ere THE UNIVERSITY OF ILLINOIS LIBRARY SIX-f Sm 3% “PW EMATIGS LABRARSS Return this book on or before the coe. Latest Date stamped below. = University of Illinois Library r Te voy e RACE et TO SAE nal ALGEBRA FOR BEGINNERS ie Wes: BY ae DAVID EUGENE SALITH, Px.D. PROFESSOR OF M EM NICS IN TEACHERS COLLEGE COLUMBI NIVERSITY, NEW YORK A \ 5 ws cy 1 | GINN. & COMPANY BOSTON - NEW YORK - CHICAGO » LONDON ' ENTERED AT STATIONERS’ HALL COPYRIGHT, 1904, 1905 By DAVID EUGENE SMITH ALL RIGHTS RESERVED 35.10 The Atheneum Press GINN & COMPANY > PRO- PRIETORS + BOSTON: U.S.A. PREFACKH This book is intended, as the title indicates, for pupils beginning the study of algebra. There is a growing dispo- sition to introduce this subject somewhat earlier than was formerly the case, and with this has come a demand for a simple, interesting, and sufficiently scientific text-book for beginners. Such a text-book should show the utility of algebra, should form a connecting link between arithmetic and the more scientific works to be studied later, and should stimulate a desire to proceed further in mathe- matics. It is to meet this demand in the spirit described that this book has been prepared. The time for introducing the work depends upon circum- stances. In some cases elementary algebra of this nature should be begun in the first year of the high school, while other conditions make it advisable to take it up in the latter part of the grammar-school course. In either event it is desirable that pupils should have some knowledge of algebra before they leave school. For those who are not to pursue the subject further this book furnishes such algebra as is necessary for the intelligent reading of form- ulas and the solution of equations found in elementary industrial manuals. Those who continue their school work will find the subject treated in this book in such a way as to stimulate an interest in their later work, and will meet no obsolete forms that must be unlearned before proceeding. In sequence of topics the author has continued the plan adopted in his arithmetics, that of recognizing the value of the various courses of study in use in different parts of the country. Modern curricula no longer sanction for lll 4 9OeK 492K De $ ooo Lore PREFACE beginners the plan of treating each topic but once. On the contrary, they suggest the repetition of the most important portions of algebra, although favoring a some- what exhaustive treatment of each subject whenever it is under discussion. Of the three chapters of this book, the second covers some of the ground of the first, and the third reviews some of the topics treated in the second. ‘The first two chapters furnish sufficient work for schools that devote part of a year to algebra and part to arithmetic. The third chapter may be used if a full year is given to the subject. The work seeks to interest the pupil in the subject at once by showing him its utilities. The formula which the artisan meets in his trade journals and the equation which throws so much hght upon business arithmetic find place in the early pages. With these applications is combined the recreation element, as seen for example in the finding of numbers which satisfy given conditions, — an element which lends much interest to mathematics. Oral algebra, like oral arithmetic, 1s necessary to lead to rapidity and to an understanding of general processes. Hence enough types have been suggested to form a basis for the best of all oral work, that which comes sponta- neously from the teacher and the class. While a large number of genuine applications have been made in the domain of the pupil’s present and prospective experiences, scientific and financial problems in which he has no interest have been omitted. With the applications has gone a large number of those abstract, formal prob- lems so necessary for drill in rapid algebraic work. ‘These “problems without content” have an interest in them- selves, and give to the elementary pupil some of that pleasure which comes to the more advanced student in the discovery of positive truth in the domain of pure science. DAVID EUGENE SMITH. CONTENTS CHAPTER I THE USES OF ALGEBRA; THE OPERATIONS WITH INTEGERS AND FRACTIONS ; THE EQUATION PAGE Some OF THE USES OF ALGEBRA 1 ForMvULAsS . : : . ; ‘ : : : : 2 EQuaTIONS : d : : : ‘ : : : ; 4 LETTERS IN SOLVING PROBLEMS . : ‘ ; > : 6 Tue USE OF & ; : F : , : . : 4 9 SoME OF THE TERMS USED IN ALGEBRA 4 : ; : 16 Soms Usges or Monomiats. , : : : ‘ 5 AN Some Uses oF POLYNOMIALS . : : ; ‘ : Pat THe NeGaATIvE NuMBER . é 4 : : : : Oy y- CurvE TRACING . ‘ : : : : : 2 ; 25 ADDITION . . , A : d ; : , : a As) SUBTRACTION . : : : : ‘ : f ; , 34 How To USE PARENTHESES s 4 : ; 2 ; . 40 MULTIPLICATION. P ° : : ; : : . 42 DIvIsION. : ; : 4 ‘ : ‘ . : og Bb Factors AND MULTIPLES : . ; : - : , 50 Facrors : : : ; ; ; , : : fe 60 MULTIPLES . 53 FRACTIONS ; 55 Uses OF FRACTIONS . : ; : ; as ; : 57 REDUCTION OF FRACTIONS é : ; ; : A = OS IMPROPER FRACTIONS . 5 ; : ; ; : : 61 FRACTIONAL EQUATIONS . : F : 4 : 2 Oe ADDITION OF FRACTIONS ; ; , A ; A ; 65 SUBTRACTION OF FRACTIONS ; : . d : 5 rea 8 MULTIPLICATION OF FRACTIONS : : : ae ae : 68 DIvisION OF FRACTIONS. : ’ ; ; , : a uae LINEAR EQUATIONS ‘ i ‘ . , ‘ é : 74 vi CONTENTS CHAPTER II OPERATIONS CONTINUED; FACTORING; PROPORTION ; EQUATIONS PAGE MULTIPLICATION 5 ; , ; : E ; : Geo FacTORING . : ; : ; ; : ; : ; 88 Division. : : F . f : 4 ; : we OF FRACTIONS : ; ? : , : : : : : 99 REDUCTION OF FRACTIONS ( : d : : : be OO ADDITION OF FRACTIONS . : 3 : : : : 106 SUBTRACTION OF FRACTIONS . : : : : ; . 108 MULTIPLICATION OF FRACTIONS . : é : ; ; 110 Division Or FRACTIONS . : ; i : . 4 ae |) EQUATIONS INVOLVING FRACTIONS . ; : : ’ q 112 PROPORTION : ; : ; : , : : : ragltg G2) SquaRE Root : : : ‘ : ‘ : , : 129 QUADRATIC EQUATIONS ; : ; : ; ; : -| Loo CHAPTER III FRACTIONS CONTINUED ; ROOTS ; SIMULTANEOUS EQUA- TIONS ; THE COMPLETE QUADRATIC FRACTIONS ; : : 4 ; ? ‘ , : e186 FRACTIONAL EQUATIONS. , , 2 : f : . 140 SIMULTANEOUS EQUATIONS ; : : : ; 2 : 143 QuapDRATIC EQuaTIons : : : i ; : : +. “163 ALGEBRA FOR BEGINNERS CHAPTER I THE USES OF ALGEBRA; THE OPERATIONS WITH INTEGERS AND FRACTIONS; THE EQUATION SOME OF THE USES OF ALGEBRA 1. Numbers represented by letters. — In arithmetic we often represented numbers by letters. We learned that If one thing costs d@d dollars, 5 things will cost 5 x d dollars, which we write $5d; and n things will cost $ nd. 2. How we indicate multiplication. —In algebra the absence of a sign indicates multiplication. It is not so in arithmetic, for 51 means 50+1; but in algebra ab means a x 0. ORAL EXERCISE 1. If a rectangle is 12 ft. long and 7 ft. wide, what is its area? Ifitis 7 ft. long and w ft. wide, what is its area? 2. If a train travels at the rate of 30 mi. an hour, how far will it travel in 10hr.? If it travels m miles an hour, how far will it travel in 2 hours? 3. While the hour hand of a clock passes over 5 1-min. spaces, how many does the minute hand pass over? While the hour hand passes over ” spaces, how many does the minute hand pass over? 1 ko USES OF ALGEBRA 3. Rules stated by letters.— We have just seen that the area of a rectangle / long and w wide is Jw. If 7 and w are numbers of feet, 7w is the number of square feet in the rectangle ; if inches, dw is the number of square inches. If a represents the area, then the statement a = lw is a simple rule for finding the area of a rectangle. 4. Formulas. — . 2a = 201Y 2. 5x = 60. 3. Tx = 84. mpl eon: gr Zo t= 120. 6. 125% = 250. . 82+5=26. (What does 3xequal? What does « equal?) Meo 108 69. oe lie 41. 10, 6% +. 9 = 15, 11. @tceao= lis 12. $¢-+-4=—60.. 13. 94-10 = 109: WRITTEN EXERCISE Find the value of x in the following: 1. 12¢%4 3= 1365. 2. lla+3=1385. 3. 102 4+ 2 — 162." 4.10”%+12= 162. Be oe dl 10. 6. 312+ 12 = 260. toe == 140, 8. 17x +11 = 300. 9. 42%74+4= 130. 10. 512+ 45 = 300. 11. 10+ 652 = 140. 12. 20 + 30% = 470. Make up problems to fit the following, writing them out like those on page 8; then solve: 15. 52 +16 — 56: 16. 22+ 3a = 265. 10 USES OF ALGEBRA 11. Equations involving subtraction. — If from 3 times the number of which I am thinking I subtract 7, the remainder is 29. What is the number? 1. I am thinking of a number which I call z. 2. Then Bat = 29; 3. Then 3x2 = 86, by adding 7 to these equals. 4, And x = 12, by dividing these equals by 3. Check. 3 x 12 — 7 = 29. ORAL EXERCISE 1. What number less 2 equals 6? 2. If from a certain number I subtract D, the result is 45. Whatis the number? , | ‘ 3. If from 4 times a certain number I subtract 4, the result is 20. What is the number? Such problems, like all oral drill, have their greatest value when given extempore by the teacher. Rapidly given, with small numbers, the work is not difficult, and it is interesting. A little daily drill of this kind soon makes pupils very ready in equation work. WRITTEN EXERCISE 1. If to 25 times a certain number I add 42, the result is 667. What is the number? 2. If to 125 times a certain number I add 50, the result is 800. What is the number? 3. If from 25 times a certain number I subtract 42, the result is 583. What is the number? 4. If from 75 times a certain number I subtract 25, the result is 200. What is the number? 5. If from 37 times a certain number I subtract 33, the result is 300. What is the number? EQUATIONS 11 12. Equations involving per cents. — After deducting 10% from the marked price of some goods, a dealer sold them for $13.50. What was the marked price? 1. Let x represent the number of dollars of marked price. (We need not, then, trouble ourselves to write the sign $ each time, as we should if x represented merely the price.) 2. Then x = 10% = 135-50; or D0 se = 15.00: because any number less 7; of itself is 7% of itself. 3. Therefore x = 13.50 + .90, by dividing equals by .90, alts 4. Therefore the marked price was $15. Check. $15 — 10% of $15 = $15 — $1.50 = $13.50. WRITTEN EXERCISE ‘1. What number less 10% of itself equals 72? 2. What number less 17% of itself equals 166 ? _ 3. Jack now weighs 84 lbs., which is 12% more than he weighed a year ago. How much did he weigh then? 4. A certain school gained 15% this year over the number last year. It now has 161 pupils. How many had it last yeager tl low = 161.) 5. A dealer saved $1968 this year from his store. This is 18% less than he saved last year. How much did he save last year? 6. A dealer was obliged to sell some damaged furniture at 10% less than cost. He sold it for $85.50. How much did it cost? How much did he lose? 7. A village having a population of 2040 at the last census found that it had lost 15% from the number at the preceding census. How many had it before? How many had it lost? 13. 3. Lhen USES OF ALGEBRA ORAL EXERCISE Find the value of x in each of the following: ee we Oates LT a cite aed be mer ge Be .2e@+ 121. Pee ain 59, we oe 16. oO + 44 = 25. de 3 = 56. 2. 4. 16. x—3=18. xe —12 = 40, . e—14 = 60. , 2e—1=A49, 2 =O Ok: Oe =O =i AO { ==00: lil« —3= 380. Illustrative problem.— Find the value of x that makes 4-+--17 2 = 140. Lelia lia — 140; 2. Then 17 «= 1386, by subtracting 4 from these equals. x =1386 +17 = 8, by dividing these equals by 17. Check. 17 X 8+4=186+4=140. WRITTEN EXERCISE Find the value of x in each of the following: L On ee awl LOT. . 1.06” = 424, WASa Seb Ree 2713 . 152% +47 = 962. . 025 + 14a = 367. . 411 E11 ae 774. . 231+ 23a = 392. . 125% 4+ 125 = 1000. O FF ot WwW Zia = 42. 2. Ol tea ie GOi7 = (D9: . 60% = 144,90, WILT 2 19s 287. 19n = 97189. Six — $= 1520. . 419 x — 88 = 800. , 125% — 125 = 750. , 1012 — 111 = 1000. FORMULAS 13 ORAL EXERCISE 1. At 5% a year, how much is the interest on $200 for one year? for two years? for 7 years? 2. At r% a year, how much is the interest on $p for one year? for ¢ years? If the rate of interest is r, and the principal is p, and the number of years is ¢, what is the interest ? 3. At 10% discount, how much is the discount on $50 worth of goods? At r% discount, how much is the dis- count on $» worth of goods ? 14. Further use of formulas. — We have just seen that if 1= interest, r=rate, ¢=time (in years), and p= the principal, ; 4= rp. WRITTEN EXERCISE 1. If i= trp, find the value of « when ¢=6, r=49%, p = $300; when ¢ = 3}, r= 6%, p = $500. 2. If i= trp, find the value of « when ¢=4, r=6%, p= $50; when t= 27,4, r= 4%, p = $600. 8. If d = discount, r= rate of discount, and p = price of goods, write the formula for d, as in Ex. 3 above. 4. From the formula of Ex. 3, find the value of d if r = 331%, p = $270; also if r= 25%, p = $240. 5. If a train goes m miles an hour, how many miles will it go in ¢ hours, at this rate? If d =the distance, write the formula. Find the value of d if m = 387}, ¢ = 33. 6. If the population of this country increases r% every ten years, and is now p, how much will it increase in the next ten years? What will the population then be? 14 USES OF ALGEBRA Roots. — The letters of,an equa- tion for which values are to be found are called ynknown quantities. \ These values are called the roots of the equation. To solve an ‘equation means to find its el 15. Unknown quantities. For example, the unknown quantity in the equation + 3=7 is x. The root of the equation is 4. 16. How to represent known quantities. — The first letters of the alphabet, in an equation, represent numbers Supposed to be known. Cull gt For example, if told to solve the equation x + h= pre would take b from these equals, leaving 2 =a—b. Here the a and bd are supposed to stand for known numbers. 17. The members of an equation. — The quantity to the left of the sign of equality is called the first member; that to the right, the second member. 18. Symbol of deduction. — The word therefore is so often used in algebra that it has a special symbol (.’.). Since 20 =ao.e et aoe WRITTEN EXERCISE Solve the following equations : 1: r+b=2a. 2.%—a=3b. Bay Sar a1 49. 4. 2—17=69. 5. a+a= 125. 6. x — oc = 127. 7, 19a” — 71 = 5. 8. 82 +92 = 190. 9. 92 — 243 = 4238. 10. 17 x — 62 = 57. 11. 142 +32 = 265. 12. 23a — 86 = 98. 13. 261 +7 x = '303: 14. 9x + 126 = 621. . 426 +1382 = 501. Mal hile 25.2. bin EQUATIONS 15 WRITTEN EXERCISE Find the value of the unknown quantity in Exs. 1-10: 1. 7x = 609. 2. 122 = 204 ft. 3. 132 = $221. fee a 7b. 5. 1742 = 1218, GUIS 5 = 85.75, 7. 123 y = 1353. 8.0 + 3 ft. = 7 ft. Bie 107 = 236, 10. $2 = 255 sq. ft. 11. The average population per square mile in Africa is 11 and the population is 126,654,000. How many square miles are there? (11a = how many?) 12. Mt. McKinley is 20,464 ft. high, and it is 1606 ft. more than 3 times the height of Mt. Washington. How high is Mt. Washington? (3w +1606 = how many?) 13. If r represents the mean (average) annual rainfall in inches at St. Paul, 7 + 33 represents it at New Orleans. If the mean annual rainfall at New Orleans is 60.5 in., what is it at St. Paul? 14. The average balance of each savings-bank depositor in this country in a certain year was $417.21. This was $54.97 less than twice the balance of each depositor in Hungary. What was the average there? 15. The number of English-speaking people in the world is 8.7 million more than twice the number of French- speaking people. There are 111.1 million English-speak- ing people. How many French-speaking people are there ? 16. The average amount paid by each person in the United States for the general expenses of the government in a certain year was $5.96, which was $1.04 less than 4 times the average amount each paid for our national pension fund. What was the average for pensions? 16 TERMS USED SOME OF THE TERMS USED IN ALGEBRA 19. Names of certain terms. —— We have seen some of the uses of algebra and have found that it is often helpful to represent numbers by letters. We now need to know the names of a few of the most important terms in algebra, especially those which we shall be using at once. 20. Coefficient illustrated. ear the expression 2a, 2 is called the CURE AGTS ap Just as 2 apples means 2 times 1 apple, or 1 apple + 1 apple, sO 2x means 2 times 12, orlx+1z. That is, in 22, 2 shows how many times z is taken as an addend. Just as $2 means 2 of $1, So 24 means 2 of lz. The expression x means the same as 1 z. Liebe a 2 Xo = 10 an oe ean. ORAL EXERCISE ‘Write the values of the letters on the blackboard before asking the questions. Give other examples of the same kind. 1. If a=5 and b = 2, tell the value of the following: ab, 3a, 5b, Tab, a+b, a—b, 2a+b, 2a—bd. 2. Ifa=2 and x = 4, tell the value of the following: ax, 5a,10%, 8ax,a+4,x—a,Ta+a2, 10a— dz. 3. Ifa=2, b=3, and c=4, tell the value of the following: abc, a+b+c¢, ab, a+b, cb, c—b, e+a, 4b6—3e. 4. If p=3,¢=7,r=9, and s = 12, tell the value of the following: ptEOndtr—-ss—r7r—3p,q—2p,S+p,p+r. COEFFICIENTS di ORAL EXERCISE 1. If I have $x and you have $x, how many dollars have we together? 2. If I have 2 dollars and you have twice as much, how much have you? 3. What is the coefficient of x in each of the following? 3x, 17 x, x (what coefficient is understood ?), 2a, 4a. 4. If there are 8 classes in this school, and 0 pupils in each class, how many pupils are there in all? 21. Coefficient defined. — A numerical factor written before a letter is called the coefficient of that letter. 22. Letters may be coefficients. — We sometimes speak of letters as coefficients. If there are a pupils in this class, -and each one has @ dollars, they all have aw dollars. Here ais called the coefficient of x. Instead of one letter we may have several letters. Thus, in 2 axy, 2 is the coefficient of ary, and 2 a is the coefficient of zy. WRITTEN EXERCISE 1. If one bag of flour weighs x pounds, how much do a bags weigh? (Find the value for a = 25, x = 96.) 2. If a glass jar of milk weighs y pounds, and there are xz of them in a basket, how much will the jars in a such baskets weigh? (Find the value for a= 6, #=12, y=1}.) 3. If one chair costs a dealer c¢ dollars, and there are 6 such chairs in a set, and the dealer buys a sets, how much do they all cost? (Find the value for a = 3, 6=6, ¢= 4.) 4. Copy the following, writing beneath each the numer- ical coefficient : Zax, ¢ayz, 40% 2, 0.5ab, 652mn, 12.5 abcd. 18 TERMS USED 23. Factors. — The quantities which, when multiplied together, form a product are called the factors of the ert thi product. 24. Squares. — If two factors are equal, their product is called the sguare of either. The product 2 x 2 may be written 22, called the square of 2. In the same way the product 25 x 25 = 25°, or 625, is called the square of 25. 20. Powers. —The product arising from taking a quantity a certain number of times as a factor is called a power. — For example, the products Dire hae OTe, 3% oO KOKO seo Oreo k are powers of 2 and 3. That is, 25 = 32, the fifth power of 2, 34 = 81, the fourth power of 35, 6 = aaaaaa, the sixth power of a, 48 = 64, the third power, or cube, of 4, 6? = 36, the second power, or square, of 6, 8 26. Exponent. —In the expression «a°, 6 is called the exponent of ay and indicates the power to which a is raised. Tnid’, aina = 2, 0° G4 all =e and 0s) ae eee ORAL EXERCISE 1. If a = 3, b = 2, tell the value of the following : A mM PCR raat tact “hae rhe lak) Be 2. [fia = 2, b= 8, oe yb tell etheavaluemoL the following : a, 0b 860%, ba, a) RO eet, EE) 3. In a’ and 2a, name the coefficient of a; the exponent of a. ‘Tell what each indicates. ys Abundant rapid oral drill, as in such examples, should be given. ALGEBRAIC EXPRESSIONS 19 27. Algebraic expression. — A collection of letters, or of letters and other number symbols, connected by any of the signs of operation (+, —, X, +, ete.) is called an algebraic eXPVeESSION. For example, 3 a (since the absence of a sign between 3 and a indicates multiplication), 5+ 62, 2a?+ 367+ c¢. 28. Term or monomial. — An algebraic expression con- taining neither the + nor the — sign of operation is called a eters a term or monomial. Gay orcas en RAED AP ABLE LD LLELLES AED : For example, the terms of 22? — 3a2y are 222 and 3ay. The expression 4 abz? is a monomial. 29. Polynomial. — An algebraic expression composed of several terms or numbers connected by the sign + or — is called a polynomial. 30. Binomial and trinomial.— A polynomial of two terms is called a binomial; one of three terms is called a trinomial. For example, a + 0 is a binomial, x — 3 y? + 22 is a trinomial. WRITTEN EXERCISE 1. lia =2; 6 =5, c=1, find the value of each binomial in the following lst: a+ 07, ab-+ be, b> —40?, a+b. 2. With the values given in Ex. 1, find the value of each trinomial in the following list: 3a? + 4 6c? + 3c’, §a?+267—-100c,a+b—c¢, 6a—b+4e. 3. Ifx=7, y=5, 2 =2, write a monomial, containing one or more of these letters, that shall have the value 70; PoeeG0s 100; 25. i 4. If m=2,n = 3,¢ =7T, write a polynomial, containing one or more of these letters, that shall have the value 12; 15; 30; 100. (For example, 5m +n" + 2 = 15.) 