on 39 .K23 Copy 1 QA X23 Class ., Book Gopiglit]^^ GCPmiGHT DEFOSm ^ EIGHTH GRADE MATHEMATICS By Harry M. If^EAL Head of the Mathematics Department Cass Technical High School Detroit, Michigan and Nancy S. Phelps Grade Principal Southeastern High School Detroit. Michigan ATKINSON, MENTZER ^ COMPANY NEW YORK CHICAGO ATLANTA DALLAS ^^31 K: COPYRIGHT, 1917, BY' ATKINSON, MENTZER & COMPANY / OCT 24 1917 ©CU477199 Introduction THE growth of this series of Mathematics for Secondary Schools, has covered a period of seven years, and has been simultaneous with the growth and development of the shop, laboratory, and drawing courses in Cass Technical High day school, as well as in the evening and continuation classes. The authors have had clearly in mind the necessity of first developing a sequence of mathematics that would enable the student to recognize fundamental principles and apply them in the shop, drawing room, and laboratory; and, second to so develop the course that each year's work would be a unit and not depend upon subsequent development for inteUigent application. It has been assumed that the school work-shop, drawing room, and laboratory would furnish opportunity to apply mathematics and that it was not necessary to exhaust every possible application in the mathematics class. The authors have been aware of the popular demand for a closer union of algebra and geometry, but have recognized that demand only when the union came about naturally and would assist the mathematical sequence desired. Instructors in the wood shop, pattern shops, machine shop, drawing rooms, chemistry, physics, and electrical laboratories, etc., have furnished examples of mathematical appUcation incident to the respective subjects. Hundreds of problems arising in the industries, have been brought in by the machinists, sheet metal workers, carpenters, electrical workers, pattern makers, draughtsmen, etc., etc., coming to the evening and continuation classes. Complete charts of machine shop work and electrical distribution requirements have been made, including a statement of the required sequence of mathematics. All of this material has been classified, with a view to the mathematical sequence. The net result is a series of Mathematics so organized that a mastery of the text makes it possible for a student to use mathematics intelli- gently in the various departments of the school, in the industries, and at the same time prepare for college mathematics. E. G. ALLEN, Director Mechanical Department, Cass Technical High School, Detroit, Mich. TABLE OF CONTENTS PAGE CHAPTER I The E quation , 1 CHAPTER II Evaluation 15 CHAPTER III The Equation Applied to Angles 25 CHAPTER IV fT, SUBTRACTIO> tion and Division 38 Algebraic Addition, Subtraction, ]\Iultiplica CHAPTER V Ratio, Proportion and Variation 79 CHAPTER VI Pulleys, Gears and Speed 96 CHAPTER VII Squares and Square Roots 107 CHAPTER VIII Formulas 123 V CHAPTER I THE EQUATION ll irA 10 ^ Fig. 1 ^ 1 In order to find the weight of an object, it was placed on one pan of perfectly balanced scales (Fig. 1). It, together with a 3-lb. weight, balanced a 10-lb. weight on the other pan. If 3 lbs. could be taken from each pan, the object would be balanced by 7 lbs. This may be expressed by the equation, x+3 = 10, in which the expressions xH-3 and 10 denote the weights in the pans, the sign ( = ) of equality denotes the per- fect balance of the scales, and x is to be found. 2 Equation: An equation is a statement that two expres- sions are equal. The two expressions are the members of the equation, the one at the left of the equality sign being called the first member, and the one at the right, the second member. 3 From the explanatory problem, it will be seen that the same number may be subtracted from both members of an equation. Oral Problems: Solve for x: 1. x+7 = 21 3. x+1. 1 = 3.5 2. x+2 = 3 4. x+2t = 7| x+f 5 _ J_i 1 2 THE EQUATION ^1\^ .f^pni^,^ o ^ Fig. 2 ^ 4 It is required to find the weight of a casting. It is found that 3 of them exactly balance a 10-lb. weight (Fig. 2). If the weight in each pan could be divided by 3, one casting would be balanced by 3^ lbs. This may be expressed by the equation, 3x=10, X = 3i. (dividing both members by 3) 5 From this explanatory problem, it will be seen that both members of an equation may be divided by the same number. Oral Problems : Solve for x : 1. 4x=12 2. 2x=16 3. 5x = 9 4. llx = 33 5. l.lx=12.1 Example : Solve f or x : 5x + 1 2 = 37 5x = 25 Why? x= 5 Why? THE EQUATION Exercise 1 Solve for the unknown: 1. x+l = 5 11. 9x+8 = 116 2. x+7 = 9 12. 7w+5f = 12f 3. 2a+6 = 16 13. 28t+14=158 4. 3x+7 = 28 14. 3x+4 = 9 5. 5s+17 = 62 15. 15s+.5 = 26 6. 9x+12 = 93 16. llx+J=8^9 7. 2x+l = 6 17. 1.2x+2=14 8. 5y+3 = 15 18. 4.6x-|-8 = 100 9. 4n+3.2=15.2 19. 6.3x+2.4=15 10. 12m+8 = 98 20. 7.1m+.55 = 9.07 Q ,1^™, ^ tim|iinf 10 Fig. 3 ^ 6 If an apparatus is arranged as in Fig. 3, it is seen that if the upward pull of 2 lbs. be removed, 2 lbs. would have to be 4 THE EQUATION put upon the other pan to keep the scales balanced. This may be expressed by the equation, 4x-2 = 10 4x= 12 (Adding 2 to both members) X = 3 (Dividing both members by 4) 7 From this problem, it will be seen that the same number may he added to both members of an equation. Oral Problems: Solve for x: 1. 3x-4 = 8 2. 7x-l = 15 3. 4x-i = 7i 4. 5x-.l = .9 5. 2x-i = 6i Exercise 2 Solve for the unknown: 1. x-7 = 10 11. 13r-21=44 2. 2x-13 = ll 12. 12s -35 = 41 3. 5x-17 = 13 13. 7f-4 = 26 4. 4x-ll = 25 14. 4x-3 = 16 5. 3x-7 = 15 15. 9x-3.2=14.8 6. 12x-4 = 44 16. 3m~2=3.1 7. 7m-5 = 31 17. 14x-5 = 21 8. 4x-18 = 18 18. 2.1x-3.2 = 3.1 9. 17t-3i=13t 19. .5y-4 = 5.5 10. llx-9 = 90 20. 3x-9i = 8.5 SIMILAR TERMS ExeYcise 3. Review Solve for the unknown : 1. 9x-8=46 6. 3w-lJ=lf 2. 8x-7 = 53 7.. 19t-.2 = 3.6 3. 5x+7 = 28 8. 6.37n+3.92 = 73.99 4. 28m-9 = 251 9. .4x+.02=.076 5. 16y+13 = 73 10. 2s+2i = 9f 11. Two times a number increased by 43 equals 63. Find the number. 12. If 10 be added to 3 times a number, the result is 50. What is the number? 13. Five times a number decreased by 6 equals 39. Find the number. 14. If 55 be subtracted from 7 times a number, the result is 22. . What is the number? 15. If to 57 I add twice a certain number, the result is 171. What is the number? 18. State the first five problems in this exercise in words. How many yards of cloth are 7 yds. and 5 yds.? How many dozens of eggs are 12 doz. and 3 doz.? How many bushels of wheat are 8 bushels and 1 1 bushels? How many b's are 4 b's and 7j b\s? How many x's are 3x and 9x? In such expressions as 2a+3x+4+2x+7+3a, 2a and 3a may be combined, 3x and 2x, and also 4 and 7, making the expression equal to 5a+5xH-ll. 2a and 3a, 3x and 2x, 4 and 7 are called similar terms. b THE EQUATION Example 1. Solve for x: 4x+13x— 7x = 40 lOx = 40 (combining similar terms) X = 4. Why? Example 2. Solve for x : 14x+7-2x=43 12x+7 = 43 Why? 12x=36 Why? x = 3 Why? 9 8-7+3 = ? 8x-7x+3x = ? 8+3-7 = ? Similarly 8x+3x-7x = ? 3-7+8 = ? 3x-7x+8x = ? • 10 These problems illustrate the principle that the value of an expression is unchanged if the order of its terms is changed, provided each term carries with it the sign at its left. NOTE: If no sign is expressed at the left of the first term, the sign (+) is understood. Example 1. 15-3x+llx = 39 8x+15 = 39 Why? 8x = 24 Why? X = 3 Why? Example 2: lly-4+21 = 50 lly+17 = 50 Why? lly = 33 Why? y = 3 Why? ORDER OF TERMS 7 ^ Exercise 4 Solve : 1. 4x-x = 12 2. llx+3x = 35 3. 14x-3x=44 4. 3x+7x = 90 5. 9y-9y+8y = 40 6. 4s+3s-2s = 17 7. 3.2x+2.3x = 110 8. 1.3y-2.7y+3.3y = 57 9. 11.2x+7.8x = 57 10. l.ls-1.4s+lls = 26.75 Exercise 5 Solve: 1. x-18=17 9. 12x-8x+6+3x = 8+12 2. x+18 = 21 10. 25x+20-7x-5+5x=56+5 3. 2y-16 = 30 11. 8x+60+4x-50+3x-7x = 20 4. 3m-m = 21 12. 2-2x+7x=42.5 - 6. 3m-l = 23 13. 3y+1.2+2y=46 6. 6.5x~l.l = 50.9 14. x-L25x+12.7+3.5x = 38.7 7. 4x+3x-3 = 25 16. 2x+15.8-2.3x+14.5x = 186.2 8. lly-4y-7 = 28 16. 6.15y-1.65y+7.8 = 57.3 17. 8y+6.875+2y=46.875 18. z-8.73+5.37z = 61.34 19. 5t-8.75t+6.87+8t = 57.87 20. 3.73x-9.23+15x = 65.69 8 THE EQUATION 11 Equations often arise in which the unknown appears in both members. In that case, aim to make the term containing the unknown disappear from one member, and the one contain- ing the known, from the other member. Example 1: 3x — l=x+3 3x = x+4 (adding 1 to both members). 2x = 4 (subtracting x from both members). X = 2 Why? Note that in adding or subtracting a term from both members, it must be combined with a similar term. Example 2: 5x+4-3x-l = 7-x+2 2x+3 = 9 — X (combining similar terms in each member). 2x = 6-x Why? . 3x = 6 Why? x = 2 Why? Exercise 6 Solve: 1. 2x-6 = x 2. 2x+3 = x+5 3. 13x-40 = 8+x 4. 7y-7 = 3y+21 5. 9x-8 = 25-2x 6. 20+10x = 38+4x 7. 3x+9+2x+6 = 18+4x 8. 5x+3-x = x+18 9. 7m-18+3m=12+2m+2 10. 18+6m-l-30+6m = 4m+8+12+3m+3+m+29 CLEARING OF FRACTIONS \) 11. 25x+20-7x-5 = 56-5x+5 12. 10x-61-12x+27x = 8x-41+20+4x+25 13. 25§+5x+6x+9i-2x=180-8x-8f 14. 2.8x+39.33+x = 180-1.2x+32.09-7.16 15. 5xH-26f+9x = 360-5x-143f 12 If an object in one pan of scales will balance a 4-lb. weight in the other, it will be readily seen that 5 objects of the same kind would need 20 lbs. to balance them. This may be expressed by the equation, x = 4 5x = 20 (multiplying both members by 5). 13 From this problem, it will be seen that both members of an equation may be multiplied by the same number. This principle is needed when the equation contains frac- tions. The process of making fractions disappear from an equation is called clearing of fractions. IJf, RULE: To clear an equation of fractions, multiply both members by the lowest common denominator (L. C. D.) of all the fractions contained in the equation. Example 1: -+3 = 4 A X+6 = 8 (multiplying both members by 2). X = 2 Why? T. 1 r» r r 16 Example 2: - — = — 3 7 3 7r — 3r=112 (multiplying both members by 2 1 ) . 4r=112 Why? r = 28 Why? 10 THE EQUATION Example 3: — —3x77+ — = — 4 ^0 5 5 3 15m-198+12m = 84-20m Why? 27m -198 = 84 -20m Why? 47m -198 = 84 Why? 47m = 282 Why? m = 6 Why? 15 The four principles used thus far may be more generally stated as follows : 1. If equals are added to equals, the results are equal. 2. // equals are subtracted from equals, the results are equxil. 3. If equals are multiplied by equals, the results are equxil, 4 . // equals are divided by equals, the results are equM. Solve: Exercise 7 1. ^-^ = 10 3 6 6. X 2_x 2 3~6 2. ?+'?^ = 9 5 4 7. i-^- 3. 3 4 8. i--i« 4. xH-|x = 6 9. 2x x_x 1 9 6 18 3 5. x-Jx = 7 10. 3x 1 X ,_ — — -j-O 7 3 21 PRINCIPLES OF EQUATIONS 11 11. l|s+fs = s+13 13. -+4r--=26+l|r 12. |x+2/2=3+- 14. -+---+- = 82 ^ ^^ 4 2 3 4 10 15. 7x+y+-+23=-+5ix+113 16 Sometimes it is convenient to make the term containing the unknown disappear from the first member, and the one containing the known, from the second. Example 1: x+6 = 3x-2 6 = 2x-2 Why? 8 = 2x Why? X = 4 Why? 1 A Example 2: — =4 • 3x 16 = 12x Why? (L. C. D. is 3x) x = = 14 Why? Exercise 8 Solve: 1. L5 = 5 a 4. 3x7 4 2 2. 5=15 a 6. 6. 16_2x 5 3 14 = x+9 3. 1 = 2 4x 7. 17 = 2x-3 12 THE EQUATION 8 x+10 = 2x-9 12. ??+47=-+4n 2 7 9. 2x-2i = 5x-17i " 13. ^-l=iI?-?-2ia 2 3 3 10. 7x+20-3x = 60+4x-50+8x 14. .lx+6.2 = .3x+.2 11. 3m+60=15mH-3-2m+7 15. 10+.lx = 5+ix Exercise 9 Solve: 6 12 3 9 6 2 2. 7x-8 = 6x+ix 6. ^■i-'i-^ = '^-^ ^ 2 5 6 3 4 3. ??-^=25i-^ 7. Z-^+2f = -^-^+^ 56 3 12 8484 4. ??+3 = ^x-2 8. ^x-^x+4f = 3x+- 3 6 3 5 ^ . 15 9. lly^x-lix-302 = 60+lJx+183 10. x-3f+ix = 9i-^ o I PROBLEMS 13 s Exercise 10 1. Five times a certain number equals 155. What is the number? 2. Four times a number increased by 7 equals 43. Find the number. 3. Twelve times a number decreased by .18 is equal to 17.82. Find the number. 4. There are three numbers whose sum is 72. The second number is three times the first, and the third is four times the first. What are the numbers? 5. The sum of two numbers is 12 and the first is 4 more than the second. What are the numbers? 6. If 10 is subtracted from three times a number, the result is twice the number. Find the number. 7. If I" of a number is increased by 6, the result is 30. Find the number. 8. The sum of ^, f and J of a number is 26. What is the number? 9. Divide 19 into two parts so that one part is 5 more than the other. 10. Divide 19 into two parts so that one part is 5 times the other. 11. Divide $24 between two persons so that one shall receive $2 J more than the other. 12. A farmer has 4 times as many sheep as his neighbor. After selling 14, he has 3j times as many. How many had each before the sale? 14 THE EQUATION 13. Two men divide $2123 between them so that one receives $8 more than 4 times as much as the other. How much does each receive? 