Cornell University Library The original of tliis bool< is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031257987 arV17990*^""*" ""'"^^fy Library olin.anx 3 1924 031 257 987 THE FRANKLIN WRITTEN ARITHMETIC "With (Examples fov ©ml |viuticc EDWIN P. SEAVER. A.M. SUPERINTENDENT OF PUBLIC SCHOOLS, BOSTON GEORGE A. WALTON, A.M. AUTHOR OF WALTON'S ARITHMETICS, ARITHMETICAL TABLES, ETC TAINTOR BROTHERS & CO. NEW YORK AND CHICAGO BOSTON: WILLIAM WARE & CO. COPTKIOHT By E. p. SEAVEE AMD Q. A. VTALtOTS 1878. University Press: John Wilsom and Son, Cambridge. PREFACE. The Franklin Written Arithnjetic contains a full course of arithmetical Lnstructioii and drill for pupils in the common schools. The definitions and principles are thoroughly illustrated and explained, so that the learner may work intelligently ; while the range of applications is hroad and varied anough to afford him good preparation for ordinary business affairs. Topics of a merely theoretical interest, antiquated or curious matter, and puzzling problems are omitted altogether ; while parts of the subject not very necessary to the greater number of pupils are given in the Appendix, 10 which references are made in the body of the book. To avoid a multiplicity of rules, decimals and integers have been treated together whenever that could easily be done. For the same purpose, the various problems in Percentage have all been referred to a few fundamental principles, stated and illustrated at the outset. The Metric System has been treated in a way to indicate the most prac- tical course to pursue in teaching it. The topics that follow Simple Interest in this book, as in most arith- metics, can wisely be deferred till the last years of the common-school course. In the arrangement of work it will be noticed that Oral Exercises precede Examples for the Slate. For convenience in the class-room, the lattri- are numbered consecutively through the whole section, with the exception of four pages of typical examples (pages 17, 25, 35, and 48), which are let- tered. The Oral Examples are designated by letters. All answers to Examples for the Slate, except those to Illustrative and Typical Examples, are omitted from the body of the book. Miscellaneous Examples are given in great number and variety, and each section is supplemented by a set of questions for review. A special feature of the book is the Drill Exercises. In general character these are like those previously published by Walton and Cogswell in their Book of Problems ; and they have been introduced in this book by consent of Mr. Cogswell. They give a large number of miscellaneous examples, with answers, on all the topics treated in the Arithmetic ; and the teacher will be spared the trouble of selecting from other books examples for class drill. The fact that these exercises have been extensively imitated in books published of late shows the high estimation in which they art held by teachers and authors. TABLE OF OOI^TEIS'TS. Page Reading and Writing Numbers 1 Addition 12 Subtraction 20 Multiplication 28 Division 38 United States Money 64 Addition, Subtraction, Multi- plication, and Division .... 65 Coins and Paper Currency. ... 67 Accounts and Bills 68 Factors 75 Symbols of Operation 79 Cancellation 80 Greatest Common Factor 82 Multiples 84 Least Common Multiple 85 Common Fractions 88 Seduction 90 Addition 94 Subtraction 97 Multiplication 99 Division 105 To find the Whole when a, Part is given Ill To find what Fraction one Number is of another 113 Aliquot Parts 114 Decimal Fractions 124 Beading and Writing 8, 124, 300 Reduction 125 Addition 15, 17, 19 Subtraction 24, 25, 27 Multiplication 34, 35, 36, 130 Division... 43, 44, 45, 48, 49, 132 Weights and Measures 136 Compound Numbers 146 Reduction 147 Fundamental Operations 162 Longitude and Time 158 Mensuration of Surfaces and Solids 160 Metric System 172 Percentage 185 Profit and Loss 192 Commission 194 Stocks, Dividends, and Brok- erage 197 Insurance 199 Taxes 201 Customs or Duties 204 Simple Interest 209 Accurate Interest 215 Partial Payments 216 Problems in Interest 220 Present Worth and Discount 224 Bank Discount 226 Commercial Discount 230 Compound Interest 231 Average of Payments 234 Average of Accounts 238 Exchange 241 Bonds 245 Ratio and Proportion 254 Partnership 263 Powers and Roots 265 Square Root 267 Cube Root 272 Mensuration 277 Appendix 299 Drill Exercises 57, 59, 63, 73, 123, 135, 171, 253 General Reviews 50, 117, 165, 205, 247, 293 Questions for Review 11, 56, 73, 115, 135, 168, 250, 264, 294 Miscellaneous Examples... 27, 37, 52, 74, 118, 119, 166, 206, 229, 248, 295, 315 AEITHMETIC. SEOTIO]^ I. READING AND VT'RITING NUMBERS. Aeticle 1. A collection of single things or ones is a number. By common usage one is also called a number. 2. A knowledge of numbers is Arithmetic. 3. Some numbers have simple names. These are o?ie, two, three, four, five, six, seven, eight, nine, ten ; also a hun- dred, a thousand, a m,illion, etc. All other numbers have compound names. (See Appendix, p. 299.) Exercises. 1. Ifame the numbers in regular order, or count, from one to fifty; from fifiy to one. 2. Count to a hundred by twos ; by fives ; by tens. Count from a hundred downward by twos ; by fives ; by tens. 3. Name the number that is made up of two tens and five ones ; of one ten and seven ones ; of one ten and a one ; of one ten and two ones ; of six tens and six ; of eight tens and five ; of nine tens ; of nine tens and nine ; of ten tens. 4. Name the number that is made up of one hundred, one ten, and a one ; of two hundreds, seven tens, and three ones ; of six hundreds and three tens; of five hundreds and four ones; of four hundreds, three tens, and three ones; of nine hundreds, nine tens, and nine ones. ' I BEADING AND WRITING 4. From their names we see that small numbers are reckoned by ones, larger numbers by tens, and still larger numbers by hundreds, as far as ten hundred. To ten hun- dred we give the simple name thousaTid. Aboye a thousand numbers are reckoned by thousands, by tens of thousands, and by hundreds of thousands, \ip to ten hundred thousands, or a thousand thousands, which we call a million. Above' a million numbers are reckoned by mOlions, by tens of millions, and by hundreds of millions, up to a thousand millions, which we call a billion. Above a billion numbers are reckoned by billions, by tens of billions, and by hundreds of billions, up to a thousand billions, which we call a trillion. And so we go on with higher numbers as far as we choose. 5. One, Ten, a Hundred, a Thousand, Ten-thousand, a Hundred-thousand, a Million, etc., are called units, because they are used in reckoning or measuring other numbers.* 6. To distinguish these units, we call one a unit of the first order, ten a unit of the second order, a hundred a unit of the third order, a thousand a unit of the fonrtL order, ten thousand a unit of the fifth order, and so. on. When we speak of a unit without mentioning the order, we usually mean a unit of the first order, or one. 7. These units form a scale ; and because ten units of any order make a unit of the next higher order, the scale is called a scale of tens, or a decimal scale. * A unit is a fixed quantity of any kind used to measure other quan- tities of the same kind. Thus, a foot, a yard, a meter, are units, being fixed lengths used to measure other lengths ; a pound, an ounce, a dollar, a cent, an hour, a second, are units, used to reckon or measure weight, value, -or time. The word unit is also much used aa a name for one, and units for ones. SIMPLE NUMBERS. 3 8. A system of numbers whose successive units form a scale of tens is a decima,! system ot numbers. The sys- tem of numbers in common use is a decimal system. 9. Table of Units of the Different Orders. Ten ones (or units) . make a Ten, Ten tens make a Hundred, Ten hundreds .... make a Thousand, Ten thousands .... make a Ten-thousand, Ten ten- thousands . make a Hundred-thousand, Ten hundred-thousands . make a Million, and so on. 10. It wUl be convenient to remember that A thousand ones (or units) . . are a Thousand, A thousand thousands . . are a Million, A thousand millions . . are a Billion, A thousand billions . . . are a Trillion and so on. 11. Exercises. 6. Count by hundreds to a thousand ; to two thousand ; to *;wo thousand five hundred. 6. Count by thousands to ten thousand ; by tens of thou- sands to a hundred thousand ; by hundreds of thousands to a million. 7. Count by millions to ten mill?on ; by tens of millions to a hundred million ; by hundreds of millions to a billion. 8. How many units of each order are there in twenty-five ? seventeen? eleven? ninety? ninety-nine? four hundred? five hundred forty-four ? one thousand eight hundred ? » Note. This kind of exercise may be extended at the discretion of the teacher. 4 BEADING AND WRITING 'Writing Numbers. 12. Besides being expressed in words, numbers are ex- pressed by writing the signs 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, which are called tigures. These signs are also called Arabic numerals, because they were first made known to Europeans by the Arabs.* 13. The first of these signs, 0, is called zero, or ciphsr and is used to stand for no number; the others are used to stand for the first nine numbers, and take their names, thus ' 1, 2, 3, 4, 5, 6, 7, 8, 9. one, two, three, four, five, six, seven, eight, nine 14. Numbers higher than nine have no single signs fox themselves, but are expressed by writing side by side two or more of the figures above. 15. Tens are expressed by writing a figure to tell how many tens, and then writing a zero at the right of it. The tens' figure is then said to stand in the second place, the first or units' place being filled by a zero. Thus, we write Ten (owe ten), 10. Forty (/owr tens), 40. Seventy (sewn tens), 70. Twenty (two tens), 20. Fifty {five tens), 50. Eighty (eight tens), 80. Thirty (three tens), 30. Sixty (six tens), 60. Ninety (nine tens), 90. 16. Numbers that are made up of tens and ones are expressed by writing a figure in the second place for the tens, and a figure in the first place for the ones. Thus, Eleven (ten and one), 11. Twenty-one (two tens and one), 21, Twelve (ten and two), 12. Twenty-two (two tens and two), 22. Let the teacher dictate numbers between ten and a hundred for the pupil to write. * For an account of the Eoman numerals, which were displaced by the Arabic, see Appendix, p. 299. SIMPLE NUMBERS. 5 17. Hundreds are expressed by writing a figure in the third place, the second and first places being filled by zeros. Thus, One hundred, 100. Four hundred, 400. Seven hundred, 700. Two hundred, 200. Five hundred, 500. Eight hundred, 800. Three hundred, 300. Six hundred, 600. Nine hundred, 900. 18. Numbers made up of hundreds, tens, and ones are expressed by writing a figure in the third place for the hundreds, a figure in the second place for the tens, and a figure in the first place for the ones. Thus, Four hundred eighty-three (4 hundreds, 8 tens, 3 ones), 483. Nine hundred sixty (9 hundreds, 6 tens, no ones), 960. Nine hundred six (9 hundreds, no tens, 6 ones), 906. Let the teacher dictate numbers between a hundred and a thousand' for the pupil to write. 19. Thousands, tens of thousands, and hundreds of thousands are expressed by writing figures in the fourth, fifth, and sixth places respectively. The figures in these three places taken together form the Thousands' group ; while the figures in the hundreds', tens', and units' places taken together form the Units' group. These groups are usually separated by a comma. Thus, One thousand, . . . 1,000. Three thousand, . . 3,000. Ten thousand, . . . 10,000. Twenty thousand, . . 20,000. A hundred thousand, 100,000. Five hundred thousand, 500,000. Five hundred twenty-three thousand, 523,000. Six hundred eight thousand, seven hundred twenty-eight, . 608,728. 20. Exercises. Write in figures: 9. Four thousand. 10. Four thousand four hundred. 11. Four thousand forty. 12. Four thousand four. 6 BEADING AND WRITING 13. Eight thousand, four hundred twenty-two. 14. Three hundred fifty-six thousand, eight hundred ninety. 15. Sixty thousand, sixty-five. 16. Eighteen hundred seventy-eight. Let the teacher dictate other numbers, to a million, for the pupil to write. 21. The examples above given illustrate the principle on which all numbers are written, and which is this : Units of any order an expressed ly vjritin^ a figure in the flace corresponding to that order. If the units of any orders are wanting in the number, the corresponding places are filled by zeros. 22. The general method of writing numbers on the principle above stated is shown by the following TABLIi. i ^ III Hi I i •a g ■3 .a'fl |:i| III |:?| -a^ji 4 4 4 4 4 4 4 4 4 4 4 4 -a -a -i piaces. S^m ^^5 Soor- CDU3.d4 CCMtH 480,297,034,508,672 o 5th group, 4th group, 8d group, 2d group, 1st group, l q^ " Trillions. Billions. Millions. Thousands. Units. ' Note. For the names of higher numbers, see Appendix, page 300. 23. This table shows that the figures used to express a number fall into groups of three figures each. The first group expresses simple units, tens, and hundreds ; the sec- ond, units, tens, and hundreds of thousands ; the third, units, tens, and hundreds of millions ; and so on. SIMPLE NUMBERS. 7 These groups are called the units' group, the thousands' group, the millions' group, etc., from the lowest order of units which they express. Note. The units themselves are grouped as the figures are. (Arts. 4 and 10.) 24. In writing large numbers it will be found con- venient to think chiefly of the groups as above described. Thus, let it be required to write the number Forty-nine lillion, three hundred seven million, seventy thousand, six hundred forty-three. The groups are 49 billion, 307 million, 10 thousand, 643. and the number itself is written 49,307,070,643. 25. Exercises. I. Beginning with the units' group, repeat the names of the groups to trillions ; repeat the names from trillions to units. II. Write the groups of figures required to express the fol- lowing numbers, with the names of the groups : 17. Forty-six thousand, five hundred twenty. 18. Four hundred six thousand, five hundred two. 19. One million, one thousand, one hundred ten. 26. Ezercises. I. Write in figures the following numbers : 20. Eighty-five million, five hundred three thousand, seven. 21. Nine hundred six million, two hundred eighteen thou- sand, twenty-eight. 22. Three billion, thirty-seven million, nine hundred thou- sand, two hundred. 23. Eighteen billion, four. 24. Forty million, seven hundred thousand. 25. Thirty-seven trillion, ninety-nine billion, nine million. 8 READING AND WBITIW& II. Write in figures as many of the numbers named on page 62 as the teacher may indicate. Reading Numbers. 27. Let it be required to read the number 53869214. To prepare this expression for reading, we begin at the right, and point off three figures for the units' group, three more for the thousands' group, leaving two for the millions' group, thus : " ^ 53,869,214. Now beginning at the left, we name the number ex- pressed by each group, adding the name of the group, thus : Fifty-three million, eight hundred sixty-nine thousand, two hundred fourteen [iroiisj. Note. The name of the units' group is usually omitted in reading. 28. Exercises. I. Eead the following : (26.) 361. (30.) 9000200. (34.) 3670980347. (27.) 3261. (31.) 86320029. (35.) 9008007006. (28.) 9301. (32.) 81402020. (36.) 7676767676. (29.) 654327. (33.) 89743208. (37.) 90002000. II. Read across the page as many of the numbers expressed on page 60 as the teacher may indicate. 29. It is frequently convenient to separate a number into parts, each part containing only the units of a single order. Thus, the number 734 may be separated into 7 hun- dreds, 3 tens, and 4 units. Such parts are called the terms of a number. Decimal Fractions. 30. As a hundred is made up of ten equal parts, each of which is a ten, and as a ten is made up of ten equal DECIMAL FBAGTIONS. « parts, each of which is one, so we may consider one to be made up of ten equal parts, each of which is a tenth; a tenth to be made up of ten equal parts, each of which is a hjindredth ; a hundredth to be made up of ten equal parts, each of which is a thousandth; and so on. Now a hundred is written 100 ; the tenth part of a hun- dred (ten) is written 10, the figure 1 being moved one place to the right ; and the tenth part of ten (one) is written 1, the figure 1 being moved one place further to the right; so, fol- lowing the same plan, the tenth part of one (one tenth) is written 0.1 ; the tenth part of a tenth (one hundredth) is written 0.01 ; the tenth part of a hundredth (one thousandth) is written 0.001 ; and so on. Tenths, hundredths, thousandths, etc., are fractional units, or fractions; and, as thSy form a decimal scale (Art. 7), collections of such units are called decimal tractions. 31. The dot put at the right of the units' place is called the decimal point. 32. The relations of these fractional units to the higher units are shown by the following table, which may be ex- tended both ways as far as we please : A thousand 1000. A hundred 100. Ten 10. One . 1. A tenth . 0.1 A hundredth 0.01 A thousandth. .... . 0.001 33. We see then that decimal fractions may be written on the principle stated in Art. 21. Thus, we write Two tenths 0.2 Three thousandths 0.003 Five hundredths . . . 0.05 Thirty-two thousandths . . . . 0.032 Twenty-five hundredths 0.25 Threehundredsixteen thousandths 0.316 10 BEADING AND WBITINO 34. The method of. writing decimal fractions is shown by the following table, which is merely an extension of the table given in Art. 22. TABLE. CM Mil , 3 S i m Flaixs. '^'''5'a535555-S ftffMJ-es. 0.708963432... Note. In writing decimal fractions it is well to fill the units' place with d zero when there is no other figure to he ■H^tten there. 35. To read a decimal fraction, name the number ex- pressed hy the figures, and then add the name of the units expressed hy the right-hand figure. Thus, 0.0739 is read " seven hundred thirty-nine ten- thousandths." See Appendix, p. 300. When a whole number and a decimal fraction are written together, read first the whole number and then the fraction. Thus, 56076.028 is read "fifty-six thousand seventy-six, and twenty-eight thousandths." 36. Exercises. . Eead the following : (38.) 0.7 (43.) 0.072 (48.) 2548. (39.) 0.03 (44.) 0.0806 (49.) 254.8 (40.) 0.25 (45.) 5.05 (50.) 25.48 (41.) 0.83 (46.) 4.056 (51.) 2.548 (42.) 0.005 (47.) 7.0056 (52.) 0.02548 DECIMAL FRACTIONS. 11 II. Write in figures the following numbers : (53.) Seven tenths. (58.) 7 units and 5 thousandths. (64.) Seven hundredths. (69.) 25 units and 49 ten-thou- (55.) Seven thousandths. sandths. (56.) Twenty-five hundredths. (60.) 306 hundred-thousandths. (67.) Thirty-nine thousandths. (61.) 5047 hundred-thousandths. Let the teacher dictate other numbers between units and millionths for the pupil to write. 37. Questions for Revie'w. What is a number ? How are numbers reckoned ? (Art. 4.) What general name do you give to one, a ten, a hundred, a thousand, etc. ? How do you distinguish the different units ? What kind of a scale do they form ? What system of numbers is in common use ? Why is it so called? What is the meaning of the word thirteen f eleven f twelve f twenty f thirty-seven ? (Appendix, page 299.) How many units make a thousand ? How many thousands make a million ? How many millions make a billion ? What is the use of figures ? How are numbers higher than nine written ? On what principle are all numbers written ? (Art. 21.) What is the use of zeros ? How are the figures used to express a number grouped? Name the first five groups. How do you write large numbers ? (Art. 24.) Illustrate. How do you read a number ? (Art. 27.) Illustrate. What are the terms of a number ? Name the terms of the number 6725. What is the largest number that can be expressed by one figure ? by two figures ? by three figures ? What is the least number of figures that will express units ? thou- sands? millions? In 100, how many tens ? how many units ? In 15000, how many hundreds? units? tens? In 18462, how many tens, and how many units remain ? how many hundreds, and how many units remain ? What is the effect of placing zeros at the right of an expression for whole numbers ? at the left ? 12 ADDITION. SEOTIOW II. ADDITION. 38. "If you have 5 cents and 3 cents and 2 cents, and count them together, how many cents do you find there are ? Counting them together, you find there are 10 cents. 39. The process of counting numbers together is ad- dition, 40. The result found by addition is the sum or amount of the numbers added. Thus, the sum of 5 and 3 and 2 is 10. 41. The addition of numbers is indicated by the sign +, which is read plus. The sign = indicates equality, and is read equals, or is equal to. Thus, the expression 5 + 3 + 2 = 10 means that the sum of 5 and 3 and 2 is 10, and is read "five plus three plus two equals ten." 42. Oral Ezercises. I. Name the sums of the pairs of numbers expressed below till you can give them rapidly at sight : a. b. c. d. e. f. g. h. i. j. k. 1. 44646323473 6 233426346232 6 7 8 4 8 7 5 3 6 9 6 2 8 4 8 8 3 8 4 8 7 2 9 7 '— — — — — — — — — — — — 8 5 3 3 9 6 4 6 6 2 9 7 4 8 5 7 6 4 7 1 6 9 4 5 ORAL EXERCISES. 13 a. b. 0. d. e. f. g- h. i. i- k. 1. 2 9 3 8 5 7 8 2 9 8 2 7 8 7 9 6 6 1 7 2 5 9 5 6 7 8 9 1 2 5 4 5 8 6 8 1 7 2 3 8 4 3 9 5 5 9 1 7 3 2 5 7 4 9 7 6 9 9 5 8 2 6 7 3 5 9 9 5 1 8 9 2 II. Count to a hundred or more, m. By twos, beginning with 2 ; with 1. n. By threes, beginning with 2. O. By fours, beginning with 3 ; with 2. P- By fives, beginning with 4 ; with 3 ; with 2 ; with 1. S- By sixes, beginning with 5 ; with 4. T. By sevens, beginning with 6. S. By eights, beginning with 7 ; with 6. t. By nines, beginning with 8. III. Add the numbers expressed in the following columns: (.!■) (^•) [S.) (^•) (5.) (6.) {7.) (8.) 2 9 4 4 6 60 600 6000 3 6 2 3 3 30 300 3000 5 3 6 7 8 80 600 9000 7 2 8 4 8 80 800 8000 6 6 8 5 3 30 400 6000 3 8 4 8 5 50 500 5000 9 4 2 9 3 30 300 3000 1 7 8 7 9 90 700 9000 Begin at the bottom and add upward, naming only the results. Thus, in the first column, say 1, 10, 13, 19, 26, 31, 34 36 ; sum, 36. Now, to see if you are right, begin at the 14 ADDITION. top and add downward. Thus, 2, 5, 10, 17, 23, 2&, 35, 36; sum again, 36. Practise exercises of this kind till you can add with great rapidity. For further drill of this sort the teacher is referred to exercises on pages 59 and 61. Examples for the Slate. 43. Illustrative Example I. What is the sum of 413, 102, and 134? WRITTEN WORK. Explanation. ■ — To find the sum of large numbers ... o like these, we add the units, the tens, and the hun- ^ . _ dreds separately ; hence, for convenience, we write the numbers so that units of the same order may be expressed in the same column. (Art. 6.) Sum, 649 Drawing a line beneath, and adding the units (thus, 4, 6, 9), we find there are 9 units, which we write under the line in the units' place. Adding the tens (thus, 3, 4), we find there are 4 tens, which we write under the line in the tens' place. Adding the hundreds (thus, 1, 2, 6), we find there are 6 hundreds, which we write under the line in the hundreds' place. The sum is, then, 6 hundreds 4 tens and 9 units, or 649. 44. ILLUSTBATIVE EXAMPLE II. What is the sum of 960, 748, 932, and 867 ? WRITTEN WORK. Explanation. — Writing the numbers as before, ggQ and adding the units (thus, 7, 9, 17), we find there r,AQ are 17 units, which are equal to 1 ten and 7 units. qqo ^s write the 7 units in the units' place, but keep - the 1 ten to add with the tens expressed in the next column. Adding the tens (thus, 1, 7, 10, 14, 20), Sum, 3507 ■W'e find there are 20 tens, which are equal to 2 hundreds and no tens. We write in the tens' place, to show there are no tens in the sum, but keep the 2 hundreds to add with the hundreds expressed in the next column. Adding the hundreds (thus, 2, 10, 19, 26, 35), we find there are 35 hundreds, which are equal to 3 thousands and 5 hundreds. We write 5 in the hundreds' and 3 in the thousands' place. The sum is, then, 3 thou- sands 5 hundreds tens 7 units, or 3507. EXAMPLES. 15 Keeping a number and adding it with the numbers expressed in the next column is called carrying. In working examples, use as few words as possible. Thus, in the above example, say merely, " 7, 9, 17 ; * 1, 7, 10, 14, 20; 2, 10, 19, 26, 35; sum, 3507." 1. Add together 6234, 785, and 5861. 2. Add together 582, 2, 49, and 124. 3. How many are 2356, 8004, and 987 ? 4. Find the sum of 70639, 600, and 7000. 5. Add 76, 33, 92, 53, 305, 78, 8, and 19. 6. What is 213 + 819 + 37 + 66 ? Addition of Decimals. 45. Illusteative Example III. What is the sum of 425.37, 433.126, 0.076, 442.09, 0.6, and 0.319 ? WRITTEN WORK. E a^lanatiou. — Wntmg the numbers so that units of the same order may be expressed in the i^o.oi same column, we begin with the units of the low- 433.126 est order (in this case thousandths) to add, and 0.076 proceed in the manner already explained, briefly 442.09 thus : thousandths, 9, 15, 21 ; write 1, carry 2 ; 0.6 hundredths, 2, 3, 12, 19, 21, 28 ; write 8, carry 2 ; 0.319 tenths, 2, 5, 11, 12, 15; write 5, carry 1; units, 1, 3, 6, 11 ; write 1, carry 1 ; tens, 1, 5, 8, 10 ; Sum, 1301.581 write 0, carry 1; hundreds, 1, 5, 9, 13; write 13; sum, 1301.581. 7. Add together 90.7, 43.68, 0.045, and 0.812. 8. Add together 0.005, 2.864, 0.9, and 0.25. 9. Add together forty-two thousandths, one hundred seven- teen thousandths, thirteen and twenty-two hundredths, seven and five hundredths. • Do not stop to say " write 7 and caiTy 1," but do it. 16 ADDITION. 46. From the preceding examples we may derive the following Rule for Addition. 1. Write the numbers to be added so that units of the same order may be eoepressed in the same column. Draw a line beneath. 2. Add the vmits of each order separately, beginning with those of the lowest order. 3. When the sum of the units of any order is less than ten, write it under the line in its proper place ; when ten or more, write only the units of the sum, and carry the tens to the numbers expressed in the next column. 4. Write the whole sum of the last addition. Proof. Repeat the work, adding downward instead of upward. Adding two or more columns at once. 47. Accountants often add at once the numbers ex- pressed in two, three, or more columns. The following example will illustrate the method : WRITTEN WORK. Eosplomation. — Beginning with 29, add to it first OK the 4 tens and then the 2 units of 42 ; then to the sum the 8 tens and the 7 units of 87 ; and so on. Naming the results merely, say 29, 69, 71 ; 151, 158; 188, 192; 242, 245; 315, 317; 347, 352. Add- ing downwards, say 35, 105, 107; 157, 160; 190, 87 194; 274, 281; 321, 323; 343, 352. After practice it will be found unnecessary to name all the results ; and it is by omitting s%m., 352 to name them that great rapidity is acquired. Note. The examples on the opposite page embrace the chief varieties in form of examples in Addition. After performing these, and hefore taking rhe Applications on page 18, pupils will usually need additional practice in similar work. Examples for such practice will he found on pages 59-63. 72 53 34 4U 29 EXAMPLES. 17 48. Examples in Addition. a. Add 5274, 206, 87, and 428. Am. 5995. b. Add 132, 3618, 7, and 53. Ans. 3810. c. Find the sum of 8972, 980, 5607, and 89. A-ns. 15648. d. What is the sum of 34, 4800, 147, and 675 ? Am. 5656. e. How many are 346, 4682, 64, and 798 ? Am. 5890. /. What is the amount of 6079, 416, 346, and five thou- sand one hundred sixty-four ? Ans. 12005. g. Two thousand eight hundred twenty-one + nine hun- dred nine + 376 + 43 equals what number ? Ans. 4149. i. Six thousand two hundred ten + eight thousand eight + 4743 + 259 = what number ? Am. 19220. i. Five thousand fifty plus 9782 plus seven thousand seven hundred seventy plus 842 are how many ? Ans. 23444. j. Six hundred two plus 7524 plus six thousand twenty plus 78 plus 4 are how many ? Atis. 14228. k. How many miles are 467 miles, 1349 miles, nine hun- dred seven miles, and sixty-four miles ? Ans. 2787 miles. 1. How many dollars are 7419 dollars, 864 dollars, four thousand twenty-five dollars, and ninety dollars ? Am. 12398 dollars. m. Add the numbers expressed by figures in the ex- amples /, g, and h. Am. 12262, n. Add the numbers expressed by words in examples /. g, and h. Am. 23112. Examples with Decimals. o. What is the sum of 7.62, 14.2, 120.5, 9.08, 0.875, and 2.125 ? Aivs. 154.4. p. Find the sum of twenty-three thousandths, five hun- dredths, ninety-seven hundredths, seven and eight tenths, fifteen and forty-one hundredths. Am. 24.253. For drill exercises, see pages 59-63. 18 ADDITION. 49. Applications. 10. A farmer raised 169 bushels of potatoes in one field, 262 bushels in another, 68 bushels in another, and 1827 in another. How many bushels of potatoes did he raise in all ? 11. My cow Mabel gave 1388 pounds of milk in April, 1456 pounds in May, 1440 in June, 1317 in July, and 1176 in August. How many pounds did she give in all ? 12. I paid $2400* for my farm, $155 for a horse, $26 for a cart, $ 86 for a mowing-machine, $ 10 for a horse-rake, and $ 108 for a yoke of oxen. What did I pay for all ? 13. A merchant buys 5 bales of cloth, the first containing 768 yards ; the second, 764 yards ; the third, 698 yards ; the fourth, 702 yards; and the fifth, 1003 yards. How many yards are there in all ? 14. A planter sold 6 bales of cotton, weighing as follows : the first, 496 pounds ; the second, 609 pounds ; the third, 508 pounds ; the fourth, 498 pounds ; the fifth, 526 pounds ; and the sixth, 487 pounds. What was the whole weight ? 16. A merchant bought at one time 324 barrels of flour for $2430; at another, 260 barrels for $2080; at another, 600 barrels for $3000; and at another, 107 barrels for $749. How many barrels did he buy ? How much did he pay in all ? 16. A steamship sailed 203 miles on Monday, 243 miles on Tuesday, 214 miles on Wednesday, 226 miles on Thursday, 239 miles on Friday, 241 miles on Saturday, and 238 miles on Sunday. How many miles did she sail in the week ? 17. In St. Joseph's District, Michigan, there were at one time 335630 peach-trees on 2953 acres of land, 57519 pear- trees on 758 acres, 9786 plum-trees on 602 acres, 17654 cher- ry-trees on 125 acres, 195995 apple-trees on 2958 acres, and 4988 quince-trees on 33 acres. How many acres of land were occupied by these fruit-trees ? How many fruit-trees were above in all ? * $ is the sign for dollars. EXAMPLES. 19 18. The distance from Boston to Albany is 202 miles; from Albany to Buffalo, 297 miles; from Buffalo to Toledo, 296 miles ; and from Toledo to Chicago, 243 miles. What is the distance from Boston to Chicago ? 19. A man dying left by his wiU $42000 to his wife; % 14000 to his daughter ; $ 3500 in cash, and other property worth $ 13650, to his son ; $ 750 to each of his two nieces ; and the remainder of his property, worth $ 2627, to his brother. What was the value of the whole property ? 50. Examples -with Decimals. 20. Add together 3.07, 0.096, 8.431, and 0.7. 21. What is the sum of $875.16, $638.12, and $400,875? 22. What is the sum of 0.08 of a mile, 0.39 of a mile, and 4.7 miles ? 23. A man paid $15 for a coat, $8.50 for a hat, $6.75 for a pair of boots, and $ 3.45 for other articles. How much did he pay in aU ? 24. A surveyor measures four fields, and finds in the first 1.625 acres ; in the second, 7.316 acres ; in the- third, 12.776 acres ; and in the fourth, 17.306 acres. How many acres in all? 25. How far does a man travel who walks 5.5 miles before breakfast, 17.25 miles between breakfast and dinner, 12 miles between dinner and supper, and 0.875 of a mile after supper ? (26.) (27.) (28.) (29.) (30.) (31.) $75.46 $754.60 $2.40 $476.48 $47.84 $780.00 18.72 42.87 74.09 207.42 98.76 15.85 9.47 5.30 53.67 77.99 3.69 119.68 15.08 106.84 184.76 4.44 0.49 45.45 11.80 70.00 66.67 0.85 9.84 99.99 4.55 14.76 407.99 109.98 55.00 710.00 7.67 107.34 31.08 3.17 46.50 84.37 17.38 21.95 213.67 6.51 27.75 76.85 20 SUBTRACTION. SEOTIOI^r III. SUBTRA.CTION. 51. If Charles has 9 apples and should give 4 of them away, how many apples would he have left ? To find how many he would have left, we take 4, a part of 9, away, and, by counting or otherwise, find there are 5 left ; thus we know that he would have 5 apples left. 52. The process of taking part of a number away to find how many are left is subtraction. 53. The number, part of which is to be taken away, is the zainuend. 54. The part of the minuend to be taken away is the snbtrahend. 55. The part of the minuend left after a part has been taken away is the remainder. Name the minuend in the example above ; the subtrahend ; the remainder. 56. The subtraction of numbers is indicated by the sign — , which is read minus, or less. Thus the expression 9-4=5 means 9 diminished by 4 equals 5, and is read " nine minus four equals five," oi " nine less four equals five." 57. Oral Exercises. I. Give rapidly the remainders in the following examples : a. b. o. d. e. f. g. h. i. 3 3 5 7 6 13 11 12 14 123424765 ORAL EXERCISES. 21 a. b. c. d. e. f. & h. i 5 4 7 6 13 12 13 15 13 4 3 2 4 8 5 9 8 7 4 5 7 6 12 17 12 14 16 2 2 6 3 9 9 3 9 7 9 7 9 8 15 14 11 12 16 7 6 4 7 6 7 6 8 9 8 9 8 7 11 11 14 11 12 5 8 2 3 8 4 8 9 4 9 6 9 8 13 16 15 12 11 3 6 6 6 6 8 7 3 3 9 8 8 9 11 11 14 17 15 2 3 4 5 5 2 6 8 9 II. Subtract (that is, count downward) J. By 2'3 from 50 ; from 49. k. By3'sfrom50. 1, By 4's from 50 ; from 49. m. By5'sfrom50; from 49; 48; 47; 46. n. By 6's from 100 ; from 99. o. By 7's from 100. p. By 8's from 100; from 99. q. By 9's from 100. III. What is 80 - 30 ? 50 - 20 ? 90 - 50 ? 150 - 70 ? 700-400? 1200-300? 1600-900? 1600-800? rV. From 100 subtract 25, 35, 85, 67, 39, 48, 73, 44, 78, 60, 51, 72, 13, 04, 57, 36, 53, 62, 46, 17, 77, 24, 87, 76. For additional oral drill, see pages 59-63. 22 SUBTRACTION. Ezamples for the Slate. 58. Illustrative Example I. If 147 trees are taken from a nursery of 489 trees, how mauy trees wiH be left ? WRITTEN WORK. Explanation. — To find how many will be left, Minuend 489 ^® ^^'^ ^'^^ °^ ^^^ number 489 away. In subtract- H , _ ing a large number like this we take away the SvbtraUnd, 147 ■. ^. , j ^i, u j i -^ i / units, the tens, and the hundreds separately; hence, Semamder, 342 for convenience, we write the minuend and the sub- trahend as in the margin, so that units of the same order shall be expressed in the same column. Drawing a line beneath, and beginning with the units, we subtract thus : 7 units taken from 9 units leave 2 units, which we write under the line in the units' place ; 4 tens taken from 8 tens leave 4 tens, which we write under the line in the tens' place ; 1 hundred taken from 4 hundreds leaves 3 hun- dreds, which we write under the line in the hundreds' place ; and we have for the whole remainder 3 hundreds 4 tens and 2 units, or 342. Answer, 342 trees. 1. If a man having 375 oranges in a box should sell 234 of them, how many would be left ? 2. I had a farm of 493 acres, and sold a part containing 172 acres. How many acres had I left ? 59. Illusteative Example IL If a minuend is 7592 and the subtrahend 3674, what is the remainder? WRITTEN WORK. Explanation. — We write these numbers and (6) (16) (8) (12) subtract as before. As we have but 2 units in the Mm. 7 5 9 2 minuend, we cannot now take the 4 units away, Svi. 3 6 7 4 so we change one of the 9 tens (leaving 8 tens) to Bern. 3 9 18 units. This 1 ten equals 10 units. We add the 10 units to the 2 units, making 12 units. Sub- tracting 4 units from the 12 units, we find 8 units left, which we write as part of the remainder. Subtracting 7 tens from the 8 tens we now have, we find 1 ten left, which we write. As we have but 5 hundreds in the minuend, we cannot now take 6 hundreds away, so we change one of the 7 thousands (leaving 6 thousands) to hundreds, and add the 10 hundreds thus obtained to the 5 hundreds, making 15 EXAMPLES. 23 hundreds. Subtracting 6 himdreds from 15 hundreds, we find 9 hundreds left, which we write. Subtracting 3 thousands from the 6 thousands we now have, we find 3 thousands left, which we write ; and we have for the whole remainder, 3 thousands 9 hundreds 1 ten and 8 units, or 3918. Tliis explanation may be given briefly thus : 4 from 12 leaves 8, 7 from 8 leaves 1, 6 from 15 leaves 9, 3 from 6 leaves 3 ; remainder, 3918. In actual work, however, all explanation should be omitted. Do not stop to say " 4 from 12 leaves 8," etc., but do the work, naming only results as you write them, thus : " 8, 1, 9, 3 ; remainder, 3918." In this way you will learn to work rapidly. What are the remainders in the following examples ? (3.) (4.) (5.) (6.) 849 321 8642 3089 278 219 370 2435 7. If I had $685 in a bank and withdrew $328, how many dollars remained ? 8. How old was a person in 1876 who was born in 1798 ? 60. Illustrative Example III. If a farm is bought for $ 965 and sold for $ 2000, how much is gained ? WRITTEN WORK. Explanation. — To find how much is gained, (1) (9) (9) (10) we take away a part of $ 2000 equal to $ 965. $^000 As we have no units, no tens, and no hun- 9 6 5 dreds in the minuend, we change one of the Ans. $ 1 3 5 thousands (leaving 1 thousand) to 10 hundreds ; then change one of the 10 hundreds (leavmg 9 hundreds) to 10 tens ; and one of the 10 tens (leaving 9 tens) to 10 units. 2000 is thus changed to 1 thousand 9 hundreds 9 tens and 10 units, from which taking 9 hundreds 6 tens and 5 units, we have foi the remainder 1035. Ans. $ 1035. 9. From 2000 years take 1028 years. 24 SUBTRACTION. 10. From 3000 oxen take 229 oxen. 11. How many more birds are there in a flock of 960 birds than in one of 487 birds ? Subtraction of Decimals. 61. Illustrative Example IV. What is the difference between 20.69 and 8.745 ? WRITTEN WOEK. Explanation. — Writing these numbers so that 20.69 units of the same order shall be expressed in the o rrif- same column, and beginning with the units of the lowest order (in this case thousandths) to subtract, we have for the remainder 11.945. 12. Take 20.6 from 199. 13. From $27.68 take $15.96. 14. Find the difference between one thousand and one thousandth. 62. From the examples above explained we may derive the following Rule for Subtraction. 1. Write the minuend and underneath write the subtra- hend, so that units of the same order may he expressed in the same column. Draw a line beneath. 2. Begin with the units of the lowest order to subtract, and proceed to the highest, writing each remainder under tlie line in its proper place. 3. If any term of the minuend is less than the corre^ond- ing term of the subtrahend, add ten to it and then subtract ; but consider that the next term of the minuend has been diminished by one. Proof Add the remainder to the subtrahend : the sum ought to equal the minuend. EXAMPLES. 25 63. Examples in Subtraction. a. From 7282 subtract 4815. Ans. 2467. b. Take 3084 from 6231. Ans. 3147. c. How many are 64037 less 5908 ? Ans. 58129. d. Subtract 807605 from 1740932. Atis. 933327. e. What number taken from 71287 will leave 40089 ? Ans. 31198. /. How many more than 94736 is 104083 ? Ans. 9347. g. Find the difference between 86045 and 708406. Ans. 622361. h. 2684753 - 764287 = how many ? Ans. 1920466. i. From four hundred twenty thousand six hundred eighty-three, take two hundred fifty-nine thousand seventy- five. Ans. 161608. j. Take eight hundred ten thousand twenty-three from one million sixty thousand forty-one. Ans. 250018. k. 1001001 minus 909199 equals what ? Ans. 91802. 1. Subtract the sum of the numbers in example c from the sum of the numbers in example d. Ans. 2478592. m. Find the difference between the amount of the num- bers in example a and the amount of the numbers in example h. Ans. 2782. Examples -with Decimals. 22. From $ 17.60 take $ 5.25. Ans. $ 12.35. 0. From 426.17 take 11.723. Ans. 4:U.U7. p. Subtract three hundred sixty-four thousandths from one. -4ws. 0.636. q. What must be added to 0.0476 to make 1 ? Ans. 0.9524. Note. The examples on this page embrace the chief varieties in form of examples in Subtraction. After performing these, and before taking the Applications on page 26, pupils will usually need additional practice in •similar work. Examples for such practice will be found on pages 59 - 63. 26 SUBTRACTION. 64. Applications, 15. A farmer who raised 948 bushels of corn sold all but 198 bushels. How much did he sell ? 16. The year's earnings of a family were $ 1172. If their expenses were $ 875, what was saved ? 17. A and B together own 5740 acres of land. If B owns 2964 acres, how much does A own ? 18. Mount Washington is 6234 feet high, which is 2286 feet higher than Vesuvius. How high is Vesuvius ? 19. The several items of an account amount to $9867.62; of this amount $ 7985.75 has been paid. Find the balance. 20. Franklin was born in 1706, and died in 1790. What was his age at the time of his death ? 21. The difference between A's and B's estates is $1463; B's, which is the greater, is worth $ 7638. What is A's worth ? 22. In one week a grain elevator received 984560 bushels of grain; of this 769386 bushels were delivered. How much remained in the elevator ? 23. The sailing distance from New York to Queenstown is 2890 miles. If a Cunard steamer has run 1368 miles on her course from New York, how far has she still to run ? The population of the city of New York was 60489 in the year 1800; 96373 in 1810; 123706 in 1820; 202589 in 1830; 312710 in 1840; 515547 in 1850; 813669 in 1860; and 942292 in 1870. What was the increase in population 24. From 1800 to 1810 ? 28. From 1840 to 1850 ? 25. From 1810 to 1820 ? 29. From 1850 to 1860 ? 26. From 1820 to 1830 ? 30. From 1860 to 1870 ? 27. From 1830 to 1840 ? 31. From 1800 to 1870 ? 32. The population of London in 1871 was 3266987. How many times may you subtract from this a population equal to that of New York in 1870 ? 33. The equatorial diameter of the earth is 41847194 feet, and the polar diameter 41707308 feet. What is the differei^ce ? EXAMPLES. 27 65. Examples with Decimals. 34. A person having 205.6 acres of land, sold 10.76 acres. How many acres had he left ? 35. What is the difference between 0.7 and 0.385 ? 36. How many thousandths must you add to 0.485 to make 1.? (37.) 86.67-9.8 = ? (40.) 641.34-56.345 = ? (38.) 7561.2-9.6456 = ? (41.) 101.1-90.014 = ? (39.) 961.62-54.645 = ? (42.) 970.2-86.37 = ? 66. Miscellaneous Examples. 43. James Fry has in his possession $172; he owes $28 to A, $ 36 to B, and $ 19 to C. After paying his debts, what will remain ? 44. In a certain miU 2415 persons were employed, of whom 581 were natives, 1119 were foreigners, and the rest unknown. How many were unknown ? 45. I have $462 in the savings-bank, and $2180 in gov- ernment bonds. How much more must I have that I may purchase a house worth $ 4700 ? 46. A man gave to his son $3575, to his daughter $4680, and to his nephew $2495 less than to his daughter. His whole property was worth $ 30500 ; what sum remained ? 47. Two persons who are 250 miles apart, travel towards each other, one 36 miles, the other 52 miles a day. How far apart will they be at the end of one day ? 48. If the same persons travel away from each other, how far apart will they be at the end of one day ? 49. From 9460 subtract 5466; from the result subtract 1284 ; to this add 3989, and from this subtract 5987. 50. A man bought a lot of land for $ 1296, and built upon it a house costing $ 7364. If he sold the property for $ 10000, how much did he make ? 28 MULTIPLICATION. SEOTIOT^ IT. MULTIPLICATION . 67. Unite three 7's into one number. 7 This may be done by adding them together thus : 7, 14, 21. 7 By this process we find that three 7's are 21. In the same way q we can find that seven 6's are 42, eight 9's are 72, eight 7's ■ — are 56, and in fact all the results which we commit to memory ^1 when we learn the Multiplication Table. 68. The process of uniting two or more equal numbers into one number is multiplication. 69. One of the equal numbers to be united is the m ultiplican d. 70. The number that tells how many equal numbers are to be united is the multiplier. 7 1. The result obtained by multiplication is the product. 72. The multiplicand and multiplier are called factors (makers) of the product. In the example "three 7's are 21," which is the multiplicand? th« multiplier? the product ? Name two factors of 21. 73. The multiplication of numbers is indicated by the sign X . Thus, the expression 50 x 4 = 200 means that four 50's are 200 ; and is read " 50 multiplied by 4 equals 200." 74. Oral Exercises. Turn to page 58, and multiply the numbers expressed a. In column h by 3. e. In column q by 7. b. In column j by 4. /. In column r by 8. c. In column k by 5. g. In column y by 9. d. In column o by 6. h. In column v by 12. EXAMPLES. 2y 75. Compare the product of five 4''s with that of four 5's: are they equal or unequal? Compare the products 3x4x6, 4^3x6, and 6x4x3: are they equal or un- equal ? From examples like these we learn this general principle : The product of two or more factors is the same, whateve-)^ the, order in which the factors are taJcen. To multiply mentally Numbers greater than 10. [At the optum of the Teacher.'] 76. Illustrative Example. At $34 each, what will 4 cows cost? Solution. — At $ 34 each, 4 cows will cost 4 times 1 34. Four 30's are 120, and four 4's are 16, which, added to 120, make 136. Ans. 1 136. i. If 9 men can build a wall in 25 days, how long would it take 1 man to do it ? J. How many gallons of water in 5 hogsheads of 67 gallons each? k. At $8 a month, what is the amount of a soldier's pen- sion for 1 year ? for 9 years ? 1, How many are three 27's ? four 16's ? eight times 84 ? For additional practice, multiply each number expressed in A, page 68, by such numbers from 1 to 12 as the teacher may select. See also oral exercises in multiplication, pages 59 and 63. Examples for the Slate. 77. Illustrative Example I. If a steamship goes 258 miles each day, how far does she go in 6 days ? Explanation. — If the steamshio goes 258 miles WRiTTEK WORK, ^^^.j^ ^.^^ ^^ g j^yg gjjg ^ju gg Q times 258 miles. Multiplicand, 258 We have then to multiply 258 by 6. Writing the Midtipiier, 6 multiplicand and the multiplier as in the margin, „ J Tkaq we multiply the units, teus, and hundreds sepa- Praduct, lo4o , , ^Z . .^, ', .. rately, beginning with the units. 30 MULTIPLICATION. Six 8'b are 48. The 48 units are equal to 4 tens and 8 units. "We write 8 under the line in the units' place, and carry 4 tens to the product of tens. Six 5's are 30. The 30 tens with the 4 tens carried are 34 tens, or 3 hundreds and 4 tens. We write 4 under the Une in the tens' place, and carry 3 hundreds to the product of hundreds. Six 2's are 12. The 12 hundreds with the 3 hundreds carried are 15 hundreds, or 1 thousand and 5 hundreds. We write, under the line, 5 in the hundreds' place and 1 in the thousands' place. The entire product is 1548. Am. 1548 mUes. For the sake of rapid working, use as few words as possible. Thus, in the example above say " ioity-eight ; thirty, thirty- four; twelve, fifteen " : while saying " forty-eight," write 8 ; while saying " thirty-four," write 4 ; and while saying "fifteen," write 6 and 1. 1. How many pounds of flour are there in 5 barrels, each containing 196 pounds ? 2. Hpw many pounds of cheese are there in 6 cheeses of 172 pounds each? 3. If a person earns $ 313 every year for 7 years, how many dollars does he earn ? 4. What will 9 pianos cost at $ 475 each ? 5. From Chicago to Peoria is 160 miles ; how far does a man travel who goes from Chicago to Peoria and back 8 times ? 6. If a person by working 11 hours a day can do a piece of work in 37 days, how many days will it take him if he works 1 hour a day ? 7. There are 5280 feet in a mile. How many feet long is a telegraph-wire that connects Boston with Eeading, 12 miles distant ? 78. Illustrative Example II. If 1 barrel of flour costs $ 8, what wiU 427 barrels cost ? boluhon. — If 1 barrel of flour costs $ 8, 427 barrels will cost 427 timas $ 8. But 427 times $ 8 is the same as 8 times $ 427 (Art. 75), Which is S 3416. Ans. ?, 3416. EXAMPLES. 31 8. What will 732 quarts of milk cost at 7 cents a quart t 9. What must I pay for 324 sheep at $ 9 apiece ? 10. When coal is $ 6 a ton, what must I pay for 476 tons ? 11. At 4 cents a mile, what must I pay for riding 1289' miles ? 12. If 294 persons gave $ 8 apiece for a charitable ohjectj, how much did all give ? 13. What must I pay for 626 car-fares at 5 cents apiece,, and for 87 car-faros at 9 cents apiece ? 14. Multiply 267 by 2 ; by 3 ; by 4 ; and add the products. 15. Multiply 628 by 5 ; by 6 ; by 7 ; and add the products. 16. Multiply 3401 by 8 ; by 9 ; and add the products. 17. Multiply 90021 by 10 ; by 11 ; and add the products. 18. Multiply 66285 by 12 ; by 8 ; and add the products. 19. Multiply 89079 by 7 ; by 12 ; and add the products. For additional examples in multiplication by one term only, see pages 59 and 63. 79. Illustrative Example III. Multiply 12 by 10; 12 by 100 ; 12 by 1000. WRITTEN WORK. Explanation. — 10 twelves equal 12 12 12 12 tens (Art. 75), or 120 ; 100 twelves jQ 200 1000 eiual 12 hundreds, or 1200 ; and 1000 twelves equal 12 thousands, or 12000. 120 1200 12000 ^ In multiplying by 10, 100, 1000, etc., the written work may be omitted, and the product immediately found hy annexing to the multiplicand as many zeros as there are in the multiplier. 20. What will 10 bushels of potatoes cost at 65 cents a bushel ? 21. At $ 100 a share, what will 100 shares in a whip com- pany cost ? 22. Multiply % 75 by 10 : by 100 ; and add the products. 32 MULTIPLICATION. 23. Multiply 5872 by 10 ; by 1000 ; and add the products. 24. Multiply 684 by 10 ; by 100 ; by 10000 ; by 1000 ; and add the products. 25. Multiply 3682 by 10000; by 10; by 1000 ; by 100; and add the products. 80. Illustrative Example IV. Multiply 4520 by 300. WRITTEN woEK. Explanation. — Here 4520 equals 452x10, and 4520 ^^'^ equals 3 a 100 ; hence 4520 x 300 is the same „„„ as 452 X 10 X 3 X 100, or, since the order of the fac- tors may be changed (Art. 75), the same as 452 x 3 1356000 X 10 X 100. We shall, therefore, find the product if we multiply 452 by 3 and annex three zeros (Art. 79). When the multiplicand and multiplier, or either of them, have zeros at the right hand, the. zeros may he dis- regarded in multiplying, hut there must he annexed to the product as many zeros as were disregarded. 26. I have 600 acres in my farm. What is it worth at $ 250 an acre ? 27. How many strawberry plants are there in 400 rows, if there are 280 plants in each row ? 28. What is the product of 1870 x 90 ? Of 1870 by 900 ? Of 1870 by 9000 ? 29. If 268000 is the multiplicand and 80 the multiplier, what is the product ? 30. Multiply 596 by 3 and by 40, and add the products. 31. Multiply 984 by 8 and by 60, and add the products. 32. Multiply 647 by 9 and by 20, ajid add the products. 33. Multiply 379 by 6 and by 80, and add the products. 34. Multiply 4837 by 2, by 30, and by 500, and add the products. 