•^WSBJmWi^s UC-NRLF $B M23 S7E >^' l^-"-^' IN MEMORIAM FLORIAN CAJORl EIGHTH GRADE MATHEMATICS By Harry M. Keal Head of the Mathematics Department Cass Technical High School Detroit, Michigan and Nancy S. Phelps Grade Principal Southeastern High School Detroit, Michigan 1 nfi^nius ATKINSON, MENTZER ^ COMPANY NEW YORK CHICAGO ATLANTA DALLAS ^ COPYRIGHT, 1917, BY' ATKINSON, MENTZER & COMPANY Introduc tion I 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 intelligent 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, chernistry, physics, and electrical laboratories, etc., have furnished examples of mathematical apph cation 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. Digitized by the Internet Archive in 2007 with funding from IVIicrosoft Corporation http://www.archive.org/details/eighthgradematheOOkealrich TABLE OF CONTENTS PAGE CHAPTER I The Equation 1 CHAPTER II Evaluation 15 CHAPTER III The Equation Applied to Angles 25 CHAPTER IV Algebraic Addition, Subtraction, Multiplica- tion AND Division 38 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 % 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 x+3 and 10 denote the weights in the pans, the sign ( = ) of equality denotes the per- fect balance of the scales, arid 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 f or X : • 1. x+7 = 21 3. x+1. 1=3.5 2. x+2 = 3 4. x+2|=7^ 1 5. x+f = 5 _ il 12 THE EQUATION Fig. 2 Jj. 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 3j 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 hoik members of an equation may he divided hy the same number. Oral Problems: Solve for x : 1. 4x = 12 2. 2x=16 3. 5x = 9 4. llx = 33 6. 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+4j = 9 5. 5s+17 = 62 16. 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 A ATA fxibrlfsn 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. 3i-4 = 8 2. 7x-l = 15 3. 4x-f = 7} 4. 5x-.l = .9 6. 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 6. 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=13f 19. .5y-4 = 5.5 10. llx-9 = 90 20. 3x-9j = 8.5 SIMILAR TERMS Exercise 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. Fi 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+5x+ll. 2a and 3a, 3x and 2x, 4 and 7 are called similar terms. 6 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 / 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. 3yH-1.2+2y=46 6. 6.5x-l.l = 50.9 14. x-1.25x+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 knowrij from the other member. Example 1: 3x — l=x+3 3x = xH-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 2x4-3 = 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 6. 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-f-30+6m = 4m-|-8H-12+3m+3+mH-29 CLEARING OF FRACTIONS 9 11. 25x+20-7x-5 = 56-5x+5 12. 10x-61-12x+27x = 8x-41+20+4x+25 13. 25f+5x+6x+9|-2x=180-8x-8| 14. 2.8x+39.33+x = 180-1.2xH-32.09-7.16 15. 5x+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. tJf. 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 Example 2: -— == — ^»= = 4 x+6 = = 8 (multiplying both members by 2). x = = 2 Why? . r r _ 3 Y _16 ' 3 7r-3r = = 112 (multiplying both members by 21). 4r = = 112 Why? r = = 28 Why? 10 THE EQUATION „ ,^ m_3,m7m Example 3: — — 3t7t+ — = - — — 4 1^ 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. // equals are added to equals, the results are equal. 2. // equals are subtracted from equals, the results are equal, 3. // equals are multiplied by equals, the results are equal. 4 . // equxds are divided by equals, the results are equal. Solve: Exercise 7 1. 5^-^ = 10 3 6 6. x_2_x 2 3 6 2. ?+? = 9 5 4 7. y = ?+16 3 7 3. ?i'+?r=i7 3 4 8. ?x+3 = ^+4 4. x+Jx = 6 9. 2x x_x 1 9 6 18 3 6. x-|x = 7 10. 3x 1 X ,_ 7"3 = 27+' PRINCIPLES OF EQUATIONS 11 11. l|s+fs = s+13 13. -+4r--=26+lJr ^7 4 3 ^ ^^ 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? Example 2: L^ = 4 3x 16= 12x Why? (L. C. D. is 3x) x = l| Why? Exercise 8 Solve: 1. L5 = 5 a 2. 5=15 a 3. 1=2 4x , 3x 7 4. — = - 4 2 ^ 16 2x 6. — = — 5 3 6. 14 = x+9 7. 17 = 2x-3 12 THE EQUATION 8 x+10 = 2x-9 12. ^+47=-+4n 2 7 9. 2x-2i = 5x-17} 13. ^-l=lZ?-^-2ia ^ ^ 2 3 3 10. 7x+20-3x = 60+4x-50+8x 14. .lx+6.2 = .3x+.2 11. 3m+60 = 15m-f3-2m+7 15. 10H-.lx = 5+|x Exercise 9 Solve: 1. 7m_8 = 5i-— 5. 2_t 5_t^t t ^^ 6 12 3 9 6 2 2. 7x-8 = 6x+ix 6. 1+''—'^ = '^-^ ^ 2 5 6 3 4 3. ??-^=25i-^- 7. y-^+2i = -^-?:+^ 56 3 12 8484 4. ??+3 = ^x-2 8. -x-^x+4| = 3x+- 3 6 3 5 ^ 15 9. lly^x-l^x-302 = 60+l|x+183 10. x-3|+Jx = 9j-^ PROBLEMS 13 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 1^ of a number is increased by 6, the result is 30. Find the number. 8. The sum of J, ^ and ^ 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 $2j more than the other. 12. A farmer has 4 times as many sheep as his neighbor. After selling 14, he has 3^ 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? 16. 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 1^ 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 w 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 7i^o. W Coefficient: The expression 7abm may be thought of as 7ab-m, 7 abm, 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; ^xyz; 17mxw. 2. Name the coefficients of ab in the following: S^axby; fmabz; .Obnsa. 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 n£ver 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 the numerical coefficients, bases and exponents in the following : V^x^, Sjaio, 3.7m2n^ f^r^ l|m 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 valu^ by substituting definite numbers for general numbers in the expression, and performing the operations indicated. EVALUATION OF EXPRESSIONS 17 Example 1 : Evaluate 4:ix^x^ if a = 3, x = 2. 4a2x3 = 4.32.2'3=4-9-8 = 288. a2 Sb* , m^ Example 2: Find the value of ~, ; ~^ TTt ^ m^ c2 2a3 when a=l, b = = 2, c = 5, m = 2. a2 5b4 m^ m^ c2 2a3 P 5-24 25 2^ 5^ ' 2V 1 80 32 ~ 8 25 2 =i-?-« 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) = 13-26 = 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 13. x(a+b) 17. —:-+:; X b d 14. 4b(x-c) 18. (x+a)(c-d) 15. a2+2ab+b2 19. Viad 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: 1. iaVc 11. (3x+7)(c-2) 2. x3-a3 12. Vb2+d2 3. x'+d^ 3a2 13. — (x2-c2+25) 4. 3b2-4m2 bd 6. xM-a^m 14. a3(x-c+3m)(c2+d2) e! 2a2x3(c-d) 15. ^^ 7. 4+^ 16- ^(x2+a2-b2)(c2-d2-m2) a^ d 2d 8. ^(x+a)c 17. Vx(a+b) 9. ^a3x2c(b3-d2) 18. ^d(a+b)+c c2 x2 ^^- {:^ + l^— 7 ^^- ^5x(a+b) b^ d^ a^ 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 = 21 J Example 2: Evaluate A = ^h(b+bO (Area of a trapezoid) ifh = 3A,b=12i b' = 6i A = i.3^(12H6i) A _ 1 q 3 .18^ 1 51 75 2*16' 4 3825 128 A= 29yV8 or 29.102- Perimeter Formulas 27 The perimeter of a figure enclosed by straight lines is the sum of its sides. 6 a 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 = ij. 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. 5. Find P, in P = 2(a+b), if a = |, b = |. 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, b = f, 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+b+c+d, when a=lf, b = lf, c = 1y^, d = li. 12. Find P, in P = a-f-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 xts perimeter is 82. 6. The length of a rectangle is 4 more than twice the width and its perimeter is ISS^^. 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 J 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 Fig. 8. Rectangle Fig. 9. Parallelogram b Fig. 10. Triangle I 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=ly^g^, b = 6.71. 6. The area of a triangle (Fig. 10) is \ the product of the base and altitude. A = ib.h. Find A, if b = 12.23, h. = 6.57. 6. Evaluate A = ^b.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 = 19f, b' = 12f 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 = 27rT. (Fig. 12). Find C, if ;r = 3.1416 (See art. 17) r=li 2. C = ttB. Find C, if D = 5.724. 