i!^:i^r:m:^-j^ M t:-lr Digitized by the Internet Archive in 2008 with funding from IVIicrosoft Corporation http://www.archive.org/details/elementaryalgebrOOstonrich .r\. ELEMENTARY ALGEBRA FIRST COURSE BY JOHN C. STONE, A.M. HEAD OF THE DEPARTMENT OF MATHEMATICS, STATE NORMAL SCHOOt MONTCLAIR, NEW JERSEY, CO-AUTHOR OF THE 80UTHWORTH-8TONB ARITHMETICS, THE 8TONE-MILLIS ARITHMETICS, SECONDARY ARITHMETIC, ALGEBRAS, GEOMETRIES, ETC. AND JAMES F. MILLIS, A.M. HEAD OF THE DEPARTMENT OF MATHEMATICS, FRANCIS "MT. PARKB9. SCHOOL, CHICAGO, CO-ADTHOR OF THE 8TONE-MILLIS ARITHMETICS, SECONDARY ARITHMETIC, ALGEBRAS, AND GEOMETRIES ^v TToXX' dXX.a TToXv BENJ. H. SANBORN & CO. CHICAGO NEW YORK BOSTON 1915 '■"? 3ii CJOPTBIGHT, 1911, BY JOHN C. STONE and JAMES F. MILLIS i PREFACE The work in this book is that comprised in the usual first-year course in algebra in secondary schools. The material is so arranged as to adapt the book to the use of those schools that give only a brief course in the subject, as well as schools giv- ing more extended or more difficult courses. A more advanced book, a companion to this, called the Second Course, reviews the important topics covered in the First Course, amplifies the treatment of these topics where necessary, and covers all addi- tional topics that are required for college entrance. In the preparation of this book, the authors have combined what teachers have found to be the permanently valuable features of their Essentials of Algebra with those features which charac- terize the most modern ideas in the teaching of secondary school mathematics, and which, through widespread test by application in the classroom, have become generally accepted. It is believed that this text will be found teachable and well balanced. It is free from "fads" and untried experiments, yet sufficiently in harmony with present tendencies to meet the demands of the most progressive teacher. The book contains a minimum of theory and a maximum of practice. Abundance of drill is provided throughout the book. New processes and principles are psychologically and adequately developed, but they are finally mastered by the pupil through plentiful application in drill exercises and problems. Throughout the text the work has been planned so as to intro- duce the pupil to only one new difficulty at a time, and this is mastered through copious drill before proceeding to the next. This controlling principle in the development and sequence of subject matter is one of the fundamentally important features iii :lv. './::«:; : preface .^if'ijiie^'fciook. J 0^)serve, for example, the introductory chapter, in* which'oiie hew point of notation is developed at a time. Observe, also, the entire treatment of the equation. The pupil is led to see the subject, not merely as a system of exercises to be pursued for the purpose of mental discipline, although the authors are thorough believers in the mental disci- pline feature of the study, but rather as a scientific instrument for solving certain types of problems such as are actually en- countered in the world's work. See §§ 1, 2, 15, 16, 17, 22, etc. To this end adequate use has been made of real applied prob- lems of the various types encountered in practical life. Thus, in Chapter I computation by use of practical formulae is intro- duced. In Chapter II the equation is introduced as a scientific instrument used in the solution of practical problems. In Chap- ter III many practical applications of negative number are made, etc. These real applied problems lend intense interest to the work. Through them the work is legitimately motivated. Furthermore, it is through the application of the pupil's knowl- edge of algebra in the solution of these real problems that the knowledge is led to function. The old-time useless puzzles have been eliminated from this book. Algebra has been correlated with arithmetic and geometry. Beview and use of the various process with whole numbers, common and decimal fractions, have been made in the practical computation with formulae, and in the process of checking work by evaluation of the literal expressions involved. Short methods of multiplication of arithmetical numbers are taught in connec- tion with algebraic multiplication and factoring, as in §§ 75, 77, etc. Principles and processes in arithmetic have been made the starting points in the development of many principles and processes of algebra, as in §§ 50, 51, 54, etc. Considerable use has been made of percentage in the solution of certain types of business problems. The facts of mensuration have been much used, and some empirical knowledge of simple facts in geometry with which the pupil is familiar has been drawn upon in problems. The graph has been used as a natural means of solving certain PREFACE ! ; ;« r; v types of problems and of interpretation of algebraic principles, rather than as an excrescence in the form of topics of anttlytio geometry. Supplementary exercises have been given at the end of each chapter. They adapt the text to the use of schools with different kinds of courses. These exercises, as a rule, are more difficult than those in the body of the chapters. Schools wanting a brief course may omit these entirely. They may be used for reviews or in classes wanting a more difficult course. The authors wish to acknowledge their indebtedness to all those whose timely suggestions and criticisms have helped in the preparation of this textbook, and especially to Mr. K,. G. Kinkead, Assistant Superintendent of Schools, Columbus, Ohio; who has read critically all of the manuscript. JOHN C. STONE. JAMES F. MILLia MAY. 1011. • • • ' •" CONTENTS CHAPTER »»AGB I. The Formula: General Number 1 II. The Equation 19 III. Positive and Negative Numbers 34 ^ IV. Addition and Subtraction of Literal Expressions . 60 ^^ V. Multiplication and Division of Literal Expressions 74 VI. Linear Equations : Problems 92 VII. Special Products and Quotients 117 VIIL Factors. Multiples. Equations solved by Factoring 136 <^1X. Fractions 164 ^X. Fractional Equations. Problems. Formulae . . 189 XI. Proportion. Variables 206 XII. Systems of Linear Equations 232 JXIIL Square Root. Quadratic Surds 255 XIV. Quadratic Equations 269 XV. Systems involving Quadratic Equations . . , 286 XVI. Exponents 294 Miscellaneous Exercises 300 vB ELEMEI^TARY ALGEBRA FIRST COURSE CHAPTER I THE FORMULA: GENERAL NUMBER 1. Algebra. — Of the many kinds of problems encountered in the world's work, some are solved by arithmetic alone, some by algebra^ and some by other branches of mathematics. Algebra, like arith- metic, deals with numbers. There is no clear line of distinction between these two subjects. But in addition to the Hindu sym- bols of arithmetic, algebra employs letters to represent numbers. In addition to the integers and fractions of arithmetic, it deals with new kinds of numbers. And, as we shall see, it employs some new and interesting principles in the use of numbers. 2. The Formula. — Many of the rules which the student has already encountered in arithmetic, as well as practical rules which are used in the various fields of manufacturing, the trades, the sciences, etc., are expressed often by formulae, in which the numbers are represented by letters. A formula expresses a rule in a sort of shorthand. This will be seen clearly in the follow- ing problems. EXERCISES 1. How many square feet in a floor 12 feet wide and 15 feet long ? 2. A concrete walk is 4 feet 6 inches wide and 24 feet long. Find the area of its surface. t 2 ELEMENTARY ALGEBRA 3. What is the area of a rectangle 3.4 inches wide and 7.8 inches long ? 4. In general, the rule for finding the area of any rectangle that is I feet long and w feet wide may be expressed by the formula : Area = 1 xw square feet. . Find the area when I = 3i feet and , w = 2 feet. When Z = 16 feet and w = 2^ feet. When I = 50 feet and ^y = 4 J feet. When I = 136 feet and w = 80 feet. 3. Particular and General Numbers. — In Problem 4, § 2, the let- ter I is used to represent the length of any rectangle, and hence if we consider all possible rectangles, I may have many particular values. Similarly, if we consider all possible rectangles, w may have many particular values. It is seen, then, that in express- ing a rule by a formula, letters are used to represent numbers, and to such letters may be assigned many particular values. Compared with the particular numbers of arithmetic, 1, 2, 3, 4, etc., numbers represented by letters are called general numbers. Numbers represented by letters are also called literal numbers. 4. Factors. Signs of Multiplication. — The numbers which are multiplied to form a given number are called its factors, as I and w inl X w. In addition to use of the sign x to indicate multipli- cation, as in Problem 4, § 2, multiplication is sometimes expressed by placing a dot between the two factors, halfway up from the lower edge of the number-symbols. Thus, Ixw may be written also I • w. But unless both of the numbers multiplied are represented by Hindu symbols, the multiplication usually is expressed by omitting the multiplication sign. Thus, the product I x w is written Iw, and 2 x lo is written 2w, Express in three ways the product of 5 and I, / THE FORMULA : GENERAL NUMBER 3 EXERCISES 1. Express the product of a and b; of 2, x, and y. Give the value of 2ab when a and b have the following values : 2. a = 2, 6 = 3. 5. a = 10, 6 = 30. 8. a = 25, 6 = 4. 3. a = 5, 6 = 6. 6. a = 40, 6 = 20. 9. a = 75, 6 = 6. 4. a = 3, 6 = 7. 7. a = 50, 6 = 30. 10. a = 60, 6 = 3^. 11. Kead Exercise 4, page 2, and give a meaning for J. = Iw. Find the value of A when Z and w have the following values : 12. Z = 18, w = 10. 14. I = 13^, w = 12. 16. I = 2.4, w = 1.5. 13. Z = 12.5, IV = 8. 15. I = 6.2, w; = 4. 17. Z = 8.5, w = 3.2. 18. When 6 represents the base of a triangle, ^ the altitude, and A the area, the area is found by the formula A = ^bh. Give the rule which this formula expresses. In the formula A = ^bh, give the value of A when 6 and h have the following values : 19. 6 = 12, h = 10. 21. 6 = 20, h = 16.5. 23. 6 = 16, ^ = 9.5. 20. 6 = 14, h = 12. 22. 6 = 8.5, h = 12. 24. 6 = 12.5, ^ = 8. 25. If c is the circumference and d the diameter of a circle, then c = ttcZ. Compute the circumference of a circle from the for- mula c — ird when d = 10. (tt = 3.1416, approximately.) 2-6. When r = the radius of a circle, c = 2 7rr. Find c when r = 40 ; r = 60 ; r = 7.5. 27. The volume of a prism is found by multiplying the area of the base by the height or altitude. If F= volume, 6 = area of base, and h = altitude, then F= bh. Find V when 6 = 40 and /i = 30 ; 6 = 48 and ^ = 12 ; 6 = 96 and h = lOJ. 28. If the base of a prism is a rectangle whose length is Z and whose width is w, then 6 = Iw. Hence bh becomes lich. Then we have F= Iwh. Express the meaning of this formula. ELEMENTARY ALGEBRA In the formula V= Iwh, find V when I, w, and h have the values : 31. I = 36, w 30, h 18. 29. I = 42, w = 16, A = 14. 30. / = 38, w = 20, h = 12i 32. / = 42, w = 38, /i = 12. 33. The volume of a pyramid is computed by the formula Vz= -1 5^, where F= volume, 6 = area of base, and h = altitude. State in words the rule which this formula expresses. Find V when 6 = 12 and h = 4i 34. The volume of a cylinder is equal to the product of the altitude by the area of the base. Express this by a formula. 35. The volume of a cone is one third of the volume of a cylinder of the same base and the same altitude. If F = volume, h = area of base, and h = altitude, write the formula for computing the volume of a cone. If the base of a cone is 36 square inches, and the altitude 1 foot 6 inches, find its volume. 36. The rule for finding the area of a circle whose radius is r is expressed by the formula, A = ttt^. Give the rule which this formula expresses. Find A when r = 4. When r = 10. 37. Find the area of a circle whose radius is 6 inches. 38. When there is a steam pressure of 90 pounds per square inch in the cylinder of an engine, what is the total pressure on a 12-inch piston ? (By a 12-inch piston is meant a piston whose diameter is 12 inches.) THE FORMULA: GENERAL NUMBER 6 5. Powers. — In Problem 36, § 4, r^ is a short way of writing rr. Similarly, rrr is written r^, aaaa is written a*, 10 x 10 x 10 x 10 x 10 is written 10^, etc. In general, when all of the factors of a product are equal, the product is called a power of one of the factors, and is written in this abbreviated form. x^ is read " x square," or " x second power " ; a^ is read " x cube,'' or *' x third power " ; a^ is read "cc fourth power " ; jc* is read " x fifth power " ; etc. In a power such as m^, the factor m is called the base, and the number 7 is called the exponent. The exponent tells how many times the base is used as a factor to make the power. EXERCISES 1. Name the bases and exponents in the following: a^, y"^y C, A"", 9^, x\ 2. Express with exponents : mmmmmm, bbbbbbbb, 2x2x2 X2x2x2x2x2x2x2. 3. Find the value of 2x2x2; of 2^; of 3"; of 2*; of 3*; of 2«. 4. In the formula, A = ttt^, find A when r = 6.5. 5. The area of a circle is found also by use of the formula, A = \ ircPf where d is the diameter. Express this in words as a rule. Find A when d = 4^. 6. The area of the surface of a sphere is computed by the formula A = 4:7r7^, where ^ = area, r = radius. Find the area if the radius is 12 inches. 7. The earth is approximately a sphere whose radius is 4000 miles. Find its area. 8. The rule for computing the volume of a cone of which the iase is a circle is expressed by V = i irr^h, where r = radius of base and h = altitude. Find V when r=iQ and ^ = 4. 6 ELEMENTARY ALGEBRA 9. The volume of a sphere is computed by the formula, K = 4- Trr^, where V = volume and r = radius. Find V when r = 3. 10. The diameter of the moon is 2160 miles. Find its volume. 11. The bases or parallel sides of a trapezoid are 6 inches and 10 inches, respectively, and the altitude or distance between them - is 8 inches. Find the area. il2. The area of a trapezoid is equal to the product of one half of the altitude by the sum of the bases. This is expressed by the formula, A = ^7i (a + b)^ where h is the altitude and a and 6, respectively, are the bases. Find A when h = 4: feet, a = 3 feet, and 6 = 6 feet. 6. Use of Parentheses. — In the formula in Problem 12, §5, the symbols ( ), which inclose a + 6, or the sum of the bases, are called parentheses. These symbols are used to inclose two or more numbers which are to be added or subtracted, etc., and indi- cate that those numbers are not to be used singly, but are first to be combined into one number and then the result used. Thus, ^h(a+b) indicates that a and b are first to be added, then the result multiplied by ^ h. Similarly, x(a^ — b^) indicates that a^ — b^, as one number, is to be multiplied by x ; that is, we must first square the values of a and b and subtract, then multiply the difference by x. And (m4-w)(m — w) indicates that we are first to find the sum of m and n, then their difference, then multiply these results. In using a formula containing parentheses, the operations with the numbers within the parentheses must be performed first, 7. Other Signs of Grouping. — There are other symbols having the same meaning as parentheses that are used sometimes to in- close numbers. They are brackets [ ] and braces { j . A fourth symbol, called the vinculum, is sometimes used. It consists of a straight line drawn above the numbers, as a-\-b, and is used especially in expressing roots and fractions. THE FORMULA: GENERAL NUMBER Thus, B{M- N), BIM-N}, B{M- N], and J5 • ilf- iVall indicate the same thing, viz. that N is to be subtracted from M^ and the remainder mul- tiplied by B. All of these symbols are called signs of grouping or signs of aggregation. EXERCISES 1. In the formula -4 = ^^ (a + 6), find A when ^ = 4, a = ^, and 6 = 7. i ; 2. Find the value of d{x — y) when c? = 12, a; = 5, 2/ = 1. 3. Find the value of (m — n){in + n) when m = 10 and 71 = 6. 4. Find the value of D^ JZ> - 5j when Z> = 8. 5. Find the value of [A -\- B^[A - B^^A^ -{- S';\ when A = 5 and J5 = 3. 6. Find the value of {x-^yy when a; = 4 and y = 2. 7. Find the value of 5(R-{-JS)\E- Sy when R = 10 and iS = 2. 8. Find the value of (g + 10) -^{5-g) when gr = 3. 9. In the formula, S = a(t — ^)j find the value of S when a = 32and« = 3. 10. In the formula A = 7r(R^— 1^), find the value of A when i2 = 14 and r = 6. 11. In the formula S = 2 TrR{H-\- R), find the value of S when i? = 4and^=5. Letting the hypotenuse of a right triangle = z, the base = x, and the altitude = y, it has been shown in arithmetic that z = Var^-h^'. Find the value of z when : 12. x=3,y = 4:. 14. a; = 40, ?/ = 30. 16. x=12,y = 16. 13. x = 6,y = S, 15. a;=15, ^ = 20. 17. x = lS,y=24:. In the formula x = Vz^ — ^, find the value of a; when : 18. z=5yy = 4. 20. « = 25, y = 20. 22. z = 50,y = 30. 19. 2 = 10, 2/ = 6. 21. 2 = 15, 2/ = 12. 23. 2 = 35, 2/ = 28. 8 ELEMENTARY ALGEBRA 8. Number Expressions. — Any number symbol, or combination of number symbols, indicating one or more of the operations of addition, subtraction, multiplication, etc., such as those used in the formulae in the preceding sections, is called a number expres- sion, or simply an expression. Thus, irr^^ 2 J. — 3 ^ + C, and ^h(b + b') are expressions. An expression involving one or more numbers represented hj letters is often called a literal expression. 9. Terms. — In the expression 5 a^ — 2 a^ + 7 /, the parts 5 a^, 2 xy, and 7 y'^ are called terms. In general, the parts of an ex- pression connected by the signs -}- or — are called the terms of the expression. An expression such as 6 a^, or 10 x(m — n), which is not formed by two or more parts connected by the signs + or — , consists of one term. 10. Names of Expressions. — How many terms in 5aWb? In 2P2_Q29 InAA-B-C? lnn^-2n''-\-3n-l? An expression which consists of only one term is a monomial. An expression which consists of two terms is a binomial. An expression which consists of three terms is a trinomial. The name polynomial is applied to an expression which consists of more than one term. It must be remembered that an expression within a sign of grouping is to be considered as one term. Thus, 4 a — (& —3 c) is a binomial. Name the two terms. EXERCISES Tell how many terms in each of the following expressions, and apply the name "monomial," "binomial," etc., to each: 1. 25 A'BC. 5. (m + 7i)2. ^ hq-S-- 2. a^-f. 6. a* + b\ ' ^ 3. a2-f2«& + &l _1_ ^ 62.5 QH 4. irhir + ry ^' '^W ' 55^ THE FORMULA: GENERAL NUMBER 9 10. m22^(y-vj. 12. VZ+C. ^ , ^v 13. f-^y + Q, 11. .7854Wl--^)-Tr. ^ ^^ ^^ ^\^ lOOy 14. o?-{h-cf. 15. m^+w^ + j>^ + 2 ?/m + 2 ?/ip+ 2 np. 16. I^-iy + 2DC-CP, 18. rir^ + rsra + rirj. 17. 2xy-{x' + y'^ + z\ 19. (a + &)(a-2'). 20. a,-3 + 3a;22/H-3a;y2_|_2^3^ 11. Evaluation of Polynomials. — The computations by the formulae in the preceding sections have given practice in finding the values of expressions of one term. In finding the value of an expression containing more than one term, the value of each term must he found before tlie additions or sub- tractions indicated between the terms. Thus, if a; = 2, 3x2 _ 4a: 4- 7 = 12 - 8 + 7 = 11. And, if a = 12, 6 = 3, c = 2, and d = 4, 2c(a - 6) - (6 + c)d = 2 X 2 X 9 - 5 X 4 = 36-20 = 16. Here the values of a — 6 and 6 + c are computed first. See § 6. EXERCISES 1. Find the value of 2P-3Q + 7R-2S when P= 10, Q = 6, i2 = l, /S' = 4. 2. Find the value of 3P + 2 MN-{- N^ when JW= 4, JV= 5. 3. Find the value of a6 + 6c— cd— 5 when a = 10, 6 = 6^ c = l, d = S. 4. Find the value ofic^ + 5a^ — 2a; — 3 when a; = 4. 5. Find the value of y^ — oc^ when y =4:, x = 3. 6. Find the value of (u — v)^ — (w — xf when u = 12, v = S, w = 5f x=3. 10 ELEMENTARY ALGEBRA 7. Find the value of [2p + 3 g]^ - llOp - 2 g][4i) + g] when p=l,q=4.. 8. Find the value of VA^ -|. ^2 ^ 2 — VC- — 75 when ^ = 5, ^ = 3, C=10. 9. V=i7rh(f + r'^ + rry Find Fwhenr= 12,7-' = 15,^=18. 10. In § 10, Exercise 8, find value of expression when 11= 96, /) = 15. 11. In § 10, Exercise 11, find value of expression when D = 18, ^==60, i^=10, Tr=1140. 12. The volume of the frustum of a pyramid is computed by the formula F=i/i (b + B + VbB), where V= volume, 7i = altitude, b = are^ of one base, and B = area of other base. Find V when h = 6 inches, 6 = 16 square inches, and 5 = 49 square inches. In the formula V = ^ h(b -\- B -\- ^bB), find the value of Fwhen: 13. ;i = 6, & = 16, i? = 25. 16. ^ = 12,6 = 49,5 = 64. 14. h = S,b = 9,B = 36. 17. 71 = 16,6 = 25,5 = 36. 15. /i = 10, 6 = 25, 5 = 49. 18. ^ = 18,6 = 9,5 = 25. In the formula V = ^ 7rh(7^ + ?''^ + rr'), find the value of Fwhen: 19. r = 5, r' = 6, 7i = 9. 21. r = 12, r' = 15, 7i = 18. 20. r = 8, r' = 9, A = 6. 22. r = 10, r' = 16, Ji = 12. 12. Coefficients. — If an expression is separated into two factors, either factor is called the coefficient (co-factor) of the other. If a coefficient is an arithmetical number, it is called a numerical coefficient. Otherwise, it is a literal coefficient. In 6 xy, 6 is the numerical coeflBcient ot xy ; x is the coefficient of 6 ^ ; and y the coefficient of 6x. In 4w(a — 6), what is the numerical coef- ficient ? What is the coefficient of 4 (a - 6) ? Oia-b? If no numerical coefficient is written, the numerical coefficient 1 is understood. Thus, since x is the same as 1 x, the numerical coefficient of a: isl. Like« wise, the numerical coefficient of FQ is 1, and of (m — n) it is 1. THE FORMULA : GENERAL NUMBER 11 13. Similar Terms. — In what respect do 2 A^ 7 Ay and 10 ^ differ ? Terms which do not differ at all, or which differ only in their coefficients, are called like or similar terms. Thus, x^y, 4x2y, and 20x^y are similar. The terms, ax, bx, and ex are similar terms in x. 14. Addition and Subtraction of Similar Terms. — Just as the sum of 2 apples, 5 apples, and 8 apples is 15 apples, so the sum of 2 a, 5 a, and 8 a is 15 a. And just as 3 men + 4 men = 7 men, so 3 m -}- 4m = 7m. Similarly, 6 a; + 9 a; = (6 + 9)a;, or 15a;; C+5C + 30 = (l4-5 + 3)C,or90. To add similar terms, add their coefficients and to the result attach the common letters with their exponents. Just as 12 lb. -7 lb. =5 lb., so 12Z-7Z = 5?. And 32a;!/- 18 02/= (32-18) ajy, or Uxy, 16 a^W-^ a^h^={lh-^) a^h\ oi&a^hK To subtract similar terms, subtract their coefficients and to the result attach the common letters with their exponents. EXERCISES Find the sums in 1-12: 1. 2a, 3a, 4a. 15. 60z-f25z + 52! + 2«=? 2. 5a;, a;, 2a;. 16. 12d-4d=:? 3. lOTF, Sir, 8TK 17. 9ri-3ri = ? 4. P, 6P, 4P, 2P. 18. 16m'-12m'=? 5. ^t,t,lt,&L 19. 137rP2_57rP2 = ? 6. bSyl2S,S,2S. 20. 102/-102/=? 7. 4A;, 2A;, 9A;, 5A;. 21. 25D-17D=? 8. 6A,12AjA,A. 22. 69m;-37w; = ? 9. 3a;, 5a;, 9a;. 23. 20-8 + 10 = ? 10. mn, 4 mn, 3 mn, 10 mn. 24. 12 + 3 — 9 = ? 11. abc, 9 abc, 3 abc, abc. 25. 16 — 2 — 12=? 12. 4 J5;, 3^, 15^, 10^. 26. 6n-2 n -f-3n -4w =i? 13. 2aH-6a-h4a = ? 27. 12t -^2t -7 t-t = ? 14. 46 + & + 126=? 28. 37rZ) + 57rZ>- 7rZ)-4,ri?=? 12 ELEMENTARY ALGEBRA 29. Compare the values of 3 + 5 and 5 + 3. Of6a + 3a and 3 a + 6 a. Of 4 P+7 P+2 P and 7 P+2 P+4 P, or 2 P+4 P+7 P. 30. Compare the values of 5 + 2 + 6 and 5 + (2 + 6). Of 6^+4^+2iy and 6^+ (4 i/+2iJ). Combine the similar terms in the following: 31. 3a + 2a + 4 + 7. Solution. 3a + 2a = 5a; 4 + 7 = 11. Hence, 3a + 2a + 4 + 7 = 5a + ll. 32. 5^-2J\^+9-3. 36. 10m- 8 + 10 -8 m. 33. 122 + 4 + 6:3 + 2. 37. 3P-12-P + 20. 34. 9P-3 + 2P + 7. 38. ^+10 + 7iir-3-4/f-6 35. 20 + v + 5 + 4'V. 39. 12s + 18-s-12-5«. SUPPLEMENTARY EXERCISES Evaluate the following 1. ^ = Iwj when I = 16 and w = 14. 2. A = ^ bh, when 6 = 18 and h = 141. 3. C= ird, when d = 16 and tt = 3|.* 4. = 2 irr, when r = 12, and tt = 3|. 5. A = in^, when r = 9 and tt = 3.1416. 6. ^ = J Trd^, when d = 8 and tt = 3.1416. 7. ^ = 4 ir?-^, when r = 16, and tt = 3|. 8. ^ = Trd^ when d = 25 and tt = 3.1416. 9. F= i irr^h, when r = 6, /i = 12, and tt = 3.1416. 10. F= I Trr^, when r = 8 and tt = 3.1416. 11. A = \h{a + b), when ^ = 8, a = 10, and b = 15. 12. V=ih{b -{-B + VbB), when /i = 16, & = 25, and 5 = 49. 13. F=47r^(r2 + r'2 + rr'), when 7r=3i, /i=10, r = 6, and/=9. 14. Eecall the rules you studied in arithmetic, and give the meanings of as many of the above formulae as possible. * In practical work 3|, instead of 3.1416, is often used for the value of tt. THE FORMULA : GENERAL NUMBER 13 In the formula for simple interest, i=zj)rt, find i when: 15. i) = $500, r = ^%, t = 2. 18. p = ^900, r = U%,t = 3. 16. p = $1650, r = 6%, « = 3f 19. p = $1200, r = 5 %, i = If 17. p=$780, r = 5i%,^ = 2|. 20. i) = $1500, r = 5%, « = f. 21. The sum to which p dollars placed at compound interest will amount in t years at the rate per cent r is ^j(1 -f r)'. To what sum will $ 500 amount if placed at compound interest for 2 years at 6 % ? For 3 years ? 22. To what sum will $2000 amount if placed at compound interest for 3 years at 5 % ? 23. I invest $ 100 in the stock of a building and loan associa- tion which pays 7 % interest compounded annually. To what will it amount in 3 years ? 24. The number of ways that a committee of 3 persons may be selected from a group of n persons is ^ J . In ^ow 6 many ways may a committee of 3 be selected from 4 persons ? From 5 persons ? From 6 persons ? 25. If n gymnastic exercises may be taken in any order, the total number of different ways in which they may be selected to follow each other is w(?i — l)(?i — 2) •••3 • 2 • 1. In how many different ways may a series of 5 exercises be arranged ? 6 exer- cises ? 7 exercises ? Note. — The expression n(n — 1) (« — 2) • • 3 • 2 • 1 means for any number n, the product of this number and each consecutive number to 1. Thus, if n = 4, the expression becomes 4x3x2x1. 26. A party of 8 people secure a row of seats at a theater. In how many different ways may they be seated ? (See Problem 25.) 27. The area of an equilateral triangle, each of whose sides is s, is found by the formula A = \s^ V3. Find A when s = 8. (V3 = 1.732). 28. By drawing the figure of a race track with two straight parallel sides S, and with semicircular ends each with radius r, 14 ELEMENTABY ALGEBRA you will see from your knowledge of arithmetic that the distance, J), around the track is expressed by: D = 2(S -\-7r 7^). Find D when S = 1 mi. and r = ^ mi. 29. In a figure (polygon) with n equal angles, the number of degrees in each angle is 5^-11 — I. How many degrees in each of n the equal angles of a triangle ? Of a figure with 5 equal angles (pentagon)? Of a figure with 6 equal angles (hexagon) ? Of a figure with 12 equal angles ? SUPPLEMENTARY VOCATIONAL FORMULJE Note. — The following exercises are added for those who wish to give a more extensive course in the evaluation of formulce. It is not intended that the teacher should make any attempt to explain the meaning of any of them. The uses of the various formulae are stated in the belief that students will be more interested in the process of evaluation if they know that such a thing must be done by people doing the world's work. They may be omitted with- out interfering with the subsequent work. F 1. The formula (7 = — is much used in work with electricity. Compute C Aen E = 20.5 and B = 16.75. 2. The velocity of the recoil of guns is computed by the for- mula F=— , where F= velocity of recoil, Tr= weight of gun W and carriage, in pounds, w = weight of projectile, and v = muzzle velocity of projectile. A lO-inch gun on a battleship fires a 400- pound projectile with a muzzle velocity of 1600 feet per second. Weight of gun and carriage, 22 tons. Find velocity of recoil. / 3. The force of pressure P of the wind, in pounds per square foot, is computed from P= .005 FV where F= velocity of wind in miles per hour. Find the force of the wind when blowing at 40 miles an hour. What would be the total pressure of this wind against the side of a house 20 feet high and 60 feet long ? THE FORMULA: GENERAL NUMBER 15 4. If an object, such as a brick dislodged from the wall, starts from rest and falls towards the earth, the distance that it will fall in a given length of time is computed by the formula s=^at\ where s = distance in feet, a = 32, and t = number of seconds elapsed. Find the distance an object will fall in 1 second ; 2 sec- onds ; 3 seconds j 4 seconds ; 10 seconds ; 60 seconds ; 5 minutes. Note. — This formula holds accurately only for bodies falling in a perfect vacuum. For bodies falling through the air, the velocity is somewhat di- minished by the resistance of the air. 6. How far would a body fall during the sixth second ? Suggestion. — Find the distance it would fall in 5 seconds and in 6 seconds. 6. How far would a falling body move during the thirtieth second ? 7. To measure temperature, two different kinds of thermom- eters are in use: the Fahrenheit and the Centigrade. On the former the freezing point is marked 32° and the boiling point 212°. On the latter these are marked 0° and 100° respectively. If the temperature is read on a Fahrenheit thermometer, the correspond- ing temperature on the Centigrade thermometer is computed by the formula C = |(F — 32), where C = temperature in degrees on Centigrade scale and F = temperature in degrees on Fahrenheit scale. When it is 70° by the Fahrenheit thermometer, what is the temperature on the Centigrade ther- mometer? When 64°? When 48°? When 80°? 8. The strength or capacity for work of engines is expressed by horse power. The horse power of steam engines is found by the formula C F 100 -212 -17 78 H.P. plan where p = pressure of steam in 32 I 33000' pounds per square inch, I = length of stroke in feet, a = area of piston in square inches, and n = twice the number of revolutions 16 ELEMENTARY ALGEBRA per minute. Compute tlie horse power of an engine in wliicli a test shows p = 95 pounds, ^ = 30 inches, a = 706.8 square inches, and n = 100. 9. Find the horse power of a steam engine in which p =r 110 pounds, 1 = 24: inches, w = 120, and the diameter of the piston is 16 inches. 10. The horse power of automobile engines is computed by the formula H. P. = KND(D - 1){E + 2), where K== .197 for com- mercial touring cars, JV= number of cylinders, i>= diameter of cylinders, and E = ratio of the stroke to the diameter. What is the horse power of a 4-cy Under engine of a touring car in which the diameter is 4 inches and the stroke 5 inches ? 11. If a beam L feet long is supported at both ends and loaded uniformly throughout its length with W pounds per foot, the greatest bend or deflection D at the middle, in inches, is obtained from 5 WL the formula D = , where E and 384 El' I have particular values depending upon the material used. This formula is used by architects in designing buildings. If a beam in which E = 30,000,000 and J= f is 12 feet long, and the load 200 pounds per foot, find the deflection. 12. The elevation of a point above sea level is obtained by use of the thermometer from the formula ^=513^+ f, where H=: height in feet above sea level, and t == difference (in degrees Fahrenheit) between 212° and the temperature at which water boils at the place of observation. The temperature of boiling water at a certain place is 210°. Find the elevation of the place. 13. The relation between the height of a chimney and the pressure of draft which it produces is given by the formula P=if(^-^-^), where P B= pressure of draft as measured by the height in § THE FORMULA: GENERAL NUMBER 17 inches of a column of water that it will support in a tube, H = height of chimney in feet, T = temperature outside, and t = temperature of air in chimney. If if = 150 feet, T= 50°, and t = 600°, find P. 14. In electrical work problems of the following kind are en- countered: An electric current flowing from a point u4 to a point B is divided at A into three branches, each passing through an ^ [A ^ ^^ electric bell, and the branches are united again into one current at B. The total resistance R of the circuit from ^ to 5 is computed by the r,roro , where r,, r,, and rgare the respective formula R = - resistances of the three branches. If ?*i = 1.2 ohms, r2 = 1.4 ohms, and 7'3 = 1.6 ohms, find the total resistance R. Note. — Problem 14 shows how the system of representing numbers by letters may be extended by attaching subscripts to the letters of the alphabet. Thus, by attaching subscripts to the letter a we can create any number of new symbols for representing numbers, as rti (read a sub 1), aa, as, 04, etc. The symbols ai and aa represent distinct and unrelated values, just as a and b do. Similarly, by attaching superscripts to letters, new number symbols may be formed, as x' (read x prime), x" (read x second), x'", x'^, x^, etc. Use is sometimes made of letters of the Greek alphabet, a (alpha), /3 (beta), 7 (gamma), 5 (delta), etc. 15. The horse power that may be transmitted safely by a certain kind of shafting without breaking or twisting is computed by the formula H. P. = , where n = . • 64' number of revolutions per minute, and d = diameter of shaft- ing in inches. How many horse power can be transmitted by such a shafting of 4 inch diameter, making 76 revolutions per minute? By one of 5 inch diameter making 100 revolutions per minute? 18 ELEMENTARY ALGEBRA 16. When a brick arch is supported by a tie rod to keep the walls from spreading, the "horizontal thrust," or stretching force exerted on the rod, in pounds per linear foot of arch, is obtained by the formula . ' i ' i ' ' !j^ -fnH V.ij!j!v ! F= — , where W= weight on arch H in pounds per square foot, S = span of arch in feet, H = rise of arch in inches. Find the strain on the tie rod in an arch on which the weight is 360 pounds per square foot, the span 4 feet, and rise of arch 18 inches. 17. The discharge of a pump in gallons per minute is obtained from the formula G = .03264 Td^, where G = number of gallons, T= travel (total distance traveled) of piston in feet per minute, d = diameter of cylinder in inches. Suppose that the diameter of the cylinder of a pump is 18 inches, that the stroke of the piston is 24 inches, and that it makes 40 revolutions per minute. Find the discharge. 18. The horse power of the pump required in Problem 17 is H.P. = .00001238 Td% where h = vertical distance in feet between levels of water at source and point of discharge. Find the horse power of the pump in Problem 17 required to raise the water to a standpipe through a height of 216 feet. 19. By the specific gravity of a solid substance, such as iron, is meant the ratio of the weight of that substance to the weight of an equal volume of water. The specific gravity of an object may be found by first weighing it in air, then weighing it again when suspended under water. An object seems to lose weight when weighed in water due to the buoyancy of the» water. If the specific gravity is s, then s = — — , where W= weight in air and w = weight in water. A piece of glass weighing 40 grams in air weighs 24 grams in urater. Find its specific gravity. CHAPTER II THE EQUATION 15. The Equation. — In solving some kinds of problems use is made of equations. An equation is the statement that two expres- sions are equal or that they have the same value. Thus, 2 X 6= 10, a + 6 = 6 + a, and 4 »i — 3 = 2 n + 5 are equations. The two expressions which are connected by the sign = are called the members of the equation. For example, in 3 P— 2 = 4 the expression 3 P— 2 is called the first or left member, and the 4 is called the second or right member. In some equations the members are equal for sfll particular values of the general number involved. These are called identical equations or identities. Thus, in a^ + 2 a5 + 6^ = (a -f 5)2 the members are equal for all vahies whatever that may he given to a and b. When a = 1 and 6=2, the equa- tion becomes 9 = 9; when a = 2 and 6 = 2, it becomes 16 = 16 ; when a = 5 and 6 = 1 , it becomes 36 = 36 ; etc. In other equations the members are not equal for all particular values of the general numbers involved. These are called con- ditional equations or simply equations. Thus, 5y — 2 = Sy is true only under the condition that y = 1. And 3 ^ + 4 = 10 is true only when B is 2. 16. Problems Expressed by Equations. — In every problem solved by use of equations, the values of one or more numbers are unknown. For example, in the problem, " If eggs cost 34 cents a dozen, find the cost of 5 dozen," the cost of 6 dozen is an unknown number. 19 20 ELEMENTARY ALGEBRA To solve a problem is to find the values of the unknown numbers. In solving a problem algebraically it is first expressed in a sort . of shorthand by an equation. The equation expresses the rela- tion between the known and unknown numbers. An unknown number is expressed by a letter, and later the value represented by the letter in the equation is found. \ Example 1. A man's salary was increased by 10%, or -^^ of itself. After . * the increase he received, annually, $1980. What was his salary before the fi-^ increase ? fJ 'jr Represent by d his salary before the increase. \^ Then the problem may be expressed hy d-\- ^-^0, = ^ 1980. jf Example 2. Two men form a partnership in business in which the first I I) invests twice as much as the second. Their profits are $3600. What part of the profits should each receive in settlement ? Let a represent the amount the second should receive. Then 2 a represents the amount the first should receive. Hence, the problem may be expressed by the equation 2a + a = To express a problem by an equation we take the following steps : (1) First read the problem to discover what numbers are uriknown. (2) Let some letter represent one of the unknown numbers. (3) Tlien, from statements in the problem, express all the other unknown numbers in terms of this letter. (4) Finally, from another statement of the problem, form the equation between these and the known numbers of the problem. ORAL EXERCISES 1. If n denotes a certain number, what will denote a number 10 larger? What will denote one 10 less? One 10 times as large ? One -Jg- as large? 2. The rainfall last year at a certain place was 6 inches more than on the year before. If x represents last year's rainfall, what will denote the rainfall of the year before? If x represents the rainfall of the year before, what will denote that of last year? THE EQUATION 21 3. The melting temperature of glass is 136 degrees lower than 3 times that of zinc. If t represents the melting temperature of zinc, what will represent that of glass ? 4. One boy sold 20 more than ^ as many papers as another. By use of some letter represent the number sold by each. 5. If A has $2/, B I as much, and C ^ as much, what will represent the amount that all three have ? 6. Of two candidates at an election one was defeated by 362 votes. By some letter express the number of votes received by each. 7. A rectangle is 2 inches more than 3 times as long as it is Tade. By use of a letter represent both the length and the width. 8. A merchant sold coffee at a profit of 20%. If it cost C cents a pound, what will express the selling price ? 9. If I lend $/) at 5% for 4 years, what will express the amount at the end of that time? 10. The distance from Chicago to New York by rail is about 900 miles. If a train ran an average of 40 miles an hour, how long would it require to make the run between the two cities ? If it ran V miles an hour, what will express the time required for the run? In each of the above problems one number has been expressed in terms of one or more other numbers, WRITTEN EXERCISES Express the following problems by means of equations : 1. One of two partners in a business invests twice as much as the other. How should a profit of % 1200 be divided? i 2. A real estate dealer sold a lot for $1500, and thereby made a profit of 25 %. What did it cost him ? 3. It requires 2240 feet of wire fencing to inclose a rectangular piece of ground that is three times as long as it is wide. Find its width and length. 22 ELEMENTARY ALGEBRA p 4. If I think of a number, double it, and add 8 to the result, I get 40. What is the number ? 7 5. In the latitude of Chicago, on the longest day of the year, the day is 6 hr. 8 min. longer than the night. What is the length B of each? J\6. In the triangle ABC, the sum of the three angles is 180°. Angle B is twice as large as angle A, and angle C is three times as large as angle A. How many degrees in each angle ? 17. Solving an Equation. — It has been seen that a problem may be expressed by an equation, in which the unknown numbers are represented by use of a letter. It is evident that the problem may be solved if the value of the letter in the equation may be found. This is called solving the equation. To solve an equation is to find the particular value or values of the unknown number that make the two members equal. A par- ticular value of the unknown number thus found is called a root of the equation. Thus, in 4^4-6 = 2y-\- 10, if y is 2, each member equals 14. Hence, 2 is a root. The equation is said to "be satisfied " when y = 2. Before proceeding to the solution of problems, we must discover how to solve an equation. 18. Axioms. — If the weights on the two pans of a balance are equal, they balance. If equal weights are added to, or taken from, the two pans, will the resulting weights balance ? If the weights in the two pans are made twice as great, three times as great, etc., or one half as great, one third as great, etc., will the resulting weights balance ? These facts about the balance illustrate certain general princi- ples in dealing with numbers which may be assumed. They are called axioms, and may be stated as follows : ^ THE EQUATION 23 1. If equal numbers are added to equal number Sy the sums are equal. 2. If equal numbers are subtracted from equal numbers, the re- mainders are equal. 3. If equal numbers are multiplied by equal numbers, the products are equal. 4. If equal numbers are divided by equal numbers (not zerd)^ the quotients are equal. An equation may be compared to a balance. The members of the equation correspond to the weights on the two pans of the balance. In the figure, the fact that the weights balance is expressed by the equa- tion P4-4 = 9. Just as the weights on the two pans may be equally increased, decreased, multiplied, or divided, and con- tinue to balance, so the members of an equation may be equally increased, decreased, multiplied, or divided, according to the above axioms, and remain equal. The use of these axioms in solving equations is shown in the following sections. 19, Equations Solved by a Single Addition, Subtraction, Multipli- cation, or Division. Example 1. — Solve Tr+6 = 10. Since TF+6 is greater than ir, what must be done to TT -1-6 to get W? Subtracting 6 from each member, W= 4. Axiom 2. Example 2. — Solve 8 a= 7a + 5. What must be done to 7 a + 5 to get 5 ? Subtracting 7 a fix>m each member, a = 5. Axiom 2. Example 3. — Solve S-i = 12. Since S — iisi less than S, what must be done to S-~ 4 to get S ? Adding 4 to each member, S = 10. Axiom 1. Example 4. — Solve 3 x = 21. What must be done to 3 x to get x ? Dividing each member by 3, sc = 8. Axiom 4 24 ELEMENTARY ALGEBRA Example 5. — Sdlve - = 4. 6 By what must - be multiplied to get n ? 6 Multiplying each member by 6, w = 24. ^A Solve : 1. a + 3 = 7. 2. P+6=20. 3. 'V + l = 8. 4. 72 + 9=16. ^5. 3a; = 2a; + 4. 6. 6^ = 5^ + 12. 7. 8A: = 23 + 7fc. 8. 9^ = 3 + 8^. 9. 162) = 54-152>. 10. 5-4 = 20. 11. a; -10 = 15. 12. Jf-50 = 12. ORAL EXERCISES 13. jj— 9 = 1. 14. R-2 = Q>. I 15. w-14 = 16. 16. x-^ = \. 17. m-li = |. 18. 5 2/ = 15. 19. 4ir=24. 20. 2a = 18. 21. 6Q = 48. 22. 7/S = 28. 23. 24Z> = 72. 24. 100^ = 125. Axiom 3. 25. 12ic = 25. 26. ^ = 25. 4 27. l^Z. 2S. ^^ = 4. 29. 3ij = r. 30. Q_ 31. ^=18, 20 WRITTEN EXERCISES Solve : 1. a; + 3.25 = 5. 9. ^-7f = 8i. V^ 2. i)- 0.426 = 4.32. 10. y-T6=T2' 3. 4.6^ = 3.6^ + 2.5. 11. ljr' = 6. 4. P- 4.08 = 8. 12. 2fv = 5f. 5. a. 2.16 m = .24. 3.1416^ = 9.5. 13. 3i ^ 7. 8. 4^(7=7 + 310. ^+3f = 7TV 14. ^ = 41 3f ^- THE EQUATION 25 20. Any Simple Equation Solved. — Any simple equation may be solved by using one or more of the processes employed in § 19, as shown by the following examples. Example 1. — Solve 4 w + 15 + 13 w = 5 n + 99. Adding similar terms, 4 n and 13 7i, 17 » + 15 = 5 » + 99. Subtracting 15 and 5 n from each member, 12 n = 84. Dividing each member by 12, w = 7. Notice that by subtracting 15 from each member we get the term free of n out of the first member, and by subtracting 5 n from each member we get the term containing n out of the second member. Check. — When n = 7, each member becomes 134, which shows that the answer is correct. A number symbol put in place of another in an expression, as in checking the answer in the preceding example, is said to be substituted for it. The student is familiar with substitution in the formulae in Chapter I. The root of evei'y equation should he checked^ or tested for accv^ racy, by substituting the value found in place of the unknown number in each member of the equation. Example 2. —Solve 7 F- 8 + 6 V= 28 + 4 F. By combining similar terms, what does the first member become ? What must be added to each member to free the first member of the term not containing V? What must be subtracted from each member to free the second member of the term containing V? Show that the resulting equation is 9 F = 36. By what must each member now be divided ? Check the answer by substituting its value in each member of the given equation and seeing if they are equal. Example 3. — Solve 2^+^ + 9=:^+ 16. What is the least number divisible by each denominator ? By what, then, must each member be multiplied to free each term of fractions. Show that the resulting equation is 8 a; + 9 a; + 108 = 10 x + 192. Now show that x = 12. Check by substituting 12 for x in the given equation. 26 ELEMENTARY ALGEBRA It is evident from these examples that the steps in solving such equations are as follows : (1) If the equation contains one or more fractions, multiply both members by the L. C. M. of all the denominators to free the equation effractions. (2) Unite similar terms by adding, or subtracting, as the case may require. (3) Free the first member of all terms that do not contain the un- known number, and the second member of all terms that do contain the unknown number, by adding the same number to both members or subtrojcting the same number from both members. To remove a term with a plus sign before it, subtract it, and to remove a term vnth a minus sign before it, add it to both members. (4) Divide both members of the resulting equation by the coejficient of the unknown number. (5) Check the work by substituting the value of the root found in each member of the given equation. EXERCISES Solve: 1. 