20 TERMS USED SoME Usrs or MONOMIALS ORAL EXERCISE 1. If a@=9, 6=8, what is the value of 4 ab? 2. If a=11, 6 = 30, what is the value of ab? 3. What is the area of a rectangle 3 in. by 4 in.? a in. by 6in.?. If area = a, find the area when a = 10,b= 30; Gea lO h == 20. 4. What is the area of a rectangle @ high and 6 long? How does the area of a triangle of base 6 and height a compare with this? What is its area? 5. If the area of a triangle is 4.0, find the area when a=10,6=20. If the measures are in inches, the area is. in what kind of units? 6. The volume of a box 3 in. long, 2 in. wide, and 4 in. high is how many cubic inches? What is the volume of a box Z long, w wide, and # high? of a box a" by 6" by ce’? 7. If volume = lwh, find the volume when 7? = 10, w=4, h=3. If tl, w, hostand for feet, hows willatic volume be expressed ? 8. If the area of a circle equals m7?, where 7 = 35} and 7 stands for the number of units in the radius, find the area when r= 1. 9. If the circumference of a circle equals 2 mr, find the circumference when r = 1. 10. If a man saves $d a month, how much will he save in ¢ months? How much will this be if d=15 and t= 6? if d=.25 andt =10? 11. If a boy earns ¢ cents a day, how much will he earn ind days? If he spends a cents in this time, how much will he then have? Suppose ¢ = 20, d=10,a=75? USES OF POLYNOMIALS 21 SoME Usgs oF POLYNOMIALS ORAL EXERCISE . If we let a stand for 10, how may we represent 20? 24? If « = 3, how shall we represent 3 x 10? 30? 34? . If «=7, how shall we represent 33? 70? 74? 80? . If x=2, y=3, e=1, how may we represent 200? Pw nwo 5. This room is a long, 6 wide, and ¢ high. What is the total length of all its 12 edges ? 6. Suppose your marks in arithmetic were x on Monday, y on Tuesday, and 2 on Wednesday, what is the average? Suppose @ = 9,7 — 10, = 87 WRITTEN EXERCISE 1. Ifx=3, y = 39, what is the value of 372+ 5 y? 2. A man earns a cents an hour, and 6 cents an hour- working overtime. What does he earn in an 8-hour day, working also 1 hour overtime? 3. Rob earns 7 cents an hour, Jack j cents, and Tom ¢ cents. How much will all three earn if Rob works 5 hr., Jack 3 hr., and Tom 7 hr.? 4. A room is Z ft. long and w ft. wide. In the middle is a rug r ft. square. How many square feet of the floor are not covered by the rug? 5. Texas has 32,290 sq. mi. more territory than five times the territory of Michigan. If Michigan has m sq. mi., what is the area of Texas? Suppose m = 58,915? 6. Richmond is 108 mi. north of the halfway point between New York and Savannah. If the distance from New York to Savannah is d, wae is the distance te Richmond? Suppose d= 904 mi.? THE NEGATIVE NUMBER THE NEGATIVE NUMBER 31. Numbers below zero.— The mercury in a thermometer stands at 68° to-day. If it falls 23° to-night, we say a <= ra i} | pal fe y |= x that it stands at 45°. If to-morrow it should fall 13° more, it would stand at 32°, the temperature at which water freezes. If it should then fall 32°, we =| would say that it stands at 0°, or at zero. When | it goes below zero, say 5°, we indicate this fact by using a minus sign, 5° below zero being written — 5°. 32. Positive numbers. — The ordinary numbers with which we are familiar are called positive numbers. 33. Negative numbers. — Numbers on the other side | of zero from positive numbers are called negative numbers. Negative numbers are written with a minus sign before them. Positive numbers need have no sign to distinguish them, but when it is desired to emphasize their quality a plus sign may be used, as in + 5, + 2, which means the same as 5, 2. 34. Two uses for the plus and minus signs. — Hence the signs + and — have two uses: (1) To indicate addition and subtraction, signs of operation. (2) To indicate positive and negative numbers, signs of quality. | 35. Debt and credit.— To say that a man has $100, or + $100, means that he has that amount to his credit; to say that he has $0 means that he is just even with the world; to say that he has — $100 is only another way of saying that he is $100 in debt, that he has $100 on the wrong side of zero. USES OF NEGATIVE NUMBERS 23 ORAL EXERCISE 1. If a man has $1000 and loses $900, how much has he? if he loses $100 more? $300 more? How is this written? 2. If the mercury in a thermometer stands at 40°, and then falls 40°, where does it then stand? If it falls 10° more, where does it then stand? How is this written? 3. If we call latitude north of the equator positive, how shall we designate south latitude? If we call latitude south of the equator positive, how shall we designate north latitude ? 4, If a man weighing 170 lb. steps into the basket of a balloon pulling up just 170 lb., what is the combined weight of the two? Suppose the balloon pulls up 270 lb., what is the combined weight? | 5. What is the altitude, in feet, of a point 1 mi. above the sea level? of a point 5000 feet lower? of a point 280 ft. lower still? of a point 100 ft. lower still? How shall we " indicate this altitude? 6. If a man has $400 in the bank, and draws out $300, how much has he left? Suppose he then draws out $100? If allowed to draw $50 more (to “overdraw”), how shall we designate his balance ? 7. If we call a certain point on the blackboard zero (0), and call distances to the right of it positive, how shall we designate distances to the left? If we call distances above it positive, how shall we designate distances below it? - 8. New Orleans is in 90° west longitude, which we will eall + 90°. What is the longitude of a place 80° east of it? of Greenwich, which is 90° east of New Orleans? of Paris, which is 100° east of New Orleans? How shall we indicate the longitude of Paris by using a negative number ? 24 THE NEGATIVE NUMBER 9. How much difference in price is there in selling a horse $15 below cost or $20 above cost? 10. The temperature on one January morning in Denver was + 8°, and the next day it was — 4°. What was the average ? 11. Weights of 7 lb. and 9 lb. hang over a pulley. ic] How will they move? What two methods could you use to make them balance ? 12. A man who was $70 in debt paid $50. How much was he then in debt? Suppose he earns $50 more, how much is he then worth? 13. If there is a house for every number, how many houses would you pass in going from 21 East Washing- ton Street to 5 West Washington Street, including both these houses? 14. Jefferson Street is 6 blocks east of Adams Street, and Monroe Street is 12 blocks west of Adams Street. Monroe Street is how many blocks west of Jefferson Street? 15. A game is played by throwing bean bags in the direction of the arrow. Suppose the score stands — 5, 3, 10,10, 0, — 10, 5,10,10, how much is the total score? 16. The tide at the ocean is measured by a tide gauge. At mean (average) tide a pencil points to 0 on a scale, sliding to the right 1 space for every foot of rise, and to the left 1 space for every foot of fall. How many feet does the tide fall when the pencil moves from + 8 to — 3? t Teachers should give a great deal of oral drill of this kind, using blackboard illustrations when necessary, until the idea of negative number as the opposite of positive number is well understood. CURVE TRACING 25 ORAL EXERCISE 1. If we call a force pulling upwards + 3 lb., how shall we designate an equal force pulling downwards? 2. Draw this figure on the board. Calling O zero, and the distances to the = : af right positive, point X’ O x to the distances +3; +5; —2; 0; —4;'+ 23; — 323 1. 36. Curve tracing. —In this figure is the successive days of the week a hated are represented on the line OX, / and the temperatures on the lines parallel to OY. The broken line shows that the temperature on one day, say at noon, was + 70°, the next day 60°, then 65°, 50°, 60°, Um O°. WRITTEN EXERCISE 1. On ten successive January days in Duluth the ther- mometer at noon registered 40°, 60°, 55°, 50°, 20°, 0°, —10°, — 15°, 10°, 30°. Trace the broken line or curve. 2. The increase in population in Nevada in four succes- sive census years was 36,000, 20,000, — 17,000, — 3000. Represent the time differences by 1 in., and 10,000 popu- lation by i in., and trace the curve. 3. The increase in population in Virginia in eight suc- cessive census years was 200,000, 30,000, 200,000, 170,000, — 870,000, 180,000, 140,000, 200,000. Using half the scale of Ex. 2, trace the curve. Any tables of statistics or records of temperature furnish material for curve-tracing problems. 26 ADDITION ADDITION ORAL EXERCISE . Add 2 apples + 3 apples; $2 + $3; 2a+4+ 3a. .- Add 4x+54;Txt+nx;244+9%; x+42; 24482. . Add 10x%+4+154; 25x+2; 54+22; 10%+4+ 0.252. . Add 2xe%+82+52;Ta+2a+a;3a+84a+8a+4+ 38a. . Add $8a+4a24 7x; 2n+3n+n; 4n4+2n+2n+38n; 26+4b+6+6; 8m+m+m+im; 6c+4ce4106e. a fF ww DD & 37. Like quantities. — Quantities like 2a, 4a, and 4a, that are the same except for coefficients, are called like quantities. 38. Adding like quantities. We have found that like quantities are added by adding their coefficients, writing the letters in the sum. 2 he For example, 3 abr + 2 abr + abe = (8 +2 + lL) abx 2 abs = 6abzx, where the parentheses show that the num- 1 abr bers inclosed are taken together. 6 abs The sums of the following should be studied : 2a $2 2 2abm 2 sheep 2s 4a 4 4 4 abm 4 sheep 4s 6a 6 g 6 abm 6 sheep 6s ieee ey 1,2 =4 12abm 12sheep 12s WRITTEN EXERCISE 1. 7Ta+6a+23a+ 41a; 19% + 272 + 692. 2.1744 3822%+u4+4+128e; 15a+49a+77a. 3. 1506 + 16ab + ab + 28.ad; 182? + 97a? + 89 a. 4. If a= 2, b =3, find the value of 16a+4 326. (Find the value of each separately, and then add.) ADDITION 27 ORAL EXERCISE Add the quantities in Kus. 1-9, reading the columns rapidly, as you read a word: 1. (ft. 3in. 2. 7sq.ft.4+3sq.in. 3. 7 f+37% Zits 410, 2 sq. ft.+4 sq. in. 2f+40 Meocalss gi. pt. 5. bg--2g+41p 6.562.274 2 9 gal. 1 qt. 9g+1¢q 9a+ y¥ 7 4a4+264+3c 8 844+ y 9. 2w+sua+4+4y 2a+36+5c OE ALD 2Zwtsax+tAy Ga 40 we 2y+42? 2wt38a+4y 10. In adding algebraic quantities, as in Ex, 8, how do you arrange the like terms? Then how do you proceed ? WRITTEN EXERCISE Add in Hrs. 1-6: JL AB 2,.177%+31y 3. 63047 +19 y 19a+.. 6b 92x%+ 73y a+ Ty 36a + 396 4ja+ 9y 91a? + 80y 4,.27@a¢+30+ c 5.16a+ 5b+7ece 6 a+ 64+ e 4a+9d+4+2c og +9e s6a+386+3¢ 9a+8b4+7T76e 5a+100 ba+56+5¢ Add in Exs. 7-9; then find the value of each addend and of the sum, lettinga = 2,b=3,c=5,x=1: eee 2b- Cc 8) a+ bee See Megs ra aR AG 2a+36+2¢ 20 +2 20 Oa at a + ¢ db+a b6+2c+32 b+ e¢ Za+4b s6+2c+ 24 28 ADDITION EQUATIONS INVOLVING ADDITION OF ALGEBRAIC TERMS ORAL EXERCISE 1. If twice a certain number plus 3 times that num- ber equals 50, how many times the number is 50? 2. If 24+ 382 = 50, how many times x is 50? Then what is the value of x? In the following examples, first find how many times a certain number you have; then find the number, as above: 3. 2+22=6. 4.¢+32=12. 5.2 +92. = 20. 6. 2+ 8a = 81. Y bate (Grek that 8. 2 +62 = 63. 9. «+192 = 80. 10. 2x2+ 62 = 56. lien 1 7 90. 12. 2y+13y = 45. 13. 212 +42 = 78, 14. 388k —3k=70. 15, Dia — 2 — 100; 16. 5% + 15m = 20. 17, 1445-22 = 62: 18. ©+22+32 = 30. 19. 27 2 — 7 2 = 100. 20..2+92 —22=—80. 21. Twice a certain number plus 5 times that number equals 49. What is the number? 22. A certain number plus twice the number plus 3 times the number equals 60. What is the number? 23. A certain number plus 7 times the number plus 12 times the number equals 100. What is the number ? 39. Illustrative problem.— Find the value of « when W7x+12¢%¢+2=510. 1. Since 177 +1227+2=510, 2. Then 30 x = 510, by adding like terms, 3. And xz = 17, by dividing these equals by 30. Check. 17 X17 +12 x 17417 = 289 + 204 +17 = 510. EQUATIONS 29 WRITTEN EXERCISE Find the value of x in Exs. 1-14: tT. 1924+ 27274+2= 94. 9 3oa2—2xr+7 = 168. a2 +2e2+3a2 = 102. sie Oa — 2. 126: Sow — 20 4+3.=12558 6. & 2a + Te = 30008 .3+424+2xe=669. 8. 104+9x—22e=619. S60 +7 —25 = 275. 1050-92132 207. Il. ec+82+52=108. 12. 9x—6+e2+4+1=7)5. 13. 2 --9xr+82=126. 14. 882+22+72+7 = 283. 15. On account of change of temperature a watch gains 27 sec. each day and loses 19 sec. each night. If set right to-day, in how many days will it be a minute fast? OO FF GO &w 16. If in a class of 13 boys and 15 girls each pupil has the same number of cents, and the total amount is $2.52, how much has each? (1382 + 15 = what number?) 17. A famous mathematician was once asked the time, and is said to have replied, “There remains of the day twice the number of hours already passed.” What time was it? | 18. The length of a field is 11 times its width. The distance around the field is 36 rods. Required the dimensions. Tf x is the width, what is the length? The distance around is how many times the length and width? In cases of this kind always draw a diagram when reading the example. 19. If you tell me to think of a certain number, then to multiply it by 3 (=3«), then to add to this twice the number (= 3a + 22), then to add 1 (= 3x+2x+41), and I tell you the sum is 51 (that is, 347+ 2x2+1= 51), how do you find the number? What is the number? 30 ADDITION 20. The Milwaukees won 83 games of baseball in one year, which was 3 less than twice the number they lost. How many did they lose? 21. One of the best baseball records is that of Wagner, who was 512 times at the bat in one year. This number is 27° more than 5 times the number of runs he made. How many runs did he make? 22. A man bought a suit of clothes and a hat for $36. The clothes cost 8 times as much as the hat. How much did he pay for each? If x equals the number of dollars paid for the hat, what repre- sents the amount paid for the clothes? for both? What is the equation? 23. A man paid $1430 for a horse, a carriage, and an automobile. The carriage cost twice as much as the horse, and the automobile 4 times as much as the carriage. What was the cost of each ? 24. A man paid $6600 for a village lot, for building a house, and for his furniture. The lot cost twice as much as the furniture, and the house cost as much as the lot and furniture together. What was the cost of each? » 25. At Christmas Mr. Brownson gave to each of his 3 children as many dollars as they were years old. Clara received twice as much as Helen, and Alden received as much as Clara and Helen together. All together they received $36. How much did each receive? 26. I have a farm of 175 acres, of which 15 acres are used for buildings, garden, and woodland. Of the rest, 3 times as much is devoted to oats as to wheat, twice as much to hay as to oats, and just as much to pasture as to hay. How many acres are devoted to wheat? to oats? to hay ? to pasture ? : ADDING NEGATIVE NUMBERS 31 ORAL EXERCISE 1. If a balloon pulling upwards 100 lb. is fastened to one pulling upwards 200 lb., what is the total upward pull? 2. If a balloon weighing — 300 lb. is fastened to one weighing — 200 lb., what is the total weight? What does the negative mean ? 3. If a man in debt $50 incurs a further debt of $20, how much is his debt? That is, if a man worth — $50 adds — $20 to his capital, how much is he worth? 4~ Add — $50. and — $20; — 70 and — 380; — 6 and —12. Make a problem about — 10 lb. added to — 30 lb. 5. Add — 2a, —3a, — 6a; also add — 20 ay, — 30 xy, — 50 ay; also add — 6y, —y, —Ty, —38y, —Sy, —10 y. 40. Adding negative quantities. — Therefore, to add several negative numbers, add as if they were positive, prefixing the negative sign to the sum. 6. If a balloon pulling upwards 10 lb. is fastened to a 12-lb. weight, what is the total weight ? 7. If a balloon pulling upwards 15 lb. is fastened to a 10-lb. weight, what is the total weight ? 8. If in a tug of war, one class pulls 800 lb. to the right (which we will call + 800 lb.), and another class pulls 750 lb. to the left, what is the resulting right-hand pull? 9. If a man has a capital of — $500 and adds to it $700, how much is his capital? How much is — 500 + 700? 41. Adding positive and negative quantities. — Therefore, to add a positive and a negative number, take the difference of their real values and prefix the sign of the numerically greater one, 32 ADDITION ORAL EXERCISE Add the following : 1. 6 ft. — 2 in. and 4 ft. — 3 in. 2.6%—2yand4a—3y; alsol7z and «—y. 16ab+2e 3. 2 lb. — 4 oz. 4. 2x%—A4Ay a. 3 lb. — 5 oz. on — OY —6ab+ 3c 42. Adding several quantities. — In adding several quantities, some positive and others negative, it is at first convenient to add all the positives, and then the ie a8 oe negatives, then taking the difference of their real ihe ie values and prefixing the sign of the numerically a if es greater one. a 6 wes Ee Later, however, it will be found easier to read ae the columns, as in arithmetic. In this example oe e a we should think: “2’s, 6, 3, 2, 6, 9; y’s, — 5, —3, — oe oa ie —10, —13, — 11.” Practice soon enables one to teehee, read the columns much more rapidly than this. WRITTEN EXERCISE Add in Has. 1-26: 1. o2ay + 32 2. 4m*n+ dmn? 14 xy — 22 16 m?n — Tmn? —ry+ 2 — 9m’n — 18 mn? —16xy—5z 23 mn + 9mn? 3. 142 ng + 148? 4. lba— 2+ ¢ 19 pg + 37 ? 16 a? — 330? — 9¢? — 17 pq — 6738? (2 AU D2 -- 6c? 5. 37a+ 926 —63c 6. 17abd+ 8be—4ca = O20. 1o0 pee 6ab — 5be+2ca 25a — 19 b.-- 1S¢ —3ab+26c+2ca Li: ADDITION 33 . 64arxy*z + p 8. m+ m—s 78 @ay*z +4 p 3m*+4m 81 axy?z + 9p Se 3s) fea OF 10. darwy+5z2+w a*—2Zab+ 6 ds acy +22—w 4a*—4ab—20 Tary—32+w 6pq+t+4qr+ os 129156 a 375 6/448 ¢ 5pqg—4gqr+ 3s — 29384 —5966+17¢ 9 pq +16s 489 a + 783 6 — 20¢ . 64 m?n + 92 mn? + 81, T6 mn? + 29 + 6 mn. . 28.07) + 382 ab?, 84 ab? + 29 a7b, 4207) + ab? . 82 pqr +45 grs + 4, 1296 grs + 487 — 39 pr. . 29 abed + 32 bede + 81 cdef, 64 bede + 92 cdef. . 81a—31b6 +48 0b, 696+52ab + 69a, — 385. . dla+42b—3c¢, 69a—38264+9c¢, —10b—6e. . 82p +424 — 349r — 89¢ + 47 p — 1277 + 8007. ~27e@+14y4+72, —42e2—18y4+192, 182+ 62y. . 02m+17n—43p —18p + 3m —14n+4 92p +100m. . 8x2 — d4y4 722 —148y 4+ 79% — 144% + 832 — 2. . 8640 + 92774 3432, T83y—492 2, 629% —7T3y. . 892a—993b+444c¢— 82d, 1008a—7b+4 556c—18d. . 22a+6—8c+d—49e, 99b—d—e+3c4+ 78a. . 427 x? + 298 xy + 829y?, 827 a? + 43477, 298 x? + 87/7. Add in Ers. 27-29; then find the value of each addend and of the sum when a =1, b= 38, c =2: 27. Z2a+4b— c¢ 98. 29 a7 +267 29° a7 bee 6 3a+66—4e 10a? 3364 3b6+4¢ 864+ 9e 4a*+ 76 Varese 34 SUBTRACTION SUBTRACTION ORAL EXERCISE Subtract in Hxs. 1-5: 1. $5 Zaft: BkiinG! 4. 5f 5. 5a $2 Pate 2d 2f Qa What must be added to the subtrahend in each of the following cases to make the minuend? 6. 9x (pealiges 8. 19 abx 9. 238a+85 8 8a 8 aba 38a+2b In Exs. 10-12 find the differences by seeing what must be added to the subtrahend to make the minuend: 10. 23 sq. ft. 11. 23 f? 12. 18 42 +157? 4 sq. ft. 4 f? Qx?+ Ty? 13. In subtracting algebraic quantities how do we arrange the like terms? Then how do we proceed? WRITTEN EXERCISE Subtract in Hrs. 1-4, as suggested above: 1. 142%+4 156y 2. 173 ax + 237 by 9 ae+ b68y 96 ax + 142 by 3. 648 2? + 2737/7 4, 12.46 abe + 2.83 592 x? + 183 7 6.23 abe + 1.04 Subtract in Hrs. 5-6; then find the value of minuend, subtrahend, and remainder, letting a = 2,b=1,c=8: 5. 42a+ 610 6. 82a+1564+ 20c 18a+19b 12a+15b+ 8c EQUATIONS 35° ORAL EXERCISE 1. Ten times a number minus 3 times the number is how many times the number? 2. If 10 times a number minus 8 times the number is 42, what is the number? 3. If 102 —2x”=16, what is the value of x? 4. If 62+ 8x” — 2x” = 36, what is the value of a? 5. Think of some number; multiply it by 6; to this product add 8 times the number; then subtract twice the number. Now tell me your result and I will tell you the number. How is it done? (Compare Ex. 4.) 6. Make up a problem like the one in Ex. 5, using other numbers. Write the statement on the board, using x or n to represent the number thought. This presents an interesting mathematical game. Pupils may profitably propose such questions back and forth as part of the daily oral drill in solving equations. Find the value of x in each of the following: (Ei ie uentiy & 8 13a — 10a = Ss: 9. 144 — Tx = 84. 10. 232% —182 = 65. lees Yel 0, 12; 522 — 212 = 938. 13 oO ee OF. 14.9242 —81la¢=121. WRITTEN EXERCISE Find the value of x in each of the following : ee te 154. 9. 194-2 2 136: eee l4) 17 oe 526. 4. 14e—- 20 += 6b, beow +47 +1927— 223. 6. 241+ 82 + 167=—.289. 7 13824+47274+6327=861. 8. 122+13874+1427=—429.- 36 SUBTRACTION ORAL EXERCISE 1. If a man is worth — $10, how much must be added to his capital to make him worth $0? $15? $50? 2. If a man is worth — $100, how much must be added to his capital to make him worth — $50? — $75? — $25? + $25? 3. If the thermometer registers — 20°, how many degrees must the mercury rise to register —10°? 0°? 30°? What is the difference in degrees between 10° —10° and + 15°? Show this by the help of the annexed diagram. 20° 4. Draw this picture on the board. Then show See what must be added to — 15 to equal 0; to equal5; ] to equal 15. 5. Looking at the same picture, how much is the differ- ence between —10 and +5? That is, what must be added to —10 to equal +5? What is the difference between — 5 and + 20? 