3. A = 7rr\ 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 = 7i, r = 4j. 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 = ljd+iifd = lf. 4. L=lfd+|, if d = 2i. 6. S = 2gt^, if t = 4. (g is a definite number. Its value is 32.16), 6. S = |g t^+vt, if t = 3, V = 7. 7. D= Va2+b2+c2, if_a = 3, b = 4, c=12. 8. V = |h(b'+b+ Vb-bO, if h = 2f, b=12, b' = 3. uv 9. F = u+v, 10. V = |^r3, if r = 2.3. if u = 11.5, v = 6.5. 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+?^5^^^ = 3x+l. 5 _ , 6x-2 _ , , 2xH ^- = 3x4-1. Why? o 10x+6x-2= 15x+5. Why? x = 7. Why? Check: 5 14+8 = 21 + 1. 22 = 22. Exercise 18 Solve and check: 1. 6y-7 = 3y+20. ^ 2(x+2) 2. ll=3x+9. ^ = 7. X V ^ 8. Z(?±^.^ = f+2. 3. 3-1 = 2-2. 12 6 4 4. 2(2x+5) = 13. 9. 2x-l = f(5-x)-l|. 7(5-x) 6. 6(z-6) = z+8. N^*^- i(5-x) = 6. ?^^ = 3. 10. ?(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. 14. 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 a-ngle 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 S3 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/GZl F/G 28 Exercise 20 1. Measure the angle in Fig. 21. 2. Measure the angle in Fig. 22. 3. Measure the angle in Fig. 23. 4. Measure the angle in Fig. 24. 6. Measure the angle in Fig. 26. 7. Measure the angle in Fig. 27. 8. Measure the angle in Fig. 28. 9. Measure the angle in Fig. 29. 6. 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/G34 S A r/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). Fig. 36 37 Zl +Z2 + Z3 +Z4 = ZAOB (Fig. 36). If AGE is a straight Hne, the Z AOB 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°. Fig. 37 Exercise 23 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 line are 16°, 78°, 51°, respectively, angle. Find the fourth 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 a^ 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 line 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? o o 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. |x, x+4, lix+2. 2. fx-2, iVx-f 7, 3(x+7) Jx+19. 3. 4(x+l), 7(2x-ll), 127-6X. 4. 3x-i, 2x, 2f (2x+l), |(x+6). 6. 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 J 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 22^°. Fig. 40 ANGLES ABOUT A POINT 33 39 Z1+ Z2+ Z6=180° Why? Z7+Z4+Z5 = 180° Why? Therefore, Z1+ 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-jx,ljx-13, 4(^+11). DO 6 2. 23+-, 136- -,-+93, -+17. 4 5' 3 2 3. 4(x-5), ?+5li, 3x+47|. 4. i(3x-36), i(2x+15), - +30, 82-|x, x+48j. 6 5. |x+3.15, 3(x+1.75), J(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. 6. 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. 4^ 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. 1^ of an angle, plus 55° is equal to ^ of its supplement, plus 4°. Find the supplementary angles. Let X = one angle 180 — X = other angle then fx4-55 = -|(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 f of an angle, minus 16°, is added to f 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°; 10^°; 43|°? 2. X is the complement of 32°. Find x. 3. X is the complement of x+76°. Find x and its com- plement. 4. fx+ 12°, and §xH- 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 minus 15°. Find the angle. 3. If 20° is added to five times an angle, and 20° sub- tracted from ^ of the complement, the two angles obtained, when added, will equal 114°. Find the angle. 4. f of an angle is equal to f of its complement, minus 14°. Find the angle. 5. f 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 \ 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. REVIEW OF ANGLES 37 6. The complement of an angle is equal to the sum of the angle and J of its supplement. Find the angle. 7. An angle increased by ^ of its supplement is equal to twice its complement. Find the angle. 8. ^ 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. Y^Y 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 ^ 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 hne. The second is ^ of the first, the third is ^ the supplement of the second, the fourth is f 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? 6. 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 j^S 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 multipHcation. (See Art. 19.) Exercise 31 Add .: 1. + 19, + 10 16. -if 2. -19, -10 17. -li 3. -19, + 10 18. -3i 2i 4. + 19, -10 19. 6!, -8| 5. -10, + 19 20. -7h +7f 6. + 10, -19 21. 13|, -23f 7. -75, +25 22. -llf,8f 8. +38, + 19 23. -2.32, -1.68 9. + 11, -26 24. 3.47, 5.43 10. + 10, -10 25. 8.44, -7.25 11. -40, +39 26. 8.75, -11.25 12. -4, +26 27. 5.732, -4.876 13. i-i 1 28. -18.777, -3.333 14. il 29. -173.29, 239.4 15. 1 1 16> ~ -1 30. -208.21, 171.589 42 ADDITION /fd 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. 6. -79, -21, -100. 6. -119, +1, -21, -14, +101. 7. -2.36, +4.24, 5.73, -8.66. 8. -3f , 5f , -4yV 10. 23|, -19|, 17f, -111,5^. ADDITION OF SIMILAR TERMS 43 61 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; as 2x3yz2, —^x^yz"^. 53 Only similar terms can he combined. Exercise 33 Add: 1. -16r, 18r, 8r. 2. 4.2s, ~5.7s, 2s. 3. 7|x, -4fx, -2ix, X. 4. 2jab, 4|ab, — 3|ab, ab. 5. 24abc, — 36abc, lOabc, +4abc, — abc. 6. -32a2b, 40a2b, -Qa^b, 2a2b. 7. 3vV, vV, -9v2y3, -4v2y^. 8. -3|xVz, 5fxVz, -4yVxVz. 9. 3.16xy2z5, -4.08xy2z5, QmxyHK 10. 8(x-y), -6(x-y), +4(x-y). 11. -12(x+y), -7(x+y), -(x+y). 12. -6|(c-d),3f(c-d),4|(c-d). 13. -8(x2+y2), 24(x2+y2), 17(x2+y2), +(x2+y2). 14. 8(x+y+z), 14(x+y+z), -2(x+y+z). 15. Il(x2+y)^ -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 </iree terms a trinomial. Addition of Polynomials 56 Example : Add 2a3 - 2a2b - b^, - Tab^ - 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^+2a^b+b^+3ab^ -2a3 -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-4|x+2jf, 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. f(a2-b2)-f(b2-c2)+f(c2-a2), f(a2-b2)-|(c2-a2), 4(b2_c2)_2i(a2-b2). 9. 5(x+y)-7(x2+y2)+8(x3+y3), -4(x'+y')+5{x'^+y')- 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 miles 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 + 12 - 5 -12 + 5 -12 - 5 + 12 lowing: + 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? 59 RULE: To subtract one nximber from another, change the sign of the subtrahend mentally and add. Exercise 35 Subtract: 1. +27 + 12 4. -32 +21 7. + 16 -42 10. -llf + 15i 2. -13 - 8 5. 6. + 15 +82 - 81 -127 8. 9. - 39 + 100 12| -4i 11. -fax -fax 3. +21 - 5 12. 7ibV 6fbV 13. by2 Ifby^ 16. 3a-b+c 4a — c 14. -5 (a+y) -li(a+y) 17. 3ix2+5fy3- z 2ix2-2|y3-2z 15. 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)+51a2bxVz+19ab2xy3z. -3|(ab+c) + l.Q7(x+y+z)-17a^bxVz+20abV^z. 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) 6. (8.91abc)-(-3fabc) 2. (+35)-(-2li) 6. (-2yVVz)-(3.1416x2yz) 3. (-|)-(+f) 7. (3a+2b)-(2a+3b) 4. (+2jmx)-(-f5mx) 8. (5x2-7y2)-(-2x2+y2) 9. (-9|m3n+3|mn3) - (4f m^n-Sfmn^) 10. (3ja+2b)-(3ia+7c) 11. (-1.7a2b2-2.9b4)-(-3.3ab3-4.16b4) 12. (4x-5§y+3fz2)-(2ix+6.25y-5jz2) 13. (- 11.23a2b2+4jbV- Ijc2a2) - (- 11.22a2b2+ 4jbV-1.875c2a2) 14. (6x2-9mx~ 15m2) - (Qx^- IGm^) 15. (-a^b^+a^b^+ab^) - (a^b-a^b^+b^) 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 6x^ — 7x— 4, subtract the sum of Qx^ — 8x+x' and 5 — x^+x. 4. Simphfy (im-in+fp)-(|m-fn+|p)-f (- iVm- |n-|p). 6. Subtract the sum of 6— 4x^ — x and 5x — 1 — 2x2, fj-om the sum of 2x3+7-4x-5x2 and 3x2-6x3-2+8x. 6. Find the difference between (12x4+6x5-2) + (6x4- 8x+14-8x3), and (0)-(- 10x3+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: fgt2 +v- Jt gt2 -|v+ 6t -1.3gt2 + llt+.02 -10v+1.2t+lJ 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-(— -^)=--(--4|) 3 6 2 2 ^ 15. ix+(?x-?)-(^x+l) = lf. 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-2bH-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-|3x+(-2x-x-a) | = 4x — |3x+ ( — 2x — x-fa) } (removing the vinculum) = 4x— (Sx — 2x — x+a } (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. 