14. Three candidates received in all 1020 votes. The first received 143 more than the third, and the second 49 more than the third. How many votes did each receive? 15. A man spent a certain sum of money for rent, f as much for groceries, $2 more for coal than for rent, and $28 for incidentals. In all he paid out $100.00. How much did he spend for each? 16. A farmer has 24 acres more than one neighbor and 62 acres less than another. The three together own one square mile of land. How much has each? 17. A man traveled a certain number of miles on Monday, -f- as many on Tuesday, f as many on Wednesday as on Mon- day, and on Thursday 10 miles less than twice as many as he did on Monday. How far did he travel each day if his trip covered 82 miles? 18. One man has 3 times as many acres of land as another. After the first sold 60 acres to the second, he had 40 acres more than the second then had. How many acres did each have before the transaction? 19. One boy has $10.40 and his brother has $64.80. The first saves 20 cents each day, and his brother spends 20 cents each day. In how many days will they have the same amount? 20. A man after buying 27 sheep finds that he has If times his original flock. How many sheep had he at first? CHAPTER II EVALUATION 17 Definite Numbers: The numerals used in arithmetic have definite meanings. For example, the numeral 7 is used to represent a definite thing. It may be 7 yards, 7 pounds, 7 cubic feet or 7 of any other unit. Also in finding the circum- ference of a circle, we multiply the diameter by tt which has a fixed value. Numerals and letters which represent fixed values are called definite numbers. 18 General Numbers: The area of a rectangle is found by multiplying the base by the altitude. This may be expressed by bXa, in which the value of b may be 10 ft., 6 in., 30 rds., or any number of any unit used to measure length, and a may be any number of a like unit. Letters which may represent different values in different problems are called general numbers. 19 Signs: When the multiplication of two or more factors is to be indicated, the sign of multiplication is often omitted or expressed by the sign (•). Thus 7XaXbXm is written 7-a-b-m or more often 7abm. NOTE: Care should be taken in the use of the sign (•) to distinguish it from the decimal point. 7-9 means 7X9, 7.9 means 7-i^o- 20 Coefficient: The expression 7abm may be thought of as 7ab-m, 7 a-bm, 7-abm, or 7b -am, etc. 7ab, 7a, 7, and 7b are called the coefficients of m, bm, abm, and am respectively. 1. In the following, what are the coefficients of x*^ 4abx; l^xyz; 17mxw. 2. Name the coefficients of ab in the following: 3jaxby; l^mabz; .9bnsa. 3. What is the coefficient of 17 in 17mxw? 15 16 EVALUATION The coefficient of a factor or of the product of any number of factors, is the product of all the remaining factors. In 8axy, 8 is the numerical coefficient. The numerical coefficient 1 is never written, laxy is written axy. 21 Power: If all the factors in a product are the same, as x-x-x-x, the product is called a power, x-x-x-x is read "x fourth power'' and is written x^- a- a- a- a- a is read *'a fifth power" and is written a^. b-b or b^ is ''b second power" but is more often read ^'b square." In the same way b-b-b (b^) is called ''b-cube." 22 Exponent: The small number written at the right and above a number is called its exponent and it indicates the power of the number. The exponent 1 is never written, x means x^ or '^x first power." 23 Base: The number to be raised to a power is called the base. Name tlie numerical coefficients, bases and exponents in the following: V^x^ S^aio, 3.7m2n^ ^ttt^, Ifm l|m^ 24 Sign of Grouping: The Sign of Grouping most commonly used is the parenthesis ( ) and means that the parts enclosed are to be taken as a single quantity. For example, 3(x-y) means that x-y is to be multiplied by 3 making 3x - 3y. (x -y)^ means (x-y) (x-y) (x-y). 25 Evaluation: Evaluation of an expression is the process of finding its value by substituting definite numbers for general numbers in the expression, and performing the operations indicated. EVALUATION OF EXPRESSIONS 17 Example 1 : Evaluate 4aV if a = 3, x = 2. 4a2x3=4.32.23=4-9-8 = 288. a2 5b^ m^ Example 2: Find the value of — ; + ;r", ^ m^ c2 2a3 when a=l, b = = 2, c = 5, m = 2. a2 5b4 m^ m3 ~ 72 "^ 2^3 P 5-2^ 2^ 23 52 2-13 1 80 32 8 ~ 25 "^ "2 ■ -^^'• _5- 128+640 40 40 ^^*» Example 3: Evaluate a(a — b+y^) when a = 13, b = 3, y = 4. a(a-b+y2) = 13(13-3+42) = 1326 = 338 Exercise 11 Evaluate the following if a = 8, b = 6, c = 4, d = 2, x = 9: 1. 2x 7. 3x2 2. x2 8. (3x)2 3. 3x 9. llax 4. x3 10. 2abcd 6. 4x 11. 2a2x3 6. x^ 12. x2-a2 18 EVALUATION . . .^ 111 13. xa+b) 17. — r+i X b d 14. 4b(x-c) 18. (x+a)(c-d) 15. a2+2ab+b2 19. vi^ 16. c2-2cd+d2 20. ab(c-3) Exercise 12 Find the value of the following, when a = 2, b = 3, c = 7, d = 4, m=l, x = 5: 11. (3x+7)(c-2) 1. ia^x^c 2. x^-a^ 3. x^+d^ 4. 3b2-4m2 6. xM-a^m 6. 2a2x3(c-d) 7. b m 8. J(x+a)c 9. ia^x^cCb^-d^) 12. Vb2+d2 13. ^(x2-c2+25) bd 14. a3(x-e+3m)(c2+d2) 15. ^^ ah 16. — (x2+a2-b2)(c2-d2-m2) 2d 17. Vx(a+b) 18. ^d(a+b)+c c^ x^ 20. (a+b)(b+c)-(b+c)(x+d) + (x+d)(d+m) PERIMETER FORMULAS 19 Evaluation of Formulas 26 A Formula is the statement of a rule or principle in terms of general numbers. For example, distance traveled is equal to rate times time. Formula, d = r • t Iwt Example 1 : Evaluate b = (Formula for board feet) whenl = 16', w = 8", t = 2" 168. 2 b = 12 = 2li Example 2: Evaluate A = Jh(b+b') (Area of a trapezoid) ifh = 3i3^,b = 12i,b' = 6i A = J.3A(12i+6i) A = i.33^-18 . 1 51 75 A=-. — . — 2 16 4 3825 128 = 29 iVs 0^29.102 Perimeter Formulas 27 The perimeter of a figure enclosed by straight lines is the sum of its sides. Fig. 4. Square Fig. 5. Rectangle 20 EVALUATION Fig. 6. Triangle Fig. 7. Quadrilateral Exercise 13 1. The perimeter of a square (Fig. 4) is equal to 4 times one side. P = 4a. Find P, if a = 9. 2. Find the value of P, in P = 4a, if a = lj. 3. Find the value of P, in P=4a, if a = 1.175. 4. The perimeter of a rectangle (Fig. 5) is equal to a+b+a+b = 2a+2b = 2(a+b). P = 2(a+b). Find P, if a = 3, b = 5. 6. Find P, in P = 2(a+b), if a = |, b = f . 6. Find P, in P = 2(a+b), if a = 1.7862, b = 2.1324. 7. The perimeter of a triangle (Fig. 6) is expressed by the formula, P = a+b+c. Find P, if a = 7, b = ll, c = 19. 8. Evaluate P = a-|-b+c, if a = f, h = ^, c = f. 9. Find the value of P, in P = a+b+c, if a = 7.621, b = 8.37, c=1.3. PERIMETER PROBLEMS 21 10. The perimeter of a quadrilateral (Fig. 7) is expressed by the formula, P = a+b+c+d. Find P, if a = 20, b = 15, c = 13, d=14. 11. Evaluate P = a+bH-c+d, when a=lf, b = lf, c = 1y'^, 12. Find P, in P = a+b+c+d, if a = 172.32, b = 96.3, c = 81.04, d = 56.2. Exercise 14. Equations Involving Perimeters 1. The perimeter of a square is 96. Find a side. 2. The perimeter of a triangle is 114. The first side is 6 less than the second and 24 less than the third. Find the sides. 3. Find the dimensions of a rectangle whose perimeter is 48 if the length is 3 times the width. 4. Find the dimensions of a rectangle if its length is 4 more than the width and its perimeter is 82. 5. The length of a rectangle is 4 more than twice the width and its perimeter is 1351^. Find the length. 6. The perimeter of a rectangle is 48.648. Find the width if it is J of the length. 7. The perimeter of a rectangle is 94. The width is 11.3 more than |- of the length. Find the length and the width. 8. The perimeter of a quadrilateral is 176. The first side is ^ of the second, the third is 8 more than the second, and the fourth is 3 times the first. Find the sides. 22 EVALUATION Exercise 15. Area Formulas cr Fig. 8. Rectangle Fig. 9. Parallelogram b Fig. 10. Triangle Fig. 11. Trapezoid 1. The area of a rectangle (Fig. 8) is equal to the base multiplied by the altitude. A = a-b. Find A, if a = 11.5, b = 18.6. 2. Evaluate A = a-b, if a = 2|, b = 3f. 3. Express the result of problem 2 in decimal form. 4. The area of a parallelogram (Fig. 9) is the base times the altitude. A = a-b. Find A, if a = l^, b = 6.71. 5. The area of a triangle (Fig. 10) is \ the product of the base and altitude. A = Jb.h. Find A, if b = 12.23, h. = 6.57. 6. Evaluate A = Jb.h, if b = 9f, h = 4|. 7. The area of a trapezoid (Fig. 11) is J the product of the altitude and the sum of the parallel sides. A = Jh(b+b'). Find A, if h = 10f, b = 19|, b' = 12j. 8. Express the result of problem 7 in a decimal correct to .001. CIRCLE AND GENERAL FORMULAS 23 Exercise 16. Circle and Circular Ring Formulas Fig. 12. Circle Fig. 13. Circular Ring 1. C = 2;rr. (Fig. 12). Find C, if ;7- = 3.1416 (See art. 17), r = li 2. C = ttD. Find C, if D = 5.724. 3. A = 7rr2. Find A, if r=l|. 4. A = .7854D2. Find A, if D = 5.724. . 5. A = ;r(R2-r2) (Fig. 13). Find A, if R = 7j, r = 4|. Exercise 17. General Formulas Evaluate the following formulas for the values given: 1. P = awh, if a = 120, w = .32, h = 9|. 2. W=-.p, if 1 = 25, h = 4j, p = 60. h 3. F = lid+iifd = lf. 4. L=lfd+|,ifd = 2i. 5. S = ^gt2, if t = 4. (g is a definite number. Its value is 32.16). 6. S = |g t^+vt, ift = 3, V = 7. 7. D= Va2+b2+c2, if_a = 3, b = 4, c = 12. 8. V = Jh(b'+b+Vb.bO, ifh = 2j, b = 12, b' = 3. 9. F = 10. uv u+v, 4. 3 if u=11.5, v = 6.5. Y = i7rT^, if r = 2.3. 24 EVALUATION Checking Equations 28 The solution of an equation may be tested by evaluating its members for the value of the unknown quantity found. If its members reduce to the same number, the value of the unknown is correct. Example: 2x + ^^^^~"^^ =3x+l. 5 2x+^^ = 3x+l. Why? 10x+6x-2 = 15x+5. Why? x = 7. Why? Check: 2.7+^(^-^-^) = 3.7+l. 5 14+8 = 21 + 1. 22 = 22. Exercise 18 Solve and check: 1. 6y-7 = 3y+20. ^ 2 (x+2) _y^ 2. ll = 3x+9. ^ 7(s+3) s_s 3. ^-1 = ^-2. «• -1^-6-4+'- 3 2 9. 2x-l = ^(5-x)-l|. 4. 2(2x+5) = 13. ; 5. 6(z-6) = z+8. N^*.^- *(5-x) = — ^ e. 3jH:ll = 3. 10. 5(x+l)+^^-l=4i 5 5 4 5 CHAPTER III THE EQUATION APPLIED TO ANGLES 29 Angle: If the line OA (Fig. 14) revolves about O as a center to the position OB, the two lines meeting at the point O form the angle AOB. The point O is called the vertex of the angle and the lines OA and OB are called the sides of the angle. Fig. 15. Right Angle B A Fig. 16. Straight Angle A Fig. 17. Perigon 30 Right Angle: If the line turns through one fourth of a complete rev- olution (Fig. 15), the angle is called a Right Angle. 31 Straight Angle: If the line turns through one half of a complete rev- olution (Fig. 16), the angle is called a Straight Angle. 32 Perigon: If the line turns through a complete revolution (Fig. 17), re- turning to its original position, the angle is called a Perigon. How many right angles in a straight angle? How many right angles in a perigon? How many straight angles in a perigon? 25 26 THE EQUATION Fig. 18. Protractor 33 A Protractor (Fig. 18) is an instrument used for measuring and constructing angles. On it, a straight angle is divided into 180 equal parts called degrees, written 180°. How many degrees in a right angle? How many degrees in a perigon? Fig. 19 Drawing Angles 34 Example: Draw an angle of 37°. Using the straight edge of the . protractor, draw a straight line OA. Place the straight edge of the protractor along the line OA, with the center point at O. Count 37° from the point THE PROTRACTOR 27 where the curved edge ^touches OA and mark the point B (Fig. 19). Again use the straight edge of the protractor to connect the points O and B. Exercise 19 1. Draw an angle of 30°. 2. Draw an angle of 45°. 3. Draw an angle of 60°. 4. Draw an angle of 120°. 5. Draw an angle of 135°. 6. Draw an angle of 150°. 7. Draw an angle of 18°. 8. Draw an angle of 79°. 9. Draw an angle of 126°. 10. Draw an angle of 163°, Measuring Angles Fig. 20 35 Example: Measure the angle AOB. Place the straight edge of the protractor along one side of the angle as OA, with its center at the vertex of the angle (Fig. 20). Count the number of degrees from the point where the curved edge of the protractor touches OA to the point where it crosses the line OB. The angle AOB contains 54°. 28 THE EQUATION F/GZ7 F/G 28 Exercise 20 1. Measure the angle in Fig, 21. 6. 2. Measure the angle in Fig. 22. 7. 3. Measure the angle in Fig. 23. 8. 4. Measure the angle in Fig. 24. 9. Measure the angle in Fig. 26. Measure the angle in Fig. 27. Measure the angle in Fig. 28. Measure the angle in Fig. 29. 5. Measure the angle in Fig. 25. 10. Measure the angle in Fig. 30. Reading Angles 36 Reading Angles: An angle is read with the letter at the vertex between the two letters at the ends, of the sides. The angle 1 in Fig. 31 is read BAG or CAB and is written Z BAG or Z GAB. ReadtheangleZ2;Z3. (Fig.31). Fig. 31 READ NG ANGLES 29 A r/o 34 s A F/G 35 Exercise 21 1. Read the Zs 1, 2, 3, (Fig. 32). 2. Read the Zs 1, 2, 3, 4, (Fig. 33). 