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 r^i.».»o.i^, giving au LUO 
 
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A Text-Book on Commercial Law. 
 
 A Manual of the Fundamental Principles Governing Business 
 Transactions. For the Use of Commercial Colleges, High 
 Schools arid Academies. By Saxter S. Clark, Counsellor-at- 
 Law. Reviser of Young's Government Clasa-Book. Handsomely 
 printed. 12mo. 300 pp. 
 
 The design of the author In this volume has been to present Pimply, 
 and compactly, the principles of law aflFectingr the ordinary transactions of 
 commercial life, In the form of a Class-book for Schools and Commercial 
 Colleges. 
 The plan of the book is as follows : 
 
 After a short introduction upon the relations of National and State law, 
 and of constitutional, statute, and common law, it is divided into two parts. 
 Part I. treats of principles applicable to all kinds of business, in tiiree divi- 
 sions, treating respectively of Contracts, Agency and Partnership, with a 
 fourth division embracing^the subject of Corporations, and a few others 
 general in their nature. Part II. takes up in order the most prominent 
 kinds of business transactions, paying chief attention to the subjects, Sale 
 of Goods, and Commercial Paper, and Is to a large extent an application of 
 the principles contained in the preceding part. 
 
 The chief aim has been throughout to make it a book practically use- 
 ful, and one easily taught, understood and remembered. As subserving 
 those purposes attention may be called to the following features among 
 others-— the use of schemes in graded type, which summarizing a suliject 
 Impresses it upon the mind through the eye; the summaries of leading 
 rules at different points : a table of definitions : the forms of business 
 papers most frequently met with ; and the frequent use of examples and 
 cross-references. 
 
 The work is used in nearly all of the leading Commercial Colleges of the 
 country. 
 
 RECOMMENDATIONS. 
 
 FmrnJi. F. Moore, A.M., Pre/t. Southern Business University. Atlanta, Oa. 
 I find the work fully adapted for use in buRiness schools an a text book, on 
 account of its conciseness ; also to the accountant as a book of reference on points 
 of commercial law and business forms. It is the most complete and concise work 
 on the subject that I have peen. 
 
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 Send to my address, by freight, 200 Clark's Commercial Law. 
 
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 Please forward me, by express, 100 copies Clark's Commercial Law. 
 
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 The B. and S. Davenpobt Business College, Davenport, Iowa, Nov. 25, 1882. 
 You may ship us, by freight, 120 Clark's Commercial Law. 
 
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 Please ship us 150 Clark's Commercial Law, HOWE & POWERS, Prop'rs, 
 
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 Please send us 100 copies of Clark's Commercial Law. 
 
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 Please send us, by express, 60 Clark's Commercial Law. 
 
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THOxMSON'S NEW SERIES OF MATHEMATICS. 
 
 NEW 
 
 PRACTICAL ALGEBRA; 
 
 ADAPTED TO 
 
 THE IMPROVED METHODS OF INSTRUCTION 
 
 IN 
 
 SCHOOLS, ACADEMIES, AND COLLEGES 
 WITH AN APPENDIX. 
 
 BT 
 
 JAMES B. THOMSON, LL.D., 
 
 AUTHOR OF A SERIES OF MATHEMATICS. 
 
 Lb=SSSi^ NEW YORK: 
 
 Effingham Maynard & Co., 
 
 SUCCESSORS TO 
 
 Clark & Maynakd, Publishers, 
 
 771 Broadway and 67 & 69 Ninth St. 
 
 1889. 
 
THOMSON'S Mathematical Series. 
 
 I. A Graded Series of Arithmetics, in three Books, viz. : 
 
 Naw Illustrated Table Book, or Juvenile Arithmetic. With oral 
 and slate exercises. (For beginners.) 128 pp. 
 
 New Rudiments of Arithmetic. Combining Mental with Written 
 Arithmetic. (For Intermediate Classes.) 224 pp. 
 
 New Practical Arithmetic. Adapted to a complete business education. 
 (For Grammar Departments.) 384 pp. 
 
 n. Independent Books, 
 
 Key to New Practical Arithmetic. Containing many valuable sug- 
 gestions. (For teachers only.) 168 pp. 
 
 New^ Mental Arithmetic. Containing the Simple and Compound 
 Tables. (For Primary Schools.) 144 pp. 
 
 Complete Intellectual Arithmetic. Specially adapted to Classes in 
 Grammar Schools and Academies. 168 pp. 
 
 in. Supplementary Course. 
 
 New Practical Algebra. Adapted to High Schools and Academies. 
 312 pp. 
 
 Key to New Practical Algebra. With full solutions. (For teachers 
 only.) 224 pp. 
 
 New Collegiate Algebra. Adapted to Colleges and Universities. By 
 Thomson & Quimby. 346 pp. 
 
 Complete Higher Arithmetic. (In preparation.) 
 
 \* Each book of the Series is complete in itself. 
 
 Copyright, 1879, 1880, by James B. Thomson. 
 ^lectrotyped by Smith 8c McDougal, 82 Beekman St., New Yo'k. 
 
PREFACE. 
 
 XT has loDg been a favorite plan of the author to make a 
 -*- Practical Algebra — a Book combining the important 
 principles of the Science, with their application to methods 
 of business. 
 
 Several years have elapsed since he began to gather and 
 arrange materials for this object. Many of the more 
 important parts have been written and re-written and 
 again revised, till they h^ve found embodiment in the book 
 now offered to the public. 
 
 In the execution of this plan, clearness and brevity in the 
 definitions and rules have been the constant aim. 
 
 A series of practical problems, applying the principles 
 already explained, has been introduced into the fundamental 
 rales, thus relieving the monotony of the abstract operations, 
 and illustrating their use. 
 
 The principles are gradually developed, and explained in 
 a manner calculated to lead the pupil to a full understanding 
 of the difficulties of the science, before he is aware of their 
 existence. 
 
 The rules are deduced from a careful analysis of practical 
 problems involving the principles in question — a feature so 
 extensively approved in the author's Series of Arithmetics. 
 
 184012 
 
iv PREFACE. 
 
 The arrangement of subjects is consecutive and logical, 
 their relation and mutual dependence being pointed out by 
 frequent references. 
 
 The examples are numerous, and have been selected with 
 a view to illustrate and familiarize the principles of the 
 Science; while puzzles, calculated to waste the time and 
 energy of the pupil, have been excluded. 
 
 Special attention has also been given to Factoring, 
 Generalization, and the appHcation of Algebra to business 
 Formulas. 
 
 In these and other respects, it is believed, some advance 
 is made beyond other books of the kind. While adapted 
 to beginners, it covers as much ground as the majority of 
 students master in their Mathematical course. 
 
 In presenting this book to the public, the author ventures 
 to hope it may receive the approval so generously bestowed 
 upon his former publications. 
 
 J. B. THOMSON. 
 
 Brookltn, N. Y., Sept, 1877. 
 
 NOTE. 
 
 At the suggestion of several teachers, an Appendix has been added 
 to the present edition of the Practical Algebra, containing a selection 
 of College Examination Problems used for admission to Yale, Harvard, 
 and other colleges. These are preceded by a collection of examples of 
 a similar character, calculated to make experts in Algebra. 
 
CONTENTS. 
 
 Introduction^ ....•..-9 
 Definitions, ---••-•- 9 
 
 Algebraic Notation, ----••-9 
 
 Algebraic Operations, -----. 14 
 
 Classification of Algebraic Quantities, - - • ^9 
 Force of the Signs, - - - - - - - 21 
 
 Axioms, --.---•--22 
 
 Addition, -.- 23 
 
 Subtraction, 29 
 
 Applications of the Parenthesis, • " " ' SS 
 
 Multiplication, -------35 
 
 Demonstration of the Rule for Signs, " - " 37 
 Multiplying Powers of the Same Letter, - • - 38 
 Principles and Formulas in Multiplication, • - 42 
 Problems, ---------44 
 
 Division, - 46 
 
 Cancelling a Factor, - -- • - • -46 
 Signs of the Quotient, - - • - - - 47 
 Dividing Powers of the Same Letter, - - - - 48 
 Dividing Polynomials, - - - - - • 50 
 Problems, - - - - - - - - -51 
 
 Factoriuff, ----..--53 
 Prime Factors of Monomials, - - - - - 54 
 Greatest Common Divisor of Polynomials, ^ - (>Z 
 
 Demonstration, 6^ 
 
 Least Common Multiple of Polynomials, - • 69 
 
6 CONTENTS, 
 
 PAGE 
 
 Fractions, •..yo 
 
 Signs of Fractions, --.--..71 
 Reduction of Fractions, ----- . y^ 
 Common Denominators, - - - • - - 77 
 Least Common Denominator, - - • - - 7^ 
 Addition of Fractions, - - - - - 80 
 
 Subtraction of Fractions, ---.-- 82 
 Multiplication of Fractions, - • • - - 84 
 Division of Fractions, --.••- 89 
 
 Simple Equations, 95 
 
 Transposition, -----.--96 
 Reduction of Equations, -- - - - - 97 
 
 Simiittaneons Equations, - - - - 112 
 Elimination by Comparison, - - - - -113 
 Elimination by Substitution, - - - - -114 
 Elimination by Addition or Subtraction, - - - 115 
 Three or More Unknown Quantities, - - - - 120 
 
 Generalir^ation, 124 
 
 Formation of Rules, - - - - - - -126 
 
 Generalizing Problems in Percentage, - - - 128 
 Generalizing Problems in Interest, - - - - 13^ 
 