35. Multiply 2802 by 8, by 70, and by 900, and add the products. EXAMPLES. 33 81. Illustrative Example V. Multiply 625 by 39. 625 39 5625 = product by 9. 1875 = product by 30. 24375 = product by 39. WRITTEN WORK. Explanation. — 'W % shall find the product of 625 x 39 if we multiply 625 by 9 and then by 30, and add the re- sults. We first find the product by 9, which is 5625, and write it under the line. The product of 625x30 is the same as 625 x 3 x 10. To find this we multiply 625 by 3, obtaining 1875, but instead of annexing a zero (Art. 79), we write the result as 1875 tens. We then add the partial products. 'Note. Compare this process with that of Examples 30 to 33, in the last Article. 36. How many are 34 x 25 ? 37. Multiply 49 by 98 ; then multiply 98 by 49. Are these products equal ? Why ? 38. What is the product of 2842 multiplied by 28 ? 39. Multiply 3684 by 36 and by 64, and add the products. 40. Multiply 625 by 339; by 705; by 7005. WRITTEN WORK. WRITTEN WORK. 625 625 339 705 5625 1875 1875 = product by 9. = product by 30. = product by 300, 3125 = product by 5. 4375 = product by 700. 440625 = product by 705. 211875 = product by 339. The explanation of this work is left for the pupil. (See Art. 81.) 41. How many are 743 x 657 ? 42. Multiply 237 by 195; 195 by 237. 43. Multiply 4387 by 235 ; 235 by 4387. 44. Multiply 7608 by 504 ; 504 by 7608. 45. Multiply 760500 by 307000. 46. Multiply 907200 by 420900. 34 MULTIPLICATION. MultipUoation of Decimals. 82. Illustrative Example VI. Multiply 108.67 by 48. Explanation. — 10867 hundredths mul- WKITTEN WORK. . ,. ,, „. „„^o^ l. J J^L ^/^n/.^ tiplied by 8 is 86936 hundredths. 10867 108.67 hundredths x 40 is the same as 10867 48 hundredths x 4 x 10. Now 10867 hun- RfiQSfi — n nrl hv 8 dredths x 4 is 43468 hundredths ; to ex- A^Ano J i_ j/\ P^ess this product multiplied by 10 we 43468 = prod, by 40. ^ ., ,, ^ ,^ . ;, , . '^ ■' write the figures one place to the left. 5216.16 = prod, bv 48. Adding the partial products we have 521616 hundredths (5216.16) for the en- tire product. Here, as in the preceding examples, we see that the product is of the same order of units as the multiplicand. 47. Multiply 8.648 by 5 ; 432.5 by 21. 48. Multiply 7.0909 by 6 ; 0.0005 by 18. 49. Multiply 0.626 and 0.375 each by 24, and add the results. 83. From the preceding examples may be derived the following Rule for Multiplication. 1. Write the multiplicand and underneath write the mul- tiplier. Draw a line heneath. ■ 2. If the multiplier consists of one term only, multiply each term of the multiplicand hy the multiplier, beginning with the term of the lowest order, and carrying as in addition. 3. If the multiplier consists of more than one term, multi- ply iy each term of the multiplier separately, writing tJie partial products so that units of the same order shall he expressed in the same cohimn. 4. Add the partial products thus obtained, and the result will be the entire product. Proof. Multiply the multiplier by the multiplicand : the two prod- ucts ought to be equal. For contractions in multiplication, see Appendix, page 300. EXAMPLES. 35 84. Examples in Multiplication. a. Multiply 4687 by 8. Am. 37496. 6. Find the product of 50875 by 7. Ans. 356125. c. Multiply 5872 by 10, also by 1000, and add the P^'iiicts. Ans. 5930720 d. Multiply 8756 by 300; by 500; by 7000; and add the results. Ans. 68296800. e. What is the product of 39700 by 9000 ? Ans 357300000. /. 37406 X 43 = what number ? Atis. 1608458. g. For multiplicand take 46059, for multiplier 76, and find the product. Ans. 3500484 h. How many are 309 times 46057 ? Atis. 14231613. i. Multiply thirty-seven thousand twenty-eight by 608. Atis. 18810224 j. The multiplier being 987, the multiplicand six thou- sand four hundred sixteen, required the product. Am. 6332592. k. What is the product of 908060 by five thousand four hundred ? Am. 4903524000. 1. One factor being 718151, the other seven hundred, what is the product ? Ans. 502705700. m. At 147 dollars per acre, how much will 385 acres of land cost ? Am. $ 56595. n. There are 24 hours in a day. How many hours in 476 days? Am. 11424. Examples Tvith Decimals. o. Multiply 40.27 by 87. Am. 3503.49. p. Multiply thirty-one thousandths by 25. Ans. 0.775. Note. The examples on this page embrace the chief varieties in form of examples in Multiplication. Examples for additional practice will be found on pages 59 - 63. 36 MULTIPLICATION. 85. Applications. 50. At $ 45 a month for labor, what will a man earn in a year ? In 5 years ? 51. If a man saves $ 17 a month, what wiU he save in 25 years? 52. If a sewing-machine can set 690 stitches in a minute, how many stitches can it set in 60 minutes or an hour ? In a day of 12 hours? In 6 working days or a week? In 52 weeks or a year ? 53. The first House of Representatives of the United States consisted of 65 members; if each member represented 30000 inhabitants, how many inhabitants were represented ? 54. In a certain mill, material for 65000 dresses is made in a week. Allowing 18 yards for a dress, how many yards are made in a week ? In a year ? 55. The cotton crop in Texas in one year was 450000 bales. Allowing 400 pounds to a bale, how many pounds were raised ? 56. In a day there are 24 hours, in an hour 60 minutes, in a minute 60 seconds. How many seconds in a day ? 57. Light, according to Foucault, travels at the rate of 185172 miles in a second. If it passes from the sun to the earth in 8 minutes 13 seconds (or 493 seconds), what is the distance from the sun to the earth ? 86. Examples with Decimals. 58. It took Mary 3.25 hours to learn a piece of music, and Olive 5 times as long. How many hours was Olive in learn- ing it ? 59. Mr. Green has 5.175 acres of land and buys 7 times ae much of his neighbor. How many acres does he buy of his neighbor ? 60. What will 38 barrels of flour cost at $ 11.75 a barrel ? 61. Mr. Gage sold 175 tons of refined bar-iron at $45.50 a ton. What did he receive for it ? 62. Multiply 5.4328 by 62. EXAMPLES. 37 87. Miscellaneous Ezamples. 63. I have four bins, containing severally 63 bushels, 54 bushels, 37 bushels, and 29 bushels. If there are 60 pounds of corn in a bushel, how many pounds of corn will they all hold ? 64. What is the height of an iceberg which is 376 feet above the surface of the water and 7 times as many feet below ? 65. Myron walks 847 steps of 2 feet each in going to school. How many more feet must he take to walk a mile, or 5280 feet ? 66. What do I save a year, my income being $ 1600 a year, and my expenses 1 24 a week, 52 weeks making the year ? 67. Mr. Fiske receives a salary of $ 1500 a year, pays $ 130 for clothing, f 276 for other expenses, also $ 6 a week for his board. How much money has he left at the end of the year ? 68. If 768 be one factor, and 861 — 237 the other factor, what is the product ? 69. Smith & Co. consume 74 tons of coal in a year. How much more did they pay for their coal in 1864, when coal was $ 14 a ton, than in 1877, when it was $ 7 a ton ? 70. If in one yard of cloth there are 580 fibres of warp and 432 of filling, and each fibre of warp contains 32 strands, and each of filling 48, how many strands are there in the yard ? 71. One house is valued at $6750, and another at three times as much. How much will pay for both houses ? 72. Mr. Gould had $ 2500 with which he bought 17 acres of land at 142 an acre, a house for $1500, 2 cows at $45 apiece, and a horse for $ 75. How much money had he left ? 73. Mr. Bodwell paid for labor and use of oxen on his land, the following sums : $ 135, 1 128, and $ 90 ; he also paid $ 64 for fertilizers and $ 10 for seed, and raised on the land 23 tons of hay which he sold at $ 25 a ton. What was his gain above his expenses ? 74. Add 284, 1752, 45, and 846 ; subtract 2731 from the sum ; multiply the remainder by 208 ; and find the difference between the product and 40801. 38 DIVISION. SEOTIOI^ V. DIVISION. 88. Mr. Eice has 24 bushels of sand to bring from the beach. If he brings 8 bushels at each load, how many loads must he bring? He must TDring as many loads as there are 8's in 24. We have already seen by multiplication that three 8's are 24, so we know that he must bring 3 loads. If a cheese weighing 54 pounds be divided equally among 6 persons, how many pounds wiU each receive ? Each person will receive one of the 6 equal parts into which the 54 pounds is to be divided. We have seen by multiplication that 6 nines are 54 ; hence one of the 6 equal parts of 54 is 9, and each person will receive 9 pounds. It will be noticed in the first example that we find how many equal numbers, one of which is given, there are in another number (that is, how many times one number is contained in another); and in the second that we find one of the equal parts of a number. 89. The process of finding how many times one number is contained in another or of finding one of the equal parts of a number is division. 90. The number to be divided is the dividend. 91. The number by which we divide is the divisor. ■ 92. The result obtained by division is the quotient. Note I. When the divisor is one of the given equal numbers, the quo- tient will tell Jum manf such numbers there are in the dividend. DIVISION. 39 Note II. When the divisor tells how many oqual parts the dividend is to he separated into, the quotient will tell how great one of those equal parts is. Note III. By comparing the first process with multiplication (Arts. 69 - 72), we see that the product, and muUiplicand are given, and the mul- tiplier is to be found. By comparing the second process with multiplica- tion, we see that the product and multiplier are given, and the multipli- cand is to he found. In either case the product and one of the factors are given, and the other factor is required. 93. If Mr. Eice has 31 bushels of sand to bring from the beach, and can bring but 8 bushels at a load, how- many full loads can he bring and how many bushels wiU then remain? The part of the dividend left after the equal numbers have been taken away is the remainder. In the example above, which is the dividend ? the divisor ? What is the remainder ? 94. The division of numbers is indicated by the sign -^ . Thus, the expression 24 -=- 8 = 3 means that the quotient obtained by dividing 24 by 8 is 3, and is read " 24 divided by 8 equals 3." The sign : is also used for division. Thus, 24 : 8 = 3. Sometimes the dividend is expressed above a line and the divisor below, in place of the dots. Thus, -g- = 3. This expression is called the fractional form of indicating divis- ion, and is read "24 divided by 8 equals 3," or "1 eighth of 24 equals 3." 95. When a thing or a number is divided into 2 equal parts, the parts are called halves; when divided into 3 equal parts, the parts are called thirds ; when into 4 equal parts, the parts are called fourths; and so on. What is one of the parts called when a number is divided into 5 equal parts ? 6 ? 7 ? 8 ? 10 ? 20 ? 100 ? 1000 ? 40 DIVISION. 96. Table for Oral Practice in Division. 1. 4 ■ 7 2 6 3 8 5 7 11 9 10 12 2. 16 16 21 14 20 13 22 17 23 18 19 24 3. 25 29 35 33 28 31 26 34 27 30 32 36 4. 39 46 38 42 37 43 47 40 45 41 44 48 5. 58 49 55 51 54 59 52 57 50 53 56 60 6. 62 71 70 66 61 64 69 63 68 65 67 72 7. 76 79 77 81 78 80 76 73 82 74 83 84 8. 90 88 91 87 89 94 85 93 86 95 92 96 9. 99 104 100 98 97 102 106 101 107 103 105 108 10. 110 118 111 109 117 112 115 114 119 113 116 120 11. 124 130 123 125 129 121 128 122 131 126 127 132 12. 134 142 135 140 133 139 136 143 137 141 138 144 97. Oral Exercises upon the Table. Beginning at the left of the table above, divide by 2 each number expressed in the first two lines, naming quotients and remainders at sight. In the first line the numbers to be divided are 4, 7, 2, 6, 3, 8, 5, etc. The results will be given as follows: "2; 3 and 1 over; 1; 3; 1 and 1 over; 4," etc. Divide in the same manner the numbers expressed in either* a. Of the first 3 lines by 3. /. Of lines 2 to 8 by 8. b. Of the first 4 lines by 4. c. Of the first 6 lines by 5. d. Of the first 6 lines by 6. e. Of the first 7 lines by 7. g. Of lines 2 to 9 by 9. h. Of lines 2 to 10 by 10. i. Of lines 2 to 11 by 11. j. Of lines 2 to 12 by 12. For other oral exercises in division, see pages 61 and 63. * As the teacher may indicate. •EXAMPLES. 41 SHORT DIVISION. Examples for the Slate. 98. Illustrative Example I. At $ 5 a day for work, how many days' work can be had for % 4730 ? WRITTEN WORK. Explanation. — As many days' work can (2) ,3) be had for $ 4730 as there are 5's in 4730. Divisor, 5) 4730 Dividend. For convenience, we write the dividend and divisor as in the margin, and divide the terms of the dividend separately, as Ans. 946 days' work, far as possible, beginning with the highest. If we divide the four thousands by 5, we shall have no thousands in the quotient, so we first divide 47 hundreds by 5. 5's in 47 (hundreds), 9 (hundred), and 2 hundreds remain. We write the 9 hundred under the line in the hundreds' place, and change the 2 hundreds remaining to 20 tens, which, with the other 3 tens of the dividend, make 23 tens. 5's in 23 (tens), 4 (tens), and 3 tens remain. We write the 4 tens under the line in the tens' place, and change the 3 tens remaining to 30 units. 5's in 30 (units), 6 (units), which we write under the line in the xmits' place, and have 946 for the entire quotient. Ans. 946 days' ■work. In dividing, the pupil may simply say, " 5's in 47, 9 and 2 over ; in 23, 4 and 3 over ; in 30, 6." Or, abbreviating stiU more, "5's in 47, 9; in 23, 4; in 30, 6." 1. How many cords of wood at 1 6 a cord can be bought for $ 522 ? for $ 3804 ? 1st Ans. 87 cords. 2. How many hours will it take to ride 3216 miles at 8 miles an hour ? at 12 miles ? 1st Ans. 402 hours. 3. At 7 cents an hour for work, how many hours must I work to earn 2835 cents ? 4. How many packages of tea, 9 pounds in a package, can be made from 8847 pounds ? 42 division: &9. Illusteative Example II. How many barrels of flour at $ 8 a barrel can I buy for $ 2597 ? WRITTEN WORK. Explanation. — Here, after dividing, we have ft\ 9t;Q7 K ^ remainder of $5: hence, 324 barrels can he ' bought and $ 5 remain unexpended, which 324 may be expressed as in the margin. .4ms. 324 barrels; |5 remain. The work may be proved by finding the product of the quo- tient and divisor (Art. 92, Note III.) and adding the remain- der. Thus, 324 X 8 + 5 = 2597. 6. How many weeks are there in 585 days ? in 730 days ? 1st Ans. 83 weeks ; 4 days remain. 6. How many 8 quart cans can he filled with 1865 quarts of milk ? with 2587 quarts ? 1st Ans. 233 cans ; 1 quart remains. 7. How many years of 12 months each are there in 200 months ? 8. There are in an orchard 1608 trees, 12 in a row. How many rows of trees are there ? 9. At 11 cents a yard, how many yards of cloth can I buy for 6972 cents ? 10. At 9 cents apiece, how many oranges can be bought for 29415 cents ? 100. Illustrative Example III. If 8 men buy 9675 acres of land which they are to divide equally among them- selves, what is each man's share ? WRITTEN WORK. Explanation. — Each one will have 1 eighth of ON Q^ye 9675 acres. We divide, briefly, thus : * One eighth of 9 thousand is 1 thousand, and Am.. 1209f acres. 1 thousand (equal to 10 hundreds) remains. One eighth of 16 hundreds is 2 hundreds ; of 7 tens, tens and 7 tens (equal to 70 units) remain. One eighth of 75 units is 9 units, with a remainder of 3 units yet to be divided. If 1 eighth of each of the 3 acres is taken, we shall have 3 eighths of an acre. This we express as in the margin, and have 1209f acres for the entire quotient. DIVISION OF DECIMALS. 43 11. What is the price of 1 hat if 6 hats cost 375 cents ? it 12 cost 2700 cents ? 1st Ans. 62f cents. 12. How far must a man travel each day to go 1761 miles in 4 days ? in 9 days ? 1st Ans. 4401 miles. 13. Mr. Stewart promises to sell me 5 rods of land for $ 1578. What is his price per rod ? 14. At $ 8 a thousand, how many thousands of bricks can be bought for 1 3287 ? 15. A man left by his will $ 45267 to be divided equally among his 6 children. What should each child receive ? 16. Eight times a certain number equals 324787. What is that number ? 17. How many 9's are there in 10000 ? 18. To what number is ^ e s s 7 equal ? 19. To what number is n^ser equal ? 20. How many are 10101019 - 7 ? 21. How many are 98306672 - 5 ? 22. Divide 864024 by 7. 24. Divide 369801 by 9. 23. Divide 164408 by 8. 25. Divide 120087 by 11. 101. Division of Decimals. Illusteative Example IV. What is 1 twelfth of 109.92 ? Explanation. — Briefly thus : 1 twelfth of 109 is WHITTEN WORK, g^ ^^ j remains; of 19 tenths is 1 tenth, and 7 tenths 12) 109.92 remain ; of 72 hundredths is 6 hundredths. Ans. 9.16. n-io In the example above it will be seen that we have hundredths in the quotient as there are hundredths in the dividend. In dividing or decimal by a whole number, the quotient is of the sa/me denomination as the dividend. In dividing a decimal by a whole number, fix the decimal point in ike quotient as soon as you reach the decimal foint in the dividend. 26. What is 1 fifth of 86.4055 ? (28.) $234.54-9 = ? 27. What is 1 eighth of 94076.8 ? (29.) $907.34-7=? 44 DIVISION. 102. To Divide, carrying the Division to Decimals. Illustrative Example V. Find 1 eighth of 9675 acres. WRITTEN WORK. Explcmation. — We divide as in Illustrative Ex- , ample III., until we come to the remainder, 3 acres, bj iJb76.000 rpj^jg ^g change to 30 tenths. One eighth of 30 1209.376 tenths is 3 tenths, and 6 tenths remain, which are equal to 60 hundredths. One eighth of 60 hun- dredths is 7 hundredths, and 4 hundredths remain, which are equal to 40 thousandths. One eighth of 40 thousandths is 5 thousandths. The entire quotient is 1209.375 acres. Perform Examples 11 to 16 in Article 100, carrying the division to decimals. 103. Where the divisor is not greater than 12, it is customary to divide as shown above without expressing all the operations. Such a process is short division. For other examples in short division, see pages 61 and 63. LONG DIVISION. 104. Illustrative Example VI. Divide 33075 by 82. WRITTEN WORK. Explanation. — We write the dividend and divisor as in the margin, and draw a curved 82) 33075 (403f I Une at the right of the expression for the divi- 328 dend. 'ZZZ Since the divisor 82 is a larger numher than 3 or than 33, we first divide 330 hundreds hy 82. Now 330 divided by 82 will give about the 29 same quotient as 33 diyided by 8,* which is 4. The first term of the quotient is then 4 hun- dreds, which we express by writing a figure 4 at the right of the curved line. Multiplying 82 by 4 himdreds, and subtracting the product, we find 2 hundreds remain ; uniting with these 2 hundreds the 7 tens of the dividend, we have 27 tens. Dividing the 27 tens by 82, we have no tens in the quotient ; bo we write a zero to show that there are no tens in the quotient, and unite with the 27 tens the 5 units of the dividend, making 275 units. * So we make 8 our trial divisor. LONG DIVISION. 45 Dividing the 275 imita by 82, using 8 for a trial divisor, we have 3 units in the quotient, which we write. Multiplying and subtracting as before, 29 units remain. Dividing each of the 29 units by 82, we have ^, which we write with the units, and have for the entire quotient 403 H. 105. When the divisor is larger than 12, it is usually convenient to express in full, as above, the work of dividing. The process is then called long division. To Divide, carrying the Division to Decimals. 106. Illustrative Example VII. Divide 33075 by 82. WRITTEN WORK. Explanation. — We divide as in the last 82) 33075 (403.36... illustrative example until we reach the re- 328 mainder, 29 units. We now put a decimal — ^— point in the expression for the quotient, and, changing the remainder to 290 tenths, divide as before; and so we keep on dividing 290 Tenths. as far as desirable, or until there is no re- 246 mainder. In this example we stop dividing .,r, , , , at hundredths, and indicate that the divis- 440 Hmidredths. ... ' , ion IS incomplete by writing a few dots. 107. Give answers to the following examples as in Art. 104, or with the quotient carried to thousandths, as the teacher may direct : * 30. Divide 4684 by 31. 34. Divide 12157 by 23. 31. Divide 9632 by 43. 35. Divide 24898 by 72. 32. Divide 5872 by 54. 36. Divide 36872 by 84. 33. Divide 6748 by 62. 37. Divide 36072 by 91. 108. Illustrative Example VIII. Divide 1849 by 192. WRITTEN WORK. Explanation. - As 192 is nearly 200, 1849 192) 1849 (9^;f J divided by 192 will give about the same quo- 1728 tient as 1800 divided by 200, or as 18 divided .jo-i by 2. We then make 2 our trial divisor. * The answers in the Key are given in both forms. 46 DIVISION. 38. Divide 26832 by 96. 40. Divide 232848 by 56. 39. Divide 97684 by 79. 41. Divide 682345 by 88. 109. From the preceding examples we derive the fol- lowing Rule for Division. 1. Write the dividend ; at the left draw a cwved line ■ and at the left of this line write the divisor. 2. Divide the highed term or terms of the dividend hy the divisor. 3. Express the result for the first term of the quotient at the right in long division, beneath in short division. 4. Multiply the divisor by this term. 5. Take the product thus obtained from the part of the dividend used. 6. Unite the next term of the dividend with the remainder for a new partial dividend ; divide, multiply, and subtract as before; and so continue till all the terms of the dividend are used.* 7. Express the division of the final remainder, should there be any, in the fractional form. (Or Change the remainder to tenths, humdredths, thousandths, etc., and continue the division as far as desirable.) Proof. Find the product of the quotient and divisor, and add to it the remainder, if there is one. The result ought to equal the dividend. 42. How many are 36247 h- 189 ? 43. How many are 53004 h- 398 ? 44. How many are 932480 - 287 ? 46. How many are 750010 -- 677 ? * If at any time the divisor is not contained in a partial dividend, write a zero for the next figure of the quotient, and unite with the partial dividend the next term of the given dividend. CONTRACTIONS. 47 Contractions in Division. 110. Illustrative Example IX. Divide 12367 by 10; by 100 ; by 1000. If the decimal point be moved one place 12367 -H 10 = 1236.7 to the left, each figure will express a num- 12367 ^ 100 = 123.67 ber 1 tenth as great as before (Art. 30) ; 12367 ^ 1000 = 12.367 therefore, 1 tenth of 12367 is 1236.7. For a similar reason, 1 hundredth of 12367 is 123.67, and 1 thousandth of 12367 is 12.367. Hence, wher - the divisor is 10, 100, 1000, etc., we may find the quotient hy moving the decimal point of the dividend as many places to the left as then are zeros in the divisor. 46. There are 100 cents in a dollar ; how many dollars are there in 2742 cents ? in 12367 cents ? 1st Ans. $27.42. 47. How many dollars are there in 14863 cents ? 48. There are 1000 mills in a dollar ; how many doUars are there in 56849 mills ? 49. Divide 25000 by 10, by 100, by 1000, and add the quotients. 50. Divide 380768 by 100, by 1000, by 10, by 10000, ana add the quotients. 111. Illustrative Example X. Divide 20864 by 6300. WKITTEN WORK. Explanation. — Since 6300 = 63 x 100, we 63) 208.64 (3.31... may first divide by 100, obtaining 208.64 189 (Art. 110), and then divide this quotient by -.Qf. 63, as shown in the margin (Art. 106). 189 Note. In cases where the exact remainder is wanted, — rr the common form of written work is better. It may ,1 be abbreviated, as in the writ- 63 63100) 208| 64 (S^M ten work of this note. — 189 J]_ Explanation. — Indicate first, 1964 -by a vertical line, the division by 100 ; this gives 208 for a quotient, and 64 remain. Dividing now 208 by 63, we have for a quotient 3, and 19 hundreds re- main. Uniting the first remainder 64 with the last remainder 19 hundreds, we have for the entire remainder 1964. ?or other contractions of division, see Appendix, page 302. 48 DIVISION. 112. Ezamples in Division. a. Divide 58643 by 9. Ans. 6515|, 6. At $8 apiece, how many sheep can he bought for % 2595 ? Ans. 324 sheep ; $ 3 remain. c. If the dividend is 86445 and the divisor 51, what is the quotient ? Ans. 1695. d. What is the quotient of 40076 - 98 ? Ans. 408f|. e. 48 times a certain number equals 38256. What is that number ? Ans. 797. ' /, What number multiplied by 87 gives a product of $22446? Ans.%2m. g. Divide 759000 by 10, by 1000, by 100, and add the quotients. Ans. 84249. h. What is the sum of 93600 divided by 20, and 93600 divided by 7200 ? Ans. 4693. 1. How many are 493689-^47000 ? Ans. 10|ff||. j. 37884 is 42 times what number ? Ans. 902. k. What is 1 thirty-eighth of 856406 ? ^»is. 22537. 1. The product of two factors is 5063 ; one of them is 83. What is the other ? •^»»s- 61. m. The product of three factors is 28350 ; two of them are 42 and 75. What is the third ? -Ams- 9- a. .aiaJAlA + one fourth of 2700 = what number ? Am. 261554 Examples with Decimals. o. Divide 42.8116 by 13. Ans. 3.2932. p. What is 1 ninth of $ 76.842 ? Ans. % 8.538. q. What is the cost of each chair if 25 chairs can be bought for % 247 ? Ans. % 9.88. Note. The examples on this page emhraoe the principal varieties in form of examples in division. Examples for additional practice will be found on pages 61 and 63. EXAMPLES. 49 113. Applications. 61. If I travel 42 miles a day, in how many days can I travel 273 miles ? 52. How many barrels are required to hold 5488 pounds of flour, if one flour-barrel holds 196 pounds ? 53. How many days are there in 9684 hours ? 54. How many days will it take a ship to sail 13724 miles, at the rate of 133 miles a day ? 55. There are 5280 feet in a mile. How many miles high is Mount Everest, which is 29002 feet high ? 56. A. B. bought a farm for 1 18785 at $ 95 an acre- How many acres were there in the farm ? 57. A produce dealer packed 19162 eggs in boxes containing 144 eggs each. How many boxes did he fill ? 58. If the dealer would put 19152 eggs into 84 equal-sized boxes, how many eggs should he put in a box ? 59. In one year, Missouri produced 4164 tons of lead, worth $ 353940. What was the value of a ton ? 60. There was sent to the U. S. Mint in 13 years $ 4377984 worth of gold. What was the average value sent a year ? If gold was worth 16 dollars an ounce, and 12 ounces make a pound, how many pounds were sent ? 114. Examples -with Decimals. 61. A man divided among his three sons 887.625 acres of land. What was each son's share ? , 62. What is the price of 1 comb, when 48 combs can be bought for 1 63.76 ? 63. When 234 oranges are bought for % 7.02, what is paid for 1 orange ? In the following examples continue dividing to the third order of decimals : 64. Find 1 ninth of 1.28 acres. (67.) 8.1^-21 = ? 0,5. Find 1 twelfth of 3.75 tons. (68.) 0.5-33 = ? 66. Find 1 fifteenth of 128.6 miles. (69.) 1.868 -h 215 = ? 50 MISCELLANEOUS EXERCISES. SEOTIOB" TI. MISCEL3LANEOUS EXERCISES. 115. General Review, No. 1. 1. 287 + 5 million + 36 thousand + 69481 = ? 2. Add 567 to the sum of the following numbers; 121, 232, 343, 454, 565, 676, 787, and 898. 3. The difference between two numbers is 95478. The larger number is 148769 ; what is the smaller ? 4. Which of the two numbers 15672 or 10560 is nearer to 13465, and how much ? 5. Take 987 from each of the following numbers, and add the remainders: 3644; 7573; 2432; 4001. 6. What number must be added to the difference between 58 and 7003 to equal 938000 ? 7. What number taken from the quotient of 1833000 -=-24 leaves 25 ? 8. What number equals the product of the three factors 1785, 394, and 624-48? 9. If 5872 be the multiplicand, and half that number the multiplier, what is the product ? 10. If 4832796 is the product, and 1208199 the multiplicand, what is the multiplier ? 11. If 894869 is the minuend, and the sum of the numbers in the fifth example is the subtrahend, what is the remainder ? 12. If 700150 is a dividend, and 3685 the quotient, what is the divisor ? 13. If 28936 is the divisor, and 86 is the quotient, what is the dividend ? 14. Divide 87 million by 15 thousand. For other questions in review, see pages 59 - 63. ORAL EXAMPLES. 51 116. OiaX Ezamples for Analysis. {See Appmdix, page SOS.) a. If a car runs 69 miles in 3 hours, how far can it run in 5 hours ? b. If 18 rows of potatoes yield 36 bushels, how many bush- els will 20 similar rows yield ? c. If $ 5 pay for 35 quarts of berries, how many quarts will $12 buy? d. If, when flour is $ 8 a barrel, a ten-cent loaf weighs 25 ounces, what should it weigh when flour is $ 10 a barrel ? e. If 5 oxen consume 185 pounds of hay in 2 days, how much will be required for 1 yoke of oxen for the same time ? /. If 6 cows were bought for $ 224 and sold for $ 260, what was the gain on each cow ? g. If 150 barrels of apples were bought for $ 200 and sold for $ 350, what would be gained by selling 45 barrels at the same rate ? h. I bought a lot of paint for $3.90 and sold it for $5.10, gaining 12 cents on a pound. How many pounds did I buy? i. If a quantity of hay lasts 22 oxen 10 days, how many days will it last 5 yoke ? j. A field of wheat was reaped by 10 men in 6 days ; what length of time would be required for 15 men to reap the same amount ? k. A cistern can be emptied in 15 minutes by 7 pipes ; in what time can it be emptied, if only 5 of the pipes are open ? i. If 8 operatives can do a piece of work in 12 days, in what time will 24 operatives perform the same work ? m. If a certain piece of work can be performed by 50 men in 4 weeks, how many more must be employed to perform it in a week ? n. Ten hunters have provisions to last them 6 weeks ; if 2 men be killed, how long will the provisions last the remainder ? 52 MISCELLANEOUS EXERCISES. 117. Miscellaneous Examples. 15. A merchant bought goods for $ 1084, and sold them for $ 594 more than he gare. How much did he receive for them ? 16. From a farm containing 984 acres there were sold at one time 347 acres, at another time 167 acres. How many acres remained ? 17. A merchant bought goods for $ 2467, and sold them for $ 875 less than he gave. How much did he receive for them ? 18. If I take 7642 gallons from 21002 gallons twice, what will remain ? 19. Of 30070 men who went into battle, 4564 were slain and 1675 were taken prisoners. How many were left ? 20. Bought two horses ; the first cost $ 215, the second $ 40 less than the first. How much did the two horses cost ? 21. If $ 19.74 were paid for 14 bushels of wheat, what must be paid for 25 bushels ? 22. If 19 tons of coal run an engine 798 miles, how far will 14 tons run it ? 23. The area of the New England States is as follows: Maine, 31766 square miles; New Hampshire, 9280; Ver- mont, 10212 ; Massachusetts, 7800 ; Connecticut, 4674 ; Ehode Island, 1306. How many more square miles are there in Maine than in the three States of Vermont, New Hampshire, and Massachusetts ? 24. How many States of the size of Ehode Island might be made out of Massachusetts, and how many square miles would remain ? 25. How much smaller is Connecticut than Vermont ? 26. Texas contains 237504 square miles. How many States of the size of New England might be made out of it, and how many States of the size of New Hampshire out of the remainder ? 27. If 5 bushels of wheat of 60 pounds each are required to make 1 barrel of flour, how many pounds of wheat are re- quired to make 100 barrels of flour ? EXAMPLES. 53 28. In a certain schoolhouse 9 of the rooms will seat 52 pupils each, and 4 will seat 48 pupils each. How many pupils can be seated in all ? 29. How many feet of fencing will be required to enclose a lot of land measuring on each of two sides 489 feet, on the third 548 feet, and on the fourth 596 feet ? 30. In a school there are 7 classes of 54 pupils each ; 196 of these are boys. How many are girls ? 31. A horse cost $ 262, a chaise $ 228, and a hack 3 times as much as both. What did all cost ? 32. A farmer exchanged 4 cows, worth $ 68 each, for a span of horses. What were the horses worth apiece ? 33. A merchant bought 45 bales of cotton, each bale con- taining 42 pieces, and each piece 38 yards, at 9 cents a yard, and sold the whole at 11 cents a yard. How much did he gain ? 34. A man raised in one year 364 bushels of corn, the next year twice as much as he did the first year, and the third year three times as much as the second year. How many bushels did he raise in all ? 35. A grocer bought 8 chests of tea, each chest containing 48 pounds, at 50 cents a pound. He sold one haK of the tea at 65 cents a pound and the other haM at 72 cents a pound. How much did he gain ? 36. After $ 158 were taken from a box there remained $ 15 more than twice that sum. How many dollars remained ? 37. Mrs. Keyes, haring 1 2000 to invest, bought 10 United States bonds at $ 112 each, and then as many railroad shares at $ 92 each as she could pay for. How much money was left ? 38. Mr. Oaks bought a piano for $ 376, paid f 14 for freight and cartage, and $ 2 for tuning, then let it 7 quarters at $15 a quarter, and afterwards sold it for $325. Did he gain or lose, and how much ? 39. A man paid |270 for a threshing-machine, and hired help to run it at $ 5 a day. He then let the machine at $ 8 a day, including the help he hired. How many days must he let the machine to pay its first cost ? 54 MISCELLANEOUS EXERCISES. 40. How many posts and how many rails will be required for a fence 156 feet long if the posts are set 12 feet apart and the fence is 5 rails high ? 41. A man sold three houses; for the first he received $ 3525, for the second 1 950 less than he received for the first, and for the third as much as for the other two. How much did he receive for the three ? 42. A jeweller sold 15 clocks and 22 watches ; for the clocks he received $ 12 apiece, and for each watch 7 times as much as for a clock. What did he receive for all ? 43. If 28 men can grade a road in 72 days, how long will it take 36 men to do half the work ? 44. If a man earns I ISO a month and spends $ 36 for hoard and $ 50 for clothes and other expenses, in how many months can he save $ 1410 ? 45. Mr. Brown bought 18 cords of wood for $ 110. For how much must he sell it a cord to gain 1 34 on the whole ? 46. Mr. Snow bought some land for $ 13825. He sold 100 acres at $ 55 an acre, and then found, in order not to lose on his bargain, that he must sell the remainder for $ 62 an acre. How many acres were there in the remainder ? 47. A had $ 45 ; B twice as much less $ 17 ; and C as much as A and B together. How much money had C ? 48. One half of one number is 1764, and four times another number is 5876. What is their sum ? 49. A and B, 450 miles apart, travel towards each other. A travels at the rate of 30 miles a day and B of 35 miles a day. If B rests the second day, how far apart are they at the end of the fourth day ? 60. A man bought 163 barrels of flour at $ 9 a barrel ; 15 barrels were spoiled, and the remainder he sold at $ 11 a bar- rel. Did he gain or lose, and how much ? 51. At an election the sum of the votes received by two opposing candidates was 4324; the successful candidate re- ceived 218 more votes than his opponent. How many votes did each receive ? EXAMPLES. 55 52. On commencing business a merchant had $ 7852 in cash, $ 7919 in real estate, goods valued at f 9728, a lot of lumber valued at $ 6930, a ship valued at 1 16834 ; during the first year he was in trade he gained above all his expenses $ 3195. What was he worth at the end of the year ? 53. The GuK Stream carries 2787840000000 cubic feet of water past a given point every hour, which is 1200 times as much as the hourly discharge of the Mississippi. What is the hourly discharge of the Mississippi ? 54. There were 1032467 cigars made in Westfield in 1 month. If these were bought for 5 cents apiece, how many families would the money thus spent supply with bread for a year (365 days) if each family should consume two 8-cent loaves a day ? 55. The distance from Boston to Albany is 202 miles, from Albany to Buffalo, 298 miles. How long will it take a train to pass over the road at the rate of 28 miles an hour, allowing 2 hours for detentions between Boston and Albany, 1 hour at Albany, and 3 between Albany and Buffalo ? 56. If it takes 5 yards of cloth to make a pair of shirts, what will 24 pairs cost at 15 cents per yard for the cloth, 45 cents apiece for bosoms, wristbands, and buttons, and 95 cents apiece for making ? 57. In how manv days, of 6 hours each, can the president of a bank sign 90000 bank-notes, if he signs 5 in a minute ? 58. If 8 presses can coin 19200 pieces of money in an hour, how many pieces can one press coin in a minute, 60 minutes making an hour ? Papyrus is said to have been used to write upon 2000 years befori^ Christ, and parchment to have been invented 1810 years later; from the invention of parchment to that of paper in China was 20 years ; to printing by movable types was 1608 years more ; stereotyping was invented 273 years still later. 59. How many years from the first use of papyrus, as given above, to stereotyping ? 60. In what year before Christ was paper invented in China ? 61. In what year after Christ was printing invented ? 56 MISCELLANEOUS EXERCISES. 62. There are in a certain school 47 pupils 14 years old ; 96 pupils 12 years old ; 114, 11 years ; 149, 10 years ; and 168, 9 years old. What is their average age ? 63. If the earth is 92000000 of miles from the sun, and the moon at its fuU is 224000 miles farther on, and light travels at the rate of 185172 miles a second, how many seconds is it in passing from the sun to the moon and back to the earth ? 118. Questions for Review. What is Addition ? What is the amount ? Add orally 64 and 87. How do you write niunbers to be added? Is this absolutely necessary ? Add five numbers expressed by four figures each, and ex- plain. Give the rule ; the proof. Illustrate adding at once numbers expressed in two or more columns. What is Subtraction ? What is the minuend ? the subtrahend ? the remainder? Take orally 28 from 91. Find the difference be- tween 368 and 7006, and explain. Give the rule; the proof. When the minuend and difference are given, how can you find the sub- trahend ? When the subtrahend and difference are given, how can you find the minuend? What is Multiplication ? What is the multiplicand ? the mul- tiplier ? the product ? What are factors ? Multiply orally 45 by 6. Perform and explain, an example in which the multiplier has at least two terms. Give the rule ; the proof. How do you multiply by 10, 100, 1000, etc. ? How do you proceed if there are zeros at the right of the expression of the multiplicand or the multiplier, or both? Tens X units = what ? Units x tens ? Thousands x tens ? Tens x hundreds? Ten-thousands x hundreds? What is Division ? What is the dividend ? the divisor? the quotient ? the remainder ? Perform and explain an example in short division ; prove the work. Perform and explain an example in long division. Give the rule ; the proof. How do you divide by 10, 100, 1000, etc. ? How do you divide when the expression of the divisor contains zeros at the right ? When the dividend and quotient are given, how can you find the divisor ? When the divisor and quotient are given, how can you find the dividend ? When the multiplier and product are given, how can you find the multiplicand ? When the multiplicand and product are given, how can you find the multiplier ? DRILL EXERCISES. 57 DRILL EXERCISES. 119. Explanation of the Use op the Drill Tables. The object of the Drill Tables and Exercises which are found on the six following pages is to extend indefinitely practice in arithmetical operations without additional labor on the part of the teacher. The exercises are not to be assigned in order, nor is any one pupil expected to perform them all ; they may be used, however, like other examples. (See Notes on pages 16, 25, 35, and 48.) The following illustration shows how they may be used for class drill, and each pupil have a different example. Addition. 1. Let the members of the class number themselves 1, 2, 3, etc., to any given number up to 25 ; and let each member find his number in the left-hand margin of the table. 2. The teacher then gives a direction in this form : "Add A, B, and C." (See Exercise 1, page 59.) 3. In obedience to this direction, each pupil will add the numbers that he finds expressed under the letters A, B, and C, and in the line of his own number. Thus, pupil No. 1 will add 65, 512, and 7901 ; No. 2 will add 34, 724, and 3053 ; and so on. Thus a series of examples is given out at a single dicta- tion, and the pupils are taught to work independently. 4. The key contains answers to all these examples. Subtraction, Multiplication, and Division. By changing slightly the form of direction described above, the same table wiU afford abundant practice in the other fundamental operations. (See pages 59, 61, and 63.) 68 MISCELLANEOUS EXERCISES. 120. DRILL TABLE No. 1. Simple Numberg. A 65 B hij 512 c k Imn 7 901 D opq 274 E r s tu 2 865 F VW xy z 40 912 34 724 3 053 613 3 742 83 640 79 235 6 360 769 8 604 20 357 46 941 1 604 133 6 821 67 904 98 858 8 029 486 4 930 13 580 56 467 7 940 918 5 439 80 439 32 673 4 809 675 7 108 50 672 48 388 6 580 436 4 583 38 204 87 747 2 096 577 8 057 29 073 76 599 8 920 814 6 974 68 370 54 252' 2 031 239 9 107 90 521 95 381 6 150 721 1 580 15 023 82 817 4 706 544 7 362 56 807 71 426 9 059 715 2 115 80 195 23 794 4 270 645 4 276 41 730 37 261 3 Oil 978 6 583 67 054 93 638 5 490 851 2 941 57 026 28 372 1 705 327 8 724 20 687 62 919 9 630 163 6 239 34 610 57 485 5 108 784 4 037 25 048 89 591 7 502 516 5 482 90 572 45 183 2 610 297 9 372 72 056 92 868 3 703 466 6 048 46 190 64 655 6 207 349 3 659 20 839 25 942 8 054 922 4 176 81 704 1 2 3 4 6 6 7 DRILL EXERGISEH. 59 121. Exercises upon the Table. Addition, 1. Add A,* B, and C. g. Add B, C, and D. 3. C plus D plus E plus JF equals what number ? 4. A + B + C + D + 16042 = ! 5. What is the sum of B, C, D, E, F, and 61375 ? 6. Find the amount of A, B, C, D, E, F, and 23456. In each column indicated by figures at the bottom of pages 58, 60, and 63, 7. Add the upper six numbers. 8. Add the upper ten numbers. 9. Add the upper fifteen num- bers. 10. Add aU the numbers. Subtraction. 11. From C take B. W. Subtract D from E. 13. Take E from F. 14. Find the diflFerence between C and E. 15. F minus C equals what number ? Multiplication. 16. Multiply B by 6. 17. Multiply C by 7. 18. Multiply D by 8. 19. Multiply E by 9. 50. Multiply B by A. 51. Multiply by B. SS. Multiply C by D. 53. Multiply E by D. 54. Find the product of F by D. 55. Find the product of F by C. • See the explanation, page 57. Eeview. 56. What number added to the amount of A and B will equal C ? 57. Add together C, E, and the difference between B and D. 58. Subtract C from 12304, and from the remainder take B. 59. Multiply D by 1002, and from the product take F. 30. Multiply C by 6 ; D by 7 ; E by 8 ; and find the sum of the products. 31. Multiply B by 10; Dbyll; and add the products with C plus E. 3^. A man having F dollars paid E dol- lars to one man and D dollars to another. How much did he have left? 33. Bought a house for C dollars ; paid B dollars for repairs ; then sold it at a loss of D dollars. How much did I receive for the house ? 34. A merchant had B barrels of flour. He sold A barrels at 1 12 a barrel, and the remainder at 1 9 a bawel. How much did he receive for the flour? Oral Practice. 35. How many are 8 -I- f -I- g -I- h, etc. to z ? 36. How many are 27-1-h-l-i, etc. to z ? 37. Howmany are 55-f-g-h-i-j? 38. How many are 100 - - p, etc. to z ? 39. How many are 7 -t- f to n less less p less q ? 40. How many are h times i less j plus k to z ? 41. How many are 100 less A ? 4S. Find the difference between 43 and A. 60 MISCELLANEOUS EXERCISES. 122. DRILL TABLE No. 2. Simple Numbers. A ef 45 B ghi 648 C j klm 6 068 68 473 5 406 74 835 8 049 56 592 7 250 48 726 3 087 64 954 4 503 78 367 2 790 83 289 5 608 54 635 9 160 76 489 6 085 58 375 8 607 47 689 4 130 65 764 7 082 53 865 3 706 49 796 8 154 84 347 4 370 79 586 5 480 63 627 1 094 57 396 8 406 69 738 6 530 75 579 7 209 67 i86 8 660 59 942 4 308 46 278 3 960 73 587 6 805 8 9 10 D nop q r s 469 007 743 600 620 085 800 659 389 700 956 800 285 004 500 376 675 400 732 900 486 300 840 016 570 068 962 400 435 600 700 843 669 700 800 876 643 007 350 092 800 765 469 007 782 500 946 007 465 300 U 12 E t uvw xyz 8 046 362 6 530 781 4 654 380 7 820 463 9 068 318 6 782 630 3 905 746 2 849 370 6 470 866 5 783 062 7 660 849 3 672 083 4 921 608 9 430 572 4 187 063 6 410 342 1 698 706 7 421 089 6 460 713 2 835 068 5 306 739 3 640 687 7 390 468 6 083 746 7 408 663 13 14 DRILL EXERCISES. 61 (3. u. 45. 46. 47. 50. 51. 5Z. 63. 54. 65. 56. 57. S8. 59. 60. 61. 6S1. 65. 66. 123. Division, Divide D by 4.» Divide D by 5. Divide E by 6. Divide E by 7. Divide D by 8. Divide D by 9. Divide C by 12. Divide C by 15. Divide D by 16. Divide D by 18. Divide E by 27. Divide C by A. Divide D by A. Divide E by B. Divide D by C. Divide E by C. Divide D by 800. Divide E by 4200. Addition. How many are 46872 + A to D? How many are 65478 + A to E? Subtraction. From E take D. Find the difference tween E and D x Multiplication. Multiply D by C. Multiply E by D. Exercises upon the Table. Keview. 67. How many more than C are B times B ? 68. What number added to ten times the amount of B and G wUl equal D ? 69. A man owns three tracts of land ; the first is valued at C dollars, the second at B doUars, and the third is worth twice as much as the second. How much is the land worth ? 70. By selling a house at C doUars I gained 12 times A dollars. What was the cost? 71. If a farmer should purchase B acres of land at A dollars per acre, and pay down C dollars, how much would he then owe for the land ? 7S. A man having C dollars spent B dollars and lost A dollars. How much would one third of the remainder be ? 73. How many cows, at A dollars apiece, can be bought for one fifth of ten times B doUars, and how many dollars will Oral Practice. 74- How many are 6 + e + f +g, etc. to z? 75. How many are 15+g + h, etc. to z? 76. How many are 29 + j + k, etc. to z ? 77. How many are exf-g-h? 78. How many are g x h -f- i ? 79. How many are h x i -r- j ? 80. Divide A by 2 ; by 3 ; by 4 ; 5 ; 6 ; 7; 8; 9. 81. Divide gh (64, 47, 83, etc.) by 3 ; by 4 ; etc. be- 10. Other dividends and divisors can be indicated, aa jkby 7 ; no by 8 ; tuby 9; etc. * See page 67, for Explanation of the Use of the Drill Tables. 62 MISCELLANEOUS EXERCISES. 124. DRILI. TABLE No. 3. Simple NumbeTS. Szamples. A 1 . Nine hundred fourteen thousand, forty-one. 2> One million, forty thousand, fourteen. 3a Nine hundred seventy-six thousand, sixty-seven. 4. Sixteen hundred seventy-eight. 5< Sixty-three million, three hundred six thousand. 6> Nineteen million, nine hundred thousand, 19. 7. One hundred seventy million, seven. 8. Ten million, one thousand, one hundred one. 9. Three hundred five million, fifty thousand. 10. Twelve million, two hundred thousand, two. 11. One million, eighteen thousand, eight. 12. One billion, six hundred thousand, six. 13. Nine hundred four million, ninety-four. 14. Three billion, thirty million, three hundred three. 15. Two billion, four hundred twelve thousand, 14. 16. One hundred one million, one thousand, one. 17. Six billion, sixty thousand, six hundred. 18. Four hundred throe billion, thirty. 19. One trillion, seventeen million. 20. 506 billion, sixty-five thousand, five. 21 . Four trillion, four billion, four thousand. 22. Six hundred eighty-nine billion, six thousand, 89. 23. Forty-two trillion, forty thousand, two hundred. 24. Eighteen trillion, 108 million, eighteen. 25. Four trillion, forty-seven billion, 4700. BRILL EXERCISES. 63 DRILL TABLE No. 3 iamtinimfy E w X y z 3 276 4 368 9 234 2 592 7 488 6 545 4 368 7 616 4 896 7 128 8 448 8 232 2 736 2 925 6 912 4158 8 778 9 996 6 435 6 376 6188 8 736 7 425 9 282 3 744 17 Ex- amples. B q c r s o t uv 1. 7 63 819 2. 4 62 364 3. 9 81 486 4. 3 36 324 5. 8 64 576 6. 5 36 595 7. 2 28 728 8. 4 68 952 9. 3 51 612 10. 6 72 648 11. 8 88 352 12. 7 56 392 13. 2 48 912 14. 5 45 585 15. 9 64 864 16. 6 66 594 17. 3 67 627 18. 7 49 588 19. 6 65 715 20. 8 96 768 21. 2 34 884 22. 4 84 672 23. 9 99 495 24. 7 91 273 25. 6 78 624 IS 16 125. Ezercises upon the Table. SIS. Express A by figures.* 83. Add, in A, the 1st and 2d; 2d and 3d; etc. 84. Add, in A, from 1 to 6; 2 to 7; etc. 85. Find, in A, the difference between tlib 1st and 2d; the 2d and 3d; etc. 86. Multiply A by 6. 87. Multiply A by 7. 88. Multiply A by 8. 89. Multiply A by 9. 90. Divide A by 6. 9^. Divide A by 8. 91. Divide A by 7. 9S. Divide A by 9. 94. Multiply C by B ; add D to the product; and find the difference between the amount and E. 95. Divide D by B ; also divide E by B ; and find the difference between the sum of the quotients and D. 96. Divide E by D; subtract the quotient from C; and multiply the remainder byB. 97. Subtract C from D; divide the remain- der by B ; and with the quotient divide E. Oral Practice. 98. How many are 9 + q to x less y less z ? 99. Howmanyare34-l-r to V, divided by q? 100. How many are 45 + 1 to z, divided by q ? 101. How many are r times s + t toz, divided byq? lOS. How many are t times u-i-v to z, di- vided by q ? See Explanation of Table, page 57. 64 UNITED STATES MONEY. SEOTIOl^ VII. UNITED STATES MONEY. 126. The picture above represents pieces of metal weighed and stamped by authority of government, and used in buying and selling. Such pieces of metal are coins. Each coin represents a unit of value. 127. Dollars and cents are the units of value chiefly used in business. Eagles, dimes, and mills are also used, but there is no coin to represent a mill. TABLIi. 10 mills = 1 cent, marked ci. or ^. 10 cents = 1 dime. 10 dimes or') 100 oents 10 dollars = 1 eagle. ^ = 1 dollar, marked $. To read and -write numbers in XTnited States Money. 128. The dollar, being the principal unit of United States money, iii expressed at the left of the decimal point ; dimes, cents, and mills, being tenths, hundredths, and thousandths of a dollar, are expressed cil Llie right of the decimal point. EXAMPLES. 65 Thus, 11 dollars, 2 dimes, 3 cents, and 4 miUs are written $ 11.234 ; and the expression is read, "Eleven dollars twenty-three cents four mills." For exercises in reading and writing, turn to page 73. 129. Oral Ezercises in Reduction. a. How many mills in 1 cent ? in 18 cents 5 mills ? By what do you multiply to change cents to mills ? dollars to cents ? dollars to mills ? b. Change $ 14.08 to cents. e. Change $ 2.625 to mills. e. Change $ 1.62 to cents. i. Change 1 5.02 to mills. d. Change $ 0.48 to cents. g. Change $4 to mills. h. How many dollars are there in 500 cents ? By what do you divide to change cents to dollars ? mills to dollars ? 1, How many dollars are there in 170 cents ? in 3689 cents ? j. How many dollars are there in 1876 mills ? in 4728 mills ? 130. In performing the examples above, you have changed numbers expressing a certain amount of money to numbers whose units are larger or smaller, but without changing the amount itself. Such a process is called leduction. For other examples in reduction of United States money, see page 73. 131. Szamples for the Slate. ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION. How do you write dollars, cents, and mills, when you are to add or subtract them ? (Art. 46.) 1. My deposits in a hank were 1 192.92 and $ 155.37 ; of this I have withdrawn $79.48, $71.62, and $78.21. What is the balance in the bank ? 2. What must I pay for 23 yards of silk at $ 2.37 a yard, and 6 yards of lace at $ 1.68 a yard ? 66 UNITED STATES MONEY. 3. What is 1 fifteenth of $287.40? 1 seventeenth of $722.50? In the foUowing examples continue the division to cents and mills. (Art. 102.) In the answers, reject mills when less than 5, and call 5 mills or more 1 cent. 4. What is 1 fifth of $ 17 ? of $ 83 ? 1st Am. $ 3.40. 5. What is 1 sixteenth of $ 981 ? 6. Eight men chartered a schooner for $ 295. What was each man's share of the cost ? 7. When 32 lawn-mowers were bought for $ 696, what was the price of each ? 8. Mr. Eice paid $ 198.45 for 35 school desks. What would 168 desks cost at the same price ? To divide one sum of money by another. 132. Illusteative Example. At $ 2.12 per pair, how many pairs of slippers can be bought for $ 100 ? WRITTEN WORK. Explanation. — To divide one sum of money by 212) 10000 (47 another, loth dividend and divisor must he expressed 848 i'^ the same denomination. Here the divisor being ■AKnn cents, the dividend must be changed to cents. (Art. 129.) Dividing 10000 cents by 212 cents, we have 47 for a quotient, with a remainder of 1484 36 36 cents. Ans. 47 pairs ; 36 cents remain. 9. I paid $80 for turkeys at $2.50 apiece. How many turkeys did I buy ? Divide $ 42 by $ 1.75. 1st Ans. 32. 10. A conductor took up $ 1224 worth of railroad tickets from Springfield to New York at $ 4.25 apiece. How many tickets did he take ? 11. How many boxes at 33 cents a box can be bought for $ 20, and how many cents will be left ? 12. How many veils at 92/* each can be bought for $ 30 ? 13. How many dinners at $ 0.625 each will $ 22 pay for ? For additional examples, see page 73. RECKONING MONEY. 67 Coins and Paper Currency. 133. The legal coins of the United States are : Gold. SaveT. Double-Eagle = $20.00 Dollar = fl.00 Eagle = 10.00 Half-dollar =: 0.50 Half-Eagle = 5.00 Quarter-dollar = -0.25 Quairter-Eagle = 2.50 Twenty-cent piece = 0.20 Three-dollar piece = 3.00 Dime = 0.10 One-dollar piece = 1.00 Copper and nickel 3-cent and 5-cent pieces and bronze 1-cent piece. Note. The gold coin is hardened by an alloy of 1 tenth copper and silver {the sUver not to exceed 1 tenth of the whole alloy). The silver coin is hardened by 1 tenth copper. The bronze cent has 95 parts of copper to 5 parts of tin and zinc. The 3-cent and 5-cent pieces have 75 parts of copper to 25 parts of nickel. The silver 5-oent and 3-cent pieces, the bronze 2-cent piece, and the old copper coins, are no longer issued. Bank bOls and United States Treasury notes (greenbacks) are largely used in place of coins. These represent the values of $1, $2, $5, 1 10, 120, $50, $100, $500, and $1000. 134. Exercises in reckoning Money. Perform as many as possible of the following examples without written work : How much money in a. Two 20-dollar bills, three 10' s, four 5's, and seven I's ? h. Eight 5-dollar bills, seven 2's, five lO's, and three I's ? c. One 50-dollar bill, six 5's, two I's, with 3 half-dollars, 5 quarters and 4 dimes ? How much more money must you receive to have % 60 if you now have d. Three 5-doUar biUs, seven 2's, four I's, with 2 half-dol- lars, 3 quarters, and four 5-cent pieces ? e. Two 5-dollar bills, three 2's, and one 10, with 6 quar- ters, 4 dimes, two 3-cent, and tbyee 5-cent pieces ? 68 UNITED STATES MONEY. 1. How muct money shall I have left of six 5-dollar bills, and 2 quarters, after paying for 6 yards of brilliant at 65/ a yard, for Silesia, 28/', and for buttons $ 1.16 ? g. What must you pay for 2 dozen eggs, 5 pounds of sugar, 2 gallons of vinegar, and 2 bushels of apples at the present prices where you live ? 135. Accounts and Bills. [EXTEACT FKOM THE AOOOONT-BOOK OF T. SMITH & Cc] EDWABD WHLIAMS, Dr. 1876. Q'^iiie/ ^ (SToff a^. "We^i^Ti S^^ui,, @ /§ 40 oo Q^May B " SS!/dt.Q/^a./a^.Ma^i^na.," /<^/ S 7& ^une / " ^t^ /da. J'a^a ^o^, " SOi^ 4 §ff II 2§ " / aay'a ^CH^iM c^f ^ciea tnan / 7^ 136. Above is a record kept by T. Smitli & Co. of ar- ticles sold and services rendered by them to Mr. Williams. 137. It is customary for persons who buy or sell goods or services to keep a record of the articles bought or sold, the kind and amount of services rendered, their value, etc., as above. Such a record is an account. 138. The person to whom a debt is owed is a creditor. 139. The person who owes a debt is a debtor. Who is debtor in the account stated above? Who is creditor? 140. A written statement of an account prepared for the debtor by the creditor is a bill. 141. When the bill is paid, the creditor, or some one authorized by him, signs his name to the bill, with the words "Eeceived payment." The bill is thus receipted, (See bills Nos. 2 and 3.) BILLS. 69 Examples for the Slate. 142. Find the cost of each article in the followiag bills, and their several amounts : 14 (1) New York, Nov. ISS, 1877. |8ongI)t of fowle, peatt, & ca ■f4 ^. '^oi'n, " §7.. § // // // It // // S^^. J" If ff It // ■^7.00 4.00 " 7 " MAe m^uo/. VTfioma, ^.7ff Beceived paymmt, CHARLES DAT. 18. (5) Bristol, Jam. 1, 1877. '^0 A. £. PEASE, Dr. 18 It &Szy Ju/y S tl It s :/0 iTO (STa p^^. M^, @ 7f " SO " (Mmcd, " ^f " 41^ " Miea4^^(S7e^ Q^. M. Maae Receimed payment, * This means that Mr. Butler is credited for goods or cash delivered. Cr. is read 'creditor." EXAMPLES. 71 Examples for Bills. 143. Find the amounts due in the following oxamples, and make out the bills, supplying dates, etc., when wanting. 19. Charles Miller bought of James Gibhs, Jan. 4, 1877, 1 horse for % 95.00, 2 cows at $ 50 apiece, 1 wagon for $ 62.00, 2 shovels at $ 1.12 apiece, 30 bushels of corn at 65 / per bushel, and 17 bushels of wheat at $ 1.62 per bushel. 20. Samuel Briggs sold to Alfred Loomis 2 pieces flannel, of 62 yards each, at 49 / per yard ; 38 yards ticking, at 29 / ; 86 yards brown sheeting, at 27/ ; and 42 yards broadcloth at $3.65. 21. Dr. Holland bought of John Avery 9 pounds oil of pep- permint at $2.50; 4 pounds oil of cassia at $1.62; 4 pounds oil of orange at $ 3 ; 6 pounds oil of lemon at $ 3.26 ; 5 pounds oxalic acid at 13 f ; and 5 pounds Seneca root at 95/. 22. Banks & Searles, of Cleveland, bought of Snow & Rising, Albany, 24 sack coats at $ 15.75 ; 36 vests at 1 3.50 ; 9 dozen felt hats at $ 36 per dozen ; 4 dozen pairs suspenders at 42 »* per pair ; and 23 dozen pairs gloves at 68 / per pair. 23. J. D. Turber bought of C. 0. Clement, Nov. 8, 1876, 2 Dictionaries, at 87/ apiece; 9 Vocal Cultures, at 90/, and 24 Spellers, at 20/. Dec. 2, he bought 2 reams of paper at $2.12, 3 dozen pencils at 60/, and 12 slates at 17/. Dec. 10, he paid Mr. Clement I 20.00, and Jan. 1, 1877, Mr. Clement made out his bill. Required the balance due. 24. Sell to your neighbor 4 pear-trees at $ 1.76 each, 9 to- mato-plants at 7/ each, 5 geraniums at 30/ each, and make out the bill. 25. Sell three different articles from a dry-goods store, and make out the bill. 26. Make out a bill for 3 days' work at 76/ a day, 4 days' work at $1.50 a day, and 2 bushels of cranberries at $4 a bushel, crediting the person against whom you make the bill with 6 hours' work at 36/ an hour. 72 UNITED STATES MONET. 144. DRUJi TABLE No. 4. United States Money. A B $ 18.40 Twelve dollars, twenty-five cents. 183.22 Seventy-one dollars, ninety cents. $ 36.41 Twenty-five dollars, sixty-two cents. $30.05 Eighteen dollars, nine cents. $ 204.75 One hundred thirty dollars, six cents. $ 9.208 Five dollars, seven cents, five mills. $ 5.632 One dollar, ninety cents, eight mills. $ 876. One hundred dollars, twenty cents. $ 100.35 Forty-nine dollars, seventy-two cents. 1 15.207 Six dollars, seven cents, three mills. $ 1.36 Twenty-seven cents, five mills. % 20.95 Twelve dollars, nineteen cents. $ 0.402 Twenty-five cents, five mills. $19,005 Sixteen dollars, six mills. $63,072 Forty-nine dollars, twenty-four cents. $ 7.645 Five dollars, sixty-seven cents, five mills. $419.28 Ninety-nine dollars, fifty -six cents. $0,626 Seventeen cents, eight mills. $600.57 Thirty-eight dollars, five mills. $268.06 89 dollars, fifty cents, three mills. $ 29.70 Ninety-two cents, five mills. $ 11.005 Seventy-five cents, five mills. $100.02 Fifty-four doUars, nine cents. $ 444.44 Four dollars, forty-four cents, four mills. $ 100.10 Nine dollars, nine cents, nine mills. DRILL EXERCISES. 73 145. Exercises on Table No. 4. 103. Read as dollars, cents, and mills, the numbers expressed in A. § lOJf. Read decimally the numbers expressed in A. 105. Write in figures the numbers expressed in B. 106. Disregarding the mills, change the numbers expressed in A to cents. 107. Change the numbers expressed in B to mills. 108. Add the numbers from 1 to 8 * in A to each number expressed in B. 109. Add the numbers expressed in A and B, (1st) from 1 to 4*; (2d) from 2 to 5 ; (3d) from 3 to 6, etc. 110. $900-A = ? 114. Ax9 = ? ii5.fA-=-7-? iii. A-B = ? 7i5.tA-T-10=? ii9.l A-f-$0.25=? IW. Ax6 = ? ' iie.tB^ll = ? 2;?0.l A^$0.16 = t 113. Bx7 = ? ii7.tB^12 = ? ISl. B-^S 0.005 = ? 1^^. If a person saves a sum equal to A in one month, how much yri] he save in 13 months ? 1S3. How many pounds of sugar, at 8 cents a pound, can be bought for each sum of money expressed in A ? t 146. Questions for Revle'w. What are the units of United States money? Give the table. How are dollars, cents, and mills expressed by figures ? What is con- sidered the principal unit ? Give the sign for dollars. How do you change doUais to cents ? dollars to mills ? cents to mills ? mills to dol- lars? cents to dollars? How do yon add numbers in United States money? How do you subtract ? When you multiply, where do you put the decimal point in the product? Divide $185 by 7, continue the division to mills, and explain. What is necessary in order to divide one sum of money by another? Divide $900 by 36 cents. What are coins ? Why is paper money sometimes used in place of coins ? Name the gold coins ; the silver coins. What is a creditor ? a debtor ? an account ? a bill ? How is a bill receipted? » IneluBiTe. t See page 66, note, J Reject mills. § See page 57, for Explanation of the Use of the Drill Tables. 74 UNITED STATES MONEY. 147. nnsceUaneoua Ezamples. 27. A girl bought a pair of boots for $ 2.37, another pair for $1.65, slippers for $1.25, and shoes for 82/. What was the whole cost ? 28. I bought a horse for $ 95.00, a wagon for $ 63.00, and a harness for $15.00; kept them a week, paying $2.50 for board of the horse, then sold them for $ 175.00. Did I gaiu or lose, and how much ? 29. What should I pay for 2 dozen pigeons at 85/ per dozen, 2 dozen at $1.10 per dozen, and 1 dozen for 90/. 30. There were sold in one week 8874 sheep at $ 4.13 per head. What did they bring ? 31. There were sold 4778 beeves, averaging 874 pounds apiece, at 7/ per pound. What was received for them? 32. What did I gain by buying 2 pieces of cambric, each containing 62 yards, for $ 39.68, and selling them for 40 cents per yard ? 33. A man paid 1 16.25 for 13 days' work. What was that a day ? 34. Among how many boys must $ 12 be distributed, that each may receive 75 cents ? 35. I sold 35 barrels Pippins at $ 1.75 per barrel, 17 barrels Pome Eoyals at $ 1.80 per barrel, 13 barrels Golden Sweets at $ 1.25 per barrel, and 25 of Russets at 1 2.25 per barrel. Paid 17 cents a barrel for picking, and $ 12.00 for freight. What remained after my expenses were paid ? 36. Paid $ 3.00 for 1 dozen apple-trees, $ 3.36 for 1 dozen peach-trees, $3.30 for haK a dozen pear-trees. What did I pay for the whole, and how much apiece for each kind ? 37. A carpenter paid for stock and work for a barn, $ 450.75 ; for mason's work, $ 38.25 ; for digging and stoning cellar, $ 47.18 ; for painting, $ 40.00 ; to the plumber, $ 8.12. He then sold the barn, and lost, in so doing, t 14.30 ; how much did he sell it for ? FACTORS. 75 SEOTIOIf VIII. FACTORS. 148. What numbers multiplied together will produce 10 ? Answer, 2 and 5 ; also 1 and 10 ; thus, 2x5 = 10 and 1x10 = 10. A number that may be used as multiplicand or as multi- plier to make another number is a factor of that number. Name two factors of 15 ; of 16 ; of 18 ; of 24 ; of 36 ; of 45. Note I. The word factor ■will be used in this Arithmetic to denote only such factors as are not fractional. KoTE II. If a numher be divided by any of its factors there will be no remainder. Hence a factor of a number is also called a divisor or a medsvire of that number. 149. Name .some factors of 12 besides the number itself and 1. Has the number 13 factors besides itself and 1 ? Has the number 14 ? 15? 17? 18? 19? A number that has other factors besides itself and one is a composite number. Which of the numbers 12, 13, 14, 15, 17, 18, 19 are com- posite numbers ? 150. A number that has no other factors besides itself and one is a prime number. Which of the numbers 12, 13, 14, 15, 17, 18, 19 are prime numbers ? Name the composite numbers from 1 to 40. Name the prime numbers from 1 to 40. IfOTE. In speaking of the factors of a number, we do not usually include the number itself and one. Thus, we frequently say that a piime number has no factors. 76 FACTORS. 151. Name the factors of 12 that are prime numbers. Name those that are not prime numbers. A factor that is a prime number is a prime factor. 152. Oral Exercises. e. What are the prime factors of 6 ? 8 ? 14 ? 24 ? 27 ? b. What are the prime factors of 22 ? 36 ? 28 ? 20 ? 35 ? c. What are the prime factors of 16 ? 21 ? 15 ? 33 ? 26 ? 153. In seeking for the factors of a number we may use certain tests, the more convenient of which are the follow- ing: 1. A number whose miits' figure is 0, 2, 4, 6, or 8, is divisible by 2. Note. A number that is divisible by 2 is an even nwrriber; a number that is not divisible by 2 is an odd number. 2. A number is divisible by 3 if the sum of its digits* is divisible by 3. Thus, 285 is divisible by 3, for 2 + 8 + 5 = 15 is divisible by 3. 3. A number is divisible by 4 if its tens and units together are divisible by 4. Thus, 6724 is divisible by 4, while 6731 is not. 4. A number is divisible by 5 if the units' figure is either or 5. 5. A number is divisible by 6 if it is an even number and divisible by 3. 6. A number is divisible by 8 if its hundreds, tens, and units are divisible by 8. Thus, 6728 is divisible by 8, while 6724 is not. 7. A number is divisible by 9 if the sum of its digits is divisible by 9. 8. A number is divisible by 11 if the sums of its alternate digits are equal, or if their difference is divisible by 11. Thus, 1782 and 1859 are divisible by 11, while 4987 is not. 9. A number is divisible by a composite nvmiber, if it is divisible by all the factors of the composite number. Thus 3555 is divisible by 15, for it is divisible by 3 and by 5. Note. For the reasons of these tests, see Appendix, page 303. * A digit here means the number denoted by a figure without regard to its place. PRIME FACTORS. 77 154. Oral Iizercise& Using the tests described above, a. Name the numbers expressed in B, page 58, that contain the factor 2 ; 4 ; 6. b. Name the numbers in C, page 68, that contain the factor 3; 6; 9. c. Name the numbers in D, page 58, that contain the factor 8; 9; 10; 100. To find the Prime Factors of a Number. 155. Illustkative Example I. What are the prime factors of 2205 ? Explanation. — Applying the tests (Art. 153) to the given number, we find that 2 is not, but that 3 is, a factor of 2205 ; and, by dividing, see that 2205 = 3 X 735. Seeking, in the same way, a prime factor of 735, we find that 735 = 3 x 245. Continuing this process, we find that 245 = 5 x 49, and that 49 = 7 X 7. Therefore, 2205 = 3 x 3 x 5 x 7 x 7, and Ans. 3, 3, 5, 7, 7. the prime factors are 3, 3, 5, 7, and 7. 156. Illustrative Example II. What are the prime factors of 409 ? WRiTTBi. WOEK. ^a;p?ana«M«. - Applying the oQ^ AaanT *«'*' (^'^- ^^'^' ^^ ^^ *^'''* ^°^ Z6) 4Uy i^LI jg ^^^ divisible by 2, 3, or 5. We WRITTEN WORK 3 2205 3 735 5 245 7 49 7 19) 409 (21 38 29 19 10 . then try to divide by the other 179 prime numbers in order until we 161 reach 23, when we see that the ~~~ quotient is less than the divisor. There can then be no prime factor in 409 greater than 23, for if there were, there would be another factor (the quotient) less than 23, which we should have found before reach- ing 23. The number 409 is therefore prime. 157. As we have found in Art. 155 that 2205 equals the product of all its prime factors, so we shall always iind that A composite number equals the prod/wet of all its prime factors. 78 FACTORS. 158. When a composite number is expressed as a prod- uct of prime factors, it is said to be separated into its prime factors. 159. From the above examples may be derived the fol- lowing Rule. To separate a number into its prime factors : 1. Divide the given number ly one of its prime factors. 2. Divide the quotient thus oitained iy one of its prime factors; and so continue dividing until a quotient is ob- tained that is a prime number. 3. This q%iotient and the several divisors are the prime factors sought. Proof. Multiply together the prime factors thus found. The product ought to equal the given number. Note. If no prime factor is readily found by whicli to divide, we try to divide by the several prime numbers in order. If no prime factor is found before the quotient becomes less than the trial divisor, the given number ia prime. See Illustrative Example II. 160. Examples for the Slate. Separate into prime factors the following numbers : (1.) 180. (4.) 208. (7.) 329. (10.) 644. (2.) 192. (5.) 260. (8.) 338. (11.) 684. (3.) 176. (6.) 169. (9.) 357. (12.) 2500. Select the prime numbers and find the prime factors of the composite numbers among the following : (13.) 341. (18.) 450. (23.) 704. (28.) 945. (14.) 344. (19.) 590. (24.) 711. (29.) 972. (15.) 362. (20.) 560. (25.) 762. (30.) 2688. (16.) 367. (21.) 596. (26.) 808. (31.) 1164. (17.) 408. (22.) 689. (27.) 836. (32.) 3248. SYMBOLS OF OPERATION. 79 SYMBOLS OF OPERATION. 161. The signs +, -, y- , and -^, since they indicate that certain operations (adding, subtracting, multiplying, and dividing) are to be performed, are called symbols ot operation, 162. In expressing a series of operations by aid of these signs, it is often necessary to indicate that an operation is to be performed on two or more numbers combined. This is done by writing the numbers to be operated upon, with the proper signs, and enclosing the whole expression in marks of parenthesis or brackets. The expression so en- closed is then treated as if it denoted a single number. Thus, (7 + 2) X 5 means that the sum of 7 and 2 is to be multiplied by 5 ; but 7 + 2x5 means that 7 i^ to be increased by 5 times 2. (7 - 2) X 3 means that the difference between 7 and 2 is to be multiplied by 3 ; but 7-2x3 means 7 diminished by 3 times 2. 7 + 2 (7 + 2) + 5, or — ^>* means that the sum of 7 and 2 is to be divided by 5. [(2 + 3) X 5 - 11] X 2 means that the sum of 2 and 3 is to be multiplied by 5, the product diminished by 11, and the re- mainder multiplied by 2. 163. In performing a series of operations indicated by signs. First, operate on the numbers that are written within parentheses as indicated by the signs. Next, multiply and divide as indicated by the signs x and -J-. Finally, add and subtract as indicated by the signs + and —. * The horizontal line here drawn between 7 + 2 and 5 is equivalent to marks of parenthesis. J- 14- 3x 4- 2 2x3^? k. 8 + 3 8 2 ^- -3 2 = ? 1. [(4 + 6): k4- 5 X 3] X 3 = = ? 80 FACTORS. 164. Oral Exercises. \The Key contains answers to the following examples.] 3. (6 + 8)x5=? A. 3x8-4x3=? 6.6 + 8x5=? i. 3x8-(4x3)=? c. (8-3) X 2= ? d. 8-3x2= ? e. 8 + 12-4= ? /. (8 + 12)+4=? g-. (2 + l)x(r-2)=? CANCELLATION. 165. Illustrative Example I. If 4 be multiplied by 3 and the product divided by 3, what is the result ? From this example we see that If a given number be multiplied by a number, and the product be divided by the same number, the result will be the given number. In such cases, both the multiplication and the division may be omitted. Note. This omission is indicated in the written work ahove by draw- ing a mark through the 3 thus, ?. 166. Illustrative Example II. What is the result of dividing the product of 4 and 6 by 3 ? Explanation. — As 6 = 2 x 3 the dividend in this WRITTEN WORK „ " example is 4 x 2 x 3, and the divisor is 3, so that we . r, may strike out the factor 3 in both dividend and — = 8 divisor, and multiply by 2 only, thus shortening the P work. The process of shortening work by striking out equal factors in dividend and divisor is cancellation. WBITTEN WORK. 4x|5 = 4 Ans. 4. GANCELLA TION. 8 1 167. Examples for the Slate. Ml operations upon numbers should first be indicated, as far as possible, by signs, that the work to be done may be shortened, if possible, by cancellation. 33. Divide 81 x 42 by 99 x 7. 34. Multiply 75 x 10 by 3 x 6, and divide that product by 15 X 25 X 12. 35. Divide 7 x 8 x 48 by 63 x 4 x 5 x 17, and multiply the quotient by 51. 36. If 5 sets of chairs, 6 in a set, cost $ 75, what did 1 chair cost ? 37. If it requires 13 bushels of wheat to make 3 barrels of flour, how many bushels will be required to make 78 barrels of flour ? 38. If a tree 54 feet high casts a shadow of 90 feet, what length of shadow will be cast by a flag-staff 105 feet high? 39. A grocer exchanged 561 pounds of sugar, at 12 cents per pound, for eggs at 22 cents per dozen. How many dozen were received ? 40. If 12 pieces of cloth, each piece containing 62 yards, cost $372, what do 24 yards cost ? 41. If the work of 7 men is equal to the work of 9 boys, how many men's work will equal the work of 90 boys ? 42. If 15 men consume a barrel of flour in 6 weeks, how long would it last 9 men ? 43. If 12 men can build a wall in 42 days, how many days will be required for 21 men to build it ? 44. If $15 purchase 12 yards of cloth, how many yards will 148 purchase? 45. A ship has provision for 15 men 12 months. How long wiU it last 45 men ? 46. How many overcoats, each containing 4 yards, can be made from 10 bales of cloth, 12 pieces each, 42 yards in each piece ? 82 FACTORS. COMMON FACTORS. 168. Illustkative Example I. What numbers are fac- tors of both 18 and 24 ? WRITTEN WORK. Explanation. — Separating 18 and 24 into their prime factors, we find 2 and 3, and conse- 18 = i X d X 3 quently 6 (which is the product of 2 and 3), to 24 = 2x2x2x3 ijg f^gtojg of ijoth 18 and 24. ^ Its 2 3 and 6. Name any common factor of 12 and 15 ; of 12 and 18; of 30 and 40. 169. Numbers that have no common factors are said to be prime to each other. Thus, 14 and 16 are prime to each other, though they are not prime numbers. 170. The greatest factor which is common to two or more numbers is their greatest common factor. What is the greatest factor which is common to 18 and 24 ? to 40 and 60 ? to 46 and 64 ? 171. We have seen that 6, the greatest common factor of 18 and 24, is the product of 2 and 3, the only prime factors common to 18 and 24. The greatest common factor of any two or more numbers is the product of all the prime fac- tors which are common to those numbers. Note. The letters g. c. f. are used for greatest common factor. To find the Greatest Common Factor. 172. Illustrative Example II. Find the greatest com- mon factor of 12, 30, and 48. WRITTEN WORK. Explanation. — The prime factors of 12 are 2, 12 = 2 X 2 X 3 2, and 3. The product of such of these as are s c f =2x3 = 6 common to 30 and 48 must be the g. c. f. re- quired. We find that 2 is a factor of both 30 and 48 ; therefore 2 is a factor of the g. c. i. We find that but one 2 is a factor of 30; therefore only GREATEST COMMON FACTOR. 83 one 2 is used as a factor of the g. c. f. We find that 3 is a factor of both 30 and 48 ; therefore 3 is a factor of the g. o. f. Thus the g. o. f. sought is 2 X 3, equal to 6. Hence the following Rule. 173. To find the greatest common factor of two or more numbers : Separate one, of the nuTnhers into its prime factors, and fiTid the. product of such of them as are common to the other numhers. 174. Examples for the Slate. Find the greatest common factor 47. Of 48, 66, and 60. 48. Of 24, 42, and 54. 49. Of 108, 45, 18, and 63. 50. Of 18, 36, 12, 48, and 42. Note. In Example 50, 18 is a factor of 36, and 12 of 48. The g. c. f. of 18 and 12 must be the g. c. f. of 18, 12, and their multiples 36 and 48 ; hence we need only find the g. c. f. of 18, 12, and 42. Find the greatest common fa. tor 51. Of 42, 28, and 84. 63. Of 32, 18, 108, and 25. 52. Of 26, 52, and 65. 54. Of 114, 102, 78, and 66. 55. What is the width of the widest carpeting that will ex- actly fit either of two halls, 45 feet and 33 feet wide, respec- tively ? 56. A has a piece of ground 90 feet long and 42 feet wide. What is the length of the longest rails that will exactly suit both its length and its width ? 57. What is the length of the longest stepping-stones that will exactly fit across each of three streets, 72, 51, and 87 feet wide, respectively ? 58. What is the length of the longest curb-stones that will exactly fit each of four strips of sidewalk, the first being 273 feet long, the second 294, the third 567, the fourth 651 ? 84 FACTORS. 175. When numbers cannot readily be separated into their fiactors, the following method for finding, the greatest common factor may be adopted. Illustrative Example. Find the greatest common fac- tor of 52 and 91. WRITTEN WORK. Dwtde the greater number hy the less, ^;9^ Q1 n '^'"^ then divide the less number hy the Ko remainder, if there he any. Continue — dividing the last divisor by the last re- dy) oz (1 mainder until nothing remains. The last 39 . . divisor will be the g. c. f. sought. 13) 39 (o Note. As tlie explanation of this method is some- 39 what difficult for younger pupils, it is not given here, but will be found in the Appendix, page 804. To find the g. c. f. of more than two numbers, find the g. c.f. of any two of them and then of that common factor and a third number, and so on till all the numbers are taken. 176. Find the greatest common factor 59. Of 323 and 663. 61. Of 6581 and 1127. 60. Of 147 and 966. 62. Of 187, 442, and 969. For other examples in factoring, see page 123. MULTIPLES. 177. Name some numbers which are made by using 3 as a factor. Ans. 3, 6, 9, 12, etc. Any number made by using another number as a factor is a multiple of the number thus used. 178. Name the multiples of 4 and of ,6 to 36. ^^^ I Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36.' i Multiples of 6 are 6, 12, 18, 24, 30, 36. Which of these numbers are multiples of both 4 and 6 ? Numbers which are multiples of two or more numbers are common multiples of these numbers. LEAST COMMON MULTIPLE. 85 Thus 12, 24, and 36 are common multiples of 4 and 6. Name a common multiple of 3 and 5 ; name two more. 179. Oral Exercises. Name any six multiples of 5. Name three multiples of 12. Name all the multiples of 11 up to 140. Name any common multiple of 10 and 6. Of 3, 6, and 6. Least Common Multiple. 180. Name the least number which is a multiple of both 4 and 6. Ans. 12. The least number which is a multiple of two or more numbers, is the least common multiple of those numbers. Name the least common multiple of 2 and 6 ; of 6 and 9. Note. The letters 1. c. m. are used for least common multiple. 181. As any number contains aU its prime factors, a multiple of any number must contain all the prime factors of that number. A common multiple of two or more numbers must con- tain all the prime factors of those numbers, and The least common multiple of two or more numbers is the least number which contains all the prime factors of those numbers. 182. Illustrative Example I. What is the least com- mon multiple of 6, 9, and 15 ? WRITTEN WORK. Explanation. — The least multiple of g ^ 2 X 3 6 is 6, which may be expressed in the 9 = 3x3 iorrm 2x3. -,K_o K The least multiple of 9 is 9, which may ~ be expressed in the form 3x3. But in 1. c. m. = 2 X 3 X 3 X 5 = 90 ^ ^^'"5 ^^'^^ already one of the factors (3) of 9 ; hence if we put with the prime factors of 6 the remaining factor (3) of 9, we shall have 2x3x3, which are all the factors necessary to produce the 1. c. m. of 6 and 9. 86 FACTORS. The least multiple of 15 is 15, which may be expressed in the form 3x5. In the 1. c. m. of 6 and 9 we have one of the prime factors (3) of 15; hence if we put with the prime factors of 6 and 9 the remain- ing factor (5) of 15, we shall have 2x3x3x5, which are all the prime factors necessary to produce the 1. c. m. of 6, 9, and 15. The product of these factors is 90, which is the 1. c. m. sought. Note. In finding the least common multiple, the factors of the given numbers seldom need to be expressed, and the written work may be gi-eatly reduced. Thus, in this example the written work may be simply !. c: m. = 2 X 3 X 3 X 5 = 90. 183. From the explanation above may be derived Rule I. To find the least common multiple of two or more num- bers : Take the priime factors of OTie of the numbers ; with these take such prime factors of each of the other nv/mhers in succession as are not contained in any •preceding num- ler, and find the product of all these prime factors. 184. Oral Exercises. What is the least common multiple a. Of4, 5, andS? c. Of 6, 14,- and 21 ? v b. Of 6, 8, and 12 ? d. Of 3, 4, and 6 ? When several numbers are prime to each other, what must their least common multiple equal ? 185. Examples for the Slate. Find the least common multiple 63. Of 8, 18, 20, and 21. 64. Of 3, 5, 12, 36, and 45. Note. When one of the given numbers is contained in another, the smaller may be disregarded in t}ie operation ; thus, in the preceding ex- ample, 3, 5, and 12 may be rejected. Why? Find the least common multiple 65. Of 18, 36, 60 and 72. 68. Of 18, 32, 48, and 52. 66. Of 12, 42, 56, and 70. 69. Of 16, 28, 35, and 63. 67. Of 13, 28, 39, and 49. 70. Of the nine digits. LEAST COMMON MULTIPLE. 87 186. The atove is a good method for finding the least common multiple when the numbers are easily separated into their prime factors. For larger numbers observe the following method: Illustrative Example II. Find the 1. c. m. of 18, 56, 38, and 30. WRITTEN WORK. Explanation. — Here, 2) 18, 56, 38, 30 ^y repeated divisions, we . take out all the factors 6) 9, ^8, 19, 15 ^.jjg^j ^j,g gojjjjjjon to t^.Q 3, 28, 19, 5 or more of the given num- 1. c. m. = 2 . 3 X 3 X 28 X 19 X 5 = 47880 JT' /?' ^",f"""\ t these factors (2 and 3) and those that are not common must be the 1. c. m. sought. Rule II. 187. To find the least common multiple of two or more numbers : 1. Write the given numbers in a line as dividends. Make any prime number which is a factor of two or more of tJie given numbers a divisor of those numbers. 2. Write the quotients and undivided numbers beneath as new dividends, and so continue dividing till the last quo- tients and undivided numbers are prime to each other. 3. The product of all the divisors, last quotients, and undivided numbers is the least common multiple required-. 188. Examples for the Slate. Find the least common multiple 71. Of 338, 364, and 448. 75. Of 165, 9500, and 855. 72. Of 184, 390, and 552. 76. Of 1146, 484, and 24. 73. Of 308, 616, and 77. 77. Of 880, 9680, and 176. 74. Of 84, 336, and 472. 78. Of 187, 539, and 8470. For other examples in multiples, see page 123. 88 COMMON FRACTIONS. SECTION" IX. COMMON FRA.CTIONS. 189. If a unit, as 1 inch, is divided into two equal parts, I 1 1 one of the parts is called one haE If the unit is divided into three equal parts, one of the I I 1 1 parts is called one third ; two of the parts are called two thirds. One of the equal parts of a unit is a fraction, or fractional unit. A collection of fractional units is a fractional number. Note I. For the sake of brevity, fractional units and fractional numbers are both called fractions. Note II. A number whose units are entire things is an integral num- ber, or an in Name a fraetional unit ; a fractional number ; an integer. 190. The unit of which the fraction is a part is the unit of the fraction. 191. The number of equal parts into which the unit of the fraction is divided is the denominator of the fraction. Thus, in the fraction two thirds the denominator is three. 192. The number of equal parts taken is the numera- tor of the fraction. Thus, in the fraction two thirds the numerator is two. 193. The numerator and denominator are called the terms of the fraction. Note. Decimal fractions have been treated of in previous articles. All fractions except decimal fractions are called common fractions. EXAMPLES. 89 Writing Common Fractions. 194. The terms of a fraction are written, the numera- fiitmerator, 8 . , tor above and the denominator below a written as in the margin. 195. Exercises. Write in figures the following : a. One half of a mile. d. Twenty twenty-fifths. b. One third of a day. e. Twelve thirds. c. Seven tenths of a dollar. /. Seven sevenths. g. Write any fraction you please, having for a denominator five ; seven ; ten ; seventeen ; one hundred. h. Write any fraction you please, having for a numerator six ; eight ; sixty ; one hundred. i. Where is the denominator of a fraction written ? Where is the numerator written ? J. Which is the greater part of a thing, J or J ? J^ or 3*5 ? 196. The form of writing fractions as shown above is the same as the fractional form used to indicate division. (Art. 94.) Thus the expression f may mean two thirds of one or one third of two. illusteation. The fact that * of 1 I 1 1 1 equals J of 2 may be illus- ' ' ' ' 1 1 1 j trated as in the margin. g ^^ ^ ^^^^ j ^^ 2_ 197. Exercises. a. What is meant by the expression | ? Ans. It means 5 of the 9 equal parts into which a unit is divided, or it means 1 ninth of 6 units. 6. What is meant by the expression i? -^^ ^? tf? i^? c. Illustrate the fact that f of 1 equals i of 3 ; that | of 1 equals J of 2. 90 COMMON FRACTIONS. REDUCTION. To change a Fraction to smaller or larger terms. 198. Illustrative Example I. Change ^f to equiva- lent fractions of smaller terms. WEITTEN WORK. Explanation. — By dividing both terms of \^ by 1 2 _ « _ a 2, we make the terms haK as large, and have the fraction ^. Now dividing both terms of the frac- tion f by 3, we make its terms one third as large, and have the fraction f. If we had divided both terms of ^f by 6, we should have made the terms one sixth as large, and obtained at once the fraction J. The illustration shows that ILLUSTRATION. tf 1 1 1 1 1 1 1 1 1 1 1 1 1 . 1 . 1 the same part of the vmit is expressed by |f, f , and |. In obtaining f and | from ^|, the 1 I I I I I I I — I — 1 — I number of parts taken has 2 I I I I been diminished as the size of the parts has been increased. If both terms of a fraction are divided hy the same num- her, the value of the fraction mil not be changed. 199. Illusteative Example II. Change | to equivalent fractions of larger terms. WRITTEN WORK. Explwnation.— By multiplj - % = i; § = +f ^S both terms of f by 2, we make the terms twice as large, ILLUSTRATION. and have the fraction |. By multiplying both terms of | by 6, we make the terms six H k I I I I 1 1 1 times as large, and have the if I I I I I I I I I I I I I ,, I ,, I fraction f|. Here the num- ber of parts in each case is increased as the size of the parts is diminished. If both terms of a fraction are multiplied by the same number, the value of the fraction will not be changed. 200. When the terms of a fraction have no common fac- tor, the fraction is said to be expressed in its smallest terms. EXAMPLES. 91 201. Oral Exercises. Perform mentally the examples given below, naming results merely ; thus, " ^ ; ^ ; \; |," and so on. a. Change to their smallest terms :|;f;f;t;f;-ft^;-^; *; I; -h; A; A; I; f; A; A; -A; A; A- fe. Change to their smallest terms: -j^; /^; -j^; ^j; ^„ ; :i^? j A ! A ; A ; A j A ; . A ; A ; A ; A ; A > A ; A ; A j A- c. Change to their smallest terms: |§; ^; ^§ ; ^|; ^± ; if; H; M; M; M; M; if; M; A^; M; H; I'A'iy d. Change f to equivalent fractions, having 12, 16, 28, 44, 100, and 120 for denominators. e. Change ^ to equivalent fractions, having 27, 54, 99, and 900 for denominators. /. Change f , f, f, ■j'^, \^, ^, each to an equivalent fraction having 120 for a denominator. g. How many thirtieths in ^ ? in ^ ? in f ? in f ? in f ? inf? in J? h. How many 24ths in f ? in f ? in ^ ? in J ? in | ? in | ? To change a Fraction to its smallest terms. 202. From previous illustrations we may derive the fol- lowing Rule. To change a fraction to an equivalent fraction of the smallest terms : Strike out all the factors which are common to the numerator and denominator; or divide both tervis hy their greatest common factor. 203. Examples for the Slate. Cliange to equivalent fractions of smallest terms : (1.) ^^. (4.) m- (7.) A*A- (10.) mi- (2.) m- (5.) m- (8.) AA- (11-) im- (3.) ^^. (6.) /^. (9.) ^^. (12.) f fff. For other examples, see page 123. 92 COMMON FRACTIONS. To chauige Improper Fractions to Integers or to Mized Numbers. 204. A fractional number, the numerator of which equals or exceeds the denominator, is called an Improper fraction. 205. Illustrative Example III. Change ff and -^J as far as possible to integers. WEiTTEN WORK. Explanation. — (1.) Since 12 twelfths make a (1.) 12) 60 miit, in 60 twelfths there are as many units as — there are 12's in 60, which is 5. Ans. 5. -^'"' ^ (2.) In ^ there are as many imits as there are ■\o\ AT -^^'^ "^ '^'^' which is 3 and \^. Ans. 3f^. (2.) i.^) 47 ~;;, , 206. The number 34-4- consists of an integer and a fraction. A number con- sisting of an integer and a fraction is a mixed number. 207. Oral Exercises. a. Change to integral numbers: f ; J^-; -^i J^; i^; |; ¥; -¥ ; ¥; ¥; ¥; ¥; -¥; !f ; ¥; ft; ¥-; If 6. Change to mixed numbers : ^ ; | ; ^- ; i^- ; J^- ; ^ ; ¥; ¥-; ¥; ¥; ¥; ¥; H; M; ¥; ¥; ¥• c. Change to integers or to mixed numbers : f ; %'- ; | : ¥;¥;¥;¥; fi; W; H^; ¥; ¥• 208. From previous illustrations we may derive the fol- lowing Rule. To change an improper fraction to an integer or a mixed number : Divide the numerator ly the denomirmtor. 209. Examples for the Slate. Change to integers or to mixed numbers : (13.) f f . (16.) W- (17.) !¥*• (19.) ¥/- days. (14.) i^. (16.) W. (18.) y^^^^. (20.) ^ years. EXAMPLES. 93 To change an Integer or a Mixed Number to an Improper Fraction. 210. ILLUSTEATIVE EXAMPLE IV. Change 23^ to fourths. WRITTEN woEK. Explanation. — Since in 1 there axe 4 fourtlia, in 23j = -^ Ans. 23 there are 23 times 4 fourths, or 02 fourths, _4 which, with 1 fourth added, are 93 fourths. 93 Am. af. 211. Oral Exercises. a. Change to improper fractions : 2\; 3f ; 2| ; 6j ; 2| ; 3|; 6J; 6§; 5i; 7|; 7^; 8| ; 8|; 9J; 9f; 10^. 6. Change to improper fractions : 2| ; 2f ; 3^\ ; ^■, ^\ 4f; 5|; 9f; 6|; 7^; 8|; 9f ; 4f; 4f ; 8^ ; 7^. c. Change 5 to ninths ; 11 to fifths ; 14 to thirds ; 8 to tweKths ; 15 to fourths ; 1 to sevenths. d. Among how many persons must 7 melons he divided that each may receive ^ of a melon ? J ? -rV ? e. How many persons will 5J cords of wood supply if each person receives J of a cord ? J of a cord ? \ oi a, cord ? 212. From previous illustrations may be derived the following Rule. To change an integer or a mixed number to an improper fraction : Multiply the integer 'by the denominator of the fraction, and to the product add the numerator; the result will he the numerator of the required fraction. 213. Examples for the Slate. Change the following to improper fractions : (21.) 69f . (24.) 76f i-. (27.) Change 48 to ninths. (22.) 272i. (25.) 10^. (28.) Change 667 to tenths. (23.) 109^^. (26.) 66f. (29.) Change 93 to forty-thirds. For other examples in reduction of fractions, see page 123. 94 COMMON FRACTIONS. ADDITION OF FRACTIONS. To add Fractions having a Common Denominator. 214. Illustkative Example I. Add f of an apple, 1 of an apple, and \ of an apple. Ans. ^ of an apple. These fractions are like parts (eighths) of the same or similar units (apples). Such fractions are like fractions. 215. Like fractions have the same denominator, which, because it belongs to several fractions, is called a common denominator. 216. Oral Exercises. a. Add /y, -^, and ■^. e. Add ^, ^, and f ^. b. Add tV^, tI w and T^^. /. Add H, f fe and ^. c. Add I, f, I, and f . g-. Add ^g^, ^^, and 5^/^^. d. Add /^, .^^, iJ, and ^. h. Add ^, ^, and ^J^. How do you add fractions which have a common denominator ? To add Fractions not having a Common Denominator. 217. Illustkative Example. Add |, |, and ■^. WEITTEN WORK. Explanation. — To be added, these 2 X 3 X 3 X 5 = 90 l.c. dmm. fractions must be changed to Kke frac- tions, or to fractions having a common f ~ t X 1 5 = f ^ denominator. (Art. 215.) The new t = txio=f^ denominator must be some multiple of ■^■g = -/^x 6 ~ tf ^^^ given denominators. A convenient Ant JLAJ = Ifi-Z multiple is their least common multiple, ■ ^^ **• which is 90. (Art. 182.) To change | to 90ths, the denominator 6 must be multiplied by 3 X 5, or 15 ; hence the numerator 5 must be multiplied by 15. (Art. 199.) Thus, f is found to equal ^. In a similar way f will be found to equal -fg, and ^ to equal ^. Adding these fractions, we have J^, or 1-|^, for the sum. ADDITION. 95 218. Oral Exercises. a. Add \, \, and ^. c. Add \, |, and f . Ans. T^j = f . d. Add f, f , and i^. b. Add f, f, and |. e. Add f, i, and |. ^m Jf = 23>ff. /. Add ^^, ^, and f . Note. When tlie denominators are prime to each other, the new denomi- nator ivill be the product of all the denominators, and the new numerators will be found hy multiplyi7i,g each numerator by the product of all the de- nominators except its own. g. Add \ and \; J and ^ ; \ and ^ ; i and J ; ^ and \ ; i and f ; J and ,1,^ ; ^ and \. h. Add f and f ; J and J ; | and f ; f and | ; \ and |. i. Add \, i, and \; %, \, and f ; J, f , and ^ ; t^, |, and \. j. If you should spend J of your time in school, ^ in practising music, and J in sewing and studying, what time would you spend in all ? k. Owning f of a paper-mill, I bought the shares of two' other persons who owned -^ and f respectively. What part of the miU did I then own ? 219. From the above examples may be derived the fol- lowing Rule. To change fractions to equivalent fractions having the least common denominator : 1. For the common denominator, find the least common multiple of the given denominators. 2. For the new numerators, multiple/ the numerator of each fraction hy the number hy which you multiply its de- nominator to produce the common denominator. ^ Note. If the number to multiply the numerator hy is not readily seen, it may he found hy dividing the common denominator hy the denominator of the given fraction. 96 COMMON FRACTIONS. 220. From what we have now learned of the addition of fractions, we may derive the following Rule. To add fractions : 1. If they have a common denominator, add their numerators. 2. If they have not a common denominator, change them to equivalent fractions that have a common denominator, and then add their numerators. 221. Examples for the Slate. (30.) f + A+i+f=? (35.) A + A + *f = ? (31.) t + lf* + H + A=? (36.) M + t'^ + A=? (32.) ^ + i^ + i + i = ? (37.) f+f + M = ? (33.) /j + f + |=? (38.) ^^ + ^ + f^=? (34.) A + A + T^ = ? (39.) T^^ + ^ + A + H = ? Add the integers and fractions of the following, and similar ex- amples, separately : 40. In my furnace there were burned 2f- tons of coal in December, 2f tons in January, and 3J in February. How many tons were burned in all ? (41.) 72^ + 16| + 18| + 23f + 37/y = ? 42. A horse travelled 4t3^ miles in one day, 52^ the next, 36^ the third, and 40|^ the fourth. How far did he travel in all ? 43. A merchant had three barrels of sugar, the first contain- ing 247^ pounds ; the second, 22%^ pounds ; and the third, 260J pounds. What was the weight of the whole ? For other examples in addition of fractions, see page 123. * What operation should first be performed on this fraction ? SUBTRACTION. 97 SUBTRACTION. 222.' Oral Exercises. * a. f less f are how many ninths ? Ans. f . 6- H-i?T = what? d. e-M = what? c. f§-/^=--what? e. lf-A-A- = what? /. IFind the difference between ^^ of a day and %\ of a day. g. What must be added to /^ to make ^ ? ^^ ? When the minuend and subtrahend are like fractions, how do you subtract ? 223. Illustkative Example. If | of a yard of velvet is cut from a piece containing | of a yard, what part of a yard will be left ? WBITTEN WORK. Explcmation. — That the subtraction may be 3 „ _ o^a _ J, performed, these fractions must be changed to * ^ ' equivalent fl&ctions having a common denomi- nator. The least common denominator is 12. f = i^ and f = ^. h. i-i? i-^? i-i? i-^? i-i? 4-T^y? i. §-f? f-f? f-t? f-i? A-l? ^-§? j. 2-§? 8-1? 11 -f? 9-3i? 7-2f? 8-3|? k. How many yards will be left if from a piece containing 6/^ yards there be taken H yards ? 1. What is the difference in the height of two boys, one being 3^ feet, the other 2f feet high ? m. A pole is standing so that f of it is in the water, ^ in the mild, and the rest in the air. What part is in the air ? n. How much will be left of a piece of cloth containing 7 yards, after cutting from it 2 vests and a coat, allowing f of a yard for a vest and 4^ yards for a coat ? 6. From a bin containing 23^ bushels of wheat there were taken out 3 J bushels at one time and 4^ bushels at another. How much remained ? 98 COMMON FRACTIONS. 224. rrom the previous illustrations we may derive the following Rule. To subtract one fraction from another : 1. If they have a common denominator, find the differ- ence of their nwmerators. 2. If they have not a common denominator, change them to equivalent fractions which have a comimon denominator, and then find the difference of their numerators. 225. Examples for the Slate. (44.) A-A = ? (50.) 12i-| = ? (45.) T^^-^ = ? (61.) 17i-,12^ = ? (46.) #^-/t = ? (52.) 26t-lf = ? (47.) 36-1=? (53.) 10f-5i| = ? (48.) 19 -2^ = ? * (64.) 17i-2| = ? (49.) 76-15i=? (66.) 18^-16115 = ? For other examples in subtraction of fractions, see page 123. 226. Examples in Addition and Subtraction. (56.) f-f + | = ? (60.) 20-5i + A=? (67.) § + A-A = ? (61-) 8^-2§ + 7f = ? (68.) f-i + A-Tf^=? (62.) 7-(^^-^) = ? (59.) f-4-A-TtF = ? (63.) 6-(t + /5) = ? 64. Two men start at the same place and travel in opposite directions, one at the rate of 3^^^ miles per hour, the other at the rate of 4JJ miles per hour. How far apart were they at the end of an hour ? 66. Two boats are 5280 feet apart and rowing towards eacli other, one at the rate of 320/^ feet per minute, the other at the rate 309^^ feet per minute. How far apart are they at the end of one minute ? MULTIPLICATION. 99 66. From 8 trees I gathered apples as follows : 2^ barrels, 3J barrels, 6| barrels, 4,\ barrels, 3f barrels. If barrels, 3J bar- rels, and 2^ barrels. If I sold 15^ barrels of the apples to one man and 2^ to another, how many had I left ? 67. A lady who had $ 50 received $ 8J more, spent $ 17|, lost 1 4/^, and collected $ 16^ of a debt. How much money in dollars and cents had she ? 68. A man having a sum of money spent -^ of it for a house, i*^2? /o^^? T^^x5? /^x9? ^^x20? d. If 2^ pounds of cane are required to seat 1 chair, how many pounds will be required to seat 12 chairs ? Note. Multiply the integer and the fraction separately. 100 COMMON FRACTIONS. e. At $ 18| a dozen, what is the cost of 5 dozen lamps ? /. If 7 men can build a dam in 4| days, in what time can 1 man build it ? g. At $ 10^ each in currency, what is the value of 5 gold eagles ? h. In a piece of land 1 foot long and 1 foot wide there is 1 square foot. H'ow many square feet are there in a piece 8| feet long and 1 foot wide ? in a piece 18| feet long and 5 feet wide ? i. If a man receives $ % for shoeing a horse and $ J for shoe- ing an ox, how much will he receive for shoeing 4 horses and a yoke of oxen ? 229. Examples for the Slate. Illustrative Example II. Multiply -^ by 56. WRITTEN WORK. 7 2 70. If a man can mow -^ of an acre of meadow in 1 hour, now much can he mow in 38 hours ? 71. How many yards of cloth are required for 6 suits, each suit requiring 7^ yards ? 72. What is the width of 18 house lots, each 6| rods wide ? 73. What distance can a vessel sail in 33 hours, going at the rate of 5f miles an hour ? 74. There are 16^ feet in a rod. How many feet are there in 40 rods ? in 320 rods, or 1 mile ? 75. How much ivory worth $ 1 a pound can be bought for the same sum that will pay for 15| pounds worth 1 12 a pound ? 76. One quart dry measure contains 67^ cubic inches. How many cubic inches are there in a bushel, or 32 quarts ? 77. If by working 11 hours a day a piece of work can be done in 45f days, in what time can it be done by working 1 hour a day ? MULTIPLICATION. 101 78. If 17 men can shear a lot of sheep in 9^^ days, in what time can 1 man shear the lot ? 79. Multiply 14f hy 9. 82. Multiply 365|: by 39. 80. Multiply 16^3^ by 7. 83. Multiply 256j by 18. 81. Multiply 23J by 11. 84. Multiply 37611 by 21. To multiply an Integer by a Fraction. 230. Illustrative Example TIL What is ^ of 2 inches ? ILLUSTRATION. , I , , , If |- of each of the 2 inches is taken, * ( |. we shall have f of an inch. Ans. | of an I 1 1 1 i ) inch. (See illustration ; also Art. 196.) iof2 = §. 231. Oral Exercises. a. What is ^ of 2? |of8? J^of4? ^of7? ^ofG? b. What is -iiiy of 3 ? J of 2? ^oi 11 ^'^^\^- H divided by {^equals 12 divided \%^\% = 1Z^1V) = 1^ by 10, or If Ans. l\. 1. Divide I by f ; f by i; ^ by t; f by f ; tV by I- When fractions have different denominators, how do you prepare them to divide ? In the written work of Illustrative Example V., after obtaining a common denominator we have 12 -=- 10, or the new numerator of the dividend divided by the new numerator of the divisor. If, in the place of these, numbers, we put the factors which formed them, we shall have (4 X 3) -^ (5 X 2) or i-^ or ^ x |, in which the expression for the divisor, f, is inverted, becoming |, and the answer, found by nvdti- plying f by f, is f, or 1-J-, as before. 249. To divide one fraction by another, we may then invert the divisor and proceed as in the multiplication of * For other explanations of division of fractious, see Appendix, p. 304. DIVISION. lOd fractions. The written work of Illustrative Example V. will then be merely ^ = f = Ij.. Perform the following examples by either of the methods illustrated above : m. How many aref-nj? fn-f? f^-ft^? f^^? n. How many are f-f? §-|? f^fj? ^--1? 250. From the previous illustrations may he derived the following Rules. 1. To divide a fraction by an integer, Divide the numera- tor or multiply the denominator hy the integer. 2. To divide an integer or a fraction by a fraction, Change the dividend and divisor to fractions having a com- mon denominator, and then divide the numerator of tlie dividend hy the, numerator of the divisor. Or, 2. Invert the divisor, and proceed as in the multiplica- tion of fractions. 251. Examples for the Slate. 135. Divide iff by 6. 138. Divide 181 by ^. 136. Divide T^ by 7. 139. Divide 96 by t^j. 137. Divide ^^f by 18. 140. Divide 108 by ^. 