3. Read the Zs 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, (Fig. 34). 4. Read the Zs 1, 2, 3, 4, 5, (Fig. 35). 30 THE EQUATION Exercise 22 1. Measure the ZCAD (Fig. 31), 2. Measure the Z ACB (Fig. 32), 3. Measure the ZCDA (Fig. 33). 4. Measure the ZEFA (Fig. 34). 5. Measure the ZBGF (Fig. 35). 87 Zl +Z2 + Z3 +Z4 = ZAOB (Fig. 36). If AOB is a straight line, the ZAOB contains 180°. Therefore Zl+Z2+Z3+Z4=180°. 38 The sum of all the angles about a point on one side of a straight line is 180°. Exercise 23 Fig. 37 Fig. 38 Fig. 39 1. Find X in Fig. 37. Check with a protractor. 2. Find X in Fig. 38. Check. 3. Find the unknown angle in Fig. 39. Check. 4. Three of the four angles about a point on one side of a straight hne are 16°, 78°, 51°, respectively. Find the fourth angle. ANGLE EQUATIONS 31 5. Find the three angles about a point on one side of a straight line if the first is twice the second, and the third is three times the first. 6. Draw with a protractor the angles of problem 5 as in Figs. 37, 38, 39. 7. Find the three angles about a point on one side of a straight line if the first is twice the third, and the second is a right angle. 8. Draw the angles of problem 7. 9. Find the four angles about a point on one side of a straight fine if the second is 5° less than the first, the third is 6° more than the first, and the fourth is 68°. 10. Draw the angles of problem 9. Exercise 24 Example : The three angles about a point on one side of a straight line 4 X are represented by x+6°, ^x — 12°, and 78°—^. Find x and the angles. x+6+|x-12+78-| = 180°. Why? 3x+18+4x-36+234-x = 540. Why? 6x+216 = 540. Why? 6x = 324. Why? x = 54. Why? x+6 = 54+6 = 60° 1st angle. |x- 12 = 72 -12 = 60° 2nd angle. 78-1 = 78-18 = 60° 3rd angle, o NOTE: The fact that the sum of the angles found is 180° checks the problem. 32 THE EQUATION If tne angles about a point on one side of a line are repre- sented by the following, find x and the angles: 1. fx, x+4, lix+2. 2. fx-2, iVx+7, 3(x+7), |x+19. 3. 4(x+l), 7(2x-ll), 127-6X. 4. 3x-i 2x, 2f (2x+l), |(x+6). 5. ix+40, 2x-9, 129.18-2X. 6. Find the angles about a point on one side of a straight line if the first is 25° more than the second, and the third is three times the first. 7. Find the angles about a point on one side of a straight line if the first is 6 times the second, plus 16°, and the third is ^ of the first, minus 4°. 8. Find the five angles about a point on one side of a straight line if the second is J of the first, the third is 5° more than f of the first, the fourth is 10° less than twice the first, and the fifth is 22i°. Fig. 40 ANGLES ABOUT A POINT 33 39 . Zl+Z2+Z6=180° Why? Z7+Z4+Z5 = 180°Why? Therefore, Zl+Z2+Z3+Z4+Z5 = 360°. 40 The sum of all the angles about a point is 360°. Exercise 25 If all the angles about a point are represented by the fol- lowing, find X and the angles: 1. |x,88-lx, lix-13, 4(^+li). 2. 23+-, 136- -, -+93, -+17. 4 5' 3 2 3. 4(x-5), ^+5li3x+47|. 4. i(3x-36), i(2x+15), ^ +30, S2-^x, x+48i. 6 6. fx+3.15, 3(x+1.75), i(x+94.05). 6. The sum of four angles is a perigon. One is 18° more than three times the smallest, another is 59° more than the smallest, and the last is 18° less than twice the smallest. Find the four angles. Supplementary Angles 41 Supplementary Angles: If the sum of two angles is a straight angle or 180°, they are called supplementary angles. Each is the supplement of the other. Exercise 26 1. What is the supplement of 16°; 92°; 24°; 13|°; 15lf°? 2. X is the supplement of 80°. Find x. 34 THE EQUATION 3. X is the supplement of x+32°. Find x and its supple- ment. 4. 2x — 20° and 7x+47° are supplementary angles. Find X and the angles. 5. One of two supplementary angles is 24° larger than the other. Find them. 6. The difference between two supplementary angles is 98°. Find them. 7. One of two supplementary angles is 4 times the other. Find the angles. 8. How many degrees in an angle which is the supplement of 3j times itself? 9. One of two supplementary angles is 27° less than 3 times the other. Find the angles. 10. One of two supplementary angles is y of the sum of the other and 63°. Find the angles. Jf.2 The supplement of an unknown angle may be indicated by 180° -X. Indicate the supplement of y°; d°; fx°; |-y°. When a problem involves two supplementary angles, but is such that one is not readily expressed in terms of the other, let X equal one angle, and 180— x the other. Exercise 27 1. f of an angle, plus 55° is equal to |- of its supplement, plus 4°. Find the supplementary angles. Let X = one angle 1 80 — X = other angle then fx+55 = -|(180-x)+4. 2. The sum of double an angle and 12j° is equal to ^ the supplement of the angle. Find the supplementary angles. COMPLIMENTARY ANGLES . 35 3. If an angle is trebled, it is 30° more than its supplement. Find the supplementary angles. 4. If an angle is added to J its supplement, the result is 128°. Find the supplementary angles. 5. If I" of an angle, minus 16°, is added to |- of its supple- ment, plus 72°, the result is 190°. Find the supplementary angles. Complementary Angles 4S Complementary Angles: If the sum of two angles is a right angle or 90°, they are called complementary angles. Each is the complement of the other. Exercise 28 1. What is the complement of 82°; 9°; 71°; 10j°; 43-|°? 2. X is the complement of 32°. Find x. 3. X is the complement of x4-76°. Find x and its com- plement. 4. fx+12°, and fx+10° are complementary angles. Find x and the angles. 5. One of two complementary angles is 25° larger than the other. Find them. 6. The difference between two complementary angles is 37f °. Find them. 7. One of two complementary angles is three times the other. Find the angles. 8. How many degrees in an angle that is the complement of 2^ times itself? 9. One of two complementary angles is 7° more than twice the other. Find the angles. 10. One of two complementary angles is f of the sum of the other and 23°. Find the angles. 36 THE EQUATION 44 The complement of an unknown angle may be indicated by 90— X. Indicate the complement of y°; m°; fx°; |-y° When a problem involves two complementary angles, but is such that one is not readily expressed in terms of the other, let X equal one angle, and 90— x the other. Exercise 29 1. The sum of an angle and \ of its complement is 46°. Find the angle. 2. The complement of an angle is equal to twice the angle riiinus 15°. Find the angle. 3. If 20° is added to five times an angle, and 20° sub- tracted from -5- of the complement, the two angles obtained, when added, will equal 114°. Find the angle. 4. Y of an angle is equal to f of its complement, minus 14°. Find the angle. 5. |- of the complement of an angle, plus 15° is equal to treble the angle. Find the angle. Exercise 30 1. The sum of J, ^, and f of a certain angle is 126°. Find the number of degrees in the angle. 2. The supplement of an angle is equal to four times its complement. Find the angle, its supplement and complement. 3. The sum of the supplement and complement of an angle is 98° more than twice the angle. Find the angle. 4. The complement of an angle is 20° more than f of its supplement. Find the angle. 5. The sum of an angle, J of the angle, its supplement, and its complement is 243°. Find the angle. KEVIEW OF ANGLES 37 6. The complement of an angle is equal to the sum of the angle and f of its supplement. Find the angle. 7. An angle increased by ^ of its supplement is equal to twice its complement. Find the angle. 8. -f-the supplement of an angle is equal to 3 times its complement, plus 20°. Find the angle. 9. The sum of treble an angle, f of its complement, and -| of its supplement is equal to 62° less than a perigon. Find the angle. 10. Yx of the complement of an angle is equal to J the supplement, plus 3°. Find the angle. 11. The three angles about a point on one side of a straight line are such that the second is 89° more than |- of the sup- plement of the first, and the third is f of the complement of the first. Find the three angles. 12. The sum of four angles is 223°. The second is twice the first, the third is J the supplement of the second, and the fourth is the complement of the first. Find the four angles. 13. There are four angles about a point. The second is -| the first, the third is the supplement of the second, and the fourth is the complement of the second, plus 30°. Find the four angles. 14. There are five angles about a point on one side of a straight line. The second is ^ of the first, the third is ^ the supplement of the second, the fourth is ^ the complement of the second, the fifth is 10°. Find the five angles. 15. Express by an equation that the supplement of an angle is equal to its complement, plus 90°. Does 41° for x check the equation? Does 25°? Does 153°? What values may x have? CHAPTER IV ALGEBRAIC ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION Positive and Negative Numbers 45 1. The top of a mercury column of a thermometer stands at 0°. During the next hour it rises 4°, and the next 5°! What does the thermometer read at the end of the second hour? 2. The top of a mercury column stands at 0°. During the next hour it falls 4°, and the next, it falls 5°. What does it read at the end of the second hour? 3. If the mercury stands at 0°, rises 4°, and then falls 5°, what does the thermometer read? 4. If the thermometer stands at 0°, falls 4°, and then rises 5°, what does the thermometer read? 5. If the mercury stands at 0°, rises 4°, and then falls 4°, what does the thermometer read? 6. A traveler starts from a point and goes north 17 miles, and then north 15 miles. How far and in which direction is he from the starting point? 7. A traveler starts from a point and goes south 17 miles, and then south 15 miles. How far and in which direction is he from the starting point? 8. A traveler goes 17 miles south, and then 15 miles north. How far and in which direction is he from the starting point? 38 POSITIVE AND NEGATIVE NUMBERS 39 9. A traveler goes 17 miles north, and then 15 miles south. How far and in which direction is he from the starting point? 10. A traveler goes 17 miles south, and then 17 miles north. How far is he from the starting point? 11. An automobile travels 35 miles east, and then 40 miles east. How far and in which direction is it from the starting point? 12. An automobile travels 35 miles west and then 40 miles west. How far and in which direction is it from the starting point? 13. An automobile travels 35 miles west, and then 40 miles east. How far and in which direction is it from the starting point? 14. An automobile travels 35 miles east, and then 40 miles west. How far and in which direction is it from the starting point? 16. An automobile goes 35 miles east, and then 35 miles west. How far is it from the starting point? 16. A boy starts to work with no money. The first day he earns $.75, and the second $.50. How much money has he at the end of the second day? 17. A boy has to forfeit for damages $.75 more than his wages the first day, and $.50 more the second day. What is his financial condition at the end of the second day? 18. A boy earns $.75 the first day, and forfeits $.50 the second day. How much money has he? 19. A boy forfeits $.75 the first day, and earns $.50 the second. How much money has he? 20. A boy earns $.75 the first day, and forfeits $.75 the second. How much money has he? 40 ADDITION 46 Such problems as these show the necessity of making a distinction between numbers of opposite nature. This can be done conveniently by plus (+) and minus ( — ). If a number representing a certain thing is considered positive (plus), then a thing of the opposite nature must be negative (minus). Thus, if north 10 miles is written +10, south 10 miles must be written — 10. If east 25 feet is written +25, west 25 feet must be written —25. 47 If such numbers as these are to be combined, their signs must be considered. Thus a rise of 19° in temperature fol- lowed by a rise of 9° may be expressed as follows: ( + 19°) + (+9°) = +28°. A trip 15 miles south followed by one 25 miles south may be expressed: ( — 15) + ( — 25) = — 40. A trip 42 miles east followed by one 26 miles west is expressed: (+42) +(-26) = +16. A saving of $1.75 followed by an expenditure of $2.00 is expressed: (+1.75) +(-2.00) = -.25. These four problems may also be written: 1. 19+9 = 28 2. -15-25= -40 or 3. 42-26 = 16 4. 1.75-2.00= -.25 - .25 This combination of positive and negative numbers is called Algebraic Addition. 