6. State rapidly the following differences; that is, the quantity which added to the subtrahend makes the min- uend : $15 20° — 15° —102y 25 a*y — $10 aI BSS — 20° — 30 xy — 25 27 7. What is the number which added to — 10 makes 0? makes — 20? makes —5? makes —15? (Use the above picture if necessary.) 43. How to find the difference. —— Hence in algebra, as in arithmetic, 3 The difference is found by finding the number which added to the subtrahend makes the minuend. SUBTRACTING NEGATIVE NUMBERS ot 44. Illustrative problems. —1. From 327 —3y subtract 22x" —2y. What quantity added to 222 makes 322? 824 — 3y Evidently 10z. What quantity added to — 2 y 224—2y makes —38y? Evidently — y. 10 gy 2. From — 1027+ 7 y* subtract 62? — 5 7. What quantity added to 6 z? makes 0? Evi- —1022+ 7y dently — 622. How much more must be added 622— 57? to make —102?? Evidently — 1022. Hence — 16274 127? their sum is — 16 2?. What quantity added to — 5y? makes 0? Evidently 572. How much more must be added to make 7 y?? Hence their sum is how much? What is the entire difference? If pupils are not taught any artificial rules, but subtract in the way suggested, stopping at 0, as in the last example, if necessary for clearness, they will soon come to subtract negative quantities without difficulty. This is less confusing than to change signs and add. WRITTEN EXERCISE Subtract in Exs. 1-4: 1. t4ay +17 yz+5 2. 15 ab — 20 be + ed Gay— 8yz—2 19 ab — 20 be — cd 3: 17 m? + 1677+ 7 4. 62m?+15n?— jp? —17m?—167n?— 3 14 m? — 20 n? — 8p? Subtract in Exs. 5-8; then find the values of minuend, subtrahend, and difference, whena=38,b=1,c=4: 5. 25a —176+ 3c 6. 35 a? + 406? 32a— 9b—4¢ — 5a?— 300? 7. 2744+ T56+ce¢+7 8. Fa-90' =33 4 20a+70b—c— 3 = 20 SETESS 38 SUBTRACTION Subtract: 9. Gabe +4 10, 38m? — 192" 9abe — 7 4m? — 17 2? 11. 42°-— 3y 12. 8p? + 4Apq 9 x? — 27 y? 7 p? — 39 pq 13. — 1243 + x* 14. a? — 147 xyz Be A ie lie x? — 298 xyz 15. 23 a? + 91 a? 16. — 829 x? — 9 49 a? — 37 a? 792 x? — 8 17. 18 a7) + 4 al? 18. 439 a — 9276 6 ab — 2 ab? 291 a — 2966 19. 8m?+ Smnp 20. — 21.82%+0.9y 6 m? — 11 mnp —42.9%—2 y 91. =U60 + 0 ——c DO et ean ae —2Z1a—b6+6¢ 6a = 4a wa O30 ome eed 24. 42n*94+2¢4+7r 6a? —5x2—1 —29p?—38¢q4+r 25. —9x—Sy—9z 26. —92x+3y— 2 —Tx—I9y—8z —T9x%—38y—9z 27. From 8927 + 42 yz + 7 zw — 6 subtract 75 — 49 zw - 93 xy. 28. From 822° — 4827+ 17 w — 37 subtract 342 — 93 x? + AS 7? 2 3: Subtract, and then find the value of the minuend, sub- trahend, and remainder, whena=1,b=1,c=1: 27a+92b—6c 30. —93a+486 —Te 29. av+2ab+ 8? 3a*—4ab—T7 0? REVIEW 39 WRITTEN EXERCISE . Add v?+ 2ay+y*, v—2aey+y’, 827+ 877. . Add m?4+2m?—38m—1, m—2m?+m+1, m?+2m. Add 3abe—4bed + 5eda, — 6abe+ 5 bed — 5 eda. . From 3224+ 42?—5a”+6 subtract 42?— 16a” + 16. . From 2m?4+5n?—2x2+38y subtract n?-4xu+2y. seErom @ 26 — oe a suvtract a 40 + 6 ol ad. 7. From 62?— Tay +44? subtract 5a? — Say + 3y?; then find the value of minuend, subtrahend, and remainder, tie and = 3. 8. Add 3a7+4bc—6d, 4a*?—5bc4+T7d, 3a°+2 bc; then find the value of each addend and of the sum, ifa=1, Peas Coby. = 1 Subtract, and then find the value of each minuend, subtrahend, and remainder, if a= 2, b=5,c = 83: G26 3070 2 a0 07 103590745 5c ola oe a0" Ge 0 Oe Add, and then find the value of each addend and sum, ufx=10,y=1,z=5: 11. 84—57? +22 12. 5a? 4+ 2y7?—382? x—2y? a va + 32? 9y?—4z e+ y+ 2 13. 2a?4+ 4y?— 32? 14. xe*— Say+ 47? 4o?— 6y?— 22? —427?+1bay— 37 8a*7— B5y?+ 42? W7x?*— Bbry+52¥ ee — 132 1627-17 xy — 417 —5a?+ 27? — 9x? — 137? 16y?— 72 39 xy +17 4? 40 PARENTHESES HOW TO USE PARENTHESES 45. Use of parentheses. — The product of 2+ 3 by 7 is indicated by either 7 (2 + 3) or (2+ 3)7, the two having the same value, 35. The sign x is evidently unnecessary. If a=4, b=7, and z = 10, then (a + 6)z =11x 10 = 110. The difference between 7 and 2+ 4 is indicated by 7 — (2+ 4), which equals 7 — 6, or 1. Ifia=9,7=5,andy= 2, thena—(@+y)=9—-—7T7=2. But a-z+ty—9—5 +2=4 72-6: The sum of 2+5 and 4+5 is evidently the same whether written (24 3) + (4+ 5)=549=14, or 243 +4-+5=14; so the parentheses are not necessary. The quotient of 5 + 9 divided by 2 + 5 is evidently the (5 + 9) 5+9 (245) or O45 But (64+9)+(2+4+5)=14+7, and this is not the same as 5+ 9+ 2-4 5, because 46. It is the custom to perform multiplications and divi- sions before additions and subtractions unless the parentheses show otherwise. For example, 2+4 +2 means2+4=2 but (2 + 4) +2 means6+2=8 1 1 same whether written > 2+4 x2 means2+8= but (2 + 4) x 2 means 6 xk 2 = WRITTEN EXERCISE. 1. (xv + y) +m, where « = 10, Y= VOTO; 2. (a+ 6)(c+d), wherea=1,b=2,c=3,d=4., 3. 24+8~x 2, (2+ 8) 2, 2(2+4 8), 13 + 165 + 15. 4. (8+7)+2,44+6+2, (446) +2, (7419) + (95 —82). REMOVING PARENTHESES 4] ORAL EXERCISE 1. What must be added to 2a to make 5a? Then how much is 5a—2a? 2. What must be added to 6 to make 0? to this to make a? Then what must be added to 6 to make a? 3. What must be added to —é to make 0? to this to make a? Then how much is a — (— 0)? 47. Subtracting negative quantities. — Let us consider the difference resulting from taking 6 —c from a. at+t0O+0 We may writea + 0+ 0 for aif we wish. Then Rashes c what must be added to —c to make 0? to 8 to eee make 0? to 0 to make a? Then what is the value Ota — (0 — 0) Because a — (6 —c) =a —b + ¢, we see that 48. If a quantity in parentheses is preceded by a negative _sign, the parentheses may be removed provided the signs of the terms within are changed. Because a + (6 —c) =a+b —e, we see that 49. If a quantity in parentheses is preceded by a positive sign, the parentheses may be removed without change of sign. WRITTEN EXERCISE —G 2 f Remove the parentheses, changing signs where necessary : .@+2ab4+ 0? — (a —2ab +d’). m*? —2mn — 38 n? + (— m+ 2 mn + bn’). et Pade 9") = ag"): . @& — 8a) + (3 ab? — 0) — (8 +3 ab —3 ab? — 0°), a*b + be + ca — (ab — b?e + 0a) + (a 4-2 b?r — 3 ca), . 6a? + (3 y? —4 2?) — (a2 + 27) +. (yy? 42%) — (464+3 y 42 MULTIPLICATION MULTIPLICATION ORAL EXERCISE », Multiply ybye2: 3b. 3462 soo ue een a . Multiply by 8: 30 mi.; 31 in.; 21 sq. mi.; 40 x; 50 aa. Multiply by.3: Yt: 2 in-3°7 squitezsqiin fia ay. Multiply by 7: a+26; 9a°?+50?; 8ab+c; 6abxe+d3by. Multiply by 6: 9 sq. ft. 2sq.in.; 9474 2y?; 2? + 67%. Multiply x* by 3; by 25; bya; by 5; by ab; by 2 abe. Multiply x+y by 3; by 17; by 200; by a; by 2a”. me) peek EA Lee OP heal oe 50. Multiplying by monomials.— The multiplication of x+y by a is indicated by a (x+y), or by (w+ y)a. Hence a (x + y) =axz-+ ay, as in the following cases: Att Dani 400+ 2 dea+ 2y ot y 5 5 5 a 20 ft.10in. 2000+10 20%+10y ax+tay 51. Therefore, to multiply a polynomial by a monomial, multiply each term separately and add the products. WRITTEN EXERCISE Multiply as indicated: , 23a by 125. . 43a by 27 y. u . A21 a by 23. 3 5. 16.ad by 15 2y. 7 9 . 39x by 46 ay. . 16ax by 15 by, . 25 mn by 25 xy. ono - ww . 25 mx by 25 ny. . abe(a+y); 5(a+6). 10. ab(x?+y?); mn(a+b?). 11. x(a+b); x(2a—6b). 12. b(p? +9"); 5a(m +2). 13. pq (a+); b(@+y). 14. 8(2da.3hr.); 3(22+43y). LAW OF COEFFICIENTS AND EXPONENTS 48 ORAL EXERCISE 1. How many times is a taken asa factor in aa? in a?? mace ain a’? in.a*? in oes 2. How many times is x taken as a factor inw?? in 2? iieceexere: (in 0708 Veiner ees 3. Then what is the product of 070°? of 0904? of mm? Oe we OL Mm? OL mar Ber NOL meen 4. If factors are the same quantity with the same or with different exponents, what do we do with the ex- ponents to obtain the exponent in the product? 52. Law of coefficients and exponents. — Because 2x? x 3x? =2xx x 3xxx =62x°, we see that if factors are the same quantities with various coefficients and exponents, we Multiply the coefficients and add the exponents. 53. The dot as a sign of multiplication.— In multiplying in algebra we not only indicate multiplication by the absence Ofeae sion bit also» by a, dot, thus: ax &= a-b'= ab, EXO ca os. 0: Multiply «? + 7? by xy. Here xy (x? + y?) = aya? + ayy? = 0°y +1xry*. WRITTEN EXERCISE Multiply as indicated in Exs. 1-10: ea 0-0, 2. ab-be-ca. 3. xy(a + y). fare a O72 0. 5. m?-m*®. m?. 6. 4a (x? + y*). 7. 2m (m®+ 3 n’). Son tn 360 9. am: 2am-3am. 10. 82a?(2a+ 3a). 11. Multiply 22y(a@+y). Then let r=2, y=38, and find the value of each factor and of the product. 44 MULTIPLICATION ORAL EXERCISE 1. How much more shall I weigh with two 3-lb. weights in my hands? 2. How much more shall I weigh with two — 3-lb. bal- loons in my hands ? 3. How much more shall I weigh if I am relieved of the two 3-lb. weights ? 4. How much more shall I weigh if I am relieved of the two — 3-lb. balloons? 54. The law of signs in multiplication. — The addition of two 3-lb. weights adds 6 lb. That is, 2.3 1b. = 6 lb. The addition of two — 3-lb. balloons adds — 6 lb. That is, 2-— 3 lb. =— 6 lb. The subtraction of two 3-lb. weights adds —61b. That is, — 2-3 lb. =— 6]b. The subtraction of two — 8-lb balloons adds 6 lb. That is, —2-— 3 |b. = 6 lb. We therefore see that 55. The product of two numbers with like signs ts positive; with unlike signs, negative. 2%—3y Thus, to multiply 272 —3y by — 22, we have =p —224-—3y=62zy, and — 24-22 =— 42%. — 422 + 6 xy WRITTEN EXERCISE 1. 3a(4a —2y). 2. — 4a(a? — 6). 3. 2lay(a—12y). 4. 52a (a? + be). 5. —15.a7b (a? + Dd). 6. — 41x? (a? — y?). 7. 17am(—a +m), 8. loxy(— 3% — Ty), 9. —a(—a—b—c—d—e). 10. — 21abe(— 36a — 425), 11. —13m?(— 2m —38n). 12. wberd(— a? +b —c? +d). ao oa fF WH WD -& DIVISION 45 DIVISION ORAL EXERCISE . Divide by 2: 4 ft.; $4; 4/; 4d; 400; 4h; 42. . Divide by 5: 25sq. ft.; 25 f?; 2527; 35 2?y; 45(x+y). : Divide by 7: 14; 2°739-(;2-7;7y; 63(@7+y' +2). SRI ICOM Ves Od Od aa ncaa eae Cred rey MUvIde Dy LYE OY, Oe CY OLY Fey ay ay, 1 Lou, » Divide 14 a’x by 14; by a7; by w; by 7; by 2; by 28. 56. Law of coefficients and exponents. — Because Sie bl PAT gir oat je ee 3x 3-4 we see that in division involving the same quantities with various coefficients and exponents we Divide the coefficients and subtract the exponents. Oo mo Oo 11. WRITTEN EXERCISE . TO p¢ + o pg. 2. 36 a7x77y? + 4a. 32 a*b®ct + 2 ct. 4. 39 piq’ + 13 pig. 27 a*x® + 3 ax? 6. 45 mn’ + 15 n°. 28 men’ -- 14 n?. 8. 96 ax?z* + 4 azz", . 144 abed + 6 ae. 10. 1728 x®yoz? + 144 237328 If 162*y* dollars are divided equally among «?z* people, how much has each? Suppose x=1, y= 2? 12. If p=7+rt andi = 3007t, find the value of p. (Do you know what rule of interest is expressed by 7 = prt?) 13 . If c=187 (m being the Greek letter pz), and if c-+27=r7, find the value of r. (Have you learned what truth about the circle is expressed by ¢ = 2 7r?) 46 DIVISION ORAL EXERCISE 1. Divide 18 ft. 6 in. by 2; 18 f+ 67 by 2. 2. Divide 28 lb. 8 oz. by 4; 28%+4+8y by 4. 3. Divide 10 mi. 25 rd. by 5; 10m + 25r by 5. 57. Dividing a polynomial ih a monomial. — Divide 16 a? +8a+6 by 2. Dividing each term separately, we my 2 have 8a’ + 4a + 3. EAS ek ah 8a@+4a+3 Divide 35 a?y + 21 by 7a. 7 2 Dividing each term separately, we 12)35 xy + 21x have 5ay + 3. Sxy + 3 Therefore, to divide a polynomial by a monomial, divide each term separately and add the quotients. a WRITTEN EXERCISE Divide: 1. ax? + ay’ by a. 2. 25a? + 35 ¥ by 5. 3. 8a + 245 by 4. 4. mnx? + mpy? by m. 5. abe? + abd? by ab. 6. @+4a?+ 5a by a. 7. 77 m+121 by 11. 8. 51 a? + 187 0? by 17. 9. 2+ 223+ 32t by a 10. p’q?r? + 2 p?q?2* by p’q?. 11. 327+ 396 ay by 3% = 12. 4a?y? + T6277? by 4x7, 13. a§+38a2+4a?+ da dy a. 14. 750° 4+ 25a?4 1254+ 625 by 25. 15. aty3z? + 2 x8y2z + 3 xy?2® + 5 xy®e? by xyz. 16. 125 x7y?z? + 600 xyz + 275 a®y%28 4+ 25 by 25. 17. 21 m®> + 35 m1n + 56 mén? + 105 mn? by 7 m?. 18. 33254 51 aty + 54257? + 69 x77? + 111 zy by 32. PARENTHESES 47 58. The parentheses. — It should be remembered that the operations indicated within the parentheses are to be per- formed first. ee For example, x (x? + xy + y*) means that the sum of x? + xy + y? is to be multiplied by z. 14 — (8 — 6) means that 8 — 6, or 2, is to be subtracted from 14. That is, 14 — (8 —6)= 14-2 = 12. WRITTEN EXERCISE Perform the operations indicated: 1. 2ax(a* + ax + x). 2. 4 mn? (mn? + 3). 3. (1627y?+8xy)+4ay 4. (at*+a*)+a?+a(at+l1). 5. (36 x8y?2 + 9 xy?z®) + 3 axyz. Grea) 07 0 6 a 7. 32axy — 2axy+y(2ax + daz). 8. (m> + m+ m) + m+m?(m?+1)+1. 9. 25 mnz + dmnx — 38mnx + T mnx — 4mnz. 10. 21 p°g + 42 p°¢.t+ p* (49 +39) — 2p’¢ — p’Q). 11. (ax? + ax + 1) a*x? — (a*a?® + atx* + a®x®) + an. 12. 6pg(p+q). Find the value when p = 2, ¢g = 3. 13. 15 pq’ + 10 py? — 5 pg’ + 4 pq? — (6 py? — 3 pq’). 14. 4a%b(a*°+ 0°). Find the value when a = 2, 6 = 1. 15. x? (x? + y?+ 27). Find the value when «= 1, y = 2, 16. (1628+ 82741224 20)+4. Find the value when e= ae : 17. (a#+a*+a?+a)+a. Find the value of dividend, | divisor, and quotient when a = 2. 48 | DIVISION ORAL EXERCISE .2u-8y; Gay+2x; 6ay+ dy. 38a-—2b; —6ab+3a; —6ab+—205. —4a-5b; — 20ab+— 4a; — 20ab+ 50d. —3m-—4n; 12m7+— 35m; 12mn+—4n. ee hos ame 59. Law of signs. — We see that Because + a-+6=+ ab, therefore + ab ++a=+0. Because + a-— 6 =— ab, therefore — ab ++a=— Bb. Because — a-+ 6 =— ad, therefore —ab+—a=+b. Because — a: — 6 =+ ab, therefore + ab +—a=-— J. That is, ; 60. The quotient of two numbers with like signs is positive ; with unlike signs, negative. Thus, to divide 8222?—16ay by —162, — 1627)32 2° — 16 xy we have so Carte 8242+—-16%=-—22, and —l6zy +-16z7=y. WRITTEN EXERCISE . 45 a7) +90. 2. 39 a°y? +— 13 ry. . 275 mn +— 25 n. 4. — 625 a7y?z? + — 125 xyz, . —1001L ay +Tay. 6. — 8535 p?q?r? + — 101 p*r*. . Axe — 32%)+-—2@. 8. (9 m2—18 mn 4+.27m)-+—9m. . (— 825 p? + 125 gq?) + — 25. 10. (17 « — 512?4 1538 2”)+—172. 11. Divide 32a*—8ab by —8a. Then find the value of dividend, divisor, and quotient if a =— 2, b=1. 12. Divide 49 m3 — 63 m? — 84m by —7m. Then find the value of dividend, divisor, and quotient if m= — 1. onwynw Ww -_ REVIEW 49 WRITTEN EXERCISE Add 45:22 + 19 2 2a endsed il (2 a? — 34 23. 2. Add 78 p’q — 134 p77 + 7 p® — 139° and 432 pq’ + 48 9? — 92 p’¢. Oo lrom ol + 1f a =346e¢-- 4072 © subtract — 42 24 + Sl 7 — 78: - 37 x: 4. From 923 a + 431 «°y — 78 vy? + 127 subtract 34 — 9244+ 78 xy? — 227 wy + 8 xy’. 5. Multiply 2° + 422° — 3a* 4 8123 — 1252741732 — 144 by 2a. Divide the product by 2. 6. Multiply 2° —32*y4 4 ay? — 81 xy? + 26 yt —17 7° by —38ay. Divide the product by — 3ay. 7. Divide 2m’ + 6 m®n — 18 m'n? — 292 m1n? + 38 m3n4 —16m?n* by 2m*. Multiply the quotient by 2 m?. 8. Divide 125 ay? + 625 xty* — 325 x8y? — 275 ay? + 425 xy? by 25 x7y*?, Multiply the quotient by 25 xy?. Find the value of x in Exs. 9-21: O85 Lai ol. LO gal ae LY: ite) (Om 1.0 ab: 12.-” + 243 = 17. 13 243 2. 14. 422 — 4 = 500. 16°20 2S = 380. 16. 275 —2z2= 3. 17, 34a +19 = 257, 18. 47 « — 58 = 600. 19. “ ay 20. = oOo} raat a ma Ou 22. If 110% of x is 84.70, what is the value of x? 23. If 92% of x is $156.40, what is the value of x? 24. What number increased by 10% of itself equals 143 ? 25. What number decreased by 7% of itself equals 1953 ? 26. What number decreased by 9% of itself equals 2912? 00 FACTORS AND MULTIPLES FACTORS AND MULTIPLES ORAL EXERCISE 1. What are the factors of 6? of 10? of 15? 2. What are the factors of 2a? of 8x? of aw? of a‘? 3. What are the factors of 9? of 37? of a?? of 352°? 61. Factor. — The word factor is used in algebra as it is in arithmetic, the process of separating a quantity into its factors being called factoring. 62. How to factor quantities. — We always factor a quan- tity by thinking of the quantities which must be multiplied together to make it. The factors of 21 are 3 and 7, because we remember that Sade at ONS: The factors of 3 a? are 3, a, and a, because we remember that DDO = do1G". The factors of abx + acx are a, z, and b+. Monomials are easily factored at sight. 63. Factoring polynomials. — To factor a polynomial, ex- amine each term to find the greatest common factor. Then divide to find the other factor. For example, to factor 4 ay + 8 2?y? + 202y’, each term con- tains the factors 2, 2, x, and y. ; Hence, dividing by 4 zy, the factors are evidently 2, 2, z, y, and x? + 2 xy +5 y’. 4 xy )4 xy + 8 x7y? + 20xy° eet ean y vt Oe ye WRITTEN EXERCISE Factor the following : Lu aa yoo ey =. 2. 14 m?nx®y; 91 m'n®z. 3. ma+my;150°4+85a%y. 4. p2¢+pq?; 51 m>+17 mtx? 5. 8 p®g +15 pp? +21 p77. 6. vy + vy? + v2? + 2? + a8. FACTORS 51: ORAL EXERCISE 1. If a man is worth d dollars and loses rd dollars, how much is he worth? Factor the result. 2. If a man is worth d dollars and gains 7d dollars, how much is he worth? Factor the result. 3. If a man has p dollars and gains pr¢ dollars interest, how much has he? Factor the result. 4. If you had w dollars in the bank a year ago, and have gained rv dollars, how much have you in the bank now ? 64. Factoring certain formulas. — It is often more conven- ient to use formulas in factored form. For example, if some goods are marked m and the rate of discount is 7, the discount is rm and the selling price is m—~rm, or a m(1—r). That 1s, Veit m = marked price, and r = rate of discount, 2. Then rm = the discount, and m — rm = the selling price, s. “} Agi. 3. That is, s=m—rm / 77) (1 = r). j 4. If, now, m = $200, and r= 10%, we have s = $200(1 — .10) = $200 x .90 = $180. WRITTEN EXERCISE 1. If 7, the interest, = prt, then the sum of the principal and interest equals p+ prt. Factor this. Find its value when p= $200, Dice PP ms 5%. 2.) 1t's'—'selling price, ¢ = cost, and 7 ='rate’ of gain} write a formula for s. Factor it. Find the value of s when ¢ = $175, r = 20%. 52 FACTORS AND MULTIPLES ORAL EXERCISE Name the greatest monomial factor in each polynomial Hy dU Rod PIG) 1. ax + aby. 2. 2abe + 4 bry. 3. 3pq7? + 6 qrs?. 4. 5m?n + 15 mn’. 5. a®x + aby + az. 6. por + qrs + rst. . am +a'n+ ax + ay. 8. mx + nay + paz. . Sxyt+10xz4+2527?+50we. 10. Txyz + 2lwey + 3d5vwe. Oo a State the products of the quantities in Exs. 11-14: ll. «(@-+ Bd). 12. a(m* — n’). 13. ab(a + 0). 14. m?n?(m? + n’). State the factors of the following: 15. ax + bx + cx. 16. mx + my + mz. 17. xy + ye + way. 18. 24°4+ 4ay + 6x2. 19. abe + bed + cde. 20. 2 —3a?+4ax4 axyz. WRITTEN EXERCISE Factor the following: 1, 2. m® — m?n. Soe 4. a®*b?x + aby. 5. a? + ab + ab? 6. p®g?r — pq”. Vee ah ae 8. t+ 3épg +p? 9. 82° + 427 + 2a. 10. 22+ 1527 + 16-2. 11.3 mina 6 mina, 12. 27 ay + 9x7? + 38y. 13. abc +- a°b%c + abc*. 14. 16 m? + 12 m2n + 8 m2. 15. 5abed +10 cdef + 15c?d?. 16. 17 a? + 5labe + 153 a¥, 17. 21 p79 +35 q’%p + 56pqr. 18. a*+322+ 4074 121 «x. MULTIPLES 53 ORAL EXERCISE 1. Name two multiples of 5; of 7; of a; of pq. 2. Name two common multiples of 5 and 7; of 2 and 8; of a and 0b. 3. Name the least common multiple of 7 and 9; of 6 aneeee: Ol. 8 and 12 oredgandso: of ab and, bc. 65. Multiple. — As in arithmetic, the product of two quantities 1s called a multiple of either. For example, ab is a multiple of a and of 6. 66. Least common multiple.—The least multiple common to two or more quantities is called their least common multiple (1.c.m.). For example, abc is the least common multiple of ab and be. In algebra this is often called the lowest common multiple. 67. Finding the least common multiple. — The l.c.m. is usually found by factoring. For example, to find the l.c.m. of az + ay and bz + by, we have: az-+ay=a(x+y), and br + by=b(z-+ y); therefore. the l].c.m. must contain the factors a, b, x + y, and is ab(x + y), or abx + aby. WRITTEN EXERCISE Find the l.c.m. of the following: 1. abz, bey. Pde Coie HO 3. px, gary. 4. pgr, q’s. 5. mnw*, npry. 6. ab?c, a*be?. 7 at, ax + be. 8. yp? + pq, 9 + Gp. 9. abx + aby, abe. 10. 27 a® — 27 0°, 9 ab. 11. a(x — y), bx — by. 12. cx(m+n), y(m+n). 13. 41 a? + 82 b?, 123 ad. 14. m®n + mn®, mp + np. 15. am + abn, m* + bmn. 16. 2°43 2%y, 3 my + mx. o4 FACTORS AND MULTIPLES WRITTEN EXERCISE Factor in Exs. 1-22: . 39 abe + 65 abc*. 2. m>n*p + mn7p'. . 17 u’v — 119 we? 4. pgr — gqrs + rst. e+ 3xe*y+5 ay”. 6. 252 ay? + 84 ay. .a+5a% —4 ab, 8. 86 ay + 129 x7. . abed + bede + defg. 10. « — 32a*y 4+ 3a”. 11. 27 amn + 108 mnp*q. 12. 57 m'n? + 95 m?n’. 13. m?np + mn?p — mnp*. 14. m* + m?n + 4 mn”. 15. pi—-3p?—Apt+ 5p. 16. 627+ 82y + 102°. 17. — p’g?r — py’? — par. 18. 82 p®g?r — 123 p?q??*. 19. 42a7b —65.ab?+168 abe. 20. 32 22 + 72 xy? — 128 x. 21. 37 p’¢ +185 p7r—TA4pg. 22. —82°y2—627y22—4ayz Oo F oO WO — Find any multiple of the quantities in Exs. 25-31: 23. pr. 24. 3¢q*re. PA Leder Pies 26. a+. 27. “1 — y. 28. m? — n?. Pas ahi vinta bo mt 30.-07 Zab b* 3h. a — o0--< find any common multiple in Hxs. 32-37: 32. ab, 3 be. 33. 2 pq, 3 qr. 34..°, n°, 2:7. 35. a*, a? — 6. 36. m*, m + n. 3722070, 6 abt Find the lem. in Exs. 88-49: 38. 2 a7b, 4 ab? 39. a* — b, 15 ab. 40. a+b, 25 ab. 41. p+ 2p, 27 p*. 42. ab, be, ed, de. 43. 15 pq?r, 20 nar. 44. 32abe, 2a+6. 45. 2a, 36, 4e, 5d. 46. abc, bed,a+b+e. 47. 15 p??r,ptatr. 48. a? + 2ab + 0, 3 a? 49. 2a? — 6ab+ 26% 2a. FRACTIONS aY9) FRACTIONS ORAL EXERCISE 1. What are the terms of the fraction 2? Which is 3 the numerator? Which is the denominator? Answer the same questions for the fraction A 2. In the fraction 3, into how many equal parts has the unit been divided, and how many have been taken? Answer the same questions for the fraction = 3. If we think of 18 as an expression of division, which number is the dividend? Which is the divisor? Answer the same questions for the fraction ; 68. A fraction.