10-[(7-4)-(9-7)] 3. 4x-[2x-(x+y)+y] 4. -llb+[8b-(2b-fb)-3b] 5. 8kz-[7kz-3kz-5kz] 6. x2-{'3x2-2F+l} 7. a' - ( - 6a2 - 12a + 8) - (a^^- 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+ 10x)] = 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, four 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 hy 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 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 pulling 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 to represent the product of (+5) (—4). In this case there is an upward-puWing force (+5), on a left t-^ 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 ^ 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. (+3f)(+li) 2. (-f)(-i) 9. (-6f)(-6j) 3. (+!)(-i) 10. (7|)(-A) 4. (-f)(+i) 11. (-i.i)(+i.i) 6. (-§)(+■!) 12. (-2.03)(-4.2) 6. (+|)(-|) 13. (+.3)(-.03) 7. (-|)(-t\) 14. (+8.75)(+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*)(y3)=x^y'. Example: Multiply (Ta^bx^) by (-3ab3y2) 7a2bx3 = 7-a2.b-x3 -3ab3y2=-3-a-b3-y2 (7a2bx3)(-3abV)=7.a2-b-x3.(-3).a.b3.y2 which may be arranged 7-( — 3)-a2-a-b-b^-x^-y2= — 21a^b^xy. 7^ RULE: To multiply monomials, multiply the numerical 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 MONOMIALS Exercise 40 Multiply: 1. (3a3)(7a«) 11. (+2ic)(-2id) 2. (4x4) (-6x5) 12. (-9m2n3)(-7mV) 3. (-5jmio)(-2m) 13. (-7)(+m2n2) 4. (-13a2b)(-2ab3) 14. (+7|)(-gt2) 5. (+5a2bc2)(-4abV) 15. (3xy)(xy) 6. (-6a2b)(+3b2c) 16. (6r2)(-|r2) 7. (+2ab)(-3cd) 17. (_x2)(-x3) 8. (+3)(-x) 18. (-6.241)(+3.48m) 9. (+5)(+|) 19. (x + y)3.(x+y)4 10. ("DC-pq) 20. (m2-n2)5.(m2-n2)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 25. (+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)(-2i)-(8x)(2) = (4i)(6) + (-7)(4x) + (6x)(0) 30. (-ll)(-3x)-(4|)(+6) = (20j)(-2)-(-3)(17x) 56 MULTIPLICATION 75 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 +3f on an arm +6? 4. What is the leverage caused by a force —27 on an arm -2i? 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 3 J? 8. What is the leverage caused by an upward force of 3 J, on a right arm of 1^? 76 Suppose two or more forces are acting on the lever at the same time as in Figs. 46, 47, 48. -« — 2 — *-+-* 3 *- ; i 6 4 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)H-(-4)(+3) = + 12-12 = LAW OF LEVERAGES '^ ! J 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. ^ 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. ^ Fig. 49 7 i ^ Fig. 54 r^-^ rr^^i — i Fig. 60 Fig. 55 F /^ r^i* Fig. 61 A5" -J* r~^ — AT -« 10 ^0 Fig. 56 6 — -4— J -^ /^ JW Fig. 52 260 Fig. 67 n ^ r 'i^-2i-\ 3i-*\ Fig. 53 Fig. 68 zyc 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)(-3a2)(+4a^)(-7a3) (-2a)(-3a2) = +6a3 (+6a3)(+4a5) = +24a8 (+24a8)(-7a3)=-168aii or (-2a)(-3a2)(+4a5)(-7a3)= -168a" Exercise 42 Multiply: 1. (-3)(-4)(+5) 2. (-|)(+3i)(-i) 3. (+6)(-li)(-ft)(-7) 4. (Ila)(-7ab)(+4abc)(-9b2c2) 5. (aV)(-4a2b)(-lla3bO 60 MULTIPLICATION 6. (-4jab)(-3fac)(-3^bc)(Jabc) 7. (1 .25m2x) ( - 2.4m3x2y) ( - 4.63mxy2) 8. (-3.57)(+a^2)(_i|.aV). 9. 4(x-y)3. (x-y)2.{-7(x~7)} 10. 3 (m2-n3) {-4(m2-n3)4} . (m2-n'^)2 Multiplication of Polynomials by Monomials a ..J-. — 1~ 2D 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. 80 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 Sm^ — 5mH-7 by — 9m^ -9mH3m2-5m+7)= -27m5-l-45m4-63m^ Exercise 43 1. a3-7a2b+9ab2 by 3a2b3 2. 6x5- 5x6 -7x* by -7x» 3. -3m2-n2+5mn by 4m2n3 .4. a3-2a2x+4ax2--8x3 by -2ax 6. ia2-iab+ib2 by -ib MULTIPLICATION OP POLYNOMIALS BY MONOMIALS 61 6. 3x^-15x2+24 by -^x^ 7. ^+1-1 by 12 2^3 6 8. -L-1+- by -30 10 5 6 ^ 2m 3p 7 , ^^ 9. — - — — by 10 5 2 15 Simplify : 10. -2.4a3zH2ja2x-3|xz+.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) ,5. 16(^+1) -24(|+| 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. 15y2+lly(2-5y)+4(10y2-9) = 19 6. ^_^ = 3 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 8. 1 x-1 9. 6a-— -3(2a-l)=i a a 10. 5x-+2x+6_,,i 5x Multiplication of Polynomials by Polynomials —Cf — M- d za ac 2b be 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) = acH-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 emd combine like terms. Example : Multiply x^— x+lbyx— 3 X +1 x -3 x^- 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 5a?+4:V> 6. a+1 by a^-l 7. a-3 by b+7 8. 2x2-5x+7 by 3x-l 9. 4m2-3ms- s^ by m2-3s2 10. x^-x^+x^-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)(|x+iy) 4. (x+3)(x-4)(x+2) 6. (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) = 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(x2H-3x+7) 11. 12. 13. 14. 16. = 2x 2x4-7 x+3 "^""^^ (3x4-2) (2x+3) ^ (2x-l)(x-f4) (x-2)(x-3) _ (x4-3)(x- 4) (x-4)(x-5) 3 4 ~ 12 1 x(2x+l) _(2x -3)(3x-f4) 2x+17 4+2 6 ~ 4 Exercise 47 Mu] 1. itiply: 2m2-m-l by 3m2+m-2 2. 2p3-3p2q+7pq2+4q5 by 4p-3q 3. a^+b^+ab^-f-a^b by a^b-ab^ 4. 5-3a+7a2 by 4+12a2 6. -4mn4-3m2-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 2b3H-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-|b+fc-|d by ^a+fb-Jc+|d 16. |a2-fab-|b2 by ija^-fb^ 17. 2fm2n2-4jn3 by Im^-fn 18. 1.25a+2.375bH-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)(5^+^) 6. (x+y)(x3+y3){x2-y(x-y)} 6. (x+yH-z)(x-y+z)(x+y-z)(y+z-x) 7. (2a+5b-c-4d)2 8. (fa3-|b2)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)3 17. (a+b+c)3-3(a+b+c)(a2+b2+c2) 18. (2in2-3mn+4n2)K^-^)^ 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 |) + (-14)(-|) + (-10f)(+i|; 2. (+15)(-^) + (-14)(-^) + (-10f)(+±|) = 3. (-4f)(5x) + (+7|)(7x) + (-8f)(0) + (7)(-12)=0 4. 3x-3(ix-7)=35 5. (2x-l)(3x+7)-3x2=(x-l)(3x-12)+20 ^ 3x+5 , x-7 ^- "^— ^"~6~ 7. (4x+f)(fx-i)=|i ^ 3(3 -2x) 2(x-3) 2 4(x+4) . 1 ^' 10 ~ 5 '^'^^-~^ +10 9. t(x+5)-|-(x+7)+V(x+l)-|(2x-5)=i(x+22) EQUATIONS INVOLVING MULTIPLICATION 67 (x-l)(x+2) _ (2x+l)(x+2) (2x+l)(x-l) 2 12 ~ 6 11. If I" 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 -f of the supplement of an angle is decreased by f of the complement, the result is 53°. Find the angle. 14. J 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^ 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. Tiiese may be derived from the same laws for multiplication, 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 (+5) = +4 (+4) = ? (-5)= -4 (-4) = ? (+5) =-4 (-4) = ? (-5) = +4 (+4) = ? Therefore, from 1. from 2 l(+20)- l(+20)- /(+20)- l(+20)- (-20)- (-20)- (-20)- (-20)- 84 Law of Signs for Division: their quotient is plus. If two numbers have unlike signs, their quotient is minus. from 3 from 4 // two numbers have like signs. Divide : 1. (+f)-(+i) 2. (-i)-(+i) 3. (+i)^(-i) 4. (-l)-(-l) (-f+)-(+3V) Exercise 50 6. 7. 8. 9. 10. 68 (+2i) ( 16\ 2 l) (+3f) ■( (+7i)-( 10) 9) (-42) + (-fi) (+72) ^(-41) DIVISION OF SIGNED NUMBERS 69 11. (-8.5)-^(-1.7) 16. (+3.6)^(-2i) 12. (+3.2)-(-.8) 17. (-3i-56)^(-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 e 12 3x i9_4x 172 x+3 _x-2 3x-5 1 2 ~ 3 ~ 12 "^"4 2x±5 x-3 x_ 1 ®- 9 ~ 5 -3-^^+173 9. f(x+2)-T'5(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^) 4- (x^) = x^ (x8)^(x3) = ? 86 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. (x^)-(y3)=^ Example: Divide 48 a^b^c^x^V by -Sa^b^c^x^ — •; = -6.a5.b.l.x3.y4 — Sa^b^c^x^ "^ = — 6a%xy c* c" NOTE : — = 1 • Also— = c" by the law of exponents. 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. Exercise 52 Divide: 1. -91a 2. -32x6 3. +22a^b3c7 4. +6jm3n3 6. -8jpi3qi'r'^ 6. -4.24xV^z7 7. 1.75an)«x5 8. -.SSm^n^ 9. -.OOlxVm^ 10. +3.1416a2b3mi« by +13 by -8x^ by -lla^b^c^ by — l^mn^ by Sfqior^ by — .4xy z^ by -.35x^ by 17n* by — lOOxym by +4a2b3mi« DIVISION OF MONOMIALS 71 Simplify : 1155a^x7z5 3.1416r^ -231a2x«z .7854r2 -1.732t^uV +a^bVQ +2u6s2 -2a3bc» 3.1416xyV7 -a^(x+y)« ". _iix26y225 "• +2-fa(x+y)4 27m^n^x7 (a+b)^(a+b)^(x+y) ^ 1.125m2n3x7 .0625(aH-b)3(x+y)^ 32.16t^ 18.75a(m^-n^)^Q -.08t2 ^"* 2i(m2-n^)7 Exercise 53 Solve for x and check: 1. -ax=-ab 6. 