7a-t-3=a + 21. 9. 6Jf-5 = 4Jf+l. 3. 4 6 + 4 = 25 + &. 10. 52/ + 5 = 8-32/. 3. 5P-4 = 12-3P. 11. 20P-25 = 5P + 5. 4. 2^-5 = 7-^. 12. 13/i + 15 = ll^ + 35. 5. 3Z>-7 = 14-4D. 13. 4:^-3 = ^. 6. 5a; + 3 = 3a;+9. 14. 12c- 13 + c = 35 + 7c. 7. 200^.-50=50^+250. 15. 13 F-7 = 5 F + 2- F. 8. 5 A; — 5 = A; + 3. 16. 3 a; = 10 + «. 17. What is the least number divisible by both 2 and 3 ? By what one number may both of the fractions - and - be multiplied to change them to whole numbers ? THE EQUATION 21 18. By what one number may each of the fractions ---, -, and f» 3 4 — ^ be multiplied to change all of them to whole numbers ? 19. What is 6 times 4 apples? 6 times 4a? 3 times 6«? 8 times In? 10 times 3^? 20. What is 6 times?? 8 times?? 10 times ?^? 12 times —? Solve: 4 3 2 6 23. l£ + i? = ^ + 7. 28. 3Tr-^=^+13. 3 4 6 3 2 24. 5 + 6 = 17-?:?. 29. 10-^ = - + ^ + 3. 2 ' 6 8 4 ' 2 ^-17 = 0. 30. i2-10+- = ^-^. 6 7 2 14 21. Problems Solved by Equations. — The following example shows the complete process of solving a problem by use of an eqviation. Example. — At an election two candidates, A and B, together received 2245 votes. A was elected by a majority of 286 votes. How many votee were cast for each ? Let n = number cast for A . Then n — 285 = number cast for B. Hence, n + n - 285 = 2245. Solving, 2 n - 285 = 2245. 2 n = 2530. n = 1265, number cast for A. Hence, «— 285 = 980, number CEist for B* 28 ELEMENTARY ALGEBRA EXERCISES 1. The initiation fee of a certain organization is $40, and the annual dues $5. In how many years will a member have paid into the treasury $ 100 ? 2. It costs $ 4 to have made the plates for printing a circular, and the cost of paper and printing is \ cent a copy. How many copies can I have made for $8? 3. One partner has 4 times as much money invested in an en- terprise as the other. How should a profit of $2500 be divided? 4. Two boys agree to mow a man's lawn during a season for $ 6. One mows it 3 times and the other 5 times. How should the money be divided between them ? j^'^- 5. A grocer bought some eggs at 16 cents a dozen. Thirty L' K V were broken, and he sold the remainder at 18 cents a dozen. He found that he got back just the cost of the whole lot. How many did he buy ? 6. Divide $ 75 between A and B so that A shall receive $ 3 more than twice as much as B. 7. Two boys sold 45 papers. One sold ^ as many as the other. How many did each boy sell ? 't 8. A boy has $1.05 in dimes and nickels. He has just as many dimes as nickels. How many has he of each ? 9. Two trains start at the same time from stations 450 miles apart, one running at the rate of 35 miles per hour, and the other at the rate of 40 miles per hour. In how many hours do they meet ? 10. The resistance of an electrical battery and the wire at- tached to it is found to be 3 ohms. By connecting to the battery a wire of the same size and material and three times as long, the resistance is found to be 6 ohms. Find the resistance of the battery. 11. The resistance of a battery and the wire attached to it is 5^ ohms. When a wire of the same kind and 4 times as long is \ THE EQUATION f9 :3onnected to the battery, the resistance is found to be 16 ohms. Find the resistance of the battery. ^ A 12. The total number of admission tickets to a circus was 836. The number of tickets sold for adults was 136 less than twice the number of children's tickets. How many were sold of each ? d*" 13. The distance from Chicago to San Francisco by rail is 2563 miles, which is 397 miles more than twice as far as from Chicago to Denver. How far is it from Chicago to Denver ? ^^14. In a class containing 24 boys and girls there are 6 more girls than boys. How many of each are there ? '\15. It takes 360 feet of wire fencing to inclose a rectangular lot 60 feet wide. How long is the lot ? 1 16. A boat steams against the current of a stream at a speed of 3 miles per hour, and with the current at a speed of 15 miles per hour. Find the speed of the current. y 17. A train left a station and traveled at the rate of 30 miles/ per hour, and three hours later a second train left the station, fol- lowing on the same track at the rate of 40 miles per hour. In how many hours from the time that the first train started was it overtaken by the second train ? 18. A bar 40 inches long is to be cut into two pieces. The shorter piece is to be | as long as the longer piece. Find the lengths of the pieces. 19. A board 62 inches long is to be sawed into two pieces so that one piece is 8 inches longer than the other. How long will each piece be ? 20. Surveyors, in order to mark off a right angle on the ground, sometimes set three stakes so that when a tapeline is stretched around them so as to form a triangle, it is divided at the stakes into parts whose lengths are as 3, 4, and 5. The ancient Egyptians used this method in laying out their pyramids, etc. If a 100-foot tapeline is 80 ELEMENTARY ALGEBRA used, find the lengths of the parts into which it must be divided at the stakes. Suggestion. — Let 3 n represent the number of feet in the shortest part. 21. If, in Problem 20, the surveyors use a 50-foot tapeline, find the lengths of the parts into which it must be divided at the stakes. 22. A surveyor's chain which contains 60 links is divided into three parts whose lengths are as 3, 4, and 5, and used instead of the tapeline in Problem 20. How many links in each part ? Note. — Students who have time will find it interesting to use the method of Problems 20-22 in making out-of-door measurements. 23.. A merchant sells tea for 70 cents a pound, and thereby makes a profit of 40% on the cost. What did it cost him ? 24. How much money must I invest in a business yielding 20% profit in order that the investment and profit together may amount to $1000? 25. The sum of the angles a, b, and c is 180°. If angle b equals angle a, and angle c equals one-half of angle a, find the value of — each. 26. The sum of the angles of a triangle is 180°. If two of the angles are equal, and the other one twice as large as either of them, how many degrees in each ? 27. An angle of 75° is made up of the sum of an angle of 15° and 5 equal angles. How many degrees in each of the equal angles ? 28. Two angles are called complementary when their sum is 90°. If one of two complementary angles is 15° more than twice the other, how many degrees in each ? 29. Three angles just cover all of the plane around a point The difference between the second and the first is 60° and be- tween the third and the second 60°. How many degrees in each ? THE EQUATION 31 30. If a polygon has n sides, the sum of all of its angles is 180° n- 360°. How many sides has a polygon the sum of whose angles is 720° ? 31. Whole numbers such as 5 and 6 are consecutive whole numbers. The sum of two consecutive whole numbers is 71. What are the numbers ? 32. The sum of two consecutive odd numbers is 32. What are the numbers ? 33. One number is twice as large as another. If I take 4 from the smaller and 16 from the larger, the remainders are equal. What are the numbers ? 34. A number, its half, its third, and its fourth make 100. Find the number. 35. In the papyrus written by Ahmes, the Egyptian, about 1700 B.C., the unknown number was called " hau." This ancient book contained the following problem : " Hau, its \, its whole, it makes 19 ; " i.e. \ of the unknown number and the unknown number make 19. What was the unknown number ? SUPPLEMENTARY EXERCISES 1. A certain number exceeds n times the number x by m. What will express the number? 2. D dollars are invested at p per cent interest. What will represent the sum to which this will amount in 1 year? In 3 years ? In ^ years ? 3. A merchant sells sugar costing c cents a pound at a profit of 12%. What will represent the selling price? If it sells at a profit of p %, what will represent the selling price ? 4. Solve ^ + | = 2i. 32 ELEMENTARY ALGEBRA 6. Solve— + — = — + 6J. 6. Solve 7 = !-?! + !• 6 9 4 7. I paid $6 for an advertisement of 6 lines, as follows: 20 cents a line for the first insertion, 10 cents a line for each of the next 6 insertions, and 2 cents a line after that. Find the number of insertions at 2 cents a line. 8. How much per bushel must a merchant pay farmers for wheat in order to market it at $ 1.14 and make a profit of 4 % ? 9. How much per bushel must a wholesale house pay for potatoes in order to sell them to retailers at 53 cents and make a profit of 6 % ? 10. How much must a country merchant pay farmers for tur- keys in order to make 6 %, if he can get 17 cents per pound foi them in the city market ? 11. In the city market, on a certain date, unwashed wool sold for 25 cents per pound. How mucTi per pound must a wool buyer pay farmers for their unwashed wool in order to make a profit oi 10% when selling it at this price, if it costs him 3 cents pel pound to market it ? 12. On a certain date hogs sold for $9.75 per hundred pounds in the Chicago market. What must a buyer pay for hogs in the country in order to market them at this price and make a profit of 20 %, making no allowance for cost of delivery ? 13. At what market price must 1-year 4 % bonds be offered for sale, in order that the buyer, by holding them until maturity, may make 8 % on his investment? Suggestion. — The profit made must come from two sources, the interest on the par vakie, which is $4, and the excess of the maturity value over the price paid for the bonds. If x is the price paid, 100 — x + 4 = .08 a;. 14. At what market price must 1-year 7 % bonds be purchased, if by holding them until maturity the purchaser makes 10 % on his investment? THE EQUATION 33 15. At what price must 3-year 5 % bonds (bonds that mature in 3 years) be purchased, if by holding them until maturity the purchaser makes 6 % on his investment ? Suggestion. — If x is the cost required, the annual profit due to excess of maturity value over cost is ^^^. o 16. At what price were Lake Shore 4's bought in 1910, which, if held until maturity in 1931, would earn the buyer 4| % on his investment ? 17. Lackawanna 5's bought in 1910 and maturing in 1923, if held until maturity, would pay the purchaser 6-f^ % on his invest- ment. What was the price of Lackawanna 5's when bought ? 18. Chicago & N. W. stock bought May 13, 1913, and sold May 13, 1913, at 173, after paying three 12 % dividends, would yield the purchaser 12J % on his investment. At what was the stock bought May 13, 1913? Suggestion. — If n = amount paid for a $ 100 share, show that 115-=-^ +12 = . 126 n. o 19. A merchant sold an article at a gain of 40%. When the cost increases $ 90, the same selling price will be a loss of 50 % , Find the cost. 20. A weight W is lifted by means of a rope running over two pulleys, A and B. The friction in each pulley increases the ten- sion in the rope 2%. Find ]r if the force required to lift it is 400 pounds. * This method of distributing the loss or gain at maturity does not consider interest upon the annual loss or gain. For a full discussion, see the Stone' Millis Secondary Arithmetic, page 90. CHAPTER III POSITIVE AND NEGATIVE NUMBERS 22. Extension of the Number System. — We have seen in Chap- ter I a class of practical problems which are solved by use of the formulay and which require a knowledge of literal notation. In Chapter II we have seen another class of problems to solve which requires a knowledge of the equation. There are still other prob- lems, now to be considered, which require us to extend our idea of number to include a new hind of number. If a man deposits $ 500 in a bank, and then withdraws $ 300, the amount of his balance is $ 500 — $300, or $ 200. It sometimes happens that a depos- itor withdraws more money than he has deposited^ or makes an overdraft. For example, suppose that he has deposited only $250, and withdraws $300. There is now a balance of $ 50 against him, showing that he owes the bank and not that the bank owes him. The balance is obtained by subtracting $300, the amount withdrawn, from $250; i.e. the balance is $250 — This problem, then, requires us to subtract a number from a smaller number. In order to solve such problems, our idea of number must be extended and this subtraction made possible. 23. Negative Number. — The remainder resulting from subtract- ing a number from a smaller number is called a negative number. We must now see how to represent a negative number. If a person wishes to withdraw $ 100 from a bank, he may withdraw part at a time. Thus, he may withdraw $75 with one check and then $25 with another. What are some ways that he may withdraw $250 from a bank, by withdrawing only part at a time ? Similarly, in general, one number may be subtracted from another by separating the subtrahend into two or more parts, and subtracting the parts one at a time. 34 POSITIVE AND NEGATIVE NUMBERS 35 Hence, in attempting to subtract 7 from 4, by separating 7 into 4 and 3, we get ^ 4-7 = 4-4-3 = 0-3. In practice the zero is dropped, and the remainder — 3 written simply —3. Similarly, 5- 9 = 5- 5 - 4 = -4 or -4. 8- 14 = 8-8-6 = 0-6 or -6. Hence, a negative number is expressed by a number preceded by the minus sign. It is clear that a negative number, such as — $ 25, indicates a reserved subtraction, there being nothing, when it stands alone, from which to subtract it. Thus, in $ 100 - $ 150 = $ 100 - $ 100 — $50 = — $50, the remainder, — $50, is a part of the subtrahend that is not subtracted because there is nothing from which to subtract it. Hence, a negative number is in nature a subtrahend. 24. Positive Numbers. — For the sake of distinguishing from negative numbers, the arithmetical numbers with which we have always dealt up to this time are called positive numbers. All numbers with which we are familiar, therefore, are to be classified as either positive or negative numbers. 25. Use of Signs. — For distinction in writing pos-itive and nega- tive numbers, the positive or arithmetical numbers are often pre- ceded by the sign + when standing alone. Thus, 6 is written +6; a is written + a. When clearness would not be sacrificed, however, the sign -f may be omitted from a positive number. If no sign is written before a number, the number is understood to be positive. It is clear, then, that the signs -h and — have two uses: to indicate addition or subtraction, and to indicate the positive or negative quality or character of numbers. They may always be considered as signs of addition or subtraction; and when conven- ient, as signs of quality. 36 ELEMENTARY ALGEBRA When standing alone, the positive numbers +1, +2, etc., are read either ''plus 1," "plus 2," etc., or "positive 1," "positive 2," etc. ; and the negative numbers — 1, — 2, etc., are read either " minus 1," " minus 2/' etc., or " negative 1," " negative 2," etc. EXERCISES Read the following numbers : 1. 5; 90; -1; -12; x; - n; \, - i; -2. 2 n What negative numbers result from the following subtractions ? 2. 3-4. 3. 8-13. 4. 10-12. 5. 7-16. 6. 32-40. 7. $100 -$125. 8. $500 -$650. 9. $400 deposited and $475 withdrawn. 10. $250 deposited and $325 withdrawn. 26. Opposite Numbers. — Since a negative number, such as — 5, always implies a reserved subtraction, if it is combined with a positive number, into one number, it tends to subtract from, or destroy of, the positive number a part equal to itself. Thus, when 6 and — 4 are combined into one number, — 4 destroys 4 of the 6, leaving 2 as the result. And when $10 and —$15 are combined, -- $ 10 of the — 1 15 destroys the $ 10, leaving — $ 5 as the result. Because of this neutralizing tendency of positive and negative numbers, they are sometimes called opposite numbers. If two numbers such as +8 and— 8, which differ only in their signs, are combined, they completely neutralize each other. Such numbers are said to have the same absolute value. Thus, 8 is the absolute value of both + 8 and — 8. 27. Positive and Negative Magnitudes. — As seen in the above illustrations, deposits and withdrawals of money in a bank tend to neutralize each other, and hence are positive and negative quantities, respectively. There are many other concrete quanti- ties or magnitudes which are capable of existing in two opposite states or senses such that one tends to neutralize or destroy an POSITIVE AND NEGATrVE NUMBERS 37 equal amount of the other. Consequently, their numerical values are represented by positive and negative numbers. Several of these are illustrated in the following exercises. EXERCISES 1. If in one of two sales of goods I lose $10, and in the other I make a profit of $ 15, what is the net result ? Loss and profit are thus neutralizing quantities, each tending to destroy an equal part of the other. If profit is called positive, what will loss be called ? If $15 profit is written +15, what will express $10 loss? 2. Express with + and — signs the following: $25 loss; $40 profit; $15 loss; $6 loss; $20 profit. 3. If I had $20 in my pocket, then spent $25 for a suit of clothes, I would have left $20- $25, or - $5. What does the — $ 5 in this case mean ? 4. Express with + and — signs the following: $50 credit; $18 debt; $25 debt; $60 cash in hand; $100 borrowed; $20 loaned ; $ 15 spent. 5. The number of people entering a store during a certain hour was 216, and the number coming out was 250. Express the rela- tion of these numbers by + and — signs. Also express the excess of the number coming out over the number entering. 6. The amount of water pumped into a tank during an hour was 1900 gallons; and during the same hour three locomotives withdrew, respectively, 675 gallons, 750 gallons, and 650 gallons. Express with + and — signs these quantities, and also the excess withdrawn during the hour. 7. A locomotive pulls a train with a force of 35 tons. The resistance of the train is 20 tons. If the pull of the locomotive is positive, represent by + and — signs the two forces. 8. A balloon lifts up with a force of 200 pounds. How much weight must be attached to keep the balloon from rising? If the lifting force is -|-, what represents the weight? 38 ELEMENTARY ALGEBRA 9. A balloon lifts with a force of 250 pounds, and a weight of 175 pounds is attached to it. If the weight is called positive, what will represent the lifting force ? What will represent the result when the weight is attached ? 10. A battleship has a displacement of 20,000 tons (sinks into the water until it displaces 20,000 tons of water). Express with + and — signs the weight of the ship and the upward pressure of the water. 11. A diver who weighs 165 pounds displaces 156 pounds of water (is buoyed up with a force of 156 pounds). Kepresent by -f- and — signs these forces and the resultant force with which he tends to sink to the bottom. 12. If the thermometer is 5° above zero at noon, and falls 15° by night, what is the temperature by night? 1 212* 5°— 15° = what ? How else may 10° below zero be written ? How would temperature above zero be written ? 13. On Jan. 6, 1910, the Weather Bureau reported the temperature at several different places as follows Chicago, -2°; Davenport, -18°; Des Moines, -12° Kansas City, — 2° ; Madison, — 14° ; Milwaukee, — 8° Peoria, -6°; Pueblo, -6°; St. Paul, -16°. Were these temperatures above or below zero ? What would they be if the signs were + ? 14. Certain railroad stock that sold at 7 above par in January fell to 3 below par in June. Since the value of the stock was reduced 10 points, how may the 3 below par be represented algebraically ? 15. Eepresent by -h and — signs the following prices of stock which show the change in 6 months : January, 10 below par ; February, 6 below par ; March, 1 below par ; April, 5 above par ; May, 8 above par ; June, 2 above par. 16. Augustus Caesar was ruler of the Roman Empire from — 31 to 4- 14. Express these dates in other terms. POSITIVE AND NEGATIVE NUMBERS 39 17. Arcliiniedes, the greatest mathematician of antiquity, was born about the year —287, and was slain by a Koman soldier while studying a geometrical figure that he had drawn in the sand in — 212. Express these dates in other terms. 18. If a file of soldiers moves 10 paces to the front, then 6 paces to the rear, it advances how many paces from its first position ? If it moves 6 paces to the front, then 10 paces to the rear, it ad- vances 6 — 10, or how many paces ? If motion forward is positive, what is motion backward ? 19. A switch engine, in making up a train, moves forward 100 yards, then backward 150 yards. The two are equivalent to what single motion ? Represent by appropriate signs these two motions and the one equivalent motion. 20. If longitude west is positive, longitude east negative, lati- tude north positive, and latitude south negative, in what country is a city whose longitude is + 106° and latitude -f 40°? Longitude -16° and latitude -f 48? Longitude -40° and latitude - 15°? Consult a map. 21. The longitude of the city of St. Petersburg, Russia, is given as — 31°. This means that it is 31° in what direction from the Prime Meridian? Washington, D.C., is in 77° west longitude. How else may this be expressed ? 22. The latitude of New York City is about -f- 41°. What will express the latitude of Cape Town, Africa, which is about 34° south of the equator ? 28. Numbers Represented by Distances. — In the preceding ex- ercises, it has been seen that many magnitudes may be represented numerically by positive and negative numbers. These include bank deposits and withdrawals, loss and gain in business, income and expense, credits and debts, increases and decreases of physical quantities, temperature above and below zero, dates in history before and after the birth of Christ, stocks above and below par, forces acting in opposite directions, motions in opposite directions, 40 ELEMENTARY ALGEBRA and longitude east and west and latitude north and south. Many others might be suggested. Since motions in opposite directions tend to neutralize each other, the distances moved through in opposite directions may themselves be considered as positive and negative. In represent' ing distances as positive and negative, the signs serve to tell the directions in which the distances are measured, as in the case of longitude and latitude. It follows that positive and negative numbers may be represented to the eye by distances measured on -9-8 -7-6-5-4 -3 -2-1 1 2 3 4 5 6 r 8 9 I I I t I I I I I ., I I— J li. I. I 1 1, 1 1 I — I I , A a line. Positive numbers are represented by distances measured to the right from a starting point A, and negative numbers by dis- tances measured to the left, EXERCISES 1. On the above line, how will — 12 be represented? +10? -21? +31? -40? +100? 2. Draw a horizontal line and mark the middle point O. Then mark on it distances representing the following numbers, a unit being represented by a quarter of an inch : + 2, + 5, + 8, + 10, + 61, -3, -4, -7, -12, -41, -lOi 3. On a horizontal line represent the following dates (i.e. years) : — 31, beginning of Augustus Caesar's reign ; +14, end of Au- gustus Caesar's reign ; — 146, fall of the Eoman Empire ; + 70, destruction of Jerusalem by Titus ; — 275, death of Euclid, who wrote the first great book on geometry ; — 287, birth of Archi- medes, the greatest mathematician of antiquity. 4. On a vertical line, represent the following temperatures: + 5°; +10°; +15°; +20°; -5°; -10°; -15°; -20°. 5. The minimum temperatures at St. Paul on the six days be- ginning Eeb. 9, 1910, were + le**, - 10°, + 6°, + 4°, + 22°. Eep- resent these on a line. POSITIVE AND NEGATIVE NUMBERS 41 6. The longitude of New York is + 74°, of Berlin - 13°, of Pekin - 116°, of Calcutta - 88°, of Pittsburg + 80°, of St. Louis -f 90°, of Rome —12°. Draw a horizontal line and mark off on it the longitudes of these places. 29. Graphs. — The following table gives the minimum temper- ature at Chicago on each of the seven consecutive days beginning with Jan. 2, 1910; Date Jan. 2 Jan. 8 Jan. 4 Jan. 5 Jan. 6 Jan. 7 Jan. 8 Temp. 21° 1° 1° 4° -4° -6° 14° Suppose that a sheet of paper is ruled by horizontal and verti- cal lines, as shown in the following diagram, each vertical line representing the thermometer scale. By marking the temper- ature for each date on the corresponding vertical line, and joining the consecutive points thus marked by straight lines, the heavy broken line is obtained, showing to the eye the variation from day to day. 25 Jan. 2 Jan.3 Jan.4 Jan.5 Jan.6 Jan.7 Jcxn.8 Such diagrams as this are called graphs. They are much used in practical work to depict to the eye graphically the changes in quantities, such as temperature, death rates, population, imports and exports, cost of living, etc. Note. — Ruled paper, often called ''squared paper" or "coordinate paper," may be obtained cheaply for drawing graphs. It is iniled accu- rately into small squares, and saves much time when used. This paper will be needed in other exercises later. 42 ELEMENTARY ALGEBRA EXERCISES 1. The following table gives the minimum temperature at St. Paul, Minn., on each of seven consecutive days, beginning Jan. 5, 1910 : Date Jan. 5 Jan. 6 Jan. 7 Jan. 8 Jan. 9 Jan. 10 Jan. 11 Temp. 6° -16° -12° -4^ -12° 2° 20° Draw a graph showing the variation in temperature during the period. 2. According to the report of the Weather Bureau, on Jan. 5, 1910, the height of the Ohio River above low-water mark at Pittsburg was 11.3 feet. The amounts in feet by which it rose or fell on the seven consecutive days thereafter were as follows, the negative sign indicating fall : — 2.1, — 2.8, 1.2, 1.3, — 2,2, - 2.7, - 1.4. Draw a graph showing the rise and fall of the river from day to day. Measure the heights of the river on vertical lines, mark- ing the lowest horizontal line low-water mark. 3. The latitude of a ship going east at noon on each of six con- secutive days was as follows : — 1°, equator, -|- 1°, -f 3°, + 1°, — 1°. Draw a graph showing the course of the ship. 4. The highest points reached by certain railroad stock during the 12 months of a year were as follows : 10 above par, 6 above par, 2 below par, 5 below par, par, 2 below par, 3 above par, 7 above par, 10 above par, 15 above par, 18 above par, 12 above par. Draw a graph showing the variation in price of the stock dur- ing the year. 30. Addition of Positive and Negative Numbers. — The addition of positive and negative numbers may be indicated by writing them in succession with their signs. Thus, to indicate the addition of + 6, + 2, — 4, -f- 3, and — 5, we write -f-6 + 2 — 4 + 3 — 5. The first term being positive, we may omit its sign, giving 6 + 2-4 + 3-6. POSITIVE AND NEGATIVE NUMBERS 43 Positive and negative numbers to be added may be placed _ „ also in columns, with the signs attached, as in the margin. « Since positive numbers are in nature ordinary arithmet- /, leal numbers, their sum is obtained by adding their absolute values, and is positive. Thus, a bank deposit of .$ 125 and another of $ 100 together make a de- posit of $225. The sum of + 4 and -f 8 is 4- 12. Also, to subtract each of two numbers in succession is equiva- lent to subtracting their sum. Hence, two negative numbers, since each is in nature a subtracted number, may be combined by adding their absolute values, and the sum is negative. Thus, a withdrawal from the bank of $ 25 and an additional withdrawal of $75 together make a withdrawal of $ 100. The sum of —6 and —10 is —16. Therefore, (1) To add two numbers with like signs, find the sum of their absolute values, and prefix the sign common. In adding two numbers with unlike signs, the one with the greater absolute value may be separated into two parts, one of which just neutralizes the number with the smaller absolute value, leaving as result the other part of the numerically greater number. Thus, to add + 7 and — 3, by separating + 7 into + 3 and + 4, we have -1-3 + 4 or 4- 7 -3 -3 + 4 +4 And, to add — 9 and + 4, by separating — 9 into — 4 and — 6, we have — 4 — 5 or — ^ + 4 +4 0-5 =. 8 J Hence, the following rule is evident : (2) To add two numbers with unlike signs, find the difference of their absolute values, and, prefix the sign of the one ivith the greater absolute value. 44 ELEMENTARY ALGEBRA Evidently three or more positive and negative numbers may be added by adding two at a time, taking them in the order written. In the steps of the addition use rules (1) and (2). A second method sometimes used when there are both positive and negative numbers to be added is to add the positive and the negative numbers separately, and then combine these two sums. ~- ^ + 8 Thus, in this column, adding upwards, the partial sums are _ g — 2, 4- 6 , and + 3, the result. Or, the sum of — 3 and — 6 is ^_ 4 — 9, and that of + 8 and + 4 is +12, and the sum of these Vs sums is + 3. Note. — If desirable, the numbers in the preceding illustration — might be rewritten in the order shown in the margin. This fun- — damental principle that numbers to be added may be arranged in + 8 ) any order is known as the Law of Order in Addition. -f- 4 j The principle that numbers to be added may be grouped in any manner — by which we find the sum of the — 3 and the — 6 and of the + 8 and + 4 and then the sum of these sums, as here indicated — is known as the Law of Grouping in Addition. EXERCISES Find the sums of the following : 1. A profit of $12 and a profit of $16. 2. A loss of $10 and a loss of $23. 3. A profit of $ 15 and a loss of $ 6. 4. A profit of $ 18 and a loss of $ 30. 5. A rise in temperature of 14° and a rise of 6°. 6. A fall in temperature of 8° and a fall of 17°. 7. A rise in temperature of 7° and a fall of 16°. 8. A credit of $ 250 and a debt of $ 325. 9. A debt of $ 136 and a debt of $ 224. 10. +6 -8 +9 -12 -20 +8 _2 -7 +5 +17 -&^ -16 POSITIVE AND NEGATIVE NUMBERS 45 11. -25 + 32 - 12 +13 - 17 +17 - 1 + 14 + 1 -98 - 30 +16 12. -14 + 21 + 7 - 8 -18 +7 + 3 -12 - 14 +20 -3 +16 13. - $12 +$27 - 7 1b. + - $ 6 -$36 +12 lb. + - 8° longitude + 6° latitude - 17° longitude - 14° latitude 13 lb. + 14 mi. 19 lb. - 18 mi. 14. 25 ft. forward 36 ft. backward 15. -2 16. + 6 + 3 -8 -4 + 12 17. - 4 - 7 - 9 - 3 - 3 + 6 - 1 + 5 +12 18. + 6 - 3 - 4 + 12 + 7 19. — 5 20. + 8 + 8 -16 + 14 -21 - 3 - 7 + 6 - 1 21. -3.25 + 4.09 - 12.08 - 10.37 + .62 -6.07 - 4.125 + 1.450 22. -^ + 6f - i +1| _4| -51 + J 23. 5, -. 8, 7, - 4, - 12, 3. 24. -9,-6,-2, 10, - 8, 9, - 1. 25. 16,. 5, 9, - 5, 3, - 7, - 2, 6. 26. The Ahraes papyrus, the earliest mathematical book of which there is any record, was written about — 1700. The next great mathematical book was Euclid's Elements, written about 1400 years later. What was the date that Euclid wrote the Elements? 27. Euclid was born about —330. Biophantus, who wrote one of the first books on algebra, was born 750 years later. At what date was Diophantus born ? 46 ELEMENTARY ALGEBRA 28. A freight car is running at the rate of 20 feet per second. A brakeman walks to the rear on the top of the car at the rate of 5 feet per second. Express by + and — signs these rates and the rate and direction of his motion with reference to the ground. 29. If the basket of a balloon weighs 280 pounds, the instru- ments 57 pounds, the sandbags 800 pounds, and the balloon itself — 1400 pounds, what is the total weight of the balloon and contents ? 30. A boat using both sails and steam is driven against a cur- rent by its steam power at the rate of 10 miles per hour, and by the wind at the rate of 3 miles per hour. The current flows at the rate of 4 miles per hour. Express these rates by + and — signs, and indicate their sum. What is the boat's rate of progress ? 31. A work train travels 12 miles north, then 35 miles south, then 8 miles north. By use of -f- and — signs express these dis- tances, their sum, and hence its final distance and direction from the starting point. 32. The temperature falls 15°, rises 18°, falls 10°, then rises 6°. By use of -{- and — signs express these changes and their net result. 33. The rise and fall in feet of the Mississippi River at Yicks- burg on each of the seven days beginning Jan. 6, 1910, were given by the Weather Bureau as follows: — .2, — .2, — .4, — .2, -f .4, -h .2, + .4. The sign — indicated fall. Find the net change during that period. 34. The average of two or more numbers is their sum divided by the number of them. Find the average of the following temperatures: 8 a.m., —7°; 9 A.M., - 6° ; 10 A.M., - 3° ; 11 a.m., - 1° ; 12 Noon, + 1° ; 1 p-m.^ -f-4°; 2 P.M., -f-7°; 3 p.m., 4-8°; 4 p.m., +8°; 5 p.m., -f 5°, 6 p.m., +3°; 7 P.M., -f 1°. POSITIVE AND NEGATIVE NUMBERS 47 35. At Madison, Wis., the minimum temperatures on the seven days beginning Jan. 4, 1910, were as follows: —12°; +10°; -14°; -18°; +6°; -6°; -10°. Find the average tempera- ture for the week. 36. At Cheyenne, Wyo., the temperatures for six days, begin- ning Jan. 3, 1910, were as follows: -10°; -2°; 0°; —2°; + 14° ; + 10°. Find the average temperature for the six days. 37. If one place is midway between two other places, its lati- tude or longitude equals one-half the sum of their latitudes or longitudes. The latitude of Panama is approximately + 8°, and of Chicago + 42°. Find the latitude of Key West, Florida, which is approxi- mately midway between these two. 38. The longitude of New Orleans is -f 90°, and of Pekin, China, — 116°. Berlin, Germany, is approximately midway be- tween these two places. What is its longitude ? 31. Subtraction of Positive and Negative Numbers. — To subtract a number (subtrahend), is to find a number (remainder) which added to it gives the minuend. Thus, to subtract 6 from 9 is to find a number (3) which added to 6 gives 9. Hence, the rule for subtracting positive and negative numbers may be obtained from addition. (1) What number added to + 3 gives + 6 ? + 5 +5 Hence, by subtracting, + 3 But, by adding, — 3 + 2 +1 Ilence, to subtract + 3 from + 6 is equivalent to adding — 3 to + 6. (2) What number added to + 3 gives — 5 ? -6 -5 Hence, by subtracting, + 3 But, by adding, — 3 Hence, to subtract + 3 from - 5 is equivalent to adding — 3 to — 6. 48 ELEMENTARY ALGEBRA (3) What number added to — 3 gives + 5 ? Show, as above, that to subtract -^ 3 from + 5 is equivalent to adding -I- 3 to + 5. (4) What number added to — 3 gives — 5 ? Show, as above, that to subtract — 3 from — 5 is equivalent to adding + 3 to - 5. From these examples it is evident that to subtract one number (subtrahend.) from another (minuend) is equivalent to adding the subtrahend, with its sign changed, to the minuend. Therefore, To suhtr act, first change the sign of the subtrahend, then proceed as in addition. — 4 — 4 Thus, to subtract — 7 change to + 7 then add. Note. — The pupil should learn not to rewrite the problem with the sign changed in the subtrahend, but to make the change mentally, and add at once. EXERCISES 1. What number added to — 6 gives — 4 ? 2. What number added to 8 gives — 2 ? 3. What number added to — 4 gives 6 ? 4. What number added to 12 gives 7 ? 5. What number added to — 15 gives — 8 ? 6. What number added to — 5 gives ? 7. What added to $ 7 loss will give $ 9 gain ? 8. AVhat added to a withdrav^^al of $ 125 from a bank will give a balance of $ 150 ? 9. What added to a rise of 12° temperature vrill give a net fall of 18°? 10. What added to a fall of 15° temperature will give a net fall of 8° ? Subtract the following : 11. -h 7 +12 +2 +20 +9 +16 -6 -3 -10 -8 -14 -4 -6 POSITIVE AND NEGATIVE NUMBERS 49 12. - 3 + 5 - 7 + 6 -25 + 15 -14 + 16 - 9 + 6 - 7 + 8 + 15 13. +12 + 3 + 9 + 5 + 7 + 9 + 15 + 20 + 21 + 16 + 12 + 20 + 40 + 55 14. - 9 - 6 - 8 - 4 - 3 - 5 -12 - -15 - 9 - 4 - 7 -10 -16 -17 15. — 3 + 7 + 6 -15 - 5 - 7 + 10 + 6 -16 + 4 + 5 + 12 + 9 - 6 16. What is indicated by — 3 — (— 2) ? Find the value. 17. What is indicated by + 8 — (+ 12) ? Find the value. 18. Find the value of the following : 5 -(-5); -3 -(-10); 6-(+8); -9-(-12); 6-(+12); -7-(+8); 9-(-6). 19. Subtract : -6.25 + .23 - .2 ft. +3.5 pk. -12.3 lb. +3.85 -4.17 -1.6 ft. +7.4 pk. -15.61b. 20. Subtract : +2 +5f + f -2i -i\ -16J + 6J 21. A thermometer registered one day — 15°, and the next day — 9°. What was the change in temperature ? 22. A thermometer registered one day + 10°, and the next day — 7°. What was the change in temperature ? 23. Give the range of temperature in the following places (Jan. 1, 1908 to Jan. 1, 1909) : Place Extremes Place Extremes Montgomery, Ala. . . Little Rock, Ark. . . Denver, Col Washington, D.C. . . Bois6, Ida Des Moines, la. . . . 107 106 106 104 111 109 - 5 -12 -29 -15 -28 -30 Louisville, Ky. . . . Boston, Mass. . . . Duluth, Minn. . . . Havre, Mont. . . . New York, N.Y. . . Chicago, 111. . . . 107 102 99 108 100 103 -20 -13 -41 -65 - 6 -23 50 ELEMENTARY ALGEBRA 24. What is the difference in time between the year 4- 36 and the year — 40 ? 25. Augustus Caesar lived from —60 to +14. How old was he when he died ? 26. Thales, the founder of the first Greek school of mathe- matics and philosophy, was born in the year — 640 and died in the year — 550. At what age did he die ? 27. Pythagoras, who is believed to have given the first proof that in a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides, was born in — 569 and murdered in — 500. At what age was he murdered ? 28. Euclid wrote his geometry, called the Elements, in about — 300. By + 1300 the book had been carried into western Europe and translated there. How long was this after the book had been written ? 29. What is the difference in longitude between Chicago and Berlin, the longitude of the former being approximately + 88° and of the later approximately — 13° ? 30. During the month of December, 1909, the maximum tem- perature at Chicago, as recorded by the Weather Bureau, was -f 55°, and the minimum — 7°. What was the amount of variar tion during the month ? 32. Multiplication of Positive and Negative Numbers. — Multi- plication as defined in arithmetic is a short form of addition, where the addends are all equal. Eor example, 3 X $ 25 means $ 25 + $ 25 + $ 25. This definition is only a special case of a more general definition which may be stated in the following way: To multiply one number (multiplicand) by a second number (mul- tiplier) is to use the first as we use 1 (unity) to obtain the second. Thus, in multiplying 3 by 5, to obtain 5 by using 1, we take 1 + 1 + 1 + 1 + 1 = 5. Hence, to obtain 6 x 3, we take 34-3 + 3 + 3 + 3 = 16. POSITIVE AND NEGATIVE NUMBERS 51 From this definition are obtained the rules for multiplying positive and negative numbers. (1) To obtain the multiplier + 3 we must add three I's j i.e, + 3 = 1 + 1 + 1. Hence, to obtain (+ 3)(+ 5) we must add three + 5's. That is, (+ 3)(+ 5) = + 5 + 5 + 5 = + 15, the product. And to obtain (+ 3)(— 5) we must add three — 5's. That is, (+ 3) ( - 5) = - 5 — 5 - 5 = — 16, the product. (2) To obtain the multiplier — 3 we must subtract three I's from ; i.e. _3 = 0-l-l-l. Hence, to obtain (— 3)(+ 5) we must subtract three + 5's from 0; i.e. change their signs and add them. Hence, (— 3)(+ 6) = — 6 — 6 - 5 = — 15, the product. And to obtain ( — 3) (— 5) we must subtract three — 6's from ; i.e. change their signs and add them. Hence, (— 3) (— 5) = + 5 + 5 + 5 = + 15, the product. From these examples the following principles are evident : (1) The product of two numbers with like signs is positive. (2) Tlie product of two numbers with unlike signs is negative. (3) Tlie absolute value of the product of two numbers is the prod* net of the absolute values of the numbers. In symbols : (+ a)(+ &) = + a6. (— a)(-&)= +a6. (+a)(-6)=-a6. (-a)(+6) = -a6. EXERCISES Give orally the products of the following : 1. +3, +7. 5. -2,-10. 9. +5,-7. 2. -6,-2. 6. +7, +-3. 10. +8,-3. 3. +4, +5. 7. -2, +5. 11. -2,-15. 4. - 8, - 3. 8. - 3, + 6. 12. - 7, + 4. 52 ELEMENTARY ALGEBRA 13. + 6,-8. 19. - 5, - 15. 25. -i-f 14. -5, -12. 20. + 6, +20. 26. + H, -|. 15. + 12, +4, 21. -10, +14 27. - 2.5, - 2.5. 16. -9,-8. 22. -h-h 28. + 1.6, - 1.6. 17. + 12,-8. 23. + 1, -|. 29. -.8, +.7. 18. -7, +12. 24. -i+|. 30. -1.3, -1.1. 31. If the lifting force of a balloon is +, the weight of the ballast is — . Adding weights is indicated by +, and removing weights by — . When 5 weights, each of 20 ponnds, are thrown overboard, what is the effect on the upward pull of the balloon ? Show that this is equivalent to saying that (— 5)(— 20) = + 100. 32. When four weights, each of 15 pounds, are added to a balloon, what is the effect on the upward pull of the balloon? Show that this is equivalent to saying that (+ 4)(— 15) = — 60. 33. If 10 boxes of goods, each weighing 250 pounds, are removed from a freight boat, what is the effect of the downward pressure of the boat in the water ? Show that this is equivalent to saying that (- 10) (+ 250) = - 2500. 33. The Product of More than Two Factors. — The product of more than two factors is obtained by performing one multiplication at a time, according to the principles in § 32. Thus, to find (- 2)(- 3)(+ 4)(- 5), we have (-2)C-3) = +6 (-H6)(+4) =+24 (+ 24)(- 5) = - 120, the product. Show that the following rule holds for finding the product of more than two factors : Find the product of the factors regardless of signs, and prefix -\- or — according as the number of negative factors is even or odd. 34. Powers of Positive and Negative Numbers. — If the factors are all equal, what is the product called ? POSITIVE AND NEGATIVE NUMBERS 53 The rule in § 33 will serve for finding powers of positive and negative numbers. Thus, (+ 5)2 = (+ 5)(+ 5) = + 25. (+3)3 = (+3)( + 3)(+3)= + 27. (-f 2)* = (+ 2)(+ 2)(+ 2) (+ 2) = + 16. Evidently, any power of a positive number is positive. Again, (_ 9)2 = (_ 9)( - 9) = + 81. (_6)8=(-5)(-5)(-5) = -125. (_4)4=(-4)(-4)C-4)(-4) = + 256. (- 2)6 =(- 2)(- 2)(- 2)(- 2)(- 2) = -32. Evidently, any even power of a negative number is positive, and any odd power of a negative number is negative. EXERCISES Find the products of the following : 1. -1,2,-3. 7. -1,-1,-2,-2. 2. -2,-4,-1. 8. 2,-7,4,-2,-1. 3. 3,-2,-5,2. 9. -1,-1,-1,-1,-1. 4. 2,-5,-8,- 1. 10. -4,2,-1,-1,-1,-2. 5. -6,-2,-1, 3. 11. -1,-2,-3,4,1,-5. 6. 4,-5,-2,3. 12. 2,3, -1,-2, 1,-2. Find the values of: 13. (+2/. 19. (-2/. 25. (-iy(-2y. 14. (-2)^ 20. (-3)^ 26. (-2)2(-3)« 15. i-Sy, 21. (+4/. 27. (-5)2(+2)« 16. (4-1)'. 22. (-1)». 28. (_1)«(_2)^ 17. (-1>'. 23. (-2)«. 29. (-1)5 (-2)3 (-3)2. 18. (-4)^ 24. (-3Y. 30. ( + 2)«(-l)V-3)». 64 ELEMENTARY ALGEBRA If a = 2, 6 = — 3, c = — 1, find the value of : 31. h^. 34. 2ah\ 37. hh. 32. a^h\ 35. 3 ac^. 38. a^ + h\ 33. a6c. 36. a^bh\ 39. a^ + ft^ 4. c^ _ a6c. 35. Division of Positive and Negative Numbers. — Division is the inverse of multiplication. For example, to divide 24 by 8 is to find a number (3) by which 8 must be multiplied to give 24. Hence, the rules for division of positive and negative numbers come directly from the rules for multiplication. Since (+ 5) ( + 3) = + 15, then ( + 15) Since (+, 5)(- 3) = - 15, then (- 16) Since (_ 5)(+ 3) =- 15, then (-15) Since (- 5)(- 3) = + 15, then (+ 15) (+3) = + 5. (-3) = + 6. ■(+3) = -5. (-3) = -5. Evidently, from these examples we get the same laws of signs in division as in multiplication : (1) If the dividend and divisor have like signs, the quotient is positive. (2) If the dividend and divisor have unlike signs, the quotient is negative, ' EXERCISES Div 1. ide: - 12 by 3. 11. 4. 14 by -14. 21. + iby-i. 2. - 20 by - 4. 12. - 20 by - 20. 22. -4by-|. 3. 4- 18 by - 3. 13. + 50 by -5. 23. -|by+4. 4. -27 by +9. 14. -100 by +25. 24. + iby-f 5. + 64 by -16. 15. + 66 by -11. 25. 4-fby+}. 6. -81 by -9. 16. -96 by -8. 26. -fby-i. 7. + 42 by -7. 17. -82 by +41. 27. + 6.4 by -.8. 8. -54 by -6. 18. + 63 by -7. 28. - 2.1 by - .3 9. + 72 by + 8. 19. _84by -12. 29. - .63 by + .7 10. - 60 by - 6. 20. + 132 by +11. 30. -1.44 by -1.2 POSITIVE AND NEGATIVE NUMBERS 55 If a = — 2, 6 = 3, and c = — 4, find the value of : 31. ^. 33. l^. 35. 2j. 37. ^. c c cr 6 32. 5!^. 34. ^. 36. 4. 38. i+*±^. 36. Positive and Negative Numbers in Equations. — In Chapter II we learned how to find the roots of equations by means of axioms. We may now consider the members of equations as made up of positive and negative terms. Let us solve 8x — 7=3x + 3. To remove — 7 from the first member add + 7 to both members. Then 8x = 3a; + 3+7. To remove 3 x from the second member add — 3 x to both members. Then 8x — 3x = 3 + 7. Uniting terms, 5 x = 10. Dividing by 5, x = 2, the root. Observe that to remove the known term from the first member and the unknown term from the second member, we proceed as follows : (1) TJie known term in the first member y with its sign changed^ was added to both members. (2) The unknown term in the second member, with its sign changed, was added to both 7nembers. Note. — The effect of adding + 7 to both members in the equation above was to make the term — 7 disappear from one member and reappear in the other, with its sign changed. Adding — 3 x to both members had a similar effect. Evidently, adding such terms to both members of an equation is equivalent to moving terms from one member to the other and changing the signs of the terms moved. This mechanical process of moving terms from one member to the other by changing their signs is called transposition. The terms are said to be trans- posed. Thus, by transposing + 2 and — 2 a, a + 2 = 6 — 2 a, becomes a + 2a = 5-2. The process of transposition was discovered by the Arabs. About + 830 an Arab named Mohammed ben Musa Al Khowarazmi wrote a mathematical book with the title Aljabr Walmuqabalah. The word Aljabr meant restora* 66 ELEMENTARY ALGEBRA tion or transposition, as the term is now called. From this word came the name algebra. The pupil should use the correct phraseology of "adding equals to both members of the equation " until the thing actually done is firmly fixed in mind. 37. Equations with Negative Roots. — The roots of some equa tions are negative numbers. Example 1. — Solve 7^ + 15 = 4^ + 3. Adding — 15 and — 4 ^ (transposing), Uniting like terms, Dividing by 3, Example 2. — Solve 3F + 3 = 8 + 4 F. Adding — 3 and — 4 F (transposing), 3 F Uniting like terms, Dividing by — 1, It- -4t = 3- -15. St = — 12. t = — 4. V- -4 F = 8 -3. - V = 5. F = : _ ■5. EXERCISES Solve: 1. 