darneli Unioeraita 2Iibtarg . Strata. Nem lark BOUGHT WITH THE INCOME OF THE SAGE ENDOWMENT FUND THE GIFT OF HENRY W. SAGE IS9I Cornell University Library arV192g9 V.I -2 Commercial algebra 3 1924 031 252 376 olin.anx gg \4 Cornell University f Library The original of tliis book 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/cu31924031252376 COMMERCIAL ALGEBRA BOOK I BY GEORGE WENTWOETH DAVID EUGENE SMITH AND WILLIAM S. SCHLAUCH GINN A^D COMPANY BOSTON NEW YORK CHICAGO LONDON ATLANTA • DALLAS ■ COLUMBUS SAN FBAITCISCO COPYRIGHT, 1917, BY GEORGE WENTWOKTH DAVID EUGENE SMITH, AND WILLIAM S. SCHLAUCH ENTBKED AT STATIONEKS' HALL ALL BIQHT8 BSSERVED 419.9 CINN AND COMPANY • PRO- PRIETORS ■ BOSTON • U.S.A. PREFACE This book is intended for use in the first year of a commercial high school. It lays the foundation for that work in mathematics without which it is impossible for a student to attain the highest success in the technique of commerce. A man may have a genius for organization, he may employ others to do his mathematical work for him, and he may advance through a combination of circumstances, but if he is to master the technique of commercial life he must be able to think in algebra. There is not to-day a single line of commercial work in which the formula is not, or may not be, profitably used ; and without a knowledge of the formula, of its value, and of its manipulation by the aid of the equation, a man cannot be a real master of any great commercial field. This knowledge of the formula is, however, no more vital than a knowledge of other parts of algebra, as, for example, the graph in its varied forms. Here is one of the several tools of mathematics which every business man needs and actually uses in his everyday business life. Algebra, then, may be said to be vital to successful training for commercial life. It must not be thought, however, that the necessary algebra can be taught in a few lessons, or that only the actual utilities will suffice. A student need not attempt to master all the formalism which has found place in our traditional courses, but it is essential that he should acquire that power of manipulating simple algebraic expressions which comes only by actually doing the work. For this reason, while the authors have eliminated a large amount of non- essential work in factoring, fractions, and complicated equations, they have retained a sufficient amount of abstract work to assure the student of the drill which is necessary to his success. iv PEEFACE Students who may be preparing for college will find that the work which is given in this book is a sufficient foundation for a subsequent half year of review, drill, and study of special topics like the quadratic, that will assure them of ability easily to enter any such! institution. The second book in this series is devoted largely to the higher technical topics of commercial algebra. The pursuit of such a work is essential to a complete mastery of the modern treatment of subjects like investments, depreciation funds, insurance, and exchange. Such subjects cannot be treated properly in a one-year course, and hence cannot be successfully presented in a book like this. Book II therefore includes the topics of powers and roots, logarithms, the slide rule, series, compound interest, equation of payments, annuities, amortization, depreciation, bond valuation, life insurance, and alignment charts ; these being the ones that the student is most apt to need in general commercial work. It is the hope of the authors that teachers who seek to raise our commercial high schools to the level of the best to be found in other countries will find in this book the material they seek for an elementary course in the important field of algebra. CONTENTS CHAPTER PAGE I. Letters used in Foemulas . . 1 II. The Equation 29 III. Elementary Graphs .... . ... 43 IV. Negative Numbers 57 V. Addition and Subtraction 71 VI. Multiplication 93 VII. Division 119 VIII. Simple Equations . . 131 IX. Fractions . : 149 X. Fractional Equations ... . ... 163 XI. Equations applied to Commerce 189 XII. Simultaneous Simple Equations 221 XIII. Graphs 245 INDEX 265 COMMERCIAL ALGEBRA BOOK I CHAPTER I LETTERS USED IN FORMULAS 1. Use of Abbreviations. If we add 3 feet, 5 feet, and 7 feet, the result is 15 feet. Practically we use the abbre- viation ft. for feet, and write 3 ft. + 5 ft. + 7 ft. = 15 ft. A carpenter will write this in a still simpler way, thus: 3'+ 5'+ 7'= 15'. In the same way we- may write 3 apples -f- 5 apples + 7 apples = 15 apples, or we may use the letter a to mean apples and write 3a-)-5«-|-7a=15a. 3 feet 3 ft. 3' 3 apples 3 a 5 feet 5 ft. 5' 5 apples 5 a 7 feet 7 ft. J[ 7 apples 7 a 15 feet 15 ft. 15' 15 apples 15 a 2. Symbols used for Numbers. We also commonly use symbols for numbers, as in the following examples : 3doz. 3gr. 3M 3C 3R 5doz. 5gr. 5M 5C • 5R 7 doz. 7gr. ■ 7M _7C J_R 15doz. 15 gr. 15 M 15 C 15 R Here we use doz. for dozen, or 12; gr. for gross, or 144; M for 1000, a custom of bond dealers and lumber dealers ; C for 100, a custom in putting up envelopes ; and R for 480 or 500, that is, a ream, a custom in the paper trade. 2 LETTEES USED IN EOEMULAS 3. Letters used for Numbers. We have seen that symbols like doz., gr., M, C, and R are conunonly used for particular numbers. But just as 3M + 5M + 7M = 15M, where M stands for 1000, so 5x + 5x+7x = 15x, whatever number x represents. li. x= 10, it is still true that ^x+bx+lx = lbx, for 3 tens + 5 tens + 7 tens = 15 tens; and similarly for any other number that x may represent. In business it is convenient to represent numbers by letters in marking goods, but more important uses will be found as we proceed with the study of algebra. Exercise 1. Letters used for Numbers All work oral Add the following : 1. 2. 3. 4. 5. 6. 2a bx 1 m ^y 8J 25 c 3 a 4a; 3m ^y 66 5c SiJihtract the following : 7. 8. 9. 10. 11. 12. 9m 8d Is 9a; %y 36 e 4m 2d 3s 2a; 3y lie 13. If a; = 10, what is the value of Sx+7x? We see that 3 a; + 7 a; = 10 a;, or 3 tens + 7 tens = 10 tens = ( ). 14. If w = 9, what is the value of 8n—2n? 15. If s = 7, what is the value of 16 s — 4 « ? 16. If e = 11, what is the value of 16 e — 8 e ? 17. From 17 doz. subtract 9 doz. ; from 17 ft. subtract 9 ft. ; from 17 M subtract 9 M ; from 17 a; subtract 9 x. SIMPLIFYING AND EVALUATING 3 4. Simplifying. If we have the expression 15 a; + 6 a; — 3 a;, we may first add 15 a; and 6 x, the sum being 21 x, and then subtract 3 x, the result being 18 a;. We say that the expression has now been simplified. Teachers should not ask for formal definitions at this time. It is merely necessary to use such a term as " simplify " intelligently. 5. Evaluating. If we have the expression 3w and know that the value of m is 7, we see that the value of 3 m is 3x7, or 21. We have now evaluated 3 m for m = 7. To evaluate 3 m for m = 4 we have 3 x 4 = 12. Exercise 2. Simplifying and Evaluating Examples 1 to 10, oral Simplify the following expressions : 1. 3a + 2a + 7a. 11. 15x + Qx—7x. 2. 6b + 7b-2b. 12. 182/ + 93/-6t/. 3. 8m + 9m-3m. 13. 27 w + 13m + 10m. 4. 5 a; + 8 a; + 9 a;. 14. 62 w + 12^1 -15 w. 5. 9«/ + 9y-7y. 15. 75a + 5a- 2a. 6. Sp + 7p + 9p. 16. 80 a -10 a -10 a. 7. 5^ + 9^-8 g. 17. 32 5 + 32 J + 32 6. 8. 7r + 7r— 6n 18. 68c + 32c -90c. 9. 9n + 9n-2n. 19. 56 d + Ud + 11 d. 10. 6c + 7c+8c. 20. 128 m + 275m. 21. If a; = 15, find the value of 25 a; + 20 a; + 5 a;. 22. Evaluate the expression 27ar+3a; — 15a;for a; = 12. 23. Evaluate the expression 81 h- a; for a; = 9. 24. Evaluate the expression a; -h 5 for a; = 75. 4 LETTERS USED IN FORMULAS 6. Uses of Other Symbols. Many other symbols besides letters are used in business records, and a commercial student must become familiar with the advantage of such use. For "example, there is shown below a daily record card sent by a salesman to the jobbing house which employs him. FiEMs Visited ►J B P5 00 og n (§1 Ed a m 10/6 Jones & Co n' K s 5 Coe & Conover . . . + J. & J. Straus . . . 2 L K L. B. Smith .... L t A. L. Bowles 1 N 4 6 T. Trask & Bro. + K Johns & Johns . . . = K This jobbing house instructs its salesmen to use the follow- ing key : =, called ; +, out ; t, intend ; O, order ; N, not in the line ; L, lost order ; K, left catalogue ; 1, leg ; 2, base ; 3, oak; 4, chestnut; 5, porcelain; 6, zinc. This cai-d shows at once that the salesman made seven calls on Oct. 5 ; that Jones & Co. are not dealing in coal ranges, that they gave an order for gas ranges, that a catalogue of oil stoves was left, and that they gave an order for oak refrigerators lined with porcelain ; that the buyer of coal ranges for Coe & Conover was out ; and so on. Thus, by the aid of a few symbols a large amount of information is given on a small card, the salesman using a kind of shorthand. COMMERCIAL SYMBOLS 5 Exercise 3. Commercial Symbols 1. The salesman referred to on page 4 sent in five other cards for the rest of the week beginning Oct. 5. On Oct. 6 he called on White & Weld, secured an order for coal ranges with legs, and found that they did not handle gas ranges, oil stoves, or refrigerators. Wolcott & Go. intended to buy some coal ranges and oil stoves, but refrigerators were not in their line. Leach & Co. gave an order for coal ranges with- base, intended to order some gas ranges, did not handle oil stoves, and ordered some chestnut refrigerators with zinc lining. The American Hardware Go. ordered some coal ranges with base and some gas ranges, and took a catalogue on oil stoves, but they did not handle refrigerators. The salesman called on G. R. Weeks & Son, left a catalogue of gas ranges, but made no sales. He then called on the J. M. Caldwell Co. and secured an order for oil stoves. Make out a card as on page 4, giving all this information in condensed form. 2. Make out cards for the other four days of the week, as in Ex. 1, giving imaginary names, sales, and information. 3. From the card given on page 4 and from those prepared in Exs. 1 and 2, add the +'s, t's,. O's, N's, L's, and K's, give the results, and write out a statement telling what each means to the jobbiug house. 4. On the following week the cards showed, under coal ranges,- O, L, =, +, K, O, O, +, K, O, O, t, O, =, O, O, t, O, = , K, O, L, = , + , K. Insert the O's, L's, +'s, and so on, give the results, and tell what the record means. 5. In the same week as in Ex. 4 the cards showed, under refrigerators, O, K, O, N, N, O, N, N, K, 0, 0, 0, N, =, O, 0, O, O, t, K, t, N. Add the hke sjonbols and tell what the record means. 6 LETTEES USED IN FORMULAS Exercise 4, Review Examples 1 to 16, oral Evaluate the following for = 100 : 1. 2C. 2. 7C. 3. JC. 4. 25 C. 5. l^C. Evaluate the following for M= 1000 : 6. 4 M. 7. 6 M. 8. \ M. 9. 10 M. 10. 2i M. Simplify the following : 11. 2 a; + 7 a;. 13. 3wi + 9m. 15. 17 a + 3 a. 12. 9c-4c. 14. 12a- 9a. 16. 27m-10w. 17. Evaluate the expression 7 a; +9 a; for a; =12. 18. Simplify the expression 17a; + 29a; — 3a;. 19. Simplify 25 a + 17 a — 12 a and evaluate for a = 5. 20. Evaluate 15 R f or R = 480. 21. A dealer buys 150 M envelopes. Taking M to mean 1000, as usual, how many envelopes did he buy, and how much did they cost at |2.75 per M ? 22. A dealer sells four lots of bonds, as follows : 25 M, 15 M, 30 M, and 12 M. If his commission is |1.25 per M, find his total commission on the four lots. 23. In Ex. 22 find the dealer's commission on each lot separately, add the results, and thus check the answer. 24. A contractor buys 12 M bricks at |8.50 per M, 15 M at $8.67 per M, and 8 M at |9.20 per M. How many bricks did he buy ? How much did he pay in all ? 25. A hardware dealer bought 2 gr. (2 gross) of hinges at 112.60, 3 gr. at |8.20, and 7 gr. at |4.20. How many hinges did he buy? How much did he pay in all? NATURE OF ALGEBEA 7 7. Nature of Arithmetic. In arithmetic we usually represent numbers by figures. If the page of a book is seven inches long and four inches ■wide, we represent these dimensions by 7 in. and 4 in. respectively. In finding the area, we say that area = 4 x 7 sq. iu. = 28 sq. in. 8. Nature of Algebra. We might state the process of find- ing the area of the page as follows : area = width x length, meaning that the number of square inches of area is equal to the product of the number of ruches of width and the immber of inches of length. We do this in algebra, often using initial letters, thus: a = wxl. Usually in algebra we indicate the multiplication of one letter by another by writing the letters without the sign of multiplication ; thus wl means w xl. Arithmetically stated: The number of units of area of a rectangle is equal to the product of the number of units of width and the number of units of length. Algebraically stated: a = wl. One use of algebra, therefore, is to make short statements of the rules of arithmetic. It is thus seen that algebra is a kind of shorthand, enabling us to give a large amount of information in a small space. We have now learned a new fact about the use of letters in algebra, namely, that if two letters are written side by side, as in a word, the product of their values is to be taken. That is : If a = 4 and 6 = 3, then a6 = 4 x 3 = 12 ; if a = 7 and 6 = 9, then a6 = 7 X 9 = 63 ; and if a = f and 5= 36, then a6 = | x 36 = 27. 8 LETTERS USED IN FORMULAS 9. Arithmetic and Algebraic Statements. We may compare arithmetic and algebraic statements as follows : Arithmetic Algebra If 1 book costs $2, 3 books If 1 book costs d dollars, cost 3 X $2. n books cost nd dollars. The cost of any number of If c represents the total cost, books is equal to the cost of one then s = nd. book multiplied by the given num- ber of books. If a train travels 35 mi. If a train travels m miles an hour, in 4 hr. it travels an hour, in h hours it travels 4 X 35 mi., or 140 mi. hm miles. To find the distance traveled If d represents the total dis- bya train in any number of hours, tance, then d = lim. multiply the number of miles per hour by the number of hours. 10. Formula. A rule stated algebraically, in letters, is called a formula. For example, c = nd and d. = hm, given above, are formulas. In arithmetic we learn that the volume of a box is equal to the product of the length, width, and height. In algebra we represent this by the formula V = Iwh. In this case, if Z = 3, «« = 2, and A = 1, we have !' = 3x2x1 = 6. In writing algebraic formulas it is not customary to express denominations like feet and inches. Before considering commercial formulas we shall take up a number of formulas relating to measurements with which all students are familiar. In this way we shall introduce the commercial work more easily. FORMULAS 9 Exercise 5. Formulas JSxaTTvples 1 to 9, oral 1. If a rectangle is 8 in. long and 4 in. wide, how many square inches of area does it contain? 2. If a rectangle has a base of 12 in. and a height of 4 in., what is its area ? 3. If a rectangle has a base of h inches and a height of h inches, what is its area? 4. If a rectangle is I inches long and w inches wide, how many square inches of area does it contain ? 5. If 1 yd. of velvet costs |2, how much will 9 yd. cost ? If 1 yd. costs d dollars, how much will y yards cost ? 6. At the rate of 3 mi. an hour, how far will a man walk in 2 hr.? At the rate of m miles an hour, how far will he walk in h hours ? 7. Read from the formula a = mn the rule for finding the area of a rectangular field m rods wide and n rods long.. 8. Read from the formula v =pqr the rule for finding the volume of a rectangular box p inches long, q inches wide, and r inches deep. 9. Read from the formula n = c-^e the rule for finding the number of articles purchased when c, the cost of all, and e, the cost of each, are given. 10. Write a formula for d, the average cost of each of n things, when they cost c dollars in all. 11. Write a formula for c, the cost of /feet of iron pipe, at n cents a foot. Write the formula so as to express the result in cents ; in dollars. 12. If n is any integer, is 2 w an even or an odd number ? Why ? What is the value of 2 w when w = 197 ? 276 ? 997 ? 10 LETTEES USED IN FORMULAS 11. Formula for the Area of a Parallelogram. It is shown in arithmetic that the area of a parallelogram, like the area of a rectangle, is equal to the product of the base and height. We may therefore express this statement in algebraic form thus : a = hh. For the triangle T may be cut off and placed at X, so as to make a rectangle of area hh. If J = 5 and A = 3, then a = 6A = 5 x 3 = 15. If & and h. represent itiches, then a represents square inches. When we speak of the product of two lines we mean the product of their numerical values. 12. Formula for the Area of a Triangle. It is also shown in arithmetic that the area of a triangle is equal to half the product of the base and height. This ^ c statement may be expressed in algebraic l form thus : a = ^hh. / This triangle may be cut as here shown so that it is geen to be half of the parallelogram of base b and height h. If & = 7 and A = 10 J^, then a = ^ Wi = J of 7 x 10^ = 36|. Exercise 6. Parallelograms and Triangles Examples 1 to 6, oral 1. If 5 = 4 and A = 3, what is the value of bh? oi^bh? 2. If 5 = 60 and A = 5, what is the value of bh? of 1 5A ? Given a = bh, find the value of a when : 3.6 = 60,^=11. 5. b = 20,h = 12. 4. J = 45,A = 10. 6.6 = 22,^ = 10. Given a = ^bh, fitid the value of a when : 7. J = 48, A = 25. 9. b = 24.8, h = 4.75. 8. & = 36, A = 19^. 10. J = 63.2, A = 19.65. FORMULAS IN MEASUREMENT 11 13. Rectangular Solid. A solid bounded by six rectangles is called a rectangular- solid. A rectangular solid all of whose faces are squares is called a cube. The six rectangles are called the faces of the solid, and their sides form the edges of the solid. A rectangular solid has length, breadth, and thickness, and these are called the dimensions of the solid. 14. Formula for the Volume of a Rectangular Solid. If this figure represents a rectangular solid 5 in. long, 3 in. wide, and 7 in. high, it is evident that in the column of cvibes shown there are 7 cu. in. It is also evident that on the base we can place 3x5 such columns. Therefore the volume is 3 X 5 X 7 cu. in., or 105 cu. in. Hence, The volume of a rectangular solid is equal to the product of the three dimensions. We may express this rule in the compact algebraic formula V = Iwh. Exercise 7. Rectangular Solids Given V = Iwh, find the value of v when : 1. 1 = 8, w = 5, h = B. 3. 1 = 17, w = 8.4, h = 7.9. 2. 1=6, w = 5.2, A = 2.3. 4. l = S4:,w = 171, h = 12|. 5. A rectangular solid is 33 ft. long, 5 yd. wide, and 16 in. high. Find the volume in cubic feet ; in cubic yards. 6. A cellar is 24 ft. by 36 ft. by 71 ft. Find the volume in cubic feet ; in cubic yards. 7. A room is 22 ft. 8 in. long, 16 ft. 4 in. wide, and 9 ft. 3 in. high. Find the volume in cubic feet. ^'--\\ y ISt::^:^:, ■'-/ / 12 LETTERS USED IN FORMULAS 15. Board Measure. A hoard foot is the measure of a piece of lumber 1 sq. ft. on one face and 1 in. or less in thickness. 1 BOABD POOT 4 BOAKD FEET 6 BOAED FEET For example, a board 2' long, 1' wide, and 1" (or less) thick contains 2 board feet (usually written as 2 bd. ft.). A joist 12' X 6" X ^" contains 21 x 12 x | bd. ft., or 15 bd. ft. A joist 12' X 6" X 2^" means a joist 12 ft. by 6 in. by 2\ in. In such cases usually use cancellation ; thus, since 2^ = f , we have 3 2J X 12 X i = ^-^ = 15. ^ 2 ^ X ;2 If we let F represent the number of board feet, I the length in feet, w the width in inches, and t the thickness in inches, we have F = J^- ^wt. Exercise 8. Formula for Board Measure Copy the following table, supply the missing numbers marked x, and find the total value at $48 per M: Lot Number I Feet w Inches ■ t- Inches F IN Each Piece NnMBER or Pieces F Total 1 16 6 3 24 50 1200 2 12 9 4 X 40 X 3 16 18 6 X 20 X 4 12 9 1 X 500 X 5 16 3 1 X 300 X Tot?'^ miTYiViPr nf hnn.rfl fp et . . . X DEFINITIONS 13 16. Terms used in Algebra and Arithmetic. As already seen, many of the terms of mathematics are used in algebra exactly as in arithmetic. Since they are well known, such terms usually do not require further definition, although a few are formally defined at this time for future reference. It should always be remembered that the letters of algebra represent numbers. This is the reason why the terms used in arithmetic may properly be employed in connection with algebraic expressions. 17. Factor. Any one of two or more numbers is called a factor of the product of those numbers. Thus just as 2 and 3 are factors of 6, so a and b are factors of ab. A factor that has itself no factors is called a prime factor. Thus 4 and 3 are factors of 12, but the prime factors are 2, 2, and 3. Neither the number itself nor 1 is considered as a factor of a number. I 18. Literal and Numerical Factors. A factor that contains a letter is called a literal factor ; a factor that is expressed by a numeral is called a numerical factor. • Factors are generally considered to be integers ; but in an expression like 2 ah, for example, a and b are factors although their numerical values may be fractional. 19. CoeflScient. If an expression is the product of two factors, either factor is called the coefficient of the other. Thus in the expression ah, a is the coefficient of b, and h is the coeffi- cient of a. The factor that is considered the coefficient is usually written first, however. Thus in 2 ah, 2 is the coefficient of ah, and 2 a is the coefficient of b. The coefficient 1 is omitted, x being the same as 1 x. 20. Power. The product of several equal factors is called a power of any one of the equal factors. Thus 2 X 2 = 2^^, or 4 ; and 2^, or 4, is called the square or second power of 2. Also 2 x 2 x 2 = 2^, or 8 ; and 2^, or 8, is called the cvhe or third power of 2. Similarly, aa is the square or second power of a, and is written a^; and aaa is the cvhe or third power of a, and is written a'. 14 LETTERS USED IN FORMULAS 21. Exponent. The number or letter placed to the right and slightly above another to indicate how many times that number or letter is taken as a factor is called an expoTient. Thus in 2 r', 2 is the coefficient of r^, and 3 is the exponent of r. The two should be carefully distinguished. The coefficient shows the number of equal addends ; the exponent shows the number of equal factors. 22. Root. Any one of the equal factors whose product is a given expression is called a root of the expression. One of the two equal factors whose product is a given expression is called the square root of the expression; one of the three equal factors is called the cube root; one of the four equal factors is called the fourth root; and so on. Exercise 9. Terms Used All work oral State the factors of the following numbers : 1. 15. 2. 77. 3. 35. 4. 14. 5. 33. State any two factors of the following numbers : 6. 70. 7. 80. 8. 100. 9. 120. 10. 1250. State the two equal factors of each of the following numbers : 11. 16. 12. 36. 13. 81. 14. 121. 15. 144. State the prime factors of each of the following numbers : 16. 32. 17. 20. 18. 48. 19. 50. 20. 75. State the coefficient of x in each of the following cases : 21. 17 a;. 22. x. 23. ^x. 24. 1.5 a;. 25. 2aa;. State the exponent of a in each of the following cases : 26. 2 aK 27. 3 a\ 28. l aK 29. ai". 30. 371 a^ TERMS AND SYMBOLS 15 23. Symbols. We may now summarize the symbols of operation which are most commonly used to indicate the operations used in commercial algebra: Akithmetic Algebra Addition 4 + 3 a + b Subtraction 4 — 3 a — b Multiplication 4x3 a xb, a ■ b, or ab Division 4 -;- 3 a-i-b, a/b, a:b, or - b Second power 5^ a^ Third power 5^ a^ A letter without an exponent is considered as having the exponent 1. That is, a^ and a have the same meaning'. Square root V4 means the square Va means the square root of 4. root of a. Cube root 'v^ means the cube V6 means the cube root of 27. root of b. Since the square root of a number is one of the two equal factors of the number, it follows that the square root of 9 is 3 ; similarly, the third root (always called the cube root) of 8 is 2, and since, 2* = 16, it follows that 2 is the fourth root of 16. If a is not a perfect square, as 3, 5, or 7, then Va can be expressed only approximately as a decimal. This is always done in practice, and similarly for other roots. Of the various symbols of division, the form a/h is coming to be very common in print and in typewritten wort. It is customary to use the symbol .". for " therefore," = for " is equal to," < for " is less than," and > for " is greater than" whenever it is necessary to save space. Such commercial symbols as a/c, f.o.b., JV/20, C.O.D., and the like either are already familiar to the student or will be explained as they are needed in the progress of the work. 16 LETTERS USED IN FORMULAS 24. Monomial. An algebraic expression in which the parts are not separated by the signs + or — is called a monomial. Thus a, 3 ah, a", and va are monomials. 25. Poljmomial. An algebraic expression consisting of two or more monomials separated by the signs + or — is called a polynomial. Thus a + J, 3 a — 4 6, 2 n^ + 1, and a^ — ^}fl + c are polynomials. 26. Terms of a Polynomial. The monomials that make up a polynomial are called the terms of the polynomial. A polynomial is called a binomial if it has only two terms, and a trinomial if it has only three terms. Thus a + 3 6 is a binomial and its terms are a and 3i; a + 36 + c is a trinomial and its terms are a, 3 h, and c. 27. Symbols of Aggregation. Symbols used to indicate that certain terms are to be treated as one number or one quantity are called symbols of aggregation. The most common of these are the parentheses, brackets, and bar. Others will be given when needed. Thus 2 X (5 + 7), 2 x [5 + 7], or 2x5+7 means that we are to add 5 and 7 before we multiply by 2 ; each is read "twice the sum of 5 and 7," and each is equal to 24. 28. Order of Operations. Unless symbols of aggregation direct otherwise, the operations indicated in an algebraic expression are performed in the following order: 1. Powers and roots. 2. Multiplications and divisions in the order in which they occur. 3. Additions and subtractions in the order in which they occur or in any other order. In the case of a'b + c -i- d we first square a and take the product of a^ and b; we then divide chj d; and finally we add the results. Thus 22 x3+ 10 -=-5 = 4x3+ 10 ^5= 12 + 2= 14. Similarly, V3 + 6 +15 -=-5-2+7x6 = 3+3-2 + 42 = 46. DEFINITIONS 17 Exercise 10. Algebraic Expressions Examples 1 to 13, oral 1. Read 4 + 7; a + h; 3a + 6; 2>a + bb. 2. Read 9 — 5; a — h; 1 a — h; 7a — 4 6. 3. Read 2x7; ax J; ab; 3 aJ ; a(h + c); x(a + l + c). 4. Read 9-^3; a^Z; a^h; ?,a^h; ^ a/l; 9 a/7; IdaB/c. 5. Read 2w; 2w + l; 2w — 1. What is the value of each when w = 3 ? when w = 5 ? when w = 10 ? 6. Read J-w; ^n; ^n; ^n. What is the value of each when n = 12? when w=24? when n = 10? 7. How many feet in a yard ? in 5 yd. ? in w yards ? 8. How many inches in a foot ? in 2 ft. ? in >; feet ? 9. How many ounces in a pound ? in 10 lb. ? in w pounds? 10. Read a + a=2a; a<2a; 2a>a; 1 gal. > 1 qt. 11. If 1 lb. of tea costs |^, how much will n pounds cost ? 12. Read 2ft.; 2yd.; 2k; 2x; 2mi.; 2m; 2A.; 2a. 13. Read 3ft. 2in.; Sx + 2^-; da + 2b; |3 + 2(f,. 14. Write the sum of 2 and w ; of a; and y ; of 2 a; and y. 15. Indicate the subtraction of y from 7; of y from x. 16. Write these products: 2 times x; 2 times i/; x times ^. 17. Write the quotient of x divided by 2, both as a fraction and with some other sign of division. 18. In the same two ways as in Ex. 17 write the quotient of 2 X divided by 3 ; of 2 a; divided by y ; of 2 a; divided by 3 i/. 19. Write the sum of 5 a?, 7 x, and 2. What is its value when a; = 10 ? What is its value when a; = 20 ? 20. Write the sum of 7a^, 2a^, 8 x, and 6. What is its value when a; = 10 ? What is its value when a; = 20 ? 18 LETTERS USED IN EOEMULAS 29. Trapezoid. A plane figure of four sides whiclj lias^tvvo of its sides parallel is called a trapezoid, i 30. Formula for the Trapezoid. It is usually shown in arithmetic that the area of a trapezoid is equal to half the product of the height by the sum of the two parallel sides. For a trapezoid D, equal to the given trapezoid T, may be turned over and put down by the side of T, as here shown. The whole figure, or twice the trapezoid, is then equal to a parallelogram whose base is the sum of the parallel sides of the ■ r j trapezoid. The trapezoid is there- / T \ JJ / fore half this parallelogram. / J^ / We indicate this in algebraic form as follows: where a is the area, h the height, and b and ¥ ("6 prime") the two parallel sides, usually called the bases. For example, if A = 4, J = 6, and ft' = 5, we have a = J A(ft + 6;) = i X 4 X (6 + 5) = i of 4 X 11 = 22. Exercise 11. Trapezoids Examples 1 to 5, oral 1. Find the value of b + b' when J = 7, J' = 9. 2. Find the value of A(J + J') when A = 3, 6 = 4, J' = 5. Given a = ^h(b + b'^, find the value of a when : 3. A = 3, S = 6, 6' = 4. 6. A = 7.2,5 = 24, 6' = 9.5. 4.^ = 7, J = 9, 5' = 5. 7. A = 19, 6 = 34, 6' =16. 5. A = 6, 6 = 10, 6' = 8. 8. A = 8.6, b = 3.8, b' = 9.8. 9. A playground is in the form of a trapezoid, with bases x and y and with height z. What is the area ? How many square rods are there in the playground if a;=30rd., ?^=26rd., and 2=24 rd.? if a;=34rd., ?/=28rd., and a=30rd.? COMMERCIAL FORMULAS 19 Exercise 12. Commercial Formulas JExamples 1 to 4, oral 1. At 6% a year, what is the interest on |200 for 1 yr.? 2. If r is the rate for 1 yr., what is the interest on jo dollars for 1 yr. ? for 2 yr. ? for 21 yr. ? 3. At 6% a year, what is the interest on $200 for 2-| yr. ? 4. If r is the rate for 1 yr., what is the interest on p dollars for t years ? PT f^ 5. If the list price of some goods is $575 and a discount of 10% is allowed, what is the discount? 6. If the list price of some goods is I dollars and the rate of discount is r, what is the discount ? ^ K 'j»o 7. If some stock is selling at 7% below par, what is the cost of $1500 worth, par value, brokerage not' considered ? ' 8. If some stock is selling at r% below par, what is the cost of p dollars' worth, par value, brokerage not considered ? 9. What is the amount of principal and interest of $1700 for 2yr. at 5%? '7<^ >^ ^T^-^ +• H^^. 10. What is the amount of principal and interest of ^ dollars for t years, the rate of interest being r ? ' iJt^ ■+ f" 11. A trade price list gives the cost in dollars per fopt of sewer pipe of diameter d inches, as follows: c— 0.004 c?2+ 0.14. Find the cost of 300 ft. of 16-inch pipe. 12. From the formula of Ex. 11 find the cost of half a mile of 18-inch pipe. Mfii^^- 13. As in Ex. 11 find the cost of 30 rd. of SJ-inch pipe. 14. If the amount of d dollars on interest for n years at the rate r is d(\ + nr), find the amount of |750 for 3 yr. 6 mo. at 5% and Ihe amount of $625 for 2 yr. 9 mo. at 6%. 20 LETTEES USED IN FORMULAS 31. Table of Squares and Cubes. We frequently need the squares and cubes of numbers in our evaluation of expressions, and hence the following table should be committed to memory: No. Square Cube No. Square Cube No. Square Cube 1 1 1 6 36 216 11 121 1331 2 4 8 7 49 343 12 144 1728 3 9 27 8 64 512 13 169 4 16 64 9 81 729 14 196 5 25 125 10 100 1000 15 225 32. Raising Numbers to Powers. In commercial work we frequently need to raise a number like 1.06 to some power, say to find 1.066. Since 2^ = 2 x 2 x 2 x 2 x 2 = 2^ x 2= = 4 x 8 = 32, we see that we can find the value of 1.06* as follows : 1.06 1.191016 = = 1.063 1.06 1.1236 = = 1.062 6 36 7146096 106 3573048 1.1236 = = 1.06? 2382032 1.06 1191016 67416 1191016 11236 1.3382255776 = = 1.066 1.191016 = = 1.063 In elementary business matters no such number of decimal places would be needed, but in large commercial and banking transactions involving millions of dollars they are frequently found. The student will also find it convenient to accustom himself to mul- tiplying from left to right instead of following the elementary plan. EVALUATION 21 Exercise 13. Evaluation Examples 1 to 15, oral State the value of each of the following : 1. 112. 3. 122. 5. 132. 7. 142. 9, 152. 2. 113. 4. 123. 6. 103. 8. 102. 10. 1003. Evaluate the following expressions for a = 4, b = B, c = 3 : 11. a\ 12. c\ 13. a%. 14. 2ab. 15. c^. FiTid the values of the following : 16. 2*. 17. 2^ • 18. 7*. 19. 12*. 20. 1202. Evaluate the following expressions for x = B, y = 0, z = S : 21. 2y. 22. y\ 23. xz.. 24. xz\ 25. xH\ 26. If a; = 3 and y = 2, find the value of 5 a?y^. Evaluate the following expressions for x = 2, y = 3, z = 4 : 27. 3 2;2«/. 28. Ixyz. 29. lba?yz\ 30. &x^y^z. Find the values of the following : 31.23x25. 32.29. 33.1.04s. 34. 1.045*. 35. To what power of 2 is 2^ x 2^ equal ? Find the value. 36. To what power of x is 0^3? equal ? Why is this ? 37. By carrying each multiplication to three decimal places only, find the value of 1.03^ to the nearest thousandth. Proceed as in § 32 ; l.OS" x l.OS^ = 1.03* and 1.03* x 1.03* = 1.038. 38. What is the error to the nearest thousandth in retain- ing only three decimal places in each multiplication used in finding 1.05* instead of keeping all the figures? 22 LETTERS USED IN FORMULAS 33. Compound Interest. We are now prepared to under- stand another very important formula used in business. In finding compound interest the simple interest is added to the principal and becomes part of it whenever it is due. The sum of the principal and compound interest is called the compound amount. Savings banks usually add the interest to the principal every 6 mo. Banks of deposit continually lend their interest as it is paid in and thus receive the benefit of compound interest. For example, if $1 is put at interest for 1 yr. at 6%, it amounts to |1.06 at the end of the year. Now if you started the second. year with, say, |1, the amount at the end of the second year would again be |1.06, and if you started it with |20 it would be 20 x $1.06. But we have seen that you really start the second year with |1.06, and so its amount at the end of the second year is 1.06 X 11.06, or |1.062. Hence we see that the amount of $1 at the end of 1 yr. is |1.06, the amount of $1 at the end of 2 yr. is $1.06^, the amount of |1 at the end of 3 yr. is $1.06^, and the amount of |1 at the end of n years is $1.06" ; or, if r is the rate per cent, the amount of d doUaxs at the end of n years is c?(l + r)". Hence, if p is the principal, A the amount, and r the rate, A=p(l + ry. For example, to find the compound amount of $900 for 5 yr. at 4%, we have ^ ^ ^^^ ^ ^^^^ ^ ^^^ ^ 1.21665 = 1094.99. Hence the compound amount is 11094.99. Notice that, in this case, it is nqt necessary to find 1.04'' to more than five decimal places to obtain the answer correct to the nearest cent. COMPOUND INTEEEST FOEMULAS 23 Exercise 14. Compound Interest Using the formula A =p(l + r)", jiTid the amount of : 1. 1450, for 2 yr., at 3%. 3. |1200, for 3 yr., at 3%. 2. $920, for 3 yr., at 4%. 4. $1500, for 4 yr., at 4%. 5. A certain savings bank compounds the iaterest on its depositors' accounts semiannually. If the rate is 3J% a year, the rate for 6 mo. is 1|%. Therefore the compound amount for 2 yr. at 3^% compounded semiannually is the same as the compound amount for 4 yr. at 1| % compounded annu- ally. If a man deposits |500 on Jan. 1, how much is the compound amount 2 yr. hence ? We have |500 multiplied by 1.0175*. 6. Find the compound amount of |850 for 3 yr. at 4%, the interest bemg compounded semiannually. 7. Since the compound interest is equal to the compound amount less the principal, we have i = jo (1 + r)" — p. Find the value of i when p = 400, r = 4%, and m = 3. Vind the compound interest on the following : 8. $800, for 3 yr., at 5%. 9. $1200, for 4 yr., at 4%. 10. Make a table of compound amounts of $1 to the nearest mill, at the rates specified below, from 1 yr. to 6 yr. The first part of the table is as follows: Yeaks 2% 3% ^% 4% 6% 6% 1 1.020 1.030 1.035 1.040 1.050 1.060 2 1.040 1.061 1.071 1.082 1.103 1.124 3 1.061 1.093 1.109 1.125 1.158 4 1.082 1.126 1.148 24 LETTEES USED EST FOEMULAS 34. Formula for the Circumference. We shall now consider a few further formulas of mensuration needed in commercial work. It is usually shown in arithmetic that the circum- ference of a circle is equal to nearly 3.1416 x the diameter. In mathematics the number 3.1416 — , or 3.14159 +, which is nearly 3^, is represented by the Greek letter tt ' (pronounced pi^. We may therefore express this law as follows : f ..^oji^"'"^! = Trd, where c is the circumference and d the diameter. In solving problems, take 3^^ for ir unless otherwise directed. Since the diameter is twice the radius, we may write 2r for d, and have c = 7rx2r, ore=2 irr. Thus if rf = 7, and we take 3|, or ^^, as the value of tt, we have c = n-d = 3| X 7 = 22. If r = 5, we have c = 2 7rr = 2x8|x5=10 x 3-f = 31f. Exercise 15. The Circumference Given c^ird^B irr, find the value of c when : 1. ci = 14. 2. c?=3.5. 3. r = n. 4. »- = 6.3. Taking "tt = 3.1416, find the value of c when : 5. d = 10. , 6. d = 50. 7. r = 30. 8. r=2|-. 9. A boy measures the diameter of his bicycle wheel and finds it to be 28 in. Find the circumference. ^ 10. A workman measures the diameter of a steel shaft and finds it to be S-Jg^ in. Find the circumference. 11. The basin of a fountain has a diameter of 30 ft. At 90^ a foot, how much will it cost for an iron railing to inclose it ? FORMULAS FOR THE CIRCLE 25 35. Formula for the Area of a Circle. It is usually shown LQ arithmetic that 1. The area of a circle is equal to half the product of the circumference and radius ; that is, a — jcr. , , 2. The area of a circle is equal to ir times the square of the radius ; that is, a =jrr^. Thus if ?■ = 10, we have a = Trr^ = 3.1416 x 10^ = 314.16, taking 3.1416 for jr. If r represents the number of feet, the area is in square feet. Exercise 16. Area of a Circle Examples 1 to S, oral 1. If r = l and c= 6|, what is 0ie value of «•? of |-cr? 2. If ?• = 10 and c = 63, what is the value of cr ? of |^ er ? 3. If r = 1, what is the value of r"^ ? of irr^ ? Given a = jcr, find a when : 4. r = 5, c == 31.416. 5. r=2^, c = 15.708. Griven a = 7rr\ find a when : 6. r = 7. ' 7. r = 14. 8. r = 2.8. 9. r = 4.9. 10. How many square feet in the area of a circle whose radius is 2 ft.? Take tt = 3.1416. 11. How many square inches are there in the z' \ area of this kite ? Take tt = 3| ; and also find V f~~i the result when 3.1416 is taken for it. \ J, / \ ^c* / 12. Find the area covered by the basin of a \ l / • \ ' / circular fountain, the diameter being 30 ft. v/ 13. The basin in Ex. 12 is surrounded by a walk 10 ft. wide. Find the cost of laying this walk at the rate of fl.lO per square yard. 26 LETTERS USED IN FOliMULAS Exercise 17. Formulas of Measurement Examples 1 to 6, oral 1. It is usually shown in arithmetic that the volume v of a cylinder is equal to the product of the base h and height h. Express this in a formula. 2. It is also shown that the volume is equal to the product of the height and ir times the square of the radius of the base. Express this in a formula. Express the following statements in formulas : 3. The volume v of a cube is' equal to the third power of an edge e. 4. The entire surface s of a cube is equal to six times the second power of an edge e. 5. The lateral (or side) area of a cylinder is equal to the product of the circumference and height. Use I for lateral area, c for circumference, and A for height. 6. The lateral area of a cylinder is equal to 3.1416 times the diameter multiplied by the height. Use d for diameter, and in general use the initial letter for a word in a formula. 7. The volume of a prism is equal to the product of the base and height. Evaluate for J = 17.5, A =8.5. V 5_ /y 8. The volume of a cylinder is equal to the product of the base and height. Evaluate for 6 = 23.2, A = 14.3. 9. The volume of a cylinder divided by the base is equal to the height. Evaluate for w = 28.8, 5 = 2.4. ' 10. The volume of a pyramid is equal to one third the prod- uct of the base and height. Evaluate for h = 16.4, h = 8.4. 