—One or more of the equal parts of any unit is called a fraction. A fraction may also be considered as an expression of division. | Thus, 2 means 2 of the 3 equal parts of 1, or it means that 2 has been divided into 3 equal parts. ; means a of the b equal parts of 1, or it means that a has been divided into b equal parts. 69. Terms of a fraction.— The terms numerator and denom- inator are used as in arithmetic. 70. Integer. — An algebraic quantity in which no fraction is expressed is said to be an integer, or to be integral. For example, az + 0 is an integer, and is integral. 71. Sign of the fraction.— The sign written before the fraction is called the sign of the fraction. a a ; : bp For example, — 2 is a negative fraction, and 4p 5 positive fraction. 56 FRACTIONS ORAL EXERCISE 1. In reducing }8 to lowest terms what factor is canceled? What is the result? Answer these questions for ari 2. Reduce each of these fractions to lowest terms: mx abu? A x*y 14 pgr my aby 6 xy 21 p*qs 72. Reducing fractions to lowest terms. — Fractions are treated in algebra just as in arithmetic. Jractions are reduced to lowest terms by canceling all SORES common to numerator and denominator, me pe ~_ 18 reduced to Bihay Thus, the fraction : by canceling the only common factor, z. WRITTEN EXERCISE Reduce the following to lowest terms: 2h a? 2 3 hy ee get eee gue a b?a bx2y? wy? arx*y a abc? 5. eseedh # 16 a*x*y | a®b’ec 3d any? 36 alyz? aa? + ay? ary + ye par + girs 1. — 8. ———_—_-- 9. ————"—__- Bs axy pars 2G 0 51 m?n+ 85 mn? ote oO OU au 4a at 17 mn te axe) 13 a? ag 8 a2h =k 3 ab? 1 pg + p97 + p97 +. py ab par mn + mn + mn? 81 vty +108 x74? + 207 x y* ao mn i 27 xy USES OF FRACTIONS OT WRITTEN EXERCISE 1. If the perimeter of a square is 16, what is each side? What isthearea? If the perimeter is p, what is each side? What is then the area? 2. If the perimeter of a rectangle 16 in. long is 50 in., what is the area? Suppose the perimeter is p and the length is 2? | 3. If a piece of silk of a yards is worth 6 dollars, how much will an employee of the store have to pay for a yard, allowing him a discount of ¢ cents a yard? What is the resulvaita = 40.-bi=2:60,6 = 257 4. A car wheel is 6ft. in circumference. How many revolutions does it make in going 60 ft.? in going d ft. ? If it is ¢ ft. in circumference, the number of revolutions, 7, in going d ft.,is how many? That is, what does 7 equal ? 5. If a barrel weighing w lb. is rolled up an incline s ft. long, to a point d ft. high, a power of we lb. is exerted; that is, pa. How much power at f be used to roll a 200-lb. barrel up a 10-ft. ae incline to a height of 4 ft.? 6. How much power must be used to roll a 100-1b. barrel up an 8-ft. incline to a height of 4 ft.? also a 150-lb. barrel up a 12-ft. incline to a height of 2 ft.? also a 300-lb. barrel up a 15-ft. incline to a height of 5 ft. ? 7. Two cogwheels, one having 9 cogs and the other 27, are fitted together. How many times will the smaller wheel turn for each turn of the larger? How many times, if the larger has 10 cogs and the smaller 5? if the larger has a and the smaller 5? 08 FRACTIONS ORAL EXERCISE 2 ; 1. How do we reduce 3 to sixths ? . to dcths ? 2. Reduce to fractions with denominator abzx: =) 5 = x a 3. Reduce to fractions having the least common denom- = and 35 andi: ; 2 4 15 — inator: > and:= 53 3 5 5 5) 73. Reduction of fractions. — As in arithmetic, Both terms of a fraction may be multiplied, or both divided, by the same quantity without changing the value of the fraction. 2 LO Ue ee cleus OC me See For example, -~=—, —=-, -=—, —=-- See Eee AI ee i ere] 74. Least common denominator.— The least denominator common to several fractions is called their least common denominator (1.c.d.). This is also, in algebra, called the lowest common denominator. For example, the l.c.d. of oe ot and ms is bac?. pars: ce 75. Finding the l.c.d.— As in arithmetic, the l.c.d. is evidently. the l.c.m. of the denominators. | a For example, to reduce ; the l.c.d. ahd 2b The l.c.d. must evidently contain the factors x + y, 2, b, 4 (which contains 2), and 6? (which contains b). It is there- fore 4 b? aoe a). peices both terms 55 Lie yy of by 4 8, we = by 2 2b(x + y), and m m(z+y) as 402 40? (rey) of = by x + y, we have the results. and ar to fractions having 2 Oye ae) 4.007 aty 4P(+y) a 2ab(x+ y) REDUCTION 59 WRITTEN EXERCISE Reduce to fractions having the denominator indicated in parentheses in Hxs. 1-10: a p * e SS 4 8 e e 2 2 e 1 3° § b°) 2 apr (L297) m ‘ory x 3. ep (m?n®p*), 4. Bo (2 pq’). & 2 ap fe 2 5. pare (d -t be). 6. oa Fay (2 (a 2 qr). a+b taal 2 Qayd pplad 7. mas. be ence 8. Pate (x?y72? + yz), m 9. Apna (ab?m? — ab?n’?), x 1035. oe + Oe 6 wt). w— 2 we + 2 Reduce to fractions having the Le.d.: Gh ¢ Cee Tia es: ee ee we bc a 2b 2n bn 1 1 1 13. Ce ae ee 14. ie 2 2,2 a a2 a? gq? 7? jemi fi : fishy 2 eee i oy o Tr ee kee ts ee 3 ah ee 7. ———, ieee Ae Wet | 1 pqr. ars. rst GoatD (ae 0 (ieee Dad gn oe 19, ee 72 oe hah, cA a+b b Bb ak BLT & a i 22. af ) we ’ ie . a—b+e b e¢ dspqr Oagrs Ipqs AS ae ip 6 Ui eee Vee eae CREST. Fu 2ab 4c Ors 60 FRACTIONS as ORAL EXERCISE | 1. How many fifths in 1? in 12? in 2? in 24? 2. How many Uths in 1? in 3? inm? inm+ ae 3. Reduce a to aths; a + ; to yths; 2° + . to y*ths. 76, Mixed quantities. — As in arithmetic we speak of 23 a mixed number, so in algebra a +5 is a mixed quantity. 77. Reduction to fractional forms. — Mixed quantities in algebra are reduced in the same way as in arithmetic. 1 2 * 25. Arithmetic: Algebra: Reduce 3¢ to fifths. Reduce a2 + to dths. Since 1 = 2, therefore3 =15. .1. Sincel = us therefore x= me seg Goes sae, Te 1s S a>va d WRITTEN EXERCISE Reduce to fractional forms: 1 3. 5. it; 2 +7 to gths. 2. 3m? + = to kths. 2a% +0+5 to bths. 4. a? —3ab + 0 to dths. m+ 3mn+2n* to nths. 6. + Bary +9? +7 to yths. 7 mn Spi 4. .4mn+—-: p +39 8 4 min + a .4a4+2d4-. 10. 16.2? + 15.0y +2 BD garetts ante 12. 37 a? — 1508 4 464. 25 y 5a IMPROPER FRACTIONS 61 ORAL EXERCISE Zz 2 1. Reduce to whole numbers: 5; ue Bat. om 2a 8 a 5 3 2. Reduce to whole numbers: OL SO y SEOs nO a 3m? 3. Reduce to mixed numbers: 5. iF Cts 1, x3 ye ns a a 78. Remainders in division. — In algebra, as in arithmetic, a remainder in division leads to a fraction in the quotient. \ Arithmetic: Algebra: Divide 124 by 5. Divide «7+ 38x41 by a. 5) 124. LY eer aby 24, 4 remainder, x +3, 1 remainder, or 244. or xz (oe eee ex 79. Reduction of improper fractions. — Zo reduce an im- proper fraction to a whole or a mixed expression, divide the numerator by the denominator. For example, 2) +2a?+3¢+4+1 Spo We] } 1 Geteodet St tel es ho waa, w+2e4 +38 a ZL x WRITTEN EXERCISE Reduce to whole or mixed quantities: L 20+ 52° +1 n Pages % Pp 16m? + 32m+7_ 4 4dat+6a°+3 16 m DW fe Sa + 407+ 20+ 6. 6 12 a? + 60% +176 22 Ga? 62 FRACTIONS WRITTEN EXERCISE Reduce to fractions having the denominators indicated: 2m aa 2: 2 AC aoe) 3. ae (15). ae, (81). 5. se axa 6. imey rey b) (18 LYZ), eee é ee | atte, (49). 8, wat, (25a). 9. » (ayz). Reduce to fractions in their lowest terms in Exs. 10-18: A ng 21 a*b Lianne Ne 16 pq’ te 49 be a 51 mn? 125 abc? 2p? +3p 2 x Loboonathe eng pt2p Lae eens 2s " 3s0x%+ 35y 38m? $c eo Sear th 49 ot 9m? + 27 m Reduce to whole or mixed quantities in Kxs, 19-27: 2,,2 or 87,2 AS) 54 19, ey’. 225 alb™ 91, O28 RT. 9 xy 25 ab 125 p*q?* 4 b 2 3 = 5 5a ath a 81 a*y +19 04 32 +5. 9a 9 xy 32x 24 x? — 2: Ole e 5 PAT 9 24% ier 32 Py? + 15° 27, (x? +81 0+25 | 6% 8 xy 9 x Write the Sea in fractional form: 28. iy anes = 30. oboe Bt d y 32. 21 ot 17at + 33. 82 xy + 21 xy? + —. x y FRACTIONAL EQUATIONS 63 ORAL EXERCISE 1. If half of your weight is 40 1b., how much do you weigh? If 4 of x is 40, how much is x? 2. Find the value of x in each of the following statements : is ay 5 ec ed 5 = 15 3 = 50 3 = 10 Pier at : re 5 +1=11, what does 5 5 equal? What does x equal? 2 If = Bei i = 4, what does 2 equal? What does a equal? 4. If you have equal weights in the two pans of the scales, will they balance if you add 2 oz. to.each? subtract 2 oz. from each? multiply each by 2? divide each by 2? State four operations which you may perform on the two members of an equation without destroying the equality. 80. Multiplying equals by equals to avoid fractions. — In a= 1, if we ee these equals by 3 we the equation = vw ae e@= 21. Inthe equation * <= +6=8we have ~- oie = 2, ors =1; ; multiplying by 5, x = oe 5 81. Clearing an equation of fractions. — Multiplying both members by such a number as to make fractional terms integral is called clearing the equation of fractions. WRITTEN EXERCISE Find the value of x: 1 Z+2=9. 2. = +2= 30. 3. 45 =15, 32x oe 9a a 4. 7 t4= 1s. et, 2 6. aq 8 19. 20 5a 64 FRACTIONS J am thinking of a number whose half added to 71 amounts to 97. What is the number? tL x = the number, 2. Then > = half of the number. 3. Then = + “1 = 97, by the statement. 4. Then : = 26, by taking 71 from these two equals, and 5. : = 52, by multiplying these equals by 2. Check. If I add 71 to 26 (half of 52), the sum is 97. 82. Checking the result.— Always check by placing the result in the original statement. You may have made a mis- take in getting your equation as well as in solving. WRITTEN EXERCISE 1. What is that number 2 of which, plus 5, is 45? 2. What is that number 4 of which, less 6, is 10? 3. If I add 10 to 5% of a certain number, the result is 30. What is the number? 4. I'am thinking of a number such that its seventh less 6is 27. What is the number? 5. If I subtract 6 from 7% of a certain number, the result is 15. What is the number? 6. If to acertain number I add half of the same number, and then add 7, the result is 10. What is the number? 7. If from a certain number I take 4 of the same number, and then add 10, the result is 30. What is the number? 8. Make up a problem somewhat like the above, and then solve it. Teachers will find that the pupils will derive much benefit from making original problems as here suggested. ADDITION OF FRACTIONS 65 ADDITION OF FRACTIONS ORAL EXERCISE Dan wi 32. cee b il Add ~ and ~; 7 and = ; and BeAdd ahd <: - “vende eee and x wx x x 83. Addition of fractions. — Fractions are added in algebra in the same way as in arithmetic, by first reducing to the least common denominator. Arithmetic: Algebra: 3 1 a c Add — and =: Add — and —: tices G pen Gol 1. The least common denom- 1. The least common denom- inator is evidently 12. inator is evidently bed. 2. a 9. eae uh iy. be bed 1 2 2 and == =~. and ee 6 11 bd bed it 1+ 2 oO. Lhe. sum, = ——- 3. The sum = AOS : 12 bed WRITTEN EXERCISE a6 mn ne oe ai 8 abe abe e abe b?ed 3. — . reo one =: yz way 2py 8 py 5 pimp 4 ab 6 C04 4b? ‘4m%a 3 mn? " 4 a¢ a? b ee ty 5 ebay? = 17 82/? ij SEE 0, aati pees fee Cc 2.0 6 mn 36 mn 66 SUBTRACTION OF FRACTIONS SUBTRACTION OF FRACTIONS ORAL EXERCISE Tie at ome eed Ae EER GR Fe p¢4yl 18 8a a tbe 3 ae w24eh OLIVE 26. 6° 2a y°>m 3m 84. Subtraction of fractions. — Fractions are subtracted in algebra in the same way as in arithmetic. Arithmetic: Algebra: 3 1 a+b het | From — take —=- From take . 4. 6 « 2¢ be 1. The least common de- 1. The least common denomina- nominator is evidently 12. tor is evidently 2 be. * 3. -9 Git0 7 Gab eb? 7 me Vee ov Te eRe. 1 2 a—b 2a-—-2b and i, d SS ee ee Tb) ee he Dihe 7 wb + b2—-2a42b 3. The difference = —. 3. The diff. =~ — : Bee 2 2 be 85. It should be noticed that the fraction bar has the same effect as parentheses, the 2a—2b in the above example being subtracted as explained on page 37. WRITTEN EXERCISE eee e TE oye ar jee 3 2S tS 4 Sab dab Sikes 107, 6 070 seca? 5 a@_a—db Re ered wi) b '3mn? 2 mn 21. 23. 25. 27. ADDITION AND SUBTRACTION WRITTEN EXERCISE Add and subtract as indicated: xr 2 ah ab? ey gale Ge 0d —— 0 Bab?) 3a a b a—b a? pec ee im aie sale abe aa Hig ON a LV a0 Dg log iran Yomneee = 1s abe bed tO yee hOncke Cctak th abe bed Clans 21 soot ard Siglo x y Zz a+2ab+b? a?—2ab+l? ab ab eee (eee eee ieee rad ee y Zz x pee a ee se PY qr UVW VUWX WLY LYe 2,2 Goi yz abe Migd SEe ee ripe ie tap py mg fi mn mn? TE Fae bx x oer ar mtn m—n 3s —— : im 30 nN 2 2 Bs 2 10. Porgy Ga! Pq’ qr Ce be eae Le En pie Se " a + ¥? a? — ¥? Ve AD Ay Cae Ore Co ade ne Ean ee eae 18. Car AM wie ab be Cu Qo (i= Lee a ey 20. erie ioe a ie 29. Le Wg ee b a 24. Garnnal TU TMA C 02 ac Oo 0 ee a ab? Nabe c7a? (U4 LY TROND View an cea ciel Se ETE 22w? 68 MULTIPLICATION OF FRACTIONS MULTIPLICATION OF FRACTIONS ORAL EXERCISE 1 er ae ee) Me 1. How much is twice 3° a times —? m times —? 2. How much is 5 times “eo 15 y b (Cancel.) m times = ve 3. How much is 10 times i (Cancel.) mn times ae ? 15 my 4. Tell how to multiply a fraction by an integer. 86. Multiplication of unit fractions. — It appears from the above exercise that we multiply fractions in algebra in the same way as in arithmetic. For example, consider unit functions: Arithmetic: Lee Take 3 of 5 We think of 1 as divided into 5 equal parts, and of each part as divided into 3 equal parts, so that 1 is divided into 5 x 3, or 15, equal parts. Algebra: yy i 1 Take b of rE We think of 1 as divided into d equal parts, and of each part as divided into 6 equal parts, so that 1 is divided into b- d, or bd, equal parts. Therefore x of M = rate Therefore A of L = sat OF SOLO bees bd WRITTEN EXERCISE 1 —b 2 loa: 2. — 2. —. 3. at hy LY e*d* 4 a a 1a) a—b Le re n 16ay cd MULTIPLICATION OF FRACTIONS 69 ORAL EXERCISE 1 How much is 5 of 5? 5g ? = b a Co , of 5 2. How do you multiply one fraction o another ? 87. Multiplication of fractions. — Compare arithmetic and algebra again, as on page 68. Arithmetic: nen Take 3 of 5 Because 4 has been divided into 5 equal parts, and each of these into 3, therefore 4 has been divided into 15 equal parts. 4. 1 Therefore 3 of — 5 Soe Beek 0 Oe Therefore = 0 5 Algebra: a ,e€ Take 5 of 7 Because c has been divided into d equal parts, and each of these into 6, therefore c has been divided into bd equal parts. Therefore — : of | = rake) b Phd here ore it —-=a: pep cae b ad bd bd 88. Therefore in algebra, as in arithmetic, To multiply one fraction by another, multiply the numera- tors for a new numerator and the denominators for a new denominator. 89. Cancellation should be employed whenever possible, as in ET es we “the case of. ts Hews a b WRITTEN EXERCISE a c? —@ —€ Cau ec Us Be de 2 ak nm nn A abe abc? Sap 3 YS abe xyz " m?nx mn? Saye -2ab eye abe ae 70 MULTIPLICATION OF FRACTIONS ORAL EXERCISE 1. At m miles in # hours, how far will a train go in GRU ge phakere Sina" Abaya Yipiininign pRIVI/!1ehe 2. At m miles in / hours, how far can you walk in ; Of an Hour ino years: IN 2 nr s 3. If y yards cost d dollars, what will 1 yd. cost? = yd.? m yd? 4. A man earns d dollars a week for working 6 days, h hours a day. How much is this per day? per wo _ 22 5. In a circle of circumference ce and diameter d, — a 7 foUnoo rer ue ? d 6. If a acres of land are worth d ee how much is 1 acre worth? 2 ofanacre? 3aacres? 25a? acres? What is the value of ae O WRITTEN EXERCISE 1. A man had of an acre in a village lot, and bought = ¢ of an acre adjoining. How much was his land then worth at d dollars an acre? 2. What is the value of the land in Ex. 1,if a=1, d = 2, ec = 300, d= 400? (Simply substitute in the result.) 3. One automobile can go m miles in f# hours, and another can go : as fast. What is the rate of the second one per hour? 4, What is the value of the result in Ex. 3, if m = 63, I reap Des CS SS 5. If m pupils are divided equally in ¢ classes, and if half are boys, how many boys in each class? 11. i 15. 18. a1. 522, 23. 24, MULTIPLICATION OF FRACTIONS Tl WRITTEN EXERCISE ab ce dt 9 (DE CO28 6): bc d at " @d ef? ab? mn pg 4 Ue eenee W -1 co in ns? pes wa ez a+b « xy*z . a—b ab mint wy yz 2 “mn mn? 8 Old (a0 00°C: : (meee Ce aec Oo Aab’c® 5 pq?r? eh Sai ae ae ee 10 17 a°b'c® §=259 min? Ty Ra ATI " 37m?n 119 a"! ie mene pane ge Re Oem Ey DT Te Te iG, i i ae ae LI ac MATA GIO yoke aoc ga Sh Thee aie ie TT LY z Se ieee et L 1 —-1 = 16. a%-— 19, — 5 mn mn n? ie 7 1 if —1 -—-1 1 1 abcd? a*be*d DOA a 007% Cay ap ay Go DD. 6 0 OO. ii eod oanbte. 87h p+atrts rs py —7s py pg 7st 8 e—2ey+y? ays — abc bei ihe — abe xyz ee Ot abe LYZ iat 12 DIVISION OF FRACTIONS DIVISION OF FRACTIONS ORAL EXERCISE 1. Because 2.3 = 6, we know that 6 + 2 = how many? Because a:b = ab, we know that ab + a = how many? © 2. Because — in = = at we know that at > 5 = = how many? 90. Division of fractions. — Because = = = e and also noe Js 4 we see that wa 6 a The result of dividing by a fraction is the same as that of multiplying by the aa enverted. me _m y Bet to divide = Wee —, we may mul- 7 7 e ao. tiply ~ by the result being — pds ee Nx First indicate the work and then cancel if possible, as follows: pease IESG OS EV oo 83) @ pt Sp @-pt-6q 2p? WRITTEN EXERCISE Th Ged “q_. mn —a’be — bed? ib, eee ee 2 ee Sy Sepa e be 6? mn pq? x ye Yz Cpa | eile en ee Stabe: = bea? 4. oe Oe Oe Sy reer gma c ab Py m: pqr pqr Eee ene pe gd ek ' 8m 2 xy? sais a(a—26) _ a®b a? +h? m* 2 2b 4o Mera. Se iL —mn —n?*m 12. a+ 2ab + UF ab. Lp) {Her ab a DIVISION OF FRACTIONS | 13 Ci Cometc meen?) m® 13. ---+-——: —— 5 —— De Tithe Si wm n ab be | abe mn np mnp Beeta Pe de 16g mg e ace. abc? " aryrz? ale? abe UE! Be ho Be Cee Cyan 7/2 x). Pg 3 gps ee Ve ald oes ee (; ‘) Bez Re eae 8 8° ab? 93 able ah a o4 27 a*b?c? 119 m?n? _3 ab Cae UG pd AW SES SY 25. see Ve YF, 26 tag! aL Ui Ps 8 6 TN OE py py apy ares ai A 98. (ett vir) 1 abed 4 Pq gr par (me Ce ke \OLUG facta ge (; c+$) Me. 4 30 vy —ye ww, — ww rae ut ay ye 31 ey te Pod Oo ; ry ye ape re Vee 0 Ce Semen mG ge cog ne 33. One train travels m miles in f# hours; another, d miles in ¢ hours. The first rate is how many times the second? 34. One man earns d dollars in w weeks, and another earns m dollars in ¢ weeks. The weekly wages of the first are how many times those of the second? 14 LINEAR EQUATIONS LINEAR EQUATIONS 91. Various kinds of equations. —There are many kinds of equations. The equation «+2=5 is very simple; 2%—7=6 is more difficult; 27+ 7x2=18 is still more difficult. 92. Linear equation. — An equation like 2 x — 7 = 6, in which the unknown quantity has no exponent except 1, is called a linear equation or a simple equation. The name linear is the more common in advanced work, but both names are used. The expression x! means the same as 2. 93. Solution of an equation. — To find the value of the unknown quantity is to solve the equation. 94. Axiom. — A statement assumed to be true is called an axiom. 95. Axioms needed in algebra. — The following include the axioms already used (page 4) in solving equations. Axiom 1. Jf equals are added to equals, the sums are equal. Thatiis, 1f ¢ — 3.— 7, then 7 = 7 + 30rd). Axiom 2. If equals are subtracted from equals, the remain- ders are equal. That 1s,1f ¢ + 3 = 9, then z = 9 — 3, or 6G. Axiom 3. If equals are multiplied by equals, the products are equal. That is, jf 1a = 5, then 7 = 4X 5, or 20. Axiom 4. If equals are divided by equals, the quotients are equal. Lhatis;i1i0 ¢ = 39, theiereeo 0 1, Orne Axiom 5. Quantities which are equal to the same quan- tity, or to equal quantities, are equal to each other. EQUATIONS ies) ORAL EXERCISE 1. Solve the equation 2 +2=5; alsor+2=08. Isb considered as known or as unknown? (See page 14, § 16.) 2. Solve the equationa +4=6; alsox+a=06. Solve the following equations, finding the value of x: 3. en = 4,.%—n=™M. 5. 3a = 6. 62.0 Gea. rE Cay Te See Onis. See = 14g. 105 na 3 i. Lio 4 2 a7, iA Se TR atten, 1B yon eek ies WE BE Pm hans eh WRITTEN EXERCISE 1. Solve lle+F=7 +46. moOlvert ae — oe oe + 21. . Solve 2e¢+3e+7=224 33. se 5 — 3000. Find the value of x: 2 3 4, 5. Solve 3(2 « + 200)= 1200. (First use Axiom 4.) 6. Find the value of « in the equation 52+10=32+4+ 32. 7. Find the number whose fifth and seventh together make 24. 8. Find the number whose half, third, and fourth together make 13. 9. Find the number which added to 4 equals 7 more than half the number. 10. A man saved half of his wages for 5 years. He then had saved $3000. What were his annual wages ? 11. A man by saving $200 more than } of his wages _ annually for 3 years, saved $1200. What were his annual wages ? : } 76 LINEAR EQUATIONS 96. How to solve equations. — We have seen that the solu- tion of a linear equation consists in arranging the x’s in one member and the known quantities in the other, and dividing by the coefficient of a. 97. Transposition. — The subtracting of terms from both sides of an equation so as to carry them from one side to the other, changing the sign, is called transposition. Thus, in the equation 8 — 2 = 12 — 42, we subtract 3 from both sides, leaving —x=12-—427-3. We then subtract — 42, or, what is the same thing, add 42, and —x+4xr=12 -3, or One so. whence ioe ts In this solution we transposed the 3 and the — 4z. 98. Solving by transposition. — Since we have now solved | so many equations that we can use the word understand- ingly, we may say that to solve a linear equation, Transpose the terms involving x to the left side, and the known terms to the right, and divide by the coefficient of x. ites x Thus, if — + 5. =~ " us, 1 oie rere Leenene 1 then, transposing, 5 = 2, or ge Le The coefficient of zis}. We may divide by }, or, what amounts to the same thing, multiply by 2, and ae Check, 2°-4+5=43-447. WRITTEN EXERCISE 1. 