9y=72/-28. 2. 3^-4 = 2^-7. 3. 5 s +- 20 = 4 - 3 s. 4. 3m + 27 = 6-4m. 5. 5/r-15 = /r-35. 6. 2^ + 3 = 4^ + 9. 7. 7P-8 = llP+12. 8. 13 = 3ic + 40. 9. 6 - a = 12 +- a. 10. 17 = 19 + Jf. 11. 5 = 5 + 95. 12. 3-^ = 3^ + 8. 13. Tr+7=5 Tf +-20. 14. «-S = 3. 5 2 15. 8-^ = 10. 4 16. . = 1-9. 17. f-f = f-' 18. .+|h-io = |. 19. ^^^^H- 20. ^-ttf 4: p. 38. Interpretation of Negative Roots. — In solving a problem by means of an equation, a negative answer may have a natural inter- pretation, or it may indicate that the problem is impossible. POSITIVE AND NEGATIVE NUMBERS 57 EXERCISES Solve the following problems and interpret the answers : 1. On the second of three days the thermometer rose 20° more than on the first, and on the third day 13° more than on the second. The total rise during the three days was 35°. What was the change of temperature on the first day ? 2. In a certain year a merchant, on the average, lost $200 in each of the first 5 months, and made a profit of $450 in each of the next 6 months. His net profit for the year was $ 1550. Did he have a loss or profit in the last month, and how much ? 3. A man gained $1300 during three months. During the second month he gained $650 more than the first month, and during the third month $ 350 more than the second. Did he gain or lose during the first month, and how much ? 4. A farmer sowed 18 acres of wheat more the second day than the firsts 12 more the third than the second, and 5 more the fourth than the second. He sowed 59 acres in all. How many acres did he sow each day ? 5. In a triangle the sum of the angles is 180°. The second is 12° less than one twelfth of the first, and the third 15° more than the first. How many degrees in each angle ? SUPPLEMENTARY EXERCISES 1. Herodotus, the Greek historian, commonly called the " Father of History," was born in — 484 and died in — 424. At what age did he die ? 2. Livy, one of the most famous of the Eoman historians, was born in — 59 and lived to be 76 years old. In what year did he die ? 3. A ship w^hich leaves port at 42° north latitude goes one day 4° south, the next 3° north, the next 2° south, the next 3° south, and the next 1° south. If sailing south is called — and sailing north -|-, express the sum of these motions, and find the latitude of the ship at the end of the fifth day. 68 ELEMENTARY ALGEBRA 4. A switch engine, making up a freight train, moves forward 30 yards, back 120 yards, forward 150 yards, forward 40 yards, then back 300 yards. Kepresent these distances by 4- and — numbers, and find the position of the engine at the end of the work. 5. Eind the average of the following temperatures: —10°, - 6°, 2°, - 4°, 3°, r, - 4°, 5°, 16°, and - 14°. 6. Add -I, -J, + J, -i, and -\-\. 7. Add + 14|, - 8|, - H, and - 5^^. 8. rind the values of ; (-ly; (-2)«; (-2/(-3)3; (- 5)(-4)X- 3)1 9. If ^ = -2 and 5 = -3, find the value of (A-B)\ 10. If 0? = 4 and 2/ = — 2, find the value of (ic^ + a^ + ?/2) (ic2 — ici/ + 2/2). 11. Solve ^-| = |-|. 12. Solve 5-^=82/4-95. 13. Positive and negative numbers may be added graphically as follows : Let positive and negative numbers be represented by distances measured to the right and left, respectively, of a point -4 on a line, as in § 28. -9-8~?-6-5-4.-3-2-1 ! 2 3 4 5 6 ^ 8 9- » I.- I l i A I I I I ,1 I I . ■ l,riJ I I ■ » ■ I > I k. To add a positive number count to the right as many spaces as there are units in the number, and to add a negative number count to the left as many spaces as there are units in the number. Thus, to add — 3 and + 7, begin at — 3 and count to the right 7 spaces. What is the result? To add —2 and —6, begin at — 2 and count to the left 6 spaces. What does it give ? POSITIVE AND NEGATIVE NUMBERS 59 Add graphically the following : 14. —5 and +8. 20. —2, —3, and +4. 15. + 6 and — 4. 21. +5, +2, and — 3. 16. +2 and +7. 22. -6, +8, and -4. 17. — 4 and — 3. 23. — 1, +5, — 7, and + 4. 18. +4 and -9. 24. + 4, + 3, - 12, and + 1. 19. - 7 and + 12. 25. +8, - 8, - 4, and + 10. 26. Show why a positive number is subtracted graphically by counting to the left the number of units in the subtrahend, and a negative number is subtracted by counting to the right the num- ber of units in the subtrahend. 27. Subtract graphically the following: —3 from —7; -f 2 from — 4 ; +6 from — 8 ; — 5 from -f- 5 ; -f 1 from — 1 ; — 1 from +1; - 6 from -6; +8 from -2. 28. Viewing multiplication as a. special form of repeated ad- dition, where the addends are equal, show graphically that the product of two positive numbers is positive, and that the product of a negative number by a positive number is negative. CHAPTER IV ADDITION AND SUBTRACTION OF LITERAL EXPRESSIONS 39. Addition of Monomials. — In § 9 it was shown that similar terms are added or subtracted by adding or subtracting the coefficients, and attaching to the results the common letters with their exponents. Thus, 3 lb. + 4 lb. = ? 3 ? + 4 Z = ? 6 sq. yd. + 10 sq. yd. = ? 6 2/2 4- 10 !/2 = ? 9 men - 4 men = ? 9 m — 4 wi = ? 26x^y- Ux^y =? Since positive and negative terms are by nature added and sub- tracted terms, respectively, they may be added by the same rule : that is, To add two or more positive or negative terms, add their coefficients according to the rules in § 30, and attach to the result the common letters with their exponents. For example, the sum of — 4 a-6, + 7 a^ft, -f 3 a^ft, and — 4 a^ft - 9 a26 is _ 4 a% + 7 a^fe + 3 a2?, _ 9 ^25 ^ _ 3 ^25. The + 7 a^h terms may be written in a column and added, as in the + 3 a^b margin. — 9 a% - 3 a^h EXERCISES Add: 1. 5a, 4a, -2a. 7. -2>x\ -x\V^x\ -5x\ 2. 9n, -12n, -Sn. 8. PQ,-4:FQ,16PQ, -12 PQ. 3. 10 ab, 4 ab, - ab. 9. - F^ 9 V, 7 V, -S y\ 4. 6 P, - 8 P, - 10 P. 10. feV, - 4 6V, -&V, -6 6V. 5.-7 mhi, 4 m^n, - 5 m^n. 11. hB\-^ B\ -12 B% 17 B\ 6. 12^, -4^, -7^, -9 J.. 12. -^xif,-lx'if,x'if,-2xy\ 60 ADDITION AND SUBTRACTION 61 13. -4a 3a? -10^ 15 f -20«w Qa -7aj - 8/f - 6f 13 zw -8a -9a; 12 K f ZIO 14. l^xyz - a* -12 pq 100 M 35 F^ - 10 xyz 7 a* -10 pq - 30M -10 F^ — Qxyz -4 a* Upq - 4.M -25 F — xyz - a' - PQ M - F 15. orhc - ^H^ 2ab^ - IT -10iV^« — 4 a-hc - 4s^2 - 3ab^ 9r 4^« 10 a^hc 20 s¥ (50 ab' -16 r 16iV^« -15a'bc - 8s«^ - 32 ab' - T - 3^« - a'bc ^¥ - 15 ab' 12 T -14^« 16. -2MA, 3.25 A 20. ^ yy -\yy- fy. 17. 0.36 a;, -3.42 a;. 21. 1 «, - i «, - ■ia. 18. -1.2 7?, - - 3.4 n, 7.1 n. 22. 3 muj — ^mrt I, 2^ mw. 19. -0.75P«, - 2.15 P8,P«. 23. - -2iv", -i;«, 3|v«. Simplify : 24. -3a; + 7a; + a;-6a;. 26. — 5 at^ - at^ -\- 10 atK 25. 12c*-{-c*-Sc*-c\ 27. 6v'-8v'-10v'4-v'. Simplify by adding similar terms: 28. 4 a — 2 a - 12 a; + 6 a + 3 a;. Suggestion. — Adding the terms in a and the terms in x separately gives 8a- Ox. 29. x^ - ab -^10 ab -6 x^ -2 ab, 30. 3 m- — 2 mn — m^ — mn, 31. a' + 2ab + b'-a' + 2ab-W. 32. y^-3yh-it3yz''-:^-2f-^z^y-4.z^. i ei ELEMENTARY ALGEBRA Add: 33. 4 a and — 7 6. Solution. — Since these terms are not simUar, their coefficients cannot be added. The addition may be merely indicated, giving 4 a — 7 6. Indicating the addition of dissimilar terms is called adding them. 34. — 2 a, 3 6, and — c. 35. h% —2,Kk, and A:*. 36. a^6, — ah^j a^, and — h\ Note. — The process of adding monomials by adding the coefficients may be used to shorten computations in arithmetical work. For example, 27 X 54 + 73 X 54 = 5400, which is determined without the aid of a pencil. At sight give the results of : 37. 6x48+4x48. 43. 53x84-13x84. 38. 8x27 + 12x27. 44. 128x96-28x96. 39. 17x25 + 13x25. 45. 42x28 + 16x28 + 2x28. 40. 84x63 + 16x63. 46. 34x17 + 27x17-11x17. 41. 46x35-6x35. 47. 16x38-4x38 + 18x38. 42. 32x46 + 18x46. 48. 16 tt + 48 tt + 36 tt. 40. Addition of Polynomials. — It is evident that two or more polynomials are added if all of their terms are added. Hence, two or more polynomials may he added by adding their similar terms. The process of adding polynomials is similar to that of adding denominate numbers involving two or more units of measure. For example : 12 bu. 1 pk. 2 qt. 12 6 + p + 2 g Just as 5 bu. 2 pk. 5 qt. so 6& + 2jo + 5g 17 bu. 3pk. 7 qt. 17 6 + 3j9 + 7 ^ To add polynomials, ari'ange the similar terms in columns and add each column as in § 39. The sums of the columns, connected by their signs, constitute the sum of the polynomials. ADDITION AND SUBTRACTION 63 As pointed out in § 30 (See Law of Order in Addition, § 30, Note), numbers to be added may be arranged in any order with- out changing the value of their sum. In polynomials to be added, if the similar terms do not come in the same order in all the polynomials, they should be rearrarged so that they do come in the same order. Thus, to add 2a + 36-4c, 5c-4a-&, and 2a + 36-4c 6 4- 5 a — 2 c, the terms in the last two expressions --4a— b + be are rearranged, and the terms in a air^ put first, the 5 a 4- b — 2c terms in b second, and the terms in c hird, as in the 3 a + 3 6 — c margin. EXERCISES Add: 1. 3a-2b 6. 4r-5/-2r" _2rH-3r'-7r" _9r- r' + 4r" 2. 4a + 56 -ex+ y * 4tx-ly 3. 4:U — 2v-\-Zw — 2?t-f V — 6w 7. SA- B + 2C 5A-2B-{- C _ ^4-3^-4(7 8. 2P + 4Q4- E-78 4. -3^+ K-12L -SP- Q-\-SR-{-2S SH-AK-^ SL 7P-6Q-9R + SS ^23-5K- L 9. _20a+3a6-106 B. — 8m-f-6n- p 16a- ab-\- 4:b 5m — 2n4-3p — 7a — 5a6+66 -6m-8n-8p - 2a-8a&-12& 10. 2x — yj—Sx-^y, and Ay — Tx. 11. 5a + 3c — 6, c — a + 6, and 2 6 — 3a-f 4 c. 12. S3x^-5xy-^2y^, Sxy- / + 6a^, and 4/- ar'- ay. 13. l-2m + 3m2, 5m-6-i-4m2, andm2-10m-f-5. 64 ELEMENTARY ALGEBRA 14. Show that the process of adding polynomials by adding the similar terms is used in adding numbers in arithmetic. Thus, to add 723, 422, and 614, show 723 = 7 x 100 + 2 x 10 + 3 that each number may be written as a poly- 422 = 4 x 100 + 2 x 10 + 2 noraial, as in the margin. In what does 614 = 6 x 100 + 1 x 10 + 4 this differ from the ordinary process of 17 x 100 + 5 x 10 + 9 arithmetic ? or 1700 + 50+9 or 1759 15. By writing them as polynomials, find by the method in Problem 14 the sums of the following : 831 123 302 915 614 502 824 201 743 954 553 672 41. Checking Work. — In much of the work of algebra one can easily check the results obtained; that is, determine whether or not mistakes have been made in the work. ^ A good method of checking addition of polynomials consists of assigning particular values to all of the general numbers in- volved in the polynomials and in the sum, and seeing if the sum of the values thus obtained for the polynomials is equal to the value of the sum of the polynomials. Example. — Add and check : 2 a— 5 6 + 3 c, 3 a+4 6 — 2 c, and 5 a— 2 b—4c. 2a — 5& + 3c=— 5" 3a + 4&-2c= 9 - -, r. » -, ^ „i . -V when a = l, & = 2, c= 1. 5a — 26 — 4c=— 3 10 a - 36- 3c = 1 When a = 1, 6 = 2, c = 1, the values of the polynomials are — 5, 9, and — 3, respectively, and the sum of these values is 1. The value of the sum of the polynomials also is 1, which shows that the work is accurate. Any other values of a, 6, and c might have been used. Check this work by giving a, 6, and c other values. Note. — This method may be used in checking much of the work in algebra that follows. While other methods of checking work may be practiced, it will be found that this method has the special advantage of helping the pupil to master more thoroughly the meanings of the symbols of algebra. The pupil should form the habit of carefully checking all of his work in some way. ADDITION AND SUBTRACTION Q^ EXERCISES Add and check : 1. 2aj2-3a; + l 5a?4.7aj-4 — 3a^-6a; + 5 2. a3-3a2 + 3a 4a2- a -a'' + 2a2 + 2a -1 + 3 6. 6. 7. 8. 5w — 6^+ 7w7 — 3tt-2v+ 9m? -8w-5v + 11m; 5r-6s+8 _7r_9.s-5 4r + 8s 3. a2 + 2a6+ h^ a^^2ab+ b^ 2a' -2b' 4. 26m + 10n + 14|) — 12m + 15n-20i) -12n- 5/) — 5 ci?i + 3 6w + 4 — 3 an — 5 hm — 1 an + bm + 1 7 a7i - 2 6w — 2 9. 4s + 3«-4, 3s-2« + 3, -s + <+5. 10. 4n2-3ri + 7, - 2^2- 9 ?i-3, 6 n^ + 4 w-9. 11. 6a2 + 5a6-252, a2-a6-86^ 3a2 + 7a6 + 62. 12. 6a;»-4ar'-3a; + 8, 6a^ + 3a;+2, 8a^-7a;+9, a:2_2a;. 13. m^— 71^, 2m*— 5mri"+6w', 3?n^n— 7n^, — m^— m^n+mn^— n^ 14. .4«-2^^ + 5^2_3^ A'-5A'-6A', 9^^ + 7, - 6 A' - ^2 _ 9 15. v^-vt-t', Sir^ + 4:vt-t', t'-Svt + v^, 7vt-St'. 16. iax-Tby-cz, ax-2by, — 3aa; + 3c2:, 7by — 4:CZ. 17. pq + 2qr —pr, — 2pq — 5 qi- + 4;)r, 7)7 — pr. 18. A:-2F+5F-3, -3 F+2Ar^+6;fc-3, 2A:2-5, 12+3A:2-^, 19. -12r + 15s-8«, 9r-7s-7^, — 9r + 4^, _54-5r, Qr-7t 66 ELEMENTARY ALGEBRA 20. 3v'-5v" + v'", 3-4?;', _ 9 v" + v'", 7 'y' - t;^ _6'(;"-hv"'. 21. 24P+Tr+3^, 5Tr-9^, 6F + 7H, - JI - 5 P, 2H-h8W-P. 22. 5.6a-2.76, 3.5a + 4.2&, -2.2a + &, a-6.56. 23. .25a;2_i 33.^81^ _ 4.5a;2_i 9^.4^ 6.3 -5.8 a; 4-3.5 ic^^ 24. 5 i22 _ 2.5 m, - .5 i?^ + 4 m, 1.45 i?^ + 3.2 m, - .75 R^-9m. 25. 1 - 2 2> - 2.6 i/ + 7.5 Z>^, !>' + 5.5 i)^ _ 7.2, - 1.5 + 6.5 i> 4- 8 B". 26. ia;-i2/, i^j + i^/, aj-y.^ 27. \m — ^n, 2^m + \nj —\m-\-\n, 28. ^H+^K, 1\H + ^K, 2H-^K, 42. Subtraction of Monomials. — What rule for subtracting posi- tive and negative numbers was discovered in § 31 ? Since a mo- nomial is itself a number, this rule holds equally in the subtraction of monomials. To subtract one monomial from another, change the sign of the subtrahend and add the result to the minuend. The change of sign of the subtrahend should be made mentally, the written sign being unchanged. Thus, to subtract 9 x^y from — 18 x^y, let the written work be „ / as shown in the margin. — ^7 2 — Zi X y EXERCISES Subtract : 1. 2a; -3a -6jr 8.V» -5ab -3a; -7a 4^r -2f -2ab 2. 4 m^n -21xyz 16^ - 6P -Sa¥ — m.^n — ^xyz -7wH -14^ - hk -14P - ab' 3. 2^ 2abc -3^ 2wH -4:hk -22 Q 12 abc ADDITION AND SUBTRACTION 67 4. ^wv* 5by^ -6wv'' -8 6/ 5. From 2 a take — 11 a. 6. From 7 a; take 10 x. 7. From -6^ take -4«. 1. 4,x-(-Sx)=? 2. -Sy-{-5y)=? 21x'yh 11 R^ -4V - SxYz - 3i?3 -Sv' 8. From - 20 P take - 9 P. 9. From 32 a^^ take - 16 af. 10. From - 16 ttt^ take - 7 Trr*. 13. 5m-(-2?7i)=? 14. 16^-(-4^)=? 43. Subtraction of Polynomials. — It is evident that one poly- nomial is subtracted from another wlien all of its terms are sub- tracted. Consequently for subtraction of polynomials we have the following rule ; Add to the min'oend the subtrahend with the sign before each oj its terms changed. Similar terms should be written in columns, as in addition. The method of checking work used in § 41 may be used also for checking subtraction. The change of signs should only be made mentally. Example 1.— From Sx^-2xy + y^ take 6 y"^ + 4 xy -{■ S y^. Sx^ — 2xy-\- y^ = 2 1 when 6x^ + 4xy-Sy^- = 6 i a; = 1 - 2 x^ - 6 xy + 4 y'^ =- 4 i y = l If there is a term in the minuend and no term similar to it in the subtrahend, such a term will be written in a column by itself, and, in subtracting, it will be placed in the answer unchanged. If such a term occur in the subtrahend instead, in subtracting, it •»;'iU be placed in the answer with its sign changed. BlXAMPLE 2. —From 4 x* — 3x4-2 x^ + 7 take x^-Sx^-{-2x'^ — x. 2 x« + 4 x* 3 x3 + 2 x2 3x + 7 X 10 1 1 «:6+4x* + 3x8-2xa-2x + 7 = 11 when x = l 68 ELEMENTARY ALGEBRA EXERCISES Subtract and check: 1. 4ic— 5i/ IZ. x^ + x^ -\-x^l x—ly a^ — o^-fo: — 1 2. O + 7 i) 3. 4r-12A 4. — m + 15 a — 3m— 4a 5. 86- 13d 96+ 8d 6. r — 2s + 5i r + 3g-2^ 7. 4a;+ 2/— 102? — X — Q>y-[- 11 z 8. — 12j9-6g' + 4r — 8y9+ q-2r 9. ?^ + V — w u — 2v-\- Sw 10. 3/- c^ + 2^ 5/-2gr+ h 11. 20p — g + r 2p+ 5g — 7r 12. a^ + 2ab-\-b^ a^-2ab + b^ 14. i^^+ 4.F' + 1 SF'-'2F'-1 15. 2ab-^bc-[-5ac 6ab-^7bc — 2ac 16. 2P-h4Q-7i? _5P_ Q + 3i? 17. 3'y-4'^;'4-10^'" _2'y-5'y' + 12v" 18. 7 W-5S-h9H 2W-i-5S-9H 19. a^^a' + a'-l 20. 2a^ -5 21. o3_ a2^_|_ ab''-b' 22. 6-2m+9m2-m3-7W* 9+ m — 9m^-\-m^—m^ 23. a' + 2ab + b' 24. _ 6 TFF^ - F* ADDITION AND SUBTRACTION 25. 4-^2_,_7^4 -e-^ if- 6^ 26. n«- 3n2 - n2 -2n + l 0-5 27. 8a; 9a; -2a;' -9 a;" a;" 28. Wf - WH - 10 - WH + 15 29. 1.5 a 4- 3.5 6 .7rt-1.36 30. 31. From 6a-76-10c take 12 a + 4 6 - 7 c. 32. From 12 ar^ - 3a; + 1 take - 2 a;^ + 9 .t - 7. 33. From m^ — 2 mn + ?i* take 4 m* — mn 4- 3 n^ 34. From 2-7^2^4^take3^-5^2^8. 35. From 2 ^ -h 6 ^2 _^ 1 take 9 i^ ^ 4 ^ + 3. 36. From 15 v' - 6 - 2 v" take _ 5 v" + 9 - v'. 37. From 6 a - 4 a^ + 7 a^ take 10 - 3 a + a^ - 8 a\ 38. From 6 P- 5 P^ + 2 P^ take P^ + 2P2 _ 7 p^ 3 39. From D"" -\- R^ -{- 2 BE take 5 P^ _ 7 jr>/^. 40. From a;« - 3 a;^^, ^ 3 ^2/2 ^^-^^ a^_^r^y_f, 41. From 1.5 r - s + 2.8 1 take .9 i - 3 r — s. 42. From 20 ^ - 12.5 E^ + 4.2 take G - 2.4 ^ + 9 jE;». 43. From 5.6ab — Sk take 4.5 A: — 25. 44. From :^x^ — :^-x — \ take 2 a;2 _ ^ ^ ^ 3, 45. m2-5m4-7-(-2m2-3m + 2)= ? 46. W^-\-24.W-10-(16-5W+W^ = ? 47. «= - Z>2 - (2 a6 - 62) == ? 48. 6 - (.S - 4) = ? 44. Removal of Signs of Grouping. — When the sign — precedes a sign of grouping which incloses an expression, it indicates that the expression is a subtracted expression. 70 ELEMENTARY ALGEBRA f Subtraction is performed by changiDg the sign of each term of the subtrahend and adding the resulting expression to the minu- end. Hence, A sign of grouping preceded by the sign — 7nay be removed from an expression^ if the sign before each term inclosed is changed. Eor example, 3E - (5 B -2 S) =3 R - 6 R + 2 S. And - (- 7 1/ 4- 2 2/2) = 7 y - 2y2. Since the sign + expresses addition, or, when convenient, may- be considered merely as a sign of distinction, therefore, A sign of grouping preceded by the sign -f- ^^ot?/ be removed from an expression, without changing the sign of any of the terms inclosed. For example, 4w+(3w — 7) = 4w + 3w— 7. And + (8 - 2 ^) = 8 - 2 ^. When signs of grouping are inclosed within other signs of grouping, it is wise for the beginner to remove the innermost sign first, then the innermost remaining sign, etc. Thus, 3a-(9a-{26- [6a-&]}) = 3 a — (9 a —{2 6 — 6 a + 6}), removing brackets, = 3a— (9 a — 26 + 6a— 6), removing braces, = 3a— 9a4-26 — 6a + 6, removing parentheses, = 3 6- 12 a. EXERCISES Kemove the signs of grouping and add similar terms : 1. 2r+(5r-3). 8. 5 -(4 tt - l)-2 tt. 2. -%T-(l-2Ty 9. 6i + (9-X)+8. 3. 12 4-{-m;4-w^J. 10. 2 M -\-t-{b M-t). 4. C-[30a-2]. 11. a2-(62_2a&). 5. 27-(i^-32). 12. l^f-lyJ^\^f-y\-y. 6. a— (6 — 2 a — 5), 13. 6 w — 4mn-f (3 m^i— 2 w). 7. 2s + M^. 14. 2 52_(jg2_4^C'> 15. (2r-7^)-(r-4^). Suggestion. — Since neither of the signs of grouping incloses the other, the tvro may be removed at the same time, independently of each other. ADDITION AND SUBTRACTION 71 16. (l-a; + a^ + (3a?-8). 17. (2i)-5g) + (4i? + g)+6g'. 18. W-X^w-2.^)-(w-l.bW). 19. p_j3Q-li4-J4Q-6i25. 20. x' + \2x-{y?-^x)\. 23. -(-(-1)). 21. 2i;-[«-v+(3^-4^;)]. 24. _(_(_(_(_(- a))))). 22. i2D-\^-D\ + l). 25. 2^-(cZ-(2i-(Z)). 26. s-^at+mat-{s-at)\-\at. 27. m — (5n — (m — (2w — 6m))). 28. a2~(2 ah - (-(- 60))-(«' +(^'-2 ah)). 29. Solve6y-(4-2 2/)=12. 30. Solve F+(3 + 5 F)-7 = 0. 31. Solve 12.5 k - (10 - 2 fc) = 19 - (A; - 2). 32. Solve 32- (8-2) = ^-(2 2-10). 33. Solvew = 8-(3i^-14-{6-w}). 34. The sum of two numbers is 20 and their difference 6. What are the numbers ? Suggestion. — Let n = the larger number. Then 20 — n = the smaller. 35. The sum of two numbers is 7 and their difference 17. What are the numbers ? 36. The sum of the angles a and h is 180° and their difference 30°. How many degrees in each ? 37. My living expenses for two years amounted to $2500. The expenses during the second year were $280 more than during the first year. Find the expenses for each year. 45. Insertion of Signs of Grouping. — By reversing the pro- cesses in § 44, it follows that terms of a polynomial may he inclosed 72 ELEMENTARY ALGEBRA within a sign of grouping, ivhen this sign of grouping is preceded by the sign +, without changing the signs of the terms; and, when pre- ceded by the sign —, by changing the signs of the terms. Thus, to inclose the last three terms of a^ + 2bc—b^ — c'^ in parentheses preceded by the sign +, we have a^ _^ (2 6c — &2 _ c2). If preceded by the sign — , it becomes a^ ■— (b^ — 2 be + c^). EXERCISES Inclose the last three terms within a sign of grouping preceded by the sign + : 1. A + B-C-D. 6. x' + 2xy +y' -z' + 2zw-wK 2. xy + yz-zw-{-wx. 7. 16-8P+P2_^12Jf-9Jf2_4. 3. 5_2n-9n2 + n3. 8. k-{-T-\-10k-l-25k\ 4. r._4s + 4s2 4.1. 9. IS-B'-b'-Bb. 5. m^-}-6pq-q^-9p\ 10. 1 + 6rr' - r^ - 9r'^ 11-20. Inclose the last three terms in each of the above ex pressions within a sign of grouping preceded by the sign — . 21. In an — 2t — 3n-}-bt, place the terms involving n in paren- theses preceded by the sign + and those involving t in parentheses preceded by — . 22. In 10 P— 2 W-{- mP— n W, place the terms involving P in parentheses preceded by + and those involving IF" in parentheses preceded by — . SUPPLEMENTARY EXERCISES Simplify : ♦ 1. 5V^ — 8V^ + 6V^ + Vm — lOVn. 2. Va + 6 — 4 Va -h ^ + 6 Va + b — 9 Va + &• 3. 12(x-y)-3{x-y)-l&(x-y) + {x-y). 4. _ 9(s - af^ - 6(s - af) + (s - at~) + 7(s - at"^. 5. (2 irR + If + 14(2 ttP 4- ly - 8(2 ttP + If + (27rP + 7)^. 6. -20^/8 + i%'+6-^8+i%2_^18^8+iTF^2_9^^^|1^^2 ADDITION AND SUBTRACTION 73 Simplify by combining similar terms : 7. 16-2VA-^5VA-24.-7VA. 8. D + Vw + -Vi — Ww — 5 D -{- 7 Vw — eVi. 9. 5(W-V) -SW' + (W-V)-\-17 W^-4.{W~-V), 10. x^-2xy-\-y^-3x^-6xy-Sy^-{-4:Xy. 11. ri^ + 4rir2 -^r,'-2 r,' -\-Sr,r, -Sr^ Simplify by writing with a polynomial coefficient: 12. ax-\-bx + ex. Solution. — These terms are similar in ic, and hence are added by adding the coefficients of x. Since the coefficients a, b, and c cannot be added into one term, but their sum can only be indicated, we get (a + 6 + c)x. 13. my-{-ny — py. 14. al^ -hR^-cBr-\-dR\ 15. Mf + 2Nt^-^ Pe 4- Qt\ 16. 5Z)2 + ^i)2-/iZ)2-7Z>2. 17. 4:a{x — y)—3h{x — y)-\-^(x~y). 18. Add nA -\- mB -\- pC and xA — yB-\- zC. 19. From all- hV - cir subtract gU + hV- JcWi Remove signs of grouping and combine like terms : 20. a + [2 a - (3 a - 2 6)] - (4 ;> - 3 a). 21. [m - (2 m - n)] - [ - 3 ?i + (4 m - ?i)]. 22. (A-5B)- \3A-l4.B + (6A-B)-(4.B-2A)-]\. In the following remove parentheses, leaving brackets, an^l Attn plify the results as far as possible : 23. [(x -2y) + (x + y)X(x-2y)- (x + y)l 24. [r-(s— «)][r + (s-0]. 25. [5 A + (3 + A)^ [5 ^ - (3 + A)]. 26. [(4 r - 3) + (3 ?•' - 4)][(4 r - 3) - (3 r' - 4)]. 27. 1(7 a -2b)- (c -|- 3 d)][(7 a - 2 6) + (c + 3 d)'j CHAPTER V MULTIPLICATION AND DIVISION OF LITERAL EXPRESSIONS 46. Order of Factors. — Is there any difference between the values of 3 X 4 and 4 X 3 ? Of 2 x 3 x 4 and 3 x 4 x 2 or 4x2x3 or 2x4x3? Does it change the value of the product to change the order of the factors ? In general, has ab the same value as ba ? Has xyz the same value as xzy or yzx ? Has 2 x a X 3 x 6 the same value as 2x3xax6? Numbers to be multiplied may be taken in any order without changing the value of the product. Note. — This fundamental principle is known as the Law of Order in Multiplication. 47. Grouping of Factors. — In 3 x (2 x 4) the parentheses indi- cate what ? Is there any difference between the values of 3 X 2 X 4 and 3 X (2 X 4) ? Of 2 x 3 x 4 x 5 and (2 x 3) X (4 x 5)? Does it change the value of the product to group the factors so that the multiplications are performed in different ways ? In general, has abc the same value as a (be) ? Has sc^ocFy^y^ the same value as (afx^) {y^y^) ? Numbers to be multiplied may be grouped in any way without changing the value of the product. Note. — This fundamental principle is known as the Law of Grouping in Multiplication. 48. Law of Exponents in Multiplication. — How else is AA written ? How else is nnnn written ? In a^ what does the ex- ponent show ? a^= aaaaa. a^ = ? i?^ = ? 74 MULTIPLICATION AND DIVISION lb Multiplying a? by a^, we get o} xa^ = aax aaa or aaaaa = a*. Similarly, m* xm^ = mmmmmm = m^. v^ XifX'v^ = vvvvvvvvvv = -y^". D'xD' = D*+' = D\ In multiplying powers of the same base, the exponent of the product is obtained by adding the exponents of the factors. In symbols, the above principle is expressed by 49. Multiplication of Monomials. — By use of the principles stated in the preceding sections the product of any two or more monomials may be found. Example 1. — To multiply 2 x'^y by 3 3?y^^ we have (2 xhi) (3 7?t) = (2x3) (a;2a^) (yy^) = 6 xV- Example 2. —The product of - 5 M'^N^ and - 4 M^N^ \a (- 6if2JVr6)(-4ilf4iV2) = (~ 5)(-4)(JJf2JJf*)(iV6iV2) = 20 M^N-. To multiply monomials, find the product of the numerical coeffi- cients, using the laws of signs ; then annex to the result the products of the like literal factors, using the law of exponents. — Iv^t^ In performing multiplication, the work is often ^ - arranged in column form, as here shown. ZT^fiw^ EXERCISES Find the product of : 1. a^ and a^. 5. 'i^ and ?-*. 9. y and i/^. 2. ar'anda;^ 6. Fand F*. 10. n« and ti^. 3. b^ and b\ 7. U and L\ 11. TFand W^. 4. M^ and M\ 8. P and f. 12. T' and T". 16. z and 2;^^ 19. p^, p2^ and p^. 17. A\ A\ and A\ 20. ic, CC-, and x^. 18. C\ C\ and C\ 21. w, 71^, w^, and n 29. c, 8c2^ , 3 c^ and 2 cl 76 ELEMENTARY ALGEBRA 13. (2^ and (f. 14. s^^ and s^ 15. WsindU^ 22. _B, ^2^ 5«, and B^. 23. 2 x^ 3 a^, and 2 ar\ 30. -Sd\ -6d, d, and 2 d^. 24. 4??i, 2 m'*, and m". 31. ic^ and x^-if. 25. - 2 ^, - 3 ^^ and 4 ^9. 32. - 2 a?n^ and 5 an^. 26. 3 jK^ - 7 i?, and - 2 R\ 33. - 4 ^(7 and 3 A^B. 27. - 5 2/, - 9 2/^ and - 2 2/''. 34. 7 s, - 2 sr, and 6 s^r^. 28. 5,-6 5", and 4 ^. 35. - 9 a^2/V ^j^^j _ 4 ^^4^ 36. 3 Z)W, - 2 I>W, and 8 D'^W. 37. - 6 M^N^ 9 Jf 2jvr and - 5 iV^^P. 38. 16 a;*^^^ - 2 a^;2^ and 7 fz\ 39. — 2 mV, 8 my, and — 5 ii^p^. 40. a^ -2a«62^ 7 a^c, and -^h(?. 41. |- a^2/> ~~ i ^^^ ^11^ ^2/^- 42. - 1 HD, i H^D, and 3 Z)^. 43. f rs2, -^2rV, and ir^s^ 44. lp(f, fp^q\ and 2pV. 45. F, iF, -4F2, andiF^. 46. - f cW^, I c^d^, and cV. 47. tV ^^^'^ ^ TF'^'^ and 4 TF^^ 48. -MV% -\MV\ and -6il^F2. 49. 3,-5 Qt, 8 e% and - \ Q^t\ 50. Multiplication of a Polynomial by a Monomial. — The multi- plication of a polynomial by a monomial is similar to the multi- plication of denominate numbers containing two or more units of measure. 9 yd. 2 ft. 6 in. 9 2/-f2/+ 6i Just as 4 so 4_ 36 yd. 8 ft. 24 in. 36 ?/ + 8/+ 24 i MULTIPLICATION AND DIVISION 77 In general, since the whole of a quantity is multiplied by a number when each of its parts is multiplied by that number, we have the following rule : To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial^ and add the 2>CLrii(^l products obtained. Note. —This fundamental principle, that a polynomial is multiplied by multiplying each of its terms separately and adding the partial products ob- tained, is known as the Law of Distribution. It is expressed in symbols by n(a + b)= na + nb. Example. — The multiplication of a^ — 2ab + b'^ a^ — 2 ah 4- b"^ by 3 a^b'^ may be indicated either in the form g ^3^3 3a«63(a2 _ 2 a6 + b'^) = 3 a^b^ - 6 a'b^ + 3 a^b^ S a^b^ - 6 a^b* + 3 a^b^ or as in the margin. It is always more convenient in algebra to perform multiplica- tions from left to right. This may also be done in arithmetic, but for convenience in " carrying " it is usually more convenient in arithmetic to begin at the right. EXERCISES Multiply : 1. a; -h 2/ by 5. 11. a;^ — 2x-|-l by 4. 2. a — 6 by 3. 12. a^ -{- 2 ab -\- b^ by ab. 3. n-4by7. 13. 3-8 m + 4 m^ by 5 m. 4 P4-lby6 14. 7 a- -^12 b'^ hy -2 a^b. 5. 2 ^ - 3 by 4. 15. 15 Rt-A f by 4 R\ 6. 6a + 5&by2. 16 4.^ + 1 fhj 2s?y\ 7. 8Jf-9iVby3. 17. 3 s^- 5 s^ + Q s by 4s2. 8. -5JE; + 7 7v^by 10. 18. B" -A.AGhy ^ AC, 9. 4 a; - 9 2/ by ic. \2. 5-2 v -ir-1 v^hy —^v\ 10. 2 a - 7 6 + c by 6. 20. X^ - 2 L^ -f- 3 by 4 LK 21. 3 ?i - 4 + 12/^ by - 2 n-p\ 78 ELEMENTARY ALGEBRA 22. TF^-6 TF2+2 TF-3 by -5 W\ 23. 2 c^ + 15 - 4 c^ by - 3 c\ • 24. A' + 2AB + B'hySA':^. 25. a6 + 6c 4- ca by a6c. 26. 2iy-3DW-\-W'hj -9L^WK 27. ^3-^3 by 2AB. 28. Sx^-ea^ + ii' + Sx-lhjSa^. 29. 7a6 + 3&c — 4cd by 2ac. 33. ^%5p + 3g). 30. 5(4- a -a^). 34. ^(^^ - 5 ^5 + 6 5^)^ 31. 4:X(3x + Sy). 35. -2 7)1^(3 - 9 m- Sm^). 32. Tt(f-at). 36. 3 2/^(a^-3a;^ + T/). 37. - 8a262(_ 15 a^b 4. a?)^ - b*). 38. 4 m^n-(m^ — m^n + 3 m/i^ — n^). 39. i(4s2-6.s + 2). 42. -2a^+^a^?/-ia;?/2+j2/3 40. -^w(iw'-iwv+iv^. -i^y' 41. ^a'-^^ab + ib' 43, 1.2 m2- 2.5 mn + 3.4 ii2 jaft -1.5m^ 44. 2.15p3_. 05^ + 1.22) -4 .5p^ Simplify by removing signs of grouping and uniting similar terras : 46. 2(2a-36)-3a. 46. 5(A-B)-^2(B- A). 47. 7(^-2st-\-f)+3{2s^ + st-^t^, 48. a(ab-b^-^b(a^-\-ab). 49. 3P(5-2P) + 4(3P+P2). 50. iS(2S'-iS)-i-i(2S'-^S). MULTIPLICATION AND DIVISION 79 51. 4(2 ^-3 5)- 2(4^ + ^). Note. — In this expression the term — 2(4 A-{- B) may be looked upon in two ways, either as indicating that 4 J. -f 5 is to be multiplied by 2 and the result subtracted from the part that precedes, or as indicating that ^ A + B is to be multiplied by — 2 and the result added algebraically to the part that precedes. In either case the terms within the parentheses (i: A-\- B) will have their signs changed in the process. 52. 6(c — 2 A;) — 4(2 c— 3 A:). 54. m{n^—mn) — n{mn-\-m^). 53.- 6P-4Q-3(7P + 2Q). 55. -5(10r+3 0-2(2 «-r). 56. -d(x'y-xf)-(^xy^-o?y). 57. -3{^ + 2v^w-2vf)-2{-2i^-\-3v^w-w% 58. This rectangle is maxie up of three parts whose common width is a and whose lengths are ar, y, and z, respectively. Show that the areas of the parts are ax, ay, and az, respec- tively. What does a{x -\- y -\- z) represent ? ^_^_^_:^ Hence, show that a {x -{- y -^ z) = ax -\- ay -{- az, which illustrates the Law of Distribution. 59. By a rectangle show in the same way that 2x{a-\-2'b -\- c)=2ax -\-4.hx-\-2cx. 60. Show that the process of multiplying a polynomial by mul- tiplying each of its terms and adding the partial products obtained is used in arithmetic in the multiplication of a number of two or more figures by a number of one figure. Thus, to multiply 432 by 8, we see that 432 is in 432 — 400 4. 30 + 2 nature a polynomial, and may be written as the sum 3 3 3 3 of three terms. 432 is multiplied by 3 by multiply- i296 = 1200 + 90 + 6 ing each of these terms by 3 and adding the results. Note. — The ancients employed this principle in multiplication. Thus, by the Greeks the method of calculating 7 x 326 was equivalent (in modern nota- tion) to that shown here. Because of the principle of place value in our notation, we perform our multi- 7x326 = 7(300 + 20-1-6) plications mechanically, without stopping to think =2100-}- 140 -|-42 of this Law of Distribution underlying the work. — ^^^^ 61. By the use of a rectangle as in Ex. 58, illustrate the Law of Distribution in finding 3 X 268 fp^ ^ fz 80 ELEMENTARY ALGEBRA 51. Multiplication of a Polynomial by a Polynomial. — The pro- cess of multiplying one polynomial by another is similar to the process iu arithmetic of multiplying one number of two or more figures by another number of two or more figures. Just as in arithmetic the multiplicand is multiplied by the number repre- sented by each figure of the multiplier, and all the partial prod- ucts added, so irf multiplication of polynomials in algebra the ^uultiplicand is multiplied by each term of the multiplier, and all ^he partial prod»^.ts added. For example : 26 2a;+5y 34 3a;-f4 y ^ust as 100 = 4 X 25 so 6 x^ + \^xy = 3 a;(2 a; -}- 5y) 750 = -SO X 25 8 xy + 20 y~ ^4.y(2x -\- by) 850 = S4 x25 6x2 + 23icy-F20?/-^ = (3a: + 4?/)(2x + 5?/) Each term of 2 x -|- 5 ?/ is multiplied by 3 a; and then by 4 y, and the similar ^rms of the partial products are written in columns and added. Note.— See that pupils clearly understand that we use this process abridged f.n arithmetical multiplication. Then have them see that algebraic multipli- cation, is the same. It is easily seen that this process is only an application of the Law of Distribution noted in § 50. Hence, T'o obtain the product of two polynomials, multiply the multipli- cand by each term of the multiplierj and add the resulting partial products. The work in multiplication can be checked by assigning particu- lar values to the general numbers involved, as in addition and subtraction, and seeing if the product of the particular values of the factors equals the particular value of the product. Example 1. — Multiply 2a— 36+5cby3a+6— 2 c. ^a~+ h -2c =^ 6 a2 - 9 a& + 15 ac 2a6 _ 3 62+ 5 5c - 4ac + 6 6c - 10 c2 6a2 - 7 a6 + 11 ac- 3 62 + 11 6c - 10c2 = 6 MULTIPLICATION AND DIVISION 81 When a = 1, & = 1, and c = 1, the multiplicand = 4, multiplier = 2, and product =8, as it should. Observe that the terms of the first partial product are written in a line by themselves, those of the second in a line by themselves, etc., and that the new kinds of terms, when obtained, are spaced out so that only similar terms will fall in columns. Note. — A polynomial is said to be arranged according to the powers of a letter when the exponents of that letter either increase or decrease in the suc- cessive terms as we pass from left to right. Thus, o^ — 2 x^ + x^-f- x — 5 is arranged according to the descending powers of x ; while 2 y* + xy^ + x'^y^ -f x^y + x^ is arranged according to the ascending powers of x. How is the latter arranged as to y ? It will be found an advantage in multiplication of polynomials to arrange, if possible, both the multiplicand and the multiplier alike according to the powers of some letter, before multiplying. Example 2. — Multiply A^ - 2 ^ 2 A\)j A- Q + A^. Arranging both trinomials according to the descending powers of A^ we have A^^2A -2 = 1 A^-{- A -Q =-4 ^* + 2 ^3 - 2 ^2 when A^ + 2A^- 2 A ^ = 1 -QA^-\2A + \2 ^4 + 3 ^a _ ^-2 _ 14 ^ + 12 = _ 4 Note. — Substituting 1 for each of the letters in the two preceding examples tests the coefficients and signs only. Why ? To test the exponents other values than 1 must be assigned to the letters. EXERCISES Multiply and test : 1. ri + 2byn + 2. 7. 8& + lby36 + l. 2. A-^hy A + 2. 8. 3 72 + 1 by 4 i? + 1. 3. a;-4bya;-3. . 9. 2F+5by3F+7. 4. 2a-f3 by 3a-2. 10. 5^/- 8 by 6?/- 1. 5. 5 ;z 4- 6 by 2 + 4. 11. 4 Z) + 2 by - 3 Z> + 5. 6. 4 i - 7 by 4 ^ 4- 3. 12.2 m + 6 by 2 m - 6. 82 ELEMENTARY ALGEBRA 13. a-lObya + lO. 21. 2A: - 3c by 2 A:- 3 c. 14. 2 TF + 3 by 2 TF+ 3. 22. x" -2 x -^Ihy x -{- 2. 15. 3 + 4a;by 2 + 3a;. 23. A^ -^ A + 4.hy A -h. 16. 7 — /S' by 3 + /S'. 24. '?;2 + 'yw + w^ i^y ^ ^ ^^ 17. 5 _ 4 5 by 5 - 4 6. 25. Jf^ _ ^2 ^y jj^2 ^ ^2^ 18. 2a + 36 by 5a — 2&. 26. 2a;— 32/ + 1 by3x+22/— 1 19. Q>p + 4:thj p — 2t. 211. a;2 — 4 + 3 ic by 2 + 5 a;. 20. ^B + GhybB+G. 28. 6- 5 72 by 4 i2 + 7. 29. a^-a- + 4a-6by 2a2-3a + 2. 30. 3 71^ — 71 + 4 by 4 71 - 3. 31. l-2>S + 3>S'2by4^ + 5. 32. 24-3r+7''by 2 + 3r+T2. 33. 4a + 66 + 10cby 2a-36+4c. 34. R-bS+'dWhj3R-\-2W-S. 35. ia + ibyia + i- 36. 2 71 + 6^- 5r by 2p — 3n. 37. a — 1 by 1 + 2 a + a^. 38. 36^-562 + &_4by 3-2&. 39. D-2D^ + 4.hy%-\-2D-D\ 40. 2w3-5w;2^4w-l by 3-2^2 — 2(;. 41. (a? + 5) (2 a; -5). 46. (3 - v) (v^ + 6 - v). 42. (2 ^ + 4) (5^-1). 47. {a^^Jf){a^-W). 43. (1-40(1+40. 48. (7i2-4)(7i^ + 6 7i2 + 5). 44. (S2-4^(7)(jB2+4^0). 49. {a - ^ t) {2 a' -\- at - f), 45. (a^-2a; + l)(aj-l). 50. {\x + \y){^x-\y). Find the product of ; 51. a; — 1, 0/' + 2, and a; -}- 1. Suggestion. — Multiply x — 1 by a; + 2, and that product by x + 1. .. MULTIPLICATION AND DIVISION 83 52. a + 3, a — 2, 2 a + 1. 53. 2J5-1, J5 + 2, 3JB-2. 54. 3F-4, 3 F-4, 1 + F. 55. m 4- n, m — w, m^ + n^. 56. 2 4-d, 2-d, 4 + (2l 57. 2p — ^t,^p-\-2t,t-\rp, 58. ^2^^4-1, ^_ 1,^3 _^i^ 59. a^ — 2^, (c^ 4- z^y ic^ + 2^. 60. (^ + l)(^ + 2X^ + 3)(^ + 4). Bemove signs of grouping and simplify by combining terms : 61. (i4-2)(^ + 3) + (2i-l)(«-4). 62. (2 4-F)(3i^-l)+i^(5i^-2). 63. h''-2h + {l-h){\+h). 64. {x 4- «;) (a; -H to) -\-(x — w)(x — w), 65. (a + 2)(a4-5)-(2a4-l)(a-2). Suggestion. — The product of (2 a + l)(a — 2) is to be subtracted from (a + 2) (a + 6) . What operation must be performed first ? 66. (A-h B)(A + E) - (A- B){A- B). 67. 2(2/^-32/ + !)- (2/ -f4)(2/-l). 68. (v-^t)(v-2t)-(v-t)(v-{-2t). 69. n2-3w + 4-/2-w^(3?i4-2). 70. (2 a 4- 3 6)(2 a4-3&)-(2a-3 6)(2 a - 3 6). o? + ^ 71. Show by the accompanying diagram that (a. 4- b) (c -\- d) = ac -\- be + ad -\- bd. 72. Show by a diagram, as in Problem 71, that (a; 4- 5) (a; 4- 7) = ar^ 4- 12 a; + 35. 73. Show by a diagram, as in Problem 71, what the product of (a-4- & 4- c) (a; -h i/ + 2^) is. 74. Show how the Law of Distribution is used in multiplying the following : { c?5 by - 1 a;Z)3. 54. Division of a Polynomial by a Monomial. — The division of a polynomial by a monomial is similar to the division of denominate numbers containing two or more units of measure. 5 ft. 3 in. 5/+3i •^^^'^ 8 )15 ft. 9 in. '^ 3 )16/+ 9^ 86 ELEMENTARY ALGEBRA In general, since divisor x quotient = dividend^ the quotient wiust be such that the product of its terms by the divisor will give the terms of the dividend. Hence the rule : To divide a polynomial by a monomial, divide each term of the dividend by the dioisor, and add the partial quotients. Example. — Divide 4:ofiy — S xV + 6 x?/^ by — 2 xy. We divide each term of the dividend hj — 2xy and set the results down in a line with their signs, which indicates their sum. The work may be written in either the form (4 x^y - 8 x2?/2 4- 6 xy'^) -i-(- 2 xy) = - 2 x^ + 4 xy - Sy^ -2x2 + 4xy- 3y2^ or — 2 x?/)4 x^y — 8 x'-^?/"-^ + 6 xy^ EXERCISES Divide : 1. 4m-67iby2. 6. -24P+ 72 Q by - 12. 2. 12D-{-9FhjS. 7. 81 a6 - 18 ac by 9a. 3. 16a; + 24by4. 8. a; + 2a;2bya;. 4. 21a-286by7. 9. A'-SA'hjA\ 5. 36 + 422/ by 6. 10. 30 m^ + 40 m^w by 5 m*. 11. 2pq — 4:pt-]-6pvhj 2p. 12. _ 6 TTV + 18 Wy- 12 Wy by 6 W^g. 13. af-x^bja^. 15. 27^-18^ by - 9^. 14. a* — a^b -j- a%^ hy a^. 16. — x — y-\-zhj—l. 17. -A-^2B-Chj -1. 18. 4 w; V + 8 w;3^3 _ 12 wh* by 4 w;^?;^. 19. 12 pH - 8i>¥ + 24p2 by 4^1 20. -21a^-35a^-l4aj^by -7a?*. 21. (8a6-46c-206c?)--4&. 22. (2;^a;?/ — 2 giya^ -f- 3 2?/^) -=- 21/. 23. (2P¥-3P¥-f-P^^)--P2^. MULTIPLICATION AND DIVISION 87 24. {-a^-to?Jrfa)^{-a). 25. {M^N^-^MN^)^{-MN^). 26. {a?K-lh^K+2cK)^{-K). 28. {-^xy-\Jy)^{-\xy). 29- (i%-iTr/)^(i%). 30. (.5« + «'-1.5^)-i-(.5 0. 55. Division of One Polynomial by Another. — Let us divide a^ + a;4 + 7a^-6a; + 8bya;--|-2a; + 8. First, arrange both dividend and divisor according to the descending powers of x (see § 51). The work may be arranged ag below. 2,1 x2 + 2 X + 8)a;* + x« + 7x2_0a; + 8 a^ + 2 a^ + 8 x^ - x*- x2-6x + 8 - x3 - 2 x2 - 8 X x2 + 2 X + 8 x2 + 2x + 8 Explanation. — The product of the term of highest power in x in the quotient and the term of higliest power in the divisor must give the term of highest power in the dividend. Hence, the highest term of the quotient is obtained by dividing the highest term of the dividend x* by the highest term of the divisor x^. This gives x-, the first term of the quotient. Multiply the whole divisor by the term of the quotient just found. This gives X* + 2 x^ + 8 x2, which is placed below the dividend. The dividend is the product of the divisor by the whole quotient. Hence, subtracting the product x* + 2 x^ + 8 0-2 from the dividend, the remainder — x' — x2 — 6x + 8 must be the product of the divisor by the part of the quotient to be found. Therefore, the product of the next highest term of the quotient by the highest term of the divisor must equal the highest term of the remainder. Hence, dividing — x^ of the remainder by x'-^ of the divisor gives — x, the second term of the quotient. Multiply the whole divisor by the new term, — x : subtract the product from the remainder. This leaves x- + 2 x + 8. 88 ELEMENTARY ALGEBRA Evidently the third term of the quotient will be obtained from this second remainder just as the second term was obtained from the first remainder. By continuing this process, all of the terms of the quotient may be found. If in any problem the divisor is an exact divisor of the dividend, the work may be carried on until a remainder zero is found. Otherwise the work may be continued until a remainder is ob- tained in which the highest term is of lower power than the highest term of the divisor. This is a t7me remainder. Hence the rule for dividing one polynomial by another : 1. Arrange both the dividend and divisor according to the descend- ing or ascending poivers of some letter. 2. Divide the first term of the dividend by the first term of the divisor to obtain the first term of the quotient. 3. Multiply the whole divisor by this term of the quotient, arid subtract the result from the dividend. 4. Treat the remainder as a new dividend (having the terms arranged as before), and repeat the process, continuing until either the remainder zero, or a true remainder, is found. Example 1. — Divide a^ — 11 a + 30 by a — 6. a-6 a - 6)a2 - 11 a + 30 a2- 5a - 6a + 30 - 6a + 30 Check. — When a = 1, dividend = 20, divisor = — 4, and quotient = — 5, as it should. Example 2. — Divide x^ -{■ ij'^ hy x + y. 3-3 _ x^y _j_ 3-^2 _ yZ X + y)x^ + y' x^ + x^y -x^y + y' — x^y- - T^Y x^y' + y" x2y2 + xy^ -xy^ + y' ^xy^- jl 2 1/4 MULTIPLICATION AND DIVISION 89 In this example there is a true remainder 2 y^ obtained. By using the fractional form to indicate the division of the remainder, as in arithmetic, the entire quotient in this problem may be expressed thus: x-\-y x + y Check. — When x = 2 and y = 1, dividend = 17, divisor = 3, quotient = 6|, as it should. Note. — Observe that the terms in the dividend, divisor, and all the prod- ucts and remainders in the above example are arranged in descending powers of X. Similar terms should be kept in columns in the work. For this pur- pose it is often necessary to leave vacant spaces between the terms, as in this example. EXERCISES Divide and check : 1. a^ + 3a + 2hy a-{-l. 14. 4 ^2_ g ^^ 2^-f 3. 2. iV2 + 3i\r_40 by JV^+8. 15. 25 R'- A by 5 M -2. 3. a^-5x + 6hyx-2. 16. 4.p^ -Ihy 2 p-{-l. 4. p2-f-7P-f-12by P+3. 17. l-da'hyl-Sa. 5. t^-6t-\-Shy t-4:, 18. 16 - 25 D^ by 4 + 5 Z). 6. y^-9y-\-20hy y-4:. 19. a^-b'^hy a-b. 7. H'^H-SOhy H-S. 20. x'-4:fhyx-^2y. 8. &=» + &-20by 6 + 5. 21. TF^- 25 / by TT-S^. 9. m^ + 3 m — 40 by m — 10. 22. 7n^-\-5 mw +4 ?i- by m + 4 n. 10. 2a^ + 3a: + lby 2a; + l. 23. P'-9 PQ-\-20Q- by P-5 Q. 11. 3a^ + 16fl;-12by a;-|-6. 24. (c^ ^ 21 -f 10 a; by 3 + «. 12. 12Jr2-^-6by 4/r-6. 25. 1 + 2 a-f a^ by a + 1. 13. 6 F2+29F+35by 3 F+7. 26. 15 2/ + 36 + / by 2/ + 4. 27. 12^-30 J5:2^2J5;-5by 2^-5. 28. 2r3-3r' + 3 + 4rby 3 + 2r. 29. ^ - 1 by 5 - 1. 30. f-^by y-\-z. 32. H'' -fby H^ -\-f, 31. W^-V'by W^+V^. ^^. 14AR''-lbyl2R + l. 90 ELEMENTARY ALGEBRA 34. 63^3&2^_36 + iby62 + 2& + l. 35. ic* 4- x^ + 1 hj x^ — x^ + 1. 36. n^ — a^ by )i^ — a^. 37. 22/^-92/^^-14 2/4-17 2/' by /-2y. 38. 8d^ + 27«^by 2d-3^. 39. 4g*-9g2_|_20g-25 by 2^2 _^Sg-5. 40. ^3-2.42 + 5 by A-5. 41. ^^-3/A: + 6A;2by^2_3;^^ 42. a'* + 7i^ + a^n^ by n^ -}- a^ — an. 43. 16 7i^-96/i3 + 81-216/H-216/i2 by 94-4/^2-12^. 44. A rectangular field whose area is oj^ 4- 3 a^2/ + ^ ^V^ + 2/^ ^^s a length of ic^ 4- 2 ^2/ 4- 2/'. What is its width ? ^N^^ Show that the problem 682 -;- 31 may be written in the f6£hl (6^2_^8«4-2)-^(3f 4-1). Eind the quotient in this form, and compare the answer with that obtained by the arithmetical process. 46. In the problem 882-5-42 express the given numbers as polynomials and find the quotient. Compare the answer found with that obtained by the arithmetical process. 47. Divide 1 -J- a; by 1 — a;. How many terms are there in the quotient? 48. Divide 1 4- 2 n by 1 4- n, carrying the quotient to four terms. 49. Divide 1 4- 2/ by 1 — y, carrying the quotient to four terms. ^/J SUPPLEMENTARY EXERCISES Find the product of: 1. a^ —2a^, and — 3 a. 6. a^ft^+i and a^+^ft". 2. x'^ and o^". 7.-8 W^v and 3 TfV. 3. 2/^ 2/^^ and 2/*"*. 8. 52*^, -^22"+^ and 2-^ 4. 3 r, — 4 r+i, and — 2 r-\ 9. a'' - a"" - oT- ■ a"" - a". 5.-5 7^«+^ — 2 i^'*-^ and R^. 10. aj*" • a?*" • a;"* • a?"* • • • to w factors MULTIPLICATION AND DIVISION 91 Divide : 11. w^'' by w\ 15. 12a^ by -Q^x''. 12. Jtf « by ilf*. 16. - 63 ^^ by - 7 A\ 13. 2/'"' by 2/2«». 17. 43 a;«+^6'-'* by - 6 a;*5»-^ 14. JB«+^ by ^. 18. — 80 vVT by 16 v'^-H''-\ Multiply : 19. a" + &'' + c'' by a^ 22. P"4- Q" by P + Q". 20. a;'^^ — 2/" + 2;""^^ by m^f. 23. a^ + h^ by a^ — 6". 21. ic« — I/* by af — 1^/^ 24. a;"— ?/", a;" + 1/", and a^ H- 2/**. 25. F'4-1, F*-l, F^ + 1, F^ + 1. Divide : 26 a^ — a^ by a*. 27. 10a"+2-4a''+»by 2a2. 28. Jf^ + ^f+^ + itf^+^by iJf'. 29. 2/"+ V*+^ + 2/2*»+ix'*+^ by a;«+V^*« 30. a^ — 2/" by a^ — 2/^. 31. "nt^ _ ];^4« by T^" — F**. 82. a«"»4-6^by a"' + &"*. 83. /S'*«- T^-by S"- T". 84. a;3^3a,2y_^3a^^2/3_|.^bya; + 2/ + 2. Suggestion. — Arrange the dividend, divisor, all partial products, and remainders according to the descending powers of x, without reference to the other letters involved. If two or more terms in any case contain the same power of x, it does not matter which precedes. 35. a^-\-b^-c^-^3abchy a-{-b-c. 36. m^ -\- n^ -\- p^ — 3 mnp hj m -\- n -{- p. 37. u^ + v^ + iv^ -\- 2 uv -{- 2 uw -\-2vwhj u-\-v-\-w. 38. 4 a^ + 4 a6 + 6^ - 12 ac- 6 6c + 9 c^ by 2a. + 6 -3c. CHAPTER VI 1/ LINEAR EQUATIONS: PROBLEMS 56. In Chapters II and III we found how to solve problems by means of equations containing one unknown number. With a knowledge of the principles discovered in Chapters IV and V at command, we are now able to consider additional points connected with the solutions of problems, including the solutions of problems by means of equations containing more than one unknown number. 57. Degree of an Equation. — The degree of a term is the sum of the exponents of its literal factors. If there is only one literal factor, its exponent is the degree of the term. Per example : aj is of the Jirst degree^ 5 x'-^ is of the second degree^ 2 a% is of the third degree., 3 vfiv^ is of the fourth degree. The degree is expressed sometimes with respect to some one letter. Thus, 4 AnH^ is of the first degree with respect to ^, of the second degree with respect to M,'and of the third degree with respect to t. The degree of an equation is the degree of its term of highest degree with respect to the unknown number or numbers. Thus, w + 3 = 4 w is of the first degree., ic2 — 3 aj = 8 ?/ is of the second degree., ^^ + i?2 _ 6 — is of the third degree., y^z'^ — y^ = 2 2/ + 3 is of the fourth degree. An equation of the first degree is called a linear equation. One of the second degree is called a quadratic equation. One of the third degree is called a cubic equation. LINEAR EQUATIONS: PROBLEMS 93 58. Linear Equations with One Unknown Number. — The process of solving a linear equation with one unknown number that does not contain a sign of grouping was given in § 15 and § 36. In solv- ing linear equations that contain signs of grouping, the latter should first be removed by use of the principles in Chapters IV and V. Example 1.