11. The volume of a pyramid divided by one third the height is equal to the base. Evaluate for v = 7.5, h = 4.5. SHOP FORMULAS 27 Exercise 18. Shop Formulas JExam/ples 1 to 4, oral 1. Given b=7, h=9, what is the value of bh? 2. Given b = 12, A= 20, what is the value of IM? 3. Given r = 10, what is the value of r^ ? oi r'? 4. Given r = 1, tt = 3^, what is the value of 7rr^ ? 5. A bar of metal has for cross section an equilateral tri- angle each side of which is s. The area of the cross section is given by the formula a = ^s^V3. Taking ^/3 as equal to 1.732, find to two decimal places the area of the cross section when s = 7. i 6. The foreman of a shop knows that the safe load (Z) in pounds that can be hoisted by a rope c inches in circumference is given by the formula I = 100 c^. How many pounds can he safely allow for a rope that is 2| in. in circumference ? 7. A carpenter wishes to put up a circular arch of height h and span 2 s. It is necessary to find the radius of the circle so that he may make his pattern. He knows that s' + ¥ Find the radius, given A = 2, s = 4 ; 2A given h = l, s=l. 8. The area of a circle with radius a being ttos^ and of one with radius b hemi^-fdi^, the area of the ring between the two circumferences is tto' — irb^. What is the area of the ring if a = 7, 6 = 4, and tt = 3^ ? 9. A workman needs to find the area of the metal in a cross section of iron pip^ the exterior diameter of which is 10 in. and the interior diameter 9 in. Find this from the formula tj^^ — 7^S^ where a and b are radii (semidiameters) and vr = 3^. 28 LETTERS USED IN FORMULAS Exercise 19. Review of Chapter I 1. A brick dealer delivered to a contractor the following consignments of bricks : 36 M, 21 M, 45 M, 72 M, 36 M, 19 M, and 27 M. How many bricks were delivered ? 2. Find the cost of the bricks in Ex. 1 at $18 per M. 3. If hemlock timber is worth $38 per M, find the cost of a stick in which 1—12 ft., w = 8 in., ^ = 5 in. 4. A ventilating shaft on a school building has a circular cover in which r = 16 in. A new copper cover is purchased, the copper weighing 1 lb. per 10 sq. in. Find the weight. 5. Find the interest on $1250 compounded annually for 3yr. at 5%._ 6. The cost in dollars per foot of sewer pipe of diameter d inches is given by the formula e= 0.004 c^^H- 0.14 in a trade price list. Find the cost of 2400 ft. of 14-inch pipe. 7. A real-estate dealer purchases for a customer a field in the form of a trapezoid of dimensions & = 30 rd., b' = 20 rd., A = 12rd. Given that an acre is 160 square rods, find the cost of the field at $400 per acre. 8. How much greater is the compound interest on $2400 for 4 yr. at 4%, the iuterest being compounded semiannually, than the simple interest on $2400 for 4 yr. at 4% ? Taking the letters to stand for the words as given in this chapter, write each of the following statements in words : 9. a = bh. 12. c = 2'7rr. 15. i=prt. 10. a = \ hh. 13. c = ird. 16. a=p(\ + rt}. 11. a = ^h(b + b'). 14. a = Trr^. 17. A=p(l + ry. 18. Find the value oip(l+ r)" when p = $1500, r = 0.03, and n = 4. CHAPTER II THE EQUATION 36. Balance Sheet. The following is a condensed balance sheet of the affairs of an individual in business : Balance Sheet of J. P. Dutton Assets Liabilities Cash . . . Mdse. . . . Personal debits Bills receivable 3 35 7 4 800 500 400 726 75 80 50 Bills payable . . Personal credits Proprietorship J. P. Button . 6 7 37 572 245 611 05 51 428 05 51 428 05 It will be seen that the two sides of this sheet balance, the totals being equal. Expressed more briefly, Assets = Liabilities + Proprietorship, or ■ Proprietorship = Assets — Liabilities. If we wish to ascertain the state of a business at any given time, we must find the above three elements, the proprietor- ship or available resources being found from this equation. The second equation is called the equation of accountancy, and upon it depends the whole practice of bookkeeping. The relation expressed by it must hold throughout all the changes of a business, and the student ■whose training leads him to see what columns to add in order that the sum of the totals may check the total of some other column will increase his power in accountancy. 29 30 THi; EQUATION 37. Equation, An expression of equality between two expressions of number is called an equation. For example, x + 2 = 8 is an equation. In this equation a; + 2 is called the first member and 8 is called the second member. In this equation x is usually called the unknown quantity. It is i-eally a nnmber to be determined. Unknown quantities are usually represented by X, y, or z, or else by initial letters, such as jo for pounds or d for dollars. Exercise 20. Using Subtraction Examples 1 to 8, oral 1. What number increased by 5 is equal to 9 ? 2. What weight increased by 8 lb. is equal to 13 lb.? In the equation |) + 8 = 13, what is the value of jo ? 3. What number increased by 9 is equal to 20 ? What is the value of a; in the equation a: + 9 = 20 ? 4. If these scales balance when we place a; + 4 oz. on one side and 16 oz. on the other side, how much weight will be left on each of the sides if we take 4 oz. from each ? What is the value of a; ? 5. If equals are subtracted from equals, what can be said of the remainders ? Find the value of x in each of the following : 6. a; + 9 = 21. 9. a; + 1.7 = 9.1. 12. x + 9^ = 10\. 7. a; + 19 = 21. 10. a; + 5.7 = 9.2. 13. a; + 3| = 4|-. 8. a; + 19 = 49. 11. a; + 0.7 = 0.9. 14. a;+5| = 7|. Pages .30-33 consider informally four simple methods of solving equations. The necessary axioms are given on page 34. SOLUTIONS 31 Exercise 21. Using Division Exarruples 1 to 16, oral 1. If twice the cost price of a table is $16, what is the cost price ? If 2 c = |16, what is the value of e ? 2. If 5 times the cost of an article is |35, what is the cost ? If 5 c = $35, what is the value of c ? 3. At 60 each, how many oranges can I buy for 480? If 6 w = 48, what is the value of n ? 4. At 120 a yard, how many yards of ribbon can I buy for 480 ? If 12 »• = 48, what is the value of r ? 5. At 400 a dozen, how many dozen eggs can I buy for $1.60 ? If 40 e = 160, what is the value of e ? 6. If I put equal weights on the two sides of some scales like' those shown on page 30, then ^ of the weight on one side will just balance what part of the weight on the other side ? 7. If equals are divided by equals, what can be said of the quotients? Find the value of x in each of ike following : 8. 3« = 6. 11. 421=8. 14. 7 a; =35. 9. 3 2; =9. 12.5 2^=10. 15. 9 a; =72. 10. 2 2; = 6. 13. 6 2; =24. 16. 7 a; =91. 17. A wholesale dealer offers Mr. Michaels 16 tubs of butter containing 52 lb. each for $325. How much is that per pound to the nearest cent ? 18. A man paid $780 for a certain number of cattle at an average price of $65 a head. How many did he buy ? 19. A dealer paid $10,325 for some automobiles at an average price of $1475 each. How many did he buy? 32 THE EQUATION Exercise 22. Using Addition Examples 1 to 13, oral 1. What must be added to 2 to m^rke 9 ? to 9 — 7 to make 9 ? to a; — 7 to make a; ? to n — 7 to make n ? 2. If a; — 7 = 30, what must be added to these equals to make the first member xT What does the second member then become ? What is the value of a; ? 3. If w — 8 = 20, what must be added to these equals to make the first member n ? What is the. value of w ? 4. If n — 6 = 40, how will you proceed to find the value of M ? What is the value of m ? 5. If |6 is subtracted from the price of a certain article, the result is |12. What is the price ? 6. If equals are added to equals, what can be said of the results ? Find the value of x in each of the following : 7. a;-l=7. 14. X - - 27 = 72. 21. a;- 21 = 31. 8. a;- 2 = 7. 15. X- - 34 = 62. 22. a;-2i=5f. 9. a; — 4 = 6. 16. X- - 71 = 89. 23. — 6f=7|. 10. a; - 5 = 8. 17. X- - 8.7.= 8.7. 24. a. --4^=53. 11. aj - 6 = 9. 18. X - - 4.9 = 5.7. 25. ^-3| = 8^. 12. a;- 7 = 3. 19. X - - 0.78 = 0.35. 26. a:-6| = 4i. 13. a;- 8 = 5. 20. X- -0.87 = 0.63. 27. a.-93=8f. 28. Joseph Ellis has paid me $125 on account and still owes me $84.75. How much did he owe me at first? 29. A coal company reduced its surplus $128,000 this year to help make up its usual dividend. Its surplus is now $3,400,000. How much was it last year ? SOLUTIONS 33 Exercise 23. Using Multiplication Examples 1 to 12, oral 1. By what must ^ in. be multiplied to make 1 in. ? 2. By what must -^ a; be multiplied to make a? ? ^ 3. By what must -I be multiplied to make 3 ? 4. By what must -I a; be multiplied to make 3 a;? 5. By what must both members of the equation - = 7 be multiplied to give the value of a; ? What is the value of a; ? 6. If g = 9j what is the value of a; ? Prove it. o 7. If equals are multiplied by equals, what can be said of the products ? Find the value of x in each of the following : 8. \x=%. 13. \x = \^. 18. iV^=26. 9. h'- 14. ^ = 16. 19. ^ = - 10. £.8. 15. fi = ^^- 20. ^-•^• 11. I-'- 16. ft = l^- 21. fi = ^-^- 12. i=^ 17. T,-^'-- 22. 0.^1 = ^-^^' 23. If ^^ of the thickness of a steel plate is \ in., what is the thickness of the plate ? 24. If -^-^ of the diameter of a steel rod is -gL in., what is the diameter of the rod ? 25. If 1% of the cost of a house is $47.50, what is the cost of the house? 34 THE EQUATION 38. Axioms. Statements like the following, admitted to be true without proof, are called axioms. 1. If equals are added to equals, the results are equal. For example, 7 = 7 6-2 = 4 3a;-5= 7 Adding 2 = 2 Adding 2 = 2 Adding 5 = _5 we have 9=9 we have 6' = 6 we have Zx =12 2. If equals are subtracted from equals, the remits are equal. For example, 7 = 7 9 + 4 = 13 2a; + 3 = 9 Subtracting 2 = 2 Subtracting 4=4 Subtracting 3 = 3 we have 5 = 5 we have 9 = 9 we have 2 a; =6 3. If eqvMls are multiplied by equals, the results are equal. For example, 7=7 3 + 2=5 ^a; + 3=7 Multiplying^ = _2 Multiplying 2 = 2 Multiplying 2 = 2 we have 14 = 14 we have 6 + 4 = 10 we have a; + 6 = 14 4. If equals are divided by equals, the results are equal. For example, 7 = 7 3 + 2 = 5 3a; + 6 = 15 Dividing 2 = 2 Dividing 2 = 2 Dividing 3=3 we have i = i '^^ have f + 1 = f we have x + 2 = 5 5r Iiike powers or like roots of equal numbers are equal. For example. 3 = 3 27 = 27 a; = 2 a;2 = 25 . . 32 = 32 .-.^^ = ^^27 .-. x= = 4 .-. X =V25 or" 9 = 9 or 3 = 3 and a:^ = 8 or a- = 5 6. Numbers equul to the same number are equal to each other. If John's money = |5' If a; + 5000 = A's share and Henry's money = |5, and 7000 = A's share, then John's money = Henry's money. then x + 5000 = 7000. APPLICATIONS 35 39. Applications. The application of the equation to com- mercial problems is best understood from a few illustrations. 1. The number 20 is 10% of what other number? From the statement 20 is 10 % of what number, we have 20 = 0.10 x. That is, the word " is " becomes the sj^mbol = ; the word " of " be- comes the absence of a sign, and this absence signifies multiplication ; and the. words " what number " become the symbol x. The equation is easily solved by dividing each of the equals by 0.10. 2. A's salary and his share of profits together amount to |2400. If his salary is |1500, how much is his profit ? From the statement that salary and profits amount to f 2400, we have 1500 + x = 2400. This equation is easily solved by subtracting 1500 from each m'ember. Exercise 24. Problems Express the followhig statements as equations, and solve : 1. The double of A's money is |9000. 2. Half of B's money is |1875. 3. The double of C'S money, together with |30, is |900. 4. The number 80 is what per cent of 960 ? . 5. The number 75 is 5% of what number? 6. The number 940 is what per cent of 7250 ? 7. A man's assets are $17,275 and his liabilities are |4750. Find his proprietorship. 8. If A's money is decreased by |750, it becomes |1725. How much money has A ? 9. If twice B's money is increased by $42, it becomes $684. How much money has B ? 36 THE EQUATION 40. Further Applications. 1. A's share of the profits of a business is twice B's share, and together their profits are $9000. Find the profit of each. If we let X = the number of dollars in B's share, then we must let 2x = the number of dollars in A's share. Then x + 2x = 9000, or 3 a; = 9000 ; whence x = 3000. That is, B's share is $3000, and therefore A's share is $6000. 2. A and B are partners, A furnishing a building worth |1200 a year and also supplying $35,000 of capital, and B supplying $27,000 of capital. If $32,200 is available to cover the profit and the rent, find each man's share of this fund. Since every $1000 of capital should receive the same profit, we may let 35 a; = the number of dollars in A's return on his capita], and 27 s = the number of dollars in B's return on his capital. Since B's profit + A's profit + A's rent = total amount divided, we have 27 a; + 35 x + 1200 = 82,200. Hence we have 62 a; + 1200 = 32,200, or 62 a; = 31,000 ; whence x = 500. Therefore 27 a; = 13,500 and 35 a; + 1200 = 18,700. That is, B's share is $13,500, and A's is $18,700. 3. For a crop of 450 bu. of apples packed in barrels A received an offer of $600. The barrels cost 30^ each, and each barrel contained 2 bu. What was the net offer per bushel ? Since value of apples + value of barrels = $600, total value, we have 450 x + 67.5p = 600. APPLICATIONS 37 Exercise 25. Problems 1. Complete the solution of Ex. 3 on page 36. 2. A's money is twice B's, and the sum of A's and B's money is $4800. How much money has each ? 3. B's money is twice A's, and the difference between A's and B's money is $800. How much money has each ? 4. The number 90 is what per cent of 720 ? 5. A standard gold coin is 9 parts pure gold and 1 part alloy. What per cent of the coin is pure gold ? 6. A's share of a profit of $7200 is three times B's. Find the share of each. 7. A owns 31 of the 50 shares of stock in a business, B owning the rest. If a profit of $7500 is to be divided between them, how much should each receive ? 8. A, B, and C own respectively 21, 17, and 2 shares of the 40 shares of stock in a busii;iess. How should they divide a profit bi $8400^? 9. There are 560 shares of stock in a business, of which A owns 190 shares and B owns the rest. B furnishes a building which is worth $1800 a year. If the profits and rent are together $15,800 this year, how should this sum be divided ? 10. Divide $900 among three partners, A, B, and C, so that A shall receive twice as much as B, and B' shall receive three times as much as C. 11. A farmer receives a bid of $700 f.o.b. New York for a potato crop of 700 bu. If the freight charges are 10^ a bushel, how much does the farmer receive per bushel? The letters f.o.b. mean "free on board"; that is, the farmer is to deliver the potatoes on board the train at New York with all transpor- tation charges paid. 38 THE EQUATION 41. Problems relating to Per Cents. The following problems are typical of those met in percentage. 1. A man receives 6% on some money invested and adds |60 to the amount received, thus making $300. How much has he invested ? Let X = the number of dollars invested. Then 0.06 x = the number of dollars received, and 0.06 x + 60 = the number of dollars stated, or 300. .. 0.06 a; + 60 = 300. Subtracting 60, ' 0.06 x == 240. Dividing by 0.06, x =4000. Therefore the man has $4000 invested. Check. 6% of |4000 = $240, and |240 + $60 It should be observed that we do nofrjet x equal the money, but we let X = the number of dollars. Then when we find that x = 4000, we know that this is the number of dollars, and that |4000 is the sum invested. 2. What per cent above cost must a man mark his goods so as to allow a discount of 20 % and still make a profit of 20 % ? The question is to find what per cent the marked price is of the cost, and then to find how much this is above cost. Let c = the number of dollars of cost. Then 1.20 c = the number of dollars of selling price. Let m = the number of dollars of marked price. Then 0.80 m = the number of dollars of selling price. Therefore 0.80 m = 1.20 c. Dividingby 0.80, m = 1.50 c. That is, the goods must be marked 50% above cost. Check. If from 1.50 c we take 20% of it, we have left 80% of 1.50 c, or 1.20 c. This 1.20 c is 20% more than c. For example, if the goods cost the dealer $80, he must mark them at 1.50 X $80, or $120. The selling price is then 20% less, or $96, and this is 20% more than |80. PROBLEMS RELATING TO PEE CENTS 39 Exercise 26. Problems relating to Per Cents Examples 1 to 12, oral X. The number 6 is 6% of what number? 2. The number 3 is 6 % of what number ? 3. The number 6 is 3% of what number? Solve the following equfitions : 4. 6% a; = 18. 7. 3% a; = 9. 10. a; + 6%2; = 106. 5. 5% a; = 15. 8. 2%x=Q. 11. x + 5%x = 105. 6. 4% a; = 12. 9. 12% a; =24. 12. a: + 5% a; = 210. 13. If 18% by weight of wheat is lost in grinding it into flour, how much wheat is used if the loss is 360 lb.? 14. From Ex. 13 how many pounds of wheat are needed to produce 820 lb. of flour? 15. From Ex. 13 how many pounds of wheat are needed to produce 779 lb. of flour ? 16. A dealer sells for $32.20 a suit of clothes which cost him .|28. What is his per cent of gain on the cost ? 17. A suit of clothes was sold for |30.80 after a discount of 12% was allowed on the marked price. What was the marked price ? 18. The profits on a business this year are |6344, or 22% more than they were last year. What were they last year ? 19. Water in freezing expands 9% of its volume: How many cubic inches of water will make 501.4 cu. in. of ice ? How many gallons (231 cu. in.) of water are needed ? ft 20. At -v^hat per cent above cost must a dealer mark his goods so that he can deduct 25% and make a profit of 25% ? What will be the marked price if the goods cost |800 ? 40 THE EQUATION 21. The experience of the Markham Grocery Co. shows that 15% of the sales are bad debts. At what price must it sell an article costing $3 so as to make a profit of 25% on the cost and allow for the bad-debt factor ? / The word " factor " is commonly used in this sense. 22. The Lesher Company bought broadcloth at $2.80 a yard, and wishes to mark it so as to allow the cash customers 5% discount and still realize a gain of 20% on the cost. At what price per yard must the broadcloth be marked? / 23. A bankrupt firm having assets of $27,500 and liabili- ties of $36,000 will pay what per cent of its indebtedness ? "^ 24. A firm of coffee roasters receives an order for 600 lb. of roasted Java coffee. How many pounds of green Java must it roast if the process of roasting causes the green coffee to lose 12% of its weight? 25. What should be the catalogue price of a library table costing $25 in order to insure a gain of 25% on the cost and still allow a discount of 20% to the purchaser? 26. Farmer Jones has 800 bu. of winter wheat and 300 bu. of spring wheat. An agent tells him that the spring wheat is worth 20% more per bushel than the winter wheat, and offers to take the entire crop for $2000. Assuming the . agent's estimate to be correct, what price per bushel, to the nearest cent, is he offering Mr. Jones for each kind of wheat ? 27. The A and B Company estimates that 15% of its re- ceipts covers the cost of doing business, and 10% of its receipts represents profits. How much must it receive for a sprayer that cost $3.40 if it is to be sold on the above basis ? 28. If an article cost $5 and is to be sold sp that 22% of the price received shall cover other items than cost, find the selling price. REVIEW 41 Exercise 27. Review of Chapter II 1. My accounts show that J. C. Ford owes me $275. He agrees to pay me $25 a month until the amount is paid. How many payments must he make ? 2. A buyer offers a manufacturer $1950 f.o.b. the buyer's town for 16,000 yd. of grade A muslin. The manufacturer estimates the freight at $30. How much does the offer yield him per yard ? Solve the following equations : 3. 3^ = 16.2. 9. %^ = 171. ^ 4. a; -5 =11. 10. 3rc-7i = 16J. - 5. a; + 4 = 22.3. 11. 5 2^ + 6.1 = 19.6./ 6. 1 + 4 = 11. 12. ^;+5=6i. y 7. 2a;- 8 = 45. 13. 7a;-f = » + 12f. 8. 1500 « + 21 = 4521. 14. 28.4 a; + 3 2 = 62.8. 15. A and B in their articles of partnership agree that A shall have a salary of $2500 for managing the business, and the balance of profits shall be divided so that A's share shall be -I of B's share. Find each man's share of a total of $4500 available for salary and profit. 16. From C's balance sheet we learn that his assets are $75,000, and his proprietorship $40,500. Find his liabilities. 17. Four men. A, B, C, and D, own respectively 27, 35, 42, and 64 shares of a business of which they are the sole owners. If they divide $84,000 profits, find the amount received by each. 18. If I Z = 350, find I. What axiom or axioms are used ? 19. $35,000 is 7% of what sum of money ? 42 THE EQUATION Exercise 28. Review of Chapters I and II 1. A triangular lot in which J = 85 ft. and A = 60 ft. is offered for sale at $3 per square foot. Find the asking price. 2. A brick wall, whose front is a rectangle in which h = 30 ft. and h = 20 ft., is estimated to require 22 bricks for each square foot. How many bricks will it take, and what will be the cost of these bricks at |27 per M ? 3. Find the compound interest on |900 for Syr. at 4%, the interest being compounded annually. ^ 4. Find the value of jo(l + ry iip = 8000, r = 0.04, w = 4. 5. Find the value of ^gt^ if ^= 32.2, t = 7. 6. Find the value of ^h(b + b') ii h = 40, b = 80, b'=7Q. 7. Find the value of 3 x^i/^ +1xy'^ '\i x=2>, y =1. 8. A peach crop of 942 baskets is sold by the grower for $500. If the baskets cost 10^ each, how much will he realize per basket on the peaches,, to the nearest cent? Write a formula for every such case. 9. If 2 a; + 5 = 11, find the value of x. 10. A circular room in a public building has a diameter of 100 ft. Find the area of the floor. 11. Find each man's share of a $5000 profit if A's share is twice B's share and B's share is twice C's. 12. A druggist used 11 oz. of alcohol in preparing a 10% solution. How many ounces in the entire solution? 13. A salesman offers to sell Mr. A for |611.25, f.o.b. Mr. A's station, 300 cans of best white lead, each can con- taining 10 lb. A dozen cans are packed in each box. Each box costs 20^ and the total freight is |6.25. How much is the salesman charging Mr. A per pound for the white lead ? CHAPTER III ELEMENTARY GRAPHS 42. Appeal to the Eye. In modern commercial ILfe it is found that there is a great advantage in appealing to the eye in statistical matters. We see this in the newspapers and magazines, so that the simpler forms of graphs are already fa- miliar to you. As an example of these sunpler forms we may con- sider this graph, which shows the growth in population of the United States from 1830 to 1910, the numbers at the left indicating the millions of population. Such a graph shows much more clearly than the figures the rapidity of S^ilili increase every ten years. Similarly, the following graph shows at a glance that the rate of growth of sales of a business house from 1918 to 1919 was not maintained in the following year. Such graphs as those shown above are often called^icto^ra«i«. The second one is called a bar pictogram because the numbers in the statistics are represented by bars. ci 43 44 ELEMENTARY GRAPHS Exercise 29. Pictograms 1. This graph shows by hours the number of calls which pass through one of the larger telephone exchanges handling Wall Street business in New York. Write a statement of what the graph tells you as to peaks of business, and explain the cause of the depression. 2. The following graph shows the value of iron and steel prod- ucts of several states in millions of dollars in a certain year. Write a statement of the four most important facts which the graph tells you. 25,000 20,000 15,000 lorning Afternoon r \ \ r ^ \ 16 000 > V 5,000' 1 \ 1 ^ HoxirB i } 10 11 12 L 2 3 16 6 FennBylTania Ohio niinolB New Tork Al&liama New jDVaey Indiana Weat Vlrglnik WlMoniiQ Maijliud Mioliigan , Tenneiaee Viifinia 40 . 80 ISO 160 800 380 360 400 440 3. Represent the following statistics graphically: The number of national banks in the United States in 1910 was 6996; in 1911, 7163; in 1912, 7307; in 1913, 7404; in 1914, 7473 ; in 1915, 7560 ; in 1916, 7578. 4. The total exchanges of clearing houses of our cities for certain years, stated in billions of dollars, were as follows: in 1911, 159 ; in 1912, 169 ; in 1913, 174 ; in 1914, 164 ; in 1915, 163 ; in 1916, 241. Represent these facts graphically. 5. The value of gold mined in the United States for certain years, stated in millions of dollars, was as follows : in 1910, 96; in 1911, 97; in 1912, 93; in 1913, 89; in 1914, 95; in 1915, 101 ; in 1916, 92. Represent these facts graphically. PICTOGRAMS 45 43. Value to Directors. An example of the- value of the pictogram is seen in this summary of the report of expenses of a certain establishment. On preceding years the figures Selling Advertising Xemaging Telephone Equipment Postage,etc. had been presented without exciting any comment, but when the treasurer set forth the facts by means of this graph, the directors saw at once that the amount allotted to advertising was too small in proportion to that given to management. 44. Value in Advertising. Advertisers are very quick to see the value of the appeal to the eye. A baker advertised without much effect that he sold a 16-ounce loaf for the same price that his rival charged for a 12-ounce loaf ; but when he resorted to the graphic method here shown, the effect was instantaneous. 45. Multiple Pictograms. In pictograms relating to statis- tics the bar diagram is often used to compare several features at once. For example, this pictogram shows the number of American ships afloat in a certain year before we began ex- tensive building, each square indicating 1000. We see at once the great preponderance of our coastwise trade over our foreign trade, and the small number of ships engaged in fishing. Steam I Coastwise trade Sail Foreign trade Canal Fisheries Bareea 46 ELEMENTARY GRAPHS vagB City A City B 46. Solid Pictogram. It is sometimes convenient to repre- sent a solid by means of a pictogram, for the purpose of showing' three related sets of statistics. Thus to represent graphically the total wages paid in the clothing industry of two cities, as well as to compare the wages per hour, the hours per week, and the total number of workers, we may make use of a rectan- gular solid. Here the number of workers is shown in one dimension, the wage per hour in another, and the number of hours per week in another. We see at once that city A has ^ as many workers in this industry as city B, that the workers in A are paid only |- as much per hour, and that the number of hours per week in A is |^ as many as in B. Furthermore, the total weekly wages are represented by the volumes of the blocks, which are respectively 24 and 22i. 47. Circular Pictogram. The circle is a favorite means of comparing statistics. For example, these circles represent the comparative business that has been done by a certain house in two consecu- tive years. We see at a glance that the sales of refrigerators and furnaces have in- creased in rela- tive_ importance, and that, although there is an increase from $61,140 to |72,000 in the sales of stoves, there is a relative shrinkage. The sale of stoves has dropped from the position of about ^ the total business to ^, and the sale of ranges has shrunk both absolutely and relatively. PICTOGEAMS 47 Exercise 30. Pictograms 1. Using bars of the same width to represent the number of million bushels of wheat received in New York in five con- secutive years, construct a bar diagram to set forth these facts : first year, 98; second, 119; third, 130; fourth, 140; fifth, 159. In this case it is convenient to use squared paper, which can be pur- chased at any stationer's, representing 10 by one space. The use of squared paper in more advanced work in algebra is explained later. 2. Usmg bars of the same width to represent the number of wage earners receiving the salaries mentioned below in the manufacturing industries of this country in a certain year, con- struct a bar diagram to set forth the facts approximately. Divide eacli bar to represent the proportion of men, women, and children in each class, shading the divisions differently. Weekly Men 16 Women 16 CniLDltEN Earnings AND Over AND Over Total Under |3 92,535 77,826 55,432 225,793 $3-$4 96,569 115,741 52,316 264,626 $4-15 149,581 158,926 31,656 340,113 $5-$6 177,550 173,713 12,430 363,693 *6-$7 272,288 176,224 5,773 454,285 $7-$8 327,726 124,061 1,416 453,203 $8-$9 336,669 86,467 553 423,689 $9-$10 557,046 62,193 226 619,465 $10-$12 654,435 54,340 83 708,858 $12-$15 714,816 26,207 13 741,036 $15-$20 609,797 8,516 1 618,314 $20-$25 170,571 1,273 . 171,844 $25-over 85,005 397 85,402 Total 4,244,538 1,065,884 159,899 5,470,321 48 ELEMENTAEY GRAPHS 3. Tliis pictogram shows the percentage of the world's mill supply of cotton contributed by each country in a recent year. Write the statement in sta- tistical form. 4. In Ex. 3 write a statement as to four important facts which the graph would tell you even if the per cents were omitted. For exam- ple, about how would you rank the United States in comparison with India or with Egypt? 5. This pictogram shows the relative quantities of the six most important of the world's textile fibers. Write a statement as to four important facts which the graph tells you. 6. From Ex. 5 find the per cent which the quantity of flax is of the quantity of jute ; of the quantity of hemp. 7. This pictogram shows the comparative sizes of a train- load on a prominent Eastern railroad in 1900 and ui 1916. Estimate the relative sizes 1900 and reduce the result to per cent. Write several reasons 1916 j=j I t JDOl n_IL ' ' 00 "u u^' OO uu<^ n XL ' ' 00 "o o" OO oo k to explain why the size of the trainload has increased. 8. Draw a pictogram in any form showing the relative space given in this book to the first three chapters. PICTOGEAMS 49 9. The Metropolitan Sewerage Commission in its report gives the following as the number of bacteria per cubic centi- meter in the waters near New York : Atlantic Ocean, 120 ; Lower New York Bay, 1600 ; Harlem River, 15,600. Draw a pictogram representing these facts, using three circles of the same size, the relative number of bacteria being shown by dots, one dot being used to represent 120 bacteria. 10. Representing total acreage either by circles or by rec- tangles with equal bases but different altitudes, show by a pictogram the number of acres of land of the United States devoted to crops of various kinds, and then estimate from the pictogram the relative sizes of the various crops in each of the two years represented in the following table of statistics : Crops FiuST Teak Second Ykak 1. Cereals (corn, wheat, etc.) 2. Other grains and seeds . 3. Tobacco, hay, and cotton 4. SuQ'ar croos .... 184,982,220 4,075,120 87,067,630 790,308 286,213 5,628,220 309,770 6-8,799 199,395,963 5,157,374 105,619,525 1,285,031 390,784 7,073,379 272,460 98,866 5. Minor field crops . . . 6. Vegetables 7. Small fruits (strawberries, etc.) 8. Flowers and nursery products 283,208,280 319,293,382 11. In Ex. 10 find from the graphs what crop used a smaller acreage in the second year than in the first year. 12. How does the relative increase in acreage of the sugar crop comp'are with that of vegetables? with that of tobacco, hay, and cotton ? with that of flowers and nursery products? It is desirable to take the latest available statistics and to consider what causes bring about the changes. 50 ELEMENTARY GRAPHS 48. Cartogram. This map, or cartogram, shows the distri- bution of the manufacture of boots and shoes in this country. ] — c ^^£ ^^v4^ _A f C/ 1 "^^ \^ 1 IlieBBthandOtoBCi.inile \ ^^tlOtoJlOO "" " V/" ^^tlOOtotlOOO " " " ^^^tlOOO and over" « " The following cartogram shows the number of thousands of sheep in the United States in a certain year. ITED STATES 52,206 THOUSAND SHEEP CARTOGEAMS 51 Exercise 31. Cartograms 1. Using the first map on page 50, write the names of the states in which the manufacture of boots and shoes was $1000 and over per square mile. 2. Give a reason for the concentration of boot and shoe manufacture in the states indicated in Ex. 1. 3. From page 50 write in order of number of sheep the names of the ten states producmg the largest number. 4. Write a statement in the way of a report to a shipping house as to the mean annual rainfall in inches, containing four important facts and based on the following cartogram: Such information is of great commercial importance, particularly to houses that sell goods which deteriorate quickly in a damp climate. 5. Sketch a map of the city or village in which you live, and shade it to show approximately the area occupied chiefly by stores, that occupied chiefly by manufacturing plants, and that occupied chiefly by residences. 52 EI.EMENTARY GRAPHS 49. Graphs of Functions. In business it is recognized that the demand for an article changes with the price, that the grocery trade ui a neighborhood (3hanges with the change in population, and that in normal times the price of wheat changes with the amount which is grown. In each case one number varies when another one does; that is, one number depends upon another number for its value. If a number x depends upon another number y for its value, a; which expression is greater, x^ + 2x or x^ +1. 4. Find the value of (l-fr)" when r = 0.03 and w = 4. 5. A merchant desiring to build a branch store near the residential section of a certain city discovers two sites equally well located and costing the same amount. One is a square plot 65 ft. by 65 ft. ; the other is a trapezoid in which 6 = 80 ft., b' = 60 ft., and A = 65 ft. Which plot will give him the greater floor space for his money ? 6. Wv^Q competing lumber companies. A, B, C, D, and E, formed a pool and established a combined selling agency. The equalizer was instructed to assign orders to the member firms in proportion to the average sales of each for the five-year period preceding the formation of the pool. These averages were respectively: A, |120,000 ; B, $150,000; C, $200,000 ; D, |250,000 ; E, $300,000. What amount of business should he assign to each company if he sold for the pool $1,632,000 worth of lumber ? 7. What is the value of a + 2b + 4:o when a = 52, b = 50, and c = 12? 8. Draw a cartogram of your city or village and indicate by dots the general distribution of the retail stores. CHAPTER IV NEGATIVE NUMBERS 50. Directed Numbers. In certain surveys the level of the ocean is taken as the level of reference. Distances above this level are called positive elevations, and distances below it are called negative elevations. The numbers representing these dis- tances may be considered as counted in opposite directions from the level of reference, and hence they are known as directed numbers. The following diagram represents the cross section of some land near the ocean, with elevations in feet : The elevation above sea level of the points marked /I is + 5 ft., read ■■ plus five feet,'' and the elevation of the points marked (2 is — 10 ft., read "minus ten feet." Similarly, the elevation of the point 7? is — 1.') ft. Exercise 35. Directed Numbers All work oral From the above diagram state the difference in level between the following points : 1. 0, A. 3. B, C. 5. 0, B. 7. P, 0- 9. B, 0. 2. A, B. 4. A, C. 6. 0, C. 8. Q,p. - 10. P, Q. 57 68 NEGATIVE NUMBERS 51. Directed Numbers on the Thermometier. When it is necessary to distinguish between temperature below zero and temperature above zero, we write +10° for 10° above zero and —10° for 10° below zero. If the temperature is 20° above zero and it decreases 15°, it is then 5° above zero, or + 5°. If it decreases 5° more, it is then 0°. If it decreases 5° more, it is 5° below zero, or — 5°. If it decreases 20° more, it is then 25° below zero, or — 25°. 52. Two Uses for Signs. We therefore find a new meaning for the signs + and — . They not only are used to indicate addition and subtraction (signs of operation) but they tell on which side of zero a number is situated (signs of qiiality). 53. Positive and Negative Numbers. The ordi- nary numbers which we use in arithmetic are called positive numbers. Thus 3°, 3 in., |, Vs, are all positive numbers. If we wish to make this fact emphatic, we may write them thus : + 3°, + 3 in., + f , + VB, but otherwise the + sign is not necessary here. The expression + 3 is read " positive 3 " or "plus 3." Numbers on the other side of zero from posi- tive numbers are caWed. negative numbers. Thus — 3° is a negative number. If distance eastward from a certain village is called positive, distance west- ward from the village is called negative, so that we may have + 10 mi. and — 10 mi. The expression — 3 is read " negative 3 " or " minus 3." We may think of zero as being either positive or negative, since it divides the two classes of numbers. 54. Absolute Value. The numerical value of a quantity, without reference to its sign, is called its absolute value. The absolute value of — 4 is 4, and that of — a is a. > ; 130-= I 120H \ 110-; \ 100-; :blood theat 90 -i r- 80 -E ^ ™H \- 60 H \ 60-1 '- «-E \. 30 -[ iWATER FREEZES ao-^ \- 10 -| V -o-j E-ZERO 10-^ =- 20 -| \- 80-; \ 40-; h CURVE TEACING 59 55. Curve Tracing with Negative Numbers. If the tempera- ture ill St. Paul in the winter falls from +40° at noon to — 20° at 2 A.M., and then rises again to + 10° at 8 A.M., as here shown, the curve tells us that it was below zero from 8 P.M. until about 7 A.M. In this case we represent negative numbers below the horizontal line, which represents zero. - V V 1 \ - s 1 s f k ti V N 2 1 ; i 10 12 2 4 6 8 Exercise 36. Curve Tracing Examples 1 to 6, oral 1. By the above curve, when was the temperature 35° ? 2. At what time was it - 10° ? + 10° ? + 5° ? - 5° ? 3. What was tlie temperature at 4 a.m. ? at 8 a.m. ? 4. At what time did the temperature cease falling? 5. How much did it rise from 2 A.M. to 6 A.M. ? to 8 A.m. ? 6. What was the difference between the temperature at 6 P.M. and 10 p.m. ? 8 a.m. and 2 a.m. ? 10 p.m. and 2 a.m. ? Trace the curves to show the following variations in tempera- ture for the twenty-four hours : 7. Noon, 23°; 2 p.m.,. 25°; 4p.m., 18°; 6p.m., 7°; 8p.m., 0°; 10 p.m., -4°; midnight, -8°; 2 A.M., -10°; 4 A.M., -5°; 6 a.m., 0°; 8 A.M., 12°; 10 a.m., 16°; noon, 24°. 8. Noon, 28°; 2p.m., 25°; 4 p.m., 16°; 6 p.m., 5°; 8p.m., 0°; 10 p.m., -5°; midnight, -8°; 2 a.m., -5°; 4 a.m., 0°; 6 a.m., 3°; 8 a.m., 8°; 10 a.m., 17°; noon, 22°. 9. Noon, 0°; 2 p.m., 5°; 4 p.m., 2°; 6 p.m., -1°; 8 p.m., -3°; 10 p.m., -7°; midnight, -10°; 2 a.m., -5°; 4 a.m., -2°; 6 a.m., 1°; 8 a.m., 7°; 10 a.m., 14°; noon, 22°. 60 NEGATIVE NUMBERS T + + X' X - ¥ 56. Other Uses of Negative Numbers. If we call some special point on a line zero (0), we usually call distances measured to the right positive and distances measured to the left negative, just as we call distances up (as on the thermometer) positive and distances down negative. Again, because we usually call weight positive, we speak of the weight of a balloon (which pulls upward) negative. The following are some additional illus- trations of negative numbers: If a man is worth $1000, we may say that he has + $1000 ; but if he is $1000 in debt, we may say that he is worth — $1000. If we have 25 on a score in a game, . we have + 25 ; but if we are 25 worse off than nothing, we have — 25. If we call latitude north of the equator positive, we should call south latitude negative. If we call longitude west of Greenwich positive, we should call east longitude negative; and if we call west longitude negative, we should call east longitude positive. If we call the motion of a piston rod of an engine positive when it is -to the right, we should call it negative when it is to the left. If we call downward pressure positive, we should speak of upward pressure as negative. If we call distance above the earth's surface positive, we should call distance below the earth's surface negative. We therefore see that negative numbers are just as real as positive munbers, for a man's debts are just as real as his capital, and the temper- ature is just as real when the thermometer indicates that the temperature is below zero as it is when the mercury rises above zero. In ancient times people used only whole numbers (integers). Other kinds of numbers, like J, Vs, and — 3, were invented as they became necessary, and such numbers are sometimes called artificial numbers. ADDITION 61 57. Adding Negative Numbers. If we tie to a 10-ounce weight a toy balloon that pulls upward 1 oz., what will the weight and balloon together weigh ? From the answer to this question we have the following: To add a positive number to a negative number, find the difference between their absolute values and prefix the sign of the numerically greater number. Thus + 6 oz. and — 1 oz. are + 5 oz. ; + 6 oz. and — 6 oz. are oz. ; + 6 oz. and — 9 oz. are — 3 oz. Similarly, to add a negative number to a negative number, find the sum of their absolute values and prefix the negative sign. As a good illustration of this principle, use two balloons, one pulling upward 5 oz. and the other pulling upward 6 oz. Exercise 37. Addition Examples 1 to 5, oral 1. What is the combined weight of + 25 lb. and — 5 lb.? 2. What is the combined weight of 30 lb. and — 60 lb.? 3. A man who was |450 m debt contracted another debt of $250 and then earned flOOO. How much was he then worth ? 