of the bill ? 17 . A’s sales for the year are 13% less than B’s, and B’s sales are llf% more than C’s. A’s sales amount to $6289.17. Find B’s and C’s sales. 18 . Smith’s share of a certain claim was 35%. After paying 5 % for having the claim collected, Smith received $2182.93. What was the amount of the entire claim ? 19 . In a certain orchard there are 25% more cherry trees than peach trees, 20% fewer plum trees than cherry trees, 25% more pear trees than plum trees, and 20% more apple trees than pear trees. There are 60 apple trees. How many trees are there in all ? 148 PERCENTAGE 30. A bankrupt pays liis creditors 51yf cents on the dollar. Barnes & Co. receive $822.18 ; what was the amount of their claim ? SI. A man marked his goods 25% more than cost; he sold them for $376, which was 6% less than the marked price. Find his gain or loss. SS. A dealer marked his goods 30% above cost. The goods becoming damaged, he sold them for 20% less than the marked price; his customer failed and he lost 20% of his selling price. If he received $800, find his net loss. S3. A bought a house, paying $1610 cash and giving his note for the remainder, which was 12J% of the cost. What did the house cost, and what was the amount of the note? SI/.. In building a house, 45% of the cost was for brickwork, 25% for the carpenter work, 15% for the mason work, and the remainder, amounting to $3840, was for painting and plastering; what did the bouse cost, and what was the amount paid for each part of the work? 35. Of a cargo, 35% was sold to one buyer, 60% of the remainder to another, and the remainder, amounting to 1716 tons, to a third. How many tons in the cargo ? 36. A, B and C formed a partnership. A invested $25200, which is 20% more than B invested, and 20% less than C invested. Find the capital of the firm. 37. A owns 15% of a business; B, 25%; C, 28%; and D, the remainder. What is the value of each one’s share, if D’s share is $34464? 38. A paid $3402 for a house which was 20% less than it cost to build it ; what did it cost to build the house? 39. Mr. Jones withdrew 30% of his deposit from bank, and invested 30% of what he withdrew, in a house which cost $3000. What amount of money has he remaining in bank ? 30. A bought 75 bbls. of flour, and B increased his stock 22% by buying 12% fewer than A. How many barrels has B now ? 520. To find the rate, the percentage and base being given. Formula. — Percentage-!- Base=Rate (expressed decimally). Example. — Bread made from 120 pounds of flour weighs 162 pounds. What per cent, more than the flour does the bread weigh? Explanation. — More than follows the words per cent, and precedes the base (the. flour). Hence, the flour, 120 pounds, is the base. Subtracting the weight of the flour from the weight of the bread to find how much more the bread weighs than the flour, gives 42 pounds as the percentage. Dividing the percentage 42 pounds by the base 120 pounds, carrying the quotient to hundredths, will give .35 or 35%. 521. Rule. — To find the rate divide the percentage by the base. Or, find equivalent fraction and change to a decimal. 162 120 1.20) 42.00 .35 or 35 % PERCENTAGE 149 MENTAL, PROBLEMS 522 . 1 . A man bought a bicycle for $50 and sold it for $60 ; what was his gain per cent. ? Solution. — He gained $10, which is \ of the cost ; this is the equivalent fraction for 20 per cent. 2. A merchant sold knives at 30 cents each which cost him 25 cents each ; what was his gain per cent.? 3. A merchant sold shawls for $8 which cost him $6 ; what was his gain per cent. ? If. A boatman paid $20 for a boat and sold it for $15 ; what was his loss per cent.? 5. A farmer bought a horse for $120 and sold it for f- of the cost; required the loss per cent. 6. A dairyman bought 10 cows for $240, and sold 8 of them for what all cost ; what was the gain per cent.? 7. A boy having 50 cents spent 20 per cent, of it ; what per cent, remained ? 8. I own property worth $5000, the yearly repairs of which cost me $150 ; what per cent, of the value is expended in repairs? 9. To close out a stock of goods a merchant offers $500 worth for $400 ; what is the per cent, of discount ? 10. The marked price on chairs is $2 each, and they are sold at $1.75 ; what is the per cent, of discount ? 11. On a debt of $750 a man paid $375 ; what per cent, of his debt did he pay ? 12. A merchant buys a bill of goods for $400, and receives a discount of $20 for cash ; what is the rate of cash discount? 13. An agent sells a consignment of goods for $1200; if his commission is $30, what is the rate of commission ? Ilf. A stockholder of a bank, who owns $1200 worth of stock, is assessed $240 ; what is the rate of assessment? 15. A stockholder of a railroad company, owning $12000 worth of stock, received a dividend of $360 ; what is the rate of dividend ? 16. In a roll of carpet of 40 yards, 4 yards are wasted in matching ; what is the per cent, of waste? 17. Linoleum costing 60 cents a yard is sold for 54 cents a yard ; what is the loss per cent. ? 18. Overcoats marked at $25 have been reduced to $15 ; what is the per cent, of reduction ? 19. A merchant who has a stock of $6000 gains $2000 per year upon his stock ; what is his average rate of gain ? 20. A house valued at $4000, rents for $500 a year ; what per cent, of its value is received in rent? 150 PERCENTAGE WRITTEN EXERCISE 523 . 1. $309.68 is what % of $13272? 2. 49 lb. 8 oz. is what % of 450 lb.? 3. $823 is what % more than $712 ? l h What % less than 1262 is 928? 5. 554f lbs. is what % of 3 long tons ? 6. 10 hr. 57 min. is what % of a year? 7. $854.44 is what °/ c of $6252? 8. $2906.86 is what % less than $3620? 9. $32899.40 is what °/ 0 more than $24390 ? 10. 19 yd. 2 ft. 44 in. is what % of a mile? 11. What % of $120 is $18? 12. What % of 2000 pounds is 600 pounds? 13. What % of 200 acres is 10 acres, 20 sq. rods ? Ilf.. What °/o of a day is 5 hr. 30 min. ? 15. What % of a short ton is 196 pounds? 16. What °/ 0 of 3.1416 is .7854? 17. What % of 3.1416 is .5236? 18. What °/ 0 of .7854 is .5236? 19. What % of 2156.42 is 231? 20 What % of $3000 is $150? WRITTEN PROBLEMS 524 . 1. Sold goods for $2770 that cost $2400; what was the rate per cent, of profit ? 2. What % of £12 6s. 4 d. is £2 3s. 2d.? 3. A loss of 2s. on a pound sterling is what rate per cent, of loss? If. Goods costing $840 were sold for $720; at what rate above the selling price should they have been sold to gain $120? 5. Sold goods for $3200 that cost $2600 ; what was the per cent, of gain ? 6. What per cent, of a short ton is a long ton ? 7. What per cent, of a meter is a yard? 8. What per cent, of a mile is a rod ? 9. What per cent, of a dollar is a franc? 10. What per cent, of a pound sterling is a dollar? 11. What per cent, commission does a collector charge, if he remits me $277.68 after collecting a bill of $284.80? 12. Sold goods for $878.22 that cost $714. What was my gain per cent.? 13. A baseball team lost 12 games out of 32. What per cent, did they win ? Ilf Jones pays $152.32 for $4760 insurance. What rate does he pay? PERCENTAGE 151 15. The population of a certain town was 2764 in 1880. In 1890 it was 3173. What was the per cent, of increase? 16. Imported 1230 yards of dress goods at $1.32, and paid $649.44 duty. What per cent, was the duty ? 17. Of a lot of 435 crates of fruit, 60 crates spoiled. What per cent, must the price of the remainder be advanced to cover the loss ? 18. If 12340 lbs. of an alloy containing 4% nickel is mixed with 4560 lbs. of another alloy containing 7f% nickel, what per cent, of nickel does the mixture contain ? 19. If A receives a commission of for selling goods, and B receives $54.16§ more than A on each $5000 worth sold, what is the rate of B’s commission ? 50. What per cent, more than $3875.60 is 93J% of $8972.10? 51. A house costs $4000, and rents for $25 per month ; the expenses for the year are, water tax, $15 ; city taxes, $80 ; and repairs, $25. What net yearly rate per cent, on the investment does the house pay ? SS. A has $6500, B has 28% more money than A. What per cent, less than B has A ? S3. A father receives a yearly salary of $1250 and his son a yearly salary of $1000. What per cent, does the father get more than the son ? Slf.. The list price of a piano is $550. It can be bought for $539. What per cent, is deducted from the list price ? 55. A man’s salary is $2750 per year; he pays $343.75 for rent and $412.50 for other expenses. What per cent, of his salary does he save? 56. A jeweler bought a watch for $160 and asked such a price for it that after falling $18, he still made 20 per cent. What per cent, did he deduct from his asking price ? 57. A house which costs $9000 rents for $50 per month. What is the net annual rate per cent, on the investment if the expenses are, $130 for taxes, $25 for insurance, $175 for ground rent and $45 for repairs ? 58. Bought 450 bushels of wheat at $1.25 a bushel. If 10 bushels spoiled, and I sold the remainder at $1.40 a bushel, what was my gain per cent.? 59. If I sell f of an article for what f- of it cost, what is my gain or loss per cent. ? 30. If an article bought at 20% below asking price is sold for 20% more than asking price, what per cent, is gained? PROFIT AND LOSS 525. Calculations in Profit and Loss are computations to determine the gain or loss resulting from buying merchandise at one price and selling at another. 526. The cost of goods is the amount paid to purchase them, or to produce them. 527. The prime cost of goods is the first or original cost. 528. The gross cost is the prime cost increased by all incidental expenses, such as freight, commission, insurance, duty, packing, drayage, etc. 529. The selling price of goods is the price at which they are sold. 530. The net selling price is the original selling price diminished by any deductions or allowances made to the customer. 531. Profit is the difference between cost and selling price, the selling price being more than the cost. 532. Loss is the difference between cost and selling price, the selling price being less than the cost. 533. Gain or loss in commercial transactions is reckoned as a certain per cent, of the gross cost. 534. To find the profit or loss when the cost and rate are given. Example. — For how much must goods that cost §34 a dozen be sold to realize a profit of 15% ? $34 .15 170 34 $5.10 profit 535. Rule. — Multiply the cost by the rate expressed decimally. MENTAL, PROBLEMS 536. 1. A desk costing $60 was sold at 33^ per cent, profit; what was the amount of gain ? Solution. — A gain of 33} per cent, is a gain of 1 of the cost, or $20. 2. A watch costing $20 was sold at a gain of 20 per cent. ; what was the selling price ? 3. A horse costing $80 was sold at a loss of 25 per cent.; what was the amount of loss ? 4- Land costing $70 an acre increases in value 14f per cent. ; what is it worth an acre ? 5. A clock costing $36 is sold at a loss of 16f percent.; what is the amount of loss ? $34 cost 5.10 profit $39.10 selling price 152 PROFIT AND LOSS 153 6. Bank stock whose par value is $50 is quoted at 60 per cent. ; what is a share worth ? 7. Carpet marked 85 cents a yard is sold at a discount of 20 per cent. ; what is the selling price per yard? 8. A man who pays $16 a month rent finds it increased 12J percent. ; what must he now pay per month ? 9. A rug costing $32 is marked 37J per cent, above cost ; what is its selling price ? 10. Stock costing me $35 a share increases in value 10 percent. ; how much shall I gain per share by selling? 11. Bought a horse for $100 and asked for him 20 per cent, above cost ; what would I gain by accepting 10 per cent, less than I ask ? 12. I buy flour at $5 a barrel and sell for 20 per cent, more ; what do I get per barrel ? 13. Silk bought at $2 a yard is marked 50 per cent, above cost and sold 33 J per cent, below marked price ; what is gained on a yard ? ll^. Lamps costing $5 each are marked 60 per cent, above cost and sold at a discount of 25 per cent, off marked price; what is gained on a lamp? 15. Cloth marked $6 a yard is bought at a discount of 16§ per cent, and sold at a gain of 20 per cent. ; what is the gain on a yard ? 16. Tea costing 45 cents a pound is sold 20 per cent, above cost ; what is the selling price of a pound ? 17. Wheat quoted at 80 cents a bushel drops 5 per cent. ; if I buy, and sell again when it reaches 80 cents, how much do I make on a bushel? 18. Bought coffee worth 32 cents a pound at a discount of 12J per cent. ; what is saved on a pound ? 19. A piano listed at $300 is sold at 33J per cent, below list ; what is received for it ? 20. A piano costing $200 to build is listed at $400 and sold 40 per cent. below 1 list; what is the amount of gain ? ORAL EXERCISE 537 . 1 . For how much must goods be sold that cost $120 to gain 25% ? 2. Cost $220 to gain 50% ? 12. Cost $90 to lose 2J% ? 3. Cost $375 to gain 33J%? 13. Cost $90 to gain 13^% ? 4- Cost $144 to gain 16f % ? n. Cost $120 to lose 12J % ? 5. Cost $250 to gain 12% ? 15. Cost $144 to gain 6f% ? 6. Cost $117 to gain 5% ? 16. Cost $30 to lose 2J % ? 7. Cost $345 to gain 20 %? 17. Cost $70 to gain 14f % ? 8. Cost $110 to gain 15% ? 18. Cost $90 to lose ll-g-% ? 9. Cost $400 to lose '• 30% ? 19. Cost $300 to lose 33J % ? 10. Cost $1100 to lose ! 2% ? 20. Cost $200 to gain 40 % ? 11. Cost $120 to gain 8i%? 154 PROFIT AND LOSS WRITTEN PROBLEMS 538 . 1 . Fish bought for $640 were sold at an average gain of 24% ; what was received for them ? 2. A fur dealer marked down his stock 15% ; what would he the reduced price of a set formerly marked $117.50? 3. The list price of a lot of goods is $960 ; if I buy at 10% off and sell at a profit of 30%, what do I gain on the goods? 1^. A vessel cost its owner to build $21000, and was sold by him at a loss of 15% ; the buyer sold it again at 15% above what he paid for it; how did the price he received for it compare with its original cost? 5. A merchant began business with a capital of $17500 and gained 16|%> which he added to his capital ; the second year he lost 2% ; what is his capital at the end of the second year? 6. A man doing a business of $6500 a year finds his expenses are 35% of his business; what is his net business? 7. A man’s inventory of furniture and fixtures at the beginning of his business year is $1550; at the end of the year he marks off 16| % for wear and tear ; what does the new inventory show ? 8. A man buys merchandise to the amount of $72500 and averages a profit of 15% ; what is the amount of sales? 9. A company having a capital of $250000 has gained 4% on its capital and divides 2J% of its capital in dividends; how much of the profit is undi- vided ? 10. Goods that cost $375.84 were marked 30% more than cost and were sold for 15% less than the marked price. What was the net gain? 11. What must be the selling price of goods that cost $3642.70, to net a profit of 18% ? 12. A merchant bought goods for $4286.91 and sold them at a loss of 224%. How much did he receive for the goods ? 13. Sold goods that cost $237.28 at a profit of 25%, but my customer failed and paid only 75 cents on the dollar; did I gain or lose on the goods, and how much ? Ilf. Bought three lots of goods for $2780.14, $3628.32 and $4245.79, respect- ively. The first I sold at a profit of 12J%, the second at a loss of 374% and the third at a profit of 30%. What was my gain or loss on the whole ? 15. A hardware dealer bought 6 gross of locks at $4.80 a dozen and marked them 40% above cost. If he sells them at 12% less than the marked price, what will be his profit on the lot ? 16. A bought a house and lot for $6500 and sold it to B at a loss of 124% ; B sold it for 22J% more than he paid for it; how much did he receive? 17. A dry goods merchant bought 348 yards of cloth for $131.37 and sold 4 of it at a profit of 11J cents a yard ; he then sold the remainder at such a price as to gain 35 % on the whole. At what price per yard did he sell the remainder? PROFIT AND LOSS 155 18. A bought goods for $2138.75 and sold them to B at a profit of 18f % ; B sold them to C at a profit of 14-f-% ; C sold them to D at a loss of 20J% and D sold them at a profit of 8%. How much did D receive for them? 19. A cargo of wheat, weighing 632897 lbs. was bought for 68f cents a bushel. One-third of it was sold at a profit of 8%, three-eighths of the remainder at a profit of 12£% and the balance at a loss of 3J%. The proceeds were invested in coffee, which was sold at a profit of 18f-%. What was the entire gain ? SO. Jones bought a house for $3285 and sold it to Smith at a profit of 94%. Smith spent $394.28 for repairs and then sold the house at a profit of 15%. How much did he gain ? 539. To find the rate per cent, when the profit or loss and the cost are given. Example. — What is the rate per cent, of profit on goods bought for $6520 and sold for $7458.88 ? $7458.88 selling price 6520. cost 14f or 14f % 6520)938188 6520 $ 938.88 profit 28688 26080 1304 j 26.08 6520 2 5 540. Rule. — Divide the profit or loss by the cost. MENTAL PROBLEMS 541. 1 . Damaged merchandise costing $240 is sold for $180; what is the per cent, of loss ? Solution. — T here is a loss of $60, or of or 1 of the value, which is equivalent to 25 per cent. loss. S. What per cent, is gained by selling an article costing 50 cents for 60 cents ? 3. What per cent, is lost by selling an article costing 60 cents for 50 cents? J. If wheat is bought for 80 cents and sold the same day for 84 cents, what is the rate of gain ? 5. If wheat is bought at 80 cents and sold the same day for 76 cents, what is the rate of loss ? 6. In an invoice of eggs two out of a dozen are broken; what is the per cent, of breakage ? 7. In a box of oranges one out of a dozen is unsalable ; what is the per cent, of loss from this source ? 8. In sponging cloth there is a shrinkage of 3 inches to the yard; what is the per cent, of shrinkage ? 156 PROFIT AND LOSS 9. If an article costing 10 cents is marked 50 per cent, above cost and sold for 3 cents less than the marked price, what is the per cent, of gain on the sale ? 10. A horse costing $200 is sold for $250 ; what is the gain per cent.? 11. A horse costing $200 is sold for $150 ; what is the loss per cent.? 18- What per cent, is gained by selling an article costing 70 cents for 77 cents ? 13. If one-half of an invoice be sold at 25 per cent, gain and the other half at 20 per cent, loss, what is the rate per cent, of loss or gain on the whole invoice? H- What is gained or lost per cent, by asking 30 per cent, more than cost for goods and selling them for 10 per cent, less than the asking price ? 15. What do I gain per cent, by selling chairs at $3 each which cost $2.50 each ? 16. If an article be marked to gain 20 per cent, and ajdiscount of 10 per cent, be allowed, what will be the rate per cent, of gain ? 17. If a share of stock costs $50 and the market price is $40, what has been the rate per cent, of shrinkage in its value? 18. If a share of stock whose par was 100 is bought at 70 and sold at 80, what is the gain per cent.? 19. What do I gain per cent, by selling stock at 60 which I bought at 56 ? 20. What do I earn per cent, as a commission agent, by selling $120 worth of goods, if I receive $6 commission ? ORAL EXERCISE 542 . Tell the rate per cent, of profit or loss on the following : 1 . Cost $150, sold for $200. 11. Cost $500, sold for $625. 2. Cost $66-§, sold for $133|. 12. Cost $600, sold for $720. 3. Cost $375, sold for $625. 13. Cost $100, sold for $90. 4- Cost $600, sold for $450. n. Cost $200, sold for $175. 5. Cost $875, sold for $750. 15. Cost $300, sold for $250. 6. Cost $144, sold for $216. 16. Cost $600, sold for $642. 7. Cost $225, sold for $150. 17. Cost $850, sold for $765. 8. Cost $80, sold for | 588.80. 18. Cost $680, sold for $510. 9. Cost $510, sold for $680. 19. Cost $216, sold for $270. 10. Cost $1200, sold for $1176. 20. Cost $350, sold for $525. WRITTEN PROBLEMS 543 . 1. Merchandise bought for $6250 was sold $782.50 above cost; wha was the per cent, of gain ? 2. Merchandise bought for $6250 was sold for $5467.50 ; what was the per cent, of loss ? 3. A business takes in $45200 a year ; its expenses are $6780 ; what is the rate per cent, of expense ? PROFIT AND LOSS 157 4 . From a tank of oil containing 2250 gallons there is a leakage of 45 gallons a week ; what per cent, is the leakage ? 5 . A creditor gave a collector $50 to collect $1000; what was the rate of commission ? 6 . Carpets bought at 85 cents a yard are marked down to 68 cents ; what is the per cent, of loss ? 7. What is gained per cent, by marking at 33J% above list and selling at 25% off the marked price? 8 . What is the gain per cent, in buying at 10% off list price and selling at 20 % above list price? 9 . A house was offered for sale at 25% of its cost, but finding no takers at that price, was sold at 25% off the price first asked ; what was the rate per cent, of gain or loss ? 10 . A man paid $650 for a diamond and sold it for $910; what did he gain per cent. ? 11 . A lot of goods bought for $3786 was marked 25% above cost, but afterwards sold for 25% less than the marked price. What per cent, was gained or lost ? 12 . If an investment of $7463.80 yields annually $559.78, what is the rate per cent, of income ? 13 . Bought goods for $4987.36, and sold them at an advance of 28f %, but was unable to collect $374.50 of the sales. What per cent, did I gain ? 14 - AVhat per cent, is gained by buving cigars at $65 a thousand and selling them at three for a quarter ? 15 . Merchandise costing $6387.50 was sold as follows: 3 - at a profit of 42%, $1846.25 worth (cost price) at a profit of 27%, and the remainder at cost. What per cent, was gained ? 16 . A grocer bought a quantity of sugar at 22 pounds for a dollar, and sold half of it at 18 pounds for a dollar, and half of it at 19 pounds for a dollar ; what per cent, did he gain on the whole ? 17 . What per cent, is gained by buying coal at $4.87J per ton of 2210 lbs. and selling it at $5.50 per ton of 2000 lbs. ? 18 . What per cent, profit is gained by buying flour at $5.60 a barrel and selling it at 85 cents per sack of 24 lbs. ? 19 . What per cent, profit does a druggist make by buying a powder at 38 cents a pound avoirdupois and selling it at 10 cents an ounce apothecary? 20 . What per cent, profit does an importer gain on cloth that he buys for 3.25 francs a meter and sells at 67 J cents a yard? 158 PROFIT AND LOSS 544. To find the cost when the profit or loss and the rate are given. Example. — I f $249.36 is gained by selling goods at 4% profit, what is the cost ? .04 )_ 2 49.36 Or, 04 = of base, $6234. and $249.36 X 25 = $6234. 545. Rule. — Divide the profit or loss by the rate expressed decimally ; or reduce rate to equivalent fraction and solve by analysis. MENTAL PROBLEMS 546. 1. A furniture dealer gained $1 by selling chairs at a profit of 20 per cent. ; what did the chairs cost ? Solution. — A profit of 20 per cent, is equal to 4 of the cost, which equals $1 ; the entire cost equals 5 times $1, or $5. 2. A man sells a lot of goods at $4 gain, which is 25 per cent, of the cost ; what is the cost ? 3. A loss of $20 is sustained by selling a horse at 10 per cent, below cost ; what is received for the horse ? If. A boat builder gains $20 on the sale of a boat, which is equal to 16f per cent, of the cost; what does he get for the boat? 5. A carriage is sold at a gain of $25, which is 12J per cent, of the cost; ■what is the selling price of the carriage? 6. A carriage is sold at a loss of $25, which is 12J per cent, of the cost; what is the selling price of the carriage? 7. I gain 10 per cent., or $15, on the sale of a w r atch ; what did the watch cost me ? 8. A jeweler gained $10 on the sale of a ring, which was 20 per cent, of the cost; what was the ring marked, if it w r as marked 10 per cent, above the selling price ? 9. A cow was sold at a loss of $6, which was equal to 5 per cent, of its value ; what should have been obtained for the cow so as to neither gain nor lose ? 10. A loss of 50 cents on a barrel of flour is equivalent to a loss of 121 per cent. ; what is the flour worth a barrel ? 11. A man gained $25 by selling a boat at 25 per cent, above its cost ; wbat would he have gained by selling it at 15 per cent, above its cost? 12. $60 less was asked for a piano at a discount of 30 per cent, from list ; what was the list price? 13. $32 is 4 per cent, of a young man’s money, whose amount is equal to 50 per cent, of his companion’s money ; what amount have both? Ilf. A dealer gains 20 per cent., or $12, by the sale of a bicycle ; what would he have obtained for it if he had sold it at a gain of $1S? 15. A watch sold at $50 more than cost gives a gain of 25 per cent. ; what was its cost ? PROFIT AND LOSS 159 16. A watch sold at $50 less than cost gives a loss of 16-| per cent. ; what was its cost? 17. A desk sold at a discount of $10, or 16§ per cent, less than its marked price, realizes cost ; what was its cost ? 18. A man bought a watch for $12 less than its cost, which was 12J per cent, of its cost ; what did he pay for the watch ? 19. Desks marked 25 per cent., or $5 above cost, are sold at cost; what is the cost of a desk ? 20. A bicycle sold at $15 above cost gains 30 per cent, on cost; what was its cost ? ORAL EXERCISE 547 . Find the cost if 1. $40 = 5 % gain. 2. $25 = Ql% gain. 3. $75 = 37J% gain. J. $84 = 12% loss. 5. $125 = 62 J% gain. 6. $96 = 8% loss. 7. $225 = 62 \°/ 0 gain. 8. $1728 = 144% gain. 9. $1100 = S3i% g a i n . 10. $570 = 19% loss. 11. $240 — S-g-% gain. 12. $160 = 16§ % gain. 13 . $120 = 12J% i oss . Ilf.. $300 = 33^-% gain. 15. $150 = 15 % loss. 16. $10 = 34% gain. 17. $125 = 25% gain. 18. $200 = 66§ % gain WRITTEN PROBLEMS 548 . 1 . By selling goods at an advance of 16f%, a merchant gained $324.68 ; what did the goods cost him ? 2. James Clark bought a carriage and sold it at a loss of 27J%. With the amount received he bought another carriage, which he sold at a profit of 30%, gaining $27 on this sale. What did he pay for the first carriage? 3. Sold a property for $287.17 less than it cost, losing thereby 13%. What was the cost ? J. A dealer sold a quantity of goods, 4 at a profit of 20% and the other \ at a loss of 18%. His net gain was $322. Find the cost of the goods. 5. Brown sold a cargo of lumber at a profit of 29%. 47 per cent, of his gain was $4211.67. How much did he pay for the cargo? 6. Goods sold at an advance of $350, realized a gain of 35% ; what did the goods cost ? 7. A house was sold for $270 less than its value, at a loss of 3% ; what was the price obtained for it? 8. A property w T as sold at a gain of $550, which wms IS % of its cost ; what was obtained for it ? 9. Merchandise sold during a year showed an average gain of 30% ; if the amount gained was $3250, what was the amount sold? 10. A corporation set aside a dividend fund of $12500, which was 24 % of the capital stock ; what was the capital stock ? 160 PROFIT AND LOSS 549. To find the cost when the selling price and the rate per cent, of profit or loss are given. Example. — By selling goods for $5239.36, 12% was gained. What did the goods cost ? 46 78 1.12 ) 5239.36 448 759 672 873 784 gQg Result, $4678. 550. Rule. — Divide the selling price by 1 {100%) plus the rate of gain, or by 1 minus the rate of loss. MENTAL PROBLEMS 551. 1. A horse sold at $110 yields a gain of 10% ; what was the cost of the horse ? Solution. — A gain of 10 per cent, is a gain of T \ T of the cost ; the cost and T U of the cost or -J-J of the cost equals $110 ; W °f the cost equals of $110, or $10, and the cost is §100. 1. By selling a horse at $90, a loss of 10 per cent, is sustained; what was the cost of the horse? 3. By selling merchandise for $60, a gain of 20 per cent, is realized ; what is the cost? Ip. Chairs sold for $30 a dozen realize a gain of 20 per cent.; what did they cost ? 5. Coffee sold at 35 cents a pound yields a gain of 16§ per cent. ; what was its cost per pound ? 6. 33J per cent, is gained by selling a wheel for $40 ; what did it cost? 7. Furniture inventoried at $180 shows a depreciation of 10 per cent.; what did it cost? 8. $105 in money returned to a lender includes 5 per cent, interest; what was the amount loaned ? 9. Land sold at $66 an acre shows again of 10 percent.; what was its cost? 10. $160 is obtained for a carriage, which is a loss of 20 per cent. ; what was the cost ? ORAL EXERCISE 552. Find the cost if 1. $600 sale nets 20% gain. 2. $600 sale nets 20 % loss. 3 $350 sale nets 16§ % gain 4~ $350 sale nets 16f% loss. 5. $950 sale nets 5% loss. 6. $1190 sale nets 19% gain 7. $264 sale nets 10% gain. 8. $333 sale nets 11% gain. 9. $111 sale nets 200% gain. 10 $315 sale nets 25% loss. PROFIT AND LOSS 161 WRITTEN PROBLEMS 553 . 1. Two horses were sold for $100 each, one at a gain of 20%, the other at a loss of 20%. How r much was gained or lost on both? 2. A sold a house to B for 33J% less than it cost him. B sold it to C for 25 % more than he paid for it. C paid $2500 for the house. What did it cost A ? 3. Sold a lot of merchandise at a gain of 21|-%, and with the sum received purchased another lot which 1 sold for $4535.17, at a loss of 3%. Find the cost of the first lot. A A jobber offered a lot of goods at 18% more than they cost him, but afterward sold them for $1237.76, which was 7f% less than his first offer. How much had the jobber paid for the goods? What rate of profit did he make on them ? 5. If goods which were marked 31 % above cost, were afterward sold for 90% of the marked price, yielding a profit of $9392.89, what was their cost? 6. A merchant’s capital at the end of a business year is $16619.80 ; during the year he lost 8 % ; what was his capital at the beginning of the year ? 7. A shipment realized $6887.50, at a loss of 5% ; what was the value of the shipment? 8. Goods are insured for $8287.50, which is 2J% less than their value; what are the goods worth ? 9. An agent obtained $956.25 for goods, which was a gain of 12|% ; what was the cost of the goods? 10. Goods marked $1620 were sold for $1350, w r hich "was a loss of 10% ; how much were they marked above cost? REVIEW PROBLEMS IN PROFIT AND LOSS 554 . 1. A hardware dealer buys locks at $4.80 per dozen, list price, less 25 % and 20%, and sells at the same list price less 20% and 16f %. What is his gain on 125 dozen, and his gain per cent. ? 2. A clothier gains 25% by selling cloth at $5 per yard, but a bale of 80 yards being damaged, he has to reduce the selling price 10%. What is his profit on this damaged bale, and his gain per cent.? 3. If green coffee costs 25 cents a pound, and 1 cent a pound for roasting, for what must a pound of roasted coffee sell to gain 16§ %, if green coffee shrinks 15 per cent, in roasting ? A A drover sold a horse for $180 and lost 25% ; with this money he bought another horse, which he sold at a gain of 25%. What was his gain or loss on the transaction and gain or loss per cent.? 5. A man bought a horse and carriage for $450 paying 25% more for the horse than for the carnage. He sold the horse at a gain of 40 %, and the carriage at a loss of 30%. What did he gain on the transaction? 162 PROFIT AND LOSS 6. A merchant bought dry goods for $6000; he sold 20% of them at a gain of 10% ; 40% of them at a gain of 20% ; 40% of the remainder at a gain of 5%, and the remainder at cost. What did he gain or lose? 7. My retail price of fans is $2.50 each ; by selling them at this price I would gain 25%. I sell the same fans wholesale, at $40 per dozen, less 124% and 20%. Do I gain or lose at the wholesale price, and how much a fan ? 8. A merchant marked his goods 60% above cost. He gave one of his customers a discount of 15% off the marked price; what was his gain on $6.80 received from that customer ? 9. A merchant sold 20 % of an invoice at 30 % profit ; 25 % of the remainder at 20% profit; what was his total gain, if the cost of the goods unsold is $3600? 10. How should goods be marked that cost $5 so as to offer a discount of 20% and 10% and still make a profit of 25% ? 11. A man bought a lot of apples and lost 25% of them. What must be his per cent, of gain on the remainder to net a gain of 20% on the cost of the lot? 12. A man sold a house at 30% profit, and with this money bought another house which he sold at 25% profit. What did he pay for each house if the total gain was $2500 ? 13. A merchant buys cloth at $2 50 a yard. How should it be marked to gain 25% if the merchant loses 5% of his sales in bad debts? Ij, l. I sold a lot for $102.25 more than cost and gained 5%. I sold another lot which cost the same for $3000. What was my gain per cent, on the second sale ? 15. The retail price of an article was $262.50, which was 25% more than cost to the retailer. The retailer bought of a jobber who sold it at a gain of 20% above the manufacturer’s selling price and the manufacturer made a profit of 16f%. What did it cost to manufacture this article, and what was each one’s profit? 16. A bought 80 yards of broadcloth at $3.40 per yard, and 74 yards of cassimere at $2.50 per yard. He sold the cassimere at a loss of 20%. What should he ask per yard for the broadcloth to net a gain of 25% on the cost of both ? 17. A sold a wagon to B and lost 25%. B sold it to C at a loss of 5% ; C spent $80 for repairs and sold it for $472.25 gaining 20%. What did the wagon cost A, B and C respectively ? 18. A real estate dealer bought three houses at $1800 each, and sold them at a profit of 8%, 12% and 15% respectively. What was his gain on the three houses ? 19. A real estate agent sold three houses for $2400 each, at a gain of 20%, 124% and 15% respectively. What w T as his entire gain? 20. How much should be asked for coffee which costs 18 cents a pound to gain 10%, allowing 10% for loss in roasting? INVOICE EXTENSION AND TRADE DISCOUNT 555. Invoice Extension, or Bill Work, consists in multiplying the items of a bill or invoice by their respective prices and placing the results in the money-column to the right. Note. — It is not necessary that students be advanced in arithmetic to take the drill in invoice extension. Any student sufficiently advanced to take a business course can, in a short time, be taught to make the extensions, and we advise that it be made a matter of daily practise. For this purpose we recommend bills of five items for beginners, the quantities and prices to contain fractions frequently met with in business, as £, J, £, §, f, f, -§, f, etc., together with a series of three trade discounts on each item. It is well to have a number of such invoices which should always be dictated and the work timed. A bill of five items should be correctly extended and the discounts taken off in from 15 to 20 minutes, the latter being the proper time-limit for a class. Beginners are excused from taking off the discounts until they have acquired the ability to make the gross extensions within this time-limit, when they should begin taking off the easiest discounts. More advanced students may be given bills of ten items which should be extended and discounted in from 25 to 40 minutes. A specimen bill such as is used in this work is given on page 167. 556. A discount is any deduction from the face of a bill or debt. It is usually reckoned as a certain rate per cent. 557. Commercial discounts are of two kinds : (1) Trade discounts, which are deductions from the fixed or list price of goods, allowed to the “trade” — (those in the same line of business) by manufacturers, jobbers and wholesalers (2) Cash discounts — deductions for immediate payment, made from the net amount of a bill for payment within a definite time. It will thus be seen that trade discounts are absolute, while cash and time dis- counts are conditional — the first being deducted from the list price when the goods are billed, and the others from the amount of the bill when payment is made in accordance with the expressed conditions. 558. The list price is also called the marked price, asking price, offer- ing price, or gross price ; and the price after the trade discount has been deducted, the net price or simply the “ net.” 559. Business men usually announce their terms on their billheads ; as, “ Terms : Net 3 months, or 5% off for cash ” ; “ Terms : Net 60 days, or 2% off in 10 days.” 560. Trade discounts arise principally from two different causes: 1. In many lines of business it is customary for merchants to issue cata- logues and price-lists of their goods, with the different articles listed at fixed prices higher than the actual selling prices. They then issue discount sheets from time to time, varying the discounts as the market prices change, instead of altering the list prices. £. In almost all lines of business, the larger quantity a dealer buys at one time the lower price he can get ; and in many cases this concession is made- in the form of an extra discount. 163 164 TRADE DISCOUNT 561. A series of discounts consists of two or more discounts to be taken off successively, the first discount being reckoned on the list price, the second on the remainder left after subtracting the first, the next on the remainder left after subtracting the second, and so on. Note. — Observe that $100 less 20% is $80; but $100 less 10% aud 10% is $81. $100 less 40% is $00; but $100 less 20% and 20% is $64. 562. To find the net amount of a bill. Example 1. — Find the net amount of a bill of $467.50, subject to discounts of 7%, 3% and 2%. 7 3 2 // 3 7 . 3-0 3 2.7 2 z/3 7-7 7 / 3.0 2 <77 f.7*/ 7.23 44 / 3. 3 / Explanation. — Multiply by 7, reject two right-hand figures and subtract; multiply the remainder by 3 and reject the two right- hand figures; multiply the second remainder by 2, rejecting and sub- tracting as before. The remainder is the net amount. The difference between the amount of the bill and the net amount is the total discount. 563. Rule. — Deduct the first discount from the list price, and each subsequent discount from the respective remainders. Note. — In taking off discounts, usage regarding fractions of a cent differs. For uniformity, students should drop all such fractions. Before beginning to discount, add two ciphers for cents to sums expressed in dollars only. Example 2 — Find the net amount of $467.50 less 25%, 14-f- % and SJ%. 7%T / //V 3 3// r/j 73 7. so / / 3. 7 7 3 S 0. 3 3 ,7 O. O 7 3 0 O.S7 2 S. O 7 27 S.SO ^esT- Explanation. — Since 25% equals 4, divide by 4 for the first discount; divide the remainder by 7, since 144% equals 4 ; divide what now remains by 12, since 8J% equals T V. The remainder is the net amount. Example 3. — Find the net result of $467.50 less 22-|%, 624% and 18J %. 3 2 '/is / <2 3 / / 7 /v 447 / O 3.77 3 3 3,32 < ? J e?3S~Q o rj / r/ r/o 227.23 / 3 3.3 3 y- ) 17 otfo r 2 S.S 3 4JLL0Z2J- / / 0 , 7 Explanation. — 22|% equals f, hence multiply by 2 (placing the product to the right), and divide it by 9 for the discount; since 621 % equals f, multiply the remainder by 5 (placing the product to the right as before), and divide it by 8; since 18|% equals T \, multiply the remainder by 3 aud divide by 16, or by 4 twice. The remainder is the net amount desired. Note. — In discounts requiring two or more operations, place to the right all results except the last. TRADE DISCOUNT 165 Example 4. — Find the net result of $467.50 less 2| -%, 3J% and 6J%. J2^ t>'/¥ 4 3 7. SO / / .3 s 4 ss. 2 2 / s. / 4 44 0. 3 3 2 7.43 4 / 3 . / 0 yjjjjLJJL Explanation. — 2£% equals 4 ; divide by 4 (writting quotient under, but one place to the right); since 3 J % equals 4 divide the remainder by 3 ( writing quotient under, but one place to the right); for 6J-% (J 5 ) divide the remainder by 4, placing the entire quotient to the right; divide this quotient by 4 for the discount ; the second quotient subtracted from the second remainder leaves the net result. Example 5. — Find the net result of $467.50 less 15%, 74% and 6f %. /s 77 3 ^3 4 & 7. SO 7 0./ 4 3 4 7-32 2 4. ro 3 3 '7.sr 2 4. SO 3 4 3.0 2 x) /y 0 2.J-# ¥ j / / ^-2./y 3) 73S/ $ Explanation. — 15% equals 4 hence mul- tiply by 3, and, after rejecting right-hand figure,, divide the product by 2 ; multiply the remainder by 3, and, after rejecting the right-hand figure, divide by 4, since 7\ % equals 4> multiply the remainder by 2, and divide the product, exclusive of right-hand figure, by 3, because 6j% equals 4- The remainder is the result. Example 6. — Find the net result of $467.50 less 87J%, 75% and 50%. 77 7 s SO 437,40 S2. 43 / 44 0 7 • 3 0 syz^.4' Explanation. — If 874-%[be taken away, only 12£% is left, hence^divide by 8j; since 75% discount leaves only 25%, divide the result by 4 ; likewise, 50% off leaves 50%, hence divide by 2 ;: the last result is the net. Example 7. — Find the net result'of $832 less 70%, 60% and 30%. 7 o 3 0 3 0 23 2.00 247 S O 7 7 . 24 3 7 ’ 2 2 Explanation.— 70% taken off leaves 30% ; multiply by 3 and reject one figure for the net after this discount ; 60% off leaves 40%, hence multiply result by 4 and reject one figure for the net after the second discount ; 30% off leaves 70%, therefore multiply former result by 7, rejecting one figure. The product is the net result of the bill. The same result can be obtained by multiplying each decimally. 166 TRADE DISCOUNT TABLE OF SHORT METHODS 564 . The short methods below apply to sums in dollars and cents ; for sums in dollars only, the student should annex ciphers to avoid falling into the error of rejecting fractions of a dollar. Operations should be performed mentally, if possible. 1. To get 2%, 3%, 4%, 5%, 6%, 7%, or any discount not an easy aliquot part of 100%, multiply and reject two right-hand figures. 2. Toget50%,33i%,25%,20%,16|%,14f%,m%,lli%,10%,9 1 i r %,8i%. Divide by (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) S. To get 22** (|), 37i% (|), 62** (*), 44** (*), 18** (A). Multiply by 2 ande-9 ; 3 and-4-8 ; 5 and-t-8 ; 4 and^-9 ; 3 ande-16, or 4 twice. T To get 5 % (i), divide by 2 and set quotient one place to the right. To get 3J% (-g 1 ^), divide by 3 and set quotient one place to the right. To get 2J% (^-), divide by 4 and set quotient one place to the right. To get (re), divide by 16, or by 4 twice. 5. To get 6f % ( ¥ 2 -q), multiply by 2, reject right-hand figure and divide by 3. To get multiply by 3, reject right-hand figure and divide by 4. To get 13j-% multiply by 4, reject right-hand figure and divide by 3. To get 15 % (yq), multiply by 3, reject right-hand figure and divide by 2. To get 35 % (-^o), multiply by 7, reject right-hand figure and divide by 2. 6. To get net results if discounts are 50% (4), 66f % (§), 75% (f), 80% (f), ■% (f), 85f% (f), 87 \% (|), 90% (gSg), which leave but A 1 A 1 1 1 1 1 2) 3) i) 5) 6’ T> 8 ) 1 0 > divide the sum by 2, 3, 4, 5, etc. 7. To get 30%, 40%, 60%, 70%, multiply by left-hand figure (3, 4, 6, 7) and reject right-hand figure in product ; or, to get the “ net,” use as a multiplier the left-hand figure of the difference between the rate and 100%. Note. — E xamples, corresponding in number to the above groups and illustrating how the work shall be done, are found on the preceding pages. 8. A series may have an easj r equivalent; as, 40 % , 33 J- % and 25% equals '70% off; 33J%, 25% and 20% equals 60% off. Example S. — Find the net result of $652.70 less 40%, 33J% and 25%. / 0 0°/° 2 0 33/3 (0 0 / 2 O 2S 2 0°/° 3 0 C>S2.70 / ^ < 2 . 2 / Explanation.— 40 % off 100 leaves 60 % ; 334% (i) off 60% leaves 40%; 25% (1) off 40% leaves 30% ; 30% of $652.70 equals $195.81. TRADE DISCOUNT 167 565. Form of invoice showing extensions, discounts and net extensions. Note. — The small figures at the right-hand in the quantity and price are fourths of a yard and fourths of a cent, the numerator alone being written. All other fractions have both numerator and denominator expressed. The operations of items 1 and 2 illustrate the method and form to be followed. / 0 S 2 ' 2 /. 3'A / 3 / S 2 3 y A 2 t, 3 'S' - S/ */. 6 S S / .4 6 / 3 4 L. 4 6> 3./ f 2 3./ S 4 4 0.04 / / .0 o 4 2 y • Q 4 ^y^e4- Operation 1 (1) 263X1-95. (2) | of 263=131 j. (3) i of 195=48|. (4) i of i=b (5) i+f+l=lf. °rlc. Operation 2 (1) 357X2.18. (2) f of 357=267|. (3) f of 218=145h (4) f of f= x V (5) t 6 2+1+|— TTi 0r 2c. -2dT / 2 - 6' 3 S 2 ./ 2 fS 6 3 sy 7/4 2 6>7 3 // ~ 2 >43 6 , '4 ) /o 7 / 6>/ — h 7/2 / 4 2 = f = 4, 42-. / z y 7 2.4 0 / 4 s. k 0 sf6.ro 7 3.3 s S / 3.4 S 3 2.0 4 4) / 2.?3& 4 f / • 3 /a Note — Less than 5 cent in a final result is rejected ; more than 1 cent is counted a cent. 168 TRADE DISCOUNT 566 . The multiplications may often be shortened by the use of aliquots, as explained under Quantity, Price and Cost, (page 90) and as illustrated by items 4 and 5 in the model invoice on preceding page. V6 2, 37 s / 2-// / 2S / / 6 SO 2 f 7 . so =, s 14 3 7 so = 3 S f 3 JS // 40 / stss Note. — I t should be noted here (1) that while the items given for practise are from the dry goods business, the discounts are, for the most part, those met with in the wholesale hardware, wooden ware, and similar businesses, where single discounts will sometimes run as high as 90% or more ; (2) that while three discounts are given here for uniform practise, in business the number will vary from one to a series of five or more ; (3) that two or more items in a bill may be discounted at the same rate, in which case the gross extensions are added and the discounts taken oft' at one time. MENTAL PROBLEMS 567 . Example. — What is the net price of an article listed at $8, 25 per cent, off? Solution.— 25 per cent, off is f off or $2 off; $8 less $2 is $6 net. 2. What is the net wholesale price of tables listed at $15 less 20 per cent.? 3. What is the net price of chairs listed at $15 a dozen less 33 J per cent.? If.. What is the list price of goods sold at $21 net, discount 124 per cent.? 5. A merchant buys at wholesale for $10, which is 16f per cent, off list* what sum would he gain by selling retail at list? 6. What is the net cost of goods listed at $150, sold 20 and 10 per cent, off? 7. State the net cost of merchandise listed at $200, 25 and 10 per cent, off- 8. What is the retail price of a piano listed at $450, sold less 60 per cent.? 9. A book retailing at list price was bought at $3, less 20 per cent, and 10 per cent.; how much was gained on the book ? 10. A buyer bought hats at $60 a dozen less 33£ per cent, and 25 per cent,, and was given an additional discount of 5 per cent, for cash ; find net cost to the buyer. 11. How much better is a single discount of 30 per cent, than a series of 20 per cent, and 10 per cent, off? 12. What is the difference in favor of a single discount of 40 per cent., over a series of 20 per cent, and 20 per cent.? 13. Which would you prefer — to buy goods at 40 per cent, off list, or to buy them 25 per cent, and 20 per cent, off? Ilf. Which is the better and how much — list less 10 per cent, and 20 per cent., or list less 30 per cent, ? 15. Which is the better — goods bought at 30 per cent, and 20 per cent, off, or at 25 per cent., 20 per cent, and 5 per cent, off? TRADE DISCOUNT 169 ORAL PROBLEMS 568 . 1 ■ What per cent, of the list price is the net selling price of goods sold at 20%, 12 4% and 10% off? 2. At 50% and 10% off? 3. At 30% and 10% off? If. At 40% and 25% off? 5. At 25%, 20% and 10% off? 6. At 50%, 40% and 20% off? 7. At 20%, 10% and 84% off? 8. At 30% and 10% off? 9. At 40%, 16$ % and 10% off? 10. At 374% and 20% off? WRITTEN EXERCISE 569 . Find the net amount of 1. $120 less 25%, 10% and 10%. 2. $372.80 less 30%, 124% and 2%, 3. $732.17 less 334% and 3%. If. $84.60 less 10%, 4% and 2%. 5. $497.83 less 40%, 224 % and 1% 6. $1200.62 less 17% and 7%. 7. $696.34 less 75 %, 324% and 34% 8. $754.55 less 20%, 10% and 6f%. 9. $3651.48 less 60% 13% and 5%. 10. $587.30 less 80% and 114%. WRITTEN PROBLEMS 570 . 1 . A bill of goods amounting to $347.60 is sold May 10, subject to discounts of 30%, 10% and 5%, terms, net 60 days ; or 2% off if paid in ten days. How much will pay the bill May 18? 2. How much must be the list price of goods subject to discounts of 20% and 10% in order to net $288? 3. What is the difference on a bill of $582.50 between a discount of 45% and discounts of 25%, 10% and 10% ? If. Find the net cost of 40 doz. locks at $3.10, less 20% and 5%; 56 doz. hinges at $2.20, less 334%; 90 doz. bolts at $2.55, less 60%, 10% and 10%; with an allowance of 2% for cash payment. 5. What must be the list price of goods that cost $97.80 in order to gain 20%, if they are sold at 25%, 10% and 2% off? 6. What is the net cost of 1286 yards of cloth at 874 cents per yard, less 10% and 5%, with a discount of 2% for cash ; case and packing $2.25? 7. If goods are bought at discounts of 25%, 20% and 5% from list price, and sold at 20% and 10% from list, what is the gain per cent.? 8. How much per dozen is gained by buying brushes at $5 per dozen, less 30%, 10% and 10%, and selling them at 35 cents each? 9. What advance on cost is necessary in order to give a discount of 25% and still make a profit of 334% ? 10. A jobber buys 5 doz. lawn mowers at $50 per doz., less 40%, 124% and 3%, and sells them to a retailer at a price that gives him a net profit of $52.65 on the lot. What price per dozen does the retailer pay, and for how much apiece must be sell the mowers to gain 25% ? 170 TRADE DISCOUNT Exercise in Invoice Extension and Discounting 1. 571. Items. 352 2 yds. Prices. $2.62 2 Discounts. 9-6-2 2. 532f 2.37 1 8-6-4 3. 389 3 a 3.22 2 11—7—3 1 516f a 1-33* 6—4—1 5. 193 2 a 1.16| 8—5—2 6. 272 2 a 1.12 2 20-10-5 7. 266* a 2.33* 25— 12 2 — 10 8. 203 3 a 218 3 33i_16|-8i 9. 192 2 a 2.87 2 20—10—10 10. 263 3 a 3.25 50— 25— 12 2 11. 297f a 1.37 2 37 2 — 18 3 — 10 12. 368 2 a 1.50 221-111-5 13. 183f a 1.62 2 62 2 — 37 2 — 12 2 n 486 3 a 2.16f 441-221-111 15. 393* a 2.33| 372— is 3 — 142 16. 218 3 a 1 .73 3 10— 5-2 2 17. 382 2 a 1.25 10— 3i— 2 2 18. 238* It 5.16| 61—5—31 19. 196 1 a 1.20 6 1 — 3i— 2 2 20. 188| a 1.87 2 5— 3i— 2 2 21. 303 3 a 2.81 3 15— 72— 2 2 22. 361 1 Ci 1.32 2 l 3 i_6|_3i 23. 245 2 a 2.1 2 2 35— 15— 7 2 21f. 243f a 2.66| 15—72—62 25. 753f a 1.75 13i_6!_5 26. 615 3 a 3.33* 75—25—10 27. 179 2 a 2.37 2 872 — 122—61 28. 432* a 4.18 3 90—10—5 29. 357f a 2.50 66|-33J-3i 30. 247 2 a 5.66f 80—50—20 31. 595 2 a 1.17 2 40—30—10 32. 125 a 7.56 2 70—60—30 33. 312 2 a 3.75 60—40—20 31 975 a 1.27 2 30-10-3 35. 477 3 a 1.55 70—40—30 36. 133* a 4.63 2 331—25—20 37. 250 a 2.95f 25—20—161 38. 428f a 2.62 2 40—331-25 39. 333 3 a 3.331 50— 20-12 2 4.0. 750 a 1.47f 60—331—25 Items. Prices. DLscounts. 41. 261 1 yds. $2.06 x 5—10—20 42. 192 2 u 2.87 2 25— 12 2 — 6 1 43. 203 3 (( 2.18 3 33*— 16f— 8* 44- 191| a 2.33i 15— 7 2 — 2 2 45. 225i a 1.62 2 3*—10—30 46. 102 2 a 1.12 2 20—10—5 47. 121 1 a 2.06 1 10— 5— 2 2 48. 133 3 a 1.33i 15 — 10 — 5 49. 155f a 1.37 2 25— 12 2 —6 1 50. 163f a 1.62 2 33*— 16|— 8* 51. 177| a 1.87 2 15 — 7 2 — 5 52. 191i a 1.66f l0—3*—2 53. 221 1 “ 2.06 1 5—10—20 54- 25 2 2 a 2.12 2 6 1 — 12 2 — 2* 55. 263 3 a 2.1 8 3 15— 7 2 — 2 2 56. 271i (( 2.331 66* — 33* — 16* 57. 283f u 2.62 2 30—15—5 58. 295| a 2.66-1 10 — 13* — 3* 59. 297i a 2.87 2 14* — 16* — 20 60. 303 3 a 3.37 2 11*— 37 2 — 50 61. 368 2 a 2.16f 25 — 15 — 5 62. 393 3 a 5.12 2 5 2 2 2 2 63. 263J a 4.18 3 6—3—2 64. 567f a 2.66| 3*— 5— 10 65. 389 3 a 1.331 13*— 7 2 — 12 2 66. 373 2 a 1.95f 50— 14*— 7 67. 737| a 4.22 2 44|— 12 2 — 5 68. 149* a 1-58* 55* 13| — 3* 69. 2811 u 1-83* 66* — 281 — 6* 70. 193 2 a 2 43* 1 1 * — 11* — 4 71. 387-| a 1.93* 83*— 16*— 10 72. 376 3 a 2.11* 22|— 18|—6 1 73. 152f a 2.43* 75 — 50 — 20 74. 235| u 4.33* 90—20—25 75. 627-| a 1-97* 40—10—10 76. 1841 a 5.87 2 5 q 1 q l TT y 77. 846 3 a 2.58* 55 — 12 — 6 78. 277f a 1.52 2 574 — 14* — 7 79. 384 2 a 3.45f 80—25—33* 80. 585f u 6.93 3 85 — 33* — 50 COMMISSION AND BROKERAGE 572. Commission or brokerage is the compensation received by a commis- sion merchant for selling goods, or by an agent or broker for either buying or selling for another. 573. Commissions are usually reckoned at a certain rate per cent, of the amount sold or purchased. Note. — A n agent receives his commission on the gross -proceeds of a sale, or on the prime cost of a purchase. 574. An account sales is a detailed statement rendered by a commission merchant or other consignee to his principal, the consignor, showing the sales of the consignment, the incidental charges and expenses and the net proceeds. The net proceeds is the sum left after the commission and other charges have been deducted. 575. An account purchase is a detailed statement rendered by an agent to his principal, showing the goods bought, prices paid, incidental charges and expenses, and the gross cost. 576. To find the commission when the amount of purchase or sale and rate of commission are given. Example. — At 3%, what does a commission merchant receive for selling $3275 worth of goods? $3275 .03 $98.25 577. Rule. — Multiply the gross proceeds of the sale, or the prime cost of the purchase, by the rate per cent, of commission. MENTAL PROBLEMS 578. 1 . An agent receives 12J per cent, commission for selling sewing machines ; what amount would he receive for selling one for $80? Solution. — 12£ per cent, commission is£ of the amount sold ; J of $80 is $10, the amount of commission. 2. What amount of commission would be received for selling bicycles for $75 at a rate of 33J per cent. ? 3. An agent gets a 40 per cent, commission for selling books by subscrip- tion ; what would he receive on an art work sold at $30? A commission merchant, whose rate is 5 per cent, commission and 2J per cent, guaranty, makes a sale of $200 worth of butter ; how much should he retain ? 171 172 COMMISSION 5. An agent sells wheels at a commission of 30 per cent. ; what sum would he remit to his principal for a wheel sold at $80 ? 6. Grain is sold at a commission of 2 per cent. ; what sum should be returned on a sale of $300 ? 7. An agent’s commission is 3 per cent, for buying leather ; what would be his charge on an invoice worth $500 ? 8. What would be remitted to a consignor for an invoice of 4000 pounds of sugar at 4 cents a pound, less agent’s commission of 3 per cent.? 9. What is the commission at 5 per cent, on goods of which the prime cost is $700 ? 10. If the prime cost is $420 and the rate of commission is 5 per cent., what is the amount of commission ? 11. A real estate agent receives 5 per cent, for collecting a monthly rental of $30; what would be the amount received by him in a year ? 12. A tax collector’s commission is 2 per cent. ; what would he receive on a tax of $150? 13. What amount would a broker receive on the sale of a $100 share of stock at | per cent, brokerage ? Ilf.. What would be the amount remitted by a collector who has collected an account of $350 on a commission of 10 per cent.? 15. What should I receive at a rate of 5 per cent, for collecting 25 per cent, of a claim of $800 ? 16. If I collect 66§ per cent, of a claim of $450 on a commission of 16§ per cent., what amount should I turn over? 17. What are the net proceeds of a sale of $300 worth of goods, rate of commission 5 per cent., other expenses $10? 18. What should I receive for buying $500 worth of produce on a commis- sion of 24 per cent.? How much should my principal send me to cover purchase? 19. What should my principal send me to cover a purchase of $600, commis- sion 3 per cent, and 2 per cent, for guaranty of quality? 20. I gave a collector bills to the amount of $250 to collect ; at 5 per cent, what must I pay him for collecting 80 per cent, of this amount? WRITTEN PROBLEMS 579 . 1 . A real estate broker sold a house and lot for $7245 ; what was his commission at 2% ? 2. At brokerage, how much must I pay my broker for purchasing; $10000 worth of stock ? COMMISSION 173 3. Find the net proceeds of the following account sales : Account Sales Of Merchandise received per P. It. R., July 2, 1908, from Sandusky d: Co., St. Louis, Mo., to be sold on their account and risk. July 7 8 10 13 14 500 lbs. Cheese @$0.10 2 250 “ Butter .25 2 75 doz. Eggs .14 2 125 lbs. Butter .28 475 “ Cheese .11 2 Charges. 5.25 3.50 Freight and Drayage Commission on Sales, 24% Your net proceeds Woodside & Co. E. and O. E. Philadelphia, July 15, 1908. J. Find the net proceeds of the following account sales : Account Sales Philadelphia, Aug. 15, 1908. Sold for account of J. D. Tuckey & Co., By Snodgrass, Murray & Co. July 2 3 9 15 642 yds. English Tweed 879f “ Broadcloth 48J “ Cheviot 21331 “ Satinet @ $2.35 5.62 2 5.25 .75 Charges. 6.25 2.25 Freight and Cartage Commission 3% 8 50 l Net proceeds 174 COMMISSION 5. Find the total cost of the following account purchase : Account Purchase Bought for account of Wilson, Brown & Co., Chicago, Sept. 1, 1908. By William R. Adams. 200 bbls. XX Flour @ $2.25 325 “ Corn Meal 3.12 2 150 “ XXX Flour 6.75 45 “ Flour 6.87 2 25 tons Bran 22.25 Charges. Cartage Commission 3% Total cost 580. To find the gross proceeds or prime cost when the net proceeds or gross cost and rate of commission are given. Example 1 . — My agent remitted $2388.73, proceeds of goods sold for me at 1J% commission ; how much did he receive for the goods ? 100 % Or 1.00 H % -OH .981 % .98J 2425 106 .985 ) 2388.730 1970 4187 3940 2473 1970 5030 4 925 1050 985 6500 Or, .985 ) 2388.730 ( 2425.10 1970 5 9“S7 4187 3940 2473 1970 5030 4925 1050 985 650 Result, $2425.11. Example 2. — I remit my agent $8733 with which to purchase wheat, after deducting his commission at 2J%. How many bushels can he buy at 71 cents? 100 % Or 1.00 852 0 ' 1.025 ) 8733.000 ( 8520 2i % .021 1.025 ) 8733.000| 8200 102i ^ 1.02J 8200 5330 5330 5 125 5125 2050 2050 2050 2050 12000 .71 ) 8520.00 71 142 142 000 Result, 12000 bus. 581. Rule. — Divide the net proceeds by 1 (100%) minus the rate expressed decimally , or the gross cost by 1 (100%) plus the rate. COMMISSION 175 MENTAL, PROBLEMS 582 . 