— Solve P-(6 - 2 P) = 9(P- 1). Here the first parentheses are removed by changing the signs of the terms inclosed, and the second by performing the indicated multiplication, giving : P_6 + 2P=9P-9, p + 2P-9P=6-9, - 6 P = - 3, P = h. Check. — When P — \ the equation becomes ^_(6_l)= 9(^-1), or -4i = -4^. Example 2. — Solve ( w - 2) (n - 3) = (n - 4) (n - 5) . The parentheses are removed by multiplication, giving : n^ - 5 n + 6 = n2 _ 9 n + 20, n2 - n2 - 5 n + 9 n = 20 - 6, 4 w = 14, n = 3J. Check. — When n = 3^ the equation becomes (3i-2)(3i-3) = (3J-4)(3i-5), or f = |. EXERCISES Solve and check : 1. 3(a + l)=12 + 4(a-l). 4. 5(4-3i?) = 7(3- 4i2). 2. 3(ar-2) = 2(a;-3). 5. 2t-{ot + ^) = l. 3. 5(2-42/) =4(1 -32/). 6. 3(v + 1) +5(v - 1) = 0. 7. 3(d-14) = 7(d-18). 8. 2(Tr-l)+3(Tr-2) + 4(Tr-3)=0. 9. 2P-5(20-P)-6 = 0. 10. 5(2/>4-l)-7 = 3(2i;-7) + 51. 94 ELEMENTARY ALGEBRA 11. 2(m-2) + 3(m-3) = 20-4(m-4). 12. 5A; + 6(A: + l) = 7(A; + 2) + 8(A;H-3). 13. 2c-3(c-4)+20 = 2c-hl7. 14. 85-3(2F+7) = 6F-+-4(4F+2). 15. 4(5-f 4£')-5(6 + 4^)+100 = 2^ + 36. 16. 12r-(4y-7) = 3 T-(9r-28). 17. (5-n)(l + w) = (2-n)(4 + n). 18. {K-l){K-2) = {K+^){K-4.). 19. (s - 1) (s - 4) = 2 s + (s ~ 2) (s - 3). 20. (5 r+ 7) (3 Y- 8) = (5 Y-\- 4) (3 F- 5). 21. (4^-7)(4^-7)=(2^-5)(8^ + 3). 22. (5 - 3i0 (4 w + 3) -f- 1 = (3 z^ + 7)(1 -4w). 23. (Q + 4)(e + 8)-e(Q + 4) = 128. 24. (4 - 3 ?/;) (5 + 4 w) = (8 + 2 w;) (1 - 6 w) - 82. 25. (a — 4) (a + 4) = a^ — 8 a. 26. 2(2; + 2)(2-4) = 2;(22; + l)-21. 69. Problems solved by Linear Equations. — The following ex- ample illustrates a type of problems solved by means of linear equations involving signs of grouping. Example. — A man made two investments in railroad stocks, amounting together to $ 15,000. On the first he gained 14%, and on the second he lost 5 %. His net profits from the two investments amounted to $ 1055. How many dollars in each investment ? Let a = amount of first investment, in dollars. Then, 15000 — a = amount of second investment. .14 a = gain on first investment. .05 (15000 — a) = loss on second investment. Hence, .14 a - .05(15000 - a) = 1055. .14 a -750+ .05 a = 1055. .19 a = 1805. a = 9500. 15,000 - a = 5500. Therefore the first investment was $ 9500, and the second, 1 5500. LINEAR EQUATIONS: PROBLEMS 95 For the steps in the process of expressing a problem by raeans of an equation involving one unknown letter, as illustrated n the above example, review § 16. / ^ EXERCISES 1. The sum of two numbers is 25, and twice the larger exceeds four times the smaller by 2. Find the numbers. 2. Separate 34 into two parts such that twice the greater shall be less by 12 than 5 times the smaller. 3. The sum of two numbers is 64. Three times the less is 12 more than twice the greater. Find the numbers. 4. Separate 86 into two parts such that three times the greater shall exceed 5 times the smaller by 2. 5. There are two consecutive odd numbers whose product ex- ceeds the square of the less by 34. Find the numbers. 6. When the square of a whole number is increased by 29 the result is found equal to the product of the two next larger con- secutive whole numbers. Find the three consecutive numbers. (^ I bought i lb. of coffee and 5 lb. of tea for $5.40. The tea cost 45 cents a pound more than the coffee. Find the cost of each per pound. (^ A newsboy sold 51 papers for 70 cents, of which some were one-cent papers and the others two-cent papers. How many papers of each kind were there ? 9. A newsboy sold two-cent evening papers and Saturday Evening Posts (price 5 cents), receiving in all 92 cents. The total number of papers and Posts together was 34. How many were there of each ? 10. Some boys had a refreshment booth at which they sold lemonade and cider. The price of lemonade was 2 cents a glass and of cider 5 cents a glass. In an afternoon they sold 43 glasses, and in counting their money found that they had $1.31. How many glasses of each did they sell ? \ 96 ELEMENTARY ALGEBRA 11. Two tanks contained equal amounts of oil. But after 75 gallons had been taken from one, and 50 gallons added to the other, one contained twice as much as the other. How many gallons did each contain originally ? 12. What amount must be subtracted from each of the numbers 12, 14, 18, 10, so that the product of the first two remainders shall equal the product of the last two ? ^^3. Two electric lights have together a strength of 128 candle- *^wer. One has twice as much and 16 candle-power more than the other. HoW many candle-power has each ? 14. Two grades of coffee costing the dealer 28 cents and 36 cents per pound are to be mixed so that the mixture shall cost 30 cents per pound. What parts must he take to make 50 pounds of the mixture ? 15. How can a merchant mix 10 pounds of tea, one kind cost- ing 50 cents and the other 65 cents per pound, so that the mixture shall cost 60 cents per pound ? 16. Two grades of spice worth 25 cents and 45 cents a pound are to be mixed so that the mixture can be sold for 50 cents a pound and at a profit of 25 %. What parts must be taken to make 8 pounds of the mixture ? 17. A grocer received a shipment of 80 dozen eggs. Part of these were sold at 28 cents per dozen and the rest at 30 cents per dozen. The total receipts from the sale of the eggs amounted to $ 23.12. How many dozens were sold at each price ? 18. Listerine contains 25 % alcohol. When used to spray the throat, it should be diluted by adding water to it. How much water must be added to 100 parts of listerine so that the mixture contains only 15 % alcohol ? 19. If a medicine contains 40 % alcohol, how much of other in- gredients must be added to 10 quarts of it so that the mixture shall contain only 25 % alcohol ? 20. How many quarts of water must be mixed with 40 quarts of alcohol, 80 % pure, to make a mixture 75 % pure ? LINEAR EQUATIONS: PROBLEMS 97 21. How many quarts of water must be added to 60 quarts of alcohol, 85 % pure, to make a mixture 75 % pure ? 22. How many gallons of cream containing 28 % butter fat and milk containing 4 % butter fat must be mixed to make 10 gallons of cream containing 25 % butter fat ? 23. How many gallons of cream containing 30 % butter fat and milk containing 4.5 % butter fat must be mixed to make 20 gallons of cream containing 20 % butter fat ? 24. In an alloy of gold and silver weighing 60 ounces, there are 6 ounces of gold. How much silver must be added in order that 10 ounces of the new alloy shall contain only |^ ounce of gold ? 25. In an alloy of copper and tin weighing 36 ounces, there are 12 ounces of copper. How many ounces of copper must be added to this in order that the new alloy may have 4 ounces of copper to every 10 ounces of the alloy ? 26. A man has two investments together amounting to $ 1800. On the first investment he gets 5 % annually, and on the other he gets 6 (fo annually. His annual income from the two investments is $ 95. Find the amount of each investment. 27. $5000 is invested in two places, part at 4 % and the rest at 5 %. The annual income from the two investments is $222. How much is each investment ? 28. One sum of money exceeds another by $ 1400. The first at 8 % and the second at 6 % give an annual income of $ 280. Find the value of each. 29. There are 51 coins in a money drawer, consisting of nickels and dimes. The total value is $3.35. How many coins of each kind are there ? 30. The value of 29 coins, consisting of quarters and dimes, is $4.55. Find the number of each. 31. A is 42 years of age, and B is 12. In how many years will A be only twice as old as B ? 98 ELEMENTARY ALGEBRA 32. Eight years ago a man was just 16 times as old as his son, and now he is only 4 times as old. What are their present ages? 33. Find where to cut a board which is 50 inches long into two parts whose difference is 26 inches. 34. A bar of iron 60 inches long is to be cut into two parts such that twice one part is equal to 5 times the other. Find where to cut it. 35. A belt runs over a pulley 48 inches in diameter, making 180 revolutions a minute. It is desired to reduce the size of the pulley so that by making 216 revolutions a minute, the belt will move with the same speed. By how much must the diameter of the pulley be reduced ? 36. A rectangular fiel^ is 6 rods longer than it is wide; and if the length and breadth were each 4 rods more, the area would be 120 square rods more than it is. What are the dimensions of the field ? 37. A square court has the same area as a rectangular court whose length is 18 feet greater and width 9 feet less than the side of the square court. Find the side of the square court. 38. A tennis court is 6 feet more than 2 times as long as it is wide, and its area exceeds by 216 square feet twice the area of the square whose side is its width. Find the dimensions of the court. 60. Business Problems. — Certain kinds of business problems are easily solved by use of linear equations. EXERCISES 1. A university has an endowment which invested at 4 % yields an annual income of $ 140,000. What is the endowment ? Suggestion. — If d dollars is the endowment, .04(? = 140,000. 2. What sum of money invested at 6 % will yield $510 simple interest per year ? 3. A certain investment at 4} % simple interest yields $ 3240 in 6 years. What is the investment ? LINEAR EQUATIONS: PROBI.EMS i i ■ "• ' J : s^dV, 4. A certain sum of money was invested at 5 % simple interest. In 3 years the principal and interest together amounted to $ 6900. What was the amount of the investment ? 5. Five years ago I invested a certain sum of money at 6 % simple interest. It now amounts to $1560. How much did I invest ? 6. A number of years ago I invested $ 1500 at 5 % simple in- terest. The amount at present is $2025. How many years ago was the investment made ? 7. A commission merchant charged $ 10.80 for selling a car load of fruit, and remitted $ 349.20. What was his rate of commission ? 8. Merchants sometimes mark goods to sell at an advance, and then allow a discount on the marked price. At what advance must a merchant mark goods costing $12 in order that he may sell them at a reduction of 20 % from the marked price, and yet make a profit of 25 % on the goods ? SuGGESTiov. — If a = the advance, in dollars, the marked price will be 12 + a, and the selling price 80 % of this, or .80(12 + a). Hence, .80(r2 + a) = 12 + .25 X 12. 9. At what advance must a merchant mark an article costing him $3.50 in order that he may sell it at a reduction of 10 % from the marked price, and yet make a profit of 30 % on it ? 10. A clothing merchant puts on sale a lot of boys' suits costing him $8 each, and advertises them to sell at a reduction of 25 % from the marked price. At what advance must he first mark them to sell in order that he may make this reduction and still make a profit of 20% on the cost of the suits ? 11. A milliner advertises a lot of hats that cost her $ 2.85 each for special sale at " ^ off." At what advance must she first mark them in order that by allowing this discount she may make a profit of 75 cents on each hat ? 12. The proprietor of a china store marks for sale a lot of dishes that cost him $ 16.50. At what advance must he mark them so that he may then mark them down 15 % and make a profit of 20 % ? if)>J :'\'\\\ \ /.mCEMENTARY ALGEBRA 13. A ladies' tailoring establishment marked a suit at an advance of 50%, then reduced the selling price by 20% of the marked price, and got $16.80 for it. What was the cost of the suit? 14. A druggist had marked perfume to sell at a profit of 40 % . He then advertised it for special sale at 42 cents an ounce, which was a reduction of 14f % from the marked price. What did it cost the druggist per ounce ? 15. Men's suits that had been marked to sell at a gain of 25 % were damaged and disposed of at a fire sale at half price, which was a loss of $ 7.50 on each suit. What was the cost of the suits ? 61. Problems involving Motion. — When an object moves, the distance that it goes in a unit of time, as a second or an hour, is called its rate of motion, speed, or velocity. For example : If a train runs 60 miles in 2 hours, its velocity is 30 miles an hour. If the velocity does not change throughout the motion, the motion is said to be uniform. In case a body moves with uniform motion, the distance that it goes in any length of time is obtained by multiplying the velocity by the number of units of time ; that is, distance = time x velocity. EXERCISES 1. An express train runs with a velocity of 45 miles an hour. How far does it go in 4 hours ? 2. The " Kocky Mountain Limited," on the Kock Island Koad, runs from Omaha to Denver, a distance of 580 miles, in 14 hours and 48 minutes. What is the average speed of the train, making no allowances for stops ? 3. Sound travels through the air 1080 feet a second. If a flash of lightning is a mile away, how long after the flash should one hear the thunder ? LINEAR EQUATIONS: PROBLJ^MS iOl 4. A train leaves a station and runs at a speed of 40 miles an hour. Two hours later a second train leaves the same station and runs over the same track at the rate of 55 miles an hour. In how many hours will the second train pass the first ? Suggestion. — Let t = number of hours after second train starts until it passes the first. Then, since the first train has a start of two hours, 55« = 40(e + 2). 5. A messenger, going 6 miles per hour, has been gone for two hours when it is found that the message is wrong. A second messenger, riding at an average of 10 miles per hour, is sent to overtake him. In how many hours will the second messenger overtake the first ? 6. Two motor cyclists start at the same time from points 280 miles apart, and travel towards each other. One rides 24 miles per hour, but is delayed 3 hours on the road because of a break in his machine. The other rides 20 miles per hour uninter- ruptedly. In how many hours will they meet ? 7. Two pedestrians started at the same time from points 44| miles apart, one walking at the rate of 2^ miles an hour, and the other at the rate of 2| miles an hour. When and where did they meet? 8. Two automobile parties started from the same place, one going north at 20 miles an hour and the other south at 18 miles an hour. In what time will they be 120 miles apart ? 9. A motor cyclist rode 75 miles in 4 hours. Part of the dis- tance was on a country road at a speed of 20 miles an hour, and the rest within the city limits at 10 miles an hour. Find how many hours of his ride were in the country. 10. A train runs from Joliet, 111., to Chicago, 39.6 miles, in 1.1 hours. The run from Joliet to Englewood is made at 40 miles an hour, and the run from Englewood to Chicago at 24 miles an hour. Find the time required to make the run from Englewood to Chicago. Find the distance from Englewood tO Chicago. ^Wa*.* -•'.••;; : cEEe^mentary algebra ** i ,*'** *t ," * * • * •*, « 11. A train running from Chicago to Denver at the average speed of 40 miles an hour takes three hours longer to make the run than one running at 45 miles an hour. Find the distance from Chicago to Denver. Suggestion. — First find the time required at a speed of 40 miles an hour. 12. An ocean liner going 20 knots an hour leaves New York when a freighter going 6 knots an hour is already 90 knots out. How long will it take the liner to overtake the freighter ? 13. A travels 8 hours at a rate of 3 miles per hour less than the rate of B. B travels an equal distance in 6 hours. What is the rate of each ? 14. Two trains approach each other, leaving stations 149 miles apart at the same time. One goes 10 miles per hour faster than the other, and they meet in 2 hours. What is the rate of each ? 15. The longest steel bridge in the world is said to be one of the Spokane, Portland, & Seattle E-ailroad, over the Columbia River. Including approaches, it is 2 miles long. It would take a freight train 1144 feet long, running 15 miles per hour (1320 feet per minute), just 3 minutes to cross the ten main spans. Find the length of the ten main spans. 16. A man shooting at a target heard the bullet strike the tar- get 3| seconds after he fired. The bullet was known to travel 1375 feet a second ; and sound travels approximately 1100 feet a second. How long after he fired did the bullet hit the target ? How far away was the target ? 17. A bullet going 1650 feet per second is heard to strike a tar- get 2 seconds after it is fired. How long after it is fired does it hit the target ? How far away is the target ? 18. Find the time between 4 and 5 o'clock when the hands of a clock are together. Suggestion. — Let x = the number of minute spaces which the minute hand has traveled from 4 o'clock on until it overtook the hour hand. Then ■^ X will be the number of minute spaces which the hour hand has traveled meanwhile. Why ? The difference is 20. Why ? LINEAR EQUATIONS: PROBLEMS 103 19. Find the time between 7 and 8 o'clock when the hands of a clock are together. 20. At what time between 6 and 7 o'clock is the minnte hand 15 minute spaces behind the hour hand ? Suggestion. — Starting at 6 o'clock, how many minute spaces has the minute hand had to gain on the hour hand ? 21. Find the time between 4 and 5 o'clock when the hands of a clock are directly opposite each other. 22. At what time between 8 and 9 o'clock are the hands of a clock directly opposite each other ? 23. How long is it from the time that the hour and minute hands of a clock are together until they are together again ? Suggestion. — Let t = the number of minutes required. Since the minute hand makes one revolution in 60 minutes, it makes ^^ of a revolution in one minute, and ^ t revolutions in t minutes. Simi- larly, the hour hand makes 7^5 of a revolution in one minute, and ^q t revolutions in t min- utes. Hence, ^ < — ^i© < = 1. 24. The earth makes a circuit around the sun in 12 months, and Mercury makes a circuit around the sun in 3 months. If the earth and Mercury are " in conjunction," as in the figure, how long will it be until they are in conjunction again? 25. Venus makes a circuit around the sun in 225 days, and Mars in 687 days. How many days elapse between two consecu- tive conjunctions of these two planets ? 62. Problems on Specific Gravities of Substances. — A cubic foot of steel weighs 7.8 times as much as a cubic foot of water. This number, 7.8, is called the specific gravity of steel. In general, 104 ELEMENTARY ALGEBRA The ratio of the weight of a given volume of any solid or liquid substance to the weight of a7i equal volume of water at the freezing point of temperature is the specific gravity of the substance. What does it mean to say that the specific gravity of alcohol is .79? A cubic centimeter of distilled water at the freezing point weighs just 1 gram. Since the specific gravity of pure gold is 19.36, 1 cubic centimeter of gold weighs 19.36 grams; 2 cubic centimeters weigh 2 x 19.36 grams ; 3 cubic centimeters weigh 3 X 19.36 grams ; etc. Evidently, The weight of an object in grams equals the product of its volume in cubic centimeters and its specific gravity. Thus, if the specific gravity of copper is 8.9, 1 cubic centimeter of cop- per weighs 8.9 grams. Hence, 10 cubic centimeters of copper weigh 10 X 8.9 grams, or 89 grams. Certain problems in the specific gravities of mixtures of sub- stances are solved by use of linear equations. EXERCISES 1. The specific gravity of cast iron is 7.2. Find the weight of 10 cubic centimeters of cast iron. Of 200 cubic centimeters. 2. The specific gravity of glass is 2.89. Find the weight of 100 cubic centimeters of glass. Of 25 cubic centimeters. 3. Brass is made of copper and zinc, and its specific gravity is 8.4. How many cubic centimeters of copper, of which the specific gravity is 8.9, must be used with 200 cubic centimeters of zinc, of which the specific gravity is 6.9, to make brass ? Suggestion. — The volume of the brass is the sum of tlie volumes of the copper and zinc, and the weight of the brass is the sum of the weights of the copper and zinc. Let v = volume of copper in cubic centimeters. Then, weight of copper = 8.9 u, weight of zinc = 6.9 x 200, and weight of brass = 8.4(i? + 200). Hence, 8.9 tj +6.9 x 200 = 8.4(u + 200). 4. How much zinc must be combined with 250 cubic centi« meters of copper to form brass ? LINEAR EQUATIONS: PROBLEMS 105 6. How many cubic centimeters of water (specific gravity 1) must be mixed with 500 cubic centimeters of alcohol (specific gravity .79) so that the specific gravity of the mixture shall be .9? 6. A piece of ice containing 1,000,000 cubic centimeters, spe- cific gravity .92, floats in water. How many cubic centimeters in an oak beam, specific gravity 1.17, that may be placed upon the ice without making it sink ? Suggestion. — The specific gravity of the combination of ice and oak must equal 1, the specific gravity of water. 7. How much steel, specific gravity 7.8, must be attached to a piece of white pine, specific gravity .42, containing 10,000 cubic centimeters, in order that the specific gravity of the combined materials may be 2 ? 8. In Problem 7, how much steel must be used so that the combined materials will just float in water? 9. A piece of glass containing 850 cubic centimeters, specific gravity 2.89, is made to float by attaching cork to it. How much cork, specific gravity .24, must be used ? 10. My watch case is made of 14-karat gold, specific gravity 14.88. How much pure gold, specific gravity 19.36, must be combined with 20 cubic centimeters of nickel, specific gravity 8.57, to make 14-karat gold ? 11. When 100 cubic centimeters of mercury and 10 cubic centi- meters of gold (specific gravity 19.36) are combined, it is found that the specific gravity of the combined materials is 14.1. Find the specific gravity of mercury. 12. It is found that when 90 cubic centimeters of copper (spe- cific gravity 8.9) and 150 cubic centimeters of tin are combined, the specific gravity of the combination is 7.9. What is the spe- cific gravity of tin ? 106 ELEMENTARY ALGEBRA 63. Problems involving the Lever. — In every form of lever, such as the steelyard, crowbar, nutcracker, teeter board, etc., it is found that the two weights or forces ex- f j T g!^ erted balance when they are placed at ' ^■^"'J _g) ^^^Y\ distances from the point of ^\xi)- Y port or fulcrum that the products of the |w| 1 weights or forces by their distances from the fulcrum are equal. Thus, in case of the steelyard, wd = WD. Some problems on the use of the lever may be solved by linear equations. EXERCISES 1. On a steelyard the distance D is 2 inches and the weight w is ^ pound. If d is 8 inches, what is the weight W? 2. A nutcracker is held 4 inches from the hinge or fulcrum, and a nut is placed 1 inch from the hinge. A squeezing force of 3 pounds is required to crack the nut. What is its resistance ? 3. Two children play at teeter. One weighs 80 pounds, and sits 5 feet from the point of support of the teeter board. If the board balances when the other sits 6 feet from the point of sup- port, what is the weight of the second child ? 4. Two boys play at teeter. One weighs 100 pounds, and sits 6 feet from the point of support. The other weighs 120 pounds. How far from the point of support must he sit in order to make the board balance ? 5. A board 12 feet long is to be used as a teeter board. If the people weigh 95 pounds and 110 pounds, respectively, and sit at the ends of the board, find the point at which it must be sup- ported in order to balance. 6. A workman lifts a stone weighing 420 pounds by means of a crowbar. If the fulcrum is placed 6 inches from the point at EFNEAR EQUATIONS: PROBLEMS 107 which the crowbar is applied to the stone, how far from the ful- crum must the workman grasp the crowbar to lift the stone with a force of 100 pounds ? 7. In Problem 6, if the man grasps the „^ crowbar at a point 48 inches from the point where the crowbar is applied to the stone, where must the fulcrum be placed in order that he may lift the stone with a pressure of 70 pounds ? 8. Weights of 240 pounds and 300 pounds, respectively, are applied at the ends of a beam 40 inches long. At what point must the beam be supported in order to balance ? 9. A and B carry an object suspended from a horizontal rod 60 inches long held between them. If one is to lift only three fourths as much as the other, at what point of the rod must the object be suspended ? 10. A man has a team of which one horse weighs 1200 pounds and the other 1500 pounds. If their draft power is proportional to their weight, how must he divide the 50-inch doubletree in order to justly distribute the load ? 11. A farmer has a team of which one horse weighs 1400 pounds and the other 1600 pounds. If their draft power is proportional to their weight, where must he place the clevis on the 4-foot doubletree so as to justly distribute the load ? 64. Problems on the Decimal Number System. — In any number such as 4258, the 8 is called the ones' digits the 5 the tens' digit, and 2 the hundreds' digit, etc. Any number of two or more digits in our decimal system of notation is in nature a polynomial. Thus, 483 = 400 + 80 + 3, or 100 X 4 + 10 X 8 + 3. Similarly, any number whose hundreds', tens', and ones' digits are respectively A, t^ and w, may be written 108 ELEMENTARY ALGEBRA in the polynomial form 100 h -[- 10 1 -\- u. In writing a number in the polynomial form, each digit must be multiplied by 10, 100, 1000, etc., according to the position that it occupies in the number. EXERCISES 1. Write in the polynomial form the following numbers : 25; 347; 4196. 2. Write the number whose ones' digit is a, tens' digit 6, hun- dreds' digit c, and thousands' digit d. 3. In a number of two digits, the sum of the digits is 9. If the digits are interchanged in position, the new number obtained exceeds the given number by 27. Find the number. Suggestion. — If a; = the tens' digit, 9 — a; = the ones' digit. Hence, 10a;4-9 — ic=: the number. Hence, 10(9 — a;) + x = the number with the digits interchanged. Hence, 10(9 - a:)+ x = lOx + 9 - a; + 27. 4. A number is composed of two digits whose sum is 12. This number exceeds by 36 the number obtained by interchanging the digits. Find the number. 5. In a certain number of two digits, the tens' digit exceeds the ones' digit by 2. The sum of the given number and that obtained by interchanging the digits is 154. What is the number? 6. A number consists of two digits of which the ones' digit is 3 more than the tens' digit, and the number is 3 more than 4 times the sum of its digits. What is the number ? 7. What is the number of which the tens' digit is 2 more than the ones' digit, and which is 7 times the sum of the digits ? 8. In a number of two digits the ones' digit is 3 less than the tens' digit, and the number is 20 less than twice that obtained by interchanging the digits. What is the number ?. 9. In a number of three digits^ the tens' digit is 1 more than the ones' digit, and the . hundreds' digit is 1 more than the tens' digit. The number is 21 more than 50 times the sum of its digits. Find the number. LINEAR EQUATIONS: PROBLEMS 109 65. Linear Equations with Two Unknown Numbers. — Problems in which the values of more than one unknown quantity are to be found are often most easily solved by use of equations containing more than one unknown number. Example. — A merchant sold 10 suits for $162. He received |15 for each of one kind and $ 18 for each of the other kind. How many were there of each kind ? Let n = the number sold at 1 15 each, and w = the number sold at $ 18 each. Then \ n + m = \Q, (1) 1 15 n + 18 w = 162. • (2) Multiplying (1) by 15, 15 » + 15 m = 150. (3) Subtractmg (3) from (2), 3 w = 12. (4) Hence, w = 4. (5) Replacing m by 4 in (1), n + 4 = 10. (6) Hence, n = 6. (7) Therefore, 6 suits were sold at $ 16 each and 4 suits at $ 18 each. 66. Simultaneous Equations. — In the example in § 65, we have seen two equations concerning two unknown numbers whose values satisfy both equations. Thus, in the example, when w=6 and m=4, equation (1) becomes 6 + 4 = 10, and equation (2) becomes 90 + 72 = 162. Two or more equations containing two or more unknown num- bers which satisfy all of the equations are called simultaneous. 67. Systems of Simultaneous Equations. — Two or more simul- taneous equations are said to constitute a system. To solve a system of equations is to find the sets of values of the unknown numbers which satisfy all of the equations. Thus, in the system of equations in § 65, the set of values w = 6 and m =4 constitute a solution. 68. Elimination. — The solution of a system of two linear equations is most easily found by so combining the two equations as to obtain a new equation in which one of the unknown num- bers does not appear. This process is called elimination. A sim no ELEMENTARY ALGEBRA pie method of elimination is shown in the example in § 65. Compare the steps in the process of solving the following system with those in the example in § 65. Example.— Solve the system i ^+4^ = 12, (1) l2^- J5 = 6. (2) Let us first eliminate B. Multiplying (2) by 4, ^A-^B= 24. (3) Adding (3) to (1), 9 ^ = 36. (4) Hence, ^ = 4. (5) The value of B may now be found in like manner by eliminating A be- tween equations (1) and (2), but is more easily found by replacing A by its value 4 in equation (1) or equation (2). From (1), when ^ = 4, 4 + 4 5 = 12. (6) Whence, B = 2. (7) See if the solution A = 4, B = 2 satisfies both equations of the system. A study of this example and that in § 65 will reveal the follow- ing rule : To eliminate one of the unknown numbers, multiply the members of each equation, if necessary, by such a number as will make the numerical coefficients of that unknown number the same in both of the resulting equations. Add or subtract {according to signs) the corresponding members of the resulting equations. After finding the value of one unknown number, substitute its value in either of the given equations and solve the resulting equation for the value of the other unknown number. Note. — Other methods of elimination will be discussed in Chapter XII. EXERCISES Solve: \2n [3n |3a;_42/ = 6. 2 f 5 Jf+4iV^=22, g f2n + 5^ = 15, 3Jf+ ^=9. [3n-4AL=ll. {QE- d = 10, r P+4i? = 12, \l E-2d = \^. '■ [2P- E = 6. LINEAR EQUATIONS : PROBLEMS 111 3;2_c = 8. ' |5m-2n = 14. 9. {bt-2a: 5T-4:H=2. '" [st = 5a-ll. r5 Tr-2F-35 = 0, j2 ri06?-llv + l |4 TF- F-25 = 0. ■ [10v-llG = 69. Problems solved by Use of Systems of Equations. — In a problem in which the values of two unknown quantities are sought it always will be found that the problem expresses or implies two facts or suppositions. Represent each unknown num- ber by a letter. Then each of the facts or suppositions expressed in the problem gives an equation containing the unknown num- bers. These two equations constitute a system, which may be solved by the method of § 68. Many of the problems in the exercises in the earlier part of this chapter might have been solved by use of two equations con- taining two unknown numbers. ExAiMPLE. —I have $1.50 in dimes and nickels. There are 25 coins in all. How many dimes and how many nickels are there ? Let d = number of dimes, and n = number of nickels. Then I ^+ *^ = 2^' ^^) ^ ' ll0d+5n = 150. (2) Note that the fact that there are 25 coins in all gives equation (1), and the fact that their total value is $ 1.50 gives equation (2). Solving this system, ^f = 5, and n = 20. Hence, there are 5 dimes and 20 nickels. EXERCISES Solve by using two equations with two -unknown numbers: 1. A man invested S5600, part in bonds that paid 4%, and the remainder in a business enterprise that yielded 8% on the investment. The total yearly earnings from the two investments amounted to $328. Find the amount of each investment. 112 ELEMENTARY ALGEBRA Suggestion. — Let a = amount of investment in bonds, and b = amount of investment in business enterprise. „, , a + 6 = 5600, ^^^"» ^04 a + .08 6 = 328. 1.. 2. A man had $2150 to invest. Part of this sum he loaned at 6 % interest, and the remainder he invested in stock of a build- ing and loan association that paid 7 % dividends. From the two investments his annual income was $142.50. What was the amount of each investment ? 3. A money drawer contains $3.50 in dimes and nickels. There are 50 coins in all. How many coins of each kind are there ? 4. In playing teeter, two boys use a board 12 feet long. • One boy weighs 80 pounds and the other 110 pounds. At what point must the board be supported to balance ? 5. A teamster has a team of which one horse weighs 1250 pounds and the other 1500 pqunds. Assuming that draft power is proportional to weight, how should he divide the 50-inch double- tree in order to properly distribute the load ? 6. A merchant mixes 28-cent coffee and 36-cent coffee to sell at 30 cents a pound. What quantities of the two grades of coffee should he take to make 40 pounds of the mixture ? 7. The specific gravity of copper is 8.9, that of zinc 6.9, and that of brass 8.4. Find the number of cubic centimeters of copper a.nd the number of cubic centimeters of zinc that must be com- bined to produce 600 cubic centimeters of brass. Suggestion. — Let c cubic centimeters copper and z cubic centimeters zinc be the amounts required. f c + z = 600, ' I 8.9c + 6.dz = 8.4 (c + z). 8. How much gold, specific gravity 19.36, and how much nickel, specific gravity 8.54, must be combined to make 10 cubic centimeters of an alloy of which the specific gravity is 12 ? LINEAR EQUATIONS: PROBLEMS 113 9. In a number of two digits, the sum of the digits is 10, and when the digits are interchanged, the number is increased by 36. Find the number. 10. A belt runs over two pulleys. The circumference of one pulley is 2 feet more than the circumference of the other. One pulley makes 3 revolutions while the other makes 2. Find the circumference of each pulley. 11. When the Panama Canal was finished, the distance from New York to San Francisco by boat was reduced by 7796 miles. The distance by the old route exceeded twice the distance by the Panama route by 2502 miles. Find the distance by the old route and the distance by the Panama route. 12. The distance from New York to Yokohama, Japan, was reduced by 2952 miles when the Panama Canal was completed. If the distance by the old route were twice as great, and the dis- tance by the Panama Canal three times as great, the latter would exceed the former by 4184 miles. Find the distance by each route. 13. The ship Mauretania, built in 1907, is 583 feet longer than the ship Britannica, built in 1840. The Mauretania is 169 feet more than three times as long as the Britannica. Find the length of each. 14. In 1847 the Deutschlandf the first ship of the Hamburg- American Line, arrived in New York. The present Deutschland carries 15,283 tons more than the first Deutschlayid. The tonnage of the present Deutschland exceeds by 226 tons 22 times the tonnage of the old Deutschland. Find the tonnage of each. 15. It is stated that the daily ration for a laboring man should contain 4 ounces each of fat and protein. In pork the per cent of fat is 26 and the per cent of protein is 13 ; and in beans the per cent of fat is 2 and the per cent of protein is 22. How many ounces each of pork and beans are required to make a ration for one day ? 114 ELEMENTARY ALGEBRA SUPPLEMENTARY EXERCISES Note. — As we have seen, many practical formulae contain two or more unknown numbers. It often is found necessary to solve such formulae for one of the unknown numbers in terms of the others. Thus, the circumference of a circle is computed by the formula = 2 irR. If this is solved for i? in C terms of C, we get R = Such formulae are solved by the same processes 27r as equations with one unknown number. 1. In the simple interest formula, I=prt, solve for t in terms of 7, p, and r. 2. The distance an object moves in t seconds at v feet per second is s = tv. Solve for t. 3. The area of a triangle is expressed by ^ = ^ BH. Solve iovH. 4. The area of a trapezoid is expressed by A = ^II(B-\- B'). Solve for B'. 5. The relation between the readings on the Fahrenheit and Centigrade thermometers for any temperature is expressed by (7=1 {F- 32). Solve for F. 6. The formula for finding the horse power of a steam engine is H. P. = ^ !\ , where _p = the mean effective pressure in pounds 33000 per square inch, I = the length of stroke in feet, a = area of piston in square inches, n = twice the number of revolutions. Solve for Z, the length of stroke. 7. In a polygon with n sides the sum of all the angles, in degrees, is given by s = 180 (n — 2). Solve for n. 8. Solve 2ax —b = cy for x. 9. Solve A(n — 1) + ^ = n f or ^. 10. Solve (F- S)(V-}- 2S)=V'-{-S' for F. 11. Solve 2(2^-1) +3 = a(^ + 2) for f. 12. Solve R{R + Jc)-k(l-Ji) = Ii' + l for B. LINEAR EQUATIONS: PROBLEMS 116 In each of the following systems solve for x and y : ^3 r8x+ 2/ = 60a, jg r3aj + 762/ = 166, \lx-10y=^ 9a. ' |2a; + 562/=136. 14 f a; 4- ?/ = 30 w, -g f 5maj — 2ny=63mn, I 3 a* — 2 y = 25 n. | 2 mx + n?/ = 18 m/i. 17. A square has the same area as a rectangle whose length is 3 inches greater and width 2 inches less than the side of the square. Find the dimensions of the square and of the rectangle. 18. In a certain family each daughter has as many sisters as brothers; but each son has twice as many sisters as brothers. How many children are in the family ? 19. The leader in a " guessing game" tells each of the others to add 6 years to his age, multiply the result by 4, subtract 24 from the product, and to the remainder add his age. When the results are announced, the leader tells at once the age of each individual. If an individual gives 80 as the result, what is his age ? ' 20. Find the longitude of a place in the Central Time belt at which it is observed that the local or sun time is 15 minutes faster than standard time. 21. Find the longitude of a place in the New York Time belt whose local time is 10 minutes slower than standard time. 22. A crop of 60 bushels per acre of corn takes from one acre l>f soil 58 pounds less of phosphoric acid than of nitrogen, and ii3 pounds less of potash than of nitrogen. Five times the amount of phosphoric acid exceeds twice the amount of potash by 8 pounds. How many pounds of each of these fertilizers are taken from an acre of soil ? 23. Fifteen pounds of tin weigh 13 pounds in water, and 15 pounds of zinc weigh 13.5 pounds in water. How much tin and how much zinc in an alloy which weighs 56 pounds in air and 49 pounds in water? Note. — The method given in this problem of finding the proportional parts of metals in an alloy by first weighing the pure metals and the alloy 116 ELEMENTARY ALGEBRA each in air, then in water, is said to have been first discovered and used by Archimedes, about 220 b.c, in determining for King Hieron of Syracuse whether a crown, claimed by the maker to be pure gold, was not alloyed with silver. 24. Two seconds after a marksman fires he hears the bullet hit the target, which is 440 yards distant. If sound travels through the air at a velocity of 1100 feet a second, find the average veloc- ity of the bullet. 25. Two weights balance when one is 12 inches and the other 15 inches from the point of support. If the first is replaced by a weight 6 pounds greater, the second must be moved 3 inches far- ^rom the point of support to balance it. Find the weights. A crew that can row 6 miles an hour down a stream can row 2 miles an hour up the stream. Find the sjjeed of the current and the speed at which the crew can row in still water. 27. A local train 820 feet long and an express train 500 feet long run on parallel tracks. When running in the same direction it requires 88 seconds for the express to pass the local, but when running in opposite directions it requires only 17.6 seconds for them to pass. Find the rates of the two trains. CHAPTER VII SPECIAL PRODUCTS AND QUOTIENTS 70. Special Rules. — The products or quotients of expressions of many forms, called type forms, may be written down at sight by special rules, without performing in detail the complete pro- cesses of multiplication or division. By discovering and learning these rules, much time and labor in multiplications and divisions may be saved. They reveal also many " short cuts " in the mul- tiplication of arithmetical numbers that are valuable to know, some of the most important of which are given in this chapter. The need for use of these special rules presents itself so often in algebra that they should be thoroughly mastered before proceed- ing farther. 71. Squares and Cubes of Monomials. — What is the product of a^xa^? Of 5 mhi' x 5 mV ? Of ( - 3 xy^^{- 3 xy^^) ? Since (7 My means 7 M*x7 M\ what is its value ? Find the value of (9 P'(^y. Of (- 4 h^lc^f. From these examples, the following rule is evident : To square a monomial, square the numerical coefficient , and mul- tiply each exponent by 2. Since (2 ijt^f means 2 ty^ X 2 w;^ X 2 w^, what is its value ? Find the value of (3 HHy. Of (- 4 A^B^CTf. These examples reveal the following rule : To cube a monomial, cube the numerical coefficient, and multiply each exponent by 3. Similar rules may be discovered for raising monomials to the fourth power, the fifth power, etc. 117 118 ELEMENTARY ALGEBRA EXERCISES Give at sight the values of the indicated powers 1. {aj. 11. (Ly. 21. (xyy. 31. (a^s^vy. 2. (E^y. 12. W. 22. (aby. 32. (Sa^fy. 3. (wy. 13. (sy. 23o (yhy. 33. (5 MNy. 4. (j^y. 14. (xy. 24. (r.W^ 2 34. (2 a'b'cy. 5. {py. 15. oy. 25. (W'gy. 35. (8 w'%y. 6. (ny. 16. (Qy. 26. {Hyy. 36. (11 Any. 7. (B^y, 17. (vy. 27. (a^^^)^ 37. (IBxy'zy. 8. (vr^ 18. (Ny. 28. (vyy. 38. (6 vHy. 9. (apy. 19. (py. 29. (A^ny. 39. (9 S'Hy, 10. {Ay, 20. (by. 30. (w^ny. 40. (16 mhiyy 41. (SD^wy. 48. (- -SE'Hy. 55. 3 &Ty. 42. (5 x*Ty. 49. (- - 2 a^'cy. 56. 4 HYvy. 43. (2a^n'by. 50. (- -lOES'vy. 57. 2P'v'iiy. 44. {^pH'yy. 51. (- -M^wy. 58. 3 Z2y^)l 45. (6 V'n'gJ. 52. (- - 3 a'b'(^\ 59. 2 why. 46. (- 2 Ay. 53. (- -x^yy. 60. Say {2 ay. 47. (-3mW)2 54. (- -2 any 61. (2 ny(-sny. 72. Square Roots and Cube Roots of Monomials. — What is the number which squared gives 9 ? 25? a^? The number whose square is a given number is called the square root of the given number. Thus, since 7^ = 49, the square root of 49 is 7. Similarly, the number whose cube is a given number is called the cube root of the given number. Thus, since (2 ^2)3 = 8 v^, the cube root of 8 ??« is 2 v^. The numbers whose fourth powers, fifth powers, etc., are a given number are called the fourth root, fifth root, etc., of the given number. SPECIAL PRODUCTS AND QUOTIENTS 119 A root of a number is indicated by placing before it tiie sign -^^ called the radical sign. Usually a vinculum is attached to the radical sign, to show how far its effect is to extend. To indicate what root it is, except in the case of a square root, a little num- ber called the index is written above the radical sign. For example, the square root of n is written Vn ; the cube root of n is written Vn ; the fourth root of n is written Vn; etc. In the case of square root the index 2 is understood and left off. In most simple practical work, one needs only a knowledge of square roots and cube roots. Since (+3)2 = 9 and (-3)2 = 9, V9 is either +3 or -3. Similarly, since (+6)2=36 and (-6)2 = 36, V36 is either +6 or — 6. In general. Every positive number has two square roots, which have the same absolute value, one positive and the other negative. The two square roots of a positive number are often written together, with a double sign ± . Thus, \/8l = ± 9 ; Vlil = ± 12. Since no positive or negative numbers squared can give a nega- tive number, A negative number has no square root that can be expressed as a positive or negative number. Note. — What to do in a problem where the square root of a negative number is required to be found will be discussed later. Since (+2)^ = 8 and (-2)3 = -8, ■v/8 = 2 and V^ = -2. In general, Any number has only one cube root that can be expressed as a positive or negative number. This cube root of a positive number is positive, and of a negative number it is negative. Note. — It will be shown later that every number has, in addition to the one positive or negative cube root, two other cube roots that cannot be ex- pressed as simple positive or negative numbers. In this chapter only the positive or negative cube root of a number will be considered. 120 ELEMENTARY ALGEBRA Since the square root of a monomial, when squared, must give that monomial, it follows from § 71 that : To find the square roots of a monomial, take the square roots of the numerical coefficient and divide the exponent of each letter by 2. Similarly, it follows, from § 71, that : To find the cube root of a monomial, take the cube root of the numerical coefficient, and divide the exponent of each letter by 3. Thus, \/64 a«64 = ± 8 a%^ ; VlOO x'^y^ = ± 10 ot^y. And V8 u^t^ = 2 wH^ ; v'-27mi%6 = _ 3 „i4^2. EXERCISES Give at sight the square roots of; 1. 1, 4, 16, 81, 121, 49, 144, 36. 2. a*, TF«, »«, P^, n"', A^\ 2/'"- 3. M'\ L^, jS^% k^', 9x% 16 a\ 4. 49 JBi«, 25 v', 100 M^\ S6a''b% 81 a^y\ 5. 16 TT^F^^ 9 m'n% 25 dH\ 121 hy, 225riV/. 6. 144 a¥", 49 w^'d', 169 a^y^^ 625 R^m'^ 7. 196 a^o^V, 256 H^T^a^ 441 ^"a^/, 400 m^w^. Give at sight the cube roots of : 8. 1, 8, 27, 64, 125, 216, 512, 1000. 9. - 1, - 8, - 27, - 64, - 125, - 1000. 10. A', n^% t% W, x'\ P^, r^'. 11. v'\ y', M^\ a^, -D\ -z'\ -L\ 12. _ 10^% - T^, - 8 x% - 27 b'', - 125 P". 13. 64 i?«, 27 vH\ - 8 m'^^ - 125 nV, - A^tH\ 14. - pYr^, 216 M\i'', - 27 aj^-^/V, 343 V^^K - 8 i^3/)i2 SPECIAL PRODUCTS AND QUOTIENTS 121 15. The area of a square is 49 d^. What is the length of one 3ide? 16. The base and altitude of a right tri- angle are 8 x and 6 ic, respectively. Find the length of the hypotenuse. Suggestion. — The square of the hypotenuse equals the sum of the squares of the base and altitude. 17. The volume of a cube is 64 E^. Find the length of one edge. 18. The diameter D, in inches, necessary of a steel shaft, upon which are fastened pulleys that drive machines in a factory, in order to impart H horse power to the machinery when making N 3/8O X H revolutions a minute, is computed by the formula D = y\ — . Find the diameter of the shaft necessary to impart 80 horse power when making 100 revolutions a minute. 73. Equations solved by finding Square Roots. — Many equations may be solved by extracting roots. The following illustrate the solutions of equations of the second degree by finding square roots. Example 1. — Solve P^ = 9. Since P2 = 9, P = V9 = ±3. Example 2. — Solve y'^-A^-0. Transposing, y- = 49. Hence, y = \/49 = ±7. EXERCISES Solve: 1. a^ = 25. 