4. A game is played by throwing bean bags in the direction of the arrow. Suppose that the score stands - 5, 5, 3, 10, - 10, 5, 10, 10, 3, 3, - 5, how much is the total score ? 5. If this board without any weights at the^ ^ ends just balances, and if I put 5 lb. at one end and 8 lb. at the other end, how much must I add to ^^^ ^^^ the 5 lb. to make it balance ? Instead ^ of adding to the 5 lb., how much must I add to the 8 lb.? 10 -10 10 62 NEGATIVE NUMBEES 6. A boat that can make 14.1 mi. an hour in still water is going against a stream that retards it 1.8 mi. an hour. What is the rate at which the boat will travel? 14.1 mi. and — 1.8 mi. are how many miles ? 7. Find the average noon temperature for the week in which the noon temperatures were 15°, 3°, 0°, - 7°, - 20°, 6°, 25°. To find the average of seven numbers, divide their sum by 7, and similarly to find the average of any other set of numbers. 8. A mass of iron and wood is placed in a tank of water. The iron tends to sink the mass with a force of 20 lb., and the wood tends to buoy it up with a force of 16 lb. What does the mass weigh under water ? 9. If my watch is 5 min. faster than the schoolroom clock, and the clock is 7 min. slower than the correct time, how near is my watch to the correct time ? 10. In a tug of war one group of boys pulls to the north with a force of 256-| lb., and the other group to the south with a force of 252^ lb. What is the resulting force ? 11. An airplane that can fly 58.2 mi. an hour in stiU air is flying against a wind that retards it 9.7 mi. an hour. At what rate does the airplane fly ? 12. An office building contains 37 stories above the street and 4 stories below. An elevator starting at the street level ascends 28 stories, descends 30, ascends 35, descends 37, and ascends 26. Where is the elevator then ? Represent the move- ments by 'a diagram, using positive and negative signs. 13. A man having $374.75 in the bank deposits |176.50 on Monday, checks out $482.60 on Tuesday, deposits $243.85 on Wednesday, and checks out $281.45 on Thursday. Add the positive numbers, then the negative numbers, and then the two suras, and thus find his balance. SUBTRACTION 63 58. Subtracting a Negative Number. If the temperature is — 10° at midnight and + 40° at noon, the diEEerence in tem- perature is evidently 50°, for the mercury must rise 10° to reach 0°, and 40° more - + to reach +40°. Like- ^' -^ o +4 x wise, in this figure the difference between — 2 and + 4 is 6 ; for a point must move 2 spaces to get from — 2 to 0, and 4 more to reach + 4 ; that is, 6 must be added to — 2 to make 4. To subtract a negative number we may add the positive num- ber which has the same absolute value. That is, 4 - (- 2) = 4 + 2 = 6. Likewise -5-(-3) = -5-|-3=-2. This is the usual rule for subtracting a negative number, but it is never necessary actually to change the sign of the subtrahend. Exercise 38. Subtraction Examples 1 to 10, oral X. How much difference in price is there in selling a horse at $25 below cost or at |30 above cost ? 2. One morning the temperature was +11°, a,nd the next morning — 7°. What was the difference in temperature ? 3. If there is a house for every number, how many houses will you pass in going from 48 East Washington Street to 17 West Washington Street, including both these houses? 4. Jefferson Street is 6 blocks east of Adams Street, and Monroe Street is 9 blocks west of Adams Street. How many blocks west of Jefferson Street is Monroe Street ? Find the value of each of the following : 5. 4 -(-3). 8. 6 -(-3). 11. 13.7 -(-27.8). 6. 5 -(-7). 9. 7 -(-7). 12. 16.5 -r (- 43.4). 7. 6 - (- 6). 10. 8 - (- 9). 13. 43.4 - (- 16.5). 64 NEGATIVE NUMBERS 59. Subtracting a Positive Number. If the temperature is — 10° at midnight and falls 2° during the next hour, it is then —12°. That is, to subtract 2° from —10° is the same as to add - 2° to -10°. Therefore To subtract a positive number we may add the negative num- ber which has the same absolute value. Hence (+ a) — (+ 6) = (+ a) + (— 6) = a — h. It is Bot necessary to remember such a rule, for if we are taking a smaller number from a larger, the result is positive, and in the opposite case it is negative. Exercise 39. Subtraction Examples 1 to 13, oral- 1. How much is 10° - 5° ? 0° - 5° ? - 5° - 5° ? 2. How much is |20 - $10 ? *0 - *10 ? t5 - |10 ? 3. How much is 9 ft. - 6 ft. ? ft. - 6 ft.? - 9 ft. - 6 ft.? 4. How much is 12(^ - 7(^ ? 7^ - 7^ ? 5(f - 7^ ? 5. How much is $25 - 115 ? |15-|15? *5-|15? State the value of each of the following : 6. 17-7. 8. 6-5. 10. 7-3. 12.-4-7. 7. 17-17. 9. 4-5. 11. 1-3. 13. -5-6. 14. If a man is worth |178 and incurs a debt of |275, how much is he then worth? 15. If a man is worth — |178 (is $178 in debt) and incurs a debt of $276, how much is he then worth ? Find the value of each of the following : 16. 69-32. 17. 69-72. 18. 48-75. 19. If two trains meet at 9 45 a.m., one running 47.6 mi. an hour and the other 39.7 mi. an hour, how far apart wiU they be at 10 15 a.m. if these rates are maintained ? MULTIPLICATION AND DIVISION 65 60. Multiplying and Dividing Negative Numbers. We multi- ply and divide negative numbers just as we multiply and divide positive numbers. If a man has — $5 (is $5 in debt), he will have — $10 if he is twice as much in debt. That is, 2 X (- |5) = - $10, and - |10 H- 2 = - |5. Exercise 40. Multiplication and Division Eocamples 1 to 4, oral 1. If one balloon pulls upward with a force of 400 lb. (weighs — 400 lb.), what will be the upward pull of 3 such balloons ? Represent the weight as a negative number. 2. If the thermometer indicates — 8°, what is the tempera- ture when it indicates half as much below zero ? when it indicates twice as much below zero ? 3. If a carrier pigeon can fly 78 mi. an hour in still air, at what rate will it fly against a wind that retards it 15 mi. an hour? against a wind that retards twice as much? 4. A checker of bales of cotton finds one bale 16 lb. short, a second bale twice as much short, and a third bale half as much short in weight. Express these shortages of weight in algebraic language. 5. Last year a man's debts amounted to |475. The year before they were 4 times as much. This year he has paid his debts and has $825 in the bank. What is the difference between his financial standing year before last and now ? Find the value of each of the following^ : 6. 3 X (- 86). 9. 7 X (- 89). 12. 2.8 x (- 4.9). 7. 4x(-49). 10. 8x(-96). 13.-125^25. 8. 6x(-73). 11. L7x(-3.2). 14. -493 h- 17. 66 NEGATIVE NUMBERS 61. Multiplying by a Negative Number. We cannot pick up a book — 2 times, but we may define what we mean by multiplying by — 2, and then we may use the word " times " with negative numbers as we do with positive integers. Because 3 x (— 2) = — 6, it should follow that — 2 x 3 = — 6. Therefore we define multiplication by a negative number to mean multiplication by the positive number having the same absolute value, the sign of the product being then changed. That is, - 2 X 3 =- 6, and - 2 x (- 3) = 6. We know that 2x3 = 6, and 2 x (- 3) = - 6. Therefore a • b = ab, a • (— J) = — ab. If two nwmhers have like signs, their product is positive; if they have unlike signs, their product is negative. Exercise 41. Multiplication Examples 1 to 4, oral 1. If each of 3 men spends |2 in a store, how much does the store receive ? How much is 3 x $2 ? 2. If each of 3 men steals |2 from a store, how much does, the store gain or lose ? How much is 3 x (— |2)? 3. If 3 men who would have spent |2 each in a store are persuaded to trade elsewhere, how much does the store gain or lose in gross receipts ? How much is — 3 x $2 ? 4. If 3 men who would have stolen $2 each from a store are arrested before they can make the theft, how much does the store gain or lose ? How much is — 3 x (— $2) ? Find the results of the following : 5. 3 . (- 9). 7. - 3 . (- 9). 9. - 42 • (- 76). 6. - 3 . 9. 8. - 27. 63. 10. - 29 • (- 38). MULTIPLICATION AND DIVISION 67 62. Dividing by a Negative Number. Division being the inverse of multiplication, we have the following: 2x3 = 6, 6-=-3 = 2; 2x(-3) = -6, _6h-(-3)=2; -2x3=-6, -6h-3=-2; -2x(-3)=6, 6h-(-3) = -2. If two numbers have like signs, their quotient is positive; if they have unlike signs, their quotient is negative. Exercise 42. Division Exwrrvples 1 to 16, oral 1. How much is 2. (-7)? -14 -f- 2? -14 ^(-7)? 2. How much is - 3 • 8 ? - 24 ^-(- 3)? - 24 -^ 8 ? ■ 3. How much is -7. (-9)? 63 -h(- 7)? 63 h-(- 9)? 4. How much is -a- (-'6)? a6^(-a)? ah^{-h^'> State the results of the following : 5. 25^5. 9. 36 -=-(-4). 13. 56 ^(-8). 6. 25h-(-5). 10. -36 -=-4. 14. 56-f-(-7). 7. -25^-5. 11. -36 ^(-4). 15. -56-;- (-8). 8. -25 ^(-5). 12. -36-^(-9). 16. -56-h(-7). 17. By what must - 22 be multiplied to. make 770? - 770? Find the remits of the following : 18. 625-4-25. 23. -3367 ^(-3?). 19. 625-f-(-25). 24. -3367 -^(-91). 20. -625 -J- 25. 25. 34.3 -f-(- 7). 21. 3367^37. 26. -34.3h-(-7). 22. 3367 + (-37). 27. - ^4.3 +(- 0.7). 68 NEGATIVE NUMBERS 63. System of Integers. We now see that our system of integers extends indefinitely on both sides of zero, thus : ■ .., - 6, - 5, - 4, - 3, - 2, -1, 0, 1, 2, 3, 4, 5, 6, • • • 64. Properties of Zero. The following are the important properties of zero: a+0 = a, — a = — a, • a=0, a — = a, a-0 = 0, 0-5-a=0. Since division by zero is not allowed, a -t- has no meaning. Exercise 43. Review Exam/pies 1 to 17, oral State the results of the following : 1. 4 + (-3), 6.-5 + 7x3. 11.-8x9. 2. 6 + (- 9). 7. - 5 - 9 -^ 9. 12. - 7 x (- 8). 3. 9 + (- 3). 8. - 7 - 2 X 4. 13. 8 -r- (- 2). 4. 1 + (- 9). 9. 2 X (- 7) X 3. 14. -8-5- 2. 5. 7 + (-3). 10. 5x(-3)-2. 15. -6-h2. 16. How much is - 3 x (- 3)? (- 3)2? (- 7)^? (-' 9)2? 17. How much is -5x7-4? -8-j-4-t-2x3? Find the results of the following : 18. (-1)2. 20. (-2)*. 22. (-5)2. 24. (-1)«. 19. (-l)S- 21. (-2)5. 23. (-6)2. 25. (-1)1 26* Jix = -2, what is the value of a;2 ? of - a; ? of (- xy? If a = 4 and h = — 3, find the values of: 27. a + h. 29. -h. 31. (- J)2. 33. - 3 a6. 28. a^-2l. '30. a^-\h. 32. ZaW. 34. d?-i^. EEVIEW 69 Exercise 44. Review of Chapter IV 1. Draw a straight line to represent the equator of the earth as shown on a map, and draw two parallel lines above it and two below it, at equal distances, to represent degrees of north and south latitude. Trace on the drawing the course of a ship which starts at the equator, sails west, and is blown by a storm, successively 1° N., 2° S., l° N., 1° S. 2. Representing gains as positive numbers and losses as negative numbers, find the net change in the price of stock whose changes in value for a month were +3, — 2, +5, -li, + 5, -3,-2,+i. 3. A cash-register attendant whose wages were $3.50 per day found that she lacked $5 of the cash called for by the sales slips in her possession. This amount was deducted from her wages. What was the effect on her week's wages of this day's operations ? Which of these two amounts may be called positive and which negative ? If a = 5 and b = — 3, find the values of: 4. (a + 6) (a -6). 6. a^ + Sa^ + b. 5. a^ + 2ab + I^. 7. a^ + 3a% + S ab^ + lfi. 8. In a battle lasting 5 da. an army division numbering 20,000 men was engaged and was affected as follows : 1st day, loss 2000, reenforcements 1000 ; 2d day, loss 2500, reenforcements 3000 ; 3d day, loss 3400, detached by commander 5000 ; 4th day, loss 2000, reenforcements 7000; 5th day, loss 2200. Represent these numbers algebraically, and find the effect on the strength of the division for each day and the total effect of the actions of the 5 da. 70 NEGATIVE NUMBEES Exercise 45. Review of Chapters I-IV 1. In the formula i = prt it is given that i = |80, p = $700, r = 0.04. Substitute these values and iind the value of t. 2. Using the formula of Ex. 1, if i=$90, r=0.05, and t=2^ jr., find the value of p. 3. Write the formulas for finding the area of a rectangle, the area of a triangle, the area of a trapezoid, and the area of a circle. Make problems which can be solved by these formulas and solve them. Find the value of x in each of the following equations : 4. a; + 3 = 111. 8. 7 a; + 2 = 30. 5. a; - 0.85 = 0.375. 9. 5 a;- 2.7= 8.3. 6. — = -9. 10. — = 2.4. 3.2 1.1 7. a; + 5 = - 3. 11. | a; - 2 = i- a; + 6. 12. By what number must 17 be multiplied to make 544 ? to make - 544 ? to make - 1088 ? 13. By what number must — 19 be multiplied to make - 399 ? to make 798 ? 14. On every article sold in a stationery store during the month of May there was a gain of 18% on the cost. If the total sales amounted to $3360, find the cost of the goods sold. 15. If increases of sales for a month, as compared with the previous month, are marked plus and decreases minus, find the sales for December, those of January of the same year being f 47,000 and the changes for the other months being: February, - $500 ; March, - |800 ; April, + |2000 ; May, -$400;. June, +$1500; July, - $1000 ; August, +$50; September, +$1100; October, +$450; November, +$1200; December, + $1850. CHAPTER V ADDITION AND SUBTRACTION 65. Addition of Similar Monomials. In arithmetic we learned that 5 apples + 3 apples = 8 apples, but that 5 horses + |3 gives no simple sum expressed in tertns of a single unit. That is, only monomials which are expressed in the same unit can be added- so as to give a simple sum. Monomials which have a common factor are called similar terms or similar, monomials with respect to that factor. Thus a, 3 a, and —7a are similar with respect to a ; 2 v5 and — § Vo are similar with respect to V5 ; 2 • (— 7) and 4 • (— 7) are similar with respect to — 7 ; and ax^ and fix" are similar with respect to x^. Monomials that are not similar are said to be dissimilar. Exercise 46. Addition of Similar Monomials All work oral 1. Add 2 ft. and 3 ft.; 2/ and 3/; 2-5 and 3 . 5. 2. Add7(^and9(^;7c'and9c;7-2and9.2; 7- 10 and 9 -10. 3. Add $15 and $11; 15 c? and 11 c?; 15 • 2 and 11 ■ 2. 4. How do you proceed to add similar monomials ? Add the following : 5. 7 mi. + 14 mi. 8. 6 bu. + 4bu.+ 9 bu. 6. Im+lim. . 9. 6 J + 4 5 + 9 6, 7.7-2+14.2. 10.6-7 + 4.7+9.7. 71 72 ADDITIOJST AND SUBTEACTIOJST 66. Algebraic Sum. The result obtained by adding two or more numbers considered with respect to their signs as well as their values is called their algebraic sum. Thus the algebraic sum of 2 and — 8 is — 1, although 2 + 3 = 5. 67. Addition of Monomials. To add similar monomials, find the algebraic sum of the coefficients of the common factor and prefix this sum to the common factor. For this purpose we may consider an expression like 27 (a — i) as a monomial, as in Ex. 15, below. In case a letter has no coefficient, 1 is understood. Thus 5 x + x = Q x. To add dissimilar monomials, write the terms one after the other, each with its proper sign. Thus the sum of o, 2 h, and — c is a + 2 J — c. Exercise 47. Addition of Monomials Examples 1 to 5, oral 1. Add $2 and f3 ; 2 c? and 3 t^; 2 • 25 and 3 • 25. 2. Add 3 yd., 2 ft., and 7 in., expressing the result as a compound number. Add 3 y, 2/, and 7 i; 3 a, 2 5, and 7 c. 3. Add a, — 2b, and 3 c; x,—2y, and Zz; m, —In, and dpi- 4. Add —2a,b, and 4c; —2x,y, and 4 g ; —Zx,y, and z. 5. Add 5 a, 6 J, and —la\ hp,^ q, and — 7 r ; a, 4 a, and c. Add the following : 6. a, -a,2,- 2. 11. 18 a^, 36 a% - 7 a%. 7. —a, a, - 2, 2. 12. 2V^, 13va, 15 Va. 8. 19 a, -Sx,3x. 13. 16 V^, - 23 Vm, Vwi. 9. -4 a, 42 a;, 4 a. 14. 92 a6c, 37 aJc, - 75 aJc. 10. 17 a, -19 2;, 19a;. 15. 27(a - 5)+ 3(a - J). ADDITION OF MONOMIALS 73 68. Addition continued. In commercial work we frequently find it necessary to make additions like the following: 7x25 112 X sh X ^-^^ 975 X 0.04 X 8 4x25 238 X s^-g X 0.05 654 X 0.04 X 8 9x25 196 X gi^ X 0.05 781 X 0.04 X 8 20x25 546 X 31^x0.05 2410 X 0.04 X 8 = 500 = 7.48% = 771.2 In the first case we add exactly as if we had 7 x + ix + Qx, only that we have 25 instead of x. In the second case we add exactly as if we had 112 x + 238 x + 196 x, only that we have ^^-g x 0.05 instead of X, and similarly for the third case. We see how much easier it is to add in this way than to perform all the multiplications and then add. Exercise 48. Addition of Monomials JExaTivples 1 to 5, oral Add the following : 1. 2. 3. 4. 5. 2«2i2 12xyz^ 2pqr - 2abo 3x10 baa?' l^xyz^ ISpqr - Sabe 9x10 7 aa? %xy^ Vifqr ^ babe 3x10 3aa^ ^xys^ bpqr — 15 abe 5 x 10 Add the following and find the value of each result : 6. 7. 8. 5 X 73 198 X 3I5 x 0.06 975 x 0.04 x 8 9 X 73 256 X ^Ig X 0.06 654 x 0.04 x 8 6 X 73 458 X 3!^ x 0.06 781 x 0.04 x 8 15 X 73 954 X ^\^ x 0.06 963 x 0.04 x 8 42 X 73 845 x ^l^ x 0.06 777 x 0.04 x 8 81 X 73 976 X 3-1^ X 0.06 549 x 0.04 x 8 91 x73 721 X 3^-^ X 0.06 1245 x 0.04 x 8 74 ADDITION AND SUBTRACTION Exercise 49. Addition of Pol3niomials Examples 1 and 2, oral 1. Add 2 ft. 3 in. and 3 in. ; 2/+ 3 « and 3 i. 2. Add 7yd. 2 ft. and 6 yd. ; ly + 2f and 6 y. Add the following : 3. 9. 6 ft. + 7 in. 2 ft. + 3 in. X^+ 4:X1/-10Z^ 4. ' 10. 6f+7i 2f+Si 3 w + 2 a: + 3 «/ + 4 a 2 w - 3 a; + 2 «/ + 3 2 5. 11. 6.5+ 7.8 2.5+ 3.8 (?).5 + (?).8 72+ 5.7.3 72- 2.7.3 (?)-72+(?).7.3 6. 12. 2rd.+ 8ft.+ 9in. • 4rd.+ 3ft.-2in. a;2 + 10«/ + 7s + 9 a;2- 32^+4s-9 7. 13. 2r+8/+9i 4r + 3/-2z a;*+2;y + /+a:2+t/2 x^-xY+y^-ay^-f 8. 14. 2.3 + 8.7+9.4 4.3 + 3.7-2.4 a + 35-e + cZ-e+/ a - 3 S + c - c? + e - / 15. If A = x^+2x^-7x + l and 5= 2iBS + a;2 + 10a;+ 2, what is the value oi A + B? ADDITION OF POLYNOMIALS 75 69. Addition of Polynomials. In adding polynomials it is usually desirable to proceed as follows: Write similar tenns in the same column and add these terms, writbiy their sums as a polynomial. 70. Check. An operation that tends to prove the correct- ness of another operation is called a check upon that operation. 71. Checks in Algebra. One of the best checks is the substi- tution of an}' values we please for the letters, as in the following example, where we \&t x =1, y = 1, and « = 1. Operation Check 2x + Zy-4:Z + y and — Zx — 7y. 3. Add a + h + c and a — h + o; a — h — c and a + l + c. Add the following and check the results : 4. x^+xy + y\x^—2xy + y% x^+2xy + y^- 5. x^+ «/2, x^ - 2/2, 2 a;2+ y^, x^+2 y\ x\ 3 y\ 6. x^ + x^y -t- xy^ + y^, 2x^— Zx'^y + 7xy^ — ^ ^- 76 ADDITION AND SUBTRACTION Add the following and check the results : 7. bx^ + 2,x-\-l,Qa?-Zx+l,x + 2. 8. 72?-9a: + 2, 62^-8a; + 3, a;-9. 9. 23? + 2,x-^,4.a?-r . 1.66|a; = 275. Dividing by 1.66|, x = 165. 3. Aftet deducting JO % from the marked price of a table, a dealer sold it for $13.50. What was the marked price ? Let X = the number of doUars of the marked price. Then x - 0.10 x = 13.50, or 0.90 a; = 13.50. Dividing by 0.90, x = 15. Therefore the marked price was $15. PROBLEMS IN PERCENTAGE 87 Exercise 59. Percentage Examples 1 to 10, oral 1. A number less 10% of itself is what per cent of itself ? 2. A number plus 10% of itself is what per cent of itself ? State the per cent of x in each of the following : 3. 2;+0.05 2T. 5. a;+20%a:. 7. a;+0.50a;. 9. x+75%x. 4. a;- 0.05 a;. 6. a;- 20% a;. 8. a;- 0.50 a;. 10. a;-75%a;. 11. What number less 10% of itself is equal to 72? 12. What number less 17% of itself is equal t6 166 ? 13. $435 is 6% of what sum of money? 14. 119.75 is 5% of what sum of money? 15. $38.25 is 4^% of what sum of money? 16. What is the number of which 14.4 is 66|% ? 17. What is the sum of which $41.25 is 15%? 18. What is the sum of \Yhich 331% is $2.25 ? 19. A certain number increased by 12^% of itself is equal to 819. What is the number ? 20. A boy now weighs 84 lb., which is 12% more than he weighed a year ago. How much did he weigh a year ago ? 21. A dealer made a profit of $7380 this year in his store. This amount is 18% less than the profit made last year. How much was his profit last year ? 22. A dealer was obliged to sell some damaged furniture at 10% less than cost and sold it for $85.50. How much did it cost? How much did he lose? 23. A farmer sold his milk to a dairy company and received credit for 1045 lb. of butter fat. If his milk tested 3.8% butter fat, how many pounds of milk did he sell ? 88 ADDITION AJSTD SUBTRACTION 80. Removal of Parentheses preceded by the Positive Sign. If to $4 we add $3 + $2, we have in all $4 + $3 + $2, or $9. It is evidently of no consequence whether we add |2 to the sum of $4 and $3 or add $5 to H- Hence H + (|3 + |2) = #4 + #3 + $2. Similarly, 5(f + (7(^- 2(^)= 5(^ + 76 - 2<^, Therefore a + (b + c) = cp+ 6 + c, and a + (b—c) = a + b — c. If the parentheses inclosing an expression are preceded hy the positive sign, the parentheses may he removed without any change in the signs of the terms. We may treat in a similar way any other sign of aggregation. Thus, 7 + in = 7 + 3 + 4, and 12 + [5 - 2] = 12 + 5 - 2. Exercise 6Q. Removal of Parentheses Examples 1 to 6, oral 1. How much is $4 plus the sum of $3 and |9? How much is the sum of |4, |3, and |9 ? Remove the parentheses and simplify : 2. 7 + (9 + 6). 4. 4a + (3a + a). 3. 8 +(15 -5). 5. Qa?y^ + (9la?y'^-x^y^-). 6. How much is the sum of 7 and 3 increased by the sum of 4 and 2 ? How much is the sum of 7, 3, 4, and 2 ? Remove the parentheses and simplify : 7. (2a + 3a) + (7a + a). 11. a-b + (a + b). 8. (2a + 3a) + (7a-a). 12. a^ - P + (a^ + P). 9. (2a-3a) + (7a + a). 13. 5a+17 + (5 a-17). 10. (2a-3a) + (7a-a). 14. (6x + 9y) + (5x~ By^. PARENTHESES 89 81. Removal of Parentheses preceded by the Negative Sign. If we subtract b + c from a and subtract b — c from a, we have the following results: a a b + c b — c a — b — c a — b -{- Therefore a — (b + c) = a—b — c, and ' a—(b—c) = a—b + c. If the parentheses inclosing an expression are preceded by the negative sign, the parentheses may be removed, provided the sign before each term is changed. That is, x + y — (x — y') = x-\-y — x + y = 2y; 3 5-4 3-5+4 2 , and — = = 2=1- the fraction bar having the force of parentheses. If 4 = a:^ + 2 a; - 1, and B = a;" _ 7 a; _ 8, then A-B = x'^ + 2x-l-x^+7x-\-% = ^x + 7, and B- A=x'^-7x-?>-x'^-2x + l=-^x-T. Exercise 61. Removal of Parentheses Examples 1 to 8, oral 1. Subtract 7 — 5 from 10. Subtract 7 from 10 and add 5. How do the two answers compare ? 2. Subtract 7 a— 5 a from 10 a. HowmuchislOa- 7a + 5a? How do the two answers compare ? Remove the parentheses and simplify : 3. 20 -(8 + 2). 6. 4a-(3a-a). 4. 20 -(8 -2). 7. 7x-(4:x-2x'). 5. 30 -(9 -4). 8. 9a;-(5a;-3a;). 90 ADDITION AND SUBTRACTION 9. li A = a?-Bx-15,a.ndB = x^-9x- 11, what is the value oi A-B? of B - A? 10. In Ex. 9 what is the value oix^-A? oix^ + 7 -A? 11. In Ex. 9 what is the value of A -(x^ - 5j- + 7)-B? 12. In Ex. 9 what is the value of -B - (s^^ + 9 a; - 6) - ^ ? 13. Simplify (17 a + 15 5) - (12 a - b). Remove the parentheses and simplify : 14. 12x-(9x+7x:). 19. 39ab + (ab+l}. 15. 17 2/ -(12?/ -4^). 20. 56a2_(3a2 + 7). 16. 39 a - (15 a + 6). 21. 48^ + (27p - 2). 17. 47a-(15&-H7a). 22. 63^-(49 4-4g). 18. 56w + (27m-4«). 23. 79 a;- (79 - 79a;). 24. Simplify 25 a2_ 2 a2+l; 25a^ + 2a^ + l. Remove the bars and simplify : 25. 4 a — 3 a — a. 28. 9 r — 7 a; - 26. 7 a — 4 a + 2 a. 29. ab'+ 3 — ab. 27. 8 a + 6 a - 3 a. 30. a;y - 7 + xy. Remove the brackets and simplify : 31. 27 a2 _ [4 - 27 a^]. 35. 80 mw + [75 - 80 ww]. 32. 35 aJ - [5 + 35 aJ]. 36. [35a-+l]-l. 33. 4:2xi/-l-xi/ + 4:2}. - Z1. [42-a:] + a:. 34. lb^y^ + \-o^y^-\-l\ 38. -[27 + a6]-a6. Remove the parentheses and brackets and simplify : 39. [a + 6] -(a -6). 42. -[a + 6]-(- a-i). 40. [a-6]-(a + J). 43. - [a- i]-(- a + 6). 41. [a + 6] + (-«-&). 44. [x^ + y^-z^']-Qi?-y^ + z'^~). EEVIEW 91 Exercise 62. Review of Chapter V 1. Add 7500 H, 475 H, 9650 H, 8754 rt, 4965 H. 2. In Ex. 1, if r= 0.05, and t = ^^-^, find the sum. 3. Representing money on which interest is allowed as positive ; sums on which interest is charged as negative ; the rate of interest allowed, by r; the rate of interest charged, by r' ; one day in interest allowed, by ^l-g- ; one day in interest charged, by -^^ ; find the sum of the interest debits and credits on the following daily balances for 11 da.: W- |35,000, + 137,500, - $15,000, - $125,000, + $27,800, + $250,000, -$18,500, -$30,000, -$12,000, +$96,000, +$63,000. 4. Find the value of the last daily balance in Ex. 3 if it is given that r = 0.02, r' = 0.04. 5. If C = gross cost, O = overhead charges, P = profit, and S = selling price, write the meaning of the following formula : C = 8-{0+P'). 6. Write the formula in Ex. 5 so that it shall have the same value, but without using the parentheses. 7. Add +P to each side of the formula derived in Ex. 6, and explain the meaning of the result in retail business. 8. Find the cost of an article which a dealer sells for $38.13, if the overhead charges are 18% of the cost and the profit is 5% of the cost. Add the following : 9. 4 gal. + 1 qt. 9 gal. + 2 qt. 8 gal. — 1 qt. 5 gal. + 1 qt. 6 gal. — 1 qt. 10. 11. 720 X 0.06 X ^\^ £18 + 5s 4560 X 0.06 X ^1^ £19 + 7s 540 X 0.06 X ^i-g- £ 5-3s 7200 X 0.06 X ^1^ £10 -4s 673 X 0.06 X ^t; £ 9 + 8« 92 ADDITION AND SUBTRACTION Exercise 63. Review of Chapters I-V 1. Using the formula i=prt, find the interest on $4000 for 35 da. at 5%, using 365 da. for 1 yr. 2. Fiud the cost of a piece of oak timber 18 ft. long, Bin. wide, and 3 in. thick at 10^ a board foot. Write the formula used. 3. Sheet copper that weighs 3 lb. to the square foot is to be used to cover a circular surface the radius of which is 5 ft. At 35^ a pound how much will the copper cost? 4. A, B, and C have a partnership contract by which they agree to allow A and B $1200 a year each for conducting the business, and to divide the remainder of the available funds by allowing 11 shares to A, 15 shares to B, and 20 shares to C. How should they divide $11,600 of available funds ? 5. Solve these equations: Bx + 0.5x = 70; •|x = 15. 6. I have 60 bu. of potatoes in barrels which hold 2 bu. each and cost 30^ each. A dealer offers me $35 for the lot. If I accept the offer, how much shall I realize per bushel ? 7. Subtract B8a^- 5xy + 9 i/" from 11 2^ +llxy-9 y\ 8. In the following algebraic and numerical expressions a 2456 h 3148 a+ I 5604 a + 2b 8752 2 a + 3 6 14356 each number after the first two is derived by adding the two immediately above it. Continue this process till 10 lines are formed, and then add. What does the algebraic work show is always the relation of- the sum of the ten addends to the seventh addend? CHAPTER VI MULTIPLICATION Exercise 64. Multiplication of Monomials Examples 1 to 2S, oral 1. Multiply by 5: 2mi.; 2ft.; 2m; 2/; 2 • 3; ^x. 2. Multiply by 7: 3 yd.; 3«/; 3 times a given number ; 3w. 3. Multiply by9: 4in.; 4i; *4; 4cZ; 40; 4e; 42?/; 4 • 7. 4. If the temperature is 3° below zero, what will it be when it is twice as much below zero ? 5. How much is 2 x(- 3°)? 2x(-4°)? 2x(-3<^)? State the following products : 6. 2x12 a. 10.3 x20 a;. 14. 4x50 w. 7. 2x12 ah. 11. 3 X 20 xi/. 15. 4 x 50 m^ 8. 2x(-12). 12. 3x(-20). 16. 4 x (- 50). 9. 2x(-12a2). 13. 3 X(- 20 a?). 17. 4 x (- 50 Va). 18. How may a X a be written more briefly ? a x a x a? a X a^? a^Xa? a x a^? a^xa^? a^X a^7 19. State a rule for multiplying a^ by a\ Find the following products : 20. 3x4 a2. 24. a? x 6 afi. 28. 8 x 36 a^. 21. ax4a2. 25.5x'^x6a?. 29. 8 a; x 36 a;^. 22. 3ax4a2. 26. 1 xll m\ 30. 9x^x36 a?. 23. 5x6a;3. 27. 7^x17^2. 31. 9a^x48a:3. 9S 94 MULTIPLICATION 82. Multiplication. Expressions like " 2 times 3 " show that the operation of multiplication had its origin in addition, 2x3 meaning 3 + 3. When we come to extend the idea of multiplication to include negative and fractional multipliers, however, we have also to extend the meaning of the word " times," for it is just as impossible to add a number 2^ times, — 2 times, or ^ times, as it is impossible to go to a store 2^ times, — 2 times, or |^ times. So we give new meanings to the word " times " to cover cases like those mentioned. For example, 2^ times 6 means 2 times 6 plus ^ of 6 ; — 2 times 3 means — 3 — 3, or — 6 ; ■| times 12 means ^ of 12, or 3 x ^ of 12. 83. Laws of Signs. The case of — 2 x 3 has already been explained on page 66. We may now state in different words the law of signs which is there given and which appUes to all algebraic expressions: In multiplication two like signs produce plus and two unlike signs produce minus. That is, +a.(+6) = +aA, +a-(-6) = -aJ, — a . (— J) = + aS, — a . (+ J) = — aJ. Evidently, therefore, the product of an even number of negative factors is positive, and the product of an odd number of negative factors is negative. These laws lead to the consideration of powers of negative numbers. Thus, (— 1)^ = + 1, and (— a) to any even power is positive ; (— ly = — 1, and (— a) to any odd power is negative. That is, even powers of negative nurnbers are positive ; odd powers of negative numbers are negative. MULTIPLICATION OF MONOMIALS 95 84. Laws of Exponents. Since a^ means a • a, and a? means a . a . <3s, we see that, as we found in the exercise on page 93, Furthermore, a^=a taken m times as a factor and «"= a taken n times as a factor. Therefore cS^ • a^—a taken m-\-n times as a factor. That is, a"" •a''=a'"+"- For example, LOG" x 1.06^ = 1.06^ as we found on page 20. It should be observed, however, that 1.04^ and 1.06^ cannot be multiplied together in this way. In rmdtiplying monomials the exponent of any letter in the product is equal to the sum of the exponents of that letter in the factors. Since (a^)^ means a^a^a^, or a% and (a™)" means a^ce^a^ • . . to n of the factors a", therefore (a"")" = a'"". For example, (a*)*" — a^^, (x'y = x^, and so on. 85. Multiplication of Monomials. As we have seen in the exercise on page 93, to find the product of two monomials we may proceed as follows: Find the product of the numerical coefficients, writing after this product the letters, each letter having an exponent equal to the sum of its exponents in the factors. It will be found better to write the product in this order : the sign of the product, the product of the numerical coefficients, the letters in their alphabetical order, each with its proper exponent. For example, 3 a™6 • 4 a^h = 3 ■ 4 • am • a" ■ 6 • J = 12 a™ + '1 &2 ; (- 7xY)2 = - 7 ■ (- 7) -a:' -a' -y' •/ = 49 x'^Y^; 6 ahc ■ (- 7 a2JV2) = 6 ■ (- 7) • a • a" • 5 • J2 • c • c^ = - 42 a%^cK In practice the result should usually be written rapidly, or stated orally, without the intermediate step given here. 96 MULTIPLICATION Exercise 65. Multiplication of Monomials Examples 1 to 42, oral Perform the following multiplications : 1. a^a^. 12. a^ajftx 23. 3 a . (- 4 a). 2. bW. 13. a^^a^". 24. 3 a ■ (- 4 a2). 3. cV. 14. ah ■ ah. 25. 3 a™ . (- 5 a«). 4. d'cf. 15. a% . a%. 26. ab . (- ab). 5. e\\ 16. aW . a%\ 27. — ah • ah. 6. m^m^K 17. a%'' . aW. 28. -ab.(- ah-). 7. pipVi_ 18. p^(f •p'c^. 29. a?f.(-2xy). 8. (a^^y. 19. 5 a^ . 6 a\ 30. -3 2/3. (-72/). 9. (ai2)2. 20. 8 (ay. 31. (2 aJ2c3)3. 10. («™a)3. 21. 7(x2)3. 32. (- 2 «2J8)4 11. (a™a4)2. 22. (8 x^y. 33. (aSe • a6e)3. Express the following products as powers : 34.2^x2^. 36.1.062x1.061 38. (a+5)*(a+J)i<'. 35. 3* X 31 37. 1.038x1.039. 39. 20^ x 20 x 20^. 40. If the circumference of a circle is ttcI, what is the sum of the circumferences of seven circles of diameter d? 41. If the circumference Trd of a circle is multiplied by d, what is the result ? What if it is multiplied hj ^d? 42. If the edge of a cube is 4 x, what is the volume ? Perform the following multiplications : 43. - 27 a;'" • (- 23 x''). 46. 37 a^™ • 67 3?'^. 44. - 42 a;" • (- 34 a^). 47. 83 a;™+i • 75 a?"-i. 45. - 68j9™ . (- 27^»). 48. iSaP + ^ -72 aP-^ MULTIPLICATION OF A POLYNOMIAL 97 86. Multiplication of a Polynomial by a Monomial. If we multiply 5 ft. 2 in. by 3, we have 15 ft. 6 in. Also, if we mul- tiply 5 times one number plus 2 times another number by 3, we have 15 times the first plus 6 times the second. That is, 5 ft. 2 in. 5/+2« 5a; + 2«/ J _3 J 15 ft. 6 in. 15/+ 6 1 15a; + 6«/ In the same way we have the following: Operation Check a2 -2a6 +3 62 1-2 + 3 = 2 a6 1 =1 a%-2aW + '^ab^ 1-2 + 3 = 2 In algebra it is more convenient to write the multiplier at the left. In this check we let a = 1 and 6 = 1. This checks the coefficients, where the error is most liable to occur, but it does not check the expo- nents, since any power of 1 is 1. If a check upon the exponents is desired, let a = 2 and i = 3, or take any other convenient values. To multiply a polynomial hy a monomial, multiply each term of the polynomial hy the monomial and add these products. Exercise 66. Multiplying by Monomials Examples 1 to 7, oral Perform the following multiplications : 1. a(6 + c). 8. 2x^(x^-2xy + y^). 2. -a(h-c'). 9. 4 2)3(2^- 3 a; + 27). 3. 2(22+23). 10. -7ab(a^-2a^ + P). 4. 2 7r?-(r-l). 11. -9a%(a^-2ab + p-). 5. aCb + c-d). 12. 4:xy^(2!t?-xy -By^). 6. -a(b-c + d}. 13. 35 aV (^a'' - 3 aV + 2 ax^'). 7. J2(a2_52+e2). 14. - 8 »V(2^y-7««/ -13)- 98 MULTIPLICATION 87, Factors. Since S x(P x +1} = 9 a? + S x, we can see at once that the factors of 9 a^ + 3 a; are 3, x, and 3 a; + 1. When asked to factor 9 ar" + 3 a;, we express the result thus : 9x^ + Sx=3x(Sx + l^. Exercise 67. Products and Factors Mcamj)les 1 to 6, oral Express the following as algebraic sums of powers : 1. a^(a^ + 0,^ + 0*-). 4. 73(78-7^ + 7*). 2. a!3(a^-a^ + a;0. 5. 410(46 + 43 + 42). 3. 53(5^ + 53 + 5). 6. 210(2 + 220 _g30)^ Find the factors of the following binomials : 7. a'^ + aK 10. 3m^ + 9tn. 13. Ax + 4 1/. 8. x-a?. U. 3to2_9to. li. ix-Si/. 9. 2a;2-2a^. 12. Zm^-9m\ 15. 12x-%y\ Multiply the following : ' 16. 39(100+10 + 2). 19. 785(1000-1). 17. 78(100+10 + 5). 20. 45(9000-2). 18. 7(60 + 20-10). 21.846(3000+100 + 2). Find the products in Exs. 22, 23; the factors in Exs. 24, 25: 22. 2;(a;3 + a?2 + a;-4). 24. a:* + a^ + a;^ _ 4 j.. 23. 2»n(m2 + 3m-4). 25. 2«i3 + 4m2-8m. Find the factors of the following binomials : 26. 7 a? +14 a;. 30. ■7ri^-2r. 34. ^bh + ^Bh. 27. 7*2- 14 a^. 31. ^g^-gt. 35. irB^-irr^. 28. 8a3+16a. 32. gt-^g. 36. 2'rrIi-2-irr. 29. ■7rr^-2t^. 33. p +prt. 37. Kx^ + ^x. MULTIPLICATION OF POLYNOMIALS 99 88. Multiplication of a Polynomial by a Polynomial. If we multiply 43 by 21, we proceed as follows: Multiplicand, 43 ' 40 + 3 Multiplier, 21 20 + 1 Multiplying by 1 unit, 43 40 + 3 Multiplying by 2 tens, 86_ 800+ 60 Sum of partial products, 903 800 +100+3 In a similar manner we multiply Aa^ — b by a^ + h, thus : Operation Check 4a2_ h 4-1 = 3 a2+ b 1+1 = ? Multiplying by a^, 4 a* — a% 6 Multiplying by b, 4 a% — V^ 4a* + 3a26_S2 4 + 3_l = 6 To multiply a polynomial by a polynomial, multiply the multiplicand by each term of the multiplier and add the partial products. Exercise 68. Multiplying by a Binomial Perform the following multiplications: 1. (a + h)(a + 2by 10. (a + J)(a-S). 2. (a + S)(a-2 6). 11. (« + 6)(a + S). ■ 3. (2 a + J) (a - i). 12. (a^ - 6^) (« - by 4. (2a-J)(«-J). 13. (a2-52)(a2 + J2). 5., (a + J) (a2 - 62). 14. (ahc -1~) {aWe^ + by 6. (2xy-V,(2xy+ly 15. (^a%^c - 2~) (c^V^c + 2y 7. (2xy + T)(2xy-2y 16. (xy^+^')(xy^^ -^y 8.' (4 x^y + 1) (4 a?y - 3). 17. (4 «2 _ 5 J2) (3 a2 + 4 J^). 9. (a^2/2 + 1) (72^2^2 _ 9). 18. (x^ + a;») (5 a:*" - 3 a;«). 100 MULTIPLICATION 89. Arranging a Polynomial. If the exponents of a certain letter in the successive terms of a given polynomial decrease or increase from left to right the polynomial is said to be arranged according to the descending powers or ascending powers of that letter respectively. Thus, ofi + Zx^ — iiX +1 is arranged according to the descending powers of x, and o' — 3 a^6 + 3 aV^ — b^ is arranged according to the ascending powers of 6 and also according to the descending powers of a. In multiplying, it simplifies the work greatly if both the polynomials are arranged according to the ascending or accord- ing to the descending powers of the same letter. Multiply 4:a^-2a^ + 7-ahj 2, + a^-Za. Rearranging the polynomials, we multiply thus : Operation Check a^~ 3ffl + 3 '_ =1 -12«4+ 6a3+3a2_21a 12a3-6a2- 3a + 21 4a6-14a*+17a3 + 4a2-24« + 21 =8 In the operation we multiply first by a^, then by — 8 a, and finally by 3. We then add the partial products. In the check we let a = 1. We have, then, only to add the coeffi- cients, having 4 — 2— l-f-7=8, and similarly for the others. Multiply x^—^ixy+^y^hyy^ + x^. Rearranging the multiplier, we multiply thus: i^-%xy + 2,y^ =1 a^+ y^ =2 a^ ~ ^ 3^y + S a?y^ 2 x^y^ -Sxf + Sy^ itii-Sx^y + 4:a?y^-dxf + 3f =2 MULTIPLICATION OF POLYNOMIALS 101 Exercise 69. Multiplying by a Polynomial ExaTnples 1 to 4, oral 1. Arrange/ a^ + a^ +1 + a for multiplying. 2. Arrange x + l+a^ — aP' + ^i^ + x^ + x^ + x' according to the ascending powers of x. • 3. Arrange o^-\-y'''-\-1x'y according to the descending powers of x. How is it then arranged with respect to ^? 4. Apply the check of a; = 1, 2/ = 1; a^ncl thus find whether 0^ -\-Zxy + y^ can be the product oi x-\-y and x + y. Multiply and check : 5. ip + q)-q). 7. (m + n) (m + ri). 10. (x + y')(x — «/). 11. cfi — h'^hja — h; hj a + h; by a^ + 5^. 12. a^ + lab + H^hY a + h; \>j a — h; hj a^ + 2ah + h\ 13. a2 _ 2 ai + 62 by a - 6 ; by a + J ; by a^ _ 2 aft + J2. 14. 7A + 4*+3w by 5« + 7m. Teachers should call attention to the relation of Ex. 14 to 57 x 743. 15. 7 A + 6 t + 2 M by 5 A + 7w ; by 5 A + 3 « + 7m. 16. 4 2^ + 2 a;2^ - 3 a;/ + 2 ^^ by 3,2 _ 2 a;y + 4 «/2. 17. 3 m^ + 2 m^ — 4 mw^ + n^ by m^ + w^. 18. 5m* + 2m^ + m+l hj m^ + 2m-l. 19. 8 j93 4- 2/ +^ _ 4 by / - 2^ + 3. 20. 42;3_3 2;a + 22;-3bya?-23;-5. 21. a2J2 + 3 a6 - 2 by a^ja _ 3 «§ + 2. 22. a^ + x^-Sax+2 by a^ _ -^a + 3 ^sa; - 2. 23. afi + y^ — a^y + xy^ by x^ — xy + y\ 102 MULTIPLICATION 90. Check of Nines. One of the first things to learn in commercial work is the necessity for absolute accuracy. Speed is desirable, but accuracy is absolutely necessary. To insure this accuracy various checks on the number work are used, and among them is the check of nines. This check is used chiefly in multipUcation and division, and we shall first give the method, using algebra to explain it later. To find the remainder in the division of a number hy 9, add the digits of the nwmher, casting out the 9's as they appear. For example, to find the remainder in the division of 1,478,603 by 9, add the digits and cast out the 9's, thus : 1 + 4 + 7 = 12 ; cast out the 9 that is in 12 and there remains 3 ; then 3 + 8 = 11, from which we east out 9 and there remains 2 ; then 2 + 6 + 3 = 11, from which we cast out 9 and there remains 2. If we divide 1,478,603 by 9, we shall find that the remainder is 2, as here shown. Practically, it is not necessary to add all the digits, for we can easily see that 1+8 = 9 and 6 + 3 = 9, so we need only cast out the 9 in 4 + 7. The remainder arising from dividing a number by 9, or from dividing the sum of the digits by 9, is called the excess of nines in the number. Thus 2 is the jexcess of 9's in 1,478,603. Multiplication may he checked hy finding the excess of nines in a product and in each factor, the excess in the product being equal to the excess in the product of the excesses in the factors. For example, consider the case of 38 X 27,394. Here the excess in 27,394 is 7; that in 38 is 2 ; and that in the product is 5. If the multiplication is correct, the excess in 2 X 7 must be 5. This is the case, because the excess in 14 is 5. The check of nines is only a check, not an absolute proof. 