1 . The gross cost of a desk is $80 and the agent’s commission is 33 J per cent.; what is the prime cost? Solution. — T he prime cost plus 33J per cent, of the prime cost isf of the prime cost or the gross cost of $80 ; J of the prime cost is £ of $80 or $20, and the prime cost is $60. 2 . The cost of an organ was $100, which included the agent’s commission of 25 per cent.; what was the price to the agent ? 3 . What were the gross proceeds of a sale which yielded $80 net proceeds, agent’s commission 20 per cent.? 1 /. Find the gross proceeds of a sale yielding $70 net, charges $2, commis- sion 10 per cent. 5 . A manufacturer furnishes bicycles to an agent who returns him $60 each after taking out his commission of 16§ per cent, and deducting $5 charges; what is the gross cost to the purchaser? 6 . An agent’s commission is $20 and his rate of commission is 5 per cent. ; what is the amount of sales ? 7. A dealer averages a daily commission of $40 on a 5 per cent, rate ; what amount of sales does he average daily ? 8 . How many sewing machines at $50 each must an agent sell to realize a commission of $100 weekly upon a commission of 20 per cent.? 9 . What amount of books must an agent sell to earn $200 a month at a commission of 40 per cent.? 10 . A man can earn $80 a month on a 20 per cent, commission; what rate per cent, of commission must he receive on the same amount of sales to earn $100 a month ? 11 . What sum was obtained by a collector who received $45 at a rate of 5 per cent. ? 12 . What amount of stocks was sold by a broker whose brokerage was $10 at a rate of \ per cent.? 13 . A landlord received from his agent a net monthly rental of $45 after $3 was deducted for repairs and a commission of 4 per cent.; what did the tenant pay per month ? 11 /.. A real estate agent receives $72 for the sale of a property at 5 per cent, commission ; what was the value of the property? 15 . What amount of sales would a commission merchant need to make at 2J per cent, to bring in $300 commission a month ? 16 . The gross proceeds are $1275, the rate of commission 4 per cent, and the other expenses $18.30. What are the net proceeds? 17 . The net proceeds are $77.90 and the rate of commission 5 per cent. What are the gross proceeds ? 18 . How much do I realize on a house and lot sold on commission of 3 per cent., the total charges, including $25 for advertising, being $130? 176 COMMISSION WRITTEN PROBLEMS 583. 1. An agent retained $11.25 as bis commission at 2^% ; what amount did lie send his principal? 2. What amount of goods must a salesman sell in a year at 3% to yield him an income of $5000? 3. A collector received $7500 of doubtlul debts to collect and obtained 75% of the amount; if be charges 7J% for collecting, what amount shall be hand his employer? A broker charged $152.40 at 3|% for selling goods; what was the amount of sales? 5. If the brokerage for buying is $166, rate 2J%, what is the value of the purchase ? What is the cost to the principal ? 6. How many pounds of tea at 42 cents, did an agent sell if he remitted his principal $953.30 after deducting his commission at 3J% ? 7. How many dollars’ worth of goods can an agent buy with $14000, allowing him a commission of 3J % ? 8. The net proceeds of an account sales were $3060.89 ; what were the gross proceeds, if the commission was 5%, cartage $13.60, storage $S.30 aud insurance $3 ? 9. How many barrels of flour, at $5.12 2 , can my agent buy for me with $6340.96, if his commission is 2% ? 10. A consigned to a commission merchant 346 barrels of potatoes and 218 barrels of onions with instructions to sell and invest the proceeds in hay. The commission merchant paid $14 60 freight and $6.20 cartage, and sold the potatoes at $1.40 per barrel and the onions at $3.20 per barrel, charging 5% commission for selling. He then purchased the hay at $12.25 per ton, charging 2% for buy- ing. How much hay did A receive? 584. To find the rate per cent, of commission when the commission and the gross proceeds or the prime cost are given. Example. — Received from my agent an account sales, showing my net pro- ceeds to be $2649.89 and his commission $96.11 ; what rate did he charge me? $2649.89 net proceeds 96.11 commission 035 2746)9611 82 38 Or, 2746) 96.110 (.035 8238 13730 13730 $2746.00 gross proceeds 13730 13730 Result, 34%. 585 . Rule. — Divide the commission by the gross proceeds or by the prime cost. COMMISSION’ 177 MENTAL PROBLEMS 586 . 1 . What rate per cent, of commission would I receive if I earned $5 by selling a wheel for $50 ? Solution. — $ 5 is r \, of the selling price, which is equivalent to 10 per cent. 2 . I am offered $3 to guarantee a debt of $100; what is the rate of guaranty ? 3 . An agent returned $45 on a sale of $50 ; what rate of commission did he charge ? If. An agent sold a book for $10 and returned $6 to the publisher; what was the agent’s rate of commission ? 5 . A broker returned $399 for the sale of a $400 bond at its face value ; what was the rate of brokerage? 6. An agent gets $12 for every set of cyclopedias he sells for $60; what is his rate of commission? 7. A merchant offers a collector $25 to collect $300 worth of accounts ; what rate does he offer? 8. What rate does a real estate agent receive for selling a property worth $1600 and retaining $80 commission ? 9 . A drover is offered $20 to sell. a horse for $160; what rate of commission would he receive ? 10 . A man sells wheels on commission and receives $10 on a $60 wheel ; what rate of commission does he receive ? 11 . What rate of commission is charged by an auctioneer who receives $5 for selling a carriage for $125? 12 . A consignor’s net proceeds of a sale were $380 ; the commission was $10 and other charges $10 ; what was the rate of commission? 13 . What rate of commission does an agent retain who remits his principal $350 from a sale of $400 ? Ilf. If an agent earns $30 by selling $180 worth of goods, what is his rate of commission ? 15 . A salesman receives $S for selling $160 worth of goods; what rate does he receive for selling? 16 . What is the rate of commission if the first cost of merchandise is $480 and the commission $4.20 ? 17 . If goods were sold for $500, and only $490 are returned to the principal, what per cent, is charged ? 18 . On a purchase of $900 worth of goods, $913.50 was paid as the total cost. What rate was charged for buying? 178 COMMISSION WRITTEN PROBLEMS 587 . 1 . A lawyer took a fee of $35 for collecting a claim of $700; what was his rate for collecting ? 2 . A collector turned over to his employer $2443.59, retaining a commission of $128.61 ; wdiat was his rate for collecting? What was the amount collected? 3 . If a merchant receives $62.50 for selling goods to the amount of $2500, what amount of commission should he receive when he sells to the amount of $3300 ? J. A collector receives $16 as his commission for collecting $320, and, later on, $28 for collecting $560 ; what is his rate for collecting ? 5 . On an estate of $32500 an inheritance tax of $162.50 was paid ; what was the tax rate ? 6 . What per cent, commission does an agent receive if he renders an account purchase showing total cost $1428.91, his commission $34.37, and other charges $19.74? 7. A commission merchant rendered an account sales showing $334.08 net proceeds and his commission $13.92. What rate did he charge? 8 . An agent received a consignment of goods which lie sold for $32414.61, charging 1J% commission. He invested the proceeds in other goods, buying $31312 worth. What per cent, did he receive for buying? 9 . Jones consigned to a commission merchant 563 bus. of wheat and 1224 bus. of corn. The merchant sold the wheat at 82§ cents a bushel and the corn at 67f cents a bushel. His charges were $90.72, including bis commission of $51.66. What was the rate of his commission, and what were the net proceeds? 10 . I purchase through an agent in London 1244 yds. of cloth at 14s. 3d. a yard, and remit to him in payment a draft which costs me $4548.10. at the rate of $4.88 for £1. What is the rate of his commission, if the other charges amount to £26 8s. 6 d? GENERAL PROBLEMS IN COMMISSION 588 . 1 . A lawyer received a claim of $624.50 to collect. He succeeded in collecting $447.25. What per cent, did the holder of the claim receive, if the lawyer’s commission for collecting was 5% ? 2 . What is the commission, at 2J%, on a sale of 268 bus. of oats at 3S 2 cents a bushel ? 3 . What will 7200 doz. eggs cost at 18f cents a dozen, commission 3% ? J. An agent remits $319.60 proceeds of a sale amounting to $33S 20 : what rate of commission does he receive ? 5 . A consignor’s net proceeds are $2314.80, the charges being $67.12 and 5% commission. Find amount of sale. 6 . A broker charges $72 for selling 80 bales of cotton, averaging 480 lbs. each, at 7J cents a pound. What is the rate of his commission? COMMISSION 179 7. Sold 750 bus. com at 56 cents per bushel, and 460 bus. rye at 82 cents per bushel. Commission 3%, freight $264, storage $86. Find net proceeds. 8. How much must I remit to my agent for a purchase of $378.90 worth of goods, at 2% commission? 9. Received $598.55 as net proceeds of a consignment on which the com- mission amounted to $20.37 and the expenses $60. Find rate of commission. 10. A real estate broker charged me $382.50 as his commission at 5% for selling a house. How much did I receive? 11. Remitted my agent $690.45 to invest in cotton at 8| cents per pound ; how many pounds did he buy, his commission being 2J% ? 12. What are the net proceeds of a consignment of 174 barrels of potatoes, of which 67 barrels are sold at $3.50 per barrel, 24 barrels at $3.40 per barrel, and 83 barrels at $3.20 per barrel, the charges being $37.40, and commission 3% ? 13. What is the rate of commission on a consignment which yields $339.68 net proceeds, the commission being $12.62 and the other charges $26.30? 74- A commission merchant remitted $408.98 as net proceeds of a consign- ment on which the charges were $8.90 and his commission 24%. How much was his commission? 15. If I remit to my agent $4000 to invest in flour, how many barrels can he buy at $4.90, and what is the balance left over, if his commission is 2f% ? 16. What is a stock broker’s commission, at §%, for purchasing $10000 worth of bonds ? 17. Find the net proceeds of the following : Account Sales Philadelphia, July 15, 1908. Sold for account of E. N. Welsh, By J. T. Rodman & Co. 350 bbls. Greening Apples 175 “ Russet “ 412 “ Baldwin “ Charges. Freight Cartage Commission and guaranty, 5% Net proceeds @ $3.70 3.85 3.65 595 58 60 50 18. Machines that cost $123.25 are listed at $200. If a discount of 12|%, 10% and 5% is allowed, and a commission of 10% paid the salesman, what per cent, is gained ? 180 COMMISSION 19. An attorney collected part of a claim, charging $12.30 for his commis- sion of 5%. If the part collected was but 35% of the entire claim, what was the amount of the loss ? 50. A Minneapolis merchant shipped to his agent in New York 700 barrels of wheat flour and 2000 barrels of rye flour. The agent sold the wheat flour at $3.50 per barrel and the rye flour at $1.75 per barrel. After taking out his com- mission he remitted $5801.25. Find per cent, of commission. 51. An agent receives $7500 to invest in wheat at 72f cents a bushel, allow- ing 3% commission. What amount of wheat can he buy ? SS. An importer buys through his agent in Paris 1217 meters of silk at 17.28 francs per meter, paying the agent 5% commission and expenses 213 francs. If the duty is $1.70 per yard, at what price per yard must the silk he sold in Philadelphia to gain 20% ? S3. A broker charged $17 as his commission at \ % for buying U. S. govern- ment bonds. What amount of bonds did he purchase? Slf.. I send an agent $2472.80 to invest in cotton at $5.23 per bale, allowing 5% commission. He pays 14 cents a bale freight, 2 cents a bale storage and 34 cents a bale drayage. How many hales does he purchase, and what is the unexpended balance? 55. An agent sells a consignment of 12000 bushels of wheat at 70^ cents a bushel, deducts his commission of 2%, and invests the net proceeds in flour at $3.72 per barrel, charging 2% commission for buying. How many barrels does he buy, and what is the balance unspent ? 56. Find the net proceeds of the following: Account Sales Sold for account of Brown & Sharp, Philadelphia, May 4. 1908. By Marshall Brothers. 1908 Jan. 29 387 yds Broadcloth @ $4.63 Feb. 13 249 “ 3.78 Mar. 10 426 “ 6.20 Apr. 29 512 “ 5.50 Charges. §27-12 $6.50 Freight and Drayage 33 62 Commission, 5% i Net proceeds INSURANCE 589. Insurance is “ a contract by which one party, for an agreed consider- ation (which is proportioned to the risk involved), undertakes to compensate the other for loss on a specified thing from specified causes. The party agreeing to make the compensation is usually called the insurer or underm' iter ; the other, the insured or assured ; the agreed consideration, the premium ; the written contract, a policy ; the events insured against, risks or perils; and the subject, right, or interest to be protected, the insurable interest — Bouvier. 590. The principal kinds of insurance are fire insurance, marine insurance, accident insurance, and life insurance. 591. Fire insurance provides indemnity for loss of or damage to property by fire or the efforts to extinguish fire. 592. Marine insurance provides indemnity for loss by shipwreck or disaster at sea. 593. Accident insurance provides indemnity for loss occasioned by explo- sion, breakage, or other accidents to property, or loss of future earnings through personal disablement. 594. Life insurance is a contract by which a company, in consideration of certain premiums, agrees to pay to the heirs of a person when he dies, or to himself if living at a specified time, a certain sum of money. 595. Adjustment of losses. Fire insurance companies pay the full amount of the loss, up to the limit of the policy, unless the contract contains the “ average clause,” which provides that the company shall pay only such propor- tion of the loss as the sum insured bears to the full value of the property. Marine insurance policies usually contain the ‘‘ average clause”. Note. — U nder the “ average clause, ” if property worth $8000 is insured for f of its value, or $6000, the company, in case of loss, will pay f of the loss; that is, in case of a loss of $4000, the com- pany would pay $3000; in case of a loss of $3000, it would pay $2250, etc. If the property were insured for \ of its value, the company would pay but I of the loss, etc. 596. To find the premium. Example. — What is the premium on $12000 insurance atlj%? The other cases under insurance are more theoretical than practical, and are simply applications of percentage principles already explained. 597. Rule. — Multiply amount of insurance named in policy by the rate per cent, of premium. // 2 0 O 0 .0 / / 2 O O fo.o 0 181 182 INSURANCE MENTAL PROBLEMS 598. 1. What is the premium on an insurance policy of $2000 at 2 per cent.? Solution. — The premiun is T § 7 or fa of $2000, which is $40. 2. What is the premium on a policy of $1200 at 2J per cent,? 3. What is the premium on a policy of $5000 at 1J per cent.? I/.. What premium must I pay for insuring a vessel worth $8000 for f- of its value, at 2 per cent.? 5. My store, worth $3600, is insured for of its value at 1 J per cent, and its contents, valued at $8000, for f of their value at 2 per cent ; what is the entire premium ? 6. My charge for insuring a house worth $6000 for § of its value is 11 per cent., what is the amount of my bill ? 7. My property, worth $9000, is insured for J of its value in a company at 1J percent, and for J of its value in a company at 2 per cent.; what amount of insurance do I carry, and what does it cost me? 8. A merchant has a stock valued at $12000 which one company offers to insure for f of its value at 11 per cent., and another for i of its value at 1 J per cent.; what will be the difference in cost between the premiums? 9. Which is better and how much — an insurance on $1200 at 21 per cent. • for three years, or $1200 at 1 per cent, per annum ? 10. What is the amount of premium on furniture worth $4200 at 11 per cent. ? 11. At 2 per cent, what will be the face of a policy whose premium is $30? Solution. — 2 per cent, or fa of policy equals $30, hence face of policy is 50 times $30, or $1500. 12. At 50 cents for $100, how much insurance can be bought for $20 ? 13. If the rate of insurance is 2J per cent, and premium $50, what is the face of the policy ? Ilf. An insurance policy cost $45, premium at 21 per cent. What was amount named in policy ? 15. If I pay $50 premium quarterly on insurance at 11 per cent., how much insurance do I carry? 16. What is the face of a policy whose premium is $30 a quarter, at 11 per cent.? 17. $23 is charged me upon my obtaining an insurance policy at f per cent, premium ; what is the face of my policy? 18. If an agent’s commissions for writing insurance during a week amount to $100 at 1 per cent, commission, what amount has he written ? 19. If a company has received premiums during a week amounting to $200, written at I per cent., what amount of insurance has it issued ? 20. I write insurance for a company which charges 2 per cent, and allows me 1 per cent. If I earn $60 a week, what amount must the company issue ? INSURANCE 183 WRITTEN PROBLEMS IN INSURANCE 599 . 1 ■ What is the premium at If % on $6250 insurance? 2. If $105 is paid as premium for five years on a policy of $3500, what is the rate per annum ? 3. At the rate of 2f%, $264.69 was paid as premium; what was the amount of the risk ? 4- What is the cost of insuring a cargo valued at $5674.50 for 75% of its value, at the rate of If % ? 5. If the cargo mentioned in the preceding problem were damaged to the extent of $820, how much would the policy holder receive from the insurance company ? 6. A insured his house on March 1, 1906, for $7500 for 5 years, at 3J%. What was the amount of “unearned premium” on Sept. 1, 1908? 7. A cargo worth $22587.36 was insured for 80% of its value at lf% premium. The ship was lost at sea. How much did the insurance company lose ? 8. A merchant insured his stock of goods with one company for $5000, with another for $3000, and with a third for $2500. If the goods should be dam- aged by fire to the extent of $4000, how much should each company pay ? 9. What is an insurance broker’s commission at 20%, for placing $40000 insurance at f % ? 10. What per cent, of loss would a marine insurance company pay on goods valued at $4678.18 and insured at $3500? 11. A man had his life insured for $5000, paying an annual premium of $162.85. After paying premiums for 12 years he died. What per cent, more did the company pay the beneficiary than had been received in premiums? 12. B insured his house in one company at If % premium, and in another at f% premium. His total premium was $105. If the amount of the second policy was half that of the first, what was the amount of each? 13. A cargo was insured for equal amounts in three different companies, at a premium of If % in the first, 1% in the second, and lf% in the third, the total insurance being f of the value of the cargo. The cargo was lost by ship- wreck, and the owner thereby lost $7135 (including the premiums he had paid). For what amount were the policies drawn ? 14. What is merchandise worth that is insured for f- of its value at a premium of ^ 0 %, if the premium amounts to $57.04? 15. A building and its contents are separately insured, the building for f of its value at f % premium, and the contents for -4 of their value at f-% premium. The premium for insuring both is $1091.67. If the value of the contents is $85000, how much is the building worth? 16. For how much must goods worth $25000 be insured, at f % premium, in order that the owner may lose nothing if they are totally destroyed by fire? 184 INSURANCE 17 . Certain insurance policies contain the following co-insurance clause: If at the time of fire the whole amount of insurance on the property covered by each separate item of this policy on property as described in such item shall be less than 80% of the actual cash value thereof, this company shall in case of loss or damage be liable for only such proportion of such loss or damage as the amount insured under said item shall bear to the said 80% of the actual cash value of the property covered by such item.” Under the provisions of this clause, how much would each company pay in case of damage amounting to $420 on property worth $12000, insured for $3000 in each of three companies? How much would each pay if the property had been insured for $3500 in each company? 18 . Under the 80% co-insurance clause, as above, how much would be recovered, in case of total loss, on property valued at $60000 and insured to the extent of $48000? How much on property valued at $60000 and insured for $40000 ? 19 . A factory (worth $3000) and its contents were insured for $10000, as follows: $2000 on building, $3000 on machinery (worth $5000), and $5000 on stock (worth $8000). The building was damaged by fire to the amount of $1000, the machinery $4000, and the stock was a total loss. How much was the claim against the insurance company? What was the premium at 1J% ? What was the owner’s loss? What was the company’s loss, if the risk was covered (1) by an “ordinary policy;” (2) if the policy contained the “average clause;” (3) if the policy contained the “co-insurance clause?” REVIEW PROBLEMS IN PERCENTAGE Remark. — It will be noticed that no attempt has been made to construct problems that would produce certain cut-and-dried results. Problems are presented just as they arise in business and the same kinds of results are obtained as would be obtained in actual business. While it would be a curious thing to find that it required ff of a man to plow a certain field of a certain extent with a team walking a certain gait, still such fractional part of a man is a perfectly logical and reasonable mathe- matical fact. In rates per cent., which come out so nicely in arithmetics, it is possible, and, in truth, more than likely, that a very unusual fraction may appear. Where an approximate result will answer, we suggest that the rate be carried to two places beyond the usual number and the rate be expressed as an integer and decimal. Thus, say $4.78 is 1.59% of $299 ; or the decimal part may be reduced to its approximate common fraction, which in this case is or f, so that If % very nearly expresses the rate. Wherever exactness is required, however, as in distributing moneys among a large number of persons, it may be necessary to carry the rate out to four or more decimal places. 600 . 1 . A railway company’s earnings for the years 1907 and 1908 were $3648729.13. The earnings for 1908 were 14% more than for 1907. Fmd earnings for each year. REVIEW PROBLEMS 185 2. What per cent, above cost must goods be marked to allow a discount of 20 % and 5 % and still yield a profit of 16§ % ? 3. One-fourth of a consignment of goods is sold at a loss of 18%, and one- third at a gain of 6J%. For how much above cost must the balance be sold to net a profit of 20% on the whole consignment ? If.. In a certain compound the weights of the ingredients are, respectively, 4 oz. ; 2 oz. 13 pwt. ; 20 gr. ; and 1 oz. 4 gr. Find the per cent, of each ingredient ? 5. Sold a bicycle for 10% less than it cost me. If I had sold it for $60, my gain would have been 20%. How much did I lose? 6. If A sells goods at list priceless 40%, 25% and 10%, and B sells at same list less 30%, 10% and 10%, what per cent, of A’s net selling price is B’s ? 7. How much is Jones’s investment if his income, at 8%, is $1893.76? 8. Consigned merchandise to an agent, which he sold for $3211.22, charging 2 \f 0 commission. According to instructions, he invested the proceeds in wheat after taking out his commission of 2% for purchasing. How much did he have to invest ? 9. A sold B a horse for $114, which is 5% less than A paid for him. B sells the horse for 15% more than it cost A. How much did B receive for the horse? 10. What will be the cost of insuring a house for $7500 at f %, and the furniture, etc., for $3000 at f % ? 11. After retaining 3J% commission for selling a consignment of cotton, my agent paid me $4174.59. How much was his commission ? 12. Sold a farm for $3904.32, thereby losing 17%. The sum I paid for the farm was what I received as my one-third share of a legacy, after the executor had received 2% for settling the estate. What was the value of the entire estate? 13. A property being sold at 23|% above cost, the gain amounted to $1741.92 ; what was the cost ? Ilf. In a school examination, James obtained the correct results to 7 prob- lems out of 10 in arithmetic, spelled correctly 34 out of 40 words, answered correctly 14 out of 15 questions in geography, 19 out of 23 in grammar, and 13 out of 18 in history; what per cent, general average did he make? 15. If the above examination had been averaged according to the U. S. civil service method, and arithmetic had been given a value of 6, spelling 5, geography 3, grammar 4, and history 2, what would have been his general average ? 16. What per cent, profit is made by buying wine at 2 francs a liter and selling it at $2 a gallon ? 17. Bought goods for $382.75 ; what price must I ask for them so as to be able to allow a discount of 25%, 10% and 5%, and net a profit of 20% ? 18. If the par value of the stock of a certain railway is $50 per share, and it is selling for 97f % of its par value, what will 150 shares cost if purchased through a broker who receives $9.38 commission ? 186 REVIEW PROBLEMS 19. On June 8, 1908, E. S. Grayson & Bro., commission merchants, of Phila- delphia, received from L. N. Clark, of Chester, Pa., a consignment of 1437 lbs. of butter, to be sold for his account and risk. They paid $6.70 charges, and sold the butter as follows: On June 10, 329 lbs. @ 35 cents; June 12, 412 lbs. @ 41 J cents ; June 14, 224 lbs. @ 37| cents ; June 15, 118 lbs. @ 36J cents ; June 17, 272 lbs. @ 39 cents; June 18, 82 lbs. @ 38J cents. On June 21 they rendered an account sales, charging 5% commission. Make out the account sales in proper form, showing Clark’s net proceeds. 20. A man having a certain sum of mone} r , invested J of it in wheat, J of it in oil, and the remainder in cotton. He sold the wheat at a profit of 6%, the oil at a loss of 2%, and the cotton at a profit of 9%. If the total amount received from the sales was $6230, how much had he at first? HI. If a grocer pays $57.60 for 180 lbs. of Mocha coffee, how many pounds of Java coffee costing 26 cents per pound must he mix with it in order to gain 20% by selling the mixture at 36 cents per pound? 22. What is 6% of £43 13s. 8 d. ? 23. A sold goods to B for $967.20 and gained 3%. B sold them to C and lost 3%. How much more or less did C pay than A ? 24 -. What per cent, larger than a gallon is a half peck ? 25. If I lose 10% by selling cloth at 72 cents per yard, at what price must I sell it to gain 25 % ? 26. The net amount of a bill of goods is $3783.18, the discounts allowed being 35%, 15% and 10%, and 2% for cash. What is the gross amount of the bill? 27. The capital stock of a railroad is $28500000. Its gross earnings for the year 1907 were $3642819.76, and its operating expenses $1938743.29. After paying $825000 interest on its bonded debt, what per cent, of its earnings remains ? 28. If an agent’s commission for selling $3264 worth of goods is $96.92, how much would he receive, at the same rate, for selling $7492.85 ? 29. Bought two houses, 28% of the cost of the first being equal to 20% of the cost of the second. If the second cost $2000 more than the first, what was the cost of each ? 30. By selling 40% of a purchase at 20% profit, 65% of the remainder at 15% profit, and what then remained at 10% loss, a net gain of $468.69 was made. Find amount of purchase. 31. A, B and C are partners. 30% of A’s share in the business equals 18% of B’s share, and 12% of B’s share equals 40% of C’s share. C’s share is $5400. Find A’s and B’s shares. 32. Robertson withdrew his money from a business that was yielding him an annual income of 8g-%, and invested in another business that turned out less profitable than he expected, yielding only 74% on his investment. His annual income was reduced $106.25 by the change. How much capital had he invested? REVIEW PROBLEMS 187 33. An attorney succeeded in collecting 70% of a claim placed in his hands, and received $55.26 as his commission at 2%. What was the whole amount of the claim ? 34-. Half of a cargo that cost $32468.18 was sold at 12% profit, and 20% of it at a loss of 7|%. For how much must the remainder be sold to yield a net gain of 10 % on the whole ? 35. If a dealer sells f of a pound for what of a pound cost, what per cent, does he gain ? 36. What will it cost me in U. S. money to buy, through my agent at Brus- sels, 580 meters of lace at 6.15 francs per meter, if the agent’s commission is 3% ? 37. What per cent, profit would I make on the lace in the above problem if, after paying freight and other expenses amounting to $89.20, and $290 duty, I sell the lace at $3 per yard ? 38. What per cent, of 10 francs is | of a pound sterling? 39. How must I mark goods that cost $4.17 a yard, so as to be able to allow a discount of 20% and 5% and still gain 25% ? 40. A real estate broker sold two houses for $3200 and $4500 respectively. The first house brought 23J% more than cost, and the second 10% less than cost. If the broker’s commission for selling was 3%, how much did his principal gain or lose on both houses ? 41. My agent in Pittsburg purchased for me 1200 bbls. of oil at $3 per barrel. He sold 500 bbls. of it at $3.75 per barrel, and the remainder at $2.90. His com- mission was 2% for buying and 3% for selling. How much should he remit in settlement of my account ? 4S. X bought a house for 33£% more than its assessed value. He made improvements costing $680, and then sold the house at a net gain of 20%, receiving $11280 for it. Find how much the taxes on the house amount to, at $1.80 on $100. 43. After paying an agent’s commission of 24%, $3247.80 were received as proceeds of a sale. For how much were the goods sold ? 44 • H goods are bought at a discount of 25%, 10% and 5% off list, and sold at same list price, less 15% and 10%, what is the gain per cent.? 45. A grocer bought five dozen eggs at the rate of 3 for 5 cents, and five dozen at the rate of 2 for 5 cents. He sold them at the rate of 5 for 10 cents under the impression that he was selling them at cost; what per cent, did he lose? 46. What percent, of £8 17 s. 5 d. is 80.75 francs, if £1=25.25 francs? 47. From a cask containing 217.50 liters of wine, 5% was lost by leakage. How many quarts remained? 48. How much will it cost to excavate a cellar 40 ft. X60 ft. and 8 ft. deep, 35%of it being rock at 94 cents a cubic yard, and the remainder clay at 44 cents a cubic yard ? INTEREST 601. Interest is a charge made for the use of money. 602. The principal is the sum for the use of which interest is charged. 603. The rate is the per cent, of the principal charged for its use for one year. 604. The amount is the sum of the principal and interest. 605. Simple interest is interest on the principal only. 606. Compound interest is interest on the sum of the principal and unpaid interest ; it is interest on interest. 607. Common interest is interest computed on the basis of 12 months of 30 days each, or 360 days to the year. 608. Accurate or exact interest is interest computed on the basis of 365 days to a year. 609. Legal interest is interest at the rate per cent, established by law for cases in which the rate is not specified. 610. Accrued interest is the unpaid interest on an obligation, usually not yet due. 611. Usury is the crime of charging a higher rate of interest than is allowed by law. 612. To find the interest on any principal. GENERAL FORMULAS For one year. Principal X Rate (expressed decimally) = Interest. For part of a year. Principal X Rate (expressed decimally) x interest. Example 1. — Find the interest of $360 for 3 mo. 13 da. at 6% per annum. Explanation. — The interest for 1 year is .06 times the principal, or $21.60. 3 mos. are equal to 90 days ; adding the 13 days makes 103 days, or Iff of a year. The interest for of a year is ||f of $21.60, or $6.18. Or, by cancelation : 30OX.O6X1O3 3 Z 3 & 0 , O //? 2 / & O * / 03 2 / FO o 6 0)2 2 2 U.F O ' 2 / £> 0 6 u r . / F t /03 eids 3 4, O 2 rro 2 F F O 188 INTEREST 189 Example 2. — What is the interest of $360 for 2 yr. 3 mo. 13 da. at 6% per annum ? $360 principal $21.60X2 .06 rate 1.80X3 12)$21.60 interest 1 yr. .06X13 30) 1.80 int. 1 mo. .06 int. 1 da. $43.20 int. 2 yr. 5.40 int. 3 mo. .78 int. 13 da. $49.38 int. 2 yr. 3 mo. 13 da. Explanation. — The interest for one year is .06 of the principal, or $21.60; the interest for one month is T ’ 5 of a year’s interest, or $1.80; the interest for one day is of a month’s interest, or $.06. The interest for two years is twice one year’s interest or $43.20; for 3 mos. it is three times one month’s interest, or $5.40; for 13 das. it is thirteen times one day’s interest, or $.78. Adding, we have $49.38, the interest for 2 yr. 3 mo. 13 da. Remark. — There is no one application of the general interest formulas that is best for all cases. The student who has mastered the general formulas should be led to see clearly the special applications and to discriminate in their uses. Let the instructor receive from the student his opinion as to which particular application best solves any particular problem under discussion. The business man should be skilful in using several applications and use the best method for each case. 613 . Rule. — To find one year’s interest, multiply the principal by the rate expressed decimally. To find the interest for part of a year , multiply one year’s interest by as many 360ths as there are days. (Or, for each month, take of a year's interest ; and for each day, of a month’s interest. MENTAL PROBLEMS 614 . 1 . I lend Jones $100 for one year, interest at 6 per cent. ; what sum should he pay me for the use of the monej’ ? Solution. — A t 6 per cent, for 1 year, the interest would equal .06 of the principal, which is $6; hence he should pay me $6 interest. < 2. What should I receive for lending Borrower $200 for 6 months, at 6 per cent. ? 3. What should I return to Lender, principal and interest, for the use of $200 for 3 months at 6 per cent.? f What would be the yearly interest on a mortgage of $400 at 5 per cent.? 5. The holder of a note for $400, bearing interest at 6 per cent., receives the interest in quarterly payments. What sum does he receive at each payment? 6. For what amount should I draw a check in full payment of a loan of $200 obtained 90 days ago, interest at 6 per cent.? 7. Shaner lends me $300 less the interest on my 6 months note, interest at 6 per cent.; what sum does he lend me? 8. Find the interest on $360 for 4 months at 4 per cent. 9. What amount of income shall I receive semiannually on a $500 bond bearing 4 per cent, interest? 10. What amount must I return in 6 months for a loan of $350 at 4 per cent.? 190 INTEREST ORAL EXERCISE 615 . 1. What is the interest on $100 for 1 yr. at 6% ? 2. On $200 for 1 yr. at 6% ? 3. On $100 for 2 yrs. at 6% ? J. On $200 for 2 yrs. at 6 % ? 5. On $100 for 2 yrs. at 5% ? 6. On $200 for 2 yrs. at 5 % ? 7. On $200 for 2 yrs. at 4% ? 8. On $200 for 1 yr. 6 mo. (If yrs.) at 6% ? 9. On $300 for 1 yr. 3 mo. at 4% ? 10. On $100 for 6 mos. at 6 % ? 11. On $200 for 3 mos. at 4% ? I®. On $100 for 60 das. at 6% ? 13. On $200 for 60 das. at 5 % ? 7J. On $300 for 90 das. at 6% ? 15. On $100 for 120 das. at 6% ? 16. What is the interest of $200 for 2 yrs. at 5% ? 17. Of $150 for 1 yr. 6 mo. at 4% ? 18. Of $50 for 2 yr. 4 mo. at 6 % ? 19. Of $3600 for 20 das. at 2% ? 20. Of $1000 for 36 das. at 8% ? WRITTEN EXERCISE 616 . 1. Find the interest on $1530 for 2 yr. 6 mo. at 6%. 2. Find the interest on $890 for 3 yr. 3 mo. at 6%. 3. What is the amount of $364 for 2 yr. 9 mo. at 6% ? Note. — P rincipal + Interest = Amount. J. What is the interest on $1000 from Jan. 1, 1908, to April 1, 1909? Note. — W hen the time extends beyond a year, find time by compound subtraction. 5. Find the amount of $2520 from May 1, 1908, to November 1, 1909, interest at 5 % . 6. Find the interest of $378 for 1 yr. 4 mo. at 6%. 7. Of $756 for 2 yr. 2 mo. 15 da. at 4%. 8. Of $2425.60 for 3 yr. 5 mo. 18 da. at 5%. 9. Of $589.70 for 84 days at 7 %. 10. Of $642.50 for 7 mo. 19 da. at 3%. 11. Find the amount of $342.42 from Feb. 5, 1907, to March 15, 1908, at 7 12. I borrowed $360.50 Aug. 1, 1907, and returned it March 5, 1908, with interest at 6J%. How much was paid? 13. If $150 was loaned Aug. 5, 1906, and returned with interest at 7 % on Mar. 17, 1908, how much was paid? INTEREST 191 WRITTEN PROBLEMS 617 . 1 ■ What amount will take up January 1, 1909, a note for $2000 given July 1, 1908, bearing interest at 6% ? 2 . I buy a house subject to a 5% mortgage of $2500, payable half-yearly. What is the amount of my semiannual payment? 3. What semiannual income do I receive from a $1200 bond bearing interest at 3J % ? If.. A ground rent of $1500 bears interest at 6%, payable quarterly ; what is the amount of the quarterly payment? 5. My property has against it a 5% mortgage of $1200 and a 6% ground rent of $800. What is the amount of my yearly interest payments? 6 . Borrowed $750 on January 18, 1908, at 4 per cent. What amount must I pay September 5, 1908? 7. Bought a house on August 20, 1906, for $8000, paying $2500 cash and giving a mortgage at 5% for the balance. Paid for taxes, repairs, etc., $218.30. Collected rents amounting to $780. Sold the house September 12, 1908, for $8600. How much did I gain ? 8 . I have a lot of ground which I offer for sale at $1300. How much should I charge a man for it, who offers to pay $300 down and settle the balance by paying $200 every three months, in order to get 6% on my money? 9. On March 13, 1908, at Philadelphia, Pa., A sells B a bill of goods amounting to $492.83, on three months credit. How much should B pay A on October 26, 1908? 10 . I have an account with a firm of bankers and brokers, and have an understanding with them that they are to allow me 4 per cent, interest on the money I have on deposit with them, but, whenever my account is overdrawn, I am to pay them 6 per cent, interest on the debit balance. On February 21, 1908, I deposit $3216.36 ; on April 13, 1908, they purchased stock on my account, amounting to $9218.75 ; on July 7, 1908, they sell the stock for me for $9462.25. What is the balance to my credit on November 10, 1908? 11 . I borrowed $327 on May 24, 1907, interest at 4f%, and $725.75 on Jan- uary 15, 1908, interest at 5%. What was due on the two debts September 1, 1908? 12. On May 4, 1907, a merchant bought goods for $780.16 on 90 days credit, but did not pay for them until April 7, 1908. What amount did he pay, interest being reckoned at 6% from the expiration of the 90 days? 13. John Smith borrowed $752, June 23, 1907, interest at 5%, and $634.40 on December 23, 1907, interest at 5J%. What total amount did he owe on Octo- ber 1,1908? 192 INTEREST SIX PER CENT. METHOD 618 . At six per cent, the interest for one year is .06 of the principal. For one month, or ^ of a year, it is of .06, or .005 of the principal. For one day, or of a month, it is of .005 or ,000| of the principal. Example. — What is the interest of $1348 at 6% for 3 yr. 6 mo. 18 da.? 3X.06 = .18 $1348 6X.005 = .030 .213 18 X. 000| = .003 4044 .213 13 48 269 6 $287,124 $1348 .033 ~ 4 044 40 44 $44,484 interest for 6 mo. 18 da. $1348 .06 ' 80.88 3 $242.64 interest for 3 vr. 44.48 $287.12 819 . Rule. — Find what per cent, of the principal the interest will be for the whole time required, and multiply the principal by this decimal, as folloius : Take as many times .06 as there are years, as many times .005 as there are months, and as many times .000\ as there are days ; add these rates, and multiply the principal by their sum. Note. — Another way to state this rule is : First find the interest of one dollar for the given time, and then take as many times this amount as there are dollars in the principal. Or, take i of the months and call the result cents, and take 4 of the days and call the result mills. 620 . The product by the above rule is the interest at 6%. To find the interest at any other rate, increase or diminish this amount proportionally. As, to find the interest at 7 %, add ^ ; to find the interest at 5%, subtract £ ; for 8%, add J ; for 4%, subtract etc. ORAL EXERCISE 621 . 1. What is the interest of $250 for 5 yrs. at 6 % ? 2. Of $1200 for 1 yr. 8 mo. at 6 % ? 3. Of $500 for 2 yr. 6 mo. at 4 % ? 4. Of $100 for 2 yr. 2 mo. 12 da. at 6% ? 5. Of $800 for 1 yr. 5 mo. 15 da. at 6 % ? 6. Of $300 for 2 yr. 3 mo. at 5 % ? 7. Of $600 for 1 yr. 6 mo. at 4 J % ? 8. Of $400 for 6 mo. 10 da. at 6% ? 9. Of $1000 for 4 mos. at 6% ? 10. Of $200 for 1 yr. 4 mo. at 8% ? INTEREST 193 WRITTEN EXERCISE 622. What is the interest of 1. $356.17 for 2 yr. 7 mo. 12 da. at 6% ? 2. $3947 for 1 yr. 5 mo. 6 da. at 6 % ? 3. $758.34 for 4 yr. 8 mo. at 6 % ? J. $1263.50 for 3 yr. 4 mo. 24 da. at 6% ? 5. $478.64 for 5 yr. 9 mo. 18 da. at 6% ? 6. $592.75 for 10 mo. 15 da. at 5% ? 7. $6438.92 for 3 yr. 3 mo. 3 da. at 7 % ? 5“. $1873.80 for 2 yr. 11 mo. 21 da. at 8 % ? 9. $237.57 for 6 yr. 8 mo. 27 da. at 4% ? 10. $5645.20 for 1 yr. 2 mo. 2 da. at 3% ? DAY METHOD 623. To find the interest of any principal for any number of days at 6 %. Example. — Find the interest of June 14 days. July 31 “ Aug. 31 “ Sept. 7 “ 83 days. June 16 to Sept. 7 = 83 days. 624. Rule. — Multiply the principal pointing off three places if the principal be and cents. 60 at 6%, from June 16 to September 7. $2560 83 7680 20480 6)212480 3541 3f $35.41, interest for 83 days. by the number of days, and divide by six, dollars, or five, if the principal be dollars 625. In Pennsylvania, the legal rate of interest being 6% and money loans being usually for terms not exceeding four months, this method is the basis for nearly all interest calculations. The formula is usually modified as below and the operation shortened by cancelation. Multiply the principal by the days, divide by 60 and point off two places if the principal be dollars only, four places if there be cents in the principal. Example. — Find the interest on $360 for 61 days at 6%. 6 $^X61 n = $3.66. This may be modified for different rates by changing the divisor. For 5% the divisor is 72; for 9%, 40; for 4%, 90; for4J%, 80; and, in general, the divisor is the number obtained by dividing 360 by the rate per cent. 194 INTEREST ORAL EXERCISE 626 . 1 - What is the interest of $600 for 20 days at 6%? 2. Of $900 for 70 days at 6% ? 7. Of $600 for 90 days 5% ? 3. Of $720 for 100 days at 6% ? 8. Of $200 for 120 days at 3% ? If.. Of $1000 for 20 days at 6 % ? 9. Of $400 for 30 days at 7 % ? 5. Of $200 for 33 days at 4 % ? 10. Of $240 for 60 days at 6% ? 6. Of $300 for 80 days at 4| % ? 11. Of $450 for 50 days at 8% ? WRITTEN EXERCISE 627 . Find the interest of 1. $892 for 79 days at 6%. 2. $1763 for 125 days at 6%. 3. $4367 for 217 days at 6%. If.. $3271.38 for 1 yr. 4 mo. 17 da. at 6%. 5. $2794.72 for 2 yr. 5 mo. 23 da. at 5%. 6. $745 from March 16 to June 19 at 6%. 7. $987 from May 3 to October 14 at 6%. 8. $723.93 from August 3, 1907, to June 17,1908, at 5%. 60-DAY METHOD 628 . The interest on any principal for 60 days at 6% is one per cent, of the principal. Example — Find the interest of $3564 for 95 days at 6%. 1 % $35.64 = 60 days int. i of 60 da. 17.82 = 30 “ “ -jt of 30 da. 2.97 = 5 “ “ $56.43 = 95 days int. 629 . Rule. — Find one per cent, of the principal by pointing off two decimal places ; this gives the interest for 60 days. Then find the interest for the required time by taking aliquot parts of 60-days interest. ALIQUOT PARTS OF 60 630 . 5 = i 1 2 of 60 40 = i 3 less than 60 63 = -W more than 60 6 = 1 1 0 U 60 45 = 1 4 << 60 65 = it 60 10 = 1 6 u 60 48 = 1 T U 60 70 = i 60 12 = 1 ~s u 60 50 = l 6 u 60 72 = i 60 15 = 1 ¥ u 60 54 = 1 1 0 u 60 75 = i 60 20 = 1 S' u 60 55 = 1 1 2 a 60 so = i 60 30 = 1 2 a 60 57 = 1 2 0 u 60 CO O II k«|M 60 Note. — These numbers may also be used as the basis for the aliquot parts of 600 and 6000. INTEREST 195 ORAL EXERCISE 631 . 1 • What is the interest of $2374 for 60 daj'-s at 6% ? 2. Of $1266 for 20 days at 6%? 7. Of $1200 for 40 days at 6%? 3. Of $240 for 90 days at 6%? 8. Of $1700 for 60 days at 5%? 4. Of $720 for 65 days at 6%? 9. Of $600 for 70 days at 7 %? 5. Of $3000 for 50 days at 3%? 10. Of $300 for 80 days at 8 %? 6. Of $1500 for 30 days at 6%? 11. Of $1700 for 93 days at 4J%? WRITTEN EXERCISE 632 . Find the interest of 1. $437.56 for 57 days at 6fi. 2. $5278 for 66 days at 6%. 3. $726.32 for 8 mo. 20 da. at 6%. 4 . $3258 for 1 yr. 2 mo. 18 da. at 4%. 5. $493.72 for 2 yr. 7 mo. 16 da. at 6%. 6. $1267.50 for 183 days at 6%. 7. $2784.10 for 9 mo. 8 da. at 7%. 8. $521.56 for 1 yr. 5 mo. 23 da. at 8%. $972.83 for 231 days at 5%. 10. $6238 for 7 mo. 29 da. at 6%. “ $6000 RULE ” 633 . The interest of $6000 at 6% is $1 a day. Example. — What is the interest of $8000 at 6 % for 210 days ? Interest of $6000 for 210 days = $210 “ “ $2000 (i of $6000) = 70 “ “ $8000 for 210 days = $280 ORAL EXERCISE 634 . 1. What is the interest of $3000 for 280 days at 6%? 2. Of $500 for 72 days at 6%? 3. Of $2000 for 315 days at 6%? 4. Of $9000 for 120 days at 7 %? 5. Of $7500 for 84 days at 4%? WRITTEN EXERCISE 635 . Find the interest of 1. $3000 for 2 yr. 3 mo. 18 da. at 4%. 2. $2400 for 1 yr. 9 mo. 16 da, at 6%. 3. $4500 for 3 yr. 4 mo. 22 da. at 6%. 4 ■ $1200 for 1 yr. 1 mo. 3 da. at 2%. 5. $12500 for 2 yr. 7 mo. 8 da. at 5%. 6. $1000 for 172 days at 6%. 7. $1500 for 312 days at 6%. 8. $2000 for 1 yr. 27 da. at 6% 9. $4000 for 278 days at 7 %. 10. $9600 for 78 days at 6%. 196 INTEREST Common Year — 360 Days 636 . The interest on $1 for 360 days at 1%, or on |360 for 1 day at 1% is one cent. In proportion as the rate is increased, the time or principal to yield one cent of interest is decreased. Thus $1 at 3%, 4%, or 6%, will yield one cent of interest in respectively 120, 90, 60 days, or on respectively $120, $90, or $60 for one day. This applies to any rate divisible evenly into 360. Note. — I n the illustrative table it will be seen that pointing off two places in the principal gives the interest for as many days as the rate is contained times in 360 ; or, conversely, pointing off two places in the days gives the interest on a like number of dollars. Also, that pointing off ODe place gives the interest for ten times as many days or on ten times as many dollars ; while three places pointed off gives the interest for one-tenth as many days or on one-tenth as many dollars. Illustrative Table $1 at 2 % for 180 days, or $180 at 2 % for 1 day $.01 $1 at 3 % “ 120 “ or $120 at 3 % “ “ “ .01 $1 at 6 % “ 60 “ or $60 at 6 % “ “ “ .01 $1 at 4 % “ 90 “ or $90 at 4 % “ “ “ .01 $1 at 4i% “ 80 “ or $80 at 4J% “ “ “ .01 $1 at 5 % “ 72 “ or $72 at 5 % “ “ “ .01 $1 at 6 % “ 6 “ or $6 at 6 % “ “ “ .001 $1 at 6 % “ 600 “ or $600 at 6 % “ “ “ .10 Example 1. — Find interest on $720 at 4% for 135 days. ( 1 ) '( 2 ) $7.20 int. for 90 days $1.35 int. on $90 _3.60 “ “ 45 “ (| of 90 ) 8 8 $10.80 ““ 135 “ $10.80 “ “$720 (90X8) Example 2. — Find interest on $960 at 4|% for 64 days. (1) ** (2) $9.60 int. for 80 days .64 int. on $80 1. 92 “ “ 16 “ Q- les s) .12 12 $7.68 “ “ 64 “ $7.68 $960 (80X12) Example 3. — Find the interest on $780 for 19 days at 6%. $.780 3 2.340 130 (!) int. 6 days 3 “ 18 “ (6X3) ( 2 ) $.019 int. on $6 130 130 $2,470 $780 (6X130) $2.4700 “ 19 u INTEREST 197 WRITTEN EXERCISE 638 . 1 . Find the interest on $475.84 for 96 days at 5%. 2. What is the interest on $286.80 for 126 days at 6% ? 3. Find the interest on $876.92 for 69 days at 8%. I/.. What is the interest on $345.75 for 136 days at 4|% ? 5. Find the interest on $595 for 288 days at 4%. 6. What is the interest on $1350 for 298 days at 4% ? 7. Find the interest on $330 for 129 days at 6%. 8. What is the interest on $1320 for 217 days at 5% ? REVIEW PROBLEMS IN SIMPLE INTEREST In the problems given below let the student use the special rule that seems best suited to the problem to be worked. In general, we suggest that for terms extending beyond a year the time be expressed in years, months and days, and that the interest be found for the years and aliquot parts of a year. For terms of less than a year, let the time be expressed in days, and the day method, or its modification, the 60 -day method be used. 639 . 1 ■ Find the interest on $4500 for 1 yr. 1 mo. 1 da. at 6%. 2. What is the amount of $2250 for 3mo. 3 da. at 6% ? 3. Find the interest on $405.50 for 66 days at 7%. If. What is the interest on $1606 from March 2, 1908, to May 17, 1909, at 4%? 5. John Hamaker took up his note for $350 to-day, January 27, 1909, which was dated April 1, 1908, and bore interest at 5% ; what amount did he pay ? 6. What is the interest on $778.75 at 6% from September 1, 1908, to Janu- ary 1, 1909? 7. Find the amount of $500 for 5 mo. 5 da. at 5%. 8. What is the interest on $1755 for 1 yr. 9 mo. 9 da. at 6% ? 9. What is the interest on $299.30 for 192 days at 7 % ? 10. What is the interest on $1666 for 6 mo. 6 da. at 6 % ? 11. What is the interest on $777 for 77 days at 7 % ? 12. Find the amount of $2200 from January 1, 1908, to September 1, 1908, at 4J%. 13. What is the interest on $920 for 8 mo. 8 da. at 8% ? Ilf. What is the amount of $444 for 44 days at 4% ? 15. What is the interest on $333 for 3 yr. 3 mo. 3 da. at 34% ? 16. Find the interest on $508.50 for 11 mo. 15 da. at 9% ? 17. Find the interest on $1532.75 from February 1, 1908, to October 1, 1908, at 10%. 18. What is the amount of $1410.18 for 10 mo. 26 da. at 12% ? 19. Sept. 4, 1907, I bought a quantity of grain for $840.70; Nov. 7, I sold part of it for $580.90; Jan. 27, 1908, the remainder for $325.40. Money being worth 6%, how much did I gain by the transaction? 20. May 9, 1907, a dealer borrowed sufficient money at 5 % to pay for 280 bushels corn, at 65c. a bushel and for 300 bushels oats at 40c. a bushel. Corn advanced 8c. a bushel, and oats 4c. a bushel. He sold at these advanced prices on July 27. After returning the money borrowed and the accrued interest, what was his gain ? 198 INTEREST ACCURATE INTEREST 640. Accurate Interest is found by taking the exact number of days, and reckoning them as 365tbs of a year. Example. — What is the accurate interest of $721 at 5% for 213 days? . 0 / 7-2 / 7 2/ X.SS X 2 /3 .0 S 3~£S 3 • O 3 7*3 2 / 3 2 / 3 72 / 2 / .3 / o f / s 3 7 0 S 7 2/Q 3 7 SJf 7 C, 7 f. 6 s (/ 2 /. 037 73 0 K 37 r 3 3S / S / S / 73) / S3 S.73/2 /.037 / V 6 , K 7S JIA 2 7J? 2/0 / 3 7 S / 0 7 s S+/0 2 7 0 0 Note. — In ordinary interest a year consists of 360 days ; in accnrate interest it consists of 365 days. Since 360 days is or 7 a j less than 365 days, it follows that the interest for a year of 360 days is A- more than the interest for a year of 365 days ; hence accurate interest may be reckoned by first finding the ordinary interest and then deducting ^ of this amount. 641. Rule. — Find the interest for one year by multiply iny the principal by the rate expressed decimally. Multiply the interest for one year by the number of days, and divide the product by 365. WRITTEN EXERCISE 642. Find the accurate interest of 1. $623 for 26 days at 6%. 2. $937.25 for 123 days at 5%. 3. $685 for 73 days at 4%. J. $427.60 for 92 days at 6%. 5. $1272.50 for 34 days at 3%. 6. $561 from Jan. 13, 1908, to June 3, 190S, at 7%. 7 . $743.25 from April 17, 1908, to Aug. 13, 190S, at 5%. 8. $1864.73 from Feb. 1, 1908, to Dec. 21, 1908, at 6%. 9. $324 from April 20, 1908, to Oct. 5, 1908, at 2J%. 10. $911.75 from June 27, 1908, to Nov. 5, 1908, at 44%. INTEREST 199 INTEREST PROBLEMS 643. To find the rate, when the principal, interest and time are given. Example. — At what rate will $480 produce $4.50 interest in 75 days? $480 principal. $4.80 int. for 60 days at 6%. 1-20 “ “ 15 “ “ “ (4 of 60 days). $6.00 “ “ 75 “ “ “ "fLOO “ “ 75 “ “1% (iof6%). $4.50-?-$l=4J Result 44%. 644. Rule. — Divide the given interest hy the interest of the given principal at 1 °/ P for the given time. MENTAL PROBLEMS 645. 1. At what rate per cent, will $100 produce $6 interest in 1 year? Solution. — A s 1 per cent., which is of the principal, will produce $1 in one year, to produce $6 will require as many times 1 per cent, as $1 is contained times in $6, or 6 per cent. 2. At what rate per cent, will $200 produce $12 in 1 year? 3. At what rate per cent, will $300 produce $20 in 2 years ? If.. Find what rate per cent, will produce $7 interest on $200 in 6 months. 5. I received $3 interest by investing $180 for 60 days ; what rate per cent, was paid ? 6. Having $200 on interest for 3 years, I received principal and interest, $236 ; what rate per cent, did I get for my money ? 7. A loan of $300, which had been out for 4 years, was returned with $60 interest ; what was the rate per cent. ? 8. At what rate per cent, will $160 in 5 years give $64 interest ? 9. Find at what rate per cent. $100 will in 4 years amount to $128. 10. At what rate per cent, will $500 amount to $510 in 90 da) r s? WRITTEN EXERCISE 646. At what rate will 1. $725 produce $8.82 interest in 73 days? 2. $2430 produce $25.92 interest in 6 mo. 12 da.? 3. $718.75 produce $39.53 interest in 1 yr. 2 mo. 20 da.? If. $522.50 produce $96.41 interest in 2 yr. 7 mo. 19 da.? 5. $358.60 produce $9.40 interest in 5 mo. 27 da. ? 6. $2332 produce $282.43 interest in 3 yr. 10 da.? 7. $2765.10 produce $230.23 interest in 1 yr. 6 mo. 5 da. ? 8. $563.25 produce $23.31 interest in 213 days ? 9. $917 amount to $989.81 in 2 yr. 5 mo. 17 da.? 10. $3223.96 amount to $3250.60 in 10 mo. 25 da.? 200 INTEREST 647. To find the time, when the principal, interest and rate are given. Example. — In what time will $375 amount to $412.38, at 5% per annum? $412.38 amt. 375.00 prin. 37.38 int. 1 9936 18.75 )37.38 18 75 0000 or, $375 .05 $18.75 int. for 1 yr. 18.75 ) 37.38000 ( 1.9936 yr. 12 18 630 1 6 875 1 7550 1 6875 6750 5625 11250 18 75 18 630 16 875 1 7550 1 6875 ~ 6750 5625 11.9232 mo. 30 27.6960 11250 Result 1 yr. 11 mo. 28 da. 648. Rule. — Divide the given interest by the interest of the given principal for one year at the given rate. The result will be years ; reduce the fraction, if any, to months and days. MENTAL EXERCISE 649. 1. In what time will $100 at 6 percent, earn $18 interest ? Solution. — I n 1 year the interest will equal T or of $100, which is $6, and to earn $18 will require as many years as $6 is contained times in $18, or 3 years. 2. In what time will $200 at 5 per cent, earn $50 interest? 3. In what time will $200 at 7 per cent, earn $28 interest? If.. In what time will $200 at 6 per cent, earn $2 interest ? 5. In what time will $300 at 10 per cent, earn $90 interest? 6. In what time will $80 at 5 per cent, earn $10 interest? 7. In what time will $300 at 6 per cent, amount to $309 ? 8. In what time will $50 at 6 per cent, earn $1J interest? WRITTEN EXERCISE 650. 1 . In what time will $5000 produce $112.70 at 6% ? 2. $2230 produce $64.27 at 6% ? 3. $3917.25 produce $910 at 5% ? f. $325 produce $72.18 at 4% ? 5. $1897.10 produce $321.37 at 3% ? 6. $376.75 produce $1S4.12 at 8% ? 7. $632.80 amount to $711.13 at 6% ? 8. $421.89 amount to $500 at 54% ? 9. $1124 produce $321.17 at 2 J% ? 10. $789.73 amount to $821.50 at 7 % ? 11. In what time will $4263.13 produce $289.10 interest at 54% per annum? 12. In what time will $750 produce $150 interest at 54 %? INTEREST 201 651. To find the principal, when the interest, time and rate are given. Example. — What principal will produce $42.99 interest in 2 yr. 3 mo. 15 da. at 5^>? $1 .1145f)42.9900( .05 6 6 $.05 int. for 1 yr. .10 int. for 2 yr. .0125 int. for 3 mo. of 1 yr.) .0020f- int. for 15 da. of 3 mo.) $.1145f int. for 2 yr. 3 mo. 15 da. Result $375.19 .6875 )257.9400( 375.186 206 25 51 690 4 8125 3 5650 3 4375 12750 6875 58750 55000 37500 652. Rule. — Divide the given interest by the interest of $ 1 at the given rate for the given time. 653. To find the principal, when the amount, time and rate are given. Example. — What principal will amount to $594.04 in 2 yr. 3 mo. 18 da. at 6 % ? $1 1.138)594.040(522 .06 $ 06 int. for 1 yr. .12 int. for 2 yr. .015 int. for 3 mo. .003 int. for 18 da. $1,138 amt. of $L00. $.138 int. for 2 yr. 3 mo. 18 da. 654. Rule. — Divide the given amount by the amount of $1 at the given rate for the given time. 569 0 25 04 22 76 2 280 2 276 4 Result $522. MENTAL EXERCISE 655. 1. What principal will in two years, at 6 per cent., amount to $112 ? Solution. — At 6 per cent, for 2 years T W of the principal equals the interest. plus T Vo or ijg equals the amount or $112, and x^o equals x j 2 of the amount or $1; hence the principal is $100. 2. What principal will in 7 years, at 5 per cent., amount to $270? 3. What principal will in 4 years, at 10 per cent., amount to $140? 202 INTEREST 1 /.. AVhat principal will in 3J years, at 6 per cent., amount to $242 ? 5 . What principal will in 2 years, at 4 per cent., produce $120 interest? 6 . What principal will in 1 year 6 months, at 7 per cent., amount to $221 ? 7 . What principal will in 6 months, at 6 per cent., amount to $412? 8 . What principal will in 60 days, at 6 percent., produce $6 interest? 9 . What principal will in 3 months, at 12 per cent., amount to $206 ? 10 . What sum of money was put at interest 3 years ago at 5 per cent., if it now amounts to $575 ? 11 . What principal, put at interest at 8 per cent, for 2 years and 6 months, will produce $60 interest ? 12 . What principal, put at interest for 120 days at 6 per cent., will produce $16 interest ? 13 . What principal, put at interest for 3 years at 5 per cent., will amount to $805? 