141. At $ iV per pound, how many pounds of rice can be bought for $ 11 ? 142. At I ^ per foot for rubber hose, how many feet can be bought for 141? (143.) 18^^=? (144.) 21-t3^ = ? (145.) 98^/^=? (146.) 54^ A=? 151. How many bushels of peas at $| a bushel can be bought for $ 18 ? for $ 12 J ? 152. At $ f^ per thousand ems for setting type, how many thousand ems can be set for $ 75 ? (147.) f- -H = ? (148.) U- -H = ? (149.) If- -A=? (150.) il- -H=? 110 COMMON FRACTIONS. 163. If 1 yard of cloth can be made from \% of a pound of wool, how many yards can he made from 5 tons of 2000 pounds each ? 154. One rod equals 16^ feet. How many rods in 100 feet? 155. How many hreadths of paper, each f f of a yard wide, will reach around a room, the distance being 21^ yards ? 156. At $2| per yard, how many yards of cloth can be bought for $45|? 157. How many lengths of 1-^ feet are there in a fence 1706i feet long ? 158. How many square rods, each containing 30^^ square yards, are there in 15^ square yards ? 159. A man had $ 1.60, which he exchanged for francs at 18f cents each. How many francs did he receive ? (160.) 3f^4A = ? (162.) 26i^3T7T=? (161.) 6T^-6i = ? (163.) 1-541| = ? 252. Illustkative Example VI. Change the expres- sion HT to its simplest form. WEITTEN WORK. 9f^2i=V-^¥ = #|^=¥=3f ^«. 3f Expressions like that above are sometimes called com- plete tractions. But they merely indicate division. Change the form of the following expressions, and perform the division indicated : (164.) -i„ (167.)^ (1^0-^ ^ (1^3-) I (165.) -i (168.) i- (171.) g (174.) gi (166.) I (169.)^ (172.)^ (l^^-)!^! For other examples in division of fractions, see page 123. ORAL EXAMPLES. 11] TO FIND THE WHOLE WHEN A PART IS GIVEN. Oral Exercises. 253. Illustrative Example I. If | of a ton of hay costs $16, what will -^ of a ton cost ? what will 1 ton cost ? a. If f of a certain number is 28, what is the entire number ? b. 81 is -j^ of what number ? c. A man bought a harness for $ 75, which was f of what he paid for his carriage. What did he pay for his carriage ? d. I paid $ 6 a week for board in Albany, which was f of what I paid in Buffalo ; this was f of what I paid in Chicago ; and this was f of what I paid in San Francisco. What did I pay in San Francisco ? e. An exploring party having lost \ of their bread, are obliged to subsist on 14 ounces a day. What were they allowed at first ? Note. If J is lost, J remain. /. If f of a piece of work be performed in 24 days, how many days wiU it take to do the remainder ? g. A vessel, having lost ^ of her cable, has 200 feet remain- ing. How many feet had she at first ? h. Mary is 24 years old, and her age is equal to once and \ the age of her brother. How old is her brother ? Note. Mary's age is f of that of her trotlier. i. A mother and her son have $ 45 in a purse ; the son's part is f as great as the mother's. What is each one's part ? Solution. — The mother's part must be f of itself, and her son's part added to her part must be | of her part. But the two together have $ 45 ; then $ 45 is ■§■ of the mother's part. j. If I seU an article for $ 80, and thereby gain a sum equal to J of the cost, what is the cost ? k. If I seU an article for $ 80, and thereby lose a sum equal to J of the cost, what is the cost ? 112 COMMON FRACTIONS. 254. Examples for the Slate. (176.) 16^ is ^ of what number ? (177.) 2^ is I of what number ? (178.) I of if is f of what number ? (179.) I of 6§ is I of what number ? (180.) -ft of -^ is I of what number ? (181.) 2 J X 7§ is 3^ times what number (or | of what num- ber) ? (182.) 182 -H (12 X 20 is 3 J times what number ? 183. An author's copyright on a book was $ 54.57. If this was ^^ of the whole profit, what was the whole profit ? 184. Mr. Smith owns f f of an acre of land ; his neighbor Mr. French owns § as much, which is ^ of what Mr. Brown owns. What does Mr. Brown own ? 185. If ^ of my property is in real estate, f in trade, and the balance, which is $33000, is in stocks: what is the value of my property ? 186. A man sold a lot of land for $ 1440, which was 2 J times what it cost him. What did it cost him ? 187. Having lost ^ of my money in trade, I now have $ 2476.50. What had I at first ? 188. A person against whom I had an account has failed, and I have lost f of what he owed me. If I receive $ 1584,72, how much did he owe me ? 189. A body of 4800 troops had \ as many cavalry as infantry. What was the number of each ? 190. A lot of land yielded 4140 bushels of grain in tWo years, yielding f as much the second year as the first. What was the yield each year ? 191. What number is that to which if § of itself be added the sum will equal 275 ? 192. In counting his fowls, -a farmer finds that he has 396 in all, which is \ more than he had the previous year. How many had he then ? EXAMPLES. 113 TO FIND WHAT FRACTION ONE NUMBER IS OP ANOTHER. Oral Exercises. 255. Illustkativb Example I. 1 is what part of 5 ? Answer. 1 is ^ of 5 because it is one of the five equal parts into which 5 may be divided. a. Lis what part of 7 ? of 9 ? of 10 ? Why ? In comparing 1 with any number to see what fraction it is of that number, what do you take as the numerator ? as the denominator ? b. 1 is what part of 7 ? 2 is what part of 7 ? Why ? c. What part of 9 is 2 ? What part of 10 is 7 ? d. What part of 200 is 20 ? 50 ? 25 ? 40 ? e. 1 peach is what part of 7 peaches ? 3 pears of 13 pears ? /. \ is what part of ^ ? ^^ is what part of ^ ? g. J is what part of ^. Note. Change J and J to sixths. h. What part of 10 is 3^? is 2^ ? 256. From the foregoing illustrations we may derive the following Rule. To find what fraction one number is of another. Make the number which is the part the nwmerator of a fraction, and the number with which it is compared the denominator. i. If a piece of work can be performed in 9 days, what part of the work can be performed in 7 days ? J. If Mr. Chase has $ 54 and spends $ 18 for a coat, what part of his money does he spend ? k. Stock originally worth $ 60 a share now sells for 1 40. What part of the original value does it bring ? 1. When goods which cost 76 cents sell for $ 1, what is the gain ? What part of the cost is the gain ? m.. A and B hired a pasture together ; A pastured 12 cows in it and B 13 cows. What part of the price should each pay ? 114 COMMON FMAOTIONS. 257. Examples for the Slate. 193. A man owing $316, paid $84 of the debt. What part of the debt did he pay ? 194. What part of 272| square feet is 9 square feet ? 195. I bQught a house for $3000 and sold it for $4500. What part of the original cost was the gain ? 196. Four men were hired to work on a farm. A worked 7 days, B worked 5 days, 8 days, and D 4 days. They received $ 72. What was each man's share ? (201.) 12f is what part of 19 ? What part (197.) Of 75 is 30? (202.) (198.) Of 267 is 89? (203.) (199.) Of 8 is i? (204.) (200.) Of 11 is If? (205.) 206. What part of 100 is 33^? 62^? 6i? 56i? 1 is what part of 2| ? 1 is what part of l-j^ ? \^ is what part oi ^? 2f is what part of 3f ? 66§? 87i? 37^? 12^? To solve Ezamples by using Aliquot Farts of STumbers. 258. What is one of the three equal parts of 9 ? of 10 ? One of the equal parts of a number is an aliquot part of the number. Thus, 3|- is an aliquot part of 10. 259. Oral E^cercises. Knd such aliquot parts of the following numbers as are indicated below: a. Of the number 30 find J ; J ; J ; J ; ^ ; ^ b. Of the number 60 find I ; i; i; i; i; ^ 0. Of the number 100 find J ; i ; i d. Of the number 100 find § ; | ; | e. Of the number 144 find \; \; I t. Of the number 200 find J ; J ; \ \ i; i; 4 A) 1*5; §• g-. Of the number 1000 find i; J; J; ^; J; \; ^; §; |. ALIQUOT PARTS OF NUMBERS. 115 260. By using the aliquot parts of numbers, the work of multiplying and dividing may often be shortened, thus : Illustrative Example. What is the cost of 25350 ft. of gas at 3 J mills per foot ? Operation. — 3^m. = ^ of 10 mills, or J of a cent. 25350 ft. at 1 cent a foot costs $253.50, and at J of a cent a foot it must cost ^ of $253.50, or 184.50. Ans. 184.50. Oral Exercises. Find the cost a. Of 1872 lbs. of butter at $ 0.33^ per lb. ? b. Of 64 bu. potatoes at $ 0.87^ per bu. ? c. Of 44 yds. of silk at 1 1.12J per yd. ? d. Of fencing 50 rods of road, both sides, at $ 3.75 per rod ? e. Of insuring a house 5 years at $ 6.66| per year ? /. Of 750 feet of boards at $ 12 per thousand ? g. Of 80 pounds of butter at 37|^per pound ? h. How many pounds of cheese at 16 1)* a pound can be bought for $10? i. For $ 20 how many yards of cloth can be bought at 1 1 a yard ? at 12^/ ? at 16f / ? at 25/ ? at 50/ ? at 37^/ ? 261. Examples for the Slate. (207.) 987--16f= ? (210.) 496^-12^=? (208.) 864 -33^= ? (211.) 684 x 66§= ? (209.) 572 X &2l= ? (212.) 487 x 37J= ? 262. Questions for Review^. What is a factor of a number ? What is a composite number ? a prime number ? a prime factor ? What is an even number ? an odd number ? What numbers are divisible by 2 ? 3 ? 4 ? 5 ? 6 ? 8 ? 9 ? 11 ? How can you find the prime factors of a number? A composite number equals what product ? Find the prime factors of 180 and explain the process. How can you make sure that a number is prime ? What is CANCELLATION ? Why should arithmetical processes first be indicated by signs ? Explain the use of the parenthesis. 116 COMMON FRACTIONS. When are numbers prime to each other ? What is a common fac- tor of two or more numbers ? the urbatest common factor ? Find the g. e. f. of three numbers by the first method given ; explain and give the rule. Find the g. c. f. of two numbers by the second method given ; give the rule. In what cases would you find the g. c. f. by the second method ? When do we make use of the g. c. f. of numbers V What is a multiple ? a common multiple of two or more numbers ? the LEAST common multiple ? When do we make use of the 1. c. m. ? Explain the first method of finding it ; the second. What does the 1. c. m. of numbers prime to each other equal ? What is a fractional unit? a fractional number? What name is applied to both ? Name and define the terms of a fraction. ~Ex- plain the expression -|. How do you change fractions to smaller terms ? to larger terms ? When is a fraction expressed in its smallest terms ? How do you change improper fractions to integers or mixed numbers . How do you change integers or mixed numbers to fractions ? When are fractions said to have a common denominator? For what operations upon fractions do we first change them to others having a common denominator ? Change -^j -f^, and -^^ to fractions having a common denominator, and explain. gow do you add fractions? Take three fractions of different denominators, add and explain. How do you add mixed numbers ? How do you subtract fractions ? Give a general rule for the addi- tion of fractions. Give a general rule for the subtraction of fractions. Let 4^ be the minuend and 1^ the subtrahend ; subtract and explain. How do you multiply a fraction by an integer ? a mixed num. ber by an integer ? Explain, by an example, the method of multi- plying an integer by a fraction. Multiply a fraction by a fraction ; explain and give the rule. How do you multiply a mixed number by a mixed number or a fraction ? How can you simplify the expres- sions called compound fractions? How do you divide a fraction by an integer ? a mixed number by an integer ? an integer by a fraction ? Explain, by an example, the method of dividing a fraction by a fraction, and give the rule. How can you simplify the expressions called complex fractions ? How do you find what fraction one number is of another? What is an aliquot part of a number ? What effect does multiplying both terms of a fraction by the same number have upon it? Why? What effect does dividing both terms GENERAL REVIEW. 117 of a fraction have upon it ? Why ? What effect does multiplying the numerator of a fraction have upon the fraction 1 Why ? In what other way could you produce the same effect, and why ? What effect does dividing the numerator have upon a fraction 1 Why 1 In what other way could yoli produce the same effect, and why 1 263. General Review, No. 2. 213. What are the prime factors of 420 ? 214. Divide 18 x 7 x 15 x 6 by 28 x 10 x 3 x 4. 215. Knd the greatest common factor of 35, 84, and 56. 216. Find the least common multiple of 63, 18, 14, and 28. 217. Cliange ^f f and JJf to their smallest terms. 218. Change 466|^ and 84 to improper fractions, having 8 for their denominator ? 219. Change ^f & and ^^ to mixed or integral numbers. 220. Change f, ^^y, and f to fractions having a common de- nominator. 221. Change ■^■^, -f^, and 9J to fractions having the least common denominator. 222. Add I of f, §, and 8^. Add 25|, 6f, and 46^. 223. From 24 take 12f Subtract ~^^ from ^5 of /^. 224. Multiply 7| by 4 ; 7f by 6 ; A by 8§. 225. Simplify the expression ^^5 of ^ of f| of 2f. 226. Divide A ^'J f ; « of /it % 1 A- 227. Simplify the expressions -|-, ^, and 5-. 228. What part of 4j is 3^ ? 229. Two trains which are 75 miles apart are running towards each other, one 30| miles an hour, tho other 40| miles an hour. How far apart will they be in half an hour ? 230. A man paid $ 18f for a load of hay weighing 1^ tons. At the same rate what should he pay for f of a ton ? 231. Having spent f of his money, Fred has 1 13^. How much had he at first ? 232. Make out a bill of sale for three barrels of sugar, weighing respectively 235 pounds, 241 pounds, and 264 pounds, at llf / a pound. 118 COMMON FRACTIONS. 264. Miscellaneous Oral Examples. a. If I of a pound of candles cost 35 cents, what is the price of 1 pound ? of \^ of a pound ? b. In f of an acre of land there are 120 square rods. How- many square rods are there in J of an acre ? c. When 1^ bushels of oats wiU feed 10 horses for a cer- tain time, how many horses will 2\ bushels feed for the same time? d. At I f each, how many cedar posts can be bought for $12? for $7i? e. At 1 2\ per day, how many days' work can be paid for with $20? with$37i? /. At $2 per day, how many days' work can be paid for with $7|? with $9i? g. If it requires 12 yards of carpeting f of a yard wide to carpet a hall, how much will be required of that which is Ij yards wide ? h. What number is that, \ of which exceeds ^ of it by 2 ? i. If f of the distance from Springfield to Albany is 80 miles, what is f of the distance ? J. If $ 5J pays for the lodging and breakfast of 7 persons, for how many persons wiU $ 11^ pay ? k. What is that number to which if f of itself be added the sum will equal 64 ? 1. I sold my watch for $ 72, which was \ more than I gave for it. What did it cost me ? m. Bought a horse and saddle for $ 75, giving f as much for the saddle as for the horse. What was the cost of each ? n. A can build a wall in 3 days, and B can do the same work in 4 days. What part of the work can each do in one day ? What part can both do in one day ? In how many days can both do it working together ? 0. C can do a piece of work in 5 days, and D in 8 days. What time will be required for both to do it ? MISCELLANEOUS EXAMPLES. 119 265. Miscellaneous Examples for the Slate. 233. What will 16^ yards of cloth cost at 63/ a yard ? 234. What will 9J bushels of corn cost at 87^/ a bushel ? 236. What wiU 271f acres of land cost at $ 31| per acre ? 236. I paid 65/ for 2 boxes of strawberries. What wiU be the cost of 45 boxes at the same rate ? 237. What is my bill for 7 pear-trees at 87J- cents apiece for the trees, and $ 2 a dozen for setting ? 238. What do I receive per pound by selling 16 pounds of coffee for 1 3.76? 239. If ^ of a man's property is in land, valued at $ 2324f , what is the value of his whole property ? 240. Boughtf of ashipfor $4075. What would the whole ship cost at the same rate ? 241. What is the cost of 3 pieces of calico, 37J yards in a piece, at 19^^ cents per yard ? 242. Sold my house and farm of 47f acres for $ 6150. Allow- ing $ 3500 for the house, what did I receive per acre for the land? 243. How long wiU a quantity of ilour last a family of 8 persons if it lasts 3 persons 14^ months ? 244. If in 32^1^ years a man saved $1694, what was his average saving per year ? 246. What number is that which diminished by 1-^ wiU leave a remainder of 1^ ? 246. What number is that to which if you add 9^ the sum will be 124f ? 247. What is that number to which if you add f of 26^ the sum willbel47i^? 248. If you buy 7 J yards of silk at $ 6 a yard, 14 J yards of cashmere at $ 1.25 per yard, 4|^ yards of silk at 75 cents per yard, and f of a yard of velvet at $ 4.60 per yard, giving in payment a $ 100 bill, what balance will be your due ? 249. What will 50 oranges cost at 62^/ a dozen ? 120 COMMON FBAGTIONS. 250. How long will 200 pounds of meat last 9 persons at the rate of |^ of a pound a day for each person ? 251. A farmer has sold his eggs at an average of 2Z\ cents per dozen, which is ^ higher than they averaged the previous year. What did they average then ? 252. He is paid for grain 1 1.80 per bag, which is ^ less than he was paid last year. What was he paid last year ? 253. Mr. Stevens, dying, left $ 75000 to his wife and two sons. To his wife he left $ 30000 ; to his oldest son just as large a part of the remainder as his wife's portion was of the entire property ; and to his youngest son the rest. What was each son's share ? 254. A man sold 54| yards of cloth at the rate of 3 yards for 2 dollars. What did he receive for it ? 255. Mr. Day bought a house and barn for $ 4060, giving \ as much for the barn as for the house. What did he pay for each ? 256. If a body faUs ICy'^^ feet in the first second of time, 3 times I63JJ feet in the next second, and 6 times IB^Jj feet in the third second, how far will it fall in the three seconds ? 267. What length of time would a man require to travel around the earth if the distance is 25000 miles and he travels at the rate of 31 J miles per day ? 268. If a man can build 2f rods of wall in a day, how much can he build in &^ days ? 259. What number is that f of which exceeds \ of it by llj ? 260. If I buy 1250 bushels of corn at 41 cents per bushel, and sell it at 52 J cents per bushel, how much do I gain ? 261. What number divided by f equals 126f ? 262. What are the contents of 3 floors measuring as foUows : 13f square yards, 32/5- square yards, and 49f^ square yards ? 263. The product of three numbers is 63^ ; two of them are 8J and 6-,^. What is the third ? 264. I exchanged 42 tubs of butter, averaging 48^ pounds, at 21 J cents per pound, for 42 barrels of flour, at $ 9f per barrel, and received the balance in cash. What was the balance ? MJSCELLANEOXfS EXAMPLES. 121 265. Owing a man in Paris 1325 francs, I have shipped to him $ 375 worth of rice. If the franc is worth 19:^ cents, how much have I overpaid him in United States money ? in francs ? 266. I have three hoxes, each containing 12 pieces of cloth, each piece 4f yards in length, and weighing 3| pounds to the yard. What is the weight of the whole ? 267. What wiU 42^ quires of paper weigh at f pound per quire ? 268. Owning f of a flour-mill, I sold f of my share for $ 1750. What is the value of the whole mill at the same rate ? 269. When hay was $ 15 per ton, I gave | of a ton for If tons of coal. What was the coal worth per ton ? 270. If a man walks 9J miles in 2J hours, how far will he walk in 4f hours ? 271. At the rate of 4J^ mUes an hour, what time will he required to walk 122 miles ? 272. In 1860 I purchased cotton at 8^ cents a pound, which I sold in 1862 at 90| cents. What did I gain on 1000 lbs. ? 273. If a man can earn $2.30 per day, how many days' work will he have to give for a suit of clothes, of which the coat costs $ 25 J, the trousers $ 8, and the vest $ 5:^ ? 274. If I of I of a ship cost $ 42000, what is f of it worth ? 275. In a certain manufactory J of the operatives are Ger- mans, \ French, ^ Scotch, \ English, t^-^ Swedes, and the remainder, 140, native Americans. What is the whole num- ber, and the number of each nationality ? 276. If :J of my money is in gold, ^ of the remainder in silver, and the balance, $ 360, in bank-notes, how much money have I in all ? 277. A certain piece of work can be performed by A in 8 days, by B in 10 days, and by C in 16 days. In what time can an do it working together ? 278. In what time can A and B do it together ? 279. In what time can A and C do it together ? 280. In what time can B and C do it together ? 122 COMMON FRACTIONS. 266. DRILL TABLE No. 5. Examplei. 1. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 23. 24. A B c D E F G f A 1 m n 9 12f A A f IM 8f 15 24^ If A hi ^^ 4i 6 18| A ^TT 1 Hi 9f 12 28# f A if A% 6f 14 42| if A A ^ms 6f 8 32f A A it Iff 4f 10 15f A A 1 fff If 7 21f f f 1^ ifi 2* 23 54i H 1 M fit 5| 20 48f i M i iiif 9f 21 35| \h A A tWh 3* 24 36f 1 li if ^^ 2| 5 45f f A A ifii 6f 16 66f A i A lit H 18 54f f A if m 2i 22 55i i f f m 3f 17 51i 'a f A m 7i 13 40^ * i A m' 8f 19 38i A * 1 m ^ 11 _ 44f i* A A m n 24 36^ i* A f Iff 3f 12 18f ii A A AW 8f 28 63f A A if AA 6| 15 334 M A T^r n 26 39§ DMILL EXERCISES. 123 267. Exercises upon the Table. W5. 1^. 130. 131. 132. 133. 135. 136. 137. 138. 139. llfi. 141. m. IJiS. lU. us. 161. 163. Find the prime factors of each numerator in D. t Find the prime factors of each denominator in D. Find the g. o. f. of the terms of each fraction in D. Find the 1. c. m. of F, G,* and H. Change D to lowest terms. Change the mixed numbers in G to improper fractions. Find the sum of A and C. 146. 147. 148. 149. 150. 161. ISZ 153. 154. 155. A + B + C. C + D + E. E + F + G + H. A-B. H-G. 6-E. H + A-G. A-Bof C. Difference of C and D. Cof E -AofB. Simplify AofBof C. Simplify A of B of Cof E. AxB. CxF. CxE. If G is B of some number, what is C of the same number ? H is C of how many times E ? C of E is B of how many times A? GxF. GxE. A-=-B. C-=-E. C-=-F. H^A. E-=-F. G-=-E. A of F-T-E. A of C -^ B of E. 156. (A + B) -f-(B X C). 157. (A-B) -T-(B-^C). 158. A + C-B. 159. ExF + G. 160. A of E ^ A-B. B + G 6-=-F. 164. What number is that from which if you take A the remainder wUlbeB? 165. What number is that to which if you add C of F the sum will beG? 166. What number multiplied by F win give G for a product ? 167. What number divided by E wUl give D for a quotient ? 168. What divisor will give E for a quotient, H being the divi- dend? 169. What number is that to which if A of itself be added the sum win equal H 1 170. What number is that from which if B of itself be subtracted, thb remainder wiU be F ? 171. Divide H into three such part9 that the 2d shall be twice the Ist, and the 3d F more than the 2d. What is the 3d part ? 172. At E dollars a yard, what will F yards of cloth cost ? 173. At E dollars a yard, how many yards of cloth can be bought for F dollars? 174. If B pounds of tea cost H cents, what will E pounds cost ? 175. John can do a piece of work in E days, and James can do the same work in F days. In what time can both together do it ? 176. If George and Albert can do a piece of work in E days, and Albert can do it alone in F days, in what time can George do it alone ? * Omitting fractions. t See page 57, for Explanation of the Use of the Drill Tables. 124 DECIMAL FRACTIONS. SEOTIOIT X. DECIMAL FRACTIONS. 268. Articles 30 to 36 treat of a series of fractions, — tenths, hundredths, thousandths, etc., — each of which has for a denominator 10, or a number made by using lO's only as factors. Such fractions are decimal fractions. Note. Decimal fractions are usually called decimals. To read and write Decimals. 269. The method of reading and of writing decimals has been explained in Articles 34 to 36. These the pupil may review. 270. Exercisea. 3. Eead 5.368; 0.406; 2.007; 0.039; 106.105. b. Eead 0.4721; 7.0497; 10.010; 15.0015. Eead the following : c. 30.0094 e. 120.250049 g. 200.005 d. 17.01845 /. 1.001025 h. 0.205 Note. To distinguisli 200. 005 (Example g) from 0. 205 (Example h), use the word dedTnal before reading the decimal part. Thus, 200. 005 may be read "two hundred and the decimal fiye thousandths "; while 0.205 may be read " decimal two hundred five thousandths." Eead the following : i. 0.315 m. . 500.0074 Of- 1000.00001 j. 300.015 72. 4700.0065 r. 14.00375 k. 36000.00018 0. 430.06 s. 0.0000027 1. 0.36018 P- 43000.06 t. 0.1000012 REDUCTION OF DEOIMALS. 125 271. To write a decimal : Write the number as an integer, and place the decimal point so that the right-hand figure shall stand in the place required hy the denomina- tion of the decimal. Note. When the given nmnher does not fill all the decimal places, sup- ply the deficiency with zeros. For other exercises in reading and for exercises in writing decimals, see page 135. The pupil may now review addition and subtraction of deci- mals (Articles 45, 50, 61, and 65). REDUCTION OF DECIMALS. To change the Denomination of a Decimal Fraction. 272. Exercises. a. What is the denominator of the fraction 0.5? 0.25? 0.075? 7.3? 4.86? b. What is the numerator of the fraction 0.4 ? 0.04 ? 0.075 ? 0.0101? 0.000007? 0.25? 0.1125? c. Write as a common fraction 0.3; 0.08; 0.375; 0.0204. 273. Illusteative Example. Change 0.5 to thousandths. WRITTEN WORK. Explanation. — Multiplying toth numerator and 5 — '500 denominator of ^ by 100, we have -^^, which is expressed decimally by writing C.500. 274. From the written work above we derive the fol- lowing Rule. To express a decimal fraction in any lower denomina- tion : Annex zeros to the given expression until the place of the required denomination is filled. 126 DECIMAL FRACTIONS. 275. Examples for the Slate. 1. Change 0.07 to thousandths. 2. Change 0.4, 0.75, 2.5, and 1.06 to thousandths. 3. Express 0.003, 1.75, and 0.006 as ten-thousandths. 4. Express 3 as tenths ; as hundredths ; as thousandths ; as ten-thousandths; etc. Answers. 3.0; 3.00; etc. Note. Eead the above answers: " Thirty tenths ; three hundred hun- dredths"; etc. 5. Express 7, 40, and 37 as tenths ; as hundredths ; as ten- thousandths. To change a Decimal Fraction to a Common Fraction. 276. Illusteatitb Examples. Change 0.25 and 0.33^ to common fractions in their simplest forms. WRITTEN WORK. Explanation. — After writing these frae- 0.25 = T^jAr = i tions with their denominators, we find that 33i = -A?''^ = 4*ft = i ^'^^ ^™* '^^^ ^® changed to smaller terms (Art. 198), and that the second may be changed to a simple fraction (Art. 352) and then to its smallest terms. 277. From the examples above we derive the following Rule. To change a decimal fraction to a common fraction: Write the decimal in the form of a common fraction, and then change the result, if necessary, to its simplest form. 278. Examples for the Slate. Change the following to common fractions in their simplest forms : (6.) 0.4 (11.) 0.3J (16.) 0.750 (21.) 0.0625 (7.) 0.80 (12.) 0.37i (17.) 0.368 (22.) 0.0338 (8.) 0.35 (13.) 0.62i (18.) 0.66§ (23.) 0.14f (9.) 0.75 (14.) 0.87i (19.) 0.666f (24.) -7.5 (10.) o.n (15.) 0.875 (20.) 0.072 (25.) 1.16f REDUCTION OF DECIMALS. 127 To change Commoa Fractions to Decimal Fractions 279. Illusteative Example. Change | to a decimal fraction. WRITTEN WORK. Explanation. — The fraction -I is the same as ^ of 8) 3.000 2' °r i °^ 3.000 (3000 thousandths), which is found by — — dividing 3.000 by 8 in the usual way (Art. 102). yj.oto 280. From the example above we derive the following Kule. To change a common fraction to a decimal fraction : Eocpress the numerator as tenths, hundredths, thcmsandths, etc., h/ annexing as many zeros as may be required, and then divide it hy the denominutor. 281. Examples for the Slate. Change to decimals : (26.) f. (29.) ff. (32.) 1^. (35.) 1.06^. (27.) i^. (30.) 5i. (33.) 8f. (36.) 0.04f. (28.) tIf. (31.) W- (34.) n^^. (37.) 0.03^. Change to decimals and add (Art. 45) the following : (38.) i, f , I, and ^^. (40.) f , |, ^, and ^. (39.) \, i, li, and ^^. (41.) 1%, 15f, and 1^^. 42. A carpenter paid for a mantel-piece $27f, for a grate $ 22f , and for a hearth $ 4-^. How much did he pay in aU ? (43.) 2^ + 3 J + 87^ + 18f = what ? 44. A drover bought a cow and a calf for $ 38.85, and sold the cow for $ 32f , and the calf for $ lOf. How much did he gain? 45. A man owning 17.635 acres of land, sold 1^ acres to one person, and 5*5^ of an acre to another. How much had he left? 46. Change to seven decimal places, and add 1.82^, O.OOQtI^, and O.IO^^. 128 DECIMAL FRAOTIOXS. 282. Illustrative Example. Wlmt is the sum of 5^ j^ards, 2| yards, and 7^ yards ? WRITTEN WORK. Explanation. — In this example there are g, _ K -1 25 fractions which cannot be completely ex- 2* — 2 666* pressed as decimals ; for, however far the rr,.~rTA^A\ dlvlsion be carried, there will still he a ** ■^ '^ •* remainder. Exact sum, 15.215^^ If we choose to stop dividing at thou- sandths, the quotients are expressed accu- O^ = O.J.Z0 rately by writing f of a thousandth and ^ 2§ = 2.667 of a thousandth, as in the margin. But 7^f = 7.424 these results are no more convenient to add Approximate mm. 15.216 ^'^^'i *^e original numbers ; hence nothing has been gained by changing the latter to the decimal form if our object was to find the exact sum. There are, however, many cases in which the error arising from the neglect of such small fractions as parts of a thousandth is of no importance. For such cases the second form of written work given in the margin is to be adopted. Here 'the decimal values are expressed to the nearest thousandth. This is done by increasing the last term of the decimal by 1 whenever the neglected fraction is ^ or more. Greater accuracy would be attained by carrying out the decimal to the nearest ten-thousandth, or to a still lower denomination. 283. Examples for the Slate. Note. Unless some other direction is given, the pupil will hereafter understand that decimal values are to be expressed to the nearest ten- thousandth. 47. Find the decimal values of |, ^^, f , and add the lesiilt!). 48. Change to ten thousandths, and add 9t|, 16|, and 33-j\. 49. Change -^ and 0.68 to ten thousandths, and find their difference. 50. Mr. Carpenter has worked for Mr. Bates 2f hours, 3f hours, and 6.5 hours. How naany hours has he worked for liim in all? 51. How many rods are there in 25f rods, 0.48^^^ rods, 1054 rods, and 8.62 J rods ? Other examples in addition and subtraction may be found on page 135. CIRCULATING DECIMALS. 129 Circulating Decimals. 284. We have seen (Art. 282) that in expressing ^ deci- mally (0.666 . . .) the figure 6 is repeated again and again. So in expressing ^ decimally (0.4242 . . .) the figures 4 and 2 are repeated again and again. Decimal fractions that are expressed by the same figures repeated again and again are called repeating or circulat- ing decimals. Note. Circulating decimals arise from the reduction of common frac- tions who.se denominators contain prime factors other than 2 and 5. 285. The repeating figures of a circulating decimal are called a rep et end. A repetend is marked by placing dots over the first and last of the figures that repeat. Thus, l| = 0.297297 . . . = 0.297 ; if = 0.4242 . . . = 0.42 ; 3^ = 3.166 . . . =3.16. 286. Change the following fractions to decimals till the figures repeat, and mark the repetends : (52.) J. (55.) |. (58.) T^^. (61.) ^. (53.) i. (56.) \\. (59.) ^. (62.) 1t^. (54.) f. (57.) f. (60.) T^x. (63.) S/^- To change a Circulating Decimal to a Common Traction. 287. Illustrative Example I. Change 0.63 to a com- mon fraction. To change a circulating decimal to a . ■ ' ^ ' common fraction : Take the repetend foi- ~ *^ "" ^^' the figures of the numerator, and for the figures of the denominator as many 9's as there are figures in the repetend. Change the fraction thus expressed to its smallest terms. For an explanation of this mle, see Appendix, page 305. 130 DECIMAL FRACTIONS. Change to common fractions in their smallest terms : (64.) 0.3 (67.) 0.39 (70.) 0.016 (73.) O.iSSi (65.) 0.6 (68.) 0.27 (71.) 0.62i (74.) 0.42857i (66.) 0.42 (69.) 0.648 (72.) 0.i08 (75.) 0.571428 To change a IMixed Circulate to a Common Fraction. 288. Illustrative Example II. Change 0.263 to a common fraction. To change a mixed circulate to a com- WRITTEN "WORK mon fraction : Take for the numerator the "^ tti^Tny difference between the mixed circulate and the part which does not repeat, both regarded as integers, and take for the figures of the denominator as many 9's as there are figures in the repetend, with as many zeros annexed as there are figures in that part of the circulate which does not repeat. (See Appendix, p. 305.) Change to common fractions in their smallest terms : (76.) 1.86 (78.) 0.033 (80.) 0.016 (82.) 2.07671 (77.) 2.73 (79.) 0.027 (81.) 0.042 (83.) 7.16i88i MULTIPLICATION. In Articles 82 and 86 the multiplication of decimals by integers has been taught. These the pupil may now review. 289. Illustrative Example I. Multiply 175 by 0.01. Multiply 175 by 0.5. Explomation. — (1.) To multiply 175 by WKITTEN WOKK. 0.01 IS to take 1 hundredth of it, which we (l ^ 175 X 01 = 1 75 ®-^P''®^^ ^y placing the decimal point so that the figures 175 may express hundredths ; /2.) 175 thus, 1.75. QQQ (2-) To multiply 175 by 0.05 is to take 5 _! hundredths of it. One hundredth of 175 is 8.75 1-75, and 5 hundredths is 5 times 1.75, which equals 8.75. Ans. 8.75. MULTIPLICATION. 131 290. Illtjsteative Example II. Multiply 0.4 by 0.9. WRITTEN WORK. „ . Explanation. — To multiply 0.4 by 0.9 is to take 9 tenths of 4 tenths. One tenth of 0.4 is 4 hundredths, and 9 tenths of 4 tenths is 9 times 0.04, which equals 0.36 0.36. Am. 0.36. 291. From the written work above may be derived the following Rule. To multiply by decimals : Multiply as in integers, and point off as many places for decimals in the prodzi^t as there are decimal places in the multiplicand and the mul- tiplier counted together. N'oTB. If there are not figures enough in the product, prefix zeros. 292. Iisamples for the Slate. Multiply (84.) 0.048 by 9. (93.) 40.5 by 0.016 (85.) 0.027 by 34. (94.) 1842 by 0.07 (86.) 0.076 by 20. (95.) 0.0758 by 20. (87.) 84 by 0.056 (96.) 6.6 by 33^ (88.) 600 by 0.07 (97.) 10.75 by 8f. (89.) 8.4 by 0.56 (98.) 18f by 0.054 (90.) 4.65 by 2.2 (99.) 56i by 2.73 (91.) 0.8 by 0.0206 (100.) 1.7 by 272^ (92.) 7.06 by 0.053 (101.) 66.6§ by 5.7 102. What is the sum of 76 x 100 and 0.001 x 1000 ? 103. What is the sum of 7.5 x 1000 and 0.0001 x 0.001 ? 104. How many are 66.8 x 0.01 + 6.29 x 1000 + 0.7 x 0.001 ? 105. How many are 48.125 x 8.33^ + 8169.5 x 0.09 ? 106. What is the cost of paving 146.74 squares at $ 16.84 per square ? For other examples in multiplication of decimals, see page 135. 132 DECIMAL FRACTIONS. DIVISION. In Articles 101, 102, and 114 the division of decimals by integers has been taught. These the pupil may review. To divide an Integer or a Decimal by a Decimal. 293. Illusteativb Example" I. Divide 72 by 1.2 Explcmation. — Before dividing by a fraction, the WRITTEN WORK. j. -j -, . x. j ■ il, n aividend must be expressed m tne same denomma- 1.2) 72.0 tion as the divisor. The divisor is a number of go tenths ; the dividend expressed in tenths is 72.0 (720 tenths). 720 tenths divided by 12 tenths gives the same quotient as 720 divided by 12, which is 60. Ans. 60. 294. Illusteativb Example II. Divide 1.935 by 0.45 WRITTEN WORK. Bxplanation. — Here the divisor is a num- ber of hundredths ; the dividend expressed as 0.45) 1.93^5 (4.3 hundredths is 193.5 hundredths (the denomi- l"^ nation may be indicated in the written work 135 by a caret). ]^35 193.5 hundredths divided by 45 hundredths — — gives the same quotient as 193.5 divided by 45, which is 4.3. Ains. 4.3. 295. From tbe preceding examples may be derived the following Rule. 1. To divide by decimals : Express the dividend in the same denomination as the divisor by putting a mark as many places to the right of the decimal point as there an decimal placed in the divisor. 2. Divide as if the divisor were an integer, making a decimal point in the quotient, when the terms of the divi- dend have been used as far as the mark. Note. When there is a remainder after all the terms of the dividend have been used, the division may be continued, as in Articles 102 and 106. DIVISION. 133 296. Xizamples for the Slate. 107. How many books at $ 0.08 each can be bought for $3.84? 108. At $2.80 per yard, how many yards of muslin can be bought for $65.30? 109. How many rods, each 16.5 feet, are there in 99 feet? 110. One quart dry measure equals 67.2 cubic inches ; one quart liquid measure equals 57.75 cubic inches : in a keg whose capacity is 5040 cubic inches, how many quarts dry measure ? How many quarts liquid measure ? 111. If coal is $6.67 per ton, how much coal can be bought for $3,335? 112. At $ 0.125 per yard for cotton cloth, how many yards can be bought for $ 26 ? Divide Divide (113.) 14.91 by 7. (126.) 56.28 by 0.0056 (114.) 8.25 by 1.6 (127.) 0.417196 by 58.76 (115.) 3.24 by 0.81 (128.) 0.08 by 1.611 (116.) 0.00468 by 0.013 (129.) 1.3 by 197.59 (117.) 180.375 by 1.626 (130.) 1203.488 by 28.6 (118.) 679 by 0.075 (131.) 49.2654756 by 0.0759 (119.) 6.9705 by 0.45 (132.) 2464.176 by 57.2 (120.) 0.0033 by 0.011 (133.) 164.6156 by 1334. (121.) 0.705 by 7.5 (134.) 0.789 by 0.03f (122.) 3 by 29.9 Note. First multiply both diyi- (123.) 20 by 0.013 dead and divisor by 7. (124.) 4066.2 by 0.648 (136.) 1.36 by 6.807f (125.) 68077 by 71.66 (136.) 43.2| by 0.58J^ 137. Multiply 0.648 by 100 ; divide the product by ^^ ; divide this quotient by 0.001 ; and multiply the result by J of 0.0362. 138. Find the sum of the following : 756.02 x ^ ; 18.3 x 100 ; 0.7 ^ 0.001 ; ^.16 H- T-^^ ; and 0.24 - 16. For other examples in division of decimals, see page 135. 134 DECIMAL FRACTIONS. 297. DRILL TABLE No. 6. 0.025 25.75 0.3604 1.01 250. 0.0008 16.005 8000. 5.6 0.708 19.364 0.0516 1.732 8016. 4.95 0.012 12.007 45.9 8.621 0.00562 1002. 1.87i 0.12i 1.0l5 8.33f Decimals. F G Twenty-five, and two thousayhdths. 5000. Two hundred six tevrthow^andths. 1.648 Seven hundred, and eight tenths. 0.657 404 hundred-thousandths. 0.002i 505050 ten-thousandths. 0.005 Eight, and ninety-six hundredths. 448, Five thousand two, and 5 hundredths. 2.5 Sixteen hundred-thousandths. 0.008 One hundred twelve millionths. 1.68 Two, and twenty-five hundredths. 607. Seven hundred one, and six tenths. 18.08 Two, and 206 thousandths. 9.16 936 ten-thousandths. 800.1 54, and 54 thousandths. 0.0052 806, and 1047 millionths. 2.0763 5 hundred, and 26 hundredths. 18.26^ One thousand millionths. 0.0101 Twenty-nine millionths. 3.0712 846291 hundred-thousandths. 80.06 Five hundred eleven thousandths. 5.4 4271, and 4271 ten-millionths. 0.6805 68 thousand, and 4J^ tenths. 4.071 One hundred twenty-two thousandths. 24.40 Eight, and ^ hundredtlis. -* 0.0002 Five hundred, and f tenths. 1.071 DRILL EXERCISES. 135 298. Ezercises upon the Table. 177. Read tlie numbers expressed in E.* 178. Eead the numbers expressed in G. 179. "Write in figures tlie numbers expressed in F. 180. Change E to ec[uiv. common fractions in lowest terms. 181. Change G to eq[uiT. common fractions in lowest terms. 182. Change H to equiv. decimals (4 places). 183. Add E and F. 184. Add F and G. 185. AddEFandG. 186. Find the difference of E and F. 187. Find the difference of E and G. 188. Find the difference of F and G. 189. Multiply E by F. 192. Divide E by F. 190. Multiply E by G. 193. Divide G by E. 191. Multiply F by G. 19^. Divide G by F. 195. Multiply E by 10 ; divide F by 100 ; add the results to 6. 299. Questions for Review. Wliat are Decimal Fractions ? How are their written expressions distinguished from those of integral numbers? What indicates the denomination of the decimal ? How do you read a decimal expression ? Eead 7.05 as a mixed num- ber; as a fraction. Eead 0.504 and 500.004 so that they may be distinguished. How do you write decimals ? What is the effect of annexing ciphers to a decimal expression ? How do you change decimals to common fractions ? Com- mon fractions to decimals ? What fractions cannot be changed wholly to a decimal form ? What are they called when ex- pressed decimally ? How do you change a circulate to a com- mon fraction ? How do you add and subtract decimals ? Perform an example in multiplication by a decimal, explain and give the rule. Perform an example in division by a decimal, explain and give the rule. How do you express the multiplication of a decimal by 10; 100; 1000; 0.1; 0.01; 0.001? How do you express the division of a decimal by 10; 100; 1000; 0.1 ; 0.01 ; 0.001 ? * See page 57, for Explanation of the Use of the Drill Tables. 136 WEIGHTS AND MEASURES. SEOTIOl^ XI. "WEIGHTS AND MEASURES. MEASURES OF LENGTH. 300. We measure the length of anything by applying to it a line of known length, as 1 foot, 1 yard, and finding how many such lengths it contains. The line of known length so used is a linear nnit. 301. The length of a line is reckoned in linear units. 302. In measuring length we employ the mile (m.), rod (rd.), yard (yd.), foot (ft.), and irich (in.). These are the units of Long Measure. 12 inches = 1 foot. 3 feet = 1 yard. 5 J yards or 16J feet = 1 rod. 320 rods or 5280 feet = 1 mile. Note I. The standard unit of length is the yard. From this the other units of length are derived. Note II. For sm-veyors' and mariners' measures, see Appendix, page 306. 303. Oral Exercises. a. How many inches are there in 1 yard ? in 2 yards ? in half a yard ? in a quarter ? in an eighth ? in a sixteenth ? b. What will it cost to grade a mile of road at $ 1 a rod ? c. What is the length in feet of a hall that is 15 yards 2 feet long ? d. How many feet in the length of a fence 5 rods long ? e. One eighth of a mile is sometimes called a furlong. How many rods in a furlong ? MEASURES OF SURFACE. 137 An Angle. MEASURES OF SURFACE. 304. Two liiias meeting at a point form an angle. Thus tlie lines a b and b c form the angle ab c. 305. The lines are the sides of the angle, and the point where they meet is the vertex. 306. The size of an angle is the amount by which one side is turned away from the other. Thus the angle d e / is greater than the angle ab c, for the side ef is turned away from e d more than & c is turned away from b a. « 307. When one line meets another so An Angle. as to form two equal angles, each of these angles is a light angle, and the lines are per- pendicular to each other. Thus, the line 6 c is turned away equally from b a and 6 d, making the angles abc and cbd equal to each ■*" other. 'Right Angles. 308. A flat surface, as the surface of a slate or the top of a table, is a plane surface. 309. A rectangle is a plane surface bounded by four straight lines, and hav- ing all its angles right angles. 310. A square is a rectangle all of whose sides are equal. 311. A square each of whose sides is 1 inch long is a square inch; a square each of whose sides is 1 foot long is a square foot, etc. 312. The area of a surface is its contents reckoned in square units. a sqmre. A Bectangle. 138 WEIGHTS AND MEASURES. ILLUSTRATION. To find the Area of a Rectangle. 313. Illusteative Example. If the, length of a rec- tangle is 4 inches and its breadth is 3 inches, how many square inches does it contain ? Explanation. — A rectangle whicli is 4 in. long and 1 in. wide will contain 4 square inches, and a rectangle of the same length and 3 in. wide must contain 3 times 4, or 12 square inches. (See illustration.) In the same way it can be shown that the area of any rectangle is found hj multi- plying the nvmiber of units in the length by the number of like units in the breadth. This is expressed, for brevity, as multiplying the length by the breadth. 314. Oral Exercises. a. How many square inches are there in a rectangle 8 in. long and 5 in. wide ? 11 in. long and 10 in. wide ? 12 in. (1 foot) long and 12 in. (1 foot) wide ? Then how many square inches are there in a square foot ? b. How many square feet are there in a rectangle 8 ft. long and 7 ft. wide ? How many square feet are there in a square whose sides are each 3 ft. (1 yard) long ? c. How would you find the number of square yards in » square whose sides are each 5| yards or 1 rod long ? 315. In measuring surface we employ the square mile (sq. m.), acre (A.), square rod (sq. rd.), square yard (sq. yd.), square foot (sq. ft.), and square inch (sq. in.). These are the units of Square Measure. 144 square inches ^ 1 square foot. 9 square feet = i square yard. 30J square yards, or 272J square feet = 1 square rod. 160 square rods = i acre. 640 acres = 1 square mile. MEASURES OF VOLUME. 139 MEASURES OF VOLUME. 316. A rectangular solid is a solid bounded by six rectangles. A Rectangular Solid. 317. The rectangles are the faces of the solid, and, together, make its surface. The bounding lines of the solid are its edges. 318. A cube is a rectangular solid bounded by six equal squares. A cube each of whose edges is 1 inch long is a cubic inch. A cube each of whose edges is 1 foot long is a cubic foot, etc. 319. The volume of a solid is its contents reckoned in cubic units. A Cube. To find the Volume of a Rectangular Solid. 320. Illustrative Example. What is the volume of a block of marble 4 ft. long, 2 ft. wide, and 3 ft. thick ? Explanation. — If the block fe 4 feet long and 2 feet wide, its lower base must contain 4X2, or 8 square feet (Art. 313). A solid 1 foot thick upon these 8 square feet will contain 8 cubic feet, and a solid 3 feet thick will contain 3 times 8 or 24 cubic feet. In the same way it can be shown that the voliune of any rectangular solid is found by multiplying the number of units in the length hy the nwmber of like units in the breadth, and this product by the number of like units in the thickness. This is expressed, for brevity, as multiplying together the length, breadth, and thickness. ILLUSTRATION. ^ / A y y y y / y y y y y y / 140 WEIGHTS AND MEASURES. 321, Oral Exercises. a. How would you find the number of cubic inches in a cube 12 inches long, 12 inches wide, and 12 inches thick, or in 1 cubic foot ? b. How many cubic feet in a cube 3 feet long, 3 feet wide, and 3 feet thick, or in 1 cubic yard ? 322. In measuring solids we employ the cuhic yard (cu. yd.), cubic foot (cu. ft.), and cubic inch (cu. in.). These are the units of Cubic Measure. 1728 cubic inches = 1 cubic foot. 27 cubic feet = 1 cubic yard. 128 cubic feet = 1 cord (cd.), used m measurimg wood. »8 FEET LONC"'W^'r^ Wood is generally cut for the market into sticks 4 feet long, and laid in piles, so that the length of the sticks becomes the width of the pile. A pile 4 feet wide, 4 feet high, and 8 feet long, contains 1 cord. One eighth of a cord is called 1 cord foot. 1 cord foot contaiIl^ 16 cubic feet. (See illustration above.) c. How many cords are there in a pile of wood 4 feet wide, 4 feet high, and 20 feet long ? 32 feet long ? 90 feet long ? d. What is the cost of a pile of wood 2 feet wide, 4 feet high, and 10 feet long, at $ 8 a cord ? e. What must I pay for 3 cords of hard wood at $ 9.50 a cord, and J of a cord of pine at $ 5 a cord ? MEASURES OF WEIGHT. 141 MEASURES OF WEIGHT. 323. In weigMng grocer- ies and most other common goods, we use the ton (T.), pound (lb.), and ounce (oz.). These are the units of Avoirdupois Weight. 16 oz. = 1 lb. 2000 lb. = 1 T. Note I. In weighing some articles, as iron and coal at the mines, and goods on which duties are paid at the United States custom-houses, the long ton of 2240 lbs. is used. In this weight 28 lb. =1 quarter (qtr.). 4 ctr. =1 hundredweiglit (cwt). 20 cwt. = 1 T. 324. In weighing silver, gold, precious stones, etc., we use the pound, ounce, penny- weight (pwt.), and grain (gr.). ' These are the units of Troy "Weight. 24 gr. =1 pwt. 20 pwt. = 1 oz. 12 oz. = 1 lb. 325. Comparison of Weights. 175 lb. Troy = 144 lb. av. 175 oz. " = 192 oz. av. 7000 gr. " =1 lb. av. Which is heavier, a pound Troy or a pound avoirdupois ? an ounce Troy or an ounce avoirdupois ? Note II. For apothecaries' weight, see Appendix, page 307. Note III. The standard unit of weight is the Teoy pound. From this the other units of weight are derived. Note IV. A cubic foot of water weighs 62Jlbs., or 1000 oz. avoirdupois. 326. Oral Exercises. a. How many ounces in 1 lb. avoirdupois ? in 2 lb. 1 oz. ? b. How many ounces in 1 lb. Troy ? in 4 lb. 6 oz. ? c. Change 60 gr. to pennyweights ; 90 pwt. to ounces. d. How many ounces in 3 pounds of silver ? e. What is the value of a gold chain weighing 1\ ounces at 90/ a pwt.? /. At 80/ a pound for camphor, what is the cost of an ounce ? g. How many more pounds in a long ton than in a common ton? 142 WEIGHTS AND MEASURES. MEASURES OF CAPACITY. 327. In measuring liquids we use the gallon (gal.), quart (qt.), pint (pt.), and gill (gi.). These are the units of Liquid Measure. 4 gi. = 1 pt. 2 pt. = 1 qt. 4 qt. = 1 gal. Note I. A pint of water weighs about a pound avoirdupois. Note II. The standard unit for liquid measure is the gallon. Note III. The standard unit for dry measure is the bushel. Note IV. In buying and selling grain and many other kinds of produce, the bushel is reckoned at a certain number of pounds. Thus, potatoes have 60 paunds to a bushel and com has 66 pounds to a bushel. 329. Comparison of Liquid and Dry Measures. 328. In measuring dry articles, as grain, small fruits, seeds, etc., we use the iushel (bu.), peck (pk.), quart, pint, and gill. These are the units of Dry Measure. 4 gi. = 1 pt. 2 pt. = 1 qt. 8 qt. = 1 pk. 4 pk. = 1 bu. Liquid Measure. Cu. In. 1 quart = 57J 1 gallon = 231 Dry Measure. Cu. In. 1 quart = 67i 1 bushel = 2150.42 330. Oral Exercises. a. How many half-pint tumhlers can a person fill with 1 gallon of jelly ? b. How many quart measures can be filled with 1 bushel of cranberries ? c. What does a vender receive for 1 peck of peanuts which he sells at 5 cents a pint ? d. Which is larger, 1 quart of milk, or 1 quart of berries ? e. How many pints in a bushel ? How many gills in a gallon ? /. I bought 3 bushels of pears for $ 2 a bushel, and sold them at 10 cents a quart ; what did I gain by the sale ? A jCixcle. CinaULAR AND ANGULAR MEASURES. 143 CIRCULAR AND ANGULAR MEASURES. 331. A plane surface bounded by a line every point of which is eq^ually distant from a point within, called the centre, is a circle. 332. The bounding Line of a circle is the circumference. Any part of the circumference is an arc. 333. The circumference of a circle is divided into 360 equal arcs, called degrees (°), each degree into 60 minutes ('), and each minute into 60 seconds ("). These are the units of Circular Measure. 60 seconds = 1 minute. 60 minutes = 1 degree. 360 degrees = 1 circumference. Note. As the circumference of every circle has 360 degrees, the length of the degree differs in different circles. 334. A degree of the circumference of the earth at the equator is about 69.16 common miles in length. 335. A minute of the circumference of the earth at the equator is a geographical or nautical mile, and equals about 1.15 common miles. 336. If the centre of a circle is placed at the vertex of an angle, the arc included between the sides is the meas- wce of the angle. Thus, if the arc contains 30 degrees, the angle is called an angle of 30 degrees. (See illustration.) Note. An angle of one degree has always the same size, but the arc that measures it differs in different circles. 144 WEIGHTS AND MEASURES. 