 Conjunction of the Hands of a Clock, - - - 133 
 
 Involution, 134 
 
 Reciprocal Powers, - -135 
 
 Negative Exponents, - - - - - - -135 
 
 Zero Power, - - - - - - - -136 
 
 Formation of I^owers, i3<^ 
 
 Formation of Binomial Squares, - - - - 139 
 
 Binomial Theorem, - - - • - - -140 
 
 General Rule, ---141 
 
 Addition and Subtraction of Powers, - - - - 144 
 
 Multiplication and Division of Powers, - - - 145 
 
 Changing Sign of Exponent, - - • - - 146 
 
 Evolution, 147 
 
 Decimal Exponents, -- - - • - -149 
 
CON^TENTS. 7 
 
 PAGE 
 
 Signs of Roots, -------- 150 
 
 Square Root of the Square of a Binomial, - - 151 
 
 Square Root of a Polynomial, 152 
 
 General Rule, 153 
 
 JRadical Quantities^ 154 
 
 Reduction of Radicals, 155 
 
 Addition of Radicals, 159 
 
 Subtraction of Radicals, - - - - - -160 
 
 Multiplication of Radicals, - ----- 161 
 
 Division of Radicals, - - - - - -163 
 
 Involution of Radicals, - - - - - - 164 
 
 Evolution of Radicals, - - - -- -165 
 
 Changing Radicals to Rational Quantities, - - 166 
 Radical Equations, 169 
 
 Quadratic Equations^ 171 
 
 Pure Quadratics. - - - - - - -172 
 
 Affected Quadratics, - - - - - - -175 
 
 First Method of Completing a Square, - - - 176 
 Second Method of Completing a Square, - - - 179 
 Third Method of Completing a Square, - - - 180 
 Problems, -------- 184 
 
 Simultaneous Quadratics, - - - - - 187 
 
 Hatio, - 192 
 
 Proiwrtion, --..--- 196 
 Theorems, ------- 198-203 
 
 Problems, -------- 204 
 
 Arithmetical I^rof/ression, - - - - 205 
 The Last Term of an Arithmetical Series, - - 207 
 The Sum of an Arithmetical Series, - - - - 208 
 Miscellaneous Formulas in Arith. Progression, - 211 
 Inserting Arithmetical Means, - - - - -212 
 Problems, - - - - - - - -212 
 
 Geometrical JProffression^ - - - - 215 
 The Last Term of a Geometrical Series, - - 216 
 
 The Sum of a Geometrical Series, - - - - 2 1 7 
 
8 CONTENTS 
 
 PAGE 
 
 Miscellaneous Formulas in Geometrical Progression, - 220 
 Inserting Geometrical Means, - - - - 221 
 
 Problems, -------. 221 
 
 Harmonical Progression, - - - - - 223 
 
 Infinite Series, - - - - - . . -226 
 
 })Ogarithnis, 232 
 
 Finding the Logarithm of a Number, - - - 235 
 
 To find the Number belonging to a Logarithm, - - 236 
 Multiplication by Logarithms, - - - - - 237 
 Division by Logarithms, - - - . - 238 
 Involution by Logarithms, - - - . . 238 
 
 Evolution by Logarithms, - - - - -239 
 Compound Interest by Logarithms, - - - - 240 
 Table of Logarithms, - - -.- - -241 
 
 Mathematical Induction^ - - - - 243 
 
 Business Formulas, 245 
 
 Formulas for Profit and Loss, - - - - - 245 
 
 Formulas for Simple Interest, - - - - 247. 
 
 Formulas for Compound Interest, - - - - 248 
 
 Formulas for Discount, - - - - - -250 
 
 Formulas for Compound Discount, - - - - 251 
 
 Formulas for Commercial Discount, - - - 252 
 Formulas for Investments, - - - - _ 253 
 
 Formulas for Sinking Funds, - - - - 255 
 
 Formulas for Annuities, - - - - - -257 
 
 Discussion of ProhlemSy - - - - 261 
 
 Problem of the Couriers, - - - - - -262 
 
 Imaginary Quantities, - • • • - -265 
 Iiidsterminate and Impossible Problems, - • - 267 
 Negative Solutions, ._.--- 268 
 
 Horner's Method of Approximation, - - - - 269 
 
 Test Examples for Review, - - - - - 274 
 
 Appendix, --------- 283 
 
 Collegiate Examination Problems, - - - - 291 
 
 Answers, --------- 295 
 
ALGEBRA 
 
 CHAPTEE I. 
 
 INTRODUCTION. 
 
 Art. 1. Algebra* is the art of computing by letters and 
 signs. These letters and signs are called Symbols, 
 
 2. Quantity is anything which can be measured; as 
 distance, weight, time, number, &c. 
 
 3. A Measure of a quantity is a unit of that quantity 
 established by law or custom, as the Standard Unit. 
 
 Thus, the measure of distance is the yard ; of weight, the Troy 
 pound; of time, the mean solar day, etc. 
 
 NOTATION. 
 
 4. Quantities in Algebra are expressed by letters, or 
 by a combination of letters and figures j as, a, h, c, z^, 
 4y, Sz, etc. 
 
 The first letters of the alphabet are used to express known 
 quantities; the last letters, those which are unktiown. 
 
 Questions.— I. What is algebra ? Letters and signis called ? 2. Quantity? 3. A 
 measure ? 4. How are quantities expressed ? 
 
 * From the Arabic al and gdbron, reduction of parts to a whole. 
 
10 INTEODUCTION-. 
 
 5. The Letters employed have no fixed numerical 
 
 value of themselves. Any letter may represent any num- 
 ber, and the same letter may represent clijfere7it numbers, 
 subject to one limitation; the same letter must always stand 
 for the same number throughout the same problem. 
 
 6. The delations of quantities, and the operations to 
 be performed, are expressed by the same signs as in Arith- 
 metic. 
 
 7. The Sign of Addition is a perpendicular cross, 
 cdXlQdi plus ; * as, +. 
 
 Thus, a+& denotes the sum of a and &, and is read, " a plus &," or 
 ••a added to &." 
 
 8. The Sign of Subtraction is a short, horizontal 
 line, called mi?ius j f as, — . 
 
 Thus, a — b shows that the quantity after the sign is to be subtract- 
 ed from the one before it, and is read, " a minus b," or "a less b." 
 
 9. The Sign of Multiplication is an oblique 
 cross; as, x. 
 
 Thus, a X 6 shows that a and b are to be multiplied together, and is 
 read, *'a times &," " a into 6," or "a multiplied by 6.'* 
 
 10. Multiplication is also denoted by a period be- 
 tween the factors ; as, a • h. 
 
 But the multiplication of letters is more commonly ex- 
 pressed by writing them together, the signs being omitted. 
 Thus, sa&c is equivalent to 5 x a x & x c. 
 
 11. The Sign of Division is a short, horizontal 
 iine between the points of a colon ; as, -^. 
 
 Thus, n-i-b shows that the quantity before the sign is to be divided 
 by the one after it, and is read, " a divided by b." 
 
 5. Value of the letters ? 6. Relations of quantities expressed ? 7. Describe the 
 sign of addition. 8. Subtraction. 9. Mntiplication. 10. How else denoted ? 
 
 ♦ The Latin term plus, signifies more. 
 \ The Latin minus, signifies it«*t 
 
DEFIN^ITIONS. 11 
 
 12. Division is also denoted by writing the divisoi 
 under the divide7id, with a short line between them. 
 
 Thus, T shows that a is to be divided by 6, and is equivalent to a-i-h. 
 
 13. The Sign of Equality is two short, horizontal 
 lines, equal and parallel; as, =. 
 
 Thus, a = 6 shows that the quantity before the sign is equal to tht 
 quantity after it, and is read, " a equals h" or *' a is equal to W* 
 
 14. The Sign of Inequality is an acute angle, with 
 the opening turned toward the greater quantity; as, ><. 
 
 Thus, a>h shows that a is greater than 6, and a<.h shows that a 
 IS less than h. 
 
 15. The Parenthesis ( ), or Vinculum , 
 
 indicates that the included quantities are taken collectively, 
 or as one quantity. 
 
 Thus, 3 (05 + 6) and a + 6 x 3, each denote that the sum of a and & 
 is multiplied by 3. 
 
 16. The Double or Ambiguous Sign is a combi- 
 nation of the ^[gn^plus and minus; as, ±. 
 
 Thus, a±J) shows that 6 is to be added to or subtracted from a, and 
 is read, " a plus or minus h.'* 
 
 17. The character ,*, , denotes hejice, therefore, 
 
 18. Every quantity is supposed to be preceded by the 
 sign plus or minus. When no sign is prefixed, the sign -f 
 is always understood. 
 
 19. Like Signs are those which are all plus, or oL 
 minus ; as, + « + J + c, or —x — y — z, 
 
 20. Unlike Signs include both plus and minus; as, 
 a — b-\-c and —x + y — z, 
 
 II. Describe the sign of division ? 12. How else denoted ? 13. The sign of 
 equality? 14. Of inequality? 15. Use of a parenthesis or vinculum ? 16. Double 
 sign ? 17. Sign for " hence," etc. ? 18. By what is every quantity preceded ? When 
 none is expressed, what is understood ? 10, Like signs ? 20. Unlike * 
 
12 INTRODUCTION. 
 
 21. A Coefficient * is a number or letter prefixed to a 
 quantity, to show liow many times the quantity is to be 
 taken. Hence, a coefiicient is a multiplier ox factor. 
 
 Coefficients may be numeral, literal, or mixed. 
 
 Thus, in 5«, 5 is a numeral coefficient of a ; in he, 6 is a literal co- 
 efficient of c ; in 2>dx, ^d is a mixed coefficient of x. 
 
 When no numeral coefficient is expressed, i is always 
 understood. 
 
 Thus, a^ means lajy. 
 
 EXERCISES IN NOTATION. 
 
 22. To express a Statement by Algebraic Symbols, 
 
 It is required to express the following statement in 
 algebraic symbols: 
 
 1. The product of a, I, and c, divided by the sum of c and 
 d, is equal to the difference of x and y, increased by the 
 product of a multiplied by 7. 
 
 Ans. axbxc-T-{c + d) = {x — y) -{- ja. 
 
 Or —77^ = {^ — y) + 7«- Hence, the 
 
 Rule. — For the words, substitute the signs which indicate 
 the relations of the quantities and the oj)erations to be per- 
 formed. 
 
 Express the following by algebraic symbols : 
 
 2. The sum of 4c, d, and m, diminished by $x, equals the 
 product of a and b. 
 
 3. The product of 5c and d, increased by the quotient of 
 a divided by b, equals the product of x and y. 
 
 21. A (Coefficient? When no coefficient is expressed, what is understood f 
 22. How translate a statement from common lan^age into algebraic symbols ? 
 
 * Coefficient, Latin, con, with, and efficere, to effect ; literally, a 
 eo-operator. 
 
EXEKCISES IK KOTATIOK. 13 
 
 4. The quotient of 3^ divided by 5^, increased by 4m, 
 equals the sum of c and 6d, diminished by the product of 
 7« and x. 
 
 5. If to the difference between a and I, we add the 
 product of X into y, the sum will be equal to m multiplied 
 by 6n. 
 