42¢%—1=2- 81. 2.1342 — 21 —427 — 3: 3. 122—-6=92+4 27. * 4.15¢—9=112+439. 5. To 2 — 32=—162-+ 86. 6. 252+8=162 + 89. EQUATIONS {yf (hs 2 Bo path 9. E42 = 69. 10 nef 11 M5 +3=249. 12 2FF 13 =48 13, = _=7 _ 49 14. 5+ 5+5=18 15, FF Sa 16 22 2 _ 135 lie 1 9 — bo.) 18) 9 F631 19, 24° == 31. 20, “= 4 2th 21. x +10 %x = 605. 22. « +120," = 14.56. gg, =F _ ett __o ag, “EE _ Sty a5, “Hy ts ag, 4 FSS _ 88, 27. Find a number whose seventh part minus its eleventh part equals 4. 28. Find the number whose third plus 7 equals the number less 3. 29. Find a number which, when subtracted from 88, equals the number less 17. 30. A man lost 32% of his capital and then had $4896. How much had he at first ? 31. A man gained 15% on his capital and then had $8625. How much had he at first? 32. A man has 27% of his capital invested in a farm. The farm is worth $1917. How much is his capital? . 78 LINEAR EQUATIONS 33. Find a number such that $ of it less 2 of it equals 19. 34. What sum increased by 1% of itself ‘amounts to $2585.60 ? | 35. Find a number which increased by 17% of itself, and then decreased by 36, equals 198. 36. A dealer sold some goods for $2802.40, thus making a profit of 13%. What did the goods cost him? 37. A man lost 30% of his library by fire. He had 630 books left. How many had he before the fire? 38. A collection agency charges 4% for its services in col- lecting a debt, and remits $912. How much did it collect? 39. The sum of a certain number, a third of the number, a fourth of the number, less 7, equals 4,4. What is the number ? 40. A bank charges 0.1% exchange on a draft. The entire cost of draft and exchange is $1751.75. What is the face of the draft? 41. An agent charges 5% for collecting rents for Mr. Glover. He deducts his commission and remits $332.50. How much did he collect? | 42. Half of the remainder found by subtracting 7 from a certain number equals a fourth of the sum of the number and 7. What is the number? 43. An agent bought a building lot for Mr. Roberts, charging him 3% commission. Mr. Roberts sent him the price of the lot and commission, amounting to $2626.50. What did the lot cost? 44. The income of a certain store increased 163% the second year it was open, and the income the third year was 25% more than the second year. The income being $3500 the third year, what was it the first year ? CHAPTER II OPERATIONS CONTINUED. FACTORING. PROPORTION. EQUATIONS MULTIPLICATION ORAL EXERCISE 1. Multiply x by a; by 6. What is the sum? 2. Multiply m+n by a; by &. The sum is the product of m+n by what expression ? 3. How can you multiply any polynomial by a+ 6? (Multiply first by a; then by what term? Then what should be done?) 99. Multiplying by a binomial. — Multiplication by a bino- mial in algebra is much like multiplication by a two-figure number in arithmetic. Arithmetic: Algebra: Multiply 45 by 23. Multiply a+ 2b bya+6 45 Cited 25 ‘ Oa 20 135 product by 3 a* + 2ab product by a 900 « 20 ab + 2 62% eee, M55,“ 28 Oo a0 2.02 Sata 4. Why do you begin at the right to multiply in arith- metic? Why may you begin at the left in algebra? Could you as easily begin at the right in algebra? Try both plans on the blackboard. 79 80 MULTIPLICATION ORAL EXERCISE 1. Multiply 4a° by 5a*; —3a* by 6a°; —5a? by — 42%. 2. State the product of a -—a’?-2a-—38a; also of 327-—4a4".—2@. 3. Tell how you proceed to multiply by a+ 6; by 2a—); by 4a? — 36°; by any binomial. WRITTEN EXERCISE Multiply in Evs. 1-20: l.a+bdby a+b. 2.a+ybya+y. 3. m+n? by m+ n?. 4.2a+bby 2a+0. 5. a—bby a—d. 6.2—ybyx—y. — 7. 3m—n by 3m—n. Sasa by Ota 9. a+b by a—b. 10.a—bbya+o. ll. x—2ybyx+2y. 12. m?+ 3n by m?—3n. 13. 2a+36b by a—26. 14. Tx?y?+1 by 2 277?—3. 15. 627+1 by 52? —3. 16. 4mn+ayby3mn+4ay. 17.a+b+cbya+b. 18. 77+ 2ay+y* by «+y. 19. m?—2mn+n? by m—n. 20. 4a?+4ab+0? by 2a+b. 21. What is the product of 27 and 23? of a+6 and a—b? Suppose a = 25 and b= 2? 22. How many square feet in a square that is 42 ft. ona side? How many square feet in one that is Vos t ft. on a side? Suppose f=40 and t=2? 23. If a man earns $27 a week for 27 weeks, how much does he earn in all? If he earns ¢+s dollars a week, how much does he earn in ¢+s weeks? 24. The product («@+y) («—y)=2?—y*. Hence write down, without multiplying, the following products : (10+ 2) (10 — 2); 12.8; (20+ 1)(20 —1); 21-19. SQUARE OF A BINOMIAL 81 ORAL EXERCISE 1. State the product of a+ 6 by a; by &. Add them. 2. Write on the board the product of x+ybya+y. How are the terms formed from x and y? 100. Squaring a + b, a — b.— Consider the squares of a-_o and of a — 0. The square of a + b. The square of a — 6. a+ 06 se Say Gare 0 Cano arts ab product by a a? — ab productby a ab + O26 “0 — ab+h « 2 Bap Aa GUT * a0 C2 00S Dae « a—b 101. Square of any binomial. — It is therefore seen that the first term of the product is the square of the first term of the binomial. The second is twice the product of the two terms (negative when one of them is). The third is the square of the last term of the binomial. Therefore The square of a binomial equals the sum of the square of the first term, twice the product of the two terms, and the square of the second term. 3. State the squares of b+c; of d+c;o0f 2+a; of 27+1. 4. State the squares of a—a; of x—y; of a—2; of a? —-y’. WRITTEN EXERCISE Write out the squares of the following without multiplying : 1. p+g@. Pane y ieee 18 Oe ea ty 4, x7? —t. 5. be — x. 6. 26—d. 7 ap. 8. 2a+0. 9. mn +1. 10. mn? + p. 11. ayz —1. 12. por + 2. 13. 2a+ 36. 14. 3a—20. 15. 5ay+1. 82 MULTIPLICATION ORAL EXERCISE State rapidly the squares of the following: l. ptr. 20 at 3. 0 +a. 4.9g—h. Spice SES Ta 6. an 9. 7. «2 — 7. 8. d? — b?. vas OT a 10. xyz + 1. 11. xyz — 3. 12ers a WRITTEN EXERCISE Write out the results without stopping to multiply: 1. (ab +c). 2. (38 — ¥°). Sue (ied) oo 4. (2— 2°)? 5. (a+ 5c)? 6. 4(a@+y)”. Cate). 8. 1 —7q)’. 1 el (ic bia bak LO(pe tL) 11. (8a + 2)?. 12. (6a — b)?. 13. (@ —11)*. 14. (xy? + 1)? 15. (x —Ty)?. 16. (2a — b)*. Ling 22) See Bate), 19. (a+ 7b)? 20. 10—3y)%, 21. (way +2)? 22. (20% +41). 23. (80x%—1)7. 24. (8m? +4 5)” 25. (xyz +10)% 26. (abed +1)% 27. 11 +22)%. 28. 100(@+y)% 29. 2Qu+2y)?, 30. d0x+ 10y)? 102. Pictures of squares. — In the figure point to the line that equals «+ y. Point to the area «?; to vad vt an area xy; to another area xy; to y*. What then does the square on « + y equal? 31. Draw a figure showing the square on 104 2. 32. Draw a figure showing the square on 10 + 1. 33. Draw a figure showing the square on 2” + y. 34. Draw a figure showing the square on 3x + a, PRODUCT OF SUM AND DIFFERENCE 83 103. Product of a + b and a — b. — Another product fre- quently met is that of the sum and difference of twe quantities. a+b @—b GD 0 ie 0 C00 — ab — b? C0 ie — } a? — } That is, the product of the sum and difference of two quantities equals the difference of their squares. WRITTEN EXERCISE Write the following products without stopping to mul- tiply : 1. («+1)@—1). 2. (1+a)(1—a). 3. (a? —1) (a? +1). 4. (ab —1) (ab +1). 5. (abe + 2) (abe — 2). 6. 22+") (2x2 —y?"). 7. (1+ mx?) (1 — mz’). 8. (p?>+4q)(p?—49). 9. (8 —42°)(8 + 42°). 10. (Sabe — d) (5abe + d). 11. 10a?—1)(10a?+1). 12. 2a?+11) (12a? —11). 13. (Sayz +7) (Sxyz —T). 14. 100m + 3) 100 m — 3). 15. 42+7)42 —7);19-5. 16. d1 —4) (11+ 4); 7-15. 17. (xyz + 1)”. 18. (7 —2 xy)’. 19. (327+ 4) 20. (84 2 pr)’. 21. (8mn—T)*. 22. (2 mn — 3). 23. (104+ 3mn)*, 24. (aye? 2)?, 25. (0274 3 y)?. 26. First write the product of the two binomials; then square the result: [(a + 0) (a — d)}?. 27. In the same way write the results of the following: [ (m?+1) (m?—1)]?; [(2a—3) (224+3)]?; [(1—-5a)A+5a)/. 84 MULTIPLICATION 104. The product of two binomials. — The product of two binomials like « + 2 and 2 + 5 is so frequently needed as to require attention. Consider two such cases: ge +2 a2 —T ee) z+3 ue? +24 a? — 7x 52+ 10 32 — 21 a’+T7Ta2+10 g—4y— 21 ORAL EXERCISE 1. In the first product how is the 7 (the coefficient of x) formed from the 2 and 5? How is the coefficient of x formed in the second product? 2. How is the last term formed in each of the products ? Can you now tell the product of «+5 and «+ 5 without actually multiplying? © 105. Absolute term.— In the expression «?+ 7x + 10, 10 is called the absolute term. 106. (x + a)(x + b).— Zhe product of x +a and x + b is x? plus (a + b)x plus ab. WRITTEN EXERCISE Write out the products without stopping to multiply: 1. (0 +7) (x +1). 2. (7@+1)(@—1). © 3. (a + 10) (a —7). 4. (pg + 8) (pq + 9). 5. (xy — 3) (ay — 3). 6. (8a + 5) (ba — 5). 7. (mn — 11) (mn + 1). 8. (mn + 6) (mn + 6). 9. (p’¢ + 7) (p’¢ + 10). 10. (abc + 6) (abe — 7). 11. (xyz + 20) (xyz — 5). 12. (xyz + 15) (ayz + 2). CHECKS 85 107. Checks on multiplication. — Because the product of a+aand «+6 is «74+(a+6)x+ ab, whatever values may be given to 2, a, and 6, we may check our work by giving any convenient values to these letters. For example, to check (# + 3) (# — 2) = 22 + x — 6. Let c=1. Then (1+3)(1—-2)=4--1=~—4, and 2 +1 — 6 =— 4; so the work is probably correct. ' Try alsox=5, Then (5+ 38)(5—2)=8-3 = 24, and 52+ 5 —6=25+5-6= 24. 108. Ease of checks. —It is usually easier to check the work than to look at an answer in a book. Work: Chieti ete yi be 2e¢+ 3y == 5 do — Ty =— 9 8x? + 12 xy —15 —14ay— 217 8a27— 2aey—21y =—165 WRITTEN EXERCISE Multiply and check: 1. (23 x + 21) (« — 17). 2. (21a + 37) (9a — 14). 3. (x? + 15 y) (a? + 14y). 4, (15%? + 7) (17 x? — 8). 5. (l7x+2y)A5xe—6y). 6. (85%+4 23y) (15% —Ty). 7. ily—22)(2y—112). 8. (31 xyz + 2) AT xyz — 7). 9. (15 a*b?c + 7) (9 ab*c — 8). 10. (27 xy? + 1) (80 xy? — 3). 11. (322 + 3y)(— Alx+8y). 12. (231 a + 147) (3829 a — 176). 13. (321 a? + 17 6) (151 a — 55). 86 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. MULTIPLICATION (a + 0) (a? — ab + 0°). (a — b) (a? + ab + 0”). (a + b) (a? + 2 ab + 0%). (a — b) (a? — 2ab + 0°). (a + b) (a — b) (a? + 0°). (81 abe — 1) (91 abe + 17). (62 abe + 1) (78 abe + 3). (2% — 3) (427—Tx-+ 2). (4a —1)(82?— 22443). (42 a — 37 b) (51a — 170). (111 ax +- 21) (97 a*x + 2). (4x? —4 ay + y”) (2% — y). (9a* + Gay +") (Sa +). (a+ 3b) (a+4b)(a+ 50). (172 a?y? + 27) (172 ay? — 2%). (42 p?q?r? + 8?) (17 p?q?r? — s?). (a+ 6) (a+ b)(a+ 6) (a+). (Tx + 2y)(9a? +4 xy —6y?"). (a*b?c? — 2 abed + d?) (abe — d). (ax? + by?) (ax* + abx?y? + by). (a + b) (a + 30% + 3 ab? + 0°), (a®b’c + 1) (a%b*c? + 2 a®b?c + 1). (8% —3y) (Gx? — Tay + 50y?). (x — y) (#® — 3 xy + 3axy? — y’). (21 x? — 0 (152 a* + 73 x? — 41). (— 27 a?y? — 2?) (— 21 x7? — 2). (208 —Ty)(9a° + 15 ay + 8 y?). (6a + 7b) (36 a + 84 ab + 4907). REVIEW 87 109. Illustrative problem. — At what rate will $320 pro- duce $19.20 interest in two years? 1. Ii the rate for 1 yr. is r%, for 2 yr. it is 27%. 2. Since 27r% of $320 = $19.20, 9) 1 = 4 of 819-20 $320 Sie WRITTEN EXERCISE 1. What per cent of $75 is $3.75? 2. What per cent of $15.50 is $0.62? 3. If x% of $156 is $4.68, find the value of z. 4. If x% of $730 is $65.70, find the value of x. 5. What sum increased by 6% of itself equals $1007? 6. On what sum is $11.25 the interest for 1 year at 41%? at 5%? at 3%? at 2%? 7. At what rate of interest will $450 produce $15.75 in 1 year? in 2 years? in 9 months? 8. At what rate will $240 produce $28.80 interest in 2 years? in 3 years? in 4 years? 9. How long will it take $350 and interest to amount to $386.75 at 34%? at 5%? at 4? 10. At what rate will $260 and interest amount to $282.75 in 3} years? in 5 years? in 2} years? 11. What sum will amount, with interest, to $381.50 in Z years at 44%? at 5%? at 6%? at 2%? 12. A man lends $250 for a year at a certain per cent, and the next year he lends the $250 and the first year’s interest at the same rate. The sum of principal and interest at the end of the first year is $260. What is the rate? What is the sum at the end of the second year? 88 FACTORING FACTORING ORAL EXERCISE 1. What are the factors of 15? of ab? of 25? of a?? 2. Name two factors of 12. Are they prime factors ? If not, state the prime factors of 12. 3. What is the product of a+ 6 and a+b? What are the factors of a? ++ 2ab+ 67? of a+ 2axy + 7? 4. What is the product of «—y and x—y? What are the factors of a? —2ay-+ 47? of m*—2mn-+n?? 5. What is the product of m+n and m—n? What are the factors of m? — n?? of p? — q?? of 4 — a?? 6. What is the product of «+2 and «+3? What are the factors of 274+ 54 4.6? of a7b7+ 5 ab +6? 110. How to factor expressions. — Factoring always depends upon remembering types of products. 111. Monomial factors. — Because we remember that x(y + 2)= ay + xz, it follows that the factors of xy + xz are candy+z2. That is, xy + xz = x(y + 2). WRITTEN EXERCISE Factor the following : laa: 2. am + mb. 3. e7+ 2 xy. 4. py — qr. 5. a*y + yx. 6. p* — 3p. 7. abe + bed. 8. ax* + axy. 9. ab+ dbe. 10. 3pq? + 6¢*. 11. mx? — nxy. 12. c6 + 3c?d?. 13. 2 x*y + 6 2. 14. abry + wxyz. 15. may + nxz + que. 16. 32 ay? 4+ 8ayz +4 xyz. USES OF FACTORING 89 ORAL EXERCISE 5 16 ab ib atin OE Ge ; 29 a(atb) wx(a+1) ab(m+n BuPATso the following: at aah aniy ee. 3. How do you ordinarily reduce fractions to lowest terms? Give three illustrations. 1. Reduce to lowest terms these fractions : 112. Uses of factoring. — One of the chief applications of factoring is in the reduction of fractions to lowest terms, so that the fractions may be used more easily. e+3nr e+4z CeO er L(g apt 9 e+4e x(@t+4) 2th = Ts to lowest terms. q to lowest terms. For example, to reduce the fraction Also, to reduce the fraction ie ee a ing AN ay—Ty y(e—7T) ¥ by factoring and then canceling « — 7 from both terms. WRITTEN EXERCISE Reduce to lowest terms the following fractions: L eae 9°. mas coe 3. an ab m* — ™m x“ — 2x a+a x + xy m? + mn 427+ 62 4, ———__+: So oe Uo aecewersapar so xy + zx n? + mn 2ay+8u 2pqa+¢q? 3mx + x 81 — 27y ‘1 8. ——-———_ 9, ———___-.. 2pr+ rq y +3 my 63 +18 y ae _ 2 22 i oie CRN PUT pegs abe — 7 be ' ga —3 yz wt 3 pq? + par 90 FACTORING ORAL EXERCISE 1. What is the square of a+ 0? of a—b? 2. Squarea+y; x—y; 2x+1; 1—3y; ab+cd. 3. What quantity squared equals a+ 2ab+0?? 2? —2Zay+y?? 4¢+4e44+1? at4+22?41? 4. What are the two equal factors of m*—2m4+1? of xy? +2ay+1? a@+t4ab4+4b?? at —2a?+41? 113. Squares of binomials. — Because we know that (a + y)? =2+2aey+y’, and («—y)?=2?—2ay+y’, it follows that the factors of these trinomials are known: a ree ae pS ae Napa x? — 2xy + y® = (x — y)’. For example, the factors of 4922+ 14241 are 7x +1 and Ve 1 or A) to Ae gen ( da) Are ( Cae == (1-214). So the factors of 9 22 — 12 ay + 4y2 are 3x — 2yand 32 —- 2y. For a2 eyed? (0 2) (OD) 2) ee a CMF oe PO be WRITTEN EXERCISE Factor the following : Gr 8 gehen Pee 2 a2? + 2aen + n’. 3. ee 41. 2 ae. 5, Agee 4 el |e 6, 145744 * 4. 9276741; 8. 1627 — Sze + 1. ° 9. 277? + 2 aye + 2 10. a*b? — 2 abed + ¢?d?. 11. pt? + 20 pg + 100. 12. 6%? +121 + 22 abe. 13. 9a* + 42 a*l? + 49 4. 14. 36.x* + 60 xy? -+ 25 y*, DIFFERENCE OF SQUARES 91 ORAL EXERCISE 1. What is the product of a+ banda—b? What are the factors of a? — b?? of a? — y?? of m*—n?? 2. What is the product of 2~7+1and2x2—1? What are the factors of 4a?—1? of 4m?—1? of 1—4a?? 3. What are the factors of the difference of two squares ? 114. Difference of squares. — Because we know that (a + 7) (x — y) =x? — y’, it follows that the factors of the difference of the squares of two quantities are the sum and difference of the quantities. x? — y? = (x + y)(x—y). For example, the factors of 25 #4 — 121 y? are5 22+ 11y and pe Ly HOT De gtee 1Aley ipa (eA TRG yy (ae ey) ee hy) Similarly, (e2—y)?-4=@-yt+2)@-y-2). WRITTEN EXERCISE Factor the following : 1. p?—4¢". 2 2a- = 4b? 3. 1644-7”. 4. 64 m‘n? — 1. diye sake 6. 1 — 100 abc? 7. (a+ db)? — e’. 8. Zax +a*?+ 27. 9. 36 a? — 25 0". 10. 144 m? — 25 n?. 11. 36 p?q* — 121. 12.0 Slt 25 7 07e70 13a? 4-2 ab +- 07. 14. 25 ab? — 25 b’c?. 15. va* + be + cry. 16. 4m2n+ n?+ 4 m* 17/564 a4? — 31 o7d*. 18. a2 = 4 ab + 407. 16: 19. a? +2ab4 0? —4. 20. 1+6(@+y)+9(@+y)*. 92 FACTORING ORAL EXERCISE 1. What is the product of ¢+1anda+2? What are the factors of 27+ 3a” + 2? 2. What is the product of x—3anda+4? What are the factors of x7 + #—12? of w+ ay —12y?? 3. Of what expression are x + 5 and « + 4 the factors? alsoxz—VTande+44? alsox+l and a—8? 115. The product of two binomials. — «a + b Because we know that (x + a) (x + 6) x 4 a =x*+(a+ 6)x + ab, it follows that a bx we can often tell the factors of expres- ax + ab sions like x? + (a + b)x + ab e7-- 7¢ —18 and 27+ ba +6. For example, the factors of 2? + 7x—18 are (x+ 9)(x— 2). For w+ Te—18= 22+ (9 — 2)\a -- 9(— 2) ee ED G3 ta PE That is, 9 and —2 are two numbers which added make the coefficient of x, and multiplied make the third term. WRITTEN EXERCISE Factor the expressions in Hrs. 1-14: Vea A 2a st eee 10: 3. 2 —5a4+6. AG ee os 5. p?>+18yp-+ 81. 653 ga 8, 1. TRL 2 ee, 8. p* — 20 p + 100. 9. pg? +13 pq + 36. 10. a*b* + 6 ab — 40. 11. a?y?z? + 11 wyz + 24. 12. m?n? +17 mn + 72. 13. mn? — mp* — g?m. 14. (a + b)? —(e + d)*. 15. Of what expression are ab? — 4cd? and ab? + 17 ed? the factors ? REDUCTION OF FRACTIONS ORAL Reduce to lowest terms: 1. a+26 ; G4 O m+n m? — n? a(b+ Cc) 6? — ¢? he ae I es Apes ite oy x2—2ey+y EXERCISE 2. aaa m(m — 1) m= —" 1. peeks eee "e+ Q2ay + y? ae (tary aie x?—2aey+y? a? — ¥? roam 93 116. Reduction of fractions. — As usual, in reducing frac- tions to lowest terms, jist factor, then cancel common factors. WRITTEN EXERCISE Reduce to lowest terms: or 10) be Coe 2ab+06? O70 = ab ie re x? — 52+ 6. ee Oe — 3 a 00 OF pt —8p+12 ara + ya? + are (a + y)c? + cz 1— 15% + 562° 1—17 24+ 7227 a + a —12 gt ig 220 a+2a— 8 a*7t+6a+8 (4 + 6)*—1- eee eee. ipa a ea 72 —— 3 e+ 2 ge? — 22% + 40 4m? —n* 10. ag Aon? eed mn 94 REVIEW WRITTEN EXERCISE 1. Solve the equation 3 «+9 = 5a —65. 2. What sum increased by 38% of itself amounts to $785.89 ? 3. What sum decreased by 7% of itself is reduced to $709.59 ? 4. The cost of a draft, less the discount at 0.1%, is $9740.25. What is the face? 5. The cost of a draft, including the premium at 0.1%, is $10,760.75. What is the face? 6. A man offered $7790 for a farm, which was 5% less than the asking price. What was the asking price? 7. A man pays an agent $7725 for buying a house, which includes the agent’s commission of $225. What was the per cent of commission? 8. A man pays an agent $5610 for buying a house, which includes the agent’s commission of 2%. What did the agent pay for the house? 9. A man bought two horses at the same price each. He sold the two for $198.90, gaining 11% on one and 10% on the other. What did he pay for them? 10. A man bought two horses at the same price each. He sold the two for $163.20, gaining 10% on one and los- ing 6% on the other. What did he pay for them? 11. A man increases his original capital by 7%, and the next year he decreases what he then had by 10%. He then had $8667. What was his original capital? 12. A mine increased its income 11% in one year, and the next year it increased this new income 10%. The income then amounted to $335,775 a year. What was its income at first ? DIVISION | 95 DIVISION ORAL EXERCISE 1. Divide 4 a7 by a’; by 6; by 2a. 2. Divide wa + ab by x; bya+6. Divide m? — mn by m. 3. What are the factors of a?—2abd406?? Divide a*— 2ab+b? by a—bd. 4. Divide a? — 0? by a—b; by a+b. Divide m?+ 2m +1bym+1. Divide a —1 by a?—1. 117. Dividing by binomials. We have just seen some examples of dividing by binomials. Dividing in algebra is quite like dividing in arithmetic. Arithmetic: Algebra: Divide 3772 by 46. Divide 2a?+ 7ab4+ 386? bya+30. 82 2a te a) Check: 46) 3772 a+3b)2a4+7a+388 a=1,b=1. 3680 = 80 times 46 2a7+ 6 ab 12+4=8. 92 ab + 3 6? ag A ab +3 0b? 118. The work is easier in algebra, however, since we need, after once arranging both dividend and divisor in the same order as to some letter, to divide only the first term of the dividend by the first term of the divisor to find the first term of the quotient. Here 2a?=+a=2a; subtracting 2a(a+ 35), there remains ab + 3b? to be divided. Then ab +a=b; subtracting b(a+ 3b), there remains nothing. Therefore 2 a? + 7ab+30b?=(2a+6)(a+3b). Inother words, the quotient is 2a + 6. 119. We check, as in multiplication, by putting numbers for letters,ias a—1,b0—1. 96 DIVISION WRITTEN EXERCISE Divide in Hxs. 1-15: l. w+2ey+y by x+y. 2. 2m? — mn — 3n? by m +n. 3. 3x°?4+ ry —2y? by 3u— 2y. 4. 6p? + pq —2¢@7 by 8p+2¢. 5. 42a?—13ay+y* by Tx — y. 6. 2+3a%0+38al?+ by a+b. 7. 10a? — 29 ab + 100? by 2a — 5b. 8. 20 p?q?r? + 41 par + 2 by par + 2. 9. 40 27y? + 58 ay — 21 by 10 xy — 3. 10. 30 m* — 229 m? + 30 by 15 m? — 2. 11. 3 —47 ay + 170277? by 1 — 102. 12. 32 2?y? + 46 xyz — 52? by 162xy — z. 13. 1 —18 por + 77 p’q?r? by 1 — 11 par. 14. 