14 -. What principal, put at interest for 30 days at 6 per cent., will produce $6 interest ? 15 . I received to-day $1080, which is the amount of a sum of money piaced on interest for 2 years at 4 per cent.; what was the sum ? WRITTEN EXERCISE 656. What principal will produce 1 . $37.27 interest in 179 days at 6% ? 2 . $184.29 interest in 1 yr. 3 mo. 3 da. at 4% ? 3 . $274.50 interest in 8 mo. 21 da. at 6% ? 4 $381.75 interest in 175 days at5%? 5 . $92.17 interest in 2 yr. 7 mo. 11 da. at 6% ? 6 . $188.77 interest in 5 mo. 12 da. at 7 % ? 7. $941.15 interest in 11 mo. 18 da, at 34% ? 8 . $62.75 interest in 3 yr. 1 mo. 27 da. at 6% ? 9 . $713.41 interest in 4 mo. 18 da. at 2% ? 10 . $825 interest in 1 yr. 10 mo. 2 da. at 54% ? What principal will amount to 11 . $4692.10 in 7 mo. 8 da. at 6% ? 12 . $378.12 in 2 yr. 1 mo. 5 da. at 4% ? 13 . $927.38 in 1 yr. 7 mo. 4 da. at 7 % ? 11 /.. $1250 in 5 mo. 23 da, at 6 % ? 15 . $3565.15 in 11 mo. 14 da. at 8% ? 16 . $2732.44 in 3 yr. 8 mo. at 5% ? 17 . $5723.92 in 4 yr. 2 mo. 7 da. at 4 4% ? 18 . $125.78 in 10 mo. 28 da. at 3% ? ~ 19 . $3716.17 in 1 yr. 3 mo. 13 da. at 2% ? 20 . $2244.39 in 2 yr. 4 mo. 24 da. at 6% ? COMPOUND INTEREST 657. Compound Interest is interest on the principal, and on the unpaid interest after it becomes due. Note. — Compound interest will not be allowed legally unless there has been an account settled between the parties, or a judgment has been obtained, thus producing a new principal, aggregating principal and interest which has fallen due, or where there is a special agreement in valid form to pay compound interest. It will also be allowed upon coupons which are overdue, or upon interest wrongfully withheld by a trustee from a beneficiary. Example. — Find the compound interest of $650 for 3 yr. 7 mo. 12 days at 6%. Explanation. — Since the interest is to be compounded annually, the amount due at the end of the first year, which is $689, will be the basis of interest for the second year ; and the amount due at end of second year, $730.34, will be the basis of interest for the third year ; the amount due at the end of 3d year, $774.16, will be the basis of the interest for the remaining 7 mo. 12 days of the time ; and since the compound amount thus found, $802.80, is made up of the compound inter- est and the principal, if from this amount the principal be subtracted, the remainder, $152.80, will be the compound interest. $650 prin. 39. int. for 1st yr. 689. amt. at end of 1st yr. 4134 int. for 2nd yr. 730.34 amt. at end of 2d yr- 43.82 int. for 3d yr. 774.16 amt. at end of 3d yr. 28.64 int. for 7 mo. 12 days. 802.80 amt. for full time. 650.00 prin. $152.80 compound int. 658. Rule. — Find, the amount of the principal for the first period of time for which interest is to be reckoned, and make this the principal for the second. Find the amount of this principal for the second period, and thus continue to the end of the given time. The last amount will be the required amount. To find the compound interest, subtract the given principal from the last amount. Note. — If the time contains fractional parts of a period, as months and days, find the amount due for the full periods, and to this add its interest for the months and days. Remarks. — 1. For interest to be compounded semiannually, take one-half the rate for twice the time. 2. For interest to be compounded quarterly, take one-fourth the rate for four times the time. 3. For interest to be compounded bi-monthly, take one-sixth the rate for six times the time. 4. For periods beyond the scope of the table, multiply together the amounts shown for periods, the sum of which will equal the time required. Thus, the amount of $1 at compound interest for 65 years at 4 %, is equal to the product of $1 at 4% for 30 years, and the amount of $1 at 4% for 35 years, that is, 3.24339751 X 3.94608899, or 12.79873520. WRITTEN RROBLEMS 659. 1. What is the compound interest of $340 for 3 years at 5 % ? £. What is the compound interest of $750 for 2 years 6 mouths at 6%, pay- able semiannually ? 3. I lent a contractor $5400 for 7 years at 10% per annum, interest payable quarterly, and took a bond and mortgage to secure the debt and its interest. Not having been paid until the end of the 7 years, how much was required in full settlement ? 203 204 INTEREST If. Aug. 1, 190S, I paid in full a note for $1275, dated April 18, 1903, bearing 10% interest. If the interest was compounded annually, what was the amount due at settlement ? 5. Money being worth 6%, compounded semiannually, which would be the better, and how much, for a capitalist to lend $20000 for 5 years and 6 months, or to invest it in land that, at the end of the time named, will sell for $50000 above all expenses for taxes? 6. At what rate per annum will $5000 amount to $20580.678, if compounded annually for 29 years ? 7. At what rate per annum will $3500 amount to $8007.74688, if com- pounded semiannually for 14 years ? 8. A invested $3000 at 7 % when he was 25 years of age. How old was he when the investment, with its interest compounded semiannually, amounted to $16754.78? 9. If compounded annually at 5%, what principal will amount to $2626.674 in 34 years ? 10. If I deposit $275 in a savings bank, and the interest thereon is com- pounded semiannually at 6% per annum, how much should I receive at the end of 20 years, if nothing has been previously withdrawn ? 11. Mr. Smith borrowed $2400 on Apr. 1, 1900, at 8% compound interest, payable quarterly. What sum was due Feb. 1, 1908, if the first five payments were made when due, and no subsequent payments were made? 12. Afather invested $3000 at 6% per annum, interest payable semiannually, and on the same terms promptly invested the interest as collected. How much should his son receive, when he attains his majority, if he was 6 years old when the investment was made? 13. Find the amount of $575.80 in 6 years at 5% compound interest. Ilf. What is the compound interest of $672 for 3 years at 4% payable semi- annually ? 15. What principal will in 4 years produce $180.40 compound interest at 6% ? COMPOUND INTEREST TABLE 205 680 . The labor of computing compound interest may be greatly shortened by the use of the following Compound Interest Table Showing the amount of $1 at compound interest at various rates per cent, for any number of years, from 1 year to 50 years, inclusive: Yrs. i per ct. 1 1-2 per ct. 2 per ct. 2 i-2 per ct. 3 per ct. 3 1-2 per ct. 4 per ct. 1 1.0100 000 1.0150 000 1.0200 0000 1.0250 0000 1.0300 0000 1.0350 0000 1.0400 0000 2 1.0201 000 1.0302 250 1.0404 0000 1.0506 2500 1.0609 0000 1.0712 2500 1.0816 0000 3 1.0303 010 1.0456 784 1.0612 0800 1.0768 9062 1.0927 2700 1.1087 1787 1.1248 6400 4 1.0406 040 1.0613 636 1.0824 3216 1.1038 1289 1.1255 0881 1.1475 2300 1.1698 5856 5 1.0510 101 1.0772 840 1.1040 8080 1.1314 0821 1.1592 7407 1.1876 8631 1.2166 5290 6 1.0615 202 1.0934 433 1.1261 6242 1.1596 9342 1.1940 5230 1.2292 5533 1.2653 1902 7 1.0721 354 1.1098 450 1.1486 8567 1.1886 8575 1.2298 7387 1.2722 7926 1.3159 3178 8 1.0828 567 1.1264 926 1.1716 5938 1.2184 0290 1.2667 7008 1.3168 0904 1.3685 6905 9 1.0936 853 1.1433 900 1.1950 9257 1.2488 6297 1.3047 7318 1.3628 9735 1.4233 1181 10 1.1046 221 1.1605 408 1.2189 9442 1.2800 8454 1.3439 1638 1.4105 9876 1.4802 4428 11 1.1156 683 1.1779 489 1.2433 7431 1.3120 8666 1.3842 3387 1.4599 6972 1.5394 5406 12 1.1268 250 1.1956 182 1.2682 4179 1.3448 8882 1.4257 6089 1.5110 6866 1.6010 3222 13 1.1380 933 1.2135 524 1.2936 0663 1.3785 1104 1.4685 3371 1.5639 5606 1.6650 7351 14 1.1494 742 1.2317 557 1.3194 7876 1.4129 7382 1.5125 8972 1.6186 9452 1.7316 7645 15 1.1609 690 1.2502 321 1.3458 6834 1.4482 9817 1.5579 6742 1.6753 4883 1.8009 4351 16 1.1725 786 1.2689 855 1.3727 8570 1.4845 0562 1.6047 0644 1.7339 8601 1.8729 8125 17 1.1843 044 1.2880 203 1.4002 4142 1.5216 1826 1.6528 4763 1.7946 7555 1.9479 0050 18 1.1961 475 1.3073 406 1.4282 4625 1.5596 5872 1.7024 3306 1.8574 8920 2.0258 1652 19 1.2081 090 1.3269 507 1.4568 1117 1.5986 5019 1.7535 0605 1.9225 0132 2.1068 4918 20 1.2201 900 1.3468 550 1.4859 4740 1.6386 1644 1.8061 1123 1.9897 8886 2.1911 2314 21 1.2323 919 1.3670 578 1.5156 6634 1.6795 8185 1.8602 9457 2.0594 3147 2.2787 6807 22 1.2447 159 1.3875 637 1.5459 7967 1.7215 7140 1.9161 0341 2.1315 1158 2.3699 1879 23 1.2571 630 1.4083 772 1.5768 9926 1.7646 1068 1.9735 8651 2.2061 1448 2.4647 1555 24 1.2697 346 1.4295 028 1.6084 3725 1.8087 2595 2.0327 9411 2.2833 2849 2.5633 0417 25 1.2824 320 1 . 4509 454 1.6406 0599 1.8539 4410 2.0937 7793 2.3632 4498 2.6658 3633 26 1.2952 563 1.4727 095 1.6734 1811 1.9002 9270 2.1565 9127 2.4459 5856 2.7724 6979 27 1.3082 089 1.4948 002 1.7068 8648 1.9478 0002 2.2212 8901 2.5315 6711 2.8833 6858 28 1.3212 910 1.5172 222 1.7410 2421 1.9964 9502 2.2879 2768 2.6201 7196 2.9987 0332 29 1.3345 039 1.5399 805 1.7758 4469 2.0464 0739 2.3565 0551 2.7118 7798 3.1186 5145 30 1.3478 490 1.5630 802 1.8113 6158 2.0975 6758 2.4272 6247 2.8067 9370 3.2433 9751 31 1.3613 274 1.5865 264 1.8475 8882 2.1500 0677 2.5000 8035 2.9050 3148 3.3731 3341 32 1.3749 407 1.6103 243 1.8845 4059 2.2037 5694 2.5750 8276 3.0067 0759 3.5080 5875 33 1.3886 901 1.6344 792 1.9222 3140 2.2588 5086 2.6523 3524 3.1119 4235 3.6483 8110 34 1.4025 770 1.6589 964 1.9606 7603 2.3153 2213 2.7319 0530 3.2208 6033 3.7943 1634 35 1.4166 028 1.6838 813 1.9998 8955 2.3732 0519 2.8138 6245 3.3335 9045 3.9460 8899 36 1.4307 688 1.7091 395 2.0398 8734 2.4325 3532 2.8982 7833 3.4502 6611 4.1039 3255 37 1.4450 765 1.7347 766 2.0806 8509 2.4933 4870 2.9852 2668 3.5710 2543 4.2680 8986 38 1.4595 272 1.7607 983 2.1222 9879 2.5556 8242 3.0747 8348 3.6960 1132 4.4388 1345 39 1.4741 225 1.7872 103 2.1647 4477 2.6195 7448 3.1670 2698 3.8253 7171 4.6163 6599 40 1.4888 637 1.8140 184 2.2080 3966 2.6850 6384 3.2620 3779 3.9592 5972 4.8010 2063 41 1.5037 524 1.8412 287 2.2522 0046 2.7521 9043 3.3598 9893 4.0978 3381 4.9930 6145 42 1.5187 899 1.8688 471 2.2972 4447 2.8209 9520 3.4606 9589 4.2412 5799 5.1927 8391 43 1.5339 778 1.8968 798 2.3431 8936 2.8915 2008 3.5645 1677 4.3897 0202 5.4004 9527 44 1.5493 176 1.9253 330 2.3900 5314 2.9638 0808 3.6714 5227 4.5433 4160 5.6165 1508 ! 45 1.5648 107 1.9542 130 2.4378 5421 3.0379 0328 3.7815 9584 4.7023 5855 5.8411 7568 46 1.5804 589 1.9835 262 2.4866 1129 3.1138 5086 3.8950 4372 4.8669 4110 6.0748 2271 47 1.5962 634 2.0132 791 2.5363 4351 3.1916 9713 4.0118 9503 5.0372 8404 6.3178 1562 , 48 1.6122 261 2.0434 783 2.5870 7039 3.2714 8956 4.1322 5188 5.2135 8898 6.5705 2824 1 49 1.6283 483 2.0741 305 2.6388 1179 3.3532 7680 4.2562 1944 5.3960 6459 6.8333 4937 50 1 1.6446 318 2.1052 424 2.6915 8803 3.4371 0872 4.3839 0602 5.5849 2686 7.1066 8335 206 COMPOUND INTEREST TABLE Compound Interest Table Showing the amount of $1 at compound interest, at various rates per cent, for any number of years, from 1 year to 50 } r ears, inclusive. Yrs. 4 1-2 per ct. 5 per ct. 6 per ct. 7 per ct. 8 per ct. 9 per ct. 10 per ct. 1 1.0450 0000 1.0500 000 1.0600 000 1.0700 000 1.0800 000 1.0900 000 1.1000 000 2 1.0920 2500 1.1025 000 1.1236 000 1.1449 000 1.1664 000 1.1881 000 1.2100 000 3 1.1411 6612 1.1576 250 1.1910 160 1.2250 430 1.2597 120 1.2950 290 1.3310 000 4 1.1925 1860 1.2155 063 1.2624 770 1.3107 960 1.3604 890 1.4115816 1.4641000 5 1.2461 8194 1.2762 816 1.3382 256 1.4025 517 1.4693 281 1.5386 240 1.6105 100 6 1.3022 6012 1.3400 956 1.4185 191 1.5007 304 1.5668 743 1.6771 001 1.7715 610 7 1.3608 6183 1.4071 004 1.5036 303 1.6057 815 1.7138 243 1.8280 391 1.9487 171 8 1.4221 0061 1.4774 554 1.5938 481 1.7181 862 1.8509 302 1.9925 626 2.1435 888 9 1 . 4860 9514 1.5513 282 1.6894 790 1.8384 592 1.9990 046 2.1718933 2.3579477 10 1.5529 6942 1.6288 946 1.7908 477 1.9671 514 2.1589 250 2.3673 637 2.5937 425 11 1.6228 5305 1.7103 394 1.8982 986 2.1048 520 2.3316 390 2.5804 264 2.8531 167 12 1.6958 8143 1.7958 563 2.0121 965 2.2521 916 2.5181 701 2.8126 648 3.1384 284 13 1.7721 9610 1 .8856 491 2.1329 283 2.4098 450 2.7196 237 3.0658 046 3.4522 712 14 1.8519 4492 1.9799 316 2.2609 040 2.5785 342 2.9371 936 3.3417 270 3.7974 983 15 1.9352 8244 2.0789 282 2.3965 582 2.7590 315 3.1721 691 3.6424825 4.1772482 16 2.0223 7015 2.1828 746 2.5403 517 2.9521 638 3.4259 426 3.9703 059 4.5949 730 17 2.1133 7681 2.2920 183 2.6927 728 3.1588 152 3.7000 181 4.3276 334 5.0544 703 18 2.2084 7877 2.4066 192 2.8543 392 3.3799 323 3.9960 195 4.7171 204 5.5599 173 19 2.3078 6031 2.5269 502 3.0255 995 3.6165 275 4.3157 Oil 5.1416 613 6.1159 390 20 2.4117 1402 2.6532 977 3.2071 355 3.8696 845 4.6609 571 5.6044 108 6.7275 000 21 2.5202 4116 2.7859 626 3.3995 636 4.1405 624 5.0338 337 6.1088 077 7.4002 499 22 2.6336 5201 2.9252 607 3.6035 374 4.4304 017 5.4365 404 6.6586 004 8.1402 749 23 2.7521 6635 3.0715 238 3.8197 497 4.7405 299 5.8714 637 7.2578 745 8.9543 024 24 2.8760 1383 3.2250 999 4.0489 346 5.0723 670 6.3411 807 7.9110 832 9.8497 327 25 3.0054 3446 3.3863 549 4.2918 707 5.4274 326 6.8484 752 8.6230 807 10.8347 059 26 3.1406 7901 3.5556 727 4.5493 830 5.8073 529 7.3963 532 9.3991 579 11.9181 765 27 3.2820 0956 3.7334 563 4.8223 459 6.2138 676 7.9880 615 10.2450 821 13.1099 942 28 3.4296 9999 3.9201 291 5.1116 867 6.6488 384 8.6271 064 11.1671 395 14.4209 936 29 3.5840 3649 4.1161 356 5.4183 879 7.1142 571 9.3172 749 12.1721 821 15.8630 930 30 3.7453 1813 4.3219 424 5.7434 912 7.6122 550 10.0626 569 13.2676 785 17.4494 023 31 3.9138 5745 4 . 5380 395 6.0881 006 8.1451 129 10.8676 694 14.4617 695 19.1943 425 32 4.0899 8104 4.7649 415 6.4533 867 8.7152 708 11.7370 830 15.7633 288 21.1137 768 33 4.2740 3018 5.0031 885 6.8405 899 9.3253 398 12.6760 496 17.1820 284 23.2251 544 34 4.4663 6154 5.2533 480 7.2510 253 9.9781 135 13.6901 336 18.7284 109 25.5476 699 35 4.6673 4781 5.5160 154 7.6860 868 10.6765 815 14.7853 443 20.4139 679 28.1024 369 36 4.8773 7846 5.7918 161 8.1472 520 11.4239 422 15.9681 718 22.2512 250 30.9126 805 37 5.0968 6049 6.0814 069 8.6360 871 12.2236 181 17.2456 256 24.2538 353 34.0039 486 38 5.3262 1921 6.3854 773 9.1542 524 13.0792 714 18.6252 756 26.4366 805 37.4043 434 39 5.5658 9908 6.7047 512 9.7035 075 13.9948 204 20.1152 977 28.8159 817 41.1447 778 40 5.8163 6454 7.0399 887 10.2857 179 14.9744 578 21.7245 215 31.4094 200 45.2592 556 41 6.0781 0094 7.3919 882 10.9028 610 16.0226 699 23.4624 832 34.2362 679 49.7851 811 42 6.3516 1548 7.7615 876 11.5570 327 17.1442 568 25.3394 819 37.3175 320 54.7636 992 43 6.6374 3818 8.1496 669 12.2504 546 18.3443 548 27.3666 404 40.6761 09S 60.2400 692 44 6.9361 2290 8.5571 503 12.9854 819 19.6284 596 29.5559 717 44.3369 597 66.2640 761 45 7.2482 4843 8.9850 078 13.7646 108 21.0024 518 31.9204 494 48.3272 S61 72.8904 837 46 7.5744 1961 9.4342 582 14.5904 875 22.4726 234 34.4740 S53 52.6767 419 80.1795 321 47 7.9152 6849 9.9059 711 15.4659 167 24.0457 070 37.2320 122 57.4176 486 SS.1974 853 48 8.2714 5557 10.4012 697 16.3938 717 25.7289 065 40.2105 731 62.5S52 370 97.0172 33S 49 8.6436 7107 10.9213 331 17.3775 040 27.5299 300 43.4274 190 6S.2179 0S3 106.71S9 572 50 9.0326 3627 11.4673 998 18.4201 543 29.4570 251 46.9016 125 74.3575 201 117.390S 529 INTEREST 207 REVIEW PROBLEMS IN INTEREST 661. 1. Find the interest of $4723.69 for 2 yr. 8 mo. 14 da. at 6%. 2. What will $528.50 amount to in 1 yr. 4 mo. 22 da. at 4%? 3. Loaned $4500 on February 16, 1907, at 5%. Received the amount due me on May 4, 1908 ; how much did I receive ? J. What sum must be invested at 4% for a child 10 years old, that he may receive $5000 when he is 21 years of age? 5. What principal will amount to $3273.40, at 6%, if loaned January 15, 1906, and paid August 3, 1908? 6. What sum of money will amount to $3630 in 3 years at 7 %? 7. What principal will produce $7 interest, at 5%, in 90 days? 8. What sum must be invested in a property that pays 7 \ c /o per annum, to produce an income of $300 a year? 9. What sum of money will amount to $2562 in 9 months at 9 % ? 10. At what rate will $120 gain $21, if placed on interest for 3 yr. 6mo? 11. If $800 amounts to $832 in 180 days, what is the rate per annum ? 12. In what time will $900 amount to $1005 at 5 per cent, per annum? 13. In what time will $880 produce $55 interest at 5% ? In what time will $500, placed on interest at 4%, double itself? 15. Jan. 1, 1908, I borrowed $1500 at 6% ; in what time will I owe $1770? 16. If a man buys a bill of goods amounting to $2762.48, terms 60 days or 3% off for cash, how much does he save by borrowing the money at 6% and paying cash ? 17. A buys a house for $6000, paying $2000 cash, and giving a mortgage at 5% for the balance. At the end of two years he finds he has paid $224 in taxes and $97.50 for repairs, and has received $800 for rent. What per cent, has he gained on his investment of $2000 (reckoning the house to be still worth $6000)? 18. At -what rate will $2475, loaned April 6, 1907, amount to $2559.94 on March 25, 1908 ? 19. A man buys a piano for $275 on instalments, paying $35 down and $10 a month. The price of the piano would have been $250 cash. What rate of interest is he paying ? 20. On Jan. 13, 1908, a man borrowed $5500, at 5%, with which he purchased a piece of land. He afterwards sold the land for $5950, paid the loan, and found that he had gained $172.44. On what date did he sell? 21. April 1, 1904, I borrowed $10500 at 4J% interest, and invested it in a farm at $75 an acre. Aug. 15, 1906, my agent sold 50 acres at $105 an acre, charging me 3%. The money was deposited in a bank paying 2J% on deposits. April 1, 1908, I sold the remainder of the farm at $95 an acre. After paying the interest, what was my gain? 208 INTEREST 22 . How much money, invested at 4%, will amount to $10000 in 8 yr. 5 mo. 29 da.? 23 . A firm bought goods on credit, and agreed to pay 7 % interest on each purchase from its date. Oct. 16, 1907, goods were bought to the amount of $268 j Dec. 31, 1907, to the amount of $765.80 ; Feb. 29, 1908, to the amount of $600 ; Apr. 1, 1908, to the amount of $325.25. If full settlement was made Aug. 25, 1908, how much cash was paid ? 21 ^. On August 27, 1906, Smith sold his farm for $16000 ; the terms were, $4000 cash on delivery, $5000 on May 27, 1907, $3000 on Feb. 28, 1908, and the remainder in two years from date of purchase, with 6% interest on all deferred payments. What was the total amount paid ? 25 . Oct. 16, 1907, goods were bought to the amount of $268 ; Dec. 31, 1907, to the amount of $567.90 ; on Feb. 27, 1908, to the amount of $575 ; on May 1, 1908, to the amount of $235.75. The firm agreed to pay 7% interest on each purchase from its date. If full settlement was made Sept. 5, 1908, how much cash was paid ? 26 . How much money must I deposit in a savings bank during each of four years, in order to be able to draw out $440, if the bank pays 4% interest on its deposits? 27 . I bought a house and lot on speculation for $14325; 4 months 23 days from date of purchase, I sold the property for $15840. If money was worth 74% per annum, how much more did the transaction yield me than if I had lent the purchase money at interest? 28 . An attorney collects a claim of $875 with ordinary interest thereon from August 29, 1908, to December 14, 1908, at 7 %. If the attorney’s rate for collect- ing is 10%, what net proceeds should be paid to the creditor? 29 . Find the difference between the accurate interest and the common interest on $8750 from May 12, 1908, to October 15, 1908, at 6f % per annum. 30 . A contractor borrowed a sum of money for 3 months 24 days at 6%. Not having sufficient funds to meet this obligation when due, he paid 60% of the debt and accrued interest by giving his check for $5054, and agreed to con- tinue the balance at 8% per annum until paid. The balance was paid 3 months 24 days later, with interest. What was the amount of the second pa} r meut? 31 . A man engaged in business was making 12J% annually on his capital of $16840. He quit his business and loaned his money at 74%. What did he lose in 2 yr. 3 mo. 18 da. by the change? 32 . A tract of land containing 516 acres was bought at $36 an acre, the money being loaned at 5J%. At the end of 2 yr. 9 mo. 15 da., f of the land was sold at $42 an acre and the money loaned at 5% ; 1 yr. 2 mo. IS da. later the remainder was sold at $384 an acre and the money loaned at 6%. Five years after borrowing the original sum, the loans were collected and the original sum returned with interest. What was gained? BANK DISCOUNT 662. Bank Discount is a deduction made from the face of a promissory note or draft for cashing such negotiable paper before maturity. This deduction is the interest of the sum due at the maturity of the note or draft for the number of days from the date on which it is discounted to the date of maturity. 663. It is called bank discount because one of the chief functions of a bank is the cashing, or buying, of such commercial paper. When such notes are cashed by individuals, however, the same rules are observed. 664. The methods of bank discount depend upon State laws and in some cases upon local customs. The methods which follow comply with the statute laws of Pennsylvania and the practise in its chief money center, Philadelphia, and banking circles contributory to it. 665. Since finding bank discount is precisely the same operation as finding interest, such method of operation may be used as is best suited to the particular problem under consideration. In Pennsylvania the legal and usual rate is 6%, and as the term rarely exceeds four months, some modification of the “ Day Method ” is usually preferable. 666. Asa calendar may not always be at hand to determine upon what day of the week the day of maturity may fall, we give a method of determining that fact. What is the date of maturity of a note dated Thursday, September 3, 1908, drawn for three months. On what day of the week does it fall ? Three months after September 3, 1908, is December 3, 1908. 27 days in September 31 “ “ October 30 “ “ November 3 “ “ December 91 days in the term to maturity. 91-w=13 weeks. There being no remainder, the date of maturity, December 3, 1908, falls on the same day of the week as the day of the date from which reckoning is made, Thursday. 667. The method is to divide the number of days from the date of reckon- ing, usually the date of the note, to the date of maturity, by 7, the number of days in a week ; then count as many days of the week from the day of the week upon which the day of reckoning falls as there are days in the remainder ; this will give the day of the week upon which the date of maturity falls. If this be Sat- urday or Sunday, the true date of maturity will be the next business day. 668. Days of grace are abolished in many States, but where recognized by statute the date of maturity falls upon the last day of grace. 669. It is not common in Philadelphia to offer interest-bearing notes for discount, but if the banks accept such notes for discount, the discount is reckoned on the face of the note ; country banks reckon the discount on the 209 210 BANK DISCOUNT amount of the note, i. e., the face plus the interest on the face for the full interest term of the note. 670. The due date of a note is the date upon which the maker is legally bound to make payment at some designated bank or place of business, and is found by adding to the date of the note the time expressed in it. 671. Should the due date of a note fall upon a Sunday, a Saturday, or any legal holiday, the note is due and payable on the next business day following, and the additional day or days must be added to the term of discount. 672. The term of discount is the number of days from the date of dis- count to the date on which the note is payable, inclusive of both days. 673. The legal holidays in the State of Pennsylvania are: New Year’s Day (Jan. 1), Lincoln’s Birthday (Feb. 12), Washington’s Birthday (Feb. 22), Spring Election Day (third Tuesday in February), Good Friday, Memorial Day (May 30), Independence Day (July 4), Labor Day (first Monday in September), General Election Day ( Tuesday after the first Monday in November), Thanks- giving Day (by custom the last Thursday in November), Christmas Day (Dec. 25). Should any of these legal holidays fall on Sunday, they are observed on the Monday following. Saturday is a legal holiday in Pennsylvania, so far as com- mercial paper is concerned, and paper nominally due on that day is legally due on the following business day. 674. The face of a note, less the interest for the term of discount, is called the proceeds. 675. To find the proceeds of a note. Example 1. — A note dated Monday, June 29, 190S, at 90 days, for $1450 is discounted July 1, 1908. What are the proceeds? / ^ S O X .0 b X O 3 b O / O . O O 2 /. 7 S / 1/ 2 tf. 2 S BANK DISCOUNT 211 Explanation. — Operation 1. — To find on what date the 90 days will expire, or the nominal due date. If the note was dated June 29, one of the 90 days will be in June, leaving 89 days ; 31 days will be in July, leaving 58 days ; 31 days will be in August leaving 27 days to be in September. Hence the 90 days will end September 27, which is the nominal due date. Operation 2 — To find on what day of the week the nominal due date will fall, or to find the legal due date. Using Monday, June 29, as the day of working the example, we find 1 day left in June ; 31 days in July, 31 days in August and 27 days in September or 90 days from the date of working the example to the due date. Dividing 90 by 7 we get 12 weeks and 6 days ; 6 days from Monday is Sunday. Hence September 27 is Sunday, and the legal due date is Monday, September 28. Operation 3. — To find the term of discount. This note was discounted July 1, hence the bank would have it 30 days in July plus 1 day for the day of discount, 31 days in August and 28 days in September, or 90 days, which is the term of discount. Operation 4. — Find the interest on $1450 at 6fo for 90 days, which is $21.75, or the bank dis- count, and the proceeds are the difference between the face of the note $1450 and $21.75 the discount, or $1428.25. Example 2.- — A note at three months, dated Monday, May 1, 1908, for $1290, is discounted June 29, 1908. Find the proceeds. Explanation. — Operation 1. — To find on what date the three months will end. One month from May 1 is June 1, two mouths is July 1 and three months is August 1. Operation 2. — To find on what day of the week Aug. 1 will fall. Using Monday June 29 as the day of working the example, we find 1 day left in June ; 31 days in July and 1 day in August or 33 days from the date of working the example to the due date. Dividing 33 by 7 we get 4 weeks and 5 days ; 5 days from Monday is Saturday. Hence August 1 is Saturday and the legal due date is Monday, August 3. Operation 3. — To find the term of discount. This note was discounted June 29, hence the bank would have it 1 day in June plus 1 day for the day of discount, 31 days in July and 3 in August or 36 days, which is the term of discount. Operation 4. — Find the interest on $1290 at 6% for 36 days, which is $7.74 or the bank dis- count, and the proceeds are the difference between the face of the note $1290 and $7.74, the discount, or $1282.26. 212 BANK DISCOUNT 676. Rule. — First find the due date of the note by adding to the date of the note the time expressed in it; if the resulting date falls on Saturday , Sunday, or a legal holiday, extend the time to the next business day. When the legal du.e date has been determined, find the interest on the face of the note for the number of days from the date of discount to the legal due date, including both. Note. — The local custom observed in Philadelphia and some other cities, of including both the day of discount and the day of maturity in the term of discount, gives one day more than the difference found by subtracting the dates. As, January 5 to January 15=10 days ; but January 5 to January 15, inclusive=ll days. EXERCISE 677. Find date of maturity and term of discount. Date of Note. Day of Week. Time. Discounted. When Due. 1. Jan. 6, ’08 Monday 60 da. Jan. 6, ’08 2. Feb. 24, ’08 Monday 2 mo. Mar. 9, ’08 3. Dec. 26, ’08 Saturday 2 mo. Jan. 15, ’09 J. May 14, ’08 Thursday 90 da. June 2, ’OS 5. July 11, ’08 Saturday 60 da. Aug. 6, ’08 6. Oct. 14, ’08 Wednesday 4 mo. Dec. 30, ’08 7. May 19, ’08 Tuesday 3 mo. June 18, ’08 8. Jan. 7, ’09 Thursday 90 da. Feb. 26, ’09 9. Dec. 19, ’08 Saturday 15 da. Dec. 19, ’08 10. Nov. 25, ’08 Wednesday 3 mo. Feb. 19, ’09 11. Aug. 11, ’08 Tuesday 2 mo. Sept. 7, ’08 12. Sept. 19, ’08 Saturday 30 da. Sept. 30, ’08 13. May 4, ’08 Monday 120 da. July 6, ’08 If. Sept. 25, ’08 Friday 4 mo. Nov. 2, ’08 15. Jan. 15, ’08 Wednesday 15 da. Jan. 24, ’08 16. Mar. 5, ’08 Thursday 4 mo. Apr. 18, ’08 17. May 18, ’08 Monday 2 mo. June 16, ’08 18. Jan. 15, ’08 Wednesday 90 da. Feb. 1, ’08 19. July 11, ’08 Saturday 4 mo. Aug. 18, ’08 20. Mar. 19, ’08 Thursday 6 mo. June 9, ’OS 21. May 21, ’08 Thursday 30 da. June 1, ’08 22. Apr. 20, ’08 Monday 15 da. Apr. IS, ’08 23. Dec. 18, ’08 Friday 90 da. Jan. 2, ’09 2f. Aug. 14, ’08 Friday 4 mo. Nov. 3, ’08 25. Dec. 31, ’08 Thursday 2 mo. Jan. 18, ’09 EXERCISE 678. Find the bank discount. Date of Note and Day of Week. Face. Time. ,, When Date of a e ' Discounted. Maturity. 1. May 18, ’08, Mon. $400.00 60 da. 6% May 18, ’08 2 Apr. 9, ’08, Thurs. $500.00 90 da. 6% Apr. 10, ’OS 3. Oct. 20, ’08, Tues. $1000.00 30 da. 6% Oct. 20, ’08 If. Dec. 21, ’08, Mon. $387.25 2 mo. 6% Dec. 21, ’08 Term of Discount. Bank Discount. BANK DISCOUNT 213 5. May 9, ’08, Sat. $487.37 3 mo. 6 % May 9, ’08 6. Aug. 13, ’08, Thurs. $587.00 1 mo. 6% Aug. 13, ’08 7. Nov. 5, ’08, Thurs. $1250.00 90 da. 6% Dec. 5, ’08 8. June 11, ’08, Thurs. $1187.50 3 mo. 6% July 14, ’08 9. July 15, ’08, Wed. $958.75 4 mo. 6% Aug. 29, ’08 10. Jan. 6, ’08, Mon. $546.27 90 da. 6% Feb. 10, ’08 11. Dec. 26, ’08, Sat. $787.37 2 mo. 6% Dec. 26, ’08 12. Nov. 25, ’08, Wed. $1358.68 3 mo. 6% Dec. 12, ’08 13. May 14, ’08, Thurs. $186.75 90 da. 6% June 20, ’08 U. Aug. 21, ’08, Fri. $50.76 2 mo. 6% Sept, 9, ’08 15. Dec. 28, ’08, Mon. $75.80 4 mo. 6 % Dec. 31, ’08 16. Dec. 30, ’08, Wed. $75.86 120 da. 6% Dec. 31, ’08 17. Jan. 10, ’08, Fri. $47.50 60 da. 6% Feb. 3, ’08 18. July 14, ’08, Tues. $57.67 3 mo. 6% Aug. 10, ’08 19. Sept. 11, ’08, Fri. $463.87 20 da. 6% Sept. 14, ’08 20. July 13, ’08, Mon. $50.27 10 da. 6% July 13, ’08 21. Apr. 15, ’08, Wed. $408.75 8 da. 6% Apr. 15, ’08 22. June 11, ’08, Thurs. $568.70 18 da. 6% June 17, ’08 EXERCISE 679 . Find the proceeds. Date of Note and Face. Time. When Bank Day of Week. Discounted. Discount. 1 . •Jan. 9, ’08, Thurs. $450.00 90 da. •Jan. 9, ’08 2. May 14, ’08, Thurs. $750.00 30 da. May 14, ’08 3. Mar. 13, ’08, Fri. $950.00 60 da. iMar. 14, ’08 If.. Apr. 20, ’08, Mon. $868.00 1 mo. May 9, ’08 5. Dec. 21, ’08, Mon. $567.27 3 mo. Jan. 4, ’09 6. Jan. 20, ’09, Wed. $50.70 10 da. Jan. 20, ’09 7. Nov. 19, ’08, Thurs. $487.70 20 da. Nov. 28, ’08 8. Dec. 26, ’08, Sat. $460.75 2 mo. Jan. 9, ’09 9. Dec. 26, ’08, Sat. $467.50 60 da. Jan. 9, ’09 10. June 11, ’08, Thurs. $1275.00 120 da. July 6, ’08 11. Oct. 20, ’08, Tues. $1100.00 30 da. Nov. 2, ’08 12. Dec. 21, ’08, Mon. $50.00 1 mo. Jan. 4, ’09 13. Sept. 5, ’08, Sat. $10.25 10 da. Sept. 5, - 0S Ilf. July 15, ’08, Wed. $1375.87 20 da. Aug. 1, ’08 15. May 9, ’08, Sat. $987.70 3 mo. June 1, ’08 16. Dec. 21, ’08, Mon. $787.70 90 da. Feb. 1, ’09 17. Sept. 21, ’08, Mon. $1500.00 10 da. Sept. 22, ’08 18. Oct. 21, ’08, Wed. $1476.80 15 da. Oct. 28, ’OS 19. Dec. 31, ’08, Thurs. $1156.76 3 mo. Feb. 17, ’09 20. Jan. 4, ’09, Mon. $987.40 60 da. Feb. 9, ’09 21. Feb. 20, ’09, Sat. $568.95 4 mo. Mar. 20, ’09 Proceeds. 214 BANK DISCOUNT MENTAL. PROBLEMS 680. 1 . What is the bank discount of a note of $60, discounted for 90 days, at 6 per cent.? Solution. — For 60 days the discount is 60 cents, and for 90 days, which is a half more than 60 days, the discount is a half more than 60 cents, or 90 cents. 2 . What is the bank discount of $200 for GO days? 3 . What is the bank discount of $S0 for 60 days? J. What is the bank discount of $120 for 30 days? 5 . What is the bank discount of $200 for 120 days ? What is the bank discount of $90 for 30 days? What is the bank discount of $180 for 60 days? What is the bank discount of $240 for 63 days? What is the bank discount of $600 for 120 days ? What is the bank discount of $300 for 20 days ? What are the proceeds of a note for $800 discounted for 30 days? What are the proceeds of a note for $240 discounted for 30 days? What are the proceeds of a note for $400 discounted for 90 days ? What are the proceeds of a note for $200 discounted for 40 days ? What are the proceeds of a note for $600 discounted for 15 days? What are the proceeds of a note for $60 discounted for 20 days ? 17 . What are the proceeds of a note for $120 discounted for 12 days? 18 . What are the proceeds of a note for $200 discounted for 15 days? What are the proceeds of a note for $300 discounted for 36 days ? What are the proceeds of a note for $540 discounted for 180 days? 681. To find the face of a note. Example. — What is the face of a note at 90 days, dated June 29, 1908, and discounted on that date, that will yield $1200 proceeds? June 29+90 days=Sunday, September 27,=Monday, September 28. From June 29 to September 28, inclusive=92 days. $1 * ,984f ) 1200 92 3 3 1)2 6 . 7 . 8 . 9 . 10 . 11 . 12 . 13 . n. 15 . 16 . 19 . 20 . 6PPJJ ) 2.954 ) •015* 3600.0000 ( 2954 6460 590S 1218.68 face of note $ 1,000 •015* $.984§ = Proceeds of $1 5520 2954 25660 23632 20280 17724 25560 23632 19280 Result S121S.69. 682 . Divide the given proceeds by the proceeds of $ 1 . BANK DISCOUNT 215 EXERCISE 883 . Find the face of note. Date of Note. Face. Time. Date of Discount. Proceeds. 1. Jan. 10, ’08, Fri. 60 da. Jan. 10, ’08 $450.00 2. Feb. 14, ’08, Fri. 30 da. Feb. 14, ’08 $500.00 3 Feb. 12, ’08, Wed. 90 da. Feb. 12, ’08 $650.00 If.. Mar. 10, ’08, Tues. 2 mo. Mar. 10, ’08 $450.25 5. Apr. 16, ’08, Thurs. 3 mo. Apr. 16, ’08 $375.60 6. June 9, ’08, Tues. 4 mo. June 9, ’08 $2000.00 7. Jan. 27, ’08, Mon. 1 mo. Jan. 27, ’08 $3000.00 8. Dec. 30, ’08, Wed. 2 mo. Dec. 31, ’08 $400.75 9. Dec. 26, ’08, Sat. 2 mo. Jan. 4, ’09 $786.50 10. Nov. 25, ’08, Wed. 90 da. Nov. 25, ’08 $10000.00 11. July 7, ’08, Tues. 4 mo. July 7, ’08 $100.00 12. Aug. 14, ’08, Fri. 60 da. Aug. 15, ’08 $50.00 13. July 18, ’08, Sat. 90 da. July 20, ’08 $10.00 Ilf.. Nov. 17, ’08, Tues. 4 mo. Nov. 17, ’08 $487.75 15. May 19, ’08, Tues. 30 da. May 20, ’08 $568.46 16. Dec. 3, ’08, Thurs. 4 mo. Dec. 5, ’08 $787.80 17. Oct. 10, ’08, Sat. 30 da. Oct. 12, ’08 $1160.75 18. Jan. 10, '08, Fri. 4 mo. Jan. 14, ’08 $1230.50 19. Aug. 31, ’08, Mon. 30 da. Aug. 31, ’08 $5.00 20. July 20, ’08, Mon. 60 da. July 20, ’08 $1250.00 - 21. Feb. 21, ’08, Fri. 90 da. Feb. 21, ’08 $1178.80 22. Jan. 21, ’08, Tues. 120 da. Jan. 23, ’08 $125.75 23. Oct. 23, ’08, Fri. 3 mo. Oct. 29, ’08 $499.60 2If. Jan. 4, ’09, Mon. 90 da. Jan. 5, ’09 $568.90 25. June 19, ’08, Fri. 30 da. June 30, ’08 $5000.00 WRITTEN PROBLEMS 684 . 1. Find the proceeds of a note for $1190, at 90 days, dated and dis- counted to-day. 2. I send to bank two notes ; the first dated to-day, at three months, for $2200; the second dated to-day, at sixty days, for $1190.25. If both these notes are discounted and the proceeds placed to my credit, what is my bank balance? 3. Find the proceeds of a note dated and discounted to-day, at three months, the face of which is $975.62. J. What is the discount of a note at five months, dated four weeks ago and discounted to day, if the face of the note is $1500? 5. What were the proceeds of a note for $1600, dated January 18, 1908, Saturday, at 90 days, which was discounted February 17, 1908? 6. Sent to bank John Alpine’s 60-day note, dated 7 days ago, for $1500, and bad it discounted to-day. Find the proceeds. 216 BANK DISCOUNT 7. Sold John Halter 925 yards broadcloth at $2.16, terms ninety days. Received His note which I immediately had discounted. Find the discount. 8 . Hiram Hillegas to-day discounted his note in my favor, given 14 days ago, for $1190 at four months. Find the proceeds. 9 . Fourteen days ago I sold Abram Levis 1280 yards linen at 18 cents, on a credit of sixty days. I received his note and to-day had it discounted. Find the proceeds. 10 . A merchant sends to bank $320 cash ; John Evans’s note dated seven days ago, at 90 days, for $350 ; Abram Ivinkler’s note for $1250 dated seven days ago, at two months ; William Witherspoon’s note for $1500, dated to-day, at 90 days ; and John Lardon’s note for $390 at 2 months, dated to-day. If all these notes are discounted, find the total amount placed to the merchant’s credit. 11 . I drew a draft on A. Bearer, of New York City, dated to-day, at four months, for $3000, and offered it for discount at my bank, and it was accepted. What amount should be placed to my credit? 12 . What is the net amount of a note, dated December 30, 1908, Wednes- day, at four months, for $1200, discounted February 1, 1909? 13 . What amount will a sixty-day sight draft for $5000, dated June 5, 1908, Friday, yield if discounted June 10, 1908? 7J. I hold A. Buyer’s 90-day note, for $1875, dated one week ago. If my bank discounts it for me to-day, what sum will be placed to my credit? 15 . I want to secure $1000 cash to-day, and to do so I have discounted my note at 90 days, dated to-dav. Find the face of the note. 16 . I have in bank a balance of $98.50. Wishing to give a check for $700, I have discounted my note dated to-day, at three months, for a sum sufficient to secure the balance. Find the face of my note. 17 . John Banderlin has in bank a balance of $19.20. He sends to bank $250 cash, Warren Baton’s note dated to-day, at three months, for $1120, and his own note at ninety days, dated to-day, for a sum sufficient to enable him to write a check for $4500 and have $1500 remaining in bank. Find the face of his own note. 18 . For what amount must I write a note at four months that will yield me ' $2211 if dated and discounted to-day? 19 . For what amount must I write a note, dated to-day, which I intend to have discounted the next business day, the time of the note being three months, to yield $900 ? 20 . I have in bank to-day a balance of $750.22. I owe a debt of $1200, and in order to secure the balance I give a note with collateral, at three months, dated to-day. What must be the face of the note? 21 . Find the face of a note that will yield $1720, if the time of the note is five months and it is dated and discounted to-day. 22 . The proceeds of a 90-day note, dated and discounted to-day, are $1115.42. Find its face. BANK DISCOUNT 217 33. I owe John Harley. $1800 to-day, and lie agrees to take in payment my note made payable to William Kessler, at three months. For what amount must I write the note? Slf.. I have in bank to-day a balance of $950. I send to bank four notes : the first for $1125, dated to-day, at 90 days; the second for $1050, dated to-day, at two months; the third for $975, dated 7 days ago, at 60 days; and the fourth is my own note at 30 days, dated to-day, for such an amount as will make my bank balance $5000. Find the face of my own note. 35. Raise $1000 to-day, January 5, 1909, Tuesday, by having Merchant’s note, dated December 19, 1908, for $375, at four months, discounted and by giving your own 90-day note, dated to-day, for such an amount that its proceeds will make up the sum needed. 36. Borrow at bank $1800 on three notes at one, two, and three months, respectively. The notes are all to be dated and discounted to-day. For what sums shall they be drawn so that the proceeds of the notes shall be equal ? 37. My bank discounts for me to-day an interest-bearing note, at four months, dated a week ago. The proceeds are $540 ; what is the face of the note ? 38. For what amount must a note at four months, dated to-day, bearing interest at 5%, be drawn so that, discounted to-day at 6%, the proceeds will be $900 9 39. If I buy goods for $1200 cash and immediately dispose of them for $1300, receiving a note at four months, which I have discounted at once, receiving the proceeds ; what is my per cent, of gain on the sale? 30. A man has a 60-day note discounted, term of discount 61 days, and receives $12.20 less than the face of the note. What sum does he receive? 31. I desire to raise $4200 to-day. I have two notes, the first for $2722.46, dated 10 days ago, at 90 days, and the other for $1200, dated 15 days ago, at 60 days; if I have these notes discounted, what additional sum must I borrow to make the required amount? 33. I received notice Oct. 6, 1908, that my account was overdrawn to the extent of $18.75. To adjust the matter and leave me a balance in bank, I send to bank two notes, one for $451.75, dated Aug. 20, 1908, Thursday, at four months, and the other for $60, dated Sept. 3, 1908, Wednesday, at 90 days; if ‘the proceeds are placed to my credit, what amount may I check out and still leave my balance $35.57 ? 33. I sold a merchant 958 pounds of sugar at 5f cents per pound ; 3852 pounds of tobacco at $7.35 per pound. In settlement I take his note at 60 days for this amount and have the note discounted immediately at 5%; what did I receive for the goods? PARTIAL PAYMENTS 685. A partial payment is a payment of a part of a debt and its accrued interest. 686. It is customary to acknowledge the partial payment of a note by writing on its back the amount of payment and the date. These payments are sometimes called indorsements. 687. There are a number of rules for determining the amount due at any time on a debt on which partial payments have been made. The most impor- tant are the United States Rule and the Mercantile Rule. UNITED STATES RULE 688. The United States Rule is the method adopted by the United States Supreme Court, and is used by courts in general. Example. — On a note of $2800 bearing interest at 5 %, dated July 8, 1907, the following payments were made : August 9, 1907, $25 ; October 24. 1907, 815 ; Dec. 18, 1907, $10; Feb. 7, 1908, $500; April 4, 1908, $1000; June 3, 1908, $300. What is due July 8, 1908? Operation Face of note July 8, 1907, $2800.00 First payment Aug. 9, 1907, $25.00 Interest on face of note to first payment (31 da. ), 12.06 12.94 New principal Aug. 9, 1907, 2787.06 Second payment Oct. 24, 1907, 15.00 Interest on new principal to Oct. 24, 1907 (75 da.), 529.03 •Third payment December 18, 1907, 10.00 Interest on same principal to Dec. 18, 1907 (54 da.), 20.90 Fourth payment Feb. 7, 1908, 500.00 525.00 Interest on same principal to Feb. 7, 1908 (49 da.), 18.97 68.90 456 10 New principal Feb. 7, 1908, 2330.96 Fifth payment April 4, 1908, 1000.00 Interest on new principal to April 4, 1908 (57 da. ), 18.46 931.54 New principal April 4, 1908, 1349.42 Sixth i^ayment June 3, 1908, 300.00 Interest on new principal to June 3, 1908 (59 da.), 11.06 283.94 New principal June 3, 1908, 1060.43 Interest to July 8, 1908 (35 da. ), 5.16 Amount due July 8, 1908, $1065.64 218 PARTIAL PAYMENTS 219 689. United States Rule. — Find the interest on the face of the note from the date of the note to the time of the first payment ( time by compound subtraction). From the first payment substract this interest and reduce the face of the note by the balance. Find the interest on this balance for the time from the first payment to the second pay- ment. From the second payment subtract this interest and reduce the debt by the bal- ance. Proceed in this way until all the payments have been disposed of. Then add the interest from the last payment to the date of settlement. Note. — Should any interest item exceed a payment, simply make a note of the payment and accumulated interest, and find interest on same amount to next payment. If the sum of both pay- ments exceeds both interest items, reduce the face of the note by the difference ; if not, continue in this way until the sum of the payments does exceed the sum of the interest items, and then reduce the debt by the difference. Interest must never be taken on a sum larger than that on which the preceding interest was taken. MERCANTILE RULE 690. The Mercantile Rule is the method commonly used by merchants. Example. — On a note of $2800 bearing interest at 5%, dated July 8, 1907, the following payments were made: Aug. 10, 1907, $25 ; Oct. 25, 1907, $15 ; Dec. 19, 1907, $10; Feb. 8,1908, $500; Apr. 4, 1908, $1000; June 3, 1908, $300. What is due July 8, 1908? Operation Face of note July 8, 1907, $2800.00 Interest to July 8, 1908, 140.00 Amount due July 8, 1908. First payment Aug. 10, 1907, $25.00 $2940.00 Interest to July 8, 1908 (333 da.), 1.16 Second payment, Oct. 25, 1907, 15.00 Interest to July 8, 1908 (257 da.), .54 Third payment, Dec. 19, 1907, 10.00 Interest to July 8, 1908 (202 da.), .28 Fourth payment, Feb. 8, 1908, 500.00 Interest to July 8, 1908 (151 da.), 10.49 Fifth payment, April 4, 1908, 1000.00 Interest to July 8, 1908 (95 da. ), 13.19 Sixth payment, June 3, 1908, 300.00 Interest to July 8, 1908 (35 da.), 1.46 1877.12 Amount due July 8, 1908, $1062.88 691. Mercantile Rule. — Find the interest on the face of the note to the date of settlement, if the date of settlement is not later than one year from date. Add this interest to the face of the note. Find the interest on each payment from the date on which it was made to the date of settlement ( exact days). Add the several payments and interest items, and subtract the sum from the amount of the note. Note. — S hould the date of settlement extend beyond a year from the date of the note, settlement of the note must be made at the end of every year from date, then for the fraction of the year, if any. 220 PARTIAL PAYMENTS We present a problem worked out by both methods with the object of showing that for large amounts there is a considerable difference between the results obtained by the United States and the Mercantile Rules. When worked by the Mercantile Rule, the difference is in favor of the payer of the note. It will be seen that the Mercantile Rule method is used for small amounts and short periods of time. 692. $12000. Philadelphia. January 3, 1906. On demand I promise to pay John Doe or order Twelve thousand with interest at 6%, without defalcation for value received. xx Dollars, 100 Richard Roe. Payments indorsed: March 3, 1906, $2000; May 3, 1906, $2000; July 3, 1906, $2000 ; Sept. 3, 1906, $2000 ; Nov. 3, 1906, $2000. What amount will take up the note January 3, 1907 ? United States Rule Face of note January 3, 1906, $12000.00 Payment, March 3, 1906, $2000.00 Interest on face, Jan. 3, 1906, to March 3, 1906, 120.00 1880.00 Balance, March 3, 1906, 10120.00 Payment, May 3, 1906, 2000.00 Interest on balance, March 3, 1906, to May 3, 1906, 101.20 1898.80 Balance, May 3, 1906, 8221.20 Payment, July 3, 1906, 2000.00 Interest on balance, May 3, 1906, to July 3, 1906, 82.21 1917.79 Balance, July 3, 1906, 6303.41 Payment, Sept. 3, 1906, 2000.00 Interest on balance, July 3, 1906, to Sept. 3, 1906, 63.03 1936.97 Balance, Sept. 3, 1906, 4366.44 Payment, November 3, 1906, 2000.00 Interest on balance, Sept. 3, 1906, to Nov. 3, 1906, 43.66 1956.34 Balance, November 3, 1906, 2410.10 Interest on balance, Nov. 3, 1906, to Jan. 3, 1907, 24.10 Amount to take up note, January 3, 1907, $2434.20 Mercantile Rule Face of note, January 3, 1906, $12000.00 Interest for one year, 720.00 Amount due Jan. 3, 1907, 12720.00 Payment, March 3, 1906, $2000.00 Interest on payment to January 3, 1907, 306 da., 102.00 Payment, May 3, 1906, 2000.00 Interest on payment to Jan. 3, 1907, 245 da., 81.67 Payment, July 3, 1906, 2000.00 Interest on payment to Jan. 3, 1907, 184 da., 61.33 Payment, Sept. 3, 1906, 2000.00 Interest on payment to Jan. 3, 1907, 122 da., 40.66 Payment, November 3, 1906, 2000.00 Interest on payment to Jan. 3, 1907, 61 da., 20.33 10306.00 $2414.00 Amounts needed to take up note : By United States Rule, $2434.20 By Mercantile Rule, 2414.00 Difference in favor of payer by Mercantile Rule, $20.20 PARTIAL PAYMENTS 221 WRITTEN PROBLEMS 693 . 1 . A note of $1460, dated June 3, 1907, with interest at 4J%, has the following indorsements: Aug. 6, 1907, $150; Nov. 19, 1907, $250 ; Jan. 24, 1908, $210; March 19, 1908, $290; May 20, 1908, $100. What was due June 3, 1908, by United States Rule? 2. On a mortgage of $5000, dated March 6, 1907, with interest at 5%,' the following payments have been made: May 23, 1907, $950; July 19, 1907, $290; Oct. 17, 1907, $450; Dec. 18, 1907, $1000;' Feb. 12, 1908, $1500. What is due March 6, 1908, by Mercantile Rule? 3. A demand note dated February 5, 1907, bearing interest at 5J%, for $2200, has the following indorsements : May 15, 1907, $25; June 11, 1907, $200 ; Aug. 20, 1907, $10 ; Nov. 5, 1907, $250 ; Jan. 9, 1908, $300. What is due Feb. 5, 1908, by United States and Mercantile Rules? J. On a note dated August 6, 1907, for $5200, with interest at 6%, the following payments have been made : Nov. 19, 1907, $550 ; Jan. 2, 1908, $30 ; March 27, 1908, $33; May 22, 1908, $38; July 23, 1908, $42. What is due Aug. 6, 1908, by United States Rule? 5. $5000. Philadelphia, June 14, 1907. On demand, I promise to pay to the order of Morris and Lewis, Five thousand Dollars with interest at 5%, without defalcation, value received. H. H. Ettkr, 104 S. Eighteenth St. On the above note the following indorsements have been made : July 30, 1907, $100; Sept. 16, 1907, $10; Nov. 18, 1907, $25; Feb. 11, 1908, $1000; March 18, 1908, $90; May 13, 1908, $25. What is due one year from date by United States and Mercantile Rules? 6. A note of $2500, dated Aug. 12, 1907, with interest at 6%, has the following indorsements : Oct. 26, 1907, $400 ; Dec. 9, 1907, $40 ; Jan. 24, 1908, $350; May 25, 1908, $25; July 15, 1908, $250; Sept. 16, 1908, $500; Nov. 5, 1908, $200. How much is due Jan. 1, 1909, bv United States Rule? 7. What is due Mar. 7, 1908, on a note for $1878.90, with interest at 6%, dated Jan. 12, 1907, on which the following payments have been made: Mar. 13, 1907, $100 ; May 18, 1907, $200 ; June 10, 1907, $75 ; Aug. 19, 1907, $150 ; Oct. 14, 1907, $250 ? (Mercantile Rule.) 8. What sum will take up January 1, 1909, a note for $10000, interest at 4J%, bearing date of September 5, 1907, upon which the following indorsements have been made: January 2, 1908, $2000; April 1, 1908, $2000; July 1, 1908, $2000: Oct. 2, 1908, $2000? STOCKS AND BONDS 694. Stock is the share capital of a corporation or commercial company ; that is, the fund employed in the carrying on of the business. 695. The capital stock of a company is divided into shares of equal amount, which are owned by the individuals who jointly form the corporation. 696. Certificates of stock are papers issued by a company, signed by the proper officers, indicating the number of shares each stockholder is entitled to, and are an evidence of ownership ; they are transferable by assignment and may be bought or sold like any other property. 697. The par value of a share of stock is the amount named in the certifi- cate — usually $100, though it may be any amount, as $50, $25, $10, etc. Shares of $50 are sometimes called half-stock , and those of $25, quarter-stock. 698. The market value of a share of stock is the price at which it is sold. 699. When the market value of shares is greater than the par value, they are said to be above par or at a premium , and when the market value is less than the par value, they are said to be below par or at a discount. 700. Quotations of the market value are generally a percentage of the par value, though they sometimes indicate the number of dollars per share. 701. The gross earnings of a company are the total receipts before expenses have been paid. 702. The net earnings are what is left of the gross earnings after paying expenses. 703. The accumulated profits which are distributed among the shareholders once or twice a year are called dividends, and are “ declared ” at a certain per cent, of the par value of their shares. 704. An assessment is a sum levied upon stockholders to make up losses or to provide for extensive improvements. 705. Preferred stocks are shares entitled to a dividend of a certain per cent, before the common stock can receive any dividend. Preferred stock is sometimes issued as a special inducement to raise money during financial embarrassment. 706. The term watered stock is applied to stock which has been increased by shares issued in excess of the capital stock subscribed for and actually paid in. 707. A bond is the written obligation of a Corporation, City, County, State, or Government, to pay a certain sum of mone} r at a certain time with a fixed rate of interest, payable at certain periods. The bonds of business corporations are usually secured by a mortgage on the whole or some specified portion of their property. 708. Coupon bonds are those with small certificates of interest attached, which are to be cut off and presented for payment as they become due. These bonds and coupons are signed by the proper officers, and, being payable to bearer, are transferable by delivery. 222 STOCKS AND BONDS 223 709. Registered bonds are those payable to the registered owner or to his order; they are transferable by assignment. The interest is paid by check or in cash to the owner. 710. Bonds are issued in various denominations, $1000 being the most common. They are always quoted at so much per cent, of the par value. 711. The Stock Exchange is a place where stock brokers meet daily to execute their customers’ orders for the purchase and sale of railroad stocks and bonds, Government securities, and such other stocks and bonds of various cor- porations as may be listed at the Exchange. 712. The regular commission for buying or selling at the New York Stock Exchange is one-eighth of one per cent, on the par value of all stocks and bonds; that is, $12.50 on 100 shares of stock of the par value of $100 each, or $1.25 on a $1000 bond. At the Philadelphia Stock Exchange the rates of commission areas follows : Rates of Commission PHILADELPHIA STOCK EXCHANGE United States Loans on par value, 1 per cent. Other Bonds and Loans on par value, 1 per cent. Bank, Insurance, and Trust Company Stocks selling at or -i over one hundred dollars j ^ ei s ^ are > cents. On all other Shares, selling at or over ten dollars per share, 12i cents. Selling under ten dollars per share, 61 cents. The charge on a single share varies from 25 cents to $1. EXCEPTIONS TO THE FOREGOIXG COMMISSION RULES Reading com. and 1st and 2nd pref., P. R. R., and Del., Lack a- -i wanna and Western, without regard to selling price . . . . J P er s ^ iare ’ *h cents - 713. A margin is a sum of money, or its value in securities, deposited with a broker to protect him against loss in buying or selling for a customer’s account. It is usually 10% of the par value of the stocks bought or sold. Note. — Brokers allow their customers interest on money deposited as margin ; and when a cus- tomer buys stock ou margin, since the broker has to pay the whole cost of the stock in cash, he charges his customer interest on the amount so expended for his account. 714. Stocks are usually bought or sold either “regular,” “cash,” “seller three” (s3), or “buyer three” (b3). Stock sold in the regular way is to be delivered the next day ; if sold “ cash ” it is to be delivered on the day sold. When stock is bought “seller three ” it means that the seller can deliver it on either of three days at his option — the day sold, the next day, or the day after next. “ Buyer three” means that the buyer may demand delivery of the stock at any time within three days, but must take and pay for it before the end of the third day. When stock is sold at either buyer’s or seller’s option for more than three days, (as blO, b20, b30, b60 ; slO, s20, s30, s60), the buyer pays six per cent, interest, unless “ flat” is specified in the bargain. 224 STOCKS AND BONDS 715. Selling short means selling stock which one does not own — either by seller’s option (in which case the seller buys and delivers the stock before the expiration of the time specified) or by borrowing the stock in order to make delivery, and afterwards buying to repay the loan. In either case the person making the “short ’’sale expects to be able to buy at a lower price, and soon enough, to make a profit. 710. In stock-exchange slang, a “ bull ” is one who endeavors to effect arise in the price of stock, and a “ bear” is one who endeavors to bring down prices. A “ bull ” buys stock, expecting to sell it at a higher price ; while a “ bear ” sells stock “ short,” expecting to buy it at a lower price. 717. A corner is a combination to buy all the available supply of a stock, so that those who have sold short may be unable to fulfil their contracts except by buying of the combination at an exorbitant price. 718. To hypothecate stocks or bonds is to deposit them as collateral security for money borrowed. STOCKS 719. To find cost when buying or proceeds when selling. Example 1. — What must I pay for 150 shares C. B. & Q. R. R stock, at 102J ; usual brokerage ? Quotation 1021 plus brokerage 1 equals 102f. // 0 zf* / ,6- O ’ S. / 0 o / 0 2 #3 .7, r S3 J3 . Example 2. — What do I obtain from the sale of 150 shares of N. Y. C. R. R. stock, at 118J ; usual brokerage? Quotation 118f less brokerage 1 equals 118|. / /Jo Sq o o / / T -f- 7/ / 7 7 3 3 .7 S 720. R ule. — Add the brokerage to the quotation and multiply this sum ( expressed as dollars ) by the number of shares to find the cost of purchase ; subtract the brokerage from the quotation and multiply this difference {expressed as dollars) by the number of shares to find the proceeds of sale. Note. — W hen the par value is other than 100, multiply by quotation expressed as dollars and afterwards add or subtract brokerage, as the case may require. A list of the stocks named in the problems of this book whose par value is other than §100 per share follows : Catawissa R. R, $50; Lehigh Valley R. R., $50 ; Lehigh Navigation, 850; North Penn. R. R., $50 ; Pennsylvania R. R., $50; Reading R. R.. $50 ; P., 11 il. & Balto., $50 ; Pitts., Gin. & St. L., $50. STOCKS AND BONDS 225 WRITTEN PROBLEMS 721 . 