337. Oral Exercises. a. How many degrees are there in a semi-circumference? in I of a circumference, or a quadrant ? in J- of a circumference, or a sextant ? b. The torrid zone is 47° wide. How would you find its width in nautical miles ? in common miles ? c. Through how many degrees does the hour hand of a clock move in 3 hours ? in 1 hour ? in 2 hours ? d. Through how many degrees does the minute hand of a clock move in 5 minutes of time ? in 1 minute ? in a quarter of an hour ? in half an hour ? e. How many degrees in a right angle ? /. How long does it take the hour hand of a clock to move through a right angle ? How long does it take the minute hand ? g. The hour and minute hand of a clock form an angle of how many degrees at 3 o'clock ? at 4 o'clock ? at 10 o'clock ? at 7 o'clock ? at 12 o'clock ? MEASURES OF TIME. 338. In measuring time we employ the centv/ry, year, month (mo.), week (w.), day (d.), hour (h.), minute (m.), and second (s.). These are the units of Time Measure. 60 seconds - 1 minute. 60 minutes = 1 hour. 24 hours = 1 day. 7 days = 1 week. 365 days or 52 weeks 1 day = 1 common year (c 366 days = 1 leap year (1. y.). 100 years = 1 century (C). MEASURES OF TIME. 145 339. Any year is a leap-year when the number denot- ing the year is divisible by Jf, and not by 100, and when it is divisible by JfiO. (See Appendix, page 307.) Which of the following named years are leap-years : 1878? 1892? 1888? 1900? 2000? 1864? 1880? 340. The year begins with the first of January, and is divided into four seasons of three months each, as follows : The winter months are December, January, and February., The spring months are March, April, and May. The summer months are June, July, and August. The autumn months are September, October, and November. 341. April, June, September, and November have 30 days each. February has 28 days, in leap year 29. The other months have 31 days each. 342. Oral Sxercises. 3. What date is three months from Jan. 5 ? July 10 ? b. What date is 6 months from May 2 ? Feb. 11 ? Nov. 1 ? c. What months contain 30 days each ? 31 days each ? d. At 10 cents an hour for 6 hours of every working day, how much can you earn in 4 weeks ? e. 8 years and 9 months are how many months ? /. How many years are there in 100 mo. ? in 200 mo. ? g. What date is 30 days from May 5 ? from Apr. 4 ? h. How many days from May 3 to June 5 ? IVIiscellaneous Measures. 343. Nnnabers. 344. Paper. 12 units = 1 dozen. 24 sheets = 1 quire. 12 dozen 12 gross 20 units = 1 gross. = 1 great gross. = 1 score. 20 quires 2 reams 5 bundles = 1 ream. = 1 bundle = 1 bale. Note. For other measures sometimes used, see Appendix, page 307. 146 COMPOUND NUMBERS. SEOTIOIST XII. COMPOUND NUMBERS. 345. In 2 feet 7 inclies, how many inches ? Ans. 31 inches. The number 31 inches expresses a quantity by reference to a single integral unit. Such a number is a simple number. 346. The number 2 feet 7 inches expresses a quantity by reference to two units of different denominations. A number expressing a quantity by reference to two or more units of different denominations is a compound number. The compound number 2 feet 7 inches expresses the same quantity that the simple number 31 inches does. 347. When the name of the units is given, the number is a denominate number. Thus, 31 inches and 2 feet 7 inches are both denominate numbers. 348. When the name of the unit is not given, the num- ber is a general number. Thus, 31 is a general number. Note. Denominate numbers are sometimes called concrete numbers, and general numbers are called abstract numbers. Name a simple number ; a compound number ; a denominate num- ber ; a general number. Is 5 feet 2 inches a denominate or general num.ber ? a simple or compound number ? Is 25 a denominate or a general number ? a simple or a compound number ? 349. Written Exercises. Write from memory the table for Long Measure, Square Measure, Cubic Measure, Liquid Measure, Dry Measure, Avoirdupois Weight, Troy Weight, Circular or Angular Measure, Numbers, Paper. reduction: 147 reduction. To change a Compound Number to a Simple Number. 350. Illustrative Example. Change 2 bu. 3 pk. 4qt.' to quarts. Explanation. — Since in 1 bushel there are WRITTEN WORK. 4 peoks, in 2 bushels there are 2 times 4, or 8 2 bu. 3 pk. 4 qt. pecks, which with 3 pecks added are 11 pecks. 4 Since in 1 peck there are 8 quarts, in 11 — pecks there are 11 times 8 quarts, etc. 11 pk. ^ g Rule. 7^ . 351. To change a compound number 92 qt. ^m. , ? ^ f n . to a simple number of a lower denomi- nation : Multiply the number of the highest denomination hy the number of units it takes of the next lower denomination to make one of that higher, and to the 'product add the given number of the next lower denomination. Multiply this sum in like manner, and so proceed till the given number is changed to units of the required denomination. 352. Examples for the Slate. 1. Change 4 T. 350 lb. 8 oz. to ounces. 2. What is the value of 2 lb. 8 oz. of gold at $ 20 an ounce ? 3. Change 3 rd. 4 yd. 1 ft. to feet. 4. What will it cost to fence both sides of a road 26 rd. 6 ft. long, at 22/ afoot? 5. How many square feet are there in an acre ? 6. Change 3 sq. m. 35 A. to acres. 7. In a cubic yard, how many cubic inches ? 8. What shall I receive for 25 gal. 3 qt. of milk at 7 cents a quart ? 9. Mr. Russell sold 4 bu. Ipk. 2 qt. of cherries at 12 cents a qt. ; what did he receive for them ? 148 COMPOUND NUMBERS. 10. If the pulse beats 80 times in 1 minute, how many- times will it heat in a common year ? 11. If a child sleeps J of his time, how many hours will he sleep in 5 years, allowing for 1 leap year ? 12. How many minutes were there in the first century ? 13. The Tropic of Cancer is 23° 30' north of the equator. What is the distance in geographical miles ? in common miles ? 353. Changing numbers to numbers of lower denomi- nations is called rednctioa descending. For otlier examples in reduction descending, see page 171. To change a Simple Number to a Compound Number. 364. Illustrative Example. Change 4354 feet to rods, yards, etc. Explanation. — Since 3 ft. make a yard, in 4354 ft. there are as o) 4o54 — 1 it. Rem. many yards as there are 3's in 50 1451 4354, which are 1451, and 1 ft. re- 2 2 mams. Since 5i- yards make a rod, in 11) 2902 - I yd. = 4iyd. Rem. ^^g^ ^^^^ ^^^^^ ^^^ ^ ^^^^ ^^ 263 as there are times 5^ in 1451, Ans. 263 rd. 4| yd. 1 ft, or 263 rd. 4 yd. 2 ft. 6 in. which are 263, and ^ yd., or 4J yd., remain, etc. Rule. 355. To change a simple number to a compound num- ber of higher denominations : Divide the given number by the number of units it takes of its denomination to make one of the Tiext higher. Set a^ide the remainder, and divide, as he/ore, the quotient thus obtained; and so proceed till the required denomination is reached. The la^st quotient unth the several remainders is the number sought. REDUCTION. 149 356. Examples for the Slate. Change to compound numbers : (14.) 3268 yards. (20.) 9328 lb. of soap. (15.) 4687 feet. (21.) 19547 oz. of salt. (16.) 9687 sq. rd. (22.) 9321 pwt. of silver. (17.) 5692 sq. yd. (23.) 2089 gr. of gold. (18.) 4791 sq. in. (24.) 5087 qt. of berries. (19.) 63684" (25.) 1127793 minutes. 26. What will 20 old silver dollars weigh in oz. pwt. etc., each doUar weighing 412^ grains ? 27. The trade-dollar weighs 420 grains. What will 20 trade-dollars weigh ? 28. How many miles is it through the earth from pole to pole, the distance being 41707308 feet ? 29. In a certain pasture 973 quarts of berries were picked in one week. How many bushels were picked ? 357. Changing numbers to numbers of higher denomi- nations is called reduction ascending: For other examples in reduction ascending, see page 171. REDUCTION" OF DENOMINATE FRACTIONS. To change a Denominate Fraction to Integers of Lower Denominations. 358. Illustrative Example I. Change | rd. to yards, feet, and inches. WRITTEN WORK. Explanation. — We iirst change | of a f of ^ yd. = 4/j yd. rod to yards, and have 4^-^ yards for the -^ of I? ft =1^ ft result. We then change -^^ of a yard to "^ ^ ' ' feet, and have If feet for the result, f of I of 12 in. = 9 in. a foot is 9 inches. Ans. 4 yd. 1 ft. 9 in. Atix. 4 yd. 1 ft. 9 in. 150 COMPOUND NUMBERS. 359. Illustrative Example II. Change 0.62 rd. to yards and feet. WRITTEN WORK. Explanation. — We first change 0.62 rd. or 0.62 rd. 0.62 of a rod to yards, and have 5 1 5.5 3.41 yards. We next change 0.41 „-l„ o-\r\ of a yard to feet, and have 1.2:j o-i o-i n feet. Ans. 3 yd. 1.23 ft. 3.41 yd. 3.410 yd. 3 3 360. From the preceding operations we derive the fol- 1.23 ft. 1.23 ft. Am. 3 yd. 1.23 ft. ' lowing Rule. To change a fraction of one denomination to integers of lower denominations : Change the fraction, as far as pos- sible, to an integer of the next lower denomination. If a fraction occurs in the result, proceed with it as with the first fraction, and so continue as far as required. 361. Examples for the Slate. Change to units of lower denominations : (30.) f of a rod. (36.) -/^ of a cu. yard. (31.) fT-ofamile. (37.) 0.15625 of a gal. (32.) ^^ of a sq. mile. (38.) 0.6 of a bushel. (33.) ^ of 1 of an acre. (39.) 1 of a degree. (34.) 1 of a ton. (40.) f of a c. year. (35.) 0.875 of a lb. Troy. (41.) 0.75 of a 1. year. 42. At 20/ a foot, what is the cost of f of an acre of land ? 43. In f of an ounce of Dover's powder, how many doses of 5 grains each ? 44. How many planks 8 inches wide will cover the roadway of a bridge | of a mile long, each plank reaching from side to side? REDUCTION. 151 To change Integers of Lower Denominatdons to a Fraction of a Higher. 362. Illustrative Examples. (I.) Change 5 oz. 6 pwt. 16 gr. to the fraction of a pound. (II.) Change 2 pk. 6 qt. to the decimal of a bushel. WRITTEN WORK. (I.) 16 gr. = Jl pwt. = § pwt. (11.), 8) 6 qt. 4) 2.76 pk. 0.6875 bu. ^m Explanation (I.). — Since 24 grains make a pennyweight, 16 gr. are •^ pwt., or -f pwt., which, added to the 6 pwt. given, are 6f pwt. 6f pwt. = ^ pwt. Since 20 pwt. make an ounce, ^ pwt. equals ^ as large a part of an ounce, or ^ oz., etc. Explanation (II.). — Since 8 qts. make a peck, 6 qts. are equal to 0.75 pk., which, added to the 2 pecks given, are 2.75 pecks. Since 4 pks. make a bushel, 2.75 pk. are equal to 0.6875 bu. Ans. 0.6875 bu. 363. From the preceding operations we derive the fol- lowing „ , ^ Rule. To change integers of lower denominations to a fraction of a higher denomination : Change the number of the low- est given denomination to a fraction of the next higher. Unite this fraction with the number of that higher de- nomination. Change, in like manner, the number thus formed, and so continue as far as required. 45. Change 1 qt. pt. 1 gi. to the fraction of a gallon. 46. Change 242 rd. 2 yd. to the fraction of a mile. 47. What part of a rod is 4 yd. ft. 4^ in. ? 48. What part of an acre is 81 sq. rd. 24 sq. ft ? 49. What part of a cu. yd. is 13 cu. ft. 864 cu. in. ? 50. What part of a year are the three winter months ? 51. Change to the decimal of a mile 87 rd. 10' ft. 152 COMPOUND NUMBERS. 52. Change to the decimal of an acre 135 sq. rd. 64 sq. ft. 53. Regarding a year as 12 months of 30 days each, what decimal of a year is 6 mo. 18 d. ? 8 mo. 24 d. ? 5 mo. 27 d. ? For other examples in reduction of denominate fractions, see page 171. ADDITION. 364. The operations upon compound numbers are simi- lar to those upon simple numbers, the principal difference being that in operations upon compound numbers we use irregular scales, instead of the scale of tens. No special rules, therefore, are necessary for addition, subtraction, mul- tiplication, and division. 365. Illustrative Examples. (I.) What is the sum of 11° 4' 58", 37° 30' 27", and 27° 24' 54" ? (I.) Explwnation. — (I.) We write ttese numbers WRITTEN WORK. SO that units of the same denomination shall 11 ° 4' 58" ^^ expressed in the same column. Adding the ^7° ^0' 97" seconds, we have 139". Dividing 139" by 60 27° 24' 54" ^^°"= ■^'^' ^^ ^""^^ ^' ^^"- ^^ ^^^ ^^' ^^" under the line in the seconds' place. Adding the Am. 76° 0' 19" 2' with the minutes of the given numbers, and dividing the sum by 60 (60' = 1°), we have 1° 0'. We write 0' under the line in the minutes' place. Adding the 1° with the degrees of the given number, we have 76°. Ans. 76° C 19". (III.) (11.) bu. pk. qt. 85 3 7 9 2 5 98 6 2 3 1 m. rd. yd. ft. 3 192 4 2 316 1 5 76 4 2 Ans. 9 m. 265 rd. 3^ yd. 2 ft or 9 m. 265rd. 4 yd. Oft. Ans. 196 bu. 2 pk. 3 qt. Note. Change any denominate fraction which occurs in an answer, or in an example, to units Gf the lower denominations given. (See examples 56, 57, and 58.) SUBTRACTION. 153 366. Examples for the Slate. 54. What are the contents of three barrels which contain respectively, 45 gal. 2 qt., 42 gal. 3 qt., and 47 gal. 1 qt. ? 55. How much land in four lots which contain as follows : 7 A. 83 sq. rd. 31 sq. ft., 15 A. 146 sq. rd., 22 A. 52 sq. rd. 13 sq. ft., and 5 A. 9 sq. rd. ? 56. What is the length of three roads measuring respec- tively 15 m. 87 rd., 28 m. 40 rd., and 35^ miles ? 57. To 8° 17' 32" add 4.735° and f . 58. Add together 7 d. 6 h., 6f d., and 0.375 of a week. 367. Illustrative Example II. To -^^ of a gaUon add f of a quart. Explanation. — That these fractions WRITTEN WORK. , jjj,, iC^i, may be added they must first be ex- ■i^ gal. = ^5^ of 4 qt. = 1^ qt. pressed in the same denomination, f qt. They may be so expressed by chang- Am. Yl qt. "^g A gal- to liarts, etc. 59. Add § of a quart to ^^ of a bushel. 60. Add 45| rods to ^ of a mile. 61. Add 64 J pounds to J of a ton. Perform such examples in exercises 206-208, page 171, as the teacher may indicate. SUBTRACTION. 368. Illustrative Example. What is the difference between 5 rd. 3 yd. 1 ft. and 1 rd. 4 yd. 2 ft. ? Expkmation. — We write these num- bers as in simple subtraction, and subtract first the 2 feet of the subtrahend. As we have but 1 foot in the minuend, we can- not now take 2 feet away. So we change 1 of the 3 yards (leaving 2 yards) to feet. This 1 yard equals 3 feet. We add the WRITTEN WORK. 5rcL 3 yd. 1ft. 14 2 Ans. 3rd. 3jyd.2ft. or 3rd. 4yd. Oft. 154 COMPOUND NUMBERS. 3 feet to the 1 foot, making 4 feet. Subtracting 2 feet from 4 feet We have 2 feet left, which we write as part of the remainder. As we have but 2 yards left ia the minuend, we cannot now take 4 yards away, so we change 1 of the 5 rods to yards. This equals 5j yards, which, added to 2 yards, make 7^ yards. Subtracting 4 yards from 7^ yards, we have 3^ yards left, etc. 369. Examples for the Slate. (62.) (63.) (64.) bu. pk. qt. oz. pwt. - gr. o / // 5 3 2 6 10 13 35 47 28 2 17 3 15 18 19 64 48 65. 1 m. 80 rd. 2 yd. less 315 rd. 3 yd. equals what ? 66. What is the difference between 5 ft. 6 in. and f rd. ? 67. A man who had f of a square mile of woodland sold 5^ square rods. How much had he left ? 68. A man having f of a pound of silver ore, gave away 3J pennyweights. How much had he left ? 69. What is the difference between 0.378 of a day and 44.55 of a minute ? 70. Cape Horn is in 55° 58' 4" south latitude, and the Cape of Good Hope is in 34° 22' south latitude. Which is farther south, and how much ? The difference of latitude between places on opposite sides of the equator is found by adding the latitudes. The difference of longitude between places on opposite sides of the first meridian is found by adding the longi- tudes. If their sum exceeds 180°, the difference of longitude equals 360° minus that sum. For a table of longitudes, see page 159. What is the difference of longitude between 71. Albany and Chicago ? 73. Kome and New York ? 72. Berlin and Paris ? 74. San Francisco and Calcutta ? 76. What is the difference in latitude between Philadelphia 39° 57' north latitude, and Buenos Ayres 34° 3' south latitude ? SXIBTBAOTION. 155 To find the Number of Tears, Months, and Days from one Date to another. Note. The following method of finding the time is generally used in computing interest. 370. Illusteative Example I. "What is the time in years, months, and days from Jan. 11, 1877, to May 5, 1881? Exclamation. — Frem Jan. 11, 1877, to Jan. 11, 1881, is 4 years ; from Jan. 11, 1881, to April 11, 1881, is 3 months ; from April 11 to April 30 is 19 days, and from April 30 to May 5 is 5 days more. Ans. 4 y. 3 m. 24 d. Rule. 371. To find the difference in time between two dates : First find the number of entire years hetvjeen the two dates, then the number of calendar months remaining, and lastly, the remaining days. 372. Oral Exercises. a. How many years, months, and days are there from Feb. 3, 1875, to Oct. 17, 1878 ? b. How many years, months, and days are there from Sept. 25, 1874, to Jan. 4, 1882 ? c. Mozart was born Jan. 27, 1756, and died Dec. 5, 1791 ; at what age did he die ? d. Goethe died March 22, 1832, and Bryant was born Nov. 3, 1794 ; what was Bryant's age when Goethe died ? 373. Illusteative Example II. How many days are there from Nov. 12, 1875, to March 10, 1876 ? Explanation. — There are 18 days remaining in November, 31 days in December, 31 in January, 29 in February, and 10 in March. 18 + 31 + 31 + 29 + 10 = 119. Ans. 119 days. e. How many days from March 7 to July 1, 1878 ? /. How many days from Oct. 9, 1876, to Feb. 11, 1877 ? g. How many days from January 15 to August 7, 1875 ? 156 COMPOUND NUMBERS. 374. A Table showiHg the Number of Days From any Day of To the corresponding Day of the following Jan. Feb. Mar. Apr. May. June. July. Aug. Sept, Oct Nov. Dec. January . . 365 31 59 90 120 151 181 212 243 273 304 334 February. 334 365 28 69 89 120 150 181 212 242 273 303 March .... 306 337 365 31 61 92 122 153 184 214 245 275 April 275 306 334 365 30 61 91 122 153 183 214 244 May 245 276 304 335 365 31 61 92 123 153 184 214 June 214 245 273 304 334 365 30 61 92 122 153 183 July 184 215 243 274 304 335 365 31 62 92 123 153 August . . . 153 184 212 243 273 304 334 365 31 61 92 122 September 122 153 181 212 242 273 303 334 365 30 61 91 October .. 92 123 151 182 212 243 273 304 335 365 31 61 November 61 92 120 151 181 212 242 273 304 334 365 30 December 31 62 90 121 151 182 212 243 274 304 335 365 Note. In leap years, if the last day of February is included in the time, a day must be added to the number obtained from the table. Find from the table above the number of days h. From April 19 to June 19. j. From Dec. 6 to Feb. 5. i. From Jan. 1 to March 4. k. From Oct. 12 to Feb. 15. Perform such examples of exercises 209 and 210, page 171, as the teacher may indicate. MULTIPLICATION. 375. Illustkative Example. How much land is there in 4 gardens, each containing 13 sq. rd. 72 sq. ft. ? Explanation. — Multiplying 72 sq. ft. by 4, we have 288 sq. ft. for a product, which equals 1 sq. rd. and 15f sq. ft. We write the 15| sq. ft. and carry the 1 sq. rd. to the square rods in the product. 13 sq. rd. multiplied by 4 are 52 sq. rd., which, with the 1 sq. rd, Ans. 53 sq. rd. 15f sq. ft. WRITTEN WORK. 13sq. rd. 72 sq.ft. 4 53 sq. rd. 15§ sq. ft Ans. carried, are 53 sq. rd. DIVISION. 157 376. Examples for the Slate. 76. How much syrup will 7 jars contain if each, jar holds 1 pt. 3 gi. ? 77. How much wheat is contained in 5 bins if each bin con- tains 7 bu. 4 pk. 3 qt. ? 78. If a car runs 18 m. 149 rd. in half an hour, how far will it run in 7 hours ? Perform such examples in exercises 211 and 212, page 171, as the teacher may indicate. DIVISION". 377. Illusteative Example. Divide 47° 18' 36" by 11. WRITTEN WORK. Explanation. — T)viid.mg 47° by 11, we ^^^ .„o u jj, nf,„ have 4° for a quotient, with a remaindei of ' , 3°. We write the 4° under the line, and 4° 18' 3^" Ans. change the 3° remaining to minutes, ob- taining 180'. Adding 180' to the 18' in the dividend, we have 198'. Dividing 198' by 11, we have 18' for a quotient, etc. 378. Examples for the Slate. 79. A farmer brought 5 bu. 3 pk. of corn to mill. How much corn did the miller take as toll, if he took -^ part ? 80. If 65 A. 125 sq. rd. be divided into 50 house-lots, what is the size of each ? 81. How long will it take to travel 1 mile, at the rate of 75 miles in 10 h. 18 min. ? 82. Among how many men may 624 gal. 3 qt. be divided, that each man may receive 12 gal. 3 qt. ? Note. Change both nmnhers to quarts before dividing. 83. How many bins, each containing 5 bu. 3 pk., will be re- quired to hold 885 bu. 2 pk. of potatoes ? 84. If a man walks 3 m. 264 rd. in one hour, how long wiU it take him to walk 23 m. 273 rd. ? Perform such examples in exercises 213 to 215, page 171, as the teacher may indicate. 15S COMPOUND NUMBERS. LONGITUDE AND TIME.* 379. As the earth turns upon its axis once in 24 hours, it follows that ^ of 360°, or 15° of longitude, must pass under the sun in 1 hour, and -^ of 15°, or 15', must pass under the sun in 1 min. of time, and -^ of 15', or 15", must pass under the sun in 1 sec. of time. Hence the following TABLE. A difference of 15° ) (A difference of 1 hour in longitude ) ™^ es | ^ time. A difference of 15' ) (A difference of 1 minnte in longitude ) raa^aa. ^ ^ ^:^^ A difference of 15" ) (A difference of 1 second in longitude \ ^^^^^ \ m time. 380. From the table above we derive the foUowing Rule. To find the difference of longitude between any two places when the difference of time is known : Multiply the difference of time between the two places, expressed in hours, minutes, and seconds, hy 15. The product will express the number of degrees, minutes, and seconds required. Note. As the earth turns from west to east, midday occurs sooner in places east and later in places west of any given point. Hence the time shown by a clock is later in all places east, and earlier in all places west, of any given point than it is at that point. 381. Ezamples for the Slate. What is the difference in longitude between two places, the difference in their time being (85.) 4 h. 17 m. ? (87.) 6 h. 12 m. 10 s. ? (86.) 2 h. 9 m.? (88.) 1 h. 5 m. 26 s. ? In what longitude from Greenwich is a place whose time compared with that of Greenwich is (89.) 3 hours earlier ? (91.) 1 hour 12 minutes later ? (90.) 6 minutes later ? (92.) 4 hours 8 minutes earlier ? * See Appendix, page 308. LONGITUDE AND TIME. 159 93. The time in St. Louis is 1 h. 5 min. \% s. slower than in New York; what is the difference in longitude between these places, and what is the longitude of St. Louis, that of New York being 74° 0' 3" west ? 94. A and B sailed together from San Francisco. A kept his watch by San Francisco time, and B set his by the sun every day. After 10 days, A's watch was 4 hours 39 minutes faster than B's : in what longitude were they then, the longi- tude of San Francisco being 122° 26' 15" west ? 382. From Art. 379 we may also derive the following Rule. To find the difference in time between any two places when the difference in longitude is known : Divide the difference in longitude, expressed in degrees, minutes, and seconds, hy 15. The quotient will express the nuniber of hours, minutes, and seconds required. 383.- The names of a few important cities are given below, with the longitude of each from Greenwich. Places. Longltadea. Places. Long''tudes. Albany Boston Berlin Calcutta Chicago London Montreal 73° 44' 53" W. 71° 3' 30" W. 13° 23' 43" E. 88° 19' 2" E. 87° 35' W. 0° 5' 38" W. 73° 25' "W. New Orleans New York ... Paris.... Philadelphia Rome (Italy) San Francisco Washington 90° 7' W. 74° 0' S'W. 2° 20' 22" E. 75° 10' W. 12° 27' 14" E. 122° 26' 15" W. 77° 2' 48" W. 384. Using the longitudes given above, find the difference in time between 95. Albany and Boston. 97. Montreal and New Orleans. 96. London and New York. 98. Philadelphia and Chicago. 160 COMPOUND NffMB^SS. When it ia noon in Washington, what is the time 99. In Philadelphia ? 103. In Kome ? 100. In New Orleans ? 104. In Berlin ? 101. In Chicago ? 105. In Paris ? 102. In San Prancisco ? 106. In Calcutta ? MENSURATION OF SURFACES AND SOLIDS. 385. Oral Exercises. a. How many square feet are there in the top of a table that is 7 feet long and 3 feet wide ? (Art. 313.) b. How many square yards are there in a concrete walk 16 J feet long and 4 feet wide ? c. How do you find the area of any rectangle or square ? 386. Prom Art. 313, it follows that when the area and one dimension of a rectangle or a square ara given, the other dimension is found by dividing the number of units of area by the number of units in the given difnension. d. There are 15 square yards in a piece of carpeting 5 yards long ; what is its width ? e. What must he the length of a walk 2^ feet wide to con- tain 17 square feet ? /. How many cubic feet will a box contain that measures on the inside 7 feet in length, 3 feet in width, and 2 feet in height ? (Art. 320.) g. How do you find the volume of any rectangular solid ? 387. From Art. 320, it follows that when the volume and two dimensions of a rectangular solid are given, the other dimension is found by dividing the number of units of volume by the product of the number of units in each of two given dimensions. SURFACES AND SOLIDS 161 h. What must be the deptli of a cistern 6 feet long and 4 feet wide to contain 80 cubic feet ? i. What must be the height of a room 6 yards long and 5 yards wide to contain 90 cubic yards ? j. A box 4 inches square must be how deep to contain a quart dry measure ? Examples for the Slate. 388. Squares and Rectangles. 107. How many yards of carpeting 1 yard wide' will cover a floor 17 feet long and 15 feet wide ? 108. How many yards of carpeting 27 inches wide will be required to cover the same floor ? 109. What must I pay for laying a sidewalk 5 rods long and 5 feet wide at 90/ per square yard ? 110. If one side of a square field is 4 rd. 8 ft. long, how many square feet are there in the field ? 111. What must I pay for a building lot in St. Louis, 90 feet long and 2 rods wide, at $ 1.75 per square foot ? 112. My building lot contains 1 quarter of an acre, is rec- tangular, and measures on the street 90 feet, how far back does it extend ? 113. What must I pay for a quarter of an acre of land at 20/ per square foot ? 114. How many more square rods are there in a field 42 rods square than in a 10-acre lot ? 115. My neighbor's garden is 2 rods square, and mine con- tains 2 square rods ; what is their difference in size ? 116. How many acres were covered by the main Centennial building in Philadelphia, which was 1880 feet long and 464 feet wide ? 117. What would it cost to make the floor of the above- named building, the boards costing % 37 per thousand feet, square measure, and the work costing 25/ per hundred, square measure ? 162 COMPOUND NUMBERS. w 6 7 18 19 30 31 5 8 17 20 29 32 4 9 16 21 28 33 3 10 15 22 27 34 2 11 14 23 26 35 1 12 13 24 25 36 389. Government Lands. Before being brought into market, the public lands of the United States are usually divided by parallels and 1^ meridians into townships, each being as nearly as pos- sible six miles square. Each township is divided in the same way into 36 sections, and each section into 4 quar- ter-sections. The township and sections are numbered and referred to special me- ridians and base lines, so as A TownsUp. to be easily designated and pointed out on government maps. a. How many square miles in a township ? in a section ? b. How many acres in a section ? in a quarter-section ? 118. What must I pay for the IST. W. quarter of section No. 9 of township 5 Korth, 20 West, meridian Michigan, at $ 2.60 an acre ? 390. RBCTANauLAR Solids. 119. How many cubic inches are there in a beam 4 ft. 5 in. long, 8 in. wide, and 4 in. thick ? 120. What is the weight of a block of Quincy granite 15 ft. long, \\ ft. wide, and 6 in. high, if 1 cubic foot weighs 1G5 pounds ? 121. How high must a block of freestone be to contain 84 cubic feet, if its length is 4^ ft. and its width 3 ft. ? 122. If a bin contains llf cubic yards, and its height is 2 ft., what is the area of its base ? 123. There being 112^ cubic feet in a shaft of marble which is 27 in. square at each end, what is its length ? WOOD MEASURE. 163 391. "Wood Measure. 124. If a pile of wood is 3 ft. 8 in. high and 4 ft. wide, how long must it be to contain 1 cord ? 125. At $ 6 a cord, what is the cost of a pile of wood 33 ft. long, 8 ft. 10 in. high, and 4 ft. wide ? 126. If wood is cut in lengths of 3|- ft. and piled to a height of 4 ft., how long must the pile be to contain 1 cord ? 127. On measuring what I bought for a cord of wood, I found it 8 feet long, 4 feet wide, and only 3 feet 8 inches high. At $ 6 a cord, how much money should be deducted from the original price ? 392. Lumber and Boards. Sawed timber and boards, when 1 inch or less in thickness, are generally reckoned by the square foot of surface measure. When more than 1 inch in thickness, they are reckoned in proportion to their thickness. Thus, 2000 scL. ft., 1 incii or les3 in thickness, == 2000 ft., board measure, 2000 sq. ft., H inches thick, = 3000 ft., hoard measure, 2000 sq. ft., 2 inches thick, = 4000 ft., board measure, and so on. 128. How many feet of boards | of an inch thick will be required to make a fence 2 rods long and 3 feet high ? 129. How many feet, board measure, are there in a piece of square timber 10 in. wide, 6 in. thick, and 9 ft. long ? 130. How many feet, board measure, are there in 200 pieces of scantling, each 18 ft. long, 4 in. wide, and 2 in. thick ? 131. How many feet in 8 boards, each 15 ft. long, 8 in. wide, and IJ in. thick ? 132. How many feet, board measure, in a plank 24 ft. long, 3 in. thick, 11 in. wide at one end, and 16 in. wide at the other ? Note. First find the average width, which equals one half the suTn of ths widths at the ends. 164 COMPOUND NUMBERS. 133. At $ 22 a thousand, what is the cost of 20 boards, each 18 ft. long, 1 in. thick, 20 in. wide at one end, and 17 in. wide at the other ? 134. Arthur bought wood for Sorrento carving, each piece being 2 feet long and \ of an inch thick, as follows : white holly, 12 inches wide at 8/ a foot, board measure ; black wal- nut, 18 inches wide at 6/ ; ebony, 9 inches wide at 25/ ; red cedar, 14 inches wide at 10/. What was the cost of the whole ? 393. Capacity op Cisterns, Bins, etc. 135. I have a cask that contains 2 cu. ft. ; how many quarts of berries will it hold ? (See Art. 329.) 136. How many gallons of water will a cistern hold that is 3 ft. long, 3 ft. wide, and 2J ft. deep ? 137. If a jar weighs 10 pounds when empty and 74 pounds when full of water, what is its capacity in cubic feet ? How many gallons will it hold ? (See page 141, Note IV.) 138. How many bushels of wheat can be put into a bin 8 ft. long, 3 ft. 2 in. wide, and 2 ft. 3 in. deep ? In measuring bulky fruits and vegetables, as apples and potatoes, the measures are heaped. Heaped measures fill about J more space than the even measures. 139. If 24 bushels of wheat can be put into a certain bin, how many bushels of apples might be put into the same bin ? 140. How many bushels of beets can be put into a barrel that holds 47 gallons ? 141. What is the difference in inches between f of a bushel and 1 cubic foot ? Note. The difference being so slight, for rough estimates of the con- tents of bins, etc., it is sufficiently accurate to call every cubic foot f of a bushel, even measure, or if of a bushel, heaped measure. 142. A box whose capacity is 50 cubic feet, will contain ■how many bushels of rye? how many bushels of pears,? GENERAL REVIEW. • 165 394. General Review, No. 3. 143. Change 5 m. 42 rd. 8 ft. to feet. 144. Change 4865 gr. to Troy pounds, ounces, etc. 145. Change f cu. yd. to feet and inches. 146. What cost 12 bu. 2pk. of pkims at 6 »* a pint? 147. What cost 2 qt. Ijpt. of oil at $ 1.12 a gallon ? 148. Change 41' 42" to the decimal of a degree. 149. Change |§ yd. to the decimal of a rod. 150. What part of anA.i8ll6sq.r.88|sq.ft. + 17sq.r.2sq.ft.? 151. Add 0.44 c. y. to 26 d. 5 h. 4 m. 162. Divide 12 A. less 7 A. 16 r. by 9. 153. Change 2 lb. av. to integers of Troy weight. 154. How many square feet in a garden 4 rd. long and 1 rd. 15ft. wide? 155. How many cu. ft. of space in a cellar measuring on the inside of the wall 5 yd. 1 ft. in length, 4 yards in width, and 10 feet in depth ? 156. What must be the depth of a cistern to contain 420 gallons of water, the base being a square covering 12;! sq. ft. ? 167. When a cistern 4 feet high is full of water, what weight is supported by every square inch of the base ? (See page 141, Note IV.) 158. How many bricks 4 in. by 8 in. will be required to pave a court 20 ft. long and 10 ft. wide ? Find the cost at $ 9 per M. 159. Divide 0.006 by 0.06, multiply the quotient by 0.05, and divide that product by 0.006. 160. Change 0.0625 to a common fraction in smallest terms ? 161. How many yards of carpeting f of a yard wide must be bought to cover a floor 13 feet square, no allowance being made for matching and no breadth to be divided ? 162. What is the difference of time in two places whose longitudes differ 7° 8' 4" ? 163. When the difference of time between two places is 3 h. 4 m. 6 s., what is the difference of longitude ? 164. How many days from Jan. 6, 1876, to March 3, 1877 ? 166 < COMPOUND NUMBERS. 396. Miscellaneous Examples. 165. If eggs are worth 30 cents a dozen, and 10 weigh a pound, what are eggs worth by the pound ? 166. If I burn 30 lbs. of coal a day, and buy my coal by the long ton, at I 7 a ton, what is the cost of my coal for December ? 167. How many furrows, each 20 inches wide, will be made in ploughing lengthwise a lot of land which is 6 rd. 1 ft. wide ? 168. A quantity of silver weighed 4 lb. 10 oz. 3 pwt. before refining, and 3 lb. 11 oz. 2 pwt. 9 gr. afterwards ; what weight was lost in the process ? 169. How many square feet on the top and sides of a box that is 3 ft. long, 2 ft. wide, and 2 ft. 6 in. high ? 170. What will be the cost of fencing a lot of land 20 rods by 26 rods at 26 cents a foot ? 171. Change f of a great gross to units of lower denomina- tions. 172. Divide an angle of 20° 4' 5" by 9. 173. Dr. Smith's wagon-wheel, which is 3 ft. 4 in. in circum- ference, turns round 200 times in going from his house to the post-office ; how far does he live from the post-office ? 174. If a bird can fly 1° in 1 h. 8 m. 15 s., in what time can it fly around the world at the same rate ? 175. What is the cost of 137 gal. 2 qt. of vinegar at 50 cts. per gal. ? 176. How many bushels of grain will a bin contain which is 10 ft. long, 8 ft. wide, and 5 ft. deep ? 177. What is the cost of oil-cloth to cover a floor 12 feet by 16J feet, at 75 cents per square yard ? 178. A farmer divided one half of his estate of 350 A. 140 rd. equally between his two daughters, and the balance, after set- ting off 17f A., equally between his two sons. What was the share of each son and daughter ? 179. How many yards of carpeting | yd. wide will cover s) floor 18 ft. sq. ? MISCELLANEOUS EXAMPLES. 167 180. If a cotton-mill can make 1200 yds. of cloth per hour, how many yards could be made by working 10 hours a day from July 7th to January 4th, allowing for 26 Sundays ? 181. Change 16 lb. 8 oz. Av. to pounds and ounces Troy. 182. How many sq. ft. does the surface of a block contain, which is 3 ft. long, 2 ft. wide, and 6 ft. thick ? 183. When 2 dozen grape-vines can be bought for $ 6.50, what is the cost of each vine ? 184. From a pile of wood 58 ft. long, 4 ft. high, and 4 ft. wide, was sold at one time 3| cords, at another 2^ cords. What is the remainder worth at $ 4 a cord ? 185. I have a shed which measures on the inside 18 ft. 7 in. by 8 ft. by 10 ft. in height. How many cords of wood can be put in it ? 186. A man purchased 75 cords of wood for $ 360 ; he sold the following lots, 10 J^ cd., 15 cd., and llf cd., all at $ 5 per cord. What did he gain on what he sold ? 187. What would be the cost of sawing the remainder of the 75 cords, at $ 1 a cord ? 188. How many gals, of water will be contained in a tank 3 ft. square, if the water is 4 ft. 3 in. deep ? 189. At 15 cents per pound, what was the cost for lead, 6 lbs. to the sq. ft., to line the above tank, it being 6 feet deep ? 190. What must I pay for a dozen silver spoons, each weigh- ing 2 oz. 9^ pwt., at $ 1.50 per ounce ? 191. Add I of the month of February, 1876, to f of the days from March 21st to June 17th, 1877. 192. How much carpeting f yd. wide will cover the top and sides of a block 3 ft. long, 8 inches wide, and 6 inches high ? 193. Estimate the cost of feeding a pair of oxen through the winter of 1879 and 1880, if 1 ox weighed 1772 lbs. and the other 1431 lbs., and hay was $ 13.75 per ton, and the oxen were allowed ^ of their weight in hay each day. 194. How many paving-stones 6 in. by 8 in. will be required to pave a street 27 rods long by 50 ft. wide ? 1&8 COMPOUND NUMBERS. 195. At 9 o'clock p. M. in Boston, what is the time in Paris ? 196. If a druggist sells 1 gross 2 doz. powders a day, how many will he sell from the 19th of Dec, 1877, to 15th Mar., 1878, deducting 12 Sundays ? 197. In what time will a vessel go through a strait 2 miles long, if she is carried ahead by tide 30 feet a minute, by wind 26 feet a minute, and by steam 100 feet a minute ? In what time can she go through the strait against wind and tide ? 396. Questions for Review. Repeat the table of Long Measure. Draw a line an inch long. Hold your hands a foot apart. What do you think the height of your school-room to be? In some convenient place mark off and walk 100 feet, counting your steps as you walk, and find their average length. By counting your steps, find how far you live from school. What is the standard unit of length ? How is an angle formed ? Upon what does its size depend? What is a right angle ? a rectangle ? a square ? area ? How do you find the area of a rectangle or a square? Illustrate. Repeat the table of Square Measures. From what are the units of square measure derived ? What is the principal unit of land measure ? What is a rectangular solid ? a cube ? How many faces has a cube ? how many edges ? How do you find the volume of any rectangular solid ? Illustrate. Repeat the table of Cubic Measure. From what are the units of cubic measure derived ? Repeat the table of Liquid Measures ; of Dry Measures. Which is larger, 1 quart liquid measure, or 1 quart dry measure ? What is the standard unit of liquid measure ? of dry measure ? How many cubic inches are there in a gallon ? in a bushel ? How do we ascer- tain the WEIGHT of anything ? Repeat the table of Avoirdupois weights ; of Troy weights. By which would you buy iron ? silver ? salt ? emeralds ? flour ? What is a long ton ? How many grains Troy make a pound Avoirdupois ? Which is heavier, 1 lb. Avoirdu- pois, or 1 lb. Troy ? 1 oz. Avoirdupois, or 1 oz. Troy ? What is the standard unit of weight ? What is a circle ? the circumference ? an arc ? Repeat the table of Circular Measures. Are all degrees of the same length ? What QUESTIONS FOR REVIEVf . 169 is the length of a degree of the circumference of the earth at the equator? What is a nautical mile ? What is its length in English miles? How is an angle measured ? Are all angles of one degree of the same size ? Repeat the table of Time. How do you know what years are leap years ? Name the months which contain 30 days each ; name the months which contain 31 days each. What is- a compound number ? a denominate number ? a general number ? How do the units of different denominations in compound numbers increase ? Give the rule for Reduction Descending; for Reduction Ascend- ing. How do you change a denominate fraction to integers of lower denominations? How do you change integers of lower denomina- tions to the fraction of a higher ? How are compound numbers added, subtracted, multiplied, and divided ? How do you find the number of years, months, and days between two dates ? (Art. 371.) How do you. find the time in days between two dates ? When the difference in time between two places is given, how do yoii find their difference in longitude ? When the difference in longitude is given, how do you find the difference in time ? Find the area of the top of your desk. Draw a square 1 inch each way ; J inch each way. What part of the first square is the second ? Difference between 5 square inches and 5 inches square ? When the length of one side of a rectangle is given in feet, and the other in rods, how do you find the surface ? When the area and one dimen- sion are given, how do you find the other ? How are the public lands of the United States divided ? When the volume and two dimensions of a rectangular solid are given, how do you find the third ? How is WOOD generally cut for market ? How many cubic feet are there in 1 cord ? How would you estimate the contents of bawbd TiMBEK and BOARDS ? How do you find the average width of a board that decreases regularly in width from end to end ? How are bulky fruits and vegetables measured ? How does a HEAPED measure Compare in bulk with an even measure ? A cubic foot is equal to what part of a bushel, even measure ? What part of ft bushel, heaped measure ? 170 DRILL TABLE. 397. BRILL TABLE No. 7. A B C 1. T. lb. 4T. 625 '"■ 12''^- 2'"- 1428""- 6« 2. I.T. lb. gi.T. 12 ■="'• 3 ■''■• 18™'- 2'^'- Jib. 3. lb.* pwt. gib. g oz. g p»t. o lb. y oz. 10P»t- 4. m. ft. lO-"- 200 "■■ 4>"'- 34 rd. 3 yd. 2". 5. sc[. m. 5q. rd. j^sq.m ggQA. 4sq.rd 4gA. gsq.rd. 6. cu. yd. eu. in. g cu. yd. 4 cu. ft. QQQ cu. in. gen. yd. igcu.ft.igo6c».ia 7. cd. cu. ft. lOO'^''- 2 cd. ft. 14 cu. ft. 92 cd. Qc±tL 12cu,ft. 8. gal. gi- 25^'- 3 It- p'- 4 gaL 2 I'. ipt 9. bu. pt. gbu. 3pk. iqt. gbu. ipk. 2 It. 10. circ. (') ■t circ 90° 40' 280° 2' 28" 11. c. y. hours 3c.y 4d. 1211. 2c.y. 7d. 18"- 12. ly- min. 11. y. 65"- 18 "■ 7*- 20"- gmia. 13. rd. in. 6-"- 310 rd. 2>"'- igrd. lift. gin. 14. A. sq. yd. gA. 29sq.rd 66="^ gyd. 1ft. 4 m. 19. 3C[. rd ft. gsi. rd. 29 sq. yd. 4 sq. ft 40 sq. yd. g sq. ft. 9 sq. in 20. qt.t gi- 3* pt- 2 «■■ 2 qt. 1 pt 2^^ 21. w. min. 28"- 3^ 121^ 3" 6^ 18 -^ 22. sq. yd sq. in 21 sq. yd. ^1- 't. 12 ^''- '" 4sq.yd. 3 sq.ft. 36»i-!» 23. d. see. 284*- 13 h. 9 min. 169"- 19 "• 42™°. 24. sq. ft. sq. in 11 sq. rd. 4 sq. ft. 90 sq. in 3=1"' 204^1- **■ 10sq.in 25. gross units 12 gross ydoz. 2 g gross 3 doz. 9 *Troy. t Liquid. DRILL EXERCISES. 171 DRILL TABLE No. 7 {contvimed). 1. (Tib. E goz. }. 4<:wt. gq.. 111b. \. 11-- 14Pwt V. 5 yd. 1"^ 1. O sq. rd. 4 sq. yd. s. 14cu.ft. 329cu,in. r. J^gcdft. 14cu.ft. i. 3qt ipt- (liquid) ». 4pi'. 7 qt. 1 pt. ). 98' 14" 1. 348'^ 3- 2. 21" 10 '"'"• 3. 2''- gin. 1. gsq yd 110^""- 5. iPt 3^- (dry) 6. 68' 58" 7. 22 P"'- 23^'-- 8. 2" 3 in. ». 9 sq- yd 8 sq.ft. 0. iPt 3^'- (liquid) 1. 23''- 41min. J. 27 =1- "■ 28 ="• '"■ 3. 4g ™in. 18 sec. 1. 178sq.f.. 108 =''■ " 5. -£ gross gdoz. 10 398. Exercises upon the Table. 196. Change five A to B.« 197. Change E to units of the lowest denomination in the example. 198. Change D to units of the lowest denomination in the example. 199. Change 3284 B to A. ZOO. Change 132687 B to units of higher denominations. SOI. Change -ft- A to B. 20S. Change 0.4627 A to B. 505. Change the numbers of lower de- nominations in D to a fraction of the highest. 204- Change the numbers of lower de- nominations in C to a decimal of the highest. (4 places.) SOB. What part of A is E ? 506. Add C, D, and E. 507. Add f A to D. SOS. Add 0.5784 A to K 509. Take E from D. 510. Take D from C. 511. Multiply C by 6. SIS. Multiply E by 15. SIS. Divide C by 10. SU. Divide D by 7. SIS. Divide D by 4 of the lowest de- nomination in the example. * See page 67, for Explanation of the Use of the Drill Tables. 172 THE METRIC SYSTEM. SECTIOIsr XIII. THE METRIC SYSTEM OF WEIGHTS AND MEASURES. 399. The me'-.ric system of weights and measures, now used in the greater part of Europe and coming into use in the United States, is derived from the standard meter. Note. The word meter means a measure. The standard meter is a certain bar of platinum carefully preserved at Paris. Copies of this har, made with the utmost precision, have been procured and are carefully pre- served by the nations that have adopted the Metric System. The standard meter of the United States is such a copy, and it is kept at Washington. The meter-sticks made for ordinary use are copies of the standard meter. MEASURES OF LENGTH. 400. The standard unit of length in the metric system is the meter. Note. The teacher should show the pupil a meter and its subdivisions. If none can readily be obtained, one can easily be made from the decimeter represented on the next page. This meter may be divided into decimeters and centimeters. From this measure the pupils can easily make their own of paper or wood. 401. One tenth of a meter is a dec'i-meter. Note. The prefix deoi- means one tenth of. 402. One hundredth of a meter is a cSn'ti- meter. Note. The prefix centi- means one hundredih of. 403. One thousandth of a meter is a mil'li-meter. Note. The prefix milli- means one thoitsandth of. MEASURES OF LENGTH. 173 (4 m a M O P !zi O 404. Exercises on the Meter and its subdivisions. a. How many meters long is the room ? How many meters wide? b. How many decimeters long is the table ? c. How many decimeters wide is the door? d. How many centimeters long and wide is your slate ? the window-pane ? etc. e. How many millimeters apart are two lines on a sheet of writing-paper ? /. How many millimeters thick is your slate-frame ? your ruler ? etc. g. How many millimeters are there in one centimeter ? h. How many centimeters are there in one decimeter ? i. How many decimeters are there in one meter? j. How many millimeters are there in one decimeter ? in one meter ? k. How many centimeters are there in 37 millimeters, and how many millimeters remain ? 1. How many decimeters are there in 84 centimeters, and how many centimeters re- main ? m.. How many meters are there in 347 centimeters, and how many centimeters remain ? n. In measuring the length of the room, did you find it to be an exact number of meters long ? 0. If not, how many decimeters do you find in the remain- der ? Do you fmd an exact number of decimeters ? p. If there is still g. remainder, how many centimeters do you find in it ? 174 THE METRIC SYSTEM. To vtrrite Numbers in the Metric System. 405. To express a length in meters and parts of a meter, we write whole meters in the units' place, deci- meters in the tenths' place, centimeters in the hundredths' place, and millimeters in the thousandths' place. Thus, if a room is found to be 8 meters 6 decimeters 9 centimeters long, we write : Length of the room = 8.69 meters. 2 decimeters 3 centimeters 5 millimeters is written : 0.235 meters. - 406. The abbre-viations used in writing expressions of length are : For meters, m ; for decimeters, dm ; for centi- meters, cm ; and for millimeters, mm. 407. Lengths may be expressed in other denominations as well as in meters, hy putting the decimal point at the right of the place of the required denomination, and writing the proper name or abbreviation after the figures. Thus, 0.235" may be written 2.35 '^■", 23.5'="', or 235"'"'. So also 728™° may he written 72.8 ■="■, 7.28 '^"■, or 0.728"'. 408. Eziercises in reading Numbers. Read the following : a. 5" e. 5.926"' i. 6.58''"' b. 47" /. 36 dm j. 3.4'-' c. 3.9 ■" S- 428'="' k. 43.7="' d. 4.21"' h. 23"'" 1. 2.5 "^ 409. Examples for the Slate. Change the following to meters : (1.) l'"" (4.) 1™ (7.) 1° (2.) 13"" (5.) 38=" (8.) 48"" (3.) 214"" (6.) 529=" (9.) 3675" mm MEASURES OF LENGTH. 175 Multiples of the Meter. 410. Besides the meter and its subdivisions, there are longer measures, which are multiples of the meter. 411. The delta-meter is ten times as long as the meter. Note. The prefix deka- means tenfold. 412. The hek'to-meter is a hundred times as long as the meter. Note. The prefix bekto- means a hundred/old. 413. The kil'o-meter is a thousand times as long as the meter. Note. The prefix kilo- means a tJumsandfold. 414. The myirla-meteT is ten thousand times as long as the meter. Note. The prefix myrla- means fere thousandfold. Note. Of these longer measures, the kilometer is used in measuring distances on. roads, canals, rivers, etc. The other measures are much less frecLuently used ; the myriameter hardly ever. 415. Exercises on the Multiples of the Meter. a. Measure off a string ten meters long. What name is given to the length of this string? Note. The string may be used in measuring distances. For this pur- pose it will be well to make knots at the end of each meter. b. Measure in dekameters and meters the length and breadth of the school-yard; of a garden; of a field, etc. c. Measure off in the street, or other convenient place, a distance of 10 dekameters. What name is given to this distance? d. Walk from the beginning to the end of the distance thus measured off, and count your paces. How many of your paces make a hektometer ? e. How many of your paces would make a kilometer ? /. How many kilometers from your home to the school-house? 176 THE METMIG SYSTEM. g. How long does it take you to walk a kilometer ? h. How many kilometers can you walk in an hour ? i. If 1600 of your paces make a kilometer, how many make a dekameter ? 416. To express distances in meters and multiples of a meter, we write meters in the units' place, dekameters in the tens' place, hektometers in the hundreds' place, and so on. 417. To express a distance in kilometers, we write kilometers in the units' place, and then hektometers, dekameters, and meters will be written in the tenths', hun- dredths', and thousandths' places respectively. Thus, if the distance from one town to another is found to be 9780 meters, the usual form of writing would be 9.78 kUometers. Note. The greatest distances are usually expressed in kilometers. Thus, the distance of the earth from the sun is about 149000000 kilometers. 418. The abbreviations used in writing are : For the dekameter. Dm; for the hektometer, Hm; and for the kilometer Km. 419. Table of Long Measure. 10 millimeters (mm) = 1 centimeter (cm). 10 centimeters = 1 decimeter (dm). 10 decimeters = 1 meter (m). 10 meters — 1 dekameter (Dm). 10 deka,meters = 1 hektometer (TTm). 10 hektometers = 1 kilometer (Km). 10 kilometers = 1 myriameter (Mm). 420. Oral Exercises. Eead the following : 3. 123-" d. 42°"" g. 49 Km b. 497.6"" e. 36.7°"" h. 593.7'^"' c. 346"" /. 57.5""' i, 6000'"" MEASURES OF LENGTH. 177 421. Examples for the Slate. Change the following to meters : (10.) 425°-" (13.) 94.6 «"■ (16.) 0.72 (11.) 35""" (14.) 9.24"^"" (17.) 0.073 (12.) 23.5''"' (16.) 39.7°" (18.) 0.05 Km Hn Km Addition, Subtraction, Multiplication, and Division of Metric Numbers. 422. Illustrative Example. Change to meters and add 14.83°-", 75.6 »", and 948°". WRITTEN WORK. Eo!!plomation. — 'Whea. these expreaaions 14.83°"'= 148.3"" have been changed to meters, they are all 75.6 ""■ = 7560. of the same denomination, and the sum is 948 ™ = 9.48 found in the same way as in the addition 771 7 78 " °^ simple numbers. 423. Numbers expressing metric measures and weights are added, subtracted, multiplied, and divided by the same rules as apply to simple numbers. 424. Examples for the Slate. 19. Add 5.6 ">, 24.07 "', 30.5 "", and 7.508 ■" 20. Express as meters and add 582 ■=■", 6428""', and 495""". 21. Express as meters and add 369 °"", 4073 "■", and 5 ''r 22. Add 48.06% 709.63"", 3708.9"', 800.9", and express the answer in kilometers. 23. If 7 "*■'" be taken from 42 ^™, how many meters remain ? 