27394 Excess = 7 38 Excess = 2 219152 14 82182 1040972 Excess = 5 CHECK OF NINES 103 Exact division may he checked by the check of multiplication, because the dividend rnay be taken as the product, and the divisor and the quotient may be taken as the factors. For example, in the case of 1/40 -^ 12 = 145, we see that 1740 = 12 x 145. Taking the excesses of 9's, we see that in 1740 it is 3, in 12 it is 3, and in 145 it is 1. Furthermore, 3 (excess in 1740) = 3x1 (product of excesses in 12 and 145). If a division is not exact, find the product of the excess in the divisor by the excess in the quotient, and add the excess in the remainder. The sum, or the excess in the sum, should be equal to the excess in the dividend. For example, in the case of 118,784 -h 4892, we see that the quotient is 24 and the remainder is 1376. Taking the excesses of 9's, we see that in 118,784 it is 2, in 4892 it is 5, in 24 it is 6, and in 1376 it is 8. We can now check the result as follows ; Excess in 4892 = 5 Excess in 24 = _6 Product of excesses = 80, excess = 3 Excess in 1876 =_8 Sum of excesses = 11, excess = 2 Excess in 118,784 = 2 24 4892)118784. 9784 20944 19568 1376 Exercise 70. Check of Nines Perform the following operations and check the work : 1. 275 X 398. 2. 309 X 588. 3. 644 X 966. 4. 722 X 880. 5. 4.29 X 6.77. 6. 5.39 X 7268. 7. 68,781-5-51. 8. 54,227 -f- 83. 9. 52,870-^85. 10. 68,725-5-77. 11. 87,316-^-57. 12. 46,817^-67. 13. 6296 X 5287. 14. 5865 X 7963. 15. 8559 X 6279. 16. 22,712-5-641. 17. 38,499 -f- 827. 18. 86,225 -V 653, 104 MULTIPLICATION 91. Proof of Check of Nines. Algebra allows us to prove the statements on pages 102 and 103. Any number of three figures may be represented by 100 a +10 b + c, because a, b, and c may represent any digits. For example, 874 may be represented by 100 a + 10 6 + c by letting a = 8, 6 = 7, c = 4. But 100a+10b + c = (99a + a) + (9b + b)+c, and this may be written (99 a + 9 6) + (a + 5 + c), where every part of the first parenthesis is divisible by 9. Therefore the only remainder possible comes from dividing a + b + c (the sum of the digits) by 9. For example, 874 = 800 + 70 + 4 = 8 x 99 + 7 x 9 + 8 + 7 + 4. Hence the only remainder we can obtain is the remainder which comes from dividing 8 + 7 + 4 (the sum of the digits) by 9. The same result as to divisibility by 9 is evidently obtained if we take a number of four or more figures. It is evident that any number is equal to some multiple of 9 plus the excess of nines. That is, 58 = 9 x 6 + 4, 343 = 9x38+1, and in general every number is of the form 9 m + e. If we represent two numbers by 9 m + e and Q m' + e', we find that their product is 9^ mm' + 9 (me' + m'e) + ee'. The teacher should explain that 9 m + e means any multiple of 9 plus the excess, and that 9 m' + e' means any other multi- ple of 9 plus the excess ; that is, that 9 m + e and 9 m' + e' repre- sent any two numbers whatever. This product is evidently divisible by 9, except the last term, ee', which is the product of the excesses. That is, as stated on page 102, the excess of ffs in the product is equal to the excess in the product of the excesses of 9's in the factors. 9 m + e 9m' + e' 9^mm' + 9 (me' + m'e) + ee' SPECIAL PRODUCTS 105 92. Special Products. There are certain products which are required so often in algebra that it is helpful to be able to write them at once without the labor of multiplying. 93. Square of the Sum. or Difference of Two Numbers. If we multiply a + bhy a + b, the product is a^ + 2ab + b^; and if we multiply a — b by a — b, the product is a^ — 2ab + P. a + b a—b a + b a—b c? + ab a? — ab ab + b^ - ah + b^ ab b-" a' ab a^ + ^ab + b^ a^-1ah + b^ The square of the sum of two numbers is the square of the first, plus twice their product, plus the square of the second. That is, (a + &)'' = a^ + 2 a& + V. The expression a' + 2 ab + b^ is called the expansion of (a + by. For example, 1.32 = (10 + 3)2 = 10^ + 2 X 10 X 3 + 32 = 169; 352 = (30 + 5)2 = 302 + 2 X 30 X 5 + 52 = 1225; 1252 = (100 + 25)2 = 1002 + 2 X 100 X 25 + 252 ^ = 10,000 + 5000 + 625 = 15,625. Thus we see that we can mentally square certain numbers, saving ourselves the trouble of writing out the work. It is easily seen that the figure representing the square on a + 6 is made up of a% ab, ab, b^, and therefore is a2 + 2 a6 + 62. The square of the difference of two numbers is the square of the first, minus twice their product, plus the square of the second. That is, {a-bf = (^-2ab+V'. Therefore 372 = (40 _ 3)2 = 402 - 2 x 40 x 3 + 3^ = 1309. Of course we might also square 37 in this way : 372 = (30 + 7)2 = 900 + 2 X 30 X 7 + 49 = 1369. 106 MULTIPLICATION Exercise 71. Square of the Sum or the Difference Examples 1 to 37, oral Expand the following : 1. Cx + yy. 10 (x -yy- 19. (2 a + 6)2. 2. ip + qy. 11. dp -qy. 20. (2 a -by. 3. (m + w)2. 12. (m -ny. 21. (2 a +1)2. 4. (a+xy. 13. (a -xy. 22. (2 a -1)2. 5. (b + yy. 14. Q> -2yy. 23. (4:x + yy. 6. (a +1)2 15. (a -2)2. 24. (px-yy. 7. (a + 2)2 16. (a -7)2. 25. (x + Qyy. 8. (a; + 3)2. 17. (X -6)2. 26. ix-7yy. 9. (4+w)2. 18. (8 -ny. 27. (2 a + 3)2. 28. How would you proceed if you wished to write the square of bx + Sy without multiplying in the usual way ? Expand the following . 29. 112. 3g. 622. 47. 1022. 56. (72^ + 6)2. 30. 122. 39. 712 48. 5032. 57. (7a? -6)2. 31. 212. 40. 882. 49. 9022. 58. (5a^ + 2y2)2. 32. 222. 41. 922. 50. 9982. 59. (5a^-2«/2)2. 33. 162. 42. 982. 51. 50062. 60. (a2J2c2+i2)2. 34. 312. 43. 792. 52. 49942. 61. (a2J2c2_12)2, 35. 142. 44. 782. 53. 50052. 62. (x^yi + Sny. 36. 412. 45. 912, 54. 70072. 63. (X^f-Sny. 37. 512 46. 522. 55. 10032. 64. (7aWc+liy. 65. Square (^a + b')+ c as if a + b were one term and then remove the parentheses in the result. SPECIAL PRODUCTS 107 94. Product of the Sum and the Difference of Two Numbers. If we multiply a — bhja + b, or a + b hj a — b, the product is a^ — 6^, as here shown : a—b a + b a + b a - b a^ — ab a^ + ab ab- -62 -ab- -52 a2 — 52 (j[2 _ 52 Therefore the product of the sum and the difference of two nwmhers is the difference of their squares. That is, {a+h)(a-h) = a^-W. For example, (5 a + 3 6) (5 a - 3 6) = 25 a^ - 9 A^, and 33 X 27 = (30 + 3) (30 - 3) = 900 - 9 = 891. Exercise 72. Product of the Sum and the Difference 1 Examples 1 to 13, oral Multiply as indicated: 1. {x + y){x-yy 5. (2 a^ -)- 1) (2 a;2 _ i). 2. (« + 7)(a-7). 6. (5a;7+l)(l-5a;0• 3. (3 2; + 8) (8 - 3 a;). 7. (4 ^2 + 2) (4 a2 _ 2). 4. (/ -I- 3) (/ - 3). 8. (a™6™-f-l)(a'»6™-l> ^ind the products of the following : 9.32x28. 14.61x59. 19.99x101. 10. 34 X 26. 15. 92 x 88. 20. 98 x 102. 11.41x39. 16.52x68. 21.117x123. 12.42x38. 17.29x31. 22.994x1006. 13. 51 X 49. 18. 44 X 56. 23. 2493 x 2507. 108 MULTIPLICATION 95. Factoring the Difference of Two Squares. Reversing the work on page 10.7, we see that The difference of the squares of two numbers is the product of the sum and the difference of the numbers. That is, a^-li' = (a+b)(a-b). Factor the binomials a* — b^ and afi — y^. ai-b^ = (a2 + 6) (a2 - b). x'~y^ = (x* + 'f) (a* - ■f) = (a;* + 3^) (ji? + y'^) (x + y)(x~ y). Exercise 73. Factoring the Difference of Two Squares JSxavvples 1 to IB, oral factor the following expressions : 1. a^-x\ 6. 82-72. 2. a;2-a2. 7. S^ - a\ 3. a2_l. 8. 92-52. 4. m2-4. 9. 492 -1. 5..w2_9. 10. 752-152. Factor the following numbers : 16. 143. 17. 120. 18. 80. 19. 624. 20: 9999. In Ex. 16, 148 = 144 - 1 = 12^ - 1 ; in Ex. 19, 624 = 25" - 1. Factor the following expressions: 21. 2;* -16/. 25. x^-OMf. 29. (x + t/y~4:z\ 22. l-afif. 26. 81a;^-l. 30. (x-yf-z\ 23. T^f-l. 27. a%^(^-d\ 31. (a - bf - (c - d[:f. 24. s^y^-a%\ 28. a%^ - 1)2 - (?• - 1)2. 11. 252 -a2. 12. ^2-102. 13. 4a2_j2, 14. 0^-16. 15. aW-(?. APPLICATIONS OF FACTORnSTG 109 Exercise 74. Applications of Factoring 1. In the trapezoid here shown, the area of triangle T is 1 5A, and the area of triangle T' is l Vli. What is the area of the trapezoid? Factor the result, letting lA be one of the factors. 1 2. In a cylinder the area of each base is irr^, and the area of the curved surface is ^irrli. This makes the total surface how much? Factor the result, thus simplifying the formula. 3. The area of the outside circle here shown is Tra^, and the area of the inside circle is ttW: What is the area of the ring formed by the two circles ? ' ' Factor the result, thus simplifying the formula. 4. In the figure here shown we have found that if AB= b, BC—a, and AG=h, then h^ = a^-i-P, or 1P'=W— a^. Write the equation h'^=}fi— a^ with the second member factored. 5. Using the factored form found in Ex. 4, find the value of 6^ when A= 5, a= 3; when A = 5, a = 4 ; when h = 35, a = 28 ; when h = 45, a = 36. 6. The edge of the larger of two cubes is a, and that of the smaller is h. The area of the base of one is how much greater than the area of the base of the other? Factor the result and evaluate for a == 30, h = 20. 7. In Ex. 6 what is the area of the entire surface of the six faces of each cube ? What is the difference in area ? Factor the result and evaluate for a = 80, h = 20. Evaluate the result also for a = 40, h = 10. B 110 MULTIPLICATION 8. The amount of principal and interest on a note is given by the formula a=p+prt. Factor the second mem- ber of the equation and evaluate for p = 350, r = 6%, t = S. ' 9. From a square of side a ' is put a square of side b, as shown in the figure. What is the area of the remaming part? Factor the result and evaluate for « = 60, b = 20. 10. The area of the surface of a sphere of radius r is 4:-7rr\ What is the difference between the surfaces of two spheres of radius a and b respectively? Factor the result and evaluate for a = 5, b = 4, and tt = 3|. Factor the following expressions : 11. (2 a; + 3 ?/)2 - a2. 17. (x + yy~(a-by. 12. (3a-2&)2_«/2. 18. (x-yy-(a + b-)\ 13. (4 m + Tif - «2. 19. (jjc _ yy -(a- by. 14. a^-(p-qy. 20. (2 a +1)^ - (2 6 +1)2. 15. 9a2_(a + J)2. 21. (1 - a)2 - (5 a - 6)2. 16. 16 62 _ (c - dy. 22. (ffl + 2 6)2 - {p - qy. 23. Find by the formula of § 95 the value of a^ — 62 when a =37.1, 6=15.3. 24. By actual multiplication find the values of 37.12 and 15.32, and then find the difference between the two results. Exs. 23 and 24 show the superiority of algebra over arithmetic. 25. Multiply 1002 by 998. Also find orally the product of 1000 + 2 and 1000-2.' 26. One square tile is 8.8 in. on a side and another is 7.2 in. on a side. What is the difference in area of the tiles ? 27. Find the difference in area of two circles of radius 7-3 in. and 5.7 in. respectively. SPECIAL PRODUCTS 111 96. Product of Two Binomials. If two binomials have a com- mon term, their product can easily be written. For example, X + 7 X + a X + 5 X +b a? + 7 X a?+ ax 5 a; + 35 hx-\-ah «2 + 12a;-|-35 x^-\-{a-^h')x-\-ah The product of two hinomials having a common term is eqaal to the square of the common term, plus the product of the com- mon term hy the algebraic sum of the other terms, phis the product of the other terms. That is, {x-\-a){x-\-b) = ^"+(0-1- b)x-\- ab. Thus (x + 7) (a; - 3) = a;2 + 4 a; - 21, because + 7 + ( - 3) = 4, and + 7 x ( - 3) =- 21. Similarly, (x ~ Q)(x - 6) = x^ -15 x + 54, and (a + 7) (a + 7) = a^ + 14 a + 49. Exercise 75. Product of Two Binomials All work oral Multiply as indicated : 1. (a-|-2)(a + 3). 10. (a-|-2)(a-3). 19. (a-2)(a-3). 2. (a-|-5)(a+7). 11. (a-5)(a+7). 20. (a- 5) (a- 7). 3. (x + 9)(x+S-). 12. (x + 9)(x-S}. 21. (a;-9)(a;+l). 4. («+4)(a-|-5). 13. (a-4)(«+5). 22. (a-4)(a-3). 5. (a-|-7)(a + 3). 14. (a+7)(a-3). 23. (a-7)(a+3). 6. (^ + 9)(i? + 4). 15. (^-9)0 + 4). 24. 0-9)(^-4). 7. (a;-|-6)(2;+2). 16. (a;+6)(a;-2). 25. (a;-6)(a;-|-4). 8. (x+5-)(x+9-). 17. (a;- 5) (2;+ 9). 26. (a;-5) (a;-9). 9. (x+2-)(x+7). 18. (a;+2)(2;-7). 27. (a;-2) (a;+7). 112 MULTIPLICATION 97. Application of Products. When two numbers differ but little from a common number whose square is easily found mentally, as in the case of 105 and 103, their product can be found mentally. Consider the following: X +5 100 + 5 X +3 100 + 3 3? + 5x 10,000 + 5x100 3a: + 15 3x100+15 x^ + Sx + 15 10,000 + 8x100+15 Practically, therefore, to multiply 105 by 103 we simply think of 10,000 + 800+15, or 10,815. Exercise 76. Products of Numbers All work oral * 1. What is the cost of 31 yd. of dimity at 32^ a yard? Think of 31 x 32 as (30 + 1) (30 + 2), or 900 + 3 x 30 + 2, or 992. 2. What is the cost of 45 lb. of butter at 42^ a pound? 3. If I pay 85% of 82% of the list price of an article, what per cent of the Hst price do I pay ? 4. A grocer purchased 42 crates of eggs, each containing 4 doz. How many eggs did he purchase ? State the following products : 5. 84 X 88. 12. 71 x 73 19. 102 x 103. 6. 43 x 44. 13. 83 x 84. 20. 107 X 109. 7. 52 X 57. 14. 61 X 64. 21. 106 x 108. 8. 21 x 24. 15. 51 X 58. 22. 203 x 204. 9. 92 X 92. 16. 91 X 92. 23. 302 x 303. 10. 90 X 96. 17. 87 x 82. 24. 302 x 304. 11. 82 X 83. 18. 95 x 92. 25. 501 x 502. APPLICATIONS OF FOEMULAS 113 98. Further Applications of Formulas. The simple formula f a(h — c)= ah — ao has an important application in computation, as in the case of 945 X 998. From the above formula we see that 945 X 998 = 945 x (1000 - 2) = 945,000 - 1890. That is, in this case we simply annex three zeros to 945, and from the result we subtract 2 x 945, obtaining 943,110. This is simpler than to multiply 998 by 945 in the usual way. This method wiU be found to be of advantage when 6 = 100, 1000, • • •, and c is a small number. A similar application of algebra to com- mercial computations may be seen in the case of a(cx + e)= aox + ac. If, for example, we wish to multiply 586 by 246, we may note that 246 = 6 x 40 + 6, and hence that we may multiply 586 by 6, and then multiply this product by 40 and add the results. Here a = 586, c = 6, and x = 40, so that ac = 3516, acx = 140,640, and the product is acx + ac, or 144,156. the formula Exercise 77. Products of Numbers Multiply as indicated : 7. 9970 x 6487. 1. 97 X 845. 2. 98 X 945. 3. 99 X 784. 4. 96 X 856. 5. 94 X 973. 6. 99 X 877. 8. 9995 X 5485. 9. 9990 X 8456. 10. 9980 X 7565. 11. 9998 X 8767. 12. 9997 X 7564. 13. 426 X 486. 14. 936 X 785. 15. 728 X 943. 16. 355 X 658. 17. 972 X *5.98. 18. 357 X *52.40. 114 MULTIPLICATION 99. Further Applications of Formulas. Since the product is the same, however we group the factors, we know that aho = (a5) c = a (he) = h (ac). If, then, we wish to find the product of 684, \, and 100, we may multiply 684 by ^ of 100, or we may just as well multiply 684 by 100 and then by \. This is the principle of aliquot parts, 25 being an ahquot part of 100. That is, if the multiplier can be expressed as an easy fractional part of 10, of 100, of 1000, • • •, we may first multiply by 10, 100, 1000, • ■ ■, as the case may be, and then by the fraction. Thus, in the case of 25 x 684, we have 25 X 684 = i of 100 x 684 = J of 68,400 = 17,100. Exercise 78. Products of Numbers Examples 1 to 12, oral Multiply as indicated : 1. 50 X 784. 3. 75 x 888. 5. 37J x 880. 2. 25 X 680. 4. 33| x 960. 6. 14f x 280. Find the cost of each of the following : 7. 32 yd. at 37^^. 10. 96 T. at *6.25. 8. 84yd. at 62lf 11. SOT. at $3.76. 9. 96 yd. at 87|-f 12. 8.8 T. at $6.25. Multiply as indicated: 13. 309 X 333^. 17. 125 x 9664. 21. 1250 x $40.56. 14. 28|. X 280. 18. 0.83^ x 7548. 22. 750 x $94.40. 15. 250 X 536. 19. 0.187|^ x 7564. 23. 625 x $140.80. 16. 785 X 25f 20. 1875 x 95.75. 24. 0.0625 x 840. APPLICATIONS OF FORMULAS 115 100. Further Applications of Formulas. Since the product is the same, whatever the order of the factors, we, know that ah = ha, and hence that 89% of. $25 is the same in value as 25% of |89, which is equal to \ of $89. The product \ of $89 is more easily found than the product 89% of $25. In commercial work the student should take advantage of this important principle, since he will thereby save himself a great deal of labor. It is a waste of time to find, for example, 96 % of $625 when we need only take f of $960, or 5 x $120, or ^ of 10 x |120, or ^ of $1200. Consider the case of 88% of $625. This is equal to 621% of $880, or |- of $880. If the position of the decimal point is not clear, this example may be written 0.88 X $625 = 0.880 x $625 = 0.625 x $880 = f of $880. That is, the decimal point should he moved as many places to the left in one factor as it is moved to the right in the other. Exercise 79. Products of Numbers All work oral State the value of each of the following : 1. 17% of $50. 9. 104% of $250. 17. 112% of $250. 2. 16% of $125. 10. 108% of $250. 18. 104% of $250. 3. 24% of $125. 11. 18.6% of $50. 19. 68% of $1250. 4. 88% of $750. 12. 19.6% of $25. 20. 77% of $5000. 5. 92% of $7500. 13. 8% of $1875. 21. 18% of $12.50. 6. 45% of $5000. 14. 32% of $125. 22. 29% of $12.50. 7. 106% of $250. 15. 64% of $1250. 23. 8% of $8750. 8. 16% of $2500; 16. 24% of $625. 24. 16% of $6250. 116 MULTIPLICATION 101. Arrangement of Partial Products. The student may have wondered why, in algebra, we arrange the partial products by multiplying from left to right instead of from right to left as in arithmetic. Our work in algebra shows us that it makes no difference, since (« + S) e = ac + Je = Jc + ac. That is, we may have any of these arrangements : 2t+ 7 27 2t + 7 27 t+ 3 13 t + 3 13 6t + 21 81 2«2+ 7t 27 2t^+ It 27 6i?+21 81 2*2 + 13^ + 21 351 2f2+i3i + 21 351 Here !; = 10, so that 2«+7 = 27 and « + 3=13. We see that the result is the same whatever the order of multiply- ing, and it is merely a matter of habit and convenience as to which order we take. In commercial work, computers commonly begin with the left-hand figure, a plan especially valuable when unnecessary decimals are to be eliminated. Exercise 80. Arranging Partial Products 1. Multiply BA + 2« + 7by2A + 3« + 5and multiply 327 by 235, arranging the partial products in different orders. As in Ex. 1, perform the following multipliocdions : 2. (7 « + 8) (8 « 4- 2) and 78 x 82. 3. (5 A -f 6) (7 A + 4) and 506 x 704. 4. (5A'-F2< + 3)(8A + 7« + 5) and 523x875. 5. (7«3 + 8!!2+4i + 3)(8!? + 7i52-|-9<-f 7) and 7843x8797. 6. Consider algebraically the case of multiplying 4575 by 3600, finding why it is that the zeros may be neglected in the multiplication if they are annexed to the product. REVIEW 117 Exercise 81. Review of Chapter VI Examples 1 to 14, oral State the following products : 1. 79 X 81. 3. 71 X 72. 5. 85 x 85. 7. 480 x 998. 2. 58 X 62. 4. 81 x 84. 6. 41 x 41. 8. 565 x 999. Express as powers the following products : 9. aS x a^. 11. 352 X 353. 13. (1 + ry X (1 + ry. 10. stfixa^. 12. 1.06* X 1.068. 14. i.04e x 1.041 15. Multiply 8a^ + 5x^-7hjl5 a^. 16. Multiply 9 a2 + 2 a6 - 3 J2 by 4 a2 _ 5 aj + 6 52. 17. Multiply p(l + r)5 by (l + ry-, p(l + ry by (1 + ry. 18. Multiply 878 X (10,000 -2); 5765 x (10,000 +100 + 3). 19. The amount of a sum at simple interest is expressed by the equation a=p+prt. Factor the second member. 20. Prove that the excess of 9's in the product of any two numbers is the same as the excess of 9's ia the product of the excesses of the numbers. Write the factors of the following : 21. a^-64. 23. a? - 81. 22. IQm^-n^. 24. a^-1. 25. a;2_225. 26. 169 a^- 42/2. 27. Using the short methods of this chapter, extend the following items and find the total of each column: 42 yd. at 25^ 3751b. at 11.60 371b. at31(^ 46 bbl. at $1.25 51 bbl. at $51 18 T. at 16.25 19 yd. at 25 (^ 96 T. at*8.75 118 MULTIPLICATION Exercise 82. Review of Chapters I- VI 1. Find the amount of $9000 for 4 jr. at 6%, the iaterest being compounded annually. 2. Two concentric circles have radii r and r' respectively. Derive an expression for the area between the circumferences, and factor this expression. 3. By the aid of an equation find the sum of which $675 is 18% ; the sum of which |675 is 27%. 4. Six coal companies enter into an agreement to sell only through a central selling agency. The " equalizer " is to apportion the orders to A, B, C, D, E, and F in propor- tion to 27, 20, 13, 13, 6, 5. What will be each company's share of an order for 1680 T. of coal? 5. Add the following and explain the algebraic principle Which is involved: $800 x 0.06 x ^l^, $1950 x 0.06 x ^l^, $2400 X 0.06 X 3!^, $5400 x 0.06 x ^i^, $750 x 0.06 x ^ip $950 X 0.06 X ^i-j. 6. Usiug negative numbers when convenient in shorten- iug the work, add the following : 31 gal. 1 qt., 52 gal. 3 qt., 18 gal. 1 qt., 15 gal. 3 qt., 16 gal. 2 qt., 18gal. 3 qt. 7. Using the formula for the volume of a rectangular solid, V = hh, where h = area of base and h = height, find the cost of cutting and storing ^ A. of ice that is 18 in. thick, if 1 cu. ft. of ice weighs 60 lb. and the cost is 30^ a ton. Remember that 1 A. = 160 sq. rd., and 1 rd. =16^ ft. 8. Multiply 8a^+3a^-5a; + 7by2a^-3a; + 4. 9. Factor: 121 /-I; 49 2??/2_i21; 4a^-25; 2^-144. 10. Factor : p+prt; g^+gt; 4 irr^ — 4 Trr'^. 11. Using the first result in Ex. 10, find the amount of $900 at simple interest for 11 yr. at 4%. CHAPTER VII DIVISION Exercise 83. Division of Monomials All work oral 1. Divide by 2 : 4 mi. ; 4 m ; 4 • 5 ; 4 a-; 4.3.5; 4:xy. 2. Divide by 5: 10 1b.; lOZ; lOw; lOn^; lOxy; 10 abc. 3. Divide by 9: 27 rd.; 27 r; 36 in.; 36 z; 45 acres; 45 a. 4. If the temperature is 8° below zero, what will it be when it is half as much below zero ? How much is — 8° ^ 2 ? Perform the followmg divisions : 5. 64^ ^8. 9. -4° ^2. 13. 75ab^25. 6. 64c-h8. 10. -4 a; -^2. 14. 60a%-H30. 7. 64a;«/-=-8. 11. -4aJcH-2. 15. ibp^-i-lB. 8. 64Va^8. 12. -16a?-:- 2. 16. 90V^h-45. 17. How do you divide a monomial by a positive integer? 18. Divide *8 by $2 ; Sdhj 2d; 8 ft. by 2 ft.; 8/ by 2/. Divide the following : 19. 8 a; by 2 a;. 25. 40 a by 10 a. 20. 10 a; by 2 x. 26. 75 b by 25 b. 2t. 15 1/ by 5 y. 27. 125 a^ by 5 a\ 22. 16 3 by 8 a. 28. 275 a? by 25 a?. 23. 25 w by 5 m. 29. 250^2 by 50^2. 24. S6p by ip. 30. 175 w by 25 m. 119 120 DIVISION 102. Law of Exponents. Since division is the inverse of multiplication, we have the following: From a2 . ^3 _ ^jS ^g have a^ -i-a^ = a?, and from d^ . dP = a'^v we have a^'^y -^ a^ = a^ ; or, in general a"" -i- a" = a""" ". The exponent of any factor of the quotient is eqwal to the eooponeni, of that factor of the dividend minus the exponent of that factor of the divisor. For example, — 21 ft'^ -f- 7 x^y = — 3 a;*^'. In algebra, division is usually expressed in the fractional form, thus : — 4 ahc 103. Law of Signs. The law of signs, page 67, is as follows: In division two like signs produce plus; two unlike signs produce minus. 4 -4 -4 4 That is, ^ = 2, 32==^' ~2~=~^' ^'"^ 32=~^- Exercise 84. Division of Monomials Examples 1 to 5, oral Perform the indicated divisions : 36 , 1.04* „ -119 mV , 32.53.7* 9. — - — :— 13. 32 1.042 Im^n^ 52.7* m^n* „ -9.6 m* ,„ 144 m%6 ,, 60Sa%''(fi 6. 10. 14. mn — 0.4 m — 4 m*n —9ab xY ^ 135 a^c ^^ -175 3?y*z^ - 504 2A0 xg 5 ac — 5 3?g^ 3 x^g^ — ^ • 8. —• 12. ^' 16. x^g^ —Qxy — 25 mp — 5 a6c DIVISION BY A MONOMIAL 121 104. Division of a Polynomial by a Monomial. If we divide 10 ft. 8 in. by 2, we have 5 ft. 4 in. Similarly, we have 2 )10 ft. 4 in. 4 )40 yd. 16 in. 9 )9 tens +9 5 ft. 2 in. 10 yd. 4 in. 1 ten + 1 2)10/+jU' 4 )40.3 + 16.5 9 )9^ + 9 5f+2i 10.3+4.5 t + 1 That is, to divide a polynomial by a monomial, Divide each term of the dividend by the divisor and add the partial quotients. If we divide zero by any number, the result we obtain is zero. Division of any number by zero has no meaning. Exercise 85. Division of a Poljmomial by a Monomial Examples 1 to 9, oral 1. Divide by 2: 8ft. Bin.; 8/+6«; 8tens + 6; 8t+6; 86. 2. Divide by 3: 9nii. 15rd. ; 9m+15r; Qxy + lBxy. 3. Divide by a: ax + ay; ahc + axy; aVx + a^y; a + a^ Divide the following : 4. px^ +py^ by p. 1. — ax + by a. 5. axy + amn by a. 8. — ax + ay by — a. 6. mpq — mxy hy m. 9. am^ — ar? + aa^ by a. Perform the indicated divisions : 35 p*m + 75 phn^ 25 m^ p + 35 m^g — 15 mV 10. p — 13. J — J • op^m —om^ 64 a26V - 48 a*63e2 a}2c3_«352e + 11. : — ::77r^ • " - 4 aWc^ aPc 75 a^SV - 65 a3J4c5 - 4 aS^z; - 6 «% + 8 a6a;w -baW(? ' -2ab 122 DIVISION 105. Division of a Polynomial by a Polynomial. The fol- lowing typical case of the division of a polynomial by a polynomial should be carefully studied: Divide a^ — 5 a; — 80 by « + 7. Operation Check a^- 5 a: -80 cc + 7 Leta; = l. Then x^+ 7x x-12 -!-.-„ -12 2;-80 -12 2;-84 ■ Dividing x\ the first term of the dividend, by x, the first term of the divisor, we have x, the first term of the quotient. Multiplying the divisor by x, we have a;^ + 7 1, which we subtract from the dividend, leaving — 12 ar — 80 still to be divided. Dividing — 12 a; by x, we have — 12, the second term of the quotient. Multiplying the divisor by — 12, we have — 12 a; — 84, which we subtract, leaving a remainder 4. Hence the complete quotient is a; — 12 and the remainder is 4. We 4- often write the result in the compact form a; — 12 H a: — 12 We may check the result (1) by carefully reviewing the work, (2) by multiplying the quotient by the divisor and adding the remainder, or (3) by substituting some convenient values for the letters. Applying the second of these checks, we can easily show that (a: + 7) (a; - 12) + 4 = a;2 _ 5 J. _ go. Applying the third check, letting a; = 1, we have 1 — 5 — 80 = — 84, 1 + 7 = 8, (— 84 — 4) -^ 8 = — 11 ; and the quotient, x — 12, becomes — 11. In the check we cannot put — 7 for x, because this would make the divisor zero, but we can use any other number. Since we cannot divide hy zero, in the ohech for division we use for the letters values that do not make the divisor zero. In commercial algebra we rarely need to divide by a polynomial of more than two terms. The principle is, however, the same, and a few examples are given on page 123. DIVISION OF POLYNOMIALS 123 Exercise 86, Division of a Polynomial by a Polynomial ExaTnples 1 to 5, oral 1. In dividing a? + ?> x^y + 2 xy^ by a^ + 2 xy, what is the first term of the quotient ? 2. How do you check the work in division ? In the following examples state the first term of each quotient : 3. a?- — bx + & divided by a; — 3 ; by a; — 2. 4. 8 m^ — 10 mn — 3 w^ divided by 4 w + w ; by 2 w — 3 w. 5. p^-Sp-2, divided by -^ + 3 ; by ^^ ^ 3^3 _,_1_ Divide the following, checking the results : 6. a2 + 3 a + 2 by « + 1. 11. a^ -5 ah -%V^hy a + h. 7. J2 + 5 5 + 4 by 5 + 1. 12. a^ -^ ab + bh^hj a-l. 8. c2 + 8 c + 12 by c + 2. 13. c^ + cd - Q d^hj c + Z d. 9. d^-2d-\bhj d+2,. li. p^-8pq+22q^hy p- 5q. 10. e^ +11 e + 28 by e + 4. 15. a^ + xy -22 y^ hy x - -i y. Arrange according to the descending powers of x, and divide : 16. a^ + a^-16x-4:-9x^hy i + x^ + ix. 17. .6+ar*-12a:+lla^-5a^by 3-3^ + 2?. 18. 2a^ + 9a;-12-52^-7a^by l-a; + a:2_ 19. -73fi-10a? + xi-5 + 21xhy x + x^-S. 20. 3a? + a^-7x-ix^ + 5hy Sx + x^-2. 21. Divide 2h + 8t + 8uhy h + it + iii; 288 by 144. 22. Divide 7T+Sh + 7t + 5uhy2h + 2t + 5u. As in Ex. 21, the teacher should show the relation of this work to arithmetic. Indeed, the chief object of teaching the division of polyno- mials in commercial algebra is to throw light upon the numef'ical work. ci 124 DIVISION 106. Short Methods with Numbers. Algebra enables us to see the reasons for many of the short methods used in commercial work. The following are some of the cases: 1. To divide hy 5, multiply hy 2 and move the decimal point one place to the left. For a-i-5=- = — -,80 that to divide a (any number) by 5 we may o \.\J * simply multiply by 2 and then divide by 10. It is much easier to multiply by 2 and move the decimal point than it is to divide by 5. 2. To divide hy 12^, which is =^, multiply hy 8 and move the decimal point two places to the left. 3. To divide hy a number, divide hy any one. of its factors, divide the quotient thus found hy any other of the factors, and so continue until all the factors of the divisor have heen used. For it is evident that ahc ^ ah = ■- b = he -i- b = c. That is, to a divide by ab we may first divide by a and then divide the quotient by b. This is often convenient, as in the case of 1146.70 ^ 45. We may first divide by 9 and then by 5, this being a little easier than to divide by 45. In the case explained above there may be re- mainders. Thus, in dividing 14,699 by 45, the first quotient is 1633 and the remainder is 2 units. The second quotient is 326, and since we are dividing the number of 9's in 14,669 by 5, the remainder is 3 nines, or 27. Hence the total remainder is 2 + 8 X 9, or 29. To find the remainder in dividing hy factors, multi- ply each partial remainder hy all the divisors preceding the one which produced it. 9 )14699 5 )1633 9's + 2 326 45's + 3 9's Quotient = 326 Remainder =2 + 3x9= 29 The sum of these products, together with the "hmainder from the first division, is the remainder sought SHOET METHODS 125 Exercise 87. Short Methods Examples 1 to 7, oral 1. Since a -:- 25 = 4 a -^ 100, what is a short method for di- viding a number by 25 ? Illustrate by the case of 15,475 -i- 25. State the short methods suggested hy the following equations : ^.-125=^. 5.a.66| = ^. 3. «^50 = — . 6. a^75 = ^^. 4. «^33i = l^. I.a^m- ^" + *" ^~100' .. - . ".2 ^QQ 8. Find a short method for dividing a number by 6 21-. 9. Find a short method for dividing a number by 871. Perform the following divisions hy the method of dividing hy factors : 10. 357-^21. 13. 1024-^32. 16. 2352-:- 42. 11. 725-^25. 14. 2135 H- 35. 17. 2352 h- 49. 12. 486 H- 27. 15. 4270^35. 18. 2352 h- 48. Divide the following hy dividing hy factors and find the remainders : 19. 721-5-35. 21. 1000-^42. 23. 1111^-25. 20. 642-^21. 22. 2000-^36. 24. 2002-^-27. Using short processes, divide the following : 25. $282.36 ^75. 27. $35.75-^*0.621. 26. $5.63^*0.121. 28. 1185-^*0.20. 29. At the rate of 6 lb. for *1, how many pounds of white grapes can be bought for *4.50 ? 126 DIVISION 107. Divisibility of Numbers. In making the computations needed in commercial work we frequently have to determine whether or not one number is divisible by another, by which we mean divisible without a remainder. For example, we frequently wish to know if, in a large number of articles, we have exact dozens, exact half dozens, or exact lots of 15 or 25. You learned something in arithmetic about this work, but you knew little of the reason for it and may have for- gotten much of it. Algebra makes the whole matter very- clear and enables us to work out the rules for ourselves if we happen to forget them. The following are some of the tests for divisibility: A number is divisible by B if the units' figure is divisible by 2. This is a short way of saying that it is divisible by 2 if the number Represented by the units' figure is divisible by 2. Such short expressions are allowable if the meaning is clear, and will be used when convenient. Any number is equal to its tens plus its units. For example, 1275 is equal to 127 tens + 5 units, and 7 is equal to tens + 7 units. Hence any number may be represented by t tens + u units, or by 10 ( + «. Now 10 1 is certainly divisible by 2, and so the number 10 i + « is divisible by 2 if « is divisible by 2. A number that is divisible by 2 is called an even numher ; a number that is not divisible by 2 is called an odd number. A number is divisible by 5 if the units' figure is divisible by 5. The proof, which is practically the same as the proof for divisibility by 2, is left to the student. A numher is divisible by 4 if the numher represented by the last two digits at the right is divisible by 4. Any number is equal to its hundreds plus its tens plus its units and hence may be represented by h hundreds + t tens + u units, or by 100 ^ + 10 ( + «. Now 100 h, is certainly divisible by 4, and so the number is divisible by 4 if 10 « + m is divisible by 4 ; that is, if the number represented by the last two digits is divisible by 4. DIVISIBILITY OF NUMBERS 127 A number is divisible hy 8 if the number represented by the last three digits is divisible by 8. Any number is equal to 1000 T + 100 A + 10 « + «. But 1000 T is certainly divisible by 8, and so the whole number is divisible by 8 if 100 ;* + 10 1 + u is divisible by 8. A number is divisible by 9 or by 3 if the sum of its digits is divisible by 9 or by 3 respectively. Anj number may be represented by calling its units a, its tens b, its hundreds c, and so on, and then writing it by beginning with the units, thus : a + 10b + 100 c + 1000 d + • • •, as far as we wish to go. It may also be written a + 9b + b + 99e + c + Q99d + d+--- or (a + b + c + d + • ■ ■) + (9 b + 99 c + QQ9 ,1 + ■ ■ ■). Now the second parenthesis is certainly divisible by either 9 or 3. Hence the number is divisible by 9 or by 3 if the first parenthesis, which is the sum of. the digits, is divisible by 9 or by 3 respectively. For example, the number 30,003 is divisible by 3 but not by 9, and the number 864 is divisible by either 9 or 3. A number is divisible by 11 if the difference between the sums of its digits of odd order and of even order is divisible by 11. Thus 207,988 is divisible by 11. We have 8 + 9 + 0, or 17, the sum of the digits of odd order (that is, in the 1st, 3d, and 5th places) and 8 + 7 + 2, or 17, the sum of the digits of even order. Now 17 — 17 = 0, and is divisible by 11, the quotient being 0. As in the case of 9, we may represent any number by a + 10 J + 100 c + 1000 d + 10,000 e+ ■■■, or by a + 11 J - 6 + 99 c + c + 1001 d-d + 9999 e + e+ ■■■, OThj (a-b + c-d + e ) + (11 6 + 99 c + 1001 d + 9999 e + • • •). The second parenthesis is certainly divisible by 11. Hence the number is divisible by 11 if the first parenthesis, the difference between the sums of the digits of odd order and even order, is divisible by 11. A numiher is divisible by 6 if it is even and if the sum of the digits is divisible by 3. For we need simply to apply the tests for divisibility by 2 and by 3. 128 DIVISION Exercise 88. Divisibility 1. State and prove the test for divisibility by 10. 2. Prove that a number is divisible by 12 if the sum of the digits is divisible by 3 and if the number represented by the last two digits is divisible by 4. 3. Prove that a number is divisible by 7 if the sum of the odd periods minus that of the even periods is so divisible. Let X, y, z represent the third, second, and first periods respectively of a number of three periods. Then the number will be represented by 1,000,000 X + 1000 y + z, or by 999,999 x -^ x + 1001 y-y + z, or by (999,999 X + 1001 y) + (x — y + z). But the first parenthesis is certainly divisible by 7, and hence the whole number is divisible hy 7 ii x — y + z, or x + z — y, is divisible by 7. But x + z — y is the sum of the odd periods minus the even period. The proof is similar to this if more than three periods are taken. The student should write the proof for the case of four periods. Find all possible divisors from 2 to 12 of the following : 4. 1158. 7. 1337. 10. 1810. 13. 9033. 16. 10,002. 5. 1055. 8. 2583. 11. 3091. 14. 2809. 17. 10,010. 6. 4809. 9. 1384. 12. 5001. 15. 7007. 18. 18,402. Without written division, write those of the following num- bers that are divisible by 7 : 19. 7014. 21. 13,021. 23. 701,707. 25. 201,714. 20. 14,721. 22. 700,707. 24. 700,714. 26. 722,001. Write such of the followirig numbers as are divisible by 9 and such as are divisible by 11, together with the work showing the short process used : 27. 69,876. 29. 88,088. 31. 37,466. 33. 54,045. 28. 76,747. 30. 42,020. 32. 80,280. 34. 54,560. EEVIEW 129 Exercise 89. Review of Chapter VII 1. Divide 800 xl.06s by 1.063; 300 xl.046 + 200 x 1.04^ by 1.042. 2. Find the value of 30x73 + 5x13^x7^-5x62x0 . 5x72 3. Divide 64 x^y - 72 x'y^ + 40 ot^f - 32 a;?/* by - 8 xy. 4. Divide 9-a^-27a?+12«*+3a: by -3+4^2-3 a;, and check the result. 5. Derive a short method for dividing any number a by 621; by37l; byl6|; byl4f 6. Resolve into factors, using the principles of divisibility of numbers : 555, 609, 1449, 489, 7014. 7. How many tubs, holding 75 lb. each, will be needed to ship a consignment of 9600 lb. of butter ? What short method should you employ in solving this example ? 8. Write two exact divisors of 4899, using the principle that a2 _ J2 jg exactly divisible by « + 5 and by a — h. 9. Divide 20 a^ + 18 am — 18m^ by 5 a— Sm, and check the result. 10. Divide |7854.85 by 1.0612, and check the result by the check of 9's. 11. Divide a^ — B a^b + 3 ab^ — b^ by a— b, and check the result. 12. Divide a^ — 81 by a; — 3, and check the result. 13. Divide a^—¥ by a — b, and check the result. 14. At 37|-^ a yard, how many yards of madras can a girl buy for $4.50 ? Use short methods in solving Exs. 14 and 15. 15. If a keg of nails weighs 62^ lb., how many kegs can safely be stored on a platform built to carry 2 T. ? 130 DIVISION Exercise 90. Review of Chapters I-VII 1. A trapezoid has bases whose lengths are m and n and whose altitude is h. What is the area? 2. What is the value of V3 ah'^d^ if a= 3, 6= 2, c=5? 3. Solve the equations 3a:+7 = 13; ^a;+2=5; a;— 2=9. 4. A mixture of 18 parts, by weight, of alcohol and 11 parts of water is to be prepared, the total to be 15 oz. How much alcohol and how much water must be used ? 5. Find the average temperature for a day if the records every 3 hr. are + 8°, +11°, +12°, + 5°, 0°, - 3°, - 5°, - 9°. 6. From the equation a — i + 5 = a, write a rule for testing the result of subtractioii in arithmetic. 7. By how much does j» (1 + r€) exceed p ? 8. By how much does cfi — IP- exceed 2 6^ ? 9. What number must be added to a to produce h ? 10. What number must be added to a—h to produce 2a-35? 11. If a dealer's profit on a certain grade of shoes is $1.80 per pair, what is his profit on 350 pairs ? 12. If a dealer's profit on a pair of shoes is p doUai's, what is his profit on n pairs ? 13. If the cost of a pair of shoes is c dollars, the profit desired is p dollars per pair, and the number of pairs sold is M, write the equation expressing the total receipts, r dollars, from the sale of the n pairs of shoes. 14. Multiply 3 a? +17 a; + 10 by 2 a; -7, and then divide the result by 3 a; + 2. 15. The selling price is « dollars, cost c dollars, overhead dollars, and profit p dollars. Write the formula for p. CHAPTER VIII SIMPLE EQUATIONS 108. Simple Equation. An equation which contains the first power and no higher power of the unknown quantity is called a simple equation. Such an equation is also said to be of the _first degree. It is also called, particularly in higher mathematics, a linear equation. We have already considered in Chapter II the easier kinds of simple equations. 109. Identity. An equation which is true for any value whatsoever of any letter or letters is called an identity. Thus (a + by = a^ + 2 ah + h' is an identity, for it is true if a — 3, 6 = 4, or if the letters have any other values vphatsoever. An identity is sometitnes expressed by th'e symbol = , as in a + ft = 6 + n, read " a + h is identical to 6 + a." The term is also used to represent such numerical equalities as 2 + 3 = 5. 110. Satisfying an Equation. The number which, substi- tuted for the unknown quantity, reduces an equation to an identity is said to satisfy the equation. This number is the value of the unknown quantity. Thus, if we have X + 7 = 9, then x = 2, and 2 satisfies the equation. 111. Solving an Equation. To find the value or values of the unknown quantity which will satisfy an equation is to solve the equation. Thus, if a; +. 3 = 5, then a; = 2 ; for if we put 2 for x, the two mem- bers of the equation become 2-1-3 and 5, which are the same in value. In other words, the equation then becomes an identity. Similarly, if a: H- a = 3 a, then x = 2a\ for if we put 2 a for x, the two members become 2a + a and 3 a, which are the same in value. Many equations can be solved by simple inspection. 131 132 SIMPLE EQUATIONS 112. Root. A value of the unknown quantity which satis- fies an equation is called a root of the equation. Thus, in a: — 4 = 9, the root of the equation is 13. A simple equation has only one root. There are, however, equations that have more than one root. For example, the equation a:"— 3a; + 2 = has two roots because it is satisfied if a; = 1 and also if a; = 2. 