1 . What must I pay for 750 shares Catawissa preferred at 53? Note. — W hen no brokerage is given, the usual brokerage of 1%, or 121 cents per share, par $100, is understood. S. Through my broker I sold 1250 shares L. V. R. R. at 24J ; what should I receive from him? 3. Bought 600 shares Lehigh Navigation at 40, and sold them at 42, usual brokerage ; what amount of money do I gain ? 4- My broker buys for me 800 shares P. R. R. at 59, and 350 shares North Penn, at 91 J ; what is the full cost to me? 5. What will it cost me to buy 250 shares United of N. J. at 249J-? 6. I bought 500 shares Un. Gas Imp. at 87§, and to-day it is quoted at 107f ; what would I gain in net money by selling at once? 7. Bought Pacific Mail at 57f, and sold it during a decline at 41 f ; what did I lose per share ? 8. Sold 750 shares Welsbach Light at 38f ; what was the amount of my proceeds? 9. Plow many shares of Chi. R. I. & P. R. R. at 92J can I buy with $10000, and what surplus will remain? 10. I sold 350 shares Lehigh Nav. at 47U and invested in P. R. R. at 594; how many shares did I buy, and what is the amount of surplus? 11. How much will 200 shares of Pennsylvania Railroad cost at 53f, if purchased through a Philadelphia broker? IS. How much will 200 shares of Pennsylvania Railroad cost, if purchased through a New York broker at 107? 13. How many shares of Reading stock at llff- can I buy, through a Phila- delphia broker, for $2375? 14- What is the cost of 100 Reading 1st preferred at 23-ff, 200 N. Pacific preferred at 41f, 300 St. Paul at 87f, and 200 Atchison preferred at 25§ ? 15. Find the cost of 400 shares Texas & Pac. at 10|, brokerage \%. 16. Find the proceeds of 250 Omaha preferred at 145, brokerage \ 17. Purchased, through a New York broker, 100 Cent, of N. J. at 88J, 500 Del. & Hudson at 1124, 300 Illinois Cent, at 99f, and 200 Lake Shore at 171J. Find cost. 18. Sold, through a Philadelphia broker, 200 St. Paul at 86f, 150 Lehigh Yal. at 30J, 425 Reading at Ilf, and 300 West. N. Y. & Pa. at 2f. Find proceeds. 19. A broker bought, on his own account, 500 St. L. & San F. 1st preferred at 47f and afterwards sold at 52J-; what was his gain? SO. Find the proceeds of the following stocks if sold through a New York broker: 100 Balt, and Ohio at 114, 250 Wabash preferred at 15J, 200 Chi. R. I. & P. R. R. at 76f , 150 West. U. Tel. at 844, 25 Del., Lack. & W. at 158, and 300 Southern Pac. Co. at 16|-. 226 STOCKS AND EONDS 722. Buying and selling on margin. Example. — On June 4, 1908, A. Lamb instructed his broker to buy for him 200 shares of a certain stock and deposited with him $2000 margin. On June 8, he bought at 89§. On June 24 he sold the stock at 87J. What is the balance of Lamb’s account July 2? What is his loss? Dr. A. A. Lamb Cr. 1908 1908 June 8 Bot. 200 shares CL i. June 4 Cash deposit 2000 00 R. I. & P.R.R. at 89f 17875 00 “ 24 Sold 200 shares Chi. U 8 Brokerage 25 00 R, I. & P.R.R, at 87 4 17500 00 i ( 24 Brokerage 25 00 July 2 Int, on casli 9 33 July 2 Interest 71 50 “ 2 Int, on proceeds 23 33- Interest 10 Interest 03 2 Balance 1536 03 19532 66 19532 66 B. Dr. A. Lamb Cr. 1908 1908 June 8 Bot. 200 shares Chi. June 4 Cash deposit 2000 00 R. I. & P.R.R. at 894 17900 00 “ 8 Int. on bal. 133 “ 24 Int. on bal. 42 40 “ 24 Sold 200 shares Chi. July 2 Balance 1536 03 R. I. & P.R.R, at 87f 17475 00 July 2 Int. on bal. 210 19478 43 1947813 Margin deposited $2000.00 Balance received 1536.03 Loss $463.97 Notb. — Interest is charged on the cost of stock carried and is allowed on each deposit. Form A. is given as showing more in detail the charges and credits of a transaction in stocks bought through a broker on “margin.” A. Lamb deposits cash to the amount of 10 % of the par value of the shares he instructs his broker to buy. On this “ margin ” Lamb is entitled to interest until the time of settlement, in this case, from June 4, 1908, to July 2, 1908. The broker advances the money to buy, upon which he is entitled to interest until time of settlement, from June 8, 1908, to July 2. 1908. He charges interest on brokerage until time of settlement. From June 24, 1908, to July 2, 1908, the proceeds are in the hands of the broker and he must pay interest to Lamb until settlement, from June 24, 1908, to July 2, 1908. The brokerage is shown both on the purchase and on the sale. The account shows the balance due Lamb to be 81536.03, a loss of $463.97. Form B. shows account as commonly kept. The brokerage being added to purchase and deducted from sale, interest being calculated on daily balances. There is a credit balance of $2000 from June 4 to June 8, on which the interest is $1.33 ; a debit balance of $15900 from June 8 to June 24. on which the interest is $42.40, and a credit balance of $1575 from June 24 to July 2, on which the interest is $ 2 . 10 . STOCKS AND BONDS 227 WRITTEN PROBLEMS 723. 1 . A speculator directed his broker to purchase 500 shares of stock, par $50, and deposited 10% margin on April 6, 1908. On April 8 the stock was purchased at 52J. On May 1 a dividend of 3% was collected. On May 12 the stock was sold at 54. How much did the speculator gain, and what was the balance due him by the broker on May 14 ? 2. October 5, 1908, my broker buys for me 500 shares of L. V. R. R. at 29|, brokerage and holds it till February 25, 1909, then sells it at 34J, brokerage as before. There were no dividends on the stock. I put up $2500 as a margin in the hands of the broker at the time of purchase. What does he owe me at the time of sale ? 3. Bought 150 shares of P. R. R. at 55f, received a semiannual dividend of 2J per cent, and then sold the stock for 55J. How much did I gain ? 4~ On April 3, 1908, a speculator deposited $1000 with his broker, who purchased for him 100 shares of stock at 89, charging \ % brokerage. On May 1, 1908, he received a dividend of $175. On October 14, 1908, he sold the stock at 92, charging |% brokerage, and made settlement, charging 6% interest. How much did the speculator receive ? 724. Buying and selling short. Example. — Aaron Bear sold “ short ” December 1, 1908, through his broker, 250 shares Northern Pacific R. R. pref. at 60§, and “ covered ” his “short” December-15, 1908, at 56. Allowing \ % brokerage for buying and selling, what was his net profit? Dr. A. Bear Or. 1908 — 1908 Dec. 1 Commission 31 25 Dec. 1 Deposit 2500 00 U 15 Bought 250 N. P. (< 1 Sold 250 N. P. pref. pref. at 56 14000 00 at 60| 15093 75 u 15 Commission 31 25 (C 15 Interest 5 83 Ba lance 3537 08 17599, 58 17599 58 Balance $3537.08 Deposit 2500.00 Gain $1037.08 228 STOCKS AND BONDS Dr. A. Bear. Cr. 1908 ~ 1908 Dec. 15 Bot. 250 N. P. pref. Dec. 1 Cash deposit 2500 00 at 56j 14031 25 U 1 Sold 250 N. P. pref. (C 15 Balance 3537 08 at 604 15062 50 u 15 Int. on cash 5 83 ,17568 33J 17568 33 Balance $3537.08 Margin 2500.00 Gain $1037.08 Note. — On “short” sales the stock is borrowed for delivery and replaced when purchase is made. No interest is charged for this. The margin is allowed interest as in any other stock trans- action. PROBLEM A broker sold “short ’’for I. Flyer, February 1,1908,700 shares Illinois Steel at 55 and “covered” sale February 14, 1908, at 52f. What was his net profit ? BONDS 725. To find cost when bought or proceeds when sold. Example 1. — What must I pay for $12000, par value, of U. S. 4s regis- tered, at 113J ? 1134 + 1=1131 $12000 1-13| 1356U00 7500 $13635 00 cost Example 2. — What sum should my broker return to me from the sale of $8500, par value, of U. S. 4s coupon at 114f ? 114f_ i=114| $8500 1 - 14 | 34000 93500 _ 5313 $9743.13 proceeds 726. Rule. — Add rate of brokerage to quotation and multiply it by tie par value of the bonds to find cost; subtract rate of brokerage from quotation and multiply it by the par value of the bonds to find proceeds. STOCKS AND RONDS 229 WRITTEN PROBLEMS 727. 1 ■ What will be the cost to me of $12000, par value, of U. S. 4s coupon, at 128f ? 2. What must I pay for $8500 U. S. 4s registered, at 127|? 3. Find the proceeds of $7000 U. S. os coupon, at 113§. 4- I sell $5500 U. S. 5s registered at 1134 and invest proceeds in U. S. Currency 6s at 103. What is the par value of bonds bought, to the nearest $100, and what the surplus ? 5. AVhat will $8000 Lehigh Coal & Nav. 5s cost at 96 (Philadelphia rate of brokerage) ? 6. Bought $15000 extended 2s at 99J and sold them at once at 102 ; what was my gain ? 728. To find rate of income. Example.— Bought U. S. 4s registered, at 113. What is rate of income? 4% on 1134% 1.131 ) 4.00 ( 8 8 9.05 ) 32.00 ( 3.53 -f — 3.54% nearly. 27 15 4 850 4 525 3250 2715 535 729. Rule . — Divide the interest rate by the cost ( quotation plus brokerage). WRITTEN PROBLEMS 730. 1. AVhat is my rate of income from U. S. new 4s registered bought at 1284? 2. AVhat is my rate of income from City of Cincinnati 7^-s bought at 127|? 3. AVhat would be my rate of income from P. & R. Gen. Mort. 4s at 86f ? 4- AVhat net rate do Catawissa R. R. 6s pay an investor if bought at 105f ? 5. Which is the better investment — City of Camden 7s at 114, or City of Harrisburg 6s at 112|? 731. To find the cost of an interest-bearing bond that will yield a given rate of income. Example. — At what price must I buy 5% bonds to get 7 % on my invest- ment? No brokerage. As quotations run in eighths, it is evident that the result obtained must be modified to nearest 1 ) 5.00 eighth, which would be a business result, but not 743. cos j-,. Qr, 71 f business result, a mathematical one. 230 STOCKS AND BONDS 732. R ule. — Divide the rate expressed in bond by the rate of income to be obtained and the result will be the cost. Note. — This is an application of the percentage formula : Percentage divided by rate gives base. WRITTEN PROBLEMS 733. 1. What would be the cost of bank stock paying annual dividends of 10%, so that the holder would realize 6% on his investment? No brokerage. 2. A bank stock whose par is $50 sells at 140 ; semiannual dividends of 10% each are declared each year. At what price should it be sold to make it an 8 % investment. No brokerage. 3. At what price must I buy U. S. new 4s to obtain 3% on my investment? No brokerage. Jf.. Southwark Bank stock is $50 per share at par. It pays 20% dividends. What price may be paid for it per share to realize 8% on the cost? 5. At what price must I buy U. S. Currenc\ r 6s to make 5% on my invest- ment ? 734. To find average rate of income derived from a bond bought and held until redeemed. Example. — "What will be the average rate of income from a 4% bond bought at 118, maturing in 30 years? No brokerage. 4% for 30 years 120% Price rec’d on redemp’n 100% Full return 220% Cost 118 Received above cost 102 Average annual income 1_0_2 3 0 yj -4-118 = 2.88 3540 ) 102.00 ( 2.88 70 80 31 200 28 320 2 8800 r concretely, Income on $1000 bond 30 yr. at 4 % $1200 Bond redeemed 1000 Full return 2200 Cost 1180 Received above cost 1020 Average annual income 34 34^1180 = 2.88 or 2^ % nearly Result 2.88 approximately. 735. Rule.-- -A dd the income derived from the bond and the redemption value, subtract the cost from this sum and divide the difference by the term of years ; this mil give the average annual income ; the average annual income divided by cost will give the average rate of income. STOCKS AND BONDS 231 736. To find price that may be paid for a bond drawing a certain rate of interest to produce a certain rate of income. Example. — What must be paid for a 4% bond maturing in 30 years to produce 3% income? 90 120 100 100 190 ) 220 ( 115.78 190 300 190 1100 950 1500 1330 1700 1520 Or concretely, $1000 bond at 4% in 30 years will produce $1200 Redemption value 1000 Result 115.78, Full return 2200 approximately ; A3 % income on $1000 or 1154, nearly. bond would give in 30 years $900 Redemption value 1000 Desired return 1900 1900 ) 2200.00 ( 115.78, or 115-f nearly. Remake:. — There are various methods in use for determining the average annual income in cases of this kind, some being based upon the condition that the annual income might be at once re-invested (compound interest) ; others that the income is put to other uses than investment. The above method is simply a business method of getting a closely approximate result. 737. Rule. — Divide full return received from bond , income and redemption, by full return acceptable, and the result will be the approximate cost. PROBLEMS IN STOCKS AND BONDS 738. 1 . Ho w much must be invested in Lehigh Valley 6s at 103J to produce a yearly income of $1200? 2. What quarterly income will $29345.63 yield, if invested in U. S. new 4s coupon at 124f, usual brokerage? 3. On July 2, I bought 300 Lehigh Valley at 28J, b30, and called for the stock July 21. What was the cost, including interest? What was my profit if I sold on July 21 at 31f ? J. What was paid for 400 Lehigh Nav., bought July 6 at 41f, s20, and delivered July 22? 5. How much invested in U. S. new 5s registered at 1144, will yield an annual income of $1500? 6. A man having $45000 to invest, bought $10000 N. Y. P. & N. 6s at 104, $15000 Lehigh Nav. cons. 7s at 129f, and $12000 S. H. & W. 1st 5s at 106, and deposited the balance with a trust company which paid 2% interest on deposits. Find his annual income. What rate per cent, is it on his investment? 7. Which is the better investment, stock paying 4J% annual dividends bought at 75, or stock paying 7 % dividends bought at 120? 8. What per cent, income on the investment do 6 per cent, bonds yield, if purchased at 104f (brokerage J%)? 232 STOCKS AN 13 BONDS 9. Which will produce the greater annual income, and how much, $35000 invested instock at 297 paying 5% quarterly dividends, or the same amount invested in stock at 118 paying 3% semiannually? 10. What did I pay for 6% stock that yields me 4J% on my investment? 11. Bought through a broker sufficient 4% bonds to produce an annual income of $1000. What did they cost at 123§ and what is the per cent, of income on the investment ? 12. A invested $25000 as follows : 100 shares of 5% stock at 102, 50 shares of 6% stock at 136f, 25 shares of 3% stock at 80, brokerage |%, and loaned the balance on mortgage at 5%. What is his total income? 13. How much invested in 5% bonds at 122J will produce an annual income of $1600 ? Ilf.. At what price must 4% bonds be purchased to yield 3|% on the investment? 15. Which yields the better income on the investment, and how much per cent., 4% stock bought at 121 J, or 3% stock bought at 92J? 16. What is paid for 4-|- % bonds if 3| % is realized from the investment ? 17. If I pay 124| for a 4 %bond maturing in 20 years, what per cent, income do I receive on my investment? 18. June 2, 1908, a broker purchased for the account of a customer 600 shares of stock at 62|, the customer depositing $3000 margin. July 8, the stock was sold at 71. What was the gain? (Usual commissions and interest.) 19. My broker sold “short” for me on July 3, 1908, 500 shares of stock at 128|, and “ covered ” on July 22 at 126. What was my gain ? 20. What per cent, on the investment do 6% bonds pay, maturing in 10 years, if bought at 112§-? 21. Stock bought through a broker at 79f yields 4% on the investment. What per cent, dividend does the stock pay? 22. What per cent, on the investment do 5 per cent, bonds pay, maturing in 20 years, if bought at 104f ? 23. What per cent, is realized on the investment by buying 8% stock at 122 ? 21^. Smith purchased, through a Philadelphia broker, 500 Reading at 12, 200 Lake Shore at 170§, and 100 Illinois Cent, at 99^. He sold the Reading at 12|, the Lake Shore at 172, and the Illinois Cent, at 98f, investing the proceeds in U. S. 4s at 124J. What per cent, annual income does he realize on his original investment ? 25. How much must be paid for 5% bonds maturing in 12 years, in order to realize 44% on the investment (allowing §% brokerage)? 26. How much must be paid for 7 % bonds to realize 3% on the investment if the bonds mature in 6 years ? No brokerage. 27. If stock bought at 10% premium will pay 6% on the investment, what per cent, will it pay if bought at 20% discount? STOCKS AND BONDS 233 28. At what rate should stock paying annual dividends of 15% be bought to realize 5J% on the investment? 29. Chemical Bank Stock is quoted at 4000 ; what rate of dividend should it pay to give the investor a 6% income ? 30. A man buys Bank of North America at 260, which pays 20% dividend (par $100). What is the rate of income to the investor ? No brokerage. 31. Bought 150 shares of P. R. R. stock at 55f, received a semiannual divi- dend of 2J%, and then sold the stock at 55 \. How much did I gain ? 32. Which pays the better interest, railway stock at 147 (par $50), which pays 20% dividends, or bank stock at 237 (par $100), which pays 16% dividends? No brokerage. 33. A market company has a capital stock of $100000 divided into shares of $100 each. They have a bonded debt of $250000. Their receipts for the year are $50000. Expenses, exclusive of interest, $11000. They, of right, will pay the interest on their bonded debt and divide the balance among the stockholders. What rate of dividend will they pay the stockholders? Slf.. Which pays the better, to buy a New Jersey 7% mortgage at 5% discount, or a Pennsylvania 6% mortgage at 10% discount? No brokerage. 35. What is my rate of income from Reading Terminal 5s bought at 118J? Usual brokerage. 36. If I buy $10000 (par value) U. S. Gov. 4% bonds at 112, payable in ten years, and receive $400 each year, and the principal, $10000, at the end of the tenth year, w T hat rate of interest have I received on my investment ? 37. I bought stock, par $50, at 6J% discount, and received two semiannual dividends of 2J% each ; I then sold the stock at 7 % discount, netting a gain of $50, after paying a broker \ % f° r buying, and the same rate for selling. How many shares were bought? 38. A owned sufficient U. S. 4s, 1907, to pay him an annual income of $300 ; be sold them when they were quoted at 113R brokerage \°/o and invested the proceeds in 5% R. R. stock at 10% premium, brokerage \ °/ 0 ; find his change of income, and the surplus from the sale of the bonds. 39. If stock bought at 110 pays 4 1 1 q% on the investment, what rate of divi- dend does the stock pay, and what rate of interest would it yield if the price should advance 20% (brokerage not considered)? IpO. What rate of interest does an investor receive on his money who buys through a broker, P. R. R. stock, par value $50, at $61, if this stock pays a semi- annual dividend of 2 J % , and the rate of brokerage is J % ? J7. I own sufficient mining stock, par $50, paying a semiannual dividend of 2J%, to yield me $400 annually. The rate of brokerage being ^%, and the market value of the stock being $67^-, how many U. S. 3% bonds could I get if I were to sell the stock and invest the proceeds in bonds at 107|? What would be my surplus? What change will be made in my income? EXCHANGE 739. Exchange is the method or system by which debts are settled between persons in different places without the actual transmission of the money. It consists in the giving or receiving of a sum of money in one place for a bill ordering the payment of an equivalent sum in another. 740. Bank drafts are the principal means employed by merchants in making remittances from one part of the country to another. Banks located in the smaller cities and towns of the country keep money deposited in the great financial centers, such as New York, Chicago, Boston, Philadelphia, St. Louis, Baltimore, New Orleans and San Francisco. The banks draw upon their accounts in distant cities and sell their drafts to their customers, making a profit on the charge for “ exchange.” They also buy commercial drafts, drawn bv one merchant upon another who owes him money, and transmit them to their corre- spondents for collection, thus keeping their accounts replenished. 741. Drafts upon New York are extensively used in effecting exchange between the other financial centers of the country. New York is thus the ultimate center of exchange for the whole country. 742. The principal financial centers of Europe are London. Paris, Amster- dam, Antwerp, Hamburg, Vienna, Geneva, Berlin, Frankfort and Bremen. London is the center of exchange for the commerce of the world. 743. A draft, or bill of exchange, is an order drawn by one person (the drciiver ) upon another (the drawee) living in a different place, directing the drawee to pay a sum of money to the order of the drawer or to a third person (the payee). 744. Bills of exchange are of two kinds — foreign and domestic. 745. A foreign bill of exchange is one payable in a country other than the one in which it is drawn. 746. A domestic, or inland bill of exchange, is one drawn and payable in the same country. 747. When, in the course of exchange between two financial centers, the drafts drawn by the one are in excess of those drawn by the other, the balances must be settled by shipment of actual funds. In such cases, exchange is said to be against the one place and in favor of the other ; and, on account of the expense of shipping money, drafts drawn in the one city (which would tend to increase such expense) are at a premium, wdiile drafts drawn in the other city (which would tend to decrease shipment of funds) are at a discount. 234 EXCHANGE 235 748. With reference to time, drafts are drawn in three ways — at sight, a certain number of days after sight, and a certain number of days after date. The first are called sight drafts ; the other two are called time drafts. 749. Time drafts are discounted in the same manner as promissory notes. 750. To find the cost of a draft, the face being given. Example 1. — Find the cost of a draft for $1200, exchange \ % premium. $1200 face of draft, Or, 1.00J cost of $1. 1200.00 3.00 $1203.00 cost of draft. $1200 face. 3 premium. $1203 total cost. Example 2. — Find the cost of a draft for $800, exchange \% 0 discount. 100 Or, $800 face. •00j 2 discount. $800X.99f = $798 $798 net cost. 751. Rule. — Multiply face by par {100%) plus the rate of premium, or by par less the discount. Or, Add the premium to the face, or subtract the discount from the face. WRITTEN PROBLEMS pr 752. Find the cost of the following 1. $1000, \ % 0 discount. $2000, \ °/ 0 premium. 3. $5000, \ % 0 discount. If. $2385, f % 0 premium. 5. $3697, discount. 753. To find the face of a draft, Example. — What is the face of a emium ? drafts : 6 $3875.50,^% premium. 7. $9217.84, discount. 8. $7382.19, yV % premium. 9. $2468.36, \% discount. 10. $8696.78, \ % 0 premium. e cost and rate being given. draft that cost $6015, exchange \% $1.00 face. .0025 $1.0025 cost of $1. 6000 1.0025 ) 6015.0000 6015 0 face. entire cost. 000 Result $6000. 754. Rule. — Divide cost by par {100%) plus the rate of premium, or by par less the discount. WRITTEN PROBLEMS ,755. Find the face of the following drafts (amount given being the cost, and the rate expressed in dollars and cents per $1000) : 1. $6234.18, $1.25 premium. 2. $3697.25, $2.50 discount. 3. $6800.00, $1.00 premium. If.. $4675.39, 75 cents discount. 5. $2500.00, 50 cents premium. 6. $5675.00, 25 cents discount. 7. $S322.12, $2.00 premium. 8. $4725.80, $1.75 discount. 9. $9786.45, 15 cents premium. 10. $3375.95, $1.25 discount. 236 EXCHANGE WRITTEN PROBLEMS 756 . Find the cost of the following drafts : 1. A 30-day draft for $5000, \f 0 premium. Explanation. — The bank discount on $1 of the draft for 30 days is $.005 ; then $1 of the face will cost $1 — $.005, or $.995 if exchange were at par ; but at premium $1 of the face will cost $.995+$. 00125 or $.99625, and $5000 will cost 5000X$-99625 or $.4981 25. 2. A 60-day draft for $7500, f % discount. 3. A 20-day draft for $9873.25, § % premium. A 12-day draft for $6589.62, \ % discount. 5. A 33-day draft for $4437.18, f % premium. Find the face of the following drafts : 6. A 30-day draft that cost $7289, $1.25 premium. 7. A 15-day draft that cost $6427.73, discount. 8. A 60-day draft that cost $9745.56, § % premium. 9. A 36-day draft that cost $4578.27, 50 cents discount. 10. A 63-day draft that cost $5539.65, f % premium. 11. I wish my agent in Chicago to buy 2000 bushels of wheat at 68J cents a bushel. If his commission is 2%, and exchange \ % premium, what will be the cost of the sight draft I send him ? 12. A cotton broker in New York sells for a dealer in New Orleans 300 bales of cotton averaging 470 pounds each at 7§ cents per pound, his commission being 5%. The dealer in New Orleans draws a 60-day draft for the amount due him; how much can he get for the draft if exchange is $2.05 premium? 13. A of New York owes B of San Francisco $15000. If exchange is f% premium, how much must he pay for a 30-day draft, the rate of interest being 6 % ? A t- A man in St. Louis sends to a New York broker 500 shares of stock, with instructions to sell and remit proceeds by bank draft. If the broker sells the stock at 87f, and exchange is \ % premium, how much does the St. Louis man receive ? 15. A commission merchant of Boston sold a consignment of 9S8200 pounds of wheat at $1.20 per bushel, charging 2^% commission, and $85 for other expenses. If exchange is at discount, and interest 6% per annum, how large a draft, payable 20 days after sight, can he buy with the net proceeds, allowing 4 days for the acceptance of the draft? FOREIGN EXCHANGE 757. Foreign exchange is the settlement of debts by means of bills drawn in one country and payable in another. 758. The sum of a foreign bill of exchange is expressed in the money of the country on which they are drawn. Note. — Foreign bills are usually drawn at sight or at 60 days after sight. No discount is reck- oned on 60-day bills, as the quotation includes the allowance for time. 759. The intrinsic par of exchange is the value of the monetary unit of one country in that of another, based on the comparative weight and fineness of the coins as determined by government assay. 760. The commercial par of exchange is the market value of the coins of one country when sold in another. 761. The rate of exchange is the market value in one country of bills of exchange drawn on the other. While this value depends primarily on the balance of trade, it is also affected by the ownership in one country of the invest- ment securities of another, and tbe disbursement of interest and dividends. The rate can never rise higher than the actual cost of exporting or importing gold. 762. Commercial quotations of foreign exchange are given by means of equivalents, without reference to the par value. 763. Sterling exchange is quoted by giving the value of £l in dollars and cents ; as, “ Sterling exchange, 4.86J @ 4.88 ” means $4.86J = £1 for 60-day bills and $4.88 = £1 for sight bills. 764. Exchange on France, Belgium and Switzerland is quoted by giving the value of $1 in francs and centimes ; as, “ Paris exchange, 5.15 @ 5.13^ ” means $1 = 5.15 francs for 60-day bills and $1 — 5.13| francs for sight bills. 765. Exchange on Holland is quoted by giving the value of 1 guilder in cents. 766. Exchange on Germany is quoted by giving the value of 4 marks in cents ; as, 94f @ 94f means 94f cents = 4 marks for 60-day bills and 94f cents = 4 marks for sight bills. 767. It is provided by an Act of Congress that the values of the standard coins of the nations of the world shall be estimated at stated intervals by the Director of the Mint, and be proclaimed by the Secretary of the Treasury. The following table shows the values of the foreign moneys of account as published July 1, 1908. 237 238 FOKEIGN EXCHANGE Table of Foreign Coins, July i, 1908 VALUES OF FOREIGN COINS COUNTRY Standard Monetary Unit Value in terms of U. S. gold dollar COINS Argentine Republic . . Gold . Peso . . . $0,965 Gold : argentine ($4,824) and % argentine. Sil- ver ; peso and divisions. Austria-Hungary . . . Gold . Crown . . .203 Gold. 10 and 20 crowns. Silver: land 5 crowns. Belgium .... Gold . I ranc .193 Gold : 10 and 20 francs. Silver : 5 francs. Bolivia Silver Boliviano .393 Silver : boliviano and divisions. Brazil Gold . Milreis . . .546 Gold: 5, 10 and 20 milreis. Silver: %, 1 and 2 milreis. British Possessions, N. A. (except Newi’nd) . Gold . Dollar . . 1.000 Central Amer. States— Costa Rica Gold . Colon . . .465 Gold : 2, 5. 10 and 20 colons ($9,307). Silver : 5, 10, 25 and 50 centimos. British Honduras . . Gold Dollar. . . 1.000 Guatemala ... 1 Honduras .... ( Nicaragua ... f Silver Peso . . . .393 Silver : peso and divisions. Salvador . . . j Chile China Gold . Silver •{ Peso . . . Tael || . . Dollar H . .365 Gold: escudo ($1,825), doubloon ($.3650), and con- dor ($7,300). Silver: peso and divisions. Colombia Gold . Dollar. . . 1.000 Gold : condor ($9,647) and double-condor. Sil- ver ; peso. Denmark Gold . Crown . . .268 Gold : 10 and 20 crowns. Ecuador Gold . Sucre . . .487 Gold : 10 sucres ($4.8665). Silver : sucre and divisions. Egypt Gold . Pound 100 piasters 4.943 Gold ; pound (100 piasters), 5, 10, 20 and 50 pias- ters. Silver : 1, 2, 5, 10 and 20 piasters. Finland Gold . Mark .193 Gold ; 20 marks ($3,859), 10 marks ($1.93). France Gold . Franc . . .193 Gold: 5, 10, 20, 50 and 100 francs. Silver: 5 francs. German Empire . . . Gold . Mark . . . .238 Gold : 5, 10 and 20 marks. Great Britain Gold Pound sterling 4.866% Gold: sovereign (pound sterling) and % sov- ereign. Greece Gold Drachma . .193 Gold : 5, 10, 20, 50 and 100 drachmas. Silver : 5 drachmas. Haiti ... Gold . Gourde . . .965 Gold: 1, 2, 5 and 10 gourdes. Silver: gourde and divisions. India (British) Gold . Pound sterling* 4.866% Gold: sovereign (pound sterling). Silver: ru- pee and divisions. Italy Gold . Lira . . . .193 Gold : 5, 10, 20, 5u and 100 lire. Silver ; 5 lire. Japan Gold . Yen . . . .498 Gold : 5, 10 and 20 yen. Silver: 10, 20 and 50 yen. Liberia Gold Dollar.. . 1.000 Mexico Gold . Peso f . . .498 Gold: 5 and 10 pesos. Silver: dollar! or (peso) and divisions. Netherlands Gold Florin . . .402 Gold : 10 florins. Silver : 2%, 1 florin and divi- sions. Newfoundland Gold Dollar. . . 1.014 Gold ; 2 dollars ($2,028). Nor wav Gold Crown . . .268 Gold : 10 and 20-crowns. Panama Gold . Balboa. . 1.000 Gold : 1. 2%, 5, 10 and 20 balboas. Silver : peso and divisions. Persia Silver Kran . . . .072 Gold : %, 1 and 2 tomans ($3,409). Silver: %, %, 1, 2 and 5 krans Peru Gold . Libra. . . 4.866% Gold : % and 1 libra. Silver : sol and divisions. Philippine Islands . . . Gold . Peso . . . .500 Silver peso : 10. 20 and 50 centavos. Portugal Gold . Milreis . . 1.080 Gold : 1, 2, 5 and 10 milreis. Russia Gold . Ruble . . .515 Gold : 5, 7%, 10 and 15 rubles. Silver: 5, 10, 15, 1 20, 25, 50 and 1U0 copeks. Spain Gold . Peseta . . . .193 Gold : 25 pesetas. Silver : 5 pesetas. Straits Settlements . . . Gold . Pound sterling § 4.866% Gold: sovereign (pound sterling). Silver: dol- lar and divisions. Sweden Gold Crown . . .268 Gold : 10 and 20 crowns. Switzerland Gold . Franc . .193 Gold : 5, 10, 20, 50 and 100 francs. Silver : 5 : francs. Turkey Gold . Piaster. . .044 Gold : 25, 50, 100. 250 and 500 piasters. Uruguay Gold Peso . . . 1.034 Gold : peso. Silver : peso and divisions. Venezuela Gold . Bolivar . .193 Gold : 5, 10, 20, 50 and 100 bolivars. Silver : 5 bolivars. Notes.— The coins of silver-standard countries are valued by their pure silver contents, at the average market price of silver for the three months preceding. *The sovereign is the standard coin of India, but the rupee ($0.3244%) is the current coin, valued at 15 to the sovereign. +Seventy-five centigrams fine gold. JValue in Mexico, $0,498. gThe current coin of the Straiis Settlements is the silver dollar issued on Government account and has been given a tentative value of $0.567758%. ||The value of the tael varies in the different provinces. The values are as follows : Amoy $ 644, Canton .642, Cheefo'o .616, Chin Kiang .629, Fuchau .595, Haikwan (customs) .655, Hanko-v .602, Kiaochow .624, Nankin .637, Niuchwang .604, Ningpo .619, Peking .628, Shanghai .5S8, Swatow .595, Takau .648. Tientsin .624. HThe dollar has the following values in the respective provinces : Hongkdng $.423, British .423. Mexican .427. FOREIGN EXCHANGE 239 The rates in the foregoing table are used at the Custom House in estimating the value of foreign merchandise on invoices made out in foreign currencies. 768. To find the cost or value of a foreign bill. Example. — Find the value of a draft for £120 15s. at 4.867. $4,867 4 ) 14601 120f 36504 584 040 3 6504 $587,690' 769. Rule. — Multiply face of bill by quotation of exchange. WRITTEN PROBLEMS 770. Find the value in United States money of the following: 6 £782 15s. at 4.874. 1. £375 at 4.88. 2. 2340 francs at 5.15. 3. 978 guilders at 414. f. 1200 marks at 964. 5. £562 8s. 6 cl. at 4.864. 7. £524 7-s. 4 d. at 4.86f. 8. 7643.12 francs a 5.134- 9. £1142 18s. 3d. at 4.884. 10. 3789 guilders at 41|. 771. To find the face of a foreign bill. Example. — Find the face of a draft on London costing $587.69, exchange 4.867. £120 7499+ 4.867 ) 587.690 ( 120 486 7 100 99 97 34 3 650 20 4.867 ) 7 3000 ( 15— 4 867 2 4330 2 4335 4.867 ) 587.690 0000 486 7 100 99 97 34 3 6500 3 4069 24310 19468 48420 43803 46170 £ .7499 20 14.9980 Result £120 15s. Note. — T he exact result is not obtained owing to the ignoring of the small fraction in finding cost. 772. Rule . — Divide vcdue of bill by quotation of exchange. WRITTEN PROBLEMS 773. Find the face of hills that cost as follows : 1. $738.72 at 4.S9 (London). 2. $975.18 at 5.144 (Paris). 3. $840.75 at 41f (Amsterdam). f. $1238.29 at 94§ (Berlin). 5. $3472.30 at .517 (Antwerp). 6. $2575.84 at 4.85| (London). 7. $395.97 at 4.88f (London). 8. $1892.33 at 954 (Hamburg). 9. $5698.77 at 5.164 (Geneva). 10. $7535.44 at 96§ (Bremen). 240 TAXES WHITTEN PROBLEMS 774. Find the value of the following hills of exchange : 1. A draft on Vera Cruz for 3426 dollars, at .662. 2. A draft on Shanghai for 6847 taels, at .901. 3. A draft on St. Petersburg for 13625 rubles, at .488. J. A draft on Yokohama for 24677 yen, at .658. 5. A draft on Havana for 36355 pesos, at .926. 6. A Liverpool merchant buys 40000 bushels of American wheat at 69 cents a bushel ; how much does it cost him to remit by draft in settlement, exchange 4.88J ? 7. If U. S. 4s are quoted in New York at 112, what is the equivalent London quotation, exchange 4.88? Note. — A merican securities are quoted in London on a fixed basis of $5 = £1. 8. What is the equivalent New York quotation of stock quoted in London at 87 f, exchange 4.88J ? 9. How much must a London merchant pay for a draft on Bombay of 30000 rupees, exchange at Is. 3f d. (for 1 rupee) ? 10. A cargo of wine invoiced at 23642.80 milreis in Lisbon is worth more than a cargo of coffee invoiced at 25000 milreis in Rio Janeiro. How mueh more in U. S. money ? TAXES 775. A tax is a sum of money assessed on the person or property of a citizen by the government to defray public expenses ; as, a city, borough or town tax, county tax, state tax, etc. 776. The public revenues of the various state and local governments in the United States are raised by direct taxation upon the property, and in some of the states upon the polls and personal incomes. 777. The public revenues of the National government are raised by indirect taxation in the form of duties on imported goods and the internal revenue taxes on liquors and tobacco. 778. A capitation or poll tax is a specified sum levied on the person of every adult male citizen. 779. Property taxes are divided into two classes — (a) taxes on real property or real estate , i. e., lands, houses, etc.; and (b) taxes on personal property or personal estate, i. e., horses, cattle, vehicles, furniture, money, stocks, bonds, mortgages, merchandise, etc. TAXES 241 780. To find amount of tax. Example — What is the amount of tax on a property assessed at $2250, rate 13 mills? $2250 .013 $29,250 781. Rule . — Multiply valuation by rate , expressed decimally. ORAL EXERCISE 782. 1. What is the tax on $2500, at the rate of 9 mills on the dollar? 2. On $17000, at 4 mills? 3 . On $4000, at 6J mills? 4. On $24000, at 8 ^ mills ? 5 . What is the tax on $3000, at the rate of $1.75 on $100 ? 6. On $4000, at $1.50 ? 7. On $12000, at $1.25? 8. What is the tax on $20000, at the rate of $18.50 on $1000 ? 9. On $11000, at $17? 10. On $6875, at $16? 783. Tax Duplicate Valuation. 1 . $5000 2 . $3000 3 . $2500 4- $12500 5 . $8000 6 . $6500 7. $800 8 . $7250 9 . $4300 10 . $1350 11 . $950 12 . $4000 13 . $2000 n. $1700 15 . $3900 County Tax Road Tax School Tax Poor Tax Total. 15 mills 8 mills. 10 mills. 2 mills. 242 TAXES 784. To find rate of tax required to raise given amount of tax. Example. — Required tax rate on assessed valuation of $1700000 to raise a school tax of $18000, 10% of levy being uncollectable, and cost of collection being 2%. $1.00 of tax levied. .882 )18000.0000( 20408.163 total amount .10 of tax un collectable. .90 of tax collectable. .018 amount paid collector on each $1 of tax levied. .882 net amount received on each $1 of tax levied. 1700000 ) 20408.16 ( .012 1 7000 00 3408 16 3400 00 Result 12 mills. The result is nearly 12 mills, which would be the rate to levy to cover the amount needed. 785. Rule. — Divide amount necessary to levy by assessed valuation. WRITTEN PROBLEMS 786. 1. A city whose assessed valuation is $2500000 must raise 830000 for city purposes ; $12500 for highways; $20000 for schools; $5000 for the support of the poor; these sums include collector’s fee. What rate must be levied for each purpose? 2. What is the tax on $14500 of real estate, and $23275 of personal prop- erty, at $2.07 per $100, less a discount of 2% for prompt payment? 3. If A’s tax is $541.64, at the rate of $19.75 on $1000, what is the assessed valuation of his property ? It. What is the rate of taxation per $100, if property assessed at $3850 pays a tax of $73.42 ? 5. The assessed valuation of all the taxable property in a certain town is $1987690. The number of polls is 418, at 50 cents each. The estimated expenses of the town are $33210. What rate of taxation will raise this amount (assuming that all the taxes can be collected) ? 6. What must be the rate of taxation to yield $5289.70, after paying col- lector’s commission of 2%, if the assessed value of the property is $321500? Note. — T he tax colleotor receives his commission upon the gross amount of tax collected. 7. Mr. Brown was assessed as follows : Real estate, $30000 ; personal prop- erty, $3500 ; money at interest, $25000; income from occupation, $3000; and two gold watches. He obtains an abatement of one-third on real estate, one- fourth on personal property, $3000 on money at interest, two-fifths for occupation, and one gold watch. The tax rate was 2J mills, and $1.50 for each watch. What was Brown’s tax after the abatement, and how much was it lessened ? 1764 to be levied. 3600 3528 7200 7 056 1440 882 5580 5292 2880 2646 DUTIES 787. Duties or customs are taxes levied by the Government upon imported goods for revenue for the support of the general government and for the protec- tion of home industries. 788- Duties are of two classes — ad valorem duties and specific duties. 789. An ad valorem duty is a tax assessed at a certain per cent, on the cost of the goods in the country from which they are imported. Note. — Ad valorem duties are computed on the invoice cost of goods “ packed and ready for ship- ment,” exclusive of subsequent expenses, such as freight, insurance, etc. In custom house calculations, duties are not reckoned on fractions of a dollar ; a fraction of a dollar if less than one-half is rejected, if over one-half is counted as another dollar. 790. A specific duty is a tax assessed at a certain sum per pound, ton, gallon, foot, yard, or other weight or measure, without regard to value. 791. Tare is an allowance made, in estimating specific duties, by way of deduction from the gross weight of goods on account of the weight of the box, cask, etc., in which they are contained. 792. Leakage is an allowance made for waste of liquids in barrels or casks. 793 Breakage is an allowance for waste of liquids in bottles. 794. To find ad valorem duty. /3S3 72 & 3 0 2 f / O // OS, f 3 3 tr r o /3 S 3.3 07 f Note. — T ake duty on dollars only, dropping cents if less than 50, adding $1 if 50 cents or more. 795. Rule. — Find value of goods in United States money and multiply by rate of duty. Note. — Specific duty is simply a tax on quantity and is found accordingly. Ad valorem duty is always expressed at a certain rate per cent. WRITTEN PROBLEMS 796. 1. What is the duty on an invoice amounting to 12673.10 francs at 40% ad valorem ? 2. Find the duty at 7 cents per pound on an invoice of goods weighing 32478 lbs., tare 2%. Example. — What is the duty on an invoice of goods invoiced at £72 12s. at 30% ad valorem ? 243 244 DUTIES 3. What is the duty on an importation of goods invoiced at 12643 marks at 40% ? Ip. Find the duty on goods invoiced at 7846 francs at 60%. 5. At 50%, what is the duty on an invoice amounting to 6955 lire? 6. Find the duty on goods invoiced at 23632 florins at 25%. 7. Find the duty on an invoice of £593 17s. 10d. at 45%. 8. At 35 cents per gallon, what is the duty on 100 cases of olive oil, each case containing 2 dozen quarts? 9. Find the duty on 40 blocks of marble, each 1J X 2 X 4 ft. at 50 cents per cubic foot. 10. At 10%, what is the duty on an invoice of 12684 lbs. of leather, im- ported from Liverpool at Is. 2 ^d. per pound? 11. At 35%, what is the duty on an invoice of chinaware from Paris, valued at 14274 francs? 12. Find the duty on 300 dozen penknives, valued at 3s. per dozen, at 25 cents per dozen and 25% ad valorem; and 400 dozen penknives, valued at 4s. 6d. per dozen, at 40 cents per dozen and 25% ad valorem. 13. What is the duty on 26548 lbs. of tobacco at 40 cents per pound ? lip Find the duty on 14400 spools of cotton thread at 5J cents per dozen spools ? 15. At 30 cents per square yard, what is the duty on 5694 yards of cloth 27 in. wide? 16. At 25%, find the duty on an importation of goods from Buenos Ayres, invoiced at 7642 pesos. 17. At 30%, find the duty on an importation of goods from Havana, invoiced at 12638 pesos. 18. At 40%, find the duty on an importation of goods from Montevideo, invoiced at 9847.60 pesos. 19. At 20%, find the duty on an importation of goods from Valparaiso, invoiced at 32624.80 pesos. 20. At 30%, find the duty on an importation of goods from Bogota, invoiced at 2978.25 pesos. 21. What is the duty on an importation of goods from Paris, invoiced at 38734 milreis at 35% ? 22. What is. the duty on an importation of goods from Oporto, invoiced at 19875 milreis at 45% ? 23. What is the duty on an importation of goods from Shanghai, invoiced at 24653 taels at 30 % ? 2 Ip. What is the duty on an importation of goods from Yokohama, invoiced at 64882 yen at 25% ? 25. What is the duty on an importation of goods from Constantinople invoiced at 527643 piasters at 20% ? RATIO AND PROPORTION 245 36. Make the extensions and calculate the duty on the following invoice : 876J yds. Broadcloth 278f “ Discount 2 % Boxes and packing Cartage Duty, 50 %. 15s. 8 d. IS s. 6d. £ s. 17 6 d. RATIO AND PROPORTION RATIO 797. Ratio is a measure of relation between quantities of the same kind There are two kinds of ratio — arithmetical and geometrical. 798. Arithmetical ratio expresses the difference between two quantities. 799. Geometrical ratio is the division of one term by another. The usual way of expressing a geometrical ratio is by placing two points, one above the other, between the quantities compared. Thus 2 : 3 signifies the ratio of 2 to 3, and is read 2 is to 3. The quantities compared are called the tervis of the ratio. The first is called the antecedent, the second the consequent. 800. The value of a ratio is found by dividing the antecedent by the conse- quent. Since the antecedent of the ratio is always the dividend and the conse- quent the divisor, the ratio may be written in the form of a fraction, the antecedent being the numerator and the consequent the denominator. It follows that the terms of a ratio may be multiplied or divided in the same manner as the terms of a fraction, without changing the value of the ratio. 801. A simple ratio consists of one antecedent and one consequent ; as, 10 : 12. 802. A compound ratio consists of two or more simple ratios ; as, 9 : 12 | 8 : 14 i ‘ 803. The value of a compound ratio is found by dividing the product of the antecedents by the product of the consequents ; thus, in the above illus- tration, 3 2 9 X 'f> 3 7 P 7 246 RATIO AND PROPORTION 804. To find the value of a simple ratio. Example. — What is the ratio of $1000 to $300? $1000 : $300 - = ty- or 3*. 805. Rule. — Divide the antecedent by the consequent. ORAL EXERCISE 806. 1. What is the ratio of 9 : 6 ? 9 : 6 = f or 11. Find the values of each of the following ratios: 2. 5 : 8. 5. 15 : 5. 8. 25 : 6. 11 . 4f : 2|. 3. 7:6. 6 . 3J : 2R 9. 151 : 4J. IS. 8i : 171 4. 12 : 26. 7. 17 : 4. 10. 6i : 9|. 13 . 5f : 14f. 14 . $8 : $10. horses horses mo. da. mo. da. men men 17. 54 : 18. 19. 4 6 : 8 12 15 . 27 : 9. bu. bu. mi. mi. yd. yd. 18. 3J : 20. 20 . 25 : 75. 16. 16f : 33 J. MENTAL PROBLEMS 807. 1 . A pole 60 feet long is sunk 15 feet in the earth ; what part of it is in the air ? Solution. — T he part in the earth, 15 feet, is 1 of 60 feet, the length of the pole ; hence \ of the pole is in the air. 2. Two men engage in a business venture, the gains or losses to be shared in proportion to money furnished ; A puts in $60 and B $90 ; how should they share the gain ? 3. In a mixture of clover and timothy 30 pounds are clover and 10 pounds are timothy; what is the proportion of each in the mixture? J. In a blend of coffee 39 pounds are Java and 13 are Mocha ; what is the proportion of each in the blend ? 5. A farmer made an insecticide of 99 parts water and 1 part London purple ; what is the ratio of the London purple to the water? Of the water to the mixture ? 6. If Java and Mocha are blended in the proportion of 4 to 1, how many pounds of each in a mixture of 60 pounds? 7. In a. blend of tea, the black is to the green as 5 to 3 ; how many pounds of each in a mixture of 24 pounds ? 8. In a certain school, the boys are to the girls as 8 to 5 ; how many of each, the total number on roll being 52? 9. Doe’s money is to Roe’s as 7 to 5 ; how much has each if both have $96 ? 10. In a mucilage mixture 3 pounds of water are used and 4 pound of gum- arabic; how much of each must be taken to make 70 pounds? RATIO AND PROPORTION 247 PROPORTION 808- A proportion is the expression of equality between two ratios. Proportion is indicated by placing four points between the ratios — thus, 6 : 12 : : 24 : 48. 809. The first and last terms of a proportion are called the extremes ; the second and third, the means. 810. In any proportion, the product of the extremes equals the product of the means. It follows from this that either extreme is found by multiplying the means and dividing this product by the given extreme; or, either mean is found by multiplying the extremes, and dividing by the given mean. 811. A simple proportion expresses the equality of two simple ratios. 812. To find the fourth term of a proportion. Example 1 — 25 : 42 : : 70 : ? 14 42 X ,70 588 =— = 11 Ip o 5 Example 2— § : : : ££ : ? 2 0 8 X11X21 1 2 ,4X22 2 Note — Since tlie first term is the divisor, we simply invert it and multiply. 813. Rule. — Divide the product of the means by the first term. Note. — The operation may frequently be abridged by cancelation. 814. WRITTEN EXERCISE 1. 17 : 32 : : 51 : ? 7. 7 : 4i : : 94 : ? 2. 1 5 1 6 . 7 . . • 64 • ’ f :? 8. 15 : 27 : : 8 : ? 3. 9 : 24 : : 7 : ? 9. 144 : 5 : : 94 : ? 4- 33 : 46 : : 3 : ? 10. 3 : 7i : : 21 : ? 5. 5 : 7 : : 9 : ? 11. 4 : 26 : : : 9 : ? 6. 74 : 134: :3J:? 12. 7f : 15 • 73.9 815. To determine the statement of a simple proportion. Example 1. — If 15 men can do a piece of work in 22 days, in how many days can 24 men do the same work ? If 15 men can do the work in 22 days, 24 men can do the work in less time ; in this case the smaller term of the complete ratio is placed as the second term. The proportion is 24 : 15 : : 22 : ? 5 11 pX22 55 ~ = — or 13f days. 2£ 4 J 4 248 RATIO AND PROPORTION Example 2. If 19 men can earn $410 in a given time, how much could 28 men earn in the same time? If 19 men can earn $410, 28 men can earn more ; in this case the larger term of the complete ratio is placed as the second term. The proportion is 19 : 28 : : 410 : ? 28X410 11480 19 19 or $604.21 816 . R ule. — Place the term which is of the same kind as the required result as the third term. Then from the conditions of the problem, determine whether the fourth term, or result, should he more or less than the given third term. If the fourth term should be less than the third , place the smaller of the remaining terms second ; if more, place the larger of the remaining terms second. WRITTEN PROBLEMS 817. 1. If 29 tons of coal cost $130.50, what would 35 tons cost at the same rate ? 2. If 32 horses can do a given amount of work in 46 days, in what time could 54 horses do the same amount of work ? 3. If $326 produce $1980 interest in a given time, how much interest would $1920 produce in the same time at the same rate? f. If $49.75 is the interest of a given principal at 5%, what is the interest on the same principal for the same time at 1\% ? 5. If $462 produce $29.75 interest in a given time, what principal for the same time and rate will produce $92.64 interest? 6. If 15f- pounds of butter cost $5.04. what will 23§ pounds cost at the same rate ? 7. If the liabilities of a bankrupt amount to $11726, and the assets to 89427, how much should A receive to whom he owes $1247 ? Note. — T he entire liability is to the liability of each creditor as the entire assets are to that creditor’s share. 8. The collectable accounts of an insolvent merchant are: notes, 84675 ; personal accounts, $11625 ; he has $2362 worth of stock on hand. His liabilities are : notes, $13654 ; and other debts amounting to $15065. How much can he pay A to whom he owes $1290; B, to whom he owes $2265, and C, to whom he owes $4475 ? 9. The total assessment of a borough is $3457500 ; the tax to be raised for 1909 is $14750. How much of this should a property holder pay whose property is assessed at $18500? Note. — T he total assessment is to each individual assessment as the tax to be raised is to the amount to be paid by that individual. 10. The total tax to be raised by a certain town for the year 1908 is $17630. The assessed valuation of the town is $4724000. What tax should A pay whose property is assessed at $7500 ; B whose property is assessed at $12300 ; and C whose property is assessed to the amount of $17500? RATIO AND PROPORTION 249 11 . A and B form a partnership. A invests $12500 and B invests $16750. They gain during the year $7250. Find each partner’s share of the gain. Note. — The total investment is to each partner’s investment as the whole gain is to each partner’s gain. 12 . Three boys buy a bicycle, each paying as follows : the first $36, the sec- ond $45, the third $49. How long should each use it every day of ten hours? How long should each keep it in a period of 3 weeks? 13 . If § of a yard of silk cost $1.85, what should If yards cost at the same rate? 14 -. If a merchant gains 16§ % by selling goods for $14, what would be his gain per cent, by selling them for $17 ? Suggestion. — The selling price contains cost and gain, hence, before a proportion can he formed, the proper conditions must be obtained. Remark. — The student should carefully observe the conditions of problems to see that they are proportional. Some of these problems are formed with the object of testing the student’s percep- tion in this respect, and in others, the operation of proportion is only incidental and something remains to be done to obtain the required result. This is very frequently the case in the actual arithmetic of business. 15 . If by selling goods for f of the marked price I gain f of the cost, what part of the cost would I gain by selling them for f of the marked price? 16 . If 25 men can do a piece of work in 34 days, in what time could 42 boys do the same work, if 7 boys can do as much as 5 men ? 17 . A works 15 hours at a certain job, B 22 hours, C 28 hours. If the amount received for the work is $42.50, what should each receive? 18 . A traveler purchased a railroad ticket for $24. To get to his destination he travels over 125 miles of one company’s road, 216 miles of another and 475 miles of a third. Find the sum due each road. 19 . If 42 men or 54 boy's can do a piece of work in 24 days, how long would it take 18 men and 26 boys to do the same work ? 20 . If 44f yards of cloth cost $145.03, what should 37f- yards cost? 21 . If the freight on a car load of coal from one point to another is $8.75, and it is carried 36 J miles on one company’s road and 46-J miles on another, what share of the freight should each receive? 22 . If by selling goods for $97 there is a loss of 15%, what would be the per cent, of loss or gain by selling them for $102 ? 23 . If a grocer’s pound weight is f of a pound light, what would his sales amount to in a week, if during this time he has defrauded his customers of $52? 21 ^. If I buy coal for $4.25 a long ton, and wish to sell by the short ton so as to gain 25%, what must I ask per ton? 250 KATIO AND PROPORTION COMPOUND PROPORTION 818. To find the value of a compound ratio. Example. — What is the value of 9 : 6 6 : 18 ?X0 1 2 2 819. Rule. — Divide the product of the antecedents by the product of the consequents. ORAL EXERCISE 820. Find the value of the following compound ratios : 1. 4 : 6 5 : 9 l. 2 . 4* 3. 8 : 12 15 : 9 74:14 15 : 4 8 : 12 r \ r 5. 6f : CO 124 : CO 5 : 8 6. 7 : 6 men men 6 : 8 7. da. da. 5 : 6 $4: SI 2 8. hr. hr. 8 : 10 9. 10 . 11 . 12:7 1 acres acres / 34 : 6 J 12 . rods rods 12 : 15 men men 8 : 6 821. A compound proportion is the expression of equality between ratios, one of which is compound. Thus: 4:5 v 8 : 12 l 35 : 15 j 25 : 20.V Note. — The fourth term in a compound proportion is found by multiplying the means, and dividing this product by the product of the extremes. 5X42X 15x25 . 8X# — 822. To find the fourth term in a compound proportion. Example.- 3f : 44 8 134 9 13# 32 . ? °9 ’ • 4xSx 2x9x5x68x29 15x7x27x2x9X 5X 9 0 12074 “'25 5 15 RATIO AND PROPORTION 251 823. Rule. — Multiply together the means, and divide by the product of the given extremes. WRITTEN EXERCISE 824. Find the fourth term in each of the following : 1. 4 : 9 7 : 6 8 : 5 1 20 : 21 J : : 9 : ? 9 : 14 ) 7:6 > : : 24 : ? 15: 36 i 3. 4- 74 : 35 1 3i : 12 - : : 6 :? 5f : 22 ) 14:26 1 8:28 l • • 71 7J : 45 { ' ' ‘ 2 3| : 15 j 14 : 26 25 : 65 5. 33 : 72 ■ 25 : 40 I 125 : 140 J 44:18 ) 6 - 5i : 16 V 6f : 42 J : : 16 :? : : 15 : ? 825. To determine the statement of a compound proportion. Example. — If 14 men, working 9 hours a day, can build a wall 27 feet long, 18 inches thick and 28 feet high in 18 days, in how many days of 10 hours each could 22 men build a wall 38 feet long, 2 feet thick and 12 feet high ? Statement : 22 : 14 1 10 : 9 27 : 38 18 : 24 28 : 12 f 1 14 22 : : 18 :? brs. 9 10 long 27 ft. 38 ft. thick high da. 18 in. 28 ft. 18 2 ft. 12 ft. ? 2 19 3 4 7 4X9X^X2 4 Xf 2x,18 22x49X2, 7x4$X2$ 11 5 9 J[ 826. Rule. — Write as the third term the quantity which is of the same kind as the result. Then, as in simple proportion, determine whether the answer depending upon each ratio should be more or less than the third term ; if more, place the larger term of ratio nearest tlte third term ; if less, place the smaller term of the ratio nearest the third term. Proceed in this way, reasoning with each ratio entirely independent of the others, until all have been disposed of. WRITTEN PROBLEMS 827. 1. If the interest of $367 at 5 % for 4 years, 7 months, 18 days, is $85.02, what is the interest of $19.72 at 4|% for 2 years, 5 months and 18 days ? 2. If the freight on 32 barrels of sugar for a distance of 54 miles is $4.25, what should be the freight on 63 barrels for 28 miles ? 3. If a 3-pound loaf of bread cost 10 cents when flour is worth $4.75 a barrel, what should be the price of a 4-pound loaf when flour is worth $5.85 a barrel ? 252 RATIO AND PROPORTION If 15 apples can be bought for 25 cents when apples are worth §4.25 a barrel, what should be the cost of 45 apples when a barrel costs §4? 5. If the tax on a property assessed at $2700 is $128.25 at a tax rate of 4f %, what should be the tax on a property assessed at $3800 if the rate is 2J% ? 6. If I pay $2.25 a yard for cloth f of a yard wide, what should I pay for 22 yards of a similar quality J of a yard wide ? 7. If 26 men, working 10 hours a day, can dig a ditch 142 feet long, 4 feet wide and 12 feet deep in 96 days, in how many days of 8 hours each could 92 men dig a ditch 108 feet long, 3 feet wide and 16 feet deep, if the digging in the second case is estimated to be 20% harder? 8. If 4 horses, working 9 hours a day, can plow a field 372 feet long, 348 feet wide, in 2J days, the walking gait of the horses being 4 miles an hour, in bow many days, working 10 hours a day, could 6 horses, whose walking gait is 3 miles an hour, plow a field containing 14 acres, 142 perches? 9. If $765 produce $144.59 interest in 2 years, 8 months and 12 days at 7 %, what principal would produce $146.75 in 3 years, 2 months and 18 days at 5% ? 10. If $1365, in 3 years, 9 months and 14 days, at 6%, produce $310.31 interest, at what rate would $1065 produce $78.41 in 1 year, 7 months and 19 days? 11. If $1426, at 5%, will produce $163.99 interest in 2 years, 3 months and 18 days, in what time will $1820 produce $726.80 interest at 44% ? 