3a2b3x=-12a3b3 2. +|bx=-8b 7. 7a-3ax = 28a 3. — .3inx = 2.4m 8. 4mx— 7mx=12m — 18m 4. -4x=-12(a+b) 9. 6a2b-7ax= -29a2b .n X 3 1 5. -fx = 2m 10. 3^-- = e^ lOx . 5x 5m 12. ^b_3(x-2b)^^^ o o 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. 7Z 71 = :; • 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 lm«-35m^+7m3 by -7m2 21m«-35m4+7m2 -7m2 = -3m*+5m2-m Exercise 54 Divide: a^x'c* — ax^c^y^ + a^xc^z axc^ 4xVV- 12xVz^-24xVz^H-16xyz — 4xyz 2.31m2n2+7.7m3n3- .33m%^ l.lm^n^ - ijt V - 9.81tV - . 378tv^ -9tv2 1 ■ 125a^x^z^ - .375aVz^ - 4.2a^x^z^ .25aVz2 -3fabcd+2|bcdeH-7|acde -l|cd Solve for x and check: 7. ax = 2ab — 3ac+4ae 8. 3a2m3x = I.lla3m3-3.3a2m* 9. 4m2s2x - 3.2m3s2 = ISam^s^ 6. EQUATIONS INVOLVING DIVISION 73 10. 3jxyz-1.4y2z = .35yz2-70yz 11. 4m2x-7m3n4-3m2x+8m2n5 = 5m3n4-2m2x+2m2n5 12. il^-T^B^^^.lOlm^ o u ^^ X a2b+2bx 13. - — ^ = 3ab2 a ab nx 3(n2x-m2n2) 4n 14. 8mn + -^ = hm^n m mn m 2mx+a2m^ 5(b^n^+nx) _^ 2m2n -3mn^ 15. — — <jX m n mn Division of Polynomials by Polynomials 90 By Art. 81, (a+b)(c+2) =ac+bc+2a+2b 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 9t 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 same way. 74 DIVISION Example (1): Divide a3-6a2-19a+84 by a-7. a^+a-12 a-7)a3-6a2-19a+84 a^-7a^ +a2-19a +a-— 7a -12a+84 -12a+84 Example (2): Divide 24+26x3+120x4- 14x- 11 Ix^ 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- - 111x2- 14x -1-24 120x^-10x3- -60x2 +36x3- - 51x2- 14x +36x3- - 3x2 -18x - - 48x2+ 4x+24 - - 48x2+ 4x+24 Exam ple (3): Divide i x^+xV- f y4 by x2+xy+y' x2- xy +y2 x2+xy+y2)x* +xV +y^ x4+ x3y+xV - x3y +y^ - X3y_x2y2 — xy3 +xy +xy3+y4 +xV +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 6. 12x2+31xy-15y2 by x+3y 6. m3+3m2-13m-15 W m+1 7. 10x3- 19x2y+26xy2-8y3 by 2x2-3xy+4y2 8. 3m4 - lOm^ - 16m2 - 10m - 3 by 3m2+2m+l 9. 2x^-x3y+4xy+xy3+12y4 by x2-2xy+3y2 10. 4x4-24x3+51x2-46x+15 by 2x2-7x+5 11. 9xV - 15x3y2+ 13xy - 3xy4 by 3x2y-xy2 12. Sm^n - 22m6n2 - 7m^n3+ 53m4n4-30m3n5 by 4Tn4+3m3n-5m2n2 13. 10x3 _ 29.3x2y +37xy2 - 20y3 by 2.5x2 -4.2xy+4y2 14. 14x3+17x2y+39xy2+17y3 by 3.5x2+2.5xy+8.5y2 15. 18x3-53x2+27x+14 by 4x-8 16. 13.12m5n+1.36m%2_ 7.15m3n3+2.35m2n4-.125mn ' by 3.2in2n-2.4mn2+.5n3 17. a2+ab+2ac+bc+c2 by a+c 18. a^bx + abcx + a^cx + ab V + b^cy+abcy by ax+by 19. x2+8-10x+x3 by 2+x2-3x 20. m^ + n^ — 4m3n — 4mn3 + Gm^n^ ' by m2+n2 — 2mn 21. a5-9a3+7a2-19a+10 by a2+3a-2 22. 16m4-72m2n2+81n4 by 4m2-12mn+9n2 23. m^— 64n3 by m— 4n 24. 32as+243b5 by 2a+3b 25. a2+2ab+b2-c2 by a+b— c 26. a2-_x2-2xy-y2 by a— x-y 27. a^+4+3a2 by a2+2-a 76 DIVISION 28. 4a2-b2-6b-9 by 2a+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-fa+3b+3 32. (a2-4ab+3b2)x = a3-8a2b+19ab2-12b' 33. (2m-3n)x = 8m3-22m2n+mn2+21n3 34. (a+bH-2)x = a2+2ab+b2+4a+4b+4 35. (y-f-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 ^ of the larger and \ 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 yof the smaller number. Find the numbers. 6. Divide 24 into two parts such that if ^ 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 a house for 3 times that amount. If the lot had cost $240 less and the house $280 more, the lot would have cost i 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 1^ 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 -j^y of its supplement + 131°. Find the angle. 13. If 3 times an angle is subtracted from J its supplement, the result is yt 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, 4j' 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 |; 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 2j 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 577- 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 § 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 S''. 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 5j" and 2f ". (See Exercise 16, problem 2.) 22. Find the ratio of the areas of two circles whose diame- ters are respectively IV 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 = 9j, 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 94 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: 1. A 8 3. 1 25 5. ?-^ 32 7. '1 64 9. .72 1* 2. ^ 16 4. 1? 20 6. !i 30 8 2f 10. 3.24 129.e 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. 9 10 ^19 ^25 , 37.5 — 3. — 4. — 6. 11 16 7| 5.15 Percentage is found by reducing a ratio to a decimal correct to .01, and multiplying it by 100. 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 62j 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 3j parts of soda, if parts of cream of tartar, and 6.5 parts of starch. Find the percentage of cream of tartar. 16. 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 . . $1200 Painting . . • . . $300 Plastering . . $ 250 What per cent of the total cost is represented by each item? Check by adding the per cents. 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. Example. The dimensions of a block of cast iron are 3j"X 2f "X 1", and its weight is 37.5 oz. Find its specific gravity. 3jX2f X 1 =8.94 cu. in. (the volume of the block) .0361 lbs. = .5776 oz. (weight of 1 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 fts 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 Separating in a given ratio. Example: Divide 17 into two parts which shall be in the ratio f . Let 2x = one part. 3x = other part. mott? ^'^-^ Then2x+3x = 17 ^^ ii^S 5x = 17 x = 3f 2x = 6|-, one part. 3x=10|-, other part. Check: 6|+10i=17, T^ = ^ = f 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 -f-. 4. Divide 200 in the ratio ^. 6. 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 ^. 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 -J, 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 16. 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 : y^ = ii 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 1^ = 9 15 = 4x (clearing of fractions.) x = 3f Check: A=— ^^ 9 5 _ 5 T2— T2 PROPORTION Exercise 62* Solve and check : 1. ^ = i? 25 14 6. 11 18 12 X 2. ^ = i? 7 17 6. 125 206 X 305 3 ^-^ ^- x'll 7. 144 3x 195 25 4 ^ = ^ 9 14 8. X 3j 24 41 87 9. The ratio of x+1 to 9 is equal to the ratio of x+5 to 15. Find x. 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 tof . 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. 16. 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 draving to represent 32'? 88 RATIO, PROPORTION, AND VARIATION 17. If the scale of a drawing is f " to 1', how long should a line be made to represent 10"? 18. If the scale of a drawing is ij" 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^" 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 g^'s in y^. 3^ 2^ s in Y 5^. Let X = number of -oV's in — — 32 15 15x = 352 x = 23y5^, approximately 23^. Exercise 63 1. How many J's in -^q? 2. How many y^'^ i^ i'^ 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 — rT = "r x-j-l 5 Check: 4^4 4+l~ 5 5 5 5x = 4x+4 iL. C. D. is5(x+l)> X = 4 Why? X 1 Example: 2. Solve ^, ^ ,, =^ o(x — 1) 6 2x = x-l |l. C. D. is6(x-l)| x=-l Why? Check; -1 1 3(-l-l) 6 -6 6 6 6 Example: 3. Solve x+l x-3 ::heck: x+2 x-4 x2-3x-4 = x2-x-6 I L. C. D. is (x+2) (x-4) — 2x=-2 Why? X = 1 Why? 1 + 1 ^ 1-3 1 + 2 1-4 3-3 1 = 1 3 3 90 RATIO, PROPORTION, AND VARIATION Exercise 64 Solve and check: 1 _^=1 7 -^ = i x-1 4 3(5x-6) 9 _x ^4 7(3x-7) ^23 ^* 3(x-l) 9 4(x+3) 12 y ^ 1 2 ^ 3 ^* 5(y+2)"l0 3x+l 5x+2 4. ^-±?4 10. ' ' x-5 6 4x-3 3x+4 X 2 ^^ x+5 x+25 3x+l 7 x-4 x-2 2y+3 ^3 5x-7 ^ 10x+ll 3y+7 4 3x-5 6x+ 7 13. 2x-3 3x 2(x-3) 3x-4 14. The ratio of an angle to its supplement is J. Find the angle. 15. The ratio of an angle to its complement is y. Find the angle. 16. The ratio of the supplement of an angle to the comple- ment is f . Find the angle. 17. If an angle is increased by 3° and its complement de- creased by 13°, the ratio of the two angles will then be -|. Find the original angle. 