113. Transposition. We have learned that if we subtract equals from equals, we have equals left. We have also seen that this amounts to taking a number from one side of the equation, changing the sign, and putting it on the other side. Taking a number from one side of an equation, changing its sign, and putting it on the other side is called transposition. For example, if 2a; — 4 = 8 + 1, we may subtract x from each side, or transpose x, and have a: - 4 = 8. We may then subtract — 4 from each side (or add + 4 to each side) and have ^ _ 12. The word " transpose," while sometimes convenient, is evidently not necessary, since we can use " subtract " or " add " in its stead. 114. Numerical Equation. An equation containing no letters except those representing roots is called a numerical equation. Thus 2a; + 3 = 5isa numerical equation, but aa; + J = 4 is not. Solve the equation 4:x — 7 + x = 10 — dx — l. Combining like terms, 5a; — 7 = 9 — 3a;. Subtracting — 7, 5 a; = 16 — 3 a:. Subtracting — 3 a;, 8 x = 16. Dividing by 8, x = 2. Check. 4x2-7 + 2= 10 -3x2-1 = 3. The solution may be shortened, thus : Combining like terms, 5a; — 7=9 — 3 x. Adding 7 and S x, 8 x = 16. Dividing by 8, x = 2. NUMERICAL EQUATIONS 133 Exercise 91. Numerical Equations Examples 1 to 9, oral 1. Is a; + 3 = 4 an identity ? State the reason. 2. Is a; + 3 = 3 + a; an identity ? State the reason. 3. What value of x will satisfy the equation a; — 4 = 5 ? Prove it. 4. What is the root of the equation a;+ 7= 12 ? Prove it. 5. What right have you to transpose — 3 in the equation a; — 3 = 9 ? Explain why you must change the sign. Solve the following equations : "6. a;-4 = 6. 7. a; + 5 = 9. 8. a- + 7 = 15. 9. In solving the equation 4x + 5 — 2r = 15 what is the first step ? the second step ? the third step ? Solve the following equations : 10. 5a:-7 + 3a;=9. 12. 8 + 2a;-19 = a;. 11. 7 a;- 4 -5a; =10. 13. 5-a;-8 + 6a; = 17. 14. If to 5 times a certain number we add 2, the sum is 20 diminished by 4 times the number. What is the number ? Solve the following equations : 15. 5«-7+3a; + 6 + 8a; = 13 + 6a; + 40. 16. 7a;-2 + 9a;-3 + 5a; = 4 + a; + 31. 17. 8a; + 7-2a; + 5 = 4a;+12-(a;-30). 18. If 4 is increased by three times a certain number, the result is 19. Find the number. 19. Four times a certain number is diminished by 3, the result being 29. Find the number. 134 SIMPLE EQUATIONS 115. Two Unknowns, One Letter. A problem may involve the finding of the values of two unknown quantities and still require the use of only a single letter if there are two statements either expressed or implied. As will be seen, one statement will give a relation which makes it possible to represent the two unknowns by the aid of only a single letter, but with different coefficients ; the other statement will be one which can be translated into an equation. 1. On June 21 daylight in a certain place lasts 8 hr. longer than darkness. How many hours are daylight? Statement I. Daylight = darkness + 8 hr. Statement II (implied). Daylight + darkness = 24 hr. Let X = the number of hours of darkness. .Then, from I, a; + 8 = the number of hours of daylight. But because, from Statement II, Daylight + darkness = 24 hr., we have x + 8 + x = 24, or 2 a; + 8 = 24. Solving, X = 8, the number of hours of darkness. Hence a; + 8 = 16, the number of hours of daylight. 2. A clerk counting change consisting of dimes and quar- ters remembers that there were 45 coins and that their total value was |8.40. How many were there of each ? Statement I. Number of quarters + number of dimes = 45. Statement II. Value of quarters + value of dimes = $8.40. Let X = the number of quarters. Then, from Statement I, 45 — a; = the number of dimes. Multiplying 25 by the number of quarters and 10 by the number of dimes, to find the values in cents, we have 25a; + 10(45-a;) = 840. Solving, X = 26, the number of quarters. Hence 45 — a; = 19, the number of dimes. PEOBLEMS 135 Exercise 92, Problems ExaTTvples 1 to 9, oral 1. Five more than three times a certain number is 20. What is the number? 2. The sum of two numbers is 34 and one of the numbers is 19. What is the other number ? 3. The difference between two numbers is 17 and the larger number is 32. What is the smaller number ? 4. The difference between two numbers is 41 and the smaller number is 19. What is the larger number? 5. The product of two numbers is 51 and one of the numbers is 17. What is the other number? 6. The quotient of two numbers is 27 and the divisor is 3. What is the dividend? 7. The quotient of two numbers is 32 and the dividend is 96. What is the divisor? 8. The difference between two numbers is 12 and one of the niimbers is 20. What is the other number ? I Notice that there are two possible answers to Ex. 8, and also to Ex. 9. 9. The quotient of two numbers is 10 and one of the numbers is 20. What is the other number? 10. The sum of two numbers is 32 and one of the numbers is three times the other. Find the numbers. 11. The sum of two numbers is 36 and one of the num- bers is 2 more than the. other. Find the numbers. 12. The sum of two numbers is 36 and one of the numbers is 2 less than the other. Find the numbers. 13. The sum of two numbers is 5.6 and one of the numbers is 4.2 more than the other. Find the numbers. 136 SIMPLE EQUATIONS 14. Two partners, A and B, have an agreement to allow A to receive from the profits $1200 as rent for the building which he owns and which they occupy, and then to divide the remainder of the profits so that A shall receive twice as much of this remainder as B. If the profits for the year, less all overhead charges excepting rent, are |7500, how much does each receive, including the rent in A's share ? 15. If A owns 9 shares of a business and B owns the remaining 11 shares, and if they divide profits accordingly, how shall they divide a profit of |6000 ? 16. In a commercial transaction two men made $4500. By their articles of agreement the second was to receive 50 °Jo more than the first. How much did each receive ? 17. A merchant put in an order for a lot of goods and a week later gave an order for 121% as much. He paid $2700 for all the goods. How much was each order? 18. A grocer makes a blend of coffee by using 2 parts of Mocha and 5 parts of Rio. How many pounds of each must he take to make 100 lb. of the blend ? 19. A fireman's experience has shown that 1 T. of anthra- cite coal fills 34 cu. ft. of space, and 1 T. of bituminous fills 42 cu. ft. He has 3200 cu. ft. of space available for storage, and needs three times as many tons of bituminous coal as of anthracite. How many tons of each should he order? 20. A mountain railroad, X, has a freight tariff per mile 11 times that of another road, Y. Some freight " billed through" for $87.50 was hauled 85 miles by X and 120 miles by Y. How should they divide the' $87.50 ? 21. A invests $5000 more capital than B in a business. After realizing a profit of 20 % on the investment, their free assets are $75,000. What is each man's share of the assets? PROBLEMS RELATING TO MIXTURES 137 116. Problems relating to Mixtures. In various kinds of business it becomes necessary to mix certain ingredients, and the quantities are often found easily by algebra. How much water must be added to 12 qt. of a 25% solution of alcohol to reduce it to a 10% solution? Let X = the number of quarts to be added. Then 12 + a: = the total number of quarts, and 10 % (12 + x) = the number of quarts of alcohol. But 25 % of 12 = the number of quarts of alcohol. Therefore 10% (12 + a;) = 25 % of 12. Combining, 1.2 + 10% x = 3. Subtracting 1.2, ' 10 % a; = 1.8 Dividing by 10%, a: = 18. Therefore 18 qt. of water must be added. Check. 25 % of 12 qt. = 3 qt., the amount of alcohol. 12 qt. + 18 qt. = 30 qt., the amount of the mixture. Furthermore, 3 qt. is 10 % of 30 qt., as required. Exercise 93. Problems relating to Mixtures 1. How much water must be added to a gallon of a 22% solution of alcohol to reduce it to an 11% solution? 2. How much water must be added to 63 gal. of alcohol 95% pure to reduce it to a 66|% solution? 3. How many pounds of water must be added to 320 lb. of a 4% solution of salt to reduce it to a 2|-% solution? How many gallons, allowing 8.35 lb. of water to a gallon ? 4. How much water must be added to a pint of a 10% solution of a certain medicine to reduce it to a 2 % solution ? 5. How much water must be added to a barrel of vinegar 80% pure to reduce it to a 60% solution? 138 SIMPLE EQUATIONS 117. Problems relating to Motion. Algebra is often useful in solving problems relating to motion. 1. A train leaves Pittsburgh for the West at the same time that another train leaves Pittsburgh for the East. The former travels at the average rate of 42 mi. an hour and the latter at the average rate of 38 mi. an hour. In how many hours will the trains be 240 mi. apart ? Let X = the number of hours required. In 1 hr. the trains are (42 + 38) mi. apart, or 80 mi. apart. In X hours the trains are x • 80 miles apart, or 80 x miles. Therefore 80 a; = 240. Dividing by 80, x = 8. Therefore the trains will be 240 mi. apart in 3 hr. Check. 3 X (42 + 38) mi. = 240 mi. 2. A man starts from a certain place and rides on his bicycle at the rate of 16 mi. an hour.- Forty-five minutes later an automobile starts after him from the same place and travels at the rate of 24 mi. an hour. How long will it take the automobile to overtake him ? In 45 min. the bicycle has gone -| of 16 mi., or 12 mi. Let X = the number of hours required. Since the automobile gains in an hour 24 mi. — 16 mi., or 8 mi., 8 a; = the number of miles gained in x hours. But 12 = the number of miles to be gained. Therefore 8 a: = 12. Dividing by 8, ^ = 1^- Therefore the automobile will overtake the bicycle in 1^ hr. Check. In l^hr. the automobile will travel 1^ x 24 mi., or 36 mi., and the bicycle 1^ x 16 mi., or 24 mi., and 24 mi. + 12 mi. = 36 mi. In all problems relating to motion the average rate is to be under- stood unless the contrary is stated. PROBLEMS RELATING TO MOTION 139 Exercise 94. Problems relating to Motion Examjjles 1 to 4, oral 1. At 3 mi. an hour, how far will a man walk in 4 hr. ? in -|- hr. ? in t hours ? m (a + J) hours ? 2. At r miles an hour, how far will an airplane go in t hours ? in 2 ^ hours ? in |^ hr. ? 3. If two men travel from the same place, one going east at the rate of 3 mi. an hour and the other west at the rate of 40 mi. an hour, how far apart are they in 2 hr. ? in h hours ? 4. If two men start from Chicago and New York at the same time and travel toward each other, the first at the rate of a miles an hour and the second at the rate of h miles an hour, how much nearer each other will they be in i hours ? 5. Two men start from Denver at the same time, one traveling south 35 mi. an hour, and the other north 38 mi. an hour. How many miles apart will they be in 3 hr. ? In how many hours will they be 292 mi. apart? 6. Two men start from the same place, one going east and the other going west, the former traveling twice as fast as the latter. In 4 hr. they are 300 mi. apart. Find the rate of each. 7. Two men start from the same place, one to the north and the other to the south, the former traveling 5 mi. an hour faster than the latter. In 2 hr. they are 150 mi. apart. Find the rate of each. 8. A man leaves a friend at the railway station and starts to drive north just as his friend leaves on the train for the south. A half hour later they are 20 mi. apart. If the man drives one fourth as fast as the train travels, what is the rate of each ? 140 SIMPLE EQUATIONS 118. Three Unknowns, One Letter. 1. Of the 50 shares of a busmess A owns 13 shares, B owns 17, and C owns 20. If they divide on the basis of the number of shares owned by each, how should they divide a net profit of $25,000 ? Statement I. A's share is ^§ of C's. Statement II. B's share is ^^ of C's. Statement III. The sum of the shares is |25,000. Since each share receives the same proiit, let x = the number of dollars profit on one share. Then, referring to the number of dollars in. each partner's share of the total profit, ' 20 X = C's share, 17 a; = B's share, and ISx = A's share. Hence 13 x + 17 a; + 20 a; = 25,000. Solving, X = 500. Therefore 13 a; = 6500, 17 a; = 8500, and 20 a; = 10,000. That is, A's share is $6500, B's $8500, and C's $10,000. 2. A dealer has charged 10^ a peck more for first grade potatoes than for second grade and 15^ a peck more for sec- ond grade than for third grade. After sorting a consignment of 10 bu. that cost him $15, he finds that he has 5 bu., 3 bu., and 2 bu. respectively of the three grades. To maintain the above differentials and to make a profit of $5 on the consign- ment, how much must he charge for a peck of each grade ? Statement I. Price per peck of second grade = price per peck of third grade + 15^. Statement II. Price per peck of first grade = price per peck of second grade -HO^. Statement III. Total receipts = $20. Let X = the number of cents per peck of third grade. Then a; -I- 15 = the number of cents per peck of second grade, and X -F 25 = the number of cents per peck of first grade. Hence 20 (x + 25) -H2 (x -1- 15) -h 8 a; = 2000. Solving, X = 33. Therefore x -I- 15 = 48, and x + 25 = 58. Hence the dealer must charge 58^, 48^, and 33^ respectively. THREE UNKNOWNS, ONE LETTER 141 Exercise 95. Three Unknowns, One Letter 1. Three partners, A, B, and C, own respectively 20, 25, and 55 shares of the 100 shares of stock of their business. A owns the building and is allowed a yearly rental of |2400. After paying all overhead charges except the rent, they have $14,400. How should this |14,400 be divided among them ? 2. A fruit dealer has charged 25^ per bushel more for apples of grade A than for those of grade B, and 20^ more for those of grade B than for those of grade C. After sort- ing 90 bu. that cost him |150, he finds that he has 20 bu., 30 bu., and 40 bu. respectively of the three grades. If he wishes to maintain the above differentials and realize a profit of |30, how much must he charge per bushel for each grade? 3. A taster has estimated a blend of tea as worth 80^ per pound. The blend is made of 3 parts of a 50^ tea, 2 parts of a 60 (^ tea, and 1 part of a $1 tea. How many pounds of each kind must be taken to make 30 lb. of the same blend, and what profit will be rea,lized on its sale? 4. An insurance company has found it expedient to adopt 'this rule in investing its funds: to place 60% of its funds in long-time securities, 15% in mortgages, and the remaining 25% in loans to policy holders. On a certain date the com- pany receives $75,000, which it desires to invest. What amount shall it put into each kind of investment? 5. An experience table shows that approximately |% of persons reaching the age of 21 years die during the following year; of those reaching the age of 22 years, -|% die during the year; and of those reaching the age of 23 years, |^% die during the year. If a group of persons insured at the age of 21 number 194,000 at the close of the age 23, how many were probably in the original group at the age of 21 ? 142 SIMPLE EQUATIONS 119. Tabular Arrangement. 1. A rectangular assembly hall is 10 ft. longer than wide, and i£ each dimension is increased 10 ft. the area is increased 800 sq. ft. Find the dimensions of the hall and the area of the floor. Given Conditions Length Width Area First condition . . iB+lO X 3?+10x Second condition . . a; +20 :b + 10 a:' +30 2; +200 Third condition . . a; +20 a; + 10 x' + lQx + 800 Such a tabular arrangement is often helpful. Since a;? + 30 a; + 200 = a-^ + lo x + 800, we have a; = 30 ; whence a; + 10 = 40, and x^^ 10x = a;(a; + 10) = 30 X 40 =1200. That is, the hall is 30 ft. wide, 40 ft. long, and 1200 sq. ft. in floor area. 2. A sum of money invested at 4% yields an income $300 larger than half the sum yields at 5%. Find the amount in- vested at each rate and the income from the first amount. Given Conditions ^UM Rate Income First condition . . 2a; 4% 0.08 a; Second condition . . X 5% 0.06 a; Third condition . 2x 4% 0.05 a; +300 Since 0.08 x = 0.05 x + 300, we have 0.03 x = 300. Dividing by 0.08, x = 10,000 ; whence 2 a; = 20,000. That is, the investment at 5% is $10,000, and the investment at 4% is |20,000, the income from the latter being |800. TABULAR ARRANGEMENT 143 Exercise 96. Tabular Arrangement Using tabular arrangemevi, solve the following problems : 1. The cost of some silk at $2."20 per yard is the same as the cost of 20 yd. more of wool at |1.76'per yard. Find the number of yards of each and the total cost of the goods. 2. A workman receives an offer of a position paying an increase of 80^ per day over his present wages. He calcu- lates that in 30 da. in the new position he will receive as much as he now earns in 40 da. Fiad his present rate of wages per day and his rate if he accepts the offer. 3. A man invests |9000, part at 4% and part at 5%. If he gets an equal income from the two investments, how much does he mvest in each, and what is the income from each ? 4. A rectangle is 5 yd. longer than wide. If the length is increased 10 yd., and the width is increased 5 yd., the area is increased 450 sq. yd. Find the length, width, and area before the increase is made. 5. A man invested a certain sum of money in United States Liberty 4% bonds, buying them at par. If he had invested $1500 more, and if the rate were 1% larger, he would be receiving in 1 yr. the same interest he now receives in 2 yr. How much did he invest in the bonds ? 6. During the month of January, 1918, the total receipts at 16^ a yard from the sale of galatea in a dry goods store amounted to half as much as the receipts from the same source in Jan., 1919, at 20^ a yard. The number of yards sold in Jan., 1919, was 3000 more than in Jan., 1918. How many yards were sold in Jan., 1918, and in Jan., 1919? 7. A certain principal invested at 4% and a second prin- cipal twice as large, invested at 41%, together bring in $260 interest in 1 yr. Find the amount of each principal. 144 SIMPLE EQUATIONS 120. Miscellaneous Problems. Before attempting the next exercise the following solutions should be studied. 1. The sum of three consecutive numbers is 48. Find the numbers. Let n = the middle number. Then n — 1 = the smallest number, n + 1 = the largest number, and (n ■ - 1) + n + (n + 1) = the sum. But 48 = the sum. Therefore (n - - 1) + n + (» + 1) = 48 ; whence 3n = 48. Dividing by 3, n =16. Therefore the numbers are 15, 16, 17. Check. 15 + 16 + 17 = 48. The solution is simpler if we take » — 1, », and n + 1 for the numbers, instead of n, n + 1, and n + 2, although the latter plan is also correct. 2. A man is now three times as old as his son. Five years ago he was four times as old as his son. Find the age of each. Let X = the number of years in the son's age. Then 8 x = the number of years in the father's age. Also X — 5 = the number of years in the son's age 5 yr. ago, and 3 a: — 5 = the number of years in the father's age 5 yr. ago. But 5 yr. ago the father was four times as old as the son. Therefore 3a; — 5=4 (a; — 5). Multiplying, 3a: — 5 = 4a; — 20. Subtracting — 5 and 4 a:, — x = — 15. Dividing by — 1, a; = 15. Therefore 3 x = 45. Hence the man is 45 yr. old and his son is 15 yr. old. Check. 45 = 8 X 15. Also, 5 yr. ago they were 40 yr. and 10 yr. of age respectively, and 40 = 4 x 10. MISCELLANEOUS PEOBLEMS 145 Exercise 97. Miscellaneous Problems JExamples 1 to 6, oral 1. Solve the equations 5a; + 7=42; 5a; — 7=43. 2. Solve the equations 2 a; = 7; 22;— 1=10; 2a; + l = 10. 3. What number is one more than half itself ? 4. A rectangle 5 in. wide has an area of 40 sq. in. What is its length? 5. A rectangular box 5 ft. long and 4 ft. wide has a capac- ity of 40 cu. ft. What is its depth ? 6. A rectangular tank 5 ft. deep and 8 ft. wide has a capacity of 480 cu. ft. What is its length ? 7. If it takes a steamer 5 n days to go 3355 mi., what is its average rate per day? 8. After playing 20 games a baseball team had won three times as many games as it had lost. How many games had it lost ? How many had it won ? 9. A freight train running 35 mi. an hour leaves a station 3 hr. ahead of an express train that makes 50 mi. an hour. How soon will the express train overtake the freight?' 10. The three angles of a triangle are together equal to 180°. In a certain right triangle one acute angle is three times as large as the other. Find the number of degrees in each angle. 11. The ground floor of a school building is in the form of a rectangle 60 ft. long and 49 it. wide. An addition 35 ft. wide is to be made to cover an area equal to half the area of the original building. How long will the addition be ? 12. If we can send a ten-word telegram to a certain place for 85^, and a 24-word telegram for 63^, what is the charge for each additional word above ten? 146 SIMPLE EQUATIONS 13. A miner succeeded in obtaining 675 oz. of silver from some ore, this being |% of the weight of the ore. How much did the ore weigh ? 14. The top of a tree 120 ft. high is broken off. The length of the part broken off is four times the length of the part left standing. Find the length of each part. 15. Three men buy a summer cottage together for $3000. The second pays twice as much as the first, and the third pays as much as the first two together. How much does each pay ? One summer they plan to use the cottage for 3 mo., only one family occupying it at a time. How much of the time should each family be allowed to use it? 16. In 30 yr. from now a boy will be three times as old as he is at present. How old is he now ? 17. Ten years ago a boy was 2 yr. more than one third as old as he is at present. How old is he now '? 18. Four years ago a man was seven times as old as his son, and his son is now 8 yr. old. Find the age of the father at the present time. 19. A man has $9 in half dollars and quarters, having four times as many quarters as half dollars. How many coins of each kind has he ? 20. A man has $3.85 in quarters and dimes, having three times as many dimes as quarters. How many coins of each kind has he ? 21. The sum of the ages of a father and his son is 60 yr., and the father's age is 3 yr. more than twice the age of his son. Find the age of each. 22. Some boys are laying out a circular running track that is to be ^ mi. in length. What radius must they use in describing the circle ? EEVIEW 147 Exercise 98. Review of Chapter VIII Solve the following equations : 1. 8 a; +30 = 5 a; +60. 3. (a; +1) (»-!) + 6 = a;' + a;. 2. 9t/+l=5-3«/. 4. (a; + 3y- (a; -1)^-10 = 6 a:. 5. A grocer buys 6 bbl. of apples at |3.50 a barrel, 2 bu. to the barrel. If 3 bu. spoil, at what price per bushel must he sell the remainder to realize a profit of |5 on his investment ? 6. Two stations are 90 mi. apart. A train leaves the first station traveling 5 mi. per hour faster than a train which leaves the second station at the same time. If the trains travel toward each other and are 22^ mi. apart at the end of 1^ hr., find the rate of each train. 7. Two stations are m miles apart. A train leaves the first station traveling e miles per hour faster than a train which leaves the second station at the same time. If the trains travel toward each other and are d miles apart at the end of h hours, find the rate of each train. 8. Substitute in the formula of Ex. 7 the values given in Ex. 6, and compare results with those found in Ex. 6. 9. How many ounces of water must be added to k ounces of an r^'^o solution of alcohol to reduce it to an r^^ solution? 10. Using the result of Ex. 9 as a formula, find the num- ber of ounces of water that must be added to 11 oz. of a 24% solution of alcohol to reduce it to a 10% solution. 11. A dealer estimates that to make his usual percentage of profit he must receive $33 for 90 lb. of chuck, 30 lb. of sirloin steak, and 20 lb. of porterhouse steak. He wishes to charge 2^ per pound more for the porterhouse than for the sirloin steak, and 5^ per pound less for the chuck than for the sirloin. What price should he charge for each ? 148 SIMPLE EQUATIONS Exercise 99. Review of Chapters I-VIII 1. The observed temperatures for a day, taken at equal intervals of time, being +6°, +9°, +11°, +12°, +6°, +3°, 0°, - 2°, - 4°, - 5°, - 6°, - 7°, find the average temperature. 2. Using the formula C=|(J?'— 32), find in centigrade degrees the average temperature folind in Ex. 1. 3. Subtract p(l + r)» from p' (1 + r)". 4. Subtract 500 (1.05)* from 1300 (1.05)* and find the value of the result. 5. Multiply a + b by a + c. 6. By the method of Ex. 5 multiply 91 by 92. 7. Two men start at the same time from cities which are 50 mi. apart, and travel away from each other, one traveling 5 mi. an hour faster than the other. At the end of 3 hr. they are 245 mi. apart. Find the rate of travel of each. 8. Two men start at the same time from cities which are d miles apart, and travel away from each other, one traveling k miles an hour faster than the other. At the end of t hours they are e miles apart. Find the rate of travel of each. 9. Divide 6h^ + 7 ht+2t^ hj 3h + 2t. 10. At what price must I mark goods which cost me $800, so that I may discount the marked price 20%, 10% and still realize a profit of 20% on the cost? 11. At what price must I catalogue goods which cost $80, so that I may allow a discount series of 20, 10, and still realize a profit of 15% on the selling price? 12. Derive a formula for operations like that of Ex. 11 by using for cost, cZ% for the equivalent discount rate of the series, and ^% for the profit rate. CHAPTER IX FRACTIONS 121. Algebraic Fraction. An expression in the form -, in which either a or 6 is an algebraic expression or both a and b are algebraic expressions, is called an algebraic fraction. In commercial work we frequently need to use algebraic fractions in formulas. Instead of the above form we often use the form a/b. Each form is commonly read "a over 6." For example, 2 x/y, -, -, and — — -^ are algebraic fractions. X X — y 2 Since we cannot divide by zero, the denominator cannot be zero. 122. Terms of a Fraction. In the fraction -» a is called the numerator, b is called the dcTiominator, and the two' to- gether are called the terms of the fraction. The numerator represents the dividend and the denominator repre- sents the divisor of an indicfited division, just as in arithmetic. 123. Reduction of Fractions. We reduce algebraic fractions to fractions having lower terms or to fractions having higher terms just as in arithmetfc. -, ^ 10 2 a + b a + b 1 Just as tt: = ;: ' so ^ — 15 3 a2-J2 (a + 6) (a -6) a-b „. ., , a ax , a a- ax cfix Similarly, v = r- ' ^nd - = = — r- b bx x X ■ ax ax' Multiplying or dividing both numerator and denominator of a fraction by the same expression does not change the value of the fraction. 149 150 FRACTIONS Exercise 100. Reduction to Lowest Terms Examples 1 to 3, oral 3 4 tt 0^ c^ 1. Reduce to lowest terms: -—zi — -> — r' -^t' -57' —^• 12 16 ah oR) a% a^o „ -r, 1 ^1 ^ , 5 ax a^x 2 a^x 5 a^a^ 2. Reduce to lowest terms 15 ay ay^ ^ay'- 10 a^ _ „ , , , , ,' ahc ahc dbc^ a^W-^ 3. Reduce to lowest terms: — , ^^, ^^, -^^. Reduce the following to lowest terms : 4. 6. 8. ^_. 10. 12a453c2 4.54 49a;«/g 455 15 a'^¥<^ 36 gioyo , 50 a^z/V 63a;V ■ 20a36Vo' ■^48ai2ji5' ' ^^x^Y'^z' ' 81a;'y" 12. Reduce ^^-^ — rj^ to lowest terms, and cheek the result a^ — V by letting a = 3, 6=2. Reduce the following to lowest terms : 13 "^~^ 15 ^^ + '^ 17 P^ + g^ 19 ^-y^ ' (a-hy ' a^-1 ' V^-t ' 3^-f' 4a2_62 m^-M* ,„ a^-16 „„ 9a^-w2 14. 16. 18. 20. —• 2 a — b m^ + n^ a^ — 4 Sx — y d + 2 aV 21. Reduce ^ — ^ to lowest terms, and check the result 1— 4 a'* by letting a = 2. Reduce the following to lowest terms: „„ a^+ah „„ c?—ah „. ah — l? 22. — — !— — r- 23. -; r-- 24. (a + 6)2 ■ a3_aJ2 (a - 6)2 LOWEST COMMON DENOMINATOR 151 124. Sign of a Fraction. The plus sign or the minus sign before a fraction is called the dgn of the fraction. If there is no sign expressed, the plus sign is understood as usual. 125. Changing Signs in the Terms. Since, from the law of signs in division, ° a _ — a _ — a _ a ' The value of a fraction is not altered by changing the signs of both the numerator and denominator ; by changing the signs of both the fraction and numerator ; or by changing the signs of both the fraction and denominator. To change the sign of the numerator means that we must change the sign of every term of the numerator, and similarly for the denomi- nator. Failure to do this is the cause of many of the errors that arise. 126. Common Denominator. Two or more fractions that have the same denominator are said to have a common denominator. For example, — i — t and have a common denominator. ax ax ax 127. Lowest Common Denominator. If the common de- nominator of several fractions is of the lowest degree possible, that is, if it contains no unnecessary factors, the fractions are said to have the lowest common denominator (L.C.D.). Reduce ^— and — to equivalent fractions having the L.C.D. be cd The L.C.D. must contain all the factors of he and cd, and no other factors ; that is, it must be of the lowest possible degree. The L.C.D. must therefore contain 6, c, and d, and no other factors, and hence is hcd. , Evidently if we multiply both terms of — by d, and both terms of — by h, each fraction has the L.C.D. hcd. Therefore the result is — and - — - ■ hcd bed The student should express fractions in lowest terms before begin- ning to find their L.C.D. 152 FRACTIONS 128. Finding the Algebraic Sum of Fractions. As in aritK metic, to add fractions we proceed as follows : If the fractions have the same denominators, write the alge- braic sum of the numerators over the common denominator. If the fractions have different denominators, express the frac- tions as equivalent fractions having the L. C. D., and write the algebraic sum of the new numerators over the common denominator. In either case rediwe the resulting fraction to lowest terms. Exercise 101. Addition and Subtraction of Fractions Examples 1 to 7, oral .,,« , c a T b a ^ b 1. Add - and -; — and — ; — and 6 b or of m m Aii* + 5 -.a — b a — h , a — b 2. Add and ; and X X m m 3. Add J and | ; ^ and ^ ; | and ^. Add the following : ^2_J2 «2 + 52 x+y x+y a2-25 a2 + 25 x—y x—y a b ®- 2' 2' c h ~2'~2 ,.^, b — c 4 '4- 8. Add and — — — . Check the result. a — h a^ — tr Add the following and check the results : n ^ — y y — ^ ,i2 mn m^ — 2 mn + n^ ^•^- + 12- ^^-"5 15 aj + y ^ — y m^ — n^m^-\-2 mn + n^ ^ 8~' "^5 "^ 10 "' ADDITION AND SUBTRACTION 153 13. Show that - + - = — ;— , aud from this formula write a ab a rule for adding two such fractions. Apply the formula to the addition of J and ^ ; of ^ and ^ ; of |^ and \. 14. Show that = — j— > and from this formula write ah ab a rule for subtraction in the case of two such fractions. Apply the formula to \ — \; J — i? 2~i- n. Of] €(i X 15. Find the sum of - + - and also of From the 6 «/ by formulas write rules as in Exs. 13 and 14, illustrating each. The student should see how valuable algebra is in enabling him to discover rules of this kind which can be used in commercial work. Add the following and check the results : a+b a+b 16. a ^ ia 5a 3"^ 4 12 17. X 2x 7 X 5 3''"15' 18. 2 a aba 7 3 '''21 19. 4w 1 m 4m 5 10 15 20. 3 4 5 a 0? c? 21. 3 2 1 c? Ba^ a 22. 2 3 5 3Z/ + 42/2 12 23. ^ + 1 + 1. y z X 24. 25. a — b a^ _ J2 X y 26. a^ a x-y ,2 (a - 1)2 a - 1 3 4 27. 1 a — 3 a + 3 28. 5 3 a— 2 a+2 29. -A-^^^ 30. 31. ^ — y^ x — y X y 3? — y^ x + y a;+ 5 x—b X— 5 x+b 154 FRACTIONS 129. Mixed Expression. An algebraic expression which is the indicated algebraic sum of an integer and a fraction is called a mixed expression. For example, 2 a + h/c is a mixed expression. In arithmetic the plus sign is omitted between the integer and fraction in a mixed number, 2^ meaning 2 + J. In algebra a- means ax-, and hence the plus sign c c or the minus sign must be used in a mixed expression. 130. Reduction of a Fraction to a Mixed Expression. If the degree of the numerator of a fraction equals or exceeds that of the denominator, in a given letter, the fraction may be reduced to an integer or a mixed expression by dividing the numerator by the denominator. 5 a^ ■\- h^ jTust as we reduce - to 1| by dividing, so we may reduce — to 2 J2 ^ x" + y^ o. + b a — b -{ We cannot, however, reduce — to a mixed expres- a+b a+b ^ sion, because different letters are involved. ^ j^ . H '117/ —1— 3 9J 1. Reduce — — — — — to a mixed expression. 2x-y Dividing the numerator by the denominator, we have i x^ - 6 xti + S y" „ „ y" ^ 2. = 2x — 2y ■{ ^ 2x — y 2x — y „ _ , a3+3a26_6a52+2 63 2. Keauce — - — ; — to a mixed expression. Dividing, — — — — = a + 5, with remainder. —%(ib^ + J'. a'' + 2 ao + 6^ We may therefore write the result in the form ° + ' + a^ + 2aft + fe^ '°'^"+^- a^ + 2a5 + i^ '°^° + ^- (a + t)^ ' the last form being the most convenient form for computation. The work may be checked by substituting any value for the letters that will not make the denominator zero. For example, if a = 1 and 5 = 1, we obtain for the original fraction and for the result. REDUCTION TO MIXED EXPRESSIONS 155 Exercise 102. Reduction to Mixed Expressions Examples 1 to 11, oral , ^ , ^ . ^ 10 10 a2 10 a2 a% (a + by 1. Keduce to integers : — - ; ; ; - — : -i^ — ■ — ^ . ^ 2' 2 ' 2a ' ab' a + b 2. Keduce to integers: — ; ; — ■ a—b a+b a+b 3. Reduce to mixed expressions: -; -; X X a Reduce to integral or mixed expressions : ax — ay abx + aby mr? + 3 10.^ ah mn rrfi — n^ a^ — ^ abx + 1 abc + c a + b ' a? + y^ ' ab ' abc 'a — b 12. Reduce 7 to a mixed expression, and check the result by letting a = 2 and 6 = 3. Reduce the following to integral or mixed expressions : . x^ + Bxy + y^ ^ m^ + S nfin + 3 mn^ + n? m^ + n^ +2 mn m^ + 3 mhi + 4 mrfi + 5 n^ x + y 11 ^ — 2>xy -y"^ x-y 15 a^ + 15a; + 56 x+1 Iff a?-15x+56 x-8 17 aW + Sab + 2 ab + 2 18. f + ^pq + f 20. 21. 22. 23. m(m + 2n) + r^ a^ + a% + aW' + ¥ a + b 2^ + 3 3?y — xy"^ — ^ x + y a3 + 53 + ^3 - 3 aSc a + b + c 24. ^+^y'+/ p + q ' ' ^ + xy+y^ 156 FEACTIONS 131. Reduction of a Mixed Expression to a Fraction. Since the value of an expression is not changed if it is both multi- plied and divided by the same expression, it follows that ah , a ab + a ^. » a = —, and a + - = — ; Therefore To reduce a mixed expression to a fraction, multiply the inte- gral part hy the denominator, to the product add the numerator, and under the result write the denominator. This is practically the same as adding two fractions. For example, , c ad , c ad + c a + — = 1 — = d d d d Keduce a + 6 o -— to a traction. a — So Reduce a + 3 J to a fraction with a — 3 6 for its denominator. Then a^ - 9 &^ a^-2ab + 4cb^ _ a'-9b^-a' + 2ab-4:b^ _ 2ab-iab^ a-Sb a-3b ~ a-3b ~ a-Sb Exercise 103. Reduction of Mixed Expressions to Fractions Examples 1 to S, oral 1. Reduce to improper fractions : 2|^ ; 4| ; 9^. 2. Reduce to fractions: a + -; a ; a + h + -. c c d €bG CfiC C 3. Reduce to fractions: a + - ; a — - ; a+J4- J + c h-\-c a — b Reduce each of the following to a fraction : a — b 8. a^-ab + h^- a — 6 5. x + 2y + -^^. x-ly 6. 2 S3 a + 6 a — b x-\-y MULTIPLICATION 157 132. Multiplication of Fractions. We multiply fractions in algebra in the same manner as in arithmetic, and hence the process need not be explained at length. To multiply a fraction hy an integral expression, multiply the numerator hy the integral expression, writing the product over the denominator. To multiply a fraction hy a fraction, midtiply the numerators together for a new numerator, and the denominators together for a new denominator. Cancel factors common to any numerator and any denominator. Exercise 104. Multiplication of Fractions Examples 1 to 9, oral Multiply the following : , . a . , a „ 3^ 52 1. 0X-- 4. mn 7. -55 •■55- 0, mn o<* 6'^ ^ A C' esI „ m n 2. 4x-rT- 5. »2g, 8. 4 pq n m 3. 46x-rT- 6. a?^2 9 . 4 xy w mf- _,.,,, T , r- 2aa^ Scg^ 45«2 , , , 10. Fmd the product of 7— — r • - — - • — -^ , and check. 3 %2 4 aa? 5 cz'' Multiply the following and check : aP hc^ ccP c^d (Pa €?h ^-5_2_4a__ ^^ a2 + 52 a + 5 13.^:^.^V- 16. (2a)2 (3 6)2 (4c)2 h G a 4a^ + 8a^ 9 x — a? x-3 2+x (a + 5)2 a2_ -J2 a^ + ¥ Zh + Ba a-h «2 + j2 158 FRACTIONS 133. Reciprocal. If the product of two expressions is 1, each expression is called the reciprocal of the other. Since = 1, the reciprocal of - is -, and conversely. ha ha 134. Division of Fractions. Just as in arithmetic, To divide hy a fraction, multiply by its reciprocal. Exercise 105. Division of Fractions Examples 1 to 8, oral , „..,«, a a y a 2 , 4 4, 2 4, a 1. Divide g by - ; -r by - ; - by - ; - by - ; - by -• 2 44 2 a a a a a I ^ ^. .-, a + b , a — b a + b, a — I a + h , a — b 2. Divide by ; -- — by —^ — ; by X X zx Zx x+y x+y Divide the following as indicated : ^ ab cd ^ 2x Bx „ m n ' xy xy ' y y ' n m . abc bed „ 2 a; 3 a;w ^ ah cd 4. 5 6. i -• 8. : xy xy y y^ cd ah 9. Divide by — — ^ , and check the result by letting a = 2, S = 3, c = 4, d = 5. Divide the following as indicated : 25 abc . 15aW<^ , a^+b^ a^+b^ ■ 32xyz ■ 28a?«/V* 17 3?yz _ S4=xy^^ ^^ 24: mV ' 7bm*n^' 35aW^28a3^ ■ 27a^^a« '' 81a?«/V" 25m^w*z 35«iw2!* ■ 33a*6V"^44a8J3c3" * m^-n^ a — b a + b a' -IP' a + b a2+J2 ■ a2+62 a^-¥ a^-W- aH6* ■ a^+lP 7r?+r? 2m^+2n^ 13. i- -i- 14. 1 . 3^ -i- 15. i- _ 1 FORMULAS 169 Exercise 106. Formulas Examples 1 to 28, oral j-r . 1,1 a-\-h jl 1 h — a „ Using - + - = — -— and = — -- as formulas, perform ah ah ah ah ^ •' the following additions and muhtractions : 2-i + i- 6-i-f 10.1 + 1. 3-i + f 7-i-i- ll-i+i- 4.1 + 1. 8. J-i. 12. 1 + f 16. ^-i- TT ■ a c ad + he , a c ad — he > 6's^w^ - + - = — _— and - — - = — — — as formulas, d bd h d hd perform the following additions and suhtractions : 17. l + f. 20. 3_2. 23. f + |. 26. f-|. 18. i+|. 21. 1-1. 24. 1+1. 27. 1-1. 19. | + |. 22. f-|. 25. l + f. 28. f-f Examining the problem a -■ = — ^> we see that any number a is divided by - of 100 by multiplying a by ^ and moving the decimal P . 84 X 8 point two places to the left. Thus 84-;- 12^ = ———7 or 6.72. In like manner perform the indicated operations in Exs. 29-40. Divide the following as indicated: 29. 65H-12J. 33. 76^25. 37. 79^331. 30. 4.83 ^ 16f . 34. 85 -h 33f 38. 436 -r- 12|.. 31. 45-f-25. 35. 94^8^. 39. 57^16|. 32. 66 -5- 50. 36. 150 -^ 12|., 40. 85 -s- 8f 41. Simplify a -. , and write the rule for dividing any 1 ^ number a by - of 1000. P 160 FEACTIONS 42. If the correct value of a fraction is - and an estimate 6 of its value is given as , bv how much is the estimate too small? Use particular values for the letters involved, such as a = 3, 6 = 5, a; = 1, and «/ = 1, and substitute in your result as a check. 43. If the correct value of a fraction is - and an estimate is given as , by how much is the estimate too large ? h-y Illustrate by using particular numbers for a, h, x, and y. 44. If X and y represent the greatest possible error in estimating dividend a and divisor d respectively, the great- est possible error of the result of division is approximately -J 01 . i md its value. 2 ■ d—y d+y ih. Using the result of Ex. 44, find the greatest possible error of the answer obtained by correctly dividing 4300 by 510, if the greatest possible error of the dividend is 50 and that of the divisor is 5. 46. The formula for the present value of an annuity is 7? r 11 ^ = — 1 — -r: . Express this formula so that the part in brackets shall be an improper fraction. 47. If a series of items whose average is to be found is represented by m-^ ± e-^, m^ ± e^, m^ ± eg, m^ ± e^, and m^ ± e^, the greatest possible average will be obtained when all the errors (e's) are positive, and the least possible average when all the errors are negative. Find the fraction representing the average in each case, and separate each result into two fractions, one in which the numerator consists of mi's and the other in which the numerator consists of e's. EEVIEW 161 Exercise 107. Review of Chapter IX , T, ■, ,1 ^ , aa; 2 X 3 23 X 3S 3 x 1.053 1. Keduce to lowest terms : — , - — - , — , ay 2x5 23x7^ 2x1.05* 2. The ratio of the amounts of two principals, p^ and p^, in t years at r per cent is found by dividing the equations ^\ =^1 (1 + ^0 ^^*i "^2 —P-iO-'^ ''0' member by member. Find the value of — in terms of p^ and p^. 