 6. The difference between x and y, added to the sum of 
 \a and h minus m, equals the product of c and d, increased 
 by 15 times m. 
 
 These and the following exercises should be supplemented by 
 dictation, until the learner becomes familiar with them. 
 
 23. To translate Algebraic Expressions Into Common 
 Language. 
 
 Express the following statement in common language : 
 
 Substituting words for signs, we have the sum of a and h, 
 divided by d, equals twice the product of «, b, and c, dimin- 
 ished by the sum of x and y, increased by the quotient of 
 d divided by the product of a and h, Ans. Hence, the 
 
 EuLE. — For the signs indicating the given relations and 
 operations, substitute words. 
 
 Express the following in common language: 
 
 2. \- a — b =: 1- axy — ^cd, 
 
 X c 
 
 Sa ^a — hcd 
 
 4o ax -\- be = 7,x, 
 
 5 X ^ ^ 
 
 aic — x^ . cdh + x 
 
 Zd --^- ' ^^- 2a 
 xy a—b _x -\- y 2 
 Sa X a 3c 
 
 6 4Q^^y , a—b _x-\- y 2a -\- d 
 
 23. How translate algebraic exprespions into common language J 
 
14 Il^^TRODUOTION'. 
 
 ALGEBRAIC OPERATIONS. 
 
 24. An Algebraic Operation is combining quanti- 
 ties according to the principles of algebra. 
 
 25. A Theorem is a statement of a principle to be 
 proved. 
 
 25. a. A Problem is something proposed to be done, as 
 a question to be solved. 
 
 26. The Equality between two quantities id denoted 
 by the sign = . (Art. 13.) 
 
 27. The Expression of Equality between two 
 quantities is called an Equation, Thus, 15 — 3 = 7 +5 
 is an equation. 
 
 PROBLEMS. 
 
 28. The following problems are solved by combining the 
 preceding principles with those of Arithmetic. 
 
 I. A and B found a purse containing 12 dollars, and 
 divided it in such a manner that B's share was three times 
 as much as A's. How many dollars did each have ? 
 
 By Arithmetic. — A had i share and B 3 shares ; now i share + 
 3 shares are 4 shares, which are equal to 12 dollars. If 4 shares equal 
 12 dollars, i share is equal to as many dollars as 4 is contained times 
 in 12, which is 3. Therefore, A had 3 dollars, and B had 3 times as 
 much, or 9 dollars. 
 
 By ALGEBRA.~We represent opebatiok. 
 
 A's share by a?, and form an -t-iGt o; = A s share, 
 
 equation by treating this letter then ^X = B's share, 
 
 as we treat the answer in proving and X -{- 2>^ ^= 12 dollars, 
 
 an operation. If x represent A's ^^^^t is,' 4:^: = 1 2 dollars, 
 
 share, 3a; will represent B's, and xt ^ i a 
 
 , ^ n .1 !> Hence, a; = 3 doL, A. 
 
 a;+3a;=i2 dollars, the sum of ' a ^ n 
 
 both. Uniting the terms, we 3^ = 9 ^oL, B. 
 
 have the equation, 4a? = 12 dollars. To remove the coefficient 4, we 
 
 24. What Is an algebraic operation ? 25. A problem ? A solution ? 26. Equality 
 denoted ? 27- Tlie expression of equality called ? 
 
ALGEBRAIC OPERATIONS. " 15 
 
 divide both sides of tlie equation by it. For, if equals are divided by- 
 equals, the quotients are equal. Therefore, :c = 3 dollars, A's share, 
 and 3a; = 9 dollars, B's share. (Ax. 5.) 
 
 Proof. — By the first condition, 9 dollars, B's share = 3 times 3 dol- 
 lars, A's share. By the second condition, 9 doDars + 3 dollars = 
 12 dollars, the sum found. Hence, 
 
 29. When a quantity on either side of the equation has a 
 coefficient, that coefficient may be removed, hy dividing 
 every term on both sides of the equation hy it. 
 
 2. A and B together have 15 pears, and A has twice as 
 many as B : how many has each ? 
 
 By Algebra. — If x represents 
 B's number, 2.x will represent A's, 
 and X+2X, or 3a;, will represent 
 the number of both. Dividing 
 both sides by the coefficient 3, we 
 have a; = 5 pears, B's number, and 
 2X = 10 pears, A's. 
 
 Note— It is advisable for the learner to solve each of the follow- 
 ing problems by Arithmetic and by Algebra. 
 
 3. A lad bought an apple and an orange for 8 cents, pay- 
 ing 3 times as much for the orange as for the apple. What 
 was the price of each ? 
 
 4. A farmer sold a cow and a ton of hay for 40 dollars, 
 the cow being worth 4 times as much as the hay. What 
 was the value of each ? 
 
 5. The sum of two numbers is 36, one of which is 3 times 
 the other. What are the numbers ? 
 
 6. A, B, and C have 28 peaches; B has twice as m.any as 
 C, and A twice as many as B. How many has each ? 
 
 7. A father is 3 times the age of his son, and the sum of 
 their ages is 48 years. How old is each ? 
 
 2^. How remove a coefficieijt ? 
 
 
 OFBBATION, 
 
 Let 
 
 X = B's number; 
 
 then 
 
 2X = A's « 
 
 and 
 
 3ic = 15 pears. 
 
 , *, 
 
 x= s pears, B's. 
 
 
 22;= 10 pears, A's. 
 
16 IN^TRODUCTION". 
 
 8. A and B trade in company, and gain loo dollars. If 
 A puts in 4 times as much as B, what will be the gain of 
 each? 
 
 9. The sum of three numbers is 90. The second is twice 
 the first, and the third as many as the first and second: 
 what are the numbers ? 
 
 10. A cow and calf were sold for 6;^ dollars, the cow being 
 worth 8 times as much as the calf. What was the value of 
 each? 
 
 11. A man being asked the price of his horse, rephed 
 that his horse, saddle and bridle together were worth 
 126 dollars; that the saddle was worth twice as much as 
 the bridle, and the horse 7 times as much as both the otherr 
 What was each worth ? 
 
 12. A man bequeathed $36,000 to his wife, son and 
 daughter, giving the son twice as much as the d?jghter, 
 and the wife 3 times as much as the son and daughter. 
 What did each receive ? 
 
 13. The sum of three numbers is 1872* the second is 
 3 times the first, and the third equals the o^ner two. What 
 are the numbers ? . 
 
 POWERS AND ROOTS. 
 
 30. A JPower is the product of two or more equal 
 factors. 
 
 Thus, the product 2 x 2, is the sqttare or second power of 2 ; 
 a; X a; X a; is the cube or third power of x. 
 
 31. The Index or JEx2Jonent of a power is a figure 
 or letter placed at the right, above the quantity. 
 
 Thus, a' denotes a, or the first power. 
 
 a^ " ax a, the square, or second power. 
 a' ** ax ax a, the cube, or third power, etc. 
 
 32. A Hoot is one of the equal factors of a quantity. 
 
 30. What is a power? 31. How denoted? 
 
ALGEBRAIC EXPRESSIONS. 17 
 
 33. Roots are denoted by the Madical Slf/n ^ 
 
 prefixed to the quantity, or by a fractional exponent placed 
 after it. 
 
 Thus, -\/a, 0^, or ^a denote the square root of the quantity a ; 
 ^ a shows that the cube root of a is to be extracted, etc. 
 
 34. The Tudedc of the Hoot is the figure placed 
 over the radical sign. The index of the square root is 
 usually omitted. 
 
 (For negative indices, see Arts. 256, 258,) 
 
 Eead the following examples : 
 
 1. «2 + 3«. 7. 4(a — hf. 
 
 2. ¥ — c2. 8. «2 ^ 2ah + 52. 
 
 3. « + 52 __ ^, g^ Va + h, 
 
 4. 01^ — y + y^* 10. 's/a^ — ^. 
 
 5. 2y^ ■\- ^ — z, II. 2«^ + c^ 
 
 6. 3(a2^^>). 12. 4x^ + 2y\ 
 
 Write the following in algebraic language : 
 
 13. The square of a plus the square of b, 
 
 14. The square of the sum of a and i. 
 
 15. The sum of a and b, minus the square of c. 
 
 16. The square root of a, plus the square root of x, 
 
 17. The cube root of x, minus the fifth power of y. 
 
 18. The cube root of a, plus the square of i. 
 
 ALGEBRAIC EXPRESSIONS. 
 
 35. An Algebraic Expression is any quantity ex- 
 pressed in algebraic language ; as, 30^, 5« — 7^, etc. 
 
 36. The Terms of an algebraic expression are those 
 parts which are connected by the signs + and — . 
 
 Thus, ina+h, there are two terms ; in ic+y x z—a there are three. 
 
 32. A root? 33. How denoted? 34. What is the fi^ire placed over ft? rac^iQ^ 
 $ipi caU'-l ? 35- Wbat is an alget>r8i<? expression ? 36. Xts t^riug t 
 
18 Il^^TRODUCTION". 
 
 Note. — Letters combined by the signs x or -?- do not constitute 
 separate terms. Such a combination, to form a term, must have the 
 sign + or — prefixed to it, and the operations indicated b,v the signs 
 X or -f- must be performed before the terms can be added to or sub- 
 tracted fiom the preceding term. (Art. 36.) Thus, a + bxc has two 
 terms, hxc foiming one term and a the other. 
 
 37. The Dimensions of a term are its several literal 
 factors. 
 
 38. The Degree of a term depends on the number of 
 its literal factors, and is always equal to the sum of theii 
 exponents. 
 
 Thus, cib contains two factors, a and &, and is of the second degree. 
 a^x contains three factors, a, a, and x, and is of the third degree. 
 ¥^ contains five factors, 6, b, x, x, x, and is of the fifth degree. 
 
 39. The Numerical Value of an algebraic expres- 
 sion is the number which it represents when its terms are 
 combined as indicated b}- the signs. (Art. 36.) 
 
 40. To Find the Numerical Value of an algebraic expression. 
 
 1. If a = s, ^ = 7, and x = g, what is the value of 
 
 6« + 85 + 3^ ? 
 