6. i2--64 = 0. 11. 3p--4S=0, 2. f = Sl. 7. ^-121 = 0. 12. 7i>--28=0. 3. A^ = S(j. 8. F' -225 = 0. 13. 49 = ^2. 4. 7r^lOO. 9. 2cr = 8. 14. 144 = .S^. 5. 2/' -16 = 0. 10. ^>^f•' = 45. 15. 75= 3 m 122 ELEMENTARY ALGEBRA 16. The time required for an object starting from rest to fall a given distance is found from the equation 16 t"^ = s, where t is the time in seconds and s is the distance in feet. Find the time required for an object to fall 64 feet. SuGGESTiox. — 16^2=64. Solve for t. Has the negative answer any meaning ? 17. In the following table are given the distances that an ob- ject falls. Complete the table by finding, by the formula in Problem 16, the time required in each case. Distance 16 Ft. 144 Ft. 576 Ft. 1600 Ft. Time "~St 18. In the manufacture of lids for metal boxes and cans, a circular piece of metal, called the "blank," is cut from a flat sheet of it, and then stamped into the required shape by means of a machine called a " die." In computing the size of the blank to be cut for a lid of given size, men assume that the area of the blank is equal to the total area of the lid. Find the radius of the blank necessary to make a lid whose area is 12.5664 sq. in. Suggestion. — If i? is the radius, 3.1416 R^ = 12.5664. 19. The area of the surface of the lid of a lard bucket is 28.2744 square inches. Find the radius of the blank from which it is made. 74. Product of the Sum and Difference of Two Terms. — Find, as in Chapter V, the products : (a — 4) (a + 4) ; (P + 5)(P — 5) ; (a^ - 7)(x' + 7) ; (2E^ + 3)(2 R^ - 3). How many terms in each product ? Why not more ? Why is the last term negative in each product ? SPECIAL PRODUCTS AND QUOTIENTS 128 In general, where a and h are any two terms, it is found that : (a -h b){a -b) = d'- b\ Hence, To find the x)roduct of the sum and the difference of the same two terms, take the difference between their squares. Example. — {'iw^ + S)(iw^-3) = (4w^)^- 32 = 16w;6-9. EXERCISES Give at sight the products of : 1. n-{-3 and ri — 3. 18. 2. A + 6 and A -6. 19. 3. -v + 2 and v — 2. 20. 4. a; 4- 5 and a; — 5. 21. 5. fe — 4 and fc + 4. 22. 6. w — 7 and n-\-7. 23. 7. w -{- V and i« — v. 24. 8. 2 a + 3 and 2 a - 3. 25. 9. 5P + 1 and5P-l. 26. 10. 4:8-7 and AS + 7. 27. 11. 10-3i?andl0+3ie. 28. 12. 1-a^ and 1 + ar^. 29. 13. 6 + 2vand 6 — 2v. 30. 14. 2a;4-5 2;and 2 a; — 5z. 31. 15. 9M-\-3Tsind9M-3T. 32. 16. 5a — 4 6 and 5a + 4 6. 33. 17. (6 W-{-7 V){6 W-7 V). 34. 12^-5a)(125^ + 5a). • a2 _ 9)(a2 + 9). aJi_4)(a:3 + 4). 2^2_i)(2.42 4-l). 7s2_4i2^(7s2^4^2)^ l-6a;^)(l + 6a;*). a8_6»)(a» + 6«). 4 - A)(A + 4). l-6a;)(6a; + l). 2M-{-7N)(-7N+2M). 12-{-5t^)(-12 + 5t^. -10Z>3+9)(10i)« + 9). a + l)(a-l)(a2 + l). 2-Tr)(2+ Tr)(4+ TT^). I_a;)(l+a;)(l4-a0(l+a;0. Give at sight the quotients of the following : 35. (a''-h'^^(a-\-b). 37. (Z)^ - 1) -?- (Z) - 1). 36. (x'-y^^(x-y). 38. (16 - P^) ^ (4 _ ij). 124 ELEMENTARY ALGEBRA 39. {^h^-2^f)-^(;6h + ^i). 42. (144?i«- l)--(12n-^ - 1). 41. (2/^_36)-(/-f6). 44. (86 0^2-49 2/0-^ (6 a; -7 7/-). Solve without the aid of pencil : 45. ^2 + n - 6 = (n - 2){n + 2). 46. a;2 _|_ 2aj 4- 3 = (a; + 3)(a; - 3). 47. 4:A^- A -^1 = (2 A + 7)(2 A- 7). 48. 9 i? (72 + 2) = (3 i? + 6) (3 i2 - 6). 75. Products of Arithmetical Numbers by the Rule in § 74. — The product of two arithmetical numbers, of which one is less than a multiple of 10 and the other exceeds this multiple by the same amount, may be giyen at sight by the rule in § 74. Thus, 58 X 62 = (60 - 2) (00 + 2) = 60'^ - 2^ = 3600 - 4 = 3596. EXERCISES Find mentally the products of the following : 1. 19x21. 7. 48x52. 13. 85x75. 2. 28x32. 8. 53x47. 14. 88x92. 3. 22 X 18. 9. 69 x 71. 15. 99 x 101. 4. 39 X 41. 10. 72 x 68. 16. 97 x 103. 5. 43 x 37. 11. 63 X 57. 17. 109 x 111. 6. 34x26. 12. 77x83. 18. 118x122. 19. Find the cost of 22 boxes of berries at 18 cents a box. 20. Find the cost of 17 dozen eggs at 23 cents a dozen. 21. Find the cost of 28 pounds of butter at 32 cents a pound. 22. Buckwheat weighs 52 pounds a bushel. Find the weight :)f 48 bushels. 23. Onions weigh 57 pounds a bushel. Find the weight of 63 bushels. 24. A field is 77 rods wide and 83 rods long. Find its area. SPECIAL PRODUCTS AND QUOTIENTS 125 76. Product of Two Binomials with One Common Term. x + 1 x — 7 x^7 x-7 a; + 5 x-5 x-5 x + 5 ar^H- Ix x"- 7x a^-\-7x x'-7x + 5a; + 35 - 5x-\-S5 -5a;-35 -\-5x-S5 x' + 12x + '6b x'-12x-hS5 x'-]-2x-35 ic2-2x-35 Tell how the first term of each of the above products is obtained. How is the coefficient in the second term of each obtained ? How is the third term of each obtained ? In general, {x + a){x -j- b) = X- -{-(a-{-b)x + ab. That is, To find the product of two binomials having a common term, take the square of the common term, 2)las the algebraic sum of the unlike terms times the common term, plus the algebraic product of the un- like terms. Example 1. — (n + 2)(n + 3)= ?i2 +(2 + 3)n + 2.3 = n2 + 6 w + 6. Example 2. — (« - 7)(« + 4) = «2 _|-(_ 7 + 4) < +(- 7 • 4) = <2 _ 3 ^ _ 28. EXERCISES Give at sight the products of: 1. (a + 4) (a + 5). 10. (?7i + T) (m + 8). 19. (d-l^(d-2). 2. {x + l)(x + 3). 11. (/ir+l)(^+10). 20. (S-6)(S-12). 3. (M-\-2)(M-\-6). 12. (a; + 8) (a; + 6). 21. (v-4:)(v-10). 4. (2/ 4. 7) (2/ + 4). 13. (a -5) (a -3). 22. (d-16)(d-10). 5. (n + 6)(n + l). 14. (W-S)(W-2). 23. (a + 5) (a -3). 6. (v-\-S){v + S). 15. {R-6)(R-1). 24. (a;+ 12)(a;- 8). 7. (t^4.)(t-9). 16. (;-3)(^-12). 25. (n + 8)(n-4). 8. (A-\-7){A-{-3). 17. (B-W){B-3). 26. (F-3)(F+9) 9. (Z) + 5)(Z> + 5). 18. {N-9)(N-7). 27. {t-5)(t + S). 126 ELEMENTARY ALGEBRA 28. (6 -12) (6 + 7). 34. (m-|-16)(m-12). 40. (y'i.7)(y'-ll). 29. (x + 3)(x-15). 35. (i6-ll)(?^ + 9). 41. (c2-10)(c2 4-4). 30. (2/_l)(2/-f6). 36. (C4-18)((7-7). 42. (2a + 5)(2a+3). 31. (^H-4)(JT-9). 37. (a;2 + 2)(a;2 + 5). 43. (3a;+7)(3aj-f2). 32. (i2-15)(i2+10). 38. (a3-l)(a3 + 9). 44. (6^-1) (6 ^-4). 33. (^ + 14) (^-4). 39. {P'-6)(P'-8). 45. (8 x+3)(8a;-2). 46. (4 TT- 2) (4 Tr+ 5). 55. (6k -{-3t)(6k-5t). 47. (5y + 7)(5y — 2). 56. (4m + 9n) (4 m- 3 w). 48. (2c-3)(2c-12). 57. (9 2) -3'y) (9Z) + 8 v). 49. (8'y + 16)(8v-9). 58. (7 v -{-2w)(Tv-9w). 50. {2x' + 5)(2x' + S). 59. (3 a^ + 4 6^) (3 a^ + 12 fc^). 51. (7 ^3 -2) (7 ^3 + 6). 60. (5t'-6s^)(5f-\-4.s^). 52. (12 TT^ - 20) (12 W + 16). 61. (SB' -4. AC) (8 B' + ^0). 53. (2a + 36)(2a+4&). 62. (4 c^ - 22 c?-') (4 c^ + 5 d^). 54. (5x-2y)(5x-6y). 63. (1 - 5 a)(l + 3 a). Find mentally the indicated quotients : 64. (a^^_5a;_^6)--(aj + 2). 69. (s^ + 4 s - 60) -- (s + 10). 65. (n'-An-\-3)^(n-l). 70. ( TT^ + TF- 72) - ( TT- 8), 66. (^2-2^-15)-(^ + 3). 71. (2/2 + 9 2/ + 20) -(2/ + 4). 67. (v2_7'y + 12)-^(v-4). 72. (r^- 16r + 63)--(r - 9). 68. (r2_y_;L2)--(r-4). 73. (m2-3m-54)-(m + 6). Solve mentally : 74. (a; + 3)(a; + 4)=a:2 + 33. 75. (v + 5)(v-3) = ('^ + 2)(?;-l> 76. (27i + 7)(2n-6)=4(w2 + 2). SPECIAL PRODUCTS AND QUOTIENTS 127 77. Products of Arithmetical Numbers by the Rule of § 76. — The products of many arithmetical numbers may be obtained, without the aid of pencil, by the rule of § 76. Thus, 87 X 92 =(90 - 3)(90 + 2)= 902 - 1 x 90 - 6 = 8100 - 90 - 6 = 8004. EXERCISES Find mentally the following indicated products : 1. 32 X 33. 6. 43 x 38. 11. 89 x 96. * 2. 51x52. 7.48x53. 12.87x91. 3. 42x43. 8. 76x74. 13. 112x113. 4. 64 X &^. 9. 87 X 83. 14. 198 x 199. 5. 39x42. 10. 98x92. 15. 251x253. 16. Find the cost of 34 dozen eggs at 32 cents a dozen. 17. If a train goes 42 miles an hour, how far will it go in 48 hours ? 18. Find the cost of 34 yards of cloth at 36 cents a yard. 19. How many square feet in a lot 92 feet wide and 98 feet long? 20. Oats weigh 32 pounds to the bushel. Find the weight of 35 bushels. 78. Square of a Binomial. — What operation is indicated by (?i + 6)^ ? Find, by multiplication, the values of : (a; + 3)^ (^-5)^ {m-\-nf', {m-n)\ What two terms in the square are positive in each of the above cases ? When is the second term of the square negative ? It is seen that for any values of a and 6, (a-f 6)2 = a2 + 2a6 + 62 and {a - bf =- a" - 2 ab -\' b\ 128 ELEMENTARY ALGEBRA These identities give the following rule : To square a binomial take the square of the first term, plus {or minus) tivo times the product of the terms, i^lus the square of the seco7id term. Example. _ (2 a^ - 3 by = (2 a^y - 2(2 ^2) (3 &) 4. (3 5)2 = 4 a* - 12 a% + 9 b^. EXERCISES Find the squares of the following : 12. 7w — 3v. 1. n + 3. 2. x-\-5. 3. P + 2. 4. a -4. 5. W-6, 6. t-3. 7. m 4- n. 8. x-y. 9. 2 a +3 6. 10. 5 r — 4 s. 11. 3(7+8i). 2 xy xy X* 13. 5J5 + 9 y. 14. 10 6 -c. 15. 9F-^Sa 16. 4/f-7A 17. 12x-\-y. 18. ^-15R 19. 11 TF+3a. 20. 4:E — 9n. 21. 5 ^ + 16 if. 22. a;2 + l. 23. tt2-3. 24. n- + 5. 25. i2_10. 26. a^-2/l 27. A'-{-2B^ 28. 3^2_4 32_ 29. 5w;-^ + 2?;3. 30. y' + l. 31. 4F-^. 32. z^-i-5. 33. 12-7a2. 34. 9 + 15i23. 35. 2p^-q\ 36. 1-8(^1 37. 1^-^AC, 38. a^/ + 5. 39. A' -4: A. -^-X- 40. Show geometrically, by the accom- panying diagram, that SPECIAL PRODUCTS AND QUOTIENTS 129 41. Show geometrically, by this diagram, that Solve : 42. (^ + 4)2 = ^(«4-3) + 36. 43. (2 F+ 3)2 = (2 F+ 1)(2 F4- 3). 44. (4A;-7)2 = (4A;-5)(4A; + 3). 45. (?t + 2)2-w2 = ^_5. h X-M >f-WH K X H 79. Squares of Arithmetical Numbers. — By the method of § 78, the squares of many arithmetical numbers are easily found with- out the aid of pencil. Thus, 622 = (60 + 2)2 = 3600 + 240 + 4 = 3844, and 792 = (80 - 1)2 = 6400 - 160 + 1 = 6241. EXERCISES Find mentally the squares of : 1. 41. 5. 68. 9. 92. 2. 59. 6. 89. 10. 99. 3. 72. 7. 78. 11. 98. 4. 38. 8. 83. 12. 101. 80. Square of a Polynomial. — By actual multiplication, it is found that : {a + b + cy = a" -\- b'' -\- c- + 2 ab +2 ac + 2 be. In general, Tlie square of any polynomial equals the sum of the squares of all of its terms, plus ttvo times the algebraic product of eoLch term into all of the terms following it. Example. — (2 n2 - 3 n + 5)2 = (2 n^y- + ( - 3 n)2 + (5)2 + 2 (2 «2) ( - 3 «) -f 2(2n2)(5) + 2(-3w)(5) = 4 n* _|_ ,j-2 ^ 25 - 12 7i3 + 20 n^ - 30 » = 4 n^ - 12 n3 + 29 n- - 30 n + 25. 13. 109. 17. 249. 14. 199. 18. 301. 15. 148. 19. 502. 16. 498. 20. 999. 130 ELEMENTARY ALGEBRA EXERCISES Write down the squares of the following polynomials : 1. a—h-\-c. 5. 2w — 6'y -I- w. 9. a + & + c -f d 2. 2a; + 3?/+2;. 6. A^ + 2A-\-l. 10. x-y-\-z-w. 3. m — 2n—p. 7. a;" — 2a; + 3. 11. a^ — a^ — a + 1. 4. ^ + 3jB-a 8. r2 + s2_i2. 12. l + 2P-3P2-f 4i*. 81. Quotients of the Form a^ — b^ Divided by a — 6. — By actual division, (a3_ ^3) ^(^a-b) = a'-\-ab + b\ That is, If the difference between the cubes of two terms is divided by the difference between those terms, the quotient equals the square of the first term) plus the product of the two terms, plus the square of the second term. Example.— ^^^~^^^^ = (2 ay + (2 a) (3 6) + (3 6)2 2a — 3 6 = 4 a2 + 6 a& + 9 62. EXERCISES Find mentally the values of the indicated quotients : ^ Z^-8 „ 27^-1 d-4.s ' 8 w^ - 27 v^ 2w-Zv hM-N ' 64. A^- 125 & 1. a^-f x-y 2. m^ — n^ m — n 3. u^-v" W — V 4. a?-l x-1' K l_p3 D- -2* ^- 27 s — 3* &- -64 B- -4 • 125 -f 5- -y a«- -86« 7. ^. 12. 13. 9. ^^^ ^. 14. 1-P' ^^' a-2b' ^^' 4.A-5B SPECIAL PRODUCTS AND QUOTIENTS 13] ,^ 1000r«-l _^ 8 - a%^ „. 8a3-276« Jo. — -— :: — . ZK). -— — . A'k. lOr-1 2-ah 2a-^h^ 2- -a6 a«- -1 a^- -1* m« -n« m' -n^" 1- -^» 19 '^y'-l 23 i:=-L' 27 ^-^-y" Find mentally the following indicated products : 28. (a - 6) (a2 + a^ + &")• 29. (m — n) (m^ + mn -f tj^. 30. {W-V){W^+WV+V^, 31. (a: - 1) (»2 ^ a; ^ 1)_ 32. (2a-l)(4a2 + 2a+l). 33. (3a;-2y)(9ic2^62.2/ + 4y«). 34. (4s-50(16s' + 20s«4-25^^. 35. (a^-l)(a;*4-ar^ + l). 36. (1-2Z)2)(1 + 2Z)2 + 4Z>*). 82. Quotients of the Form a^ + b^ Divided by a + 6. — By actual division, (flS ^ ^3^) _j. (a ^ ^) ^ ^2 _ ^^ ^ ^2^ That is, If the sum of the cubes of two terms is divided by the sum of these termSy the quotient equals the square of the first term, minus tlie (troduct of the two terms, plus the square of the second term. Example. -l^l^^i^' = (5 Sf - (5 S) (2t) + (20« = 2b S^- 10 St + 4: t\ 132 ELEMENTARY ALGEBRA EXERCISES Find mentally the values of the indicated quotients •. 1_ ^ + ^ ^Q i^' + 21v^ ^^ x'-^l m + 71 2 m + ?i W^ + B 4 «' + ! - 13 27i>^ + ^ , 22 ^/±2Il'. a + 1 * 3i>4-« ' 22/^ + 3^2 5 •^ + ^ 14 e4^ + 125:y« 12oP«4-l 2/ + 2' • 40.^ + 52/ ' . * 5P2+1 « 27 + ^ -.^ 1000 /r^ + a^ „^ 216 + a¥ D. — • 15. — -• >54. 3 + ^ lO/f + a i5-{-at 64 + J\r3 216 771^4- 27 71^ ^^ 343 + 8 v¥ ^ + N 6m + 3w 7 + 2 vH^ g 0^^ + 125 6« + 343d^ g^ + JT^ i» + 5 ■ • h + ld ' ' H^ + K^' ^ 1 + 8^^ ^g 8P^ + 125c« 27 + n^ 1 + 26 • 2P + 5c * ' 3 + w* ' Find mentally the following indicated products : 28. (m + n)(m2-mn + 7i2). 33. (4/S' + l)(16/S2- 4>S' + 1). 29. (x + l){x'-x-]-l). 34. (a + 56)(a2-5a6 + 2562) 30. (l+a)(l-a + a2). 35. {l + o^(l-x' + x'). 31. (Tr+FXT^'-TFF+F^. 36. {f^z^){y'-yh^ + z''): 32. (3P+l)(9P2_,3p_|_i), 37. (2+3^)(4-6^ + 9^2)_ SPECIAL PRODUCTS AND QUOTIENTS 133 SUPPLEMENTARY EXERCISES Give the squares of: 2. 3 X« ; 5 X«+i F«-' ; - 12 a =^6^ ; - 7 V^-^ TF^'^+i Give the square roots of : 4. 16//«+8; 25X8'*-2r«"+-; 81 a^"^-"+*. Give the cubes of : 6. 2^"; SiC+y-'; -4Jf*''iY2-; - 5 c^-^d^"+*. Give the cube roots of : 7. w^"; Q*; ^V^"; -^4«^; -Sa^^Y". 8. 27a^<»«; -64^V"; -125a«"P^ Give at sight the products of : 9. (a;'» -f 1) (a;" - 1). 13. (^^ + 3) (^'* + 7) 10. (F' + 3)(F*-3). 14. (Z^ + G)(Z>2'-4). 11. (a"+i + 2)(a«+^-2). 15. (xV- - 1) (iV^* + 8). 12. (a;- 4- 2/*) (a;- -2/*). 16. (3 /« + 6) (3 /~ - 2). Give the squares of : 17. a'* 4-1. 19. F'+T". 21. 5H^-d^. 18. af — y\ 20. 2iy3« + 3w;2«. 22. 3i92'^4-g2^ The products of polynomials often may be obtained mentally by grouping their terms so as to form binomials. Find the products of the following by writing the factors as binomials, and using the rule in § 74. 134 ELEMENTARY ALGEBRA 23. (a + h + c)(a + h — G). Solution. — (a + & + c) (a + 6 — c) = ([a + &] + c) ([a + 6] — c) = a2 + 2 a6 + 62 _ c2. 24. {x-\-y — l)(x-\-y-]-l). 27. (6- c + 6)(6- c- 6). 25. (a2 + a + l)(a' + a-l). 28. (>S'+2 ^-3)(^ + 2^ + 3). 26. (m — n-{-p)(m — n—p). 29. (a — 56 — c)(a — 5 6 + c). 30. (iV+3'y-5w)(JV4-3v + 5ty). 31. (a-2 6 + c)(a + 26 + c). Suggestion. — Group a and c. 32. (P-Q + i2)(P+Q + i2). 33. («2 + a; + l)(a;2-a; + l). 34. (a^ - a6 + 62) (a^ + «& + &")• 35. (a + 6-c)(a-6 + c). Solution. — Since h and c both have different signs in the two factors, group them. Then (a + &-c)(a- 6 + c) = (a +[6- c])(a-[6-c]) = a2-[&-cP = a2 _ [62 _ 2 6c + c2] = a2 - 62 _)_ 2 &c - c2. 36. (4n-3a;-y)(47i + 3a;4-2/). 37. (2u + 3v — w)(2u-3v + w). 38. (3P-4T+F)(3P4-4:r-F). 39. {a-\-h + c-\-d)(a-\-h—c — d). Suggestion, — Group a and h into one term, and c and d into one term. Thus, ([a + 6] + [c + d])([a + 6] - [c + tf]). 40. (x-2y — t-\-z){x-2y + t-z). 41. (P+2^-2!r+C^)(i2 + 2/S' + 2r-V). 42. (ar'-a:2_a._|_i)(a^_ic2 + a;_l). 43. (lH-a-a2-a«)(l + a + a2 + a3). SPECIAL PRODUCTS AND QUOTIENTS 135 Find the products of the following by writing the factors as bi- nomials and proceeding as in § 76 : 44. (a + & + 3)(a + 5-5). Solution. — (a + 6 +3) (a + 6 - 5) = ([a + 6] + 3) ([a + 6] - 5) = [a + &]-^ - 2[a + 6] - 15 = a2 + 2 a6 + 62 - 2 a- 2 & - 16. 45. (aj-y + 2)(a:-2/ + 3). 46. {p + q-\-lQ)){p + q-lQ). 47. (2 >F- 3 F+ 8)(2 TF- 3 F- 3). 48. (aj2 + a;4-6)(ic2^a._2). 49. (f-t + 10)(f-t-U), 50. (n2-l4-n)(w^-2-|-w). CHAPTER VIII FACTORS. MULTIPLES. EQUATIONS SOLVED BY FACTORING 83. Factoring. —What are the factors of 6 ? Of 10? Of 12 ? Of mn? Oiv{v-\-t)? The process of finding the factors of a given expression is called factoring. If a factor of a given expression does not itself have factors, it is called a prime factor of the expression. In general, to factor an expression means to find its prime factors. Facility in the factoring of many expressions depends upon an ability to recognize the special forms of the expressions and upon a knowledge of the rules in Chapter VII. 84. Polynomial with a Monomial Factor. — If each term of a given polynomial is divisible by the same monomial, the poly- nomial is divisible by the monomial, and hence the monomial is a factor. For example, in 18 a* — 9 a^ + 27 a^^ each term is divisible by 9 a^. 2 a2 _ « + 3 9a2)i8a4_9a3 + 27a2 Hence, since the dividend always equals the product of the divisor and quotient, 18 a* - 9 a^ + 27 a2 ^ 9 a^(2 d^-a-\-^). That is, the factors of 18 a^ _ 9 ^3 + 27 a^ are 9 a2 and 2 a2 _ « + 3. Hence the rule for finding the factors of a polynomial with a monomial factor : Find, by inspection, the monomial of highest power that will divide each term of the polynomial. Divide the polynomial by this mono- mial. The divisor and quotient are the monomial and polynomial factors, respectively, of the given polynomial. 136 FACTORS. MULTIPLES. EQUATIONS 137 The division should always be performed mentally^ and the work written as shown in the following example : Example. — Factor 2 wx^ — 4 ny'^ + 6 nz^. 2 /ix2 - 4 ny2 + 6 W2;2 = 2 n(a;2 - 2 2/2 4- 3 5;2) . 6. .43-4^2 7. TrR'-hTrR". 8. oi^ + 5x\ 9. n%^ — m\ 10. 2TrR^-\-27rRH. 11. sH^ + s'l^, 12. 5?^ — lOw;^^. 13. ISar'-giK^. 14. c^n-\-2<^n\ 15. 7P^ + 21P3. EXERCISES Factor : 1. 2a-\-2b. 2. 3 a; — 6?/. 3. AB'-ieAC. 4. 5?ii2 + 10nQ. 5. 8c?ic2_3^2/2. 16. a^ — a;2 + a;. 17. 4F12 + 6F1F2 + 4F1F3 18. H^-6H'b-^2H*b\ 19. 'y2^ + 2vi2^_^^^ 20. 8n«m»4-6wV + 4nm^ 21. 3a^&-6a362^3a263. 28. 4a;«-6a;y + 8fl;3/-12a.V. 29. -10M'i-12M'-16M*+SM\ 30. 56 ay -14 ay +28 ay -105 ay. 31. Show by this diagram that Qc^-{-icy^x(x + y). ^ ^ 32. Show by a diagram that 7ia 22. 2TrR^-\-27t)''-\-2irRr. 23. a^6V + a26V + a^6V. 24. 2^ li^d^- 12 yj'd + 4:2 w'^dK 25. Gy^ + Sv^O-lG-y*. 26. 27. — a^ — sc^ — x. X^ i»- OrlD—*^ nb = n(a — b). 33. Many arithmetical computa- tions may be shortened by applying the process of factoring in § 84. Thus, 4 X 7 -f 6 X 7 = 10 X 7, or 70. Find by this method the values of the following : 17 X 45 + 13 X45; 38x92 + 42x92; 64x575 + 85x575-24x575; 9x 3.1416 + 16x3.1416; 2x16x8 + 2x24x8 + 16x24; 365 X 784 - 15 x 784 + 50 x 784. 138 ELEMENTARY ALGEBRA 34. If, in two numbers of two digits each, the tens' digits are equal and the sum of the ones' digits is 10, the numbers may be expressed by 10 ^ + w and 10 ^ + 10 — «^, respectively, in which t is the tens' digit, u the ones' digit of one, and 10 — w the ones' digit of the other. Show that their product is 100 f + 100 ^ + 10 u - u\ Show that this may be written 100 t{t + 1)4-1^(10 — u). Hence, to multiply two numbers of which the tens' digits are equals and the sum of the ones^ digits is 10, take the product of the number of tens by a number one greater for the hundreds and annex the product of the ones. Thus, 74 X 76 = 5624 ; 69 x 61 = 4209 ; 43 x 47 = 2021. 35. Give at sight the products of the following: 23 x27; 16 x 14; 26 X 24; 32 X 38; 46 X 44; ^6 X 54; 68 X 62; 77 X 73; 88 x 82; 93x97; 81x89; 84x86; 79x71; 75x75; 91x99; 94x96. 36. State a rule for squaring any number ending in 5 that fol- lows as a special case of the rule in Exercise 34. 37. Give at sight the squares of the following : 25; 45; 35; 65; 85; 55; 95; 75; 105; 115. 85. Polynomials Factored by Grouping Terms. — Some polyno- mials that do not have monomial factors may be factored by the method of § 84, after first grouping the terms. Example: 1. —Factor a^ -\- ah -{■ ac -\- be. Grouping the first two terms and factoring them, then the last two, gives a(a + h) + c(a-\- b). Now the factor a + 6 is common to the two terms. Dividing by a + 6 gives a quotient a + c. Hence the factors are a + b and a ■}- c. That is, a2 + a6 +, and since (a + hf = a^ + 2ah + V and {a -by = a^-2ab + b^, evi- dently any trinomial which is a perfect square must be either of the form a^ + 2 a6 + 6^ or a^ — 2 a6 + b^. In either case, the term 2 ab is twice the product of the absolute values of Vo^ and V6^. Hence, A trinomial is a perfect square if one term is twice the product of the absolute values of the square roots of the other two terms. In 9 ic2 — 24 xy + 16 y^, what are the square roots of 9 x^ and 16 y^ ? What is twice the product of their absolute values? Is the trinomial a perfect square ? Since the binomial factors of a trinomial which is a perfect square are equal, to factor such a trinomial is equivalent to express' ing it as the square of a binomial. Example 1.— Factor 25 m^ _ 40 m + 16. V25»»2 = ± 5 m, and Vl6 = ± 4. Since twice the product of these roots must give — 40 m, the positive root of one term and the negative root of the other must be chosen, which can be done in two ways. Hence, 25 m^ - 40 m + 16 = (5 m - 4)2 or (- 5 m + 4)2. Check. — When m = 2, the trinomial is 36 and the factors 6 and 6, or — 6 and — 6. Example 2. — Factor 64 v^ + 112vt + 49 1^. V64t?2 = ± 8 V, and V4972 = ± 7 «. , FACTORS. MULTIPLES. EQUATIONS 141 Since twice the product of these roots must give + 112v«, they must be taken with like signs, which can be done in two ways. Hence, 64v2 + 112 v« + 49 <2= (8v+70^or (-gtJ-TO^. It is seen here, as in § 85, that there are two sets of factors in each problem. In practice only one of these is used, viz. the set in which the first term is positive. To find a binomial of which a given trinomial is a square, take as the terms the square roots of the two square terms of the trinomial with such signs that their product gives the sign of the remaining term, EXERCISES Determine which of the following are perfect squares : 1. a;'^-|-Ga; + 9v 9. 16 ir'-12 m + 9t\ 2. n2 + 16 + 8w. 10. 49x2^42/2_20. 3. 1-4:A + 4:AK 11. 24 TP-lOTP^ + l. 4. 25^2 ^12^ + 1. 12. a^-2a'b'-b\ 5. a:^ + ax + a\ 13. m^ + 18 ma -f 81 a*. 6. 9^2_6v + 4. 14. 62-12 6cH-36c2. 7. M-\-4:MN+4:lP. 15. 9 p^ + SS pq -{- A9 qK 8. r'-10rs + 25^. 16. Z)* + l + 2i>^ Make perfect square trinomials of the following by supplying the missing terms : 17. ar^^_?_|.2/2. 24. 3G-\-?-12x. 18. a'-2ab-^?. 25. a--? + 100. 19. ?-\-2mn+n\ 26. ?-{-l-\-x\ 20. 4^2^ ? + l. 27. 4P34-1 + ?. 21. 16-24^+?. 28. r^^- 12 rir^ + r. 22. 25v^-? +9w^. 29. ?-32JV^+?. 23. Tr^+49 + ?. 30. ? + 4262 + ?. 14^ ELEMENTARY ALGEBRA Factor : 31. a^ + 6a;-f9. 46. 4: wh -{- SQ wv^ + 81 if. 32. i22-4i2 + 4. 47. M^~50MN-\-e25]Sr''. 33. ^2 + 8^ + 16. 48. 9n^ + S0r^r2-\-25r2\ 34. 25 — lOa + al 49. a^ — 2 a^y- -^ xy*. 35. l + 12n4-36n2. 50. a* + 4 a^fe^ _^ 4 a^^*^ .36. 49TF2-28TF+4. 51. P2_6PQ3_^9Q6 37. ^2 + 2^ + 1. 52. 81m* + 36m27i4-4w2. 38. 81-18^ + ^2^ 53. 25W'-40W'g-i-16g^, 39. a;^ + 4ic2 + 4. 54. afy^ + 2x^y-{-x. 40. 16-24Z)2 + 9D*. 55. c-c?^ + 2 cdmn + m V. 41. ax^-6axy + 9ay^ 56. i^V - 8 i?r^ + 16 «l (First remove monomial factor a.) 57^ 24 v^ + 36 i;^ + 4 v. 42. 2n2 + 20nm4-50m2. 68. 25 2^ + 81 2* + 90 2^. 43. tS^ ~ 30 tSV+ 225 tV. 59. 64 ^j^ + 49 ^/ - 112 ^1^2. 44. 16£r2 + 56ird + 49(^. 60. 36^3 + 144^ + 144^2^ 45. 2562-60 6c + 36c2^ 61. -l-6a-9a2. 87. Trinomials of the Form x^-{-ax-\-b. — By the principle in § 76j the product of two binomials having a common term x is always a trinomial in the general form x^+ ax-\-b. Thus, (X + 5)(a; + 7) = x2 + 12 jc + 36. Here a equals 12 and b equals 35. It follows that a trinomial in the form x'^-{-ax-\-b may he factored into two binomials with a common term x, if for the other terms of the factors two numbers can be found whose algebraic sum is the coefficient a and whose algebraic product is b. Example 1. — Factor x^—^x + 15. For the second terms of the factors, we are to find two numbers whose sum is — 8 and whose product is + 15. These are evidently — 3 and — 5. Hence, x^ - 8 a; + 15 = (x - 3) (a; - 5). FACTORS. MULTIPLES. EQUATIONS 143 Check. — When x = 1, the trinomial is 8, and the factors - 2 and — 4» respectively ; or mentally multiplying, (x — 3)(x — 5) = x^ — 8 x + 15. Example 2. — Factor m^ — 4 mn — 45 n^. We are to find two numbers whose algebraic sum is — 4 n and product — 45 n^. These are evidently — 9 n and + 5 n. Hence, m^ — imn~-^bn^={m — Qn)(^m + 6^ n). EXERCISES Factor : 1. t^j^5t+6. 20. 2>2^14|) + 48. 39. aj* + 2aj — 3. 2. a2^5a4-4. 21. 5^ _ 5 ^ ^ g. 40. i?^ 4-372-10. 3. n^^Sn + 12. 22. iY2-5iV^+4. 41. K^-^SK-4:. 4. P2 + 9P+14. 23. w;2_8w; + 12. 42. s^-S5 + 2s. 5. a^ + 9a; + 20. 24. Q^-9Q-\-U. 43. 2/^ + 3y-18. 6. /S^ + O/S + S. 25. z2_924.20. 44. 3a-28 + a2. 7. v^ + lOv + ie. 26. «2_io^-f-9. 45. C«-12 + 4a. 8. a^-\-10a-\-9. 27. i)--10Z) + 16. 46. Sm + wi^-U. 9. i22 + 10R + 24. 28. F'- 10^+21. 47. 62.^3 6-40. 10. A:2 4_iofe + 21. 29. a^- 10 a + 24. 48. Saj-SG + ar^. 11. F=^H-11F+18, 30. T'-9T+S. 49. ^-35-18. 12. c«2 + ll(«+24. 31. 2/2-17^4-30. 50. /-3(/-40. 13. Jl[f24- 11 3f 4- 30. 32. H'-UH+^5. 51. m'-U-5m. 14. 62 4_i2&4-20. 33. TF-4TF4-3. 52. T^-4.T-12. 15. /4-12y + 27, 34. m2-12m4-32. 53. 0^2^53^4.52/2. 16. ^2^12^4-32. 35. c2 4- 35 -12 c. 54. a^ + 7ab+10b\ 17. r^ 4- 15 r 4- 54. 36. ^4-55-16 5. 55. w;2_5^^^6^2 18. m^-{-17m-{-70. 37. <2_f_7_8^. 56. r^ - 7 rs 4- 10 s^. 19. 0^4- 13 c 4- 30. 38. i^2^2i?^-15. 57. i^-3pq-2SqK 144 ELEMENTARY ALGEBRA 58. C^-^BCD^Un^, 66. w^v^-\-2w^v-99i^. 69. f + Byz-Uz^. 67. fZV^d%-56d3. 60. a'b'-^ Sab -IS. 68. 2 s^2_56 ^^^230^ 61. mV — 2mn-35. 69. 3 a^ - 90 a; + 600. 62. H^-19ir--120. 70. 6 62c 4-126 60 + 660 c. 63. F^-F- 30. 71. SI t' + 30 at' + aH\ 64. a62-3a6-70a. 72. 4: W7i' - 2S wn - 240 w. (First remove monomial factor a.) 73. 90 v^/ — 204 v + 6 vyK 65. iC2/2 ^ a;^/ — 90 a;. 74. 6-5a — a^. 88. Trinomials of the Form ax' + bx+ c. — There are different methods of factoring trinomials of the form aa;^ + 6a; + c. Two methods are given here. First Method. — If any two binomials such as mx-\-p and nx + q are multiplied, their product equals mnx^ -{- inqx + npx ■i-pq, which may be written mna^ + (mq + np) x -\- pq. This is of the general form aa^ + 6a; + c. Thus, (2 a; + 3)(3a; + 1) = 6 x2 + 11 x + 3. If a trinomial of the general form aoi^ -\-bx-\-c has binomial fac- tors of the forms 7nx -\-p and nx -f- q, these may be found by trial. Example 1. — Factor 2x^ + 5x -\-2. The first terms of the binomial fac- tors must be factors of 2 x^j and the second terms must be factors of 2. The sum of the products of the first term of each by the second of the other, called the ci'oss products^ must be 6 x. Since all signs of the trinomial are positive, only positive terms for the binomial factors need be sought. The possible trial sets of factors may be arranged as follows : 2a; + 2 a; + 2 x+l 2a;+ 1 It is seen that the second of these gives the right sum of cross products. Hence, the factors are rK + 2 and 2x + \. Check. — 'WhQTi a; = 1, the trinomial is 9, and the factors 3 and 3, respec- tively ; or multiplying the factors, (a: + 2) (2 a; + 1) = 2 x^ + 5 x + 2. Example 2, — Factor ^vfl -{■ lov —10 v^. Since the third term is negative, the factors of it chosen must have un- like signs. FACTORS. MULTIPLES. EQUATIONS 145 Four of the many possible trial sets of factors are : W — 2.V io + 2tj w—^v io+5o 3to4-5t> 3to — 5t) 810 + 2?) 3io — 2d It is seen that the second of these gives the right sum of cross products. Hence, Zvfi + wv - \Q v'^ = (w + 2v)(Z'w- bv). Each trial set should be tested as written. If done, it usually will be unnecessary to write down all possible sets. Each set should be written down as above, and the sum of cross products found mentally. . In a short time the student should be able to factor most trinomials in very few trials. The following princi- ples systematize the work : I. If the trinomial contains no monomial factor, neither factor can contain one. Thus, in Example 1, since 2x^+ 5x -\-2 contains no monomial factor, it was useless to write the first trial set, with the binomial 2 x + 2. II. If the last term of the trinomial is +, the last terms of the factors tvill be both + or both — , according as the middle term of the trinomial is + or —. III. If the last term of the trinomial is — , the last terms of the factors will have unlike signs. Second Method. — In mnx^ 4- '^V'Q^ 4- 'npx 4- PQf which is the product of mx-\-p and nx + g, the product of the terms mnx^ and pq equals the product of the terras 7nqx and npx. From this we may infer that If we may separate the second term of a trinomial in the form ax^ 4- 6jr 4- c into two terms ivhose product is equal to that of the first and third tenns, we may factor the expression as in § 85. Example 3. — Factor 3 x^ 4 11 a- + 6. 3 x2 . 6 = 18 ic2. Now 18 a;2 = 9 x . 2 X = 18 sc . x. The sum of the factors 9 X and 2 x is the middle term. Hence we write 3x2 + llx + 6 = 3x2 + 9x + 2x46 = 3x(x + 3) 4 2(x4 3) = (x + 3)(3x4 2). Note that it will be necessary only to use the coefficients in determining how to break up the middle term, as shown in the next example. 146 ELEMENTARY ALGEBRA Example 4. —Factor 12 «2 _ 5 «?, _ 2 v^. 12(- 2) = - 24. 24 = 3 X 8 = 6 X 4 = etc. Since the product - 24 has a negative sign, the two terms into which — btv \^ to be separated must have unlike signs. The required terms are —^tv and 3 tv. Hence, 12 f^ - btv - 2 v'^ = 12 t'^ - ^tv + ^tv - ^v"^ = 4 «(3 « - 2 v) + v{?j «-2 1^ = (3«-2?j)(4« + v). EXERCISES Factor : 1. 10a2 + 17a + 7. 22. 452-35-27. 2. 3^2^11^ + 10. 23. 6n2-17n-14. 3. 3a;2 + 10a; + 3. 24. ^lJ-2L-n. 4. 5m2 + 12m + 4. 25. 3i^2-i^-4. 5. 2F2_^9F+10. 26. SR^-R-2. 6. 3^2^10^ + 7. 27. 2a'^x-ax-2Sx. 7. 8 7^ + 22 r + 15. (First remove the monomial fac- 8. 3B^-11B + 10, tor.) 9. 10y"-17y-h7. 28. 6c^d-cd-S5d, 10. 3S^^10S-\-S. 29. 6z''-7z-20. 11. 5^-12^ + 4. 30. 20a;2-46a;-84. 12. 15-22a; + 8a^. 31. S6g^-15g-6. 13. 2P2-9P+10. 32. 4a2/2-10a^-36a. 14. 5-llc + 2c2. 33. WW'g-5Wg-20g, 15. 4-8A; + 3fe2^ 34. SO w^v^ - 37 wv^ - 7 v\ 16. 4jR2 + 3i2-27. 35. 2QH-5Qt-25L 17. 6m2 + 17m-14. 36. 4 02 + 32(7 + 15. 18. 3i>2 + 2Z)-5. 37. 4a^ + 23a;2/ + lS/- 19. 3a2 + a-4. 38. 2 M^ + 15 MN-['7 N\ 20. 3^2 + ^-2. .39. 2d^-3dk-l-¥. 21. 4ir2_^_5. 40. 6.£;2__23^F+20i<^2 FACTORS. MCLTIPLES. EQUATIONS 147 41. U 1^-^-2(^-11 be, 42. 7 A^ + 4.n^-29An, 43. 21i22 + 2i?^ -8^^ 44. 2ri2_5n^'2-18rs* 45. ST^S-2 TT'S~. r"S. 46. 12 G'rv'- 5 Gdn'- 2 cPn\ 47. 4: b'^x- 2 bdx- 20 cPx. 48. 12 r^-4rz-i6Z2. 49. 3 a* -a^ -2. 50. 6 2<;2^ + 15w;?;2f-54'y*f. 89. Binomials of the Form a^ — b^. Since the product of a -f 6 and a — b is a- — b^, thts factors of a^ — b^ are a + 6 and a — b. Hence, 27ie factors of the difference between the squares of two numbers are the sum and the difference of the numbers. Example. — Factor 4 <2 _ 25. 4 <2 _ 25 = (2 0* - (5)2 = (2« + 5)(2<-5). Check. — When < = 1, the trinomial is — 21, and the factors 7 and — 3 respectively. Or, check by finding the product of the factors. Observe that, by taking the negative square roots of the terms, we might also write 4 «« _ 25 = ( - 2 0^ - (-5)2 = (_2(-5)(-2« + 5). See § 84 and § 85. EXERCISES 1. Show by the accompanying diagram that 3^ — y^=(x + y) (x-y)' Factor: 2. N'-A. 8. fc2--25. 3^ 3. a^-9. 9. 10. 11. 12. 16 -S'. H'-64. 49 -wl 100 -?;2. X 1. 4. F'-l. 5. ^2-16. 1 6. 1-P\ K X- — *— y — ^ 7. 4-wK 13. ar'-81. 14. A'-IU.^ 16. 1- -49^^ 18. h'-l<^. 16. l-16a;2. 17. 9- -42 Jl 19. 8] Lm2~64?i« 148 ELEMENTARY ALGEBRA 20. a? — ^^y\ 21. 16^2__25Z>2. 22. 4 c^- 25(^2. 28. a^/-2^. 29. 'm;^ — w. 30. 6at^-'24:a, 31. 16m*-36m^ 32. HH-'2bf. 23. 36jE;2„j.2, 24. 81F2-2 — t\ Grouping terms, vfi + 2 wv ■{• v"^ - ' - : Ctp2 + 2 tc» + t?^) - ^2 Check. — When «? = 1, r=:l,« = l, the polynomial is 3, and the factors 3 and 1, respectively. Example 2. —Factor A^ - B^ + 2 SC - C'\ A^ - B^ + 2 BC - C^ = A^ -(B^ - 2 BC + C2) = A^ -{B- 0)2 = {A-\-B-C){A-B+ C). Learn to omit all except the first and last stepSy and to write out the work as follows : Example 3. — Factor x^ + 6x— m^ + 4mn — in^ + 9. a;2 + 6 a; - »w2 + 4 m7i - 4 n2 + 9 = (x2 -f 6 a; + 9) - (m2 - 4 win + 4 w'-^) = (a; + 3 + m - 2 n)(a; + 3 — w + 2 n). EXERCISES Factor : 1. (a 4- 6)^-1. 10. i9V^-(ST-S)\ 2. {x-yy-z\ 11. (x-hyy-(a + by. 3. (S-hTf- 25. 12. (M-N)'-(W-Vy. 4. (/t_A:)2-l6«». 13. (C+Dy-'(4:E'-'Fy. 5. (2m + ny-S6. 14. (l-xy-(v^ty. 6. 1-(A + By. 15. m2 + 2mw + 7i2-62. 7. 4:-(2w + vy. 16. R'-4:RA-^4:A^-B\ 8. 25a^-(b-cy. 17. l+2x + x^^yK 9. 36?f-(r2-4r3/. 18. r^^ ■}- 6 r^r^ + 9 r^^ -- r^- 150 ELEMENTARY ALGEBRA 19. x*4-4a^4-4a;2_l. 22. i -^ a" -2ab -h\ 20. W^-{-9D^-4:M''-eWD. 23. i - M' -\-6 MV-9 V^ 21. l-8a; + 16a;2-;32^ 24. 1 - A^ + 14 M - 49A;2. 25. A^ -9B'' -30 BC ~2n C^ 26. 16 C/^ + 36 FTF- 4 V- 81 TT*. 27. S^- r2 + 4ar-4a=. 28. 12i)g + 16r2-4i)2_9 52 29. A'-AC'-SAB + 16B'. 30. 6^m-«2_9^2^j^^ 31. a2_|_2a6 + 62_c2_2c(^-(2«. 32. 0/"^ — 4 ic 4- 4 — m'' + 6 ?n,w — 9 nK 33. l + 2a;2/ + 2jp-2/^-a^4-y. 34. TF2 + 12a^-10 TFc^-4a2 + 25/-9r8 + 64 F8 = (3 Wf + (4 vy = (3 Tr+4F)(9Tr2-12 TFF+16F2). Check. — When W=l and F= 1, the binomial is 91 and the factors 7 and 13, respectively. ^ V ;:« ^ ^ Factor : 1. m*-fw'. 2. A'+l, 3. l-^p^ 4. w^+S. 5. ^3 + 64. 6. 27 + ^8. 7. 125 + ^. 8. i)^ + 216. 9. 1 + 8 a\ 10. l+27a^. EXERCISES 11. 64JV3_|.l. 12. 8a3+63. 13. 2/^+272^. 14. TF^ + 64 2)^. 15. 27/^^ + 8 T^. 16. 125ri3 + 27r/. 17. 64a3+27^. 18. W 0^+54: y\ 19. a^ft^ + l. 20. 27+mV. 21. 27 P2+P^ 22. 8x*+125x. 23.- 81 Z)*-i- 24 D. 24. 'y^ + ^«. 25. 8i\r6-|,l. 26. 40 a'' + 135 63. 27. 7rZ>3 + Trd\ 28. 54Pi»+2i?/. 29. STrr^ + 81 TrrhK 30. to^ + t'^ 93. General Suggestions on Factoring. — As shown in the preced- ing sections, the method of factoring any given expression depends upon the special form of the expression. There are no general rules or principles that can be laid down for factoring all expres- sions. To factor any expression, wherever the need for it is encountered, requires first that the expression be identified as hav- ing some special form, such as one of those discussed in this chap- ter, and second the application of the special rule for factoring expressions of that form. The following miscellaneous exercises FACTORS. MULTIPLES. EQUATIONS 153 are given for practice in identifying tlie special forms of expres- sions to be factored. • General Suggestions : I. In factoring any expression, first see if there is a monomial factor; and if there is one, separate the expression into this mono- mial and the corresponding polynomial factor. II. T7ien try to identify the polynomial factor as having one of the special forms discussed in the preceding sections; and if success- ful, factor it by the appropriate rule. III. Repeat the process with each factor obtained, until the prime factors of the given expression are found. Write the given expression as the indicated product of all of the prime factors, MISCELLANEOUS EXERCISES Factor : 1. A'-hSA'-^A\ 16. 3P-{-lSM-\-42. 2. 2a^y-i-6a^f-{-4:xf. 17. 42-13i/-|-/. 3. 4i^ + 12wjv + 9v*. 18. F^_42-TK 4. m^n-^6mn-\-9n. 19. a^ — A2-\-a. 5. P'* + 8P+12. 20. ^-4^2_^4«. 6. ^2^3^^ 12. 21. 8-^. 7. 22H4i2-12. 22. iB» + 2 ar^ - 48 a?. 8. m2_4^_i2. 23. Q^-27?-i&x. 9. a3 + 8a2-fl5a. g^. W^-9Wg + 20g\ 10. M^-^M^' + l^M. Sfi. TT^H-Q %4-20^2. 11. d^'\-2d^-lbd. 26. H*-Ut^. 12. v^ — -u^ ' 27. a3 + a2-h8a + 8. 13. ^a?y-10^f + ^xy^. 28. 1 - A. m^ -{- A: mv - v^. 14. a — a\ 29. x^y -\-Zxy — 2^y. 15. D'-2D''-1^D. 30. 12 + N^ + n N. 154 ELEMENTARY ALGEBRA 31. 11 at-{-12t-\-aH. 59. V^+V^-^2V, 32.^2^5^-50. QQ. o? + 2xy-d5y\ 33. A^-^A-60, 61. a^-{a-Q>)\ 34. 2i»2 + 3a;4-l. ' 62. .^V-i>'- ^' + 1. 35. ^-125. 63. 7w;2 + 36w; + 5. 36. aV-6W. 64. 2 ^2^ 5 ^-3. 37. U4.U-M\ 65. 2yl2_5^_|.3, 38. ax — hx — a + h. 66. a2 4-4 a6 — 0^+4 ft^. 39. 'y2_i5^^50. 67. ^ + 125. 40. a^y^-lOxy-be,, 68. 5m^-^mn-2n\ 41. 2a2-3a + l. 69. 812/^-3. 42. Q^i«-9F^. 70. 12 5-2 + 5 5'- 2. 43. xy-x — y-\-l, 71. 216 + ^. 44. m^ + Ti^ 72. Saha? + 2abx—ab. 45. 2A^ + bAB-{-2BK 73. SD^ + l. 46. -v^ + S. 74. 6 6c-lls^ + 5^2^ 47. wV + 5 w;v^ + 6 ^^ 75. 81 x* - 100 v*. 48. D^-liy' + lO, 76. JV^2_f.^^_2ivr_2a. 49. a^y + 7 ar/ 4- 10. 77. 4 a^- (a + 6)2. 50. F'* + llF'-26. 78. 27^3 + 1. 51. 6a2 + lla + 3. 79. 4 ^^ _^ 32 ^5 + 39 ^2. 52. 12w'-2Sw + 10. 80. 125 v^ + ^^ 53. 27 +r^ 81. 4:(a-bf-(y + zy. 64. 36 3f2-64iV^J/2 g2. 8^^ + 15^-2. 55. Sx'^+7xy + 2y\ 83. ri^ + G rir2-l-4 r3-4rs2 + 9 r/. 56. SlJ^^-ieC*. 84. 40Jf3 + 135F«. 57. ab-3a-2b + 6. 85. H^-f + vH-vt. 58. 2m2-9m72 + 10n2. 86. 3x^-\-x-^2, FACTORS. MULTIPLES. EQUATIONS 155 87. A^-itA^n-'A-n, 94. l-\-4:aH'^ -a^ -^1*. 88. 15 + 22-22. 95. 9a2 + 66d-d2_952^. 89. xy ^z^ — xz — yz. 96. v^ — 216 f. 90. 9x2 -27 a; + 20. 97 5 Q2_ 79 Q^_ 16 ^2, 91. i>3-216. 98. oj + a^-ox^-a. 92. TT^— F^ 99. xP ■\- v^w -\- vw^ + in^. 93. mn2 + 7?7in-30m. 100. R"^ -\-2 R- Rr -'2r, HIGHEST COMMON FACTOR AND LOWEST COMMON MULTIPLE 94. Highest Common Factor. — An expression that is a factor of each of two or more expressions is called a common factor of those expressions. The expression with the greatest numerical coefficient and the highest powers of its literal factors that is a common factor of two or more expressions is called their highest common factor (H.C.F.). We shall consider here only the highest common factors of expressions whose factors may be obtained by the methods of this chapter. Example 1 . — Find the H. C. F. of 8 A^BG^ and 12 A^B^CK The largest number contained in both 8 and 12 is 4. The highest power of A contained in both A^ and A^ is AK The highest power of B contained in both B and B^ is B. The highest power of C contained in both C* and C^ is C. Hence, the H.C.F. is iA^BC^. Example 2. — Find the H. C. F. of x^ - x^y - xy^ + y^ and 2x8- ix^y-\-2xy^. x» _ a;2y _ xy^ -{■ y^ = (x - yY (x + y). 2x8 - 4x2y + 2 ary2 _ 2 X (x - yy. Hence, H. C. F. = (x - y)2, or x^ - 2 xy + y^. To find the highest common factor of two or more monomials^ take the j)roduct of the greatest common divisor of their numerical coeffi- cients and each letter raised to the lowest power to which it appears in any of the monomials. In polynomials, factor each expression and proceed as with mono- mialSf treating each factor as you would a single letter. 156 ELEMENTARY ALGEBRA EXERCISES Find the highest common factor of : 1. A.a%\Qa^h\ 7. ^9 wi^t\ 21 w^r^l^u. 2. 9 a*hh\ 12 a^feV. 8. 35 CWP', - 56 CD^P\ 3. IBWV^^OWV^IP, 9. 24m2wy, 42 mpS 18 my^. 4. 8 mnV, 96 mV^. 10. 12 (v + Q^ 8 (v + 0^- 5. 16 M'N\ 24 M^N^P". 11. 14 (^ - i^)«, 21 {E - F)\ 6. 2^Y^\f^' 12. 6(a;-l)2, 8(a;-l)(a;-2). 13. 10(2/ + l)^4(2/ + l)2(2/-l). 14. 39 (m - 7i)2 (m + nf, 26 (m - ti) (m + n)\ 15. A'-1,A'-^2A-S. 16. 1-^2, 1-^,1-^. 17. F^ + 2 F- 15, F^ - 4 F+ 3. 18. a2_52^a»-6^a2-3a& + 26«. 19. 2ar^ + 4 a;, 4 0)3 + 120^ + 8 a;. 20. i22_32^ 9_ J22 222-5i2 + 6. 21. 'm;^ _ 1^ ^3 _|_ j^^ ^ _j_ 1^ 22. 2a2 + a- 6, 6a2-7a-3. 23. 3rir2«-3riV2, 4riV + 2riV-6riV8. 24. ^''-^^, ^^ + ^'^ + -4B+-B^. 25. 6m^w + 14m27i2 + 4m7i3,4m* + 16m''n+16mV. 95. Lowest Common Multiple. — A multiple of a given expression is any expression of which the given expression is a factor. A common multiple of two or more expressions is an expression that is a multiple of each. The lowest common multiple (L. C. M.) of two or more given expressions is the expression with the least numerical coefficient and the lowest powers of its literal factors that is a common multiple of the given expressions. FACTORS. MULTIPLES. EQUATIONS 167 Example 1. — Find the L. C. M. of 24 A^B^C* and 30 A^B^G^. The least number that will contain both 24 and 30 is 120. The lowest power of A of which A^ and A^ are factors is A*. The lowest power of B of which B^ and B^ are factors is B^. The lowest power of C of which C* and C^ are factors is C*. Hence, the L. C. M. is 120 A^B^C*. Example 2. — Find the L. C. M. oi2v^ - iv ■{• 2 and 3 1;2 + 3 v - 6. 2u2_4^ _^ 2 =2(t>- 1)2. 3t)2 + 3D-6 =3(v + 2)(r - 1). Hence, L. C. M. = 6{v - iy{v -\- 2). To find the lowest common multiple of two or more monomials, take the product of the least common multiple of the numerical coefficients^ and each letter raised to the highest power to which it appears in any one of the monomials. In polynomials, factor each eocpression, and proceed as with mS2 = 64. 27. 21t«2 = 10 + 29w. 8. 4-r2 = 0. 28. «3-a; = 0. 9. 552 + 6 = 4^52+7. 29. m3 + 2m2 = 3m. 10. 7cZ2-28 = 0. 30. F^ = 3F2 + 10F. 11. ^(^ + 4) = 4^+49. 31. 4^ + 10^ + 4^ = 0. 12. c2 = 8c + 9. 32. a3 + a2 = 9a + 9. 13. <2 = 22 + 9<. 33. i2'» + 4 = i22 + 4i?. 14. F2-15F=54. 34. 3/_p2 = 27p-9. 15. 2>2 + 12i> + 20 = 0. 35. 32 + ar» = 16a; + 2a;». ":U jp2^6-p. 36. iV*-5i\r2 + 4 = 0. p- ■ ■ '} *.'i7. Tr2+14 TF+40 = 0. 37. v^ = ^v\ ;ri8. 7i2-llA + 30 = 0. 38. ^^ + 12^2^7^43. 19. iV^ + 5^=14. 39. «^-18«2 + 81 = 0. 20. 2/2 + 12 = 72/. 40. 2 D* + 18 = 20 Z)^. 160 ELEMENTARY ALGEBRA 41. Explain the fallacy in the following reasoning, by which it is shown that 2 = 1. Let w = l. Multiplying by w, w2 = n. Hence, n2_i = n-l Dividing by n — 1, 71 + 1 = 1. Substituting 1 for n, 2 = 1. 42. If to the square of a certain number five times the number is added, the sum is 104. Find the number. 43. Find two consecutive whole numbers whose product is 182. 44. The sum of the squares of two consecutive odd numbers is 202. Find the numbers. 45. The sum of the squares of three consecutive numbers is 194. Find the numbers. 46. The area of a triangle is 144 sq. in., and the altitude is twice as long as the base. Find the base and altitude. 47. The base of a triangle is 7 ft. longer than the altitude, and the area is 60 sq. ft. Find the base and altitude. 48. The base of a right triangle is 3 ft. greater than the . alti- tude, and the hypotenuse is 15 ft. Find the length of the base and altitude. 49. One base of a trapezoid is 2 in. greater than the altitude, and the other base is 4 in. greater than the altitude. The area is 54 sq. in. Find the bases and the altitude. 50. The altitude of a trapezoid equals one base and exceeds the other base by 4 in. The area is 120 sq. in. Find the bases and altitude. 51. A square box without a lid and 6 in. deep is to be made from a square piece of tin by cutting out a square from each corner, bending up the sides and soldering along the edges. The box is to hold 384 cu. in. Find how large to cut the piece of tin from which to make the box. L..iHi...J FACTORS, MULTIPLES. EQUATIONS 161 52. An open box, to be 4 in. deep, twice as long as wide, and to hold 512 cu. in., is to be made from a rectangular piece of cardboard by cutting out a square from each corner, bending up the sides and pasting along the edges. Find how large to cut the cardboard from which to make it. 53. A farmer has a field 60 rd. wide and 80 rd. long, which he is plowing to sow with wheat. How wide a strip must he plow around the field in order to have it half done ? 54. Two boys have a lawn to mow that is 60 ft. long and 32 ft. wide. The first boy is to mow half of it by cutting a strip of uniform width around it, and the other is to finish it. How wide a strip must the first boy cut ? 55. By measuring the side of a room 15 ft. square, a man found that he got an area 3500 sq. in. too small. Find the amount of his error in measuring the side of the room. Suggestion. — If e inches is the error, show that e^ — 860 e + 3500;= 0. 56. A room is 10 ft. square. What error (in inches) in meas- uring the side would make the computed area 476 sq. in. too small? 57. In the room in Problem 56, what error in measuring the side would make the computed area 729 sq. in. too large ? 58. The horse power of a gasoline engine is computed by the formula H= , where H = horse power, D = diameter of cyl- inders in inches, and N= number of cylinders. In an engine of one cylinder, find the diameter of the cylinder required to yield 40 horse power. 59. In a gasoline engine of 4 cylinders, find the diameter of the cylinders required to yield 160 horse power. 60. From a group of n children two leaders for a game may be selected in n(7i — 1) ways. If the number of ways in which the two leaders may be chosen is 240, how many children are there in the group ? . 162 ELEMENTARY ALGEBRA SUPPLEMENTARY EXERCISES Factor : \/ 1. 4^2^-8^. '10. m^ — n^ -\- mx — nx. Suggestion. — Group and factor the first two terms, then group and factor the last two terms. 11. V- W + tV-tW. 12. x — a-\-{x—ay. 13. 31? — X — y^ -f y. 14. m^n^ — m* — 71^ -f 1. ^ 2. ax"" + 6a;^ (/ 3. t;« _ ti;i;»». 1/5. P'-'-5P^Q\ • 7. 5 v«2/"~^ — 10 -?;*- V-' {/S. 8 Jtf 2»+i + 12 Jf 2«jv: />^. a262_(a-6)2. 15. n* - r^* - (rj^ - ra^)^ 16. Af-\-At-\-Bf+Bt+A-{-B. Suggestion. — Group the first, second, and fifth terms ; also group the third, fourth, and sixth terms. One factor is a trinomial. 17. 2a-2h-^ad—hd-{-2c-[-cd. 18. ny''-\-^ny-\-by^ + l^y + 2n^l0. 19. p^-\-A:p^-\-^p — p^q—4,pq — ^q. 20. x^ — y^-\-z^ — w^ — 2{xz — yw)' ; 21. W^-\- TF^+T^- y\ 22. A^-\-A^C^-ABC^B^C-W. Suggestion. — Group the first and last terms together, and the other three terms together. 23. 4 mV-(m2 4-^2 _p2)2. 24. a(a-\-c)-h(^-\-c)\ 25. w^ — v^-\-u{\i — 2w). 26. aj2n _ 2 a;«2/** + y2n^ 27. (i~«^$a''^"-j-9a''i2« 28. 6P-9P2_i. Suggestion. — First remove the monomial factor — 1. 29. -^4_j_g^2_ig^ 30. -ic2 4-2^-2/^. 31. -4Jf2-l2il[/i\r~9iV^2 32. (a-}-6)2 + 7(a + 6)+10. 33. (a;-2/)2_6(a;-y)-4a 34. ay? ■\-{a-\-h)x -{■}). 35. aa;2-f (6-a)cc-6. FACTORS. MULTIPLES. EQUATIONS 163 36. 2/ + (4a + 6)2/ + 2a6. 43. ««-/. «« o TT7-2 /o . NTTT . SUGGESTION. — Flrst factor as the 37. 2 W- (2m-{- n)W+ mn. difference of two squares, then factor 38. av^+(ab-^l)v + b. the factors. ^' 44. l-n«. 40. 4^^-9B««. 45 ^_1 41. w;3« + v3". 46, a«-64 6«. 42. 1-Q^. 47. B}^-7^. Solve the following equations for a; : 48. a,-2-3aa; + 2a2 = 0- 52. a;^ - 1(;2 ^ 2 wv + v^. 49. wV-ri^ = 0. 63. a;24.94.4a5=6a;H-a2-|-462. 50. 4va; = ic2_^4^^ . 54 ar^ = 2 rx + 63 r^. 51. 2x^-3^ = ^x1. 55. a^ + 7iar^ = 62a;+62n. 56. Any two arithmetical numbers with three figures each may be written in the polynomial forms 100 C -\-10 B-{- A and 100 c + 10 6 -I- a, respectively. Show that their product is 10000 (7c + 1000(C6 + cB) + 100(ea+B6+^c) + 10(^6+a5)H-^a. From the form of this expression may be deduced a process, called " cross-multiplication," for finding the product of two arith- metical numbers in an abbreviated form. The process is appli- cable to numbers with any number of figures. By this process practically all of the work in multiplica- 4-3 6 tion is done without the aid of pencil, but it requires great mental concentration. Thus, to find 274 x 436 the work is done mentally as follows, and only the answer writ- ten : To get the ones, take 4 x 6 = 24 and write the 4 ; to get the tens, take 2 + 4x3 + 6x7= 56 and write the 6 ; to get the hundreds, take 5 + 4x4 + 6x2 + 7x3 = 54 and write the 4 ; then take 6 + 7 x4 + 3x2 = 39, and write the 9 ; then take 3 + 2 x 4 = 11. 57. Find, by " cross-multiplication," the products of : 34 x 52 ; 61x36; 47 X54; 67x62; 52x83; 245x326; 318x241; 234 X 571 ; 384 x 615 ; 724 X 318 ; 27 x 356 (same as Oi27 X 356> CHAPTER IX FRACTIONS 97. Algebraic Fractions. — Tlie indicated quotient of two num- bers or algebraic expressions, as — or — — -^ -, is called a n or -i-2ab -{- Ir fraction. In algebra, as in arithmetic, the dividend is called the numerator, and the divisor the denominator, of the fraction. The numerator and denominator are called the terms of the fraction. 