1. + 19 + 9 +28 2. -15 -25 -40 3. +42 -26 + 16 4. + 1.75 -2.00 ADDITION OF SIGNED NUMBERS 41 . ADDITION ^5 RULE: To add two numbers with like signs, add the numbers as in arithmetic, and give to the result the common sign. To add two numbers with unlike signs, subtract the smaller number from the larger, and give to the result the sign of the larger. NOTE: If no sign is expressed with a term, + is always understood. Care should be taken not to confuse this with the absence of the sign of multipUcation. (See Art. 19.) Exercise 31 Add : 1. + 19, +10 2. -19, -10 3. -19, +10 4. + 19, -10 5. -10, +19 6. + 10, -19 7. -75, +25 8. +38, +19 9. + 11, -26 10. + 10, -10 11. -40, +39 12. -4, +26 13. 3 5 4j -8" 14. 3 7 4j 8 15. H,-i 16. -hi 17. -hi 18. -32, % 19. 6f, -8| 20. -7|, +7f 21. 13i -23t 22. -llf,8f 23. -2.32, -1.68 24. 3.47, 5.43 25. 8.44, -7.25 26. 8.75, -11.25 27. 5.732, -4.876 28. -18.777, -3.333 29. -173.29,239.4 30. -208.21, 171.589 42 ADDITION Ji9 1. Add -19, -10 2. Add +11, +26. 3. Add the results of problems 1 and 2. How does the result of problem 3 compare with the result if — 19, —10, +11, +26, were to be added in one problem as follows? -19 -10 +11 +26 =? -19 +11 -10 +26 =? -19 +26 +11 -10 =? +26 -10 -19 +11 =? (See Art. 10.) 50 RULE: To add several numbers, add all the positive numbers and all the negative numbers separately, and combine the two results. Exercise 32 Add: 1. +50, +41, -23, -7. 2. +47, -49, +2, -35. 3. +3, -40, -17, 4. 4. 82, 18, -100. 5. -79, -21, -100. 6. -119, +1, -21, -14, +101. 7. -2.36, +4.24, 5.73, -8.66. 9. 31, 7|, -4f, -6i 10. 23|, -19f, 17f, -Uf, 5i. ADDITION OF SIMILAR TERMS 43 51 Term: A term is an expression whose parts are not sepa- rated by plus (+) or minus ( — ). llx^, — 14abxy, +23f are terms. NOTE: Such expressions as 8(x+y), 3(a— b), etc., are terms because the parts enclosed in the parenthesis are to be treated as a single quantity. (See Art. 24.) 52 Similar Terms: Similar or like terms are those which differ in their numerical coefficients only; a 53 Only similar terms can he combined. in their numerical coefficients only; as 2x^yz2, — ^^x^yz^. Exercise 33 Add: 1. -16r, 18r, 8r. 2. 4.2s, -5.7s, 2s. 3. 7fx, -4fx, -2ix, X. 4. 2§ab, 4 Jab, —3 Jab, ab. 5. 24abc, — 36abc, lOabc, +4abc, — abc. 6. -32a2b, 40a2b, -Qa^b, 2a2b. 7. 3vVj vV^ — 9v2y^, — 4vV. 8. -3|xVz, 5|xyz, -4yVxVz. 9. 3.16xy2z^ -4.08xy2z^ e.GQxy^z^. 10. 8(x-y), -6(x-y), +4(x-y). 11. ~12(x+y), -7(x+y), -(x+y). 12. -6j(c-d),3|(c-d),4|(c-d). 13. -8(x2+y2), 24(x2+y2), 17(x2+y2), H-(x2+y2). 14. 8(x+y+z), 14(x+y+z), -2(x+y+z). 15. Il(x2+y)4, -5(x2+y)4, 24(x2+y)^. 44 ADDITION 54 Monomial: An expression containing one term only is called a monomial. 55 Polynomial: An expression containing more than one term is called a polynomial. A polynomial of two terms is called a binomial, and one of three terms a trinomial. Addition of Polynomials 56 Example : Add 2a3 - 2a2b - b^, - Vab^ - 1 la^, and b3+7a3+3ab2+2a2b. Since only similar terms can be combined, it is convenient to arrange the polynomials, one underneath the other with similar terms in the same vertical column, and add each column separately as follows: 2a3-2a2b-b3 -lla^ -7ab2 +7a^4-2a^b+b^+3ab2 -2q? -4ab2 Exercise 34 Add: 1. 4a+3b-5c, -2a-m+3c, 2m-9c+2b, 5a+3m-4b. 2. pq+3qr+4rs, — pq+4rs— 3qr, st— 4rs. 3. 2ax2+3ay2-4z2, ax2+7ay2-4z2, 2z2+ay2-a2x. 4. fa2-|ab-Jb2, 2b2-a2-fab, -ab-Sb^+fa^. 5. 3|m-4ix+2if, 2iVx-f+2jm. 6. 8.75d~3.125r, 2.873r+7.625f-10d, 4.29f-r+1.25d. ADDITION OF POLYNOMIALS 45 7. 3(x+y)-7(x-y), 5(x+y)+5(x-y), -2(x+y)- 3(x-y). 8. I(a2-b2)-f(b2-c2)+|(c2-a2), |(a2-b2)-|(c2-a2), 4(b2_c2)_2i(a2-b2). 9. 5(x+y)-7(x2+y2)+8(x3+y^), -4(x3+y3)+5(x2+y2)- 4(x+y), 2(x2+y2)-4(x3+y3)-(x+y). 10. 6(ab+c)+7(a-m)+a2bc3m, Sa^bc^m-SCab+c)- 5(a — m), 3(ab+c) — (a — m)— 4a2bc^m. SUBTRACTION 57 1. If a man is five miles north (+5) of a certain point, and another is 12 miles north (+12) of the same point, what is the difference between their positions (distance between them), and in what direction is the second from the first? 2. If the first man is 5 miles south ( — 5) of a point, and the second 12 iniles south ( — 12), what is the difference between their positions, and in what direction is the second from the first? 3. The first man is at ( — 5), and the second is at (+12). What is the difference between their positions, and in what direction is the second from the first? 4. The first is at (+5), and the second at ( — 12). What is the difference between their positions, and in what direction is the second from the first? 58 To find the difference between the positions of the men in the above problem, the signs of their positions must be con- sidered. Finding the difference between such numbers is called Algebraic Subtraction, 46 SUBTRACTION Find the difference between the positions and the direction of the second man from the first in each of the following: Second man First man + 12 + 5 -12 - 5 + 12 - 5 -12 + 5 + 5-5+5 + 12 -12 -12 - 5 + 12 Add the following: + 12 - 5 -12 + 5 + 12 + 5 -12 - 5 + 5 -12 - 5 + 12 + 5 + 12 - 5 -12 How do the results of the corresponding problems in the two groups compare? 5Q RUIfE: To subtract one number from another, change the sign of the subtrahend mentally and add. Exercise 35 1. tract: +27 + 12 4. -32 +21 7. + 16 10. -llf -42 +15| 2. -13 - 8 +21 - 5 5. 6. + 15 +82 - 81 -127 8. 9. - 39 11. -fax + 100 -fax 3. 12| 12. 7|b2y -4i 6fbV 13. by2 lfby2 16. 3a-b+c 4a — c 14. -5 (a+y) -li(a+y) 17. 3ix2+5iy3- z 2jx2-2|y3-2z 16. 14.92(m2 149.2 (m2 18. -a^-a^b+ab^ -a^b -b^ SUBTRACTION OF POLYNOMIALS 47 19. ^ ,3(x+y)-4.8(x2+y2) -5.7(x+y)+4.8(x^+y^) 20. -5 (ab+c)-10.7(x+y+z)+51a2bx3yz + 19ab2xy3z. -3i(ab+c) + 1.07(x+y-fz)-17a2bx3yz+20ab2xy3z. 60 A problem in subtraction is often written in the form ( — 19) — (+7). In that case it is better to actually change the sign of the subtrahend and then the problem is one of addition instead of subtraction and is written: (-19) + (--7)or -19-7= -26. Exercise 36 Subtract: 1. (-28) -(-36) 5. (8.91abc)-(-3fabc) 2. (+35)-(-2lJ) 6. (-2y%x2yz)-(3.1416x2yz) 3. (-|)-(+f) 7. (3a+2b)-(2a+3b) 4. (+2|mx)-(+5mx) 8. (5x2-7y2)-(-2x2+y2) 9. (-9|m3n+3|mn3)-(4fm3n-3|mn3) 10. (3ia+2b)-(3ja+7c) 11. (-1.7a2b2-2.9b4)-(-3.3ab3-4.16b4) 12. (4x-5jy+3fz2)-(2|x+6.25y-5jz2) 13. (-11.23a2b2+4jb2c2-l|c2a2)-(-11.22a2b2+ 4§bV-1.875c2a2) 14. (6x2-9mx-15m2)-(9x2-16m2) 15. (-a3b2+a2b3+ab4)-(a%-a2b3+b5) 48 ADDITION AND SUBTRACTION Exercise 37. (Review) 1. From the sum of x2-2hx+h2 and x^-Ghx+Qh^, sub- tract 3x2+2hx-4h2. 2. Simplify (x2-2xy+y2) + (2x2-3xy+y2)-(3x2-5xy+ 2y^). 3. From 6x3 — 7x— 4, subtract the sum of Qx^— 8x+x' and 5— x^+x. 4. SimpHfy (im-4n+fp)-(fm-in+Jp) + (- iVm- fn-fp). 5. Subtract the sum of 6— 4x3 — x g^j^^j 5x — 1 — 2x2, from the sum of 2x3+7-4x-5x2 and 3x2-6x3-2+8x. 6. Find the difference between (12x4+ 6x^-2) + (6x^- 8x+14-8x3), and (0)-(- 10x^+2- 15x2+llx5-4x). 7. Subtract: 3x2-5xy+2y2-2x+ 7y 2x2- xy+8y2-9x-14y 8. Simphfy: (5a3-2a2b+4ab2) + (-9a2b+7ab2+8b3) + (-8a3-ab2+2b3). 9. Add: |gt2 +v- Jt gt2 -fv+ 6t -1.3gt2 + llt+.02 -10v+1.2t+l| 6gt2+ 11.625V -2.25 Solve and check: 10. 12a+l-(3a-4)=2a+8+(4a+4). 11. (3x-4)-6=(x-l)-(2x-3). 12. -25-(-5+2p) = (13p-50). 13. (3x-15)-(2x-8)=0 ADDITION AND SUBTRACTION 49 14. 2k-(--^)=--(--4f) 3 6 2 2 15. -xH-(-x — ) — (-x+-) = 14. 3 5 5 6 3 ^ Signs of Grouping 61 Removal of Parentheses: By Art. 47, parentheses connected by plus signs may be used to express a problem in addition, and the parentheses can be removed without affecting the signs. For example: (3a--2b) + (2a-3b) = 3a-2b+2a-3b = 5a~5b. By Art. 62, two parentheses connected by a minus sign may be used to express a problem in subtraction, and the paren- theses can be removed by actually changing the sign of each term enclosed in the parenthesis preceded by the minus sign (the subtrahend). For example: (3a-2b)-(2a+3b)=3a-2b-2a-3b = a-5b. 62 RULE: Parentheses preceded by minus signs may be removed if the sign of each term enclosed is changed. Parentheses preceded by plus signs may be removed without any change of sign. NOTE: The sign preceding a parenthesis disappears when the paren- thesis is removed. 63 Other signs of grouping often used, are the brace { } the bracket [ ] and the vinculum . These have the same meaning as parentheses and are used to avoid confusion when several groups are needed in the same problem. 64 When several signs of grouping occur, one within the other, they are removed one at a time, the innermost one first each time. 50 REMOVAL OF PARENTHESIS Example : Simplify 4x — {3x + ( — 2x — x - a) } 4x- f3x+(-2x-x-a) } = 4x — JSX + ( — 2x — X + a) } (removing the vinculum) = 4x— {3x — 2x — x+a j- (removing the parenthesis) = 4x — 3x+2x+x — a (removing the brace) = 4x — a (combining like terms) NOTE: In the case of the vinculum, special care must be taken. — X — a is the same as — (x— a). The minus sign preceding the vinculum is not the sign of x. Exercise 38 Simplify : 1. -6+{5-(7+3) + 12} 2. io-[(7-4)-(9-7)] 3. 4x-[2x-(x+y)+y] 4. -llb+[8b-(2b+b)-3b] 5. 8kz-[7kz-3kz-5kz] 6. x2-{'3x2-2i^} 7. a3 - ( - 6a2 - 12a + 8) - (a3+ 12a) 8. [6mn2- (8P-mn2+3n3 - mn^) - (22mn2-8P)] 9. 3a-(5a-{-7a+[9a-4]]) 10. c-[2c-(6a-b)-{c-5a+2b-(-5a+6a-3b)}] Solve and check: 11. 4x-(5x-[3x-l])=5x-10 12. 12x-{8-(8x-6)-(12-3x)} = 13. -(12-20x)-{l92-(64x-36x)-12} = 96 14. 5x-[8x-{48-18x-(12-15x)}] = 6 15. 12x-[-81-(-27x-4+ I0x)] = 61 {8x-(-20-29-4x)} MULTIPLICATION Multiplication of Monomials zs Fig. 41 65 Two boys of equal weight are on a teeter-board at equal distances from the turning point (as at A and B, Fig. 41). The board balances. If one boy weighed one-half as much as the other, he would have to be twice as far from the turning point in order to balance the other. Similarly, if one weighed one-third as much as the other, he would have to be three times as far from the turning point in order to balance the other. From these illustrations, it is readily seen that a weight of one pound, four feet from the turning point, will turn the board with four times as much power as a weight of one pound, one foot from the turning point. A weight of three pounds, jour feet from the turning point, will turn the board with three times as much power as a weight of one pound, four feet from the turning point, and therefore with twelve times as much power as a weight of one pound, one foot from the turning point. The tendency of the board {lever) to turn under such conditions is called the leverage, the weights acting upon it are called forces, and the distance of the forces from the turning point {fulcrum) are called arms. From this explanation it is evident that: 66 The leverage caused by a force is equal to the force times the arm. This law affords a very convenient means of working out the law of signs for multiplication of positive and negative numbers. 51 52 MULTIPLICATION 67 Let it be required to represent the product of (+5) (+4). If the result is to be thought of as a leverage, the (+5) will be the force, and the (+4) the arm. In discussing positive and negative numbers in Arts. 45, 46, and 47, measurements up- ward and to the right were represented by (+), and measure- ments downward and to the left by ( — ). Then the (+5) will be considered an upward pulling force, and the (+4) an arm measured to the right of the fulcrum (Fig. 42). An upward -I Fig. 42 force on a right arm causes the lever to turn in a counter- clockwise (opposite the hands of a clock) direction. To be consistent with arithmetic, (+5) (+4) must be (+20). There- fore, in determining the sign of the result of multiplication, a counter-clockwise motion of the lever must be positive, and a clockwise motion, negative. 68 Let it be required to represent the product of (—5) (—4). If the result is to be thought of as a leverage, the ( — 5) will be a downward pulhng force, and the (—4), a left arm (Fig. 43). P^ Fig. 43 It is seen that the lever turns in a counter-clockwise direction which is positive. Therefore ( — 5) (—4) = +20. LAW OF SIGNS 53 69 Let it be required io represent the product of (+5) (—4). In this case there is an upward-pulling force (+5), on a left ti^ Fig.. 44 arm (~4) (Fig. 44). The lever turns in a clockwise direction which is negative. Therefore (+5)(— 4)= — 20. 70 Let it be required to represent the product of ( — 5) (+4). In this case there is a downward-pulling force ( — 5), on a right ^ =1 Fig. 45 arm (+4) (Fig. 45). The lever turns in a clockwise direction which, is negative. Therefore ( — 5) (+4)= —20. From the four preceding articles: 1. (+5)(+4) = +20 2. (-5)(-4) = +20 3. (+5)(-4)=-20 4. (~5)(+4) = -20 From these the law of signs for multiplication can be derived. 71 Law of Signs for Multiplication: If two factors have like signs, their product is plus. If two factors have unlike signs, their product is minus. 54 MULTIPLICATION Exercise 39 Multiply: 1. (+|)(+i) 8. (+3t)(+li) 2. (-f)(-i) 9. (-6i)(-6i) 3. (+f)(-i) 10. (7i)(-A) 4. (-f)(+i) 11. (-1.1)(+1.1) 5. (-f)(+f) 12. (-2.03)(-4.2) 6. (+f)(-|) 13. (-l-.3)(-.03) 7. (-|)(-t\) 14. (+8.7o)(+3i) 15. (-8.66)(-2i) 72 By Art. 21, x^ means x-x-x-x-x and x^ means x-x-x. Therefore (x^) (x^) = (x • x • x • x • x) (x • x • x) = x^. From this the law of exponents for multiplication can be derived. 