 Analysis. — Since a = 5, 6a must 
 equal 6 times 5, or 30 ; since & = 7, 8& 
 must equal 8 times 7, or 56 ; and since 
 x — g, 3X must equal 3 x g, or 27. Now 
 30+56 + 27=113. Therefore, the value 
 of the given expression is 113. Hence, 
 the 
 
 EuLE. — For the letters, substitute the figures which the 
 letters represent, and perform the operations indicated by 
 the signs. 
 
 2. If 5 = 3, c= s, and d=z8, what is the value of 
 Sb-{-7c + 6d? 
 
 Suggestion. 15 + 35 + 48 = 98, ^tw. 
 
 37. The dimensions of a term? 38. Degree? 39. Numerical valae of an alge* 
 bra ic expression ? 40. Hqw found? 
 
 OPERATION. 
 
 
 6a = 5 X 6 = 
 
 Sb = 7 xS = 
 
 30 
 56 
 
 SX = SX9 = 
 Ans, 
 
 27 
 "3 
 
ALGEBKAIC QUAI^TITIES. 19 
 
 Find the numerical value of the following expressions, 
 when a = 2, J = 3, c = 4, J = 5, and x=z6, 
 
 3. 4a + 6ah + 5c = how many ? Ans. 64. 
 
 4. (a -\-b) X c xd — (x-^ c) = how many ? 
 
 5. (:f — a) + rt2;4- (c -j- «) = how many ? 
 
 6. 2; -f- 2 + (6/ — 6') + Jf — ic ^ how many ? 
 
 7. f^x— (« X c) 4- (« X ^) + a; = how many? 
 
 8. ^ + (:i; X a) + « — ii'' + c = how many? 
 
 CLASSIFICATION OF ALGEBRAIC QUANTITIES. 
 
 41. Quantities in Algebra are primarily divided into 
 hnotvn and unknown, 
 
 42. A Known Quantity is one whose value is given. 
 An Unknown Quantity is one whose value is not 
 
 given. 
 
 These quantities are subdivided into like and unlike, posi- 
 tive and negative, simple, compound, monomials, etc. 
 
 43. IdJce Quantities are those which are expressed 
 by the same power of the same letters; as, a and 2a, 
 2X^ and x^» 
 
 44. Unlike Quantities are those which are expressed 
 by different letters, or by different powers of the same let- 
 ters ; as 2X and 3?/, 2X and x^. 
 
 Note. — An exception must be made in cases where letters are re- 
 garded as coefficients. Thus, ax^ and Ix'^ are like quantities, when a 
 and 6 are considered coeflBlcients. 
 
 45. A Positive Qnantity is one that is to be added, 
 and has the sign + prefixed to it ; as, 40^ + 3 J- 
 
 46. A Negative Quantity is one that is to be sub- 
 iracted, and has the sign — prefixed to it; as, 4a — 3Z'. 
 
 41. How are quantities in Algebra primarily divided ? 42. A known quantity? 
 Unknown? 43. Like quantities? 44. Unlike? 45. A positive quantity? 46. A 
 negative ? 
 
20 Il^TRODUCTION". 
 
 47. The terms Positive and JSegative denote oppo- 
 siteness of direction in the use of the quantities to which 
 they are applied. If lines running Nortli from any point 
 are positive, those running South are negative. If future 
 time \& positive, past time is negative; if credits ^tq positive, 
 debts are negative, etc. 
 
 48. A Simple Quantity is a single letter, or several 
 letters written together without the sign 4- or — ; as, a. 
 db, Zxy, 
 
 49. A Compound Quantity is two or more simple 
 quantities connected by the sign + or -— ; as 3« f 4J, 
 zx — y, 
 
 60. A Monomial * has but one term ; as, a, il. 
 
 61. A binomial f has two terms ; as, « -|- J, « — h. 
 Notes. — i. The expression a — 6 is often called a reddual, because 
 
 It denotes that which remains after a part is subtracted. 
 2. A binomial is sometimes called a polynomial. 
 
 62. A Trinomial % has three terms ; as, a + J — c. 
 
 63. A Polynomial \ has two or more terms ; as, 
 a -\- h — c ■\- X. 
 
 64. An Homogeneous Polynomial has all its 
 
 terms of the same degree. 
 
 Thus, 2ab + cd + s^V is homogeneous ; but ^ahc ■\-c^ + sx is not. 
 
 66. The Reciprocal of a quantity is a unit divided by 
 that quantity. 
 
 Thus, the reciprocal of a is - ; the reciprocal ofa + biB r • 
 
 47. What do the terms positive and negative denote ? 48. A simple quantity.' 
 49. A compound? 50. A monomial? 51. A binomial? 2^ote. The expression 
 — 6 called? 52. A trinomial? 53. A polynomial? 54. When homogeneous? 
 55. The reciprocal of a quantity ? 
 
 * Greek, monos, single, and nome, terra, having one term. 
 f Latin, bis, two, and nome, name (a hybrid), two terms. 
 X Greek, treis, three, and nome, name, having three terms. 
 \ Greek, polus^ manj^, and noim, name, having many terms. 
 
FOECE OF THE SIGKS. 21 
 
 FORCE OF THE SIGNS. 
 
 56. JEach term of an algebraic expression is preceded 
 by the sign + or — , expressed or understood. (Art. i8.) 
 
 The Force of each of these signs is limited to the term 
 which follows it; as, 7 + 5— 3 = 12 — 3 = 9; 15—6 + 8 
 = 9 + 8=17. 
 
 57. If a term, preceded by the sign + or — , is combined 
 with other letters by the sign x or -^, each of these let- 
 ters forms a part of that term, and the operations indicated, 
 taken in their order, must be performed before any part of 
 the term can be added to or subtracted from any other term. 
 
 Tims, the expression 12 + 4 x 2, shows that 4 is to be multiplied by 
 2 and the product added to 12, and is equal to 20, 
 
 In like manner, the expression 16 — 8 -f- 2, shows that 8 is to be 
 divided by 2 and the quotient subtracted from 16, and is equal to 12. 
 
 58. If two or more terms joined by + or — are to be 
 subjected to the same operation, they must be connected by 
 a parenthesis or vinculum. 
 
 Thus, if a + & or a — 6 is to be multiplied or divided by c, the oper- 
 ations are indicated by {a+b)x e, or c{a+b); (» — &)-*- c, or • 
 
 c 
 
 EXERCISES. 
 
 !• 50 + 5x2= what number ? 
 
 2. 50 — 5x2 = what number ? 
 
 3. ac + 4b X 2 = what ? 
 
 4. ^b — 6d -T- 3 = what ? 
 
 5- 15 + 5 X 3 + lo-^ 2 =what? 
 
 6. 18 — 2x4-^2 + 10 = what ? 
 
 7. 3X+Sy-i-4-\-axb = what ? 
 
 8. 6b — jc X X + ga -^ S = what ? 
 
 9. {b -\- c) X xy = what ? 
 
 56. By what are all algebraic terms preceded ? The force of each of these signs ? 
 57. Of the figns x and n- ? 58. Of the parenthesis and vinculuia ? 
 
22 H^TTEODUCTIOS". 
 
 la ^x X S^ -^ 2Z -{- a = what ? 
 
 11. {b — a) -T- xy -]- 2Z = what ? 
 
 12. 3a; + a;^ + 22 X 32/ = what ? 
 
 13. The difference of x and y, multiph'ed by a less b, and 
 the product divided 'bjd = what ? 
 
 Find the value of the following expressions, in which 
 a = s> ^ = 4} (^ = 2, X = 6, y = 8, and z= 10: 
 
 14. a-\-(axx)-^C'\-yxz = what ? 
 
 15. 2^ ~ (a; — Z») 4- « X ^ X y + 2;^ = what ? 
 
 AXIOMS. 
 59. An Aociom, is a self-evident truth. 
 
 1. Things which are equal to the same thing, are equal to 
 each other. 
 
 2. If equals are added to equals, the sums are equal. 
 
 3. If equals are subtracted from equals, the remainders 
 are equal. 
 
 4. If equals are multiplied bj equals, the products are 
 equal. 
 
 5. If equals are divided by equals, the quotients are equal. 
 
 6. If a quantity is multiplied and divided by the same 
 quantity, its t^^^^^^e is not altered. 
 
 7. If the same quantity is added to and subtracted from 
 another quantity, the value of the latter is not altered. 
 
 8. The whole is greater than its part. 
 
 9. The whole is equal to the sum oi all its parts. 
 
 10. Like powers and like roots of equal quantities, are 
 equal. 
 
 Note. — The importance of tliorougUy understanding the defini- 
 tions and principles cannot be too deeply impressed upon the mind of 
 the learner. The questions at the foot of the page are designed to 
 direct his attention to the more important points. Teachers, of course, 
 will not be confined to them. 
 
CHAPTER II. 
 ADDITION. 
 
 60. Addition in Algebra is uniting two or more quan 
 
 fcities and reducing them to the simplest form. 
 
 61. The Mesult is called the Sum or Amount. 
 
 62. Quantities expressed by letters are regarded as 
 concrete quantities. Hence, their coeflBcients may be added, 
 subtracted, multiplied, and divided like concrete numbers. 
 
 Thus, ja and 4« are ya, 46 and 5& are gb, as truly as 3 apples and 
 4 apples are 7 apples, or as 4 bushels and 5 bushels are 9 bushels. 
 
 PRIi\!CIPLES.* 
 
 63. 1°. Like quantities only can he united in one term. 
 2°. Tlie sum of two or more quantities ii the same in 
 
 whatever order they are added. 
 
 CASE I. 
 
 64. 10 Add like Monomials which have like signs. 
 
 I. What is the sum of i$ab + isao + igab ? 
 
 Analysis.— These terms are like quantities operation. 
 
 and have like signs. (Art. 19.) We therefore + ^S^^ 
 
 add the coefficients, to the sum annex the com- _j- i^ab 
 
 mon letters, and prefix the common sign. The 1 tq^A 
 
 result, + 4706, is the answer required. (Ax. g.) 
 
 47 « J, Ans. 
 
 60. What is addition ? 61. The result called ? 62. How are quantities expressed 
 by letters regarded ? 63. First principle ? Second ? 
 
 *■ The expressions 1°, 2°, 3°. etc.. denote first, second, third, etc. 
 
24 ADDITION. 
 
 2. What is the sum of — 142^^, — i6a;y, and — i8a:y ? 
 
 Analysis. — Since these terms are like quan- — 14^^ 
 
 tities, and have like signs, we add them as i6xil 
 
 before, and prefix the sign — to the result, for r> 
 
 the reason that all the quantities have the — 
 
 sign — . Hence, the — A^xy, Ans. 
 