2a; — 1 The fraction is read either " 2 ic — 1 divided by x^ -\-x -\-l^^ or, more briefly, " 2 a; — 1 over x^ ■{- x -\- 1.'* 98. Signs of a Fraction. — In an algebraic fraction there are three signs involved: (1) the sign before the fraction, which is the plus or minus sign written (or understood) before the line sepa- rating the numerator from the denominator; (2) the sigyi of the numerator ; (3) the sign of the denominator. Since a fraction is an indicated quotient, it follows from the laws of signs in division that i-^ and ^^^ are both positive and + 6 — h are equal, and that ^^ and -^t_^ are both negative and are equal. -{•0 —b Hence, ' ±^ == =1^ =- Zl^ = -±^. ' +6 -b +6 -6 It follows that to preserve the algebraic value of a fraction (1) If the signs of both terms of the fraction are changed, the sign before the fraction' must be left unchanged, 164 FRACTIONS 165 (2) If the sign of only one term of the fraction is changed, the sign before the fraction must be changed. + 4 -4 +4 -4 If the numerator or denominator be a polynomial, its sign is changed by changing the sign of each of its terms, because this is equivalent to multiplying the polynomial by — 1. Thus a;g-4a; + 2 ^ - a;2 + 4a;~ 2 ^^ x^-4x-{-2 * -a;2 + 3x-l x^-3x + l x^-Sx + l EXERCISES Write with positive numerators and denominators : '•^• '■^■ 7 -2a' -z1^- ..^. '■5|- -^. ". -;r- '■^. '■m- '■^ — vg* iVrite with denominators a ■ -6 : ^vh- 14. F- - X - a' 15. ^-'*. b — a In each of the following expressions write all of the fractions \* ih the same denominator : 16. _6_ + _2_. 18. 2Mil__3^ + l^. x — y y — x ar* — 1 1 — a^ l—o? yj a , b __ a + h -g bat a^ a' + f • a- 6"^ 6- a 6 - a' ' a' - t^ ^ _ a^ "^ a* - i»' „^ 1 + r ^ + 1 1 — r r— 1 r — 1 166 ELEMENTARY ALGEBRA Write for eacli of the following an equal fraction with positive denominator and the sign -f before the fraction : 21. -JL. 23. -J#-. 25. -1 "- -'^ -A' /it* -TTT^' -5 — n 28. -100* 22. -^. 24. -I^. 26. 99. Reduction of Fractions to Lowest Terms. — As in arithmetic, to change a fraction to lowest terms is to change it to a fraction with equal value, but whose numerator and denominator have no common factor. Thus, 6^2x^ = 5. 1=? 1^=.? i^ = ? 8 2 X 4 4 18 24 100 The principle involved in the process is that : Dividing both terms of a fraction by the same expression does not change the value of the fraction. . :, Example 1. — Reduce — - — to lowest terms. ISa^b* Dividing both terms by 6 a%^^ their highest common factor, 12 a^h^ ^ 2 62 18a3&* 3 a* Example 2. — Reduce ^-i — ^~ to lowest terms. ic2 + 5 X + 6 Factoring both terms, and dividing by oj + 3, their H. C. F., x2 + 2a;-3 ^ (a; + 3) (a; - 1) ^ x-\ ^ ic2 4.6x + 6 (ic + 3)(x + 2) x + 2 It is evident that : To reduce an algebraic fraction to its lowest terms, factor the numerator and denominator^ and divide both terms by their highest common factor. ^ .. FRACTIONS 167 The division may be indicated by cancellation of the commou factors of the terms. Thus ^^^ — g?^^ _. iKja^-^^^jm + n) _ m -ir n ' am^ — an^ jiif(wj^--7?J(m'^ + win + w^) wi'-^ + win + n^ Note. — It must be remembered in using cancellation that' only factors common to the numerator and denominator may be cancelled. Terms in polynomial numerators and denominators cannot be cancelled — a common 3 4-5 mistake made by students. Thus, in "^ the 5's cannot be cancelled, be- 4 + 5 cause the fraction equals |, and if the 5's were cancelled, it would become |. Similarly, in ^ "^ no cancellation is possible. EXERCISES Reduce to lowest terms : 1. 1^. 6. ?5. 11. «5. 16. ^^. 12 96 ac 10 yz — • 7. — • 12. • 17. 12 54 6ar^ 12 wv' 1^ 8. -I?-. 13. i^. 18. '^'"'^ 18 100 6m?i2 21 i^st 4 ?1. 9 ?40, 12j)V ,9 8^^ * 28 * 800 * 16j9V * 36 v^ 6. i?. 10. -5^. 15. A^. 20. i^. 20 720 Qa^hd irl^ 4.7rR(E-\-H) A'-AB^ 4 7t^4-47i + l STrH^iE + H) * ^2^2^J3+^2 • 4^2-1 22. ^'-^ 26. ^'-^ > 30 ^^^^ a2 + 2a6 + i>' 2By + 2y a^-b^ A'-{-2AB-{-B^ R'-l 2Ry + 2y ^2 + 2^-15 f-t- 6 l-p« om--xm^ ^2 + 2 ^ - 15 l-a; 24. ^^. 28. _1^I^!_. 32. i^^I^. 168 ELEMENTARY ALGEBRA 33 F^ + 8F+12 M'-M'N^ r,%-nr,\ 34. ^^±^. 36. ^-^ . 38. ^'-^' 39 ^-^-2 . 42 6a^ + 7a;-5 . • 4-i>2 • 3a^ + 17a; + 20 ^^ ac-6c-ad4-5cg ^ ^3 10 a" -S3 at-^7 t^ • ac4-ad-&c-6d' * 12 a^ - 52 a« + 35 i«* B^-Br-B + r l + P' 100. Reduction of Fractions to Mixed Expressions. — An algebraic expression containing no fractions with literal denominators is an integral expression ; an expression consisting of one or more such fractions is a fractional expression; and an expression consisting of an integral part and a fractional part is a mixed expression. /v. 1 A A Thus, x^ ■ and ^=— + A^ — 1 are mixed expressions. x2 + l ^-lu4 + l Note. — In arithmetic the mixed number 4| means 4 + f , the sign + being omitted. In algebra the sign between the integral and fractional parts of a mixed expression must never be omitted. Thus, x + - may not be written x - , because the latter implies multiplication. An algebraic fraction whose numerator is of as high degree as the denominator, or of higher degree, may often be reduced to a mixed expression. The process is similar to that in arithmetic of reducing an improper fraction to a mixed number. Example 1. — Reduce ^^-^- to a mixed number. Dividing 731 by 46 gives a quotient 15 and remainder 41. Hence, 3^%^ = 15^. Example 2. — Reduce ^-i — ^'~ to a mixed expression. JD + 3 Dividing x^ + 6x — Qlaj x + S gives a quotient oj + 2 and remainder — 15. 15 Hence, ^' + ^^-Q = x + 2 + ' x+3 x+3 15 This answer may be written x + 2 — ■ x + 3 FRACTIONS 169 To reduce a fraction to a mixed expression, divide the numerator by Uie denominator. The mixed expression equals the quotient plus the fraction whose numerator is the remainder and denominator the denominator of the given fraction. EXERCISES Reduce to mixed numbers : 1. i. 3. V. 5. -1^. 7- W- 9- ^4W- 2. ^. 4. If. 6. ^. 8- m- 10- ^w- Reduce to mixed expressions : .1. ^ + 1. 16. P' . „^ ^A'+S'AB+iB? 3A -2B 22 v" -fi V + t 23. V^ -Sn' '-\-Sn- -1 n + 1 S'- -16 jS- -2 ^-f 4^ + 3 x-1 P-3 2^-4 * * M+n' ^^ 7i^-5n + l . ig r^ + 3r^ + 5r4-2 w-2 * r + 1 14. -1^^. 19. ^+^. 24. 16. ^•^ + ^ . 20. 5^+«,\ 25. 2«2_^4-2 a-2 E'-'SE 101. Reduction of Fractions to Higher Terms. — It sometimes is necessary to change a fraction to an equivalent fraction with denominator of higher degree. The principle involved is that: If both terms of a fraction are midtiplied by the same expression, the value of the fraction is not changed. The process is the same as that in arithmetic of reducing a fraction to higher terms. Example 1. — Reduce | to 48ths. Dividing 48 by 8 gives 6. Heuce, multiplying both terms by 6, i = ih 170 ELEMENTARY ALGEBRA Example 2. — Change -^-^t — to a fraction with denominator 6 a^+aft — 6*. 2 a + 6 Dividing 6 a^ + a6 — 6^ by 2 a + 6 gives 3 a — 6. Hence, multiplying both terms by 3 a — 6 gives a + h _ 3 qi' + 2 a5 - 62 2 a + & 6 a2 + a6 - 62 * Divide the required denominator by the given denominator, and multiply both terms of the fraction by the quotient. Note. — The division may be quickly performed by factoring the required denominator and cancelling from it the factors of the given denominator. Thus, in Example 2, the required denominator6 a^+ab—b^= (2 a + 6) (3 a--6). Cancelling 2 a + 6, the quotient is 3 a — 6. EXERCISES 1. Change f to 26ths. 3. Change ^ to 64ths. 2. Change ^ to 56ths. 4. Change ^ to 64ths. 5. Change ^ to 126ths. 6. Change each of the following to 72(is : JL 3. 5 If 4> ■§"> jf T2"> a* 7. Change — - to a fraction with denominator a^6V. b^c 8. Change to a fraction with denominator x^ — 1. x—1 9. Change -— to a fraction with denominator M^ — N^, M -\- N 10. Change — — - to a fraction with denominator R^-{-3Ii-^2, M -{-1 11. Change ^^-^ to a fraction with denominator v' — ^. v — t 12. Change -— to a fraction with denominator 2x-\-6y 6V4-7icy-202/2. FRACTIONS 171 XS. Change each of the following to a fraction with denom- inator a^h — aWi ^ ^ ^ J a+h a—b a b 14. Change each of the following to a fraction with denominator a;-?/' x + 2/' {x-yf' (x + yf 102. Reduction of Fractions to a Common Denominator. — It is clear that by the process of § 101 two or more fractions may be changed to equivalent fractions having for the denominator of each the lowest common multiple of all of the denominators of the given fractions. This lowest common multiple of the denominators of the given fractions is called the lowest common denominator (L. C. D.) of the fractions. Example. — Reduce ^ "^ ' and ^^ — to equivalent fractions having a;2_i a;2_4a;_5 the lowest common denominator. a;2-l = (x + l)(a;-l). a;2 - 4 a; - 6 = (x + 1) (a; - 5). Hence, L. C. D. = (x + l)(x - \){x - 5). Cancelling the factors of the first denominator that are found in the L. C. T)., it is seen that the terms of the first fraction must be multiplied by a; — 5. Similarly, the terms of the second fraction must be multiplied by x — 1. Hence, ^±1 = {^-^){^^^) or ^^-^^-^^ .. a;2-l (x-6)(x2-l) x^-Sx^-x + S' and 1-x ^ (x-DCl^x) ^^ -^2^2x^1 x^^^x-b (X- l)(x2-4x- 6) x8-5x2-x4-6 EXERCISES Reduce to equivalent fractions having the least common de- nominator : 1- h i. i 3 I, I, f 5. I, f, ^\. 7. J, I, i^, ^. 2- i. *, f 4. h 4, «. 6. ■^, A. i- 8. A, A. h h 172 ELEMENTARY ALGEBRA Reduce to equivalent fractions having the lowest common de- nominator : 9 A J_ A. 12. i-, ^, y-. ' Sx 2x^' 6x ' xy yz xz 10. 1., A, 1 13. ^ ^ 1 Sf' 6^' 4t*' ' A + 1' A-1' A^-1' 11. :rT^, TT— T-, . .. > 14 36V 2aV 4a262- * n^ + 3n + 2' 2712 + 571 +2 15 ^-1 F+3 F' + 5F+4' F2+ F-12* 16 2ig + 7 3ig-4 • 6i22 + 13i2-5' 12i22-13jK + 3* 17. 1 ^ 1 18. a - 6' a + 6' a' + 6'* 2 c 2c2 c + r (c + if (c + i/ V i^-u — v^' w^ — v^ X y a? x—y 2x-2y 6(a^ — f)' 21 P-^ i> + l i) + 2 . 1 1 1 eS^ + S-2\ 15S^-\-10S' 12S'-6JS FRACTIONS 173 103. Addition and Subtraction of Fractions. — Algebraic fractions are added or subtracted by the same rules as fractions are added or subtracted in arithmetic. Example 1. — Add |, |, and |. Reducing all of the fractions to the least common denominator, l + l + i = M + ^ + H- Adding the numerators and placing the sum over the common denominator, M + 3^\ + il = M or n- ^Example 2.— Add ^^ + \ x^^l and -. x-\ x+\ X Reducing the fractions to their lowest common denominator, 2a; + l , x-2 3 _ 2x^ + ^x^ -\-x _^ x«-3a;2 4-2x ^x^-Z X— 1 X-\-\ X X^ — X X^ — X 3^ —x' Adding the numerators, this becomes ^ — ^^ * Example 3. — Perform the indicated addition and subtraction in 5.1 3 + a2 + 5 a + 4 a^ - a - 2 a'-' + 2 a — 8 1 3 a2 + 5a + 4 a'^-a-2 a^ + 2a-B 5 . 1 (a + l)(o + 4) (a-2)(a + l) (a+4)(a-2) 5a- 10 , q + 4 8 a + 8 (a + l)(a4-4)(a-2) (« + !)(« + 4)(a -2) (a + l)(a + 4)(a- 2) 3a-9 ^„ 3a-9 (a + l)(a + 4)(a-2) a^ + Sa'^-Ga-S After reducing the fractions to a common denominator, the first two nu- merators are added, and the third one subtracted from their sum. To add or subtract algebraic fractions, reduce them to their lowest common denominator, then find the sum or difference of their numera- tors, and place the result over the common denominator. 174 ELEMENTARY ALGEBRA EXERCISES Simplify by adding or subtracting, as indicated : 1- t + f 3. A + A-f- 5. f-i + i. 7. f-A-*- 2. i-A- 4. A+i-A- 6- A-A+A- 8. H + |-«- - 3a , 5a , a ,, 3j) , ■» ,„ a c *• T+T+3- "• 20 + r ^^' b~d 15. £ + 1. + ^. 2/2! Qsz xy 16. !L±l + J- + 2"-l. 1,. i+JL+i. n?'2 ^1^3 t\n 18 3 2TrR-\-H H+B ' 4.7rB 27r2?2 7ri22 * 19. -_^ + _X_. 23. -^ '^ ^ x^y x-y 20. -^^ -^ — 24. 1 + a 1 — a 21. m + n _^ m-n ^ 2^ m — n m-\-n 22. 3^Zll_!Htl. 26. v + 1 v-1 a2+6a + 5 tt2 + 2a-15 27 ^-2^ M-h4:N 3P-{-5MN+4:JSf'' M''-MN-2N^' «« 1 1 S^-s S2 + S 5 205 + 1 x-2 . + 2. 2a;-l Sv ^ 1 ^ tov w-}-v to — v 2<;^ -ir' 1 1 2b^-\-5bc-Sc' 4 62_i3 5c + 3c« FRACTIONS 175 29. 80. 2TrR 2'rrR rH ttR + ttH ttR 3 2 4 31. ^ + ^ + a + l 1 1-a? Suggestion. — In problems such as this it is best to arrange the terms of all denominators according to either the descending or ascending powers of some letter. In order to have the first term of the denominator of each frac- tion positive when rearranged, it may be necessary to change the signs of the terms of the fraction, as in § 4 Thus, the last term in this problem may be changed to + a2-l 32. 33. P-1 V 2uv 1_P2 Sv U-\-V U — V 'l^—V? 36. 84. 35. + T- 2x l+o; 1-ar^ o^-l Mil M^-N^ M+N N-M 3-f-2y 16y-y« 2-3y . 2-3/ f-^ 2 + y 87. 38. 89. R R R' 1+R 1-R b^ 2 6 R'^1 . 1 1-6'^ 62 + 6 + 1 6-1 x-^2a x — 2 a 4a6 2b-x 2b + x x'-Ab*' 40. + a—b a+6 a— 26 a+26 Suggestion. — In certain cases, such as this, it is best to combine only part of the fractions at a time. It avoids long multiplications. Here, com- bine the first and second fractions, then the third and fourth, then the two results obtained. 176 ELEMENTARY ALGEBRA 41. -A^+ 1 1 1 42. w-i-l wH-3 w — 1 n — 3 _j^ 1_ , _J_ L., B-r B + r S-s S + s 43.^4 ^ ' ^ 44. y—6 y+1 y+6 y—1 Jl 1 1 1 2^+1 ^-3 3^-2 104. Reduction of Mixed Expressions to Fractions. — A mixed expression may be changed to a fraction by the process of § 103. Compare the process as illustrated in the following example to that of reducing an arithmetical mixed number to an improper fraction. Example. — Reduce m + n -^ — to a fraction. m— n Writing the integral part as a fraction with the denominator 1, m + n-i = — ; 1 m — n 1 m — n m— n m — n w2 m — n To reduce a mixed expression to a fraction, write the integral part as a fraction with the denominator 1, and proceed as in addition and subtraction effractions. Reduce to a fraction : FRACTIONS 177 EXERCISES 1. 4. ^ 4-2. 7. ir4- — < B - m^ .. « r^r^ %, B + ^r^^* 5. — ^^^: m. 8. ri- 1 + 5 m — n '•i + ^2 3. r + 1 -• 6. w; + 5h — • 9. - + a-&. r — 1 w —6 a — o 10. 5-^G 14. v^-Il^-V+1. 11. l-«4 2.V' 1+2/ 1-2/ 115 ^ ■ n^ — 2 -u^ n^i; + uv^ » ^2 _ ^^ 4_ ^2 u^^v^ 12. 5 + -^ + -^. IG. _£!_ + 2 + -^. 13. a^ + 62_a!zi|!_«5. 17. l + ^ + |. a + 6 B^ B 105. Multiplication of Fractions. — The product of two or more algebraic fractions is obtained by the same process as the product of two arithmetical fractions. Thus 2^7_2x7_14.3^29_ 3x2x9 _ 54 . inus, 3 X 9 - 3 ^ 9 - 27 ' 6 >^ 7 ^ 11 - 5 X 7 X 11 " 385 Similarly, ^x^ = ^; ^^±1 ^^J^^ll = (jL±^)(^JL:zM. b n 6»a; + 5 a;-4 (x + 5)(x — 4) The product of two or more fractions is the fraction whose numer- ator is the product of their numerators and denominator the product of their denominators. As in arithmetical fractions, the product often may be reduced to lower terms. In such a case, work may be saved by factoring 178 ELEMENTARY ALGEBRA the terms of the fractions, and cancelling any factor of any numer- ator by a like factor of any denominator, before performing the multiplications. Example. - Find the product of 2 a^ + a6 - 6^ ^^^ Sa^^ab-2b^ , 2 a2 - a6 - 62 2 a^ + 5 a6 - 3 62 2a^ + ab-b-2 ^^ Sa^-ab-2b^ ^ (a + bXJ^sr'^ Cij^^^CSa + 2b) 2 a2- aft -62 2a2 + 5a6-362 (2 a + 6) j^.^-^ ^2jl^^^ {a + S b) _ (a + b)(Sa + 2b) (2a + 6)(a + 3 6) _ 3a2 + 5a6 + 262 ^ 2a2 + 7a6 + 362* Find the product of : EXERCISES 1. |xf 4. i X i X f 7. ixfxll. 2. *x|. 5. fxtfxf. 8. f X J X «. 3. ixif. 6. « X -jV X |. 9- iXAxfi- 10. 14. 8r g IS ^ "20* 11. 16. 4^3- 3d ay 12. 22 • 2Z)» 11. 9f 2v 3 12. 8Tr 9P • " 3 * 21jL^ 3^2 TF+2F 3Tr+6F 15 J6riV__^12_»^ ^g a^^-lSx + SO . ar^-15a; + 56 16. «±l-^£l=i. 20. ^-1 , ig^-12ie + 35 a a» i2*-3i?-10 i22 + 3ie + 2 184 ELEMENTARY ALGEBRA 21. -1^„^JL-. 26. ^-1 - ^'-^ a^ + t^ a + « 3ri2 + 8n-f-5 w« + 4n + 3 22. ^' + c^ . 36^ + 3c^ ^^^ a^-9y^ . x^-^xy-6 f p'—pq + q' i>' + $ 109. Complex Fractions. — Fractions whose numerators, denomi- nators, or both, contain fractions are called complex fractions. A « 1 + 1 d b W t Thus,—, -, -, and are complex fractions, d WV ""7 A complex fraction is said to be simplified when it is reduced to an equivalent fraction whose terms are integral expressions, or to an integral expression. Since a complex fraction may be con- sidered an indicated quotient, it may be simplified by dividing the numerator by the denominator. Example. — Simplify A-l -i A A A^-^^Aj^^i, 1+JL ^ + 1 A A + l A A FRA CTIONS 185 EXERCISES Simplify : 1. |. 7. ^. l + # A-l + --^„ I 1+1 12. ^^. 17. ±Z^. .. ^ w i?-l .1-2+ ^ iV^ A-6 13. 2+1. 18. _£. £lzii 1+i _ 2 « 3*' 4 8i. ^^^ 14. H H+R m + n 8. a; m? — v? 9. y ay + b' a 1 10. n-H 1 u 11. u — v «» 15. x-y x-^y 19. X y <»-y^ x-\-y y X R 1-R 20. 1 + /? ' R R 1-R. 1 + i? R X ,1 n* ^«* a* ^, w — v ^^ 1 + a; «, ^'2* ^i* 6. -. 11. 5—- 16. -~ 21. 1 + a; + a; 7*1 + -^ SUPPLEMENTARY EXERCISES Note. — In some problems in the addition or subtraction of tractions, such as the following, it is advisable to change the signs of factors of one or more denominators before finding the lowest common denominator. In this change of signs use is made of the principle that, if the sign of one factor of a product is changed, the sign of the product is changed ; and if the signs of two factors of a product are changed, the sign of the product is not changed. Thus, (a - 2) (a — 4) = a^ — 6 a + 8. If the sign of a — 2 is changed, by changing the signs of its terms, we get (2 — a) (a — 4) = — a^ + 6 a — 8. If the signs of both factors are changed, we get (2 — a) (4 — a) = a^ — 6 a + 8. 186 ELEMENTARY ALGEBRA Perform the indicated additions and subtractions : 1.1.1 1. {A-B){A^C) (B-C){B-A) (C-A){C-B) Solution. — Writing all factors of denominators as A — B., B— C, or C-A, + iA-B){A-'C) {B-C){B-A) {C-A){C-B) QA-B)iG-A) {B-G){A-B) {C-A){B-C) -\(B-C) -.i(C-A)^ iA-B)(B- G){G - A)'^ {A- B){B - C){C - A) -HA-B) {A-B)(,B-G)iG-A) __ ^B-\- G- G + A- A+B _^ (A-^B){B- G)iG-A) ,2. 1 1^+___1__ (x-yXx-z) {y-x){y-z) {x-z){z-yy 3 mn J np pm {p — m)(p — n) (m — n)(m—p) (n~-p)(n — my 4. "^ +- V. + ^ . n+rg ^ rg + y-g ^ rg + n 6. (a-6)(a-c) (6-c)(6-a)^(c-a)(c-6)" (u — v){w — u) (v — w){u — v) (w; — u) {v — w)* 8. •: TT^: ^-1 FRA CTIONS 187 1.1 (x^l)(x-2) (2-x){S-x) (a;-l)(3-a?) 9 e 3 1 Note. — In the following problems invert divisors, and find the products of all factors at one step by cancellation. Simplify : 10. ^x?^^-^. 11 ^-^ 12 n — 1 n^-{-3n-10 . n' + 5n *7i — 2 w^ + n — 2 n'* — n — 6 13. (3t.^2i;/2-^i^JJ^V^^' + ^^^^-^^. ^ \ 2u-\-vJ 2u + v 6P'-nP-10 . 7P'4-17P-12 . 10P^-27P + 5 • 3P2 + 2P-5 * 6P2 4-9P-2 * 21 P^H- 23 P- 20* Note. — A good method of simplifying a complex fraction (see § 109) consists of multiplying both terms of the fraction by the lowest common de- nominator of all simple fractions in the terms, as suggested in the following problem. 188 ELEMENTARY ALGEBRA a-\-b . a — b 18 ^ ^ Suggestion. — Multiplying both terms by the L. C. D. gives a—b a-^b' a b h(a-\-b)-{-a(a-b) _^^^ b^a-b) - aia+b) ,„ ri + 2 n-fS 19. 3 rt + 2 2y Sx n 2A'-AB-SB' 20. ^ ,. „„ SA'-AB-2B' A' + SAB-\-2B'' 4:A^-5AB-{-B' CHAPTER X FRACTIONAL EQUATIONS. PROBLEMS. FORMULA 110. Fractional Equations. — Some problems are expressed and solved by means of equations in which the unknown number appears in one or more denominators of fractions. An equation of which the unknown number appears in one or more denomi- nators is called a fractional equation. Thus, ^ "^ = — }- 5 is a fractional equation. ' n-2 n + 2 ^ 111. Clearing of Fractions. — To solve a fractional equation it is necessary first to change it so that it is free of fractions. This process is called clearing the equation of fractions. It was found in § 107 that if a mixed or fractional expression is multiplied by the lowest common denominator of all of the frac- tions involved, the product is integral. We know also, as an axiom, that if equal quantities, such as the members of an equation, are multiplied by the same quantity, the products are equal. These principles are used in clearing an equation of fractions. Example. — Solve -^ + — ^ = -^ + 2. The L. C. D. of all fractions in the equation is fi — 1. Multiplying both members of the equation by f-^ — 1, by multiplying every term of each mem- ber, gives ('' - ^>(^i) + ('^ - 'Krh) = ('^ - »^(rri) ^ ' ^''''^^ or 2t^ + t^ -\- 1 = f - t -{■ 2t^ ^2, Solving, 2f=-2, 189 190 ELEMENTARY ALGEBRA To dear an equation of fractions, multiply each member, by multi- plying every term of each member, by the lowest common denominator of all fractions in the equation.* Solve: 1. 2 = « n 2. 3 1 2y 4* 3. 4. r r r 1. 6. x + 1. 2 1 = — . X 6. 4 7 5c 10c 1 10' 7. ^-i+-i.=20. T T ST g m + 12 ^ 2m + 12 2m 3m 2 1- 1. 10. ?= 1 EXERCISES 11. S-3 12. 3 2 A: + 1 fc - 1* 13. 6 2 2^ + 3 ^-4* 14. n-3 71-5 7i-f 9 w + 5 15. X>-1 i>-5 2i>-5 2D-2' Ifi 9 7 _^ 5r + 2 3r + 4 17. ^ + ^ =1. B-2 B+2 18. 1= 2/ ^ 4 y-2 ' 2/ + 1 19. 2P+3 4P+5 ^ 3P_4 6P-1 0(\ w _ 3w; Q V V — 1 w -\-l w -{-2 * It is important that the teacher should see that no pupil gets a wrong idea of the process of clearing of fractions through careless thinking. Thus, a pupil some- times thinks that clearing an equation such as ^^^^ — = ~^ of fractions con- sists of multiplying the first member by 1 + a; and the second by 2x + 4, rather than each member by (1 + x) (2 x + 4). FRACTIONAL EQUATIONS 191 21. 4 5 A- 5 A' -^- y^- 22. t + 1 2 __1 26 c + 3_c + 9 c c + 4* 23. 3y + 62/ + 4 26. 2» + l- 4 1 n ^^•2/+3 + 1 1 2i-3 4i2_9' 28. ^=: 2 + 1 !i^^. m + 1 2 1 m-2 m + 2' 30. A + 2 16 T7- n TT-O J* F+2 F-2 F'-4 p p + 3 p — 1* 32 3a + 5 _ 3a^ + 5a-4 4a-3 4a2-3a4-2* 2a; + l 8 ^ 2a;~l ' 2aj-l 4ar»-l l + 2» 34. !^!^ = _i_ + - + l 86. 36. 1 — V^ 1 + V 1 — V 37 4__ ^ 7 i22 + 5i2 + 6 i2 + 2 [rTS* 1 ^ r-6 T T-3 T + 3 T+3' 87. ^^L^ + iL+ 3 4i-12 220 5X-15 192 ELEMENTARY ALGEBRA x — 5 7 — x 2x — 15 38. a^-10a;-f21 a^-Sx + W ar^-12a; + 35 39 9^ + 17 2A-1 ^ 2A + 1 A'-2A-A8 2A + 12 2^-16' 40. ^2-2^ + 5 ^2^3^-7 41 iV-4 N-15 ^ 2N'-10N-^1 2^-5 N+4: N''-N'-20 ' 42. 7a^ + lla + 4 ct4-3 ^ 5a + ll 112. Problems Leading to Fractional Equations. — Types of problems such as those in the following list lead to fractional equations, which are solved by the method of § 111. EXERCISES 1. The sum of two numbers is 265, and if the larger be divided by the smaller, the quotient is 14 and remainder 10. Find the numbers. Suggestion. — Let n = the smaller number. Then 265 — w = the larger number. Hence, ?6in» = 14 + 15. n n 2. The sum of two numbers is 160, and if the larger be divided by the smaller, the quotient is 4. Find the numbers. 3. The difference of two numbers is 324, and if the larger be divided by the smaller, the quotient is 3 and remainder 80. Find the numbers. 4. Separate 84 into two parts whose quotient is f . 5. Separate 148 into two parts such that if the larger be divided by the smaller, the quotient is 5 and remainder 16. FRACTIONAL EQUATIONS 193 6. Find where to divide a beam 18 ft. long into two parts such that their quotient is Z^. 7. A board is 42 in. long. A hole is to be bored in the board so that its distance from one end divided by its distance from the other end shall be |. How far from one end must the hole be bored ? 8. A surveyor has two stakes set in the ground 150 ft. apart. He wishes to drive a third stake between them such that its dis- tance to the nearer of the two divided by its distance to the other shall be -J. Where must he set the third stake ? 9. What number must be subtracted from each term of the fraction \^ in order that the result shall be equal to ^ ? 10. What number must be subtracted from each term of the fraction ^ in order that the result shall be equal to f ? 11. What number must be subtracted from each term of the fraction |-J- in order that the result shall be equal to J ? 12. What number must be added to each term of the fraction ^ so that the result shall be equal to |? 13. What number must be added to each term of the fraction •ji^g- so that the result shall be equal to 3^? 14. What number must be subtracted from each of the num- bers, 14, 20, 32, and 60, so that the quotient of the first two re- mainders shall be equal to the quotient of the last two remainders? 15. In a certain factory a large machine can turn out a number of articles in 4 days, and a smaller sized machine can turn them out in 6 days. If both machines are operated at once, how long will it take them to turn out the articles ? Suggestion. — Let n = number of days required for the machines to- gether to turn out the articles. Show that 1 = 1-1-1. n 4 6 16. In a factory which manufactures a certain kind of goods there are 3 large machines and 5 small ones. To manufacture the goods required to fill an order would take either one of the 194 ELEMENTARY ALGEBRA large machines alone 60 days, and either of the small machines alone 90 days. If all 8 machines are operated, how long , does it take to fill the order ? 17. In a city water system, two pumps pump the water into a reservoir. One pump would fill the reservoir in 16 hr., and the other would fill it in 12 hr. If both pumps were driven at once, how long would it require to fill the reservoir? 18. A tank is fitted with two pipes. One pipe alone could fill the tank in 5 hr., and the other pipe alone in 8 hr. If both pipes were opened at once, how long would it require to fill the tank ? 19. A tank can be filled by one pipe in 4 hr. and emptied by another in 5 hr. If both pipes are open, how long will it take to fill the tank ? 20. A railroad water tank can be pumped full in 4^ hr. But locomotives draw out and use, on the average, a tankful every 8 hr. At this rate, how long would it take to get the tank pumped full? 21. Of two furnaces in a school building one will burn a given quantity of coal in 8 days and the other will burn it in 10 days. If both furnaces are fired, how long will the quantity of coal last? 22. A farmer has a quantity of corn which he feeds to his horses and hogs. It would last the horses alone 160 days, and the hogs alone 60 days. How long will it last if he feeds it to both horses and hogs ? 23. Assuming that they work at constant speeds, A can do a piece of work in 7 days that it would take B 9 days to do. If they work together, how long should it require them both to do the work ? 24. A can do a piece of work in 10 days ; but after he has worked 2 days, B comes to help him, and together they finish it in 3 more days. In how many days could B alone have done the work ? 25. John could remove the snow from the sidewalk in 30 minutes. His larger brother James could do it in 20 minutes FRACTIONAL EQUATIONS 195 John began the work, but later James took his place, and the snow was all removed in 25 minutes from the beginning. How long did John work ? ' ' ; ; . - > 26. A steamboat makes 8 miles an hour against the wind on a journey of 40 miles. After it has gone 15 miles, the wind ceases. The entire time consumed on the journey is 4^ hr. Find how many miles per hour the wind retards the boat. 27. A train runs into a city from a suburban town, a distance of 40 mi., in 1 hr. 40 min. Of this distance 8 mi. are within the city limits. The train makes 50 mi. an hour outside of the city limits. What is its speed within the city limits? 28. Two trains, approaching each other, leave stations 225 miles apart at the same moment. One train runs 5 miles an hour faster than the other, and they meet at a point 120 miles from the station from which the faster train starts. Find the speeds of the trains. 29. A motorcyclist was overtaken 75 miles from his starting point by an automobile which started from the same place 2 hours later, and traveled 10 miles an hour faster. Find the speed of each. 30. A messenger was started on a journey with a message that later was found to be wrong. One hour and 20 minutes after he left, a second messenger started to overtake him, traveling 2 miles an hour faster. The first messenger was overtaken after he had gone a distance of 10 miles. Find the speed of each. 31. A man started on a journey of 72 miles. After going 42 miles, he stopped and rested 2 hours. Including the 2 hours that he rested, to finish the journey at the same speed took him as long as to go the first 42 miles. Find how fast he traveled. 32. An aviator flew to a point 60 miles away, and back, in 5 hr. 24 min. His rate going was 25 miles an hour. What was his rate returning ? , 33. An aviator started on a trip to a point 80 miles away. After going 30 miles he increased his speed 20 miles an hour, 196 ELEMENTARY ALGEBRA and made the remaining distance in the same time that it took him to fly the first 30 miles. What was his speed the first 30 miles ? 34. An aviator made 25 miles an hour against the wind on a journey of 45 miles. After he had gone 20 miles, the wind ceased. The entire time required for the trip was 1 hr. 25J min. What was the speed of the wind ? 35. In the National League of baseball teams, at one point of the season, Chicago and New York had records as follows ; Won Lost Chicago New York 64 58 30 32 If these two teams were to play a final series of 6 games together, how many would Chicago have to win in order that the quotient of the number of games won and the number lost should be greater for Chicago than for New York ? Suggestion. — Let x be the number Chicago must win in order that the quotients of the numbers of games won and the numbers lost should be the same for both teams. Show that 64 + x ^58_^6-x, 80+6- aj 32 + ic Show that ic = 1||. Hence, Chicago must win two more games in order to fin^h ahead of New York. 36. The Pittsburg and Philadelphia baseball teams of the National League had records as follows; Won Lost Pittsburg Philadelphia 54 50 34 36 FRACTIONAL EQUATIONS 197 If these two teams were to play a final series of 5 games to- gether, how many would Pittsburg have to win in order that the quotient of the number of games won and the number lost should be greater for Pittsburg than for Philadelphia? 113. Literal Equations: Formulae. — An equation that contains two or more letters, such as the practical formulae encountered in the earlier part of the book, is called a literal equation. In a literal equation the number represented by any one of the letters involved may be considered as the unknown number, and its value solved for. It is clear that in such an equation the value found for the unknown number will, in general, be an ex- pression containing the other letters involved in the equation. If a literal equation is linear with respect to the unknown num- ber, it may be solved by the method of this chapter. Example. — Solve x = ^^ + ^ for n. n Clearing of fractions, 7ix = na + 6. Transposing, nx — na = h. Uniting terms, (x — a) n = 6. 6 Dividing by a; — a, n X- a Note. — The process of uniting the similar terms in a literal equation, as in step 3 of the above example, is equivalent to factoring the terms. EXERCISES 1. Solvea; = ^^^5±^fora. 2. Solve 2a-^=^ for JB. X X 3. Solve a = ^-::^ for a. x — 1 4. Solve 2 + ^-^ = ^- ^-^ for. 198 ELEMENTARY ALGEBRA 6. Solve -^ + ^ = m -\-n for y. m n 7. Solve ^ = ^11^ ioTt. 8. Solve !^^±I--l_ = ??'i:^ forr. n-\-r ir — 'tr n — r 9. Solve i4-- + i = lforP. p q R 10. Solve -i + 1 - ~ + — ^= for d. N d dN 11. Solve ^±^ + , y = / for *. 12. Solve ^-AtiO_4_^+3^^J_f,,^^ 3 ^ 4 ^ 10 13. Solve 2Z+2Z P+TT ^ P+ TT f^, p. 14. Solve ^f + f = -^ 4- 1 for c. c^ — c^^ c + d Note. — The following formulse express important principles in science, geometry, etc. It often is necessary, in using such formulse, to express one of the quantities involved in terms of the others. The pupil will find it of decided advantage to learn that process here. 15. If O and c are the circumferences, and E and r the radii, respectively, of any two circles, then or Solve for r. FRACTIONAL EQUATIONS 199 16. The figure bounded by two radii of a circle and the arc subtended by them is a sector of the circle. If s and a are the area and arc, respectively, of a sector of a circle whose area is S and circumference (7, then £. _ ^ Solve for s ; iov S) f or a ; for C, 17. If one gear or cogwheel drives another, and S and T are the speed and number of teeth of the driving wheel, and s and t the speed and number of teeth of the driven wheel, respectively, then — = — s T Solve for S ; for s ; f or « ; for T. If the number of teeth of the driving wheel is 96, of the driven wheel 12, and the speed of the driving wheel 40 revolutions a minute, find the speed of the driven wheel. 18. The formula for computing the area iv of a lune, the por- tion of the surface of a sphere bounded by two semicircles such as the meridians on the earth's surface, is ST 4 ' Solve for L ; for A. "^ 19. In two similar right cylinders whose volumes, altitudes, and radii are F, H, R, and v, hy r, respectively, Solve for F; for y; for H; for h. 20. If T and t are the areas of the entire surfaces of the above cylinders, t ^ R(H+E) t r(Ji-\-r) Solve for T; fort; for H] for h. 800 ELEMENTARY ALGEBRA 21. If the radius of a right cylinder is denoted by i?, the area of the curved surface by S, the area of the total surface by T, the altitude by -ET, and the volume by V, (1) It = ^; (2) RH^f; (3) U^H^E; (4) R' + HR = ^', (5) R'-\-HR = ?^^ Solve (1) for F; for S. Solve (2) for i?; for H; for >S'. Solve (3) for H'y for F. Solve (4) for H; for T. Solve (5) for T; for /S; for^. 22. In the above cylinder it is known also that — = — - — • Solve for H; for R; for T; for S. MS 23. The formula F= — ^ is used in computing centrifugal r force, or the force with which an object moving in a circle tends to fly off from the center. It is the force which causes grind- stones and flywheels to burst when run at too high a speed, and carriages to overturn when turning a corner at too high a speed. Solve for M) for r. 24. Solve /= — for w : for g : for r. gr 25. The densities of some substances are computed by the cmula, Solve for w\ for w\ formula, ^ 26. The formula (7 = — is of great importance in electricity. R It gives the relation between the quantity of current C of elec- tricity flowing through a wire or other conductor, the resistance offered to the current by the conductor, and the electrical pres- sure (electromotive force) required to overcome that resistance. Solve for ^; ioi R, FRACTIONAL EQUATIONS 201 27. If the resistance within an electric battery cell is R, the resistance of the external circuit through which the current flows r, the strength of current G, and the electric pressure E, then E + r' Solve for E ; for E ; for r. 28. If n equal electric cells are connected by wires in one way (series), so as to form a battery, the current strength is found by the formula nE R-{-nr Solve for n ; for E ; for R ; for r. 29. If the cells in Problem 28 are connected in another way (parallel), the current strength of the battery is found by the formula C: E rV- n Solve for n ; for E) for R\ for r. 30. If an. electric circuit between two jjoints is divided into two branches, so that part of the current flows through one branch and the rest through the other, the total resistance of the circuit is computed by the formula R Vi r^ where R is the total resistance, ?'i the resistance of one branch, and Vz the resistance of the other branch. Solve for R. 31. If the circuit in Problem 30 is divided into three branches whose resistances are ri, o^, and r^, the formula is R ri r, ra Solve for R, 202 ELEMENTARY ALGEBRA 32. Solve /S = ^^=^^ for r; for a; f or ^. 33. The radius r of the arch of a bridge or doorway whose span is w and rise h is found from the formula Solve for r. 34. Find the radius at which the stones must be cut for an arch whose span is 16 ft. and rise 4 ft. Solve SUPPLEMENTARY EXERCISES 1. „-.^-^+ ^ ^ 2n^-n-l 2w2 4-3n + l w^-l 2. 1 ^-2 ^+2 ^-3 7^ + 2 w + 2 ^ 2 6/2-1 3r-5 . o r2-9r + 14 r-7 18^-27 11^-1 ^ 9^ + 11 . 14 3^ + 1 7 Suggestion. — In equations like this, in which some denominators are monomials and others polynomials, it often is best to clear of monomial , denominators only at first, and then simplify before clearing of the remaining denominators. Thus, multiplying both members by 14, 18 J - 27 + IMliril = 18 « + 22. This becomes IhU-U ^ ^g^ U-1 « + l FRACTIONAL EQUATIONS 203 4P4-3 7P-29 ^ 8P4-19 9 5P-12 18 5Jc + 5 ^ nic-15 33A; + 15 . * k-5 10 30 8 ±\ -^ + ^ 9iVr+3 ^ 3i^+6 * 2(3iV^-4)"^ 15 5 g a? — '^ a; — 9 _ ar — 13 x — 15 ' x — 9 ic— ll~a; — 15 ic — 17* Suggestion. — In equations of this kind much multiplication may be avoided by combining some of the fractions before clearing the equation of fractions. Thus, performing the subtractions indicated in each member, a;2 - 20 a; + 99 x^ - 32 a; + 266 10. y-5 ^ y-7^y-4 ^ y- 8 y — 7 y — 9 y-6 y — 10 Suggestion. — First transpose iu order to form differences such that the fractions when combined will give simple numerators. Thus, y-5 y— 4 __ y- 8 y — 7 y_7 y_6 y-10 y-9* 11 J3 B-tl ^ B-S .8-9 . • B-2 B-1 B-6 B-f 1111 r 12. V— 3 v—1 v— 4 V— 2 ,o w — 3 n — 4 n — 1 , n — 2 18. - = --\-- • «— 4 w— 5 n—2 6—n 14 <»-M a + 6 ^ a4-5 a-f-2 ^ a + 2 a + 7 a + 6 a-j-S' 204 ELEMENTARY ALGEBRA jg^ P-1 . P-7 P-n P-3 P-2 P-8 P-6 P-4 16 ^^ A;-5_A;-10 >[;-4 ^5"^^9 ~ Xf4 "^^+10* 17 2aH-5 2a + 2 ^ 2a4-l 2aH-6 2a + 6 2a-i-3 2aH-2 2a + 7* 18. ?:i^=_L_+ri:i?. r-2 r2-r-2^r + l 19 4y + l _ 4y-l ^ 8 42/-1 4y + l 16/-1' _z 3-2g ^ 3(2 g~ 7) ^ 3-« 3 + ;3 6(3 + »)' 21. i^ = -^+ 3 s — 2 S4-2 s + 1 22. Solve ^^±^~ ^^-^ =^' + <^ + ^)fora; 5a;H-c 6a; + 2c 6(6a; + 3c) 28. Solve - — ^^^ + £ll? = for I. V + ct — at — ao t — e 24. Solve -f — ^ = — ^ forn. a(p — n) b(c — n) a(c — n) «.. solve f-^(|-|)=^(„-?M)fo.. FRACTIONAL EQUATIONS 205 26. Solve i = i+i+i + i fori?. B Tx n n n 27. Solve E = ^ — j-r- for <, ^-m 28. Solve ^^V^ + 1= , . ^ , forr. m — w 29. In a guessing game, the leader says : " If you will add 20 years to your age, divide the sum by your age, add 3 to the quo- tient, and tell me the result, I will tell you your age." How did he find it ? Suggestion. — Let x = the age and a = the result given. Write an equa- tion and solve it for x. 30. Make up a guessing game similar to that in Problem 29, and show how the age is found. 31. If a body whose weight is W pounds, moving with a ve- locity of Ffeet per second, strike a second body whose weight is w pounds and which is at rest, so that the two bodies move on together, the velocity v with which they continue together is found from the formula W+w Solve for W; for F; for w, 32. If a freight car, weighing 40,000 lb. and moving 8 ft. per second, strike a second car that weighs 32,000 lb. and is standing still, and the two are then coupled together, with what speed will they continue? CHAPTER XI PROPORTION. VARIABLES 114. Ratio. — The quotient of one number divided by another of the same kind is sometimes called their ratio. It usually is written in the form of a fraction, and is subject to all of the rules that apply to fractions. Thus, the ratio of .$ 6 to $ 100 is written -^^. The ratio of 2 lb. to 5 lb. 2 1b ^^^^ is written '-, And, in general, the ratio of a to 5 is written -. 5 lb. b Note. — An old form of writing the ratio of a to & is a : &. This notation is less convenient for computation than the fractional form, and is used less now than formerly. The dividend or numerator of a ratio is sometimes called the antecedent, and the divisor or denominator is called the consequent. EXERCISES 1. Express the ratio of 3 qt. to 6 qt. as a fraction in its lowest terms. 2. If the rate of interest on a sum of money is 5 %, that is, the ratio of $5 to $100, express the rate as a fraction in its lowest terms. 3. The death rate in Boston in a recent year was 18 to 1000 population. Express this ratio as a fraction. 4. An alloy consists of copper and tin in the ratio of 2 to 3. What part of it is each ? 5. A solution consists of alcohol and water in the ratio of 3 to 5. What part of it is water ? 206 PROPORTION. VARIABLES 207 6. In a city of 84,000 population the number of births in a year is 560. What is the birth rate ? How many per 1000 ? 7. When a man 5 ft. 10 in. tall casts a shadow 35 ft. long, find the ratio of his height to the length of his shadow. 8. The specific gravity of a solid or liquid is the ratio of the weight of any volume of it to the weight of an equal volume of water. A cubic foot of water weighs 62.5 lb. A cubic foot of steel weighs 490 lb. Find the specific gravity of steel. Express the answer first as a common fraction, then as a decimal computed to tenths. 9. Find the specific gravity, computed decimally to tenths, of each of the following substances : Substance Weight in PorNDS OF 1 Cr. Ft, SlBSTANCE Weight in Pounds OF 1 Cu. Ft. Cast Iron Brass Gold 450.0 523.8 1200.9 Oak Ash Cork 65.0 52.8 15.0 10. The diameter of a circular plate is 6 in., and the circum- ference is measured with a tape line and found to be 18.85 in. Find the ratio of the circumference to the diameter. 11. A boy measures the wheel of his bicycle, and finds that its diameter is 28 in. and circumference 88 in. What is the ratio of the circumference to the diameter ? 12. Simplify the ratio of x-\-y to 3?^y^ by writing it as a fraction reduced to its lowest terms. 13. Simplify the ratio of - to — by writing it as a simple fraction. ^ 14. Which ratio is the greater, -^ or |f ? Suggestion. — Reduce thera to a common denominator, and compare the numerators. 15. Write in descending order of magnitude |, |f, |^. 208 ELEMENTARY ALGEBRA 16. Two partners in business, A and B, divide a profit of $ 2135 between them so that A's part and B's part are in the ratio of 3 to 4. How many dollars does each receive ? Suggestion. — Let d be the amount that A receives. Then, 2135 — (Z is the amount that B receives. Hence, ^ = -. ' 2135 -(^ 4 17. If two partners, A and B, divide a profit of $1200 in the ratio of 1 to 2, how many dollars does each receive ? 18. In a certain city with a population of 247,520, the ratio of the Germans to all other nationalities together is 2 to 15. What is the German population? 19. Separate 40 into two parts which are in the ratio of 3 to 7. 20. Separate 85 into two parts which are in the ratio of 5 to 12. 21. The sides of a triangle are 5 in., 9 in., and 7 in. Divide the side 7 in. long into two parts whose ratio equals the ratio of the other two sides. 115. Proportion. — An equation whose members consist of two ratios is called a proportion. Thus, f = |, ^ = Jl^, and f = ^ are proportions. 486oloo d The proportion - = - , is read either " a over b equals c over d," Ct/ or " a is to 6 as c is to c?." The numbers forming one of the ratios are said to be "propor- tional to" the numbers forming the other ratio; or the four num- bers forming the ratios are said to be "in proportion." • Ob C In any proportion - = - , a, 6, c, and d are called the terms. The first and fourth terms, a and d, are called the extremes ; and the second and third terms, h and c, are called the means. Since a proportion is an equation, all of the operations which PROPORTION. VARIABLES 209 may be performed upon an equation, such as clearing of fractions, etc., may be performed upon a proportion. Note. — Since the ratio of any two concrete numbers is the same as that of the corresponding abstract numbers, in performing any operation upon a proportion, confusion ma^ be avoided by using only abstract values for the terms. EXERCISES Find the value of the unknown term in each of the following proportions : , n 15 . 7 42 36_1 1- x = Tr' 4. - = 7. 2 10 9 t V 4' a; 1 3 _^ 9 _12 8. * 12 8* * 10 40* 16~P « 4 16 24 r 20 8 3. — = — -• 9. — . — — • y 21 • 35 70* 36 L Find the values of n in the proportions : 10. 1 = '-. 12. i=». 14. l-L. n V W r n 11.2 = 2. 13. »-« = « + ^. 15. an _h n z a h h ~c' 16. A family of three members and a family of four members camp out together. The total cost of provisions is $ 112, which they wish to divide in proportion to the sizes of the families. How much must each family pay ? 116. Definitions. — In any proportion - = -, the fourth term d b d is called a fourth proportional to the other three terms, a, b, and c. A proportion such as - = - , in which the means are equal, is called a mean proportion. The mean term x is called the mean proportional between the extremes a and b. In a mean proportion such as - = -,6 is called the third pro» portional to a and x. ^ o 210 ELEMENTARY ALGEBRA EXERCISES Find the fourth proportional to ; 1. 2, 3, and 10. Suggestion. —Let x be the fourth proportional. Then - = — . 3 X 2. 4, 5, and 12. 5. 1, a, and a?, 3. 7, 10, and 28. 6. m, n, and mH, 4. 16, 12, and 4. 7. 1. 1 - t, and 1 + ^. Find the third proportional to : 8. 3 and 6. Suggestion. — Let x be the third proportional. Then - = -. 6 X 9. 8 and 12. 12. a and h, 10. ajand3^. 13. 1 + ^ and 1 — w^. 11. Pandit. 14. 10 and 1. Find the mean proportional between : 15. 2 and 8. Solution. — Let x be the mean proportional. Then = = 5. Clearing of fractions, ic^ = 16. Taking square root, a; = i 4. 16. 3 and 12. 19. n and n\ 17. 7 and 28. 20. 2 A"^ and 50 A 18. 9 and 1. 21. i^ and i2 + 2 i?^ + R\ 117. Distances found by Proportion. — Two triangles that are drawn with the angles of one equal to the corresponding angles of the other have the same shape, and are called similar triangles, as ABC and DEF, ^\. X \^ Let the student draw two A-^^ ^B D^^ ^F such triangles, making the PROPORTION. VARIABLES 211 side DE of one three times as long as the corresponding side AB of the other. Measure the other sides, and compare them. It will be found that: In two similar triangles, the corresponding sides are proportional. AB AC Thus, in the above figure, — = — - , etc. This principle is much used in computing unknown distances. EXERCISES 1. In two similar triangles, the sides of the larger are 12 in., 14 in., and 20 in., and the shortest side of the smaller is 3 in. Find the other sides of the smaller triangle. 2. The sides of a triangular field are 16 rd., 24 rd., and 30 rd. The smallest side of a similar field is 40 rd. Find the other two sides. 3. A tree casts a shadow 48 ft. long when a vertical rod 6 ft. high casts a shadow 4 ft. long. How high is the tree ? /I / ^^j^ 4. A church spire casts a shadow 23 _/^J-^-'^ ft. long when a man 5 ft. 10 in. tall -j^-- - who is passing by casts a shadow 2 ft. long. Find the height of the spire. 5. The distance from ^ to ^ on opposite sides of a lake may ^^^^^ be found as follows : A 's^ j^^^^^p ^B The distances from A to R and from B ^v^^,.^^^ to T are measured off, making the triangles ^.^.^"^X^^ ASB and TSR similar. If AS is taken '^^=^— -^R 400 yd., and SR 300 yd., and TR is meas- ured and found to be 580 yd., how far is it from ^ to J5 ? 6. If A and B are two points on opposite sides of a hill, and out in the plain at the foot of the hill distances are measured as in Problem 5, so that AS is taken 4050 ft. and SR 1350 ft., and TR is found to be 1600 ft., how far is it between A and B ? 