18. The base of one rectangle is 3 less than the base of another. The altitude of the first is 3, and that of the second is 5. The ratio of the areas is f . Find the bases of the two rectangles. 19. The ratio of 3° to the complement of an angle is equal to the ratio of 21° to the supplement of the same angle. Find the angle. DIRECT VARIATION 91 20. Find three consecutive numbers such that the ratio of the first to the second is equal to the ratio of 5 times the third to 5 times the first plus 16. Variation 101 Direct Proportion: If a train travels 120 miles in 3 hours, it would travel 240 miles in 6 hours, -f is the ratio of the two times, and -J^f ^ is the ratio of the two distances, taken in the same order. Both ratios reduce to J and therefore the problem may be expressed by the proportion, -f = i^%. An increase in time produces an increase in distance. If the train travels 120 miles in 3 hours, it would travel 80 miles in 2 hours because |- = ^-^-^ . A decrease in time produces a decrease in distance. When two quantities are so related that an increase or de- crease in one produces the same kind of a change in the other, one is said to be directly proportional to the other, or to vary directly as the other. Example: If a piece of steel 3 yds. long weighs 270 lbs., how much will a piece 5 yds. long weigh? Let X = weight of the 5-yd. piece. 3 270 - = — — (the weight is directly proportional to the length.) 5 X x= ? Exercise 65 1. If 60 cu. in. of gold weighs 42 lbs., how much will 35 cu. in. weigh? 2. If the interest on a certain sum of money is $84.20 for 5 yrs., what would be the interest for 8 J yrs.? 3. If a section of I-beam 10 yds. long weighs 960 lbs., how long is a piece of the same material weighing 1280 lbs.? 92 RATIO, PROPORTION, AND VARIATION 4. An engine running at 320 revolutions per minute (R.P.M.) develops 8^ horsepower. How many horsepower would it develop at 365 R. P. M.? 5. At 40 lbs. pressure per square inch, a given pipe dis- charges 180 gallons per minute. How many gallons per minute would be discharged at 55 lbs. pressure? 6. What will be the resistance of a mile of wire if the resistance of 500 yds. of the same wire is .65 ohms? 7. A steam shovel can handle 900 cu. yds. of material in 8 hrs. At the same rate how many cu. yds. can be handled in 7 hrs.? 8. A 12 pitch gear 10" in diameter has 120 teeth. How many teeth would a 6" gear with the same pitch have? 9. An engine running at 185 R. P. M. drives a line shaft at 210 R. P. M. At what R. P. M. should an engine run to give the line shaft a speed of 240 R. P. M.? 10. If a machine can finish 65 pieces in 75 minutes, how long will it take it to finish 104 pieces? * 102 Inverse Proportion: K a train travels a given distance in 4 hrs. at the rate of 40 miles per hour, it would take 8 hrs. to travel the same distance if the rate were 20 miles per hour. ■^ is the ratio of the two times, and -|^ is the ratio of the two rates, taken in the same order. |- = J but ■f§ = |-. Therefore the problem may be expressed as a proportion if one ratio is first inverted. ^ = ^% or |- = -f^. An increase in time produces a decrease in rate. If a train travels a given distance in 4 hours at the rate of 40 miles per hour, it would take 2 hours to travel the same distance if the rate were 80 miles per hour, because |^ = -|^ or 1^ = 1^-0^. A decrease in time produces an increase in rate. INVERSE VARIATION 93 When two quantities are so related that an increase or a decrease in one produces the opposite kind of a change in the other, one is said to be inversely proportional to the other, or to vary inversely as the other. Example: If 6 men can do a piece of work in 10 days, how long will it take 5 men to do it? Let X = time it will take 5 men 6 X^ (the number of men is inversely proportional to ^ ~ jQ the number of days.) x = ? Exercise 66 1. A train traveling at the rate of 50 miles per hour covers a distance in 5 hrs. How long would it take to cover the same distance if it traveled at 40 miles per hour? 2. A man walking at 4 miles per hour can travel a dis- tance-in 3 hrs. At what rate would he have to walk to cover it in 2 hrs.? 3. If 40 men can do a piece of work in 10 days, how long will it take 25 men to do it? 4. 12 men can do a piece of work in 28 days. How many men could do it in 84 days? 5. The number of posts required for a fence is 42 when they are placed 18 ft. apart. How many would be needed if they were placed 14 ft. apart? 6. One investment of $6,000 at 3|% yields the same income as another at 3%. What is the amount of the second investment? 7. A man has two investments, one of $15,900, and the other $21,200. The first is invested at 6%. At what rate must the other be invested to produce the same income as the first? 94 RATIO, PROPORTION, AND VARIATION 8. A man planned to use 36 posts spaced 9 ft. apart in building a fence. His order was 6 posts short. How far apart should he place them? Exercise 67. (Review) 1. The circumference of a circle is directly proportional to its diameter. If the circumference of a circle whose diam- eter is 6" is 18.8496", what is the circumference of a circle whose diameter is 4"? 2. If the circumference of a circle whose diameter is 5" is 15.708", what is the diameter of a circle with a circumference of 28.2744"? 3. The area of a circle varies directly as the square of its diameter. If the area of a 2" circle is 12.5664 sq. in., find the 12.5664 4 area of a 4" circle. (Suggestion : = — ) 4. The volume of a quantity of gas varies inversely as the pressure when the temperature is constant. If the volume of a gas is 600 cubic centimeters (c. c.) when the pressure is 60 grams per square centimeter, find the pressure when the volume is 150 c. c. 5. A quantity of gas measures 423 c. c. under a pressure of 815 millimeters (m. m.). What will it measure under 760 m. m.? 6. The volume of a cube varies directly as the cube of the edge. If the volume of an 11" cube is 1331 cu. in., what is the volume of a 7" cube? (See suggestion, problem 3.) 7. The volume of a sphere is directly proportional to the cube of its diameter. Find the volume of a 6" sphere if a 10" sphere contains 523.6 cu. in. 8. The volume of a quantity of gas varies directly as the absolute temperature when the pressure is constant. If a REVIEW 95 quantity of gas occupies 3.25 cu. ft. when the absolute tem- perature is 287°, what will be its volume at 329°? 9. The velocity of a falling body varies directly as the time of falHng. If the velocity acquired in 4 seconds is 128.8 ft. per sec, what would be the velocity acquired in 7 seconds? 10. The weight of a disk of copper cut from a sheet of uniform thickness varies as the square of the diameter. Find the weight of a circular piece of copper 12" in diameter if one 7" in diameter weighs 4.42 oz. 11. A wheel 28" in diameter makes 42 revolutions in going a given distance. How many revolutions would a 48" wheel make in going the same distance? 12. If 3 men can build 91 rods of fence in a certain time, how much could 7 men build in the same time? 13. If 25 men can do a piece of work in 30 days, how long would it take 27 men to do the same work? 14. If the pressure on 230 c. c. of nitrogen is changed from 760 m. m. to 665 m. m., what will be its new volume? 15. The absolute temperature of 730 c. c. of hydrogen is changed from 353° to 273°. What is its new volume? 16. If the circumference of a circle 3 J" in diameter is 10.9956, what is the diameter of a circle whose circumference is 23.562? 17. A sum of money earns S1750 in 3j yrs. How long will it take it to earn $2750? 18. An investment of $1125 at 5% earns the same amount as another of $1250. What is the rate of the second investment? 19. If an investment at 2j% produces an income of $400, what would it produce if invested at 3f %? 