 KuLE. — Add the coefficients; to the sum annex thecoma 
 man letters, and prefix the common sign. 
 
 (3.) 
 
 (4.) 
 
 (S) 
 
 (6.) 
 
 (7.) 
 
 sab 
 
 s«y 
 
 7a' 
 
 — 'jhcd 
 
 -4^f 
 
 sal) 
 
 8x1/ 
 
 3a» 
 
 — Zbcd 
 
 -3^^^ 
 
 6ai 
 
 xy 
 
 4^2 
 
 — Sbcd 
 
 - a^y^ 
 
 yah 
 
 3«y 
 
 a^ 
 
 — Ucd 
 
 -Sa^y^ 
 
 8. Add 5«&2 ^ i>ja¥+ i^al^. 
 
 9. Add — Sabx^y^ — saix^y^— 2Sabx^y\ 
 
 10. Add s^dm^ + Wdm^ + 9l^dm^ 4- W^dm^ 
 
 11. If 3a -f 5« + « + 7a = 48, to what is a equal ? 
 Solution. 3a + sa + a+'ja=i6a ; hence, a=4S-7-i6, or 3. Ans. 
 
 12. If 4bc + ghc + 2hc + ^bc = 80, to what is he equal? 
 
 13. If xy + s^y + s^y + 4xy = 65, to what is xy equai ? 
 
 CASE II. 
 65. To Add like Monomials which have Unlike signs. 
 
 14. What is the sum of ^ab — ^ab — yah + gab + tab 
 ^Sab? 
 
 Analysis, — For convenience in operation. 
 
 adding, we write the negative terma Sab — S^b 
 
 one under another in the right- gab — jab 
 
 hand column, with the sign — be- ^^j 3^J 
 
 fore each, and the positive terms ; T" 
 
 in the next column on the left. ^^^^ " ^^^^ = ^^^^ ^^^^^ 
 We then find the sum of the coefficients of the positive and negative 
 
 64. How add lT!o^omia^• which have like signs ? 
 
ADDITION. *Z5 
 
 terms separately ; and taking the less sum from the greater, the result 
 2ab, is the answer. Hence, the 
 
 KuLE. — I. Write the positive and negative terms in sepa- 
 rate columns with their proper signs, and find the sum of the 
 coefficiefits of each column separately, 
 
 II. From the greater subtract the less j to the remainder 
 prefix the sign of the greater, and annex the common letters. 
 
 Note — If two equal quantities have opposite signs, they balance 
 each other, and may be omitted. 
 
 15. Add 4d + ^d — ^d -\- 6d — 2d. Ans. 6d. 
 
 16. Add — ^x -}- 6x -\- 8x — ^x -\- gx — yx. 
 
 17. Add 3«Jc + i2aic — 6abc + s^^^o — loaic — saic. 
 
 18. Add 2J — 5^ + 45 — 65 — yb. 
 
 19. Add —6g-\-4y^Sy — gg-\-Sy — y. 
 
 20. Add 4m -\- i6m — 8m — 9m + 5m — 10m. 
 
 21. If 6ab + i4ab — yab + i^ab — i2ab + i6ab = 32, to 
 what is ab equal ? 
 
 22. To what is bed equal, if bed — $bcd + 4bcd + 4bcd 
 
 -Sbcd=7s^ 
 
 Eemark. — The sum in Arithmetic is always greater than any of its 
 parts. But, in Algebra, it will be observed, the sum of a positive 
 and negative quantity is always less than the positive quantity. It is 
 thence called Algebraic Sum, 
 
 66. Unlike Quantities cannot be united in one term^ 
 Their sum is indicated by writing them one after another, 
 with their proper signs. (Art. 6^, Prin. i.) 
 
 Thus, the sum of yg and 3(? is neither log nor lod, any more than 
 7 guineas and 3 dollars are 10 guineas or 10 dollars. Their sum is 
 79 + 3^- (Art. 63, Prin. i.) 
 
 67. Poh/no^nials are added by uniting like quantities, 
 as in adding monomials. 
 
 65. How add monomials having unlike Bigne. ? Bern. What is true of the Bura in 
 Arithmetic ? In Algebra ? 66. How add unlike quantities ? 67. Polynomials ? 
 
26 ADDITION. 
 
 23. What is the sum of the polynomial ^ah — 35 + 46? 
 — 3^5 — 5^^ -\- ^x — c— 2d; and bg ■\- d + zab -f d ? 
 
 Al^ALYSIS. For OPERATION. 
 
 convenience, we 3^^ — 3^ + 4^ — 3^ — C 
 
 write tlie quanti- — 5«5+ 5 — 2^+ 4a; 
 
 ties so that like ^ab + d + J^ 
 
 tenns shall stand — — 
 
 one under another, - 2^> + 3^/ + x + bg - C, Ans. 
 
 and uniting those which are alike, the result is —2h + 2,d-{-x-\-l}g—c. 
 
 68. From the preceding illustrations and principles we 
 deduce the following 
 
 GENERAL RULE. 
 
 I. Write the given quantities so that like terms shall stand 
 one under another, 
 
 II. Unite the terms which are alihe, and to the result 
 annex the unlike terrns with their proper signs. (Art. 65.) 
 
 1. Add $a — ^a -{- 6a + ya -{- ga -\- 2b — ^d, 
 
 2. Add ^mn -}- ^mn — ^mn + gmn — xy ■\- be. 
 
 3. Add 2,bc — "jbc -\- xy — mn + 1 1 Jc + gbc. 
 
 ^ Add $ab — 37W?^ — ab -{• ^ab -\- 2Z — ^ab + ab. 
 
 5. Add 2t^y — xy -\- ab — "jxy ■\- b -\- Sxy — xy -{- i^xy. 
 
 69. Compound Quantities inclosed in a paren- 
 thesis, are taken collectively, or as one quantity. Hence, if 
 the quantities are alike, their coefficients and exponents are 
 treated as the coefficients and exponents of like monomials. 
 (Art. 64.) 
 
 6. What is the sum of 3 {a-{-b) + 5 (a + b) + 7 {a-\-b)? 
 Solution. 3 (« + 6) and 5 (a + &) and yia + b) are 1 5 (a + b). An». 
 
 f. Add 13 (« 4- J) -1- 15 (a + 5) - 7 (« + h). 
 
 8. Add 2>c{x — y) + ic^—y) — So{x—tj) -\-yc{x—y). 
 
 9. Add saVxy + saVxy — jaVxy + SaV^^y- 
 
 10. Add sVa + 3a/« — sVa + 9 V« — 3 V^. 
 
 11. Add S^x — y — 3^/^ — y + sVx — y. 
 
 68. The general rule for addition ? 69. How aid quaniities included in a paren- 
 thesif 'i 
 
PROBLEMS. 27 
 
 70. The sum of unlike quantities haying a common letter 
 or letters, may be expressed by inclosing the other letters, 
 with their signs and coefficients, in a parenthesis, and an- 
 nexing or prefixing the common letter or letters to the result. 
 
 12. What is the sum of 5^2; + ^ix — 4cx? 
 
 Solution. 5«ic + 3bx-4cx = (5« + 3&— 4^) aj, or x (5a + 3b— 4c]. Ans. 
 
 13. Add ja — 6ba + sda — ^ma. 
 
 14. Add ahy + sy — 2cy — ^tny. 
 
 15. Add 9m + abm — ^jcm ■\- ^dm, 
 
 16. Add iT^ax — 2fix -{-ex — ^dx + mx. 
 
 1 7. Add axy + 5a;^ — cxy, 
 
 PROBLEMS. 
 
 71. Problems requiring equal quantities to be added to each 
 
 side of the equation^ 
 
 1. A has 3 times as many marbles as B, lacking 6 ; and 
 both together have 58. How many has each ? 
 
 Analysis. — If x represents operation. 
 
 B's number, then will 3a;-6 Let X = B's No. ; 
 
 represent A's, and 37 + 32;— 6 ^Yien ^X — 6 = A's " 
 
 = 58, the sum of both. To , /- o v. ix. 
 
 . j^ , X 4- 'XX — 6 = ^S, both, 
 
 remove —6, we add. an eqiml 
 
 positive quantity to each side a; + 3a; — 6 + 6 = 58 + 6 
 
 of the equation. (Axiom 2.) 4^ ^^ 64 
 
 Uniting the terms, we have x= 16, B's NOo 
 
 4X = 64, and x = 16, B's, and ox 6 = 42 A's " 
 
 3 times 16—6, or 42 = A's No, 
 
 72. When a negative quantity occurs on either side of an 
 equation, that quantity may be removed by adding an equal 
 positive quantity to both sides. 
 
 Note. — In forming the equation, we treat x as we do the answer 
 in proving an operation. 
 
 2. A kite and a ball together cost 46 cents, and the kite 
 cost 2 cents less than twice the cost of the ball. What was 
 the cost of each ? 
 
 70. How may the sum of unlike quantities which have a common letter be ex- 
 
^8 ADDITION". 
 
 3. In a basket there are 75 peaches and pears ; the num- 
 ber of pears being double that of the peaches, wanting 3. 
 How many are there of each ? 
 
 4. The sum of two numbers is 85, and the greater is 
 5 times the less, wanting 5. What are the numbers ? 
 
 5. A certain school contains 40 pupils, and there are 
 twice as many girls, lacking 5, as boys. How many are 
 there of each ? 
 
 6. If 44a; + 6sx — 24 = 85, what is the value of x? 
 
 7. If jx — $ -{- 2X = 60, what is the value ofx? 
 
 8. If 4?/ + 2?/ + 5?/ — 7 = 70, what is the value of y? 
 
 9. The whole number of votes cast for A and B at a cer- 
 tain election was 450 ; A had 20 votes less than 4 times the 
 number for B. How many votes had each ? 
 
 10. The sum of two numbers is 177 ; the greater is 3 less 
 than 4 times the smaller. What are the numbers? 
 
 1 1. What is the value of ?/, if 41/ + 32/ + 2?/ — 1 2 = 60 ? 
 
 12. A lad bought a top and a ball for 32 cents ; the price 
 of the ball was 3 times that of the top, minus 4 cents. 
 What was the price of each ? 
 