12. If a pile of wood 18 feet long, 4 feet wide and 9 feet high cost $22.50, what should be paid for a pile 21 feet long, 3 feet wide and 11 feet high ? 13. If I pay $42.25 for a pile of wood 16 feet long, 5 feet wide and 7 feet high, how wide should a pile be to be worth $50.50 if it is 14 feet in length and 8 feet high ? 14 -. If there are 20 perches of masonry in a wall 33 feet long, 5 feet high and 3 feet thick, bow many perches are there in a wall 32 feet long, 14 feet high and 2 feet thick? 15. If a wall 122 feet long, 12 feet high, 2| feet thick contains 148 perches of masonry, how high should a wall 136 feet long and 3 feet thick be to contain 136 perches? 16. How many Belgian blocks, 3J inches wide, 7 inches long, will be required to cover a space 36 feet wide, 88 feet long, if 5120 blocks, 9 inches long, 4J inches wide, are required to cover a space 32 feet wide and 45 feet long? 17. How many hours a day must 64 men work to dig a canal 2 miles long, 24 feet wide and 7 feet deep in 528 days, if 42 men, working 10 hours a day, can dig a canal 1J miles long, 26 feet wide and 6 feet deep in 610 days? 18. If 45 tons of hay will be sufficient for S5 horses 96 days, how many tons must be added to be sufficient for 125 horses 112 days? 19. If 3 men or 5 boys can bind 5 acres of oats in 2J days, how many acres of the same average yield could 5 men and 7 boys bind in 3f days ? ALLIGATION 828. Alligation, or medial proportion, is the process of combining two or more quantities of different values so as to make a combination of a given mean value. 829. To find the average value when the values of the several ingredients are given. Example. — A grocer mixes 45 pounds of tea worth 50 cents a pound, 85 pounds worth 60 cents, 125 pounds worth 75 cents, and 90 pounds worth 81- Find the average price. 50 cents X 45=822.50 60 cents X 85= 51.00 75 centsXl25= 93.75 $1.00 X 90= 90.00 $257.25 value of the 45 lbs. “ « “ 85 • “ “ “ « 125 “ “ “ “ 90 “ 345 $257.25 - 4 - 345 = 74^-f cents, value of 1 lb. 830. To find the amount of the several ingredients taken when their respective values and the mean or average value are given. Example. — It is required to know how many pounds of coffee at the follow- ing prices, 25 cents, 32 cents, 38 cents, 40 cents and 45 cents, may be taken to form a mixture worth 35 cents a pound. 1 lb. at 25 cents 6 “ “ 32 “ 1 “ “ 38 “ 3 » « 40 « 1 “ “ 45 “ )■ Answers. Explanation, — Arrange the prices of the different ingredients in a vertical column. Then link one that is above the average price with one that is below it. In the operation given above, we first linked 25 cents with 45 cents. Now, if one pound at 25 cents be mixed and is then worth 35 cents, there is a gain of 10 cents, or to gain one cent we take -fe of a pound ; and if one pound at 45 cents be mixed so that it is worth 35 cents, we lose 10 cents, or to lose one cent, we take of a pound. The loss of one cent on T \j of a pound at 45 cents is balanced by the gain of one cent on r q of a pound at 25 cents. In the same way we reason with each of the other combinations, and find next that the loss of one cent on 1 of a pound at 40 cents is offset by the gain of one cent on 1 of a pound at 32 cents, and that the loss of one cent on J of a pound at 38 cents is offset by a gain of one cent on 1 of a pound at 32 cents. Reducing these several sets of results to whole numbers and carrying out the results, we have the answers given above. 253 254 ALLIGATION Integral Method g. and 1. on 1 lb. of 1 and 5 g. and 1. on 1 lb. of 2 and 4 g. and 1. on 1 lb. of 2 and 3 Amounts of 1 and 3 g. and 1. equal Amounts of 2 and 4 g. and 1. equal Amounts of 2 and 3 g. and 1. equal Amount of each Pboof. 25 10 1 lb. 1 lb. @ 25c. = §0.25 32 3 3 5 lbs. 1 lb. 6 lbs. @ 32c. = 1.92 38 3 1 lb. 1 lb. (& 38c. = .38 40 5 3 lbs. 3 lbs. @ 40c. = 1.20 45 10 1 lb. 1 lb. @ 45c. ■ .45 12 lbs. )§4.20 Average price, §0.35 Explanation. — Arrange the prices and link as before. Now, if one pound at 25 cents is mixed so as to be worth 35 cents, there is a gain of 10 cents, and if one pound at 45 cents is mixed so as to be worth 35 cents, there is a loss of 10 cents ; again, if one pound at 32 cents is mixed so as to be worth 35 cents, there is a gain of 3 cents, and if one pound worth 40 cents is mixed so as to be worth 35 cents, there is a loss of 5 cents ; likewise, if a pound worth 38 cents is mixed so as to be worth 35 cents, there is a loss of 3 cents, and if a pound worth 32 cents is mixed so as to be worth 35 cents, there is a gain of 3 cents. In order to have the gain and loss equal, one pound of the 25-cent kind (gain 10 cents) must be taken as often as one pound of the 45-cent kind (loss 10 cents) ; 5 pounds of the 32-cent kind (gain 15 cents) as often as 3 pounds of the 40-cent kind (loss 15 cents) ; and one pound of the 38-cent kind (loss 3 cents) as often as one pound of the 32-cent kind. Carrying out the results we have the answers as above. 831. To find the number of each ingredient when one or more of the ingredients are limited. Example. — It is required to know how many gallons of alcohol 65% proof and how many gallons of water must be mixed with 90 gallons of alcohol 80% proof to produce alcohol 60% proof. r 65 x 12 12 gals., 65% 80 ) i 2 0 3x30 = 90 90 ^00 ) ¥ (T 1 To 1 1X30 = 30 31 gals, water Explanation. — Proceed as in Art. 830, and multiply the number obtained of the limited quan- tity by such a number as will produce the required amount of that quantity. Multiply also by the same number the quantity obtained of the kind to which it was linked, and carry out the results. ALLIGATION 255 832. To find the number of pounds of each ingredient when the total amount is limited. Example. — A grocer wishes to fill an order for 680 pounds of tea which shall cost him 55 cents a pound. To produce this grade of tea, he blends the following grades: 40 cent, 50 cent, 75 pounds of each does he take ? cent, 90 cent and $1.00. How many 3X40 = 120 at 40c. 7 4 11X40 = 410 at 50c. 1 1X40 = 40 at 75c. 1 1X40= 40 at 90c. 1X40 = 40 at $1 00 17 17)680(40 68 0 Integral Method g. and 1. on 1 lb. of 1 and 5 g. and 1. on 1 lb. of 2 and 4 g. and 1 on 1 lb. of 2 and 3 Amt. of 1 and 5 g. and 1. equal Amt. of Amt. of 2 and 4 2 and 3 g. and 1 . i g. and 1 . equal eqnal Relative Amt. of each Average of lbs. Proof. 40 15 c. 3 lbs. 3 120 @40c.= 4800 50 5 5 7 lbs. 41bs. 11 440 @ 50c. =22000 55 ■ 75 20 1 lb. 1 >40X = 40 @75c.= 3000 90 35 1 lb. 1 40 @ 90c. = 3600 1.00 45 c. 1 lb. 1 40 @1.00= 4000 17 X 40 = 680 ) 37400 55c. Explanation. — P roceed as iu Art. 830. Divide the number of pounds required by the tota number obtained by comparison, and multiply each quantity by this number. WRITTEN PROBLEMS 833. 1 . What is the average price of the following mixture of tea: 95 pounds at 38 cents, 150 pounds at 45 cents, 68 pounds at 95 cents and 75 pounds at $1.00 ? 2 . In a certain school there are 150 children 6 years of age ; 105 children 7 years of age; 96 children 9 years of age ; 168 children 11 years of age, and 214 children 12 years of age. Find the average age of the children of the school. 256 ALLIGATION 3. The average price of a certain blend of tea, consisting of teas worth 50, 60, 65, 75, 80, 85, 90, 95 cents and $1, is 80 cents. Find the number of pounds of each kind required to produce this grade of tea. If. A broker bought 221 shares of stock (par $50) at an average premium of 10% ; he bought some at 6% discount, some at 2% discount, some at a pre- mium of 16% and some at a premium of 20%. How many shares of each kind did he buy ? 5. A merchant mixed 95 pounds of coffee worth 35 cents with coffees worth 40, 42 and 48 cents a pound. How many pounds of each did he take, if the average price was 45 cents a pound? 6. What relative quantities of alcohol 62%, 68%, 89%, 92%, 96% and 98% proof, are required to form alcohol 85% proof? 7. A grocer has an order for 900 pounds of tea, the price to be 80 cents a pound. To fill the order he mixes the following kinds: 100 pounds at 42 cents, some at 50 cents, some at 90 cents and some at $1. How many pounds of each kind does he take ? 8. I want to fill an order for 1000 pounds of coffee which I have agreed to sell at 66 cents a pound, a gain of 10%. I use 200 pounds at 42 cents, some at 55 cents, some at 70 cents and some at 80 cents. Find the number of pounds ■ of each kind. 9. I want to mix wine costing $2.50 a gallon with 90 gallons of wine cost- ing $3.00 a gallon and with water, so that I may be able to sell the mixture for $2.40 a gallon and gain 20%. How man}^ gallons of water and how many gallons of the wine at $2.50 must I take ? 10. A feed merchant mixes bran worth 95 cents a hundred, middlings worth $1.05 a hundred, cracked corn worth 82 cents a hundred, with oats worth 62 cents a bushel of 32 pounds. How many pounds of each of these ingredients shall he take to fill an order for 3 tons (2000 pounds each) at an average price of $20 a ton ? 11. I want to mix coffees worth, respectively, 28, 32, 35, 38 and 40 cents a pound so as to be able to sell for 40 cents a pound and gain 10%. How many pounds of each kind must I take? 12. I have on hand teas worth 65, 70, 75, 80 cents, and $1 a pound. I have an order for 2500 pounds at 72 cents a pound. I use 250 pounds of the $1 tea. How many pounds of the remaining kinds shall I take? 13. A man has $190 in 10-cent, pieces, which he wishes to exchange for 2- cent, 5-cent, 25-cent and 50-cent pieces. How many of each kind will it take ? Ilf I want to mix alcohol 98% proof, 95% proof and 80% proof with water so as to get alcohol 75% proof. Find the number of gallons of each kind I must take, if in all I want 1000 gallons ? 15. From teas worth 48, 54, 72, 80 and 85 cents a pound I want to mix 1200 pounds of tea which I can sell at 80 cents a joound and gain 20%. By taking 150 pounds of the 85-cent tea, how many pounds of the remaining kinds must I take ? GENERAL AVERAGE 834. Average is of two kinds — general average and particular average. 835. General average arises when sacrifices have been voluntarily made, or expenditures incurred, for the preservation of a ship, cargo and freight, from some peril of the sea or from its effects. It implies a subsequent contribution from all the parties concerned, ratable to the values of their respective interests, to make good the loss thus occasioned. 836. Particular average signifies the damage or partial loss happening to the ship, goods or freight, by some fortuitous or unavoidable accident. It is shared by the persons whose property is destroyed or by their insurers. 837. The fundamental principle of general average is that a loss incurred for the advantage of all the coadventurers should be made good by all in equitable proportion to their stakes in the venture. 838. All general average losses may be divided into two principal classes — {1) sacrifices of part of the cargo and freight (as when part of a cargo is thrown overboard to save the ship from foundering in a storm), or of part of the ship (masts, etc.) for the general safety; (2) extraordinary expenditures incurred with the same object (as when a ship is obliged to put into a port of refuge in conse- quence of damage received in the course of the voyage). 839. Jettison is the throwing overboard of goods or cargo, in stress of weather or to prevent foundering. 840. Salvage is the compensation allowed to person* by whose voluntary exertions a vessel, her cargo, or the lives of those belonging to her are saved from danger or loss in case of wreck, capture, or other marine misadventure. 841. A general average loss may include the following items : (a) Jettison: damage to cargo by water getting down the hatches during jettison; damage by breaking or chafing after jettison; freight on cargo jetti- soned. ( b ) Sacrifices of ship’s materials, by the cutting away of masts, spars, rigging, etc. One-third of the cost of repairs to a vessel is a special charge on the ship, the new work being considered better than the old ; this leaves the remaining two-thirds only of such cost to be included in the general average. (c) Expense of floating a stranded ship ; salvage in general. (d) Expense of entering a port of refuge, whether disability were caused by accident or sacrifice. (e) Expense of discharging cargo to make repairs, reloading, etc. (f) Wages and provisions of crew from time vessel deviates from its course until it resumes its voyage. 257 258 GENERAL AVERAGE 842. The contributory interests are as follows: (a) The ship contributes on what was its full value before the loss. (b) The cargo contributes on its net market value at port of destination, including value of part jettisoned or damaged. (c) The freight contributes on its full amount, less £ for the wages, etc., of the crew. (In New York, and some other States, less J). 843. Insurers indemnify the owners of contributory interests for such proportion of the contribution of each as the amount insured on such contrib- utory interest bears to the full value of said contributory interest. 844. An adjuster is a person who apportions the losses and expenses of a general average. 845. To apportion a loss by general average. Example. — The schooner Swallow from Savannah for Philadelphia, with a cargo of cotton valued at $37800 and a deck load of pine lumber valued at $18760, encountered heavy gales off Cape Hatteras which made it necessary to throw overboard the deck load and cut away masts and rigging, after which the vessel was rescued by a steamer and towed into Wilmington, N. C. The salvage amounted to $15000 ; repairs to the vessel, $2871 ; wages and provisions of seamen from date of the disaster, $874.60 ; the other expenses, $1486.38. The value of the vessel was $12000 ; the total freight, $6438, of which $2132 was the freight on the deck load. Apportion the settlement. GENERAL AVERAGE LOSS CONTRIBUTORY INTEREST Jettison 18760 Vessel 12000 Freight on jettison 2132 Cargo of cotton 37800 Repairs (f of $2871) 1914 Cargo of lumber 18760 Salvage 15000 Freight ($6438 less ^) 4292 Wages, etc., of seamen 874 60 Other expenses 1486 38 Total 40166 98 Total 72852 $72852 : $40166.98 : : $12000 : Vessel’s contribution = $6616.20 $72852 : $40166.98 : : $37800 : Cotton’s contribution = 20841.05 $72852 : $40166.98 : : $18760 : Lumber’s contribution = 10343.33 $72852 : $10166.98 : : $4292 : Freight’s contribution = 2366.40 $40166.98 (receives) Adjuster’s Settlement (pays) Owners of cotton 20841 05 Owners of lumber 8416 67 Freight company 234 40 Repairs 2871 Owners of vessel 7573 20 Salvage 15000 Wages, etc., of seamen 874 60 Other expenses I486 38 2864S 65 28648 65 j GENERAL AVERAGE 259 Explanation of Adjuster’s Settlement. — The adjuster settles with the owners of the cotton by receiving from them their contribution to the general average, $20841.05. He settles with the freight company (the charterers of the vessel) by receiving from them the difference between their contribu- tion and the freight on jettison, $2366.40 — $2132 = $234.40. He settles with the owners of the vessel by receiving from them their contribution plus their £ of the repairs, $6616.20 -j- $957 =$7573.20. He settles with the owners of the lumber by paying them the value of the lumber jettisoned, less their contribution, $18760 — $10343 33 = $8416.67. He pays the full amount for repairs, $2871 ; for salvage, $15000 ; and other expenses, $1486.38. He pays the general average portion of seamen’s wages, etc., the charterers paying their regular wages. Of course, the charterers receive their freight charges from the owners of the cotton, and they pay the owners of the vessel for the charter of it, entirely aside from the general average settlement. 846 . Rule. — Find the total of the contributory interests. Find the total loss subject to general average. Calculate the proportionate contribution of each contributory interest by the following formula : Total interest : Total loss : : Each owner's interest : Each oivner's contribution. WRITTEN PROBLEMS 847 . l. A steamer having on board $47923 worth of goods shipped by A, $25437 worth shipped by B, and $11926 worth shipped by C, experienced such rough weather that $20624 worth of goods were jettisoned for the safety of the ship. Of the goods jettisoned, $8370 worth were from goods owned by A, and the remainder from B’s shipment. The vessel having been damaged, the repairs amounted to $2145.38, and the maintenance of the crew during delay was $537.80. The loss of freight on cargo jettisoned amounted to $2085.50 ; the freight paid on the remainder was $6249.80. The steamer, before starting on the voyage, was valued at $57500. Apportion the loss among the different contributory interests, and state how settlement should be made. £. The bark Bonnie Jean started from Glasgow to New York with an assorted cargo. During the voyage she encountered a severe storm, driving her out of her course and causing considerable damage. For the safety of the ship her masts and rigging had to be cast away ; also a portion of her cargo, valued at $2375. Bv rigging up jury masts, she was enabled to reach the port of St. John, N. B., for repairs. Replacing the masts and rigging cost $5438 ; repairing other damage $675. The freight on cargo jettisoned amounted to $435.25. Port charges for entering, with expense of discharging cargo, etc., amounted to $835.15 ; wages and provisions for crew, $715 ; adjuster’s fee, $175. The value of the vessel on arriving at New York was $28000 ; value of cargo delivered, $50860. The total freight (including freight on cargo jettisoned) was $17150. The con- signors of the cargo were : James Thompson, $9500 ; William Irvine, $21275 ; Alexander McPherson, $7325 ; Joseph Montgomery, $15135. The jettisoned goods were part of Irvine’s consignment. How ought the settlement to be made ? Note. — The value of the vessel on arriving at New York being $28000, to find the value before it was damaged (which will be the value upon which its contribution is computed) deduct one third of the charges for repairs, this being reckoned as the superior value of the new material. That is to say, the vessel, after being repaired, is worth that much more than before the damage was sustained. 260 GENERAL AVERAGE 3. James Thompson’s consignment was insured for $8000, and Joseph Montgomery’s consignment for $12000. How much did each receive from the underwriters? J. During a storm off the cost of Maine, the schooner Sarah Jane , valued at $25000, having a cargo consigned to Frazer & Co., Baltimore, had to jettison $5000 worth of the cargo for the general safety. The cargo amounted to $50500, part of which was consigned by Jones & Brown and invoiced at $15750, of which $3000 was jettisoned. The remainder of the jettison, $2000, was taken from goods consigned by Bradley & Co. By the shifting of the cargo, and the enter- ing of water through the hatchway while throwing the goods overboard, the remainder of Jones & Brown’s consignment was damaged to the extent of 1\%. Freight expected on cargo was $12500, including that on jettison, $315. The adjuster’s fee was $175. How much do Jones & Brown and Bradley & Co. each contribute in the general average? 5. If Bradley & Co. carried $30000 insurance on their consignment, how much do they receive from the insurance company? 6. The brig Arcotta — owned by Jacob Orr, Win. Coates and Henry Fox, general traders, whose respective interests in the vessel were, Orr J, Coates f , and Fox J — having delivered cargo at several of the principal seaports of the Medi- terranean, shipped a cargo of oranges and lemons at Messina for the return voyage, each of the owners putting in his own venture. The value of the cargo was $43575, which was as follows : Jacob Orr $14525, William Coates, $7262 50. Henry Fox $9078.13, and a consignment by Wessenberg& Co. to Penn Fruit Co., Philadelphia, $12709.37. On the homeward voyage the brig was struck by a violent storm, during which the vessel was damaged to a considerable extent, Some of the cargo had to be thrown overboard, and having sprung a leak, the vessel had to put into Bordeaux for repairs. Expenses for repairing loss and damages, port charges for entering, wages and provisions for crew while detained, unloading, reloading, etc., amounted to 7532.75 francs, which was paid by the owners, Wessenberg’s share to be collected from the Penn Fruit Co. Of the total expenses, 1236.20 francs was the amount paid for repairing damage to ve-sel caused by the storm.* The goods jettisoned were valued at $8500 — $4500 being from J. Orr’s consignment and $4000 from Wessenberg’s consignment. Freight calculated at $8400, including $345 freight on jettison. Value of brig in Philadelphia, $25000. Adjuster’s fee, $225. How was settlement made? 7. The cargo of the brig Arcotta was insured for $38000. How much did the insurance company pay, and how apportioned? * This item is, of course, a particular average loss to be shared by the owners. EQUATION OF ACCOUNTS 848. Equating an account is finding a date on which settlement of two or more debts, due at different dates, can be made without loss to the debtor or creditor. 849. The date thus found is called the average date or equated time. 850. It is necessary to assume some common date of comparison. This date is called the focal date. Remark. — Any date conceivable may be taken as a focal date, and interest may be computed at any rate per cent, without varying the result ; providing only that the dates of all items be compared with such focal date, and uniformity in rate and manner of computing interest be observed throughout. In practise it is well to observe a simple method, by assuming the latest date in the account as a focal date, computing all interest at 6% by the short method on a 360-day basis. 851. The term of credit is the time allowed for the payment of a debt after the date of contracting it ; if given in days, it is counted on from the date of pur- chase or sale, the exact number of days of the term; if given in months, it is counted on the number of months regardless of the number of days thus included. 852. The average term of credit is the average time allowed for the pay- ment, in one sum, of the total of two or more debts, due at different dates. Remark. — B ook accounts bear legal interest after they become due, and notes, even if not con- taining an interest clause, bear interest after maturity. 853. The importance of a thorough knowledge of both the theory and prac- tise of Equation of Accounts, on the part of bookkeepers and accountants, can hardly be overrated, as much of this class of work is to be found in every whole- sale and commission business. 854. The equity of the settlement of an account by equation rests in the fact that, by a review of such account, one of the parties owes the other a balance to which certain interest should be added or from which certain interest (dis- count) should be subtracted. 855. Accounts having entries on but one side, either debit or credit,, are appropriately called simple accounts; accounts having both debit and credit items may appropriately be called compound accounts. 856. It is the custom in many lines of business to charge interest on items that have matured. Brokers also charge customers interest for stock carried or allow it on deposits made. Private and state banks frequently allow interest on daily balances on deposit. These various business customs lead to interest cal- culations of special kinds, with which the business man should be familiar. There are several methods in use. 261 262 EQUATION OF ACCOUNTS 857. To find the average date of sales. Example. — In the following account find the average date of sales: on June 8, 1907, John Bowen sold to William Lorigan mdse, to the amount of §960 ; on June 16, to the amount of $1240 ; and on June 30 to the amount of $497. r /& 30 *9ru^ 2 / / 20 / 73 6>0 0 oooo J 3 & 4^ ' 267 00 472 00 10. Find when due by equation, and the amount that would settle the balance on the latest date of account. Dr. Beyer Brothers Cr. 1908 Jan. 3 Mdse., 30 da. 67925 j 1908 Feb. 3 Cash o o so Feb. 18 “ 30 da. 79420 Mar. IS (( 750 EQUATION OF ACCOUNTS 275 11. When is the balance of the following account due by equation ? Dr. Frank S. M islet Cr. 1908 Mar. 3 Mdse., 30 da. 796 14 1908 Apr. 1 Note, 60 da. 750 00 it 17 “ 2 mo. 1722 11 tt 28 “ 30 “ 850 00 tt 31 “ 60 da. 1946 75 May 1 Acceptance, 2 mo. 1500 00 May 19 “ 30 da. 472 11 June 26 Cash 390 00 IS. In the following, find the amount due April 26, 1908. Dr. J. L. Lippincott Or. 1908 Jan. 7 Mdse., net 826 54 1908 Jan. 15 Cash 675 00 tt 15 “ 30 da. 1692 74| tt 28 Note, 3 mo. 1000 00 Feb. 12 “ 2 mo. 1749 89 Mar. 18 Draft, 30 da. 1200 00 “ 25 “ 90 da. 1845 29 Apr. 22 Note, 4 mo. 2000 00 Mar. 8 “ 1 mo. 749 86| • 13. In the following, find the equated date, prove it, and then find the cash balance Sept. 15, 1908. Dr. John N. Chapman Or. 1908 1908 May 6 Casli 1000 00 May 6 Mdse., net 1620 70 it 20 Note, 2 mo. 1620 00 June 12 “ 60 da. 922 40 June 15 Acceptance, 60 da. 2240 00 Aug. 19 “ 2 mo. 1452 49 J uly 22 Cash 1490 00 “ 27 1 “ 4 mo. 2975 14 U- Find cash balance June 1, 1908 ; find also the equated date. Dr. John R. Cartman & Co. Cr. 1908 j: 1908 ! Mar. 9 Mdse., net 600 00 Apr. 1 Cash 550 00 tc 25 “ 60 da. 796 49 II May 2 Note, 10 da. 695 00 Apr. 14 “ 2 mo. 1472 74 15. Find cash balance April 15, 1909. Dr. James J. King & Co. Or. 1908 1908 Aug. 12 Mdse., 30 da. 796 41 Sept. 23 Cash 1000 00 Sept. 8 1 mo. 949 56 Nov. 17 Note, 2 mo. 1200 00 Nov. 17 “ 60 da. 1264 75 Dec. 22 Acceptance, 90 da. 750 00 Dec. 15 “ 2 mo. 694 86 1909 1909 Mar. 6 Note, 30 da. 1420 00 Feb. 10 “ 90 da. 1543 75 270 EQUATION OF ACCOUNTS 16. What is the cash balance due Jan. 27, 1909; when due by equation? Dr. John Mills & Co. Cr. 1908 1908 Sept. 5 To Mdse. 60 da. 758 90 Oct, 4 By Cash 1000 00 CC 18 CC u 30 “ 387 40 Nov. 7 cc cc 520 00 Oct. 7 CC cc 4 mo. 500 00 cc 30 “ Draft, 30 da. 100 00 cc 20 cc cc 10 da. 400 50 Dec. 18 Cash 650 75 Nov. 6 cc cc 3 mo. 515 70 1909 u 29 Ci cc net 400 00 •Jan. 15 By acceptance, 60 da. 58400 Dec. 8 cc a 90 da. 274 60 Feb. 9 “ Cash 478 90 CC 27 u cc 2 mo. 309 75 Mar. 10 CC CC 35040 1909 Jan. 14 cc cc 1 mo. 357 80 May 2 cc cc 150 00 Feb. 5 cc cc 30 da. 200 00 17. When are the proceeds of the following account sales due by equation? Philadelphia, Pa., Apr. 7, 190S. Account sales of flour sold for account of J. S. Harlington & Co. 1908 Feb. 11 420 bbl. at §4.10 CC 25 245 “ “• 4.62 Mar. 18 296 “ “ 4.48 CC 31 369 “ 4.32 Apr. 7 162 “ “ 4.25 Charges Feb. 14 F reight 89 15 CC 18 Cartage 16 75 Mar. 20 Remittance 200000 Apr. 7 Storage 100 00 CC 7 Cooperage 12 00 Commission 4% Note. — D ate the commission on the last date of sales, and equate the account in the ordinary way, taking the charges as debits and the sales as credits. EQUATION OF ACCOUNTS 277 18. Account sales of shoes Sold for account of Vm. B. Munich & Co. 1908 July 7 325 pairs at $2.10, cash (C 22 456 “ “ 2.55, 3 mo. Aug. 19 267 “ “ 1.75, 60 da. U 31 412 “ “ 2.18, cash Sept. 24 564 “ “ 2.95, 90 da. Charges July 9 Freight 102 00 Aug. 10 Dravage 5 60 U 26 Remittance 1000 00 Sept. 24 Storage 110 00 Commission 5% 19. Account sales of broadcloth Sold for account of John Detts’ Sons 1908 May 5 375 yards at $3.25, cash “ 20 642 (£ u 3.92, 60 da. June 17 895 U u 4.15, 3 mo. (( 30 268 U u 3.68, net. July 15 495 u u 4.10, 30 da. Aug. 14 722 u a 4.05, 60 da. Charges May 5 Inspection 9 00 May 19 Freight 59 20 June 2 Cartage 10 50 July 10 Remittance 2500 00 Aug. 14 Storage 90 25 Commission 3% Guaranty 2% Net proceeds due 30. Account sales of apples Sold for account of Joshua Half.y 1908 Oct. 8 228 bbl. at $2.25, cash U 22 245 “ “ 2.30, 10 da. Nov. 2 280 “ “ 2.40, 30 da. U 18 175 “ “ 2.28, 10 da. Charges Oct. 5 Freight and drayage 92 50 Nov. 18 Storage 18 56 Oct. 5 Advertising and Insurance 25 00 Commission 2J% Guaranty 2% Net proceeds due | ACCOUNTS BEARING INTEREST ACCOUNTS BEARING INTEREST Daily Balances The Frugality Sayings Company In account with John Wise. 1908 1908 July 1 To Balance 155 00 July 1 By Check 33 33 U 9 “ Cash 50 00 a 15 a a 15 00 a 23 a a 50 00 Aug. 1 a a 33 33 Aug. 6 £( u 50 00 a 15 a a 12 00 U 20 a a 50 00 Sept. 1 a a 33 33 Sept. 10 a u 50 00 U 15 a a 10 00 U 24 (i a 50 00 Oct. 1 “ Balance 319 70 Oct. 1 “ lilt. 1 69 456 69 456 69 1908 Oct. 1 To Balance 319 70 Operation Balances Days $121.67 from ■July 1 to J ul\ 9 8 $968 171.67 “ “ 9 U (i 15 6 1026 156 67 “ “ 15 a C 23 8 1248 206.67 “ “ 23 “ All g- 1 9 1854 1 73 34 “ Aug. 1 a 6 5 S65 223.34 “ “ 6 a u 15 9 2007 211.34 “ “ 15 a u 20 5 1055 261.34 “ “ 20 “ Sept 1 12 3132 228.01 “ Sept. 1 U u 10 9 2052 278.01 “ “ 10 u u 15 5 1390 268.01 15 o a 24 9 2412 318.01 “ 24 “ Oct 1 7 2226 $20235 Total daily balance, $20235 for 1 day Interest at 3% $1.69 ACCOUNTS BEARING INTEREST 279 Form with Balance Columns Penn Trust Company In account with Smith & Jones. Date. Dr. Cr. Daily Average. Days. 1908 Jan. 1 8000 00 8000 00 14 U 15 2000 00 10000 00 16 a 31 5000 00 5000 00 15 Feb. 15 6000 00 11000 00 13 U 28 4500 00 6500 00 15 JMar. 15 4 00 00 11000 00 16 31 5000 00 6000 00 1 20500 00 14500 00 Apr. 1 42 75 Bal. 601$ 75 20542 75 20542 75 Apr. T 6042 75 Interest at 2% on 1769500 for 1 day=$42.75. Total Balance. 112000 00 160000 00 75000 00 143000 00 97500 00 176000 00 6000 00 769500 00 878. It is to be noticed that the interest rates differ according to the cus- toms of various lines of business. Trust companies commonly pay 2% on daily balances on active accounts ; on those that require notice to withdraw money, it is common to allow 3% on daily balances. In stock brokers’ accounts with their customers, and on other running accounts bearing interest, the legal rate is charged, unless there is a special agreement upon some other rate. It sometimes happens that one side of an account bears one rate and the other side another rate ; as, where a broker ■charges 6% on debit balances and allows 3% on credit balances. If an item is not yet due at the day of settlement, it is usually adjusted by a contra entry ; that is, by adding the interest to the other side of the account. WRITTEN PROBLEMS 879. 1. Wh at is the net amount due on the following account July 1, 1908 ? Dr. I. M. Porter in account current with Vici Brothers. Cr. 1907 1907 • July 1 Balance 1200 00 ■ Nov. 1 Mdse., 3 mo. 600 00 Sept. 12 Draft 900 00 ;j 1908 1908 Mar. 14 “ 60 da. 600 00 -J an . 3 U 900 00! Apr. 25 “ 30 da. 850 00 May 16 u 400 00|j June 4 “ net 500 00 280 ACCOUNTS BEARING INTEREST 2. What is the balance of the following account May 1, 1908, interest at 6% ? Dr. Hill & Son in account with Grove Land Company. Or. 1908 Jan. 15 Draft 950 00 1908 Jan. 1 Balance 3500 Feb. 1 a 860 00 U 22 Sales at 3 mo. 1200 Mar. 1 u 1800 00 Feb. 14 “ 3 “ 700 “ 24 u 960 00 Mar. 2 “ 3 “ 2400 3. The City Trust Company allows 2% interest on daily balances of §100 or more. What is the balance to the credit of Charles Careful on December 31, 1908, after crediting him with the interest due? Deposits: March 5, 1908, $50; April 1, 1908, $40; April 15, 1908, $60; April 30, 1908, $50 ; May 14, 1908, $50 ; June 1, 1908, $35 ; June 15, 1908, $50 : June 25, 1908, $40; June 27, 1908, $35; July 16, 1908, $52.92; July 9, 1908, $62.50; July 30, 1908, $50; August 3, 1908, $38; August 16, 1908, §50; August 31, 1908, $50; September 15, 1908, $40; September 30, 1908, $40; October 15, 190S, $50 ; October 31, 1908, $50 ; November 25, 1908, $50. Checks: March 19, 1908, $12; April 9, 1908, $10; April 15, 1908, $25 ; April 21, 1908, $20 ; April 23, 1908, $12 ; May 2, 1908, $10 ; May 7, 1908, §11 : May 14, 1908, $25 ; May 20, 1908, $10 ; May 25, 1908, $10 ; June 10, 1908, $10 ; June 15, 1908, $25 ; June 20, 1908, $10 ; June 24, 1908, $20 ; July 1, 1908, $100 ; July 5, 1908, $33.33 ; July 20, 1908, $25; July 27, 1908, $7 ; August 1, 1908, $33.33; August 5, 1908, $20; August 10, 1908, $11.50 ; August 12, 1908, $10; August 13,1908, $10; August 17, 1908, $15; August 24, 1908, $15; September 1, 1908, $33.33; September 1, 1908, $10 ; September 1, 1908, $19.55 ; September 15, 1908, $10 ; September 21, 1908, $15; September 30, 1908, $3.25; October 3, 1908, $33 33; October 7, 1908, $16.80; October 8, 1908, $10; October 14, 1908, $10; October 21,1908, $12; October 26, 1908, $20; October 28, 1908, §8; November 2, 1908, $5; November 3, 1908, $33.33; November 6, 1908, $10: November 9, 1908, $4.70; November 12, 1908, $6. 880. To calculate the interest on a saving-fund account. WRITTEN PROBLEMS 881. In the Western Saving Fund Society of Philadelphia interest is allowed at the rate of three and one-half per cent, per annum on accounts of five dollars and over. No interest on fractional parts of a dollar. Interest is calculated by calendar months, and is allowed on deposits made on or before the fifth of the month ; in the case of withdrawals, interest is allowed for one-half month on money withdrawn on or after the sixteenth of the month. At the end of the calendar year the interest is reckoned, and is added to the principal. 1. William Trask deposited in the Western Saving Fund Society of Phila- delphia, as follows: January 2, 1907, $20; February 16, 1907, $24; March 14. 1907, $18; April 26, 1907, $60; May 31, 1907, $12 ; June 3, 1907, $15; July 19, 1907, $36; August 24, 1907, $72 ; November 14, 1907, $24; December 6, 1907, $50. ACCOUNTS BEARING INTEREST 281 On March 18, 1907, he withdrew $16; April 10, 1907, $15; September 9, 1907, $25; October 3, 1907, $15. What was the balance of his account January 1, 1908? 2 . Mary Jones deposits in the Western Saving Fund Society as follows: February 3, 1904, $20; May 6, 1904, $32; July 7, 1904, $18; October 22, 1904, $22; December 15, 1904, $25; February 10, 1905, $15; April 20, 1905, $17; June 30, 1905, $24: August 17, 1905, $12 ; September 26, 1905, $20 ; November 3, 1905, $15; December 20, 1905, $21; February 9, 1906, $16; April 14, 1906, $25; June 12, 1906, $18 ; August 22, 1906, $20 ; October 12, 1906, $16; Decem- ber 5, 1906, $25; January 26, 1907, $12; March 12, 1907, $26; May 18, 1907, $17; July 16, 1907, $30; September 18, 1907, $12; November 7, 1907, $21 ; December 20, 1907, $14 ; January 24, 1908, $10 ; March 30, 1908, $20. What is the balance due her on July 1, 1908? 3 . On January 1, 1907, William Smith’s balance in a savings bank, in which interest at 3% per annum is allowed on the smallest monthly balance and compounded quarterly, was $180. During the year he made the following deposits: January 18, $24; February 6, $16; March 4, $28; March 6, $14; March 30, $15 ; April 16, $13 ; May 1, $60 ; May 8, $24 ; May 16, $25 ; June 3, $12 ; June 20, $18 ; July 10, $18 ; August 1, $14 ; August 20, $13 ; September 3, $16; October 24, $28; November 5, $22; December 3, $30. He withdrew January 21, $25 ; February 28, $30 ; April 20, $60; August 20, $25 ; September 16, $12 ; October 31, $15; December 24, $12. What was the balance of the above account January 1, 1908 ? J. Mr. W. T. Zaner made the following deposits and withdrawals at a sav- ings bank, the by-laws of which provide for an allowance of interest at three per cent, per annum on the smallest balance for interest terms of three months ending, respectively, January 1, April 1, July 1, and October 1, interest to be compounded semiannually. (That is, a deposit begins to draw interest from the first day of each quarter, and only the balance left the entire quarter draws interest in the event of withdrawals.) He deposited January 1. 1907, $600 ; February 2, 1907, $145 ; February 15, 1907, $180; March 16, 1907, $160; March 28, 1907* $240; April 20, 1907, $160; May 1, 1907, $120; June 21, 1907, $180; July 16, 1907, $125; August 26, 1907, $280; September 2, 1907, $460; October 4, 1907, $240; October 18, 1907, $160; November 12, 1907, $180. He withdrew January 22, 1907, $100 ; February 28, 1907, $200 ; March 25, 1907, $200 ; May 20, 1907, $150; July 1, 1907, $50; October 23, 1907, $20; December 24, 1907, $200. What was due Mr. Zaner January 1, 1908 ? BANKRUPTCY 882. A bankrupt or insolvent is a debtor whose property is taken to be divided among his creditors by a court under the operation of an insolvent law. 883. An insolvent law provides that when a person has been judicially ascertained to be insolvent, and has surrendered his property for distribution among his creditors under a decree of court, he is to be discharged from his indebtedness. 884. An assignee in bankruptcy, or an assignee in insolvency, is a person to whom the property of a bankrupt or insolvent is transferred in order that he may dispose of the same for the benefit of the creditors. 885. A dividend is a sum arising out of the bankrupt’s available assets, which the assignee divides among the creditors pro rata. 886. To find each creditor’s share. Example. — The statement of a bankrupt’s affairs was as follows: Assess: Cash, $2870.13; Bills Receivable, $3750, Merchandise, $8439.70 ; Real Estate, $4200 ; Stocks and Bonds, $3430.75; Personal Accounts, $7833.48. Liabilities: Due A, $9238.17; clue B, $12375; due C, $20453.38; due D, $17216.85; due E, $4114.29. The expenses of the assignment were $917.34. What was the rate of dividend, and how much should each creditor receive? $2 270 . /3 r 37s o. 3 337 . 76 ? 2 20 O, 3 23 0.7 s 7 23 3. 32 O >2 2 3 • O 6 2/ 7.3 3 6 o 6.7 2 $723227 2237s. 2 0 2 S 3 . 32 / 7 2 / 7 . 7 S 3 / 63.2 7 3 3 y y .3 7 $276,0 7 . 72 -2 $63377.67 = .267^ $ 7 23 7. 7 7 7,267 = $23/ 2.23 /237S. 7.267= S777 ./ 2 73s 2 0333.37 X .267 = 7 SS/. 73 672 / 72/ 6.237.267 = S 020. 27 As 2/ / 2.277 ,267 = / 22/. 37 $27606.72 UT‘ , / 63377.67 : $7237/7 63377.67 : /237S. OO 63 3 77.67 : 2 0 333.37 63377.67 : / 72/ 6.73 63377.67 : 27/2.27 : : $2760 6.72 : 3 /^^ :: 27606 . 72 : 732 °,, :: 2760 6.72 : 62 v :: 27606 72 : 7h „ :: 2760 6 72 : & „ = 7/23/3.23 J 377722 = 7SS/.73 _ 7030.27 = Z72/.37 Note — U sually it is necessary to carry the decimal to eight places. 282 BANKRUPTCY 283 887. Rule. — Deduct expenses of the assignment from the total assets, and divide by the total liabilities. The quotient will be the rate of dividend. Then multiply the amount of each creditor's claim by the rate of dividend. Or, by proportion, Total liabilities: Each liability : : Net assets : Each creditor's share of assets. WRITTEN PROBLEMS 888 . l. If a firm fails with liabilities amounting to $60000 and assets $42000, how much should each creditor receive on the dollar — expenses of assign- ment being $1412? What sum should a creditor receive whose claim amounted to $6489.78? 2. A bankrupt’s assets are as follows : Cash, $1263.18 ; Merchandise, $6492.72 ; Bills Receivable, $3820; Personal Accounts, $9868.47. His liabilities are : Due to Jones, $20316.24; Brown, $11624.25; Smith, $7422.96; Clark, $3127.89. Expenses of settlement, $812.70. How much does each creditor receive? 3. After converting all the available assets into cash and paying all expenses, an assignee has in his hands $73462.37 to distribute among creditors whose claims amount to $104392.64. How much does A receive, whose claim is $21622.78 ? J. An insolvent merchant owes A, $3122.75 ; B, $2646.38 ; C, $9421.67 ; D, $248.90 ; E, $647.28 ; F, $427.32. His assignee has realized $9862.48 from the assets, and has $364.77 expenses to pay. How much will each of the creditors receive ? PARTNERSHIP SETTLEMENTS 889. A partnership is an association of individuals formed by an agreement made between them to combine their money, labor and skill in some business enterprise, the profits and losses of which are divided in certain proportions. This association is generally called a firm or house. 890. The capital of a partnership consists of the money, real estate, or any other property invested. 891. The resources of a firm consist of the money and other property it owns, and the debts due the firm. In theory, the firm is distinct from the indi- viduals composing it ; hence, a partner may owe the firm, and his obligation to the firm is a resource of the firm, as in the case of withdrawals. 892. The liabilities of a firm are the debts it owes. For accounting pur- poses the firm may be considered as owing the individuals composing the firm for their investment; hence, the capital of the business is a quasi liability of the firm. 893. The net capital of a firm, or present worth, is the excess of the resources over the liabilities. While the excess of resources over liabilities is called either net capital or present worth, still, strictly speaking, the former is the appropriate term to use when opening books and the latter when closing. 894. When a firm’s resources exceed its liabilities, it is said to be solvent. When the liabilities exceed the resources, the firm is said to be insolvent. 284 PARTNERSHIP SETTLEMENTS 895. The net insolvency of a firm is the excess of liabilities over resources. 896. The net gain is the excess of gain over loss for a given period ; or the excess of present worth over the invested capital. 897. The net loss is the excess of loss over gain for a given period : or the excess of invested capital over present worth. 898. An adventure, enterprise, or venture, is usually a co-operation for a. single business arrangement for a limited time or for separated business arrangements, and they are not partnerships in the strict use of that term. They are settled in accordance with the terms of the agreement or in accordance with the custom in such cases. 899. Partnerships are settled in accordance with the terms of the agreement ; this should preferably be written, but may be oral. The acts of the partners may modify, and, in fact, entirely abrogate the terms of the agreement. In general, where there is no distinct agreement as to the sharing of gains or losses, partners share equally, no matter what the respective amounts of investments may be. Differences in investment are usually adjusted through interest allow- ances. Where one or some of the members of a firm are active in its business, devoting their entire time and energies to its interests, while others are passive, these differences are frequently adjusted by allowing the former salaries. 900. When two classes of accounts are furnished, one showing resources and liabilities, and the other gains and losses, and the results obtained do not agree, the result of the class showing resources and liabilities is to be taken, as the accounts of this class can be verified by taking inventories, sending out statements, etc. Remark. — The principles of accounting should be carefully applied in working the problems that follow, and the statement should be presented in such form that the results may be readily seen, and the process by which they are attained readily followed. Students frequently fail to get accurate results because of lack of system and method iu arranging the work. 901. To find the net gain or net loss of a business when the invest- ment is given, and the present worth. Example. — 1. What is the net gain of a business whose net capital at starting was $4000 and whose present worth is $4500? Present worth $4500 Net capital 4000 Net gain $500 Example. — 2. What is the loss of a business whose net capital was $15000, now showing a present worth of $13500 ? Net capital $15000 Present worth 13500 Net loss $1500 902. Rule, — Find the difference between the net capital and the present worth. If the present worth is the greater, the difference shows a gam ; if the capital is the greater, the result shows a loss. PARTNERSHIP SETTLEMENTS 285 WRITTEN PROBLEMS 903. Find the net gain or net loss in the following: 1. Net capital $2500, present worth $3750. S. Net capital $6000, present worth $5800. 3. Net capital $15500, present worth $15700. h Net capital $2500, insolvency $1500. 5. Insolvency $500, present worth $1200. 6. Insolvency $200, present insolvency $150. 7. Insolvency $1100, present insolvency $1500. 904. To find the net gain or net loss of a business when the items of gain and loss are given. Example. — The total gains of a business are $5400 and the total losses are $1700; what is the net gain ? Total gains $5400 Total losses 1700 Net gain $3700 905. Rule . — Find the difference between the total gains and total losses ; the excess of gain shows the net gain and the excess of loss the net loss. WRITTEN PROBLEMS 906. 1. What is the net gain or loss, if the total gain is $3500 and the total loss $1800? 2. Total gain, $7200; total loss, $3750 ? 3. Total gain, $825 ; total loss, $653? f. Total gain, $12500 ; total loss, $14200 ? 5. Total gain, $1700; total loss, $1900? 6. Items of gain : merchandise, $500 ; discount and interest, $80 — Item of loss : expense, $420 ? 7. Items of gain : merchandise, $1250; rent, $400; discount and interest, $120 — Items of loss : expense, $650 ; furniture and fixtures, $35 ? 907. To find the present worth of a business when the net capital is given, and gain or loss. Example. — A merchant invested in business $12000. Upon closing, his Loss and Gain account shows the following items : Gains, Merchandise, $3500 ; Discount and Interest, $375 . Losses, Expense, $1500. What is the present worth of the business ? Loss and Gain. Expense $1500.00 Merchandise $3500.00 Net gain 2375.00 Discount & Interest 375.00 $3875.00 $3875.00 Net capital $12000 Net gain 2375 Present worth $14375 286 PARTNERSHIP SETTLEMENTS 908. Rule. — Add the net gain to the present capital , or deduct the net loss from it. WRITTEN PROBLEMS 909. 1. What is the present worth, if net capital was $3500, gain $550 ? 2. Net capital $7500, loss $700? 3. Net capital $6500, gain $800? L Net capital $1700, gain $3500? 5. Insolvency $300, gain $600? 6. Insolvency $700, loss $300? 7. Insolvency $1200, gain $900? 8. A business was started with a capital of $17500; the totals of the Loss and Gain account show debits $5625, credits $6000. What is the present worth ? 9. The net capital of a business was $8000. The Loss and Gain account shows debits : Expense, $325 ; Discount and Interest, $78 ; Real Estate, $200 — and the following credits: Merchandise, $2250; Stocks, $250. What is the present worth of the business? 10. A invested $6000 and B $3000 in a partnership. At the end of a year the items of loss were : Expense, $1200 ; Discount and Interest, $300 ; Furniture and Fixtures, $150. The items of gain were : Merchandise, $750 ; Rent, $300. What is the present worth of the business?' 910. To find the present worth of a business when the resources and liabilities are given. Example. — The resources of a business are Cash, $1500 ; Merchandise, $2420; Bills Receivable, $3100 ; Accounts Receivable, $850 ; and the liabilities are Bills Payable, $1500 ; Accounts Payable, $2400. What is the present worth? Resources. Liabilities. Cash $1500.00 Bills Payable $1500.00 Merchandise 2420.00 Accounts Payable 2400.00 Bills Receivable 3100.00 Present Worth 3970.00 Accounts Receivable 850.00 $7870.00 $7870.00 911. Rule. — Find the difference between the resources and the liabilities. The excess of resources over liabilities shows the present worth ; the deficiency of resources below liabilities shows the insolvency. WRITTEN PROBLEM 912. The resources of a business are Merchandise, $4340 ; Cash, $2810 ; Bills Receivable, $1300 ; Accounts Receivable, $1500 ; and the liabilities are Bills Payable, $8420; Accounts Payable, $3200. What is the condition of the business ? PARTNERSHIP SETTLEMENTS 287 913. To distribute gain or loss when periods of investment are equal. That is, to find the amount of gain to which each partner is entitled, or the amount of loss each partner is to sustain, when the division of loss or gain is to be made in proportion to investments, and these investments are made for the same length of time. Example. — A, B and C form a partnership, agreeing to share the gains or losses in proportion to their investments. A invests $9500; B, $12600; C, $7300. They gain $6300. Find each partner’s share. Operation by Fractional Method A $9500 B 12600 C 7300 Total $29400 Or, 150 m 2 100 95X0400 204 9$ 14 vWA of $6300 UrU of $6300 ^VWo of $6300 14250 = 2035.71 7 150 30(3 900 73X0400 10950 ~200 7 4? 14 $2035.71 A’s gain $2700.00 B’s gain $1564.29 C’s gain 150 18 900 120X6300 g ' - = 2700 294 02 4 1564.29 Operation by Proportion The ratio of the total investment to each man’s investment equals the ratio of the total gain to each man’s gain. $29400 : $9500 : : $6300 : A’s gain = $2035.71 $29400 : $12600 : : $6300 : B’s gain = $2700.00 $29400 : $7300 : : $6300 : C’s gain = $1564.29 Operation by Percentage Method $6300-?-$29400 = 21.4285% $9500X21.4285 = $2035.71 $12600X21.4285 = $2700.00 $7300X21.4285 = $1564.29 288 PARTNERSHIP SETTLEMENTS 914. To distribute gain or loss when periods of investment are unequal. That is, to find the amount of gain to which each partner is entitled, or ihe amount of loss each partner is to sustain, when the division of loss or gain is to be made in proportion to the investments, and these investments are made for different lengths of time. Example. — A and B form a partnership, agreeing to divide the losses or gains in proportion to investments. On January 1, A invests $5200 and B invests $4800. A, on May 1, invests $2200 additional, and on July 1, $3000. He withdraws on August 1, $2800 and on November 1, $3300. B invests on March 1, $2600, and on June 1, he withdraws $3000. They gain $3860. Find each partner’s share. Dr. A. Cr. Aug. 1 Cash 2800 : Jan. 1 Cash 5200 Nov. 1 Cash 3300 May 1 Cash 2200 | July 1 Cash 3000 2800X5 = 14000 5200X12 = 62400 3300X2= 6600 2200 X 8 = 17600 20600 3000 X 6 = 18000 98000 20600 A’s net investment for 1 mo.= 77400 Dr. I Cr. June 1 Cash 3000 Jan. 1 Cash 4800 Mar. 1 Cash 2600 3000X7=21000 4800X12= 57600 2600x10 = 26000 83600 21000 B’s net investment for 1 mo.= 62600 $77400 A’s net investment for 1 mo. 62600 B’s “ “ “ 1 “ $140000 total investment for 1 mo. $140000 : $77400 : : $3860 : A’s gain = $2134.03 $140000 : $62600 : : $3860 : B’s gain = $1725.97 Explanation. — A’s investment on January 1, $5200, was invested for 12 months. An invest- ment of $5200 for 12 months is equal to an investment of $62400 for 1 month. The $2200 was invested for 8 months, which is equal to an investment of $17600 for 1 month. The $3000 was invested for 6 PARTNERSHIP SETTLEMENTS 289 months, which is equal to an investment of $18000 for 1 month. A’s total investment is equal to an investment of $98000 for 1 month. Considering his withdrawals, we find the $2800 to have been out of the business for 5 months. A’s withdrawal of $2800 for 5 months is equal to a withdrawal of $14000 for 1 month ; and the $3300 for 2 months is equal to $6600 for 1 month. His total withdrawals are equal to a withdrawal of $20600 for 1 month. Subtracting the total withdrawal for 1 month from the total investment for 1 mouth, he has a net investment for 1 month of $77400. We proceed in the same way with B’s account, and find that he has a net investment for 1 month of $62600. Now these net investments for one month will form a true basis for the division of the gain. Taking the sum of these investments as the first term of a proportion, A’s investment for the second term, the total gain for the third, we find the fourth term, or A’s gain, to be $2134.03. In the same manner we find B's gain to be $1725.97. 915. Rule. — Multiply each investment by the time it was invested, and each withdrawal by the time it was withdrawn. The difference between the sums of these products is the investment, which forms a basis for the division of gain or loss. Then by proportion find gain or loss of each partner. 916. Formula for determining each partner’s share of loss or gain. Whole investment : Each partner’s investment : : Whole gain : Each partner’s share of gain. 917. To find each partner’s capital. That is, to find each partner’s capital at time of closing the accounts, when interest is allowed on deposits and charged on withdrawals, the gain or loss to be divided equally, or in a given proportion. Example. — A and B form a partnership, agreeing to share equally the losses or gains. It is also agreed that each partner is to receive interest on his investment at 8%. The following is a statement of their investments and withdrawals. They gain $4250. Close their accounts. Dr. A. Or. 1907 Apr. 9 Cash 1800 00 Nov. 6 Cash 600 00 Dec. 31 Balance 8738 19 11138 19 Int. on $1800 for 266 da. -- $106.40 “ “ 600 “ 55 “ = 7 33 $113.73 1907 Jan. 1 Cash 2500 00 Mar. 16 Cash 3600 00 Sept. 9 Cash 2900 00 Dec. 31 Interest 391 09 (6 31 Gain 1747 10 1908 11138 19 Jan. 1 Balance 1 Oo CO Co 19 Int. on $2500 for 1 yr. = $200.00 “ “ 3600 “ 290 da. = 232.00 “ “ 2900 “ 113 “ = 72 .82 $504.82 113.73 A's net credit interest, $391.09 290 PARTNERSHIP SETTLEMENTS Dr. B. Or. 1907 Mar. 6 Cash 900 00 Sept. 19 Cash 1200 00 Dec. 31 Balance 8fll 81 1051181 u Int. of $900 for 300 da. = $60.00 “ “ 1200 “ 103 “ = 27.47 $87.47 $391.09 + $364.71 = $755.80. $4250 00 — $755.80 - $3494.20. 1907 Jan. 1 Cash 3200 00 Apr. 9 Cash 1600 00 Aug. 26 Cash 3600 00 Dec. 31 Interest 364 71 i i 31 Gain 1747 10 1051181 1908 Jan. 1 Balance 841181 Int. of $3200 for 1 yr. = $256.00 U U 1600 “ 266 da. = 94 58 U :c 3600 “ 127 “ - 101.60 $452.18 87.4 7 B’s net credit interest, $364.71 $3494.20 h- 2 = $1747.10, each partner’s share of the gain. 918 . Rule. — Find the interest on the investments for the time for which they were made. Find also the interest on the withdrawals for the time they were withdrawn. The difference between the sums of these interest items is the interest with which the account should be either credited or debited. Then divide the remainder of the gain or loss. Enter this on the proper side of the account and find the balance. WRITTEN PROBLEMS 919 . 1. A and B form a partnership, agreeing to divide the gains or losses in proportion to investments. A invests $7500 and B invests $9600. Divide a gain of $3750. 2. Three men rent a pasture for three months for $96. One puts into it 14 head of cattle, the second 22 head, the third 19 head. What should each be required to pay ? 3. A, B and C enter into partnership. A puts in $1000 for 8 months : B. $1200 for 10 months; C, $800 for 12 months. They gain $1500. What was the share of each ? J. A set of books shows the following results: Loss and Gain, Dr., $7895.00, Cr., $9873 21; A’s Capital, Cr., $3175.29, withdrawals, $S46.71 : B’s Capital, Cr., $6295.35, withdrawals, $1237.18. B is to have a salary of $2000; an interest account is to be kept, to adjust the differences in capital. After crediting salary and one year’s interest at 6% on investments, the balance is shared by A and B equally. AVhat are the proper balances for A and B to commence the new period ? 5. Loss and Gain, Dr., $38967.81, Cr., $92865.28 ; A’s account, Dr., $4567.28, Cr., $24758.62 ; B’s account, Dr., $3967.19, Cr., $16969.56. Before settling, credit A’s salary, $2500 ; credit B’s salary, $4500 ; also credit A and B interest at PARTNERSHIP SETTLEMENTS 291 6 per cent, on the original investment; then close Loss and Gain, giving each partner half of net gain. State each partner’s balance at closing. 6 . A set of books shows net gain of $8937.62; A’s investment, $2500.00; B’s investment, $3900.00 ; A drew $3016.28 ; B drew $9173.21. Interest is to be allowed on the investment of each partner, but is not to be charged on with- drawals, and the balance of net gain is to be evenly divided. What is the balance of each partner’s account? 7. Loss and gain, credit in excess of debit, $9346.21 ; Ames’s debits, $1300.00, credits, $15265.75; Bell’s debits, $900.00, credits, $6875.94; Carr’s credits, $10000.00. The agreement allows Bell $1000 for services, and each partner is to be paid with interest before the profits are divided. Bell is to pay 12 per cent, interest to Carr for the use of his money ; Carr is not a partner, but loans money to the firm of Ames & Bell, at the expense of Bell alone ; what will be the balance of Ames’s and Bell’s accounts after settlement ? 8 . Close books containing the following: Net gain, $8937.65. Partners’ capital; A — Cr., $9654.62 ; B — Cr., $6978.78 ; C — Cr., $11964.65. Partners were debited as follows : A— Dr., $2245.67 ; B— Dr., $1463.79 ; C— Dr., $1296.55. The articles of agreement provide that the partners shall receive salary for their services : A, $2000 ; B, $1500 ; and C, $3000. The\ r shall have interest on capital, but no interest is charged on withdrawals; the balance, gain or loss, shall then be equally divided. 9 . A, B and C traded in company ; A put in $1 as often as B put in $3, and B put in $2 as often as C put in $5. B’s money was in twice as long as C’s and A’s twice as long as B’s ; they gained $5250; how much was each man’s share of the gain ? 10 . Armor and Baker engaged in a partnership for four years ; Armor put in $6000 and Baker $8000. At the close of the second year Armor took out $2000, and Baker putin $2000; at the close of the fourth year they divided $8890 as net gain. What was the share of each ? 11 . A certain ledger, after all the work was posted and the accounts closed, showed the following results : Loss and Gain, debits, $2829.46, credits, $18765.40 ; A. Bard, debits, $2526.89, credits, $26345.48; C. Dash, debits, $3687.95, credits, $19875.69. The articles of copartnership require an interest account kept with the partners, crediting each with interest on capital invested, but not charging interest on withdrawals ; C. Dash is to be credited $1000 a year, because he is the outside partner, and is subjected to personal expenses on that account. Com- plete the closing of the ledger and determine the balance to the credit of each partner. 1 2 . Determine the balances to the partners’ credit in a set of books exhibit- ing results as follows ; Loss and Gain, net credit, $12365.92 ; Black’s capital at beginning, $10000.00 ; Green’s capital at beginning, $5000.00. Black drew $3305 ; Green drew $2976; Green is to have a salary of $3000, and each partner is to have interest at the rate of 5 per cent, on his capital. Gains and losses shared equally. 