73 Law of Exponents for Multiplication: To multiply powers of the same base, add their exponents. NOTE: The product of powers of different bases can be indicated only. (x^)(y^) =x^y^. Example: Multiply (Ta^bx^) by (-3abV) 7a2bx3 = 7.a2.b.x3 -3ab3y2=-3-a.b3.y2 (7a2bx3)(-3aby)=7.a2.b.x3.(-3).a-b3.y2 which may be arranged 7-(— 3)-a2-a-b-b^-x^-y2= — 21a^b^y. 7^ RULE: To multiply monomials, multiply the nimierical coefficients, and annex all the different bases, giving to each an exponent equal to the sum of the exponents of that base in the two factors. MULTIPLICATION OF MO] NTOMIALS ^Exercise 40 Multiply: 1. (3a3)(7a«) 11. (+2lc)(-2id) 2. (4x4) (-6x5) 12. (-9mW)(-7mW) 3. (-5|mio)(-2m) 13. (-7)(+mV) 4. (-13a2b)(-2ab3) 14. (+7i)(-gt^) 5. (+5a2bc2)(-4abV) 16. (3xy)(xy) 6. (-6a2b)(+3b2c) 16. (6r2)(-§r2) 7. (+2ab)(-3cd) 17. (-x^)(-x') 8. (+3)(-x) 18. (-6.241)(+3.48in) 9. (+5)(+|) 19. {x+yy-(x+yy 10. (-l)(-pq) 20. (ni2_n2)6.(^2_u2)2 55 Solve and check: 21. (x)(3) = (-3)(-6) 22. (r)(-2) + (r)(+3) = (-4)(-9) 23. (-3)(-6) = (w)(+2) + (0)(-5) 24. (-2)(d) + (d)(+3) + (2)(-3)=0 26. (+3)(-8) + (3k)(4) + (-2)(4k) = (0)(4) 26. (3s)(-6)-(-90)(-2) = (6s)(-6) 27. (3x)(+3) - (800)(+13) = (-5x)(19) 28. (5y)(-4) + (-56)(-l) = (-3y)(4) 29. ( - 9) ( - 2^) - (8x) (2) = (4i) (6) + ( - 7) (4x) + (6x) (0) 30. (-ll)(-3x)-(4i)(+6) = (20l)(-2)-(-3)(17x) 56 MULTIPLICATION 15 1. What is the leverage caused by a force +7 on an arm -3? 2. What is the leverage caused by a force —16 on an arm +4? 3. What is the leverage caused by a force H-3f on an arm +6? 4. What is the leverage caused by a force —27 on an arm -2\l 5. What is the leverage caused by a downward force of 6, on a right arm of 3?- 6. What is the leverage caused by a downward force of 12, on a left arm of 7? 7. What is the leverage caused by an upward force of 16, on a left arm of 3j? 8. What is the leverage caused by an upward force of 3^, on a right arm of li? 16 Suppose two or more forces are acting on the lever at the same time as in Figs. 46, 47, 48. Fig. 46 What is the leverage caused by the force ( — 6), Fig. 46? What is the leverage caused by the force (—4)? In which direction will the lever turn? This may be expressed by (-6)(-2) + (-4)(+3) = + 12-12 = LAW OF LEVERAGES • ^2-^1 '^ i ^ 57 Fig. 47 What is the leverage caused by the force ( — 6), Fig. 47? What is the leverage caused by the force (+4)? In which direction will the lever turn? This may be expressed by (-6)(+2) + (+4)(+3) = -12+12 = 0. =ff Fig. 48 What is the leverage caused by ( — 2), Fig. 48? What is the leverage caused by (+3)? What is the leverage caused by ( — 3)? In which direction will the lever turn? This may be expressed by(-2)(-6) + (+3)(+l) + (-3)(+5) = 12+3-15 = 0. From these illustrations the law of leverages may be derived. 77 Law of Leverages: For balance, the sum of all the leverages must equal zero. 58 MULTIPLICATION Exercise 41 1-10. Find the unknown force or arm required for balance in the levers shown in Figs. 49, 50, 51, 52, 53, 54, 55, 56, 57, 58. See Art. 16. 4 Fig. 49 7 3 i Fig. 54 Fig. 50 Fig. 55 /4 ^3i* H Fig. 51 r~^ — ST W is 60 to zo Fig. 56 A5-- 6— *s3* 7X" 3W 6—— pSI -t-'~< Fig. 52 ^ /^ JW 2fiO JW Fig. 57 zx -2/ 7X /Z Fig. 53 ] 360 3X ZX Fig. 58 11. What weight 12" to the left of the fulcrum will balance a weight of 10 lbs., 9" to the right of the fulcrum? (Draw a figure). MULTIPLICATION OF SEVERAL MONOMIALS 59 12. Two boys weighing 75 lbs. and 105 lbs. play at teeter. If the larger boy is 5' from the fulcrum, where would the smaller boy have to sit to balance the board? 13. A crowbar is 6' long. What weight could be raised by a man weighing 165 lbs., if the fulcrum is placed 8" from the other end of the bar? 14. A lever 12' long has the fulcrum at one end. How many pounds 3' from the fulcrum can be lifted by a force of 80 lbs. at the other end? 15. A man uses an 8' crowbar to lift a stone weighing 1600 lbs. If he thrusts the bar 1' under the stone, with what force must he lift to raise it? Multiplication of three or more Monomials 78 Example : Multiply ( - 2a) ( - Sa^) ( +4a^) ( - Va^) (-2a)(-3a2) = +6a3 (+6a3)(+4a5) = +24a8 (+24a8)(-7a0 = -168aii or (-2a)(-3a2)(+4a0(-7a3)= -168a" Exercise 42 Multiply: 1. (-3)(-4)(+5) 2. (-|)(+34)(-i) 3. (+6)(-li)(-e)(-7) 4. (lla)(-7ab)(+4abc)(-9bV) 5. (a2c2)(-4a2b)(-lla3b0 60 MULTIPLICATION 6. 7. 8. 9. 10. (-4iab)(-3fac)(-^bc)(|abc; (1 .25m2x) ( - 2.4m3x2y) ( - 4.63mxy2) (-3.57)(+a7b2)(-lfa%«) 4(x-y)3. (x_y)2.{_7(x~7)} 3 (m2-n3) {-4(m2-n3)4 . (m^-n^*)^ Multiplication of Polynomials by Monomials a -4— >6- za 2b Fig. 59 19 The product of 2(a+b) may be represented by a rectangle (Fig. 59) having a+b for one dimension and 2 for the other. The area of the entire rectangle is equal to the sum of the two rectangles, 2a and 2b, or 2(a+b) = 2a+2b. SO RULE: To multiply a polynomial by a monomial, multiply each term of the polynomial by the monomial, and write the result as a polynomial. Example : Multiply 3m2 - 5m + 7 by - 9ml -9m3(3m2-5m+7) = -27m5+45m4-63m3. Exercise 43 1. a3-7a2b+9ab2 by 3a2b^ 2. 6x^-5x6-7x4 by -7x3 3. — 3m2 — n^ + 5mn by 4m2n3 4. a3-2a2x+4ax2-8x3 by — 2ax 5. }a2-iab+lb2 by -4b MULTIPLICATION OF POLYNOMIALS BY MONOMIALS 61 6. 3x^-15x2+24 ' by -Jx^ 7. ^+^-1 by 12 2^3 6 8. —_!+?. by -30 10 5 6 ^ 2m 3p 7 , ,„ 9. \-^- by 10 5 2 15 Simplify : 10. -2.4a3zH2|a2x-3jxz+.125z3) 11. 5(3x+2y)+4(2x-3y) 12. 3x(4a-2y)-5x(3y-5a) 13. -5(3a-2)-3(a-6)+9(2a-l) 14. x(3x-l)-2x(7-x)-5(x2+2x-l) 15. 16(^i) -24(^+5) Exercise 44 Solve and check: 1. 2(5m+l)=3(m+7)-5 2. 3(5x+l)-4(2x+7) = 3 3. 3~(x-3)=7-2x 4. 10(m-6) = 3(m-2)-5 5. 6. 15y2+lly(2-5y)+4(10y2- x+5 x+l_ -9) = = 19 SUGGESTION: 2(x+5)-(x+l) = 12 (clearing of fractions). The line of the fraction has the same meaning as a parenthesis. See Art. 62. 62 MULTIPLICATION „ x+5 x-10 ^ 7. — =4 3 4 1 x-1 9. 6a-— -3(2a-l)=i a a 10. 5x-+2x+6_,,, 5x Multiplication of Polynomials by Polynomials — Cf — .j^ b za ac zb tc 4 Fig. 60 81 The product of (a+b)(c+2) may be represented by a rectangle whose dimensions are a+b and c+2 (Fig. 60). The area of the entire rectangle is equal to the sum of the four rectangles, ac, be, 2a, and 2b, or (a+b)(c+2) = ac+bc+2a+2b. It will be seen that the first two terms are obtained by multi- plying a+b by c, and the last two terms by multiplying a+b by 2. It is convenient to arrange the work thus: a+b c+2 ac+bc +2a+2b ac+bc+2a+2b MULTIPLICATION OF POLYNOMIALS BY POLYNOMIALS 63 S2 RULE: To multiply a polynomial by a polynomial, multiply one polynomial by each term of the other and combine like terms. Example : Multiply x^-x+lbyx-S x2- X - -X +1 -3 - X2+ X -3x2+3x- -4x2+4x- -3 -3 Exercise 45 Multiply: 1. x+2 by x+7 2. a-3 by a — 5 3. m+2 by m— 4 4. 3m -5n by 4m+3n 5. 2a2+3b5 by 5a2+4b» 6. a+1 by a^-l 7. a-3 by b+7 8. 2x2-5x+7 by 3x-l 9. 4m2 — 3ms — g by m2-3s2 10. x4-x3+x2-x+l by x+1 Exercise 46 Simplify : 1. (a — b+c)(a — b — c) 2. (2n2+m2+3mn)(2n2-3mn+m2) 3. (ix-iy)(Jx+iy) 4. (x+3)(x-4)(x+2) 5. (x2+xy+y2)(x2-xy-|-y2)(x2-y2) 64 MULTIPLICATION 6. (2a+3b)(6a-5b) + (a-4b)(3a-b) 7. 5(x-4)(x+l)-3(x-3)(x+2) + (x+l)(x-5) Solve and check: 8. (y-5)(y+6)-(y+3)(y-4)=0 9. (m+3)(m+2) = (m+7)(m-5)+50 10. 3 (2x-4)(x+7)-2 (3x-2)(x+5) = 5-(3x-l) 4(x2+3x+7) 11. 12. 13. 14. 16. 2x 2x+7 (3x+2)(2x+3) ^ (2x-l)(x+4) (x-2)(x-3) _ (x+3)(x-4 ) (x-4)(x-5) 3 4 ~ 12 1 x(2x+l) (2x -3)(3x+4) 2x+17 4+ 2 ~ 6 ~ 4 Exercise 47 Mu] 1. Itiply: 2m2-m-l by 3m2+m-2 2. 2p3-3p2q+7pq2+4q3 by 4p-3q 3. a^+b^+ab^+a^b by a^b-ab^ 4. 5-3a+7a2 by 4+12a2 6. -4mn+3ni2-lln2 by 2m2-5n2+7mr 6. -5m2+9+2m3-4m by 5m2-l+6m 7. 3ax2-4ax3-5ax5 by .l-x+2x2 8. p3-6p2+12p-8 by p3+6p2+12p+8 9. s3-2s2-s-l by s3+2s2-s+l 10. a-l+a^-a^ by 1+a MULTIPLICATION OF POLYNOMIALS BY POLYNOMIALS 65 11. 4x3-3x^+2x2-6^ by x-x^+l 12. 3b3-7b2c+8bc2-c3 by 2b3+8b2c-7bc2+3c3 13. a^+b^+c^+ab— bc+ac by a— b — c 14. a3-3+2a2-a by 3-a+a3-2a2 15. ia-Jb+fc-|d by fa+fb-^c+ld 16. fa2-fab-fb2 by ija^-fb^ 17. 2fm2n2-4|n3 by fm^-fn 18. 1.25aH-2.375b+3.5c by 8a-8b+8c 19. .35a2+.25ab+3.75b2 by 4.1a2-.02ab-.57b2 20. 3.5x2-2.1xy-1.05y2 by 4x-f Exercise 48 Simplify: 1. (a-l)(a-2)(a-3)(a-4) 2. (a-b)(a2+ab+b2)(a3+b3) 3. (3x-4y)(2x+3y)(4x-5y)(x-7y) 4. (m+n)(m-n)(^+^) 6. (x+y)(x3+y3){x2-y(x-y)} 6. (x+y+z)(x-y+z)(x+y-z)(y+z-x) 7. (2a+5b-c-4d)2 8. (fa3-fb2)3 2 3 4 10. (x+y)(x2-y2)-(x-y)(x2+y2) 66 MULTIPLICATION 11. (3a-2b)(2a2-3ab+2b2)-3a(2a2-3ab) 12. 6(m-n)(m+n)-4(m2+n2) 13. 12(x-y)-(x2+x-6)(x2+x+y) 14. 15ab-3(2a2+4b2) + (3a-2b)(5a-3b) 15. 6(a+2b-2c)2-(2a+2b-c)2 16. (x2+l)(x-l)-(x-3)(2x-5)(x+7)-(x+2)' 17. (a+b+c)3-3(a+b+c)(a2+b2H-c2) 18. (2m^-3mn+4iiOH^-^)^ a+b+c a-b+c a+b-c b+c-a 2 2 2 2~ (x-2)(2x-3) (x+2)(2x+3) (x^ +4) (4x^+9) 3*7*2 19. 20. Exercise 49. (Review) Solve and check: 1. (-16)(-x) + (-13)(+12) + (-2)(+2x)=0 2. (+15)(-|) + (-14)(-|) + (-10f)(+i|)=0 3. (-4f)(5x) + (+7i)(7x) + (-8|-)(0) + (7)(-12)=0 4. 3x-3(|x-7)=35 5. (2x-l)(3x+7)-3x2=(x-l)(3x-12)+20 ^ 3x+5_, x-7 ^- ~^— ^ 6" 7. (ix+f)(fx-i)=|^ ^ 3(3 -2x) 2(x-3)_ 2_4(x+4)_ 1 ^- 10 " 5 "^^^~ 5 +10 9. f(x+5)-|(x+7)+V(x+l)-i(2x-5)=J(x+22) EQUATIONS INVOLVING MULTIPLICATION 67 (x-l)(x+2) _(2x + l)(x + 2) (2x+l)(x-l) 2 12 6 11. If -| the supplement of an angle is subtracted from the angle, the result is 27°. Find the angle. 12. If f the complement of an angle is subtracted from three times the angle, the result is 39°. Find the angle. 13. If I of the supplement of an angle is decreased by f of the complement, the result is 53°. Find the angle. 14. I" the supplement of an angle is equal to the angle diminished by f of its complement. Find the angle. 15. Find three consecutive numbers such that the product of the second and third exceeds the product of the first and second by 40. 16. The difference of the squares of two consecutive num- bers is 43. Find the numbers. 17. The length of a rectangle is three times its width. If its length is diminished by 6, and its width increased by 3, the area of the rectangle is unchanged. Find the dimensions. 18. Two weights, 123 and 41 respectively, are placed at the ends of a bar 24 ft. long. Where should the fulcrum be placed for balance? (Suggestion: Let x = one arm, 24— x = the other.) 19. A man weighing 180 lbs. stands on one end of a steel rail 30 ft. long, and finds that it balances with a fulcrum placed 2 ft. from the center. What is the weight of the rail? (Sug- gestion: The weight of the rail may be considered a down- ward force at the middle point of the rail.) 20. An I-beam 32 ft. long weighing 60 lbs. per foot, is being moved by placing it upon an axle. How far from one end shall the axle be placed, if a force of 213 J lbs. at the other end will balance it? DIVISION Division of Monomials 83 To divide positive and negative numbers, a law of signs and a law of exponents are necessary. These may be derived from the same laws for multipHcation, from the fact that the product divided by one factor equals the other factor. By Art. 70: 1. (+5)(+4) = +20 2. (-5)(-4) = +20 3. (+5)(-4) = -20 4. (-5)(+4)=-20 Therefore, from 1. from 2. I (+20) 1 (+20) from 3. from 4. (+5) = +4 (+4) = ? /(+20)^(-5)=-4 \(+20)^(-4) = ? (-20) + (+5) =-4 (_20)-(-4) = ? (_20)-(-5) = +4 (+4) = ? // two numbers have like signs, (-20) 84 Law of Signs for Division: their quotient is plus. If two numbers have unlike signs, their quotient is minus Divide (+f) (-1) (+1) (+*) (+i) (-f) (-l)-(-l) (-fl)-(+ 3 3' Exercise 50 6. 7. 8. 9. 10. (+2i)-(+3f) (-if)-(-io) (+H)-(-9) (-42) -(+i) (+72) -^(-4i) DIVISION OF SIGNED NUMBERS 69 11. (-8.5)^(-1.7) . 16. (+3.6)^(-2|) 12. (+3.2) -(-.8) 17. (-3y5e)-(-6.25) 13. (+2.65) ^(+100) 18. (-34.56) ^(-.288) 14. (-.008) ^(-.02) 19. (+26i)^(-llf) 15. (-15)-^(+.003) 20. (-.0231) ^(-6|) Exercise 51 Solve and check : 1. 3x+14-5x+15 = 4x+ll 2. 20x+15+32x+193-12 = 36x+100-32x 3. s(2s-3)-2s(s-7)+231 = 4. (x-5)(x-6) = (x-2)(x-3) 5. (3x-l)(4x-7) = 12(x-l)2 fi 12 3x i9_4x 172 x+3 x-2 3x-5 r "^^ 2 ~ 3 " 12 +4 3 ?^^-^-^-| = 2x+17i 9. f(x+2)-A(x+5) + 10 = 2-i(x+l) s(s-2) _ s(s-9) -2s^-91 ^°- 5 ~ 3 ~ 15 85 By Art. 72, (x'){x^) = x^ Therefore (x^) -r- (x^) = x^ (x8)--(x3) = ? S6 Law of Exponents for Division: To divide powers of the same base, subtract the exponent of the divisor from that of the dividend. / U DIVISION NOTE: The quotient of powers of different bases can be indicated only. Example: Divide 48 a'b^'c^iV* by -SaVc^x' -^g^,j^=-6.a=.b.l.x^.y^ = -Ga^bxV NOTE : — = 1 . Also — = c^ bv the law of exponents, c^ c^ ' Therefore c° = 1 and may be omitted as a factor in problems like the above example. S7 RULE : To divide a monomial by a monomial, divide the numerical coefficients, and annex all the different bases, giving to each an exponent equal to the difference of the exponents of that base in the two monomials. ^iv iflp* Exercise 52 1. -91a by + 13 2. -32x« by -8x4 3. +22a%3c7 by -lla^b^c^ 4. +6im3n3 by -fmn2 5. -8|pi^qiir2i by 5|qi«r5 6. -4.24xyz7 by — Ax^yH^ 7. l.TSa^^x^ by -.35x4 8. -.85mV by I7n4 9. -.OOlxVm^ by -lOOxym 0. +3.1416a2b3mio by +4a2b3m^^ DIVISION OF MONOMIALS 71 Simplify : 1155a^x7z5 3.1416r^ -231a2x«z .7854r2 ^„ -1.732t2uV ,„ +a4bVo +2u^s2 -2a3bc9 3.1416xyV7 -a^(x+y)9 -I|x26yz25 +2|a(x+y)4 27m^n^x7 (a+b)^(a+b)^(x+y) ^ 1.125m2nV .0625(a+b)3(x+y)^ 32.16t^ 18.75a(m^-n^)^Q • -.08t2 ^°* 2i(m2-n3)7 Exercise 53 Solve for x and check: 1. -ax=-ab 6. 