20. The diameter of a sphere which contains 47.71305 cu. in. is 4|". What will a sphere contain whose diameter is 3"? CHAPTER VI . PULLEYS, GEARS, AND SPEED 103 An important problem in the running of lathes is the cal- culation of the speed at which the work should be turned, in order to complete the work in the shortest time possible, with- out injury to the work or the tools used. Similar problems arise in the use of fly-wheels, emery wheels, grindstones, etc. 104 Rim Speed: When the work in a lathe is turned through one complete revolution, a point upon the surface of the work travels a distance equal to the circumference of the work. In one minute, it would travel a distance equal to the circum- ference of the work multiplied by the number of revolutions per minute (R. P. M.). The distance in feet traveled by a point on the circumference of a wheel in one minute is called Rim Speed or Surface Speed. 105 RULE: To find the rim speed, multiply the circumference of the revolving object by the number of revolutions per minute (R. P. M.)f and express the result in feet. Example 1: The diameter of a wheel is 2". If it makes 2500 R. P. M., what is the rim speed? C = ;r.D = 3.1416 -2 = 6.2832". 6.2832 ^^ = .5236 (circumference in feet.) .5236X2500= 1309, rim speed. Example 2: The surface speed of a wheel is 3000. If the diameter is 4", what is its R. P. M.? 3.1416.4 = 12.5664" 12.5664 ^fy = 1.0472 (circumference in feet.) 96 RIM SPEED 97 Letx = R. P. M. then 1.0472x = 3000 x = 2865, R. P. M. Example 3 : What is the diameter of a wheel if its R. P. M. is 2500 and its surface speed is 1500 ft. per minute? Let x = diameter of the wheel. Then 3. 1416x = circumference of the wheel. 3.1416X. 2500 = 1500 7854x=1500 X = .19, diameter in feet. .19 • 12 = 2.28 diameter in inches. Exercise 68 1. What would be the rim speed of a 12' fly wheel running at 75 R. P. M.? 2. An emery wheel 15" in diameter runs at 1400 R. P. M. Find the surface speed. 3. A pulley 5i" in diameter runs at 1250 R. P. M. What is its rim speed? 4. A 12" circular saw runs at 2450 R. P. M. What is its cutting speed (rim speed)? 5. A 10" emery wheel has a rim speed of 5000 ft. per minute. How many R. P. M. does it make? 6. A grindstone will stand a surface speed of 800 ft. per minute. At how many R. P. M. can it run if its diameter is 4' 8"? 7. At how many R. P. M. should a 9|" shaft be turned in a lathe to give a cutting speed of 60 ft. per minute? 8. A fly wheel having a rim speed of a mile a minute runs at 120 R. P. M. What is its diameter? 98 PULLEYS, GEARS, AND SPEED 9. An emery wheel runs at 950 R. P. M. If its surface speed is 5500 ft. per minute, what is its diameter? 10. A line shaft runs at 186 R. P. M. A pulley on this shaft has a riih speed of 1350 ft. per minute. What is the diameter of the pulley? 11. The splicing of a belt connecting two equal pulleys travels through the air at the rate of 2000 ft. per minute. At what speed must the pulleys run if they are 20" in diameter? 12. A band saw runs over two pulleys each 32" in diameter. If the band saw is 16' long, and the speed of the wheels 600 R. P. M., what is the cutting speed of the band saw? Pulleys Fig. 61. Pulleys 106 When two pulleys are connected by a belt, the rim speeds of the two pulleys must be the same if there is no slipping of the belt. Suppose pulley I (Fig. 61) is 6" in diameter and pulley II is 12". The circumference of I is J as large as the circumference of II, and therefore I will revolve twice while II revolves once. In other words, the ratio of the diameter of I to the the diameter of II is J, while the ratio of the R. P. M. of I to the R. P. M. of II is y. This may be expressed as a proportion if one ratio is inverted and therefore : PULLEYS 99 107 When two pulleys are connected hy a belt, the size of the pulley varies inversely as its R. P. M. Example: One of two pulleys connected by a belt is 12" in diameter, and its R. P. M. is 400. What is the R. P. M. of the other pulley if it is 3" in diameter? Let x = R. P. M. of the second pulley. 12 X — = (the size varies inversely as the R. P. M.) 3 400 x=1600. R. P. M. Exercise 69 1. A 12" pulley running at 200 R. P. M. drives an 8" pulley. Find the R. P. M. of the 8" pulley. 2. A 14" pulley drives a 26" pulley at 175 R. P. M. What is the R. P. M. of the 14" pulley? 3. A 30" pulley running 240 R. P. M. is belted to a 12" pulley. Find the R. P. M. of the 12" pulley. 4. A pulley on a shaft running at 120 R. P. M. drives a 24" pulley at 200 R. P. M. What is the diameter of the pulley on the shaft? Lineshaft: The line shaft is the main shaft which drives the machinery of a shop by means of pulleys and belts. Counter Sfiaft: A counter shaft is an auxiliary shaft placed between the line shaft and a machine to permit a convenient location of the machine. 5. A line shaft runs at 250 R. P. M. Determine the size of the pulley on the line shaft in order to run a 6" pulley on a machine at 1550 R. P. M. 100 PULLEYS, GEARS, AND SPEED 6. It is found necessary to run a counter shaft at 310 R. P. M. If driven by an 18" pulley running at 175 R. P. M., what must be the diameter of the pulley on the counter shaft? 7. A counter shaft for a grinder is to be driven at 375 R. P. M. by a line shaft that runs at 210 R. P. M. If the pulley on the counter shaft is 12" in diameter, what size pulley should be put on the line shaft? 8. A motor running at 875 R. P. M. has a 10|" driving pulley. If the motor drives a line shaft at 180 R. P. M., what must be the size of the line shaft pulley? 9. The diameters of two pulleys connected by a belt are in the ratio f . If the R. P. M. of the larger pulley is 966, what is the R. P. M. of the smaller? Fig. 62 10. Pulley I is belted to II, and III to IV (Fig. 62). II and III are on the same shaft. If the diameter of I is 18" and its R. P. M. is 240, find the R. P. M. of II if its diameter is 8". Find the R. P. M. of IV if it has a diameter of 6", and III has one of 20". PULLEYS Step-Cone Pulleys 101 Fig. 63, Step-Cone Pulleys 108 To secure different speeds on the same machine, step-cone pulleys (Fig. 63) are used on both the driving shaft and the driven shaft. The large step of the driving pulle> may be belted to the small one of the driven for high speed, the medium one to the medium one for middle speed, and the small one to the large one for low speed. Example: A step-cone pulley having diameters 11", 8 J", and 6", running at 120 R. P. M., drives a step-cone pulley having diameters 4", 6^", and 9". Find the three speeds. Let x = R. P. M. at high speed. Why? 1320 = 4x X = 330, R. P. M. at high speed. T'>-T=i|-0 102 PULLEYS, GEARS, AND SPEED Let y = R. P. M. at middle speed. Then -4=-^ Whv? 6 J 120 1020 = 6^y. y = 157 — , R. p. M. at middle speed. Let z = R. P. M. at low speed. 6 z Then -= — • Why? 9 120 720 = 9z. Z = 80 R. P. M. at low speed. Exercise 70 1 it "'"""III"' o r J* t Fig 64 Fig. 65 1. The steps of a pair of cone pulleys are 7", 5", 3", and 4", 6", 8" in diameter (Fig. 64). If the lower pulley has a speed of 1050 R. P. M., find the three speeds of the upper pulley. 2. The diameters of the steps of a step-cone pulley on a machine are 10", 8|" and 7", and the corresponding counter shaft diameters are 5^'', 7", and 8j". Find the speed for each step on the machine if the counter shaft runs at 1190 R. P. M. GEARS 103 3. The steps of the cone pulley on a wood-turning lathe are 7|", 5f ", and 4". The corresponding diameters of the driving pulley on the motor are 2f ", 4|", and 6j". Find the three speeds on the lathe if the motor speed is 1165 R. P. M. 4. The smallest steps on a pair of cone pulleys are 2|" and 2f ". The increase in diameter of each succeeding step is 1^' (Fig. 65). The first pulley has a speed of 1000 R. P. M. Find the three speeds of the second pulley. Gears Fig. 66. Gears 109 In machines where absolute accuracy in the speed of the work is required, gears are used instead of belts to eliminate slipping. When two gears are meshed as in Fig. 66, it is evi- dent that their rim speeds are the same. Sizes of gears are measured by the number of teeth rather than their diameters. Suppose a 48-tooth gear drives one with 24 teeth. The smaller one will revolve twice, while the larger one revolves once. The ratio of the numbers of teeth is f, while the ratio of the speeds is -J. Therefore : 110 When one gear drives another, the speed is inversely propor- tional to the number of teeth. 