1 3. If contractor A can do — of a piece of work in 1 da., ■, m and contractor B can do - of the work in 1 da., what part of the work can they do in 1 da. working together ? 4. What part of the work in Ex. 3 would be done in 1 da. if A could do all the work in 3 da. and B in 5 da. ? Substitute 3 for m and 5 for h in the result of Ex. 3. , TT ■ a , ad + be , . ,,11 ,3 5. Usmg -+- = — — — twice, add -, -, and -• b a bd o 4 8 6. "Divide |84.65 by $12.50, making use of the formula 100 ap "'^■^-Too- , „ , 32 X 5 X 7 , . , 7. Reduce — - — = — to an mteger. o X 7 8. Simplify ix(^+^) and ix(| + |). „.-,.» a^— 6^ a—b ,3 5 9. Simplify —5 -; and 7 -^ 7: • ^ ^ (a + by a + b 4 2 10. If ij is the interest on p for 1 da. at 4|-%, and i^ is the interest on p for 1 da. at 3%, find the value of ■^- i^EL.^A i'^EL ' ,•' 11. If i = ^ — and i' = ^r-z: > find the values of - and -■ 360 365 i' I. 162 REACTIONS Exercise 108. Review of Chapters I-IX XT 1. The formula for the area of a triangle is a = — - If 6 = 2, what is the value of a ? 2. The area of a circle is expressed by the formula a = irr^. If r = I/tt, what is the value of a ? 3. Find the sum of 34 x $1.25, 45 x fl.25, 15 x *1.25, and 66 x $1.25. What algebraic principle did you use? 4. If 184 is :r^ of 450, find the value of x. 5. What per cent of 1440 is 92 ? 6. If 65 doz. eggs are packed in containers costing $3.90 and sold for $26.65, how much is received per dozen? ■ 7. By how much does 2x—\ exceed 3 — 2 a; ? 8. Using the principle ab = ba, find 72% of $125. 9. Using the principle ^ = ( — )-5-6, divide $162.45 , , c ab \ a / by 45. , 10. Three companies. A, B, and C, enter into a pooling agreement with a common sales agency, with the understand- ing that the agency shall assign the business to the companies in proportion to their average total business for the last 5 yr. These averages were $2,800,000, $3,500,000, and $4,000,000 respectively. What portion of an $800,000 order should be assigned to each company? 11. If goods cost $60, how should they be marked so that a discount of 20% of the marked price can be allowed and a profit of 15% of the cost be realized? 12. From the equation i=prt derive a rule based on 360 da. to 1 yr. for finding interest for 1 da. at 2^%. Use this rule to find the interest on $8400 for 32 da. at 2^%. CHAPTER X FRACTIONAL EQUATIONS 135. Fractional Equations. An equation containing a frac- tion or several fractions is called a fractional equation. Multiplying both members of a fractional equation by such an expression as shall leave no fractions in the equation is called clearing the equation of fractions. If - + ^ = aj, a multiplying by ah, bx + ax = a%'^, an equation cleared of fractions. To clear an equation of fractions, multiply both members of the equation by the lowest common denominator of the fractions. If a- fraction is preceded by a minus sign, change the sign of every term in the numerator when the denominator is removed. 1. Solve the equation -H 4 = -■ A a a ^ Multiplying by 4 a, the L. C. D. of the fractions, 2 aa; + 12 - 16 a = 4 - a. Subtracting 12, 2ax —16 a=— S — a. Subtracting —16 a, 2 ax = 15 a — 8. 15 o - 8 Dividing by 2 a, 2. Solve the equation 2a x+1 x-1 ■1 x-2 Multiplying by the L. C. D., x^-x-2 = x^-2x + l. Solving, a: = 3. 163 164 FRACTIONAL EQUATIONS Exercise 109. Fractional Equations Examples 1 to 4, oral 1. Clear of fractions: - = b; -— =-: - + - = 1. a 2a 2 3 4 2. Solve 1 + 1=2; f-l=2; |-2.= 2. o 4 5 3. Solve ^ = 4; ^=6; ^=8; ^=10. 4. Solve ^ = 3; ^ = 5; ^^ = 6 ; ^ = 2. 5. Given i=prt, solve for p in terms of i, r, i, and write the rule in interest derived from the resulting formula. 6. As in Ex. 5, solve the equation i —prt for r and for t, writing the rule in each case. 7. What sum will yield |3600 interest in 4yr. at 4%? 8. In what time will $8000 earn |2000 interest at 5% ? 9. At what rate will |5000 yield |600 intere|t in 2 jt. ? 10. Given the formula a=p(\+rt'), solve for p, then for r, and then for t. Write the rule derived from each formula. Solve the following equations : 11 " + ^-^ ... 1^.0.4. 19. X— x+1 4 -f^ = ^- 20. 2;-3 „ x-7='^- 13.^ + '=lf. X— 2 ' 17. "^i-t rt + J 4 21. x + 1 2a;-5 a;_l 2a;-7 x+2 7 "• Ie| = ^- 22. a;+2 2«-5 x-S 2 a; -10 COMBINIlSrG TEEMS 165 136. Combining Terms before Clearing. It is often better to combine terms before clearing an equation of fractions. . o T ,1 i- x+1 2x1 X - 1. bolve the equation —- h r— = t. — I- -t. V o. 2a 3a 2a S a Here it is apparently advisable to unite the fractions with the same denominators before clearing of fractions. 1 X Subtracting — — and — , 2a Sa X +1 — 1 , 2x — X _ o + —Ti = ^' 2 a 6 a or ^ + ^ = 5. 2a 3 a Combining, — = 5. Dividing by 5, - — = 1. D a Multiplying by 6 a, a,- = 6 a. „ , ,, ,. x — 1 x—2 x—5 x—Q 2. solve the equation = r • x — 2 x—6 a; — fa x—1 Combini4J.g the fractions in the respective members, (x - 1) (x - 3) - (x - 2f _ (x-5)(x-7)-(x- 6)^ (x-2)(x-&) (x-6)(x-7) Simplifying the numerators, -1 _ -1 (x - 2) (x - 3) (x - 6) (x - 7) ■ Multiplying by — 1, and then clearing of fractions, (x -6)(x-7) = (x- 2) (x - 3). . Expanding, x'^ — 13 x + 42 = x^ — 5 x + 6. Therefore -8x=-36, and X = 4^. We might first reduce to mixed fractions, thus : ^^7^~ ~ x-B~ '^ x-6~ ~ x-7' 166 FRACTIONAL EQUATIONS Exercise 110. Fractional Equations Examples 1 to S, oral 1. Solve the equations x-\-^ — \ — 1; x-\-^ = '?> — \. 2. Solve the equations a; + f — J = 21; x + ^ + ^ = 4i. 3. Solve the equations a; + | — f=5J; a! + | + | = 17. Solve the following equations : 3 a; — 5 5 a; — 1 a; — 4 _ a ■ 5a;-5 7a;-7^^^~ ' First factor the first two denommators. , Sx+2 2X-1 , Zx+2> , ;; .. ^' ^:r2'~ 3^36 + 5^^110 + ^- ^^• ^- 27^ + 3^:^9 + 4^132 + ^^-^^- „ Zx 5 152;-7 17a:-a? , 71 7- ■J^ K + TTT ^-^ TT^ r^r^^TTT-i 7T+: 2a;-2 6(a;-l) 9(a; + l) 6 (a^ - 1) ' 18 (a? -1) 4 ^ 1 _^ a^-3 a; + l 1 — a; a;— 1 1 — ai^ Wrile the first fraction -—. — and the third fraction + 1 + a; 1 — a; Solve the following equations and check the results: 9.-^ + ^ §Z =0. x + 2 S+x a!(a; + 5)+6 10. 1 1 1 1 a;— 1 a;— 2 a;— 3 a; — 4 11. One fifteenth of 6a;+ 7, divided by x, is equal to the quotient of 2 a; — 2 divided by 5 a; — 6. Find the value of x. 12. One fifteenth of 6 a; + 1, diminished by the quotient of 2 2; — 4 divided by 7a; — 16, is equal to one fifth of 2a;— 1. Find the value of x. FOKMULAS 167 137. Equations used in Formulas. 1. From the formula for the amount a=p +prt, derive a formula for the time, t. From the equation a= p ■\- prt we may obtain the value of t as follows : pr ' , That is, if we divide the difEerence between the amount and the principal by the product of principal and rate, we obtain the number of years required for the given principal to produce the given interest at the given rate. 2. From c = 2 tt?" derive a formula for the radius. Dividing each member of the given equation by 2 ir, we have c That is, if we divide the circumference by 2 x 3f we shall obtain the radius of the circle. 3. Given the formula v = ttj^A, derive a formula for r. Dividing by irh, ~T ~ '^■ Taking the square root, -2=12, and 7r = 3^, find the value of r^. 10. If from a square of side Sj there is cut out a smaller square of side Sg, the area of the part that remains is given by the formula a = (s^ + s^) (sj — s^). Find a when Sj = 11, Sj = 9 ; also find s^ when a and Sj are given. 11. In connecting two wheels by a belt it is often required to know the difference in the two citcumf erences. This is given by the formula Z) = 2 tt (r^ - Suppose that Z> and rg are known, find Vy 12. If from a circle of radius r there is cut a square of side s, the area that remains is given by the formula a = Trr^ — s^. If r = 7 and s = 4, find a. If a and s are known, find r. If a and rare known, find s. FOEMULAS AS EQUATIONS 169 The following formulas will he met hy the student in his subsequent work in mathematics or physios. Solve as directed : 13. s = vti find*. 21. W=Fs; finds. 14. s = vt; find V. 22. ViP-^ = V^P^; find F^. 15. d = rt; find r. 23. V^P-^ = V^P^ ; find P^ 16. d = rt; find t. 24. v^t = v^t + n; find v^. 17. 8 = 1^*2. find (/. 25. ■y2* = V + '*; find L 18. s = J ^^2 . find «. 26. v^t = v^t + n; find w. 19. W^L^ = W^L^ ; find L^. 27. Z = Z^?' («' - f) ; find X 20. W.^L^=W^L-i^; find JF2. 28. I>(w^-w^') = w^; find Wg- 2ffr Given s = |^ w (a + Z). Solve for n; iov a; for Z. 30. Given s = :p. Solve for r; for I; for a. r— 1 31. Given a = ^h(b + b')., Solve for b; for 5'. 32. Given C= Solve for for each mem- ber reduces to — 3.8. 3. Solve 0.7 a; + 6.5 = 8 + 0.05 x. Multiplying by 100, 70 a; + 650 = 800 + 5 x. Subtracting 650 and 5 a:, 65 a; = 150. Dividing by 65, x = 2-^^. 178 rEACTIONAL EQUATIONS Exercise 115. Equations involving Decimals Uxanvples 1 to 3, oral 1. Solve a; = 5 + 0.5 a; ; x = l +Q.bx; x = 9 + 0.bx. 2. Solve x=Z + 0.2hx; «= 6 + 0.25a;; a;=9 + 0.25a;. 3. Solve 0.22; = l-0.3 2:; 0.3a;=l-0.2a;; 0.4a:=l-0.1a;. Solve the following eqwatioTis and check: 4. 0.4a; + 5 = 9. 9. 0.5a; - 6 = 0.2a;. 5. 0.72:+3=ll. 10. 0.7 a;- 5 = 0.2 2;. 6. 0.9a; + 7=25. 11. 2.5a; - 24 = 1.3a;. 7. 0.09a;+7=25. ' 12. 3.9 a;- 4.8 = 2.7a;. 8. 0.09 a; + 11 = 10.8. 13. 5.8 a;- 0.68 = 1.9 a-. 14. Solve the equation 0.3 a; + 7.2 = 4.8 - 0.02 a;, (1) by clearing of decimal fractions before subtracting any terms from both members ; (2) by first subtracting certain terms from both members and then dividing by the decimal coefficient of x. Solve the followiifig equations and check: 15. 0.7a;-3+|a;^ = 0.3 + 0.4a;(l + a;). 3.48 a;- 8 -0.4a; g 8 + 0.01 a; ~ ' 17. 2.725 a; , ^ 1.635 + 0.1 a; -18 0.1a;-18 0.02a;- 3.6 18. If 7 is increased by 0.75 of a certain number, the sum is 9.7. What is the number? 19. If 4 is increased by 0.45 of a certain number, the sum is 9.4. What is the number ? 20. If 0.4 is added to 0.72 of a certain number, the sum is 22. What is the number ? RATIO 179 142. Ratio. The relation of one number to another number of the same Jcind, as expressed by the indicated division of the first number by the second, is called the ratio of the first number to the second. The ratio of a to J is indicated thus : ■- , a/h, ai«.:l. b Hence all ratios may be looked upon as fractions, and the subject of ratio is properly a part of the chapter on fractional equations. Indeed, it is largely a matter of tradition that we have ratios instead of ordinary fractions in any of our work. We seldom employ the word " ratio " in business, commonly speaking of dividing a sum in the proportion of 2 to 3 instead of dividing it in the ratio of 2 to 3. 143. Terms of a Ratio. The two numbers involved in a ratio are called the terms of the ratio. The dividend is called the antecedent and the divisor is called the consequent. Thus we have ri _ antecedent _ numerator _ dividend h consequent denominator divisor a 2 If - = _ , we say that a is to 6 as 2 is to 3. b 3 The following important law relating to ratios is frequently used: Both terms of a ratio may he multiplied hy the same number or both terms may he divided hy the same number without changing the value of the ratio. A ratio being a fraction, the laws of fractions hold true for ratios, and it has been seen that we may multiply both terms or divide both terms of a fraction by the same number without changing its value. For example, the ratio 20 : 25 may be written in simpler form, thus : 20 : 25 = If = f . Similarly, if we wish to express the ratio 7 to 9 so that the conse- quent shall be 45, we have 7:9 In the same manner, 1:1 = 2:2 = ! 180 FEACTIONAL EQUATIONS Exercise 116. Ratios Exam/pies 1 to 4, oral 1. Express in simplest fractional form the ratio of 10 to 20; of 32 to 40; of 45 to 54; of 49 to 56; of 26 to 39. 2. Express in simplest form, as an improper fraction, the ratio of 15 to 10; of 20 to 12; of 35 to 21; of 81 to 63. 3. How is the ratio of 9 to 5 changed by multiplying both terms by 2 ? by adding 2 to both terms ? by subtracting 2 from both terms ? 4. How is the ratio of 6 to 9 changed by multiplying both terms by 3 ? by dividing both terms by 3 ? by adding 3 to both terms ? by subtracting 3 from both terms ? Simplify the following ratios : 5. a^-P:a^-2ab + P. 6. a+1: a^+2 a+1. 7. Separate 50 into two parts having the ratio 2 : 3. Let X = the smaller part. Then 50 — x = the larger part. X 2 Therefore -r-— — = - • 50 — a; 6 Solving, X = 20, and bO - x = 30. Separate into two parts having the ratios specified : 8. 24; 1:2. 9. 91; 2:5. 10. 272; 7:9. 11. A wheel of diameter 30 in., making 300 R.P.M. (revo- lutions per minute), is belted to another wheel of diameter 15 in. Find the speed of the smaller wheel. 12. Two cogwheels are geared together and the distance between their centers is 20 in. What are their diameters, if their speeds have the ratio of 4 to 5 ? EATIO IN STATISTICS 181 144. Use of Ratio in Statistics. We have already seen the value of graphic work in commercial affairs, and particularly iu statistics, and are now prepared to see one of the principles upon which its vahdity rests. This principle is as follows : A series of numbers may he represented hy lines, rectangles, circles, or other geometric magnitudes, provided that the lengths of any two lines representing two of the numbers have the same ratio as the numbers represented ; or provided that the area^ of any two rectangles representing two of the numbers have the same ratio as the numbers they represent; and so on. If the numbers to be represented are large and the space on the paper is small, we first determine the length of the longest line that can conveniently be drawn to represent the largest number of the series. Suppose, for example, that its length is 8 cm., and the quantities to be represented, the num- ber of long tons of pig iron produced in the United States annually in recent five-year intervals, to be as follows: Year Production Fkst 9,446,000 Second 13,789,000 Third 22,992,000 Fourth 27,304,000 If a liue 8 cm. long represents 27,303,000 T., a line 1 cm. long will represent 27,304,000 T. -^ 8, or 3,413,000 T. Now the smallest division easily distinguished on the common scale is the millimeter, or 0.1 cm. Since 1 cm. represents 8,400,000 T. approximately, 1 mm. represents 340,000 T. Dividing the numbers of the above table by this number, we get 27.8, 40.6, 67.6, and 80.3 respectively. Lines of these lengths, in milli- meters, have approximately the same ratios as the numbers of tons in the table. 182 FRACTIONAL EQUATIONS Exercise 117. Ratios in Statistics Taking any convenient length as a unit, represent graphically the numbers in the following statistics relating to products in the United States : 1. Wheat: 2. Pig iron: Year Bushels Year Tons 1871 . . 300,000,000 1870 . . 1,800,000 1881 . . 449,000,000 1880 . . 4,300,000 1891 . . 476,000,000 1890 . . 9,000,000 1901 . . 522,000,000 1900 . . 13,800,000 1911 . . 635,000,000 1910 . . 27,300,000 3. Anthrac ite coal: 4. Com: Year Tons Year Bushels 1880 . . 33,000,000 1880 . 1,600,000,000 1890 . . 45,000,000 1890 . 1,700,000,000 1900 . . 54,000,000 1900 . 2,100,000,000 1910 . . 65,000,000 1910 . . 2,900,000,000 5. Cotton: 6. Gold: Year Bales Year Value 1880 . . 6,300,000 1880 . . . 36,000,000 1885 . . 6,400,000 1885 . . . 32,000,000 1890 . . 8,600,000 1890 . . . 33,000,000 1895 . . 7,100,000 " 1895 . . . 47,000,000 1900 . . 10,300,000 1900 . . . 79,000,000 1905 . . 10,800,000 1905 . . . 86,000,000 1910 . . 12,000,000 1910 . . . 96,000,000 1915 . . 11,200,000 1915 . . . 101,000,000 These statisi ics are, of course, only approximations, but they are substantially correct. PEOPOETION 183 145. Proportion. An expression of equality between, two ratios is called a proportion. Thus - = - is a proportion, and a, h, c, d are said to be in proportion. This proportion may be written a:h = c: d, and is often read, " a is to J as c is to d" or " the ratio a to J is equal to the ratio c : d." In older works the above proportion is often written a:b::c:d. 146. Terms of a Proportion. In a proportion the first and foTirtli terms are called the extremes, and the second and third terms the means. In the proportion a ib ■= c : d the extremes are a and d, and the means are 6 and c. 147. Mean Proportional. If the means of a proportion are the same, each mean is called the mean proportional between the two extremes. Thus ro is the mean proportional between x and y if x:m = m:y. 148. Laws of Proportion. The following are two of the important laws of proportion : 1. If four numbers are in proportion, the product of the extremes is eqtial to the product of the means. Let the proportion be r ~ 3 " a Multiplying by hd, ad = he. 2. If the product of two numbers is equal to the product of . two other numbers, either two mccy be made the means of a pro- portion, and the other two the extremes. Let ad = be. Dividing .by bd, ad _ bd' _ be bd a h' c 'd' 184 FRACTIOlSrAL EQUATIONS Exercise 118. Proportion McaTTuples 1 to 9, oral 1. Solve the equation J « = f • X 2 2. In the proportion ^ = -^ find the value of x. 3. In the proportion a; : 2 = 2 : 3 find the value of x. Find the value of x in each of the following proportions ; 4.^ = 1 4 8 6.2 = ^. 3 9 8 '' *2 ■ X 48 ^- 5~15 „ 3 12 '■ 5 = T- 9 ^6 ^ ■ 48 X 10. If the means of a proportion are 3 and 12, and one extreme is 4, what is the other extreme ? 11. If a; - 5 : 5 = 28 : 10, what is the value of a; ? Find the value of x in each of the following proportions : 12. x:2A= 1.21 : 1.1. 14. 2.6 : x = 1.1 : 7.7. 13. a;-12:12 = 5 :1. 15. 17 - a; : a^ = 16 : 18. 16. If each mean of a proportion is 8, and one extreme is 32, what is the other extreme ? 17. In a lever PW, if sufficient power (^) is applied at P, a weight (w) at W can be lifted. If F is the fulcrum, it is known that the ratio p:W is equal to the ratio FW : FP. If FW =10 in., FP = 30 m., and w = 240 lb., find p. 18. If two boys weigh respectively 100 lb. and 120 lb., where must the fulcrum be placed under a 10-foot board so that the boys sitting at the ends will just balance ? The law given in Ex. 17 also applies in the case of this example ; that is, we have 100 : 120 = a; : 10 — a;. PROPORTION 185 19. It is proved in geometry that a line parallel to the base of a -triangle divides the other two sides pro- portionally. In this figure, if AP = 1^, FC=6, and QC=5, find the length of BQ. Teachers should use their discretion as to the problems to be assigned on pages 185 and 186. If the students have had no geometry of any kind, most of this work should be omitted. 20. It is proved in geometry that two triangles of the same shape have their coij-esponding sides proportional. In Ex. 19, a AB = 8, AC= 71, and PC = 6, what is the length of FQ ? 21. If two sides of a triangle are 7 in. and 9 in., and a line parallel to the third side cuts the first side into the parts 3 in. and 4 in., into what lengths will it cut the second side ? 22. In this square, PQ is parallel to AB. If a side of the square is 10 in., DB = 14.14 in. If i>P= 3ia., what is the length oi DQ? 23. Two pieces of timber 1 ft. wide are fitted together at right angles as here shown. AB is 8 ft. long, ^C is 6 ft. long, and the distance BC along the dotted line is 10 ft. A carpenter finds it necessary to saw along the dotted line. Find the length of the slanting cut across the upright piece of timber; across the horizontal piece. 2 ; 10 = 1 : 6 is one proportion. 24. This figure represents a pair of proportional compasses used by draftsmen. By adjusting the screw at 0, the lengths OA and OC, and the cor- responding lengths OB and OD, may be varied proportionally. The triangle formed by 0, A, and B is the same shape as that formed by O, C, and D. If OA=B in. and 0C= 5 in., then AB is what part of CD? a. 186 EBACTIONAL EQUATIONS 25. In the following combination of levers a power of 100 lb. is applied at P. What weight can be raised at Wi Jf 26. It is proved in geometry that if AB is the diameter of the semicircle here shown, CJD is the mean proportional between AD and DB. If AB = 3 and CZ> = 4, what is the length of DB? 27. In the same figure, if DB = 4 AD and CD =12, what are the lengths of AD and DB? 28. If a perpendicular is drawn from the vertex of the right angle upon the hypotenuse of a right triangle, it divides the right triangle into two triangles similar to the original triangle and to each other. In the given figure, if AF= 5 and CF= 3.5, what is the length oi FB? 29. In the same figure, if AB — 7.45 and AC =6.1, what is the length of AF? 30. If a boy 4| ft. tall casts a shadow 41 ft. long at the same time that the school building casts a shadow 67J ft. long, how high is the school building? 31. If a boy, lying down with his eye to the ground, sights over the top of a 10-foot pole held vertically 6^ ft. from his eye, and can just see the top of a tree 37J ft. from his eye, what is the height of the tree ? a + b i 32. If a — b find a formula in a?. 33. A tree casts a shadow 48 ft. 3 in. long at the same time that a flagstaff 32 ft. high casts a shadow 28 ft. 4 in. long. Find to the nearest inch the height of the tree. REVIEW 187 Exercise 119. Review of Chapter X 1. Solve the equations -— + 3 = - + 4 and - H f- - = 1. 5 6 XXX 2. Solve for f the equation ^- — — + b=p — - — —■ 3. In the study of electricity the formula C = c en often found. Solve for n. 4. In Ex. 3, if C= 0.37, i? = 5, r = 2, and e = 1.05, find the value of n. 5. The use of a certain drug in the United States has increased 50% every 5 yr. and at the present time 8000 lb. of the drug are consumed each year. How many pounds of the drug were consumed 15 yr. ago in the United States ? 6. The records of two workmen show that A can join the parts of 500 sink mats in 24 hr., and B can do the same in 80 hr. If these are the only available men, what is the smallest number of working days of 8 hr. each in which the employing company can guarantee to finish 2000 sink mats ? 7. The total exports of the United States in a certain year were valued as follows: to Europe, $1,136,000,000; to North America, $386,000,000; to South America, $174,000,000; to Asia, $61,000,000; to Africa, $19,000,000. Represent these quantities by lines or bars having approximately the same ratios as the quantities themselves. 8. If a:b = o:d, show that a: c = b: d. 9. li x:i/=p:q, show that x + y.y =p + q: q. Add 1 to each member of the equation - = - and reduce each member of the resulting equation to a fraction. ^ " 10. If x:y=p:q, show that x — y.y =p — q: q. 11. If m:n = x:y, show that m + n: x + y = n: y. 188 FEACTIONAL EQUATIONS Exercise 120. Review of Chapters I-X 1. How much greater is 17 than 11 ? x — 1 than x—3 2 2x-b than a; + 6 ? 2. Multiply 2;3 _ 3 a;2 + 5 a; _ 11 by 2 a? - 3, and cheek the result. 3. Solve C=f(i?'— 32) for F. Use the result to change a temperature of 40° C. to Fahrenheit. 4. Multiply a + h by c + d, and from the result derive a short, practical method for multiplying 8 + ^ by 6 + 1^, that is, for multiplying 8^ by 6|. 5. Prove that if x, y, and z represent the three digits of a number 100a; + 10y + 2, it follows that the number is divisible by 3 if x + y + z is so divisible. 6. The records of a business show gross sales of $1,250,000 for the year. The cost of doing business for the year was 1200,000 and the net profit was |300,000. Assuming that the same ratios are maintained, what must be the selling price of articles that cost |300 to cover the cost of doing business and profit? 7. Derive a one-day rule for interest at 3|^%. Use this rule to find the interest on $8400 for 24 da. at 3|%. 8. If stocks of par value flOO are bought $1.75 below par and sold |2.50 above par, how much is made on, each share ? Represent the process algebraically. 9. Calculate the selling price and profit on an article if the cost of doing business is 18% of the selling price, the profit is 45% of the selling price, and the proportional cost of doing business is |3.60 for this article. 1111 10. Solve the equation -. - = -, and ,,,, 1, x—l x—% x — i x—o check the result. CHAPTER XI EQUATIONS APPLIED TO COMMERCE 149. Net Cost. The original cost of goods before freight, commission, cartage, insurance, and other overhead charges (overhead, or burden) are added, is called the net cost. Before considering further commercial problems it is necessary to define a few terms that are in common use in business. 150. Gross Cost.. The net cost of goods plus all overhead charges to the time of sale is called the gross cost. 151. List Price. The price of goods as given in a printed list or catalogue is called the list price. Retail dealers usually mark their goods instead of issuing a catalogue. The term marked price has the same meaning as the term list price. ■ 152. Net Price. The price at which goods are actually- sold, after all discounts are deducted, is called the net price. 153. Profit. The difference between the gross cost and the net selling price of goods is called the profit. If the profit is a negative number it becomes a loss. 154. Illustrative Problem. A man loses 15% on the gross cost of his house by selling it for |6290. What was the gross cost of the house ? If C is the gross cost, the selling price is C — 0.15 C, or 0.85 C Then since 0.85 C = the selling price, we have 0.85 C= 6290. Dividing by 0.85, C = 7400. Hence the gross cost of the house was |7400. 189 190 EQUATIONS APPLIED TO COMMERCE Exercise 121. Commercial Problems 1. I gain 12% on an investment of #2000. Write a.formula for finding the gain and the selling price, and then find each. 2. A man began business Jan. 1, investing $12,000. On Dec. 31 his resources and liabilities were as follows : Resources LlABIHTIKS Cash on hand 3 500 Accounts payable 1 200 Merchandise 7 200 Notes payable 1 100 Real estate 3 000 Fixtures 500 Accounts receivable 4 200 < Find his rate of gaijl or loss, based on his investment of $12,000. Write a formula for the general case. 3. The annual report of a dairy company for a recent year shows the following facts relating to the output of butter for each of its four plants: Plant Average Cost per Pound Average Selling Price PER Pound Total Out- put FOR Year IN Pounds Annual Profit Kate of Profit PER Pound A B C D 18^ 20 22 21 25 (^ 30 35 32 250,000 180,000 300,000 100,000 Tota Is Find the rate of profit per pound for each plant and enter it in the last column ; find the output and annual profit, and from these find the rate of profit on the cost and enter this in the last column. Write a formula for the general case. COMMEKCIAL PROBLEMS 191 Total Cost Rate of Gain OK Loss Gain or Loss c r c I r 9 15200 tsod In the following table use C for total cost, including overhead charges, r for rate of gain or loss, g for gain, and I for loss, and fill the vacant places : 4. 5. 6. 7. 8. A dealer sold some goods for S dollars and received r per cent commission. Find the amount of his commission. Find the amount ii S = 725 and r = 31. 9. A merchant bought a bill of goods for G dollars and received a discount of d per ceiit. Find the net cost of the goods. Find the net cost if (? = 7500 and d = Q. 10. If I gain g dollars on a set of furniture that cost me C dollars, including all overhead, what per cent do I gain ? Apply the formula to a problem involving numbers. 11. If I gain c cents on a pair of ovei^shoes that cost me S dollars, including all overhead, what per cent do I gain ? Apply the formula to a problem involving numbers. Notice the difference in denominations in Exs. 10 and 11. 12. If iV foot pounds of power are supplied to a machine, and only n foot pounds are used effectively, what is the rate of efficiency of the machine ? The teacher should explain the meaning of " foot pound " if neces- sary, or refer the pupils to the dictionary. 13. Solve Ex. 12 for i\r=400, n= 350. 192 EQUATIONS APPLIED TO COMMEECE 155. Trade Discount. It is a general custom that dealers, jobbers, and merchants who buy directly from the manu- facturers or wholesalers are allowed a discount from the catalogue or hst prices of goods. Such a discount allowed to the trade is called a trade discount. 156. Discount Series. If large quantities are ordered, a further discount from the first net price is often allowed, then a 'discount from the second net price, and so on, this resulting in a chain of discounts, or a discount series. 157. Terms. The conditions of payment for a bill of goods are called the terms, and these are usually stated on the bill. Thus a bill on which 60 da. is allowed for payment but which may be discounted for cash, say at 2 % if paid within 10 da., will have at the top the following: "Terms: 2% 10, net 60," or "Terms : 2/10, N/60." An allowance made on a bill because it is paid before the date of the bill is called anticipation. A bill of goods maybe shipped in July for the holiday trade, simply to relieve the factory, and be accompanied by the bill dated Nov. 1, terms 2/10, N/30. This means that the buyer has a discount of 2% if he pays within 10 da. from Nov. 1, and must pay within 30 da. there- from. But the buyer may anticipate the payment by sending the money any time before Nov. 1, and then the anticipation will reduce the bill. The usual anticipation is from ^ % a month to 1 % a month. 158. Discount Formulas. If the list price is L and there is a single rate of discount r%, N=L~r%L =L(l-r%). If there are two discounts, r^^ and r^%, we have iV=i(l-r-i%)(l— r2%), and so on. Since i(l- ri%) (l-r2%)=L(l-r^%) (l-r^^), it makes no difference which discount is taken first. Discard in the last result any fraction less than J^. TRADE DISCOUNT 193 Exercise 122. Trade Discount 1. If the list price is L and there is a chain of discounts ^■1%' ^'a^' ^3%' what is the net price? 2. Apply Ex. 1 when L = ?50, r^ = 6, r^ = 4, r^ = 3. 3. In Ex. 1 show that the result is the same whatever the order of the discounts. 4. Apply Ex. 3 to the case given in Ex. 2. 5. Find the net price of a book hsted at h with discounts of mofo, w%. 6. Apply Ex. 5 to the case of a book listed at 80^ with discounts of 20%, 5%, or, as it is usually written, with dis- counts of 20, 5. 7. Apply Ex. 1 to a bill for a stove listed at |30 with the chain of discounts 20, 10, 5. 8. Apply Ex. 1 to a bill for the following: 24 boxes oranges @ $5.70, discounts 20, 5, 5. 9. The Conover Hardware Company sold the following bill of goods : 2 doz. locks @ |18 less 20, 10, 5 ; li doz. locks @ $20 less 20, 5 ; 5 doz. 2^" bolts @ |1.50 less 30, 5. Find the amount of the bill. 2 doz. locks @ |18 means that 2 doz. locks have been sold at f 18 per dozen, and similarly for other cases of this kind. 10. The American Woolen Company bought the following : 1500 yd. lining @ $0.35 less 10, 10, 5 ; 850 yd. worsted suitings @ tl-70 less 20, 10, 10; 500 yd. woolens @ $1.50 less 10, 10. Find the amount of the bill. H. The National Market Company receives two bids : one offers to supply 1200 lb. butter @ $0.30 less 20, 10 ; the other 1200 lb. @ $0.31 less 10, 10, 10 for the same quality of butter. Which is the more advantageous for the company ? 194 EQUATIONS APPLIED TO COMMERCE 159. Net-Cost-Rate Factor. As explained in § 158, the net price in the case of a chain of two discounts is found by the formula -.^ ^ ,^ ^ s ^^ ^ ^ and in the case of a chain of three discounts by the formula and so on, whatever the number of discounts in the chain. But evidently it would be much easier if we had a single number by which to multiply L, instead of multiplying successively by several numbers. For example, if the list price is flOOO, and the rates of discount are 6%, 5%, we have 1 — 0.06 = 0.94 and 1 - 0.05 = 0.95, so that we must multiply $1000 by 0.94, and the result by 0.95 to find the net price. It would be easier, however, if we knew the product of 0.94 and 0.95 and could multiply by that. Now this product is 0.893, and, if we knew this, the operation would be very simple. The single multiplier which is equal to the product of all the multipliers of the form 1 — r% used in a chain of dis- counts is called the net-cost-rate factor. That is, in the case of the chain 8 %, 6 %, 4 %, the net-cost-rate factor is (1 - 0.08) (1 - 0.06) (1 - 0.04), or 0.92 x 0.94 x 0.96, which is equal to 0.830208. Using this particular net-cost-rate factor we have to make only three different multiplications, by 8, 3, and 2. Because of the convenience in using the net-cost-rate factor, salesmen usually prepare tables based on the rates which they ordinarily use in their computations. Such a table is shown on pages 195 and 196. It would be mathematically simpler to quote a single dis- count in each case, obviating the necessity for such tables, and the inertia of custom is responsible in large measure for the use of the present plan. NET-COST-EATE FACTORS 195 160. Use of Table. The use of the following table of net- cost-rate factors will be understood from a single example. If we are finding the net cost of goods listed at |85 and subject to the chain discount 121-, 10, 5, we look in the table under 12 J, the largest factor, and opposite 10, 5, and find 0.74813 as the net-cost-rate factor. We then multiply $85 by 0.7481, or practically 0.7481 by 85, only four decimals being necessary when the list price is less than flOO, and we have $63.59 as the net price. 0.7481 85 3 7405 59 848 ■ , 63.5885 163.59 Table oi' Nbt-Cost-Eate Eactoes Rate % 6 10 12^ 16f 20 25 33J Net 95 90 875 83333 80 75 66667 n 92625 8775 85313 8125 78 73125 65 5 9025 855 83125 79187 76 7125 63333 o 2i 87994 83363 81047 77188 741 69469 6175 .5 5 85738 81225 78969 75208 722 67688 60167 ■', 2i 83594 79194 76995 73328 70395 65995 58663 10 81 • 7875 75 72 675 6 10 2i 78975 76781 73125 .702 65813 585 10 5 7695 74813 7125 684 64125 57 10 5 2i 75026 72942 69469 6669 62522 55575 10 10 729 70875 675 648 6075 54 10 10 5 69255 67331 64125 6156 57713 513 20 5 608 57 50667 20 10 5 5472 513 456 In the table decimal points are to be inserted as needed. 196 EQUATIONS APPLIED TO COMMERCE The table on page 195 is here continued for the cases of 35%, 37^%, 45%, 50%, 66|%, 7.5%, and 80%. Rate % 35 37i 45 50 66f 75 80 Net 65 625 55 50- 33333 25 20 2i 63375 60938 53625 4875 325 24375 195 5 6175 59375 5225 475 31667 2375 19 5 2i 60206 57891 50944 46313 30875 23156 18525 5 5 58663 56406 49638 45125 30083 22563 1805 5 5 2i 57196 54996 48397 43997 29331 21998 17599 10 585 5625 495 45 3 225 18 10 2i 57038 54844 48263 43875 2925 21938 1755 10 5 55575 53438 47025 4275 285 21375 171 10 5 2i 54186 52102 45849 41681 27788 20841 16673 10 10 5265 50625 44 5o 405 27 2025 162 10 10 2J 51334 49359 43437 39488 26325 19744" 15795 10 10 5 50018 48094 42323 38475 2565 19238 1539 10 10 5 2^ 48767 46891 41264 37513 25009 18757 15005 20 52 5 44 4 26667 2 16 20 5 494 475 418 38 25333 19 152 20 10 468 45 396 36 24 18 144 20 10 5 4446 4275 3762 342 228 171 1368 20 10 10 5 40014 38475 33858 3078 2052 1539 12312 Of course it will be understood that this particular table is merely illustrative. For example, in some lines of business it would be neces- sary to have the table so arranged as to allow for the discounts 4, 2, or 6, 4, 2, or some othsT chain that would be used frequently. Such a table is easily made to cover all discounts that a business house ever uses, and this is what is actually done by the bookkeeper or salesman. NET-COST-EATE EAPTOES 197 Exercise 123. Net-Cost-Rate Factors 1. Using the table on pages 195 and 196, find the net amount of a bill for $92.50 less 10, 5, 21. The chain of discounts is really 10 %, 5 %, 2 J %, but business men have no time for unnecessary symbols. 2. In the table no factor is given under 5 and opposite 10. Why is this ? If the factor were given, what would it be ? Find the net amount of a bill for |75 less 5, 10. 3. In the table on page 195 certain spaces are left blank. After finding what spaces should be filled, compute and in- sert the necessary numbers. 4. In a certain business it is necessary to have the major rates as given at the top of the preceding table, and to have the factors for the minor rates 7^, 5. Compute these factors. Using the table, find the net amounts of the following hills : 5. 300 bags coffee @ |30 less "331, 10, 5 ; 50 bags coffee @ |40 less 25, 10, 10 ; 80 bags coffee @ |45 less 45, 10, 5. 6. 9 bbl. sugar XXX @ $12 less 5, 2i ; 10 bbl. sugar XX @ 111 less 5, 5. 7. A dealer remembers that the net price of an article was $2.70 after a discount of 25, 10 had been allowed, but he cannot find the list price. What was it? 8. A dealer buys galvanized iron pails in lots of 100, thereby being allowed discounts of 45, 20, 5 from the list price of 20^ each. If he sells them at the list price, what is his rate of profit on the net cost? 9. On a bill of goods listed at $2875 with discounts 25, 10, 10, 5, what difference does it make in the net price if you take the factor to four decimal places instead of five? How is it with a bill of $250 ? with a bill of $75 ? 198 EQUATIONS APPLIED TO COMMERCE 161. Fixing Retail Prices. Until recently, profits were computed as a certain per cent of the cost of the goods. As business became more complicated, the profits were computed on the cost plus the overhead charges, and this is still the common method in small lines of business. When salesmen or agents are employed to sell goods, however, their commis- sion is some per cent of the selling price, and this enters into the overhead charges, overhead, burden, or cost of doing business. In a large establishment, therefore, it is more logi- cal to estimate the cost of the goods, the overhead charges, and the profit, each as a per cent of the selling price. By keeping cai-eful recol-ds of their business, including all the kinds of overhead charges, wholesale dealers are able so to standardize their expenses as to determine what per cent of the selling price should be allowed to overhead charges to realize the expected profit, and to compute this profit on the selUnff price instead of on the gross cost. A dealer paid $24.50 for an article, plus 75^ for cartage. The cost of doing business has been found to be 18% of the selling price, and the profit is to be 10% of the selling price. Find the selling price. We have cost = |24.50 + $0.75 = $25.25. Let X = selling price. , Then 0.18 x -t- 0.10 x = 0.28 x, overhead + profit. Hence x — 0.28 x = $25.25, or 0.72 X = $25.25 ; whence x = $25.25 h- 0.72 = $35.07. From 100% subtract the sum of the rate of profit and the rate allowed to overhead, both based on the selling price as found from experience. Divide the cost of the goods by this remainder, and the quotient is the selling price. COPACTOES 199 162. Cofactor. The quotient of 1 divided by any given number is called the cofactor (or reciprocal^ of that number. We can simplify the solution given on page 198 if we know the cofactor of 0.72, because it is easier to multiply by this cofactor than to divide by 0.72. That is, it is easier to multiply by 0.04 than to divide by 25, and the result is the same because x ^ 25 = x ■ i^g = x • ^^t = 0.04 x. Similarly, it is much easier to multiply by 0.3183, Vhich is equal to -, than to divide by 3.1416. 3.1416 ' It is therefore desirable to have a table of the cofactors of numbers which we have frequent occasion to use in fixing prices. With such a table the prices may be quickly fixed, and business houses do well to recognize this fact. The average of overhead charges for- a series of years fixes this as a standard in any given establishment, at least for the time being. This and the rate of profit determine the market price. A house which can cut the per cent of overhead charges by efficiency devices and checks can either increase the per cent of profit or under- sell its competitors. Hence such a device as the cofactor table, which saves a great deal of time, has a high commercial value. A single example will illustrate the advantage of using cofactors in ascertain- ing the selling price. A house whose average overhead is 17% and rate of profit 12% buys some goods for $1775. At what price must it sell the goods ? Looking at the table on page 200, in the horizontal line to the right of 17% and under 12%, the cofactor 1.4085 is found. Multiplying #1775 by 1.4085 we obtain $2500.09 as the selling price, allowing 29% of it to cover overhead and profit. This is close enough for work with such a table. The house will naturally charge |2500, which is the exact result. 