24. From 87.04"' take 42 '="'- 25. The distance round a certain park is 2.58 kilometers. How many meters will a man go who rides around it six times ? 26. A school-hoy walked one third around the above park in 12 minutes. How many meters did he walk in 1 minute ? 27. How many kilometers in 36.68 " x 2004 ? 28. Divide 38.07 "' by 4 and by 3, and add the answers. 29. Ellen's hoop is 3.6 " around. How many times will it turn in rolling a distance of 1.08 ^'" ? 178 TBE METRIC SYSTEM. MEASURES OF SURFAOE. 425. The units used in measuring surfaces are squares, each having sides equal to a unit of long measure. Thus, a square meter is a square having sides one meter long; a square decimeter is a square having sides one decimeter long ; etc. 426. Exercises. a. How many square decimeters in a square meter ? Illus- trate by drawing a square meter on the blackboard or on the floor and dividing it into square decimeters. b. How many square centimeters in a square decimeter ? IUustra,te by drawing a square decimeter on your slate and dividing it into square centimeters. c. How many square meters in a square dekameter ? 427. The square dekameter, when used as a unit of land measure, takes a special name, and is called an ar. One hundredth of an ar, which is one square meter, is called a centar. A hundred ars, equal to one square hektometer, is called a hektar. 428. Square Measure. 100 square millimeters (sq mm) = 1 square centimeter (sq cm). 100 square centimeters = 1 square decimeter (sq dm). 100 square decimeters = 1 square meter (sq m) =1 centar (ca^. 100 square meters = 1 square dekameter = 1 ar (a). 100 square dekameters = 1 square hektometer = 1 hektar (Ha). 100 square hektometers = 1 square kilometer (sq Km). 429. As the units of square measure form a scale of hundreds, in writing numbers expressing surface two deci- mal places must be allowed for each denomination. Thus, 45="" 4'='"'"' 86 ^i^" are written 45.0486 '"'"'; and T''^ 6=6" are written 706.05 ^ MEASURES OF VOLUME. 179 430. Examples for the Slate. 30. How many square meters of carpeting will be. required to carpet a room 5.3 ™ long and 4.5 "" wide ? 31. How many meters of carpeting 0.7" wide will be re- quired to carpet a room 4" long and 3.5"" wide ? 32. What is the cost of polishing the surface of a rectangu- lar piece of marble 2.8 meters long and 1.2 meters wide, at 1 2.50 per sq. meter ? 33. In a piece of land 15 '" long and 14.5 '" wide are how many square meters or centars ? how many ars ? 34. Express the following in ars and add them : 1.3 hektars, 165.5 ars, 43 hektars, 26 centars. 35. A had 6 hektars, 7 ars, 9 centars of land, and sold 0.2 of it at $ 54 an ar. How much did he receive for what he sold ? MEASURES OF VOLUME. 431. The units used in measuring cubic contents, or volume, are cubes, each having its edges equal to a unit of long measure. Thus, the cubic meter is a cube having edges one meter long ; a cubic decimeter is a cube having its edges one decimeter long ; etc. 432. Exercises. a. How many cubic decimeters in a cubic meter ? b. How many cubic centimeters in a cubic decimeter ? Illustrate by means of a cubical block having edges one deci- meter long, marked off into centimeters. 433. The cubic meter, when used as a unit of measure for wood and stone, takes a special name, and is called a ster. 434. The cubic decimeter, when used as a unit of liquid or dry measure, is called a liter. 180 TBE METRIC SYSTEM. 435. Cubic Measure. 1000 cubic millimetera (cu mm) = 1 cubic centimeter (cu cm). lOOO cubic centimeters — 1 cubic decimeter (cu dm) = 1 liter. 1000 cubic decimeters = 1 cubic meter (cu m) =1 ster. 436. TVood Measure. 10 decisters (ds) = 1 ster (s). 10 sters = 1 dekaster (Ds). 437. As the units of cubic measure form a scale of thousands, in writing numbers expressing volume three decimal places must be allowed for each denomination. Thus, 427 '^""" 29'=''^"" 3™'="' are written 427.029003'=''°'. 438. As the units of wood measure form a scale of tens, only one decimal place is needed for each denomination. Thus, 7 dekasters 5 sters 6 decisters are written 75.6 sters. 439. Examples for the Slate. 36. Express the following in cubic meters and add them : 7 cu. meters 40 cu. decimeters ; 5 cu. meters 3 cu. decimeters 19 cu. centimeters ; 26 cu. centimeters 49 cu. millimeters. 37. How many cubic meters of earth must be removed to dig a cellar 14.5"" long, 4.6"" wide, and 2.3"" deep ? 38. At $ 1.25 a cubic meter, what will it cost to dig a trench 75.5 '^ long, 2.2 "" wide, and 1.8 ■" deep ? 39. How many loads of earth, each filling 2.25 ■=" "", wiU fiU a space 15.4 "" long, 12 "" wide, and 4.5 "" deep ? 40. If a cubic centimeter of gold is worth $ 12.50, what is the value of a brick of gold 2.4 "■" long, 1.3 """ wide, and 0.75 "^ thick? 41. If I bum 27 sters of wood in the three winter months, what must be the length of a pile 1 meter wide and f meter high to last a month, and what wOl it cost at $ 2.26 a ster ? MEASURES OF CAPACITY. 181 MEASURES OP CAPACITY. 440. The primary unit of measure for all substances that can be poured into a dish or box is the liter. 441. A liter is equal in volume to one cubic decimeter. 442. Larger and smaller measures are derived from the liter in the same way that longer and shorter measures are derived from the meter, that is, by taking decimal multiples and subdivisions. 443. Liquid and Dry Measures. 1 milliliter (ml) = 1 cu cm. 10 milliliters = 1 centiliter (cl). 10 centiliters = 1 deciliter (dl). 10 deciliters = 1 liter (1) = 1 cu dm. 10 liters = 1 dekaliter (Dl). 10 dekaliters = 1 hektoliter (HI). 10 hektoliters = 1 kiloliter (Kl). = 1 cu m. Note. The milliliter is employed in computations, but rarely, if ever, in actual measurements. Chemists and druggists use cubic centimeters instead of milliliters. 444. Examples for the Slate. 42. If one hektoliter of kerosene costs $20, what is the price of a liter ? 43. What must be paid for 2.5 liters of milk each day for a week, at 7 cents a liter ? 44. From a vessel containing 1 hektoliter of syrup, 25 liters were drawn out. How many liters remained ? 45. How many hektoliters of oats can be put into a bin that is 2"" long, 1.3" wide, and 1.5"" deep? 46. What must be the length of a bin 1 meter wide and 1 meter deep, to contain 4500 liters of grain ? 182 THE METRIC SYSTEM. WEIGHTS. 445. The primary unit of weight is the gram, 446. A gram is the weight of one cubic centimeter of pure water at the temperature of 4 degrees centigrade ( = 39.2 degrees Fahrenheit), at which temperature water has its greatest density. 447. Larger and smaller weights are derived from the gram by taking decimal multiples and subdivisions. 448. "Weights. 10 milligrams (mg) = 1 centigram (eg). 10 centigrams = 1 decigram (dg). 10 decigrams = 1 gram (g) = wt. of 1 cu cm of water. 10 grams = 1 dekagram (Dg). 10 dekagrams = 1 hektogram (Hg). 10 hektograms = 1 kilogram (K) =wt. of 1 cu dm of water. 10 kilograms = 1 myriagram (Mg). 10 myriagrams = 1 quintal (Q). 10 quintals = 1 metric ton (T.) = wt. of 1 cu m of water. Note I. The gram, kilogram, and metric ton are the only units used in actual weighing, except hy jewellers, druggists, and those who weigh very small or very expensive articles, like gold or powerful medicines. Note II. The kilogram is generally called the kilo. The kilo is the unit of weight for weighing common articles, such as sugar, tea, etc. Note III. The metric ton is used to weigh very heavy articles, like hay, coal, etc. 449. Examples for the Slate. 47. At $ 0.60 a kilo for honey, what is the cost of 6.15 kilos ? 48. At $ 11 per T. for coal, what will the coal cost to keep a fire one week if 30 kilos are burnt each day ? 49. What weight of mercury will a vessel contain whose capacity is 10°"°°", mercury being 13.5 times as heavy as water ? 50. If marble is 2.7 times as heavy as water, what is the weight of a pedestal 1 meter square at each end and 2 meters high ? EQUIVALENTS. 18?. 450. Table of Equivalents. The equivalents here given agree with those that have been established by Act of Congress for use iu legal proceedings and in the interpretation of contracts. 1 inch = 2.540 centimeters. 1 foot = 3.048 decimeters. 1 yard = 0.9144 meters. ■ 1 rod = 0.5029 dekameters. 1 mile = 1.6093 kilometers. 1 centimeter = 0.3937 inch. 1 decimeter = 0.328 foot. 1 meter = 1.0936 yds. = 39.37 in. 1 dekameter = 1.9884 rods. 1 kilometer = 0.62137 mile. 1 sq. inch = 6.452 sq. centimeters. 1 sq. centimeter = 0.1550 sq. inch. 1 sq. foot = 9.2903 sq. decimeters. 1 sq. decimeter = 0. 1076 sq. foot. 1 sq. yard = 0.8361 sq. meter. 1 sq. meter = 1.196 sq. yards. 1 sq. rod = 25.293 sq. meters. 1 ar = 3.954 sq. rods. 1 acre = 0.4047 hektar. 1 hektar = 2.471 acres. 1 sq. mile = 2.590 sq. kilometers. 1 sq. kilometer = 0.3861 sq. mile. 1 cu.inch = 16.387 cu. centimeters. 1 cu. centimeter = 0.0610 cu. inch. 1 cu. foot = 28.317 cu. decimeters. 1 cu. decimeter = 0.0353 cu. foot, 1 cu. yard = 0.7645 cu. meter. 1 cu. meter = 1.308 cu. yards. 1 cord = 3.624 sters. 1 liquid quart = 0.9463 liter. 1 gallon = 0.3785 dekaliters. 1 dry quart = 1.101 liters. 1 peck = 0.881 dekaliter. 1 bushel = 3.524 dekaliters. 1 ounce av. = 28.35 grams. 1 pound av. = 0.4536 kilogram. 1 ster = 0.2759 cord. 1 liter = 1.0567 liquid quarts. 1 dekaliter = 2.6417 gallons. 1 liter = 0.908 dry quart. 1 dekaliter = 1. 135 pecks. 1 hektoliter = 2.8375 bushels. 1 gram = 0.03527 ounce av. 1 kilogram = 2.2046 pounds av. 1 ton (2000 lbs.) = 0.9072 met. ton. 1 metric ton = 1.1023 tons. 1 grain Troy = 0.0648 gram. 1 gram = 15.432 grains Troy. 1 ounce Troy = 31.1035 grams. 1 gram = 0.03215 ounce Troy. 1 pound Troy = 0.3732 kilogram. 1 kilogram = 2.679 pounds Troy. 184 THE METRIC SYSTEM. 451. To change numbers in the metric system to equiva- lents of the old system : [Use preceding table.] Examples. 61. In 48 meters how many feet ? 62. If you travel 60 kilometers in a day, how many miles do you travel ? 63. Change 18 hektars of land to acres. 64. How many inches long is an insect that is 5.2 centi- meters long ? 55. How many pounds av. are there in 85.6 kilos of salt ? 56. How many gallons are there in 24 kiloliters ? 67. In 20 metric tons how many tons ? 452. To change numbers in the old system to equiva- lents of the metric system : [Use preceding table.] Examples. 68. Change 25 miles to kilometers. 59. In 200 acres are how many hektars ? 60. How many liters will a cistern hold that measures on the inside 5 feet in length, 4 feet in width, and 4 feet in height ? 61. In 3 rods how many meters ? 62. Change 18 qt. 1 pt. to liters. 63. In 1 lb. 7 oz. 18 pwt. of gold, how many grams ? 64. What is the weight of a barrel of flour (196 lbs.) in kilograms ? 453. Approximate Equivalents. The ec[mvalent3 here given are accurate enough for most purposes, and are easy to remember. A decimeter = 4 inches. A. meter An ar _{3 ft. 3|in., ~ / or ItV yards. A dekameter = 2 rods. A kilometer = f of a mile. !4 sq. rods, or -^ of an acre. A hektar = 2 J acres. A ster = J of a cord A liter 1.06 liquid qt., or-i^of adryqt. A dekaliter = 1 peck and 1 qt. A hektoliter = 2^ bushels. A gram = 15^ grains. A kilogram = 2| pounds av. A metric ton = 2200 pounds av. PMGENtAGB. 185 SEOTIOl^ XIY. PERCENTAGE. 454. Find yf^ of 500 men. Ans. 45 men. A number obtained by finding a number of hundredths of another number is a percentage of that number. 455. The number of which the percentage is found is the base of that percentage. In the above example what number is the percentage ? the base ? 456. If a person having $2000 should gain a sum equal to -^^ of it, how much would he then have ? 2000 X -^ = 200 ; 2000 + 200 = 2200. Ans. $ 2200. The sum of the base and percentage is the amount. 457. If a person having % 2000 should lose -^ of it, how much would he have left ? 2000 X ,1,^ = 200 ; 2000 - 200 = 1800. Ans. % 1800. The part of the base left after a percentage is taken away is ^e remainder. In the above examples what number is the amount? the remainder? 458. The number of hundredths which the percentage is of the base is the rate per cent, generally called the per cent. Thus, j-|^ of anything is 7 per cent of it. Note. Per cent is a contraction of the Latin per centimi, and means hg the Ivwndred. 459. Oral Exercises. 3. Find 1 per cent of 600 ; 7 per cent ; 20 per cent. b. Find 10 per cent of 1 250 ; 5 per cent ; 60 per cent. 186 PEMGENTAOE. To express a given Per Cent. 460. The sign % is used for the words per cent. Thus, 5 % means 5 per cent. 461. Any per cent may be expressed as a common frac- tion, as a decimal, or with the sign for per cent, %. Thus, 1 per cent may be expressed ^^-g, 0.01, or 1%. 6 per cent ^U, 0-06, or 6/«. 7J per cent T^A, 0.07i, or 7J/«. 100 per cent m, 1.00, or 100%. 120 per cent m, 1.20, or 120/.. \ per cent ■^U, O.OOi, or i %. Exercises. 462. Express the following in the three forms given above . a. 2 per cent. c. 7^^ per cent. e. 175 per cent. 6. 5 per cent. d. 200 per cent. /. ^ per cent. 463. Express the following as common fractions, and change them to their smallest terms : g. 5%. k. 50 fo. o. 4:%. s. &l%. h. 10%. 1. 100 /». • p. 75%. t. 8J%. i. 20%. m. 12^%. 3.90%. h. 83J%. j. 25%. n. 16|-%. r. 37^%. v. 125%. To find the Complement of a given Per Cent. 464. "What is the difference between 100 % and 25 % ? The difference between 100 % and any given per cent less than 100 % is the complement of the given per cent. 465. Oral Exercises. a. What is the complement of 75%? 40%? 60%? 33 J%? 6i%? 15%? b. What is the complement of 62 J%? 16%? 37^%? 18%? 87i%? 72%? EXAMPLES. 187 To change a Common Fraction to a Per Cent. 466. Illustrative Example. What per cent of a num- ber is J of it ? WRITTEN WOBK. Explanation. — Since any number equals ^N -j^QQ 100^ of itself, ^ of the number must equal i of 100^, or %b%. Am. 25 fe. 26 Ans. 25%. 467. Oral Exercises. a. What per cent of a number is J of it? ^? ^? ^^ ? SV? ^? A? A? i? i? i? I? T^? 6. What per cent is|? f? f? f? f? f? ^? |? |? F i? f ? J? f^? /ir? A? A? A? A? A? H? c. What per cent is ^^? ^V? A? A? /s? H? A? /it? The Base and Rate per cent being given, to find the Per- centage, Amount, or Remainder. 468. Oral Exercises. a. What is 7% of $ 300 ? Solution. — 7 per cent of $ 300 is ^ of $ 300, or $ 21. Am. $21. b. What is 20^ of 80 trees ? of 300 words ? of 90 ? 60 ? 240? c. What is 10% of 80 days? 12^% of 80 days? 26/.? 40/.? 50%? 90%? d. What is 6% of $ 100 ? of $200 ? of $1.50 ? of $2.60 ? of $500? of $12.50? e. What is the amount of $ 40 + 5% of $ 40 ? $16 + 25% of $16? /. What is the amount of $100 + 7% of $100? $60 + 50% of $60? g% What remains of an income of $500 after 40% of it is spent? after 25% is spent? 10%? 15%? 30%? 80%? 188 PEBCENTAOE. 469. Illustrative Ex- ample I. A had % 500. If by trading he gained a sum equal to 25 ^ of his money, what was his gain? How much money did he then have ? WEITTEN WORK. Baie, $500 Per cent, 0.25 Percentage, $125 A'sgain, $ 625 Ammtnt. 470. Illustrative Ex- ample II. B had $800. If by trading he lost 12 % of his money, what was his loss ? How much money did he then have ? WEITTEN WORK. Bom, $ 800 Per cent, 0.12 Percentage, $ 96 B's loss. $ 704 Bemavnder. In the examples above, A's amount and B's remainder might have heen considered as a percentage of the base and obtained directly thus : (I.) Base, $500 Per cent, 1.25 5 Q2o A's ammtnt. (IL) Bose, $ 800 Per cent, 0.88 $ 704 B's remainder. Explanation. — (I.) The money A had in trade was 100 fo of itself. Adding to this the 25 fo gain, his amount was 125 % or 125 hun- dredths of 1 500, equal to 1 625. (II.) The money B had in trade was 100 % of itself. Having lost 12 fo of this, he had remaining 88 fo or 88 hundredths of $800, equal to 1 704. 471. From the operations above we derive the following Rules. 1. To find the percentage : Multiply the base hy the rate per cent. 2. To find the amount : Add the percentage to the hase, or multiply the hase hy 1 plus the rate per cent. 3. To find the remainder : Subtract the percentage from the base, or multiply the base hy 1 minus the rate per cent. EXAMPLES. 189 472. Examples for the Slate. What is (1.) 12 % of 1 940 ? (6.) 85 % of 16f pounds ? (2.) 25 % of $250.60 ? (7.) 75 /« of 120 % of 486800 ? (3.) 62 % of 2000 men ? (8.) 37 J % of 4000 feet. (4.) I % of $28.80 ? (9.) 100% of $6000 + 75% of $6000 ? (5.) 120% of 75 days? (10.) 100% of 10800- 13% of 10800 ? 11. If 5% of the price of goods is deducted for cash, what deduction is made from a bill of $ 25.40 ? 12. If a piece of rubber hose 146 feet long shrinks 10% when wet, what is its length when wet ? 13. What is 25% of 125% of 75% of 50% of 384 inches ? 14. A farmer paid for shearing 104 sheep 4% of what he received for the wool ; the ileeces averaged 5 pounds each, and sold at 40/' a pound. What did he pay for shearing ? The Percentage and Rate per cent being given, to find the Base. 473. Illustrative Example. 750 bushels is 25 % of what number ? Explanation. — Since 25 % WRITTEN WORK. of the number sought is 750, ''^50 X 100 _ 1 % of the number sought is ^r- - 3000. Ans. 3000 bu. , f „,„ , 1 rvA ,rf c LI. number sought, or the number itself, is 100 times ^ of 750, or 3000. Ans. 3000 bu. Hence the fol- lowing Rule. To find the base when the percentage and rate per cent are given : Divide the percentage by the numerator of the rate per cent, and multiply the quotient hy 100. Note. Since 750 divided by 25 and multiplied by 100 equals 750 divided by 0. 25 (whiob is the rate per cent), tlie form of written work given below may be used instead of that above : WRITTEN VfORK. JL^ = 3000. Ans. 3000 bu. 0.25 Here the work is done by dividiTig the percentage ty the rate per cent. This rule agrees with formula 4, page 192. 190 PERCENTAGE. 474. Examples for the Slate. (15.) $48 is 3% of what number ? 10^ of what number ? (16.) 436. days is 24^ of what number? 8% of what number ? (17.) $ 31.35 is 6^ of what number ? 15% of what number ? . (18.) $300 is 1^% of what number? 5% of what number? (19.) $220.50 is 105/« of what number? 15% of what number ? (20.) I is 25% of what number? \% of what number ? 21. The number of children of school age in a certain town is 1275 ; if this is 20% of the whole population, what is the whole population ? 22. I drew out 25% of my deposits in a bank ; of this I have spent $468.72, which is 9% of what I drew out. What did I draw out ? What remains in the bank ? 23. If $ 240 is 20% more than some number, what is that number ? XoTE. Since $240 is 20% more than the number sought, it must be 120% of the number sought, etc. Hence when the amount is given instead of the percentage, divide by 100 plus the numerator of the rate per cent, and multiply by 100. 24. $ 1860 is 25% more than what number ? 25. A sold a horse for $ 225, which was 5% more than he paid for it. What did he pay for it ? 26. A grocer sold tea for 115% of its cost, and made 9 cents per pound. What did it cost a pound ? 27. If $450 is 10% less than some number, what is that number ? Note. Since 1 450 is 10 % less than the number sought, it must be 90 % of the number sought, etc. Hence when the remainder is given instead of the "percentage, divide by 100 minus the numerator of the rate per cent, and multiply by 100. 28. $ 1000 is 4 % less than what number ? 29. Having lost 40% of my money, I have $ 750 left. How much had I at firat ? EXAMPLES. 191 30. A son is 15 years old, which is &2\fo less than his father's age. What is his father's age ? 31. The daily attendance upon a school is 558, which is 7 % below the number belonging. What is the number belonging ? 32. After the wages of a workman were reduced 1\%, he received $ 3.70 a day. What were his wages before they were reduced ? 33. By assessing a tax of %% on the valuation, a town raised $ 75000. What was the valuation ? The Percentage and Base being given, to find the Rate per cent. 475. Illustkative Example. If a pupil is absent from school 6 days in a term of 75 days, what per cent of the time is he absent ? WRITTEN WOKK. Explanation. — If he is absent 6 days in 75 75) 6 00 days, he is absent -^ of the time. -^ changed -— — to hundredths is 0.08, or 8 %. Ans. 8 %. 0.08, or 8^ Ans. 476. From the example above may be derived the fol- lowing Rule. To find the rate per cent when the base and percentage are given : Divide the percentage ly the hose, carrying the division to hundredths. 477. Examples for the Slate. 34. What per cent of 1 104 is 1 26 ? is $ 52 ? is $ 18.20 ? 35. Whatper cent of $3isl2/? is $3.75? isl/? 36. What per cent of a dozen is a score ? 37. Out of 300 words, Charles spelled 280 correctly, Mary 284, Sarah 268, and Dwight 272. What per cent of the words did each spell correctly ? 192 PEBGENTAGE. 38. The surface of the earth contains about 144 million square miles of water, and about 63 million square miles of land. What per cent of the entire surface of the earth is water ? 39. From a cask containing 120 gal. of oil, 6 gal. 2 qt. leaked out. What % was lost ? 478. The operations in percentage, illustrated above, may be expressed by the following formulas : 1. Percentage = Base x Rate. 2. AmoTint = Base •><■ (i + Rate). 3. Remainder = Base x (1 - Rate.) 4. Base = Percentage -^ Rate. 5. Rate = Percentage -^ Base. For additional examples in percentage, see page 253. PEOPIT AND LOSS. 479. Oral Exercises. a. How much money is gained by selling goods at 25% above cost, the cost being $ 8 ? $ 10 ? 1 1.60 ? b. How much money is lost on goods which cost $ 24, by selling them at a loss of 25 f»? 50/.? 12 J%? c. At what price must paper which cost $ 2 a ream be sold to gain 10/.? 20%? 25%? 50%? 100%? d. At what price must hats which cost 80/ be sold to lose 10%? 5%? 25%? 50%? 12i%? e. What must have been paid a pound for nutmegs if by selling them at $1.00, there is a gain of 25%? Z^%? 10%? /. What was the cost of gloves which sold for $ 1.00 a pair at a loss of 20%? 50%? 33J%? 25%? g. What per cent is gained if goods costing 10/ a yard are sold for 11/? 12/? 15/? 20/? PROFIT AND LOSS. 193 h. What per cent would be lost if goods costing 1.5 f a. yard were sold for 12/ ? 10/ ? 9/ ? i. A drover bought cows at $26 a head, and paid $7 each to get them to market. If he sold them at $ 40 a head, what per cent did he gain ? j. What is the cost of goods when a gain of 20/ a yard in seUing is 10% of the cost ? 5%? 8/»? 50%? 12J%? k. What was the length of a piece of cloth before shrinking, if when shrunk 6 inches, it was shortened 1%? 2%? 3%? 4%? 480. The difference between the cost of goods and the price at which they are sold is a profit or a loss. 481. Profit and loss may be reckoned as percentage, the cost being taken as the base. Hence the rules of percentage already illustrated apply to profit and loss. 482. Szamples for the Slate. 40. A farm which cost $ 6842 was sold at a gain of 16%. What was received for it ? 41. A lot of coal was bought for $ 750. For what must it be sold to gain 33 J%? 42. If 2000 reams of paper were bought for $ 1500, at what price per ream must it be sold to gain 40%? 43. A merchant sold a cargo of wheat at 12 J % profit, and gained $ 746.25. What was the cost ? 44. By selling a farm for $ 2760, a man gained on the cost 15%. What was the cost ? 45. What was my property worth 5 years ago, if it has increased 150%, and is now worth $ 17500 ? 46. A man sold a picture for $ 275 at a loss of 16§ %. What did he pay for it ? 47. If I pay 45 ^ a pound for tea, and sell it at 56/, what per cent do I gain ? 48. What was the original value of a share in a bridge, which, selling at an advance of 35%, brings $ 780 ? 194 PERCENTAGE. 49. What is the per cent of gain if goods which cost 1 7500 sell at a gain of 11876? 50. A grocer sold 280 barrels of apples for $ 708.40. If he paid 11.40 per barrel for the apples, and 44/ a barrel for transportation, what per cent did he gain ? 51. A merchant bought carpetings at 85/, $ 1.20, and $ 1.50 a yard. At what prices must he sell them to make 20% profit ? 52. If $ 1000 be paid for goods of which one half sells for $ 640, and the remainder for $ 300, what is the per cent of loss ? 53. Bought paper at 1 1.75 per ream, and sold it at 20 cents per quire. What per cent did I gain ? 54. A dealer bought 10 gross of combs at 1 12.50 a gross. If he sold 50 of the combs at 20 cents apiece and the rest at 18 cents apiece, what per cent did he gain ? 56. If 160 beeves are bought at the rate of 1 42.50 each, and 30 at the rate of $ 45.00 each, and the lot is sold for $ 10300, what per cent is gained ? OOMMISSION. 483. One person is sometimes employed to buy goods or collect money for another, and is allowed for the service a percentage on the amount he lays out or collects. This percentage is called commission. 484. A person employed to transact business for another is an agent or tact or. 485. A person who sends goods to another for sale is a consignor, and the person to whom the goods are sent is a consignee. 486. The remainder, after the commission and other charges of a sale are deducted^ is the net proceeds. 487. Commission being a percentage, of which the money expended or received is the base, the rules of per- centage already illustrated apply to commission. COMMISSION. 195 488. Examples for the Slate. 56. At 1% commission, what is the commission on the sale of 4760 pounds of sugar at 1\ cents per pound ? 67. A factor in Mobile purchased for the Pacific Mills $ 90000 worth of cotton at 1|% commission. What was tlie bill for cotton and commission ? 68. If an auctioneer sells on a commission of 8 %, 14 chairs at ? 1.26 each, 1 table for % 10, and a miscellaneous lot for f 53.70, what is his commission, and what sum will be due the person for whom he mates the sale ? 69. A lawyer collected 25 % of an account of $ 680, charging 6% commission. What was his commission, and what sum should he pay over ? 60. What is the commission on the sale of 200 yards of cloth at $4.80 per yard, 6% being paid for selling, and 2\% for guaranteeing payment ? 61. What are the net proceeds from the sale of 1250 barrels of flour at $ 5.50 per barrel, charges for freight and storage being 40/ per barrel, commission for selling being 2%, and for guaranteeing payment 1^7o? 62. An architect charged 1139.75 for plans and for supex- intending the building of a house. If his commission was 2^%, what was the cost of the house, including his commission ? 63. What is the per cent of commission when an ageut reserves to himself $ 270.00 of $ 9270, sent him to invest ? 489. Illctsteative Example. What part of a remittance of $ 328.25 will remain to be invested after 1 % of the in- vestment has been deducted ? Solution. — -The remittance contains both the investment and the commission upon it. The commission being 1 % of the investment, the remittance must be 101 % of the investment. Hence $ 328.23 ^1.01= $ 325, the investment. 64. I send to my agent at Havana $ 1224. What part of this sum will remain to invest in sugars, after deducting his commission of 2% on what he lays out ? 196 PERCENTAGE. 65. How many barrels of flour at $ 5 each can a factor pur- chase with a remittance of $ 2575, after deducting his commis- sion of 3 % ? 66. A real estate broker received $2593.75 for the purchase of land. Reserving 3f % commission on the purchase, what number of acres of land could he purchase at $ 125 per acre ? 67. If $ 109.65 is sent to an agent to purchase 2000 pounds of sugar at 5f cents per pound, and to pay his commission on the purchase, what % is the commission ? 68. An agent sold 62 lawn-mowers at $ 20 each, and 18 at $15 each. If, after deducting his commission, he remitted $ 1057 to the manufacturer, what was the % of his commission ? 69. Find the balance of the following account of sales : B^M on account o/ Q^^^eMU. (^. Q^. ^cz//^ (^on. ISS COON, BEG., & 00. 1877. " ^4. § " — ^^^ §4. 41^7 ^1^^' /^. § ffSS §4 44P ^§f CHARGES: Taid Freight atid Cartage Commission and Guarantee, ^ % ... Philadelphia, April 15, 1877. Balance 490. Written Exercises. a. Supplying names and dates, write an account of the sales given in Example 58. b. In the same way write an account of the sales given in Example 61. * GrosB weight. t Weight of tubs. X Net weight. STOCKS, DIVIDENDS, AND BROKERAGE. 197 STOCKS, DIVIDENDS, MD BEOKEEAGE. 491. An association of individuals formed for the pur- pose of transacting business is a company or partnership. 492. An association of individuals authorized by law to transact business under a company name, to hold property and be liable for debts in that name as an individual would be, is a corporation. 493. When a corporation is formed for transacting busi- ness, the persons forming the corporation subscribe what money is needed for conducting the business. This money is called capital stock. This stock is divided into shares, usually of $ 100 each. 494. The owners of the stock are stockholders. As evidence of their ownership, they hold papers called cer- tificates ot stock. The stockholders form the corporation and elect directors, who are responsible for the business transacted. 495. A sum levied upon a stockholder to help meet the expenses or losses of the business is an assessment. 496. The gain upon the capital of a corporation is di- vided among the stockholders. Gain thus divided is called a dividend. Each stockholder's part of the dividend is the same per cent of his stock that the whole dividend is of the capital. 497. Stocks may be bought and sold like other prop- erty. Persons who make a business of buying and selling stocks are called stock-brokers. The commission paid to a broker is called brokerage. Note I. When a share of stock will sell at its nominal value, it is at par; when for more than its nominal value, it is above par, or at a premium ; when for less than its nominal value, it is helow par, or at a discount. 198 PERGENTAOE. Note 11. The market values of stocks are "quoted " daily in the prin- cipal newspapers, at given per cents of their values. Whea a stock is quoted at 90, it is worth 90 % of its face or nominal value ; it is then 10 % below par. When quoted at 105, stock is worth 105% of its face or nominal value ; it is then 5 % above par. 498. The rules of percentage already illustrated apply to stocks, dividends, and hroherage. Examples for the Slate. 499. The following quotations are taken from a daily paper : Sales of Stock tMs day at the Brokers* Board. 70 Chicago, Burlington, & Quincy R. R 103J 150 Burlington & Mo. R. R. in Neb 43i $5000 Atchison, Topeka, & Santa F^ 7's, 1st mortgage 88^ AT AUCTION. 8 American Watph Co 90^ 5 Metropolitan Bank 92} 40 Boston & Albany R. R.... 125 36 Nashua & Lowell R. R.... 94J 15 Bates Manufacturing Co. 80| 12 Neptune Insurance Co.. 122| 5 Maveriok Bank 150f 10 N. England Bank 135i At the above quotations, what is the cost 70. Of 3 shares in the Maverick bank, and 7 in the Metro- politan ? 71. Of $ 2000 Atchison, Topeka, and Santa Fe 7's ? 72. Of 8 shares in the Bates Manufacturing Co., and 7 in the Neptune ? 73. Of 75 shares in the Burlington and Missouri, including i % brokerage on the par value ? Note. Brokerage is usually J % , and reckoned on the par value. It is thus reckoned in this book, unless otherwise specified. At the above quotations, what is the cost, with brokerage, 74. Of 10 shares Boston and Albany E. E., and 25 Nashua and Lowell ? 75. Of 15 shares in the Chicago, Burlington, and Quincy E. E., 6 shares in the American Watch Co., 40 shares in the New England Bank, and 12 shares in the Neptune Inmrance Co.? INSURANCE. 199 76. What is the value of 7 shares in a gold company's stock at 4|% above par, the original value being |200 per share ? 77. A dividend of 3 % having been declared by a gas com- pany, what should a stockholder receive who owns 700 shares, the par value of each share being $ 100 ? 78. A broker sold a lot of stock for $2250, which was 10% below par. What was the par value ? 79. When stock, originally worth 130 per share, sells for $ 45, at what % above par does it sell ? 500. From January, 1862, to January, 1879, paper cur- rency was below par. The value of gold as compared with it was given from day to day in the newspapers. 80. When gold was quoted at 102|-, how much paper cur- rency could be bought for $ 200 in gold, no allowance being made for brokerage ? 81. If the passage to Liverpool was $125 in gold, when gold was at 103J, what was the cost of two tickets in paper currency ? 82. In 1878, 1 sent to Ireland 6 pounds sterling, valued at ■$4.86 each in gold, what did I pay for them in paper money, gold being at 102§, and brokerage J % ? 83. Stock at par is what per cent of stock at 103 ? INSUEANOE. 501. A, owning a house, agrees to pay B a certain per- centage on its value, B on his part agreeing to pay A the whole value of the house in case it should within a limited time be destroyed by fire. Such a contract is a contract of insurance ; and A's house is said to be insured. 502. Insurance is security against loss. 503. Fire insurance is security against loss of build- ings or goods by fire ; marine insurance is security against loss of ships or cargoes at sea ; accident insurance against 200 PERCENTAGE. loss by accident in travelling or otherwise ; health insur- ance secures a stated allowance during sickness, and life insurance secures a certain sum to one's heirs or assigns in case of death. 504. The parties that insure are called insurers or underwriters. 505. The written contract that binds the parties is the policy. 506. The sum paid for insurance is the premium. Note I. "When property is insured, the valuation or amount insured is generally made less than the value of the property. Note II. Policies are renewed yearly, or at stated periods, and the premium is paid in advance. 507. The premium is a pekcentage of which the sum insured is the base. Hence the rules of percentage already illustrated apply to insurance. 508. Examples for the Slate. 84. What is the insurance on $1500 worth of goods at f %, including $ 1 for the policy ? 85. What amount is paid for insurance on f of a store valued at $15600 at f %, including $1 for the policy? 86. A merchant insured a cargo from Liverpool worth 2000 pounds at a premium of 1^%. What was the premium, the pound being valued at $ 4.86 ? 87. A merchant insured $ 3600 worth of goods in one com- pany at 1 J % premium, and $ 2500 worth in another at 1 J % premium. What was the cost, including $ 1 for each policy ? 88. A druggist paid $ 125 for the insurance of a lot of goods in transportation. If the face of the policy was $ 10000, what vras the rate of insurance ? 89. Jan. 1, 1876, a person took out a health policy, paying $ 1.60 on the first day of each month. March 2, 1877, he was disabled by sickness, and received $ 12 a week for 3 weeks. How much did he receive more than he paid out for premiums ? TAXES. 201 The yearly rates of life insurance depend upon the age of the per- son when he begins to insure, younger persons paying less per year than older persons, because they are likely to live longer. Thus A, being 35 years old, pays $ 109.50 a year for a policy of f 5000, while B, who is 40 years old, pays $ 131.50 a year for a policy of the same amount. The number of years that a person of a given age is likely to live is called his expectation of life. 90. At the age of 38, 1 secured a policy upon my life for $ 6000, paying the first year 1 122.55, including 1 1 for the policy. What was the premium paid upon $ 1000 ? 91. Jan. 1, 1868, a man took out a policy on his life for $3000, in favor of his wife, paying $21.30 on $1000 yearly. If the man died Feb. 15, 1878, how much did the widow re- ceive more than had been paid in premiums ? TAXES. 509. The citizens of a town or city or the members of a society usually meet the expenses of their government or society by a sum assessed on their property, their income, their business, or their persons. Such a sum is called a tax. 510. A tax on the person of a citizen is called a poll tax. A tax on property is called a property tax. A tax on annual income is called an income tax. 511. Movable property, such as money, stocks, cattle, ships, etc., is called personal property. Immovable prop- erty, as lands, houses, etc., is called real estate. 512. Officers appointed to estimate the value of prop- erty and to apportion the sum to be raised among the indi- viduals are called assessors. 513. A property tax is reckoned at a certain per cent on the estimated value of each person's property, or at a given number of mills or cents on $1, $100, or $1000. 202 PERCENTAGE. 514. An income tax is reckoned at a fixed per cent on the net income of a person after certain deductions have been made. 515. Illustrative Example. The whole amount to be raised for State, county, and town taxes in a certain town is $ 10600. The property of the town is valued at $ 1250000, and there are 300 polls, each taxed $ 2. What is the tax on $ 1 ? What is the tax of E. StUes, who has $4000 worth of real estate and $1000 worth of per- sonal property, and who pays 1 poU tax ? Explanation. — $10600 less the amount of poll taxes leaves $10000 to be levied on $1250000, wMch is 8 miUs on $ 1. If E. Stiles pays 8 mills on $ 1, on $ 5000 he will pay- 5000 times 8 mills, or $40. $40 plus his poll tax of $2 is $42. Ans. 8 milla on$l; $42 tax. 516. From the above may he derived the following rules for assessment of taxes : I. To find the rate of the property tax : Deduct from the whole amount to he raised the amoimt of the poll taxes, and divide the remaiiider ly the amount of taxable property. II. To find each person's tax : Multiply each person's tax- able property ly the rate, and to the product add hii poll tax. 517. Examples for the Slate. 92. The tax levied by a certain town is $ 46800 ; the valuar tion of the town is $ 3600000, and there are 1800 polls, at $ 1 each. What is the tax on $ 1 ? What is the tax of A, who has $ 15000, and who pays a poll tax of $ 1 ? WRITTEN WORK. $10600 600 125|0000) 1|0000 0.008 6000 $40. $40+ $2 = $42 Ans. $0,008; $42. TAXES. 202 93. The valuation of a school district is % 48000. A tax of $ 120 is levied for the repairs upon a school-house. What is the tax on $ 1 ? What is assessed upon a person having $3500 of taxable property? 94. What is the net tax in a town whose taxable property is 1430000, the rate 12 mills on the dollar, 6% of the tax assessed being paid for collecting ? 95. The school-tax of a certain town being $ 6625, at the rate of 3f mills on the doUar of taxable property, what is the taxable property ? 96. The amount of money to be raised by taxes in the town of H is $212093.20; the taxable property is $11522400; there are 3350 polls, each taxed $ 1.40. Knd the tax on $ 1. Note. Assessors commonly construct a table giving tke tax on con- venient amounts of property at the determined rate. 518. TAX TABLE. Showing the tax on various sums, at the rate of 18 Tnilln on $ 1. Pkop. Tax. Prop. Tax. Prop. Tax. Prop. Tax. Prop. Tax. 1 1 $0,018 $10 $0.18 $100 $1.80 $1000 $18 $ 10000 $180 2 0.036 20 0.36 200 3.60 2000 36 20000 360 3 0.054 30 0.54 300 5.40 3000 54 30000 540 4 0.072 40 0.72 400 7.20 4000 72 40000 720 5 0.090 50 0.90 500 9.00 5000 90 50000 900 6 0.108 60 1.08 600 10.80 6000 108 60000 1080 7 0.126 70 1.26 700 12.60 7000 126 70000 1260 8 0.144 80 1.44 800 14.40 8000 144 •80000 1440 9 0.162 90 1.62 900 16.20 9000 162 90000 1620 97. Find by the above table the tax on $ 4250. Note. Find the tax on 1 4000, |200, and $50 separately, and add tJie results Find by the above table the tax 98. Of A on $3000. 102. Of Eon $9068. 99. Of B on $ 2800. 103. Of F on $ 6565. 100. Of Con $7850. 104. Of G on $5687. 101. Of Don $1565. 105. Of Hon $10793. 204 PERCENTAGE. CUSTOMS OE DUTIES. 519. The expenses of the national government are met in part by taxes upon imported goods; these taxes are called custoxas or dizzies. Note I. A tax called tonnage is laid upon a vessel according to the weight she is estimated to carry. Note II. Places are established by goyemment for the collection of cus- toms or duties ; these places are called ports of entry. Each port of entry has a ctistom house, which is in charge of an officer who collects the customs ; this officer is called the collector of customs. 520. A duty proportioned to the quantity of goods im- ported, is a specific duty. Thus a duty of 30/ a pound on yarn is a specific duty. Note. In estimating specific duties, an allowance is made (1) for waste, or impurities, called draft ; (2) for the weight of boxes,, casks, etc., called tare ; (3) for the waste of liquids, called leakage ; (4) for the breaking of bottles, called 'breakage. 521. The weight of goods, before allowances are made, is called gross weight; and the weight, after all allow- ances are made, is called net -weight. 522. A duty proportioned to the cost of goods in the country from whence they are imported, is an ad valorem duty. Thus a duty of 15 % on iron castings is an ad valorem duty. Note. A list of a ship's cargo containing a description of each package of goods imported, with the price in the currency of the country from whence imported, must be exhibited to the collector. Such a list is called an invoice or manifest. When no'invoice is received, the value of the goods is determined by appraisement. 523. Examples for the Slate. 106. What is the duty at 5/ a gallon, on 238 hogsheads of molasses, 60 gallons in a hogshead ? 107. What is the duty at 30 cents a gallon on 25 barrels of spirits of turpentine, 32 gallons in a barrel, leakage 2 % ? GENERAL REVIEW. 205 108. At 15%, what is the duty on 75 boxes of tin, 112 Iba. in each box, invoiced at 7f a pound, tare 10 pounds a box ? 109. What is the duty at 2^ cents a pound on 13 boxes of raisins, 24 lbs. in a box, tare 6J lbs. a box ? 110. At 25%, what is the duty on 100 dozen watch-crystals invoiced at $ 1.60 a dozen, breakage 3 % ? 111. At 36 %, what is the duty on 200 tons of bar-iron (2240 lbs. to a ton), invoiced at 2^f a pound, tare 5 % ? 112. At 3^ / a pound and 10 % ad valorem, what is the duty on 7147 lbs. of steel, invoiced at 20 cents a pound, damage being 8% ? 113. What is the cost at the store of 2556 lbs. of sugar bought in Havana for $ 148.92, on which is paid $ 35.75 for freight and carting, and 2^f a pound for duties, after deduct mgl5% for tare? 524. General Review, No. 4. 114. Change ^^ to a per cent. 115. Represent 1^% decimally. 116. Change 106:|% to a common fraction in its smallest terms. 117. What is ^ per cent of $ 56.49 ? 118. $ 700 is 140% of what number ? 119. If a percentage is 1 540 and the rate 3%, what is the tese ? 120. 25% of a certain number exceeds 10% of it by $75. What is that number ? 121. A schooner formerly valued at $ 7500 has depreciated 20%. What is her present value ? 122. Find the cost of goods which sell for $ 120 at a gain of 25%. 123. What per cent is 125 of 1200 ? 124. What commission must be paid for collecting $ 17380 at 3^ per cent ? 125. What amount of stock can be bought for $ 9682, allow- ing i per cent brokerage ? 206 PERCENTAGE 126. What is the value of 20 shares bank stock, at 8^ per cent discount, the par value of each share being 1 150 ? 127. How many shares of stock at 35 % advance on a par value of 1100 can be bought for $1215?* 128. Insurance was effected on the ship Susan, to Cadiz and back, for $ 10000 at 27o, and on her return cargo, worth f 7500, at 1 J % . What was the amount of premium, including $ 1 for ■ policy ? 129. What insurance may be covered by a premium of % 28 at \% ? 130. What is the insurance premium at J-% on f of a house ■worth $6000? 131. What is the duty, at 12 1* a lb. and 10% ad valorem, on 20 bags of wool, each containing 115 lbs., valued at 42 cts. per lb.? 525. Miscellaneous Examples. 132. A man paid for a house 14600, for repairs $157.50, and then sold it for 18% above the entire cost. What did he receive for it ? 133. I bought 100 railroad shares at 116| and sold them at 120J. What did I gain, the par value being $100 ? 134. A mason sold 75 barrels of lime at 27% profit, and gained $40.50. What was the cost per barrel ? 135. A broker bought 48 shares of $50-stock at 9J % dis- count and sold them at 2^% premium. How much did he make ? 136. What amount of current money will be given in ex- change for $ 450 of that which is at 5 % discount ? 137. If I buy 10 shares of stock originally worth $100 each at 18 % above par, and sell it at 7 % below par, what do I lose ? 138. A cotton-mill valued at $175000 is insured for J of its value by two companies, the first taking f of the risk at 0.9%, the second the remainder at J % . What is the total cost of the premium ? * See Art. 499, note. MISCELLANEOUS EXAMPLES. 207 139. A school-house was insured for $15600 at 2f %, $1.60 being paid for the policy and survey. What was the entire expense for insurance ? 140. If the school-house named above was lost by fire, what was the net loss to the insurance company ? 141. Suppose I buy 20 shares of stock originally worth $ 50 a share, at 10% discount, and sell at a premium of 8%, what do I make ? 142. A merchant sold some iron for $ 278, and made 15 % . What should he have ^old it for to make 26 % ? 143. Wlien 75 shares of stock originally worth $ 100 a share sell for $ 7556.25, at what per cent above par does it sell ? 144. If a company takes an accident risk of $ 8000 &il\%, and reinsures one haM of it in another company at 1J%, what will the first company gain if no accident occurs ? 145. After losing 11 % of his apples, a dealer has 133.5 bbls. of apples left ; if they cost him $2.50 per bbl., for what must they be sold per bbl. that he may lose nothing upon his pur- chase ? 146. A broker bought insurance stock at 80, and sold it at 112. What per cent did he make upon his investment ? 147. A broker sold 19 shares of stock for $ 1389.85, which was at 4 J % above par. What was the brokerage a.i \% on the par value ? 148. A factory is insured at the rate of $ 2 on $ 100. If the premium, with $ 1 for the policy, is $ 241, and the insurance is upon f of the value of the property, what is the value of the property ? 149. When an insurance stock, originally $ 100 per share, is quoted at 102f, how many shares can be bought for $ 8815, brokerage \%? 160. If a watch sells for $ 60 at a loss of 22%, what should it have sold for to gain 30%? 151. The capital of a gas company is $ 200000, and the net earnings are $10746. What rate of dividend can the com- 208 PERCENTAGE. pany declare, reserving a surplus of $2746 to meet future demands ? 152. A vessel brought into port 12000 melons. 8% proved worthless, 10% of the remainder sold for 18/ apiece, and the rest for 12 J-/ apiece. What was received for the whole? 153. At the sale of a piano, 20% was deducted from the retail price, and 5 % of the balance for cash payment. If the retail price was $ 750, and the wholesale price $ 476, for what per cent advance upon the wholesale price was it then sold ? 164. A regiment of 1000 men was reduced to 850 by sick- ness and battle, the loss by sickness being 60 % as great as by battle. What was the entire per cent of loss ? what by sick- ness ? by battle ? 155. I sold 250 lbs. of fish, gaining thereby $ 3.76, which was 42f % of the cost. What was the cost ? Tor liow much a pound was the fish sold ? 156. A grain dealer's sales amounted in one year to 1 75000 ; f of his receipts were for wheat, on which he made 10 % profit, and the balance for other grains, on which he made 20 % profit What was the cost of the whole stock ? 167. A broker bought stock at 8 % premium, and sold it at 9% discbunt, and lost $610. How many shares originally worth $ 100 each did he buy ? 168. Two horses were sold for $ 144 each ; on one there was a gain of 20%, and on the other a loss of 20%. How much was the gain or loss on both ? 159. What is the cost of 6 hhds. of molasses containing in all 2074 gallons, which was bought in Porto Rico at 42/ a gallon, and on which is paid $ 45.76 for freight and carting, and 6/ a gallon for duty, after deducting 12% for leakage? 160. A certain corporation wishing to increase its stock without multiplying the number of its shares, assessed the stockholders 40% on the par value of their stock, which was $ 500 per share. What was the par value of the stock after the assessment was made ? SIMPLE INTEREST. ''M'^ SECTION XY. SIMPLE INTEREST. 526. A had the use of $ 300 of B's money for a year. At the end of the year he paid B for its use a sum equal to 7 % of the money borrowed. What did he pay for its use ? Ans. $21.00, 527. Money paid for the use of money is interest. 528. The money for the use of which interest is paid is the principal. 529. The sum of the principal and interest is the amount. In the above example, what is the interest ? the principal ? the amount ? 530. Interest is reckoned at a certain per cent of the principal. It is, therefore, a percentage of which the base is the principal. 531. The number of hundredths of the principal taken in finding the interest for one year is the rate per cent per annum, usually called the rate. Note. When a rate of interest is given, it is understood to he the rate per year, unless a different time is stated. 532. The rate of interest established by law is the legal rate. Interest at a rate higher than the legal rate is usury. Note. Debts of all kinds draw interest from the time they become due, but not before, unless it is so specified. Interest on interest unpaid when due is sometimes, though not usually, allowed. 533. Interest on the principal alone is simple interest. Note. The laws regulating rates of interest are frequently changed, but the following is a table compiled from official sources in 1877. 210 SIMPLE INTEREST. 534. Table of Legal Rates of Interest, 1886. When two rates are given in this table, any rate not exceeding the highest is allowed, if agreed upon in writing. States. Eate %. States. Rate %. States. Eate %. States. Bate %. Ala.. Ark.. Arizona. Cal...... Conn. . . Colo Dak.... Del D. C... Fla Ga Idaho... 10- Any Any Any Any Any 10 Any 8 18 III. . . . Ind. . Iowa. Kan. . Ky. .. La — Maine Md. .. Mich. Minn. Miss.. Any Any 10 10 10 Mo. Montana N.H.. N. J.. N. Y.. N. C. Neb.. Nev.. Ohio.. Dr.... Penn. R. I.. 10 Any 10 Any 8 10 Any S.C. Tenn. Texas Utah. Vt. . . . Va.... W. Va W. T.. Wy. , 12 Any Any 10 Any Note I. In this took, when no rate is mentioned or implied, 6% is understood. Note II. In reckoning interest, it is customary to consider a year to he 12 months, and a month 30 days. 535. In reckoning tlie months and days between two dates, take the entire calendar months as months, and then the exact number of days remaining. (See Art. 371.) Note. In computing interest for short periods of time, it is cust^imaiy to take the exact numher of days. 536. Oral Exercises. What is the interest a. Of 1 100 for 1 year at 7%? for 2 years at 3%? b. Of $300 for 2 years at 6%? at 8%? at 11%? at 12%? c. Of $400 for 3^ years at 4%? at 10%? at 7%? at 8%? d. Of $40 for 3 years at 10%? at 6%? at 7%? at 6%? e. What part of a year's interest is the interest on anj' sum of money for 6 mo. ? 2 mo. ? 3 mo. ? 4 mo. ? 1 mo. ? /. At 5 % , what is the interest of $ 600 for 1 year ? for 6 mo. ? 3 mo. ? 4 mo. ? 