3a2b3x = - 12a3b3 2. +Jbx=-8b 7. 7a-3ax = 28a 3. — .3mx = 2.4m 8. 4mx — 7mx=12m — 18m 4. -4x=-12(a+b) 9. Ga^b - 7ax = - 29a2b .« X 3 1 5. -fx = 2m 10. 3^-- = 6^ 11. p.-4m = ^-'-^ 3m 6m 2 12. ^b_3_(x-2b)^^^ 13. (x-5y)(x+4y)=x2+y2 14. (x+m)(x-2m)-x(x-7m)=m(3x-5m) ^^ (x-3s)(x-2s) x(x-5s) x(x-3s) 15. ZZ := • 72 DIVISION Division of a Polynomial by a Monomial 88 By Art. 79, 2(a+b) = 2a+2b Therefore, — '- = a+b 89 RULE: To divide a polynomial by a monomial, divide each term of the polynomial by the monomial, and write the result as a poly- nomial. Example: Divide 2 Im«-35m4+7m3 by -7m2 21m«-35m4+7m3 -7m^ -3m*+5m2-m Exercise 54 Divide : aVc* — axVy^+a^xc^z axc^ 4xVV- 12xVz^-24xyz^+16xyz — 4xyz ^ 2.31m2n2+7.7m3n3-.33m%4 4. 5. l.lm^n^ - l|tV^ - 9.81tV - . 378tv^ -9tv2 1 . 125a3x2z3 - .375a2x2z2 - 4.2aVz2 6. .25a2x2z2 3f abed + 2ibcde + 7 j acde -IJcd Solve for x and check: 7. ax = 2ab — 3ac+4ae 8. 3a2m3x = I.lla^m3-3.3a2m4 9. 4m2s2x ~ 3.2m3s2 = ISam^s^ EQUATIONS INVOLVING DIVISION 73 10. 3|xy z - 1 .4y2z = .85yz2 - 70yz 11. 4m2x - Tm^n^ - Sm^x+Sm^n^ = 5m%4 - 2m2x+2m2n5 12. 4mx_ 7mn+mx ^^^Q^^, a ab nx . SCn^x — m^n^) 4n , » 14. 8mn +- = f-m^n m mn m 2mx+a2m5 5(bV+nx)_ 2m2n-3mn2 15. — — oX m n mn Division of Polynomials by Polynomials 90 By Art. 81, (a+b)(c+2)=ac+bc+2a+2b ^, , ac+bc+2a+2b Therefore, —, = c+2 a+b In multiplying a+b by c+2, the first two terms were obtained by multiplying a+b by c, and the last two by multiplying a+b by 2. In dividing, the c may be obtained by dividing ac by a, and the 2 may be obtained by dividing 2a by a. It is convenient to arrange the work as follows: c + 2 a+b)ac+bc+2a+2b ac+bc +2a+2b +2a+2b 91 RULE: To divide a polynomial by a polynomial, divide the first term in the dividend by the first term in the divisor to obtain the first term of the quotient. Multiply the divisor by the first term of the quotient, and subtract the result from the dividend. To obtain the other terms of the quotient, treat each remainder as a new dividend and proceed in the §ame way. 74 DIVISION Example (1) : Divide a3-6a2-19a+84 by a -7. a2+a-12 a-7) a3-6a2-19a+84 a3-7a2 +a2-19a +a?— 7a -12a+84 -12a+84 Example (2) : Divide 24 + 26x3 + 120x4 - 14x - 1 1 1x2 by -x-6+12x2 NOTE: Arrange the terms according to the powers of x, in both dividend and divisor. 10x2+3x-4 12x2-x-6)120x4+26x3- 120x4-10x3- -111x2-14x4-24 -60x2 +36x3- +36x3- - 51x2- 14x - 3x2- 18x - 48x2+ 4x+24 - 48x2+ 4x+24 Example (3): Divide X4_|_x2y2. f y4 by x2+xy+3 X2- x2+xy+y2)x4 x4+ xy+y2 +x2y2 x3y+x2y2 x3y x3y — x2y2 +y^ +y^ — xy3 +x2y2 +x2y2 + xy3+y4 +xy3+y4 Exercise 55 Divide the following: 1. x2-7x+12 by x-3 2. a2-2a-15 by a — 5 3. a2-3ab-28b2 by a+4b DIVISION OF POLYNOMIALS BY POLYNOMIALS 75 4. 6m4-29m2+35 by 2m2-5 5. 12x2+31xy-15y2 by x+3y 6. m3+3m2-13m-15 by m+1 7. 10x3- 19x2y+26xy2-8y3 by 2x2-3xy+4y2 8. 3m4 - lOm^ - 16m2 - 10m - 3 by 3m2+2m+l 9. 2x4-x3y+4xV+xy3+12y4 by x2-2xy+3y2 10. 4x4-24x3+51x2-46x+15 by 2x2-7x+5 11. 9xV - 15x3y2+ 13xV - Sx/ by Sx^y — xy2 12. Sm^n - 22m6n2 - 7m5n3+ 53m%4-30m3n^ by 4m^+3m3n — Sm^n^ 13. 10x3 _ 29.3x2y +37xy2 _ 20y3 by 2.5x2 -4.2xy+4y2 14. 14x3+ 17x2y +39xy2+ 17y3 by 3.5x2+2.5xy+8.5y2 16. 18x3-53x2+27x+14 by 4x-8 16. 13.12m5n+1.36m%2_ 7.15m3n3+2.35m2ii4- .125mn 5 by 3.2m2n- 2.4171 n2+.5n3 17. a2+ab+2ac+bc+c2 by a+c 18. a^bx + abcx + a^cx + abV + b^cy+abcy by ax+by 19. x2+8-10x+x3 by 2-l-x2-3x 20. m4+n^-4m3n-4mn3+6m2n^ by m^+n^— 2mn 21. a5-9a3+7a2-19a+10 by a2+3a-2 22. 16m4-72m2n2+81n4 by 4m2-12mn+9n2 23. m3-64n3 by m— 4n 24. 32a5+243b^ by 2a+3b 25. a2+2ab+b2-c2 by a+b — c 26. a2_x2-2xy-y2 by a — X — y 27. a44-4+3a2 by a2+2-a 76 DIVISION 28. 4a2-b2-6b-9 by 2aH-b+3 29. a2-b2+x2-y2+2ax+2by by a+x+b-y 30. m2-2mn+n2+3m-3n+2 by m-n+1 Solve and check: 31. (a+3)x = ab+a+3b+3 32. (a2-4ab+3b2)x = a3-8a2b+19ab2-12b3 33. (2m-3n)x = 8m3-22m2n+mn2+21n3 34. (a+b+2)x = a2+2ab+b2+4a+4b+4 35. (y+2)x = y3-y2-34y-56 Exercise 56 1. One number is 6 more than another, and the difference of their squares is 144. Find the numbers. 2. One number is 3 less than another, and the difference of their squares is 33. Find the numbers. 3. Divide 42 into two parts such that j of one is equal to ^ of the other. 4. Divide 57 into two parts such that the sum of -g- of the larger and J of the smaller is 12. 5. The difference of two numbers is 11, and, if 18 is sub- tracted from f of the larger, the result is y of the smaller number. Find the numbers. 6. Divide 24 into two parts such that if J of the smaller is subtracted from f of the larger, the result is 9. 7. If the product of the first two of three consecutive numbers is subtracted from the product of the last two, the result is 18. Find the numbers. REVIEW PROBLEMS 77 8. If the square of the first of three consecutive numbers is subtracted from the product of the last two, the result is 41. Find the numbers. 9. I paid a certain sum of money for a lot and built q. house for 3 times that amount. If the lot had cost $240 less and the house $280 more, the lot would have cost J as much as the house. What was the cost of each? 10. A boy has 2 J times as much money as his brother. After giving his brother $25.00, he has only l-J- times as much. How much had each at first? 11. The sum of J a certain angle, J of its complement and Y^Q- of its supplement is 48°. Find the angle. 12. Three times an angle, minus 4 times its complement, is equal to y^ ^^ ^^^ supplement + 131°. Find the angle. 13. If 3 times an angle is subtracted from J its supplement, the result is yy of its complement. 14. A certain rectangle contains 15 sq. in. more than a square. Its length is 7 in. more and its width 3 in. less than the side of the square. Find the dimensions of the rectangle. 15. The altitude of a triangle is 4 in. more than the base, and its area exceeds one half the square of the base by 16. Find the base and altitude. (Suggestion: See Exercise 15, problem 5.) 16. A wheelbarrow is loaded with a barrel of flour weighing 196 lbs. The center of the load is 2' from the axle of the wheel. What force at the handles, 4^' from the axle of the wheel, will be required to raise the load? 17. A wheelbarrow is loaded with 5 bars of pig iron weigh- ing 77 lbs. each. How far from the axle of the wheel should the center of the load be placed, if a force of 154 lbs. 4 ft. from the axle will raise it? 78 DIVISION 18. A timber 12" X 18" X 24' is balanced on wheels and an axle by a force of 120 lbs. at one end. How far from the center shall the axle be placed if the timber weighs 45 lbs. per cu. ft.? 19. A lever 12' long weighs 24 lbs. If a weight of 30 lbs. is hung at one end and the fulcrum is placed 4' from this end, what force is needed at the other end for balance? 20. A piece of steel 1' long, weighing 15 lbs. per foot, is resting upon one end. A weight of 1400 lbs. is placed l\' from that end. What force at the other end is necessary to balance the load? CHAPTER V RATIO, PROPORTION, AND VARIATION Ratio 92 Ratio: The relation of one quantity to another of the same kind is called a ratio. It is found by dividing the first by the second. For example: the ratio of $2 to $3 is f , written also 2:3; the ratio of 7" to 4" is I; the ratio of 18" to 6' is || = i. 93 Terms of Ratio: The numerator and denominator of a ratio are respectively the first and second terms of a ratio. The first term of a ratio is called its antecedent, and the second, its consequent. Exercise 57 1. Find the ratio of 85 to 51. 2. Find the ratio of 27 to 243. 3. Find the ratio of 2^ to 3f . 4. Find the ratio of 6.25 to 87.5. 5. Find the ratio of y\ to .3125. 6. Find the ratio of 8" to 6'. 7. Find the ratio of 12a to 16a. 8. Find the ratio of drr to Stt. 9. Find the ratio of a right angle to a straight angle. 10. Find the ratio of a right angle to a perigon. 11. Find the ratio of a straight angle to a perigon. 12. Find the ratio of f of a perigon to -| of a right angle. 79 80 RATIO, PROPORTION, AND VARIATION 13. Find the ratio of 55° to its complement. 14. Find the ratio of 55° to its supplement. 15. Find the ratio of 45° to J its supplement. 16. Find the ratio of the supplement of 48° to its comple- ment. 17. A door measures 4' X 8'. What is the ratio of the length to the width? 18. There were 25 fair days in November, while the rest were stormy. What was the ratio of the fair to the stormy days? 19. The dimensions of two rectangles are 5" X 8'^, and 6" X 8". Find the ratio of their lengths, widths, perimeters, and areas. 20. The bases of two triangles are 3.9 and 2.4, and their altitudes are respectively .8 and .7. Find the ratio of their areas. 21. Find the ratio of the circumferences of two circles whose diameters are respectively 5 J" and 2f ". (See Exercise 16, problem 2.) 22. Find the ratio of the areas of two circles whose diame- ters are respectively 11'' and 13". (See Exercise 16.) 23. Find the ratio of the two values of P in the formula P = awh, when a = 120, w = .32, h = 9f, and when a = 48, w = .38, and h = 24. 24. Find the ratio of the two values of F in F=l§d+J, when d = if, and when d = 2j. 25. Find the ratio of the two values of S in S = Jgt2, when t = 3j, and when t = 10j. (See Exercise 17.) RATIO AS DECIMALS 81 P4 To Express Ratios ^as Decimals: It is often convenient to have results in decimal rather than in fractional form. For example : the ratio i- is often written .875. Exercise 58 Find the decimal equivalents of the following ratios : 5 9 35 61 .72 1. 3. 5. 7. 9. "8 25 32 64 li 11 19 7i 2f 3.24 2. 4. 6. ^ 8. 10. 16 20 30 3| 129.6 95 Sometimes it is sufficiently accurate to express the decimal to two places only. In this case it is necessary to determine the third place, and, if this is 5 or more, it is customary to increase the second place by 1. For example: the ratio y'|=.946 +, which would be written .95 if two places only are desired. Exercise 59 Find the decimal equivalents of the following ratios, correct to .01 : 1. 1 7 Percentage is found by reducing a ratio to a decimal correct to .01, and multiplying it by 100. 10 19 25 37.5 2. 3. 4. 5. 11 16 7i 5.15 For example: ^^^ = 6.688 = 669%. .0369 82 RATIO, PROPORTION, AND VARIATION 6. In a class of 27 students, 22 passed an examination. Find the percentage of successful students. 7. A base ball player made 89 hits out of 321 times at bat. Find his batting average (percentage). 8. The total cost of manufacturing an article is $5.36 of which $2.79 represents labor. What per cent of the total cost is the labor? 9. If Q2^ tons of iron are obtained from 835 tons of ore, what per cent of the ore is iron? 10. In a class of students, 25 passed, 2 were conditioned, and 6 failed. Find the percentage of failures. 11. Babbitt metal is by weight 92 parts tin, 8 parts copper, and 4 parts antimony. Find the percentage of copper. 12. Potassium nitrate is composed of 39 parts of potassium, 14 parts of nitrogen, and 48 parts of oxygen. Find the per- centage of potassium. 13. Potassium chloride is composed of 39 parts of potassium and 35.5 parts of chlorine. Find the percentage of chlorine. 14. Baking powder is composed of 3 J parts of soda, if parts of cream of tartar, and 6.5 parts of starch. Find the percentage of cream of tartar. 15. If 12 quarts of water are added to 25 gallons of alcohol, what per cent of the mixture is alcohol? 16. If 5 lbs. of a substance loses 5 oz. in drying, what per cent of its original weight was water? 17. If 5 lbs. of a dried substance has lost 5 oz. in drying, what per cent of its original weight was water? 18. If a dried substance absorbs 5 oz. of water and then weighs 5 lbs., what per cent of its original weight is water? SPECIFIC GRAVITY 83 19. The itemized cost of a house is as follows: Masonry . . $ 750 Plumbing . . $350 Carpenter Work $ 900 Furnace . . $150 Lumber . . S1200 Painting . . . $300 Plastering . . $ 250 What per cent of the total cost is represented by each item? Check by adding the per centa 20. The population of Detroit in 1900 was 285,704, and in 1910, it was 465,776. Find the percentage of increase. 