104 PULLEYS, GEARS, AND SPEED Exercise 71 1. A 38-tooth gear is driving one with 72 teeth. If the first gear runs at 360 R. P. M., what is the speed of the second gear? 2. A 14-tooth gear running at 195 R. f*. M. is to drive another gear at 105 R. P. M. What must be the number of teeth in the second gear? 3. Two gears are to have a speed ratio of 3 to 4. If the first gear has 36 teeth, how many will the second have? 4. The ratio of the numbers of teeth in two gears is y. The R. P. M. of the first is 350. What is the speed of the second? ^6T Fig. 67 6. In Fig. 67 gear I has 72 teeth, II has 40, III has 56, and IV has 32. The R. P. M. of gear I is 60. Find the R. P. M. of II. If gear III is on the same shaft as II, find the R. P. M. of IV. REVIEW PROBLEMS Exercise 72. (Review) lO" Fig. 68 1. The gear with 72 teeth has a speed of 35 R. P. M. Find the speed of the 32-tooth gear. (Fig. 68.) 2. If the 32-tooth gear (Fig. 68) is to be replaced by one which is to have a speed of 280 R. P M., what size gear must be used? Fig. 69 3. In Fig. 69 what must be the size of the line shaft pulley (I) to run the emery wheel (V) at 1215 R. P. M., if the R. P. M. of the line shaft is 150? 106 PULLEYS, GEARS, AND SPEED 4. What would be the R. P. M. of the emery wheel (V), Fig. 69, if the line shaft pulley (I) is replaced by a 48" pulley? 6. Find the grinding speed of the emery wheel in problem 4, if its diameter is 12". 6. A wood-turning lathe is driven by a motor running at 1200 R. P. M. The smallest step of the cone pulley on the motor shaft is 2" in diameter, and its mate on the lathe is 7". All increases in the diameters of succeeding steps are 2". If the work being turned is 3" in diameter, find the cutting speed on high speed. 7. Find the cutting speed in problem 6 on middle speed. 8. Find the cutting speed in problem 6 on low speed. CHAPTER VII SQUARES AND SQUARE ROOTS 111 Square of a binomial: A few kinds of multiplication prob- lems are used so often that it is a saving of time to be able to write the result without performing the actual multiplication. One of these is the square of a binomial. Find the value of the following by multiplying, and write the results as in part 1: 1. (a+3)2 = a2H-6a4-9. 3. (m+n)2 = 2. (b+5)2= 4. (x+7y)2 = In each result, observe the following: I. There are 3 terms in the result. II. The first term of the result is the square of the first term of the binomial, and the third term of the result is the square of the second term of the binomial. III. The second term of the result is 2 times the product of the two terms of the binomial. Find the value of the following by actual multiplication and write the results as in part 1 : 1. (a-3)2 = a2-6a+9. 3. (-m+n)2 = 2. (b-10)2= 4. (-x-7y)2 = In each result observe that the same facts hold true as in the preceding case, and that the law of signs for multiplication must be used. 112 RULE: To square a binomial, square the first term, take 2 times the product of the two terms, square the second term, and write the result as a trinomial. 107 108 SQUARES AND SQUARE ROOTS Example: (2a-3bx)2 = (+2a)2+2(+2a)(-3bx) + (-3bx)^ = (-f4a2) + ( - 12abx) + (+9bV) = 4a2-12abx+9b2x2. Exercise 73 Wri te the results without written multiplication: 1. (a+l)2 16. (a2+b2)2 2. (t+u)^ 17. (2m2-3n)2 3. (d-4)2 18. (4t3-3u2)2 4. (x-y)^ 19. (a4+4a)2 6. (2a+b)2 20. (7-3m2)2 6. (3x-5)2 21. (m-J)2 7. (a-3b)2 22. (y+*)^ 8. (x+4y)2 23. (2x-i)2 9. (2m+3n)2 24. (3m+f)2 10. (5t-4u)2 26. (4x+iy)2 11. (6ab-5xy)2 26. (|x^-fy)^ 12. (5ab+4bx)2 27. (Ift2-fu3)2 13. (m2+5)2 28. (ff+sO^ 14. (x2-8)2 29. (2.3 l-5.1m)2 15. (a2-2)2 30. (.3125m3n+3|mn3)2 31. Square 32 mentally. Suggestion 322= (30+2)2 = 900+120+4 = 1024 Square the following mentally: 32. 21 36. 34 33. 22 36. 37 34. 29 (Suggestion 29 = 30-1) 37. 49 38. 19 39. 35 40. 43 SQUARE ROOT OF MONOMIALS 109 SQUARE ROOT Square Root of Monomials 113 Square Root: Problems often arise in which the reverse of squaring is necessary. For example: what must be the side of a square whose area is 25 sq. in.? The side must be such that, if multiplied by itself, the result will be 25. It is evident that 5 is the side of the square since 5^ = 25. The square root of a number is a number which if squared, will produce the given number. Finding such a number is called extracting square root, and the operation is indicated by the radical sign, V ( 1 4)2 1 16 } .-. V"l6 = +4, or - 4, written +4. (-3a)2 = 9a2\ .-^-^ , „ (+3a)2 = 9a2/-'-^^^ =±^^' (Ila2b4)2 = 121a%8.-. V121a%« = ^lla^b^. Exercise 74 Find the square root of: 1. 81 4. 144m2n2 2. 121 5. 25xV 3. 4a2 Find the value of: 8. 9. 6. V49xy V 196xV' 7. V64a2b^ V256ci«d«e^ 10. V400a2b4c8 The square root of a negative number cannot be found since, by the law of signs for multiplication, the square of either a positive or a negative number is positive. no SQUARES AND SQUARE ROOTS Square Root of Trinomials 114 (a+b)2 = a2+2ab+b2 \ . . , , ^ , , , , , , , , (_a_b)2 = a2+2ab+b/-- ^^ +2ab+b2=+aH-b,or-a-b, written + (a +b). (a-b)2 = a2-2ab+b2 \ . (_a+b)2 = a2-2ab+bV •*• Va2-2ab+b2= +a-b,or-aH-b, written +(a — b). 115 Trinomial Square: A trinomial in which two of the terms are squares and positive, and the other term is 2 times the product of the square roots of those terms, is called a trinomial square, and is the square of a binomial. Exercise 75 Select the trinomial squares in the following: 11. 64a^-176a+121 12. 49m4n2+112m2+8n4 13. x2+x+i 14. m2+fm+^ 1. x2+2xy+y2 2. m^— 4m+4 3. m2-4m+6 4. a2-4a-4 5. x2-6xy4-9y2 6. 4t2+6tu+9u2 7. 16x2+25y2 8. 169m6-26m3n+n2 9. 25x2+16y2-40xy 10. 49V-70xyz-25z2 15. x2+|x+| 16. 4a2+ab+YVb' 17. 9m2-24mn + 16 18. x2 + |x + y% 19. y^+fy+A 20. t^-it-aV^e SQUARE ROOT OF TERMINALS 111 116 By Art. 114, Va2+2ab+b2= -f(a+b). Va2-2ab+b2=-|-(a-b). Observe the following facts in each result: I. The two terms of the binomial are the square roots of the two terms of the trinomial which are squares. II. If the sign of the other term of the trinomial is plus, the terms of the binomial have like signs, and if it is minus, the terms of the binomial have unlike signs. 117 RULE: To find the square root of a trinomial square, extract the square root of the two terms which are squares, connect them with the sign of the other term of the trinomial, and prefix the Sign + to the binomial thus formed. Example : Find the square root of 25x2 _j_ j gy2 _ 40xy . V25x2+16y2-40xy=+(V25x2- V16y2). = +(5x-4y). Exercise 76 Find the square root of: 1. 9x2-24xy+16y2 2. 9+6x+x2 3. 49m2+14mn+n2 4. t2-10tu+25u2 5. a^^-2aV+y^^ 6. 4a6-4a3b2c+bV 7. 4a2-20ay+25y2 8. 9m2+42mx+49x2 9. -72xy+81x2H-16y2 10. 25x64-49a^b2-70a2bx3 112 SQUARES AND SQUARE ROOTS Find the value of: 11. V30m+25+9m2 12. V -60m2nV+25m4n4+36p8 13. V 49a4x2+ 1 12a3x3+64a2x* 14. Vx2+x-hi 15. yJisi^-\-25h^-\^sih 16. VaV'+Vtu+Qu^ 17. Vf|m^+2m2n24-||n* 18. V x^-fx^+yfo 19. Via%2_|.a2bc3H-2^3— c« 20. V|fx«+iV'z'-ixVz Square Root of Numbers i^5 By problem 31, Exercise 73. 322= (30+2)2 = 900+1204-4 = 1024. .•.V1024= 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 1 30+2 t2 = 900 2t = 60 u= 2 2t+u = 62 124 = u(2t+u) 124 ".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. 