50 xty* + 77 w?y?2? + 3 24 by 25 x?y? + 2”. 15. 3224 — 5027+ 3 by 162? —1; also by 4a +1. 16. If a body moves uniformly 42? + 3ay — 7? feet in 4x — y seconds, how far does it move per second? 17. If 3%+7 pounds ona lever will raise 12 x7+ 19% —21 pounds, how much will 1 pound raise? How much will x +2 pounds raise ? 18. If «+ y articles cost 3”7+ 2zy — 7”. dollars, how much does each cost? Supposing « = 1 and y = 1, what is the result ? 19. If there are 1082? + 144277 + 362zy? words in a book of 36x pages, how many words are there, on an aver- age, to each page? If there are 32+ lines to a page, how many words, on an average, to a line? REMAINDERS QT ORAL EXERCISE 1. Divide by x: ax + ba, ax + 0. 2. Divide by x+y: (@+y), («+ y)? +m. 3. Divide by « —y:2?—y*?, 2*7—-—2ay+y*,0—-—y—a. 4. Divide by a+1: (a@+1)’, (a4+1)?—4, (a41)?— 2’. 120. Remainders in division. — In dividing by binomials remainders are treated in the same way as in dividing by monomials. That is, they lead to fractions in the quotient. For example, divide 9 a2 Bry te + oy bye ty. ee roll Rearranging the dividend ary ; 2 » 22 and dividing in the usual way, x+y) = tery + 3y there is a remainder of 2 7. et ry Therefore the quotient is ty + 3 y? ryt ¥# 2 a2 esi OE Ly WRITTEN EXERCISE Divide in Hrs. 1-4: 1. w—2ayt+Ty by«—y; byx+y. 2.0 +3m—38m by m+1; by m—1. 8. 8+ 3a%y4+38ay—6ybyx+y; byx—y. 4. 6a—17a%+4+11ab?4+ 50? by 2a—b; by 2a+0. Rearrange the dividend and divide in Hrs. 5-9: 5. 17 ab + 21 a? + 5 0? by Ta + 6; by Ta— b, . 8797 +6 p? +7947 by 6p4+ 9; by 6p—g¢. . 17 avy — 137° + 62? by 3u—2y; by 82+ 2y. . 6 m+18mn —12n? by 2m+7n; by 2m+5n. . 138 mn + 3 n? + 20 m? by 5m%+4+ 2n; by Sm—2n. ©o © = OD 98 DIVISION 121. Arranging terms. —It often happens that in attempt- ing to arrange the terms of the dividend in the same order as those of the divisor, certain powers will be missing. In that case zeros may be inserted if desired. For example, to divide x? — y? by x — y either of these forms may be taken. After a little the second one will naturally come to be used. xt2+ xy + y* x +ay + 7? a—y)e®+ OF O-% a—y)x — a — xy a3 — ay Cpa cM) xy — yP x2y — xy Ly iy xy — yp xy — WRITTEN EXERCISE Divide in Hxs. 1-6: 1 2+yby x+y. 2. a—b by a— bd. 3. at — yt by at+y. 4. 82° —1 by 22 —1. 5. oe +y by x+y. 6.14 27a by 14 3a. In Exs. 7-10 one factor is given in parentheses ; required the other factor: 7. @&+3a?+5a4+3, (a+1). 8 1— 640%), (1 — 4 xy). 9. 32m> +1, (2m-+1). 10. a°b*c® — d®, (abe — a). Since at is equal to a?-a?, at — bt may be thought of as the difference of the squares of a? and l?. Hence a* — b* = (a? + 6?) (a® — 6?) = (a? + 0?) (a + 6) (a — B). Factor the following: 11. a* — 7%. 12. 16 m* — ni’. 13. 16 p* — 1. 14.°8lat— 169%) Set 916.81 p494s* = 413 FRACTIONS Je FRACTIONS ORAL EXERCISE a(a + 6) a(b ae) b(a+6) a(e+da) to a fraction with the denominator 1. Reduce to lowest terms: 2. Reduce au x (x + y)*; with the denominator x? — y”. 122. Polynomial denominators. — The method of reducing fractions is the same for polynomial as for monomial denominators. 22+1 For example, to reduce EOL ie al nO ce eA Viet (dee 2) =, 2. Therefore. x — 2 must be multiplied by x — 3. 3. Therefore both terms must be multiplied by « — 3 (§ 73). ‘ PR eM 3 LGR SATO To Ee VO Alito Shoe Se ae iQ (E93) (062 )) @—5at 6 to a fraction with the denomina- gL 123. Changing Signs. —It should also be remembered that, since we may multiply the terms by —1, we may, change the signs in both terms. For example, Sty kee Qt ti eB hod WRITTEN EXERCISE Reduce to fractions with the denominator x? —9x +14: e+.2 yeaa xo? + 6 a? eee ae) ee a 27 Ba ee eh iy eee With the denominator 6 a3? — 29a? 4+ 46a— 24: 5 oe A Ole 7 Boe 5 Cereal. ee) ee ae a ane ORE 100 FRACTIONS ORAL EXERCISE 1. Express as mixed numbers: %, $) 72: oe he -. oy RE a ee a a 2. Reduce to mixed expressions : 1 . A | 3. Express in fractional form: 2}, a + 7 x + 124. Mixed quantities reduced to fractions. — Required to 2 “a as a fraction.* a ——= express a — b — a +b b 1. Since 1 pepe a+b —b b 2 — fp? 2. Therefore a—-b= & Bs at Ne oF a+b at+b 3. Therefore a—-& a@—b? a-— p a—b-— = = a+b a+b atb i 9D Sn) 4, — a+b ie w—-b—-a+h a—a o. = ---_—_-_ = . ath a+b WRITTEN EXERCISE Express in fractional form: eae. 2. 3%. 3. ate. 1 | x a a Pai 1 3 ; 4. m “a 5. e+ y+ 5. PEO ae 6 m, ee ty 7 Carlen 8 en hey ‘ 10. Lr ee Bi plat 11. m—1——- 12. a+6— 13. 15. Whe 19. 21. 23." 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. REDUCTION a+ 6? a. 14. =o 1 one x wep 16. 2 {en eae A 18. Deeg ae TZ 20. a— a ipa aon er am Pepe. 5 p-l1 np? — 24 Merde etsy a? — 73 2 2 é Gielm I Sey, 26 4 x? EPR INT RIESE: 28 Wo 2 mt-n— 30 m—n hs wits 2h Pe 2 ee a?+ 6* + c7+ ara 32 272 tatgepeec ee ag tab ere et el 8x7? —3ay y+ 36 —4¥P er+2ary+ cee . 38 Ni Ra dees LS 2p? — 5 nq Z29+3 q+ ———: 40 pP q peg Dy ale yee Dip ch ype ea 3m—dn 32a — 7b? + 60? 4+ 4. 44 101 ta ti—=. 5 ae or. ow+i— rr ae Es 2 q° on Pe Han rcay, 8 2 pass ab, ie, & oe aie 3 p? tat+l——+at Te Da RUE SEE = reas ~- 6m+21 UP an Gs aoe ep re} 2 i ele ax —b a + 2 nity" Be Bs 0 bg voy 3 2 are te alg OE G0 2m? ie ts oe i i . 1d Tm ais] ae on .m+m*+m men eh Ee aee Reyne ee — Leo 2 20-4305 ——", he Bey Aye ae a—b . 91a? —62b? + 310?— 62d?, 102 FRACTIONS ORAL EXERCISE Reduce to entire or mixed quantities : 1 32 ab 48 m?n 3 x — 9? " 160 "12 mn oy 7. (a+ b)* 4a? — be . ot") a+b Za—b ' 25 (a — y) 7 a*—2ayty” 3 at(@e+y)* (4 + b)* +m ; xv—Yy a+b 21 2» 125. Illustrative problem. — Reduce 2 r a as : to an entire or a mixed quantity. : Since a fraction is an expression of division, v+3ae4+9 if WRITTEN EXERCISE Reduce to entire or mixed quantities: “L a — ues 9. at + De ety ato CT Pes MO 3. . 4. : ea p— 2 z e+ 3a+y? F a*—92+6 a+ty x —4 : op aettshie ans eiViameea Us LO: ; ZT f ay —1 e+ta+ta+ 1 Tet ae OO a 10. =? __ a+l pg —11 ey? + 11 2y 4+ 28 mn*p? — 23 mnp + 60 Li . 12. —?—_——__—___—__. ey +7 mnp — 20 ee a> — b eh a> + On 14. e+ 3sx°%y+3ay?+y Gab a+b e+y REDUCTION 1038 ORAL EXERCISE Reduce to fractions having the lowest common denomi- nator : 1. 4, 4. 2. 3, ¢. 3. 8, t. rh ae a ec i) oh aN Seat rar ~ and —~— to frac- be cp + cq tions having the lowest common denominator. 126. Illustrative problem. — Reduce 1. Since the factors of the denominators are b, c, p + q, the Pe-dsis 0¢(p.-h.d). etter ar a a(p + q) 2. Multiplying the terms of — b + q, we have ——————. plying be Bey | be(p +9) bx 3. Multiplying the terms of 2 by b, we have ah ea cp + cq bc(p + q) WRITTEN EXERCISE Reduce to fractions having the lowest common denom- nator : ee 9 Iho ee 3 ee Cget "m m+i1 Qpr—r? 7? a b aa t 1 mt-1i 1 4. aes 2 HY, 3 ae 6. Si ei Te pers ah as imal UY a’—a a” m+m m a b m iy x z « be’ ae +e? cae mn +n? gs ye y(e +y) ry 2 4 1 i, SR ER i ee eM So ee Oe ree Deen? 2 PY PY-—P Och IEP AY CH 13 we eay + yy ett tay ty" 4 m—1 m tay” vv Gite i SL 15. Solve the equation 14% —175 = 172 — 241. 104 FRACTIONS ORAL EXERCISE Reduce to lowest terms: REE . a Za+b . 3. atb- 6 way 4a* — b? a” — b? 4 x+y 5 4(a+b) 6. Wawa “e+ 2ey+y? 8(a+6)? . m* —1 aye(a + y +2)? © x.— Y ve (2(a+y)* eyet(aty+te) #2 —LZey+y? ~~ 36(a@+y)? 127. Illustrative problem. — Reduce the fraction v?+8a+15 ete) eee be to lowest terms. et Oo ti Lore bs) (Leto en at : oe . 1G eee) irk SAS (Mire ON 4 ater (5) eda ad (38 ) WRITTEN EXERCISE Reduce to lowest terms: ie ge tae erm Ltt yo 16 48) " 2? + 20a + 99 a y2+19y + 84 3 Pe lop 20 4 m* +18m +17. ' p? +18 p+ 65 mm? +19m + 34 5. ot 16 800 3 py tA py —T7 x? + 14a —120 py +2 pq —99 7 xy? +8 xy — 48 8 Orne 00 2?y? + Say — 84 | ' 2? — 17 ay — 847? 9 mn? — 26 mn + 105, 10 a*b*c? — 15 abe +- 26 nee mi 076767 18 abe + 65 pg? —18 par $17 7? 12 mn? + 20 mn + 91 ok pq? — 19 par + 847? m?n? + 21 mn + 104. REDUCTION 105 WRITTEN EXERCISE Reduce to fractional forms: 1 bl we ++. 2. id casch SA aie oe ee ee ioe mnt eae ae m—n pr nee on eae me Pe ee ee 5. 8 p*q? er aT GMO ty ey VST 17—2? m—n oy dee Wey FA : 1 ae ROE doy eee te Lay? — 17 ary ers 8. 23m IED 23m 17 Reduce to entire or mixed quantities: gl + gilt — 3 yi t+ 3y't+2y? Ls ar pa © LO encase aay a m4 Py + 1) x? + 22ey+121 xy? — wy® + 2 ee 12. ee 11. xyS +1 xt +4 a8y + 42%? 2 13.2 ee eee Ue 14. jee ANG ea ee e+ y (ie 22 ; = 15. mn 17 mn + 15° ys ut — $4a7y? + 289 ys mn — 5 x manier Reduce to fractions having the lowest common denom- inator : y Core el i ins ih SS aero Ch eae ee eee de — 4 19. Bary eer he 90. Lt38y Lys xy — yp xy 3x2 -+6y Gay 91 api 1 dpsed: 99. aes © i ar 8 din i ' @b?— ab ab? 2 (p—9q)~ 4 meee PT 4, ot 2 abe oes Oe Ang? + q° : 2q° an— be ar 106 FRACTIONS ORAL EXERCISE 1. Add ; and a3 — and =; and *. 2. Add 5 and = and 1. “— and t cea a oa a+b GSI as e+td 4. In adding fractions, what must be the nature of the denominators ? 3. Add 128. Addition of fractions. — Required to add the fractions a b en ee 1. The least common denominator is evidently ) (b + c). | Bos (OGG) Genel see ae) DO Gere) 0 ime ere b b? b? bte (b+e)o B+be ae ag 3. Therefore the sum _ ab tact b2 + be 9 me and WRITTEN EXERCISE 1 a ee Le gaye a oan peer Pee m+mn mm ola Le Slee 0 m 1 2 au ie ab 2 n' men gS ala Oa eae ee P78 “b(a+b) a+b “p(p+ay p 9. Solve the equation 3% +17 =z + 59. ADDITION OF FRACTIONS 107 10. A man bought some goods at 6% discount from the What was the marked price ? marked price, paying $1128. Ly 13. 15. 17. Solve the equations in Kxs. 21. 23. 25. 27. 29. 31. 1G 9p? + 6 pg og 39 Pon x x m2 | m? + 4m? x—y Le Wy —= Ui Aad oe mtn m? ad, 3p x 4 4 —~+———+ y 9 Suede 120. x 46 4e44+7=22+41. 32 —x = 48 — 132. Get a 182% —175 = 157 x. ee) o 5 mn? + 2 mn ge as 12. 14. 16. 18. 20. 21- 24. 28. 30. be iI ab zs ac abe 4° aoe o+e x+y Bie cee ek -{- Me eae act ra ee (a — 3 ie ye 5 vel GPE) eek te a x ye ao + 141. Gils (te A at ee atte 6) (622 a = 98 9) 20. ee Ala —76=>17x2-+ 19.43 « — 83.6 = Ze 32. lr a ee ee 33. Find the number whose half, third, fourth, and sixth together equal 15. 34. What number diminished by 3 equals the sum of the half and fifth of the number? 35. Find the number whose half, fourth, eighth, sixteenth together equal the number less 2. and 36. A man’s salary when increased 8% amounted to $1350 a year. What was it before the increase ? 108 129. Subtraction of fractions. — Required, from tract a— b a+b FRACTIONS a =~ b- ;, to su 1. The least common denominator is evidently b(a + 6). a (a+ b)a a? + ab 9 ae 4. The difference = b (a+b)b ab+b Ci Dy mb ( isbn bia a+b b(a+b) ate a+ abi— (ab — 6?) CL a 4b ab+ 6b? er ab — ab mine Met je” ab + b2 ~ ab+ b? WRITTEN EXERCISE at i 2. —— Faery “an a—w7 4,.—— . by b—y hee Bhi 6: a+ 0 ah b a—b 8. (ea “@ @+y Digs te ONL By cea 1S Re. P+tY pr a b 1 be ca abe Pad Saas ame . Solve the equation . Solve the equation 5 a + : ae ) ue a Aseitges ees Sy 4 . Solve the equation 2” + 8? = 163. . Solve the equation 3 x — 7, « = 7007. ath (a+b)? “(a-+5)* ei+a— 1 ieee atl + 10 — a Lice h SUBTRACTION OF FRACTIONS 109 15. A man borrowed a sum of money for 1 yr. at 5%. The sum of principal and interest was $262.50. How much did he borrow ? i ea ee yte y 2 ie, Se [Die ee a y 1 ie begs BS TSG T ane a4 nls he ee ee ey a DOD rat a—2ab a 27. pf g "pt +Tp 3 aa P hy 2 qe a b Ly + 2u xy oe a ary is: Cay ie a b? . 2 € C972 — 12a m? Solve the equations in Hxs. 28-35: 28. 5— 5 = 5. 30. i cae 32. 2-2-7 +2 Se | 34. = — 30=7-1 eo a oS WIS WIS WIS & “| ~] 36. What number decreased by 14% of itself equals 172 ? 37. What number increased by 8% of itself equals 37.8 ? 38. What number increased by 10% of itself equals 687.50? 39. If from a fourth of a certain number I take a seventh of the number, the result is 3. What is the number? 110 FRACTIONS ORAL EXERCISE Uy ele ae Chin 5° - of ot il a ,x+4 1 1 pa sae Pe cal Gens a+b ) a Ome a i L+Y coe a 4. How do you multiply a fraction by an integer? a frac- tion by a fraction? 5. In multiplying fractions why should you first indicate the multiplication, then factor and cancel, and finally actu- ally multiply ? ik 1. How much is 5 of 3? = of 7? 3 Oboe 2. How much is : of 3. How muchis Bx? a: 9 (0-7-0) ae 130. Multiplication of fractions. — oe ool the last ques- — ab x a piacere tion (Ex. 5) with the product ote az— ab x = atx? — abxz? —————-.= to be reduced. - m2 + xy a® ata? + ary’ If we first indicate the multiplication, and then factor as far as possible and cancel, we have a(a—b)z* (a—6b)z a(at+y)e (x+y)a , a reduction easily performed. WRITTEN EXERCISE a ba + by 9 Spy ey (e+ Y). "6b ax — ay ' 5 wyz 9p »n a+b 2m? na — n*y 31 asa eas. oe n® 4m? + 6m? atdiea 10a—15 3mn —4 2: ee se, pat Tn 10) 3mn +4 DIVISION OF FRACTIONS eal ORAL EXERCISE mer. 1 1 1. Divide = by 2; =, PY 83 pret SY. 2: Divide 1 by 33 2 by 33 a by 53 je - DY MS = 1b a 1 2 a se Be Dine b by 3 iby 2 cniby a ieby 5 ye by pe 4. Divide 5 by 2 by 3 ~ by - How do you divide one fraction by another ? 131. Division of fractions. — We have already learned that there is a short method of dividing one fraction by another. Just as we divide 3 ft. by 2 in. by reducing to the same denominator (3 ft. + 2 in. = 36 in. + 2 in. = 18), so we may divide 3 by 2 by reducing to the same denomi- nation (3 + 2= % + 8=9+8= 2). But it is easier to Cad (ad a see that 3-2 — 3-3 — 2, and Da eta ap a as already explained on page 72. .., 40—8a' 16 ab? Thus, to divide 623 — ay Vv 15 ay toring and multiplying by the reciprocal of the divisor, 15 xy -4 a? (a — 2b) a Sxy(a— 2b) 16 a*b?.3a°(2xe—s8y) 4a°b (2x —3y) » we have, by fac- WRITTEN EXERCISE Gg SR ye ie amd alee ay eat A Pe ot x PU ir ey a a+b ab abl 2) ea OS ease ean 112 EQUATIONS INVOLVING FRACTIONS EQUATIONS INVOLVING FRACTIONS ORAL EXERCISE 1. By what numbers could you multiply } and have the 25 om Be Tp varac ie 2. What is the smallest number by which we can mul- product an integer? also ? tiply 5 = to have the product an integer? also 5 z ; Jefe 5 = 7, what is the value of x? By what number do we multiply both members of the equation? 4, If 2 = 6, what is the value of x? Also find the value of w if — eepleme 2 Mik Giger pa se ame 132. Illustrative problem. — Solve the equation : +7=9. Ab Since > +7 19; 2. Therefore : = 2, by Axiom 3. (State it.) 3. Therefore x = 10, by Axiom 4. (State it.) WRITTEN EXERCISE Solve the equations in Hus. 1-9: te eee ae ot 32 lotsa i. 2. i a RE opae ak —6=3. 4.2¢4+7=15. 5. 2%4+6= 42. 6. 32—7 = 83. 7 2a+7=}a 4 23. 8. §x—6=>3¢% — 5. 9. If from 3 of the number in our class I take 9, the result is half of the class. How many have we? CLEARING OF FRACTIONS 113 QRAL EXERCISE 1. Ifi«=T7, what does x equal? 2. If 2¢@=8, what does 2x equal? a? 3. If 34 = 2, what does 3x equal? «? By what should we multiply both members in order that there shall be no fractions in the following equations ? be |) 2m 2a 4 2 4. 7 Tae aaa oS ae ages Ao pt rs eae ea ee x 3 Race tra i me i a pO a a 8. In general, by what should we multiply both mem- bers of any equation in order that there shall be no fractions in the resulting equation? 133. Clearing of fractions. — Multiplying both members of an equation by such a number as to have no fractions in the result is called clearing an equation of fractions. To clear an equation of fractions, multiply both members by the least common multiple of the denominators. 134. Illustrative problem. — Solve the equation 34+ 24=42 4 2. 1. Clearing of fractions by Axiom 3, multiplying by 15, 45 + 10¢ = 12% + 30. 2. Subtracting both 45 and 1242, so as to place all the z’s on one side and all the known terms on the other, 102 — 122 = 30 — 45, or —i2¢=— 15. 3. Dividing by — 2, to find the value of 2, ge == 48, or 7. Check. 3 + 2-18 = $-15 + 2; for each equals 8. y ~ 4 EQUATIONS INVOLVING FRACTIONS WRITTEN EXERCISE Solve the equations in Kars. 1-10: fipekes ee 2, oe URIS aS oy 1 aL a ora, aioe aera es eae i Abie ail x 5, We G. 7a ao 6.04 0 eA, 8. da+4xe=3%+4 }. 9.10+0lea=5+1e% 10. 01¢24+6.2=0.3% + 0.2. 11. Find a number whose half, third, and fourth added together equals 36. 12. A man spends every year $200 more than half his sal- ary. In 3 years he saves $900. How much is his salary ? 13. After selling $ of his farm, and then 4 of what was left, a man still had 140 acres. How many acres had he at first ? | 14. There is a certain number such that its fourth added to its fifth equals one less than its half. What is the number ? 15. A man invests half of a certain sum at 6% interest, and the other half at 5%. The total interest for a year is $66. How much did he invest ? Solve the equations in Exs. 16-21: x x x LOWE RLD 16.75 —jg=3 t+ Eee Re rece gee ao es they 2 4 x x x 22 x x x x 20 a a7 2 20. 21 6tyo77p7U- DIRECTIONS FOR SOLVING 135. Illustrative problems. —1. Solve the equation etd aN 1. Clearing of fractions by multiplying both members by z — 5, ko pe 2. Two numbers have the ratio of 2 to 3. tracted from the larger, the result is 7 more than the LS: x 473 = l.8 2 — 9. . Subtracting 1.8 z and 3 from both members, or transposing, —0.8%= — 12. Dividing both numbers by — 0.8, smaller. What are the numbers? in ax “ ° 1. Because aes ta: we may conveniently take 2x and 32 to aes represent the numbers. oo Then and 22+7= 8. Therefore 3a2—3 = 4. Therefore 32 — 22 = ing), or 5. But we wish 2 x and 3x2 —3 = the larger less 3, the smaller plus 7. 7 + 3, by Axioms 1 and 2 (transpos- ea ht) 3 x2, and these are 20 and 30. Check. 30 — 3 = 27 = 20>+ 7. WRITTEN EXERCISE Solve the equations in Hrs. 1-6: ees. “zt4 6 5 ei 9 —z2z eet a 3a 3 ee lea ye ge lara houses 120 am 4. Tae amou 6. —2_ 4+2 Frsebe sub: 116 EQUATIONS INVOLVING FRACTIONS 136. General directions. — We have now found the general plan of solving a linear equation: 1. Clear of fractions. 2. Transpose the x’s to the left side and the known terms to the right. 3. Collect the terms and divide by the coefficient of x. For example, solve the following : 2 » ie “4+ -=+7=8. UMD aH It will evidently save some work to transpose the 7 first. 2. Then = poe! G0 aa 3. Clearing of fractions by multiplying by a(a — 6), ais 2axr—2 bz = ala —D), . Simplifying and collecting terms, (3a —2b)x=a(a—b). 5. Dividing by the coefficient of 2, pee RS oN) 4 eee 3a—2b ad — 20 use WRITTEN EXERCISE Solve the equations in Hus. 1-8: 1 te+7=32 + 18. Eee eas 3 Saya oat A Bee Be, ; 5 ti eas 6 ote tam 7 <4" 41 4+a=0 8 mdse =32—1 9. A flag pole is so broken off by the wind that } of the part broken off equals } of the part left standing. The original height was 90 ft. How much was broken off ? PROBLEMS oe 137. Illustrative problem. — Rob can gather the apples on his father’s trees in 6 hours, and Tom can do it in 5 hours. Working at the same rates, how long will it take the two together to gather them ? 1. Rob can gather 4 of them in 1 hour, and Tom can gather + of them in 1 hour. 2. If it takes the two together x hours, they can gather : of them in 1 hour. 7 3. Therefore Le a Cia) mee 4. Therefore 5z + 62 = 30, by clearing of fractions, or LL on x= 2;%. Therefore it takes the two together 2,5, hours. Solve the equations in Hxs. 1-6: Piel Peds tey /) 12h) ny, ~ 1S Pe pt+tgatr pr+pg+pr Ape Le ab 1 15 a—b ath 4u+y xy e+ 2ab+? a—2ab+o? 2 6a+2y° Q2y+4a (138 FRACTIONS 172. Multiplication of fractions. — Required to multiply x — y? 2a+y 4e°+4ay+y x—y Factoring, indicating the multiplication, and canceling, (@ty@—y:Crty_ zty (Aaa) ay) 2n+y WRITTEN EXERCISE 2 yty 9 m* mn +n Ura ere “2? mn —m Hei GS Grea ea 4 3. emma: 4 . . (ete (CS set) a*t+2Z2ba atd 5 9x*—-y? x—38y A ConA o—9y? 38a+y ; 2m+1 4(m?—m)+1 ah ed* ce a " Bd we ad b “1 at + 2 0%? + bt ase. tei a ee 9 Ma Le tnd OE on ered em, = leg eM. 4 10. a*h + ab +b a tt a Sra a ea, 1 TRACI Shel tg YG, a—2b Se ee a bse b2—¢ ab b+e 0b a? b—e 16 m®— 2577 10 m?n? 20 m8n? 4m? + 5 n? ee Se Se ene . . . DIVISION OF FRACTIONS 139 173. Division of fractions. — Required to divide -a+db 2 ab Qa y a? — b2 Multiplying by the reciprocal of the divisor, as explained in § 90, we have G0 (G0) (a Pen (ait) a—b 2 ab 2 ab WRITTEN EXERCISE L Cie ee 16 aty®—-9 | day — 3 ab ah? , 2uy A a7y? ¢ OB NG ee ie et 4 Olt Pte mT " ab ' a —$? “m+ 8m+7 > m+7 F ag ee Pe aN Pe ke Z a +9e+20 2+5 Sa ae eens eg 0 CES ga CR 5 5 Parva, ett or oe ptd | pP—_q " e@+5 § #?+6x+5 Oe ee a eae + et? ie atb+e .(atb+e? “a—b—e- (a—b—ec) rr mn? + 3mn +2 : mn Le mn + 5 " mn? — 25 1 Ee Yo ty ty eee UA a? — 2ey + yf? 229 13 Cg ony, Oey aes ; at — y' “ety? ath at+tb\) . fa—b a+b se: (S ee ett) (ott 140 174. Illustrative problem. — Solve the equation 3 te, FRACTIONAL EQUATIONS FRACTIONAL EQUATIONS 12 ae see ee: 1. Multiplying both members by (x — 8) (# + 1), we have Otol +8 (4 5 Bor 2 4% —8= 12. 3. Therefore 4x2 = 20. 4. Therefore Ei 0 Check. Substituting 5 for x, }+%= 432. WRITTEN EXERCISE Solve the following: 1. OL 1 a ak E— Nea 7 2 ff baie ye ant, 1 1 9 6 a+1. «x(@+1) 20a eel ee EL e—1l-“#+1 27—1 220 Sh a Rag Os @.-- lake leet 3 1 6 Don 1 ae Ore orem Teme Page, Se etl x-—5 2—4e¢—5 3 20 — 30 2 b> pees = oe tOs 3a —4(x—1})=9 —1(5a —7). 23. 24. PROBLEMS 141 22 1—2 x 3 = 4 — 5 12 % 48 = 4 sO oy Se Oe eee il tae o— ox 2(a—1) 8 2 3 4x2+7 oe, en x — 4 dee wit aoe: 4 24 — oe 4. 6 3 12 Eee Cae ey Z2x2+a 32 — a 13° S 5 4. CIEE eet EE ' 5a+2(a —1) 19 ieee? %+-35 . 32 a= 1 + 5 +2+ 5 e=a()) 9 9 CZ a®—Qa+1 24+2a4+1 (@—1)* 142 25. 