96 Specific Gravity: The specific gravity of a substance is the ratio of the weight of a certain volume of the substance to the weight of the same volume of water. For example: if a cubic inch of copper weighs .321 lbs., and a cubic inch of water weighs 321 .0361 lbs., the specific gravity of copper is = 8.88. .0361 Example. The dimensions of a block of cast iron are 3j"X 2f"X^^ and its weight is 37.5 oz. Find its specific gravity. 3iX2| X 1 =8.94cu. in. (the volume of the block) . 0361 lbs. = . 5776 oz. (weight of l cu. in. of water) .5776 X 8 . 94 = 5 . 16 (weight of 8.94 cu. in. of water) = 7 . 27, (specific gravity of iron) 5.16 NOTE: Specific gravity is usually found correct to .01. Exercise 60 1. A cubic inch of aluminum weighs .0924 lbs. Find its specific gravity. 84 KATIO, PROPORTION, AND VARIATION 2. A cubic inch of tungsten weighs .69 lbs. Find its specific gravity. 3. A cubic inch of cast steel weighs .282 lbs. Find its specific gravity. 4. A cubic inch of lead weighs 6.56 oz. Find its specific gravity. 5. A cubic foot of bronze weighs 550 lbs. Find its specific gravity. 6. A cubic foot of cork weighs 240 oz. Find Tts specific gravity. 7. A brick 2" X 4" X 8" weighs 4.64 lbs. Find its specific gravity. 8. A cedar block 5" X 3" X 2" weighs 10.5 oz. Find its specific gravity. 9. Each edge of a cubical block is 2' . If it weighs 4450 lbs., what is its specific gravity? 10. A man weighing 185 lbs., displaces when swimming under water, 5760 cu. in. of water. Find the specific gravity of the human body. 97 Sejparating in a given ratio. Example: Divide 17 into two parts which shall be in the ratio f . Let 2x = one part. 3x = other part. 2x ^2 Then2x+3x = 17 3x 3 5x = 17 x = 3f 2x = 6|-, one part. 3x= 10+, other part. Check: 6|+10|-=17, -^='52 5 SEPARATING IN A GIVEN RATIO 85 * Exercise 61 1. Divide 20 in the ratio f . 2. Divide 18 in the ratio ^. 3. Divide 100 in the ratio j;. 4. Divide 200 in the ratio y. 5. Two supplementary angles are in the ratio ^. Find them. 6. Two complementary angles are in the ratio -J. Find them. 7. A board 18" long is to be divided in the ratio -J. How far from each end is the point of division? 8. If a line 4' 6" long is divided in the ratio ^, what is the length of each part? 9. Divide a legacy of $25,000 between two persons so that their shares shall be in the ratio ^. 10. The sides of a rectangle are in the ratio |-, and its perimeter is 100. Find the dimensions of the rectangle. 11. Bronze is composed of 11 parts tin and 39 parts copper. Find the number of pounds of tin and copper in 625 lbs. of bronze. 12. A gold medal is 18 carats fine (18 parts of pure gold in 24 parts of the whole alloy) . Find the amount of pure gold in the medal if it weighs 2.7 oz. 13. Two men purchase some property together, one paying $750 and the other $450. If the property is sold for $2,000, what will be the share of each? 14. Two men agree to do a piece of work for $45. The work is completed in 10 days, but one of them was absent 2 days. How should the pay be divided? 86 RATIO, PROPORTION, AND VARIATION 15. How much copper would there be in 208 lbs. of Babbitt metal? (See Exercise 59, problem 11.) 16. Divide a perigon into three angles in the ratio 7:8:9. 17. Divide a line 5' 3" long into four parts in the ratio 5:6:7:3. 18. The sides of a triangle are in the ratio 5 : 8 : 9, and its perimeter is 6' 5". Find the sides. 19. Divide the circumference of a circle whose diameter is. 16" into three parts in the ratio 3 : 5 : 7. 20. Five angles about a point on one side of a straight line are in the ratio 1:2:3:4:5. Find them. Proportion 98 Proportion. A proportion is an equation in which the two members are ratios. For example : -^ = ^f is a proportion, and may be read 8 is to 12 as 16 is to 24. The first and fourth terms of a proportion are called the extremes, and the second and third are called the means. In the proportion -^ = ^^, 8 and 24 are the extremes, and 12 and 16, the means. Example: Solve 5 _? 12-9 15 = 4x (clearing of fractions.) x = 3j Check -_«-3| '' 9 1_5 A=A PROPORTION . Exercise 62 Solve and check: X 13 11 18 1. 25 "14 X 12 5. 12 X 125 206 2. 7 ~17 8 5 6. X 305 144 3x 3. X 11 7. 195 25 4. 7 X 9 14 8. X 3j 24~4| 9. The ratio of x+1 to 9 is equal to the ratio of Find X 87 I 10. The ratio of the complement of an angle to the angle is equal to the ratio y. Find the angle. 11. The ratio of the supplement of an angle to the angle is equal to the ratio y^ . Find the angle. 12. The ratio of an angle to 84° is equal to the ratio of its complement to 96°. Find the angle. 13. One number is 5 larger than another, and the ratio of the larger to the smaller is equal to-f . Find the two numbers. 14. The length of a rectangle is 6 more than its width, and the ratio of the length to the width is |-. Find the dimensions of the rectangle. 15. Two numbers are in the ratio f . If 2 is added to the smaller, the ratio of that number to the larger is f . Find the numbers. (See Example, Art. 97.) 16. If the scale of a drawing is J" to 1', how long should a line be made in the drawing to represent 32'? 88 RATIO, PROPORTION, AND VARIATION 17. If the scale of a drawing is J" to 1', how long should a line be made to represent 10"? 18. If the scale of a drawing is 1^" to 1', what line would be represented by a line 3^" on the drawing? 19. If a drawing is to be reduced to f its size, what would be the length on the new drawing, of a dimension 3 J" on the original drawing? 20. If a dimension line f " on a drawing represents a line 4j" long, what is the scale of the drawing? 99 It is often necessary in shop practice to express a fraction or decimal in halves, fourths, eighths, sixteenths, etc. A proportion is a convenient means of changing to these denominators. Example : How many 3^'s in y^-. Let x = number of -gVs in y-g-. 32~15 15x = 352 x = 23y5-, approximately 23§. Exercise 63 1. How many f 's in y^^-? 2. How many y^g-'s in |-? 3. How many -^'s in .3? 4. Reduce 1.312 to eighths. 5. Reduce 1^-^ to sixty-fourths. PROPORTION 89 X 4 100 Example: 1. Solve j^irY='5 /. ^^. \ 5x = 4x+4 II. C. D. is 5 (x + l)| X = 4 Why? 4 4 Check : :j— — r = t 4+1 5 5 " 5 ' X 1 Example: 2. Solve x-. rr = — ^ 3(x— 1) 6 2x = x-l ab 16. V2V+¥tu+9u2 17. 18. Vffm4+2m2n2+ffn* Vx^-fx^+rto 19. Via%2_4a2bc3+^9^_c6 I 20. VtfxS+iV^z^-JxVz Square Root of Numbers 118 By problem 31, Exercise 73. 322 =(30 +2)2 = 900 +120 +4 = 1024. .-. V 1024= V 900+120+4= +(30+2) = +32. To extract the square root of such numbers as 1024, it is necessary to separate them into the form of a trinomial square. This can not be done by inspection. Therefore it is convenient to use the simplest form of trinomial square, t2+2tu+u2, as a formula. In that case, t2+2tu+u2 corresponds to 1024, and its square root, t+u, corresponds to the square root of 1024, or 32. The work may be arranged as follows: t+u t2+2tu+u2 = t2+u(2t+u)= 1024 130+2 t2= 900 2t = 60 u= 2 124 = u(2t+u) 2t+u = 62!l24 V 1024= +32. SQUARE ROOT OF NUMBERS 113 Example 1: Find the Square root of 5625. In order to find how many digits there are in the square root of a number, observe the following: 92 = 81. 992 = 9801. 9992 = 998001. The square of a number of one digit can not contain more than two digits, the square of a number of two digits can not contain more than four digits, etc. Therefore, the number of digits in the square root of a number may be determined by separating the given number into groups of two digits each, beginning at the decimal point. t-^u t2+u(2t+u) = 56'25 170+5- t2 = 4900 2t = 140 u= 5 725 = u(2t+u) 2t+u=145 725 .-.V 5625 = +75. Observe that t is found by extracting the square root of the greatest square in the first group, and u is the integral number found by dividing the remainder by the number equal to 2t. Example 2: Find the value of V289. t+u t2+u(2t+u)= 2'89 1 10+9 t2 = 100 2t = 20 u = 9 2t+u = 29 189 = u(2t+u) 2 61 114 SQUARES AND SQUARE ROOTS t+u t2+u(2t+u) = = 2'89 |10+8 t2 = = 100 2t = 20 189 = = u(2t+u) u= 8 2t+u = 28 2 24 ? t+u t2+u(2t+u) = :2'89 |10+7 P = 100 2t = 20 189 = = u(2t+u) u= 7 2t+u = 27 189 .-. V 289 =+17. Observe that in finding u, it is not always possible to take the largest integral number" found by dividing the remainder by the number equal to 2t. Exercise 77 Extract the square root of : 1. 1849 5. 2916 8. 4624 2. 3136 6. 961 9. 1521 3. 576 7. 256 10. 4489 4. 5184 Example: Find the square root of 60516. t+u b2+u(2t+u) = 2t = 4 00 u= 40 2t+u = 4 40 = 6'05'16 1200+40 = 4 00 00 2 05 16 = ur2t+u 176 00 29 16 SQUARE ROOT OF NUMBERS 115 The square root of *60516 will contain three digits. The first two are found in the usual way. The root is evidently 240+ ? and the amount that has been subtracted from 60516 (40000+17600) is 240^. Therefore 240 may be considered a new value of t, and 2916 a new value of u(2t+u), in finding the third digit of the root. The problem then becomes: t2+u(2t+u)=6'05'16 t2 = 5 76 00 t+u 1240 + 6 2t = 480 u= 6 2t+u = 486 29 16 = 29 16 u(2t+u) These two operations may be combined into one problem as follows: t+u t+u t2+u(2t+u) = 6'05'16 1200+40 240+6 t2 = 4 00 00 2t = 400 . u= 40 2 05 16 = u(2t+u) 2t+u = 440 176 00 2t = 480 u= 6 2t+u = 486 29 16 = u(2t+u) 29 16 /. V 60516 = ±246. Exercise 78 Find the value of 1. V 37636 2. V7344i 3. V 54756 4. V 173889 5. V 98596 6. V 233289 7. V 94249 8. V 648025 9. V 9778129 10. V 1022121 116 SQUARES AND SQUARE ROOTS 119 The operation of extracting square root may be abridged as follows : Find the the value of: V 235.6225 t + u t + u t + u t2+u(2t+u) = 1 5. 3 5 = 2' 3 5' .6 2' 2 5 t2=l = u(2t+u) 2t= 20 1 3 5 = u= 5 2t+u= 25 1 2 5 2t= 300 1 6 2 = u(2t+u) u= 3 2t+u= 303 9 9 2t = 3060 1 5 3 2 5 = u(2t+u) u= 5 2t+u = 3065 1 5 3 2 5 .-. V 235.6225 = ±15.35 NOTE: In pointing off the given number into groups of two digits each, begin at the decimal point and proceed both right and left. Exercise 79 Find the square root of: 1. 2323.24 2. .120409 3. 2.6569 4. 32.1489 5. 123.4321 6. .07557001 7. .00003481 8. 1621.6729 9. 1040400 10. 1624.251204 SQUARE ROOT OF NUMBERS 117 120 If a number is not a perfect square, the operation may be continued to as many decimal places as is desired by annexing a sufficient number of ciphers. Example: Find the value correct to .001 of: . V4 .329< )4. t + u t + u t + u t 2. -hu 8 8 = 2.081 t2+u(2t+u)=4. t2 = 4 32' 99' 40' 00 2t = 40 u= 32 = 00 = u(2t+u) 2t = 400 u= 8 32 32 99 = u(2t+u) 2t+u = 408 64 2t = 4160 u= 35 40 = u(2t+u) 00 00 2t = 41600 u= 8 35 40 00 = u(2t+u) 2t+u=41608 33 28 64 11 36 /. V 4. 32994 =+2.081 Observe that if 3 decimal places in the result are required, it is necessary to determine the digit in the 4th place, and if it is 5 or more, to add 1 to the digit in the 3rd place. 118 SQUARES AND SQUARE ROOTS Exercise 80 Find the square root of the following correct to 4 decimal places : 1. 15 3. 126 2. 38 4. 2.5 5. 634.125 Find the value of the following correct to .0001 : 6. V2 8. V5 7. V3 9. V-5 10. V 14.4 V36= ^[9A= ^f9' V4 = 3-2 = 6. 121 From this it is evident that : The square root of a number is equal to the product of the square roots of its factors. This law may be used to simplify the process of finding the square roots of numbers which contain one or more factors that are squares. For example: V12 = vlT V3 = 2 v'i = +3.4642. Exercise 81 Given >/¥= 1.4142, V^= 1.7321, v'5"= 2.2361, Find the value of the following correct to .001 : 1. V~8 6. V45 2. V18 7. V48 3. V~20 8. V50 4. V27 9. V72 5. V32 10. V108 SQUARE ROOT OF FRACTIONS 119 11. V180 16. V 98 12. V 80 17. V147 13. V125 18. V320 14. V363 19. V243 15. V512 20. V128 U^ — 3*3 — 9 •• V 9 — Xs- 122 The square root of a fraction is found by extracting the square root of the numerator and of the denominator. Exercise 82 Find the square root of: 1. ii 3. eV 2. If *• '-V 6- iff ^'ini i the value of the following correct to .001: 6. Vif 8. V3V 7. vif 9. viH •■■"• V 200 123 In fractions where the denominator is not a perfect square the operation of finding the square root may be simplified by multiplying both numerator and denominator by a number which will make the denominator a square. Example: v|= VA=-^=±?:^^=+.6124 V16 4 — 120 SQUARES AND SQUARE ROOTS Exercise 83 Find the value of the following correct to .001 : 1. Vl"' 6. vT" T 3 2. V* 7. V4 3. Vi 8. vf 4. Vj_ 9. V|_^ 5. Vf 10. V^ Quadratic Equations 12 If. Quadratic Equation: A quadratic equation is one which con- tains the square of the unknown quantity as the highest power of the unknown. X 13 3x 40 Example: -— — = — 2 3x 2 3x 3x2-26 = 9x2-80 Why? 54 = 6x2 Why? x2 = 9 Why? X = +3 (extracting the square root of both ~ members) Observe that: I. A quadratic equation of the form x2 = 9 may be trans- formed into one containing the first power of the unknown by- extracting the square root of both members. II. In extracting the square root of both members of the equation x2 = 9, the full result would be -i-x=+3, which is a condensed form of: 1. +x=+3 3. -x=+3 2. +x=-3 4. -x=-3 1 and 4, 2 and 3 are the same equations and therefore x= +3 expresses all four equations. QUADRATIC EQUATIONS 121 ^ Exercise 84 Solve (correct to .001 where necessary): 7. x2 = f 8. x2+10 = 59 9. y2-ll = 185 10. 