 13. A man being asked the price of his saddle and bridle, 
 replied that both together cost 40 dollars, the former being 
 4 times the price of the latter, minus 5 dollars. What was 
 the price of each ? 
 
 14. A lad spent a dollar during a holiday, using three 
 times as much of it in the afternoon as in the morning, 
 minus 4 cents ; how much did he spend in each part of the 
 day? 
 
 Find the value of x in the following equations : 
 
 15- 3a; -f 6a; + 4a; -f 52; — 8 = 154. Ans. 9. 
 
 16. 2X -\- sx ■{- sx — 10 = 130. 
 
 17. 4X + 3.r + 7a: — 12 = S6. 
 
 18. lox — 4X + gx — 2S = 155. 
 
 19. 15a; — 7a; — 2a; — 60 = 300. 
 fio. 18a: — 4a; -f- a;— 75 = 225. 
 
OHAPTEE III. 
 SUBTRACTION. 
 
 73. Subtraction is finding the difference between two 
 quantities. 
 
 The Minuend is the quantity from which the subtrac- 
 tion is made. 
 The Subtrahend is the quantity to be subtracted. 
 The Difference is the result found by subtraction. 
 
 74. Since quantities expressed by letters are regarded as 
 concrete, the coefficient of one letter may be subtracted from 
 that of another, like concrete numbers. (Art. 62.) 
 
 Thus, 7a — 3a = 4rtj ; 8& — 5& = 36. 
 
 PRINCIPLES. 
 
 75. I®. Like quantities only can le subtracted one from 
 another. 
 
 2°. Tlie sum of the difference and subtrahend is equal to 
 the minuend. 
 
 3"^. Subtracting a positive quantity is equivalent to add- 
 ing an equal negative one. 
 
 Thus, let it be required to subtract +4 from 6+4. 
 
 Taking +4 from 6 + 4, leaves 6. 
 Adding —4 to 6 + 4, we have 6 + 4—4. 
 But (Ax. 7) 6 + 4—4 is equal to 6. 
 
 4°. Subtracting a negative quantity is the same as adding 
 an equal positive one. 
 
 73. Define subtraction. The Minuend. Subtrahend. Difference. 75. Name the 
 Qrpt principle, Second. Illijstmte Prtn. 3 upon the blackhoar^. IlJustmte Prin. 4. 
 
30 
 
 SUBTRACTION". 
 
 Thus, let it be required to subtract —4 f jom 10—4. 
 Taking —4 from 10-4, leaves 10. 
 Adding +4 to 10—4, we have 10—4 + 4. 
 But (Ax. 7) 10—4 + 4 is equal to 10. 
 
 Again, if the assets of an estate be $500, and the liabilities $300, 
 the former being considered positive and the latter negative, the net 
 yalue of the estate will be $500— $300 = $200. Taking $50 from the 
 issets has the same effect on the balance as adding I50 to the liabilities 
 in like manner, taking $50 from the liabilities has the same effect as 
 adding $50 to the assets. 
 
 76. To Find the Difference between two like Quantities. 
 
 This proposition includes three classes of examples, as 
 will be seen in the following illustrations: 
 
 1. From 25a subtract ija. 
 
 Remark. — i. In this example the signs are 
 alike, and the subtrahend is less than the min- 
 uend. Subtracting a positive quantity is 
 equivalent to adding an equal negative one. 
 (Prin. 3.) We therefore change the sign of 
 the subtrahend, and then unite the terms 
 25a— 17« = Sa, 
 
 2. From 4a subtract ja. 
 
 Remark. — 2. In this example the signs 
 are alike, but the subtrahend is greater than 
 the minuend. Changing the sign of the sub- 
 trahend, and uniting the terms as before, the 
 subtrahend cancels the minuend, and has 
 —3a left. (Prin. 3.) 
 
 3. From 45 «5 subtract — 2gah. 
 Remark. — 3. In this example the signs 
 
 aie unlike. Subtracting a negative quantity 
 is the same as adding an equal positive one. 
 (Prin. 4.) Changing the sign of the sub- 
 trahend and proceeding as before, we have 
 iSab + 290* = 7406. Ans. 
 
 OPERATION. 
 
 2$a Minuend. 
 — lya Subtrahend. 
 
 Set Difference. 
 
 in addition. Thus. 
 
 OPERATION. 
 
 4(1 Minuend. 
 
 — 'JCt Subtrahend, 
 
 — 3a Difference. 
 
 OPERATION. 
 
 4$ab Minuend. 
 •^ 2gab Subtrahend. 
 
 74«^, Ans, 
 
 7^. How And the difference between two like quantities ? 
 
SUBTRACTION. '31 
 
 4. From ^hc + 7^ — S^r, take ^bc -{- 2d — 4X. 
 
 Analysis. — In subtraction of operation. 
 
 polynomials, for convenience, we g^c -}- jd — ^X 
 
 place like terms under each other. - j^ 2d -\- dX 
 
 Then, changing the signs of all the r 
 
 terms in the subtrahend, we unite ^^^ + 5^ ^f ^^^* 
 
 them as before. 
 
 77. From the preceding illustrations and principles we 
 deduce the following 
 
 GENERAL RULE. 
 
 1. Write the subtrahend under the minuend, placing like 
 terms one under another. 
 
 II. Change the signs of all the terms of the subtrahend, or 
 suppose them to be changed, from + to — , or from — to 
 +, and then proceed as in addition. (Art. 75, Prin. 3, 4.) 
 
 Notes. — i. Unlike quantities can be subtracted only by changing 
 the signs of all the terms of the subtrahend, and then writing them 
 after the minuend. (Art. 66.) 
 
 2. As soon as the student becomes familiar with the principles of 
 subtraction, instead of actually changing the signs of the subtrahend, 
 he may simply suppose them to be changed. 
 
 EXAMPLES. 
 
 1. From 43c + d, take 25c + d. 
 
 2. From 49a:, take 23:^ + 3. 
 
 3. From 2Sxgz, take i4xgz. 
 
 4. From — 43ab, take + igab. 
 
 5. From 4ab, take — i^al). 
 
 6. From 4s^y, take + i6xy. 
 
 Ans. 1 8c. 
 . 262; — 3. 
 
 (7.) (8.) (9.) (lo.) 
 
 From 2oaG 42aa^ 370^25 — 290:^ 
 
 Take — 23ac ^ax^ — i4«^J + 15^^ 
 
 77. General rule for subtraction ? Note. How subtract unlike quantitiee. 
 
32 SUBTRACTION. 
 
 (II.) (I2.) (13,) (14.) 
 
 From 3 1 a^J I gabx^ — ssm^x 4 1 x^y 
 
 Take — ya^b igaba^ + 44m^ ^ i2a^y 
 
 15. A is worth $100, and B owes I50 more than he is 
 worth. What is the difference in their pecuniary standing ? 
 
 16. What is the difference in temperature, when the ther- 
 mometer stands 15 degrees above zero, and when at ic 
 degrees below ? 
 
 17. By speculation, A gained on a certain day I275, and 
 B lost $145. What was the difference in the results of their 
 operations ? 
 
 (18.) (19.) (20.) 
 
 From 'jxy — Sa 8J2 -f jam 13^^ — jy^ 
 
 Take ^xy — 2a — 5^ — gam — s^^ — ^y^ — ^^ 
 
 21. From i^ab -{- d — x, subtract ^m — ^n, 
 
 22. From gcd — ab, take 2m — 37^ — 4^. 
 
 23. From 13m — 15, take — 5m -|- 8. 
 
 24. From jx^ — ^x + 15, take — ^x^ ■\-^x-\- 15. 
 
 25. From igab — 2c — 7<:7, take ^ab — 15c ~ Zd, 
 
 26. From a, take b — c, and prove the work. 
 
 27. From II (a -f b), take 5 (<^ + b). 
 
 28. From ij (a — b ■\' x), take 8 (« — J + a;). 
 
 29. Subtract — 18 (oj + Z>) from — 13 (« -|- b), 
 
 30. Subtract 21 {x^ — y) from 14 (a;^ — y). 
 
 31. A and B formed an equal partnership and made 
 $1,000. B's share by right was $1,000 — 1500; but wish- 
 ing to withdraw, he agreed to subtract $100 from his share. 
 What would A's share be ? 
 
 32. What is the difference of longitude between two 
 places, one of which is 23 degrees due east from the meridian 
 of Washington, the other 37 degrees due west? 
 
 Remaek. — The svhtraJiend, in Algebra, is often greater than the 
 minuend, and the difference between a positive and negative quantity 
 greater than either of them. It js thence called Algebraic Dif- 
 ference, 
 
SUBTRACTION, 33 
 
 78. The Difference of unlike quantities which have a 
 common letter or letters may be indicated by e?idosing all 
 the other letters, with their coefficients and signs, in a 
 parenthesis, and annexing, qy prefixing the- common letter 
 or letters to the result. 
 
 SS' From $am, take 2bm, 
 
 Analysis. 3am = 3^ times m, and 2bm = 26 times m ; therefore, 
 
 Sam—2bm = {3a— 2h) m, or m (3a— 2&). Ans. 
 
 34. From 2l)x^, take cx^ — dx\ 
 
 35. From ahy, take cy ■\- dy — xy, 
 
 36. From ic^, take W — ca\ 
 
 37. From ahx, take ^cx -\- dx ■\- mx. 
 
 38. From ^xy, take alxy — cxy -\- dxy, 
 
 39. From 5a<? + J/w<?, take 3«c — Jc. 
 
 APPLICATIONS OF THE PARENTHESIS. 
 
 79. A parenthesis, we have seen, shows that the quanti- 
 ties inclosed by it are taken collectively, and subjected to 
 the operation indicated by the sign which precedes it. 
 (Art. 15.) 
 
 80. A parenthesis having the sign + prefixed to it, may 
 be removed from an expression, if the signs of the included 
 terms remain unchanged. 
 
 Thus, a— 6+(c— (Z + e) = a— 6+c— e?+e. Hence, 
 
 81. Any number of terms may be inclosed in a parenthe- 
 sis and the sign + placed before it, if the signs of the 
 inclosed terms remain unchanged. 
 
 Thus, a + &— c + (? = « + (&— c + <f), or <? + & + (— c + <f). 
 Note. — This principle affords a convenient method of indicating 
 the addition of polynomials. (Art. 67.) 
 