26. Sx+5 14 1+ 72 a—1l pei Sa) : 3 ee bane FRACTIONAL EQUATIONS Bee eo Ree: 3—T% 42+6 ieee = 0. 2(1 + 22) 3(3 — &) _204+22) 3 ox »(02- we) Heist eaves eyo 1) -Ee@tsy=0 37. A third of the sum of a certain number and 5, plus a fifth of the difference found by taking 5 from the num- ber, equals 6. What is the number ? ; 38. If 1 less than a certain number is divided by 1 more than the number, the quotient is the number divided by 1 less than itself. What is the number ? SIMULTANEOUS EQUATIONS 148 SIMULTANEOUS EQUATIONS : ORAL EXERCISE 1. Given the two equations 7+ 3y=7,x+y=83; sub- tract member for member, the second from the first. What is the value of y in the result? | 2. Given the equations e+7y=10,%7+4y=1; how may we find an equation containing only y? Find it and solve for y. 3. Given the equations x + 9y =19, «+ 7y=18; what is the value of y? How will you now find the value of «? 175. Simultaneous equations. — Equations which have the same values for the unknown quantities are called simul- taneous equations. For example, a+2y=12,%+y=7, give by subtraction the equation y= 5. But if y= 5, we may put 5 in place of y in either equation and find the value of x. If we do this, we find that 2. 176. Illustrative problems. —1. Find two numbers such that the sum of the first and 10 times the second is 21, and the sum of the first and 3 times the second is 7. 1. Let x = the first number and y = the second. 2. Then Hig ME Be 929 i Pio Y=. 3. Therefore 7y = 14, by subtracting, and Tees 4. Substituting this value of y in the first equation, geo) = 21, and yeas Check. Substitute both values in the second equation, and 1+ 6 144 SIMULTANEOUS EQUATIONS 2. Solve the equations 27+ 7y=39,a+y=T. If we subtract at once, the x will not disappear. Hence we multiply both members of the second equation by 2. (We might have multiplied by 7, the y disappearing by subtracting.) al. 2 ey ae 2x+2y= 14, the second multiplied by 2. 5y = 25, subtracting. y= 9. x +5 = 7, substituting 5 for y in the second. 9 ot Bm oo tO . r= Check. Substituting in the first equation, 2-2 + 7-5 = 39, for 4+ 35 = 39. | WRITTEN EXERCISE Solve the following : lee yy Ss 2.2+5y=8, ey =D, x—. y=0. 32s O ia aL, 4,.%— y=3, eo = Dy = 9. “x+5y = 15. be — 1 Yo, 6. «+4y= 24, x+2y=12. e+ 2y=14. 7. 38a+4y=7, 8. 3%+2y = 20, x—y=0. x—-2y=A4. OU Die-) Yi ls 10. 42%— y=81, e413 Yas Loe e+ Ty=15. 1l. 374+ y= 40, 12. 2% —) jee) 30, x—-2y=A4., x+3y = 38d. 13. 5e+ 2y = 68, 14. 7% + 6y= 57, x+3y = 37. x—9y =18. 15. 14% —29y =1, 16. 80x%+ 4y= — 34, C— (ya), eles AW Toe 1G PROBLEMS 145 177. Illustrative problem. — Solve the equations dx—2Zy= 24, 244+ T7Ty=41. Since the coefficients of z are 3 and 2, we may make them alike by multiplying both members of the first equation by 2, and both members of the second by 3. Then Le 6x— 4y = 48, 2 Ga 2by = 1235. 3. 25 y = 75, subtracting (1) from (2). 4. y = 3, dividing by 25. 5. Hence 3x — 6 = 24, by substituting in the first, and ie eae Na vel Check. Substituting in the other equation, 20 + 21 = 41. WRITTEN EXERCISE Solve the following: 1. 4¢%4+ 3y=17, 2.5x%—2Zy=8, Darel Ye oe. oa@+ 5y = 42. 3. 5a—3sy=8, 4.7Tx—d5y=46, oSxa+ y=s0. 5x+9y = 58. 5. da — Sy = 38, 6. 82+ 56y=0, Tx —9y= O4. Tx— y=d50. i294 --* 27 = 66, 8. 8e+i11ly=— 98, Oa Liye O14 3a 2y = 0. 9. lix—8y=1, LOPS eas ye 90, Tx+5y=—T7O0. Sipe toe Wate LOO: 11. }x—d3y = 38, 129, tea+hy=3d. ye —ty=4. 13. Ly — 2x = 20, 14. iy— (=2%, La —2y=— 16. hie 4G) YH, 15. te4+1y=8, 16.1%x+2%y= Lyte 10. 146 SIMULTANEOUS EQUATIONS 178. Elimination. —To cause one of the unknown quan- tities to disappear in the treatment of simultaneous equa- tions is called edimination of the quantity. For example, if z + y = 7, x — y = 3, then, by adding, 2 z = 10, and z=5. Here we have eliminated y. 179. Methods of elimination. — Not only may an unknown quantity be eliminated by subtracting, as in §§ 176, 177, but it may be eliminated by adding, as in §178, above. We are at liberty to eliminate the ~ first if we wish, or the y first, as in § 178. If the signs of the term containing the letter to be eliminated are alike, we naturally sub- tract; if unlike, as in § 178, we add. It is sometimes better to eliminate by substituting at once, as in the case of a+ 38y=17, 8x%—y=1. Here we see from the first equation that x =17 —3y. Sub- stituting this in the second, we have or Dieta yee 2. Hence —10y¥ =— 50, y = 5. 3. Therefore z=17 —3y=17 —15 = 2. WRITTEN EXERCISE Solve the following : laa = 3 25; 2. 6x —y = 51, 20 Oy = 11s Sa+y=47. 3. e241 y = 39, 4. t= ype 19, 5x+2y = 380. 5e@+3y=19. 5. 28 yo, 6. Ta+4y=15, 4e+ty=— 59. - 4x —Ty= 565. 7. 2xe+3y=5S0, 8 «x«—sy=116, 32— y= 20. 5a+ y=100. ELIMINATION 147 180. Illustrative problem. — If to the first of two numbers I add 8 times the second, the sum is 21. If from the first number I subtract twice the second, the difference is 1. What are the numbers? 1. If x = the first number and y = the second, Cee, eyes, 2. Subtracting, LO — 20; Yeo. 3. Substituting, eae ols mee ay, Check. Substituting in the first equation, because the second one has been used in finding the value of z, Der 16 = 21, 9. Find two numbers whose sum is 99 and whose difference is 17. 10. Find two numbers such that 7 times the first equals TA 8 times the second, and such that their sum is 74. 11. The sum of two numbers is 20, and if 14 be sub- tracted from the first, the result is the second. What are the numbers? 12. What is that fraction which equals 2 when 5 is added to both terms, but equals } when 2 is subtracted from both terms? 13. What is that fraction which equals 14 when 7 is added to both terms, but equals 50% when 6 is subtracted from both terms ? 14. The sum of the faces of two promissory notes is $1000. The first draws 6% interest and the second draws 5%. The sum of the interest on each for one year is $56. Required the face of each note. 37. 39. 41. 43. SIMULTANEOUS EQUATIONS tat ty =4, gu+ty =8. _Qa+38y = 22, 3a+4y = 29. ee Oy Ens ~Fe+2y =d0, Oe 2 f= 198: > hy, Ape SUT SEY =. je 1 — ~4a+ Ly=8, se+yoy=l. 3,¢ + y=101. 385a+14y=9, 10 — ey a oe y— 1 8a+16y=18. .842—Ty=19, 3g —4y=19. txa— y=A4, fru —dy = 4. .18%—-107=9, 27” + 40 y = 82. 10 Syl, 124%—5y=—4. ta+3y=Odl, 2% —3y =-— 30. 2+ 9y=Al, 8x—lly=—1T. 16. 18. 20. 22. 24. 26. 28. 30. 32. 34. 36. 38. 40. 42. ye+ ty =4, sc+3y=3l, te — Ly 4, se+rgy=Y, 40 +4Y=0 TET ZYy=", $x — y = 18. tatiy=12, e+ 3y = 99. Ae+Ty = 22, AO t= ede 3 « — 2 y = 36, ig Pa eLOr==eL 00 Tx + 1ly=188. 14% + 334 = 22. 4xn+5y=4, 12%—dy=—2. 22+ 83y = 34, 15a — 79 y = 255. 424—60+3y=2e+ y+ 234, c=y. PROBLEMS 149 181. Illustrative problem.—In preparing some medicine a druggist wishes to know how much water he must add to a quart of alcohol, which already contains 5% water, so that the mixture may contain 50% alcohol. 1. Let « = the number of quarts of water to be added. 2. Then 50%(1 + 2)= number of quarts of alcohol in the mixture, which must be the 95% of a quart with which he started, since no alcohol has been added. o> Then 50% (1 + 2) = 95%, l+a=%, £=33 -l==% 4, Therefore he must add ;° of a quart of water. 44. How many ounces of gold must be melted with 30 oz. of gold 16 carats fine (4¢ pure) to make an ingot 18 carats fine ? 45. How many ounces of pure gold must be melted with 18 oz. of pure gold and 6 oz. of silver to make an ingot 22 carats fine ? 46. How much water must be added to a quart of a solu- tion containing 8% acid and the rest water, so that the new mixture shall contain 6% acid? 47. How many ounces of pure silver must be melted with 300 oz. of silver 800 fine (800 parts of pure silver in 1000 parts of metal) to make a bar 850 fine? 48. How many ounces of pure silver must be melted with 40) oz. of silver and 100 oz. of tin to make a bar 900 fine? 49. How much water must be added to a gallon of a solution containing 9% of a certain extract, so that the new mixture shall contain 4% of extract ? 50. How much cotton-seed 011 must be added to a pint of a mixture, which is + cotton-seed oil and the rest olive oil, so that the new mixture shall contain { cotton-seed oil? 150 SIMULTANEOUS EQUATIONS 182. Literal equations. — Equations in which some or all of the coefficients of the unknown quantities are letters are called literal equations. For example, ax + by =c, mz + ny = p are literal equations. The letters a, b, c, m, n, p in these equations are supposed to rep- resent known quantities. 183. Illustrative problems. —1. Solve the equation laa ax—Tb=e. 1. Adding 7 6 to both members, OL = 97 Doe Tb+e x =—_——_-: a Le) (Check the result.) 2. Solve the equations ax + by=c, be+cy= d. 1. Multiplying the first by 6 and the second by a, abz + b’y = be, abx + acy = ad. 2: (6? — ac)y = be — ad. be — ad ae y — = . b? — ac We may now find the value of « by substituting, or we may eliminate y instead of x. Taking the former plan, and sub- stituting in the first, b(be — ad 4, ax alae!) =r 2— ae b(be — b2 — ac b?c — ac? — b?c + abd 6. ERAS A I ot A, b2 — ac a(bd — c? é _a(bd~ of), b2 — ae bd — c? 8, e= ; b2 — ac LITERAL EQUATIONS 151 WRITTEN EXERCISE Solve the equations in Hxs. 1-16: l. axe +b=c. 2. he aT 3. abe + bex = abe. 4. px —GY= qe — p. 5. axt+b=ce+d. 6. maz — nx = ax +- 0. 1. mx + ne = px + 9g. 8) 2 me 4 NX — pr = YJ. 9. «+ ax+ bxe=cx+d. 10. (a+b)ea+c=(b+c)x+d. ll. aw+y=m, 12. c+ay= 8, bx —y=n. ax+y =e. cy ea (ae ee ath ab xe+2 x—Y- ee saieps 15. dax+3y=a4, 16. 4¢%+1y=2a, 2x+7ay= 0d. oa —2Zy=0. 17. One number is @ times another, and their sum is 2a. What are the numbers? 18. The sum of a certain number and @ equals the sum of another number and 2. The first number plus the second equals c. Find the numbers. 19. The sum of two numbers is m and the sum of the first and » times the second is ». Find the numbers. What are the numbers if »=3, m=2? 20. If to a times a certain number there be added n times another number, the sum is 0; the first number minus c times the second equals a. What are the numbers? 21. The sum of two numbers is s and their difference is d. What are the numbers? From the result write a rule for finding each of two numbers, given their sum and their difference. 152 SIMULTANEOUS EQUATIONS Solve the equations in Exs. 22-40: Pe TU NS iene y ti 23. aa +7 ab = 02. 24. abx — 8 abe = bec’. 25. aba + abe = be’. 26. 4ax — 3bc = Qbe. 27. £2 2a — ao 28) Ont — be 99. x? —4ab =a’? +467. 30. 1627 —n?#=n7-+2mn. 31. 6 076% + babe=—A1 abe. 32. ax +y=), 33. ax — by = 8, Diy —G bx — ay = 5. 34. we — by =e, 35. abx — bey = ac, ax — by =. ace — aby = be. 36. daz + 3by = Ge, 37. D008 2y ==, Sax + 2 by = Te. dace + by = 5. 38. }e@+ar+aV=0?+4 az. 39. 27 — 1247 = 360+ o- 40. a? (a? — 6?)= ¢(2ab- ¢). 41. If the sum of two numbers is 19, and their difference is 7, what are the numbers? 42. If the sum of two numbers is 251, and their difference is 75, what are the numbers? 43. If the sum of two numbers is 24, and one is 7 times the other, what are the numbers? 44. If I have 15 cents more in one hand than in the other, and the total amount is 75 cents, how much have I in each hand ? 45. Of two numbers, a times the first plus 6 times the second is k, and m times the first plus » times the second is 7. What are the numbers? 46. Of two numbers, the sum of twice the first and 3 times the second is 38, and the sum of 5 times the first and 6 times the second is 83. What are the numbers? QUADRATIC EQUATIONS 1538 QUADRATIC EQUATIONS 184. Solution of the complete or affected quadratic. — We have already learned (§ 168) how to solve the incomplete or pure quadratic equation. We shall now consider the solution of a complete quadratic equation like $9 -— 15 = 0. Peevacvorins go oho, we nave (e— 3)(7—-0) = 0. 2. It is evident that this product cannot equal 0 unless one of its factors is 0, and that if either factor is 0 the product must be 0. 3. Therefore the equation is true if x — 3 = 0, in which case x = 3, or if x — 5 = 0, in which case x = 5. Check. Substituting 3 for x in the equation, 9 — 24 +15=0. Substituting 5, 25 — 40+ 15=0. ORAL EXERCISE Solve the following equations: VY @—2)@—3)=0. 2. (« —1)(«—7)=0. 3. (x — 8)(# —10)=0. 4. (x —9)(# —11)=0. 5. («@ + 8)(@ —11)= 0. 6. («@ + 9)(@ —10)= 0. 7. (© +7)(@+12)=—0. 8. (x + 6)(« + 20)= 0. Jee) =) 010% 7a a Orel ee == 0, 125 (e+6)=—0. 13. eto On la eae a0.) 0. 15. (2a —1)(a—1)=0. If 2%a—-1=0, 2u=1; then what does x equal? What is the other value of «? 16. (32 —1)(# —2)=0. 17. (4a —1)(x —3)=0. 18. (5% —2)(x — 3)=0. 19. (7x —3)(«#—5)=0. 20. (8% —2)(4# + 6)=0. 21. 2e%—7)(3x%+ 2)=0. 22. (2a +1)(8e%+4+2)=0. 28. (8x%+5)5u+4+ 8)=0. 24. (2a —4)(38a%—9)=0. 25. (Tx —8)(8x—7)=0. 154 QUADRATIC EQUATIONS WRITTEN EXERCISE Solve the equations in Hrs. 1-20: eg a 2 0) oe ee Or: BO? gh: 4. a7? —xa2 —12=0. 5. 2? + bat Ss = 0, 6. 27+ 544+6=0. Vee Bs ee AI 8 a Oe, Sree ule ne al Bc OF 10. 27 +9272+8=0. Mea te 2, 12. 27 -— 92+ 20— 0. 13.27 — 92 + 14 — 0, 14. 27+ $2 + 12= 0, 15) 7 - 102 9 = 0. 162°97 4 2 Lipa — 12%: -- 36'=-0. 18: a? = 112-4 30 =%. 19. #7? + 12%+4+27=0. 20. +1274 35=0. 21. What number is 16% of its own reciprocal ? 22. Find a number which is 6 less than its square. Are there two such numbers? Are both positive? 23. Find a number which when multiplied by 1 more than itself equals 12. What are their signs ? 24. Find a number whose square increased by 35 is 12 - times the number itself? . What are their signs? 25. If a certain number be subtracted from 16, and the difference be multiplied by the number, the product is 55. Required the number. 26. A certain rectangle is 3 yd. longer than wide. If the width be decreased by 2 yd., and the length increased by 7 yd., the area is not changed. What are its dimensions? 27. The width of a certain rectangle is 7 ft. less than its length. If the width be decreased by 4 ft., and the length be increased by 22 ft., the area is not changed. What are its dimensions ? ANSWER BOOK TO ALGEBRA FOR BEGINNERS BY DAVID EUGENE SMITH, Pu.D. PROFESSOR OF MATHEMATICS IN TEACHERS COLLEGE COLUMBIA UNIVERSITY, NEW YORK GINN & COMPANY BOSTON - NEW YORK - CHICAGO - LONDON Copyright, 1904, by David Eugene Smith ENTERED AT STATIONERS’ HALL COPYRIGHT, 1904, 1905 By DAVID EUGENE SMITH ALL RIGHTS RESERVED 35.10 The Atheneum Press GINN & COMPANY. PRO- PRIETORS - BOSTON - U.S.A. ANSWERS TO SMITH’S GRAMMAR SCHOOL ALGEBRA Answers are given for the written exercises only. Page 5 Leese eazoe. 63: 10, 4. 8. 5. 34. Ga S20 nlon Os 8. 4. 9249. 107.840 11 212 12328. s18.188. 14, 711. Page 7 Page 8 Loe lo: 2. 4. 3. 15. 4. 14. Ory 6. 16. (he. VAS : Page 9 Lael: 2. 12. 3. 16. 4. 15 5. 7. 6. 8. Sali 9.3: 10. 5. 1S) fe OE 12. 15. 13. 4 14, 4, 15. 8 16. 53. Page 10 Loo: e205 3. 25. 4. 3. 5. 9. Page ll LOU: 2. 200. 3. 75 lb. 4, 140. 5. $2400. 6. $95, $9.50. 7. 2400, 360. Page 12 “A eye 2eclds Sipal. 4115 5. 400. 6. 230. C213; S213, 9. 61. 1035 11. 3. L219; 13. 33. 1492; LO uf; 1Giai Lien Loot OoOowPe GRAMMAR SCHOOL ALGEBRA Page 13 . $72, $105. 2. $12, $50. Fg) Amit Fs 4. $90, $60. 5. d = tm, d =140%; 6. rp, p+ rp. Page 14 20-6 ea 3d 8135.) 4. 86 5. 125 —a 127 + ¢. 1.4 8. 12. 9. 74 LOS. 4] 12ao 13. 6 14. 55 15. 510°. 16. 11 Page 15 Sete 2. 17 ft. 3. $17. 4. 7b6—3a. § aie Geo pelts 8. 4 ft. 9. 363. 10, 357 sq. it. 11: 117514,000: . 6286 ft. 13°°27-5 in: 14. $236.09. 15. 51,200,000. 16. $1.75. Page 17 . ax lb., 2400 lb. 2. ary \lb., 108 lb. . abe dollars, $72. 4. Coefficients, 3, 4, 40%, 0.5, .652, 12.5. Page 19 LOR LOS AMET 2% 115, 60;,0,.1 1: . For example, 8y=15, 3yz2=60. 4. For example, 4n¢g+8 m=100. Page 21 me oUGs 2. 8a-+ 5b cents. 8. 6r +3747; .lw—r? 5. 5m + 82, 290; 826,865 sq. mi. 6. 57 108; 344 mi. Page 26 ed Ose Lows 2. 173%, 1414. 8. 55 ab, 154272. 4. 128. Page 27 . 67a4+ 440. 2. 150%+4+118y. 8. 155272 +172y. . da + 200 + 10c. §. 24a 4150 + 16c. 6. 9a+9b+ 9c. .4a+60+4+ Be, 13+ 23474+8=51. .§6a+8b4+ 382, 645410416 = 37. . 764+ 8c+4+ 72, 10 + 22 + 16 4 20 = 68. ANSWERS 3 Pages 29, 30 ee ee 2. 156. pe MiG 4. 64. 5. 40. 6. 30. (ea G he 8. 87. a oy es 10. 9. De 12: 12. 8. ; 1G Sen 14. 3. 15. 7: days. 16. 9 cents. 17. 8 o’clock. 1S 28rd 10 1d? 19210: 20. 45. 21. 97. 22. $4, $32. 238. $1380, $260, $1040. 24. $1100, $2200, $3800. 25. $6, $12, $18. 26. 30, 10, 60, 60. Pages 32, 33 1. 29ay —z. 2. 34m?n —11mn2 38. 144 pq — 16 82. 4. 1038a2+5b2—2c%, 5. 0. 6. 20 ab. 7. 223 a2zy?z + 14 p. 8. 4m?. 9. 6a? — 4 ab. 10. ldary+4z2+w. 11. 20 pq + 20s. 12. 988a — 188b + 45c. 18. 70 m2n + 168 mn? +110. 14. 99 a2b + 67 ab?. 15. 48 rqr + 1539 grs + 491. 16. 29 abcd + 96 bede + 178 cdef. 17. 100a + 100 ab. 18. 1004. 19. 129 — 47q —176r. 20. —15%+ 63y 4+ 442. 21. 155m+ 38n-+ 31 p. 22. 12% —182y +4 155z. 28. 14932 + 1637 y — 149z. 24. 1900 a — 10006 + 1000c — 100d. 25. 100a+ 100b — 50e. 26. 1052242 + 2938 ay + 1271 7. 27. 5a+1864+4c, 12 +134 42 = 67. 28. 43a2+ 662, 47 —17 + 67 = 97. 29. 2a27+4b+ 4c, 24+124+16= 8380. Page 34 1. 45% + 88 y. 2. 77 ax + 95 by. 3. 5622 + 90 72. 4. 6.23 abe + 1.79. 5. 24a + 426, 145 — 55 = 90. 6. 20a+12c, 1389 —68 = 76. Page 35 Peelieeeee AS weeo wis. 4. be" 0,057) Goll ania ise) Seeds Pages 37, 38 1. 8ay + 20yz + 7. 2. —4ab+ 2cd. 3. 34 m2 + 32 n2 + 10. 4. 48 m?+ 35n?+ 2p?) 5. —Ta—8b+7c. 6. 40a%+ 706% 7. 7a+564+2c+10. 8. 7Ta+170?—6c. 9. —8abe +11. 10. — m2 — 222, 11, — 542+ 24y?. 12. p?+ 43 pg. GRAMMAR SCHOOL ALGEBRA . 1744. 14. 151 xyz. 15. — 26.43 4+ 128 a’. . — 162122 — 1. 17. 7 a2b + 6ab2. 18. 148a@ — 631 b. . 2m2+ 14 mnp. 20. 21.1%4+2.9y. 21. —69a4+2b—2c¢. . da2+ 11 a. 23. 77?4 2. - 24. 71 p?+ dq. .—- 24+ 6y—2. 26. —18%7+6y+4 8z. . 182a¢y + 42 yz + 56z2w — 81. - 28. 3423 — 4522-172 — 80. . 120a4 44b+4+ ¢. 30. —2a?+6ab+ 8b? Page 39 . 1042+ 10 y?. 2. 3m’, 3. — 3abe + bed. . 8434+ lla —10. 5. 2m?4+2n2?+2a+y. 6. b—4c— 30d. . e+ ary + y?, 18 —(—1) = 19. 8. 10a2+ be+d. . 8—a@b+ab?—b% 10. 3a%4+ 2024 8c?. 11. 4% 4+ 2y? —2z. . 8224+ By% 4+ 22, . 6x2 + 11 y? — 2122, 129 + 344 + 395 — 125 — 498 — 159 = 86. . 21474 29xy + 16 y?, 74 — 258 + 1702 +1889 — 913 + 407= 2406. Page 40 Ais 2. 2. 3. 18, 20, 20, 24. 4 Dt, Osta Page 41 a?+ 2ab+ 62 — a?+ 2ab—b2=4ab. .m—2mn — 8n2— m+ 2mn + bn? = 2 n?. p+ g@— p?+q274+2 p?— g@— p?+38Q@= p?4+ 42. . — 6a°b + 6 ab? is the simplified result. . @b+b2e+c2a—a2b+ b2c —c?2a + a2b+ 2 b2e —3 c2a=a2b+ 4 b2c —8 c2a. 522+ y? — 10 2? is the simplified result. Page 42 . 2875 a. 2. 9683 a. Sue LOLry. 4. 1794 ary. . 240 abay. 6. 240 abzy. 7. 625 mnzy. 8. 625 mnxy. . abex + abcy, 5a+ 5D. 10. abx? + aby?, mna + mnb?. . ae+ bx, 2ax — be. 12. bp? + bg?, 5am + 5az. . apg + b3pq, ba + by. 14. 6 da. 9hr., 6x74 Dy. Page 43 . a3d3, 2. a2b2c?2. 3. x22y + xy. 4. abbict, . on 6. 423 + 42y?. 7. 2m4*+ 6 mn’. 8. 60a’. . Baim’, 10. 16405 + 2464. 11. 2a%y + 2ay?, 12-5 — 60, ANSWERS 5) Page 44 . 1227 — 6 xy. 2. —4a3+ 4ab?. 21 x2y — 252 xy?. 4, 32 a*+ 32 a2be. — 15 atb — 15 a?b?. 6. — 41 a4 + 41 xy?. . —17a&m + 17 am. 8. — 45a2y — 105 zy?. .@t+tabt+actad+ae. 10. 756 a2be + 882 ab2c. . 26m3 + 389 m2n. 12. —atbc?d + a2b2c2d — a2bctd + a2bc2d?, Page 45 . 15 pqs. 2. 9arxy?. 3. 16.4253. 4. 3q'. . Dax: 6. 3m. 7. 2men. 8. 24. . 24 bd. 10. 12 273y3z3. 11. 16 22y?, 256 + 4 = 64. . p= 800 rt + rt = 300. 138. 9=1897 +27. Page 46 . C2 + y?, 2. 522+ Ty?. 3. 2a4+ 60D. . ne + py?. 5. c2 + a2, 6. a27+4a-+4 5B. . im+i1i. 8. 8a3 + 11063. 9. 1+ 2a 4.3 22. . 72 + 2 22, 11. 14 1382y. 12, 1+19y. .a@+3a2?+4a-+ 5. 14. 3a? + a? + 5a + 25. . wy2z + 2a2y + 8 yz? + 5 yz. . Saury2z? + 24 ayz + 11 w3y3z3 + 1. . dm+ 5m2n + 8mn?2 4+ 15 n?. . lett 17 a8y + 18 22y? + 23 ay3 + 87 yt. Page 47 . 2a%e + 2 ax? + 2 ax’. 2. 4m2nt + 12 mn?2. 3. 4ay +2. . 2a2+ 2a. 5. 12 22y + 13 yz?. 6. 16 a2x?. . dd5axy. 8. 2m* +2 m? + 2. 9. 30 mnz. . 69 pq. Lie: 12. 6p2q + 6 pq?, 36-5 = 180. 138. 21 pq?. . 405b + 4a7bt = 144. 15. x4 + wy? + x22? = 14. 42% + 224*7+324+5= 14. 17. a +a2?+a+1, 80, 15, 2. Page 48 5 a?, 2. —d3zy. 3. —llm. 5xyz. 5. — 1482. 6. 35q?. . —422 4 32. 8. —m+2n—3. = 9. 13 p?2 — 5q?. —-a7+32—-—9, Ble eee Bye ty, 12. —7m?+9m + 12. p—_ S rm or 6 O aT Cr HP CO rt oo Oo _ woneoa re GRAMMAR SCHOOL ALGEBRA Page 49 . 923 + 199 22 + 19% + 35. 2. — 14 p2q + 298 pq? + 85 g? + 7 p®. . 4204 + 35122 + 17 2? — 80x24 159. . 1015% + 658 x3y — 86 ay? — 78 xy? + 98. . 227 + 8425 — 6275 + 162 x2* — 25023 + 316 22 — 288 2. . —daeoy + 9xdy? — 12 cty3 + 243 a3yt — 78 22y5 + Ol zy?. m> + 3m4n — 9m3n2 — 146 m2n3 + 19 mnt — 8 n>. . oat + 25 ay? — 13 ay? — ll yt 4+ 17 wy. 11. LOST, Lies 1224) 6) oso 14. 12. 18. 16. 55. 17a. 18. 14. 19. 408. 20. 148. Sa ae OVP earl e 23. $170. 24. 1380. 25. 2100. 26. 8200. Page 50 ~2-7-m-em-n-@-u-e-y, 7T-18>-M-M-M-M-M-M-M-N- NN: Z. ~m(et+y), dx7(8e+ 7y). 4. pq(p+q), 17 mt (8m? + 22). . dp'g(p+5q?+ 79°). 6. ary + y2?+2+a + a). Page 51 . p(it+tr), $200 (1 + 0.10) = $220. ~8=ce+cer=c(1+7r) = $1756 (1 + 0.20) = $210. Page 52 . “(2 —8). 2 mi(m—n). 38. at(a+2b). 4. a*b(abr + y). . a(1+ b+ 0). 6. pq?r (p?q — 1). 7. e(v? +2247). . p2(p?+3pq4t1). 9. 24(402?+2241), 10. x(x?4+ 15% +16).. . 3m2n2z (m2n + 22). 12. 3y(9xy? + 3x7y + 1). . abc (ab? + ab + c). 14. 4m?(4m+38n-+ 2). . 5cd(ab + 2ef + 8cd). 16. 17a(a+ 3be + 9a?). . Tpg(sp+5qt8r). 18. x(78 + 842+ 4% +4 121). Page 53 . abcauy. 2. ab, 3. pquy. 4. pqrs. . mnpxy, 6. a2b2c?, 7. x(a-+ b). 