7m2- 175 = 11. 8s2-38 = 90 12. lla2-5 = 2+2a2 13. 3(x-2)-x = 2x(l-x) 14. (2t-|-3)(t-|-2)-(t+3)(t+4)=4t2-21 15. (t-}-4)2+(t-4)2=48 3x^+1 5(x2-l) (4x^+1) ^ 1. x2=12 2. x2 = 75 3. x2 = 55225 4. x2 = 46 5. x^ - ''' 1225 6. 2 75 108 17. z!±y+i_y!:iy+i=i5 y-l y+l 5r-3^ r+2 9r+l 2r-[-5 ^^ 2x-5 , _ 3x+10 3x-l . 3x+l 29 20. 3x+l ' 3x-l 14 122 SQUARES AND SQUARE ROOTS 21. The length of a rectangle is 3 times its width, and the area is 243 sq. in. Find the dimensions of the rectangle. 22. How long must the side of a square field be that the area of the field may be 5 acres? 23. The dimensions of a rectangle are in the ratio f , and its area is 300. Find the dimensions of the rectangle. 24. The side of one square is 3 times that of another, and its area is 96 sq. in. more than that of the other. Find the sides of the two squares. 25. If the area of a 3" circle is 28.2744, find the diameter of a circle whose area is 78.54. (See Exercise 67, problem 3.) 26. Find the diameter of a circular piece of copper whose weight is 3.01 oz. if a 10" disk weighs 9.03 oz. (See Exercise 67, problem 10.) 27. The intensity of light varies inversely as the square of the distance from the source of light. How far from a lamp should a person sit in order to receive one half as much light as he receives when sitting 3 ft. from the lamp? 28. The distance covered by a falling body varies directly as the square of the time of falling. If a ball drops 402 ft. in 5 seconds, how long will it take it to drop 600 ft.? 29. The weight of an object varies inversely as the square of the distance from the center of the earth. If an object weighs 180 lbs. at the earth's surface, at what distance from the center will it weigh 160 lbs., if the radius of the earth is 4000 miles? 30. The surface of a sphere varies directly as the square of the diameter. Find the diameter of a sphere whose surface is 78.54 sq. in., if the surface of an 11'' sphere is 380.1336 sq. in. CHAPTER VIII FORMULAS Evaluation of Formulas Containing Square Root Exercise 85 Evaluate the following formulas for the values given (correct to .001 where necessary): 1. h = ^V3 when a = 5. 2. c=Va''+b2 whena = 4, b = 5. 3. V = 2 ttVR when r = J, R = \\ 4. A = |r2V3 whenr = 3j. a" 5. V = — ^'- when a = 3.2. 6. G=Vab whena = 4, b = 5. 7. v = ^^'-'''''j" ''^'' +|;ra^ whena = 6, r=18, R = 24. ^ t= ./ when 1=1, g = 32. 9. s=|(V5-l) whenr = 2|. 10. D=Va2+b2+c2 when a = 5, b = 6, = 7. 11. b= Va2+c2-2a'c when a =14, a' = 5, c= 12. 12. l = 2VD2+a2 + ttT) when D = 16, a = 35. 13. M = iV2(a2+c2)-b- whena=15, c = 17, b = 19. -b+Vb2+4ac , o u c 14. X = when a = 3, b = 5, c = 20. 2a 123 124 FORMULAS 15. x= whena = 3, b = 5, c = 20. 16. I = y/(5^)Va24-7r(^±^)when D = 36,d = 6,a = 96. 17. s = -rV10-2V5 when 1 = 4:^. 18. s = ^.JJ-R'- J V4R2-C2 whenN = 72, R= 10, C= 13. 19. X- Vr(2r- V4r2-s2) whenr = 3,s = 2. 20. A= Vs(s-a) (s-b) (s-c) when a=15, b = 18, c = 22, s = i(a+b+c). 125 A formula is an equation and may be solved for any of the letters involved if the values of all the other letters are given. Example 1: Z = 4;rra. Find a, when Z = 502.656, r = 8. 502.656 = 4-3.1416-8.a 0. 15.708 62.832 502.656 a = =5 4-3.1416-8 Example 2: V = |57r2a. Find r, when V = 593.7624, a = 7. 1.0472 593.7624 = f 3.1416-r2.7 81 84.8232 ^,^593.7624_^^ 1.0472-7 r=+9. FORMULAS INVOLVING SQUARE ROOT 125 Example 3: b2 = a2+xj^— 2ac'. Solve for c', when a = 5, b = 6, c = 7. 36 = 25 +49 -2- 5- c' 10c' = 38 c' = 3.8 Exercise 86 Find the values (correct to .001 when necessary) : Find a, when P = 5|. Find c, when P = 7962, a = 1728, b = 3154. Find a, when P=17|, b = 2j. Find b, when A = 2.31, a = 1.1. Find h, when A = 3U, b = 3j. Find h, when A = 96, b = 18, b' = 6. Findb',when A=12.8,b=1.2, h = 8. Find r, when C = 50. Find r, when A = 50. Find p, when w = 333 J, 1 = 25, h = 4 J. Find 1, when w = 320,h = 24, p = 213§. Find h, when w = 150, 1 = 162, p = 100. Find d, when L=4Y^g^. Find t, when S= 196.98. (See Exercise 17, problem 5.) Find V, when S = 164.72, t = 3. 1. P = 4a 2. P=a+b+c. 3. P=2(a+b). 4. A = ab. 6. A = |bh. 6. A=ih(b+b'). 7. A=ih(b+b''). 8. C = 27rr. 9. A=;rr2. 10. 1 W = £.p. 11. 1 W = g.p. 12. 1 W = j^.p. 13. L=lfd+|. 14. S=igt^ 15. S = Jgt2+vt. 126 FORMULAS 16. F = -^. FmdF,whenu = ll,v = 7. IIV 17. F = -^. Findv,whenF = li,u = 3. — b— Vb--f4ac -n- i i i_ - o 18. x = z: . Findx. wbenb= — o, a = 3, c= — 2. 2a 19. A=-^. Find D, when A =115. 4 20. V=;rr2a. Find r, if V = 330, a = 7. 21. V=J7T2a. Fiiida,ifV=46.9,r = 2.3. «o v-5rr2^±5 Find r, when V=1932, H = 14.6, ^ • 2 • h = 8.2. 23. V=^r^^±5. FindH,whenV = 2246,r = 8,h = 6. 24. A = ^. Find A, when a = 2.3, b = 3.2, c = 4.L 4r r = 2.058. 25. A = ^. Find r, when a = 21, b = 2S, c = 35, A = 294. 26. A = Jr(;a+b+c). Find r, when a = 79.3, b = 94.2, c = 66.9,A = 261.012. 27. A = |r(a+b+c). Find a, when A = 27.714, r = 2.3095, b = 8, c = 8. 28. A = i(27rR+2:rr)s. Find A, when R = 8, r = 3, s = 7. 29. A = |(2j7-R+2j7t)s. Find r, when A = 439.824, R = 10, s=10. 30. A = |(2:rR+2^r)s. Fmds,whenA = 106.029,R = 7|,r = 6. Findl, whenD = l|, d = lj, a = 15. 32. h = |>'3y Find a, when h = 27.7136. FORMrL-\S IXVOLVIXG SQUARE ROOT 12^ 33. A = -5- \Y. " Find h. when A = -— v 3. o o 34. A = — V 3. Find a, when A = ^ 48. 35. V = T^r V 2. Find ^ , when a = 6. 36. A = ^ ^"3^ Find r, when A = 153. 37. c2 = a2+b2. Findb, whenc = 2.1, a = 1.7. 38. b2 = a2+c2+2a'c. Finda. whenb = 8, c = 5, a' = 2.1. 39. b- = a-+c2+2a'c. Finda', whena = 18, b = 16, c = 31. 40. b- = a-+c--2ac'. Find c, whena = o, b = 4, c' = 2.3. 41. b- = a-+c--2ac'. Find c', when a = 14, b = 15, c = 16. 42. H = rVs(s-a)(s-b)(s-c). FindH,whena = 2.1S,b = 5, c = 3.24, s = i(a-fb4-c). 43. a-+b- = 2r§y+2m-. Find m, when a = 9, b = 12, c = 15- 44. a-+b- = 2('0''+2m2. Findb, whena = 5, c = 13, m = 6i. 45. s = I( ^ "5- 1) . Find r, when s = 10.50685. — b+ \ b-+4ac. ~ , , 46. x = Findx,whena = 3,b= — /.c = -f 2. 2a . . • 47. V = 2T-r-R. Find r, when V = 98696.5056, R = 50. 48. X = ^ r(2r — V 41^ — s-). Find x, when r = s = 10. X . c 49. s = ^.J7r-- - > 4r--c-. Find X, when s = 23.1872, c = r= 16. 50. x = — ^- — "^^ "^ ^^ Findx,when a=-6,b=-9,c = +2. 128 FORMULAS Right Triangle 126 One of the formulas most commonly used is that of the right triangle. _ Fig. 70 127 Right Triangle: A right triangle is a triangle in which one angle is a right angle. The hues including the right angle are called the sides^ and the hne opposite the right angle is called the hypotenuse. It can be proved that; 128 The square of the hypotenuse is eqvxil to the sum of the squ/ires of the two sides. THE RIGHT TRIANGLE 129 This truth is stated by the formula: c2 = a2+b2 (Fig. 70). Exercise 87 Find results correct to .001 when necessary: 1. Find c, when a = 8, b = 15. 2. Find a, when b = 9, c = 41 . 3. Find b, when a = 3, c = 6. 4. Find the hypotenuse of a right triangle when the sides are 3.2 and 2.4. 6. The hypotenuse and one side of a right triangle are respectively 2f and if. Find the other side. 6. The sides of a right triangle are 5j and 12.5. Find the hypotenuse. 7. The two sides of a right triangle are equal to each other, and the hypotenuse is 18. Find the sides. (Fig. 71.) \f ^ ^s^ 7 X \ x^ X /3Z /ly Fig. 71 Fig. 72 Fig. 73 8. One side of a right triangle is 3 times the other, and the hypotenuse is 80. Find the sides. Draw a i&gure. 9. The two sides of a right triangle are in the ratio f , and the hypotenuse is 225. Find the sides. Draw a figure. 10. Find the diagonal of a square whose sides are 1.32. (Fig. 72.) 130 FORMULAS 11. Find the perimeter of a square whose diagonal is 17. Draw a j&gure. 12. Find the diagonal of a rectangle whose dimensions are 11 and 16. (Fig. 73.) 13. Find the dimensions of a rectangle whose diagonal is 91, if the length is 5 times the width. Draw a figure. 14. The perimeter of a rectangle is 70, and its sides are in the ratio -f-. Find the diagonal. 15. A ladder 36 ft. long is placed with its foot 11 ft. from the base of the building. How high is a window which the ladder just reaches? 16. A flag staff 79 ft. long is broken 29 ft. from the ground. If the parts hold together, how far from the foot of the staff will the top touch the ground? 17. How long is a guy wire which is attached to a wireless tower 227 ft. from the ground, and is anchored 362 ft. from the foot of the tower? 18. The slant height of a cone is 12", and the radius of the base is 5 J". Find the altitude of the cone. (Fig. 74.) Fij. 71 19. One side of the base of a square pyramid is 14", and the altitude is 16". Find the edge, E. (Fig. 75.) (Sugges- tion : The altitude of the pyramid meets the base at the middle point of the diagonal.) 20. Find the slant height, S. (Fig. 75.) INDEX SUBJECT PAGE Addition, Algebraic, Defini- tion 40 Addition, Algebraic, Rule 41 Addition, Algebraic, of several numbers 42 Algebraic Subtraction, Defini- tion of 45 Algebraic Subtraction Rule.— 46 Angle, Definition of 25 Angle, Right 25 Angle, Straight 25 Angles, Complementary 35, 36 Angles, Drawing of 26, 27 Angles, Measuring 27, 28 Angles, Reading 28 Angles, Sum of 30, 33 Angles, Supplementary 33, 34 Antecedent 79 Arm 51 Base 16 Binomial, Definition 44 Binomial, Square of 107 Brace 49 Bracket 49 Checking Equations 24 Clearing of Fractions 9 Clockwise 52 Coefficient 15 CoeffiiCient, Numerical 16 Complement 35 Consequent 79 Counter Clockwise 52 Counter Shaft 99 Decimals, Ratios as 81 Decimal Equivalents 81 Degrees 26 Division Law of Exponents for 68 Division Law of Sign for 68 Division of Monomials....68, 69, 70 SUBJECT PAGE Division of Polynomials by Monomials ..— 72 Division of Polynomials by Polynomials 73, 75 Equations, Definition of 1 Equation, Principles of 10 Equation, Checking 24 Equations, Quadratic Defini- tion 120 Equations, Quadratic, Solu- tion of 121, 122 Formula, Definition 19 Formulas, Area 22 Formulas, Circle 23 Formulas, Circular Ring 23 Formulas, Evaluation of 19 Formulas, General 23 Formulas, Involving Square Root 123, 124, 125, 126, 127 Formulas, Perimeter 20 Fractions, Clearing of . 9 Fulcrum 51 Gears, Size and R.P.M. of. 103 Hypotenuse 128 Law of Exponents for Divis- ion 68 Law of Exponents for Multi- plication 54 Law of Leverages 57 Law of Signs for Division 68 Law of Signs for Multiplica- tion 53 Lineshaft 99 Lever 51 Leverage 51 Means of a Proportion 8& Monomial, Definition 44 Multiplication 51 131 132 INDEX SUBJECT PAGE Multiplication of a Poly- nomial by a Monomial 60 Multiplication of a Poly- nomial by a Polynomial..-*62, 63 Multiplication of Monomials 54, 59 Multiplication Law of Expon- ents for 54 Multiplication Law of Signs for 53 Multiplication Sign 15 Negative Numbers 40 Members, Definite 15 Members, General 15 Numbers, Definite 15 Numbers, General 15 Numbers, Positive and Nega- tive ..._ 38, 39, 40 Nmnbers, Signed 40 Order of Terms 6 Parenthesis 16 Parenthesis, Removal of 49 Percentage 81 Perigon 25 Perimeter, Definition 19 Perimeter, Formulas 20 Perimeters, Equations involv- ing 21 Polynomial, Definition 44 Polynomials, Addition of 44 Positive Numbers 40 Power 16 Proportion, Definition 86 Proportion, Direct 91 Proportion, Extremes of 96 Proportion, Inverse 92, 93 Proportion, Means of 86 Protractor 26 PuUevs, R.P.M. and Size of.... 99 Pulleys, Step, Cone 101 Quadratic Equation, Defini- tion 120 Quadratic Equations. Solu- tion of 121, 122 Ratio. Definition 79 Ration, Separating in a given 84 SUBJECT PAGE Ratio, Terms of 79 Ratios, To express as Deci- mals 81 Right Triangle, Definition 128 Right Triangle, Formula 129 Right Triangle, Hvpotenuse of 128 Right Triangle, Sides of 128 Rim Speed...... 9G Separating in a Given Ratio.. 84 Sign of Multiplication.. 15 Signed Numbers 40 Signs, Law of Signs for Divis- ion 68 Signs. Law of Signs for Mul- tiplication 53 Signs of Grouping 16, 49 Similar Terms 5 Similar Terms, Combination of 43 Singular Terms, Definition 43 Specific Gravitv 83 Speed .'. 96 Speed, Cutting 97 Speed, Rim or Surface 96 Speed Rule 96 Square of a Binomial 107 Square Root, Definition 109 Square Root of a Negative No 109 Square Root of Fractions 119 Square Root of Monomials 109 Square Root of Numbers..ll2. 113 Square Root of Numbers not Perfect Squares 117, 118 Square Root Trinomials. 110 Square, Trinomial 110 Subtraction, Algebraic, Defini- tion 45 Subtraction, Algebraic, Rule.. 46 Supplement 33 Terms, Definition 43 Terms, of Ratios 79 Terms, Order of 6 Trinomial, Definition 44 Trinomial Square 110 Trinomials, Square Root of.... Ill Variation 91 Vinculium 49 ^ LIBRPIRY OF CONGRESS 005 594 586 A