1.4085 1775 7 0425 98 595 985 95 1408 5 2500.0875 $2500.09 200 EQUATIONS APPLIED TO COMMERCE Selling-Pkicb Cofactok Table Overhead Pek Cent of Profit Desired 1% 2%- 3% 4% • 5% 6% 7% 8% 15% 1.1905 1.2048 1.2195 1.2346 1.25 1.2658 1.2821 1.2987 16% 1.2048 1.2195 1.2346 1.25 1.2658 1.2821 1.2987 1.3158 17% 1.2195 1.2346 1.25 1.2668 1.2821 1.2987 1.3158 1.3333 18% 1.2346 1.25 1.2658 1.2821 1.2987 1.3158 1.3333 1.3514 19% 1.25 1.2658 1.2821 1.2987 1.3158 1.3333 1.3514 1.3699 20% 1.2658 1.2821 1.2987 1.3158 1.3333 1.3514 1.3699 1.3889 21% 1.2821 1.2987 1,3158 1.3833 1.3514 1.3699 1.3889 1.4085 22% 1.2987 1.3158 1.3333 1.3514 1.3699 1.3889 1.4085 1.4286 23% 1.3158 1.3333 1.3514 1.3699 1.3889 1.4085 1.4286 1.4493 24% 1.3333 1.S514 1.3699 1.3889 1.4085 1.4286 1.4493 1.4706 25% 1.3514 1.3699 1.3889 1.4085 1.4286 1.4493 1.4706 1.4925 9% 10% 11% 12% 13% 14% 15% 20% 15% 1.3158 1.3333 1.3514 1.3699 1.3889 1.4085 1.4286 1.5385 16% 1.3333 1.3514 1.3699 1.3889 1.4085 1.4286 1.4493 1.5625 17% 1.3514 1.3699 1.3889 1.4085 1.4286 1.4493 1.4706 1.5873 18% 1.3699 1.3889 1.4085 1.4286 1.4493 1.4706 1.4925 1.6129 19% 1.3889 1.4085 1.4286 1.4493 1.4706 1.4925 1.5152 1.6393 20% 1.4085 1.4286 1.4493 1.4706 1.4925 1.5152 1.5385 1.6667 21% 1.4286 1.4493 1.4706 1.4925 1.5152 1.5385 1.5625 1.6949 22% 1.4493 1.4706 1.4925 1.5152 1.5385 1.5625 1.5873 1.7241 23% 1.4706 1.4925 1.5152 1.5385 1.5625 1.5873 1.6129 1.7544 24% 1.4925 1..5152 1.5385 1.5625 1.5873 1.6129 1.6393 1.7857 25% 1.515a 1.5385 1.5625 1.5873 1.6129 1.6393 1.6667 1.8182 COFACTOES 201 Exercise 124. Cofactor Table 1. Calculate for the table on page 200 two additional columns headed 25% and 30% respectively. 2. Using the table, determine the selling price of goods costing $300 plus |21 for freight and |5 for cartage, the overhead being 18% and the profit 15% of the selling price. 3. J. B. Price and Co. buy some wagons for children, listed at $2.20 each, discounted at 20, 10, with total freight charge of $8. Their cost of doing business is 15%, and their profit basis is 12%. Using the chain-discount and cofactor tables, find the price at which they must mark each wagon. Fill out the last column in the following table : Cost op Goods Oyekhead Profit Selling Price 4. 1245.50 21% 8% 5. $530.75 18% 8% 6. $945 25% 20% 7. $500 20% 15% 8. $475.50 19% 13% ' 9. The Conover Company buys wagons costing $140. Their cost of doing busmess is on a 12% basis, and they make a profit of 20% on the selling price. At what price must they sell the wagons? 10. The Eureka Shoe Co. handles shoes which cost them $2.20 a pair; Their cost of doing business is on a 12^% basis, and their profit is down to a 21-% basis. At what price must they sell the shoes ? 202 EQUATIONS APPLIED TO COMMEECE 163. Commission. From his work in arithmetic the student is more or less familiar with commission. Much new light, however, is thrown on the subject, and in a very practicsil way, by algebra. Indeed, it is by means of algebra that we obtain a general view of many modern business rules. A sum charged by one person for transacting business .ior another is called commission, or brokerage. The person for whom the business is transacted is called the principal, and the agent is called a commission merchant, broker, collector, ot factor. In actual business there is a distinction in these terms, but for our present purposes they may be used interchangeably. The mathematics of the case is the same whichever term is used. Commission, or brokerage, is charged on the basis of the entire volume of business transacted. It is usually a certain per cent of the cost when a -purchase is made, and of the gross proceeds when a sale is made. . " In the case of stocks and bonds the brokerage is usually at the rate of f 1.25 on every flOOO of par value. Merchandise sent by a principal to his agent is called a shipment, but when the agent receives it he speaks of it as a consignment, and frequently the agent is called the consignee of the merchandise. An itemized statement of a sale of merchandise, showing the gross proceeds, charges, commission, and similar items, together with the balance to be remitted to the principal, is called an account sales. The balance to be remitted is called the net proceeds. An itemized statement showing the prime cost, charges, and gross cost is callpd an account purchase. Definitions of this kind are not to be learned, but they should be read understandingly and the terms used correctly. FORMULAS IN COMMISSION 203 164. Formulas in Commission. From the explanation given on page 202, it is evident that, if P represents the gross proceeds, C the prime cost, c the commission, and r the rate of commission, we have the following: Pr = c, c^ P = r, c -^r = P, Cr=c, i: -i- C = r, c-i-r= C. The student should tell how each of these formulas is obtained, and should understand clearly the significance of each. The formulas in the upper row apply to sales, those in the lower row to purchases. Exercise 125. Formulas in Commission Examples 1 to 4, oral 1. Given that Pr = c, find o when P = $200, r = 4% 2. Given that c-i-C=r, find r when o= $7.50, C = find C when c = $120, r = S%. Find the missiifig nuTnbers in the following : 3. 4. 5. 6. 7. 8. Gkoss Proceeds Eate op CoMMissiosr Commissio;n or Brokerage Net Proceeds $1000 2% $1000 $20 $8500 H% $1200 16f% 2% $40 $1500 $1200 5% $85 204 EQUATIONS APPLIED TO COMMERCE 10. Find the commission and net proceeds in the following-: New York dwcf. 27, 1920 SotD FOE THE Account of j., ?n. S^LoAeAj, ycyyiJcEA^, cA. "If. By}. L. King Co. 1(^20 3 /2 20 60 {>St. a/wifOAj 8 60 Ul. mocfa,v 8.- /OO {My. QAi^OAf ^.- Chaeges ^cym/m-Lo/QAycyyv, 6% ^00 6/0 fo on notes for the following amounts: 28. $2500, for 60 da. 30. $600, for 1 mo. 10 da. 29. $3500, for 90 da. 31. $850, for 2 mo. 15 da. ONE-DAY METHOD IN INTEREST 211 169. One-Day Method in Interest. In large city banks the discount rate varies from day to day. The discount clerks derive a short rule for fiiiding the interest for one day at the announced rate and multiply this result by the number of days to find their discount factor. 1. Derive a one-day rule for finding discount at 41%. Since i =prt, r = y^^jj, and t = ^^^, we see that .^ J ]_^ _J_ ■^ ' 200 ' ^^P -^ ' 8000 ■ 40 Hence we have this rule : To find the interest for 1 da. at 4j%, point off three places and divide by 8. 2. Find the interest on |94,600 for 24 da. at 4J%. Since, in this case, 24 da. happens to be 3 x 8 da., we may simply take 8 x $94.60, which is equal to |283.80. 3. Extend the rule found in Ex. 1 for 8 da. and 80 da. For 8 da. the formula evidently becomes ^ =P ■ s^B^xr ■ sf (J =P ■ ttsVt' and for 80 da. i= p • ^^5. Hence we have this rvde : To find the interest at 4j% for 8 da., point off three places ; for 80 da., point off two places. 4. Derive a one-day rule for finding discount at 4%. Evidently « = i' ■ t Jtt ■ ffi-o = ^ • s-^tss- Hence we have this rule: To find the interest for 1 da. at 4%, point off three places and divide hy 9. These abridged rules for finding interest are now evident : For 1 da. at 4^%, point off three places and divide hy 8. For 8 da. at 4j%, point off three places. For 1 da. at 4%, point off three places and divide hy 9. For 9 da. at 4'^o, point off three places. 212 EQUATIONS APPLIED TO COMMEECB Exercise 128. One-Day Method Find the interest at 4^% on the following : 1. t72,000, 1 da. 4. $65,000, 2 da. 7. *960, 6 da. 2. 185,400, 1 da. 5. |72,000, 4 da. 8. $7240, 16 da. 3. 190,000, 1 da. 6. $72,400, 8 da. 9. |45,700, 28 da. 10. Derive a formula for finding iaterest for 1 da. at 6%, and from this write out a one-day rule. 11. Consider Ex. 10 at 3% and also at 2%. 12. Derive a formula for finding interest for 1 da. at 5%, and write out the rule. Determine whether it is easier to use this rule or to use the Sis Per Cent Method and deduct \. 13. Show how to use the Six Per Cent Method when the" time exceeds 1 3^. 14. If y is the number of years, m the number of months, and d the number of days, show that when r = 6%, ^•=px0.06(J/ + J2..m + ^l^c^) =f (0.06 y + 0.005 m + 0.000^ c?), and from this formula derive a rule for finding interest at 6% for a given number of years, months, and days. In solving Exs. 15-20 use the rule obtained in Ex. 14. Find the interest at 6 "jo on the following : 15. $5200, 2 yr. 4 mo. 18. $275, 2 yr. 3 mo. 15 da. 16. $6000, 3 yr. 1 mo. 19. $150, 3 yr. 1 mo. 18 da. 17. $5500, 1 yr. 3 mo. 20. $350, 1 yr. 3 mo. 10 da. 21. Find the interest on $950 from 5/20/19 to 11/24/22, at 4%. Business men commonly write dates as in Ex. 21, where 5/20/19 means May 20, 1919, and 11/24/22 means Nov. 24, 1922. SIMPLE INTEREST TABLES 213 170. Simple Interest Tables. As an efficiency device a bank uses interest tables except in cases where the Six Per Cent Method or the One-Day Method is easier. The interest tables most commonly used are based on 360 da. to the year. These tables are generally used by banks in computing interest (discount) on notes payable to them, but banks often use tables based on 365 da. to the year in computing interest paid by them on accounts of their depositors, as stated on page 80. This plan gives the banks an appreciable advantage on the large sums handled by them. The following shows part of a page of an interest table at 6 % : 3 Months, 6% Total Days 1000 2000 3000 4000 5000 6000 7000 8000 9000 Days OVER 3 Months 90 15.00 30.00 45.00 60.00 75.00 90.00 105.00 120.00 135.00 91 15.17 30.33 45.50 60.67 75.83 91.00 106.17 121.33 136.50 1 92 15.33 30.67 46.00 61.33 76.67 92.00 107.33 122.67 138.00 2 93 15.50 31.00 46.50 62.00 77.50 93.00 108.50 124.00 139.50 3 94 15.67 31.33 47.00 62.67 78.33 94.00 109.67 125.33 141.00 4 95 15.83 31.67 47.50 68.33 79.17 95.00 110.83 126.67 142.50 5 For example, find from the table the interest on |2750 for 94 da. at 6%. Interest on |2000 = |81.33 Interest on 700 = 10.97 Interest on 50 = .78 Interest on $2760 = f48.08 It is not practicable to give, in a book like this, a full interest table. It is better to show part of a real table, as above, than to give a table that is not of the form generally used in large banks. 214 EQUATIONS APPLIED TO COMMERCE Exercise 129. Interest Tables Using the table on page 213, find the interest at 6% on: 1. 13500, 90 da. 4. f 5500, 90 da. 7. $6500, 91 da. 2. $3650, 90 da. 5. $4250, 90 da. 8. $3700, 92 da. 3. $4500, 90 da. 6. $8000, 90 da. 9. $5750, 93 da. 10. Make a 6% interest table for 60, 61, 62, and 63 da. similar to the table on page 213. From the table made in Ex. 10, find the interest at 6'fo on: 11. $750, 60 da. 14. $3700, 60 da. 17. $2750, 61 da. 12. $825, 60 da. 15. $4500, 60 da. 18. $3500, 62 da. 13. $975, 60 da. 16. $7500, 60 da. 19. $5000, 63 da. 20. Make a 6% interest table for 30, 31, 32, and 33 da. similar to the table on page 213. From the table made in Ex. W, find the interest at G^ on: 21. $250, 30 da. 24. $2500, 30 da. 27. $4300, 31 da. 22. $750, 30 da. 25. $3750, 30 da. 28. $5500, 32 da. 23. $825, 30 da. 26. $8500, 30 da. 29. $7500, 33 da. 30. Make a 5% interest table similar to that on page 213. From the table made in Ex. 30, find the interest at 5% on : 31. $375, 90 da. 34. $2000, 90 da. 37. $7250, 90 da. 32. $750, 90 da. 35. $2500, 91 da. 38. $4750, 92 da. 33. $825, 90 da. 36. $2750, 92 da. 39. $5250, 93 da. 40. A depositor allowed $1750 to remain in a bank of deposit, without interest, from July 1 to the following Jan. 1. How much interest would he have gained had he transferred it on July 1 to a savings bank paying 3^%? DIFFERENCE OF TIME 215 171. Table of Difference of Time. Bankers always find the difference in time between two dates by means of tables. The following is one form of the table of difference of time : Jan. Feb. Mah. Al>K. May JnNE July Aug. Sept. Oct. Nov. Dec. Januaiy . 365 31 59 90 120 151 181 212 243 273 304 334 February 334 365 28 59 89 121) 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 122 153 181 212 242 273 308 334 365 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 The exact number of days from any day of any month to the corresponding day of any month within a year is found in the table opposite the first month and under the second. For example, from Aug. 7 to Dec. 7 is 122 da. ; from Nov. 16 to Apr. 23 is 158 da., since to Apr. 16 it is 151 da., and to Apr. 23 it is 7 da. more, or 158 da. in all. Exercise 130. Difference of Time All work oral From the table state the number of days from : 1. May 7 to Aug. 7. 2. Apr. 6 to Sept. 6. 3. Jan. 5 to Aug. 20. 4. Feb. 7 to Sept. 15. 5. July 4 to Dec. 25. 6. Apr. 15 to Oct. 20. 7. Nov. 16 to July 17. 8. Sept. 20 to Mar. 25. 9. Oct. 21 to Feb. 20. 10. Nov. 25 to Apr. 15. 11. Aug. 27 to Feb. 5. 12. Aug. 19 to Jan. 7. 216 EQUATIONS APPLIED TO COMMERCE 172. Exact Interest. Since interest at 5% means that 5% of the principal is to be paid for its use for 1 yr., or 365 da., to charge 5% for only 360 da. is to charge more than 5% for a year. Nevertheless, for convenience in computing in- terest without tables, 360 da. is commonly taken as 1 yr. For example, required the exact interest on |1241 from Dec. 21 to Feb. 9 at 5%. The time, found by the table on page 215, is 50 da. Then the interest is /A X 5% of 11241. Canceling, !iiiiiLm = 111 = $8.50. n 2 The relation of common to exact interest is discussed on page 217. Exercise 131. Exact Interest Wind the exact irvterest on the following : 1. $4000 for 30 da. at 6%. 2. t2500 for 60 da. at 5%. 3. $3250 from Apr. 2 to July 1 at 5%. 4. 14500 from May 7 to June 6 at 5%. 5. $4250 from Jan. 6 to Feb. 2 at 6%. 6. 14600 from Feb. 9 to Apr. 10 at 6%. 7. 15000 from May 15 to July 14 at 6%. 8. 137,500 from Aug. 14 to Nov. 12 at 5%. 9. $45,750 from Sept. 27 to Dec. 26 at 5%. 10. $75,000 from Dec. 6 to Feb. 4 at 5%. 11. $125,000 from May 2 to May 24 at 4|%. 12. $175,000 from June 3 to July 8 at 5%. 13. $250,000 from Aug. 4 to Aug. 17 at 5%. EXACT INTEEEST 217 173. Exact and Common Interest. Having now considered both exact interest at 365 da. to 1 yr. and common interest at 360 da. to 1 yr., we may discuss algebraically the relation between exact interest and common interest. The common interest for 1 da. is evidently found by the formula » — nr . i while the formula for exact interest for 1 da., say i', is Dividing the first formula by the second, member by member, we have i _ pr • 3^0 _ 1 365_73 i'-p^._i^"360^ 1 ~72"' Therefore, by multiplying both sides by i', we have That is, the common interest is equal to the exact interest increased by ^ of itself. Similarly, i' = ^^ i. That is, the exact interest is equal to the common interest decreased by j^ of itself. Hence to find the exact interest on any sum, we may first find the common interest by any one of the standard methods and then diminish it by ^^ of itself. For example, find the exact interest to be added to taxes which were due not later than Dec. 2 and were paid Dec. 24, the taxes being |850 and the penalty 10% per annum. We may consider the penalty the same as interest at 10 %. We then have i =prt = |850 x 0.10 x ^Yb = $5-12. If we have only interest tables based on 360 da. to the year, we may find that the common interest is $5,194, which, diminished by ^ of itself, amounts to |5.12. 218 EQUATIONS APPLIED TO COMMERCE Exercise 132. Exact and Common Interest FiTid the difference between the exact interest and the common interest on $27,500 and on $126,000, for these times and rates : 1. 60 da., 6%. 3. 30 da., 5%. 5. 15 da., 4|%. 2. 90 da., 6%. 4. 2 da., 21%. 6. 18 da., 51%. 7. A man delayed paying his taxes on property assessed at $92,500, the rate being 0.023, and as a result was required to pay interest on the taxes due for 37 da. at 6%. Find the total amount which the man must pay. Cities usually calculate all interest charges by the exact method. 8. The sum of daily balances on BV deposit account for July was $225,400. He is allowed 1 day's exact interest on this total at the rate of 3% per annum. Calculate this interest by finding the common interest by the One-Day Method and then using the formula, i' = ^^ i. 9. Our government, in settling a claim for |3500, paid exact interest for 90 da. at 6%. How much more interest would have been paid if it had allowed common interest? 10. By how much does the common interest exceed the exact interest on $3000 at 4^% from July 1 to Sept. 10 ? 11. Give a simple method for changing the common inter- est on d dollars for 73 m days to the exact interest on d dollars for the same time at the same rate. 12. Explain how the formula i = |^| i' or the formula i = i' -\- yL i' makes it easy to calculate the common interest on the same sum and at the same rate for 12 da., 18 da., 24 da., 36 da., 48 da., 72 da., 144 da., and so on, when the exact interest is known. 13. Find the common interest and the exact interest on $900 for 120 da. at 4^%. REVIEW 219 Exercise 133; Review of Chapter XI 1. The Lasher Company bought 2500 yd. of linings, hsted at 30^ per yard, discounts 20, 10, 5. They sold the goods so as to cover overhead of 18% of the selling price, and a profit of 20% of the selling price. Find the selling price of the 2500 yd. and the price per yard. 2. G. K. George discounted his 90-day note for $10,000 at a bank at 4^%. With the proceeds he purchased cloth at |2 per yard, investing all but $7.50. During the 90 da. he disposed of all but 250 yd. of the cloth at an average price of 1 3. 25, depositing the proceeds of the sales in the bank. At the end of the 90 da. the bank toojc $10,000 from the amount deposited by Mr. George to meet the note. How much was his profit for the 90 da. ? 3. If the gross cost of goods is c, the cost of doing busi- ness d per cent of the selling price x, and the profit p per cent of X, derive a formula for x. 4. Using the formula a=p(l + rt'), find the value of p and state the rule for finding the principal which will amount to a given sum at a given rate in a given time. 5. Applying the formula derived in Ex. 4, find what principal invested at 6% will amount to $8000.23 in 90 da. 6. Since a=p+prt, it is possible to find r when a, p, and t are given. Solve this equation for r. 7. At what rate will $6000 amount to $6067.50 in 3 mo. ? 8. Solve for t the equation given in Ex. 6. State the rule for finding the time in which a given principal will amount to a given sum at a given rate. 9. Using the formula derived in Ex. 8, find the time, in this case the fractional part of a year, that it will take $5000 to amount to $5037.50 at 41%. Reduce the result to months. 220 EQUATIONS APPLIED TO COMMERCE Exercise 134. Review of Chapters I-XI 1. If I is the length in feet, w the width in inches, t the thickness in inches, and F the nuinber of board feet in a piece of timber, write -the formula for F- 2. Find the value of A^ in the formula Aq = — [(!+»')"— 1] when B = 500, r = 0.04, and n=2. *" 3. If coffee costs c cents per pound by the bag and loses 5% of its weight in roasting, at what price per pound must it be sold to gain 20% on the cost? 4. In Ex. 3 find the selling price when c= 24^. 5. Find the sum of prt, p'rt, p"rt, and p"'rt. 6. An investor has $850, |2500, |7000, and |3500 all at interest at 6%. Calculate ~ the total interest he will receive each half year on these investments. 7. Find the sum of pr^t, pr^t, pr^t, and pr^t. 8. Using the result of Ex. 7 as a formula, find the total interest on the following for 6 mo.: $5000 at 4%, $5000 at 41%, 15000 at 4^%, and $5000 at 5%. 9. Explain the algebraic principle involved iix multiplying as follows : 82 X 78 = 6400 - 4 = 6396. ■ ' 10. Factor the expressions p —prt and (1 + r)^ — 1. 11. Explain how the distributive law, a'(b — c) = ah — ac, affords an easy method for multiplying 784,675 by 999. 12. A discounts at a bank at 6% his 60-day note for $5000 and invests the proceeds in flour listed at $6 per barrel, less. 20, 10. He sells all but 15 bbl. in the 60 da. at |7.50,per barrel. With what gross profit can he credit himself on the transaction, provided he bought an integral number , of bar- rels and those that remained unsold are inventoried at cost ? CHAPTER XII SIMULTANEOUS SIMPLE EQUATIONS 174. Indeterminate Equations. If we have one simple equa- tion containing two unknown quantities, we can iind either unknown quantity in terms of the other. For example, if 2 a; — 3 3^ = 8, we can see that x — —^ — and that 2 a; — 8 y = , but this does not tell us the value of either. ■' 3 Indeed, if ;/ = 1 we see that x = b\; if y = 2, i = 7 ; if ?/ = 4, x= 10, and so on, there being an indefinite number of values of x and y. An equation which has an indefinite number of roots is called an indeterminate equation. An equation that is not indeterminate is said to be determinate. 175. Two Simple Equations. If we have two simple equa- tions containing two unknown quantities, we can usually derive from them a single equation containing only one unknown quantity. For example, suppose that x + y = 17, and a; — y = 9. Adding member for member, 2 a; = 26 ; whence ar = 13. And because x ■¥ y = VJ, we have 13 + ^ = 17 ; whence y = 4- 176. Simultaneous Equations. Two or more equations that have the same values for the unknown quantities are called simultaneous equations. 221 222 SIMULTANEOUS SIMPLE EQUATIONS 177. Elimination. The process by which we cause an un- known quantity to disappear in deriving one equation from a system of equations is called elimination. Thus in § 175 we derived the equation 2 a; = 26 from the system of equations x + y = 17 and a; — y = 9, thereby eliminating y. We might equally well derive the equation 2 y = 8, thereby eliminating x. The most common methods of elimination are (1) by addi- tion (or subtraction, as a special case) and (2) by substitution. 178. Elimination by Addition or Subtraction. 1. Solve the system of equations 2a: + 3y = 27 (1) 5a;-2?/=l (2) Multiplying (1) by 2, 4 a; + JS j^ = 54. Multiplying (2) by 3, 15 a; - 6 3^ = 3. Adding, 19 a; = 57. Dividing by 19, , x = 3. Substituting 3 for x in (1), 6 + 3 ?/ = 27. Subtracting 6, Zy = 21. Dividing by 3, y = 7. Check. Substituting 3 for x, and 7 for y, in (1) and (2), we have 6 + 21 = 27, and 15 - 14 :± 1. Because y was eliminated by adding two equations, member for member, we say that we have eliminated y by addition. 2. Solve the system of equations 3 a; + 2 «^ = 23 (1) 2 a; + 8 «/ = 27 (2) Multiplying (1) by 2, 6 a; + 4 y = 46. (3) Multiplying (2) by 3, 6 a: + 9 2/ = 81. (4) Subtracting (3) from (4), 5 y = 35. Dividing by 5, y = 7. Substituting in (1) or (2), a; = 3. In this solution we have eliminated x by subtraction. APPLICATIONS AND METHODS 223 179. Applications and Methods. Let us now consider a practioal case in which this work might be used. A dairy receives an order for 30 gal. of cream, to contain 25% of butter fat. To cream containing 40% of butter fat the manager wishes to add milk containing 5% of butter fat. How many gallons of each should be taken? We may solve the problem by using only one unknown. Thus, let X = the number of gallons of cream. Then 30 — a; = the number of gallons of milk to be added. From the conditions of the problem, 0.40 X + 0.05 (30 - a;) = 25 % of 80, or 0.40 a; + 1.5 - 0.05 a; = 7.5 ; whence 0.35 a; = 6, and X — m. Therefore, 30-.r = 12f. That is, 17i^ gal. of cream and 12f gal. of milk should be taken. Instead of this we may, if we choose, use two unknown quantities, and this is often the simpler plan. Thus, let X = the number of gallons of cream, and y = the number of gallons of milk. Then x + y = 30, the total number of gallons ordered, (1) and 0.40 x + 0.05 y = 7.5, the number of gallons of butter fat. (2) We now have two equations which we may solve as follows : Multiplying (1) by 0.05, 0.05 a; + 0.05 y = 1.5. (3) Subtracting (3) from (2), 0.35 x = 6; whence x = VI\, and y = 12f . In solving the equations produced by complicated com- mercial problems the second of the above methods, that of simultaneous equations, is usually the easier. 224 SIMULTANEOUS SIMPLE EQUATIONS 180. Directions for Elimination. From the preceding solu- tions we see that in the elimination of an unknown quantity by addition or subtraction we proceed as follows : Multiply both members of the equations by such numbers as will make the coefficients of one of the unknown quantities numerically equal. If these coefficients have opposite signs, add the equations member for member ; if they have the same signs, subtract. Solve the resulting equation. Substitute the result thus found in the simpler of the two given equations and solve for the other unknown quantity. Check the results by substituting in both the given equations. We commonly speak of multiplying the equation, meaning thereby that we multiply both members of the equation. Similarly, we speak of adding equations and subtracting equations, meaning that we add or subtract the equations member for member. Exercise 135. Elimination by Addition or Subtraction Examples 1 to 8, oral 1. Solve for x, 5. Solve for u, x + y = 9 2u + v = 14: x — y=% 3m — v = ll 2. Solve for y, 6. Solve for w, x + 2y = l 2w + 33 = 20 x + y = 4i w — Sz=l 3. Solve for x, 7. Solve for P, Sx + 2y=5 3P+Q = 17 5x-2y = 8 2F-Q = 8 4. Solve for m, 8. Solve for F, 7m + Sn=2S 2F+7G = 21 5m + Sn = 19 3F-7G = U ELIMESrATION BY ADDITION OK SUBTRACTION 225 9. When oats are worth $1.35 a bushel and corn is worth fl.SO-a bushel, how many bushels of each must be taken to make a mixture of 80 bu. worth |1.68 a bushel ? Here the equations are x + y = 80 and 135 x + 180 ?/ = 80 x 168. 10. Two grades of coffee worth 30^ and 38^ a pound respectively are to be combined to make a mixture of 25 lb. worth 35^ a pound. How many pounds of each kind must be taken? Such cases are actually used in large establishments, the question of the blend being considered in deciding upon the proportions. 11. A dairy having on hand 20 gal. of milk containing 4% of butter fat, a supply of milk containing 5% of butter fat, and a supply of cream containing 40%, receives an order for 50 gal. of cream to contain 20 % of butter fat. The super- intendent wishes to dispose of all the 4% milk and to add enough of the 5% milk and 40% cream to make up the 50 gal. ordered. How many gallons of each must he add ? 12. Three grades of wheat worth respectively |2, |2.10, and |2.16 a bushel are to be mixed to form 2400 bu. worth 12.12 a bushel, the number of bushels of $2.16-wheat being twice that of the $2.10-wheat. How many bushels of each kind must be taken ? 13. An importer of teas wishes to mix two grades of tea costing respectively 40 ^ and 50 ^ a pound, so as to have 80 lb. which he can sell at 60^ a pound and still make a profit of 25% on the cost. Assuming that the teas will blend satisfactorily, how many pounds of each must be taken ? 14. Having on hand a supply of alcohol 90% pure, and another supply 96% pure, a wholesale dealer wishes to fill an order for 200 gal. which shall be 95% pure. How many gallons of each kind must he take ? 226 SIMULTANEOUS SIMPLE EQUATIONS 181. Elimination by Substitution. It is often convenient to find from one equation the value of one unknown quantity in terms of the other unknown quantity and to substitute this value in the other equation. 1. Solve the system of equations 3a; + 7y = 22.4 • (1) x-by=Q ' (2) (3) From (2), a; = 6 + 5 y. Substituting in (1), 3 (6 + 5 3^) + 7?/ = 22.4, or 18+15^ + 7^ = 22.4. Subtracting 18, 22 2/ = 4.4. Dividing by 22, y = 0.2. Substituting in (3), x = 'J. Check. 3 X 7 + 7 X 0.2 = 22.4, and 7 - 5 X 0.2 = 6. 2. Solve the system of equations 5 a; + 2 «/ = 34 7x-Zy^l Frbm (1), 34 -5 a; y~ 2 ■ Substituting in (2), ..-s.«-- = . Multiplying by 2, 14 a; -102 +15 3: =14. Adding 102, '29a; =116. Dividing by 29, x = 4. Substituting in (3), 34-5-4 = 17-5-2=7. (1) (2) (3) 3. Solve the system of equations x+Zy=l x-2y = <2, We have x = 7 — %y and x = 2 -\-2y; hence 7 — 3^ = 2 + 2 y, and y = 1. Substituting, we have a; = 4. ELIMINATION BY SUBSTITUTION 227 182. Directions for Elimination. From the work on page 226 we see that in the elimiaation of an unknown quantity by substitution we proceed as follows : From one of the e'qwations find the value of either unknown quantity in terms of the other. Substitute this value in the other equation and solve. Exercise 136. Elimination by Substitution Samples 1 to 6, oral 1. If a; = 3 and x + y = 5, what is the value of y ? 2. If y = 7 and x + y=lQ, what is the value of a; ? 3. If a; = — 2 and x + y=l, what is the value of ^ ? 4. If «/ = — 2 and x + i/=0, what is the value of a; ? 5. If r = 4 and r + r' =9, what is the value of r' ? 6. If r = 7 and r — r' = 5, what is the value of r' ? Solve the following equations by substitution : 7. x + ly=2& 13. 4.5a:-7^ = 7 2a; + 3?/=19 5.5a; + 6y= 59.5 8. a;-3z/ = -l 14. 0.8a; + 0.3 3/ =11.8 3a; + 2«/=19 2 a; + 3.5 1/= 14.5 9. a; + 4 2^ = 35 15. 5m + 7w = 125 3a;-2?/ = 7 7m-n = lB 10. a;- 4 2/ = 1 16. 5^-3^=27 5x + 2y = 49 ~ 7^-3^ = 15 11. a; + 5^ = 25 17. 19a;-j/ = 18 7a; + 32^ = 47 27a; + 4^ = 31 12. a;-2y=2 18. F+7M^26 6x + 5y = S0 F-ldM=-52 228 SIMULTANEOUS SIMPLE EQUATIONS Exercise 137. Simultaneous Simple Equations Solve the following equations : 1. 7x + Sy = 2B 5x+Sy = lQ 2. 10x + B'y = 75 8x-y = iS 3. 11 a; + 21 «/ = 288 16 « - 30 «/ = - 126 7.^ + 1 = 3 X y 5-^=-l X y 8.1^ + ^ = 8 X y 21_12_^ X y 6 3 2 10. M + i = 2 2x y 25_3^^ 2x y 11. 43.2 6.2^4 bx hy 5 ^ + ^ = 2.6 bx by 4. 1.5 P + 1.25 6 = 19 1.2 P + 0.75 6 = 13.2 5. 0.5ilf-0.3Ar=2 0.3 If- 0.4 iV^= 0.1 6. x + ^y^Vl %\x + y=%l "utj first clearing offractiojm 12. 1 1 2x y'"^ 13. a h - + - = c X y X y 14. -4- ax by hx ay 15. a h j- + — = a + h bx ay * + f = «2 + J2 X y 16. a b ^ y ^ + ^=.' ■• - X y PROBLEMS 229 17. If a rectangle were 2 in. longer and 3 in. wider, its area would be increased by 35 sq. in. If the rectangle were 2 in. shorter and 2 in. wider, the area would be unchanged. Find the dimensions. Let I = the number of inches of length, and w = the number of inches of width. Then Iw = the number of square inches of area. Then (I + 2) (m + 3) = Zio + 35, and (I -2)(w + 2) = Iw. Simplifying, 3 Z + 2 w = 29, and I— w = 2. Solving, I = 6.6, and w = 4.6. Therefore the dimensions are 6.6 in. and 4.6 in. Here the initial letters of length and width have been used to repre- sent the unknown quantities, a custom that is coming into use. It is pernlissible to use x and y, or I and w, or any convenient letters. 18. If a rectangle were 5 in. shorter and 1 in. wider, its area would be decreased by 35 sq. in. If it were 5 in. longer and 1 in. -wider, its area would be increased by 65 sq. in. Find the dimensions. 19. The perimeter of a rug is 20 ft., and three times the length plus five times the width is 36 ft. Find the area. 20. The length of a room is 33-^% greater than the width, and the perimeter is 70 ft. Find the dimensions. 21. The width of a room is two thirds the length, and the length exceeds the width by 7 ft. Find the dimensions. 22. The length of a rectangle is 50% greater than the width, and the perimeter is 160 ft. Fiad the dimensions. 23. The altitude of a rectangle is 20% less than the base, and the perimeter is 18 in. Find the dimensions. Chech. 10-2 4 + 4 S'16-1 8+1 1 Exercise 138. Clearing of Fractions ExaTnples 1 to 5, oral \. If - = 3, what is the value of a; ? 1 230 SIMULTANEOUS SIMPLE EQUATIONS 183. Clearing of Fractions. If the equations contain frac- tions, it is usually better to clear of fractions before eliminating one of the unknown quantities. Solve the equations a;.-2 y + 4 ^ ' 2x-5 ^ x-1 2\ 4y-l 2y + l "^ ^ Clearing (1), a;,y + 2y + 4a; + 8=a;2'-2y + 8a;-16. (3) Clearing (2), 4a;y — 10y + 2a; — 5=4a;y — 4y — i + l. (4) Simplifying (3), x — y = Q. ■ Simplifying (4), s — 2 y = 2. Solving, X = 10, and y = 4. 10 + 2 4 + 8 12 20 - 5 10-1 1 X 3 2. If - = 7, what is the value of a; ? X 11 2 3 3. Find the value of x when - = - ; when - = -• X 1 ' X 1 1111 2 4. If — + - = 4 and = 2, what is the value of - ? X y X y ■ , X of i? of a;? of «/? Check the results. X 12 12 2 5. If - + - = 4 and = 0, what is the value of - ? X y X y X of -? of a;? of w? Check the results. ^ X CLEARING OF FRACTIONS 231 Solve the following equations : 3^4 X y _2k> 4^3~T ' 3 5 — =-^ = 2a; + 7 8 £^^y + 4 ■ 2;-3 y+7 a; + 5 _ t/ — 1 a; + 2~y-2 „ 2; + w,2; — « . '• 8 + 6 =^ x+y x-y ^. 4 3 10. 11. 7 a; + 2y 2a; + «/ 7 _ 5 32;-2 Q-y 3x-2_ Sy+7 Sa; — 1 5«^ + 16 Zx-l _ 6y-5 x+5 ~2y+B ,„ 4a; + 5w 12. J-—l = x — y 40 "^ 1x — y 1— 4«/ 13. *.+ «/ 1 ^-y 1 7 2 a; + y x-y 2 8 ^"* 14. a; + 22/ + l „ 2x-y+\ ^x-y+1 J 2;-«/ + 3 15. x+y y-x 3 ' 2 -^ 2 ' 9 16. 7a;-| = 48 5 ^^^ + ^ = 26 17. a;-2/ + l x-y + 1 18. x + y + 2. _ 3 a;-y-3 2 a;-y-3_ ^ a;-«/ + 3 19. a;-j^_ 8 x + y 15 63.-3^,-44^ 232 SIMULTANEOUS SIMPLE EQUATIONS Exercise 139. Miscellaneous Problems 1. When weighed in water, tin loses 0.137 of its weight, and copper 0.112 of its weight. If a 10-pound mass of tin' and copper loses 1.195 lb., find the weight of the tin and the weight of the copper in the mass. Let t = the number of pounds of tin, and c = the number of pounds of copper. Then t + c = 10, (1) and 0.137 1 + 0.112 c = 1.195. (2) Multiplying (1) by 0.112, 0.112 t + 0.112 c = 1.12. (3) Subtracting (3) from (2), 0.025 t = 0.075. Dividing by 0.025, t = 3. Substituting in (1), c = 7. Therefore there are 3 lb. of tin and 7 lb. of copper in the mass. 2. When weighed in water, silver loses 0.095 of its weight, and copper 0.112 of its weight. If a 12-pound mass of silver and copper loses 1.174 lb., find the weight of the silver and the weight of the copper in the mass. 3. Wheii weighed in water, gold loses 0.051 of its weight, and silver 0.095 of its weight. If a 6-ounce piece of gold and silver loses 0.35 oz., find the weight of the gold and the weight of the silver in the piece. 4. When weighed in water, tin loses 0.137 of its weight, and lead 0.089 of its weight. If a 65-pound mass of tin and lead loses 6.025 lb., find the weight of the tin and the weight of the lead in the mass. 5. An iron bar covered with brass weighs 13 lb. When weighed in water, iron loses 0.128 of its weight, and brass 0.119 of its weight. If the bar loses 1.655 lb. when weighed in water, find the weight of the brass that covers the iron. MISCELLANEOUS PROBLEMS 233 6. Twenty-seven coins, dollars and quarters, amount to $19.50. How many are there of each kind? Let d = the number of dollars, and q = the number of quarters. Then d + q = 27, and d + i= 19.50. 4 Explain the second equation and solve. 7. A man distributed |5.25 in dimes and quarters among thirty boys, each boy receiving a coin. How many boys received dimes? How many received quarters? 8. The charge for admission to an entertainment was 50^ for adults and 25 (^ for children. If the proceeds from 125 tickets were $51.25, how many tickets 'for adults and how many tickets for children were bought ? 9. A school gave an entertainment for which the tickets were sold to pupils at 40^ each, and to others at 50^ each. There were 245 tickets sold and the receipts were $108.50. How many of each kind were sold ? 10. A grocer has in his cash drawer 106 bills, some one- dollar and the rest two-dollar bills. The total amount is |138. How many bills has he of each kind? 11. A receiving teller at a bank took in 515 bills, some five-dollar and the rest two-dollar bills. The total amount was $2155. How jmany of each kind did he receive? 12. A paymaster at a shop has 210 silver coins, some quarters and the rest half dollars. The total amount is $75. How many coins has he of each kind ? 13. A dealer has 19 pieces of iron pipe, some 12 ft. long and the rest 6 ft. long. . If the total length of the pipe is 168 ft., how many pieces has he of each length ? 234 SIMULTANEOUS SIMPLE EQUATIONS 184. Three Simultaneous Equations. If three or more simul- taneous equations are given which, involve three or more un- known quantities, these quantities are eliminated by combining pairs of equations, as shown in the following solution : Solve the equations 5a; + 2«/-4g = -3 (1) 3a;-3«/ + 5a = 12 (2) 4:x + 5y + 2z = 20 (B) Using (1) and (2), we may eliminate z as follows : Multiplying (1) by 5, 25 a; + 10 ?/ - 20 z = - 15. (4) Multiplying (2) by 4, 12 a; - 12 y + 20 z = 48. (5) Adding (4) and (5), 37 x - 2 j^ = 33. (6) We may eliminate z between (1) and (3), as follows : Multiplying (3) by 2, 8x +10y + iz = 40. (7) Adding (1) and (7), 13 s + 12 2^ = 87. (8) We now have two equations, (6) and (8), involving x and y. We may eliminate y as f oUqws : Multiplying (6) by 6, 222x-12y = 198. (9) Adding (8) and (9), 235 x = 235. Dividing by 285, x = 1. Substituting in (8), 18 + 12 y = 37. Hence 12 y = 24, and y = 2. Substituting a; =1, j^ = 2, in (1), 5 + 4— 4z=— 3. Solving for z, 2 = 3. Therefore x = 1, y = 2,z = S. Check. Substituting in (1), (2), and (8), we have 5 + 4 -12 =-3, 3 - 6 + 15 = 12, 4 + 10 + 6 = 20. THEEE SIMULTANEOUS EQUATIONS 235 Exercise 140. Three Simultaneous Equations ExaTwples 1 to 4, oral 1. In the system of equations x + y + z = 6 (1) a; + «/ — 2 = 4 (2) x-y + z = 2 (3) eliminate y and 2 at the same time between (2) and (3). What is the value of a; ? 2. In the equations of Ex. 1 eliminate z between (1) and (2), and find the value of x + ^/. Then substitute the value of X found in Ex. 1, and find the value of «/. 3. In the equations of Ex. 1 substitute in (1) the values of X and y found in Exs. 1 and 2, and find the value of z. 4. In the equations of Ex. 1 check the values of x, y, and z found in Exs. 1^3, by substituting in each equation. Solve the following equations: 5. x + y + z=^lQ 9. 3a; + 2«/-4» = 15 . x-y + z = 2 5a;-3z/ + 22 = 60 x + y — z=% 2a; + 4y— 32 = 45 6. x + y + z=7 \Q. x-2>y + z = 10 Bx+y-z=3 2x-7y-5z=-2 2x + 4:y + z = 12 x + y—2z = 5 7. x + y + z = lS 11. 10a: + 8«/-92=10 3a; + y-32=5 12x + 2y -IQ z = 15 x-2y + 4:z=10 2x + lQy-2bz = 8. x + 2y + Bz = 4:l 12. x + y + z = 14.6 x—Sy + 4:Z = Q x — y + z^SA 5x + 6y-7z = 6S x + y-z = 12.2 236 SIMULTANEOUS SIMPLE EQUATIONS 13. A farmer who had kept accounts for his farm as a whole, but not for the separate crops, desired to find the average profit per acre of each crop. His records show the following acreage and profits: CoKN Wheat Oats Profit First year: 80 A. 20 A. 20 A. $2400 Second year : 70 A. 30 A. 20 A. $2500 Third year : 60 A. 30 A. 30 A. $2300 Find the average profit per acre for each crop. 14. A company announces in its circular that it makes three standard grades of fertilizers containing respectively the following ingredients and selling at the prices stated: Nitrogen Potash Phosphate Price per Ton Grade I : 501b. 110 lb. 150 lb. $23.10 Grade II : 701b. . 1001b. 150 lb. $25.50 Grade III : 901b. 901b. 120 lb. $26.10 The expert at an experiment station advises a farmer to use a fertilizer containing 80 lb. nitrogen, 80 lb. potash, and 1-60 lb. phosphate. How much should he pay the company for a ton of fertilizer made up on this formula? 15. A railroad in its annual reports for three different years, during which there was no great change in operating costs and no change in rates, gave the following statistics: Number of Number op Number of Net Passengers Tons op Tons op Operating Carr'ied Freight Express Revenue First year: 12,400,000 9,600,000 500,000 $4,500,000 Second year: 12,100,000 9,100,000 600,000 4,900,000 Third year: 13,100,000 10,200,000 600,000 6,000,000 Find to the nearest mill the average net operating revenue per passenger, per ton of freight, and per ton of express. STANDARD DIETARIES 237 185. Standard Dietaries. In preparing scientific dietaries, based upon the composition of foods and the amount of each ingredient needed by the average person of a certain class, the simultaneous equation is useful. For this purpose the foUowing.table of composition, in per cents, and of fuel values (calories) is given, with the recent average cost per pound. The per cents are given to the nearest tenth ; the calories are per pound. Food Material PROTlSIIf Fat Carbo- hydrates Mineral Matter Calories Cost Beef Loin . . . . 16.1 17.5 0.9 1025 10.26 Ribs . . . . 13.9 21.2 0.7 1135 .24 Round . . . 19.0 12.8 1.0 890 .25 Dried. . . . 26.4 6.9 8.9 790 .30 Veal Leg ... . 