212 ELEMENTARY ALGEBRA 7. To find the distance AB across a stream, measure off a dis- tance -4(7 several yards long, along the bank. Then at C measure off a distance, (7Z> at right angles to AC. By sighting across from D to J5, locate a stake at a point E of AC in line with D and B. Get the lengths of AE and EC. Since the triangles are similar, write the proportion by means of which AB may be computed. If AE is 80 yd., EC 20 yd., and DC 15 yd., what is the width of the stream ? 8. It is known in geometry that a line parallel to one side of a triangle divides the other two sides into four proportional parts. That is, if DE is parallel to AB,— = — . ^ ^ ' DA EB ^ (1) li AD = 6 in., DC = 3 in., and BE y^\ = 4 in., find EC. PX^ \ e (2) If AD = 10 in., BE =12 in., and EC ^^ X p = 20 in., find DC (3) If DC = 9 in., BE =T in., and EC = 5 in., find AD. 9. In the figure of Problem 8, if AD = S in., DC =5 in., and BC=26 in., find BE and EC. Suggestion. Hence, Let BE 8. 6 X. Then EG = 26- x. X 26- a; 10. In the figure of Problem 8, if AC = 4.0 ft., J5^^=12 ft., and EC =15 ft., find AD and DC. 11. It is known in geometry that in any triangle ABC, if the line CD divides the angle at C into two equal parts, it divides Q the opposite side into parts proportional to the AD other two sides ; that is, = AC DB BC' If AC= 15 in., BC = 10 in., and AB = 12 in., find AD and DB. 12. In the figure of Problem 11, if AC = AO yd., BG= 32 yd., and AB = 60 yd., find AD and DB. B PROPORTION. VARIABLES 213 13. In the triangle ABC, the angle at is a right angle, and CD is perpendicular to AB. It is known ill geometry that CD is a mean propor- tional between AD and DB. If AD =20 CD. in., and DB = 5 in., find 14. A method used several hundred years ago of finding the dis- tance from A to the inaccessible point B was to erect a vertical staff ACj place upon it an instrument resembling a carpenter's square, pointing one blade towards B, and note the place D on the ground to which the other blade pointed. ACsnid DA were meas- ured. If AC =52 in., and DA=6 in., find AB. 15. In the semicircle with the diameter AB, CD, a perpendic- ular to AB, is a mean proportional between AD and DB. If AD = 12 in., and DB = 3 in., find CD. ^ X If ^D = 8 in., and CD = 6 in., find DB. If AB = 20 in., and CZ>=8 in., find AD and DB. 118. Proportion in Simple Machines. — Simple machines, such as the lever, the wheel and axle, etc., are instruments by means of which a small effort exerted may be made to overcome a resistance of much greater size. Proportion is involved in the use of simple machines. EXERCISES 1. A lever is a stiff bar of wood or metal that is movable about a fixed point or pivot called the fulcrum. The resistance W to be overcome is applied at one end of the bar, and the effort w, exerted to overcome it, at the other end. If D is the dis- tance of W from the fulcrum, and d the distance of vo from the fulcrum, then 214 ELEMENTARY ALGEBRA What effort is required to lift a weight of 600 lb. by a lever, if the weight is 8 in. from the fulcrum and the effort applied 40 in. from the fulcrum ? 2. What effort is required to lift a stone weighing 840 lb. by means of a crowbar 60 in. long, if the fulcrum is placed 4 in. from the stone ? 3. A blacksmith weighing 180 lb. lifts wagons, etc., by means of a wagon jack. The weight lifted is applied at a point 6 in. from the pivot or fulcrum, and he grasps the handle 32 in. from the pivot. By throwing his whole weight upon the jack, how much can he lift ? 4. The entire length of a lever is 48 in. Where must the fulcrum be placed in order that a resistance of 360 lb. may be overcome by an effort of 120 lb. ? 5. The wheel and axle consists of a wheel and a cylindrical axle passing through its center, the two being fastened rigidly together, so that the axle is turned by revolving the wheel. The weight or resistance to be over- come is applied at tlie circumference of the axle through a cord wrapped around the axle, and the effort required to overcome it is applied at the circumference of the wheel. If the resist- ance is W, the effort required to overcome it w, the radius of the axle B, and the radius of the wheel r, then — = — . W r If the radius of the wheel is 20 in., the radius of the axle 6 in., and the resistance to be overcome 500 lb., what is the effort required ? 6. The axle of a windlass shown in the above figure, which is used for drawing water from a well, has a radius of 4i in., and the radius of the wheel 15 in. What effort is required to lift a bucket of water weighing 54 lb. ? PROPORTION. VARIABLES 216 7. A capstan, a form of wheel and axle used in lifting anchors on ships, has an axle 12 in. in diameter and a lever arm or spoke 45 in. long. How much effort is required to lift by it an anchor weighing a ton ? 8. The inclined plane is a smooth sloping surface that is used in raising a heavy object through the application of a compara- tively small effort, the object being raised by sliding or rolling it along the surface of the plane. If the effort is applied parallel to the surface of the plane, the effort applied is to the weight lifted as the height of the plane is to its length ; that is, -^= — , where w = the effort applied, W = the weight, H = the height of the plane, or distance the weight is lifted, and L = the length of the plane. If the length of an inclined plane is 12 ft., the height 3 ft., and the weight of the object lifted 400 lb., what effort is required to move it up the plane ? 9. A barrel of material weighing 180 lb. is loaded into a wagon bed 32 in. from the ground by rolling it up a board 10 ft. long with one end resting on the ground and the other on the wagon bed. What effort is required ? 10. Two men place a 580 lb. building stone upon a wagon by sliding it up a board 8 ft. long, one end of the board being on the ground and the other resting on the bed of the wagon. The wagon bed is 20 in. from the ground. What effort do they use in loading the stone in addition to that required to overcome the friction ? 11. Ice is stored in an ice house by dragging it up an inclined plane. The incline is 200 ft. long and 40 ft. high. What effort is required to pull up a block of ice weighing 250 lb. ? 12. A locomotive pulls a train weighing 720 tons up a grade with a rise of 2^ ft. to 100 ft. of grade. How much more pull must the locomotive exert than if the train were on a level road bed ? 216 ELEMENTARY ALGEBRA 119. Important Principles in Proportion. — The following im portant principles in proportion are often used in geometry, and elsewhere, where proportion is applied. Let the pupil establish each of them. (1) If four numbers are in proportion^ the product of the extremes equMs the product of the means. That is, if ^ = £ then ad = be. Suggestion. — Clear - = - of fractions. b d (2) If the product of two numbers equals the product of two other numbers J the four numbers are in proportion. That is, if ad = be, then - = - • ' b d Suggestion. — Divide both members of ad = he by hd. (3) If four numbers are in proportion, they are in proportion by inversion. That is, if 2 = «, then5 = ^. b d a G Suggestion. — Divide 1 = 1 by the members of - = - . b d (4) In any proportion^ the means may be interchanged, or the extremes interchanged, without destroying the propmi,ion. That is, if » = £, then« = * and^ = ^. b d c d b a Suggestion. — Multiply both terms of - = - by - ; by - . b d c a (5) Tlie terms of any proportion are in proportion by addition. That is, if^ = ^, then^±^ = £±^. b d' b d Suggestion. — Add 1 to each member of - = - • b a PROPORTION. VARIABLES 217 (6) The terms of any proportion are in proportion by subtraction. That is, b d' b d b d d c Suggestion. — Subtract 1 from each member of (7) Like powers of the terms of a proportion are in proportion. That is, if « = ^, then^ = |, ±' = 4 etc. Suggestion. — Raise both members of - = - to the same power. h d (8) If two or more ratios are equal, the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent. That is, •^ a c e . .■, aH-c + e4- etc. a . if - = - = — = etc., then ^ ^ ^ = -= etc. b d f b-\-d+f+ etc. b For, since - = - = - = etc., let each ratio equal r. b d f Then a = rb, c = rd, e = rf, etc. Hence, a + c-h e + etc. = rb ■}- rd -\- rf + etc. = r(6 + pressing the relation between the variable time t and the dis- tance d. 2. If the price of coffee is 30 / a pound, the cost of any quan- tity varies as the number of pounds bought. If c = cost and w = weight in pounds, express the relation between them by an equation. 3. If the price of sugar is 5 / a pound, express by an equation the relation between the number of pounds bought and the cost. 4. If the rate of interest is 6 %, the amount of interest on any sum of money varies as the time. Express by an equation the relation between the interest and the time. 5. The velocity of a body let fall toward the ground varies as the time during which it has fallen from rest, and the velocity at the end of 3 sec. is 96 ft. per second. As the time increases, the velocity increases at the same rate. Write the equation between £he velocity and time. 6. The force with which a moving body of any given velocity strikes a stationary body varies as the mass or weight of the moving body. If I strike a nail with a force of 15 lb. by using a hammer weighing i lb., with what force would a 2 lb. hammer strike it when swinging with the same speed ? 7. The resistance offered by a wire of given size to a current of electricity varies directly as the length of the wire. If a wire 50 ft. long gives a resistance of 6 ohms, what will be the resist- ance of 75ft.? 8. If R is the resistance in Problem 7 and L the length of wire, write the equation between them. 9. If X varies directly as y, and x=2 when y = 9, find x when y = 36. Write the equation between x and y. 10. If P varies directly as V, and P= 7 when V= 5, write the equation between P and V. Find V when P = 35. PROPORTION. VARIABLES 221 122. Variation shown by Graphs. — In dealing with variable quantities, their variation often is depicted to the eye by means of a chart or graph. See § 29. For example, the public debt of the United States, in millions of dollars, from 1820 to 1900 is given in the following table : Year 1820 1830 1840 1850 I860 1866 1870 1880 1890 1900 Debt 91 49 4 63 65 2773 2480 2120 1652 2136 On squared paper draw a horizontal line OX and a line OF at right angles to it. These are called the axes. The point is called the origin. 2500 2000 1500 1000 500 ^^ 1820 1830 1840 1850 I860 1870 1880 1890 1900 On the line OX let a distance of one space represent 2 years of time. Then the dates in the above table will be represented on OX as in the chart. On OF let a distance of one space represent 100 million dollars. On the 1820 vertical line measure off a distance representing 91 million dollars, and mark a point. On the 1830 vertical line measure off a distance representing 49 million dollars, and mark a point. Similarly, locate a point on each of the other date lines corresponding to the amount of the debt of that date. Draw a line connecting all of the points obtained. This curved line, called the graph of the national debt, depicts clearly to 22£ ELEMENTARY ALGEBRA the eye the variation of the national debt throughout the period of time considered. Note. — Various kinds of apparatus are made for constructing graphs automatically. The thermograph or recording thermometer is a good illus- tration. A pen connected to a thermometer moves up and down as the temperature rises and falls. At the same time a clock mechanism runs a strip of ruled paper under the pen, so that the pen traces a continuous curve or graph on the paper. Among other instruments for making graphs is the seismograph^ used for recording earthquakes. The pupil should be provided with several sheets of squared paper for use in the exercises of this and subsequent chapters. EXERCISES 1. The number of teachers in the public schools of Chicago since 1850 has been as follows : Year 1850 1860 1870 1880 1890 1900 1909 Teachers 21 123 557 898 2711 5806 6296 Draw a graph showing the growth in the size of the teaching force during that time. Suggestion. — On squared paper draw axes as in the example above. On one represent the time, letting one space represent 2 years. On the other represent the number of teachers, letting one space represent 200 teachers. 2. The following average heights of children at different ages have been determined from measurements of thousands of indi- viduals : Age JYr. 1 Yb. 2 Yr. 3 Ye. 4 Ye. 5 Yb. 6 Ye. 7 Yr. 8 Yr. Height 24 in. 29 in. 36 m. 40in. 42 in. 44 in. 46 in. 48 in. 50 in. Draw a graph showing the rate of growth of the average indi- vidual. PROPORTION. VARIABLES 223 3. The population, to the nearest million, of the United States since 1800 is given by decades as follows : 1800, 5 1840, 17 1880, 50 1810, 7 1850, 23 1890, 63 1820, 10 1860, 31 1900, 76 1830, 13 1870, 39 1910, 93 Draw a graph showing the growth of the population. 4. The population, to the nearest thousand, of the city oi Chicago siuce 18.40 is recorded as follows : 1840, 4 1860, 109 1880, 503 1900, 1699 1850, 28 1870, 299 1890, 1100 1910, 2185 Draw a graph showing the growth of the city. 5. The following table gives, in billions of dollars, the total value of the farm property and products of the United States by decades from 1850 to 1900: Year 1850 I860 1870 1880 1890 1900 Value 4 8 11 12 ' 16 21 Show by a graph the growth of agriculture in the United States. 6. The amount of the annual exports of the United States, in millions of dollars, has been as follows : 1790, 20 1820, 70 1850, 152 1880, 853 1800, 71 1830, 74 1860, 400 1890, 910 1810, 67 1840, 132 1870, 451 1900, 1499 Show the growth of our exports by a graph. 7. The temperature at a place was recorded as follows : 6 a.m. . 8°; 7 A.M., 8°; 8 a.m., 10°; 9 a.m., 11°; 10 a.m., 14°; 11 a.m., 18°; 12 m., 20°; 1 p.m., 20°; 2 p.m., 20°; 3 p.m., 18°; 4 p.m., 16°; 5 p.m., 15° ; 6 p.m., 13°. Draw a graph to show the variation of tempera^ ture during the 12 hours. 224 ELEMENTARY ALGEBRA 123. Price Curves, etc. — Graphs may be constructed and used for determining costs of different quantities of goods, interest on money for different periods of time, etc., without computation. This is shown by an example. If eggs sell at 30 ^ a dozen, the relation between the number of dozens and the cost may be expressed by the equation c = 30c?, where d is the number of dozens and c is their cost. If values are given to d, corresponding values may be found for c, as given in the table : d 2 4 6 8 10 12 G 60 120 180 240 300 360 On squared paper draw two axes, OX and Y, at right angles, as in § 122. ,-"' - -u - .» ' ' ' *» ' t ^ "^ ' ■ e ^^^' , "" ^ ■ I* 55"- -j^ \kd aid p n < Cf _ _^ _ - - ^ - ' r - _ — — — — ■ ^ »* . > .F E A ; _ _ _ _ L U L radius and the area is found to be the curve in the figure. The approximate value of the area of a circle with any given radius or of the radius of a circle with any given area, may be deter- mined at sight by this graph. Give at sight the area of a circle whose radius is 1| in. ; 21 in. Give at sight the radius of a circle whose area is 10 sq. in. Note. — Observe that the graph in Problem 16 is not a straight line. Many- problems lead to graphs that are not straight lines. Some of these will be encountered in more advanced work. PROPORTION. VARIABLES 227 124. Graphs of Linear Equations. — As shown in the preceding sections, the relation between two quantities may be expressed by an equation, also the relation may be shown by a graph. The graph is often spoken of as the graph of the equation. The two literal quantities in an equation may, in some cases, represent negative as well as positive values. The process of drawing the graph of an equation between two quantities which have both positive and negative values will be given in this section. The construction of such a graph is given below. X 2 4 5 7 10 1 -2 -6 -7 y 4 6 10 16 -2 -4 -8 -14 -18 Let the equation be 2ic = y + 4. By assigning values to y, compute the corresponding values of x and tabulate the results as shown in the table. Thus, when y = 0, the equation becomes 2 x = 4, from which a; = 2. When y = Qt 2 a; = 10, from which x = 5. When y=— 8, 2aj = — 4, from which a; = — 2, etc. Two axes, XX^ and YY', are drawn at right angles and meeting at 0. Cor- responding to each set of values of x and y, a point is located, as in § 122 and § 123, the values of x being measured along or parallel to XX, and the values of y along or parallel to YY'. Positive values of x are measured to the right of YY' and negative values to the left, as in § 28. Positive values of y are measured above XX and negative values below XX. For example, the point A corre- sponding to X = 4 and ?/ = 4 is obtained by measuring 4 spaces to the right and 4 spaces upward. The point B correspond- ing to X = 1 and y = — 2 is obtained by measuring 1 space to the right and 2 spaces downward. The point C corre- sponding to X = — 2 and y = — 8 is obtained by measuring 2 spaces to the :a^ J ?5 228 ELEMENTARY ALGEBRA left and 8 spaces downward. The point D corresponding to x = 2 and y = is obtained by measuring 2 spaces to the right on XX' ; etc. By draw- ing a line through all of the points located the graph of the equation is obtained. It is seen that the graph of the above equation is a straight line. It is shown in more advanced mathematics, and may be assumed here, that TJie graph of every linear equation containiyig two variables is a straight line. Note. — It is seen that in locating a point corresponding to a set of values of the two variables in an equation, the signs of the numbers serve to tell the directions of the point from the two axes. This is analogous to the method of locating a point on the earth's surface by knowing its longitude and latitude. K the latitude of a place is minus, the place is south of the equator, etc. EXERCISES Draw two axes and locate the following points : 1. a; = 4, 2/ = 6. 7. a? = — 6, 2/ = 5. 2. a; = 10,2/ = 7. 8. a? = - 10, i/ = 12. 3. a? = 15, 2/ = 2. 9. j» = 9, 2/ = — 4. 4. a; = 0, 2/ = 8. 10. a; = 15, 2/ = — 5. 5. a; = 12, 2/=0. 11. aj = -6, 2/ = -6. Q. x= 0,2/ = 0. 12. a; = -10, 2/ = -12. 13. By assigning eight different values to y in the equation a; — 2 2/ = 7, including at least two negative values, compute the eight corresponding values of x. Draw axes and locate the eight points corresponding to these sets of values of x and y. Are all of these points in one straight line ? Draw a line through all of the points. By assigning eight values to y in each of the following equa- tions, including at least three negative values, and computing the corresponding values of x, locate eight points of the graph of the equation, and draw the graph. PROPORTION, VARIABLES 229 14. x = y. 16. a; + 4 2/ = 10. 18. 5 a; — 4?/ = 20. 15. a; + 2^ = 0. 17. ^x-y = 12. 19. 2x + Zy = 12, 20. Since the graph of a linear equation in two variables is a straight line, how many points of it must be located in order to draw it accurately ? By locating only two points of each, draw the graphs of all of the following equations on one pair of axes : 21. Sx — y = 12. 23. 2 a; — 4 2/ = 5. 25. 5 a; — y = 4. 22. y=2x. 24. 2x-^^y = l, 26. ^x = ^y. 27. If the cost of setting the type for printing a circular is 75^, and the cost of paper and press work in printing it is ^^ a copy, then c = |w + 75, where c denotes the cost in cents of printing any number of copies and n denotes the number of copies printed. Draw the graph of this equation. Give at sight from the graph the cost of 12 copies ; 20 copies ; 100 copies. SUPPLEMENTARY EXERCISES 1. Express in simplest form the ratio ofm tol • 2. Separate 72 into three parts which are in the ratio of 2:3:4. Q I Q 3. If n is a positive number, which is the greater ratio, - — - — o+4n or|±ii;? 3 + 5w Suggestion. — Reduce to a common denominator, then compare the nu- merators. 4. If X and y are positive numbers, which is the greater ratio, a? + 5.v ^j. x±7_y^ x + 6y x-{-Sy' 230 ELEMENTARY ALGEBRA 5. Find the mean proportional between 4 a^ and y a{a ~ by. 6. Find the fourth proportional to v^ — w'\ , and v — w. 7. Write in three ways the proportion between 3, 4, 20, and 15. 8. Write as a proportion oc^ — y^ = 5xQ. 9. If £ = !-=«, show that 4±|5^±|^ = £. a' b' c' a'+3b' + 5c' a' 10. If 2 = £, show that ^jz^^e^jz^. 11. If 2 = ^, show that -5^= i±^ h d a-{-c a -\- b -\- c -\- d 12. In the similar polygons ABODE and A'B'C'D*E' it is known that AB BC CD DE EA A'B' B'C CD' D'E' E'A' Show that the ratio of the perimeters of the polygons equals the ratio of any two corresponding sides. 13. X varies as y, and when x = 2, y = 5. Find y when x — 15. 14. The distance that a body falls from rest varies as the square of the time. In 2 seconds it falls 64 ft. How far will it fall in 3 sec. ? In 4 sec. ? In 10 sec. ? 15. One quantity is said to vary inversely as another when, during all of their changes, their product remains constant. As one increases, the other decreases. If x varies inversely as y, then xy = kj where fc is a constant. PROPORTION. VARIABLES 231 If X varies inversely as ?/, and y=^2 when a; = 4, find y when a; = 32. 16. The volume of any gas varies inversely as the pressure upon it. When the pressure is 8 lb. the volume is 8 cu. in. What is the volume when the pressure is 4 lb.? 17. The number of vibrations made by the pendulum of a clock in a given time varies inversely as the square root of its length. A pendulum 39.1 inches long makes one vibration in a second. How ^ong must a pendulum be to make 4 vibrations in a second^ CHAPTER XII SYSTEMS OF LINEAR EQUATIONS 125. Systems of Equations. — In Chapter VI it was shown that some problems may be expressed by means of two linear equa- tions containing two unknown numbers. These equations were called simultaneous, and were said to form a system. It was found that the equations in each system in that chapter had one set of values of the unknown numbers that satisfied both equations, called a solution of the system. This solution was discovered through a process called elimination. There are three principal methods of elimination in common use that will now be discussed, and applied in the solution of problems. 126. Elimination by Addition or Subtraction. — The method of elimination shown in § 68 is known as the method of elimination by addition or subtraction. The student should now study again the rule on page 110. The following exercises are given for review of that process. TJie student should check every answer by seeing if the values of the unknown numbers found satisfy both of the given equations. EXERCISES Eliminate by addition or subtraction, solve, and check : 7 s + ^ = 42, r3a-6 = 21, {ls + ' l2a + & = 4. , • \Ss- 2 rm + 4?i=4, 5 H \m-2n = 16. ' [o \3A- t = S. w -{-v = 25, w-2v = S5. 2x-y = 5, f4^-3J5 = l, [5x-2y=U. ' [3A-4.B = 6. 232 'SYSTEMS OF LINEAR EQUATIONS 233 7. 9. 10. 11. 12. 13. 14, [2x = y-6, (Sp + 2q=12, [4.p = 3q-l, 5R-2r = ly SE = 5r-ll, r3Jf-2^44 = 0,- [2M-N-{-l = 0. (2h-}-k = 35y [5h-3k=2r. r 5 2/ — 5 a; = 15, [Sx + 5y = 71, J3ri + 2r2 = 4, l4r2-3ri + l = 0. -4jB + 3(7 = 46, 2(7 + 6jB = 4. 15, 16, 17. 18. 19. 20 f m = 4 — 2 w, [2n-\-12 = m. 8 W-h w = 7, W+2w = 28. fl0/4-4«=22, Uh-5/=11. r 2 a^ — ajg = 9, 1 5 a^i — 3 ajg = 14. "-4-^-9 7 2^ 6 4 ' .4^6 ^ Example. — Solve Eliminate x. Solving (1) forx, j4x + y = 34, \ 4 J/ + a; = 16. 127. Elimination by Comparison. — The method of elimination by comparison is illustrated in the following example. (1) (2) (8) (4) (6) 4 Solving (2) for x, a; = 16 - 4 2^. Comparing the two values of x given in (3) and (4), 4 Solving (6) for y, 1/ = 2. Replacing y in (2) by its value 2, 8 + X = 16. Solving, X = 8. Hence the solution is x = S, y = 2. This example illustrates the rule : Solve each of the eqtiations for the value of either one of the un- knoim numhersj expressed in terms of the other ^ and write one of tlie 234 ELEMENTARY ALGEBRA values equal to the other. Solve the resulting equation. Substitute the value found for the one unknown number in either of the origiual equations, and solve for the other unknown number. EXERCISES Eliminate by comparison, solve, and check : \m--7i= — 1. 2 ra;-5a = 7. [2 a; -15 a =9. r + 2 s = 4, 3 r - s = 5. -I l5ri-3r2 = : 14. 3^ + 25 = 2, I f3i>- (v + 2t \2t~v ^-^=9. g = l, 5g = 41. = 4, + 12 = 0. 9 Jf - 5 N= 13, 5Jf+J\^=ll. 3e-5/=5, 7e+/=265. 10 pi^-^=5> 17? + 2T=25. riO F+3n=:174, ^^- 1 3 F+10n = 125. r5a; + 2aj' = l, |l3a; + 8a;' = ll. 13. 2h-{-k = 9, 5h + Sk = 25. 14. ^ rjB + 2Q + 2=0, 15. 16. 17. 18. 19. 20. 21. 22. 8S. [4 5-7 Q= 37. r 6 2/ - 5 2; = 1, l92/ + 10;2 = 12. '7i: + 3Q + 9 = 0, . 6 Q - 9 i = 28. 3W-{-2D = 17, 4 Tr+Z) = 16. 8 a + 5 ^ = 5, 3 a - 2 «= 29. 5 a; -f 9 2/ = 8, 6?/-9a; + 7 =0. 5F+4P=22, 3 F+P=9. T 5 » 6 + 3 = ^' 2r^o 7^ 3 7_E 4 19 15' 2 TF- 7 TF 3^ 3 SYSTEMS OF LINEAR EQUATIONS ' 235 128. Elimination by Substitution. — The method of elimination by substitution is illustrated in the following example. ^ e 1 ( 4 to - 5 w = 26, (1) Example. — Solve J ^ ' ^^( [3M?-6t;=15. (2) Eliminate w Solving (1) for w, w= ?^-±A?. (3) Substituting — — — - in place of w in (2), 4 sn^y\-ev^i6. (4) Solving (4), v = 2. Replacing v by its value 2 in (1), 4 w? - 10 = 26. Solving, w = 9. Hence, the solution is w = 9, © = 2. This example illustrates the rule: Solve one of the equations for the value of either of the unknown numbers, expressed in terms of the otherf and substitute this value in place of that number in the other equation. Solve the resulting equation. Substitute the value found for the one unknown number in either of the original equations, and solve for the other un- known number, EXERCISES Eliminate by substitution, solve, and check: ( 2. [a + 36 = 9. 1F+3«=16. x-y = ^, r3a-7» = 40, a;H-22/ = 16. \4a-3;2 = 9. rm + 4n=7, f4p-5g = 26, • lm + 6n = 9. |3p-6g = 15. ra;i + ^2 = 30, \^D [3a^-2a;2 = 25. ^' {2 D i-2k = 33. + 7Q = 16, + 5Q = 13. (R-S==^, f3^ + 2A: = 26, ^ [2R + S=-14. *"• \5g-2k=^ 236 ELEMENTARY ALGEBRA 11. 12. 13. 14. 15. 16. 17. 7^-95=13, 5 ^ + 2 J5 = 10. 7ri4-4r2 = l, 9 ri + 4 ^2 = 3. \i T+t = 10, r 7 'u + s = 42, 2 M- 6 = 5, 5 Jf- 2 6 = 14. r^-8j9 = 45, 13^-2) = 20. r3a;4-22/ = 26, [5a; = 38 + 2y. 18. 19. 21. 22. r7m = 26 + 3a, I 6 a = 6 m - 20. r8i/ + rf l72/ = 9 60, 20(?. ri6-7^=3r, |2r-13 + 5^ = 0. a+ A; =30, 2 3 3' - + a— o 6—4 3 a = 2 6 + 7. r2w;_ 3 4:W -\-v _2 6w -i-v 5 = 0, 5, MISCELLANEOUS EXERCISES Eliminate by the method that seems best suited, and solve 3. [x-y = 4:. 7v — 10w=zd. (21Ii + 8t-{-66 = 0, 1 49 jR + 53 = 15 ^ r5Jf+63 = 26, l2Jf+6 = 18. r2a-26 = 6|, l7a = 264-36. r3n + 152; = 7, ^- 1 12 -n + 5 2 = 0. 10. 11. ^-7 + j-O, | = 9m-5t, 1+5 = 1 X y xy* 2x-3y = l. r2-5P=4 F, 3 ^ 4 [p-1 v+2' SYSTEMS OF LINEAR EQUATIONS 237 129. Problems solved by Systems of Equations. — For the steps in the process of expressing a problem by means of a system of equations with two unknown numbers see § 69. In the follow- ing problems use that method of elimination which seems best suited to each individual problem. EXERCISES 1. Find two numbers whose sum is 29, and difference 13. 2. Find two numbers whose sum is 9, and such that one of them exceeds twice the other by 27. 3. The difference between two numbers is 72, and one of them is 4 times the other. What are the numbers ? 4. The sum of two numbers is 64, and their quotient 7. What are the numbers? 5. The difference 'between two numbers is 21, and if the larger is divided by the smaller, the quotient is 4 and the remainder 3. Find the numbers. 6. The value of a fraction is J, and if 7 be added to each term, the value of the resulting fraction is J. Find the fraction. 7. A board 10 ft. long is to be cut into two pieces whose lengths are as 3 and 4. How long are the pieces ? 8. A surveyor wishes to set a stake in a line 320 ft. long, so that its distances from the ends of the line are as 7 is to 19. Where must he place it? 9. In building a bridge a steel bar 12 ft. long is to be bolted to another bar at a point which divides the 12 ft. bar into parts whose lengths are as 7 and 17. Where must the hole be bored for the bolt ? 10. In the steel frame in the figure the brace ^(7 is 15 ft., AD is 7 ft., and DB is 3 ft. It is desired to find where to bore the hole for the bolt E in AC. B,propoHion,|| = ^. Find where to locate E. 238 ELEMENTARY ALGEBRA y(. 11. Two boys desired to weigh themselves. They had no scales, but had a 10 lb. weight. They placed a board across a sup- port, as in playing teeter. When they sat, one 5 ft. from the ful- crum and the other 4 ft. from it, they balanced. When the smaller boy took the 10 lb. weight in his hands, he had to sit only 4 ft. 4 in. from the fulcrum to balance. What were their weights ? 12. Two weights balance when the larger is 3 ft. and the smaller 4 ft. from the fulcrum. If the smaller weight, with 3 IJ). added to it, is moved to a position 3 ft. 5^ in. from the fulcrum, they still balance. Find the weights. 13. Two weights balance when one is 4 in. and the other 6 in. from the fulcrum. If the first weight is decreased 12 lb., the other must be moved 1 in. nearer the fulcrum to balance. Find the weights. 14. Two weights balance when one is 3 ft. and the other 2 ft. from the fulcrum. The larger weight is decreased 12 oz., and the smaller weight is decreased 6 oz. and moved 2| in. nearer the ful- crum. Again they balance. Find the weights. 15. Two partners in business, A and B, are to divide profits in the ratio of 2 to 5. In dividing a profit of $ 6300, how much should each receive? 16. A man invests part of $ 12,000 at 5 % and the balance at 6%. His annual income from both investments is $645. Find the number of dollars in each investment. 17. A man invests $2000, part at 5% .and the rest at 4%. The annual income from the 5 % investment exceeds that of the 4% investment by $2.80. Find the amount of each investment. 18. An investor purchased two kinds of bonds. One kind yielded 3% and the other 4%. His yearly income from both was $ 558. Had he invested as much in 4 % bonds as in 3 % bonds, and vice versa, his yearly income would have been $576. How much did he invest in each kind of bonds ? 19. A man invested in $22,500 worth of bonds, part of them New York City 4s and part U. S. Steel 5s. His annual income SYSTEMS OF LINEAR EQUATIONS 239 was $ 1045. Had he purchased as many New York City 4s as he did U. S. Steel 5s, and vice versa, his annual income would have been only $980. What was the value of each kind of bonds bought? 20. I change $3 into dimes and nickels. There are 50 coins in all. How many dimes and how many nickels are there ? 21. I got $100 changed into $5 bills and $10 bills. I re- ceived 16 bills in all. How many bills of each kind did I get? 22. If a merchant blends 25-cent coffee and 32-cent coffee to sell at 30 cents a pound, what quantities of each grade of coffee must he take to make 50 lb. of the blend ? ^X 23. How much each of copper, specific gravity 8.9, and zinc, specific gravity 6.9, must be combined to produce 1000 cu. cm. of brass, specific gravity 8.4 ? (See Problem 7, page 112.) 24. In a number of two digits the sum of the digits is 11, and when the digits are interchanged, the number is diminished by 45. Find the number. 25. Two angles are called supplementary when their sum is 180°. If one of two supplementary angles is 20° more than twice the other, how many degrees in each ? 26. The sum of the angles of a triangle is always 180°. If each of the angles at the base of the triangle is twice the angle at the vertex, how many degrees in each angle of the triangle ? 27. The altitude of a trapezoid is 6 ft. and its area 84 sq. ft. One base is 8 ft. longer than the other. Find the lengths of the bases. 28. If the length of a rectangle is diminished 3 in. and its width increased 2 in., its area is unchanged; and if its length is increased 5 in. and its width diminished 2 in., its area is un- changed. Find the length and width. 29. Two groups of students of surveying were sent to find the area of a rectangular field. One group got the length 4 ft. too small and the width 6 ft. too large, which gave the area 2960 sq. 240 ELEMENTARY ALGEBRA ft. too large. The other group got the length 3 ft. too large and the width 3 ft. too small, which gave the area 741 sq. ft. too small. Find the dimensions of the field. 30. The circumference of the fore wheel of a carriage is 3 ft. less than that of the rear wheel. One goes the same distance in 40 revolutions that the other does in 30. Find the circumference of each. 31. A belt runs over two pulleys. One makes 10 revolutions while the other makes 3. If the larger pulley were replaced by one whose circumference was 20 in. less, it would only make 10 revolutions while the other made 4. Find the circumference of each pulley. • 32. A man who can row 6 mi. an hour downstream can row 2 mi. an hour upstream. What is the speed of the current? 33. A crew rows downstream 6| mi. in an hour, and returns in 4 hr. 20 min. What is the speed of the current, and at what rate could the crew row in still water ? 34.. An aviator flying against the wind makes 20 mi. an hour. On returning with the wind he makes 50 mi. an hour. What is the speed of the wind, and what would be his speed if there were no wind ? 35. If an aviator flies to a point 24 mi. distant, against the wind, in 40 min., and returns in 30 min., what is the speed of the wind, and what would be his speed if there were no wind ? 36. Two trains, each 300 ft. long, run on parallel tracks. If running in the same direction, it requires 20 sec. for one to pass the other. If running in opposite directions, it requires only 4 sec. for them to pass. What are the speeds of the trains? 37. A passenger train and a freight train run in opposite direc- tions on parallel tracks. The speed of the passenger train is 60 ft. per second, and the speed of the freight train is 40 ft. per sec- ond. The freight train is 180 ft. longer than the passenger train. It takes the trains 12.6 sec. to pass. Find the length of each train. SYSTEMS OF LINEAR EQUATIONS 241 38. A passenger and a freight train run on parallel tracks and in the same direction. The speed of the passenger train is 45 mi. an hour, and the speed of the freight train is 25 mi. an hour. The freight train is 96 ft. longer than the passenger train. It takes the passenger train 36 sec. to pass the freight. Find the length of each train. 39. How much milk testing 4 % butter fat and cream testing 24% butter fat must be mixed to make 20 gal. that test 20% butter fat ? 40. How much milk testing 3.8 % butter fat and cream testing 26.6 % butter fat must be mixed to make 16 gal. that test 18 % butter fat? 41. Fifteen pounds of tin weigh 13 lb. in water and 15 lb. of zinc weigh 13.5 lb. in water. How much tin and how much zinc in an alloy which weighs 66 lb. in air and 49 lb. in water ? 42. Some authorities claim that the daily ration for the aver- age adult workingman should contain 4 oz. of protein and an equal amount of fat. White bread contains 9 % protein and 1 % fat. Mutton contains 14% protein and 37% fat. Find how many ounces each. of bread and mutton would be required to make a daily ration. 43. Eggs contain 13% protein and 9% fat. From the facts given in Problem 42, can bread and eggs be used to make a stan- dard ration ? 44. A society that wished to raise $ 100 gave an entertainment. They estimated that enough children and adults would attend that they could raise this amount by charging children 10 cents and adults 25 cents admission, and that if they charged children 15 cents and adults 30 cents admission, they would raise $26 more than the amount desired. How many children and how many adults did they expect C would attend ? 45. In the triangle ABC, the line CD bisects (divides into two equal parts) the angle at C. ^rn I n It is known from geometry that VD divides AB a 242 ELEMENTARY ALGEBRA into two parts, m and n, which are proportional to the adjacent sides. If AC = 15 in., BC = 12 in., and AB = 10 in., find the lengths of m and n. Example. — Solve for x and y Multiplying (1) by 6, Multiplying (2) by a, Subtracting (4) from (3), Factoring (combining terms), Dividing by a^ _ 52^ Substituting ah for y m (1), 130. Systems of Literal Equations. — Systems of linear equations in which one or more of the known numbers are literal are solved by the methods of §§ 126, 127, 128, the solutions being expressed in terms of these literal quantities. ax — by = a\ (1) bx — ay = 63. (2) abx - h^y = a^h. (3) ahx — a^y = db^. (4) a'^y - 62y = a% - ab^. ia^-b^)y = abia^-b^), y = ab. ax — ab^ = a^ ax = a^ + ab\ x = a^+b^, :a^ + b^,y = ab. Hence, the solution is Solve for x and y : hx -\- ay=2 ah. ax -\- hy = a^ — p. 2. 3. 4. I \ay -\-hx = a^ — h\ ( my -{- x = m + rif \ mx + my = m^ -\- n, (x-\-Ey = S, \2x-Sy = M. Ax + y = B, Bx + y = A. \x-p^y = 0, [X-\-q'y = l, EXERCISES 7. 8. . 9. 10. 11. 12. rix-i-r2y = l, r^ + ^i3^ = 1. ax — b^y = a^ — h\ Q?y — hx = 0. 2pa;4-2g2/ = 4y + g«, x — 2y = 2p — q. 12x-lly = M^ + 12R2 x-hy = 2R^ + B,. (m + n)x = l —py, (m-\-n)y+px = l, vx — ty = 0, x-\-y = w. SYSTEMS OF LINEAR EQUATIONS 243 13. 14. 15. Px- Qy = B, x^y = Q- 2^3 = a, 3^4 = 6. X 4- ^ a-b a + 6 x-\-y = 2a. 16. 17. X o m — 12n Sx + 9y = 4:m-\-9n. X V V + 5 = 21, V 7y^3^ 4i; 4* 18. If m be added to the numerator of a certain fraction, the value of the resulting fraction is 4. But if n be added to the denominator, the value of the resulting fraction is 3. What is the fraction ? 19. The sum of two numbers is s, and the quotient of the first divided by the second is q. Find the numbers. 20. If a boy who weighs IF pounds and one who weighs P pounds balance at the ends of a teeter board I feet long, find the lengths of the two parts into which the board is divided at the fulcrum. 21. Part of $12,000 is invested at «% and the rest at y%. The annual income from both investments is d dollars. Find the number of dollars in each investment. 22. If milk tests m% butter fat and cream n 12' 17 3 [p'^30 4.71 12. The perimeter of a triangle is 28 in. Two of the sides are 3qual, and their sum exceeds the third side by 4 in. Find the lengths of the sides. 13. The sum of the two sides of a triangle which meet at one vertex is 44 in., at another vertex 40 in., and at the third vertex 36 in. Find the lengths of the three sides. 14. The sum of the three angles of any triangle is 180°. If two angles of a triangle are equal, and each is 5° more than the third, find the size of each. SYSTEMS OF LINEAR EQUATIONS 251 15. In triangle PQR, angle P is 16° less than angle Q, and angle Q is 4° less than angle B. How man}' degrees in each ? 16. It is known in geometry that two tangents drawn from the same point to a circle are equal. Hence, if a circle is inscribed in a triangle, as in the figure, the points of contact divide the sides of the tri- angle into three sets of equal tangents. If AB=12 in., 5(7=16 in., and AO=U in., show that x + y = 14, a; + 2; = 16, and y-{-z = 12. Find the distances of the points of Contact of the circle with the three sides from the vertices. 17. The sides of a triangle are 24 in., 32 in., and 38 in. Find the points where the inscribed circle touches the sides. 18. Three circles are to be drawn tangent to each other, and with their centers at three given points A, B, and C, respectively. If the distance between A and 5 is 10 in., between B and C 13 in., and between A and C 15 in., find the radii of the circles. 19. In accurate tool work where holes are to be bored close together in a metal plate, the centers of the holes are first marked carefully to thou- sandths of an inch. This may be done by first turning out disks on a lathe of such sizes that when placed tangent to each other their centers mark the positions of the centers of the required holes. These disks are then fastened on the metal plate in tangent positions, and the holes bored at their centers. Three holes are to be bored, the distances between whose centers shall be 0.^60 in., 0.620 in., and 0.844 in., respectively. Find the radii of the required disks. See Problem 18. 20. A certain number consists of three digits whose sum is 9. If 198 be subtracted from the number, the remainder will consist of the same digits in a reverse order ; and if the number be di- vided by the hundreds' digit, the quotient will be 108. What is the number ? 252 ELEMENTARY ALGEBRA 21. It is given that the standard daily ration for an adult laboring man should include 16 oz. starch, 4 oz. fat, and 4 oz. albumen. The amount of these materials in bread, butter, and beef are as follows : Food Starch Fat Albumen Bread Butter Beef 54% 0% 0% 1% 83% 15% 9% 1% 15% Find the quantities of these foods required to make a daily- ration. 22. Potatoes contain 20 % starch, 1 % fat, 2 % albumen ; pork contains no starch, 26 % fat, 13 % albumen ; and cream contains 4 % starch, 18 % fat, 2 % albumen. Find the quantities of these foods required to make a standard daily ration. ' "^ 23. There are three compounds composed of different metals. The first contains 7 parts (in weight) silver, 3 parts copper, and 6 parts tin; the second contains 12 parts silver, 3 parts copper, aud 1 part tin ; the third contains 4 parts silver, 7 parts copper, and 5 parts tin. How much of each of these three compounds must be taken in order to form a fourth which shall contain 8 oz. of silver, 3f oz. of copper, and 4^ oz. of tin ? 24. In an athletic contest team A won with a total of 35 points, team B got second place with a total of 32 points, and team C got third place with a total of 31 points, as shown in the table : Team ISTS 2d8 8d8 Total A 7 3 1 35 B 6 4 4 32 C 2 7 9 31 How many points does each place count ? SYSTEMS OF LINEAR EQUATIONS 253 SUPPLEMENTARY EXERCISES Solve : 1. 2. 5M,N_^ 2M N o 4 TF+w ^2 6 Tr+ w 5' ^ + ^ = ^- ri + r, + l ' rj + l ra-l' Solve for x and y : a_6_ 3a; 4y Solve: a+6 + c = 2, 6 + c + d = 4, c H- c? + « = 6, la + 6 + d = 3. r + s 4- ^ = 18, r + s + 10 = 18, s + w; + « = 18, r+w>+^=18. 9. 10 1 +^=0. a+6 a— 6 a-36 h-2a 2. 5. 6. ^ 1-^=1. l + a; 1 + 2/ _2 1_^1 1 + a; 1 + ^ 2 a:-2 y^l 1 . 6 1, .aj-2 y + 1 8 r3w . w-1 8. a? 2/ i2x y = w. = «. 11. 12. « + 2/+24-«« = 10, a; — y — 2 + w = 0, 2x + y-\-Sz — w = % ^x — y+z-2w + ^ = 0. 2p-q-r + s=lZ, p + q — r — s-\-l=zOj 3p-f-2g-r + 2s = 17. 13. Find a number of three digits, such that the sum of the digits shall be 15; the sum of the hundreds' digit and the ones' digit shall be 1 less than the tens' digit ; and the ones' digit sub- tracted from 4 times the hundreds' digit shall equal the tens' digit 254 ELEMENTARY ALGEBRA 14. A merchant has two kinds of tea. If he mix 3 pounds of the poorer with 7 pounds of the better, the mixture will be worth 76i^ a pound; but if he mix 7 pounds of the poorer with 3 pounds of the better, the mixture will be worth 68|-^ a pound. What is the price of each kind of tea ? 15. There are two alloys of copper and silver, of which one con- tains 3 times as much copper as silver, and the other contains 5 times as much silver as copper. How much must be taken of each alloy to make 7 pounds, of which half shall be silver and the other half copper ? 16. If the altitude of a rectangle be increased 4 inches, and its base diminished 2 inches, the area will be increased 22 square inches ; and if the altitude be increased 1 inch, and the base dimin- ished 1 inch, the area will be increased 2 square inches. Find the base and the altitude of the rectangle. 17. A company of men rented a yacht. When they paid their rental they found that if there had been 2 more persons to pay the same bill, each would have paid 50 cents less than he did ; and if there had been 2 fewer persons, each would have paid $ 1 more than he did. Find the number of persons and the amount that each paid. CHAPTER XIII SQUARE ROOT. QUADRATIC SURDS 135. Square Roots of Polynomials. — Since (a+6)^=a^-|-2 a6+6', Note. — As shown in § 72, every quantity has two square roots, differing only in sign. Thus, the other square root of a^ -\- 2 ab -\- b^ is — (a + 6), or — a — 6. In this chapter we shall consider only one of the square roots of any expression, viz. the one of which the first term is positive. A study of the above identity will reveal the process of finding the square root of any polynomial which is a perfect square. If the polynomial has three terms arranged according to the powers of one letter, these correspond to the terms of a^+2 ab-\-bK Evidently, the first term a of the root is a square root of al If a^ be subtracted from the trinomial, the remainder is 2ab-\- b^ The second term of the root, b, may be found by dividing 2 a6, the first term of the remainder, by 2 a, or twice the term of the root already found. The work is usually arranged as follows : a^-\-2ab + b^\ a-{-b /,2 2a 2a + 6 2ab + b^ 2ab-\-b^ The divisor 2 a, used in finding the second term, is called the trial divisor. AVhen the second term b is added to the trial divisor, it gives the true divisor, 2 a + 6, because when the latter is multi- plied by b it gives the entire remainder 2 ab-\- b\ 255 256 ELEMENTARY ALGEBRA Example 1. — Find the square root of Oic^ _}. 