2 mo. ? g. At 9%, what is the interest of $100 for 1 year? for 1 mo. or 30 days ? for 6 days ? for 1 day ? for 5 days ? h. What is the amount of $ 100 for 4 years 6 months at 8 % ? i. What is the amount of $100 for 1 year 4 months at 5%? j. Wliat is the amount of $ 200 for 3 years 3 months at 10 % ? METHODS OF COMPUTING INTEREST. 211 METHODS OP COMPUTING INTEREST. To THE Teacher. Two methods of computing interest are given m the following pages ; but the teacher is advised to have pupils use but one. The method by aliquot parts will be found on page 308 of the Appendix. GENERAL METHOD. 637. Illusteative Example. Find the interest of 1840 for 4y. 3 mo. 5d. at 8%. WRITTEN WORK. Explanation. — The interest of ,| 840 1 840 X 0.08 X 4 = 1 268.80 ^°' ^ ^^ ^* ® ^" '' -^ ®^° ' "■°^- ^ u' interest tor 4 years is 4 tunes as much, 08 X 95 *"" * 268.80. ^ — ^ = 17.73 3 mo. 5 d. ecLual 95 days. The ^^ Ans. $286.53 interest of $840 for 1 year being $ 840 X 0.08, the interest for 1 day is ^^ of this (Art. 534, Note II.), and for 95 days it is 95 times as much, or $ 17.73, which, added to $268.80, makes $ 286.53, the entire interest. 538. From the example above may be derived the fol- lowing _ ° Rule. 1. To find the interest at any per cent for any number of years : Multiply the principal by the rate for 1 year, arid that product hy the number of years. 2. To find the interest for months and days : Change tlie months to days (Art. 535) and take as many SSOths of a yearns interest as there are days in the given time. 539. This rule may be expressed by the formula : Interest = Principal x Rate x Number of years. 540. Examples for the Slate. 1. What is the interest of $ 720 for 3 y. 7 mo. 6 d. at 8% ? 2. Of $472.30 for 2 y. 2 mo. 12 d. at 4% ? 3. Of $ 400.50 for 3 y. 10 mo. 24 d. at 10% ? 4. Of 18480 for 6 y. 3 mo. 20 d. at 6% ? 5. Of 1 116.20 for 2 y. 10 mo. 16 d. at 7% ? 212 SIMPLE INTEREST. SIX PER CENT METHOD. 541. OreU Exercises. a. At 6 %, what part of the principal is the interest for 1 year ? for 2 months ? b. If the interest for 2 months is 0.01 of the principal, what part of the principal is the interest for any number of months ? A as. One half as tnany hundredths of the principal as there are months. c. At 6%, what is the interest of $500 for 2 mo. ? for 4rao.? 6mo.? 8mo.? lOmo.? 5 mo. ? 7mo.? 16mo.? d. If the interest for 2 months, or 60 days, is 0.01 of the principal, what part of the principal is the interest for 6 days ? e. If the interest for 6 days is 0.001 of the principal, what part of the principal is the interest for any number of days ? Ans. One sixth as many thousandths of the principal as there are days. f. At 6% what is the interest of $ 500 for 6 days ? 1 day ? 2 days ? 3 days ? 12 days ? 18 days ? 24 days ? 542. Illusteative Example. What is the interest of $ 480 for 1 y. 3 mo. 7 d. at 6 % ? at 7 % ? What is the amount at 7 % ? WRITTEN WORK. Explanation. — 1 y. 3 mo. equals * AOQ A QjK 15 mo. The interest for 15 mo. at 07fil 001 i 6% is O.OV-J, or 0.075 of the princi- — * — * pal. The interest for 7 days is 0.001| 2880 0.076J of the principal. Hence the interest 3360 for ly. 3 mo. 7d. at 6 % is 0.076J 80 of the principal. 0.076J of the prin- 6) $36i560 Int. at 6 % . "^P^^ ^^ * 36.56. Q QQ To find the interest at 7 % , we add '. — to the interest at 6% ^ of itself, and $42.65 Int. at 7%. have for the sum $42.65. 480. $480 + $42.65 = $522.65, the $522.65 Amt. at 7%. ^"oi'^t at 7%. Ans. $36.56 ; $42.65 ; $522.65. SIX PER CENT METHOD. 213 543. From the foregoing may be derived the following Rule. 1. To compute interest at 6 % : Take 6 times as many huTidredths as there are years, 1 half as many hundredths as chere are months, and ^ as many thousandths as there an l,ays, and by this decimal multijgly the jgrincipal. 2. To find the interest at any rate other than 6 % : Sav- ing found the interest at 6%, increase or diminish that in- terest by adding or subtracting such ■part of itself as will give the interest at the required rate. 3. To find the amount : Add the principal to the interest. Note I. Observe that l%=^of 6%; 2%= J of 6%; 3% = J of 6%; 4%=6%-2%; 5%=6%-l%; 7%=6%+l%; 7J%=6%+(iof 6%), etc. Note II. It will often be more convenient to increase or diminish the principal before taking the interest instead of increasing or diminishing the interest. Thus, in the foregoing illustrative example we might add to $480 \ of itself and then take 6 % interest on $ 560. This would be the same as the interest at 7 % on $ 480, which is $ 42.65. 544. Examples for the Slate. Find the interest on $ 1 at 6 % 6. For 1 y. 3 mo. 6 d. 10. For 1 y. 1 mo. 10 d. 7. For 4 y. 16 d. 11. For 1 y. 8 mo. 8. For 4 mo. 5 d. 12. For 16 y. 8 mo. 9. For Imo. 26 d. 13. For 7y. 10 mo. 18 d. At 6 % what is the interest 14. Of $300 for 2 y. 5 mo. ? 16. Of 136.18 for 3 y. 7 d.? 16. Of $ 872.32 for 6 y. 2 mo. 16 d. ? 17. Of $ 130.50 for 2 y. 9 mo. 13 d. ? 18. Of $800.20 for 3y. 4 mo. 12 d. ? 19. Of $1000 for 3 y. 10 mo. 2d.? 20. Of $ 25.50 for 1 y. 1 mo. Id.? 21. Of $ 400.37 for 2 y. 6 mo. 26 d. ? 214 SIMPLE INTEREST. What is the interest 22. Of 1 837.36 for 3 y. 2 mo. at 7 %? 23. Of $ 187.50 for 2 mo. 12 d. at 10 % ? 24. Of $1000 from Nov. 11, 1874, to Aug. 15, 1880, at 7%? 25. Of 1130.16 from Eeh. 7, 1874, to Dec. 1, 1878, at 8%? 26. Of 1 19.80 from Oct. 15, 1875, to April 19, 1876, at 5%? 27. Of $ 62.50 from Aug. 3, 1874, to April 11, 1876, at 7^%? Find the amount 28. Of $540 for 3y. 6 mo. at 6%. 29. Of $ 496.60 for 2 y. 2 mo. at 12%. 30. Of $ 830 for 5 y. 4 mo. at 8 % . 31. Of $ 110.10 for 3 y. 6 mo. at 9%. 32. Of $ 896 for 2 y. 6 mo. 16 d. at 6^ %'. 33. Of $ 416 for 3 y. 16 d. at 7%. ' 34. Of $ 720 for 3 y. 9 mo. 19 d. at 8%. 35. A note for $ 150, dated July 6, 1872, was paid Mar. 17, 1874, with interest at 6 % . What was the amount ? 36. I gave my note to a person, Jan. 1, 1877, for $ 387.20 with interest at 7% from date. What should I pay to dir- charge this note Oct. 20, 1877 ? 37. Chase and Fowle bought goods to the following amounts, agreeing to pay 7 % interest from the date of purchase : July 8, 1876, 1 470 ; July 28, $ 235 ; Oct. 2, $ 206. What will be the amount due Jan. 1, 1877 ? Short Method for Days ; Application of 6 per cent Method. 545. Illustrative Example. What is the interest of % 126.80 for 93 days at 6% ? WRITTEN WORK. _ Explanation. — The interest at $ 126.80 6 % for 60 days, or 2 months, Is TocQ T 4. j; en J ^-^l °f ^^^ principal. 0.01 of 1.268 Int. for 60 d. «, , oc or. t_ j i. $ 126.80 may be expressed by mov- ing the decimal point two places 0.634 " " 30 d. 0.063 " " 3d. towards the left; this gives f 1.268- Ans. $ 1.966 " " 93 d. The interest for 1 month, or 30 ACCUUATE INTEREST. 215 days, is \ of $1,268, or $0,634, and for 3 days it is ^ of $0,634, or $ 0.063. Adding these interests, $ 1.268 + $0,634 + $ 0.063 = $ 1.965. Ans. $1.97. 546. From the foregoing may be derived the following Rule. 1. Fivd the interest for 60 days at 6% hy taking 0.01 of the principal. 2. For other periods of time. Take convenient multiples or aliquot parts of the interest for 60 days. 547. Examples for the Slate. Find the interest of (38.) 1 300 for 93 d. at 6 % . (40.) $ 1000 for 33 d. at 10 % . (39.) 1 250 for 95 d. at 7 % . (41.) $ 280 for 127 d. at 12 % . (42.) 1 270.80 from Aug. 20 to Oct. 30 at 8 %. [Exact days.] (43.) $416.60 from Nov. 12, 1875, to Feb. 5, 1876, at 5%. (44.) $1560.50 from Mar. 27, 1875, to June 7, 1875, at 9%. (45.) $6000 from Nov. 15, 1875, to March 7, 1876, at 6%. ACCUEATE INTEREST. Note. The above methods of performing examples in interest heing based upon the supposition that a year equals 12 months of 30 days each, or 360 days, though in common use, are not exact. The government of the United States and that of Great Britain pay accurate interest. 548. To obtain accurate interest for months and days : Find the exact number of days between the given dates, and take as many 365ths of a year's interest as there are days. 549. Examples for the Slate. 46. Find the accurate interest of $2000 from Mar. 1 to Aug. 10 at 5%. What is the accurate interest 47. Of $ 700 from May 7 to July 9 at 7^%? 48. Of $ 20000 from April 4 to July 7 at 7 % ? 49. Of $ 1000 from Nov. 15, 1875, to April 1, 1876, at 5%? For additional examples in interest, see page 253. 216 SIMPLE INTEREST. PARTIAL PAYMENTS. 550. [demand note.] S^oi vauce iececveaf, 3^ ^ief^nide ■iit /i^y ^ ^^ oiaei o^ on t^'mana, w 551. The above is the written promise of one person, Flint, to pay another person, Gleason, or any one to whom Gleason may order it paid, a certain sum of money, % 470.60, for value received. Such a promise "is called a promissory note, or simply a note. 552. The sum named in the note (as $470.60 above) is the face of the note. To discharge the interest and in part pay the above note a payment of I 94.13 was made Nov. 1, 1874. What balance then remained due ? Ans. $ 400. Suppose the above balance of $400 to remain on interest from Nov. 1, 1874, to Nov. 1, 1875, when a payment of $ 224 was made, what sum then remained due ? Ans. $200. 553. Payments in part of a note or other debt, as the payments described above, are partial payments. 554. A record of the sum paid, with the date of the payment, is made upon the back of the note ; such a record is an indorsement. The method adopted by the Supreme Court of the United States, and by most of the States, for computing interest in case of partial pay- ments, requires (1.) That a payment be applied first to discharge accrued interest, and then, if the payment is large enough, to reduce the principal. (2.) That no unpadd interest be added to the principal to draw interest. PARTIAL PAYMENTS. 217 555. Illustrative Example. A note for $ 600, dated June 20, 1874, had payments indorsed upon it as follows : Oct. 2, 1874, 1110.20. May 23, 1876, $125.25. Feb. 29, 1876, 24.00. Dec. 11, 1876, 113.20. Find the balance due Jan. 21, 1877 ; interest 6%. WRITTEN WORK. Principal from June 20, 1874 ... $ 600.00 Interest to Oct. 2, 1874 (3 mo. 12 d.) . . 10.20 Amount 610.20 First payment, Oct. 2, 1874 .... 1 10.20 New principal from Oct. 2, 1874 .... 500.00 Interest on f 500 to Feb. 29, 1876 (1 y. 4 mo. 27 d.) f -12.25. Second payment, f 24 will not discharge interest. Interest on 1 500 from Oct. 2, 1874, to May 23, 1876 (1 y. 7 mo. 21 d.) 49.25 Amount 549.25 Second and third payments, $24 +$125.25 . . 149.25 New principal from May 23, 1876 . . . 400.00 Interest to Dec. 11, 1876 (6 mo. 18 d.) . . . 13.20 Amount 413.20 Fourth payment . . . . . .113.20 New principal from Dec. 11, 1876 . . . 300.00 Interest to Jan. 21, 1877 (1 mo. 10 d.) . . . 2.00 Balance due Jan. 21, 1877 . . . {Ans.) $ 302.00 656. The above is in accordance with The United States Rule for Partial Payments. 1. Fiifid the amount of the principal to the time when the -payment or the sum of the payments equals or exceeds the interest; take from this amount a sum equal to the pay- ment or payments. 2. With the remainder as a new principal, proceed as lefore, to the time of settlement. 218 SIMPLE INTEREST. 557. Examples for the Slate. 50. Oct. 12, 1873, I gave my note on demand, with interest at 6 %, for 1 480 ; Feb. 6, 1874, I paid $ 120. What remained due Aug. 24, 1874 ? 51. I held a note for $ 600, which bore interest at 6% from May 10, 1869 ; Sept. 16, 1870, 1 received $ 140 ; July 28, 1872, I received $ 60. What remained due Sept. 4, 1872 ? 62. June 15, 1873, George Rich borrowed of John Jones S2000, and gave his note for the same, with interest at 8%. Aug. 27, 1874, a payment of $ 1450 was made, and a new note given for the balance. For what sum was the new note given ? Write the new note in proper form, dating it at Boston. 53. A note for $ 1000, dated Oct. 5, 1874, was indorsed as follows: Dec. 8, 1874, $125; May 12, 1875, $316; Sept. 2, 1875, $417. What balance was due March 9, 1876; interest 6%? 64. What balance will be due July 1, 1881, on a note of $935 on interest from Sept. 1,, 1875, and indorsed $125.75, Jan. 15, 1876; $250, March 26, 1877; $300, May 10, 1877; interest being 6 % . (65.) $ 425. New York, July 13, 1869. Six months after date I promise to pay A. Hyde & Co. Four Hundred Twenty-five Dollars, with interest at 6 % ; value received. Stewabt E. French. Indorsetnents : Aug. 9, 1871, 1 50; Nov. 17, 1872, 1 150. What was due July 12, 1873? (56.) $ 800. St. Louis, July IS, 1870. For value received. We jointly and severally promise to pay H. Hooker, or order. Eight Hundred Dollars on demand, with interest at 7%. James Holland. Henky Holland. Indorsements: AprU 18, 1871, $100; Dec. 31, 1872, |70; June 14, 1874, $62.50. What was due J uly 14, 1875 ? partial payments. 219 558. Illustbative Example. tnfyiea^ a^ u fo / inzuie ieceovea. q^ . ^pj-^ . Indorsements: Aug. 16, 1876, |200 ; Oct. 8, 1876, |480 ; Feb. 20, 1877, 1 49.92. What balance was due July 1, 1877 ? 559. When partial payments are made upon notes on interest for short periods of time, as upon the above, inter- est is often computed by the following, caUed The Merchants' Rule. 1. Compute interest on the princvpdl from the time it begins to draw interest to the time of settlement, and also on each payment from WBITTEN WORK OP EXAMPLE ABOVE. ^^ **»"« ^* *« "^^ Principal on interest from July 7, '76 $800.00 *" ^^^ **"** "-^ **^^' Interest to July 1, '77 (11 mo. 24 d.) . 47.20 '"*'**• Amount of note .... 847.20 2. Takethediffer- Vaymsni, Aug. 16, '76 . 200.00 ence between the sum Interest to July 1, '77 (lo mo. I5d.) 10.50 of the principal and Payment, Oct. 8, '76 . . 480.00 its interest and the Interest to July 1, '77 (smo. 23 d.) 21.04 ,„„ of the payments Payment, Feb.- 20, '77 . 49.92 j L ■ ■ f . T 1 ^ 1 T T 1 !Hn /. , , \ ■, nn «»*» their interests; Interest to July 1, '77 (4 mo. lid.) 1.09 „„„ I- „ this difference will Balance due . . . . Ans. JMM ie the balance due. 560. Examples for the Slate. (57.) 1 10000/^. Washington, Oct. 3, 1875. In two months from date I promise to pay to the order of Cyrus Parsons, at Suffolk Bank, Boston, Ten Thousand ^^ Dollars, with interest at 6%; value received, -r 'TJnKTnN- Indorsements: Nov. 5, 1875, $672.41; Nov. 15, 1875, 17682.42 ; Nov. 16, 1875, $437.98; Nov. 19, 1875, $833.42. What was the balance due on the above when it became due ? 220 SIMPLE INTEREST. (58.) $ 1200. Baltimobb, April 1, 1875^ One year from date, for value received, I promise to pay B. F. Bryant, or order, Twelve Hundred Dollars, witli interest at 7%. Isaac C. Fellows. Indorsements: April 12, 1875, $161.08; July 19, 1875, $224.14; July 28, 1875, $17.90 ; Jan. 29, 1876, $ 100.25. What was due on the above note April 1, 1876 ? For annual interest, also for Vermont, New Hampshire, and Connecticut rules for partial payments, with annual interest, see Appendix, pages 309 and 310. PROBLEMS IN INTEREST. To find the Time, having the Interest, Principal, and Rate given. 561. Illustrative Example. In what time will $ 480 on interest at 5 % yield $ 36 of interest ? WRITTEN WORK. ExplaitMtion. — The interest of $ 480 for 1 1 480 X 0.05 = $ 24. year at 5 % is 1 24. $ 36 -H 1 24 = 11-. Since $ 480 at 5 % yields $24 of interest in 1^ yr. = 1 yr. 6 mo. ^ J"®*'"' *° ^^^^"^ ^ ^^ ^* "^^^^ require as many years as there are times $ 24 in $ 36, which is 1\. Ans. 1 yr. 6 mo. 562. From the above may be derived the following Sule. To find the time, having the principal, interest, and rate given : Divide the given interest hy the interest of the prin- cipal at the given rate for 1 year ; the quotient will be the number of years. This rule may be expressed by the formula : Interest 1. Number of years ■■ Principal x Rate Note. It will often he found more convenient to divide hy the interest for 1 month or 1 day, in which case the answer will be in months or in days. PROBLEMS IN INTEREST. 221 563. Examples for the Slate. In what time will (59.) $ 400 gain $ 20 at 6 % ? (62.) $ 3000 gain $ 205 at 5 % ? (60.) 1 500 gain $ 60 at 4 % ? (63.) $ 408 gain $ 170 at 7^ % ? (61 .) $ 640 gain $ 67.20 at 7 % ? (64.) $450 gain $192.30 at 8 % ? 65. In what time will $280 amount to $301 at 5%? Note. To find interest, subtract $ 280 from $ 301. 66. How long must a note of $ 7500 run to amount to $ 7800 at8%? 67. In what time will $500 double itself at 1%? at 2%? at3%? at6%? at 10%? To find the Rate, having the Interest, Principal, and Time given. 564. Illustrative Example. The interest on $200 for 10 mo. 24 d. was % 14.40 ; what was the rate % ? WEITTEN WORK. Explanation. — The interest of $ 200 for $200x0.009 = $1.80. 10mo.24d. at 1% is |1.80. * 14 40 — U 1 80 — 8 Since the interest at 1% on f 200 for r»^ ■ 10 mo. 24 d. is 1 1.80, to yield 1 14.40 the rate must be as many times 1 % as there are times $ 1.80 in $ 14.40, which is 8. Am. 8%. 565. From the above may be derived the following Rule. To find the rate, having the interest, principal, and time given: Divide the given interest hy the interest of the prin- cipal for the given time at 1 % ; the quotient will be the number of the per cent. The above rule may be expressed by the formula : Interest Principal x Number of years 566. Examples for the Slate. 68. At what rate % will $ 360 gain $ 40.80 in 1 y. 5 mo. ? 69. At what rate % will $ 100 gain 1 33 \ in 12 y. 6 mo. ? 222 SIMPLE INTEREST. At what rate % 70. Will $ 250 gain $ 3.75 in 4 mo. ? 71. Will $25 gain $ 7.87^ in 3 y. 6 mo. ? 72. Will $100 gain $25 in 1\ y.? 73. The amount of 1 75 for 2 y. 6 mo. was 1 78.75 ; wb^i was the rate % ? Note. To find the interest, deduct 1 75 from $ 78.76. 74. A note of $ 50 on interest from Feb. 29, 1872, to Feb. 28, 1874, amounted to $55.25; what was the rate %? 75. When a note of $1000 amounts to |1058.33J in 7 mo., what is the rate % ? To find the Principal, having the Interest or Amount, the Time, and the Rate given. 567. Illustrative Example I. What principal on in- terest at 6% for 3 y. 4 mo. will yield $ 80 of interest ? WRITTEN WORK. Explanation. — The. interest of |1 at 1 X 0.06 X 3^ = 0.20 6 % for 3 y. 4 mo. is 1 0.20. $ 80.00 H- $ 0.20 = 400 S™<=e 1 dollar of principal at 6 % in 3 y. A $ 400 ^ ^°' yi®-^*^^ ^^ cents of interest, to yield $ 80 of interest will require as many dol- lars of principal as there are times 20 cents in f 80, which is 400. Am. 1 400. 568. Illustrative Example II. What principal on interest at 10% for 2 y. 6 mo. will amount to $478.50 ? WRITTEN WORK. Explanation. — Ths interest of $1 1 X 0.10 X 2i = 0.25 for 2 y. 6 mo. at 10 % is 1 0.25, and the $1.25) $478.50 (382.8 amount of |1 is $1.25. 375 Since 1 1 of principal at 16 % in 2 y. 6 mo. amounts to $1.25, to amount to $478.50 will require as many dollars of principal as there are times 1 1.25 350 in $478.50, which is 382.8. 250 Ans. $ 382.80 1035 1000 1000 etc. PROBLEMS IN INTEREST. 223 569. From the foregoing may be derived the following Rules. I. To find the principal, having the interest, the time, and the rate given : Divide the given i/nterest by the interest of $1 for the given time and rate. II. To find the principal, having the amount, the time, , and the rate given : Divide the given amoimt by the amount of $1 for the given time and rate. The ahove rules may be expressed by .the formulas = Interest 3. Principal = 4. Principal = Eate X Number of years Amount 1 + Kate X Number of years 570. Examples for the Slate. What principal on interest 76. At 6% will gain $ 15 in 2 years ? 77. At 5% will gain $20 in 4 years ? 78. At 3% will gain 1 76.50 in 2 y. 6 mo. ? 79. At 4% wiU gain $ 1.705 in 7 mo. 15 d. ? 80. At 6 % will gain 1 4.128 in 11 mo. 14 d. ? Note. 4.128 -^ 0.057 J (both changed to thirds of thousandths) equals 12.384 -r- 0.172. 81. At 2% a month wiU gain $ 24 in 60 days ? 82. At 6% will amount to $ 870 in 7 y. 6 mo. ? 83. At 5% will amount to $2072.25 in 30 d. ? 84. At 1 % a month will amount to $ 412 in 90 d. ? 85. What sum on interest 3jyrs. at 5^% -rrill amount to $100? 86. What sum put upon interest Jan. 1, 1875, at 7% will amount to $ 343.75, Feb. 1, 1877 ? 87. What principal put upon interest to-day at 6% will amount to $206.25 in 7 mo. 15 d. ? 224 SIMPLE INTEREST. PRESENT WORTH AND DISCOUNT. 571. Illustrative Example. If one person owes another $ 214, to be paid 1 year hence, without interest, what sum should be paid to-day to discharge the debt, the current rate of interest being 7 per cent ? WRITTEN WORK. Explanation. — In justice to both parties, 1.07) 214.00 (200 ^^^^ ^ ^'^"^ should be paid to-day as would, 2j^4 if put at interest at 7%, in 1 year amount to $214. A f 200 Since 1 1 in 1 year at 7 % amounts to $1.07, it would require as many dollars to amount to $ 214 as there are times $ 1.07 in $ 214, which is 200. Ans. $ 200. 572. A sum which will without loss to either party discharge a debt at a given, time before the debt is due is the present worth of the debt. 573. A sum deducted from a debt or from a price is discount. The difference between the face of a debt and the present worth is the true discount. What is the present worth in the example above ? What is the true discount 1 Note. — It will be seen that the present worth is tlie principal, the true discount is the interest, and the sum due at a future time is the amount. This subject is then an application of that illustrated in Ait. 568. 574. From the illustrative example above may be de- rived the following Rules. I. To find the present worth : Divide the given delt by the amount of $1 for the given time and rate. II. To find the true discount : Subtract the present worth from the face of the debt. PRESENT WORTH AND DISCOUNT. 225 575. Examples for the Slate. The current rate of interest being 6%, what is the present worth and what is the true discount 88. Of % 27.50, due 1 year 8 months hence ? 89. Of $ 100.96, due 8 months hence ? 90. Of 1200, due in 3 months? 91. Of $ 175.80, due in 9 months 20 days ? 92. Of % 661.37^, due in 3 months 15 days ? 93. What is the present worth and true discount of $ 1609.30, due in 10 months 24 days, current rate o % ? 94. If a hill of $ 600 is payable in 3 months after May 1, without interest, what sum will discharge it June 1, current rate of interest being 10 % ? 95. Macomber «fe Earle sold goods to the amount of $ 138.48 on 6 months' credit. For how much ready money could they afford to sell the same goods, the use of the money being worth to them 2% a month? 96. A merchant bought goods to the amount of $ 1574, one half payable in 3 months and the rest in 6 months, without interest. What sum would pay the debt at the time of pur- chase, rate 7%? 97. A dealer bought $ 1500 worth of grain on 6 months' credit, and sold it immediately for 10 % advance. If with the proceeds he paid the present worth of the $1500, rate 8%, what sum remained ? 98. A bookseller bought $ 240 worth of books at a discount of 33J% on the amount of his bill, and 5% on the balance for present payment. He then sold the books on 3 months' time for the price at which they were billed to him. Money being worth 7%, and the purchaser discounting his own bill by true present worth at the time of purchase, what was the bookseller's gain ? For other examples in present worth, see page 253» 226 SIMPLE INTEREST. BANK DISCOUNT. 576. Holding a note against James Peak for foOO, dated April 1, and given for 4 months, without interest, and desiring the money April 1, 1 transfer the note to a bank, and allowing the bank to take interest on the sum named in the note for 4 months, and 3 days ($ 10.25), receive from the bank the balance ($ 489.75) in cash. The note is then said to be discounted. The sum named in the note is called the face of the note. Before transferring the above note, I endorsed it by writ- ing my name across the back and thus became responsible for the payment of the note when due. 577. The three days for which interest is taken beyond the specified time for paying a note are called days of grace. Note I. A note is nominally due at the expiration of the time specified in the note, but it is not legally due till the expiration of the 3 days of grace. A note is said to mature when it is legally due. 578. The interest upon the face of a note from the time it is discounted to the time it matures is bank discount. What is the bank discount in the example given ? 579. The face of a note, less the discount, is the pro- ceeds, avails, or cash value of the note. What are the proceeds in the example given ? Note II. The time when a note is nominally and when legally due is usually written with a line between the dates ; thus, August 1 '4. Note III. When a note is given for months, calendar months are under- stood, and the note is nominally due on the day corresponding with its date ; if the month in which it falls due has no corresponding day it is due on the last day of that month. Note IT. Notes maturing on Sunday or on a legal holiday must be paid on the business day next preceding. Note V. In computing bank discount, the more general custom is to reckon the time in days ; hence, in the examples in bank discount which follow, the time is so reckoned, when dates are given. BANK DISCOUNT. 227 580. Illustrative Example. What is the bank dis- count of a note for $ 400, payable in 90 days, discount at 7 % ? What are the proceeds ? WRITTEN WORK. Explanation. — Bank discount is in- « 400 terest for the specified time and 3 days 0.0155 "^S'^^'=^- a\ ft QAnn '^^^ interest of $ 400 for 93 days at 6) 6.2000 7% is $7.23, the discount. $400 less 1.0333 17.23 equals $392.77, the proceeds of $7^2333 the note. f 400- $723-^39277 ^™s. $7.23 discount; $ 392.77 proceeds. 581. From the above may be derived the following Rules. I. To find bank discount on a note due at a future time, without interest : Compute interest on the face of the note from the, time of diacount to maturity (including the three days of grace). II. To find the proceeds of the note : Subtract the dis- count from the face of the note. Note. When a, note drawing interest is discounted, the discount is computed upon the amount of the note at the time of its maturity. 582. Examples for the Slate. 99. What is the bank discount of a note for $ 750, payable in 30 days, discount 6 % ? What are the avails ? Find the bank discount and proceeds of a note 100. For $1000, payable in 90 d., discount 7%. 101. For $300, payable in 4 mo., discount 8%. 102. For f 700, dated Dec. 10, payable in 69 days, and dis- counted at date at 10 % . 103. For $ 500, dated Aug. 20, payable in 3 mo., and dis- counted at date at 7^%. 228 SIMPLE INTEBSST. Find the bank discount and proceeds of a note 104. For $ 290, dated Dec. 30, 1877, payable in 2 mo., and discounted at date at 9%. 105. For $ 500, dated May 10, pjiyable in 90 days, and dis- counted June 9 at 6%. 106. For % 256.84, dated Oct. 28, payable in 60 days, and discounted Nov. 12 at 12%. 107. For $ 1200, dated Jan. 31, payable in 3 months, and discounted March 8 at 5%. 108. I bought a horse and carriage for $ 324, for which I gave my note Nov. 5, payable in 1 year, with interest at 6%. What would be the avails of this note at a bank, Aug. 1, dis- count 7%?* 109. Find the bank discount and avails of the following note, discounted Feb. 12, 1876, at 10%. $ 4000. San Francisco, Nov. 7, 1875. Six months from date, with interest at 10%, I promise to pay F. Egleston & Co., or order. Four Thousand Dollars ; value received. ' Jambs Noble. 583. Illustrative Example. For what sum must a note be drawn, payable in 60 days, without interest, that the avails may equal $591.60 when the note is discounted at a bank at 8 % ? WRITTEN WORK. Explanation. — Tho Bank discount of $ 1 for 63 d. = $ 0.014 l>ank discount of 1 1 foi Avails of $1 for 63 d. = 0.986 63 days at 8% is f 0.014; $ 691.60 - $ 0.986 = 600 ^^'"^' *^^ ''^^jl' f f ] . „„ „ discounted will be 1 1 minus $0,014, whicl, equals |0.986. Since the avails of 1 1 are 1 0.986, that the avails may be $ 591.60 the note must be drawn for as many dollars as there are times 1 0.986 in $ 591.60, which is 600. Ans. 1 600. * See Art. 581, note. BANK DISCOUNT. 229 584. From the foregoing may be derived the following Rule. To find the face of a note which discounted at a bank will yield given proceeds ; Divide the given proceeds hy the proceeds of 1 dollar for the given rate and time, with S days of grace. Note. To find the face of the note when the discount is given : Divide the given discount by the discount of $1 for the given rate and time, with 3 days of grace. 585. Examples for the Slate. 110. For what sum must a 30 days' note, without interest, be drawn that the avails at 6 % discount may be $ 80 ? 111. For what must a 4 months' note, without interest, be drawn that when discounted at a bank it may yield $ 489.76 at 6% discount ? 112. What must be the face of a note given for 90 days, without interest, that the avails at a bank may be $ 1469, dis- count being 8%? 113. What was the face of a note given for 45 days, not bearing interest, on which the bank discount at 9 % was ^ 11.40 ? 586. Miscellaneous. 114. What difference does it make in the avails of a note for $200, payable without interest in 18 months, whether it be re(-koned.by true or by bank discount, rate 8% ? 115. What will be the difference between the true and the bank discount of a note for $ 9171, payable May 9, 1878, and discounted Jan. 15, 1878, at 6 % ? $500. Richmond, Oct. 5, 1876. For value received, I promise to pay Charles Towle, or order. Five Hundred Dollars in three months. James Allen. 116. What cash must be paid to discharge the above note at its date by true present worth, rate of interest 6 % ? 117. What would be the avails of it at a bank, Dec. 5, 1876 ? 230 SIMPLE INTEREST. 118. What would be the amount of it, March 17, 1877 ? 119. What would be the true discount of it, Nov. 5, 1876 ? 120. What would be the bank discount of it, Nov. 5, 1876 ? For otlier examples in bank discount, see page 253. COMMERCIAL DISCOUNT. 587. Business men are usually allowed a deduction foi making cash payment for goods purchased on time. Notes also not bearing interest are discounted by the deduction of a certain per cent, not wholly depending upon the time. Such a deduction is called business or commercial dis- count. 588. Examples for the Slate. 121. A merchant bought a lot of goods amounting to $ 124, on 30 days' credit ; 5 % discount on the price was allowed for making payment at the time of purchase. What was paid ? 122. A man having bought a bill of goods amounting to $468.20 on 6 months' time, cashed the bill for 10% off. What did he pay ? 123. What is the cash value of a bill of cloth amounting to $347.20, on the face of which a discount of 6% is made, and on the balance another of 5 % ? 124. What is the difference between discounting a bill of $1000 at 33 J % and taking 10% off from the remainder, and discounting the whole bill at 43 J % ? 125. A person paid 1 1.14 per yard for goods after a dis- count of 6 % had been made upon the invoice price. What was the invoice price ? Note. Since 5 % had been deducted, 95 % remained. 126. What was the invoice price of a lot of French plate- glass for which I paid $ 39 per pane after a discount of 40 % had been made ? 127. If from the retail price of a book 20 % is deducted, and a discount of 10 % is made upon the balance, and then the book sells for $ 1.33, what is the retail price ? COMPOUND INTEREST. 231 COMPOUND INTEREST. 589, A sum of $ 500 was loaned at 7%, interest payable annually. At the end of the first year the interest for that year was added to the principal, and upon the amount as a new principal the interest was reckoned for the second year. The amount for the second year formed a new principal, upon which interest was reckoned for the next six months, at the end of which time the note, with interest, was paid. What was the amoimt then due ? What was the interest gained ? WRITTEN WORK. Principal $500. Interest for 1st year 35. Amount, or 2d principal .... 535. Interest for 2d year 37.45 Amount, or 3d principal .... 572.45 Interest for 6 months .... 20.0357 Amount $592.49 utAns. 1st principal 500. Interest 1 92.49 M Ans. 590. Interest upon both interest and principal, the sum of the two forming a new principal for specified periods of time, is compound interest. In the example above the interest is compounded annually. It may be compounded semi-annually, or for any period of time agreed upon. 591. From the operation above may be derived the fol- lowing Rule. To compute compound interest : 1. Find the amount of the given principal for the first period of time. With this as a new principal, find the 232 COMPOUND INTEREST. amount for the second period of time, and so continue for the whole time. The last amount is the ammmt re- quired. 2. The last amount minus the given principal is the compound interest. 592. Examples for the Slate. At compound interest, what is the amount 128. Of $200 for 3 years at 6%? 129. Of $ 350.50 for 4 years at 5%? 130. Of $2000 for 3 years 11 months at 6%? 131. Of $2000 for ly. 6 mo. at 7%, interest compounded semi-annually ? Note. Take interest at 3^ % for three intervals of time. 132. What is the compound interest of $ 40 for ly. 2 mo. at 6%, interest compounded semi-annually ? 133. What is the compound interest of $ 900 for 1 y. 1 mo. at 6%, interest compounded quarterly ? 593. The work of computing compound interest may be shortened by the use of the following TABLE, Showing the amount of 9 1 at compound interest from 1 year to 10 years, at 3, i, 4J, 5, 6, and 7 per cent. Years. 3 per cent. 4 per cent. 4J per cent. 5 per cent. 6 per cent. 7 per cent. 1. 1.030000 1.040000 1 .045000 1.050000 1.060000 1.070000 2. 1 .060900 1.081600 1.092025 1.102500 1.123600 1.144900 3. 1.092727 1 .124864 1.141166 1.157625 1.191016 1.225043 4. 1.125509 1.169859 1.192519 1.215506 1.262477 1.310796 5. 1.159274 1.216653 1 .246182 1.276282 1.338226 1.402552 6. 1.194052 1.265319 1 .302260 1.340096 1.418519 1.500730 7. 1 .229874 1.315932 1 .360862 1.407100 1.50.3630 1.605781 8. 1.266770 1.368569 1.422101 1 .477455 1 ..593848 1 .718186 9. 1.304773 1.423312 1.486095 1.551328 1 .689479 1.838459 10. 1.343916 1.480244 1.5.52969 1.628895 1.790848 1.967151 EXAMPLES. 233 594. Illustrative Example. "What is the compound interest of $ 1000 for 2 y. 4 mo. at 7 % ? WRITTEN WORK. Amount of $ 1 at 7 % for 2 years . . . 1 1.1449 1000 Amount of $ 1000 for 2 years .... 1144.90 _ 1.02J Amount of $ 1 144.90 for 4 mo. Amount of $ 1000 for 2 y. 4 mo. 1171.6143 1000. Compound interest 1171.61 Am, Note. In the above operation, the amount of $ 1000 for 2 years is first found, and the amount for the months is then obtained by multiplying by 1.02J. It would be equally well to find the amount of $ 1 for the entire time, and then multiply that amount by 1000. 595. Examples for the Slate. Using the preceding table, find the amount at compound interest 134. Of $ 200 for 2 y. 4 mo. at 7%. 135. Of $580 for 7y. 10 mo. at 6%. 136. What is the compound interest of $ 300 for 3 y. 2 mo. 6 d. at 8 %, interest payable semi-annually ? 137. What is, the compound interest of $ 380 for 1 y. 10 mo. 22 d. at 6%, interest payable semi-annually? 138. If at the age of 25 years, a person puts 1 1000 on in- terest, compounding annually at 6%, what will be the amount due him when he is 40 years old ? Note. First find by the table the amount for 10 years, then find the amount of that amount for 5 years more. For additional examples in oompoimd interest, see page 253. WRITTEN WORK. Due. Items. DayB. Interest. Oct. 1, $262 0- " 10, 220 9 1 0.66 Not, 6, 260 36 3.00 234 AVERAGE OB EQUATION OF PAYMENTS. AVERAGE OR EQUATION OF BiA-YMENTS. 596. Illustrative Example. A debtor owes to one per- son the foRowing sums at the dates specified : Oct. 1, $262; Oct. 10, $ 220 ; Nov. 6, $ 250. At what date may he pay the total of these items without loss of interest to either party ? Interest Method. ETyplamation. — To do this example, we may suppose all the items to be paid at the earliest date at which any item becomes due, viz. Got. 1. This would involve a loss 1 day's int. of 732 = 0.244) 3.66 (15 to the debtor of interest on $ 220 from Oct. 1 to Oct. 10 Oct. 1 + 15 d. = Oct. 16. Am. (9 days), and on 1 250 from Oct. 1 to Nov. 6 (36 days). The interest of $ 220 for 9 days at 12 % * is 1 0. 66 " " " 250 "36 " " 12% is 3.00 Total interest . . . $3.66 That no loss may result, the total of the items, $732, should be paid as many days after Oct. 1 as will be required for $ 732 at 12 % to gain f 3.66 of interest. To find this time, we divide 1 3.66 by the interest of $ 732 for 1 day at 12% (Art. 562, note), and have for a quotient 15. 15 days after Oct. 1 is Oct. 16. Am. Oct. 16. 597. The process of finding the time when the payment of several items, due at different times, may be made at once, without loss of interest to either party, is a,veia,ge, or equation of payments. 598. The date at which several sums due at different times may be paid at once is the average date or equated time ot payment. * Any per cent may be taken, "but 12 per cent (1% a month) is taken for convenienee, the interest then being for every month 0.01 of the principal, and for every 3 days 0.001 of the principal. AVERAGE OB EQUATION OF PAYMENTS. 235 599. From the foregoing operation may be derived Rule I. To find the average time for the payment of several sums due at different times : 1. Select some convenient date ; for example, the earliest date at which any item matures. 2. Compute the interest on each item from the selected date to the date of its maturity. 3. Add the interests thus found; divide their sum by the interest of the sum of the items for one day ; the quo- tient will express the number of days from the selected date to the average date of payment. 4. Add this number to the selected date; the result will he the average date required. 600. The foregoing illustrative example performed by The Product Method. WRITTEN WORK. Explanation. — To do this example Days. Products. by the product method, we select some X 262 = 00 date, for example the earliest date at 9 X 220 = 1980 which any item becomes dite, and sup- 36 X 260 = 9000 P°^^ ^ ^^^ items to be paid at this date. This would involve a loss to the 732) 10980 (15 ^g^^^or of interest on 1 220 for 9 days, Oct. 1 + 16 d. = Oct 16. Ans. ^'^d on $ 250 for 36 days. The interest on $220 for 9 days equals the interest on f 1 for 1980 days ; the interest on f 250 for 36 days equals the interest on f 1 for 9000 days, which together equals the interest on f 1 for 10980 days, but $732 is the sum to be paid, and the time required for the interest on this sum to equal the interest on f 1 for 10980 days will be yj^ of 10980 days, which is 15 days. 15 days after Oct, 1 is Oct, 16, .4ns, Oct. 16. 236 AVERAGE OR EQUATION OF PAYMENTS. 601. From the preceding operation may be derived Rule II. To find tlie average date for the payment of several sums due at different dates : 1. Select some convenient date; for example, the earliest date at which any item matures. 2. Multiply the time each item has to run hy the num- her of dollars in the item. 3. Divide the sum of the products thus obtained by the number of dollars in the sum of the items; the quotient will express the time from the selected date to the average date of payment. 4. Add this time to the selected date ; the result mil be the average date required. 602. Proof. Find the sum of the interests on all items due bbfoee the average date, from the date at which they are severally due to the average date ; also find the sum of the interests on all items due after the average date from that date to the dates at which they are severally due. If these sums are equal, or differ by less than half a day's interest on the sum of all the items, the result is correct. Note I. The examples in tMs "book are performed by the interest method, which has the advantage of brevity when the accountant uses interest tables. The pupil will perform the work by either or by both methods, as directed by the teacher. Note II. Any date may be selected from which to average an account. The last day of the month previous to the earliest day at which any item becomes due is a convenient date. Note III. When any item contains cents, if less than 50, disregard them, if 50 or more, increase the units of dollars by 1 1. Note IV. When a quotient contains a fraction of a day, if less than J, dLsiTegard it; if J or more, call it 1 day. AVERAGE OR EQUATION OF PAYMENTS. 237 603. Examples. 139. What is tlie average date for paying three bills due as follows: March 31, $400 ; April 30, $300; May 30, $200? 140. What is the average date of maturity of three notes of $ 800 each, due respectively Nov. 5, Dec. 8, and Feb. 3 ? 141. What is the average date of maturity of the following items of account, viz., $ 900 due Sept. 10 ; $ 2250.48 due Oct. 21 ; and $ 1049.65 due Oct. 28 ? 142. Eind the equated time for paying $430 due in 5 months; $270 due in 9 months; and $300 due in 8 months? 143. Average the above, having the first item due in 3 months, the others in 9 months each. 144. A gentleman purchased a farm for $ 3600, agreeing to pay $ 600 down, and the remainder in five equal semi-annual instalments. At what time may the whole be paid at once ? 145. When shall a note to settle the following account be made payable ? J. K. INGEKSOL To B, TISH & CO., Dr. 1876. April 10 May 16 June S July 18 To Mdse on 30 days' credit. " 60 " SO " Cash. SOO S20 250 Note. First find at what time each item falls due by adding the time of credit to the date of the item. 146. What is the equated date of maturity of the following ? V. M. HURON To COLTON IKON CO., Dr. 1876. 1 Mar. 11 it 29 Feh. 29 May 8 Jtine 12 To Mdse on SO days^ credit. (I (6 a QQ it ic (t a ii QQ it Cs account ojf H g ^cfoui, S oC^aten^ aei,va?2/t. To George Flint, Esq., JameO- q4. ^U^. New Orleans. ^ / 609. Such a written order for the payment of money is a draft, or Mil of exchange. The method of making payments by drafts or bills of exchange is exchange. Ames will take this draft and send it to Smith, who, when he re- ceives it, will present it to Flint for acceptance. If Flint is willing to obey the order and pay the money, he writes the word "Accepted" across the face of the draft, adds the date, and signs his name. In due time Smith gets the money from Flint, gives up the draft, and the transaction is complete. 242 EXCHANGE. 610. The person who makes and signs a draft is the drawer. The person to whom it is addressed is the drawee. The drawee when he accepts the draft becomes the acceptor. The person to whom the draft is payable is the payee. In the case described above, who is the drawer ? the drawee ? the acceptor ? the payee ? 611. If the payee wishes to transfer the draft to another person, he writes his own name across the back of the paper; this is called an indorsement, and the payee then becomes an indorser. The person to whom the draft is so transferred is an Indorsee. If the indorsee wishes to transfer the draft to a third person, he also wi'ites his name under that of the former indorser. He thus be- comes a second indorser ; and there may be a third in- dorser, a fourth, and so on indefinitely. 612. The person who holds the draft at any time (the payee or the last indorsee) is called the holder. The holder looks for payment first to the acceptor, and then to the indorsers in their order. Each indorser is liable to pay the draft when the acceptor and previous indorsers have failed to do so. To avoid becoming liable, an indorser may write over his name the words " Without recourse." Drafts may be " at sight " or " on time " ; bankers charge less for the latter than for the former, the difference in price being equivalent to a discount for the given time. When our exports to another country, England for example, exceed in value our imports from that country, more money is due to us from the English merchants than is due to them from our merchants. The larger sum due us in England will make it easy for us to buy bills of exchange on England. They will be plenty here, and the price of them vrill fall. If they can be bought for less than their face, they are at a discount, or lelow par. On the contrary, when the value of the goods imported from Eng- land exceeds the value of those sent to England, more money is due EXAMPLES. 243 to the English merchants from us than is due to us from them. The smaller sums due us in England will then make it difficult for us to buy bills of exchange on England, and the price of them will rise. If they cost more than their face, they are at a premiwm, or above pa/r. 613. Bills of exchange are either foreign bills or inland bills. Poreign bills are those which are drawn or are pay- able in a foreign country ; and for this purpose each of the United States is foreign to the others. Inland bills are drawn and payable in the same State. 614. Examples for the Slate. 1. What is the cost in Philadelphia of a draft on San Fran- cisco for f 800 at 1 % premium ? 2. What is the cost of a draft on Detroit for $2500 at J% premium? 3. What is the cost of a draft on New York for $ 700 at 12 days after sight, premium ^%? 4. What is the cost of a sixty days' draft on New York for $ 2000 at 2 % discount ? 5. I bought a bill on Chicago for f 700 at a discount of i%. What did I pay? 615. Illustrative Example. What is the face of a draft on New York bought in St. Louis for $ 8820, when the discount is 2%? WBITTEN WOEK. $1-2% of $ 1 = $ 0.98, cost of $1. $ 8820 ~ $ 0.98 = 9000. Ans. $ 9000. 6. What is the face of a draft that may be bought for $500 at a discount of 1^%? 7. A merchant in New York bought a draft on Cincin- nati at J% premium for $275. What was the face of the draft? 244 EXCHANGE. Exchange with Europe. 616. Exchange with Europe is effected chiefly through large business centres, as London, Paris, Hamburg, etc. In computing foreign exchange, it is necessary to change the values expressed in the currency of one country to equivalent values ex- pressed in the currency of another country. On page 311 of the Appendix will be found a list of the monetary anits of foreign countries, with their values in United States money, as proclaimed by the Secretary of the Treasury, Jan. 1, 1886 ; also on page 312, tables of English, French, and German money. The rates of exchange between this country and the principal busi- ness centres are given from day to day in the newspapers. The follow- ing is an extract showing the exchange value of the pound sterling in United States money ; the number of francs and centimes which equal a dollar ; and the exchange value of 4 marks in cents : "We quote bankers' 60-day bills on London at $4.84@4.84J, and short- sight bills at I 4.86, both in gold. On Paris, francs 5.15 per dollar for short sight, and 5.18| for 60-day bills. Gossler & Co.'s rates on Hamburg for 60-day bills are 95, and short-sight bills 95f . " " In making bills on foreign countries, it is customary to write two or more of the same tenor and date, the payment of either one of which cancels the other one or two. And to provide against accident in their transmission, it is customary to send two, at least, of a set, at different times, or by different modes of conveyance. 617. Examples for the Slate. 8. What was the cost of the follovnng hill in U. S. money, the rate of exchange being $ 4.86 ? (3^ a^n^ of' '(nii -ftia^ a* Sxcncinae, iecona ana I QT'cve (S^Ui-ncAea .^Zounaa a^feiMna, vauce ieceivea, and \ To Messrs. McGalmont Bros. & Co., <^^<^e^. ..jSia/oc^ ^ ^o. 3 Crown Court, London. UNITED STATES BONDS. 245 9. T. Van Horn, of New York, bought of E. J. Birney & Co. a set of exchange payable at sight for £1000 sterling on Brown, Shipley, & Co., of Liverpool, at $ 4.84. What was the cost in gold ? 10. What is the cost in gold of a set of exchange on Paris for 550 francs, exchange being 5.15 per dollar ? 11. What is the cost of a 60 days' bill on Paris for 668.6 irancs, exchange being 5.18| ? 12. What is the cost of a draft on Hamburg for 2O0"marks when the quotation is 96 ? 13. What is the cost in New Orleans of a bill on London for £ 76 10 s, when exchange is there quoted at $ 4.85 J^ ? (See Appendix, page 312.) 14. When exchange is $ 4.86, what is the face of a bill on London which can be bought for $ 9720 ? 15. What is the face of a draft on London which can be bought for $ 1938.42, the rate of exchange being 1 4.84 ? United States Bonds. 618. Governments and corporations sometimes borrow money, giving, as evidence of the loans, certificates or notes payable at or Avithin some definite time, with interest at stated periods. Such certificates or notes are called bonds. 619. Bonds sometimes have certificates attached, prom- ising the holder certain sums of interest as they become due upon the bonds. These interest certificates are called coupons. Note I. When the interest is paid, the coupons are cut off by the holder and given up as receipts. Note II. The extraordinary expenses of the government of the United States during the civil war were met in part by the sale of bonds. 620. United States coupon bonds are issued in the de- nominations of % 50, % 100, % 500, and $ 1000. Eegistered 246 UNITED STATES BONDS. bonds are issued in the same denominations, and also in denominations of $ 5000 and % 10000. Bonds are usually named according to the rate of interest they hear. 62 1 . The following is a list of the more important United States bonds not redeemed in 1886 : Names of Bonds. Payable. Bate of Int. Int. payable. United States 3's United States 4's United States 4J's ( United States - Cur- ( reney 6's On call 3 % Quarterly in gold. Quarterly in gold. Quarterly in gold. j Semi-annually in ( currency. 1907 4 % 1891 ii% 6 %. ... ( 1895 to ] 1899 Bonds are bought and sold as other stocks. Their prices from day to day are quoted in the newspapers. 622. The rules of percentage already illustrated apply to hands. 623. Examples for the Slate. 16. When U. S. bonds are sold at 108^, what is received for eight $ 600 bonds ? 17. When U. S. bonds are worth 106^, what will $ 850 in bonds cost ? 18. What amount in bonds shall I receive for % 2675 in- vested in U. S. 4's at 107 ? 19. What shall I pay a broker for a $ 1000 U. S. 4 % bond at IIOJ, and two $ 1000 U. S. 4^ % bonds at liSf, with his brokerage of ^ % ? 20. How much money must I remit to a broker that he may purchase for me three U. S. bonds of $ 1000 each, the bonds selling at 109, and his commission being \%'i GENERAL BEVIEW. 247 21. When gold was at 102|, I received interest at the rate of 3 ^ in gold on twelve $ 1000 U. S. bonds. What was the value of my interest in paper money ? Note. See page 199, Art. 500. 22. In 1877, 1 purchased some U. S. bonds at 106, which yielded an interest of 6 % per annum payable semi-annually in gold. Gold being at a premium of 3 %, what per cent of my investment in paper money did I receive semi-annually ? 23. Which yielded the greater per cent semi-annually and how much, U. S. bonds at 110 bearing interest at 6 % payable semi-annually in gold, gold at 102J, or a mortgage on real estate paying Z^s 25. Find the height and area of an isosceles triangle whose base is 20 feet long, the other sides being each 26 feet long ? Fig. 22. SOLIDS. SOLIDS. Fig. 23. 285 Fig. 24. [v^-, \\ Prism. Cube. Rectangular Solid. 