15.5 7.9 0.9 625 .18 Loin . . . . 16.6 9.0 0.9 685 .25 Breast . . . 15.4 11.0 0.8 745 .16 Mutton Leg ... . 15.1 14.7, 0.8 890 .18 Loin . . . . 13.5 28.3 0.7 1415 .25 Pork Loin . . . . 13.4 24.2 0.8 1245 .22 Ham, smoked . 14.2 33.4 4.2 1635 .30 Bacon . . . 9.1 62.2 4.1 2715 .24 Chicken . . . 13.7 12.3 0.7 765 .22 Turkey . . . . 16.1 18.4 0.8 1060 .30 Eggs 13.1 9.3 0.9 635 .55 Cod, fresh . . . 11.1 0.2 0.8 220 .13 Mackerel, fresh . 10.2 4.2 0.7 370 .14 Salmon, tinned . 21.8 12.1 2.6 915 .18 Oysters . . . . 6.0 1.3 3.3 1.1 225 .30 Butter . . . . 1.0 85.0 3.0 3410 .44 Cheese . . . . 25.9 33.7 2.4 3.8 1885 .25 238 SIMULTANEOUS SIMPLE EQUATIONS Food Matebial Protein Fat Carbo- hydrates Mineral Matter Calories Cost Milk, whole . . . 3.3 4.0 5.0 0.7 310 |0.09 Oatmeal .... 16.7 7.3, 66.2 2.1 1800 .14 Corn meal . . . 9.2 1.9' 75.4 1.0 1635 .04 Rice 8.0 0.3 79.0 0.4 1620 .08 Wheat flour, white 11.4 1.0 751 0.5 1635 .03 Wheat,b'kf'st foods 12.1 1.8 75.2 1.3 1680 .15 Crackers .... 9.7 12.1 69.7 1.7 1925 .12 Macaroni . 13.4 0.9 74.1 1.3 1645 .14 Sugar . . 100.0 1750 .05 Beans, dried 22.5 1.8 59.6 3.5 1520 .08 Peas, dried 24.6 1.0 62.0 2.9 1565 .10 Beets . . 1.3 0.1 7.7 0.9 ,160 .05 Cabbage . 1.4 0.2 4.8 0.9 115 .05 Potatoes . 1.8 0.1 14.7 0.8 295 .02 Tomatoes 0.9 0.4 3.9 0.5 100 Apples 0.3 0.3 10.8 0.3 190 .03 Bananas . 0.8 0.4 14.3 0.6 260 .03 Grapes 1.0 1.2 14.4 0.4 295 .12 Oranges . 0.6 0.1 8.5 0.4 150 .15 Strawberries 0.9 0.6 7.0 0.6 150 Almonds . 11.5 30.2 9.5 1.1 1515 .20 The following table shows the average amount requked per adult person per day: Protein Fat Carbo- hydrates Fuel Value Ounces Ounces Ounces .Calories Man at hard work . . 1 . . 4.70 3.35 15.41 3270 Man at moderate work . . . 3.84 1.87 17.11 2965 Woman at moderate work . . 3.25 1.24 12. 2700 These standards are average findings of several investigations. Other standards differ somewhat. To obtain the amount of each required for a child, take half the amount required by a man at moderate work. STANDARD DIETAEIES 239 186. Illustrative Example. Determine the quantity of pork loin and the quantity of beans that will supply the protein and fat necessary for one day for a family consisting of a man and woman at moderate work and a child. By the table, the needs for the family fori da. are : Protein Fat Carbohydrates Calories Man . . . . 3.84 oz. 1.87 oz. 17.11 oz. 2965 Woman . . 3.25 oz. 1.24 oz. 12. oz. 2700 Child . . . . 1.92 oz. 9.01 oz. 0.94 oz. 4.05 oz. 8.56 oz. 1483 37.67 oz. 7148 Let X = the number of ounces of p'ork, and y = the number of ounces of beans. Then, from the nearest per cent in the table, 0.13 X + 0.23 y = 9, (1) and 0.24 x + 0.02 2/ = 4 ; (2) or, multiplying (1) by 100 and (2) by 50 to avoid decimals, 13 X + 23 2/ = 900, (3) and 12 a: + y = 200. (4) Multiplying (4) by 23, 276 x + 23 y = 4600. (5) Subtracting (3) from (5), 263 x = 3700 ; whence x = 14.1, approximately. Therefore y = 30.8, approximately. Hence the required amount of protein and fat can be secured from 14.1 oz. of pork and 30.8 oz. of beans. Thus far, however, the problem is not complete, since it does not determine the carbohydrates and the calories. The 14.1 oz. of pork and 30.8 oz. of beans would supply together, as the table shows, 18.4 oz. of carbohydrates and 4023 cal- ories, showing a lack of 19.3 oz. of carbohydrates and 3125 calories. This is considered on the next page. Of course the question of digestibility is also to be considered. 240 SIMULTANEOUS SIMPLE EQUATIONS 187. Full Dietary. The example on page 239 did not con- sider, as already stated, the carbohydrates and calories. These may be provided by bread and sugar, thus: Protein Fat Pork 14.1 oz. 1.89 oz. 3.41 oz. Beans 30.8 oz. 6.93 OZ. 0.55 oz. Brea4 16. oz. 1.82 oz. 0.16 oz. Butter 4. oz. 0.04 oz. 3.40 oz. Sugar 7. oz. 10.68 oz. 7:52 oz. Cakbo- htdrates Calories Cost 1097 10.194 18.86 oz. 2926 .154 12.02 oz. 1635 .03 853 .11 7. oz. 766 .022 37.38 oz. 7277 10.51 The above work does not involve any algebra; it is only a fair approximation and is added for completeness. Exercise 141. Dietaries FiTid the quantity of each of the following foods needed to supply a ration containing 9 oz. protein and 4 oz. fat : 1. Veal loin, beans; also cheese, macaroni. 2. Leg of mutton, beans. 3. Bacon, white bread made of wheat flour. 4. Mutton loin, bread as in Ex. 3. 5. Eggs, breakfast food made of wheat. 6. Taking the results of Exs. 1-5, determine the cost of each ration and arrange the rations in order of expense. 7. Using three unknowns, determine the quantity of oat- meal, milk, and eggs that will supply the necessary protein, fat, and carbohydrates for the breakfast of a man at hard work, assuming that this is to be one fourth of his day's nourishment, and find whether this will supply the necessary 817 calories for a quarter of the day's work. RATIO OF SOLVENCY 241 188. Ratio of Solvency. Two firms, A and B, failed at the same time, their accounts showing the following : Assets Liabilities A Due from B, |200,000 From other sources, 700,000 Owed to B, |300,000 To other parties, 900,000 B Due from A, $300,000 From, other sources, 800,000 Owed to A, |200,000 To other parties, 1,300,000 The court decided that it would be unfair to the other creditors if A and B settled their balance of |100,000 as if they were solvent, but that all assets and liabilities of each must be counted in fixing the ratio of solvency. Let X = the ratio of solvency of A, and y = the ratio of solvency of B. ™ Assets of A 200,000 y + 700,000 ^^^'^ Liabihties of A = 1;200:000 = ^' ^ ^ ^^*^°' Assets of B 300,000 2;+ 800,000 and ■ = y, B's ratio. Liabilities of B 1,500,000 Solving, we have x = \^\ and y = ^^. We now have : Assets Liabilities A 31 X 1200,000 = 1134,482.76 700,000.00 Total . 1834,482.76 ili X 1300,000 = $208,620.69 If J X 900,000 = 625,862.07 Total . . . 1834,482.76 B m X 1300,000 = 1208,620.69 800,000.00 Total . . . 11,008,620.69 -II X 1200,000 = 1134,482.76 f 1 X 1,300,000 = 874,137.93 Total . . 11,008,620.69 Hence A's assignees pay B's $208,620.69 less 1134,482.76, or $74,137.93 ; A's other creditors are paid $625,862.07, and B's other creditors are paid $874,137.93. 242 SIMULTANEOUS SIMPLE EQUATIONS Exercise 142. Ratio of Solvency Determine the ratio of solvency and method of settlement of the accounts in each of the following cases : 1. Assets Liabilities A Due from B, Other sources, $400,000 950,000 ToB, To others. $300,000 1,400,000 B Due from A, Other sources. *300,000 800,000 To A, To others, $400,000 1,100,000 2. Assets Liabilities A Due from B, Other sources, 1120,000 500,000 ToB, To others. $180,000 720,000 B Due from A, Other sources, *180,000 400,000 To A, To others. $120,000 500,000 3. Assets Liabilities A Due from B, Due from C, Other sources. $35,000 42,000 120,000 ToB, ToC, To others. $25,000 50,000 180,000 B Due from A, Due from C, Other sources. $25,000 45,000 80,000 To A, ToC, To others. $35,000 60,000 85,000 C Due from A, Due from B, Other sources. 150,000 60,000 130,000 To A, ToB, To others. $42,000 45,000 200,000 EEVIEW 243 Exercise 143. Review of Chapter XII 1. Solve the simultaneous equations 5a: — 3«/ = 1.9 and 4a;— 11'^ = — 0.2, checking the results. '2. Solve the equations mx + ny = a and px + qy = b. 3. Using the results in Ex. 2 as formulas, substitute the values of m, n, p, q, a, and h as given in the equations in Ex. 1, and thus check the results found in Ex. 1. 4. A confectioner made a profit of |750 in January bj selling 12,000 lb. of candy and 1000 qt. of ice cream. In February his profit on 11,500 lb. of candy and 1200 qt. of ice cream was $755. What was his profit per unit on each? 5. Solve the equations ax + y = b and 3x — cy= d, using the method of substitution. 6. A dealer shows two different samples of spring wheat, selling at #2 and |2.15 respectively. A buyer orders 1200 bu. at $2.10 per bushel, to be made up of the two kinds. How many bushels of each kind should be taken? 7. A dairy having on hand 15 gal. of milk containing 4% of butter fat, some milk containing 5% of butter fat, and some cream containing 40% of butter fat, received an order for 60 gal. of cream containihg 25% of butter fat. The manager wishes to use the 15 gal. of 4% milk, and. enough of the 5% milk and 40% cream to produce the 60 gal. of cream ordered. How many gallons of each should he use ? 8. Solve the simultaneous equations 4:x — y=ll, 3t/ — s = 2,, and 5a;+22=17, checking the results. 9. Determine the quantities of round steak, potatoes, and bread made from wheat flour that will be sufficient to give the necessary protein, fat, and carbohydrates for a day to two men at hard work and two women at moderate work. 244 SIMULTANEOUS SIMPLE EQUATIONS Exercise 144. Review of Chapters I-XII 1. Using m to represent the cost of raw material, I the cost of labor, and o the miscellaneous overhead charges in producing n phonographs, write the formula by which to find the value of c, the average cost of a phonograph. 2. Solve the equation ^ (5 a; — 3) = 4 a; — 7, and check the result. 3. Using the formula i=prt, derive a one-day rule for mterest at 2|-%. 4. If the cost of goods is c, the cost of doing business h per cent of the selling price, and the profit p per cent of the selling price, find the selling price. 5. Multiply 2a? — ^x + b by 5a? + Qx, and check. Gi C 6. Simplify j-^-^i and use the result as a formula by which to write the result of f ^- f • 7. If I buy goods for |800, and my overhead charges are 18% of the cost and my profit is 10% of the cost, find the selling price. Find the selling price if the overhead charges and the profit are 18% of the selling price and 10 % of the selling price respectively. 8. Using the formula a=p(\+rt'), find the number of years in which a principal p doubles itself at r per cent simple interest. 9. Using the result obtained in Ex. 8, find in how many years a principal will double itself at 6% simple interest. 10. Solve the equations 2,x — y = b and 2 cc + 3^ = 7. 11. Using the formula a=p(\+rt'), find the rate at which a principal p will double itself in t years at simple interest. CHAPTER XIII GRAPHS 189. Location of Points on a Map. Points are located on a map by means of latitude and longitude. Latitude is stated in degrees north or south of the equator, and longitude is stated in degrees east or west of the prime meridian through Greenwich, England. Thus, to the nearest degree, the position of New York City is 41° N., 74° W. ; that is, the latitude is 41° north of the equator, and the longitude is 74° west of Greenwich. 190. Location of Points on Paper. In a similar way we may locate points on paper. We may take two perpendicular lines, one vertical and the other horizontal, and measure dis- tances to the right and left of the vertical liue, and upwards and downwards from the horizontal line. In this figure P-^ is 3 units to the right of the vertical line YY', which is called the y-axis or axis of y, and 3 units above the horizontal line XX', which is called the x-axis or axis o/ X ; Pj is 4 units to the left of YY' and 2 units above XX'; P, is 2 units to the' left of YY' and 2 units below XX'; and P^ is 6 units to the right of YY and 1 unit above XX'. In the same way we may locate a chair on the floor with reference to the east and north walls, a spring in a field with reference to two fences meeting at right angles, or a point on the blackboard with reference to two Unas perpendicular to each other. The students fiave already studied some of this work in Chapter III. 245 ¥ — ^ — — .^ ' fj " }' j: f? Y — — — — 246 GEAPHS x; K u- 191. Coordinates, The distances of a point to the right or left of the axis of y and above or below the axis of x are called the coordinates of the point. Thus the point P is in %h.Q first quad- rant (I). It is designated as the point (4, 3), and its coordinates are 4 and 3. A point (— 4, 3) would be represented in the second quadrant (II), and so on, dis- tances measured to the right of YY' or above XX' being positive, and distances measured to the left of YY' or below XX' being negative. The coordinates of 0, which is called the origin, are and 0. 192. Abscissa. The distance of a point from the vertical axis, measured on the axis of x or parallel to that axis, is called the abscissa of the point. Thus the abscissa of P is 4. The abscissa of a point in the second quadrant is negative. The abscissa of a point in the third quadrant is also negative. The abscissa of a point in the fourth quadrant is positive. 193. Ordinate. The distance of a point from the -horizontal axis, measured on the axis of y or parallel to that axis, is called the ordinate of the point. The ordinate of P is 3. The ordinate of a point in the first or second quadrant is positive ; in the third or fourth quadrant is negative. 194. Plotting a Point, Representing a point by means of its coordinates is called plotting the point. To plot the point (— 2, — 3), take the abscissa — 2 and draw its ordinate — 3. The point is, therefore, in the third quadrant. 195. Coordinate Paper. Paper ruled in squares for con- venience in plotting points is called coordinate paper. Coordinate paper, Slsp known as cross-ruled 'paper, cross-section paper, or squared paper, will be found of use in the representation of equations and in drawing many of the figures used in geometry, as has abeady been suggested in Chapter III. PLOTTINa POINTS 247 Exercise 145. Plotting Points ExaTnples 1 to 6, oral 1. In what quadrant is the point (2, 5)? 2. In what quadrants are the points (1, 1), (— 2, — 6), (4, -2), and (-2, 4)? 3. Where is the point (0, 0)? (5, 0)? (0, 5)? (-5, 0)? 4. What is the distance from (4, 0) to (-4, 0)? 5. In what quadrant is (2, 7)? (-2, 7)? (2, -7)? 6. What is the distance from (4, 3) to (— 4, 3) ? from (4, 3) to (4, - 3)? from (0, 0) to (0, 7)? from (- 2, 5) to (- 2, - 5)? from (2, - 5) to (- 2, - 5)? 7. Plot the points (1, 3), (7, 6), (5, 2), (9, 1), (1, 9), (5, 5), (7, 0), (0, 7), (0, 0). 8. Plot the points (- 2, 3), (- 4, 6), (- 6, 4), (- 5, 2). 9. Plot the points (- 2, - 3), (- 4, - 6), (- 5, - 2), (-6, -9), (-7,-7). 10. Plot the points (2, - 3), (3, - 2), (4, - 6), (6, - 4), (7, -7), (-7, 9), (0, -4), (11, 21), (-0.5,-3). 11. Plot the points (4, 4), (2, 1), (4, 1), (- 2, - 4), (-4, -4), (-2, -1), (-4, -1), (-2, -4). 12. Plot the points (3, 5), (2, 3), (1, 5), (2, 6), (5, 3), (2, 0), (- 3, - 5), (- 2, - 3), (-1, - 5). 13. Plot the points (1, 4), (4, 5), (4, -4), (1, - 1), (1, -2), (1, 3), (-4, -5), (-1, -1), (-1, 4), (0, 4). 14. Join (0, 0) and (3, 5), (3, 5) and (6, 0), (1.2, 2) and (4.8, 2), (7, 0) and (12, 5), (7, 5) and (12, 0), (13, 0) and (13, 6), (18, 4) and (17, 5), (17, 5) and (15, 5), (15, 5) and (14, 4), (14, 4) and (14, 3), (14, 3) and (18, 2), (18, 2) and (18, 1), (18, 1) and (17, 0), (17, 0) and (15, 0), (15, 0) and (14, 1). What word is spelled by these lines ? 248 GRAPHS 196. Plotting an Equation^ Not only is it easy to plot a point, but it is also possible to plot or graph an eqtmtion. For example, consider the equation 3 a; + 4 ^ = 7. Solving for y we have 7 — 3 a; rru i. • y = — inat is, y is a function of x, and therefore for any value that we may give to X we may find 'a corresponding value of y. Thus, if x = l, 7-3x1 then V- = 1. r V ,7 \ ,^^ \ ^ fi ' in only one point. Therefore, in general, The graphs of two linear equa- tions involving two unknowns have only one point in common. Two linear equations involv- ing two unknowns have only one pair of values of the unknowns in common. Thus the graphs of the equa- tions X ■{■ y = 5 and x — y = ^ intersect at P. The coordinates of P are 4 and 1. Hence the solution of the given equations is a; = 4, !/ = 1. There are an infinite number of points on each graph, but there is only one point on both graphs. 200. Inconsistent Equations. If we plot the two equations a; — 2 y = 4 and 3 a; — 6 y = 5, we shall have two parallel lines. Such liaes have no point in common. Considering the equations, we see that the second one reduces to x—2y= 1|. The equations are there- fore inconsistent, since x—2y cannot be equal to both 4 and 1|-. 201. Equivalent Equations. If we plot the equations 2 a; -)- 4 y = 5 and a; + 2 y = 2|-, we shall find that one graph coiucides with the other. They have an infinite number of points in common, and therefore an infinite number of values of x and y satisfy the equations, every pair of roots of either being a pair of roots of the other. The equations are therefore equivalent. LINEAR EQUATIONS 251 Exercise 147. Graphs of Linear Equations Examples 1 to 4, oral 1. What is the nature of the graph oi x=2? oi y =21 2. What is the nature of the graph ofy = 4? ofz/=— 4? 3. State two points on the graph of a; + 4 t/ = 8. 4. State two points through which passes the graph of the equation x = y. Plot the following eqwations, and solve each hy measuring the coordinates of the point of intersection of the graphs : 5. z + iy = 11 9. x + 5y=0 2x-y = 4: Sx+^y=-6 6. 2x + Sy = 19 10. 7x + 2y=14:' 7x-2y = 4: 5x-Sy=-21 7. x + 5y = -S n. 2x-Sy = 7 2x-3y = 20 5x-7y = 14: 8. 2x-9y = 2d 12. 6x-Zy = 15 5x + y = -lS 2x + 7y = 4:5 13. Show by graphs that the equations a; + 4 «/ = 6 and 0.5x+2y = 4: are inconsistent. 14. Show by graphs that the equations 0.2x—0.5y=6 and X — 2.5 y = SO are equivalent. 15. Show by graphs that the three equations x + y = 6, 2x — y = 0, and 5x + Sy = 22 have a common root. From the graphs find the root. 16. Show by graphs that the three equations x + y= 5, 2x—Sy=20, and Sx + y = 2 have no root common to all. 17. If x + 5y = 2.1, y is equal to what- function of a;? X is equal to what function of 2/ ? \i2x — y = 2,y\% equal to what function of a; ? a; is equal to what function of «/ ? 252 GRAPHS 202. Wage Table. The graph will often be found to serve a very important purpose in relation to prices and wages. For example, a contractor who pays a number of work- men at the rate of 37^ an hour finds that he saves a great deal of time by using either a wage table or a wage graph. Each method is, of course, based on the formula w = 10.37 h, where w is the wages due, and A the number of hours. The wage table in this case, for the 5^ da. of 8 hr. each in the working week, is as follows: Hours DA. 1 DA. 2 DA. 3 DA. 4 DA. 5 DA. *2.96 $5.92 18.88 *11.84 $14.80 - 1 !|0.37 3.33 6.29 9.25 12.21 15.17 *2 .74 3.70 6.66 9.62 12.58 15.54 3 1.11 4.07 7.03 9.99 12.95 15.91 4 5 1.48 1.85 4.44 4.81 7.40 7.77 10.36 10.73 13.32 13.69 16.28 I hr. $0.09 6 2.22 5.18 8.14 11.10 14.06 1 hr. .19 7 2.59 5.55 8.51 11.47 14.43 f hr. .28 Find the wages due a man who worked 4 da. of 8 hr. each, and 3|^ hr. the next day. Looking in the table under 4 da. and opposite 3 hr. we find that the wages for 4 da. 3 hr. are $12.95. Looking for ^ hr. in the lower right- hand corner, we find $0.19. Hence the total wages due are $13.14. The wage table can be used with about the same facility as the wage graph described on page 253. The wage tables used by paymasters are often in book form, a page or two being devoted to a table at a given rate. The size of the entire table depends largely on the number of different rates included. It is evident that when time and overtime are paid for at different rates, two different tables must be consulted. WAGE GRAPHS 253 203. Wage Graph. If we draw the graph of the equation w = $0.37A, we shall find it to be a straight line, as stated on page 248. On account of the size, we give here only a portion of this graph, leaving it to the student to complete as directed in the exercise below. From the graph it will be seen that the wages for 8 hr. are |2.96, as shown by the dotted lines. Similarly, by laying a ruler on the page, or by following the coordinate lines, we see that the wages for 6 hr. are $2.22, for 12 hr. they are |4.44, for 14 hr. they are $5.18, and so on. A little- practice makes it very easy to use the wage graph. 6.92 5.95 6.18 4.81 4.44 4.07 3.70 3.33 2.96 2.50 2.22 1.85 1.48 1.11 .74 .37 k -- r " - - " - Y- / IV f / ^J / .'J / 4 / / / / / / / / . / 2 4 6 8 10 12 14 16 Exercise 148. Wage Graphs 1. Complete the ab»ve wage graph up to 44 hr. 2. From the completed graph of Ex. 1, fill in the following section of a pay roll, the wage per hour being $0.37: Men's Numbers Hours Amount Dub 23 181 24 33 27 371 28 40 29 42 32 43 33 44 3. Draw a wage graph to 44 hr., the wages bemg 39^ an hour ; that is, draw a graph of the equation w = 0.39 /?.. 4. Draw a wage graph to 44 hr., the wages being 42| i^. 254 GRAPHS 204. Commercial Interpolation. In commercial calculations many of the numbers vary directly, as in the case of wages and hours ; that is, as we double the number of regular hours we double the wages, the wage graph being a straight line. Many other cases occur in which the cost varies directly or approximately as some other factor, such as length, weight, horse power, or capacity. If the graph is a straight line or approximately so, the cost of inter- mediate sizes or number of articles can be determined very closely by graphic interpolation; that is, by finding corresponding intermediate points on the graph. To illustrate this, consider the case of a house which catalogued aluminum saucepans at the fol- lowing prices : 3 pt., 30 ^ ; 4 pt., 35(^; 6pt., 45(f; 8 pt., 54^, with intermediate sizes at correspond ing prices. The house received an order for 18 doz. 5-pint pans and 24 doz. 7-pint pans, much should it charge for the lot ? In the graph the four points represent the given prices, and they are nearly in the same straight line. We therefore see at a glance that the 5-pint size should cost 40 ^ and the 7-pint size 49 ^. 55 50 15 §35 .gSO «25 «ao IB 1 2 3 15 6 7 Size in pints How Exercise 149. Commercial Interpolation 1. Assuming that for a short period of years the increase in insurance rates varies directly as the increase in age, find the premium per $1000 at ages 22, 23, and 24 years, if at 21 yr. the premium is $17.90 and at 25 yr. it is $19.63. In this example it is easier to take $17.90 at the origin. COMMEEGIAL INTERPOLATION 255 2. The same company as in Ex. 1 charges $25.88 at 35 yr. and $30.55 at 40 yr. Find by graphic interpolation the rates for the four intermediate years. In Ex. 2 the results will not be as close as in Ex. 1. 3. The cost of a certain type of engine for various horse powers (H.P.) is Usted as follows: 3 H.P., |400; 6 H.P., 1600; 12 H.P., $800. Plot the three pomts representing these facts, join them by a smooth curve, and determine the approximate cost of engines of H.P. 4, 5, 7, 8, 9, 10, and 11. 4. A contractor finds that the average expenditure for a group of 50 workmen employed on a job was $630 per week, while another group of 75 averaged $945. All the workmen being of the same grade, find from a graph the wages for a week of 19 men; of 32 men; of 68 men; of 96 men; of 125 men ; of 132 men. 5. In an experiment a wooden rod was deflected as follows : Load in grams ...."..,. 50 60 70 80 Deflection in centimeters ... . . . 1.3 1.5 1.8 2.1 Drawing • a graph, find if the deflection varies uniformly with the load, and find the deflection for intermediate loads. 6. A yellow pine rod 80 in. ■ i . i long rests on two supports, , m IH m A and B, 24 in. apart. Under loads applied at the mid-point, L, it is deflected as follows: Load in pounds . . . 8 16 2L5 38 49 60.5 70 Deflection in inches . . 0.14 0.3 0.41 0.77 1 1.35 L8 Draw the graph and determine the deflection under a load of 10 lb. ; of 18 lb. ; of 25 lb. ; of 42 lb. ; of 65 lb. 256 GEAPHS 205. Net-Profit Graph. The net-profit graph is composed of the graphs of the equations of total cost and of total sales. To illustrate, consider the following: The cost of the necessary machinery and its installation before a certain toy could be manufactured was |200. Each toy produced costs 50^ for material and labor and is sold for |1. The cost equation is e= 200 + 0.50 w, and the sales- receipts equation is /S = w. Drawing both graphs on one dia- gram and to one scale, we have the figure shown below, the straight line AB being the graph of c = 200 + 0.50 n and the straight line OD that of S=n. When w = 800, it is easily seen from the diagram that c = |600 and S= $800. It is also seen from the figure that^, the net profit, is $200 or that portion of the ordinate drawn through 800 which lies between the graphs AB and OD. When n=1000, we easily see from the diagram that c — |700, S= IIOOO, and ^ = |300. In general, if we represent the total receipts from the sale of an article by S, the number sold by n, and the price per unit by p, we have S = np. / $1000 / $800 / o % '> ■0 V E ..$600 ■a /, M •^ ** S / ■^ ^ $400 'jj ^ y ** ^ E A ^ / •p ■y S n 200 400 600 800 1000 In this equation the value of n is supposed to be known, and the graph of the equation is a straight line. If k represents the initial cost before any of the units are sold, and k' the cost per unit which must be added' to the initial cost, the cost of producing n units is e = A + k'n. This is also a linear equation. By subtracting the value of e for any number n from the corresponding value of S, we evidently find the profit made in selling the n articles. NET-PROFIT GRAPHS 257 Exercise 150. Net-Profit and Other Graphs Using the diagram on page 256, consider Exs. 1-4 : 1. How many toys must be sold before the company will " break even " ? 2. What is the total cost of producing 200. toys ? the sales receipts for 200 toys ? the loss on 200 toys ? 3. How many toys must be sold before the gain is |100 ? 4. What is the gain on 1000 toys ? 5. If the cost of running a certain store for selling auto- mobile tires averages |15 per day and the profit is $1.50 per tire, graph the equation expressing the gain or loss per day if w is the number of tires sold in a day. 6. The telegraph rate between cities A and B is 20^ for any number of words up to 10, and 10 for each additional word, the equation being c = 20 + (w — 10). Construct the graph of this equation, beginning with n=10. 7. If the first cost of producing a book is |700 for plates and printing, aJid each copy represents an additional outlay of 600, write the equation of cost. If the book sells at $1.50, write the equation of sales receipts. Graph these equations, and answer questions analogous to those in Exs. 1-4. 8. The cost of a mine and the installation of machinery was 1125,000. The overhead charges in getting a ton of ore to the smelter were |1.50 per ton. If the ore was sold at the smelter at an average price of |3.50 a ton, construct the net-profit graph for this mining venture. 9. From the graph constructed in Ex. 8 find how many tons must be sold to break even ; what the net profit or net loss will be when 15,000 T. have been marketed ; how many tons must be sold to make a net profit of $300,000. 258 GRAPHS 7n ' 1 1 '" - "- "iT"± 60 " _L ; ± 40 - Z". i^'vrf ■>><'" 3SP \jA /L/f'^-\y<'- ---*>- 20 WyHl ■"fS*^ ^ ■4^ ~i-tl>r-*'TT 'Sh-t' ' ' T ~ II i-^ ' "f"" ^^ ■ 'x' x — X"" 100 200 300 400 600 600 700 PrlQclpui 206. Interest Graphs. If we take the equation i=pr, which gives tlie interest for 1 yr., and give to r the values 2%, 3%, .-., 9%, we shall have eight linear equations. The graphs of all these equations are shown in the accompanying figure and are sufficient for the purpose of checking computations in interest. Suppose, for example, we wish to find the interest on 150 for 1 yr. at 3%. Look above the point 500, which in this example may be taken to represent $50, on the axis of principals and the ordinate strikes the 3% graph at 15. That is, the interest on |50 for 1 yr. at 8 % is |1.50. Of course this graph is too small for practical purposes. If drawn to a scale four times the one here used, and if extended to include other principals, it can be used to advantage, particularly as a check. It should be noticed that the numbers on the axis of principals may be read as 1, 10, 100, • • •; 2, 20, 200, • • -, as circumstances demand ; also that the experience and intel- ligence of the student will enable him to determine whether the number read on the iaterest axis is 0.15, 1.5, 15, or 150, and so on. Graphs are very valuable as checking devices, but they cannot be used in general unless the one using them has mastered the relatioils expressed by the algebraic equation of the graph or has had experience in dealing with the num- bers whose relation is represented graphically. No device or method of calculating can eliminate the necessity for master- ing the mathematics involved in commercial transactions. INTEREST GRAPHS 259 Exercise 151. Interest Graphs 1. Draw the graphs on page 258, and continue the princi- pals to 1200 and the interests to 100, extending the interest graphs accordingly. From the graphs in Ex. 1 find the interest for 1 yr. : 2. On |75'at 6%. 6. On $105 at 3%. 3. On $50 at 4%. 7. On $108 at 4%. 4. On $85 at B%. 8. On $110 at 5%. 5. On $62 at 5%. 9. On $120 at 6%. Of course, in general, these results are merely approximate. The graph serves, however, as a ready check on more exact work. 10. Draw interest graphs as above for 1 yr. for the follow- ing rates: 21%, 31%, 41.%, 5J%. 11. Reversing the use of the graph as given on page 258, find the principal which will yield $3.60 interest in 1 yr. ^t 2%; at 3%; at 4%. 12. By interpolating between the graphs find the approxi- mate interest on $450 for 1 yr. at 4|-%. 13. By interpolating between the graphs find the approx- imate rate that wiU yield $4 interest on $90 for 1 yr. 14. By interpolatiag between the graphs find the approx- imate interest on $750 for 1 yr. at 4^%. 15. By interpolating between the graphs find the approx- imate interest on $1150 for 1 yr. at 5|-%. 16. Reversing the use of the graphs drawn in Ex. 10, find the principal that will yield $85 interest in 1 yr. at S^%; at 4^% ; at 5|%. 17. Draw the interest graph for 1 yr. represented by the equation i= 0.08^. 260 GKAPPIS 207. Interest Graphs. Another problem in interest can be quickly solved to a fair degree of approximation by means of graphs of the size shown below. This figure represents a 4% interest graph, the equation i = prt becoming i= 0.04^ pt. The curves are found by locating various points when i = 50, then when i = 100, and so on. 1. Find the time necessary for $400 to earn $100 interest at 4%. Start at the point for |400 on the a;-axis, or axis of principals; run up the ordinate to the |100 in- terest graph, and notice that this is reached 6 \ spaces up. That is, the time is approxi- mately 6 \ yr. 2. Find the principal which in 10 yr. will yield $200 in- terest at 4%. Start at the point 10 on the y-axis, or axis of time ; go to the right to the |200 interest graph; then drop to the axis of principals, the , point thus reached being that of |500. That is, the principal is $500. 3. Find the approximate interest on $900 for 9 yr. at 4%. • Follow the line leading from $900 on the axis of principals to the point of intersection with the line leading from 9 on the axis of time. A line drawn through this point perpendicular to the adjacent $300 and $400 interest curves is divided at the point, approximately as 1 to 3. We say the interest is approximately $325. ZO 1 1 ~ r" ~ ^ \ \ \ \ \ \ 1 ^ 1 \ \ V 1 , \ \ ^ \ \ \ - \ I ' \ \ s. \ V V \ s v £ ,„ > s. Vi fc>. 1 V s*> ' s \ \ s K s V \ V S^ ^ k, V ■^ s \ V ^ ^ ^ 5 - V 'a s s ■^ ^ ^ -i _ \ S' 1 1 ■* *■ r- % ./ V , ' ^ ■- ■* L. >^ — ^ - — — _ ^ p= — L = = L j—j — 1 -J L l—J —J lJ LJ t $600 $1000 $1600 Principal $2000 $2500 IXTEKEST GRAPHS 261 Exercise 152. Interest Graphs Draw interest graphs, similar to those on page 260, for the following rates : 1. 3%. 2. 5%. 3. 51%. 4. Q%. 5. From the graph on page 260 find what principal will yield |500 in 12 yr. at 4%. 6. From the graph on page 260 find the time needed for $2000 to earn |400 interest at 4%. 7. From the graph on page 260 find what prmcipal will earn as much interest in 10 yr. as $100 will earn in 7i yr., the rate in each case being 4%. 8. From the graph on page 260 find what principal will earn $300 interest in 2 yr. at 4%. 9. From your graph in Ex. 1 find the time needed for $1500 to earn $300 interest at 3%. 10. From your graph in Ex. 1 find what principal will earn $300 interest in 10 yr. at 3%. 11. From your graph in Ex. 2 find the time needed for $800 to earn $200 interest at 5%. 12. From your graph in Ex. 2 find what principal will earn $400 interest in 6 yr. at 5%. 13. From your graph in Ex. 3 find the time needed for $500 to earn $275 interest at 5i-%. 14. From your graph in Ex. 3 find what principal will earn $200 interest in 3 yr. at 51%. 15. From your graph in Ex. 4 find the time needed for $400 to earn $100 interest at 6%. 16. From your graph in Ex. 4 find what principal will earn $100 interest in 4 yr. at 6%. 262 GRAPHS 17. Draw an interest graph similar to that in Ex. 1 for 41%, and find the time needed for $1000 to earn $100 interest at this rate. 18. Draw an interest graph for 5|-% and solve the prob- lem of Ex. 17 for this rate. 19. Using the formula prt = i, let p =1000 and i = 100, so that we have 1000 rt = 100 or H = 0.1. The graph of this equation will give a curve showing the relation of the rate to the time in which $1000 will yield $100 interest. Draw the graph of rt = 0.1, and a succession of graphs for the equations 1000^ = 200, 1000 rt = 300, ■ • ., 1000 »•« = 900. State in what respects this family of graphs resembles, and in what respect it differs from, the one given on page 260. 20. Determine from the graphs constructed in Ex. 19 how much longer it will take $1000 to earn $300 at 3% than for $1000 to earn $200 at 4%. 21. Determine from the graphs of Ex. 19 how long it will take $500 to earn $400 interest at 3%. 22. Using the formula a^pCl + rf), construct the series of straight-line graphs for r = 0.06 and t = ^. 23. Explain how the graph of Ex. 4 may be used to check the amount at maturity of a 60-day note bearing interest. 24. In the formula a=p(l+rf), if a = 150 and ^=100, what will be the general shape of the curve showing the relation of r and ^? 25. Construct the graph of the formula mentioned in Ex. 24 and from it determine the time in which $100 will amount to $150 at 6%. 26. Construct on the diagram used in Ex. 25 the graph of the equations t=100 x 0.05 x t and a = 100(l-|- 0.05^- EEVIEW 263 Exercise 153. Review of Chapter XIII 1. Locate the points 2, 4, and —2, —4, and connect them by a straight Une. What is the length of the connecting line ? By what geometric principle can this length be determined without actual measurement? 2. What are the coordinates of the point of intersection of the graphs of the equations y = ^x + S and y = x? 3. The cost equations of two different manufactured articles are c = 4500 + 2.5 n and c = 3000 + B n. Find alge- braically the value of n for which the cost of producing these two articles will be the same. 4. Solve Ex. 3 by drawing the graphs of the equations and determining the coordinates of the point of intersection. 5. If the cost of a wireless message of 20 words is 60^, and that of one of 40 words is |1, find by constructing the straight-line graph the cost of the first 10 words and the rate per word for the excess over 10 words. 6. Construct the graph of the formula s = 16f, which represents the space s passed over m t seconds by a body falling freely. From the graph determine the time it will take a body near the surface of the earth to fall 70 ft. 7. By drawing the graphs, determine whether the equa- tions 2^ -I- y2 = 4 and y = a; -|- 4 are consistent. 8. Construct a series of curves whose equations are xy = 4, xi/= 5, xy = 6, and so on. Explain how these curves can be used for multiplying and dividing numbers. 9. By drawing the graphs test the consistency of the equations xi/ =6 and a^ -f- 1/^ = 9. 10. How can the graph of the second equation in Ex. 26, page 262, be easily constructed when the first one is drawn ? 264 GEAPHS Exercise 154. Review of Chapters I-XIII 1. For what value of p will i=prt become i = rt? 2. In the formula i=prt what is the value of j» in. terms of t when i = r? 3. li x = S, y = 2, z = 5, find the value of the expression a^ + 2«y-22+3v'52. 4. Find the sum of 3a^-4:ab + 5P, 6a^-7ab-9b% -11 a2 + 2 aJ - 8 b^, and ia^ + 16ab-9 b\ 5. What must be added to p +prt to make the sum 2p ? 6. Apply the principle (a + J) (a — i) = a^ _ 52 ^q ^-j^q multiplication of 2501 by 2499. 7. Using the principle ab = ba, find 87% of $125. 8. Divide a by 100/p. 9. Express the result of Ex. 8 as a rule for dividing any number by an aliquot part of 100. 10. In particular, in Ex. 9, if j? = 6, what is the rule for dividing any number by l^^, or 16| ? 11. Derive, a one-day rule for interest at 3|^%. State the rule in three parts, as usual. 12. The catalogue price of pianos is $1000 and the dealer receives discounts of 30, 10, 10. His cost of doing business is 23% of the selling price, and his rate of profit is 17% of the selling price. Find the selling price. 13. A city which owed $19,000 was given the privilege of paying in three installments of any amounts, provided the total was paid in 3 yr. The city council desired to arrange the installments so that the second installment should be 50% more than the first, and the third 50% more than the second. Find the amounts, not considering interest. INDEX PAGE Abbreviations 1 Abscissa . 24;i Absolute value 58 Addition 61, 71 Aggregation, symbols of . 16 Algebraic sum 72, 152 Aliquot parts 114 Antecedent 179 Area . 10, 18, 25 Arranging a polynomial . . 100 Axes . 245 Axioms 34 Balance sheet' . . . 20 Bar pictogram . 43 Binomial .... ... 16 Board measure 12 Brokerage .... . 202 Cartogram .50 Check .... 75, 78, 97, 102, 122 Circular pictogram ... 46 Clearing of fractions . . 163, 230 Coefficient 13 Cofactor ... 199 Commission . ... 202 Common denominator . . 151 Compound interest 22 Consequent 179 Consignee ... ... 202 Coordinates 246 Cube 11, 13, 15, 20 root 14, 15 265 PAGE Daily balances 80 Decimals, equatipns involving . 177 Denominator 149 Dietaries ... . 237 Directed irambers 57 Discount ' . . 192, 206 Divisibility of numbers . . . 126 Division . ... 65, 119, 158 Elimination 222 Equation . . 29, 30, 131, 189, 221 Equivalent equations . . 2-50 Evaluating ... . . 3- Exact interest . . 216 Exponent . . 14, 95, 120 Extremes . . .... 183 Factor' . . . 13, 98, 108, 202 Formula . 8, 10, 11, 12, 18, 22, 24, 113, 159, 167, 192, 203, 207 Fraction 140 Fractional equation . . 103, 230 Function '52 Graph . 43, 52, 245, 248, 258 Gross cost 189 Identity 131 Inconsistent equations . . . 250 Indeterminate equation . . 221 Integers, system of . .68 Interest . . 22, 80, 206, 213, 258 Interpolation 254 266 INDEX PAGE Laws of exponents .... 95, 120 of proportion 183 of signs . . 88, 89, 94, 120, 151 Linear equation .... 131, 251 List price 189 Lowest common denominator . 151 terms 150 Mean proportional . . . 183 Means 183 Members of an equation '. . . 30 Mixed expression 154 Monomial 16 Multiple pictogram 45 Multiplication . . .65, 98, 94, 157 Negative numbers .... 57, 58 Net cost 189 Nel^cost-rate factor 194 Net price 189, 192 Net proceeds 202 Net-profit graph 256 Nines, check of ... 102 Numerator 149 Numerical equation 132 One-day method 211 Order of operations 16 Ordinate 246 Origin 246 Parentheses 16; 88, 89 Percentage 38, 86 Pictogram .... ... 43 Plotting 246, 248 Polynomial 16 Positive numbers 58 Power 13, 20 Prime factor 13 Primes and subscripts .... 167 Principal 202, 206 PAGE Proceeds 207 Profit 189, 198 Proportion 183 Quadrants 246 Quantity, unknown 30 Ratio 179 Reciprocal 158, 199 Reduction of fractions . . 149, 154 Root 14, 132 Satisfying an equation . . . 131 Short methods 124 Sign of a fraction 151 Signs, laws of . . 88, 89, 94, 120, 151 Similar monomials 71 Simple equation 131 Simplifying 3 Simultaneous equations . . . 221 Six per cent method .... 209 Solid pictogram 46 Solvency, ratio of 241 Solving an equation . . . .30, 131 Special products . . 105, 107, 111 Square 13, 15, 20, 105 root 14, 15 Subtraction 63, 71, 82 Symbols 1, 4, 15, 16, 58 Tables 20, 195, 200, 213, 215, 237, 252 Terms . . 16, 149, 179, 183, 192 Three unknowns .... 140, 234 Trade discount 192 Transposition 132 Trinomial 16 Two unknowns 134, 222 Unknown quantity 30 Zero 68