4 y'i __ 12 xy. Writing the expression in descending powers of a:, the work is as follows 9 a;2 - 12 icy + 4 y2 |3 x-2y Q'»-.2 — 12 icy + 4 y^ 6x 6x-2y — 12xy + 4:y^ The first term of the root is Vdx^, or Sx. Subtracting 9a;2 leaves — 12 xy -f 4 y^. The trial divisor is 2(3 a;), or 6 x. Dividing — 12xy hj dx gives — 2«/, the second term of the root. Adding —2y to 6a;, the trial divisor, gives the true divisor 6x — 2y. Multiplying this by — 2 y gives — 12 a;y + 4 2/2 exactly, which shows that 6 x — 2 y is the entire root. If the square root of a polynomial has three or more terms, the first two may be found as above ; then by grouping terms, these two may be used as one, and the third term obtained by a repeti- tion of the process used to obtain the second. Similarly, by grouping the first three terms of the root, the fourth may be found, etc. Example 2. — Find the square root of 8 w — 4 w^ + w* + 4. First, arrange the terms in descending powers of n. 4w3 +8n + 4|w2-2w-~2 n* 2/l2 2n2-2w -4n8 +8w + 4 -4n3 +4n2 2w2-4 7i 2 ?i2 _ 4 ^ -2 -4w2 + 8w + 4 - 4 w2 _|_ 8 n 4- 4 The first term of the root is \/#, or n^. Hence the Jirst trial divisor is 2 w2. Dividing — 4 n^, first term of the remainder, by 2 n^ gives ~ 2n, second term of the root. Adding this to 2 n^ gives 2 n^ — 2 n, the first true divisor. Multiplying this by — 2n gives — 4 w^ + 4 n2. Subtracting this product from the first remainder leaves — 4n2 + 8 w + 4, the second remainder. Now using the root found, n^—2 n, as one term, the second trial divisor becomes 2(^2 _ 2 w) or 2^2 — 4 n. Dividing the first term of this into the first term of the remainder gives — 2, the third term of the root. Adding — 2 to the trial divisor gives 2 »2 _ 4 ^ _ 2, the second true divisor. This multiplied by — 2 gives the second remainder, which shows that w2 — 2 w — 2 is the entire root. SQUARE ROOT. QUADRATIC SURDS 257 Note. — Care must be taken to first arrange the terms in any problem in descending or ascending powers of some letter. Each remainder and each divisor must also be arranged like the original expression. The above examples illustrate the general rule: (1) Write the given polynomial in descending or ascending powers of some letter. (2) Take the square root of the first terin for the first term of the root, and subtract the first term, from the polynomial. (3) Double the root found for the first trial divisor, divide the first term of the remainder by this, and write the quotient as the second term of the root. (4) Add this quotient to the trial divisor to obtain the true divisor, multiply the true divisor by the second term of the root, and subtract the product from the preceding remainder. (5) If there is still a remainder, double all of the root already found, for a new trial divisor, and proceed as before. Continue this process until all terms of the root are found, EXERCISES Find the square roots of the following : 1. 4a2 + 20a + 25. 2. l-16y4-642/». 3. m* + 25 w^- 10 m^n. 4. 5ar' + a;*-2a^-|-4-4a;. 5. 4<^ + 49-3^-70« + 20^. Q. l-^2b-b^ + :^W-2b^ + b\ 7. 49 ^« + 42^« -19^^-12 ^2 _|.4^ 8. r* + 21r2 + 4-10r34-20r. 9. a* + 4a36 + 6a2&2 + 4a63+6*. 10. 4w;^-4w;'' + 17?c--8z«4-16. 11. l^M^N^-\-M^-^M^N-\2MN^-\-4.N*. 12. a;^ - 2 a^?/ - 2 a;/ 4- 3 ^y"^ + y\ 13. 9i)< - 12 p^q - 26i)¥ _^ 20p/3)(l + V2-V3) 2V2 Divide : 21. V3+V5byl4-V3 + V5. 25. V5- V3 by V5- V3- 1. 22. 1+V2by 1-V2+V5. 26. V^ by V^ + V^ + V^. 23. 2by V2+V3 + V7. 27. Va- V& by Va+ V5- Vc, 24. V2by V2 + V3+V5. 28. 1 - Vm by 1 + Vm4- V^. CHAPTER XIV QUADRATIC EQUATIONS 144. Quadratic Equations. — As shown in § 73 and § 96, some problems may be expressed and solved by use of equations of the second degree, or quadratic equations (see § 57 for definitions). The equations .discussed in § 73 and § 96 were of very special types, and the methods of solution there shown are not applicable to all quadratic equations. We shall now discover how to solve any quadratic equation whatever. A quadratic equation that contains a term of the first degree in the unknown number is called a complete quadratic equation. One that does not contain a term of the first degree in the unknown number is called a pure quadratic equation. Thus, 3 w'-^ — 4 n = 5 is a complete quadratic equation. And 1 1^ — 9 = is a pure quadratic equation. By simplifying and combining similar terms, if necessary, any quadratic equation whose unknown number is denoted by x can be written in the form Ax^ -\- Bx-\- (7= 0, where A, B, and C are known numbers. 145. Quadratics solved by Factoring. — Some quadratics can be solved hy factoring, as shown in § 96. The rule there given should now be reviewed and applied in the following exercises. Note. — It must be remembered that every quadratic equation has two roots. Both roots should be checked by seeing if they satisfy the given equation. 270 ELEMENTARY ALGEBRA EXERCISES Solve by factoring, and check : 1. a^==7a-V2. 2. P^ + 3P=10. 3. ^2 + 9^+14 = 0. 4. 2a.-2 4-5a; = 3. 5. 6 F2+13 F+6=0. 6. 10f-\-S^t = l. 7. 6?/2 = 23?/ + 4. 8. 2-R = 21E^. 9. 54^ = 9;s2_^72. 10. 4.A = A^-n, 11. ml 1 12. i_i_28 11 13. 15/r+4 = -|. 14. 5 = ^ + f. r 7^ 15. a2 + i.a = 3f. 16. 1 + ^_6P=0. 17. 18. 5r-8.a. 19. ¥s*— ■ 20. lw--«- 21 3&_10 6+120 14 h 99 1 aj-7 a;+24 _^ 2a;-2 aj+1 5a^-5 23. 24. s+3_10 2 ^ 2 3 s + 3 25. 3(^-1) _5 , 2(^ + 1) ^+1 ' ^-1 26. r-1 r+1 27. 3a = a2-18. 146. Quadratics solved by Square Root. — Any pure quadratic equation can be solved by the method of § 73. To solve a pure quadratic equation first, by clearing of fractions, transposing terms, etc., reduce the equation to the form x^ = A, where X is the unknovm number and A is known. Take the square root oj both members of the resulting equation, attaching the double sign ± to the second member. Example 1. — Solve 8 «2 = 2 «(« - 5) + 10(« + 2). QUADRATIC EQUATIONS 271 Removing parentheses, transposing, combining terms, and dividing by the coefficient of «, Taking square root, t=± V^ These roots may be expressed decimally, if desired. Thus, V30 = 5.477, approximately. Hence, « =± 1.825, approximately. Note. — In some problems in which the roots involve surds it is sufficient to express the roots with radical signs. But in many applied problems it is desirable to compute the approximate values of such roots decimally. The student should be able to compute the roots in either form when desirable. Example 2. — Solve Z>2 + 5 = 0. Transposing, D^ = -. 5. Taking square root, D = ± V— 5, imaginary numbern. EXERCISES Solve : 1. 7y2-28 = 0. 2. 10 F^- 150 = 4 V^. 3. 7^ = 845-47^. 4. a(a + 4) = 4 a + 49. 5. (M+2){2M-{-3)=7 (M+S). 6. 8«« + 72=:0. 7. 5n2 + 20 = 0. 8. 4.A^ = 7. 9. 6i22_8 = 0. 10. 5^2 + 10 = 0. 11. 20 P2 4-8 = 0. X _5 125 ""a;* St^± 4 3t 12 13. ^ = ^. 14. 15. 16. 17. 18. 19. 20. 21. y + 1 a2 + 4 8 F^ + 8 , y-l 6 = 1. 2 9^2-1 = 2. = 4. J 1,^9 2Jc' Tc" 4* 6 5 71 + 1 = L 1 w+1 n— 1 = 3, 272 ELEMENTARY ALGEBRA 22. ^Aszl^Alll=:A±l, 26. ^^+? 15- = 0. 23. 24. 25. ^2-1 ^ + 1 ^-1 2 + s 2s + 3 £±^ §_=i. 27. 2r+i+ (^^+i)^^-^) :^o 6 Q + 1 ^ ^ T4-2 g + 6 6^±l_n 28 2G^_l+^^3 6^24-1 7+6"" ' "^ 2(^ + 3 5 4^4-1 ^ 8j9-19 ^ 29. ^l±lil±l-6 = - 3i)-4 7p-24 ^-1 a; + l Eind approximate to hundredths the roots of : 30. x^^l. 36. 5^^-14 = 0. 31. 4a2-10 = 0. 37. 3«2=28. 32. 8i2_i2 = 0. ^^- (P+4)(p-4) = 4. 33. 15n2-4 = 0. 39. -1- =IZL§. r + 3 4 34. 7F^ = 26. ^^^ ^^ 35. 2i>2 = i7. 4 iT-l • 147. Quadratics solved by completing the Square. — As shown in § 86, any trinomial that is a perfect square may be written in either the form a? + 2 ah -\- h"^ ov a? — 2 ah -\- h^. Hence, if we have given either the binomial a^ + 2 a6 or a^ — 2 ah, to make it a perfect square we must add 6^, i.e. the square of one half of the coefficient of a in the term 2 ah. This process is called completing the square. Thus, to complete the square whose first two terms are n^ — 10 n, we add the square of ^ of 10, or 25. This gives w^ — 10 w + 25, which is the square of w — 5. By use of this process of completing the square any complete quadratic equation whatever may be solved, as shown in the following examples. Example 1. — Solve v^ + 3 u — 10 = 0. Transposing, v^ -{- S v = 10. Adding the square of ^ of 3 to both members, to complete the square on the left, «2 + 3 u + I = 4,9 . QUADRATIC EQUATIONS 273 Taking square root, » + f = ± |. Transposing, v = - f ± |, i.e. -| + |or~|-|. Simplifying, ?> = 2 or — 5 Example 2. — Solve 6 P2 + 12 P = 2. Dividing by 5, P2 + i^ P = |. Adding square of | of V, P^ + V ^ + if = If • Taking square root, P+ f = db ^^46. Transposing, P = — | ± i V56. If the approximate values of these roots, expressed decimally, are desired, they may be found by taking the square root of 46 and simplifying. Thus, \/46 =6.782, approximately. Hence, P= 0.15 or - 2.55, approximately. Example 3, — Solve y^ = 4y — 7. Transposing, y^ — 4y= — 7. Adding square of | of 4, ^/^ _ 4 y _}. 4 = _ 3, Taking square root, y — 2 = ± V— 3. Transposing, y = 2 ± V— 3. These roots involve imaginary numbers. Evidently, any complete quadratic equation may be solved by the following rule : i^ (1) Reduce the equation to the form x- -\-px = q, where x is the unknown number. (2) Add to each member the square of one half of the coefficient of the term in the first x>ower of the unknovm number. (3) Take the^square root of both members, attaching the double sign ± to the second member. (4) Solve the resulting linear equations. EXERCISES Solve by completing the square : 1. 2/2-62/ + 8 = 0. 6. r^ + a; = 6. 2. n24-4w = 12. 7. F'-llF+30 = 0. 3. A^-^A = ^. 8. d^ = U-hd. 4. f = 22 + ^t. 9. i1f2_^12JI^+15 = 0. 5. 'ir' + 14v + 40 = 0. 10. 2A;2 + A: = 4. 274 ELEMENTARY ALGEBRA 28. -^+-^ = 1 + 11. 2P2_5p = 3, 12. 6r2 + 6 = 13r. '" V^-^ ' i> + l " ' p-1 13. 4TF^-llTr=3. 2^ 1 I 2 15. 3 2^2 = 17^4.28. OA Q 1 1 30. o — = — • 16. 2b'-\-3b = 4., y + 2 y-2 17. ^lc' = k-\-l. ^^ 1 J 1 . 1 18. 12 6^2^3 = 14 G^. C^-1 C^-2 U-3 19. 9^2^6^ + 26. g^ c + 2 4-c^o^ 20. 16 a2_ 96 a = 1792. ' c-1 2 c ^* 21. iV^2_7j^_^i5^Q^ 33^ (;2_5)2_,.(2;_10)2 = 37. 22. -^ ^ = -1^. 34 n + 3 ,r,-4_ 9^ ^ I ^ — ^ 3 11 4 + r''^4-r' 3' 35. _,_^+^^_^ ^ 36. 24. ^-|+20^ = 42|. _ 1 1 1 m + 2 m — 2 1 — m 25. 6ir + — :^- — = 42. 2F-1 , F-l 3^-10 ±1 37. 1 = • 26. a^ + 7a;=7(a; + 3)+4. Y-^ Y-2 Y-3 2^ ^ — 3 ^ + 4 ^2 3g cg-4 (^-4 ^11 ' w + 5 w-7 ' ' 2d-12 Sd-ie 2* Find approximate to hundredths the roots of : 39. w2 = 3n+7. 44. 12 Ii' = Ii + 2. 40. u^-15u = 60. 45. 34^ = 3«2_iq5^ 41. 9^2^1-3A 46. 19 = a+4a2. 42. />2 + 8Z) + ll = 0. 47. 5M^ = 2M-{-7, 43. z^ + 6z = l. 48. 21-2v = 4vl 148. Quadratics solved by a Formula. — Since any quadratic equation may be written in the general form, QUADRATIC EQUATIONS 275 where x denotes the unknown number and A, B, and O are known numbers, the roots of this equation will give a formula by which the roots of any particular quadratic equation may be written down at once. Ax^ + Bx 4- (7 = is solved by completing the square as follows : Dividing by ^, a;^ + — x + -^ = 0. A A B C Transposing, a;^ + — a; = — - • A A B B^ B^ — 4: AC Completing square, x^ + -r ' A Taking square root, Transposing, 2A Now, by replacing A, B, and C in the formula for the roots ^"4^-2- 4^2 -^= ^/B^-4AC 2A x = B y/B^-4AC 2A 2A ^B±VB^-4AC _ -B±^B--^AC ^" 2A by the particular values which they have in any given equation, the roots of any quadratic equation may be written down at once. Thus the long process of completing the square in every equation may be avoided. Hence, this formula should he mastered and used in all future work where the equation cannot be readily solved by factoring. Example 1. — Solve 3 n2 - 4 n = 15. , Written in the form Ax^ + Bx + C = 0, this becomes * ♦ 3 n2 _ 4 n - 15 = 0. Here ^ = 3, J5 = — 4, and C = — 15. Substituting these values for A, Is. and C in the formula, 4 -j_ Vl(3 + 180 n = 4 ±14 = 3 or - |. "*• 276 ELEMENTARY ALGEBRA Example 2. — Solve 4 F2 + 2 = - 3 F. Transposing, 4 F2 + 3F-|-2 = 0. Here ^ = 4, 5 = 3, and C = 2. Substituting, ^^ -3 J:V9-32 8 _3_t-V-23 . s= ^- , imaginary expressions 8 EXERCISES Solve by use of the formula • 1. /-32/ = 10. ^^ 2^n + l 2. i)2^10p + 21=0. * ?^ 2 * 3. 2A^ + A = Q. 22. -^-^.Zd = 0. 4. 3^2_p5^ = 2. l-« 3 5. 6/^2 + ^-5=0. 23. 4 = 2+ ^ 6. 7ii2 = 50?^_7. v^ 2v 7. 4Jf2_^3f+5 = 0. 24. - + ^ = 11. 8. 6ri2+6 = 13ri. ^ . 4 ^ 25. J? + 3= * 9. 10a2 + 14 = 39a. R-1 10. 8i)2=65Z>-8. 2g s_l = 3_l. 11. 2n^ + Sn-ir4. = 0. ' s 3 12. 5^2 = 22-7. 27. 2« + 3 ^ a + 3 13. 15g2 + 3 = g. «-2 2a-l 14. ^ + 4 = 8JB. 28. ^"•^ + ^ + ^ = ^-4^^+3 3a;2-4 2a; + l' 4F 15. 2'?;2^7^_g 16. 4fc^ + 5 = 6^. 29. x + 2 17. 21c«=c + 4. 18. 9f=6 + y. 30. 1+F=^^^ 19. 12 + a; = lla:2. 1_ 20. l-a2-3a = 0. ^^^ ^ + 6-^' QUADRATIC EQUATIONS 277 33. -6- +1 = 3. 35. ^ « 8 ^ + 1 A x—4: x — 5 x—3 36 ^-2^ + 2 10^-8 _Q 37. 2 1 - w ?o 4- 1 5w-10 w-\-2 w^^4: 7-3M 1-M ^ 23f+3 . • 4-3Jf 2Jfcr+2 43f-2* • 18^2/4-1 22^ + 8 8^4a;4-4 2a.'2-2 T^+1 41. 8+Z + 4-2P^8, 44^ (n-iy n-2^g^ 8-P 4 + 2P 3 2-4 *2- ^'-733-^'+-2~ ''^' 4+r r-6-12 Find to hundredths the roots of : 46. a2-5a = 8. 48. 12(^2 = 3^4.19. 50. 6Q2 + i = 9Q. 47. 4J\r2+^_7 = 0. 49. 9-2y = y\ 51. 18 = 3 JS + 2 -B^. 149. Problems solved by Quadratic Equations. — It has been seen that the roots of quadratic equations may be fractions, negative numbers, surds, or even imaginary numbers^ In solving a problem by means of a quadratic equation, it is necessary to consider the nature of the roots to see if both roots saU^y all requirements of the given problem. Thus, a problem may require for its solution a j9os?Y/ve number. If one of the roots of the equation formed is found to he negative, it must be discarded, although it satisfies the equation. If the nature of a problem requires a real number for its spluttqn, and the roots of the equa* tion formed are found to \)Qi'*4((i(i^inary, they must be discarded. 278 ELEMENTARY ALGEBRA Example 1. — A train runs at a uniform rate between two points 280 miles apart. If it ran 5 miles per hour faster, it would make the distance in 1 hour less time. Find the rate of the train. Let V equal the rate of the train. 280 Then = number of hours required for the run. V 280 And = number of hours required at the increased speed. „ 280 280 , 1 Hence, — = + 1. V V + 5 Solving, t? = 35 or — 40. But the rate must be a positive number. Hence, the root — 40 must be discarded, and the only rate possible is 35 miles per hour. Example 2. — Find the real number whose square increased by 82 equals 8 times the number. Let n = the number. Then, w^ + 32 = 8 w. Solving, n = 4 ± V— 16. Since both roots involve imaginary numbers, the problem has no solution, i.e. it is impossible. EXERCISES By use of one unknown number solve: 1. Eind two consecutive arithmetical numbers tbe sum of whose squares is 313. 2. Eind two consecutive arithmetical numbers whose product is 240. 3. Divide 31 into two parts the sum of whose squares is 541. 4. rind two arithmetical numbers whose sum is 50 and prod- uct 336. 5. Divide 14 into two parts such that 4 times the square of the larger shall exceed 6 times the square of the smalkr by 40. 6. The denominator of a certain fraction is 5 more than the numerator. If the fraction be added to the fraction inverted, the sum will be 2|-|. . Find the fraction. QUADRATIC EQUATIONS 279 7. A pupil was to divide 12 by a certain number, but by mis- fc^ke he subtracted the number from 12. His result was 5 too great. Pind the number. 8. The side of one square exceeds that of another by 3 inches, and its area exceeds twice the area of the other by 17 square inches. Find the lengths of their sides. 9. A rectangular field is 96 ft. longer than it is wide, and it contains 298,000 sq. ft. What are its dimensions? 10. A floor can be paved with 200 square tiles of a certain size. If each tile were 1 in. shorter each way, it would require 288 tiles. Find the size of each tile. 11. In the center of a rectangular room is a rug 9 ft. by 12 ft. Around this is a border of uniform width. The area of the floor is 208 sq. ft. Find the width of the border. 12. A lawn is 40 ft. by 90 ft. Two boys agree to mow it. The first boy is to mow one half of it by cutting a strip of uniform width around it. How wide a strip must he cut ? 13. A farmer has a field of wheat 80 rd. wide and 120 rd. long. How wide a strip must he cut around the field in order to have one fourth of the wheat cut? 14. In Problem 13, how wide a strip must he cut around the field in order to have one half of the wheat cut? 15. A farmer has a field 40 rd. wide and 80 rd. long to be plowed and planted in corn. He wishes to plow 4^ acres the first day by plowing a strip of uniform width around it. How wide a strip must he plow ? 16. The hypotenuse of a right triangle is 4 in. longer than one leg and 2 in. longer than the other. Find the sides of the triangle. 17. One side of a rectangle is 4 in. longer than the other, and its diagonal is 20 in. How long are the sides? 18. Two men start at the same time from the intersection of two roads, one driving south at the rate of 4 miles an hour and 280. ELEMENTARY ALGEBRA the other- west at the rate of 3 miles an hour. In how many iiours will they be 25 miles apart? 19. Two trains are 100 miles apart on perpendicular roads, and are approaching the crossing. One train runs 10 miles an hour faster than the other. At what rates must they run if they both reach the crossing in 2 hours ? 20. A train running 6 miles an hour slower than usual, due tc a storm, was 1 hour late in making a run of 252 miles. Find its speed. 21. An engineer increased the speed of his train 5 miles an hour, and made a run of 360 miles in 1 hour less than schedule time. What was the speed when running according to schedule ? 22. A company of people arranged for a dinner, for which they contracted to pay % 45. Five of the people were unable to attend, and as a result each of the others had to pay 30 cents more than he had expected. How many were present? 23. A farmer sold his wheat for $540. A month later the price of wheat advanced 10)^ a bushel, and he found that if he had held his wheat until that time, he could have received the same money and have had 75 bu. left. How much per bushel dic^ he receive for the wheat? 24. A trader bought a flock of sheep for $1020. Two of them died, and he sold the rest at a profit of $3 a head. He made $225 on the transaction. How many did he buy? 25. A man bought a farm for $20,000. Later he sold all except 35 acres of it at a gain of $ 35 per acre over the cost, and received just what he paid for the whole farm. How many acres in the farm ? 26. One pump can fill a tank in 6 minutes less time than ia required for another pump to fill it. The two working together can fill it in 10^ minutes. Find the time required for each pump alone to fill the tank. QUADRATIC EQUATIONS 281 27. One of two pumps can fill a tank in 28 minutes, and the time required for the other pump to do it is 19|- minutes longer than is required for the two pumps together to fill it. Find the time required for the two pumps together to fill the tank. 28. With A's help B could do a piece of work in 6J hours less time than would be required for B to do it alone. A could do it alone in 21 hours. In how many hours could B do it alone ? 29. One pipe could empty a tank in 2 hours less time than would be required for a second pipe to empty it. After the first pipe has been open 4 hours, it is closed and the second pipe is opened. The second pipe finishes emptying the tank in 5 hours more. In what time cpuld each pipe alone empty it ? 30. An aviator flew 80 miles against the wind, and back again, in 6 hours. His speed against the wind was 10 miles per hour greater than the speed of the wind. Find the speed of the wind, also what his speed would have been had there been no wind. 31. A line 16 in. long is to be divided into two parts such that the ratio of the whole line to the longer part shall be equal to the ratio of the longer part to the shorter part. Find to tenths of an inch the lengths of the parts. Note. — The line in Problem 31 Is said to be divided in "extreme and mean ratio." Since the time of the Greeks the problem to divide a line in extreme and mean ratio has been called the *' Problem of the Golden Section." 32. Experience has shown that a book, photograph, front of a tall building, or other rectangular object, is most pleasing to the eye when its length and width are obtained by dividing their sum in extreme and mean ratio. The page of a book is to be 5 in. wide. How long should it be made to be most pleasing to the eye ? Suggestion. — If w denotes the length, ^ "^ = - . n 5 33. A photograph is to be mounted on cardboard 6 in. wide How long should the board be cut ? 282 ELEMENTARY ALGEBRA 34. An architect is designing a building whose frontage on the street is to be 80 ft. How high should he make it for the greatest beauty ? 35. An open box that shall hold 384 cu. in. is to be made from a square piece of cardboard by cutting out a 6-inch square from each corner and turning up the sides. Find the size of the piece of cardboard that must be used. 36. An open box that shall contain 324 cu. in. and that shall be twice as long as wide is to be made from a rectangular piece of tin by cutting out 2-inch squares from the corners and bending up the sides. Find the dimensions of the piece of tin required. 37. A flat disk of sheet metal 3 inches in radius is to be stamped by means of a die into a box lid 1 inch deep. The total area of the lid must equal the area of the flat disk. What will be the radius of the lid ? Suggestion. — If r is the radius of the lid, its total surface is composed of a circle whose area is irr^ and a cylindrical part whose area is 2 irr. Hence, 38. If the radius of the disk in Problem 37 is 2 inches and the depth of the lid ^ inch, what will be the radius of the lid ? 39. The rim of the front wheel of a carriage is 2 ft. less than the rim of the hind wheel. The front wheel makes 88 more revolutions in going a mile than the hind wheel. Find the length of the rim of each wheel. 40. The circumference of the little wheel on a sewing machine is 29 inches less than that of the big wheel which drives it. While the belt moves a distance of 1275 inches, the little wheel makes 116 more revolutions than the big one. Find the circum- ference of each wheel. 41. A park is 900 ft. wide and 1650 ft. long. It is desired to move out the boundary lines the same distance along one side and both ends so that the area inclosed will be just twice as great as it now is. Find how far the boundary lines must be moved. QUADRATIC EQUATIONS 283 42. The longer base of a trapezoid is 14 in. longer than the other base, the altitude is equal to the shorter base, and the area is 120 sq. in. Find the lengths of the bases and the altitude. 43. If any two chords AB and CD of a circle meet at 0, then AO x OB=COx OD. If 00 = 3 in., OD find AO and OB. 2 in., and AB = S in., 44. In Problem 43, if (70 = 9, 0D = 7, and,^B=22, find AO ^_ and OB. 45. AB, the span of the stone arch ACB that is to be constructed, is 40 ft. (^0 = 20 ft. and OB = 20 ft.), and the diameter CD of the arch is 50 ft. Find CO, the rise of the arch. 46. If the span of an arch is 16 ft. and the diameter 20 ft., find the rise of the arch 47. If the span of an arch is 20 ft. and the diameter 30 ft., compute the rise. 150. Literal Equations : Formulae. — If a literal equation or formula (see § 113) is of the second degree with respect to the unknown number, it may be solved by the methods of this chapter. Example. — Solve t(t — b) =a(a + 6) for t. Removing parentheses, t^— bt = a^ + ab. Transposing, t'^—bt- (a^ + ab) = 0. In formula, A = l, B = Substituting in formula, 6, C= -(,a^-\-ab). 2 _ b±(2a-\-b) 2 = a + 6 or — a. 284 ELEMENTARY ALGEBRA EXERCISES 1. Solve av? = h for n. 2. Solve aV — a'* = for v. 3. Solve mn = m^ — 4 for m. 4. Solve 2 o;^ — 3 r^ = 5 £cr for a; ; for r. 5. Solve W' + 2A' = 3WA for TT; for ^. 6. Solve y^ -{-2 my — n^-\-2 mn for y, for n. 7. Solve v^ -{-2rw-\- s — for w. 8. Solve z-\- — — n-\-— for w : for z. n z 9. SolveP2 + 2P = vP+2'yforP. 10. Solve f-{-at-\-U-\-ah=^0 for «. 11. Solve A^— (m — n)A = mn for ^. 12. Solve :i±-^ + i=^ = 1 forn; for B, 1-Rn 1+Rn 13. Solve/S' = iw[2a4-(^-l)(^] forn. 14. Solve /S^ ^^"^"-^^ forn. 2 15. If an object is let fall downward from a height s feet above the earth, the time in seconds required for it to strike the earth is computed by the formula s = 16 1^. Solve for t. 16. How long would it take a body to fall a distance of 1280 feet ? 17. If an object is thrown downward toward the earth with an initial velocity of v feet per second, the distance s in feet that it will fall in t seconds is computed by the formula s = vt-\-lQ> f. Solve for t. 18. An object is thrown downward with a velocity of 32 feet a second from a distance of 240 feet above the earth. How long will it take for it to strike the ground ? QUADRATIC EQUATIONS 285 19. The velocity of the discharge of water from a pipe is com- puted by the formula 4 1;^ + 5 v = 2 -{■— — . Solve this for- mula for v, SUPPLEMENTARY EXERCISES A second method of solving a quadratic equation by completing the square besides that given in § 147 is shown below. 1. Solve aa^ + bx-\-c = for x. Solution. — Multiply through by ,4 a, or 4 times the coefficient of x^. Then 4 a^x^ + 4 a6x + 4 ac = 0. Transposing, 4 a^x^ + 4 dbx = — 4 ac. Add h^^ the square of the coefficient of x in the original equation, to both members. Then 4 aH^ + 4 aftx + 6^ = &2 _ 4 ac. Taking square root, 2 ax + 6 = ± y/h'^ —4 ac. Transposing, 2 ax = — h± y/h'^ -4ac Dividing by 2 a, x = - & =b -/^/^ - 4 ac ^ 2a Solve by the method of Problem 1 : 2. Sx^ + 4:x + o = 0, 7. 15F'-7 = 9F. 3. 7w2_37i-4 = 0. 8. 662 = 6-1-5. 5. 12 K'-{-K= 11. 10. SM^==7M+15. 6. 9a2 = 3a + 7. 11. 6?-2 = 13r-6.' CHAPTER XV SYSTEMS INVOLVING QUADRATIC EQUATIONS 151. Systems involving Quadratic Equations. — Some problems may be expressed and solved by means of a system of two equa- tions containing two unknown numbers in which at least one of the equations is of the second or higher degree. In this chap- ter we shall discuss only systems in tchich one equation is linear and the other quadratic, these being suflB.cient for the present needs of the student. A complete treatment of systems involving quad- ratic and higher equations will be found in the Second Course. 152. Elimination by Substitution. — The easiest method of eliminating one of the unknown numbers in a system consisting of one linear and one quadratic equation is by substitution. See §128. Example— Solve f »>* - 2 n = 5, (1) 1 m2 + n2 = 10. (2) Solving (1) for m, m = 2n + 6. Replacing m by 2 w + 5 in (2), (2 ?i + 5)2 + ^^ = 10. (3) Solving (3), w = - 1 or - 3. Replacing w by — 1 in (1), to + 2 = 5, m = 3' Replacing w by — 3 in (1), m + 6 = 5, m =— 1. Hence there are two solutions : VI = 3, n=—l, or m = — 1, n=~-3. As shown in the above example, elimination of one of the un- known numbers in a system containing a linear and a quadratic equation always leads to a quadratic equation in one unknown number, which may be solved by the methods of Chapter XIV. SYSTEMS INVOLVING QUADRATIC EQUATIONS 287 By substituting each of the values thus found for one of the unknown numbers in the linear equation of the given system, a corresponding value of the other unknown number is found. Hence, a system consisting of one linear and one quadratic equa- tion lius two solutions. EXERCISES Solve : 1. 2 a -f- 6 = 4, 0" + ^ = ^. 13. \a^ + f = 7^, [3x-2y = l. 2. .3x'-y' = 23. 14. 2W-h2D = 5 WJJy l2TF+2Z>=5. 3. m 4- ^ = 5, mn = 4. 15. 2^^4-3^2 = 11 + 4^, U^5 = 3A;. 4. IA-B = 1, \AB=2, 16. E'-EF-{-F^== 31, 6. u^-\-uv-{-v^ = S9, u — v + 3 = 0. 17. a4-6 = 4, 6 + a = ab. 6. \V' = 4.t, If+2^=4. 18. ' W~3=Vy 7. fr,2 = 4r2+18, l3ri = 4r2 + 24. 19. 8F+12 = 4^, \3V' + 2t'=^S-\-t. 8. y + 3i?5 + g2 = 22, l2p = g. 20. 2x-5 = x', .x-{-3x' = 2xx'. 9. 2i? + r=:14, I 222 + 3 22,. = 49. 21. (6-2/)(7 + ^) = 80, ■ z + y = 5. 10. |c + 2(^ = 4, 22. m=3w + l, ,n^ + mn = 33. 11. M-\-N=2, M'-MN-\-N'=6. 23. \R = 3S+1. 12. y-2x = 12, ,2/2_ar2_3a;-}.6 = 0. 24. 3T-t-12 = 0, 19^2 + ^2^72. 288 ELEMENTARY ALGEBRA fa &_13 25. U"^a 6' 28. a-{-h=n. {G-\-t = L 26. M+N=2y [V.-l'^ — 'M -^4.1 = 6 ^^* U^v"^' 27. -,^_g^l9, 30. B B' B'' I p~ p' \a + B = 9, 31. Find two numbers whose difference is 2 and the sum of whose squares is 34. 32. The hypotenuse of a right triangle is 20 feet and the sum of the legs 28 feet. Find the lengths of the legs. 33. A rectangular field is 40 rods longer than it is wide, and its area is 1152 square rods. Find the dimensions of the field. 34. A rectangular field contains 18 acres, and the length of the fence around it is 232 rods. Find the length and width of the field. 35. The base of a triangle is 5 inches longer than the altitude, and the area is 42 square inches. Find the base and altitude. 36. Of two squares the side of one is 7 feet more than the side of the other, and the area of the larger is 161 square feet more than the area of the other. Find the sides of the squares. 37. A bin is to be made to hold 144 cubic feet. It must be 3 feet deep and 3 times as long as wide. Find the dimensions of the bin. 38. Of two machines in a mill one can turn out an order of goods in 3 hours less time than the other, and if both machines are operated at once, they can turn out the goods in 2 hours. How long would it require each machine alone to turn out the goods ? 39. If one machine in a mill can produce a quantity of goods in one half of the time required for a machine of smaller capacity to do it, and the two machines together can produce the goods in 4 days, how many days would it take each machine alone to produce the goods ? SYSTEMS INVOLVING QUADRATIC EQUATIONS 289 40. If one boiler would consume a tank of water in 2 hours less time than another, and the two together would exhaust it in 2 hours 55 minutes, how long would the tank of water supply each boiler alone ? 41. In any circle whose radius is R, diameter Dj and area Sj S = 7VR\ D = 2R. By eliminating R between the two equations, get the equation between S and D. 42. In any sphere whose radius is Rj diameter D, and area S, D = 2R. By eliminating R between these two equations, derive the equation between /S and D. 43. In any right circular cylinder the radius of whose base is R, altitude H, area of cylindrical surface S, and volume V, S = 7rRII, V =7r R'H. Eliminate R between these two equations and get a formula between S, V, and //. 153. Graphs of Quadratic Equations. — It was shown in § 124 that the graph of a linear equation containing two unknown numbers is always a straight line. Similarly, it will be found that in general the graph of a quadratic equation containing two unknown numbers is some kind of curve. For example, consider the graph oi y'^ = Q x -\- IQ. By assigning values to x, and computing the corresponding approximate values of y, the following sets of values of x and ij are obtained : X 30 20 10 6 2 1 -1 -n Greater neg. values y ±16.9 ±14 ± 10.3 ±8.4 ±5.8 ±5 ±4 ±2.6 Imaginary 290 ELEMENTARY ALGEBRA By locating the points corresponding to these sets of values of x and y and joining them, as in § 124, we get the curve shown below. ^'nse^ve that the graph is perfectly symmetrical with reference to the axis X-3^. By assigning greater and greater positive values to x, corresponding seal vames can always be found for y, which shows that the two branches of the curve continue indefinitely to the right. But if negative values are assigned xio x larger than 1|, the corresponding values of y are found to be imaginary, which shows that no part of the graph extends farther to the left than x —■- 1|. Thfc) ;ibove curve is called a parabola. It is the kind of path in which astronomers have found that many comets move. Other comets and all of the planets move in closed curves called ellipses. 154. Systems solved Graphically. — The solution of a system consisting of one linear and one quadratic equation may be found graphically, as in § 132, by drawing the graphs of both equations SYSTEMS INVOLVING QUADRATIC EQUATIONS 291 upon the same axes. The sets of values of the unknown numbers corresponding to the points where the graphs meet are the solu- tions of the system. -y = 4, (1) 41/2 = 400. (2) The graph of (1) is the straight line (1) in the figure below. By assign- ing values to x in (2) the following sets of values of x and y are obtained : Example. — Solve graphically X 5 -5 10 -10 15 -15 20 -20 y ±10 ±9.7 ±9.7 ±8.7 ±8.7 ±6.6 ±6.6 For either positive or negative values of x greater than 20, y is imaginary. And for either positive or negative values of y greater than 10, x is imaginary. Hence the graph lies entirely within the region bounded by a; = 20 on the right, X = — 20 on the left, y = 10 above, and y =— 10 below. The graph of (2) is seen to be the ellipse (2) in the figure. 292 ELEMENTARY ALGEBRA The straight line (1) meets the curve (2) at two points, P and Q. The set of values x = 12, ?/ = 8 corresponding to P, and the set of values x— — 5.6, y = — 9.6 corresponding to Q, are the two solutions of the system, and will be found to satisfy both equations. EXERCISES Draw the graphs of : 1. 0^2 = 4?/ + 4. 2. 0^ + 2/2 = 256. 3. 40^ + 92/2 = 324. Solve graphically: ix-y = 2, 4. x'-y^ = 16, 5. 9T'-y^ = S6. 6. xy-^Sx-10y=:S0, 7. 9. a;2 + 2/2 = 100. y-2x = 2, 4a;2 + 2/2 = 164. a^ = 22/ + l, a; + 2 2/ = 4. 10. 11. 12. 5 0^ + 22/2 = 532, 3o;-2/ + 10 = 0. 2x-5y = S, o.-2_2/2 = 171. 2 a; - 3 2/ = 10, 3i»2_5a;2/ = 2 2/2. 13. If the two graphs of a system did not meet at all, what kind of solutions would the system have ? 14. If the graph of the linear equation of a system just touched the graph of the quadratic, what would be the nature of the solutions ? SUPPLEMENTARY EXERCISES The roots of a quadratic equation in one unknown number may be obtained graphically as follows : r O X^ ~~ ?/ Eliminating y from the system J ' _^ gives the gen- eral quadratic equation in one unknown number ax^ + 6ic + c = 0. Since the roots of this equation must satisfy both equations of the above system, they may be found by solving this system graphically. SYSTEMS INVOLVING QUADRATIC EQUATIONS 293 1. -20 Solve = 0. 2a;2-f3a; Solution. — This equa- tion corresponds to the system 2y + 3a;-20 = 0. (1) (2) The graph of (1) is the parabola in the figure. The graph of (2) is the straight line meeting the parabola at P and Q. The values of x corresponding to the points P and Q are 2^ and — 4, respectively, which are the roots of the given equation. Note. — Since the equa- tion x^ = y will be the same for the systems formed from all quadratics, the parabola need be dra^vn only once and all quadratics solved by use of the one figure. To solve any quad- ratic in x, substitute y for 0:2, and find two points of the graph of the linear equation formed. Connect- ing these by a ruler, ob- serve the values of x corre- sponding to the points where the ruler crosses the parabola. These are the roots of the quadratic. Draw a large perfect parabola, and by use of it and a ruler find the roots of: 2. x^ — x--2 = 0, 3. a^~2a;=8. 4. sc'^^x + lO. 6. Q^^7x + X2==0. 6. a^ + a; = 12. 7. a;2 = 2a^ + 15. 8. 2a^-5a:-|-2=0. 9. 2a?2-}-5a; = 3. 10. 5x^=1 4- 4 a;. 11. 6a;2 = aj + 35. 12. 4.x-2=21a;-f 18. 13. 5 ar^ = 3 a; -I- 14. GHAPTEK XVI EXPONENTS 155. Positive Integral Exponents. — A positive whole number used to indicate how many times a given number is to be used as a factor was defined in § 5 as an exponent. Expressions some- times are encountered in which negative numbers, zero, and frac- tions are written in the form of exponents. In this chapter we shall discover the meanings of such expressions. In the following sections are given some laws which hold for positive integral exponents. Some of these laws have been used in earlier chapters, and are given here primarily for review. Others have been applied only in special cases. 156. Law of Exponents in Multiplication. — The law of exponents in multiplication, as stated in § 48, is expressed in symbols by fl'" X a" = a'"'*"". The general proof is as follows : a"* = axaxa"- to m factors. a" = axaxa--- to 71 factors. Hence, a"* X a" =:(a X a x a ... to m factors) (a X a X a ••• to n factors) = axaxaxa"' to m-\-n factors = a"*"*"", by definition of an exponent. EXERCISES Give orally the products of : 1. a^ X a*. 3. a^ X a^. B. v X v^. 2. N^xN". 4. «2o X ^^ g ^12 ^ ^w 294 EXPONENTS 295 7. r^** X r". 12. F^ X F' X F^. 17. a"+i x a""^ x a^^. 8. y^^xy^. 13. s^"xs'xs2. 18. p^-^y^p^+\ 9. P^ X P^'. 14. QxQ^x Q\ 19. 6i2-« X 6»+3. 10. d« X d^''. 15. A;i2 X A;!^ X A;*'. 20. F"""* X F^^+^m^ 11. a?xa^xa\ 16. a;" x a^" X a:^". 21. Q^^" X Q*"-*. 157. Law of Exponents in Division. — The law of exponents in division was given iu § 52. In symbols, Qtri ^ qh -~ Qin-n^ This follows from § 156, because the quotient a"*"** times the divisor a** equals the dividend a"*. EXERCISES Give orally the quotients of: 1. /--y. 8. (r^(^'- 15. a.4n+l _^ ^ 2. A'^^A'. 9. ^22 ^^W 16. ^2x+4^^x+l 3. iv^-h w. 10. mi^H-7ft". 17. yAt+10 _^ yS 4. B^'^R'. 11. B^-^B'^ 18. pr3a+26 _j_ y^a+b^ 5. N'*^]Sr'. 12. Jc'^^l''. 19. Zm _^ ^m-2^ 6. ^20^^12^ 13. W"^ -- W\ 20. Qt+^^Qt-2^ 7. x^'^x^. 14. R'^^-^R^. 21. Kr^^^K^-\ 158. Power of a Power. — The law by which a power of some number is itself raised to a power is expressed by. That is, the mth power of the nth power of any number equals tht mnth power of the number. For, by definition of an exponent, (a")*" = a" X a" X a" • • • to m factor* _ ^n+n+n+ .•• to m terms^ j^y § -j^gg Thus, (a^y = a2o ; (i^)^ = wjis . (^4)9 ^^m 296 ELEMENTARY ALGEBRA EXERCISES Give orally : 1. {aj. 6. (f)\ 11. (Ay. 16. (QyK 2. {p^)\ 7. (B-y. 12. (s^y. 17. (r^y-'. 3. {Py. 8. Q^y, 13. (ay. 18. (R-+y- 4. {ry. 9. (E!^y\ 14. (w»)2- 19. (^Jj2,n-ny 5. {y^y. 10. (mY- 15. (a^y-. 20. (2y. 159. Power of a Product. — The law by which a power of the product of two numbers is found is expressed by, (aby = a"b". That is, tJie nth power of the product of two numbers equals the jproduct of the nth powers of the numbers. For, by the meaning of an exponent, (aby — (ab)(ab)(ab) ••• to n factors = (ct X a X a ••• to n factors)(6 X 6 X ?> ••• to n factors) = a'^6^ By similar reasoning the law can be shown to hold for any number of factors. Thus, {ahcdy = a^h^cH^ ; (8 xyY = 3* x^y^ = 81 oc^yK 160. Power of a Fraction. — The law by which a fraction is raised to a power is expressed by, fa\" ^ a^ \bj b"' That is, the nth power of a fraction equals the nth power of the numerator divided by the nth power of the denominator. For, /^lY = ^x-X-.-.ton factors \bj b b b Thus. /2a6y^ (206)4 ' X^cdJ (3 cdy ax a X a '•' to n factors b xb X b -"to 71 factors or b^' (2 ahy _ 2* g^M ^ 16 a^b^ 3*c4d* SI c^d* EXPONENTS 297 161. Power of any Monomial. — By use of the laws of exponents in the preceding sections, any power of any monomial may be obtained. Thus, And (2 a3&5c4)6 = 26(a8)6(65)6(c4)6 by § 159 = G4 ai8630c24 by § 158. 81m^ 625^12^20 by §§ 159 and 168. EXERCISES Give orally: 1. {2ay. 2. (p^f)'. 3. (2N^]\PY' 4. {a^hhy^ 5. {P'fiff. 6. (m'vhy. 7. {BA^B'y. 3. {m'nYy\ 9. (3 v'py, 10. (J^nY- 11. 12. ©■■ 2E^. "■ (^0' 15 f N'M' \ ; 162. Root of a Power. — To find any root of any power of a hose divide the exponent by the index. That is, Va^ m an ' For, by § 158, (a^)" = a**, or a*. Thus, \^ = a^; \/^ = c« ; Vm^ = m^. EXERCISES Give orally : 1. c 31. Change x^ — ^x- ^ ^^^ ~ ^) to a fractional form. a;— 2 32. Change ah, ^~~ , and ^"^ to fractions having a common a + 6 a — 6 denominator. 33. Add2a, 3a + ^, and a- — - 5 9 34. Add ^-* , __A±^ , and ^+» (6-e)(c-a) (c-a)(a-6) (a-6)(6-c) 802 ELEMENTARY ALGEBRA 5 1 24 35. Simplify 2(0; + 1) 10(a:-l) 5(2 a; + 3) ax 36. Multiply -^ by 4: ax my""-^ 37. Multiply a;^ - a; + 1 by ^ + - + 1. or X 1 — x^ 1 — v^ 38. Find the product of , ^, and 1-}- 1 +y x-\-xr 39. Divide x* by oj^ + — • x^ or 40. Simphfyr__-,-^3J^^-±^ + -^^ 1 41. Simplify X j,^ a ^ trr 5 X — 4: S X -{- 4: OA 307-4-8 - 42. Solve 17 a; -^ — = 20 a; ^ 5. A^ cs 1 a — x 4a — X ■. o 43. Solve • = a — 6 for a;. h c 44. Solve — = 1 ^- = for x, ah — ax he — ox ac — ax 45 |3a;+72/ = 33, ^g |6a + 56 = 112, t2a; Solve the following systems : + 72/ = 33, + 42/ = 20. '" •[8a-26 = 80. 46 f7a; + 32/ = 62, 43 f 6 ^ + 11 ^^ = 115, l5a;-22/ = 36. ' j 8 ^-22?^ = -30. ^^ .12a; + 82/ = 116, 50 f 2^+ 3^ = 47 ri2a; 1 2a;- 2/ = 3. tl0;2-12^=-63. a; y 51. Solve ^l for x and «. a; , y ^ c d MISCELLANEOUS EXERCISES 303 Solve : r 8a:-92/-7z=-36, nx-\-A.y- z = 78, 52. 12a?— 2/-32!=36, 53. i 4a;-5 ?/ -3^ = - 21, [ Q>x-2y- 2 = 10. i a;-32/-42 = -37. 54. When the cost of an article to a merchant is lowered 20 %, the merchant who keeps the same selling price makes 30 % more. Find his former per cent of gain. 55. By getting a discount of 5 % off for cash in buying, a mer- chant makes a net profit of 8 % more. Find his rate of gain when no discount is received. 56. A merchant has two kinds of grain, one at 60 cents per bushel, and the other at 90 cents per bushel, of which he wishes to make a mixture of 40 bushels that may be worth 80 cents per bushel. How many bushels of each must he use ? 57. A merchant has three kinds of sugar. He can sell 3 lb. of the first quality, 4 lb. of the second quality, and 2 lb. of the third quality, for 60 cents; or, he can sell 4 lb. of the first quality, 1 lb. of the second quality, and 5 lb. of the third quality, for 69 cents ; or, he can sell 1 lb. of the first quality, 10 lb. of the second quality, and 3 lb. of the third quality, for 90 cents. Find the price of each quality. 58. How much cream testing 36 % butter fat must be mixed with 20 gallons of milk testing 3 % butter fat to have milk test- ing 4.8 % butter fat ? 59. Find the square root of 4 a* — 16 a'' + 24 a^ — 16 a + 4 60. Find the square root of m^ + 2 m — 1 — — + — . 61. Find the square root of 33.1776. 62. Find to three decimal places the square root of 2. Simplify the following: 63. V96 a^a^. 64. -s/ {x" - f){x + y). 304 ELEMENTARY ALGEBRA 65. xP^^. 66. ^-±l^E^. 67. Multiply 4 a^y/H by f^A. 68. Multiply a + 2 v& by a — 2 V&. 69. Divide V20 + Vl2 by V5 + V3. 70. Solve 4 ar^ + 16 a; = 33. 71. Solve 4 a; -5^^ =14. a; + l 72. Find to three decimal places the roots of ^2^1 3 2 X ^ X 73. A lady finds that there are just 85 square yards of floor surface to be covered in two of her square bedrooms, one of which is 3 feet longer than the other. How large is each ? .. oi f^-2/' = 23-3a;?/. ^ ^, [23^-5^^ = 20- ^1 76. A farmer has two small fields, each an exact square. It takes 200 rods of fence to inclose both. Together they contain 8i acres. Find the dimensions of each. Simplify and express with positive exponents : 77. ahhxah-h^. gO "^^x—- 78. (a'b'cf X (a^bciy. 3« X 3"+^ ^>s INDEX (Numbers refer to Sections.) A-bsolute value, 26. Addition, 14, 30, 39, 40, 103, 140. elimination by, 126. Algebra, 1. Antecedent, 114. Arrangement of terms, 61. Axioms, 18. Base, 5. Binomial, 10. Braces, 7. Brackets, 7. Cancellation, 105. Checking work, 41. Clearing of fractions, 111. Coefficient, 12. numerical, 12. literal, 12. Comparison, elimination by, 127. Complete quadratic equation, 144. Completing the square, 147. Complex fractions, 109. Conditional equations, 15. Consequent, 114. Constant, 120. • Cubic equation, 57. Degree of an equation, 57. of a term, 57. Denominator, 97. lowest common, 102. Distribution, law of, 50. Division, 35, 52, 53, 64, 56, 108, 142. Divisor, trial, 135. true, 135. Elimination, 68. by addition or subtraction, 126. by comparison, 127. by substitution, 128. Ellipse, 153. Equations, 15. complete quadratic, 144. conditional, 15. cubic, 57. degree of, 57. equivalent, 131. fractional, 110. graphs of, 124, 153. identical, 15. inconsistent, 131. linear, 57. literal, 113. members of, 15. pure quadratic, 144. quadratic, 57. roots of, 17. simultaneous, 66. systems of, 67. to solve, 17. Equivalent equations, 131. Evaluation of polynomials, 11. Exponents, 5. fractional, 165. laws of, 48, 52, 156, 157, 158, 159, 160 162. negative, 163. positive integral, 155. zero, 164. Expressions, 8. fractional, 100. integral, 100. literal, 8. mixed, 100. names of, 10. number, 8. Extremes, 116. Factoring, 83. general suggestions on, 93. Factors. ^ 305 B06 INDEX Factors, grouping of, 47. highest common, 94. order of, 4li. prime, 83. Formula, 2, for solving quadratic equations, 148. Fourth proportional, 116. Fractional equations, 110. exponents, 165. expressions, 100. Fractions, i»7. clearing equations of, 111. complex, 109. denominator of, 97. lowest common denominator of, 102. numerator of, 97. signs of, 98. terms of, 97. General numbers, 3. Graphs, 29. of linear equations, 124. of quadratic equations, 153. Grouping, laws of, 30, 47. signs of, 7. Highest common factor, 94. Identical equations, 15. Identity, 15. Imaginary numbers, 137. Inconsistent equations, 131. Index of root, 72. Integral expressions, 100. Law of distribution, 50. of grouping in addition, 30. of grouping in multiplication, 47. of order in addition, 30. of order in multiplication, 46. Laws of exponents, 48, 52, 156, 157, 158, 159, KJO, 162. Lever, 63. Like terms, 13. Linear equations, 57. graphs of, 124. Literal equations, 113. expressions, 8. numbers, 3. Lowest common denominator, 102. Lowest common multiple, 95. Mean proportion, 116, proportional, 116. Means, 115. Members of an equation, 15. Mixed expressicms, 100. Monomial, 10. Multiple, 95. lowest common, 95. Multiplication, 32, 49, 60, 51, 105, 106 107, 141. signs of, 4. Negative exponents, 163. numbers, 23. Number expression, 8. Numbers, general, 3. imaginary, 137. literal, 3. negative, 23. opposite, 26. particular, 3. positive, 24. real, 137. Numerator, 97. Numerical coefficient, 12. Opposite numbers, 26. Order, laws of, 30, 46. Parabola, 153. Parentheses, 6. Particular numbers, 3. Polynomials, 10. evaluation of, 11. Positive numbers, 24. Powers, 5. of a fraction, 160. of a monomial, 161. of a power, 158. of a product, 159. of positive and negative numbers^ 34. roots of, 162. Price curves, 123. Prime factors, 83. Proportion, 115, important principles in, 119. mean, 116. terras of, 115. Proportional, fourth, 116. mean, 116. INDEX 307 Proportional, third, 116. Pure quadratic equation, 144. Quadratic equation, 57. complete, 144. graph of, 153. pure, 144. Quadratic surd, 138. Quality, signs of, 25. Radical sign, 72. Rate of motion, 61. Ratio, 114. Real numbers, 137. Root, index of, 72. of a power, 162. of an equation, 17. square, cube, etc., 72. Signs of a fraction, 98. of quality, 25. Signs of grouping, 7. insertion of, 45. remo^ al of, 44. Similar terms, 13. Simple machines, 118. Simultaneous equations, 66. Solution of a system, 67. Special products and quotients, 70. Specific gravity, 62. Speed, 61. Square of a binomial, 78. of a polynomial, 80. of arithmetical numbers, 79. perfect, 86. Substitution, 20. elimination by, 128, 152. Subtraction, 14, 31, 42, 43, 103, 140. elimination by, 126, Surds, 138. quadratic, 138. simplest form of, 139. Systems of equations, 67. Terms, 9. arrangement of, 51. degree of, 57. of a fraction, 97. of a proportion, 115. similar, 13. Third proportional, 116. Transposition, 36. Triangles, similar, 117. Trinomial, 10. True and trial divisors, 136L Variable, 120. Velocity, 61. Vinculum, 7. Zero exponent, 164. 2^ THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $1.00 ON THE SEVENTH DAY OVERDUE. MOV 4 1938 Sfi^ I inA2 mv 5 ^13 1942 DEC 27 .M ^£^ 16194) JUN 2 6 ^^ ^ DEC 5 l&li SEP 21 tt4&^ J^N 25 1941 ^m^^ im ^.■r' '-^^^'^ AUG 3i 1942 MAR 9 1940 yqQo'^ ^A^ RETC'D LD 4 Nov'57ES MAR 1 'b5 -9 pm HT /