 ■ ttt; • 
 
 rS.^How subtract unlike quantities having a cw&mon letter or letters ? 79. What 
 is the object of a parenthesis? 80. How removed when the sign + is prefixed to U. 
 
34 SUBTRACTIOI'T. 
 
 82. A parenthesis having the sign — prefixed to it, may 
 be removed by changing the signs of all the inclosed terms 
 from + to — and — to +. 
 
 Thus, removing it from tlie equal expressions, 
 
 1 ,, . ')-=a — o + c — a. Hence, 
 « — 6 — (c2 — c) j 
 
 83. Any number of terms may be inclosed by a paren- 
 vhesis, and the sign — placed before it, if all the signs of 
 the inclosed terms are changed. 
 
 Thus, a—l-^c—d = a—(b—c+d), or a—'b—{—c+d)y etc 
 Note. — This principle enables us to express a polynomial in diffei* 
 ent forms without changing its value. 
 
 1. How express a — x -\- c, using a parenthesis? 
 
 Ans. a — X •\- c =z a — {x — c), or a — ( — c + x). 
 
 2. How express a — h — x — y-\-z, using a parenthesis ? 
 
 Ans. a — h — {x -\- y — z), or 
 a — b — (^ -\- X — z), or 
 a — h—{—z-\-x-\-y). 
 
 84. When two or more parentheses occur in the same 
 expression, they are removed by the same rule, beginning 
 with the interior parenthesis. 
 
 Thus, a—\h—c-{d + x)-\-e\=a—(h—c—d—x+e)=a—'b+c + d + x—6. 
 Note. — Quantities may be included in more than one parenthesis, 
 by observing the preceding rules. 
 
 Remo^^^ the parentheses from the following expressions-. 
 
 3. ah — {he — d). Ans, ah — hc + d. 
 
 4. h — (c — d -{- m), 
 
 5. c^x — {— y -\- ab ^ /^d). 
 
 6. 2« — [& + c — (a; + «/) — ^. 
 
 7. a — (b — c) — (a ~ c) ■\- c — (a — h). 
 
 Jg^ The principles governing the signs in the use and removal of 
 parentheses should be made familiar by practice. 
 
 82. How when the sign — is prefixed ? 83. How inclose terms in a parenthesis 
 With — prefixed to it? 
 
OHAPTEE IV. 
 MULTIPLICATION. 
 
 85. Multiplication is finding the amount of a qnaii' 
 tity taken or added to itself, a given number of times. 
 
 Thus, 3 times 4 are 12, and 4 taken 3 times (4 + 4 + 4) = 12. 
 
 The Multiplicand is the quantity to be multiplied. 
 The 3Iultiplier is the quantity by which we multiply. 
 The Product is the quantity found by multiplication. 
 
 86. The Factors of a quantity are the multiplier and 
 multiplicand which produce it. 
 
 PRINCIPLES. 
 
 87. 1°. The multiplier must he considered an abstract 
 quantity, 
 
 2°. The 'product is of the same nature as the multiplicand; 
 for, repeating a quantity does not alter its nature, 
 
 3°. T/ie product of two or more factors is the same in 
 whatever order they are multiplied. 
 
 CASE I. 
 
 88. To Multiply a Motiotnial by a MonotniaZ, 
 
 I. What is the product of « multiplied by c? 
 
 Ans. a X c, or ac. 
 
 Note. — The product of two or more letters, we have seen, is ex- 
 pressed by writing them one after another, either with or without the 
 sign of multiplication between them. (Art. 10.) 
 
 85. Define multiplication. The multiplicand. Multiplier, Product. 86, Fao 
 tors. 87. Name Prin. x. Prin. 3. Prin. 3. 
 
36 MULTIPLICATIOir. 
 
 2. If I ton of iron costs a dollars, what will x tons cost ? 
 Analysis, x tons will cost x times as much as i ton ; and x times 
 
 a dollars are ax dollars. That is, a dollars are taken x times, and are 
 equal to a + a + a . . , . , and so on to a; terms. 
 
 3. What is the product of 4a by 25 ? 
 
 Analysis. — Since each coefficient and each letter opbbation. 
 
 in the multiplier and multiplicand is a factor, it fol- 4^ 
 
 lows that the answer must be the product of the 2 J 
 
 coefficients with all the letters of both factors an- . r~T 
 nexed. Hence, the 
 
 EuLE. — Multiply the coefficients together, and prefix the 
 product to the product of the literal factors. 
 
 Multiply the following quantities : 
 
 4. 4ab by 50?, Ans, 2oabx, 
 
 5. 6bc by 7«. 9. 'jxy by 8ah» 
 
 6. ^abc by ^xy, 10. 6ac by jdx. 
 
 7. 8dm by xy, 11. gbd by 6cm, 
 S, gbcd by 'jxyz, 12, jxyz by gadf 
 
 SIGNS OF THE PRODUCT. 
 
 89. The investigation of the laws that govern the signs 
 of the product, requires attention to the following 
 
 PRINCIPLES. 
 
 1®. Repeating a quantity does not change its sign, 
 2°. The sign of the multiplier shows whether the repetitions 
 of the multiplicand are to he added, or subtracted, 
 
 90. If the Signs of the factors are alihe, the sign of the 
 product will be positive ; if unlike, the sign of the product 
 will be negative. 
 
 88. How multiply a monomial by a monora.al ? 89. Name Principle i. Prin. 2, 
 90. If signs of factorH are alike, what ia the sign ol the product? If unlike 1 
 
MULTIPLIC ATIOK. 37 
 
 91. Demoksteation. — There are four points to be 
 proved: 
 
 First, That -f- into + produces +. 
 
 I/et +a be the multiplicand and +4 the multiplier. It is plain 
 that +05 taken +4 times is +4*. (Prin. i.) The sign of the multi- 
 plier being -r, shows that the product +4^, is to be added, which is 
 donf» \}j setting it down without changing its sign. (Art. 66.) 
 
 Second. That — into -f produces — . 
 
 Let —a be multiplied by +4. Now —a taken 4 times is —4a ; for 
 a negative quantity repeated is stiU negative. (Prin. i.) But the 
 sign before the multiplier being + , shows that the negative product 
 — 4a, Is to be added. This also is done by setting it down without 
 changing its sign. (Art. 66.) 
 
 Third. That 4- into — produces — . 
 
 Let +a be multiplied by —4. We have seen above that +a taken 
 4 times is +^a. But here the sign of the multiplier being — , shows 
 that the product +4a> is to be subtracted. This is done by changing 
 its sign from + to — , on setting it down. Thus, +a x —4 = —4a. 
 (Art. 77.) 
 
 Fourth. That — into — produces +. 
 
 Let —a be multiplied by —4. It has also been shown that —a 
 taken 4 times is —4a. But the sign of the multiplier being — , shows 
 tnat this negative product —4a, is to be subtracted. This is also done 
 by changing its sign from — to + , when we set it down. Thus, 
 —a X — 4 = +4^. (Art. 77.) Hence, universally, 
 
 92. Factors having like signs produce -f, and unlike 
 signs — . 
 
 13. Multiply -I- 4ab by ~ 'jcd, Ans. — 2d>ahcd. 
 
 14. Multiply — <^xy by + ^ad. 
 
 15. Multiply + ()db by + ^dc. 
 
 16. Multiply — ^xy by — i()alc. 
 
 17. Multiply + i2>aic by — 232:^. 
 
 18. Multiply — 35^?/ by — 272*^^. 
 
 91. Prove the first point from the blackboard. The second. Third. ' Fourth 
 92. Rule for signs. 
 
38 MIJLTIPLIC ATIOH. 
 
 93. When a letter is multiplied into itself, or taken twice 
 as a factor, the product is represented hj a x a, or aa\ 
 when taken three times, by aaa, and so on, forming a series 
 of powers. But powers, we have seen, are expressed by 
 writing the letter once only, with the index above it, at the 
 right hand. (Art. 31.) 
 
 19. What is the product of aaa into ««? 
 
 Analysis, aaa y.axi=: aaaaa^ or a^, Atib. Now aaa = a^, and 
 aa = a^ ; but adding the exponents of a^ and a^ we have a^, the same 
 as before. Hence, 
 
 94. To multiply powers of the same letter together, add 
 their exponents. 
 
 Notes. — i. All powers of i are i. 
 
 2. When a letter has no exponent, i is always understood. 
 
 Multiply the following quantities : 
 
 20. aV^(? by a^c, Ans. a^^d^, 
 
 21. 2>(^Wxhj 2al^y, Ans. 6a^¥xy. 
 
 22. 32;?/2 by 5ic2, ^^. ab'^hjab^. 
 
 23. 6a^ by 4«5^. 26. $xyz by 2xy, 
 
 24. a'^x'^y by a^a^y. 27. 6a^^c by :^a^dl^c. 
 
 28. If a = 3, what is the difference between sa and a^? 
 
 29. If a; = 4, what is the difference between 4X and a^ ? 
 
 95. The preceding principles illustrating monomials may 
 be summed up in the following 
 
 EuLE. — Multiply the coefficients and letters of hoth factors 
 together; to the product prhj.^ . :.? proper sign, and give to 
 each letter its proper index. 
 
 Note. — It is immaterial in what or 2 the factors are taken, but it 
 is more convenient, and therefore cusix>mary, to arrange the letters in 
 alphabetical order. (Art. 87, Prin. 3.) 
 
 30. Multiply — z'^y ^y — 2a;. 
 
 31. Multiply 6^2^ by — ^c^lc. 
 
 94. How multiply powers of the same letter together? 05. What is the rule for 
 multiplying moaomifids ? 
 
MULTIPLICATIOK. 39 
 
 (32.) {33) (34.) (35) 
 
 Multiply 4x7/ 
 By x^y 
 
 7«^ 
 
 5^y 
 
 
 (36.) 
 Multiply s^y 
 By - -xf 
 
 (37.) 
 
 7a5c3 
 
 CASE II. 
 
 (38.) 
 
 — 7ac 
 
 (39.) 
 xyz 
 
 96. To Multiply a Polynomial by a Monomial, 
 
 1. What is the product of a + 5 multiplied by J ? 
 Analysis.— Multiplying each term of the operation. 
 
 multiplicand by the multiplier, we have axb Ct -\- 
 
 = fl*, and & X & = &2. The result, «d + &^ is the ^ 
 
 product required. Hence, the Ans, ah A- V^ 
 
 Rule. — Multiply each term of the multiplicand ly the 
 multiplier; giving each jtartial product its proper sign, and 
 each letter its proper index. 
 