292 PARTNERSHIP SETTLEMENTS 13. Loss and Gain account, Dr., $3567.89, Cr., $19656 25. Hyde’s personal account, Dr., $3467.89, Hyde’s capital account, Cr., $15624.00 ; Jones’s personal account, Dr., $4275.00, Jones’s capital account, Cr., $8767.49. Jones receives a salary of $2500. Interest is allowed on capital and the balance is equally divided. Wliat is the balance to the credit of each partner at the beginning of the new year? Ilf.. Lome and Sage engage in trade. Lome puts in $5000, and at the end of four months takes out a certain sum. Sage puts in $2500, and at the end of five months puts in $3000 more. At the end of the year Lome’s gain is $1066§, and Sage’s is $1333J. What sum did Lome take out at the end of four months ? 15. Three men take an interest in a coal mine. B invests his capital for four months, and claims one tenth of the profits ; C’s capital is in eight months; D invests $6000 for six months and claims two-fifths of the profits; how much did B and C put in? Losses and gains to be divided in proportion to invest- ments. 16. X, Y and Z entered into business as partners, each putting in $5000 as capital. At the end of two years X took out $1000, Y $2000, and Z $3000. At the end of the fourth year they closed the business with a loss of $3600. What was the loss of each ? Losses and gains to be divided in proportion to invest- ments. 17. Close a set of books which shows the following results, and state each partner’s balance. Net credit to Loss and Gain account, $25000. White, Dr., $1800, Cr., $12000. Redd, Dr., $2500, Cr., $10000 ; Redd to be allowed $1000 for services, and each partner to be allowed interest on his capital, and the net gain to be divided equally. 18. The Loss and Gain account in a set of books is credited $31684.89. Abel’s account (senior partner), debited $2937.63. credited $60758.93. Bearer’s account ( junior partner), debited $925.76, credited $17896.49. Determine and credit the interest, ignoring withdrawals of partners. Credit each partner $3000 for services. Divide the net profits equally and bring down the new balances to the credit of partners. What are they? 19. Two men purchase a house which rents for $420 a year. One of them pays $2200 of the purchase money. The other receives $295 as bis share of the rent. Find the amount of the purchase money the second one pays. If the total amount paid for taxes, repairs, etc., is $105, how much of this should each pay '? 20. Three boys, A, B and C, buy a bicycle, A paying $32, B $48 and C $20. They agree that each shall have the use of it, in even 7 week of six days, 10 hours each, for a time proportionate to their payments. If A begins the use of it at six o’clock Monday morning, when should B receive it? When should C receive it‘.' 21. A, B and C form a partnership, agreeing to share gains or losses in proportion to investments. A invests $4600. They gain $3800. B receives as his share of the gain $1500 and C receives 4 of the gain. Find B s and C’s investments. PARTNERSHIP SETTLEMENTS 293 22. A and B form a partnership, agreeing to share losses and gains in pro- portion to average net investments. On January 1, 1908, A invests $3200, and B on the same date invests $2650. A on March 1 invests $1600 and on Sep- tember 1, $2500, and withdraws on May 1, $1600 and on October 1, $550. B invests on April 1, $1800, and on August 1, $1950, and withdraws on June 1, $1000 and on December 1, $900. The present worth at the close of the year is $12500. How much is due each partner? 23. A and B form a partnership, agreeing to share gains and losses in pro- portion to average net investments. A on January 1 invests $4800, and B on the same date invests $5300. On May 1, A invests $2200, and on August 1 withdraws $1200. B on April 1 invests $4300, and on September 1 he with- draws $1900. They take in C as a partner on July 1, and lie invests a sufficient amount to entitle him to -l- of the gain. Find A’s and B’s gain and C’s invest- ment. They gain $6200. 21/.. A and B form a partnership on January 1, 1908, agreeing to share the gains and losses in proportion to their average net investments. A on January 1 invests $2600, and B $4800. A invests on July 6, $2800, and withdraws on September 1, $1200. B, on September 1, withdraws such an amount as will make his average investment sufficient to entitle him to f of the gain, and C on June 1 invests a sufficient amount for the rest of the year to entitle him to J of the profits. Find B’s withdrawal, A’s gain, and C’s investment. They gain $5800. 25. A and B form a partnership, agreeing to share losses and gains in pro- portion to average net investments. A on January 1 invests $5500, and B on the same date invests $4200. A withdraws on June 1, $1200, and he invests on August 1, $1400. B withdraws on September 1, $1500, and invests on November 1, $2500. C is admitted to the partnership and invests $3800 for a time sufficient to entitle him to £ of the gain. Find the date on which C made his investment. 26. Three men, A, B and C, rent a pasture for 4 months, for which they agree to pay $250. A puts in 25 cattle for 2 months, and 32 additional for the remaining 2 months. B puts in 18 cattle for 1 month, and for the remaining three months has in 26. C puts in 28 for 3 months, and for the remaining month has in 34. Find the amount each should pay. 27. John Rankin, of Texas, and James Moore, of Chicago, agree to form a partnership for carrying on a trade in cattle. To begin the business, Moore sends Rankin $12000. Rankin buys cattle during the year to the amount of $22500, and ships to Moore cattle of which Moore sells to the amount of $9650, and Rankin sells to the amount of $11920. They agree to dissolve partnership, and an examination of their books shows that Rankin has paid for expenses during the year $922 and Moore has paid $1150. Rankin has on hand at the time of settlement cattle valued at $5020, and Moore has on hand cattle valued at $6211. It is agreed between them that each will take the cattle he has on hand at the price given. Find their gain or loss. Find also which one owes the other, and how much. 294 PARTNERSHIP SETTLEMENTS 28. A, B, C and D engage the services of a teacher for 9 months, and agree to pay him $1200 ; each is to pay in proportion to the number of children sent. A sends three children for 180 days; B sends one child for 180 days and one for 160 days; C sends three children for 170 days and two for 130 days ; and D sends two children for 180 days, one for 170 days and two for 140 days. Find the amount due from each. 29. X, Y and Z agree to do a piece of work, for which they are to receive $970; each one is to receive the same amount per hour. X works 32 days of 9 hours each; Y works 18 days of 10 hours and 22 days of 9 hours; Z works 12 days of 8 hours and 18 days of 10 hours. Divide the sum among them. 30. Two men, M and N, hired a team to go a distance of 10 miles and return ; the charge for the team was $8. On arriving at their destination M proposed to return by another conveyance, to which X agreed. They also agreed that M should only be required to pay in proportion to the number of miles he had ridden. Find what each must pay. 31. A man had two sons and four daughters. To the 3 r oungest son he gave 4 of three times $2800, which was § of the share of the older son, and what the younger son received was J of f of the entire estate. The balance of the estate was divided among his four daughters in reciprocal proportion to their ages — 12, 16, 18, 21. Find each daughter’s share. 32. A father settled his estate, which was valued at $39000, upon his six children in the following proportion: William John l, Ann ^ , Mary 4, Samuel 1 and Howard What sum should each receive? 33. William Reid and John Day agree to form an equal partnership. They buy a planing mill, the price of which is $15000, $7000 of which Reid pays and the balance is paid by Day. They buy rough lumber to the amount of $3900, half of which is paid by each. Reid pays during the year for wages and other expenses $3950, and Day pays $2620. Their sales of finished product during the year amount to $7320, and they make an additional purchase of rough lumber to the amount of $1850. Their taxes and insurance during the year amount to $310. At the time of settlement, at the close of the year, they have on hand rough lumber to the amount of $920 and finished product to the amouut of $1750. Day is to receive $1200 for conducting the business, and Reid $1200 for managing the mill. It is required to know, first, their gain or loss ; second, in case Reid wishes to sell out, how much Day should pay him for his interest. 34- Three families agree to support a private tutor for their children, and for this purpose rent a room at an expense of $250. They agree to pay the teacher $1000 for 300 days’ service, 6 hours each, and it is agreed among them that they shall pay in proportion to the number of children sent and the hours they are taught. The first family sends five children for the full time ; the second, two children for 300 days and four children for 210 days, and the third family is PARTNERSHIP SETTLEMENTS 295 required to pay ^ of the expense ; this family sent four children for the same length of time. It is required to know how many days the children of the third family were sent. 35. In a certain school district there are seven families. The first is located 1 mile from the schoolhouse; the second, 1^ miles ; the third, 2J miles; the fourth, 2|- miles; the fifth, 2f miles; the sixth, 3J miles ; the seventh, 4 miles. The total amount of tax for school purposes to be raised by these families is $170, which it is agreed they shall pay in reciprocal proportion to the distance they are located from the school. Find the amount paid by each. 36. Reese, Wood, Harris and Mills form a partnership — Reese 4, Wood Harris Mills | ; the remainder of the gain is left in the business. To secure these proportional shares of profits, it is agreed that each partner shall invest $12500. Harris, however, invested only $9500, and Mills $10200. The net gain amounted to $1720. Since Harris and Mills did not invest the required amount, determine each one’s share of the gain. 37. A, B and C form a partnership, agreeing to share losses and gains equally. At the close of the year it is found that A has to his credit $9200, that B has overdrawn his account $1500, and that C has overdrawn his account $2150. There being no other resources or liabilities, determine an equitable dissolution. 38. A and B form an equal partnership. At the close of the year A’s account is overdrawn $2900, and B’s $2300. How may they settle? 39. A and B are doctors, and agree to combine their interests and share equally in the net amount made during a year. A owns the office which they use, and values the yearly rental at $750. B supplies the horses and carriages necessary to the business and pays $1000 for their care and keep; their value is $900. He charges 15% of this value for their use. A collects during the year $4225, and B $5006. A’s incidental expenses connected with the business are $112.50, and B’s are $212.75. Determine each one’s share of the receipts and how they shall settle. Jf.0. A company engaged an agent to do business for them for four months at a salary of $175 a month. They shipped him merchandise amounting to $3920.50, and sent him also $920 cash. During the four months the agent purchased goods amounting to $2140.50. At the end of the time the agent’s sales amounted to $3950.82, and the goods on hand amounted to $1005.26. Find gain or loss. Find also the amount due the company at the end of that time. Ifl. Lippincott, Kehm and Cressman are partners in business, and at the close of the year their books show that Lippincott has overdrawn his account $1050 and Kehm has overdrawn his $320. The firm assumes a private debt of $960 for Cressman. When their books were last closed, each partner had an equal sum to his credit. The resources at the present time are: cash, $10250 ; bills receivable, $4100 ; merchandise, $9285; book accounts, $726. Their liabili- ties are: bills payable, $9175; interest, $422; sundry accounts, $7005. What is ■each partner’s capital at closing, the gain being $5900? 296 PARTNERSHIP SETTLEMENTS 40. Mariner and Sailor are joint owners of a vessel which cost §80000, of which Mariner paid $50000 and Sailor $30000. They sell one-third interest to Merchant for $50000, each retaining one-third interest. What part of the amount paid by Merchant should each receive? 43. The firm of Sharp & Dull have just closed their books, which show a gain of $550. A statement of resources and liabilities shows the following : Cash, $8000; Merchandise, $13000 ; Real Estate, $5000 ; Bills Receivable, $3500; Accounts Receivable, $6750 ; Bills Payable, $8000; Accounts Payable, $15200. What was Sharp’s investment, Dull having invested $2500 ? 44- The firm of Platt & Son find upon closing their books a net loss of $8700. A statement of resources and liabilities shows Cash, $1800; Merchan- dise, $11000; Bills Receivable, $500 ; Accounts Receivable, $1500; Bills Par- able, $4000 ; Accounts Payable, $7500. What was the net capital of Platt & Son at the opening of the season ? 45. Henry Jones has his business incorporated. His assets are Cash, $9000 ; Real Estate, $15000 ; Mdse., $10000 ; Bills Receivable, $560; Accounts Receiv- able, $8950. He owes Bills Payable, $1200; and Accounts Payable, $3500. The business is incorporated under the title of the “ Linen Manufacturing Company,” with a capital of $200000, divided into 2000 shares at §100 each. Henry Jones takes 500 shares for his share; George Long takes 500 shares: William Day takes 500 shares; William Smith takes 400 shares; Leonard Hay receives 100 shares for his services as attorney. William Day buys his shares at par and pays one-half cash and note for balance. George Long buys his at 95 and pays cash for them. William Smith buys at 105, paying one-half cash and turns over real estate for balance. What does each pay for his interest in the corporation? What is the real value of each one’s interest? 46. A merchant is doing business and keeps his books by single entry. He began with Cash $10000. He bought Mdse, from A, $7000; B. $3000 ; C, $4000; D, $1000. Paid A $4000 and note $3000. Sold E $5000 and paid B in full. Sold F $7500, gave C on account $3000 and paid D in full. Sold G S5300. Received from E $4000. Sold H $4200. Received from F cash $5000 and note for $2500, and from G cash $3000 and note $1300. Received from H cash $1000 and note $1200. Wishing to change his books to double entry, he took an account of stock and made the change. Inventory, Mdse., $2500. What is the present worth of the business? 47. A and B are partners, A having f interest and B J interest in gains and losses. They have disposed of everything belonging to the business and have on hand $5000 cash. They wish to settle with each other. A’s account shows a net credit of $6000, while B’s account shows a net debit of $3500. What is the final and equitable settlement between them? 4.8. A Philadelphia firm opened a branch store in Bethlehem, Pa., and put J. M. Fayth in charge. He is to receive 5% commission on all sales and PARTNERSHIP SETTLEMENTS 207 incomes of the business. He employs an assistant at a salary of $1500, which is paid by himself. The firm supplied the branch as follows : Merchandise, $15000; Cash, $5000, During the year Faylh bought for the branch $175325 and the sales were $222900. He received a consignment of goods invoiced at $1200, from Wilmington, Del., upon which he paid freight $100 and storage $75. The consignment was disposed of at an advance of 10% above invoice value. A commission of 3 % was received and proceeds remitted. The running expenses of the business, rent, etc., outside of assistant’s salary, were $2300 ; these come out of the business and have been paid. The Merchandise inventory is $4500, and there is $6500 cash on hand. What is the result of the business? What has been Fayth’s income from the business? Ifd. A and B form a copartnership, agreeing to share gains or losses in pro- portion to average net investment. On January 1 each invests $6000. On February 1 A invests $2000 additional, and B withdraws $500. On March 1 A withdraws $750. On April 1 C is admitted to the firm, and invests $10000. On May 1 B invests $1000 additional. On June 1 A withdraws $500. On July 1 D is admitted to the firm, and invests $7500. On August 1 C with- draws $2500. On September 1 B withdraws $400. On October 1 D withdraws $500. On November 1 A withdraws $750, B withdraws $100, C withdraws $1500, and D withdraws $1000, leaving their investments equal for the remain- der of the year. On December 31 their books show the following resources and liabilities : Cash, $10000; Mdse., $13000 ; Bills Rec., $5000 ; Accts. Rec., $7650; Inventory Mdse., $6000 ; Bills Pay., $5000 ; Rent due by us, $1800 ; Accts. Pay., $4000. Allow interest at 6% on all investments and charge interest on all withdrawals. Find net gain of each, and find present worth of each partner. 50. June 1, 1905, Brown and Smith form a partnership, agreeing to share gains and losses in proportion to average net investments. They have a joint capital of $8400, of w 7 hich Brown furnished ^ and Smith the remainder; Apr. 1, 1906, Smith invested $1200, and Brown withdrew $700; Sept. 1, 1906, they admitted Gray to the partnership with an investment of $4800 ; Jan. 1, 1907, each partner invested $1500, and on Jan. 1, 1908, each withdrew $600. Sept. 1, 1908, on closing business, it was found they had sustained a net loss of $2800. What was each partner’s share of the loss ? INVOLUTION AND EVOLUTION INVOLUTION 920. Involution is the process of raising a number to any required power. Any power of a number is obtained by using the number as a factor as many times as are indicated by the exponent, which is a small figure written above and to the right of the number. As, 3 1 = 3, the first power of 3 ; 3 2 = 3X3 = 9, the second power, or square, of 3 ; 3 3 = 3X3X3 = 27, the third power, or cube, of 3 ; 3 4 = 3X3X3X3 = 81, the fourth power of 3 ; etc. 921. The process of raising numbers of two or more figures to a required power may be formulated into a rule by separating the number into its component parts and then indicating the multiplication instead of actually multiplying ; thus, 37 = 30+7 37 = 30+7 (30X 7)+7 2 30 2 + (3 0x7) The square = 30 2 +2(30X7 ) + 7 2 3 0+7 (30 2 x7) + 2(30X7 2 ) + 7 3 30 3 + 2(30 2 X7 ) + (30 X7 2 ) 30 3 + 3(30 2 X 7) + 3(30 X 7 2 ) + 7 3 which is the cube of 37. or t+u t+u (tXu )+u 2 t 2 +(tXu ) t 2 +2(iXu )+u 2 t+u (t 2 Xu)+2(tXu 2 )+u 3 u 3 +2(t 2 Xu)+ (tXu 2 ) u 3 + 3(t 2 Xu)+3(tX u 2 )+u 3 922. From this we derive the following principles: t. The square of a number of two figures (t+u), equals tens 2 + 2 times tens X units + units 2 . 2. The square of a number of three figures (h + t + u), equals hundreds 2 + 2 times hundreds X tens + tens 2 + 2 times ( hundreds + tens) X units + units 2 . 3. The cube of a number of two figures (t+u), equals tens 3 + 3 times tens 2 X units + 3 times tens X units 2 + units 3 . 4~ The cube of a number of three figures (h+t + u), equals hundreds 3 + 3 times hundreds 2 X tens + 3 times hundreds X tens 2 + tens 3 + 3 times (hundreds + tens) 2 X units + 3 times ( hundreds + tens) X units 2 + units 3 . Note 1. — Higher powers may be obtained by combining the lower ; thus, 4 4 =4 2 X4 2 ; 4 5 =4 S X 4 2 ; 4 6 = 4 3 X4 3 , etc. Note 2. — A fraction is raised to a required power by raising numerator and denominator sepa- rately ; as (f) 2 = f, etc. 298 EVOLUTION 299 WRITTEN EXERCISE 923. Find the value of 1. 14 2 . 9. (I) 4 - 17. .09 2 . 25. 1.672 4 . 2. 22 3 . 10. (5J) 3 . 18. ,67 4 . 26. (2.76J) 3 3. 46 2 . 11. (6f) 4 . 19. 003 8 . 27. 9.06 4 . 4. 29 3 . 12. (34) 7 . 20. (.17i) 5 . 28. 91.67 3 . 5. 16 4 . 13. (22|) 5 . 21. ,27 4 . 29 22.05 2 . 6. 17 5 . H- (194) 3 . 22. .086 6 . 30. (18f) 5 . 7. 927 3 . 15. (26f) 2 . 23. .0721 3 . 31. (66§) 3 . 8. 842 4 . 16. (2i)N 24- (.96f) 5 . 32. (33i) 4 . 33. Cube all the whole numbers from one to six inclusive and square sura of those powers. 34- Cube the three highest digits and cube the sum of those powers. 35. Find the value of (7 2 +8 3 -f-4 2 +3 4 +5 6 ) 3 . 36. Find the value of (5 4 ) 6 . EVOLUTION 924. Evolution is the process of finding the root of a number. 'sMU 925. A root of a number is one of its equal factors. The root is indicated by the use of the symbol -\/ , called the radical sign. Note. — The square root of 36 is 6 (]/36 = 6), because 6 X6 = 36. The cube root of 216 is 6 (# / 216 = 6), because 6X6X6 = 216. 926. When the radical sign is used alone, it indicates the square root. When any other root is required, the figure indicating the root is written in the angle of the radical sign. Thus, i/ — indicates square root, f/ indicates cube root, i y~ indicates fourth root, etc. 927. A perfect power is a number of which the required root can be found exactly. 928. An imperfect power is a number whose root can not be found exactly. Square Root 929. Observe the following numbers and their squares . Numbers. 1, 3, 5, 9, 10, 19, 34, 99, 100, 500. Squares. 1, 9, 25, 81, 100, 361, 1156, 9801, 10000, 250000. From this we learn : 1. That the square of a number of one figure contains one or two figures. 2. That the square of a number of two figures contains three or four figures. 3. That the square of a number of three figures contains five or six figures. 4. In general, the square of any number contains twice as many figures as the number, or twice as many less one. 5. That if a given number be separated into groups of two figures each, each group or partial group will be represented by a figure in the root. Hence the following rule : 300 EVOLUTION 930. .Rule. — Separate the number into two-figure groups. 931. To extract the square root of a whole number. Example. — Find the square root of 225. 2' 27 / O / o 2 = / O 0 7 / 0 X 2 = 20 / 27 /7 OLuTX 7 27 / 27 10 A c 10 5 — B £ Explanation. — Separate the number into periods of two figures each. This determines that the root contains 2 figures. The first or left hand period is 2, and the greatest square contained in it is 1 ten or 10 units. Place 10 in the root-place at the right, and subtract the square of 10 or 100 square units from 225, leaving 125 square units. The largest part of these 125 square units is in rectangles A and B, each of which is ten units long, or together 2X10 or 20. ThisTlivided into 125 will give the width, 5 units. The remaining square, C, is as large each way as the width of rectangles A and B, 5 units. Add this 5 units to 20 units, and we have the length of A, B, and C, 25 units. Multiply this length 25 by the width, 5, and we have the area, 125 square units and no remainder. The square root is 10+5 or 15 units. 932. Rule. — Beginning at units, separate the number into two-figure groups. Each period represents a figure in the square root. Find the largest number whose square is contained in the first or left-hand period ; write it in the first root-place, and put ciphers in the remaining root-places. Subtract the square of the entire root from the given number for a new dividend. For a trial divisor, multiply the root by 2 ; divide the product into the dividend and write the.quotient for the next figure of the root. Add the second part of the root to the trial divisor, and multiply the sum by the root just found. Subtract for the new dividend. Proceed in like manner until all the periods have been disposed of. 933. To extract the square root of a decimal. Example. — Extract the square root of .7 to four decimal places. r oo 0 s - = 7000 X 2 = / O O O 30 0 / 63 O O S' 3 O O x 2 = / 6 60 0 6 o / 6 6 6 0 73 6 o X 2 = / 6720 / 67 2 6 .7ooo 3 0 0 6 o . 73 6 / / '/ O O O 0 Of 0 & 0 0 / / 070 o / O O 376 / o o 77 70000000 67000000 6000000 777 OOOO EVOLUTION 301 934. Rule. — Separate into two-figure periods, beginning at the decimal point, and supply ciphers to make the number of periods desired. 935. To extract the square root of a common fraction. Example. — What is the square root of J-Ji ? i / 121 11 1 TT4 — TT 936. Rule. — Take the square root of the numerator and of the denominator ; or, if this cannot be done, reduce to a decimal and extract the root as above. PROBLEMS IN SQUARE ROOT 937. Extract the square root of 1. 1 369. 5. 74926. 15. .768. 2. 3136. 9. J (to 3 places). 16. 43.1. 3. 277729. 10. f (to 3 places). 17. 95.267. I 9339136. 11. 72645. 18. 43.14682. 5. f (to 3 places). 12. 669847. 19. 9.00746. 6. 1429785. 13. 224.85. 20. 10.001. 7. 984. If 76.745. 21. A square piece of ground contains 12 acres. What is the distance around it in yards ? 22. A carpenter has 9 boards, each 16 feet long and 18 inches wide. What is the largest square platform he can make of them, allowing nothing for waste? 23. A rectangular field contains 9 acres. Its length is three times its width. Find the distance around it. 21).. A rectangular field is 16.7 rods long, 12.47 rods wide. What is the side of a square field that contains the same area? 25. If it cost $320 to inclose a farm 104 rods long, 98 rods wide, how much less will it cost to inclose a square farm of the same area with the same kind of fence ? Similar Figures 938. Similar figures are those which have the same form and differ from each other only in magnitude. Circles, squares, right-angled triangles, etc., are similar figures. 939. Rule. — The areas of all similar surfaces are to each other as the squares of their like dimensions. Like dimensions of similar surfaces are to each, other as the square roots of their areas. 302 EVOLUTION WRITTEN PROBLEMS 940. 1. The area of a circle, whose diameter is 5 rods, is 19.635 sq. rd. What is the diameter of a circle whose area is 49 sq. rd.? 2. The area of a circle, whose diameter is 42 feet, is 1385.44 sq. ft. What is the area of a circle whose diameter is 18 feet? 3. If a pipe 2 inches in diameter will empty a cistern in 9 hours, in what time will six pipes, each one-half inch in diameter, empt} ? it? 1^. If a horse tied to a post by a rope 20 feet long can graze upon a certain area, how long should the rope be to allow him to graze upon an area four times as great ? 5. If a pipe 6 inches in diameter is 5 hours in running off a certain quan- tity of water, in what time will 3 pipes each 3 inches in diameter discharge the same quantity ? 6. How many pipes one-half inch in diameter are equal to one pipe 2 inches in diameter? CUBE ROOT 941. The cube root of a number is one of its three equal factors. Thus, a cubic yard equals 3x3x3, or 27 cubic feet; and since 3 is one of the three equal factors of 27, it is the cube root. The cubes of 2, 4, 5, 7, 9, 10, 25, 3.5, 5.4 are 8, 64, 125, 343, 729, 1000, 15625, 42.875, 157.464. From this it is seen : 1. That the cube of a number of a single figure will contain one, two or three figures. 2. That the cube of a number of two figures will contain four, five or six figures. 3. That the cube of any number will contain three times as many figures as the number itself, or three times as many less one or two. 4- That if a number be separated into three-figure groups, beginning at the right, or, in case of decimals, at the decimal point, the number of figures in the root will equal the number of periods or partial periods. 942. To find the cube root of a number. Example 1 . — Extract the cube root of 1728. Explanation. — Separating the number into three-figure groups, it is seen that the cube root will contain two figures (tens and units). 10 3 = 3X (10) 2 =300 3X10X2 = 60 2 2 = 4 1‘728 10 1 000 2 728 12 cube root 364 728 EVOLUTION 303 Figure 1 — t 3 The first or left-hand period is 1, and the larg- est cube in 1 is 1. This is represented in Figure 1 by the cube. Write 1 in the tens place of the root and place a cipher in the remaining root-place. This cube, having 10 units (1 ten) for each dimension, contains (10) 3 or 1000 cubic units (t 3 ). This subtracted from 1728 cubic units leaves 728 cubic units, the contents of Figure 2. Figure 2 — 3t 2 Xu The largest part of Figure 2 is three square slabs, each 10 units square. The area of each is 10X10 (t 2 ), and of the three it is 3 X (10X10) or 300 = 3(t) 2 . Place this area to the left and use it as atrial divisor. Dividing 300 into 728, the approx- imate thickness of the slabs is obtained, which is 2 units. Place 2 under the former root in units place. Figure 3—3 (tXu 2 ) Removing these three square slabs, there remain three rectangular slabs and a cube, Figure 3. Each slab is as long as the square slabs (10 units), and as wide and thick as the thickness of the square slabs, 2 units. The area of each rectangular slab is 1 0 X 2, and of the three it is 3 X 10X2 or 00, (3 tXu). Piace 60 under the trial divisor. Removing the rectangular slabs, the small cube remains, Figure 4. Each dimension of this cube is the same as the thickness of the slabs, 2 units. The area of the cube is 2X2 or 4 (u 2 ). Place 4 under the former partial divisors. The sum of these areas (364) is the true divisor. Multiplying this by the root-figure 2, gives 728. Since there is no remainder, our root is complete. Adding the partial roots we have 12, which is the cube root of 1728. Figure 4 — u 3 304 EVOLUTION Example 2. — Extract the cube root of 78402.752. Explanation. — Commencing at the decimal point and separating the number into three-figure groups, it is seen that the cube root will contain three figures (tens, units and tenths). 40.0 3 = 40. O z X 3 = 4X00.00 40.0X3X2.0 = 2 40.00 2.0 Z = 4.00 SO 44.00 42. O z X 3 = 42f2.00 42.0 X3 XX = / O O.XO .X z = 73.44 yX 402 . 7sd 04 ooo.oo o /4 402 . 742 40.0 2.0 .7 42 . 7 / 0 0 73 . 00 0 7 3 /4. 74 2 4 3 /4. 74 2 The first period is 78 and the largest cube in 78 is 64, whose cube root is 4. This is represented by the cube in Figure 1. Write 4 in the tens place of the root and place ciphers in the remaining places. This cube, having 40.0 units (4 tens) for each dimension, contains (40.0) 3 or 64000.000 cubic units (t 3 ). This subtracted from the original number leaves 14402.752 cubic units, the contents of the remainder, a portion like Figure 2, but consisting of two sets of slabs of different thickness, one within the other. The largest part of this remaining portion is made up of the three inner square slabs, seen in Figure 2, each 40.0 units long and 40.0 units wide. The area of each is 40.0 times 40.0 or 1600.00 square units, and of the three it is 3 times 1600.00 or 4800.00 (3 X tens 2 ). Place this area to the left aud use it as a trial divisor. This product (4800.00) divided into the remainder gives the approximate thickness of the three slabs (2.0 units). Place 2 under the former root in units place and place a cipher in the remaining place. Removing these three slabs, a cube and three rectangular slabs remain, each of which is as thick as the slabs removed, 2 units. Each rectangular slab is 40.0 units long and 2 units wide, and its area is 40.0 times 2.0 or 80.00 square units ; and the area of the three is 3 times 80.00 or 240.00 square units (3 X tensX units). Place this area to the left and beneath the trial divisor. Removing these rectangular slabs, there remains of the inner part of Figure 2 a cube, Figure 4, each dimension of which is the same as the thickness of the slabs removed, 2.0 units. The area of this cube is 2.0 times 2.0 or 4.00 square units (u 2 ). Place this to the left and beneath the other partial divisors. The sum of these partial divisors is the area of all parts of the inner portion of Figure 2 (the true divisor). This sum (5044.00) multiplied by the thickness 2.0, gives 10088.000 cubic units, the con- tents of the part removed This subtracted from 14402.752 cubic units, leaves 4314.752 cubic units, the contents of an outer portion like Figure 2. The largest part of this outer portion is composed of three square slabs, each as long as the cube, 40.0, plus the thickness of the former slab, 2.0, or 42.0 units. The area then is 42.0 times 42.0 times 3 or 5292.00 square units 3 (t -f- u) 2 . Place this area to the left and use it as a trial divisor. This area, 5292.00, divided into the remainder, 4314.752, gives the approximate thickness, .8 units. Place this in the proper place in the root. After removing the square slabs, the rectangular slabs of Figure 3 form the largest part. Each of these is 42.0 long and as thick as those just removed, .8 units. Hence, the area of these is 42.0 times .8 times 3 or 100.80 square units 3 ( t + u X tenths). Place this area beneath the trial divisor. EVOLUTION 305 Removing the rectangular slabs, there remains the small cube, which is of the same dimensions as the thickness of the rectangular slabs, .8 units. Its area then is .8 times .8 or .64 square units (tenths 2 ). Place this beneath the partial divisors. The total area is 5292. 00+100. 80+. 64 or 5393.44 square units (the true divisor). This multi- plied by the thickness, .8, gives 4314.752 cubic units. The cube root is the sum of the partial roots, 40.0+2.0+.8 or 42.8. 943 . Rule. — Separate the number into three-figure groups counting left and right from units. Determine the root of the largest cube in the left-hand period and write it for the first figure of the root. Place ciphers in the remaining root-places. Cube this entire root, and subtract from given number for a new dividend. For a trial divisor, square the root and multiply by 3 ; determine how often it is contained in the new dividend, and place this number under the root already found. To the trial divisor add 3 times the product of the first part of the root by the second part of the root, and to this result add the square of the second part of the root. Multiply this complete divisor by the second part of the root and subtract the product from the dividend. For the next trial divisor, square the sum of the two partial roots and multiply by three ; divide this product into the new dividend; the quotient is the next root figure. To the trial divisor add three times the product of the third part of the root multiplied by the sum of the other two parts, and to this add the square of the third part. Multiply this complete divisor by the third part of the root, and subtract for the new dividend. If there are more root figures to be found, for a trial divisor square the sum of the partial roots and multiply by 3, and proceed as before. Note. — I f there should be a remainder after all root places are filled, annex periods of ciphers, placing one cipher in the root for each period of 3 placed in the dividend. The cube root of a fraction is the cube root of the numerator and of the denominator. WRITTEN EXERCISE 944 . Find the cube root of 1. 91125. 5. 10218313. 2. 24389. 6. 131096512. 3. 274,625. 7. 187149.248. f. 103823. 8. 277167808. .9. 633.839779. 10. .997002999. 11. 118805247296. 12. 4.080659192. 13. u. 15. 16. 7 2 9 3 3 7 5 " 1 7 2 8 21952 - 1 ft 65 6 1 °TT3T- QQ 172 8 ou 2 1952- 306 SIMILAR SOLIDS WRITTEN PROBLEMS 945. 1 . A cubical box contains 64000 cubic feet. What are its dimensions ? 2. A cellar 24 feet long and 12 feet wide was excavated to a depth of 6 feet. Wt iat would be the depth of a cellar of equal capacity if it were cubical ? 3. A cubical tank holds 108.4156 barrels of water. What is the cost of lining the sides and bottom with tin costing 8 cents a square foot? 4- An open cubical bin holds 3810.24 bushels of grain. At $34 per M find cost of the 2-inch lumber required to make the bin. 5. What is the entire surface of a cube whose contents are 91125 cubic feet? 6. At 6 cents a square foot, what will it cost to plaster the sides and bottom of a cubical reservoir which contains 444.675 barrels of water? 7. What is the length of a cubical block containing 2 cubic feet, 1457 cubic inches ? 8. The length of a square stick of timber is 32 times its width or thickness. If it contains 13f cubic feet, what are its dimensions? 9. A bin in a granary is 3 times as long as deep and If- times as wide as deep. If it holds 270 bushels, find its dimensions. 10. A cubical pile of wood contains 500 cords. What decimal part of an acre does it cover ? Similar Solids 946. Similar solids are those which have the same shape and differ from each other only in volume. 947. The volumes of similar solids are to each other as the cubes of their like dimensions. 948. Like dimensions of similar solids are to each other as the cube roots of their volumes. WRITTEN PROBLEMS 949. 1. If a ball 3 inches in diameter weighs 12 pounds, how much will a ball 6 inches in diameter weigh ? 2. If a haystack 15 feet in diameter contains 20 tons, how much hay should be in a similar stack 25 feet in diameter? 3. If a cubical bin 10 feet long contains 125 bushels, how much grain should a cubical bin 5 feet long hold ? 4 . The clay in a cubical bank 10 feet long contains 50 loads. How long should a cubical clay bank be to contain 120 loads? 5. The contents of a cubical box 5 feet long weigh 120 pounds, what should be the length of a similar cubical box to contain 700 pounds? ARITHMETICAL PROGRESSION 950. When the difference between any two consecutive numbers in a series is the same, the numbers form an arithmetical series. If the numbers become greater from left to right, it is an ascending series; if they become less, it is a descending series. 951. The difference between any two consecutive numbers is the common difference. 952. The quantities considered are the first term (a), the common differ- ence (d), the number of terms (n) the last term (L) and the sum of the terms (S). 953. To find the last term of an arithmetical progression. Example. — Given a = 10, d = 5, n = 20, find L. Explanation. — First term = 10 ; second term = 10 + 5 ; third term = 10 -f 5 + 5 ; fourth term = 10 -(- 5 -(- 5 -j- 5. Then the 20th term = 10 + (19 X 5) or 105, for in any term the com- mon difference is used one time less than the number of that term. Substituting letters for terms used, the following formula is deduced : Formula. — L = a + (n — 1) d. Substituting figures for the letters used, L = 10 -)- (20 — 1) 5, or L = 105. Note. — B y the use of this formula, or the following ones, the first term, or the common differ- ence, or the number of terms may be found by substitution. Thus : a = L — (n — 1) d n —1 L — a . , n --- d +i 954. To find the sum of an arithmetical series. Example. — What is the sum of seven terms of the series 5, S, 11, 14, etc.? Explanation. — Writing the series as follows : Sum = 5 -p 8 -f- 11 -f- 14 -+- 17 -(- 20 -f- 23 : in reverse order Sum = 2 3 + 20 + 1? + 1 1 + 1 1 + 8 + 5 Adding 2S = 28 + 28 4- 28 + 28 -f- 28 -f- 28 -f- 28 ; or, twice the sum = 28 X 7. Now 28 is the sum of first term (a) + last term (L), and 7 is the number of terms (n) ; hence the formula : s = L+a x n _ Substituting figures for the letters used, S = ^ X 7 = 98. 307 308 ARITHMETICAL PROGRESSION 955 . R ule — The last term of an arithmetical series equals the first term increased by the product of the common difference multiplied by one less than the number of terms. The sum of an arithmetical series equals the sum of the first and last terms multi- plied by one-half the number vf terms. WRITTEN PROBLEMS 956 . 1. Given a = 7, d = 4, and n = 10 ; find L and S. 2. Given n = 15, a = 2J, and L = 44J ; find d. 3. Given L = 90f, n = 22, d = 4^ ; find a and S. f Given a = 10, L = 94, d = 3J ; find n. 5. Find the 8th term of the series 44, 6f, 94, etc. 6. Find the sum of 12 terms of the series 2, 5, 8, 11, etc. 7. A boy worked 6 weeks receiving 10 cents the first day, and an increase of 5 cents each day. What were his wages the last day? His total wages? 8. A man paid his debt in 15 years by paying $30 the first year, $38 the second year, $46 the third year and so on ; what was the debt? 9. A merchant added $250 annually to his capital for 20 years, and then had $6000. What was his original capital? 10. Fourteen properties were valued each $225 higher than the preceding one. The eighth property is valued at $4575. What is the total valuation? 11. A man travels 15 miles the first day and increased the distance 10 miles each succeeding day. The last day he traveled 85 miles. How many days did he travel? How far? 12. A clerk receives a salary of $15 the first month, $25 the second month, $35 the third month and so on for two years. What was his salary the last month, and the average monthly salary? 13. A man owing $3240 paid it in a year by paying $50 the first month, and increasing each monthly payment by a specified sum. What was the sum? If A bo} r , who earns 15 cents the first day, has expenses averaging 95 cents daily ; if his wages increase 10 cents daily, what will be his total net earnings in 40 days? On which day will he be most in debt? 15. Fifteen horses were purchased whose values are in arithmetical progres- sion. The fifth horse cost $149, the twelfth cost $191. What was the total cost of the horses purchased ? 16. Find the sum of 20 consecutive odd numbers, commencing with 19. 17. In selling a Persian rug a dealer numbers tickets from 1 to 200, and obtains 200 persons to draw each a ticket. Each person pays for his ticket as many cents as the number on the ticket he draws, and the owner is then deter- mined by lot. How much does he receive for his rug? GEOMETRICAL PROGRESSION 957. A geometrical series is one iti which each number after the first is derived by multiplying the preceding by a given number or by dividing the preceding by a given number. 958. The multiplier or the divisor is called the ratio. 959. When the numbers increase from left to right, the series is ascending, as 2, 6, 18, 54 ; if they decrease, it is descending, as 108, 36, 12, 4. 960. The quantities considered are the first term (a), the ratio (r), the number of terms (n), the last term (L), and the sum of the terms (S). 961. To find the last term of a geometrical progression. Example. — What is the tenth term of the series 2, 10, 50, etc.? Explanation. — First term=2 ; second term=2X5 ; third term=2X5X5 ; fourth term 2X5X 5X5 ; then the tenth term=2X5 9 or 3906250, for in any term the ratio is used one time less than the number of that term. Substituting letters for the terms used, the following formula is deduced : Subtracting (1) from (2) or 3 S=32768 — 2 or 32766 and S equals J of 32766 or 10922. 32766 is the last term, 4 is the ratio, and 3 (the divisor) is one less than the ratio. Hence the L = l n_1 X a From this formula the following formulas are deduced : L Note. — n may be found by observing how many times the ratio may be divided successively a 962. To find the sum of a geometrical series. Example. — What is the sum of 7 terms of the series 2, 8, 32, etc.? Explanation. — The sum (S)=2-f-8+32 -(-128+512+2048+8192 4 (ratio) sum ( 4S ) = 8+32+128+512+2048+8192+32768 ( 1 ) ( 2 ) formula : S=L Xr — a Substituting figures for the letters used S= 8192 4—2 4—1 or 10922 309 310 GEOMETRICAL PROGRESSION 963 . Rule. — The last term equals the first term multiplied by the ratio raised to a power one less than the number of terms. The sum of a geometrical series equals the last term times the ratio and this product divided by the ratio less one. WRITTEN PROBLEMS 964 . 1. Given a = 6, r — 6, n = 5 ; find last term. 2. Given a — 4, n = 7, L = 2916; find the ratio. 3. Given L = 640, n = 8, r = 2 ; find the first term. I/.. Given L = 6576, r = 6, a = 6 ; find the number of terms. 5. Given a = 8, r = 4, n = 4 ; find the last term and the sum of all the terms. 6. If 4 of the contents of a vat is evaporated each day for 8 days, what part of the original contents remains? 7. A milk dealer received 4 cent for the first pint of milk sold, and trebled the price on each pint he sold. What did he receive for a gallon sold in pints? 8. A man having $1000 to pay at the end of a year, began by paying $5 the first month, $10 the second month, $20 the third month and so on. At the end of the year has he overpaid or does he still owe and how much? 9. A debt of $93410 was canceled by paying $10 the first year, and treb- ling each following yearly payment ; how many years did it require to pay the debt? 10. The cost of the 20th cow was $7864.32. If the first cow cost 14 cents and the prices were in geometrical progression, what was the ratio? 11. A merchant who began business with $6000 capital increased the capi- tal annually by 4 of itself. What was his capital at the end of the sixth year? 12. A man deposited $1000 in a savings bank that pays 4% compounded semiannually. If the depositor withdrew no money from the bank for six years, what was the amount he had on deposit? 13. If a rubber ball, falling upon a marble floor, rebounds to 4 the height from which it has fallen, then falls again and rebounds 4; etc., how far will it move before coming to rest, if tossed to a height of 50 feet? Note. — Last term = 0. MENSURATION 965. Mensuration is the process of finding the lengths of lines, the areas of surfaces, and the volumes of solids. 966. A line has length without breadth or thickness. 967. A surface has length and breadth, without thickness. A plane surface or plane figure is one that is flat or level. 968. A solid has length , breadth and thickness , or depth. 969. Lines are either straight or curved. 970. A straight line is the shortest distance between two points. 971. A curved line is one that changes its direction at every points 972. Parallel lines have the same direction and are equidistant. 7 973. A line is perpendicular to another when the two angles thus formed are equal. 974. An angle is the difference in direction of two inter- secting lines, or the opening between them. 975. When one straight line intersects another so as to make the four angles so formed equal, these angles are called right angles. 976. An acute angle is an angle that is less c than a right angle ; as, CBD. 977. An obtuse angle is one that is greater than a right angle ; as, ABC. Acute or obtuse angles may A D be called oblique. B Right-angled Triangles 978. A triangle is a plane surface bounded by three straight lines. The following are the various kinds : right- angled, acute-angled, obtuse-angled and equi-angular ; or, if named according to the sides, equilateral, isosceles and scalene. 979. A right-angled triangle is a triangle which has one right angle. 980. The base of a triangle is the line upon which it rests; as, AB. Any side may become the base. c 981. The altitude of a triangle is its height perpendicular to the base ; as, BC. 982. The hypotenuse of a right-angled triangle is the side opposite the right angle ; as, AC. 311 altitude 312 MENSURATION 983 . In every right-angled triangle the square of the hypotenuse is equal to the sum of the squares of the base and altitude. 984 . The hypotenuse of a right-angled triangle is equal to the square root of the sum of the squares of the other two sides. 985 . The base of a right-angled triangle is equal to the square- root of the difference of the squares of the hypotenuse and altitude. 986 . The altitude of a right-angled triangle is 3 3 +4 2 — 5 2 equal to the square root of the differenceof the squares 9 +16 = 25 0 f f-] ie hypotenuse and base. 987 . Example.— A tree is broken in such a manner that the top rests on the ground 40 feet from the foot of the tree; the part remaining standing is 53J feet; what is the length of the broken part? 40 2 = 1600 53 f 2 - 2844f 126 1326 66.6 = 66f or 66| Explanation. — The broken part of the tree, the upright part and the ground from the broken end to the foot of tree form the three sides of a right-angled triangle, thebaseand altitude being known. Note. — A lways draw the geometrical figure. 988 . Rule. — To find the hypotenuse, extract the square root of the sum of the squares of the other two sides. To find the base or the altitude, extract the square roo't of the difference between the squares of the other two sides. WRITTEN PROBLEMS 989 . 1. A ladder leaning against a house reaches 84 feet, its base being 3S feet from the house. What is the length of the ladder? Two rafters, each 38 feet long, meet at the ridge of a roof 18 feet above the attic floor. What is the width of the house? 3. Two ships start at the same point. One goes due north 120 miles, the other due east 160 miles. How far are they apart ? i. A tree was broken 56 feet from the top and fell so that the end struck 42 feet from the foot of the tree. Find the height of the tree. 5. Find the distance between a lower corner at one end and the opposite upper corner at the other end of a room 40 feet long, 36 feet wide, 18 feet high. MENSURATION 313 6. Find the hypotenuse of a right-angled triangle whose base is 260 feet and altitude 82 feet. 7. Find the hypotenuse of a right-angled triangle whose base is 1760 yards and whose altitude is 880 yards. 8. Find the hypotenuse of a right-angled triangle whose base is 4 feet and altitude 3 feet. 9. Find the diagonal of a square which is 12 inches on a side. 10. Find the diagonal of a rectangle whose length is 4 rods and whose width is 3 rods. 11. Find the hypotenuse of a right-angled triangle whose length is 1200 yards and whose height is 1500 feet. 13. Find the base of a right-angled triangle whose hypotenuse is 360 yards and whose altitude is 240 yards. 13. Find the altitude of a right-angled triangle whose hypotenuse is 500 feet and whose base is 400 feet. 7^- Find the length of a rectangle whose width is 10 feet and diagonal 12 feet. 15. Find the side of a square whose diagonal is 20 yards. Surfaces of Triangles 990. To find the area of a triangle. whose altitude is 49 yards ? 35 Xj of 49 = 35x24| = 857J sq. yd. area. Example 2. — What is the area of a triangle whose sides are respectively 30, 32 and 40 feet ? 30 + 32 + 40 = 5 51 — 30 = 21 51 _ 32 - 19 51 — 40 = 11 51 X 21 X 19 X 11 = 223839 j/223839 = 473.116+ sq. feet. 991. Rule. — Multiply the base by one-half the altitude. Or, When the three sides are given and not the altitude, take half the sum of the three sides, from it subtract each side separately , multiply the half sum and these remainder's together and extract the square root. 314 MENSURATION WRITTEN PROBLEMS 992. 1. What is the area of a triangle whose base is 36 feet and altitude 15 feet ? 2. What is the area of a triangular field whose altitude is 50 yards and w r hose base is 70 yards? 3. What is the area of a triangular field whose altitude is 25 chains and whose base is 20 chains ? 4- What is the area of a triangle whose sides are 150 yards, 150 yards and 250 yards ? 5. What is the area of a triangle whose sides are 40 rods, 50 rods and 60 rods ? Surfaces of Quadrilaterals. 993. A parallelogram is a plane surface bounded by four straight lines, its opposite sides being parallel. 994. A rectangle is a parallelogram whose rectangle angles are all right angles. 995. A square is a rectangle whose four sides are equal in length.. 996. To find the area of a rectangle. Example. — What is the area of a rectangular field whose length is 20 chains and whose width is 15 chains? 20x15 = 300 sq. ch. or 30 A. 997. Rule. — Multiply the length by the breadth. 7 998. A rhomboid is a parallelogram, the adjoining sides of which are not equal to each other, and which contains no right angle. rhomboid 999. To find the area of a rhomboid. 1000. Rule. — Multiply the base by the altitude. 1001. A rhomb is an equilateral parallelogram having oblique angles. 1002. To find the area of a rhomb. Example. — What is the area of a rhouiboidal figure 6 inches long, whose altitude is 3 inches? 6x3 = 18 sq. in. 1003. Rule. — Multiply the base by the altitude. MENSURATION 315 WRITTEN PROBLEMS 1004. 1 . Find the area of a rectangular field 12. chains 80 links long, 10 chains wide. 2. Find the area of a rectangular field 100 yards long, 87 yards wide. 3. Find the area of a field 36 rods square. f. Find the surface of a blackboard 8 yards long and 4 feet wide. 5. What is the area of a rhomboid whose base is 12 chains and altitude 8 chains ? 6. What is the area of a rhomboid whose base is 200 yards and altitude 70 yards? 7. Find the altitude of a rhomboid whose area is 1 acre and base 95 yards. 8. Find the altitude of a rectangle whose area is 1 square mile, base 6 chains. 9. Find the altitude of a rhomboid of 200 acres, whose base is 100 rods. 10. If a road 3 miles long contains 12 acres, how wide is it ? TEAPEZOID 1005. A trapezoid is a four-sided plane figure, of which two sides are parallel and the other two are not parallel. 1006. To find the area of a trapezoid. Example. — The parallel sides of a trapezoid are respectively 20 and 26 inches and the altitude is 18 inches ; what is the area ? 20+26 2 23x18 = 414 sq. in. 1007. Rule. — Multiply J the sum of the parallel sides by the altitude. 1008. A trapezium is a four-sided plane figure of which no two sides are parallel. 1009. To find the area of a trapezium. Example. — The four sides of a trapezium are respectively 20, 10, 8, 24 ft. ; the figure is divided into two triangles by a line 28 ft. long. What is the area of the figure ? Area of A = 70.42 sq. ft. Area of B = 89. sq. ft. Area of figure = 159.42 sq. ft. 1010. Rule. — Separate the trapezium into two triangles , find the area of each , and take the sum. 316 MENSURATION Surfaces of Polygons PENTAGON HEXAGON 1011. A polygon is a figure bounded by straight lines or arcs, especially more than four ; a figure having many angles. 1012. A regular polygon is one whose sides and angles are all equal. 1013. To find the area of a regular polygon. Example. — The sides of the hexagonal base of a statue are 10 feet each and the distance from the center of the base to the middle of each side is 8.66 feet ; what is the area of the figure? 30 00X8.66 P 259.8 sq. ft. 1014. Rule . — Multiply the distance around the polygon (perimeter) by J the distance from the middle of one of the sides to the center of the polygon. The Circle 1015. A circle is a plane surface bounded by a curved line called its circumference, every point in which is equally distant from a point within, called its center. 1016. The diameter of a circle is a straight line passing through its center, beginning and ending in the circumference, and dividing the circle into two equal parts, called semicircles- 1017. The radius of a circle is a straight line from its center to its circumference. It is J the length of the diameter. 1018. To find the circumference of a circle. 1019. Rule. — Multiply the diameter by 3. If 16. 1020. To find the diameter of a circle. 1021. Rule. — Divide the circumference by 3.1 f 16. Or, Multiply the circumference by .31831. 1022. To find the area of a circle. 1023. Ru le. — Multiply the circumference by h the radius. Or, Square the diameter and multiply by .785 f. Or, Square the circumference and multiply by .07958. MENSURATION 317 WRITTEN PROBLEMS 1024. 1. Find the circumference of a circle whose diameter is 3.5 yards. 2. Find the circumference of a circle whose diameter is 12.3 meters. 3. What is the circumference of a circle whose radius is 28 inches? If.. The circumference of a circle is 10 feet; what is the diameter? 5. The circumference of a circle is 100 yards ; find the radius. 6. The diameter of a wheel is 28 inches; what is its circumference? 7. What is the circumference of a circle whose diameter is 100 rods ? 8. Find the area of a circle whose diameter is 1 yard. 9. Find the area of a circle whose circumference is 84 inches. 10. Find the area of a circle whose diameter is 1 chain. 11. Find the area of a circle whose diameter is 1 foot. 12. Find the distance traversed in an hour by a point in the circumference of a water-wheel 17 feet in diameter, if the wheel makes 1400 revolutions in an hour. 1025. To find the circumference or the diameter of a circle, the area being given. Example. — Find the diameter of a circle whose area is 200 sq. ft. Find circumference. 254 64 2513 2 7854 ) 200.0000 .07958 ) 200.00000 157 08 15916 42 920 40 840 39 270 39 790 3 6500 1 0500 3 1416 7958 50840 25420 47124 23874 37100 15460 31416 15916 254.64( 15.95 2513.20)50.13 1 25 25 | 154 1001 1320 125 1001 309 2964 Diameter — 15.95 ft. 10023 | 31900 2781 1 3185 1 18300 Circumference 50.13 ft. | 15925 1026. Rule. — Divide area by .785If and extract the square root of the quotient to find diameter. Divide area by .07958 and extract the square root of the quotient to find circumference. 318 MENSURATION WRITTEN PROBLEMS 1027. 1. The area of a circle is 10 acres ; what is the diameter ? 2. The area of a circle is 1 square mile ; what is the radius ? 3. The area of a circle is 100 square feet; what is the circumference? j. The area of a square inscribed in a circle is a square rod ; what is the diameter of the circle? 5. A square whose side is 10 feet is inscribed in a circle ; what is the area of the circle ? The Ellipse 1028. An ellipse is a plane figure bounded by a curved line, such that the sum of the distances from any point of the curve to two fixed points (the foci ) is constant. 1029. The transverse axis of an ellipse is the diameter drawn through the foci. 1030. The conjugate axis is the diameter at right angles to the transverse axis. 1031. To find the area of an ellipse. 1032. Rule. — Multiply J of the transverse axis by J of the f conjugate axis, and the resulting product by 3. If 16. f, g. foci ; pf+pg, con- stant Solids TRIANGULAR PRISM SQUARE PRISM PENTAGONAL PRISM 1033. The principal regular solids are the prism, cube, pyramid, cone and sphere. 1034. A Prism is a solid whose bases or ends are any similar equal and parallel plane figures, and whose sides are parallelograms. 