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 ^t. Example 2: Find the value of V 289. t2+u(2t+u) = t2 = 2'89 100 t+u 10+9 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 = a 00 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 cample: Find the square root of 60516. t+u t2+u(2t+u)=6'05'16 1200+40 t2 = 4 00 00 2t = 4 00 u= 40 2t+u = 4 40 2 05 16 = ur2t+u 1 76 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: t+u t2+u(2t+u) = 6'05'16 1240+6 t2 = 5 76 00 2t = 480 u= 6 2t+u = 486 29 16 = u(2t+u) 29 16 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 2 05 16 = u(2t+u) u= 40 2t+u = 440 1 76 00 2t = 480 29 16 = u(2t+u) u= 6 2t+u = 486 29 16 .-. V 60516 = ±246. Find the value of 1. V 37636 2. yllSUi 3. V 54756 Exercise 78 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: V2, t2+u(2t-hu) = t2 = 35.6225 r t + u 1 5. t + + u 3 u 5 -2' = 1 3 5' .6 2' 2 = u(2t+u) 5 2t= 20 u= 5 2t+u= 25 1 1 3 5 = 2 5 2t= 300 u= 3 2t+u= 303 1 9 6 2 = u(2t+u) 9 2t = 3060 u= 5 2t+u = 3065 1 1 5 3 2 5 3 2 5 = u(2t+u) 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< 94. t + u t + u t + u 8 t 2. + u 8 = 2.081 t2+u(2t+u)=4. t2 = 4 32' 99' = u(2t4 40' 00 2t = 40 u= 32 = 00 -u) 2t = 400 u= 8 32 32 99 = 64 = u(2t+u) 2t+u = 408 2t = 4160 u= 35 00 40 = u(2t+u) 00 2t =41600 u= 8 2t+u = 41608 35 33 40 00 = u(2t+u) 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. V14.4 V36= v"9^= VV- 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: VT2 = vT V3 = 2 V3 = +3.4642. Exercise 81 Given V2^= 1.4142, V^= 1.7321, v'5 = 2.2361, Find the value of the following correct to .001 : 1. V~8 6. V45 2. VTS 7. V48 3. V20 8. V50 4. V~27 9. V72 5. V32 10. V108 SQUARE ROOT OF FRACTIONS 119 /'2\2_2 2_£ \3J — 3*3 11. V180 16. V 98 12. V 80 17. V147 13. V125 18. V320 u. V363 19. V243 16. V512 20. V128 4 .-. vl= ■X3- 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: lie q -i- 2. "25 ^- 64 3.6. A 121 4 9 *• 9 O- 2 2^ Find the value of the following correct to .001: 6. Vif 8. VsV 7. V 8T ^' ^144 in ./ 1 62 10- V 2 123 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: vl= v"A=-^ = ±?^^^=+.6124 V16 4 — 120 SQUARES AND SQUARE ROOTS Exercise 83 Find the value of the following correct to .001 : 1. v§ 2. v4 3. vi 4. vi 6. VI 6. VI 7. vf 8. vf 9. VI 10. vV Quadratic Equations 124 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 +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): 1. x2=12 7. x2 = f 2. x2 = 75 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 ^^ 3xy-l 5(x^-l) (4x^+1) _^ ^^' —5 lO 25^"" y-l y+1 5r-3^_r+2 ^*** 9r+l 2r+5 2x-5 , - 3x+10 19. — TT- =1.5 — 3. X2 = = 55225 4. X2 = = 46 6. X2 169 1225 6. X2 = _ 75 108 20. 3 2x+ 5 3x-l 3x+1^29 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=Va2+b2 whena = 4, b = 5. 3. V = 2 7r2r2R when r = J, R= l|. 4. A = |r2V3 whenr = 3|. 5. V = ^V2 when a = 3.2. 6. G=Vab whena=4, b = 5. 7. Y = — ^ --^-^TTSi^ when a = 6, r = 18, R = 24. V g 8. t^TTA/- when 1=1, g = 32. 9. s=^(Ar5-l) whenr = 2j. 10. D=Va2+b2+c2 whena = 5, b = 6, c = 7. 11. b= Va2+c2-2a'c when a =14, a' = 5, c = 12. 12. l = 2VD2+a2 + ttD when D = 16, a = 35. 13. M = |V2(a2+c2)-b2 whena=15, c = 17, b = 19. — b+Vb2+4ac , o u c on 14. x = when a = 3, b = 5, c = 20. 2a 123 124 FORMULAS 16. x= — : >.:cn a = 3, b = 5, c = 20. 16. l = 4/(^^)Va2+7r(5±^)when D = 36, d = 6,a = 96. 17. s = -rV10-2V5 when r = 4l. N ^„ C 18. s = 5^.;rR2-yV4R2-0 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 = |(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 5. 15.708 62.832 502.656 a = 4-3.1416-8 Example 2: V = Jrfa. Find r, when V = 593.7624, a = 7. 1.0472 593.7624 = f 3. 1416- r2. 7 81 84.8232 „ 593.7624 1.04727 r=+9. = 81 FORMULAS INVOLVING SQUARE ROOT 125 Example 3: b2 = a2-f c^— 2ac'. Solve for c', when a = 5, b = 6, c = 7. 36 = 25+49 -2- 5c' 10c' = 38 c' = 3.8 Exercise 86 Find the values (correct to .001 when necessary) : P=4a P=:a+b+c. |bh. P=2(a-l-b) A = ab. A 6. A=Jh(b+b'). 7. A=ih(b+bO. 8. C = 27rr. 9. A = 7rr^. 10. 1 w = £.p. 11. W = r.p. 12. W = r.p. 13. L=lfd+J. 14. S = |gt2. 16. S = |gt2+vt. 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 = l.l. Find h, when A = 3U, b = 3|. Find h, when A = 96, b = 18, b' = 6. Find b', when A = 12.8, b= 1.2, h = 8. Findr, 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=4:yq' Find t, when S = 196.98. (See Exercise 17, problem 5.) Find V, when S = 164.72, t = 3. 126 FORMULAS IIV 16. F=— -—. FmdF,whenu=ll,v = 7. 17. F=-^. Findv,whenF=liu = 3. u+v ' ^' 18. x = — ' . Fmdx, whenb=-5, a = 3, c=-2. 19. A=-r-. Find D, when A =115. 4 ' 20. V = !rr2a. Find r, if V = 330, a = 7. 21. V = ^2a. Find a, if V = 46.9, r = 2.3. 22. V = 3nr2.?^i5. ^'^^ ^' w^^^ V=1932, H = 14.6, 23. V = ;rr2.^^±5. FindH, when V = 2246, r = 8, h = 6. 24. A = ^. Find A, when a = 2.3, b = 3.2, c = 4.L ^ r = 2.058. 25. A = ^-. Find r, when a = 21, b = 28, c = 35, A = 294. 26. A = |r(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 = J(23rR+2zrr)s. Find A, when R = 8, r = 3, s = 7. 29. A = J(2?rR+23rr)s. Find r, when A = 439.824, R = 10, s=10. 30. A=|(2jrR-f-2;rr)s. Find s,when A = 106.029, R = 7j,r = 6- 31. l = 2/(5±^)^+a^ + .(^). Find 1, when D = l|, d = 1 J, a = 15. 32. h = ^V3r Find a, when h = 27.7136. FORMULAS INVOLVING SQUARE ROOT 127 33. A = ^ V"3l Find h, when A = y vl". 34. A = ^ V"3^ Find a, when A = VlS. 35. v = :^V2^ Find V, when a = 6. Qt.2 36. A = ^V3. Find r, when A = 153. 37. c2 = a2+b2. Findb, whenc = 2.1, a = 1.7. 38. b2 = a2+c2+2a'c. Find a, whenb = 8, c = 5, a' = 2.1. 39. b2 = a2+c2+2a'c. Finda', whena=18, b = 16, c = 31. 40. b2 = a2+c2-2ac'. Find c, when a = 5, b = 4, c' = 2.3. 41. b2 = a2+c2-2ac'. Find c', when a= 14, b = 15, c = 16. 42. H = r-Vs(s-a)(s-b)(s-c). ■ FindH,whena = 2.18,b = 5,c = 3.24, s = J(a+b+c). 43. a2+b2 = 2Qy+2m2. Find m, when a = 9, b = 12, c = 15- 44. a-+b2 = 2r|y+2m2. Findb, whena = 5, c = 13, m = 6i. 46. s = I(vT-l). Find r, when 8=10.50685. 46. x= ^+^^'+^^^- Findx,whena = 3,b=-7,c=+2. ^a 47. V = 27r2r2R. Find r, when V = 98696.5056, R = 50-. 48. X = V r (2r — V 41^ — s^) . Find x, when r = s = 10. N 49. s = ^.;rr2-- V4r2-c2. Find N, when s = 23. 1872, c = r= 16. — b— Vb2+4ac -o- j , ^ , 50. x = \- — ' . Findx,when a = — 6,b= — 9,c=+2. ^a 128 FORMULAS Right Triangle 126 One of the formulas most commonly used is that of the right triangle. ^jb^^C Fig. 70 127 Right Triangle: A right triangle is a triangle in which one angle is a right angle. The lines including the right angle are called the sides, and the line opposite the right angle is called the hypotenuse. It can be proved that; 128 The square of the hypotenuse is equxil to the sum of the squxires 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 l|. 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.) \ ^ 7 \^ X /32 /tf" 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 figure. 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 figure. 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|". Find the altitude of the cone. (Fig. 74.) 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.— 4(5 Angle, Definition of 25 Angle, Right 25 Angle, Straight 25 Angles, Complementary 35, 36 Angles, Drawing of 26, 27 Angles, Measuring 27, 2S 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 Coefficient, 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.... 86 Monomial, Definition 44 Multiplication 51 132 INDEX SUBJECT PAGE Multiplication of a Poly- nomial by a Monomial GO 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 Numbers, Signed 40 Order of Terms 6 Parenthesis 16 Parenthesis, Kemoval 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 Pulleys, 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, Hypotenuse of 128 Right Triangle, Sides of 128 Rim Speed 96 Separating in a Given Ratio.. 84 Sign 6f 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 Gravity 83 Speed i 96 Speed, Cutting 97 Speed, Rim or Surface 96 Bpeed 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.. 112, 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 iv;270571 0-/A 3 9 THE UNIVERSITY OF CALIFORNIA LIBRARY