8. pgq(p+q). . abc (x + y). 10. 27 ab (a3 — 8). 11. ab(x — y). . cy(m+n). 13, 123 ab (a? + 262). 14. mnp(m + n). 15. am(m + bn). 16. mx*(x + 3y). Li: 14. . 18abc(8a+ 5c). . l7uv(u— 7). . (a2 + 8ay + by?). . a(a? + 8ab — 40"), 8. . d(abe + bce + efg). . 27mn (a + 4 pq). . mnp(m +n — p). . D(p — 8p? —.4 p3 + 5). ANSWERS Page 5 . — par (pq? + qr? + 1). 18 . 2Lab(2a—38b4 8c). 20 . of pg(p + br — 2). 22 . For example, 5 p2qr. 24. . For example, ma —mb+mec. 382 . 47d. 39. 15 ab (a? — b?). . 27p? (p+ 2). 42. abcde. . d2abe(2a+b). 45. 60 abcd. 4 . mn2p (m2 + p?). . r(pg — gs + St). . Stay (dry + 1). 43 x2y(22+5y). . “(v2 — day +d3y?). . 19m2n2 (8m + dn). . m2(m2 + n+ 4mn?). . 22(8e+4y+ 52%). . 41 p2q’r (2 p — 87°’). . 84 (422 + Oxy? — 16). . — 2Qxyz(4a2y + 3 xy? + 2). For example, 3 aq?rz. . For example, 12 a262c?. 40. 25ab(a + 0). 43. 60 p?q?r. 46. abed(a+b+ 0). . Lbp?¢r(p+qt+r). 48. 3a?(a? + 2ab+ b?). 49. 2a(a? — 3ab + 0?). Page 56 ; 2 2, a py an Pees Anes pe b y? 45 a? 5 avy 2 2 2 2 5 é 4x v+y gy eo 9 2 + qs 10. 2? 9 atz3 3a a ps 4 2 1 22 8m +5n. eee ee cee Sane fies b 3 2 2 3 ) 3 2 9/3 Paty ot Pde rae Tae eee ee 16: See + 12 ry + 23 y3 r 3 Page 57 2 — maeige el. Seidler eine) cere 4 2 b c . —- — —— dollars, $1.25. 4. 10,id,r=d-+e. ao. 100 . 80 Ib. 6. 50 Ib., 25 lb., 100 1b. (AREER ; “m(p—qtr) m(p—q+r) . Numerators, ab (a? + b?), a? (a? — b?), b? (a? — b?). . Numerators, b?(a? — b?), ab (a? + b?), a? (a? + b?). . Numerators, be(a+b+c), ac(a—b+c), Pb(a—b+¢). . Numerators, 6 abs, 3 bep, 2 cdr. . Numerators, ab, b?(a — b—c), ac(a —b—C¢). . Numerators, 2 abcx, 2cy(a + b), abz(a + DB). GRAMMAR SCHOOL ALGEBRA Page 59 2 ab? 2. Le 3. nenp? 4. 2qrt 4 b3 12 g2r2 m2n3pt 2 pg2r? ab ade 2 pq aes axy + bay 8. x2y2z2 — yar b2 + be 2q? —2qr axy — bry v2y2z2 + y2z2 ab?m 382 11. ane Les be? ab2m? — ab?n? 3 w? +322 — 6we abe abe abe an bm 2% a) ea per? pq 1 "2bn’ 2bn 2bn a a8 * p2q2r2? p2q2r2? p2q2r2- pqr PAD tT ate “araqt” ar(atr) art” LYZ xz (y2+ 27) xy (y? + 2) "yz (y2 + 22)° yz(y2+ 2) yz(y2+ 2) am AC ie ac 18. Numerators, st, 2 pt, 3 pq. Page 60 Only the numerators are given to Exs. 1-6. 1. rq+p. 2. 8m2k + n?2. 3. 2a°b? ++ be +a. 4. ab —8ab?+b3. 5. m2n4+3mn242n3. 6. xy 4+ 2ay?2+ 34+ w. " 6 p?q + p? 8 12 m2nz + mn Thee , 3x 9 4 ad? + 2bd?+ ¢ 10 16.x2y + ldxy? + 3x 7 : aia ae 2 Aree SWeinyes Sk ese 3 3 11. 32a2y — 1l5y +t 12. 185 a 75 ab + 166 y oa Page 61 1 Lad 1. 227+ 5%+4+-.- Ee 3. m+2+ ae o 16m 3 17 b 4.2@+34 ’ 5. daeacoeees 6. 2a+0+ x 2 a2 6 a? Only the numerators are given to Exs. 1-9. IP eR 13. ON. . 126 m2n. . 14a + 280. 10. 1 4 py 5¢ ZC ay +2 ay? + 4 y eel. 6. 22. _ 60. ad + be bd 3 a?beq + 2 b?cdp 6 pq? . 308. 7 00; . 380. 5. ANSWERS Page 62 -on + 9+ . 10 pq. . 4 q?rxiy?. . 2xyz + y2z. n 2 w+ ie m2 38m + on OD: = o 9 eS ea Ov 25 On ad +bd+e ; ad 82 xy? 4+ 2laryi+ ex y Pal 5. 24. OU: 231. 5. 300. abew + abcz WLYZ 6 a? — ab? + 16 b2c 24 xy. 3 . tarbcyz. 6 . 52 (p? —4q). 9 ga” 12. 7 be ees 15. 3p+2 bat by 18. i . 9a. 21 . 9x+ ae 24 9 xy 2 . Ay + ae 27 8 xy 4 5pt+ 4g 30. 5 p2 6 5 212° +172 a 33. 2 Page 63 3. 35. 4 S200: 9 Page 64 3. 400. 4. Page 65 m2 + n2 mn 9 a2bn? +16 a2b2m 12 am?n2 38a 8 47 x3y? 2¢ " 36mn 4 are 10 26. 27. GRAMMAR SCHOOL ALGEBRA ad — be bd 10 abd — 9 abc 12 202 o2 =f y? LYZ mng — mn2p pq? ay? + bz? a+b ay? ; ile ofa lard ae par 2e7y + 2y3 — x + ry? 4 xy acdef — b?def + bc2ef — bed?f + bede? Bayz + yz + wz? + xy LYZ 4. ez — az + ytxe — yx + zty — zy LYZ Page 66 . 9 a? — c? 3 2ay — 2 bx "abe Zay 5 1 6 2mxz—8nxe+ 8ny Cite ; 6 m2n2 Page 67 2 — 2 2 2 gm Da g, Hod + abrc mpg c2d? 5 a2 + 2ab — b? 6 azx + ba — x } 3 a-b? aby 8 6mn + 5 n? — m2 9 ab+b+a—-—a 35 mn a2b2¢ Dares, EE Pee thee q abed 15. bd + cd — ab—ac abcd bcdef 18 2ab +-2be + 2ca abe ab? + 6? —ab+b+a—l1 20. cae oc een 99 a—@+a}—b+b?—68 ; ab PY gaan pa er Ls rer ey pqr ac abcxyz + bedwyz + cdeuzv + defurvw UVWLYZ 2 a2c? + 2 bc? + 2 a*b? arb? az2w2 + bz2w? + cz2w? + dz2w? + bxr2w? + ca2w2 + dxr2w?2 wx? y2z2 + C%2 w2 + ca2y? 4 dx2y? +. ex2y/? + fxry? wrarry2z2 ; Page 68 asb3 x+y a—bd ab — a? " ¢3q3 ‘Tmay "16 ney edn. 21. 24. 16. 19. . $. ANSWERS ala Page 69 ac? ac 3 (b+ c) a3b3c3 b azn? b2d bd? "mn mnsx3 zg c2z2 Page 70 Be 2.9500. 3.97. 4. 14 mi. heey b bh 2¢ Page 71 3 bed 9, ade. eae hy Wace pete as bcf ns?qr? yz 2 = Shyns 2b4e ane abmn (a 0) ow. 9 a’bp?q” 8. _ &btc aces 19, nt 3 10cr det qg atbtc? 12. 1. Wap e » § Cy) aa a de w mn? 2b (q2— a*b (a®— b) 7, 1. ; deat 19 eee 50) ss. mn? LY a2b3c3d3 6 ptgtrt abary> 8d (q2—b?2 LAC sala Pp > nap esl a a 23. (a — y)2xyz. 37 8 abcxyz (a2 + b? + c?) pqr Pages 72, 73 893 2 ye 9, PT, Gopeen a mn? dx? ab(a +b) 5, m2 (p— @) rae cd? pqry ar 2m (p? — 4q) P m3 (m + n) 9 2¢(a — 2b) py ; ; ays ; a2b2 4(q2 4 p2 2 CALs salle Ties {apo ee més yn a2b2 4 2 bd gy, Ae 1500. ac m? c2d? 2 3 pat 17. ners 18. x2y2z2. pq? abc Se oA 292 802 1 G2y8 resent 920. pres* OS aid ae 21. a2e + ab?. Pq 3 2 22. —(l+a+ a). 23. a+b+c. 24, Bei abmn 25. 2(xz — y?). 26. n2— m2. Q7; ite! sree a2h2c2d2 28. pr+ar+pa+pr. 29. afd +bde — cbf. 30, “4. urv2w? 81. (x — y)2. oe apy ke, gat ab2c?d7e hd wine Page 75 1. 42, 2. 10 3. 6 4. 1200. 5. 800. Geite eel; 8. 12. 936: 10. $1200. 11. $800. Pages 76-78 1. 2. Pe yep See 4. 12. Sa: 6. 9. 7. 42. 8. 108. 9. 168. 10. 23%. loro: 12. 310. 13. 210. 142712. 15. 42. 16. 28385. 17. 70. 18. 7. 19. 90.8. 20. 5. 21. 550. 22. 13. 23. 15. 24. 3. 25. 1. 26. 90. Pi hee hb 28. 15. 29. 50. 30. 87200 31. $7500. 32. $7100. 33. 56. 34. $2560. 35. 200. 36. $2480. 37. 900. 38. $950. $9.57; 40. $1750 41. $350. 42. 21. 43. $2550. 44. $2400. Page 80 1. a2+2ab + b?. 2. 274 2ay + y?. 3. m2+2mn2-+ ni. 4. 4a? +4ab+4 bd? 5. a? —2ab + v2. 6. x? —2ay + y?. 7 9m?—Gmn+n% 8. 9 —122?+ 1624. 9. a? — 02. 10. a? — 02. 11. vw -—4y?. 12. mt — 9n?. 13. 2a? — ab — 602. 14. 14 a4y4 — 19 x2y? — 3. 15. 3024 — 1322 — 3. 16. 12 m2n? + 19 mnzy 4+ 4 2x2y2. 17. a2 4+ 2ab+4+ b2 + ac + be. 18. 224+ 822y + Say? + y?. 19. m3? — 3m2n + 38 mn? — nn’. 20. 8a? + 12a2b + 6 ab? + b. 21. 621, a? — b2, 27-23 = 252 — 22 — 625 — 4. 22. 1764 sq. ft., f2 + 2ft + t?, 4024+ 2-40.2 + 22? = 1764. 23. $729, # + 2ts + 8%. 24. 96, etc, GRAMMAR SCHOOL ALGEBRA ANSWERS 13 Page 81 Only typical answers are given for pages 81-84. 1. p? +2 pq + q. 4. vt — 227t + PP. 9. min? + 2m2n +1. 12. ptg?r? + 4 p2qr + 4. 13. 4a? + 12ab 4 96. 14. 9a? — 12 ab + 4572. 15. 252?y? + 10 ay + 1. Page 82 1. ab? + 2abe+c?, 2 9—6yF 4+ y® 3. y+ 2y> +1. 4, 4—475 + g1, 5. a2+10ac+25c2, 14. xty® + 27273 + 1. 18. 9+ 302% + 2522, 21. wey? + 2 wryz + 2. 24. 9m4 + 380m? 4+ 25. Page 83 3. at—1 5. a2b2c? — 4, 7. 1 — mxt. 9. 9 — 1626. 15. 144 —-49 = 95= 19.5. 27. (m* — 1)? = m§ — 2m4 + 1. Page 84 9. pig? + 17 p2q + 70. 12. x2y2z2 4+ 17 xyz + 30. Pages 85, 86 1. 2322 — 370% — 357. 2. 1892? + 39a — 518. 3. zt + 2922y + 210 y?. 4. 25524 — x2 — 56. 5. 25522 — 72ay — 12 y?. 6. 52522 + 100 cy — 161 7. 7. 22 y2 — 125 ay + 22 22, 8. 527 x2y2z2 — 1838 xyz — 14. 9. 1385 abbtc? — 57 a*b2c — 56. 10. 810 xty* — 51 x?2y2 — 3. 11. — 18122? + 183 zy + 24 y?. 12. 75999 a? + 7707 a — 25872. 18. 48471 at + 962 a2b — 85 b?. 14. a + b?. 15. a3 — Bb. 16. a2 4+ 8a%b 4+ 3 ab? + BD. 1%. a® — 3.a2b + 3.ab? — b3. 18. at — bt. 19. 7871 a2b2c2 + 1286 abc — 17. 20. 4836 a2b2c2 + 264 abe + 3. 21. 823 — 2622 + 25% — 6. 22. 322° —-16 242 + 142 — 3. 23. 2142 a? — 2601 ab + 629 B?. 24. 10767 atx? + 2370 atx + 63. 25. 823 — 12 a2y + 6 xy? — 7. 26. 2723 + 27 xy + Day? + ¥3. 27. a + 12 a2b + 47 ab? + 60D. 28. 29584 xty* — 24, 29. 714 ptqtrt — 25 p2q?r2s2 — st. 380. at + 403d + 6 a2b?2 + 4 ab3 + F. 31. 6323 + 46a2y — 34ay? —12y3. 32. a3b3c3 — 3 a2b2c2d + 3 abed? — d?. 33. a2x6 + arhaty? + ax2hyt + axrtby? + abx2y3 + by, Se woonrt >» = wo Ot Pp KF bob GRAMMAR SCHOOL ALGEBRA . at + 40° + 6 a2b? + 4.ab3 + DF, . 29063 + 8 aSbtc? + 3 a8b?c + 1. . 4803 — 74 x2y + 421 zy? — 150 y4. . wt —4axiy + Oxy? —4ay3 +4 yf. . 8192 x6 + 1381 x* — 934 22 + 41. . 567 xtyt + 48 w2y2z2 4+ 28, . 1829 — 83 a5y — 89 a8y? — 56 y3. . 216 a8 + 756 a2b + 882 ab? + 343 b3. Page 87 .6. 2.4. 3.3. 4.9. 5. $950. 6. $250, $225, $375, $562.50. - 34%, 13%, 42%. 8. 6%, 4%, 3%. 9.938 ¥ru, 2.2 Yolen 2 eey te . 22%, 14%, 83%. 11. $3850, $346.815%, $340.62, $366.82%. 12. 4%, $270.40. Page 88 - (a+ Dd). 2. m(a-+ Db). 3. v(v+2y). - a(p—7). 5. xy (x + 9). 6. p(p— 34). . bc(a+ a). 8. ax(x~+y). 9. b(a+ 5c). . 8q?(p+24q). 11. x(mx — ny). 12. c?(c2+ 3d). . 2y (a2 + 3 yz). 14, xy (ab + wz). 15. c(my + nz+ qu). 16. 4ay(8y +22 +4 2?). Page 89 mad g ttl. pees At eee m—1 x—1 a+1 y+2 pls eas ets es n y+4 Yr Yy eA) oe Sse eee 19, Lee 7T+2y z 3spa+tr Page 90 . (2 — m)?. 2. (x + n)?. 8. (7? + 1)2. . (1 — 2?)2, 5. (2¢% + 1)? 6. (1 — 22?)2 le eee ye: 8. (4a — 1)?. 9. (xy + z)?. (ab — cd)?. 11. (pq + 10)?. 12. (abe + 11)?. (5 a? + 7 b?)2, 14. (622+ 5 y?)2. ANSWERS 15 Page 91 - (p+ 2q)(p — 29). 2. (8a+2b) (8a — 2d). ~ (4274+ y) (427 — y). 4. (8m?n + 1) (8m2n — 1). . (7+ 11m) (7 —11m). 6. (1 +10 abc) (1 — 10 abe). -(a+b+c)(a+b—c). 8. (x + a)2. . (6a + 5b) (6a — 5D). 10. (12m + 5n) (12m — 5n). . (6 pg? + 11) (6 pq? — 11). 12. (9 + 5 abcd) (9 — 5 abcd). . (a? + b)?. 14. 25b?(a +c) (a—c). - £(a2 + b+ cy). 16. (2m? + n)?. . (a2 + 9 cd2) (8 a2b — 9 cd?). . (a—2b)2— 42 = (a — 26 + 4) (a — 20 — 4). . (a+b+ 2) (a4 b — 2). 20. [1+38(e+y)P=(14+82+3y)% Page 92 . (x + 7) (x — 6). 2. (v + 2) (x + 5S). 3. (% — 2) (x — 8). . (a— 9) (a+ 2). 5. (p+ 9)? 6. (a7 + 9) (x? — 2). . (m+ 7)(m + 5). 8. (p — 10)2. 9. (pq + 4) (pg + 9). . (ab + 10) (ab — 4). 11. (xyz + 3) (xyz + 8). . (mn + 9) (mn + 8). 13. m (n* — p? — q?). .(a+b+c+d)(a+b—c—d). 15. a?b° + 138 abscd? — 68 cds. Page 93 a+b ry mee 3. 1 re elem ab a—9d v?+1 a+2 x uss 4 ile ais pe eee aoe yea x+1 p—6 y* 2 — a aan ar 10. 4% ye 11. 1 Ta 12. 2m n c2 xz — 20 1—9z2 2m+n Page 94 mol: 2. $763. 3. $768. 4. $9750. . $10,750. 6. $8200. 12.3%, 8. $5500. . $90 each. 10. $80 each. 11. $9000. 12. $27,500. Page 96 oY. 2. 2m—38n. 3. 0+ y. 4. 2p—4q. . 6r-y,. 6. a24+2ab4+ 0% 7. 5a—2b. 8. 20pqr+1. . 4ey+7 . 1—Tpgqr. a 4 5 .m+2m— 5+ m + + 2ay + y? — . da+2b+4 — . SO) SS = - 4m+n— . 8 — xy + ey? — y?. GRAMMAR SCHOOL ALGEBRA 10. 14. Lit, 2m? — 15. 2 x2y? + 8 22. w+ y. 11. 38 —17ay. 15, 222 — 4% —38, 4027+ 527 — 6. 12. 2ay + 32. 38, 8x3 —2a%7 —127+43. 18. 38x —y, $2. 19. 3872+ 4ay 4+ y?, r+ y. Page 97 2 10 7/2 oy »x—-syt+ ee. w+ Y Ty? + oe cea Daca spr 6p —q i TM TS 5 y 10 ab + - 2 CUM nas oe ~y oh es + 5b 2a+b 3 b2 20 ab + 5b2 iD 38 pq + 7¢ ox say —13y? 3mn i: ae n2 2m+7n n2 seein SAI, 5m+2n ‘ cs Page 98 a —xy + y?. 2. 4. = xy + xy? ea xy? + yt. 6. .@+2a4+3. 8. . 16mt*— . (&+y) (4 — y) (2? + 9’). aA2p +41) (29,1) (4 p24). . (3a +2y) (82% —2y) (9224 4y%). . (cy + mn) (cy — mn) (x?y? + m2n?). . Bpgr + 1) (B par — 1) (9 p?q?r? + 1). 10. 12. 8m3 + 4m2—-2m+1. 3xr+2y 3mn — 12 n? 2m +5n— 21mn + 3 n? Bm—2n | at + asb + a2b2 + abs + dA, 4¢7 +2241. 1-—3a+4+ 9a. 1+ 4ay + 162272. a?b2c2 + abed + d?. (2m + n) (2m — n) (4m? + n?*). ANSWERS ae Page 99 Only the numerators are given. 1. 2 —52—14. 2. 2 —52+6. 3. 28 + 622. So (0 — See). 5. 6a3 — 5a? — 22a +4 24. 6. 8a? — 8la?+4+ 78a — 56. 7. 6a —19a?+41la+6. 8. — (8a’ — 7a? — 2a + 8). Pages 100, 101 Oe we D) 6 TE a et pat ee pe ee VD, 4 i) m ye 6 2x3 + ay " ary +b 8 am + an + m Oy gy m+n eye. 2 ta, Prtueeta oe 9. Y ae, 10. m Aas 11. m mk y m m ENG 72 72 Ieee ie bee ce 4 sae 19. 2b? 13. a2 + ab = ab —b Cc 14. a+oed+a le a—b a—b cl 3 g- evn? Sek) BE. pote Ot La 15 c 16. az+a 2 17, Ut# 3H + 2 x+1 a—1l Ag 9 2 © 2 6 Dee fig, aay oe ee 19, -@ —8ats Dg a—2 2 nt — 33 Dina 6 ot ipl ee 20. 3p 3p? + 21p 8 21. e4+at*+ 3a 18 3 p? 2 4 3 2 = 2 3 Diy eee oie Te. go Bteta+a-1 Ps alle ova pL ey a Pg 2 46 — 245 BRAC Sn 2 x8 2 2 94. oo 3a°+ 3a 2a - 25. 22 + ay + ty? az+1 e+ y wer} ve 2 ie — gq2y72 — 2b 26. 6m + 17m O17. 124 +22 21 28. arn 2b 2m 3 4x+7 ap 0 — 2m2 2 € Dy o 73 2 2 2 29. 2m +n? 30. SUA c 31. 2a2+ac?+a ale m—n 2ny a+l 2 Qp2 _ 33 ‘ 39. 2 a2b — Gab 33. 10 a2b? — 14 a2b + 8ab+2 a—b 1— ab 34 16 m> — 7 m4 — 15 m3 + 7m? — 2 35 8 x3y3 + 5 aty? —4ey4+1 m2 — 1 , ry+1 36 m* + m3 + m2 —m + min + m?n + mn + n m+n 18 37. 39. 41. 43, " 2 (2 p 13. GRAMMAR SCHOOL 2234+ 38x2y + 3827? ty SDA gd prd — 5n? 3m—5n 3552 a2 — 777 b2 + 666 c? + 37 111 Page 102 a OU ae cere 2. a » pg+8. . a2+ ab + b2, a2 — ab + 02, Page 103 a+od a? 2pr Pp") —r) r(2p—7) a(a?+ 1) az#—1 " @2(a2— 1)’ a2(a2—1) a(a +c) b? “be(a+e) be(atc) x(x + y) oe yz(ety) yz(ety) Li "(1 100). pay pg(l— pq) p*q2(1— pq) yore ete wy (x+y) cy(@@ty) 15. 22. 38. 40. 42. 44. ALGEBRA 3273 + 4007+ 42% —5 e+1 2ab — 6? + 8ac4+ 3be a+b , —4ab+0?+2ac a—b 28 a2 — 20b? + 10 c? — 3 20 d? 2 b4 3 — q2b + ab? — b? + ——__.- a+b 11. zy + 4. 14, 7?4+22y 4 y?. m+1 m2 3 m (m + ey m (m+ 1) a(y + 1) bx xy(y +1) ay(y+ 1) m2(m?+1) m+1 * “m3(m +1)" m3(m+1) m(m + n) n n2(m + n) "7 (m + n) DY Ae Dae p(p—4) p(p—4) BAe Dh ie: @r(l+p) Pr(+p)’ (m— 1)? m2 m(m — iby m(m = 1) 12. 14, - ANSWERS 13 Page 104 e+7 9 ¥t4 3 p+2 mare “g+9 yore Ges Bon 2 %— 4 6. pq —7 7. cy —4 g ttBy x—6 pg —9 zy —7 c+t4y mn — 21 10. abe — 2 fhe Aas 12. mn +7 mn — 22 abc — 5 pg —2r mn + 8 Page 105 v§4+273+222? +2741 9 —~e+at4+r3—g72?—24r4+1 e+] : 1 > m2 — n? — pm + pn — p 4 men? — 138 mn + 2 m2n — 26+ mn? m—n mn + 2 pg—1 c—y 99 x8y7 — 158 xoy* — 17 + x? Day 528 ml} — 391 m4n}2 + 391 m1 — 288 nl? 23 m* +17 x + 23 — 3 y—1 . £8 + 43 + —______.. 10, y3 — 3 : ep Sey ESE Re Le aera 6 aod 37/3 —_ Re es Ce) oe ane) eee g-+11 wy +1 9 y* 13 . 8+ 327%y + zy? — xy? + ——. 14. por —16— a+ y aoe eee nee a are 7 817, Ce mn — 5 w(e+y) x(x+ y) a(a—1) a—-1 VO et Py eee) s a a ay(e—y) ay(e@—y) 2ey(e+3y) («+ 2y) (4a — y) 21 ab (ab+1) —(ab—1)? “6ay(v+2y) Gay(x +2y) "a3 (ab—1)' a'b3 (ab—1) BAP Gc PG) (Pp — 2944), 4(p — 9q) Ago) Bap — Apr dg?) (4p +9) (pb? gs) 2G(4p+q) 2q3(4p +9) a(a+b)? (a —b)8 ax(a — bv)’ ax (a — b) 20 22. GRAMMAR SCHOOL ALGEBRA Pages 106, 107 . 4, 2e+y a2 + 2ab — b? 2m+n ox (e + 9) a(a — b) " m(m +n) 222—1 5 2 6 m+mn+n @ (x? — 1) ob "-n(m+n) ere Pele Py gee eat EN 9. 21. b(a + b) p(p+q) — 292 22 _ $1200. 11 6pg— 8g 12. Oe Die p*(p — q) abc (b + c) am+4a2-4+ 22 14 acde + b?de + bc?e + bed? "m2 (m + 4) bede v3 + 72 — y? 16 e2+eytatyt at — xy a? (x + y) . a? (x + 1) 2 (m2 + mn + n?) 18 222y — 2Qry? + 3 "m2 (m + 2n) xt (x — y)? 8p —q4+8p24+11pq4+6¢q? 20 5 3 + 29 7? + 98¢ 4 11 3p (3p + 29) x? (aw + 2)? 180. 22. 282. 23. 3. 24. 9. 2OpmLte 26. 52. veld 28. 4. 29. 7. 30. 5. 31. 48. 32. 5. 33. 12. 34. 10. 35. 32. 36. $1250. Pages 108, 109 patel Ye Ee ett al g Pte —py-1 x (% + y) a (% — 1) q(p — 1) abe — axy — aby + bry 5 pg —3p—3q by (b — y) pq (p + 9) a3 + a2b2 — 2 ab — b? a2 + 62 —1 8 —2b b (a2 — b) abe "(a +b)? 2 Roos cit lara 10, @ 2 2@t!, Oh. ih P(p — 4) a(a + 1) 138. 10,192. 14. 2. 15. $250. ay" + yi — xy — 2% ay euciey 1g, PY PT PT yy + 2) a (a? — 1) Oped) pg — 8p — 21 20 ayy 21 2y + vy — 227 " -p?(p +7) at (a + y) y (x2 — y) x+22-—624+9 3 axy? — by — bz 24 3 (a + 2) 22 (2% — 3) " g2y2(y + 2) " a2 (a + 8) 25. 28. 13. 19. ap ALTO ONE a—2b ANSWERS a | vty — 2+ y3 26 3ab+3 6? — a? a2m — 2 b?m + 4b xy? (43 — y?) ata 2p) 6 m2 (m — 2) 30. 29. 30. 30. 14. 31. 21. 32. 70. 33. 33. n Ad. 35. 56. 36. 200. 37. 35. 38. 625. 39. 28. Page 110 e+ 9 Pa(e+y), 3. (a +d)? Den ae c—y 15z n(2m + 5) 5. eae 6. (8mn — 4)". Page 111 a(a — b) v2+1 q x b(a + db) oes tl 6 p y (x — 1) Bip hee bts 3(a + 6) b> Page 112 28. 2. 160. 3. 33. 4. 28. 5. 54, 210. 7. 40. 8. 4. 9. 36. Page 114 8.4. 2. 37.5. 3. 15. 4, 5.5. 5. 1. 6. 14. 1s 8. 0.02. 9. 50. 10. 80. 11. 33);. 12. $1000. 320. 14. 20 15, $1200. 16. 32. 17. 5. LSS: 2 20. 51 21. 60. Page 115 ahi, a1: 0 ee 4. 8. 52 6. 6 Page 116 2 = 88 213 ; bal) Aer: 5. Gat + 6ab — 2a a+ob+1 3a+b bs ‘ 3 a2 SSO Dy tana Al) Cys iy AGP gt =sh3 ligne 2716; gp) OO O81 glee Ce ca eee d+ ac a+ob-—1 6. 15. 7. 50, 40. 8. 75 yd. 9. 60in., 40 in. 10. $60. 11. 30, 20. 12. 4. 13. 10. 14. 24. 15. $1100. 16. 10 rd., 29 rd., 290 sq. rd. 17. $1500, $8500, $5000. 18.2. 74ns, 3.0m. ee ip: 19. 450 mi. 20. $5000. Page 120 1g? re NY Sali. 4. 30. 5. 1383. Cin0, Page 121 1. 8 ft. from 56 1b. 2. 160 |b. 3. 751b. 4. 1 ft. from stone. Page 122 1. $11.20. 2. 68} min. 3. 63 mi. 4. 155 ft. 5. 1), ft. 6. 561 mi. endo; Page 123 1. 2 mo 2. 28, 3. 42 yd. Page 124 1. $1210. 2.5da. 8. 27da. 4. $168.75. 5. 1172 sec. 6. $864,000. 7. $1.05. 8. $750,000. 9. 7i da. 10. 8da., 6 da. Page 126 1. 63 pt. 2. 248cu.in. 3. 1793 cu. in. 4. 22 in.; Sbint 5. 22 sq. in. 6, 352} lb. Tam leas 8. 1.8,% in. Page 127 1. $150, $225. 2. $2037, $582. 8. 455 ft., 1183 ft. 4. $2073, $2764. 5. 361, 209. 6. 308. 7. 997.92, 3754.08. Page 128 1.718; 2. 44. 3.1472 o Seals ab 6. 4.15. CO: 8. $275, $750. 9. 28.8 sq. in. 10. 131.2 sq. in. 11. $9. 127 12* AB: 13. $1225, $2450, $3675, $4900. 14, $1000, $1500, $3500. 15. 12, 8, 12, 20, 28. GRAMMAR SCHOOL ALGEBRA Pages 117, 118 ANSWERS 23 Page 129 Each root has the sign + on pages 129-132. teob: 2. 24. 3. 15. 40213 Sa. 2 7, Geek tyne len kat Sie 9. 19. 102° 32. Page 130 Only a few types are given. 2. 3+ 22, 3. 2a2?+4+ 1. Boo ts 9. m+ 9. 12. p+ 5q. 14. 7—8y. Page 131 Ler bi. 2. 61. 3. 63: 4. 71. 5. 79 Page 132 1. 8.97. 2. 44.1. 3. 1.05. 4. 0.343. d. 0.907. Ga0-5s, ToULie 671234. Orel: 10. 4607. 11, 7008. 12. 9812. Logi 4) 14. 2.236. 15. 2.645. 16. 2.828. Lieu LUO: 18. 33, Page 133 ike) Soa 2. + 14. 3. 15 in. 4. +14. 5. + 15. 6. + 12. fie SORRY Pages 134, 135 1. +30. 2. + 40. Sen 25. 4. 5 sec. 5. 10 ft., 15 ft. 62 20 1t,, 18 16. Ted, Se i4 it 21 in. 26 tt. 9. S4in., 70 ft., 252 in. 10. 20. Lime 25: Py Bop se ish Se 14. + 89. 15. +81 16. +110. 17. +90. 18. + 221 1927-61: 20. 121. 21. §1 22. 31. 23. 31 rd., 93 rd. 24. 142 in. 25. 142 in., fin: 26. 100 rd., 25rd. 27. 12 in.,12.48 in. 28. 7 in. a*b?(a — b) + ab(a + 5) (a + ¥)2(a — b) Page 136 2 x?2y 3 4a+b “(@+y)(t-y? 40-28 10. 13. 15. 10. 13. GRAMMAR SCHOOL ALGEBRA 2 pqr —1 ” pg’r? — 1 at + yt" uy? («+ Y) a2 + b2 (a + b) (a — b)2 . m2 -—m + 2. ab — a*b — ab? a2 — b2 a3 — ab? + a2b — b3 — a3b — ab? ac — a2— b2 + be (b — e)(e —a) at — bt a> — 3.a2b — ab? + 63 ~ b2 — q2 Water ah p(p+q+r) x 6 a2b + 2 63 (a2 — b2)2 Gea Cail eee (4 m3 — 5n?). 2m . ab(a — Dd). m—1 m+1 11. we + y2z? g. wy + abx2y? - a?b? yz(w + x) ; xy? — a*b? 82n + 12m 9 207 — 2xy " 9m? — 16 n2 "(a+ y)? sr+y4+3 1 _ bayz +a g?— y2 " ab(x + y + 2) Page 137 j oes 2. 4 xy : 3. 9 3a x? — y? a? — 9 5. 4@b gt 2mn b2 — at n2 — m2 ry EL Viraie: + il 37d — 27/2 aa 10, Vay? + ty, 1 — xty? 12. Ee oo as (f=~1)? A ig 4 a?b2 — 1 82+ 6ayt+ y2— 822 — ry? 16. 2(3a + y)(2@ + y) Page 138 m(m + 1) rey, n(n — 1) 38xu-—y 6 2m+1 r+3y 2 weal a? + 0 9 (m —1)(m +1)8 a+b 4 Ao 12. b—c. 14. i : ab Page 139 . 2ay (4ary + 8). 3. (a + b) (a? + 6?). e—l 6 “+ 4 +3 ated 10. 13. ANSWERS 25 x? + y? (eae to 8. (e+ 1)(4 +2). 9. ee a ae 2 z : 11. m2n2?-—3mn—10. 12. oe) a+b-+ec x—y (xy — 1)? 14 a* + ab + B? a2 — y?2 . a2 + b2 Pages 140-142 2, 2, —b4 aud. 4. 8. 5. 10. Gal qgee.8. 8. 9, 22, 10.21, 11. 22=0..2=0. 5. 13. 4. 14. 2. 15. 5. 16. 24. Lie sh 1. 19. 4 20. 38a PE howe aces 23. 13. 2. 25, —1, 26. —10. 27. 7 28. 7 peo ahs eames ree) Ao) go eames, 634.10. a ihe Nees PSG SUM Mase ee 6 10 382 4 0 3 5 Page 144 2. 2 44 ae 4. 5,2. 10, 1. 6. 4,5 Ce gale 8. 6, 1. Oa 1; 10. 8, 1. Tito 12. 20, 5. 10, 9. [ance \Seeano vert 16g aa Page 145 2, 3. 2. 4, 6. ae Goth 4. 8, 2. jay 62 1 Ch te ey eee as —d, —7 LOD LOB: lL. 12, -— 9. 12. 6, —8 — 9,6 14.3: 30. 15. 32, 16. 16. 100, 50. Pages 146-149 8, 11. Dy ey) Per hele eo eeY Pep e Bok, Gah ve<-6, 7.010, 10, 8. 26, — 30. 58, 41 10. 4, 7. 1173 12. 3. t 14. $600, $400. 152 10,12: 16. 14, 16. <4 Re: 18. 18, 25. 19: 77, 33. 20. 36, 24, 10, 24. 22. 4, 3. 23. 54, —50. 24. 30, 40. 70, 80. 26. 63, — 4. 27. 4, 4. 28. 55, 6. 35 37 38. . &+ 3800, + 800 . 4% (4 + 1) = 9%, x = 8. GRAMMAR SCHOOL ALGEBRA 3, 6. 30. 1, 3. —15,-—16. 934. 144, 64 3, 2. 38. 2, 56 Be RBy Py i, . $8-80+2 = 12(80+ 2), 2= 10. _ c+ 1B 2 ey 1848), 48: — _850 Sa WO) oe Lif + 300), x=? Sli, 20%) eee SOG 35,°0.7.\d. eens 6h Cire $95 755.60.) 940; se 43. 7,7 46. 6% (1 +2) =8%,2=? 48, 50. 500. ie+1)=2+},0=? Pages 151, 152 GD ad? + be 3 a 4 a? ; d—b b aac m—n — 26 i paras ae) 10m 1l+a+b—c dare teh ab—¢ 13. 1,2 at—1 a—7 a 2 avs 6b ieee ab 16 Aah Gee WR = DN ae D een g pean 19. 2 2 dm +067 0.— a 21 s+d “n+tac’ n+ac ‘ 2 ; 22% 24 ae Te 24. as 2.0Gh 25. a? a: p= Nae — 2 ab 22 28 scnelee Ee Ge 32. ab Aes Ne AL cake t 34 be—a? b?— a rae e(a be) c(ac — b”) 36 a(ab — c2)’ b (ab — c2) 10—7)b Dh 25.D “a(6c—5b) 6c—5B? St VPN a, a= +2 Vb? a2, ae Aer a+e pie Ee ge m+n — Dp l—-m +n—p mtn bm—an 11. oe a+b a+b 14. Re ) 2 2 2 eae ay AE 17. “i ’ zh a+la+l TOV WE TIRE og og n—1 ese s—d 29. ab+3b 2 a c? — ac 26 3 be a a ne! leg a(t b=c ac—f eb hue b c(c — b) c(¢ — a) alas db) Bian) s Ne 26 Dane gan 39. + (6a + 5D). 40. 41. ANSWERS ab+ec ax = +(ab+c), «= +—— - 13,6. —«- 42..:«163, 88. 45. 21, 5. rer ene AL 46. 7, 8. bu —an bn—an Rage 154 . —2, 1. 2. — 3, 2. 3. —4, 3 wc A 8 Sh eS eae ay peeeney ye meeese saat eRe 1 ld = 65.2 27) [4p 6 ee el be oad 6, 6. 18. 5, 6. 19. —3, -—9 wee DP), ea 235 30234 11. 26. 4 yd.,7 yd. 44, 4. 8. 12. 16. 20. 24. 27, 134 ft., 6+ ft. 27 30 ct., 45 ct. ats Et iel: 5, 4. Bel ol ee 5, 7. -URBANA C001 ALGEBRA FOR BEGINNERS BOSTON 3 0112 017080703 UNIVERSITY OF ILLINOIS IANO UT 512.9SM5A