1035. Prisms are triangular, square, pentagonal, etc., according as the figures of their ends are triangles, squares, pentagons, etc. MENSURATION 319 PARALLELEPIPED 1036. A pri sm whose bases are parallel- ograms is called a parallelepiped. 1037. A cube is a solid bounded by six equal squares. 1038. To find the entire surface of a prism. cube 1039. Rule. — Multiply the distance around the base by the altitude, and to the product add the areas of the top and bottom. ~7 / y / 1040. To find the volume or contents of any prism. 1041. Rule. — Multiply the area of the base by the altitude of the prism. 1042. To find the volume or contents of a parallelepiped. 1043. Rule. — Multiply together the length, breadth and height. CYLINDER 1044. A cylinder is a round prism of uniform diameter with circles for its ends. 1045. The altitude of a cylinder is the distance from the center of one base to the center of the other. 1046. To find the entire surface of a cylinder. 1047. Rule. — Multiply the circumference of the base by the altitude and to the product add the areas of the tivo ends. 1048. To find the volume or contents of a cylinder. 1049. Rule. — Multiply the area of the base by the altitude. WRITTEN PROBLEMS 1050. 1. Find the entire surface of a parallelepiped whose length is 6 feet and sides of the base 4 feet. Find the volume of a granite block 6 feet high, 2 feet wide, 1 foot thick. 3. Find the volume of a triangular prism whose area is 3 square feet and wdiose length is 10 feet. f Find the entire surface of a cylinder 18 inches in diameter, 6 feet long. 5. Find the volume of a cylinder 4 feet long whose circumference is 2 feet. 1051. A pyramid is a solid bounded by a plane polygon for its base, and by triangular planes meeting in a point called the vertex. 1052. To find the entire surface of a pyramid. 1053. Rule. — Multiply the perimeter of the base by 1 the slant height, and to the product add the area of the base. 1054. To find the volume or contents of a pyramid. PYRAMID 320 MENSURATION 1055. Rule. — Multiply the area of the base by £ of the altitude. 1056. A cone is a solid that tapers uniformly from a circular base to a point called the vertex. 1057. To find the entire surface of a cone. 1058. Rule. — Multiply the circumference of the base by | the slant height , and to the product add the area of the base. 1059. To find the volume or contents of a cone. 1060. Rule. — Multiply the area of the base by J of the altitude. 1061. A sphere is a solid, every point of whose sur- face is equally distant from an interior point, called the center of the sphere. 1062. The diameter, or axis, of a sphere is a straight line passing through the center and terminated both ways by the surface. 1063. The radius of a sphere is a straight line passing from the center to any point on the surface. Its length is \ of the diameter. 1064. To find the surface of a sphere. 1065. Rule. — Multiply the circumference by the diamt CONE Multiply the square of the diameter by 3.14-16. 1066. To find the volume or contents of a sphere. 1067. Rule. — Multiply the surface by } of the diameter. Or, Multiply the cube of the diameter by .5236. SPHEROID 1068. A spheroid is a solid formed by the revolu- tion of an ellipse about one of its axes. 1069. To find the volume or contents of a spheroid. 1070. Rule. — Multiply the square of the revolving axis by the fixed axis , and the resulting product by .5236. WRITTEN PROBLEMS 1071. l. Find the whole surface of a pyramid, a side of the square base being 6 feet and the slant height 8 feet. 2. Find the whole surface of a pyramid having a rectangular base 5 feet by 4 feet and a slant height of 4 feet 3 inches. MENSURATION 321 3. Find the volume of an octagonal pyramid, the area of the base being 2 square feet and the altitude being 2 feet. If. Find the entire surface of a cone, the radius of the base being 2 feet and the slant height 5 feet 6 inches. 5. Find the volume of a cone whose diameter is 12 inches and altitude 2 feet. 6. Find the surface of a sphere 3 feet in diameter. 7. Find the volume of a sphere whose circumference is 3.1416 feet. REVIEW PROBLEMS IN MENSURATION 1072 . 1 . What is the area of a triangle whose base is 16 feet, altitude 36 feet ? 2. What is the area of an isosceles triangle, the equal sides of which are 170 yards and the base 308 yards ? 3. What is the area of a triangle, the sides of which are 205 feet, 228 feet and 281 feet ? If. If the base of a right-angled triangle is 42 feet and its altitude 48 feet, what is the hypotenuse? 5. A rectangular field is 296 feet long, 272 feet wide. Find the distance between the opposite corners. 6. A ladder placed against a wall reaches to a height of 68 feet. The bottom of the ladder is 22 feet from the foot of the wall. Find the length of the ladder. 7. The sides of a trapezium taken in order are 12, 12, 9 and 7, and the diagonal which cuts off the first two sides is 14 rods. What is the area of the trapezium ? 8. The circumference of a circle is one-half of a mile. Find its diameter. 9. The diameter of a circle is 290 feet. Find its circumference. 10. The diagonal of a square is 1000 feet. Find its area in square feet. 11. What is the area of a piece of ground in the form of a circle 225 feet in diameter? 12. Of two concentric circles, the diameter of the inner one is 25 rods, and of the outer one, 28 rods. Find the area of the space included between the two circumferences. 13. What is the total surface of a cylinder whose diameter is 12 feet and whose length is 28 feet ? Ilf. How many gallons of water will a tank hold that is 8 feet square at the base and 9 feet deep? 15. A room is 24 feet long, 15 feet high, 18 feet wide. Find the diagonal of the room. 322 MENSURATION 16. At 22 cents a square yard, what will it cost to gild a ball 9 feet in diameter ? 17. What is the weight of a ball of iron 7 inches in diameter, if iron weighs 450 pounds to the cubic foot? 18. A wagon 7 feet long, 4 feet 6 inches wide, is built to hold 3 tons of coal. Find its depth, there being 54 pounds to a cubic foot of coal. 19. A mow 28 feet long, 24 feet wide, and 12 feet high is filled with timothy hay. If the weight of a cubic foot is 7 pounds, what is the value of the hay in this mow at $15 per ton of 2000 pounds? 20. A bin 12 feet long, 7 feet wide, is built to hold 500 bushels of oats. Find its depth. 21. What is the weight of a conical piece of metal 22 inches in diameter, 15 inches in slant, if the metal of which it is composed weighs 420 pounds to the cubic foot ? 22. A field in the form of a rectangle contains 3 acres of ground. Its length is three times its width. What will it cost at 48 cents a running yard to put a fence around it ? 23. Which costs the more and how much, a fence around a field containing 10 acres whose length is twice its width, at 62 cents a running yard, or the same kind of fence around a square field containing the same area? 2 If.. A chandelier is suspended from the center of the ceiling of a room- The room is 15 feet high, 24 feet long, 22 feet wide. If the bottom of the chan- delier is 9 feet from the floor, what is the distance from a lower corner of the room to the central point of the bottom of the chandelier ? 25. A reservoir in the form of a circle is 9 feet deep and 48 feet in diameter. When full, how long will the water contained in it last a towm of 800 people, if to each inhabitant there are allowed 50 gallons every 24 hours? 26. The rain which falls on a roof 90 feet long and 85 feet wide, in 4S hours, when the rainfall averages f of an inch every two hours, will exactly fill a cylin- drical tank 22 feet in diameter. Find the depth of the tank. 27. A ball made of iron, which weighs 450 pounds to the cubic foot, has a diameter of 9 inches. How many balls whose diameters are one inch would be required to weigh as much ? 28. A field is 126 chains long and 105 chains wide. What is it worth at $490 an acre ? 29. The base of a triangle is 90 feet. Its area is 2242 square feet. Find its altitude. 30. A rectangular field contains 9 acres of ground. One of its dimensions is 42 chains. Find the other dimension in rods. 31. The dimensions of a dumb-bell are as follows : the handle is 5 inches long and 1 inch thick ; the ends are spheres 3 inches in diameter. If the dumb- bell is made of iron weighing 450 pounds to the cubic foot, what is its weight '? MENSURATION 323 32. What will it cost, at 92 cents per square yard, to macadamize a section of street 2J miles long, 80 feet wide? 33. If the diameter of a circle is 320 rods, find its circumference. 34.. If the circumference of a circle is a quarter of a mile, find its area. 35. If a pipe 3 inches in diameter will supply a certain amount of water in a given time, how many pipes one-half inch in diameter will be required to supply the same amount of water in the same time? 36. At 8£ cents a pound, what is the cost of a grindstone, the diameter of which is 18 inches and thickness 4 inches, if in the center there is a hole one inch square? A cubic foot of this kind of stone weighs 185 pounds. 37. A cabinet maker has a piece of mahogany 18 feet long, 7 feet 6 inches wide. How large a square table-top could be made from this, no allowance for waste? 38. A platform, circular in form, holds 850 people. Allowing 2 square feet for each person, what is the diameter of the platform? 39. A right-angled triangle, the base being 5 feet and altitude 4 feet, is made to turn round on the longer side; find the volume of the cone thus gen- erated. 40. The surface of a certain solid is 3 times as great as the surface of a sim- ilar solid; what is the ratio of their volumes? 41. A rectangular field is 300 rods long and 200 yards wide; what is the length of its diagonal? 42. A lot is 1 rod square; how many cubic yards of earth will it require to raise the surface of the lot 2 feet ? 43. Find the weight of a spherical iron ball whose diameter is 8 inches, a cubic foot of iron weighing 450 pounds. 44 • If a sphere 4 inches in diameter weighs 4 pounds, what will be the weight of a sphere of the same material 1 foot in diameter? 45. Find the weight of a spherical shell 1 inch thick, outside diameter 1 foot, the metal weighing 450 pounds to the cubic foot. LATITUDE AND LONGITUDE 1073. Latitude, the distance north or south of the equator, is measured in degrees, minutes and seconds ; no latitude can exceed 90 degrees, because from the equator to either pole is one-fourth of 360° or 90°. 1074. Places north of the equator are in north latitude ; those south, in south latitude. Places on the equator have no latitude. 1075. Longitude, the distance east or west from a selected or prime meri- dian (usually Greenwich), is measured in degrees, minutes and seconds; no lon- gitude can be greater than 180 degrees, because from a given meridian to that meridian is 360° ; then one-half of 360° is east and one-half is west. Note. — la the United States the meridian of Washington is sometimes selected, and maps show longitude from Greenwich at the top and from Washington at the bottom. 1076. Places east of the selected meridian are in east longitude-, those west, in west longitude. 1077. To find the difference in latitude or longitude. Examples. — (a) The latitude of Philadelphia is 39° 56' 39" N., and of Mont- real 45° 24' 28" N. ; what is the difference in latitude ? (b) The longitude of Berlin is 13° 24' 28" E., and of New York is 74° 3' W. ; what is the difference in longitude? 45° 24' 28" N. Explanation. — (a) Since Montreal and Philadelphia are both north of the equator, their difference of latitude must be the distance that Montreal is farther north than Philadelphia ; this distance is found by subtraction. 13° 24' 28" E. 74 3 00 W. 39 56 39 N. 87° 27' 28" 5° 27' 49" (b) From the meridian of Berlin to the prime meridian is 13° 24' 28", and from the prime meridian to the meridian of New York is 74° 3' ; the dis- tance from Berlin to New York, or the difference of longitude, is found by addi- tion. 1078- Rule. — The difference of latitude or longitude is found by subtraction when, both places are in like latitudes or longitudes ; by addition when the places are in different latitudes or longitudes. WRITTEN PROBLEMS 1079. 1. The latitude of New York is 40° 24' 40" N., of Cape Horn 55° 5S' 30" S., and of Washington 38° 53' 39" N. What is the difference in latitude between New York and Cape Horn? between Washington and Cape Horn? between New York and Washington? 2. The longitude of Detroit is 82° 58' W., of San Francisco 122° 26' 15" W., and of Rome 12° 27' E. What is the difference in longitude between Detroit and San Francisco? between Detroit and Rome? between San Francisco and Rome? 324 LONGITUDE AND TIME 325 3 . The latitude of Philadelphia is 39° 56' 38" N., of Callao 11° 45' 2t" S., and of Boston 42° 21' 27" N. What is the difference in latitude between Phila- delphia and Callao? between Boston and Callao? between Philadelphia and Boston ? The longitude of Philadelphia is 75° 9' 5" W., of New Orleans 90° W., and of Calcutta 88° 19' 2" E. What is the difference in longitude between Phil- adelphia and Calcutta? between New Orleans and Calcutta? between Philadel- phia and New Orleans? 5 . The longitude of Canton is 113° 15' 33" E., of St. Paul 93° 4' 55" W. of Omaha 95° 55' 15" W. What is the difference in longitude between Canton and St. Paul? between Canton and Omaha? Note. — If the difference in longitude exceeds 180 degrees, subtract this difference from 360 degrees to find the real difference in longitude. LONGITUDE AND TIME 1080. The rotation of the earth on its axis from west to east ^3b0 m 24 hours) causes midday or noon, or sunrise, or any given time, to pass westward 15° in one hour; 15' in 1 minute; 15" in one second. Note. — When it is sunrise, 6 A. M., in Philadelphia, places west of Philadelphia, as Chicago, will not have seen the sunrise, hence the hour will be before sunrise, or earlier. When Chicago sees the sunrise, Philadelphia will see the sun above the horizon, and the time indicated hy clocks will be past 6 or later than Chicago. From this we conclude that a place east of a given place has later time than the given place ; places west, earlier time. Places on the same meridian have the same time. 1081. Solar time is the sun’s time as indicated by a sun dial. 1082. Standard time is time established by the leading railroad companies of the United States. For convenience they have divided the United States into four time-belts each fifteen degrees wide, and have adopted the time of a single meridian in that belt as the standard time for the entire belt. These belts are : 1. Eastern, whose time is the local or solar time of the 75th which is 9' east of Philadelphia. 2 . Central, whose time is the local or solar time of the 90th which is 2° 15' west of Chicago. 3 . Mountain, whose time is the local or solar time of the 105th which is 44' west of Denver. It-. Pacific, whose time is the local or solar time of the 120th which is 2° 26' east of San Francisco. meridian, meridian, meridian, meridian, Note — An outline map showing time-belts and official time may be secured from a railroad company. . 326 LONGITUDE AND TIME Dials Exhibiting Comparison of Solar and Standard Time 1083. To find the difference in time between places differing in longu tude. Example. — ( a) When it is 11 A. M., solar time, at Philadelphia, 75° 9' 5" W., what time is it at Denver, 104° 44' 20" W. ? ( b ) What is the standard time at Denver when it is 11 A. M. standard time at Philadelphia ? 104° 75 44' 9 20 " 5 W W Explanation. — (a) The difference in longitude between the two places is 29° 35' 15". Divide this difference by 15, since every 15° = one hr. ; 15' = one min. ; 15 // = one sec. This gives 1 hr. 58 min. 21 sec., the difference in time between the two places. Denver being west of Phila- delphia has earlier time ; subtracting 1 hr. 58 min. 21 see. from 11 A. M. leaves 9 hr. 1 min. 39 sec. ; hence the time at Denver is 1 min. 39 sec. after 9 A. M. ( b ) The standard time at Denver (mountain time) is 2 hr. earlier than Philadelphia (eastern time), hence subtract 2 hr. from 11 A. M. 9 A. M. is the standard time at Denver. 1084. Rule 1. — Divide the num , >er of degrees, minutes and in longitude between two places ) by 15 ; the quotient will be the difference of time in 1 lours, minutes and seconds. 15) 29 ' 35 15 1 58 21 hrs. min. sec. 11 00 00 1 58 21 9 1 39 seconds ( the difference Rule 2. — Multiply the hours , minutes and seconds (the difference in time between two places ) by 15 ; the product will be the difference in degrees, minutes and seconds between two places. LONGITUDE AND TIME 327 WRITTEN PROBLEMS 1085 . 1 . The longitude of Pekin is 118°, and that of Berlin 13° 24' 28" E. What is the difference in solar time? When it is 3 P. M. at Berlin, what time is it at Pekin ? 2. The longitude of Paris is 2° 20' 17" E., and that of Philadelphia 75° 9' 5" W. What is the difference in time ? When it is half-past two o’clock P. M. in Philadelphia, what time is it in Paris? in Berlin? 3 . What is the standard time in New York when the standard time in Portland, Oregon, is 25 min. 18 sec. past 3 o’clock P. M. ? J. The longitude of Washington is 77° 3' W. If the difference in time between Washington and Edinburg is 4 hr. 53 min. 28 sec., what is the longitude of Edinburg? When it is 20 min. past 3 o’clock P. M. at Edinburg, what time is it in Washington ? 5 . The captain of a ship observed that when the sun crossed the meridian, the solar time, shown by his chronometer set to Greenwich time, was 22 min. 4 sec. after 3 o’clock P. M. In what longitude was the ship ? 6. St. Paul, whose longitude is 93° 4' 55" W., is situated in the Central belt. What is the difference between the local or sun time and the standard time at St. Paul? 7. In traveling from Philadelphia to Seattle, Washington, my watch, which shows Philadelphia time, must be changed on arriving at Seattle. How much must I change it and which way (standard time)? 8. The longitude of Manila is 121° 15" E., and of San Juan 66° 5' W.; what is the difference in longitude? When it is 5 P. M. Tuesday at San Juan, what is the time at Manila? 9 . The longitude of Berlin is 13° 24' 28" E., and of Philadelphia 75° 9' 5" W. A cablegram sent from Berlin at 12.30 noon requires 30 minutes in transit. At what hour will it be received in Philadelphia? 10. Sydney is in longitude 151° 38' 42" E., and San Francisco is 122° 26' 15" W.; when it is 4 P. M. July 4th at San Francisco, what time is it at Sydney? 11. A and B met, and on comparing watches found that A’s indicated 10.30 A. M. and B’s showed 5.10 P. M. The longitude of A’s place was 82° 58' ; what was the longitude of B’s place ? 12. Two persons are so situated that they are 14° 42' 48" apart, yet are in the same time belt. The first is 6° 14' 27" East of the standard meridian, and the other is 8° 28' 21" West. Will the local time of each be too fast or too slow, and how much ? REVIEW MENTAL PROBLEMS 1086. 1 - A man’s age increased by J of his age equals 40 years ; what is his age? 2. What number is that which if its \ be taken away, the remainder will be 33? 3. Three times a number increased by its £ equals 32 ; what is the number? 4. What number is that which being increased by its \ and ^4- equals 27? 5. A man having spent \ of his money had 45 cents remaining ; what sum had he at first? 6. Margaret spent J of her money in one store and 4 of her money in another store and had 21 cents remaining; how much had she at first? 7. One-half of the length of a pole is in the air, 4 in the water and 18 feet in the ground; required the length of the pole. 8. Two-thirds of Adam’s money diminished by 2 dollars equals J of his money ; what was his sum of money? 9. What number is that which diminished by its 4 and increased by its 4 will equal 50 ? 10. What is Benjamin’s age, if f of his age increased by 6 years equals 36 years ? 11. The cost of a horse diminished by \ of its cost and five dollars equals seventy dollars; what did the horse cost? 12. Abram’s money increased by twenty dollars equals f of his mone} T ; what is his money ? 13. Bella’s age increased by 32 years equals 3 times her age ; what is her age ? 14- Four times a number diminished by once the number equals 75 ; what is the number ? 15. If the height of a tree be increased by its f and 20 feet, the sum will be twice the height ; what is the height of the tree? 16. The cost of a horse diminished by £ of its cost and §30 equals 4 of its cost ; wbat did the horse cost ? 17. I sold J of my farm and then bought 30 acres and now have SO acres ; how much land had I at first ? 18. If the height of a monument be increased by its 4 and that sum dimin- ished by its 4, the height will equal 75 feet; wdiat is its height? 19. A father and son earned in one week $24 ; how much did each earn if the father earned three times as much as his son? 20. A pole 60 feet in length was sawed into two unequal parts so that 4 of the longer piece equals the shorter ; what was the length of each piece ? 328 REVIEW MENTAL PROBLEMS 329 21. X and Y had equal sums of money. X lost $10, Y earned $20 and they then together had $110 ; how much had each at first? 22. A cow and a horse cost $120. If the cow cost f as much as the horse, minus $10, required the cost of each. 23. A wheelman rode 130 miles in 3 days. He rode 10 miles farther the second day than the first and 10 miles less the third day than the second ; how far did he ride each day? 2 Ip. M and N together have $40 ; how many dollars has each, if three times M’s money equals N’s ? 25. P and Q can build f of a boat in a day and Q does | as much as P ; what part can each complete in a day ? 26. Divide $66 between tw 7 o persons so that as often as the first has $2, the second shall have $3J. 27. A’s money added to 4 of B’s money equals $70 ; how much money has each if B’s money is to A’s as 3 to 2 ? 28. If § of a yard of cloth cost f of a dollar, what will J of a yard cost? 29. If 5 men earn $40 in a certain time, how much will 4 men earn in twdce the time? 30. Two men hire a pasture for $5. One turns in 3 horses for 7 days and the other 7 horses for 2 days; how much should each pay ? 31. Two men had equal sums of money; one lost $7 and the other lost $4 and they then together had $19. How much did each have at first? 32. Jones and Black can build 4 of a boat in a day, and Black builds § as much as Jones; how much can each build in a day ? 33. If 5 men can build 40 rods of w r all in 16 days, how many men can build half as much wall in 4 days ? 3Ip. If a five-cent loaf of bread w r eighs 8 ounces when flour is worth $4 a barrel, how much should it weigh when flour is worth $5 a barrel ? 35. Three men hired a horse for 30 days at the rate of $2 a day. The first used it 8 days, the second 10 days and the third 12 days ; how 7 much should each pay ? 36. A and Z contract to do a piece of work in 90 days ; A sends 9 men and Z 15 boys. What part of the price should each receive, supposing 5 boys to do as much as 3 men ? 37. X, Y and Z dig a ditch for $72 ; X gets $1 a day, Y $1J a day and Z $lf a day. How many days were they employed and what did each receive? 38. Two men can mow a field of grass in 4 hours ; the first can mow it alone in 9 hours; how long would it require the second alone to mow it? 39. Brown can dig a ditch in 6 days and Black can dig it in 8 days ; in what time can they dig it working together? 330 REVIEW MENTAL PROBLEMS Ifi. A cistern can be filled by an inlet pipe in 5 hours and emptied by an outlet pipe in 4 hours ; if the cistern is full and both pipes are opened, in what time will the tank be emptied? 41. A cistern can be filled by one pipe alone in 6 hours and by a second alone in 9 hours; how long long will it require both to fill it? 42. A cistern can be filled by an inlet pipe in 4 hours and emptied by an outlet pipe in 5 hours; if the cistern is empty and both pipes are opened, bow long will be required to fill it? 43. Brown, Baker and Broad lunch together, Brown having 3 loaves, Baker 5 loaves and Broad 24 cents to divide between them. If the bread be divided equally among them, how should Brown and Baker divide the money? 44- What is the time of day, if the time since noon equals ^ of the time since midnight? 45. At what time after 3 o’clock are the hour and minute hands together? 46. What is the time of day, if the time till midnight equals -f of the time since midnight? 47. At what time after 3 o’clock are the hour and minute hands opposite? 48. A pole whose length was 64 feet was broken into two lengths so that if 4 feet be taken from the longer and added to the shorter, the pieces will be equal ; what are the lengths of the broken pieces ? 49. A lady having two watches bought a chain worth $20. If the chain be put upon the silver watch, they together will be worth § as much as the gold watch ; if the chain be placed upon the gold watch, the} T together will be worth 4 times as much as the silver watch. What is each watch worth ? 50. A boat which sails at the rate of 6 miles an hour moves down a river whose current is 3 miles an hour; how far can it go and return in 8 hours? 51. A philanthropist gave f of his money, less $12, to a poor family and then had $21 left; how much did he give awa} 7 ? 52. A toy balloon in the air fell J of the distance to the ground and then arose 1 the distance it was from the ground ; what part of the first distance is it now from the ground ? 53. If J of my weight be added to my weight and from the sum 55 pounds be taken, the result will be 200 pounds.; what is my weight? 54 ■ If the height of a monument be increased bv its 1 and from this sum be taken the difference between its J and ^ , the result will be 40 feet; what is its height ? 55. The sum of three numbers is 55 ; the second is J greater than the first and the third is twice the second ; find the numbers. 56. A man walks 120 miles in three days, walking each day 10 miles more than the preceding day ; how many miles did he walk the second day ? 57. Two pipes fill a cistern of 200 barrels ; f of what flows through one pipe being equal to \ of what flows through the other ; how much is carried in by each pipe? REVIEW MENTAL PROBLEMS 331 58. Hunter’s money is to Fisher’s money as 2 to 3; how much money has each, if Hunter’s money and J of Fisher’s money is equal to $350? 59. If 8 men can do a piece of work in 6 days, how many days will be needed to complete the work, if 4 men be added when the work is J done? 60. Jones and Smith rent a pasture for $40. Jones puts in 12 horses and Smith 8 cows. How much should each pay if 2 cows eat as much as 3 horses? 61. X, Y and Z build a wall, working an equal number of days; X gets $3 a day ; Y, $2 a day and Z, $1 a day. They receive $60 ; what is each one’s share ? 62. Brown can build a shed in 10 days and Black in 12 days; how many days would it take to build it if they work together? 63. White and Grey working together can build a boat in 4f days. White working alone will build it in 8 d4ys ; in how many days would Grey working alone build it? dj. A piece of work can be done by 4 men or 6 boys in 10 days; how long would it require 2 men and 10 boys to do it ? 65. Wood can cut a cord of wood in J of a day and Black can cut it in f of a day ; working together, how many cords can they cut in a week of 6 days? 66. William receives $2 a day and pays 50 cents a day board ; at the end of 40 days he has saved $40 ; how many days was he idle? 67. James had 3 loaves of bread and David had 5 loaves, which they shared with Benjamin, who gave them 16 cents; how shall they share the money? 68. Two men mow a field of grass for $40 ; the first mows twice as much as the second, less 4 acres, and is paid $24 ; how many acres does each mow? 69. What is the asking price of cloth, if by dropping 25% and selling it at $2.40 a yard, a gain of 20% is made ? 70. A sum of money put on interest at a certain rate for 3 years amounts to $236, and the same sum at the s ime rate for 10 years amounts to $320 ; find the sum and rate 71. If 5 dollars be taken from f of a sum of money there will remain the original sum ; what was the sum ? 72. A tank can be filled by an inlet pipe in 4 hours and emptied when full by an outlet pipe in 1J hours ; if the tank be full and both pipes opened, in what time will the tank be empty ? 73. A working alone can build a boat in 7 days, B in 8 days, and C in 6 days ; in what time can they, working together, build it? 7J. A cyclist makes a journey of 300 miles in 4 days, riding each day 10 miles farther than on the preceding day ; how many miles did he ride each day ? 75. If 5 men can do a piece of work in a certain time, how many men would be required to do double the amount of work in half the time ? GENERAL REVIEW PROBLEMS 1087 . 1 . The net proceeds of a sale are $940.10. The total expenses are, 3% commission, $14.10 storage, $16.50 drayage. Find the amount of the sale. 2. I sold a house for $2450 and gained 14%. I sold another for the same price and lost 14%. Find my net loss or gain. 3. I buy a piano for $250. I want to sell it so as to gain 15% after throw- ing off 10%. What price shall I ask for it? If.. Change £142 19s. Id. to French money. 5. What is the cost in U. S. money of 900 tons of steel, invoiced at £3 8s. 1 0 d. a ton ? 6. I sold a lot of goods for $760.25, and thereby lost 12%. At what price should the goods have been marked to gain 10% and allow a trade discount of 3%? 7. The net proceeds of a sale of hardware are $210.50, the commission 5%, the other expenses $7.42. Find the sales and the amount of the commission. 8. An insurance company insures a house valued at $42000 for f of its value at If % premium. This company reinsures f of its risk at f%. What rate per cent, is the first company making on its risk? 9. A pipe can fill a cistern in 24 hours ; another pipe in 30 hours. There is a delivery pipe running from the cistern that will empty it in 20 hours. Sup- pose the cistern to be empty, and the three pipes opened at once; in what time will it be filled ? 10. A can carry a ton of coal to the third story of a building in 5^ hours, B can do the same in 6| hours, C in 7f hours. In what time could the three together carry 2f tons ? 11. An insurance policy is written for an amount sufficient to cover an insurance of $5000 on a house, and 24% premium. Find the face of the policy. 12. A can do a piece of work in 12 days, B in 15 days ; if C "works with them, they can do the work in 4 days. In what time could C do the work alone ? 13. If it requires 3f bushels of seed to sow an acre, how many quarts will be required to sow a lot 110 feet square? H. A cistern 8 feet in diameter, 7 feet deep, has how many gallons of water in it when the water is within 3 inches of the top? 15. The hypotenuse of a right-angled triangle is 82 feet, the base 62 feet. Find the altitude. 16. What will it cost, at 48 cents a square yard, to paint the dome of a hall, the dome being in the form of a hemisphere 40 feet in diameter? 17. A has a circular garden, and B a square one; each contains 2 acres. A walk 6 feet wide is made around each garden. Which walk contains the greater area, and how much? How much greater would the area of each walk be if it surrounded the 2 acres? 332 GENERAL REVIEW PROBLEMS 333 18. A square field containing 12 acres is surrounded by a tight board fence 8 feet I iigli. What did the boards cost at $22 per M ? 19. I buy muslin at 6f cents a yard and mark it to gain 20%, but on account of a defect I conclude to sell it at 85% of the marked price. What is my loss or gain on 1000 yards ? 20. A miller bought a ton of wheat at 68 cents a bushel, which he manu- factured into flour. If each bushel yielded 36f pounds and he received $4.10 a barrel, what is his per cent, of gain or loss? 21. A bin is 9 feet 5 inches long, 4 feet 3 inches wide, 5 feet 8 inches deep, and is § full of wheat. How many bushels does it contain and what is the wheat worth at 68 cents a bushel ? 22. How deep must a circular cistern 4 feet in diameter be to hold 40 barrels of water ? 23. The fore wheel of a carriage is 3 feet 6 inches in diameter; the hind wheel 4 feet 8 inches. How many revolutions will the smaller one make while the larger one makes 900 ? 21^. An agent sold goods for $1395 and lost f of the cost. For how much should he have sold the goods to have made 20% gain? 25. A man lost $920, which was 25% of what he had remaining. What per cent, of his money did he lose? 26. A man bought four houses for $15000 and sold them as follows : on the first he gained 20% ; on the second he gained 15% ; on the third he lost 15% and on the fourth he lost 4%. If he received the same amount of money for each house, what was the cost of each? 27. A coal dealer buys a car load of coal, 35000 pounds, at $4.15 a long ton, which he retails at $5.60 a long ton. What per cent, does he gain if there is a loss of 3% in handling the coal? 28. The minute-hand of a clock is 5 inches long, the hour-hand 3f inches. Over how much more area does the minute-hand pass in 5 hours than the hour- hand ? 29. At what per cent, above the manufacturer’s price must a wholesale merchant mark goods so that he can allow a retailer a discount of 30% and 5% and still make a profit of 25% ? 30. An article is marked to gain 33-J%, but the seller throws off ^ and the collector is afterwards paid 15% for collecting the debt. What is the percent, of loss ? 31. The owner of a mill had it insured at a rate of If %. He afterwards introduced steam power, and the company took an additional risk of $1500. They, however, raised the rate f%. The extra premium amounted to $65. For what amount was the mill first insured ? 32. A book jobber buys books at 40% and 10% off list and marks his stock at an advance of 15% on list prices. He makes a sale to a private library at a discount of 15% from his marked price and makes a profit of $2500. How much did the books cost the jobber? 334 GENERAL REVIEW PROBLEMS 33. I ordered an agent to buy flour for me which I afterwards sold at 20% profit and gained $1.50 on a barrel. If my agent’s commission was 3|%, and his total commission $25, how many barrels did he buy? 34- A dairyman sold a quantity of milk, butter and cheese, receiving for all $115.25. He gained 15% on the milk, 18% on the cheese, 22% on the butter. If the amount received for each was equal, find the cost of each. 35. A grocer wishes to mix coffees worth 38, 42, 45, 50 and 55 cents a pound so as to produce a mixture worth 45 cents a pound. Find the number of pounds of each kind he must take. 36. A clerk is required to mark goods so as to make 25% after throwing off 10%. By selling a lot of these goods for $78 net, there was a gain of but 3%. How should he have marked the goods? 37. I go to bank to-day with four notes : one dated to-day, at 90 days, for $1500 ; the second dated 7 days ago, at three months, for $920; the third dated to-day, at 60 days, for $1720 ; the fourth is my own note dated to-day, at 90 days, for enough to raise my bank balance to $5000. What was the face of my own note, my balance previous to having these notes discounted being $290.62 ? 38. I buy in Paris 328 meters of silk at 27.42 francs per meter. If exchange is at 5.19, what is the cost of a draft that will pay for the invoice? 39. A river is moving at the rate of 4 miles an hour. It is 119 feet wide and has an average depth of 104 feet. How many tons of water will pass a given point on the river’s bank in 12 minutes, water weighing 62J pounds to the cubic foot ? Ifi. On May 12, 1908, I asked my broker to purchase for me 320 shares Cen- tral Railroad stock and left with him as margin $3000. He bought on May 27 at 1064 ; on June 2 he sold 150 shares at 107f, and on June 18 he sold the remainder at 107J. Our account was settled on June 30. How much was due me on that day, and what was my gain? 41. Two concentric circles have diameters of 350 feet and 390 feet, respect- ively. What is the area of the space enclosed between the circumferences? 4%. A wall 54 feet high, 2J feet thick, cost for building $2006.90, at the rate of $1.18 a perch. Find the length of the wall. 43. A can do a piece of work in 27 days ; B in 36 days; C in 42 days. They all work together until | of the work is done, when A retires, leaving B and C to finish it. In what time is the whole work done? How long do B and C work ? 44 ■ A ditch 426 feet long, 24 feet wide, 54 feet deep, cost for digging $1522.65. How wide should a ditch be that is 294 feet long, 4f feet deep, to cost $2420 ? 45. A cylinder is 7 feet in diameter, 38 feet long. Find its entire surface. 46. A C 3 r lindrical tank is 7 feet in diameter and 15 feet deep. It has in it 560 gallons of water. What per cent, is this of its capacity? 47. A man bought a house and lot, paying five times as much for the house as for the lot. If the house had cost 25% less than it did, it would have cost $3800. Find the cost of each. GENERAL REVIEW PROBLEMS 335 4 - 8 . A dealer spent equal sums in hats, coats and shoes, and sold at a profit of 20% on the hats, 22% on the coats, and 15% on the shoes. The entire sales amounted to $22500. Find the cost of each. 49 . A invested a certain amount in real estate, and B invested 4 times as much. A lost 15% and B gained 20%. The difference between the amounts received was $1250. How much did each invest? 50 . A company engaged an agent to sell for them on a commission of 5% and shipped him merchandise amounting to $5500. The agent purchased additional merchandise to the amount of $1200, and the firm sent him $500 cash. At the end of a certain period the agent had sold goods to the amount of $4920, and the goods on hand amounted to $3006 78. Find the loss or gain. Find also the amount due the firm by the agent at this time. 51 . A circular piece of ground, containing 5 acres, is enclosed by a tight board fence 7 feet high. What was the cost, at $13 per M, of 1-inch lumber required to make it ? 52 . A and B form a partnership on January 1, 1908, with the understanding that the gains and losses are to be divided in proportion to their average net investments. On January 1 A invests $3000 and B invests $2500. A on March 1 invests $1500, on July 1 $2200, and withdraws on September 1 $2000. B on April 1 invests $1800 and on September 1 $2000, and withdraws on November 1 $3300. A statement of their resources and liabilities at time of settlement is as follows : Resources : Cash, $2200 Liabilities: Bills, $1920 Mdse., 6850 Sundry Accounls, 2600 Bills, 3290 Book Accounts, 2600 Close each partner’s account. 53 . If a wall 520 feet long, 4J feet high, 2J feet thick, requires the work of 175 men, 212 days of 10 hours each in building, how high should a wall be that is 248 feet long, 2£ feet thick, to require the work of 98 men, 217 days of 9 hours each ? 54 - Sold for account of consignor on September 8, 1899, $5240.26 on 4 months’ credit; September 29, $538 cash; October 16, $2750.85, 60 days; on September 15, paid freight amounting to $95.23; my commission for selling is 5%. What is due to consignor, and when are the proceeds due by equation ? 55 . A merchant bought 22 pieces of cloth, each containing 25 yards, at $4.50, on a credit of 6 months, and sold them at $6.25 on a credit of 3 months. What was his cash gain, money being worth 6% ? 56 . Bought a check on a bank which had suspended, at 65% of its face, and exchanged it for 5% railroa l bonds at 82. What rate of interest do I receive on my investment ? 57 . Bought, at 8% discount, a 6% mortgage for $3800, with two years to run. What interest do I get on the money invested, if the mortgage is paid at maturity ? 336 GENERAL REVIEW PROBLEMS 58. A bath tub will hold 200 gallons of water. It is filled by a faucet discharging 50 gallons in 4 minutes, and emptied by a waste pipe discharging 48 gallons in 4 minutes. If both the pipes are opened at once and the waste pipe closed one hour afterwards, in what time will the tub be full? 59. A retail merchant sold a quantity of silks for $1250, thereby gaining 20%. The wholesale merchant from whom he bought them made a profit of 15%, and the importer who sold them gained 20%. What did they cost the importer? 60. A merchant in Philadelphia made the following importation from Belfast, Ireland: 1720 yds. linen at 2s. 6d. 1850 “ “ “ 3s. 3d. 512 “ “ “ 5s. 2d. 1680 “ “ “ 3s. id. Duty 22% ad valorem and 8 cents a yard specific. Find the duty. Find also the cost of a draft in payment of the invoice, exchange at 4.87. 61. A manufacturer carried on business for four years. The first year he gained 20%, which at the close of that year he put into the business. The sec- ond year he lost 15%. The third year he gained 25% of what he had at the beginning of that year, and put it into the business. The fourth year he gained 16§ %, and then his total capital was $40250.85. How much had he gained in the four years? 62. If the sales in a certain merchandise account are $256742.90, the cost of the merchandise $275420.82, and the inventory $45923.75, what per cent, is the profit? 63. The list price of a sewing machine is $60; an agent buys at 25% off and sells for $55. What is his per cent, of profit ? 6 If.. A merchant sells goods at retail 35% above cost, and at wholesale 15% less than the retail price. What is his gain per cent, at wholesale? 65. If I buy at list less 40%, will I gain or lose if I add 50% to the list and sell at | off? 66. What is the balance due June 15, 1909, on a note of $1350, dated June 15, 1908, and showing the following indorsements : August 18, 1908, $50 ; Sep- tember 23, 1908, $75; October 20, 1908, $90; November 24, 1908, $75 ; January 14, 1909, $5: March 25, 1909, $60; April 21, 1909, $350? Work by the United States rule. 67. Find the cost of 300 shares Reading Railroad stock (par $50) at Ilf, brokerage f %. 68. I bought 290 shares Philadelphia Traction at 72f, brokerage \%. What did they cost me? 69. I bought a bill of exchange on London, and paid $2892.50 for it. What was the face of the bill, exchange at 4.87f ? 70. What is the face of a bill of exchange on Paris which can be bought for $1590.62 at 5.15? GENERAL REVIEW PROBLEMS 387 71. What is the duty on 9000 yards of 28-inch dress goods at 42 cents per square yard ? 79. What is the dut} r on 720 gallons of brandy at $1.25 per gallon, the allowance for leakage being 2% ? 73. A firm in Philadelphia imports from Paris 9200 yards of 22-inch silk, the marked price of which at the time of purchase was 3.25 francs per meter. What was the duty at 10 cents per square yard and 35% ad valorem ? 7L. What is the duty on a consignment of iron invoiced at £2265 18s. 6d. at 42 % ? 75. A man’s personal property is assessed at $5000, his real estate at $13000. Find his total tax at the rate of $1.32 per $100. 76. The expenses of a town for a year, not including the tax collector’s commission of 3%, are as follows: for schools, $3500 ; for roads, bridges, etc., $1700; for incidental expenses, $800. If the total valuation of the town is $2450000, what should be A’s tax whose property is assessed at $3500 ? 77. A lumber dealer purchased two piles of wood ; one pile was 66 feet long, 7 feet high, 12 feet wide, and contained 43^- cords. The other pile was 29 feet long, 16 feet wide and of sufficient height to contain 54f cords. Find the height of the latter pile. 78. A contractor engaged to dig two cellars at the same price per cubic yard. He received $450 for one 75 feet long, 28 feet wide, 9 feet deep. What should he have received for the other, if its length was 88 feet, width 33 feet, depth 8 feet? 79. If 52 men can earn $1850 in 19 days by working 10 hours a day, how much should 29 men earn in 23 days of 8 hours each? 80. If the cost of pasturing 85 head of cattle for 13 weeks is $75, what should be the cost of pasturing 99 head of cattle for 4 weeks? 81. A note of $5000, dated July 7, 1908, with interest at 5J%, has the fol- lowing indorsements: September 15, 1908, $10; October 28, 1908, $25; Decem- ber 23, 1908, $1000!; February 25, 1909, $1500; April 28, 1909, $90; May 26, 1909, $1300. What is due July 7, 1909 ? Use United States and Mercantile rules. 82. I bought 357 shares Lehigh Navigation (par $50) at 41|. brokerage Find the cost. 83. A legacy of $15000 is invested as follows : 50 shares P. R. R. (par $50) at 53f ; 92 shares Central at 82J ; 36 Lehigh Valley (par $50) at 27f ; and the balance in Union Pacific at 41f. Brokerage at Philadelphia Stock Exchange rates. Find the number of shares of Union Pacific purchased and the balance unspent (no fraction of a share being purchased). 84- A fire insurance company had a risk of $58000 at § % premium, and reinsured -§- of the risk in another company at f% premium and £ of it in another at f% premium. What rate per cent, is the first company making on its risk? 85. If the premium paid for insuring a property was $45.20, and the rate of insurance was sixty cents on $100, for what sum was it insured ? 338 GENERAL REVIEW PROBLEMS 86. If from 220 reams of paper 3000 copies of a book containing 380 pages can be printed, bow many reams will be required to print 8000 copies of a book containing 480 pages? 87. A grocer wishes to mix 500 pounds of coffee which be can sell at 75 cents a pound and gain 25%, by using coffees worth 38, 40, 45, 65 and 70 cents a pound. How many pounds of each should he take? 88. What is the amount of $998.65 from January 16, 1908, to September 14, 1908? (Time by compound subtraction ) 89. What is the cost of a circular piece of ground 228 feet in diameter, at $300 an acre ? 90. A legacy of $22000 is invested as follows: $2500 of 4% Government Bonds at lllf; $3200 of 6% railroad bonds at 104J ; $1600 of 5% railroad bonds at 98f ; and the balance in U. S. 5s at 128f. Find the total income from the investment. Find also the balance unspent (no bond being of smaller denomination than $100), brokerage at Philadelphia Stock Exchange rates. 91. If I buy 4% stock at 97f, brokerage what per cent, income do I receive on my investment? 92. If I buy 6% bonds at 97f, brokerage \% , what per cent, income do I receive on my investment? 93. I bought a 5% bond (par $1000) at such a price as to yield me 4J% on my investment. Find the cost of the bond. 94- Bank stock paying 5 % is selling at 1124. Find the per cent, income on investment. 95. A stone wall 3 feet thick, 44 feet high, cost for its construction $1590, at 96 cents a perch. Find the length of the wall. 96. A contractor agrees to build a cellar which shall be 18 feet long, 16 feet wide and 9 feet deep, inside measure, when finished. His contract price is 62 cents per cubic yard for making the excavation, and $1.25 a perch for the wall. If the wall is 2 feet thick, find his total contract price. 97. At $18.50 per M, what is the cost of sufficient lumber 2 inches thick to make 50 boxes, each 4X4 feet, 3 feet deep, inside measure? 98. The expenses of a town for a year, not including the tax collector's com- mission of 3%, are as follows: for schools, $2500 ; for roads, bridges, etc., $1200 : for incidental expenses, $900. If the total valuation of the town is $1420000, what should be A’s tax whose property is assessed at $5500? 99. A fire insurance company had a risk of $85000 at f % premium, and reinsured j of the risk in another company at f-% premium and 4 °f it in another at f % premium. What rate per cent, is the first company making on its risk ? 100. The stock of a wholesale house is insured in a number of companies for $175000, and is damaged bv water to the amount of $57641.73. What per cent, of its risk should be paid by each company, and what amount should a company pay that had a risk of $2500 upon it? GENERAL REVIEW PROBLEMS 339 101 . The value of a ship’s cargo is $15500, and the owner desires to insure it for a sum which will cover the value of the cargo and the cost of insurance. If the rate of insurance is lg-%, what should be the amount of the policy? 102 . A roof is 42 feet long, 28 feet from eaves to ridge. How many slates are required to cover it, if the slates used are 10 inches wide, showing 12 inches to the weather, and what will they cost at $10.90 a square? Note.— A square contains 100 square feet. 103 . Find the entire surface of a cylinder 9 feet in diameter, 15 feet long. 101 /.. A garden 125 feet long, 86 feet wide, has on its border a stone wall 3 feet high, 18 inches thick. What is the cost of the wall at 92 cents a perch? 105 . If the premium paid for insuring a property was $45.20, and the rate of insurance was sixty cents on $100, for what sum was it insured ? 106 . The stock of a wholesale house is insured in a number of companies for $15000, and is damaged bv water to the amount of $900. What per cent, of its risk should be paid by each company, and what amount should a company pay that had a risk of $6500 upon it? 107 . The value of a ship’s cargo is $20750, and the owner desires to insure it for a sum which will cover the value of the cargo and the cost of insurance. If the rate of insurance is 1|%, what should be the amount of the policy? 108 . A lumber dealer purchased two piles of wood ; one pile was 72 feet long, 10 feet wide, and high enough to contain 45 cords. The other pile was 18 feet wide, 16 feet high, and long enough to contain 87f cords. Find the height of the first pile and the length of the second pile. 109 . A contractor engaged to dig two cellars at the same price per cubic yard. He received $350 for one 70 feet long, 25 feet wide, 8 feet deep. What should he have received for the other, if its length was 96 feet, width 32 feet, depth 9 feet? 110 . If 42 men can earn $1580 in 18 days by working 11 hours a day, how much should 19 men earn in 29 days of 9 hours each? 111 . If the cost of pasturing 75 head of cattle for 11 weeks is $72, what should be the cost of pasturing 92 head of cattle for 3 weeks? 112 . I go to bank to-day with two notes, one dated seven days ago for $2200, at 3 months; the second is my own note dated to-day, at 90 days, for a sum sufficient to make my balance large enough to enable me to give a check for $4500 and have a balance of $300 in bank. If my present balance is $290, what is the face of the note? 113 . If it costs $19.20 to carry 7642 pounds 129 miles, what should be the cost of carrying 12693 pounds 68 miles at the same rate? 114 .. 85% of a meter is what per cent, of a yard ? 115 . I have in bank to-day $311.72. I send to bank H. R. Reed’s note dated to-day for $726 at 3 months, and my own note dated to-day at 90 days for a sum sufficient to enable me to give a check for $1150 and have left in bank $125.10. Find the face of my note. 340 GENERAL REVIEW PROBLEMS 116. In the following account state when the balance is due by equation, and what amount of cash will settle it June 1, 1909. Dr. Frank B. Hughes Cr. 1908 Mar. 2 Balance 911 72 1908 Mar. 27 Cash 900 00 U 28 Mdse., 60 days 375 93 Apr. 22 Note, 10 days 370 00 Apr. 9 “ 2 mos. 702 10 May 9 Draft, 30 days 600 00 June 16 “ 1 mo. 397 98 117 . If 19 men can do a piece of work in 12 days, how many days should 26 men require to do the same work? 118. If 14 men or 19 boys can do a piece of work in 42 days, in what time can 12 men and 22 boys do the same work? 119. A can do a piece of work in 9 days, B in 9J days, and C in 10J days. In what time could they do it, working together? 120. I send to bank to-day William Smith’s note dated to-day, in my favor, for $950 at three months, and my own note at 60 days, dated to-day, for a sum sufficient with the proceeds of Smith’s note to allow me to draw a check for $1450 and leave in bank a balance of $375.50. If my balance before sending these notes to bank for discount was $250.10, what is the face of my own note ? 121. A’s money is 5% less than B’s, and B’s is 15% greater than C’s. If they together have $19360.72, how much does each have? 122. I send an agent $1190 to invest in apples at $1.10 a barrel, allowing him a commission of 4%. He pays 17 cents apiece for barrels and 22 cents a barrel transportation. How many barrels of apples does he buy ? What is the surplus ? 123. On a lot of merchandise which I sold for $962.50, I lost 18%. The cost price of the remaining merchandise is $2642.50. At what per cent, advance must I sell this to recover my loss? 124- In what time, at simple interest, will $1796.14 amount to $2496.12 at 4i% ? 125. What amount will settle the following account November 11, 1908? Dr. H. M. Kennedy Cr. 1908 June 6 Balance 403 19 1908 July 13 Cash 475 U 22 Mdse., 60 days 1126 92 Sept. 18 Note, 10 days 500 July 26 “ 2 mos. 472 85 ov. 4 Draft, 30 “ 375 Aug. 19 1 mo. 396 12 126. A rectangular field produces 1726 bushels of wheat, at the rate of 82 bushels to the acre. If one side of the field is 726 feet, what is the distance around the field ? 127. How many bricks 8 inches long and 4 inches wide, are required to lay a pavement 176 feet long, 9 feet wide? What will they cost at $13.50 a thousand. GENERAL REVIEW PROBLEMS 341 128. A can do a piece of work in 10J days, B in 11 J days, C in 12 \ days. In what time could they do the work together? 129. An importer buys in France 426 meters of silk at 26.14 francs per meter. He pays 40% duty. What should the entire quantity be marked in order to gain 15% and throw off 5% ? 130. I buy two houses for $9670, giving 14% more for one than for the other. How should I sell the cheaper house to gain 20% ? 131. What is the duty at 42% on 456 dozen pocket knives invoiced at 17s. 4d. a dozen? 132. The net proceeds of a consignment are $1496.72. The rate of commis- sion is4%. The other expenses are : $14.62, drayage ; $26.75, storage; $32.96, sundry expenses ; what were the total sales ? 133. At what price per yard must I sell silk that cost me 5.42 francs a meter, in order to gain 30% ? 131/.. Certain cloth shrinks 5% of its length in sponging. How many yards of the cloth did I have before sponging, if after sponging there are 672J yards ? 135. I purchased a quantity of green coffee. In roasting it there is a loss of 10% of its weight. What was the gross cost of the coffee, purchased at 18 cents a pound green, if after roasting I sell the entire quantity for $288 at 24 cents a pound ? 136. I import from Sheffield, Eng., the following : 120 dozen knives at 18s. 3d.; 215 dozen knives at £l Is. Id. ; 295 dozen knives at £1 3s. 2d. ; 82 dozen knives at £1 7s. 8 d. I pay 40% duty, £1 19s. 2d. consul fees, 15s. 3d. drayage, 3% commission for buying, and $7.90 drayage in Philadelphia. Find the total cost, exchange at 4.86J. 137. If 220 men working 9 hours a day can build a wall 550 feet long, 31- feet high, 2^ feet thick in 268 days, how long a wall should 185 men, working 10 hours a day, build, if the wall is 7J feet high, 2J feet thick, and the men work 275 days ? 138. On June 19, 1908, I give a note payable on demand for $2800 with interest at 4%. I pay $500 each quarter. What amount will take up the note one year from date, there having been three payments made? Use both United States and Mercantile rules. 139. A can do a piece of work in 15 days, B in 18 days, C in 20 days. The three work together until -fa of the work is done, when A retires, leaving B and C to finish it. In what time will the work be done? 11/0. A man invests $25000 as follows: 72 shares of 5% stock at 143| ; 27 shares of 4% stock at 105| ; 22 shares of 5% stock at 109f ; and the balance in stock paying 6% at 115J. What is his total income, brokerage \ % ? 11/1. What is the weight of an iron pipe 18 feet long, the inside diameter being 10 inches and the outside diameter 12 inches, if a cubic foot of iron weighs 450 pounds ? 11/2. A field contains 10 acres; it is in the form of a square. Find the length of the diagonal of the field. 342 GENERAL REVIEW PROBLEMS llf.3. A ball 9 inches in diameter weighs 750 pounds. How many balls of the same metal, three inches in diameter, will be required to weigh as much ? In the following statement, when is the balance due by equation? What amount would settle it September 28, 1908? Dr. John D. Bartram Gr. 1908 Mar. 5 Balance 492 50 1908 Mar. 16 Cash 150 00 U 24 Mdse., 60 days 450 92 Apr. 22 Note, 4 months 350 00 Apr. 15 “ 2 mos. 490 15 May 27 Draft, 90 days 380 00 U 22 “ 10 days 690 15 Sept. 22 Cash 270 00 May 18 “ 1 mo. 212 42 1 lJt.5. On June 22, 1908, I gave William Hafler a note for $1800 and agreed to pay $150 every Jay thereafter until the 29th of June, inclusive, when I would give a new note for the balance due on that day. If the interest is at 6%, what will be the face of the new note? United States rule. llf.6. What is the value of a shaft of iron 18 inches in diameter and 22 feet long, at $16.50 a ton (2240 lb.), a cubic foot of iron weighing 450 pounds? llf.7 . A and B form a partnership, each putting in $3000. At the end of 3 months A draws out $1500 and B $400 ; at the end of 6 months A draws out $300 and B $200 ; and at the end of 9 months A draws out $750 and B $650. At the end of the year they dissolve partnership, having a capital of $3220. How must they divide it? llf.8. Reed & Co. fail in business with liabilities amounting to $65650. The assignees sold the real estate for $19500 and the stock of goods for $2700; col- lected debts owing them amounting to $7200, and expended $1250 in settling up the business. What will William Harper receive if his claim is $12720? lJf.9. The above-mentioned firm 8 years afterwards met with sufficient suc- cess to enable them to pay off their debts in full, with interest at 5%. What would John Baker receive now, if his original claim was $7250? 150. A retail dealer’s private mark is “ Blacksmith ”. How does he mark goods which he buys for $2.40 and sells at a gain of 25% ? Note. — The letters of the private mark, in their respective order, correspond to the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 and 0. 151. A retail dealer’s private mark is “ Charleston.” He has marked a lot of goods at $h.lr per yard. Find the cost of the goods, if his gain is 20%. 