718. A prism is a solid bounded by two equal and parallel polygons and a number of parallelograms. Note I. It wiU te seen that all rectangular solids are [irisms. Note II. The equal and parallel polygons are the bases of the prism, and the parallelograms taken together form its ccmvex surface. When thft bases are regular polygons and the parallelograms are perpendicular to the bases, the prism is a regular prism. Fig. 26. Fig. 27. A Pyramid. Cylinder. Cone. 719. A pyramid is a solid bounded by one polygon, which is the base, and a number of triangles which terminate in a common point called the vertex. Note. The triangles form the convex surface of the pyramid. When the base of a pyramid is a regular polygon, and a line drawn from the vertex to the middle of the base is perpendicular to the base, the pyramid is a regular pyramid. 720. A solid formed by turning a rectangle about one of its sides as an axis is a cylinder. Note. As the rectangle is turned, the side opposite the axis describes the convex surface of the cylinder, and the other two sides describe parallel circles, which are the bases of the cylinder. 286 MENSURATION. Fig. 28. Fig. 29. 721. A solid formed by turning a right triangle about one of its shorter sides is a cone. Note. As the triangle is turned, the hypothenuse describes the convex surface of the cone, and the side perpendicular to the axis describes a circle, which is the base of the cone. 722. If the upper part of a pyramid or of a cone is. cui. off by a plane parallel to the ) base, the part that remains is a frustum of the pyramid or of the cone. Ft a Pyramid. a Cone. 723. The height of any of the solids here defined is the perpendicular distance from the highest point ^^ above the base to the plane of the base. Thus, in Fig. 30, the line A B indicates the height of the solid. Note. In a regular pyramid or in a cone, the shortest i distance from the vertex to the perimeter (boundary) of the ^ base is the slant height. In the frustum of a regular pyramid or of a cone, the shortest distance between the perimeters of the two bases is the slant height. Thus, in Fig. 30, the line o p indicates the slant height of the frustum. j,j^ g^^ 724. A globe, or sphere, is a solid Abounded by a cuiTed surface, every point of ■which is equally distant from a point vi^ithin, called the centre. Note. A circle which divides a sphere into two equal parts is a great circle of the sphere. A circum- ference, a diameter, or a radius of a gi-eat circle of a sphere is also a circumference, a diameter, or a radius of the sphere itself. Volumes of Solids and Areas of their Convex Surfaces. 725. A PRISM or a cylinder _ inch high contains as many cubic inches as there are square inches in the base. If the height be in- creased to 2, 3, or any number of inches, the volume will be increased in the same proportion. Hence the SOLIDS. 287 Rule. To fiud the volume of a prism or cylinder : Multiply the area of the base hy the height. 726. If a PRISM or a cylinder is 1 iach high, its convex surface contains as many square inches as there are inches in the perimeter of the hase. If the height be increased to 2, 3, or any number of inches, the convex surface will be increased in the same proportion. Hence the Rule. To find the convex surface of an upright prism or cylin- der : Multiply the perimeter of tlie hase hy the height. 727. It can be proved that a pyramid or a cone is equivalent to J of a prism or a cylinder of the same base and height. Hence the Rule. To find the volume of a pyramid or cone : Multiply the area of the hase by the height, and divide the product hy 3. 728. The convex surface of a regular pyramid or cone may be regarded as composed of triangles whose bases form the perimeter of the base of the solid, and whose height is the slant height of the solid. Hence the Rule. To find the convex surface of a regular pyramid or cone : Multiply the perimeter of the hase hy the slant height and divide the product hy 2. 729. It can be proved that the frustum of a pyramid or cone is equivalent to the sum of three pyramids or cones, which have for their common height the height of the frustum, and whose bases are the lower base of the frustum, the upper base, and a mean proportional between them. Hence the Rule. To find the volume of a frustum of a pyramid or cone : Multiply the s^im of the areas of the two bases, plus the square root of their product, by the height, and divide the product by 3. 288 MEN SURA TION. 730. The convex surface of the prdstum of a regular pyramid or cone may be regarded as made up of trapezoids whose parallel sides form the perimeters of the bases, and whose height is the slant height of the frustum. Hence the Rule. To find the convex surface of a frustum of a regular pyramid or cone : Multiply the sum of the perimeters of the two bases ly the slant height and divide the product by 2. 731. It can be proved that the surface of a sphere is equivalent to that of four great circles of the sphere. Hence the Rule. To find the surface of a sphere : Find the, area of a great circle of the sphere, and multiply it by li,. 732. A SPHEEE may be regarded as composed of pyramids whose bases taken together form the surface of the sphere and whose height is the radius. Hence the Rule. To find the volume of a sphere : Multiply the convex surface by the radius and divide the product by 3. 733. The last two rules may be expressed by the follow- ing formulas : Surface of a sphere = 4 Radius^ x 3.1416. Volume of a sphere = J Radius^ x 3.1416. Note. To find the capacity of a cask, see Appendix, page 314 734. Examples for the Slate. 26. How many cubic inches are there in a prism whose base is 8 inches square, and whose height is 7 inches ? 27. How many cubic feet in a prism 5 feet high and having for its base a triangle, each side of which is 10 feet long ? 28. How many cubic feet in a pyramid 10 feet high and having for the base 1 square rod ? EXAMPLES. 289 29. How many cubic inches in the frustum of a pyramid whose bases contain 12 and 108 square inches respectively, and whose height is 18 inches ? 30. How many bushels of corn can be put into a corn-crib 9 feet square at the bottom, 12 feet square at the top, and 8 feet high ? (See Art. 393, note.) 31. What is the convex surface of a prism, the perimeter of whose base is 7 yards 2 feet, and whose height is 5 yards 1 foot ? 32. How many square feet in the surface of a four-sided pyramidal roof, the slant height being 18 feet and the house 20 feet square ? 33. How many feet of boards will cover the sides of an eight-sided tower, the length of each side of the base being 2 feet 9 inches, that of each side of the top 1 foot 10 inches, and the slant height 12 feet ? 34. How many gallons will a pail contain that measures on the inside 14 inches in depth and 11 inches across ? 35. How many square feet of sheet-iron are there in a piece of stove-pipe 9 feet long and 6 inches in diameter, no allowance being made for lapping at the joints ? 36. What is the height of a conical tent, if the diameter of the base is 15 feet and the slant height is 19J feet, and how many cubic feet will the tent contain ? 37. How many square yards of canvas will be required to make this tent, no allowance being made for seams ? 38. How many gallons will a circular vat contain that meas- ures across the bottom 12 feet, across the top 15 feet, the depth being 6 feet ? 39. How many square feet in the surface of a foot-ball 1 foot in diameter ? 40. At 38/ a square foot, what is the cost of painting a globe 6 feat in diameter ? 41. How many cubic feet are there in this globe ? 42. If the diameter of the earth is 7900 miles, and 73^% of the surface of the earth is water, how many square miles of the surface is water ? 290 MENSURATION. SIMILAR SURFACES. 735. Figures which have the same shape are similar figures. Note. The corresponding sides of similar figures are proportional. Hg. 32. 736. We see from the illustration above that a figure 1 inch square contains 1 square inch, a figure 2 inches square contains 4 square inches, and a figure 3 inches square contains 9 square inches, etc. In general, The areas of similar figures are to each other as the squares of their corresponding dimensions. 737. Illustrative Example I. The area of a certain triangle is 120 square feet and its base is 24 feet. What is the area of a similar triangle whose base is 96 feet ? WRITTEN WORK. 4 4 24? : 96^ = 120 : a; ^'^^^^^^^ = 1920. ^«. 1920 sq. ft. 738. Illustrative Example II. One side of a triangle is 40 feet long; what must be the length of a side of a similar triangle containing twice the area ? WRITTEN WORK. l:2=.402:a;2 ^40x40x2 = 66.568... ^m. 56.568... ieui. 739. Examples for the Slate. 43. If a room 16 feet in length requires 22 yards of carpet- ing to cover the floor, how many yards of carpeting will be required for a room 20 feet long and of the same shape ? SIMILAR SOLIDS. 291 44. There is a public park 1320 feet long, containing 25 acres. What is the length of ^, park of the same shape con- taining 49 acres ? 45. If a circular lot of land which is 10 rods in diameter contains 78.5398 square rods, what number of rods will a cir- cular lot contain which is 6 rods in diameter ? 46. If a pipe 2 inches in diameter discharges 20 gallons of water in a given time, how many gallons will a pipe 6 inches in diameter discharge in the same time, no allowance being made for friction ? 47. If it costs 17/ for tin to make a pail 6 inches high, what will it cost for tin to make a similar pail 14 inches high ? 48. If it costs $ 72 for material to paint a spire 50 feet high, what will it cost for material to paint a similar spire 75 feet high ? 49. If the cost of plating a pitcher 6 inches high is $ 1.75, what is the cost of plating a pitcher of the same shape 10 inches high ? 50. What is the height of a pitcher similar to that described above, of which the cost of plating is $ 4.00 ? SIMILAR SOLIDS. 740. Solids which have Fig. 33. the same shape are similar solids. Note. The corresponding dimen- dions of similar solids are propor- tional. 741. We see from the illustration above that a cube whose edge is 1 inch contains 1 cubic inch, a cube whose edge is 2 inches contains 8 cubic inches, a cube whose edge is 8 inches contains 27 cubic inches, etc. In general, The volumes of similar solids are to each other as the cubes of their corresponding dimensions. 292 MENSURATION. 742. Illustrative Example I. If a cube of lead whose edge is 3 inches weighs 12 pounds, what is the weight of a cube of lead whose edge is 2 inches ? ■WRITTEN WORK. 3^ 2» = 12 lb. : X ^y>' thick ? 243. The specific gravity of honey being 1.456, how many kilos of honey may be put into a box 2 ^^ square at the bottom and LS"*" deep? 244. How many metric tons of Egyirtian marble in a rectangular block 0.75"" square at the base and 3" high, the specific gravity of the marble being 2.668 ? 245. If a bar of silver is 2 "m square at the end and 1 *"" long, and has a specific gravity of 10.474, how many kilos does it weigh ? 246. How many sters in a pile of wood 15 " long, 3.5° high, and 1 " wide, and what wiU it cost at $ 3.50 a ster ? 247. How many pounds avoirdupois in 1000 letters, each weighing 18 grams ? [Use equivalent for gram.] 248. Philadelphia being about 160 kilometers from New York, what is the distance in miles ? 330 APPENDIX. 49. Percentage. 249. The prime cost of an article being % 230, at what price must it be sold that a profit of 30 % may be made? 250. A baking powder, when analyzed, was found to contain 38.12 % of starch, 31.82 % of soda, and the remainder was burnt alum. What per cent was burnt alum? 251. If the price of gloves has declined since 1864 from % 2.25 per pair to f 1.25, what per cent has it declined? 252. If water weighs 62J pounds to the cubic foot, and milk 64i pounds, by what per cent is milk heavier than water ? 253. An estate sold for $45000, which was 37^ % below the ap- praised value. What was the appraised value ? 254. How many pounds of bread can a baker make from a barrel of flour, if the bread made is 32 % heavier than the flour used ? 255. How many pounds of flour must a baker use to make 200 pounds of bread, if the bread weighs 32 % more than the flour used ? 256. From a bill amounting to |2463, there is thrown ofi' |63 ; that is equivalent to a discou.nt of what per cent ? 257. A owns | of the copyright of a book, and B owns the remain- der ; but for services on the book they agree that C shall have 5 % of the whole copyright. What per cent will A then have ? What per cent will B have ? 258. A person has a preparation worth $3.20 a pound (A v.). What per cent does he gain by putting it up in powders of 10 grains each, and selling them at the rate of 6 powders for 25 cents ? 259. If by buying collars at the rate of 3 for 25 cents, instead of buying them singly, I save 16f %, what is the price per collar when bought singly ? 260. What is the value of the gold in a ton of ore, if 3 % is gold worth $ 16 an ounce ? [Reckon by the long ton.] 261. A seal weighing 56 pounds has been known to eat a quantity of fish equal to 25 % of its own weight in a day. At this rate, how many pounds would it eat in a common year ? 2G2. A savings-bank having suspended payment, its deposit books are sold at 23 % discount. What could be realized on a book show- ing a deposit of $ 952.17 ? 263. The entire cost of finishing a room was $115. If 5 % of this was for laths at 25 )* a hundred, how many laths were used ? PERCENTAGE. 331 264. What is the retail price of a book, which is sold at 33 J- % be- low the retaU price, and then brings 33J- cents ? 265. When chairs are sold at 1 4.80 per dozen, with a discount of 5 % for cash, what is the cash value of 200 chairs ? 266. A lamp is sold at 10 % below cost, and brings 54 f. What would it bring if sold at 25 % above cost ? 267. If a jockey seUs a horse for 90 % of his cost, he gets $45 less than if he sells him for 120 % of his cost. What was his cost ? 268. A commission merchant gets 5 % commission on his sales. His commissions during the year amount to $ 13685.50 ; what is the whole amount returned to his consignors ? 269. At 5^ % , what is the commission on the sale of 350 barrels of apples at 1 2.80 per barrel, and how much money should the con- signor receive for them ? 270. Find the cost, including \% commission, par value $100, of 17 Old Colony Railway @ 112J. 10 Atchison, Topeka & Santa Fd @ 137 J. 23 Flint & Pere Marquette @ M\. 5 Chicago, Burlington & Quincy @ 124 J. 12 Rutland preferred @ 29f. 271. If the wages paid for work on the backs of chairs is 10 f per chair, and on seats is 8/, how much must be paid for one hundred of each, if the wages are advanced 10 % ? 272. How large a bill of goods can be purchased for a remittance of f 1000, if 20 % discount is allowed for cash ? 273. I bought a lot of leather at 10 % below the asking price, and sold it for $ 2400, and by so doing I gained 33J % on the cost. What was the asking price ? 274. I built a house costing $ 3000 upon a lot which cost $400. The house being burned, the insurance company paid me 75 % of the cost of the house. I then sold the land for 1 1400 ; did I gain or lose by the transactions, and what per cent ? 275. A lot of goods was marked for sale at 40 % above the cost ; if the lot was sold at auction at 30 % below the marked price, was there a gain or a loss on the cost, and of what per cent ? 276. A house is insured for $ 4500 for 5 years, at a premium of $ 45. What is the rale per cent of insurance per year ? 277. What is a building worth if the insurance premium of | % on I of its value, including $ 1 for the policy, equals $ 43 ? 332 APPENDIX. 278. What must I pay to insure some goods worth $3500, at a premium of 1^ % on f of their value ? 279. If these goods should be destroyed by fire, what would be the loss to the owner ? 280. What would be the loss to the underwriters ? 281. What amount must be insured to cover property worth $1800 and a premium of f ^ V 282. A marine insurance company took a risk of $25000 at 3^ %, and reinsured -| of their risk in another company at 3^%. Should the property be destroyed at sea, how much would the first company lose ? 283. The taxes in a school district are assessed upon property valued at f 87000 ; the whole tax is as follows : for building and furnishing a school-house, 1 2100 ; for fuel, care of house, and wages of teacher, $ 510. What is the tax on 1 1 ? What amount of tax is paid by a person whose property is valued at f 4500 ? 284. At 20 % ad valorem, what is the duty on 18 chests of tea, the gross weight being 765 pounds, 6 pounds for tare being allowed on each chest, the tea costing 35 / per pound ? 285. When the duty on a quantity of lace at 30 % ad valorem is 1 115.80, what was the cost of the lace and duty in francs at 1 0.193 per franc ? 286. A merchant imported from Geneva 35 watches at $ 65 each, and 42 watches at 1 125 each. The duty being 35 % ad valorem, what did the watches cost, and for how much apiece must they be sold to make a profit of 15 % ? 60. Interest 287. Find the amount of a $ 300 note given the 25th of August, and paid the 19th of the next December, interest 10 % per annum. 288. A trader borrowed $ 8000 at 5 % per annum. He used this money in a way to yield him $ 1256.42 during the year. How much did he clear by borrowing this money ? 289. What is the interest of 1 cent for 1 day at 1 % a year (365 d.)' 290. Find the amount of each of the following sums at 7 % : $820 from Feb, 10, 1880, to Dec. 4, 1880. $ 1500 from Dec. 15, 1880, to May 10, 1881. 291. A due- bill is given for $ 68.50 with interest at 7^ %. What will the bill amount to if it runs 72 days ? INTEMBST. 333 292. Find the difference between the accurate interest of f 1500 at 6 % and the interest by the common method, the time being from May 24 to Oct. 15, 1881. 293. What per cent on my investment do I save by buying, April 1, $ 30 worth of goods for which I should pay $ 48 the first of the fol- lowing December, money being worth 6 % a year ? 294. A note given May 4, 1879, on demand at 7 % for 1 475, was indorsed July 2, 1879, 1 100 ; Oct. 1, 1879, 1 100 ; what sum re- mained due April 1, 1880 ? 295. Jan. 1, 1879, I gave a note for 1 356 at 6 % . When will this note amount to $ 400 ? 296. In what time will the interest on a sum of money be | of the principal at 6 % per annum ? 297. At what rate per annum will the interest on a sum of money be 0.0215 of the principal in 4 months ? 298. The use of money being worth 6 % a year, what is the pres- ent value of $ 500 due four years hence ? 299. If a merchant sells flour at $9 a barrel, and has to wait 6 months for his pay, at what price could he afford to sell for cash down, the use of money being worth to him 2 % a month ? 300. Sold goods amounting to f 4500, one half on 4 months' credit, the rest on 6 months, and got the notes discounted at a bank at 8 %. What was realized ? . 301. A trader buys 900 pairs of gloves at f 0.75 a pair cash, and immediately sells them for $ 1090 on a note payable in 4 months with- out interest. Suppose he gets the note discounted at a bank for the 4 months, what will he have made ? 302. Write a note on 3 months' time for such a sum that, when dis- counted by a bank at 7 %, the proceeds may be 1 400. 303. What is the amount, at compound interest, of f 100 from April 1, 1876, to Jan. 1, 1879, at 6 % per annum, interest payable semiannually ? 304. What is the equated time for paying the following bills ot goods bought on 30 days' credit : June 2, $ 420 ; June 8, $ 600 ; June 15,1560? 305. A person holding three promissory notes, one for $ 100 pay- able in 2 months, another for $ 300 payable in 4 months, and another for $ 200 payable in 6 months, exchanges them for a single note for the sum total. When should this be payable? 334 APPENDIX. 306. A owes B |2000 clue Oct. 5. If he should pay $ 1200 of it on the 8th of the previous September, when should the balance be paid, that there may be neither gain nor loss of interest ? 307. F. Bates owes A. Smith 1 350, due May 29. If he should pay $ 50 on the 29th of April previous, when should he pay the bal- ance? 308. Find the balance due Flint in the following account, Oct. 1, 1880, interest at 6 %, from the date of the items : — Dr. SETS DAVIS in Aoct. with AMBROSE FLINT. Cl. 1880. 1880. Mar. 29 To Mdse %m 93 Apr. 24 By Mdse $389 51 Apr. 22 " Cash 869 82 May IS " Mdse 379 84 51. Exchange. 309. The value of J 1 being $ 4.866^, find the value of $ 1 in shillings and pence. 310. The value of 1 franc being $ 0.193, what is the value of 1 1 in francs and centimes ? 311. The value of 1 franc being 1 0.193, and the value of £1 being $ 4.866^, find the value of J 1 in francs and centimes. Also find the value of 1 franc in units of English money. 312. If ^1 = 25.22 francs, find the value in English money of a French coin worth 20 francs. Note. — In solving the next four examples use the par value of money as shown on page 311. 313. A railroad has been commenced, passing from Algiers to the Niger, near Timbuctoo. It is estimated that the remaining 1700 miles will cost 400,000,000 francs. Find the estimated cost per mile in francs ; in dollars. 314. Coal was 29 shillings a ton in London in 1879. What was that in United States money ? 315. It is estimated that the actual cost of rearing a boy of one of the poorest classes in England is £ 300. What would be the cost in dollars of rearing a family of 5 boys? 316. If electric pens cost 18 francs each, what is the cost of a dozen such pens in dollars and cents ? 317. I wish to remit £ 143 10 s. to London. Exchange is quoted at $ 4.87^. What will mv bill of exchancre cost ? BONDS. — PBOPORTJON. 335 52. Bonds. 318. If Mr. Wm. H. Vanderbilt owns $31,000,000 in U. S. 4 per cent bonds, what is bis income per day from these bonds ? 319. When the use of money is worth b\% a year, what should be the price of a thousand-dollar Government 4 per cent bond, no account being made of interest upOn the quarterly interests, nor of the fact that when, the bond becomes due, it will be paid at par. 320. A man has settled on his wife 1 1200 a year. What sum must he invest in government 4 per cents at \Q1\ to produce that amount of income ? 321. By investing in the 6 per cent bonds of a city I get an annual income of 4| % on my money. At what price were the bonds bought? 322. From which investment would the larger returns be derived ; from 7 per cent bonds at 115, or 6 per cent bonds at 98 ? 323. What per cent must I make on an investment of $ 1000 to equal the income from three U. S. 4 per cents of 1 500 each ? 53. Analysis and Proportion. 324. If 112 men can do a piece of work in 10^ days, how many men can do it in 12 days ? 325. How many eggs at 24 1* per dozen wiU pay for 25 lbs. of but- ter at 42 f per pound ? 326. My bill for gas being 1 17.25 when gas is $ 1.90 per 1000 cubic feet, what would it be if gas were | 2.15 per 1000 cubic feet, and I should use f as much? 327. If the cost of keeping 25 horses is $ 57.50 a week, what is the cost of keeping 13 horses from July 1 to October 31, both days in- cluded? 328. At noon on a certain day the shadow of a man 5 ft. 11 in. tall is found to measure 4 ft. 2 in. What is the height of a tree whose shadow at the same time measures 47 feet 4 inches ? 329. The interest of 1 500 at 6 % for 3 years is equivalent to the interest of how much at 8 % for ^ years ? 330. If 450 pounds of merchandise can be carried 26 miles for 30/, how many mUes can 3 tons be carried for 1 4 ? 331. If, when flour is $ 7.50 per barrel, a 3-cent loaf weighs 2 oz., what should a 12-cent loaf weigh when flour is $ 12 per barrel ? 332. If a seamstress earns $ 9 a week by working 10 hours a day, how much can she earn in 3 weeks 2 days by working 8 hours a day ? 336 APPENDIX. 333. If 5 men can do a piece of work in 7 days of 11 hours each, how long will it take to finish the work when it is half done, if 3 more men are set to work upon it, and aU. work 8 hours a day ? • 334. If it takes 21 yards of cloth | of a yard wide to make 4 gar- ments, how many yards 1^ yards wide wiU. it take to make 2 dozen garments of the same kind ? 335. It takes 8 men 15 days to huild 28 rods of fence ; how long will it take to build 80 rods if 4 boys help, each boy doing half as much work as a man does ? 336. If the weight of a block of sandstone 3 ft. long, 2 ft. wide, and 2 in. thick, is 140 pounds, what is the weight of another block of sandstone 4 ft. long, 3 ft. 6 in. wide, and 3 ft. thick ? 337. If 3 silver spoons, each weighing 2 oz. 6 pwt. 16 gr., cost $9.00, silver being worth $ 1.25 per ounce, what should be paid for 6 dozen similar spoons, each weighing 3 oz. 3 pwt. 12 gr., when silver is worth f 1.40 per ounce, and the cost of making is the same ? 54. Partnership. 338. Pour men hire a pasture in common, paying $ 34.50. A puts in 3 cows 19 days, B 2 cows 15 days, C 3 cows 7 days, and D 1 cow 30 days. How much must each pay ? 339. Two contractors have finished work for which they have been paid $ 12500. One of them employed 50 laborers for 125 days of 12 hours each ; the other 40 laborers for 90 days of 10 hours each. Divide the money between them in proportion to the labor each furnished. 340. A, B, and C invest 1 4860 in trade : A invests twice as much as B, and C invests twice as much as A and B together ; they gain 40 per cent on their investment. What is each person's share? 341. Divide $ 9765 among 8 men, so that one half of them shall have twice as much as the other half. 342. Mr. Bacon has failed for $ 14568 more than is covered by his available resources. His creditors' claims amount to f 30548. How much must M lose, if Bacon owes him $2500? 343. T. and V. formed a partnership to trade in coal. T. furnished $3000 for the first 10 months, at the end of which time he added $ 1000 more, and at the end of the second year, f 500 more. V. put in $ 2500 for the first 18 months, at the end of which time he put in $ 3500 more. At the end of the third year they found their gain to be $ 5565 ; what should each receive ?, POWERS AND ROOTS. 337 65. Powers and Roots. Find the powers indicated below : (344.) 192. (347.) 2.5052. (350.) (2^)2 (353.) (f X 1 J)8. (345.) 1.9». (348.) 1.0048. (351.) (5^)2. (354.) {\ + \)\ (346.) 0.192. (349) 0.03*. (352.) (16^)2. (355.) 1.05*. Find the square root 356. Of 6241. 359. Of 19600. 362. Of 44.009956. 357. Of 9409. 360. Of 1960. 363. Of 0.005648. 358. Of 1633. 361. Of 640900. 364. Of 369056.25. 365. Find the difference between the sum of the square roots of 6075 and 41616 and the square root of their sum. 366. v'Io^ = ? 370. V^=? 374. v/0.4 + 25=? 367. v'9A- = ^ 371. v'l00f| = ? 375. ^OM X 0. 16= ? 368. V^l7|| = ? 372. v'lfxl|? = ? 376. V^offof 1^ = ? 369. \f^\ = 'i 373. V^l of I J = ? 377. ^0.36 H-^ X 6^ = ? 378. If a square contains 12J square feet, what is the length of one of its sides ? 379. On a roof are laid 7200 slates, the number in the length being twice the number in the width. What is the number each way ? 380. My orchard contains 7350 trees. The number of trees in width is to the number of trees in length as 2 is to 3. What is the number each way ? Find the cube root 381. Of 39304. 384. Of 4104. 387. Of 0.27. 390. (^5^=? 382. Of 103.823. 385. Of 79.507. 388. Of 6.4. 391. v'^=? 383. Of 0.00043. 386. Of 2.24. 389. Of 0.8. 392. \JY\ = ? 393. Maury estimates the annual amount of rainfall in the Missis- sippi Valley to be 620 cubic miles of water, of which 107 cubic miles is poured into the sea. What would be the depth of a cubical reser- voir that would contain the remainder ? 394. It is estimated that in 1871 the world's consumption of petro- leum was 6,000,000 barrels. If each barrel contained 50 gallons, what would be the length of a cubical tank large enough to contain itaU? 338 APPENDIX. 56. Mensuration. Rectilinear Figures. 395. Find the area of a rectangle whose base is 7 "• and whose height is 5.4 "". 396. Miss Gove has a triangular flower-bed whose sides are 10 feet, 15 feet, and 9 feet long respectively. How many square feet are there in the bed? 397. Of two triangular flower-beds, one having each of two sides 8 feet in length and the third side 12 feet, the other having a right angle enclosed by two sides each 7 feet long, which is the larger, and how much larger? 398. How many yards of lace must be used to trim the edge of a "half-handkerchief," the length of the diagonal side being 28 inches, and 4 inches being allowed for fulness around each corner ? 399. Wishing to find the distance between two points A and B, on opposite sides of a swamp, I measured two lines A C and B C, from A and B, making a right angle at C, and found them to be 60 feet and 80 feet long respectively. How far was A from B ? 400. How many ars are there in a square field whose diagonal is 100 meters ? 401. Find the area in acres of a rectangular field whose length is 10 chains and breadth 8.49 chains. [See page 306, Art. 19.] 402. If the area of a parallelogram is 500 square yards and its height is 50 feet, what is its base ? 403. Two parallel sides of a quadrilateral field are 45 chains and 36 chains respectively, the distance between them is 25.4 chains. How many acres are there in the field ? 404. The sides of a quadrilateral field are 50 chains, 43.4 chains, 26.8 chains and 43.2 chains respectively ; the length of a diagonal cutting oflF the first two sides is 53 chains. How many acres are there in the field ? ClRCIiES. 405. How far must I go to walk from one point in the circum- ference of a circular pond to the opposite point, if it is 40 feet across? 406. If it is 36 feet around the casing of a circular window, how far is it across ? 407. How many square yards of turf will be required to cover a circular plat of ground 27 feet across? MENSURATION. 339 408. If the area of a circular park is 12 acres, how long must a fence be to enclose it ? [Ans. in rods.] 409. If the area of a circle is 40 scLuare feet, what is the area of an inscribed square ? 410. If the side of a square inscribed in a circle is 40 chains, what is the area of the circle ? Solids. 411. Find the entire surface of a cube each of whose edges is 2 feet long. 412. Find the entire surface of a rectangular solid whose dimen- sions are 2 feet, 3 feet, and 4 feet respectively. 413. Find the convex surface of a regular pentagonal column whose height is 10 ft. and the width of each of whose faces is 1 ft. 3 in. 414. Find the convex surface of an octagonal block whose slant height is 6 inches, the length of each side of the upper base being 7 inches and that of the lower base being 9 inches. 415. Find the convex surface of a regular 4-sided pyramid 12 feet high, the length of one side of the base being 10 feet. ♦ 416. Find the immber of cubic feet in the pyramid named in Example 415. 417. If the pyramid named in Example 415 be cut by a plane parallel with the base and 2 feet from it, how many cubic feet will be contained in the lower part? 418. How many cubic feet are there in a piece of timber 15 feet long, 16 inches square at one end and 12 inches square at the other? 419. Find the convex surface of a pipe 2 meters long, the diameter of which is 36 centimeters. 420. Find the convex surface of a cone a meter high, the radius of the base being 3 decimeters. 421. How many square inches in the surface of the largest sphere that can be put into a cubical box whose dimensions on the inside are 14 inches each ? 422. How many cubic feet of air may be contained within a conical tent the diameter of the base being 25 feet and the slant height 20 feet 10 inches ? 423. How many square feet of canvas will cover a conical tent, the diameter of the base being 28 feet and the height 13 feet? 424. How many cubic inches of water can be contained in a globe whose diameter inside is 15 inches ? 340 APPENDIX. ^ h e 14 feet. d "\ 18 feet. 15 feet. ZSR. 1 f i 15 feet. 6 — 8 ft. (S CD 8 ft. / / a, study ; I, hall ; c, sitting-room ; d, dining-room ; c, conservatory ; /, /, /, piazza. 425.* Above is a pLan of some rooms in a hous^ If you should carpet the study with ingrain carpeting a yard wide, arid lay it down so as not to split any breadth nor turn any under, how many yards would you use, nothing being allowed for matching the figures ? 426. How many yards would you use to carpet the study if you should lay the carpeting down lengthwise of the room and should not split any breadth, nothing being allowed for matching figures? What will be the' cost of the carpeting at $ 1.25 per j'ard ? 427. How many yards of Brussels f yd. wide will it take to carpet the sitting-room, if the breadths are laid lengthwise of the room and one breadth is split into two equal parts, no allowance being made for matching figures ? What will be the cost of the carpeting at $ 2 25 per yard ? 428. If the cost of carpet for carpeting the sitting-room vritii Brussels carpet be $ 87.75, what will be saved by carpeting it with ingi-ain at $ 1.25 per yard, laying the breadths lengthwise of the room, and splitting ho breadth ? 429. How many square feet are there in the floor of the dining- room if the bay window is 2 feet deep and 9 feet wide ? 430. How many square feet are there in the floor of the hall ? Note. In solving Examples 430 and 432, no allowance is made for thickness of walk. MENSUBATIOlf. 341 431. How many square feet in the floor of the conservatory if it is 8 feet in front, 10 feet at the back, and 3 feet deep ? 432. How many square feet in the floor of the piazza if it is 5 feet wide? 433. Wishing to carpet the stairs as shown in the plan, I found the height of the second floor from the hall floor to be 10 feet. Allow- ing half a yard for the landing, etc., what would be the cost of carpet- ing at 1 2. 12^ per yard ? 434. The back stairs are 14 in mimberand carpeted with tapestry at 87^/ per yard, 5 % off for being damaged. Each stair is 8 inches high, and 8 inches wid«s and | of a yard is allowed for turnings, etc. What is the cost of the carpeting ? 435. In the dining-room is an extension-table 4 feet wide. The ends of the table are semicircular, and besides there are C leaves each 15 inches wide. How many square feet are there in the surface of the table and leaves, and what is the greatest number of people that can be seated around it, allowing 2 feet for each person ? 436. When 3 leaves arc in, how long must the table-cloth be to cover the tabic and hang a quarter of a yard over at each end ? 437. What is the cost of a concrete walk 10 feet wide extending diagonali}"^ across a square park containing 1 acre, the concrete costing 8/ a square foot? 438. At $ 3.50 a yard, what is the cost of a piece of velvet, the length of one side on the selvage being 2f yards, and of the other being 3^ yards ? 439. At $ 2.60 a yard, what is the cost of a piece of satin 27 inches wide cut off at one end at right angles with the selvage, and at the other end being cut "on a bias" (making half a right angle with the selvage), the length of the longer side of the satin being 2-|^ yards ? 440. On a piece of land 10 rods long, 7 rods wide at one end and 5 rods wide at the other, 1 bushel of oats was sown. This was at the rate of how many bushels per acre ? 441. There are 3,000,000 cu. ft. of masonry in the Victoria' Bridge at Montreal. If this masonry was put into a pyramidal form whose base was square and whose height equalled that of Bunker Hill Monu- ment (221 feet), what would be the length of one side of the base ? 442. How many hektoliters of water may be contained in a reser- voir 10 meters deep, 360 meters square at the bottom, and 400 meters square at the top ? 342 APPENDIX. 443. The diameter of the large wheel of a bicycle is 46 inches. How many times will this wheel turn in going a mile ? [Ans. to tenths.] 444. The large wheel of John's bicycle is 4^ feet in diameter, that of Burt's is 46 inches. In going 2 miles and a half and returning, how many more revolutions are made by Burt's bicj'cle than by John's ? [Ans. to tenths.] 445. If a point on the rim of a circular saw whose diameter is 40 inches goes at the rate of 1^ miles per minute, how many revolutions does the saw make a minute ? [Ans. to tenths.] 446. How many revolutions would a circular saw whose diameter is 30 inches have to make that the rim might go at the rate of IJ miles a minute ? 447. What will it cost to enclose a hectar of land in a circular form with a fence costing $ 1.25 per meter ? 448. At f 1.25 per meter for fencing, what will it cost to enclose a square piece of land containing a hectar ? 449. Wishing to ascertain the number of cubic inches in an irregu- lar piece of stone, I immersed it in water and found it displaced enough to fill to the depth of 1^ inches a cylindrical dish whose diameter was 6 inches. How many cubic inches in the stone ? 450. How many barrels of water, 31|^ gallons to a barrel, can be contained in a cylindrical tank 8 feet deep and 12 feet in diameter ? 451. How many hektoliters in a circular reservoir 6 meters deep, the diameter at the bottom being 60 meters and at the top 70 meters ? 452. Estimating the diameter of the earth to be 8000 miles, and its specific gravity to be 5.6, what is its weight in tons ? 453. A pipe 4 inches in diameter will drain a certain reservoir in 96 hours. In how many hours would a pipe 16 inches in diameter drain it ? 454. If a whale 95 feet long weighs 245 tons, what will a whale similar in form 50 feet long weigh ? 455. What must be the height of a tree to contain 125 times as much as a similar tree 30 feet high ? 456. Wishing to know the quantity of coal piled 3 meters high in a corner of a coal-bin, I weighed a pile similar in form 4.85 deci- meters high, and found it contained 85 pounds. What vraa the weight pf the larger pile ? MISCELLANEOUS EXAMPLES. 343 57. Miscellaneous Ezamplea. 457. What is the par value of stock of which 4 shares, at 72 % discount, sell for $ 112 ? 458. "When the government tax for matches was a cent for every hundred, and $ 2,000,000 was received per year for the taxes on matches, how many matches were manufactured during the year ? 459. From a bushel of fruit and 15 pounds of sugar 25 jars of pre- serves were made. What was the cost of the preserves per jar, if the cost of the fruit was |1.12J and the sugar was ll^f per pound? 460. I sent to my agent a lot of nails which he sold for 1 963 ; on this he was allowed a commission of 3J %. I directed him to invest the balance in hops, after deducting a commission of 1^ % for invest- ing. What sum remained to invest? 461. The first instalment of the fund for the Smithsonian Institute was $515169; the residuary legacy was $26210.63. Congress has since added $108620.37, and Mr. Hamilton bequeathed $1000. What is the yearly interest on this fund at 6 % ? 462. The Smithsonian Institute also holds certificates and bonds in Virginia worth at par $88125, present value $37000. What per cent has this Virginia stock depreciated ? 463. In my History of England the third chapter begins at the top of the 110th page and ends at the bottom of the 189th. How many pages are there in the chapter? 464. Four per cent per annum paid quarterly is equal- to what per cent per annum paid annually, interest at 6 % being allowed on each quarterly interest ? 465. A dealer in provisions loses 10 % by giving credit, and pays 2^ % for having his bills collected. What per cent above cost must he charge for goods, that he may clear 10 % ? 466. It is a principle in mechanics that the two arms of a lever are inversely proportional to the pressures at their extremities. What power 6 feet from the fulcrum will balance a weight of 80 pounds 5 feet from the fulcrum? 2 feet from the fulcrum? 15 feet 6 inches ? 467. How far from the support of a tilt must a boy weighing 120 pounds be placed to balance another boy weighing 96 pounds and seated 10 feet from the support ? 344 APPENDIX. 468. What is the compound interest at 6 % per annum, payable semiannually, on a note for $1000, dated September 14, 1880, and paid December 2, 1881 ? 469. A water-tank 3 feet in depth and 4 feet square at the base, is supplied by rain from a flat roof 30 feet long and 20 feet wide. What depth of rain must fall to fill the tank ? 470. Find the cost of materials and labor for 100 square yards of lath and plaster-work, 3 coats, hard finish, as follows, and make out a bill for the same. Common lime, 4 casks @ $ 1.00 ; lump lime, | cask @ $ 1.35 ; plaster of Paris, \ cask @ $ 1.50 ; laths, 2000 @ 20^ per hundred ; common sand, 7 loads @ 30/ ; white sand, 2^ bu. @ 8;« ; nails, 13 lbs. @ bf ; hair, 4 bu. @ 15/ ; mason's labor,- 3^ days @ $2.50 ; laborer, 3 days @ 1 1.25 ; cartage, $2.00. 471. A speculator had 10 rectangular lots of land, situated side by side and having their fronts in the same straight line. Each lot was 66 ft. wide in front and 132 ft. deep. What was the cost of fencing these lots at $ 20 a rod for the front fences, and $ 10 a rod for the remainder ? 472. Suppose coal to be worth $ 6 a ton, and the net cost of manu- facturing coal-gas to be 15 % of the price of the coal. If a ton of coal yields 12000 cubic feet of gas, what is the cost of gas per thousand feet? 473. North Eastland, near Spitzbergen, is said to be covered with a glacier from 2000 to 3000 feet deep. At the average depth of 2500 feet, what pressure is exerted by the iceberg upon a square foot of earth beneath, if the ice is 0.9 as heavy as water ? 474. Wishing to get 1 400 from a bank on 3 months, the cashier added the discount for 3 mo. 3 d. to the $ 400 and directed me to draw my note for the amount. When this note was discounted, how much did the proceeds fall short of $ 400 ? 475. A collector received $ 5.87 as his commission at 5 % on the amount of his collections for 1 day. If the debtors were allowed 10 % discount from the face of their bills, what was the total due on the bills collected ? 476. If Mr. Cook gets $25 a month for rent of a house worth $6000, and on which he pays $70 a year for taxes, how much does he lose a year, money being worth 6 % V At what price can Mr. Cook afford to sell the house ? MISCELLANEOUS EXAMPLES 345 477. A party of emigrants bought a township of government land at $ 1.25 an acre. Reserving 150 acres for public purposes and setting aside 200 acres which were worthless, they divided half of the re- mainder into farms which sold at 1 4 an acre, and the other half into farms which sold at $5 an acre. After selling all the farms, and paying the first cost of the township, how much money was left ? 478. A person buys coffee at $ 29 per hundred pounds, and chicory at $11.75, and mixes them in the proportion of 2 of chicory to 5 of coffee. He sells the mixture at 38 !* a pound. What is his gain per cent? 479. In Lorabardy 60,000,000 cubic yards of water are daily distrib- uted over 1,375,000 acres of land. If this water were equally distrib- uted over the surface, what would be its depth in inches ? 480. If a body moving at the rate of 40 miles per hour could go from the sun to Jupiter, whose mean distance is 475,692,000 miles, how many years of 365J days each would it require to make the passage ? 481. How many years would it take the body mentioned above to go from the sun to Neptune 27,270,308,000 miles further from the sun than Jupiter is ? 482. On a note for 1 400 having three years to run find the differ- ence between bankers' discount and true discount, at 6 % ? 483. The rate of discount at the bank being 5^ % , what would be the total proceeds on the 14th of June, 1880, of the following notes : $500, given for 6 mo., dated May 17, 1880. $750, given for 3 mo., dated May 23, 1880. $254, given for 4 mo., dated May 31, 1880. $435, given for 2 mo., dated June 6, 1880. 484. The largest known diamond in the world in its original state weighed 900 carats of 3J Troy grains each ; what was its weight in Troy units ? 485. The Kohinoor, now owned by Queen Victoria, once weighed 186^ carats, but lost in cutting 82^5g carats. What is its present weight in Troy units ? What per cent was lost in cutting ? 486. Among the crown jewels of Russia is a diamond that weighs 194 carats, which wa.s bought for $ 450000 and an annuity of $ 20000. What is the yearly expense of the jewel to Russia if money is worth 4 % per annum V 487. In the temple at Baalbec is a stone 66 feet long 12 feet wide and 12 feet thick. If its specific gravity is 2.6, what is its weight 1 346 APPENDIX. 488. What is gaintd by selling a hundred powder-kegs at \^f apiece, the cost per hundred being $1.87 for making and f 1.75 for hooping, eight hoops at $ 0.045 per hundred being required for each keg, and the value of the other stock being 8 f per keg ? 489. The rate of a clock is 0.000375 fast. How much time does it gain in one week ? 490. What is the average date for making the foUovnng payments : $ 100, due May 28, 1879 ; $ 250, due July 3, 1879 ; $ 150, due August 31, 1879 ; ¥200, due September 26, 1879 ; $400, due November 30, 1879. 491. Which is the more valuable, and how much more, $1200 paid to-day or $ 1500 paid four years hence, the use of money being worth 6 % a year ? 492. A man agreed to pay $800 with interest at 6 % annually. Suppose the payment of interest was deferred at the end of the first and second years, and that simple interest was allowed upon the de- ferred payments, what would be due at the end of the third year ? 493. Mr. Lamb offered to sell his house for $4500, but, finding no customer, he was obliged to keep it five years ; he then sold it for $ 8000. If he received $ 200 per year for rent more than he paid for taxes and repairs on the house, money being worth 6 % a year, how much did he gain by keeping the property ? What per cent on the $4500? 494. Pure iron weighs 7.79 times as much as an equal bulk of water. A cubic foot of water weighs 1000 oz. Av. How many cubic inches in a cube of iron weighing 1 lb. Av. ? 495. Find the diameter of a cast-iron ball weighing 9 lbs., supposing that the iron weighs 7.2 times as much as an equal bulk of water. 496. Intensity of light from a given object varies inversely as the squares of the distances. If your book is 27 inches from the light and mine is 6 feet 9 inches, how many times as great is the light- on your book as on mine ? 497. There is a cypress-tree in Lombardy said to have been quite a large tree 42 years b. o. Suppose it to have been planted 87 years B. c, what is its age in 1880 a. d. ? 498. This tree is 121 feet high. What must be the length of a cord that will reach from the top to the ground 50 feet in a horizontal Une from the bass ? MISCELLANEOUS EXAMPLES. 347 499. This tree is 23 feet in circumference at the base. If the trunk were a perfect cone to the top of the tree (121 feet), how many cords of timber would it contain ? 500. August 8, 1866, I insured my life, paying a premium of $ 121.30 a year. After paying my insurance for 11 years, the com- pany went into bankruptcy, and on February 14, 1880, the sum of J 162.12 was sent me in full for all demands on the company. Al- lowing the use of money to be worth 6 %, how much money did I lose by the transaction ? 501. R. Fales & Co. imported from France 220 meters of cloth costing 15.75 francs per meter, paying 15/' per yard for duties. If the cloth be sold at $4.50 per yard, what is gained ? 502. How many square yards of canvas in a circular tent of 200 feet diameter, the vertical wall standing 16 feet high, and the roof extending from the top of the waU to a height at the centre of 50 feet from the ground ? 503. A hot-air register in a school-room is 2 ft. long by 1 ft. 6 in. wide, and half the area is taken up by the grating. How much air per minute must pass through each square foot of the opening of this register into the room to supply each of 42 scholars with 4 cubic feet of fresh air every minute ? 504. A schooner beating against the wind sails S. E. 6 miles, then S. W. 12 miles, then S. E. 12 miles, then S. W. 12 miles, and finally S. E. 6 miles. How many miles is it in a straight course from the point she left to the point she arrives at ? 505. Exchange on Paris is quoted here to-day at 5.17 and on Lon- don at 4.85. At London, exchange on Paris is quoted 25.32^ francs to £ 1. Which will be the cheaper, and how much, to remit to Paris directly or to remit to my correspondent in London and let him remit to Paris ? 506. If I sell 22 rakes for as much money as I paid for 36, what per cent is gained ? 507. Find the first cost of an article of which 100 can be made with raw materials costing $350, labor 1 150, and otber fixed charges $ 200 ; and find the price to gain 25 % . How much would this price he affected by raising the wages of all the laborers 15 % ? 508. Find the average age of the pupils in the graduating class of a certain school, the ages being as follows : 14 y. 3 mo., 15 y. 2 mo., 15 y. 1 mo., 14 y. 4 mo., 14 y. 5 mo., 14 y. 1 mo., 13 y. 11 mo., 13 y. 9 mo., 14 y, 2 mo., 15 y. 6 mo., 14 y. 8 mo., 14 y. 7 mo., 15 y. 2 mo. 348 APPENDIX. 509. In the Centigrade thermometer the freezing-point of water is marked 0° and the boiling-point 100°. In the Fahrenheit thermometer the freezing-point is marked 32° and the boiling-point 212°. When the Centigrade thermometer stands at 37° at what degree will the Fahrenheit thermometer stand? What degree of the Centigrade thermometer corresponds to 92° Fahrenheit ? 510. A builder hired money at 5J % per annum, and built with it a house which, with the land, cost 1 8572. At the end of 18 months he sold the house for $ 10500. How much did he gain ? 511. In an election three candidates were voted for, and the total number of votes cast was 4214. The winning candidate, A, had a plurality of 613 over B, and 1125 over C. How many votes did each candidate receive ? 512. To meet the appropriations made in a town meeting f 158400 must be raised by taxation. What must be the amount of the tax levy if a margin of 10 % is allowed for uncollected taxes and a com- mission of 7^ % on taxes collected ? 513. The cost of making a certain book is 38 f per copy. What must be the retail price of the book that one third may be taken off for wholesale buyers, a further discount of 5 % from the face of their bills allowed, and yet a profit of 50 % remain for the publisher? 514. A travelling salesman is allowed 12 % on his sales. His employer makes a profit of 20 % on the goods sold. What is the first cost of goods which are sold by the salesman at $ 7.65 ? 515. How far from the wall of a house 24 feet high must a ladder 23 feet long be placed that a person may ascend to within 5 feet of the top of the wall ? 516. If from a point between two houses a ladder 32 feet 6 inches long will reach to a window 26 feet high in one house and 30 feet high in the other, what is the distance between the houses ? 517. What can be paid on the dollar by a bankrupt having assets worth 1 375240 and liabilities amounting to 1 681426? How much more could he pay on the dollar if 1 150000 of these liabilities should fail to be proved by the supposed creditors? [Am. to mills.] 518. A contractor has 20 days in which to do a piece of work. He hires 30 men, who after working 8 days strike for higher wages, re- maining idle 5 days and then returning to their work. But for the strike the work would have been done at the end of the 18th day. How many more men must the contractor hire that he may finish his work within the time set? 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