152. On June 1, 1908, I purchased through a broker ISO shares of Lake Shore R. R. at 84J and deposited $2000 as margin. On June 18, he sold 75 shares at 87§, and on June 26, he sold the balance for 88R On June 15. a dividend of 4J% was declared. What is due me on June 30? What is my profit? 153. A and B are equal partners in business, and at the time of settlement it is found that A has to his credit $4250.75, while B has overdrawn his account $1920.65. There are no other resources or liabilities. How shall they settle? GENERAL REVIEW PROBLEMS 343 15 If.. John R. Lamper and R. E. Butterwick are masons and form a partner- ship to do job-work. They agree to share equally. Each partner keeps an account of the labor he has performed, the expenses he has paid and the collec- tions he has made. Lamper’s labor amounted to $2850.62. Butterwick’s labor amounted to $3002.65. Lamper paid for expenses $196.15, but lost 10% of his labor accounts. Butterwick paid for expenses $212.62, and failed to collect $302.25. Adjust each partner’s account. 155. John Elarper begins a business January 1, 1908, which is to consist of preparing building lumber. He rents a factory for $300 a year. He bu} T s raw lumber amounting to $2550 and sells during the year finished lumber to the amount of $1475.22. He has paid $217.32 during the year for necessary expenses, and for labor $725.38. Find his loss or gain for the year, if the material on hand inventories $2175. 156. James Gorvin, of Kansas, and Henry Trexler, of Allentown, Pa., engaged in buying and selling western horses. Trexler advanced to Gorvin $5200 during a certain period, and Gorvin purchased horses to the amount of $9250. At one time Gorvin shipped to Trexler horses valued at $4250, and at another time horses valued at $8720. During the period Trexler made sales amounting to $6290, and Gorvin made sales amounting to $2420. Trexler spent for expenses incident to the business $290.68, and Gorvin $320.75. They agree to discontinue the partnership, and from an inventory taken it is found that Gorvin has horses on hand valued at $1720, and Trexler has horses on hand valued at $2950. If, as a part of the settlement, each agrees to take the horses he has on hand, find their loss or gain and how much is due one from the other. 157 . A and B united in the purchase of two car loads of coal to be divided between them. The invoice was as follows: Date Car No. Egg Stove Price Total Tods cwt. Tons cwt. % cts. % cts. July 9 U 44303 47363 20 00 20 00 2 2 65 75 53 55 10S 00 00 ~oo The freight was $1.70 a ton on 40 tons. The coal was weighed by the wagon load as delivered. A received 23625 lbs. of the egg coal and 23375 lbs. of the stove coal ; B received 22000 lbs. of the egg coal and 21900 lbs. of the stove coal. In settling for the coal and freight, A paid $91 and B paid $85 ; was this settlement correct ? 158. If 15% of a certain number is 2232 more than 3% of three times the number, what is the number? 159. Received an invoice of goods, 14% of which was unsalable ; at what per cent, above cost must I sell the remainder in order to clear 30% on the whole invoice? 344 GENERAL REVIEW PROBLEMS 160 . Find the net amount of the following invoice of glassware: 25 doz. goblets @ 60 cents, 40% and 10% off; 40 doz. goblets @ 55 cents, 35% and 5% off ; 50 doz. dishes @ 85 cents, 50% and 15% off; 60 doz. dishes @ $1.25, 30% and 3% off. 161 . A Georgia planter shipped 400 bales of cotton, averaging 510 lb. each, to a Philadelphia broker who sold it for him at 8J cents a pound, charging 3% commission and $73.40 for other expenses. The broker remitted the proceeds by draft, purchased at |% premium ; what was the face of the draft? 162 . The International Navigation Company insured a vessel and cargo for f of their value with the London Marine Insurance Company at 3J%. The London Marine reinsured | of the risk with the Transatlantic Insurance Com- pany at 2f %. The vessel was lost at sea, and the loss of the Transatlantic Com- pany was $1200 more than that of the London Marine. What amount did the International Navigation Company lose? 163 . A merchant bought goods as follows: Jan. 1, 1908, $2234.76; Feb. 1, 1908, $1362.84; Mar. 15, 1908, $6327; May 15, 1908, $7415.80; July 1, 1908, $4225.66. On each of these bills he was entitled to a credit of four months. What does he owe December 1, 1908? 161 ,.. William Hendrickson bought of Levi Morris merchandise amounting to $6324.78, terms 30 days, and gave in payment his note at 90 days; what was the face of the note? 165 . A note for $5000, with interest at 4%, dated Jan. 6, 1908, has the following indorsements: Apr. 1, 1908, $1000; June 1, 1908, $1000; Aug. 12, 1908, $1000 ; Oct. 1, 1908, $1000. What is due December 1, 1908, by the Mer- cantile rule? 166 . A can do a piece of work in 14 days, B can do it in 20 days. If C works with them, all three can do the work in 6 T 4 T days. A commences the work and continues for 3 days; B then relieves him, and works 5 days; B and C work together for the next two days. How long does it then take A and B to finish the work ? 167 . On January 8, 1906, I bought $12000 worth of 6% railroad bonds at 116J, brokerage \%. On May 1 and November 1 of each year I cashed the coupons, and on September 8, 1908, 1 sold the bonds at 114|, brokerage \ %. How much less did I receive than if I had loaned my money on mortgage at 44% ? 168 . A steamboat that can run 18 miles an hour with the current, and 14 miles an hour against it, requires 16 hours to go to a certain point and return. What is the distance? 169 . A Philadelphia firm of exporters and importers, having received an account sales from its agents in London showing net proceeds £784 14s. lOd. to its credit, directs the agents to remit a draft for 10248.75 francs to one of the firm’s creditors in Paris, and forward the balance by draft on Philadelphia. If Paris exchange is 25.18 at London, and Philadelphia exchange 4.89, find the face of each draft. GENERAL REVIEW PROBLEMS 345 170. If it requires 125 reams of paper to print 5000 copies of a book of 280 pages, each page being 5J X 8J inches, how many reams will be required to print 10000 copies of a book of 176 pages, each page 6J X 9J inches? 171. Henkel & Co. made an assignment for the benefit of their creditors, with the following resources and liabilities: Resources. — Cash, $3263.17; Mdse., $17324.80 ; Real Estate, $12000 ; Bills Receivable, $26340 ; Accounts Receivable, $32725.62. Liabilities. — Bills Payable, $22637.50; Jones & Hoyt, $25784.32; Clark & Co., $18737.90 ; Hanson & Adams, $34859.18 ; William E. Davis, $6284.38 ; Henry Thompson, $19374.62 ; Lavelle & Wilson Co., $27136.40 ; Morton & Haskell, $7929.82. The expenses of the assignment were $3214.85. Show how much each creditor receives. 172. Robert F. Allison and Peter J. Crosby form a copartnership, with the understanding that each is to share in gains or losses in proportion to his net investment. Allison invests $10000, January 1, 1908; $3000, March 10 ; $4000, June 1 ; and $2000, September 1. He withdraws $475, February 1 ; $350, April 1 ; $275, May 1 ; $500, July 1 ; and $675, October 1. Crosby invests $8000, January 1, 1908 ; $2000, April 1; $1000, July 1 ; and $1500, August 1. He withdraws $700, March 1 ; $300, June 1 ; $200, August 15 ; and $150, September 1. They gain during the year $3200; how should this be divided on January 1, 1909? 173. Six per cent, interest is charged and allowed on each item in the fol- lowing account. What is the balance due December 31, 1908? Dr. Arthur C. Bennett Cr. 1908 Jan. 14 Mdse., net 3418 20 1908 Feb. 19 Cash 200000 Mar. 4 U 60 days 1725 44 May 20 Note, 30 days 1500 00 Apr. 21 (C 30 “ 378 19 July 15 Cash 1000:00 Aug. 10 u 60 “ 2621 80 Sept. 24 Cash 2500,00 17 If.. What is the value of a wedge of fine gold 8 inches long, 5 inches wide and 4 inches thick at the butt (the other end being a sharp edge)? The specific gravity of gold is 19.26; a cubic foot of water weighs 1000 ounces avoirdupois; one pound avoirdupois equals 7000 grains Troy ; and 23.22 grains of fine gold are worth $1. Note. — T he specific gravity of any solid or liquid denotes the ratio of its weight to the weight of an equal volume of water. 175. The duty of 40% on a quantity of silk imported from France, invoiced at 6 francs a yard, was $520.80 ; how many yards were there in the importation? 176. A farmer bought one hundred animals for $100, consisting of calves at $10, sheep at $3, and hens at 50 cents each. How many of each did he buy? 177. What will it cost, at 34 cents a cubic yard, to make the excavation for a circular pond, 50 feet in diameter, 2 feet deep at the circumference and 5 feet deep at the center, the bottom having a uniform slope from the circumference to a point at the center (in the form of an inverted cone) ? 346 GENERAL REVIEW PROBLEMS 178 . If the pond in Problem 177 be filled within six inches of the top what per cent, of its entire capacity will it contain? 179 . An importer buys an invoice of Italian goods, amounting to 23724 lire, pays the duty at 30%, and marks the goods at 25% above cost. He sells one half of the invoice at this price, but is obliged to sell the remainder at 10% below his marked price. Find his gain on the entire invoice. 180 . A stock of goods worth $24000 is insured in four different companies as follows: in the first for $5000; in the second for $5000; in the third for $4000, and in the fourth for $3000. Each of the policies contains the 80% co-insurance clause. If the goods were to be damaged to the extent of $2368, how much would each company pay ? 181 . What is the interest at 5% on £582 12s. 6d from January 14, 1908, to September 22, 1909? 182 . A note for $3246.27, with interest at 4%, dated May 14, 1904, has the following indorsements: August 20, 1904, $46.27 ; November 12, 1904, $100; April 16, 1905, $100; July 23, 1905, $100; June 11, 1906, $50 ; August 6, 1906, $25; March 4, 1907, $200; June 17, 1907, $500; December 30, 1907, $150; April 14, 1908, $300. How much is due August 1, 1908 ? United States rule. 183 . If it takes 4 men and 7 boys 2 days to do a certain piece of work, or 3 men and 2 boys 3§|- days to do the same work, how long would it take 2 men and 1 boy to do it? 181 ^. Which is the better investment, and how much more annual income will it yield, to invest $32000 in 5% bonds at 92|%, brokerage or in 6% bonds at 117f, brokerage £%, reckoning $1000 as the smallest denomination purchased, and the balance to be deposited in a saving fund at 3% interest? 185 . An invoice of merchandise from Amsterdam, amounting to $12000 guilders, the duty on which was 40%, and freight $432.50, cost altogether $7372.10; at what rate of exchange ivas the draft in settlement purchased ? 186 . How many gallons each of wines worth $1.15, $1.25, $1.40 and $1.70 a gallon, respectively, should be added to 120 gallons of wine worth $1.45 a gallon to form a mixture worth $1.35 a gallon? 187 . A and B form a partnership, agreeing to invest equal amounts and share equally in gains or losses, and further agreeing that neither shall with- draw any money from the business for a period of five years, and that in case the yearly statement shall show a loss at the end of any year, the} 7 will make additional equal investments sufficient to make up the loss. At the end of the first year their books show a net gain of 12J% ; at the end of the second year a net loss of 8% on the capital at the beginning of this year, which they make good by additional investments; at the end of the third year, a net gain of 20% on this year’s capital; at the end of the fourth year a net loss of 5%, which the} 7 make good; at the end of the fifth year, a net gain of 18%. Each partner now has $12744 to his credit. What amount did each invest on commencing business, and what was his average profit per cent, per annum on his original investment? GENERAL REVIEW PROBLEMS 347 188. If a cistern 16 feet long, 8 feet wide and 12 feet deep can be filled in 8 hours by 2 pipes, each 1 inch in diameter, how long will it take 3 pipes, each T 6 g- of an inch in diameter, discharging twice as fast, to fill a cistern 18 feet long, 7 feet wide and 9 feet deep ? 189. What is the balance due on the following account June 30, 1908, interest at 6% ? Dr. B. F. Dodge in account with Kelly & Co. Or . T. E2 i H 02 g 02 >< <5 | 53 02 > < g 53 P a H 53 e fc § 1908 M < j 1908 Hi < Jan. 4 Mdse., net *** ** ** 2312118 I Jan. 29 Cash *** ** ** 2000 Feb. 11 ({ 30 d. *** * ** 46750 Feb. 26 i( *** * ** 350 Mar. 15 (( 60 d. *** ** ** 712 84 Mar. 1 (6 *** * ** 200 Apr. 21 a 30 d. ** * ** 327 48 Apr. 26 U ** * ** 500 May 19 a net ** * ** 674 32 May 31 it ** * ** 500 June 3 (( net ** * ** 925 75 June 30 Interest bal. ** ** « 30 Interest bal. ** ** a 30 Balance ** *** ** ** *** ** ** June 30 Balance ** 190. The specific gravity of iron is 7.84; how much will a hollow globe of iron weigh, whose diameter is 8 inches, and the thickness of the metal f of an inch ? 191. How many feet of inch lumber will be required for the floor of a plat- form 30 feet long, and 20 feet wide (from front to back), and for a roof of the same material, the roof to project one foot beyond the edge of the platform on all sides, and to be supported by posts 16 feet high above the floor in front and 12 feet above it at the back ? What will the lumber cost, at $22 per M? 192. A certain excavation consists partly of clay and partly of rock. If it were all clay, A could do the work in 20 days ; if it were all rock, he could do it in 36 days. If it were all rock, B could do the work in 32 days ; while if it were all clay he could do it in 18 days. A and B work together on the job and finish it in lO-gy-g- days ; what per cent, of the work is clay, and what per cent, rock ? 193. How many $1000 U. S. 4s at 122 J, brokerage §•%, can be bought with $32500, and what will be the balance remaining ? 191/.. If a man purchases $12000 U. S. 4s at 123§, $5000 railroad 5s at 102| and $8000 railroad 6s at llOf, through a Philadelphia stock broker, what rate of income does he receive on the total investment? 195. How many thousand ordinary bricks will be required for the walls of a round tower 12 feet in diameter, inside measurement, and 50 feet high, the walls being 2 feet thick for the first 15 feet, 1 \ feet thick for the next 15 feet, and 1 foot thick for the remainder of the height? 196. A man purchased a •§ interest in a factory, and afterwards sold 18% of his share at a loss of 7 % for $1699.95 ; what was the value of the whole factory at the rate at which he purchased his share ? 348 GENERAL REVIEW PROBLEMS 197. 8 lbs. of tea and 5 lbs. of coffee of a certain grade are worth $6.10, and 3 lbs. of tea and 7 lbs. of coffee of the same grade are worth $3.62; what is one pound of each worth ? 198. A grocer sells a blend of coffee at 34 cents a pound, which he claims to be § Java at 33 cents, and £ Mocha at 36 cents; the Java coffee costs him 30 cents a pound and the Mocha 32 cents, so that he is ostensibly making a profit ofl0-f%. But the mixture actually contains \ Rio coffee, which costs the grocer only 18 cents a pound ; what per cent, of profit does he really make ? 199. A commission merchant received a consignment of cotton, consisting of 260 bales averaging 510 lbs. to the bale, which he sold, and, after deducting his commission at 3%, remitted $10128.98 proceeds; at which price per pound did he sell the cotton? SCO. The premium paid for insuring a stock of merchandise at 2J% for £ of its value was $689.87 ; what was the value of the merchandise ? 301. A borrowed from B $1000 at 6% interest, agreeing to pay $100 at the end of each year, on account of principal and interest, until the whole obligation was canceled; how many yearly payments did he make and what was the amount of his last payment ? 302. Exchange being quoted at 5.14, what will be the cost of a draft in settlement of an importation of 6 dozen French clocks, invoiced at 22.50 francs each; for how much apiece must they be sold in Philadelphia to gain 30%, if the duty is 25% and other charges in Philadelphia amount to $37.40 ? 203. If 14 men can build a wall 60 feet long, 5 feet high. If feet thick in 20 days, how many men will be required to build a wall 340 feet long, 44 feet high, 24 feet thick in 50 days ? 201^. If a lot of goods is bought for $327.42, and f of it is sold at a gain of 30%, and £ of the remainder proves unsalable, how should the other f of the remainder be marked so that a discount of 10% may be allowed and yet enough gained to net 25% profit on the whole purchase ? 205. A piece of ground in the form of a circle has a diameter of 16£ rods ; what will it cost, at 20 cents a cubic yard, to dig a ditch around it, outside, 24 feet wide, and 3J feet deep ? 206. What per cent, is gained by buying coal at $3.75 a long ton and selling at $5 a short ton ? 207. A boy’s father deposits $300 in a savings bank w r hen the boy is 16 years old. If this bank allows 1 % interest quarterly, and balances its books quarterly but reckons no interest on fractions of a dollar, how much will there be to his credit when he reaches the age of 21 ? 208. A commission merchant in New York sells for me 1200 baskets of potatoes at 48 cents a basket; he pays freight and other charges $112.80; deducts his commission of 3% ; and invests the proceeds in flour, charging 2% for purchasing. If he buys 100 barrels of flour, and remits a balance of $7. IS, at what price does he purchase ? GENERAL REVIEW PROBLEMS 349 209. If an insurance company paid $3920, under its policy of $20000 with u average clause,” on a loss of $6272, what was the value of the property insured ? 210. A works 3 days on a certain piece of work, then B works 4 days, after which A finishes the work. If it takes A alone 21 days to do the work, or A and B together 12 days, how long will it take A to finish it ? 211. Sold $15000 U. S. 4s at 121f, and invested the proceeds in railroad stock at 48f, which was afterwards sold at 52J. What was the gain on the stock, if Philadelphia rates of brokerage were paid? '212. The assessed valuation of the property in a certain town is $2347896. The amount to be raised by taxation is $48394.50. What is the tax rate ? What is the tax on $100 ? 213. What is the duty at $1 per pound on an importation of partly manu- factured silk from France, weighing 874 pounds and invoiced at 9098.45 francs, with the proviso that “ in no case shall the duty be less than fifty per centum ad valorem of the invoiced cost” of the goods? 21/, l. What would be the cost, exchange at 5.14, of a draft to pay for the importation in Problem 213? 215. How many pounds each of teas worth 42, 48, 54, 60 and 75 cents per pound, respectively, should be taken to form a mixture of 80 lbs. at 55 cents? 216. If it takes 22 men, working 8 hours a day and 5 days a week, 4 weeks to do a certain piece of work, how many weeks will it take 40 boys, working 10 hours a day and 4 days a week, to do half as much, 9 boys doing as much as 7 men? 217. Find the contents of a triangular prism 4 feet high, the sides of the base being 14, 19 and 26 inches, respectively. 218. Bought a car load of wheat, 34263 lbs., at 83 cents a bushel, and a car load of corn, 32796 lbs., at 56 cents a bushel. Sold the wheat at a profit of 20% and the corn at a profit of 16§ %. What was the total gain? 219. A, B and C form a partnership, agreeing to divide gains or losses equally, under the following conditions : B is to be general manager at a salary of $2000 per annum, and C bookkeeper at $1500 per annum, their salaries to be credited quarterly at the end of each quarter. Interest is to be charged and allowed at 6% per annum. A is to furnish J of the capital, and B and C each \. A invests January 1, $12000 ; B $6000, and C $6000. A withdraws March 1, $675; June 1, $500, and September 1, $900. B withdraws February 1, $300; May 1, $500 ; August 1, $400, and October 1, $375. C withdraws April 1, $500; July 1, $500; September 1, $400, and November 1, $400. A invests $2000 additional on July 1. The net gain for the year, exclusive of salary and interest items, is $4760.83 ; what is the balance of each partner’s account ? 220. A pile of coal in the corner of a cellar reaches within 1 foot of the flooring above, which is 8 feet above the cellar floor, and extends 10 feet along each wall at the bottom of the pile, the pile having a plane slope (so that it forms J of a pyramid). How much coal, dong tons, is there in the pile, reckoning 80 lbs. to the bushel of 2150.4 cu. in. ? 350 GENERAL REVIEW PROBLEMS 221. What amount is due on the following account January 1, 1908, interest at 6% ? Dr. Ambrose C. Headley Or. 1908 1908 Jan. 4 Mdse. net 757 14 Mar. 11 Cash 1500 00 Feb. 10 u 30 days 1213 19 Apr. 15 (( 800 00 Mar. 18 u 60 days 472 18 May 1 Note, 30 days 215 00 Apr. 15 u 30 days 327 65 June 15 Cash 500 00 June 9 a net 782 80 July 1 U 300 00 July 22 u net 485 25 Aug. 10 U 200 00 Aug. 19 u 60 days 379 92 Sept. 1 Note, 30 days 300 00 Sept. 30 (( 3 mos. 915 20 Oct. 15 Cash 750 00 Nov. 3 u 30 days 455 70 Dec. 15 u 60 days 1114 22 222. A certain country paid 4J% interest on its debt. A war increased the debt 25%. During peace which followed, the debt was diminished $25000000, and the rate was reduced to 4%. The annual interest was then the same as at first. What was the indebtedness of the county before the war? 223. A piece of gold, alloyed with silver, is 14 carats fine and weighs 72 pennyweights. How much gold must be added to make it 18 carats fine? 22 If.. A man drew out 50% of the amount he had to his credit in bank, and gave 16§ % of the money withdrawn for a house, for w T hich he paid $4000. He had the house insured for $3000 at f %. He invested the balance of the money withdrawn from bank in wheat, which he sold for cash, J at an advance of 30%, | at an advance of 18%, and | at a loss of 71%. The house was destroyed by fire and the insurance company paid the amount of their policy, which the man deposited in bank, together with all his other receipts. How much had he in bank originally, what was the final balance of his bank account, and what was his per cent, of loss or gain ? 225. A Philadelphia commission merchant received a consignment of wheat to sell, the total w r eight of which w r as 423632 lbs. He paid freight and other charges amounting to $1080.50; sold at 98 cents per bushel; charged 21% commission; and remitted the proceeds by bank draft on Chicago, purchased at |% premium. How much did the consignor receive? 226. A and B have 15 acres to plow. At the end of 1J days A leaves, and B finishes in 3f days. If B had left instead of A, it would have taken A 21 days to finish. How long would it take each to plow the field alone? 227. If I buy 4% bonds maturing in 28 years, at 125J, how much per cent, greater or less is my annual income than if I buy 4% bonds maturing in 10 years, at 112? (Brokerage \ % in each case.) 228. A walks at the rate of 3f miles an hour and starts IS minutes before B. At what rate per hour must B walk to overtake A at the ninth mile-stone? GENERAL REVIEW PROBLEMS 351 229. Divide 175 into four such parts that the first plus 2, the second minus 3, the third multiplied by 4, and the fourth divided by 5, shall be equal to each other. 230. Find the equated date for payment of the balance of the following account : Dr. Harolb S. Warner Or. 1908 Apr. 7 Mdse., 60 days 2314 75; 1908 June 9 Cash 2000 00 June 11 “ 30 days 6218 90 July 15 U 5000 00 Aug. 19 “ net 572 20 Aug. 27 Note, 30 days 2000 00 Oct. 7 “ 30 days 376 19 I Nov. 12 Cash 250 00 Dec. 2 “ net 1042 18 231. The specific gravity of silver being 10.4, what will a spherical ball of silver weigh whose diameter is 7 inches? 232. A merchant added $1700 to his capital the first year; during the second year he further increased it by a sum equal to 10% of his original capi- tal ; during the third year he lost 40% of what he had at the end of the second year, and found that he then had just wdiat he began with. What was his original capital ? 233. If a draft on London for £2348 13s. Id. cost $11482.11, what was the rate of exchange? 234.. A man borrowed enough money at 6 per cent, to pay for a house and also for repairs amounting to 2% of the purchase money. The house was vacant for a year, and during that time he had to pay $34.60 taxes. At the end of the year he sold the house for $4600 and found his net loss to be 6§ % of the pur- chase price. What did the house cost? 235. How many square yards are there in the surface of the sidew T alk around a city square 244 feet X 480 feet on the building line, if the sidewalk is 18 feet wide ? 236. A and B have equal incomes. A’s expenses are 16f % more than his income, while B lives on 75% of his. At the end of three years, B lends A enough money to pay his debts, and has $150 left. What is the income of each ? 237. What per cent, is gained by buying silk at 4.95 francs a meter, and selling it at $1.87J a yard, if the duty is 50% and exchange 5.14J? 238. How many gallons of water will a cylindrical boiler hold, 12 feet long and 5 feet diameter, inside measurement, if there are 160 flues, each 3 inches outside diameter, passing through it lengthwise? 239. A merchant increased his capital the first year by 10% of itself ; the second year he gained 20% ; the third year he lost 25%. He then had $100 less than at first. What was his original capital ? 211.0. At his death, A’s property was valued at $42360 ; $27360 being in real estate and the remainder in personal property. His heirs were the widow, three sons and two daughters. The will provided that the widow should receive ^ of the real estate and 20% of the personal property and that the remainder should 352 GENERAL REVIEW PROBLEMS be divided among the sons and daughters, each daughter receiving 25% less than a son. What was the share of each? 241. What per cent, of a kilogram is a pound? 21$. In the following account find the balance due February 1, 1909, reck- oning by “ daily balances,” charging 6% interest on debit balances and allowing 4% interest on credit balances. Dr. R. J. Melrose Or. — E- m — E- m DAYS g W H Eh P O DAYS ® H « g Eh a 1909 Z 5 1909 £ Jan. 4 500 Penna. 26187 50 Jan. 1 Balance 42128 17 U 7 1500 Readg. 18177 50 CC 4 Interest ** CC 12 400 Erie pfd. 28900 cC 7 CC * CC 12 Interest * * 'fc ¥ CC 18 1500 Readg. 18875 CC 18 cc * % ** cc 22 400 Erie pfd. 29500 22 * jj; ** CC 26 500 Penna. 26750 Feb. 1 Interest bal. ** CC 26 Interest * ** cc 1 Balance ^ >K ^ 5^ >j< 1 Feb. 1 »C * ** ** CC 1 Interest bal. JjC JjC jfc ^ ** ** ** 1909 Feb. 1 Balance 24S. Sent 12000 bushels of wheat to my agent, which he sold at 58 cents per bushel. He paid expenses $214.52 and deducted his commission of 21%. He then invested the proceeds in sugar at 5| cents per pound, commission for buy- ing 2%. How many pounds did he purchase? 244 - A merchant’s sales increased the second year 20% over the first year; the third year 25% over the second, and the fourth year 40% over the third. During the four years he sold $131250 worth of goods. What was the amount of his sales the first year? 245. James Farley has $352 in bank and has a note for $850 coming due to-day which has been made payable at his bank. For how much must he give his note at four months, dated to-day, that when discounted the bank may pay his $850 note and leave a balance to his credit at $55.25 ? 246. How much must I invest in U. S. 4s at 122J, brokerage |%,to secure a quarterly dividend of $460? 247. When is the following account due bj r equation, and what amount is due June 1, 1909 ? Dr. L. T. Spencer Or. 1908 Jan. 1 Balance 694 87 1908 Feb. 20 Cash 550 00 CC 28 Mdse., 30 days 1400 00 a 28 Draft, 20 days 700 00 Mar. 20 “ net 510 40 Mar. 31 Cash S45 50 Apr. 11 “ 10 days 383 S3 GENERAL REVIEW PROBLEMS 353 248. What amount will be required on September 16, 1909, to pay the balance due on a note for $720, with interest at 4%, dated September 17, 1908, upon which the following payments were made : October 8, 1908, $125 ; January 15, 1909, $15 ; February 23, 1909, $333 ; March 1, 1909, $225 ? U. S. Rule. 24-9. A merchant sold a lot of goods at 124% off list and another lot of the same value at 20% and 10% off list. He then allowed, on the 'whole invoice, a discount of 3% for cash. What was the value of each lot of goods at list price, if the cash paid was $3841.20? 250. If the interest on $510 at 6% for 4 years and 9 months is $145.35, what will be the interest on $1350 for the same time at the same rate ? 251. An importation of 1312 meters of silk from Lyons, France, was invoiced at 3.5 francs per meter. The importer purchased a bill of exchange to pay for it at 5.15. The duty was 50%, and other expenses $42.50. At what price per yard must the silk be sold to gain 25 % ? 252. Suppose a globe to be inclosed in a cylinder that will exactly contain it, and the cylinder is to be inclosed in a cube that will exactly contain it ; what decimal part of the volume of the cube is the volume of the cylinder, what part of the volume of the cylinder is the volume of the sphere, and what part of the volume of the cube is the volume of the sphere? 253. If the diameter of the base of a cone and the diameter of a hemisphere are equal, and the altitude of the cone is equal to the radius of the hemisphere, what is the ratio of the volume of the cone to the volume of the hemisphere? 254 . Find the cost of a sheet-iron smoke-stack 40 feet high and 2 feet in diameter at 15 cents per square foot. 255. Merchandise weighing 8743 lbs. is transported 1472 miles at a through rate of $3.24 per 100 pounds. The first road carried it 329 miles, the second 658 miles, the third 219 miles, and the fourth forwarded it to its destination. How shall the freight be apportioned, the fourth road receiving a 2-cent terminal ? 256. An importer received an invoice of 350.5 meters of silk at 5.85 francs per meter. He paid duty at 50%, and other charges $25.60. Exchange for his remittance cost him at the rate of 5.13J. What price per yard must he receive for the silk in order to gain 25 % ? 257. Bought goods at five dollars a gross, 20% off, and sold them at fifty cents a dozen, 20% and 10% off. Is there a profit or a loss, and what is the rate per cent. ? 258. When is the balance of the following account due by equation ? Dr. Francis E. Hoyt Cr. " 1908 1908 May 8 Mdse., 30 days 0 0 CO oo 1 May 20 Cash 150 00 June 20 “ 60 days 463 29; July ; 1 Note, 1 mo. 412 50 Aug. 28 “ net CO - 1 -1 89: Sept. | 2 Cash 200 00 354 GENERAL REVIEW PROBLEMS 259. At 63 cents a square foot, what is the cost of a building lot having two parallel sides, respectively 140 feet and 116 feet long, and 83 feet apart? 260. If a draft on Philadelphia, payable 60 days after date, was bought in St. Louis for $3723.29 at \ % premium, what was its face? 261. Find the height and area of a triangle whose base is 42 feet long, the other sides being each 35 feet long. Note. — Observe that this triaDgle is equal to two rigid-angled, triangles, each having a hypot- enuse of 35 feet and a base of 21 feet. 262. A man sold two properties for $8522.50 ; on the first he lost 124%, and on the second he gained 30%. What did he gain on the whole transaction, if f of what he gave for the first property was equal to J of what he gave for the second ? 263. Having sent a New Orleans agent $1836.46 to be invested in sugar, after allowing 3% on the investment for his commission, I received 32400 pounds of sugar. At what price per pound did the agent purchase? 264.. A buys a bill of goods amounting to $6776.40, 20% and 10% off on the following terms : “Four months, or less 3% for cash.” He accepts the latter, and borrows the money at 6% to pay the bill. How much does he gain? 265. What is the weight of a pint of alcohol, if its specific gravity is .792? 266. A draft on Philadelphia for $9375.15 was purchased in Chicago at \ % discount. If the draft cost $9300.16, at how many days after date was it drawn ? 267. A and B together can do a certain piece of work in 11J days; A and C together can do it in 10|- days ; B and C together can do it in 14-f days. If A and B work together for 2 days, and then A continues alone for 3 days, how long would it take C to finish the work alone ? 268. A commission merchant in Chicago sells for me 12 bales brown sheet- ing, each bail containing 800 yards, at 7 cents per yard ; pays transportation and other charges amounting to $72; and invests the proceeds in flour. If he charges 24% for selling and for purchasing, and sends me 120 barrels of flour, at what price does he buy ? 269. What is the balance due on the following account November 1, 1908, interest at 6 % ? Dr. Walter G. Farnsworth Oi'. 1908 1908 Mar. 27 Mdse., 30 days 243748 Apr. 7 Cash 2500 U 30 “ net 51760 June 1 (( 3000 May 12 “ 30 days 3224 71 July 1 Draft, 10 days 2500 28 “ net 327 14 Aug. 1 9 Cash 4000 June 10 “ 30 days 1895 36 Sept. 17 U 1800 July 21 “ 30 “ 524307 Sept. 8 “ 30 “ 678 25 Oct. 15 “ 30 “ 132427 GENERAL REVIEW PROBLEMS 355 370. A merchant sold a bill of goods at 20%, 10% and 5% off list price, and allowed a discount of 3% for cash. What was the list price, if the cash paid was $3481.20? 371. Since the volumes of similar solids are to each other as the cubes of their corresponding dimensions, how many steel balls \ of an inch in diameter will weigh as much as one 2 inches in diameter? 373. A column 10 inches in diameter and 18 feet high is to be gilded with a spiral stripe 2J inches wide, winding around it so that the turns are 7 inches apart. What will it cost for the gold leaf needed, at $3.20 per book of 100 leaves 3X4 inches, estimating that one-sixth of the material will be wasted ? 373. An exporter shipped 400 cases of canned goods to Liverpool, invoiced at 14s. 10 d. per case and drew on the consignees for the amount of the invoice, selling the draft at 4.87. He shipped a similar lot, 400 cases, to Havre, invoiced at 18.70 francs per case, and sold his draft on consignees at 5.15. For which draft did he receive the more money, and how much was the difference? 37 It.. How many cubic inches of lead will weigh as much as 12 bushels of wheat? (The specific gravity of lead is 11.35.) 375. A note for $327.50, with interest at 5%, dated March 21, 1908, has the following indorsements: June 1, 1908, $25; July 15, 1908, $10; September 3, 1908, $50; October 20, 1908, $100 ; November 10, 1908, $20; December 8, 1908, $30. How much is due February 17, 1909? 376. If I pay $9.48 interest on $555 for 123 days, what is the rate per cent. ? 377. Loaned $2400 at 6% simple interest until it amounted to $2998.80. For what time was the loan made? 378. Two clerks have 2000 circulars to address. One .quits at the end of 4 hours, and it takes the other 10 hours to finish. If the one who left had remained 2 hours longer, the other could have finished in 5 hours. How many can each address in 1 hour? 379. Bought in Manchester, England, 8 gross of razors at £3 13s. 6d. a dozen, less 10% and 5% ; what sum in United States money is equivalent to the net amount of the invoice? 380. A commission merchant sold 83748 lbs. of cotton at 7f cents a pound, paid transportation $368.72, and cartage $8, and charged 3% commission. What per cent, of profit did the consignor make, if the cotton cost him $5600 ? 381. A shipment of 300 cases of merchandise, valued at $87.50 a case, was insured for $20000, the policy containing the “ average clause.” In consequence of a fire at the railroad depot, 42 cases of the goods were partly damaged ; and the proceeds of the damaged lot, when sold by auction, amounted to $843.50. What amount was recovered on the policy ? 383. A can do f of a certain piece of work in 4 days ; B can do -f- of it in 3 days ; C can do f of it in 7 days. If A begins the work alone and works for 3 days, and then B and C relieve him, working together for 2f days, how long would it take A and C together to finish the work ? 356 GENERAL REVIEW PROBLEMS 283. Received $1436.84 on September 10, 1908, as the amount of a loan at 4J%, made June 14, 1907 ; what was the principal? 284.. A speculator deposited $5000 witli his broker, who 10 days later pur- chased for him, 500 shares of stock at 92§- (brokerage \%). The stock was sold 23 days afterward at 94f- (brokerage \%), and settlement made the same day. How much did the speculator receive from the broker (interest at 6%)? 285. The tax on a certain property, at $18.93 on $1000, amounts to $460.32 ; what is the assessed valuation of the property ? 286. What is the duty on 680 cubic feet of marble, invoiced at 21465 lire, at $1 per cubic foot and 25% ad valorem? 287. A draft on San Francisco for $6850.75, payable 60 days after date, was purchased at f% discount; how much did it cost, money being worth 6% ? 288. How much tea at 22 cents, 28 cents and 50 cents a pound must be mixed with 45 pounds at 64 cents a pound, so that the whole may be sold at 48 cents a pound and produce a gain of 20% ? 289. X and Q are partners, sharing gains or losses in proportion to average net investment. On January 1 X invests $18000, and Q invests $15000. X draws out $350 at the end of each month during the year, and Q draws out S300 on the 15th of each month. At the end of the year they have a net gain of $4350 to divide ; how much does each receive ? 290. What is the cash balance of the following account on January 1, 1909 ? Dr. Calvin Thompson Cr. 1908 | 1908 Jan. j 5 Mdse., 60 days 389 26 Apr. 1 Cash 600 Feb. 4 U 30 “ 414 70 June , 1 “ 500 Mar. 12 (( 90 “ 853 25 Aug. 2 “ 500 Apr. 10 u net 721 19 Oct. 1 “ 1000 June 17 a 90 days 436 20 Dec. 15 “ 1500 Aug. 12 u 30 “ 525 00 Sept. 9 u net 324 50 Nov. 10 u 90 days 622 80 Dec. 122 “ 60 “ 875 70 291. An ice company has its men at work on a lake cutting 10-inch ice. If ice weighs 58 lbs. to the cubic foot, and 20300 lbs. can be loaded in a car, bow long a strip 100 feet wide will the men cut to fill 40 cars ? 292. In digging a well 4 feet in diameter, 37 cubic yards ot earth were taken out; what was the depth of the well ? 293. At what rate will $2468, in 1 yr. 1 mo. 13 da. amount to $2744.28? 294 ■ The net proceeds of a sale were $3214.78 ; the charges $212.50, and 4% commission. What were the gross proceeds ? 295. Goods that were purchased for $2319.75, less 30%, 10% and 4%, were marked 40% above net cost and sold at 20%, 124% and 3% off the marked price ; how much was the loss ? GENERAL REVIEW PROBLEMS 357 296. How many reams of paper, in sheets 24X38 inches, will be required to print 12000 copies of a book of 288 pages, each page measuring 6X9| inches (untrimmed) ; and how much will this amount of paper cost, at 6 cents per pound, if it weighs 60 pounds to the ream ? 297. What per cent, of the volume of a pound of lead is the volume of a pound of gold ? Note. — This is a problem in compound proportion. The specific gravity of lead is 11.35 ; that of gold 19-26. A pound of gold weighs 5760 grains ; a pound of lead, 7000 grains. 298. The polar diameter of the earth is 7899 miles; the equatorial diameter is 7925J miles. How many cubic miles less is the volume of the earth, than if it were a perfect sphere 8000 miles in diameter ? 299. A hexagonal prism is 5 feet 6 inches in height ; the distance from the center of the base to the center of each side of the base is 17.32 inches ; and the length of each side of the base is 20 inches. Give the contents of the prism in cubic feet. 300. A, B and C do a piece of work. The portions done by A and B together equal ^ of the whole work ; the portions done by B and C together equal T 7 T of the whole. What part of the work was done by B? 301. The sides of two square fields are to each other as 3 to 5, and the total area of the two fields is 3f acres ; what is the length of a side of each field, in yards ? 302. If the driving wheels of a locomotive are 4 feet 4 inches in diameter, at the rate of how many miles per hour is the locomotive running when the driving wheels are making 312 revolutions per minute? 303. A, B and C enter into partnership on March 1, 1907, A furnishing J of the capital and each of the others J; they agree to share gains or losses in this proportion, each partner being charged 6% interest on any sums he may with- draw during the year. A invests $10000, B and C each $5000. A withdraws during the year as follows: May 6, 1907, $212.50; July 23, 1907, $67.20; October 12, 1907, $132.70; January 20, 1908, $74.75; February 3, 1908, $65. B withdraws : April 16, 1907, $84.30 ; July 2, 1907, $105 ; September 28, 1907, $38.75 ; December 24, 1907, $165.80. C withdraws: June 10, 1907, $90; August 31, 1907, $123.40; November 27, 1907, $218.50. On March 11, 1908, before the interest on withdrawals has been computed, the books of the firm show a net loss of $1837.92 ; what is the balance of each partner’s account after the books have been closed for the year? 30 J. A commission merchant in Philadelphia sold 84723 lbs. of corn at 48|- cents a bushel, paying $32.40 charges, and receiving a commission of 2J%. He then purchased merchandise according to the consignor’s directions, the invoice amounting to $324.30, on which he received a commission of The balance due the consignor was remitted by draft at 30 days after date, purchased at f % discount ; what was the face of the draft ? 358 GENKRAL REVIEW PROBLEMS 305. A room is 27 ft. 3 in. long, 23 ft. 5 in. wide. If carpeted with a border 22 in. wide an allowance of 16 in. must be made on each strip for matching, strips laid lengthwise, carpet 27 in. wide; but if no border is used an allowance of 1 ft. 8 in. mast be made on each strip. What will it cost to carpet the room with a border and without a border, carpet and border costing $1.20 a j^ard? In each case there is a lining 1 yard wide at 10 cents a yard. 306. What will be the cost of carpet f of a yard wide, and selling at $2.25 per yard, and of lining f of a yard wide, costing 30 cents per yard, to cover a room 24 feet long and 20 feet wide; the strips of carpet are laid lengthwise, and there is a waste of 9 inches to each strip of carpet in matching, also an allowance of 10% in width and 6% in length for shrinkage of the lining? 307. The assessed valuation of the real estate of a township is $910887, and of the personal property, $521073 ; it has 3564 inhabitants, subject to a poll-tax of $1.25. The year’s expenses, not including the collector’s commission of 2%, are : for schools, $4400 ; interest, $3850 ; highways, $4560 ; salaries, $3150 ; and for contingent expenses, $8675. If $585 is in the treasury and two hotels each pay a license of $250 into the treasury and the revenue from fairs amounts to $2500, what tax must be levied on a dollar, to meet expenses, and provide a sinking fund of $5000? What would be A’s tax whose property is assessed at $7800 and who pays for two polls? Find the collector’s commission. 308. Ten cylindrical tanks of oil, each 18 inches in diameter and 30 inches long, inside measurement, and weighing 225 lbs. each, were imported from Chris- tiana. The invoice price was 10 crowms per gallon, commission 3^%, consul fees 85 crowns, freight 10c. per cwt., insurance 2% ; an allowance of 1J% was made for leakage. The ad valorem duty was 40% and the specific duty was 10J cents a gallon. A draft to cover all costs in Christiana was purchased in Philadelphia, the exchange being .274. At what price shall the oil be marked per gallon, to fall 20% of the marked price and lose 10% of all in bad sales and still make a gain of 25 % ? 309. An importer of Philadelphia owed on foreign invoices as follows: To C. Shepherd Sons, London, £1600 12s. ; -J. L. Von Buesche, Berlin, 1400 marks ; Perrie, Buzzell & Co., Paris, 3027 francs ; F. Gonzalez, Mexico, 715 pesos. Exchange on London, 4.875; on Berlin, 97J ; on Paris, 5.18J ; on Mexico, 78J. He purchased through his brokers, wdio charged him brokerage ; and issued one check to cover the total cost. What was the amount of the check ? 310. A commission merchant of San Francisco bought for a Philadelphia firm a consignment of 120 chests of green tea, each containing 50 lbs. at 22f cents a pound ; 15 cases of Muscatel raisins, each containing 20 boxes of 50 lbs. each, at 18^ cents a pound ; 10 cases of California prunes, each containing 20 boxes of 25 lbs. each, at 11 J cents a pound; charges 2% commission, 3% guar- anty, and 1J% insurance. The firm in Philadelphia remitted a 30-day draft for the amount due at f % premium. What was its face and what did the firm pay for the draft, if the rate of interest was 6% ? GENERAL REVIEW PROBLEMS 359 311. Find the cost, at $2.08 a } T ard, of carpeting a room 32J feet long, 12J feet wide and 12 feet high, the carpet (which is 22 inches wide) being laid the more economical way, and one foot being allowed to each strip for matching. The room has five windows, each 7| feet by 4J feet ; three doors, each 9J feet by 4J feet ; two fire-places, each 5 feet by 6 feet, and a baseboard 10 inches high. Find the cost of papering the room at 50 cents a double roll, and putting on a border 15 inches wide, which costs 20 cents a yard. What must be the face of a 90-day note, dated and discounted to-day, to cancel the total cost ? 312. I sent my agent in London $10000 with instructions to purchase cloth at 15s. Qd. per yard, after paying consul fees of £15 8s. and charging a commis- sion of 3%. What amount is unexpended, supposing he bought a whole number of yards ? The duty was 20 % ad valorem, 3c. a yard specific, the freight amounted to Jd. a yard, and other charges to $125. After keeping the cloth three months and charging interest at 6% on total cost, I sell it at $4.25 a yard and receive in payment two notes of equal sums, dated to-day, one at 30 days, the other at two months, both of which I have discounted immediately. What is my gain or loss on the investment? 313. Bought a consignment of wool in Buenos Ayres, the gross weight of which was 34675 lbs. @ 15 pesos per lb. ; an allowance of 8% of the gross weight was made for tare; freight charges were 9f cents per cwt. ; ad valorem duty 18f % ; specific duty 3| cents per lb. ; consul’s fee 85 pesos. What was the cost in Philadelphia at § % premium, of a sight draft to cover all expenses in Buenos Ayres ? At 2 % commission what will my agent receive? At what price per lb. must it be sold to make a clear profit of 20%, after allowing 5% for damaged wool, and 10% of my sales forbad debts? 311/.. A Philadelphia soap manufacturer makes a cake of soap 3 inches long, 2 inches wide and J inch thick. He wishes to make a sample cake of the same soap, keeping the proportions of the larger cake, but using only one-half the material. Find the dimensions of the sample size. 315. Borrowed $18360 for one year at 6%. At the end of 3 months I invested in produce through an agent charging 2% ; after keeping the goods 4 months I sold one-half of them at 18f % profit, and the remainder at 15% profit, paying 2% commission for selling. I loaned the money I received at f% a month for the remainder of the year. What was my year’s profit ? 316. A legacy of $436.85 was left to a minor ; it was invested at 4 J% com- pound interest, and when the legatee w T as 21 years old it amounted to $750. How old was the child when the bequest was made? 317. A public square 200 yards on the building line has -walks 20 feet wide around it, and two walks of the same width diagonally across the square. The cost of paving is $1.50 per square yard. For -what sum must a 60-day note be drawn to-day that when discounted will pay the debt ? 318. A factory worth $3000 and its contents are insured for $10000, as fol- lows : $2000 on the building, $3000 on machinery worth $5000, and $5000 on 360 GENERAL REVIEW PROBLEMS stock worth $8000. The building is damaged by fire to the amount of $1000, the machinery $4000, and the stock is a complete loss. What was the premium at ? If the risk is covered (1) by an ordinary policy, (2) by a policy con- taining the “average clause,” (3) by a policy containing the 80% co-insurance clause, what was the owner’s loss ? What was the company’s loss? 319 . What is the difference in volume between two wedges, each being 60 inches in altitude and 25 inches broad at the base, but the base of one being 70 inches long and its edge 50 inches, while the base of the other is 50 inches long and its edge 70 inches ? 320. A room is 35 feet long, 28 feet wide and 12 feet high. What is the shortest distance a spider may crawl in a straight line from a lower corner at one end to the opposite upper corner at the other end ? APPENDIX THE METRIC SYSTEM In 1875 a treaty was signed at Paris by seventeen of the principal nations of the world, the United States being among the number, which provided for the permanent organization of an International Bureau of Weights and Measures under the direction of an International Committee. The most important work of the International Committee was to provide for the construction of a sufficient number of platinum-iridium meters and kilograms to meet the demand of the interested nations. The comparison of all these standards with one another and with the original meter and kilogram was made at the International Bureau which had been established near Paris on neutral territory ceded to the Inter- national Committee by the French Government. This work was completed in 1889, and after selecting a certain meter and a certain kilogram as the international prototypes, the others were distributed by lot to the different countries. The international meter and kilogram have values identical with the original meter and kilogram, are preserved in a special under- ground vault at the International Bureau, and are accessible only to the International Committee. The United States secured two meters and two kilo- grams, which are now preserved at the Bureau of Standards at Washington and serve as the fundamental standards of length and mass of the United States. It is the plan of the International Committee to intercompare all the national meters and kilograms with the international prototypes at regular intervals or whenever considered necessary. At the present time the International Bureau of Weights and Measures is supported jointly by the following countries: The United States, (its use “ made lawful throughout the United States ” by Act of Congress in 1866), Great Britain, Germany, Russia, France, Austria-Hungary, Belgium, Argentine Confederation, Spain, Italy, Mexico, Peru, Portugal, Roumania, Servia, Sweden, Norway, Swit- zerland, Venezuela, Japan, and Denmark. The advantages claimed for the metric system are : (1) The decimal relation between the units. (2) The extremely simple relation of the units of length, area, volume, and weight to one another. (3) The uniform and self-defining names of units. Note. — The facts contained in this appendix are based on a pamphlet issued by the Department of Commerce and Labor, entitled “ The International System of Weights and Measures.” 361 362 THE METRIC SYSTEM SYNOPSIS OF THE SYSTEM The fundamental unit of the metric system is the Meter — the unit of length. From this the units of capacity (Liter) and of weight (Gram) were derived. All other units are the decimal subdivisions or multiples of these. These three units are simply related ; e. g., for all practical purposes one cubic decimeter equals one liter and one liter of water weighs one kilogram. The metric tables are formed by combining the words “ meter,” “ gram,” and “ liter,” with the six numerical prefixes, as in the following tables: PREFIXES MEANING UNITS milli- one thousandth 1 TOTO .001 , 1 meter, for length. centi- one hundredth 1 10 0 .01 I deci- one tenth 1 1 0 .1 Unit one 1 \ gram, for weight or mass. 1 deka- ten 1 0 1 10 hecto- one hundred 10 0 ] 100 | , liter, for capacity. kilo- one thousand 10 0 0 1 1000 MEASURES OF LENGTH The meter is the unit of length and is used for measuring dry goods, mer- chandise, engineering construction, building and other purposes where the yard and foot are used. The meter is 39.37 inches long, about a tenth longer than the yard. 10 millimeters (mm.) 10 centimeters 10 decimeters 10 meters 10 dekameters 10 hectometers = 1 centimeter (cm.) = 1 decimeter (dm.) - - 1 meter (m.) = 1 dekameter (dekam.) = 1 hectometer (hm.) = 1 kilometer (km.) The centimeter and millimeter are used in machine construction and similar work, instead of the inch and its fractions. The centimeter, as its name shows, is the hundredth of a meter. It is used in cabinet work, in expressing sizes of paper, books, and in many cases where the inch is used. The centimeter is about two-fifths of an inch and the millimeter about one twenty-fifth of an inch. The millimeter is divided for finer work into tenths, hundredths and thousandths. 1 1 1 1 1 1 1 1 i iiii|iiii iiiijim 1 1 1 1 1 1 1 1 1 iiii|iiii mijim iiii|im mijim iiiijini iiiijim 0 1 2 3 4- 5 6 7 8 9 10 cm. O 1 2 3 4 m. l I I ll I I I I I I I I I I 1 1 1 ! 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Fiq. 1. Comparison Scale: 10 Centimeters and 4 Inches. (Actual Size THE METRIC SYSTEM 363 Where miles are used in England and the United States for measuring distances, the kilometer (1000 meters) is used in metric countries. The kilometer is about 5 furlongs. There are about 1600 meters in a statute mile, 20 meters in a chain, and 5 meters in a rod. If a number of distances in millimeters, meters and kilometers are to be added, reduction is not necessary. They are added as dollars, dimes and cents are now added. For example, “1050.25 meters” is not read “1 kilometer, 5 dekameters and 5 centimeters,” but “ one thousand fifty meters, twenty-five centimeters,” just as “ $1050.25 ” is read “ one thousand fifty dollars and twenty- five cents.” SURFACE MEASUREMENT The table of areas is formed by squaring the length measures, as in our common system. For land measure 10 meters square is called an “ Are ” (meaning “ area ”). The side of one “ are ” is about 33 feet. TABLE 10 milliares (ma.) = 1 centiare (ca.) 10 centiares 10 deciares 10 ares 10 dekares 10 hectares = 1 deciare (da.) = 1 are (A. or a.) - 1 dekare — 1 hectare (ha.) = 1 kilare (ka.) The Hectare is 100 meters square, and, as its name indicates, is 100 ares, or about 2J acres. An acre is about 0.4 hectare. A standard United States quarter section contains almost exactly 64 hectares. A square kilometer contains 100 hectares. For smaller measures of surface the square meter is used. The square meter is about 20 per cent, larger than the square yard. For still smaller surfaces the square centimeter is used. A square inch contains about 6| square centimeters. MEASURES OF VOLUMES The cubic measures are the cubes of the linear units. The cubic meter (sometimes called the stere, meaning “ solid ”) is the unit of volume. It is used in place of the cubic yard and is about 30 per cent, larger. This is used for “cuts and fills” in grading land, measuring timber, expressing contents of tanks and reservoirs, flow of rivers, dimensions of stone, tonnage of ships, and other places where the cubic yard and foot are used. TABLE 1000 cubic millimeters (mm. 3 ) = 1 cubic centimeter (c.c. or cm. 3 ) 1000 cubic centimeters = 1 cubic decimeter (dm. 3 ) 1000 cubic decimeters = 1 cubic meter (in. 3 or stere) The thousandth part of the cubic meter (1 cubic decimeter) is called the Liter. (See table of capacity units.) For very small volumes the cubic centimeter is used. This volume of water weighs a gram, which is the unit of weight or mass. There are about 364 THE METRIC SYSTEM 16 cubic centimeters in a cubic inch. The cubic centimeter is the unit of volume used by chemists as well as in pharmacy, medicine, surgery, and other technical work. One thousand cubic centimeters make 1 liter. A cubic meter of water weighs a metric ton and is equal to 1 kiloliter. MEASURES OF CAPACITY The Liter is the unit of capacity. It equals a cubic decimeter. The liter is used for measurements commonly given in the gallon, the liquid and dry quarts, a liter being 5 per cent, larger than our liquid quart and 10 per cent, smaller than the dry quart. TABLE 10 milliliters (ml.) = 1 centiliter (cl.) 10 centiliters = 1 deciliter (dl.) 10 deciliters = 1 liter (1.) 10 liters = 1 dekaliter (dekal.) 10 dekaliters = 1 hectoliter (hi.) 10 hectoliters = 1 kiloliter (kl.) Fig 3 comparison of the dry quart. ut£ r Ar «0 LIQUID QUART. I ACTUAL SlZE-1 366 THE METRIC SYSTEM The hectoliter (100 liters) serves the same purpose as the United States bushel (2150.4 cubic inches), and is equal to about 3 bushels, or a barrel. A peck is about 9 liters. A liter of water weighs exactly a kilogram, i. e., 1000 grams. A thousand liters of water weigh 1 metric ton. MEASURES OF WEIGHT The Gram is the unit of weight. It is the weight of a cubic centimeter of distilled water at freezing temperature, and weighs 15.432 Troy grains. TABLE 10 milligrams (mg( 10 centigrams 10 decigrams 10 grams 1 0 dekagrams 10 hectograms = 1 centigram (eg.) = 1 decigram (dg.) = 1 S ram (g-) = 1 dekagram (dekag.) = 1 hectogram (hg.) = 1 kilogram (kg.) Measurements commonly expressed in gross tons or short tons are stated in metric tons (1000 kilograms). The metric ton comes between our long and short tons and serves the purpose of both. The kilogram and “ half kilo ” serve for every-day trade, the latter being 10 per cent, larger than the pound. The kilogram is approximately 2.2 pounds. THE METRIC SYSTEM 367 Fig. 5. Relative Size of Avoirdupois Ounce, 30-Gram, and Troy Ounce (Brass) Weights. (Actual Size.) • 0 0 Fig. 6. Relative Size of Gram and Scruple »Brass) Weights. (Actual Size.) The gram and its multiples and divisions are used for the same purposes as ounces, pennyweights, drams, scruples and grains. For foreign postage, 30 grams is the legal equivalent of the avordupois ounce. Duke University Libraries .. D00885628. 511 L957 223C32