 Multiply the following quantities : 
 
 2. he — adhj ah, Ans, al^c — a%d, 
 
 3. 3«a^ -f 4cd by 2c. 
 
 4. 5«^ — 2cd -\- xhj lax, 
 
 5. 4«2 — 3^5 _|- ^2 ]3y __ 2.hd. 
 
 6. 3^2 — 4W — 2C^ by — $a^c, 
 
 CASE III. 
 
 97. To Multiply a Polynomial by a Polynomial, 
 
 7. What is the product oi a + h into « -f- Z* ? 
 Analysis. — Since the multiplicand is to operation. 
 
 be taken as many times as there are units in a -j- h 
 
 the multiplier, the product must be equal to a -{- h 
 
 a times a+b added to 6timesa + &. *Now ~2~T k 
 
 a times a+b = a^ + ab, and b time^ a+b ^ 
 
 = +ab+b^. Hence, a+b times g;+5 must -{- ah -^ i^ 
 
 be equal to a'' + 2ab+¥. Ans. a^ + 2^5. -f W- 
 
 96. How multip'.y a r;o'»\TioTninl l)v 1 monomi:;. 
 
40 MULTIPLICATIO:^. 
 
 8. Multiply J 4- 2e? — 3c by 05 4- 5. 
 
 Analysis. — We multi- operation. 
 
 ply each term in the multi- 2a -jr i — $0 
 
 plicand by each term in the a -{- b 
 
 multiplier, giving to each 2^2+ ab -^ sac 
 
 product the proper sign. 
 (Art. 89.) Finally, we add 
 
 + 2ab + ^ — 3&g 
 
 the partial products, and the Ans, 2a^ + ^ab — ^ac -{- b^ — 2,bc 
 result is the answer required. 
 
 98. The various principles developed in the preceding 
 cases, may be summed up in one 
 
 GENERAL RULE. 
 
 Multiply each term of the multiplicand by each term of 
 the multiplier, giving each product its proper sigti, and each 
 letter its proper exponent. 
 
 The sum of the partial products will be the true ^jroduct. 
 
 Note. — For convenience in adding the partial products, like terms 
 should be placed under each other. 
 
 Multiply the following quantities: 
 
 1. 2a-\- bhj s^ + y' 5« sa + 4b — chjx—y. 
 
 2. 3^ + 4y^J a — b. 6. ^x + sy +^^ya + b. 
 
 3. /^b — c by 2>d — a, 7. jcdx — ^ab by 2m — $n. 
 
 4. 6xy — 2a hj b -{- c. 8. Sabc + 4m by $x — 4?/. 
 9. Multiply ^ab"" by Sa^. Ans. 24a^"+\ 
 
 10. Multiply — 'jacff" by — Sa^af. Ans. 56a^a;"*+". 
 
 11. Multiply 3a Jc" by xyz"^. 
 
 12. Multiply acd"^ by iibcd\ 
 
 13. Multiply — ax^ by — ax\ 
 
 14. Multiply x{a-\- by hy c{a + by. (Art. 15.) 
 
 15. Multiply c{a^ bf by 5 (a — bf. 
 
 16. Multiply a{x-\- y)"" by be {x + y)". 
 
 17. Multiply 3X {a + by by — (« + Z>). 
 
 98. fiow multiply a polynomial by a polynomial? 
 
MULTIPLICATIOiq'. 41 
 
 99. When the polynomials contain different powers of the 
 same letter, the terms should be arranged so that the first 
 term shall contain the highest power of that letter, the 
 second term the next highest power, and so on to the last 
 term. This letter is called the leading letter, 
 
 (i8.) (19.) 
 
 a ■)- b 4a^—$ab 
 
 a^^2a^+ ab^ i2a^b^-\-4a'^b^ 
 
 a^^^a^ + sal^-^l^, Ans. i2aW—sa^b^—zaW, Am. 
 
 20. Multiply a^ — ab -{- b^ hj a -^ b. 
 
 21. Multiply a? — ab + W by a^ -^ ab -^ ^. 
 
 22. Multiply x^ ■\- X ■\- I by 0^ — x -\- i. 
 
 23. Multiply ^x^ — 2xy + 5 by a;^ + 2xy — 6. 
 
 24. Multiply ^ax — 2ay by ^ax + 30!^. 
 
 25. Multiply d -{- bx \)j d ■{- ex. 
 
 100. The Multiplication of polynomials may be indi- 
 cated by inclosing each factor in a parenthesis, and writing 
 one after the other. 
 
 Thus, [a + 6) (a + 6) is equivalent to {a + h)x{a + h). 
 
 Note. — Algebraic Expressiotis are said to be developed or 
 expanded, wlien the operations indicated by their signs and exponents 
 are performed. , 
 
 26. Develop the expression (a -\- b) {c -\- d). 
 
 Ans. ac •\- be ■\- ad -\- bd, 
 
 27. Develop {x + y) {x — y). 
 
 28. Develop {a^ + i) (« + i). 
 
 29. Expand {x^ + 2xy + y'^) {x + y). 
 
 30. Expand (a"^ + b"") (a"^ + 5"). 
 
 31. Expand [x -{- y -\- z) {x -^ y -{- z). 
 
 99. How arrange different powers of the same letter ? 100. How indicate the mul- 
 tiplication of polynomials ? '' 
 
42 KTJLTIPLICATION". 
 
 THEOREMS AND FORMULAS. 
 
 101. Theorem i.—The Square of the Sum of two quan- 
 tities is equal to the square of the first, plus twice their 
 product, plus the square of the second, 
 
 1. Let it be required to multiply 
 a-\- b into itself. 
 
 Analysis. — Each term of the multipli- 
 cand being multiplied by each term of the 
 multiplier, we have a times a + 6 and b times 
 a + b, the sum of which is a^ + 2ab + &*. 
 Hence, the A71S. 
 
 Formula. (a + W = a^ -\- 2ab + l^. 
 
 102. Theorem 2.— The Square of the Difference of two 
 quantities is equal to the square of the first, minus tivice 
 their product, plus the square of the second, 
 
 2. Let it be required to multiply 
 
 a — hhya — b. a — b 
 
 Analysts. — Reasoning as before, the re- ^ 
 
 a-Jf-b 
 a-\-b 
 
 a^-\- ab 
 + ab+1^ 
 
 a^ + 2a6 + ^>2 
 
 suit is the same, except the sign of the mid- ^ — ^^ 
 
 die term 2ab, which has the sign — instead — ab ■\- 1^ 
 
 of -I-. Hence, the A7IS. a^ — 2ab + ^ 
 
 Formula. (a — bf = a? — 2ah-\- b\ 
 
 103. Theorem ^.—The Product of the Sum and Differ- 
 ence of two quantities is equal to the difference of their 
 squares, 
 
 3. Let it be required tp multiply 
 a -\-bhj a — b. a + h 
 
 Analysis. — This operation is similar to ^ " 
 
 the last two ; but the terms -f- ab and —ab, <^ + ^^ 
 
 In the partial products, being equal, balance ^b fr 
 
 each other. Hence, the Ans. cf — W' 
 
 Formula. {a + h) (a — h) = a? — l^. 
 
 loi. What is Theorem i ? 102. Thoorom 2 ? 103. Theorem 3 ? 
 
MULTIP-LI CATION". 
 
 43 
 
 104. TJie product of the sum of hvo quantities into a third, 
 is equal to the sum f^f their products. 
 
 4. Let X and y be two quantities, whose sum is to be 
 multiplied ^y a. Thus, 
 
 Tl\e product of the sum (a; + y) x a —ax-\-ay 
 
 The sum of the products ofa;xa+yxa=:aa;+ay 
 And aic + ay = aaj + ay. Hence, the 
 
 Formula. a (a? + 2/) = aoc + ay. 
 
 105. The product of the difference of two quantities into a 
 third, is equal to the difference of their products. 
 
 5. Let X and y be two quantities, whose difference is to be 
 multiplied by a. Thus, 
 
 The product of the difference (a;— y) x a = ax— ay 
 
 The difference of the products of a; x a—y x a = ax— ay 
 And ax— ay — ax— ay. Hence, the 
 
 Formula. a(x — y) = ax — ay. 
 
 Remark. — The application of the preceding principles is so frequent 
 in algebraic processes, that it is important for the learner to make 
 them very familiar. 
 
 Develop the following expressions by the preceding for- 
 mulas: 
 
 {^x — i) (4.r — 1). 
 (55+i)(5J+i). 
 (i _ x) (i - x). 
 (i +2:z;)(i + 2x). 
 {Sb — 3a) {Sb - 3a). 
 {ah + cd) {ah -f cd). 
 (3« — 2y) (3« + 2y). 
 {x^^y){x^-y). 
 {x - y^) {x — y^). 
 
 {20^ + X) (2«2 _ ^, 
 
 I. 
 
 (a+ i)(«+ i). 
 
 II. 
 
 2. 
 
 (26? 4- I) (265+ l). 
 
 12. 
 
 3. 
 
 i2a — h) {2a — h). 
 
 13. 
 
 4. 
 
 {x + y) {x + y). 
 
 14. 
 
 5- 
 
 (^ -y){x- y). 
 
 15- 
 
 6. 
 
 {i + x){i-x). 
 
 16. 
 
 7. 
 
 W-y){iy'-y)' 
 
 17. 
 
 8. 
 
 (4m — 3n) {4m + 3^)- 
 
 18. 
 
 9- 
 
 (^2_^)(^2 + ^). 
 
 19. 
 
 10. 
 
 (i _ 7:^,) (i _|_ 7:^;). 
 
 20. 
 
 104. What is the product of the sum of two quantities into 
 105. Of the difference? 
 
 third equal to? 
 
Let x=. 
 
 : No. apples ; 
 
 X 
 
 : pears. 
 
 '<- 
 
 24 
 
 ^x + x = 
 
 :72 
 
 4X = 
 
 ,', X = 
 
 :72 
 
 : 18 apples. 
 
 X _ 
 
 : 6 pears. 
 
 44 MULTIPLIC ATIONo 
 
 PROBLEMS. 
 
 106. Problems requiring each side of the equatKon to be 
 
 multiplied by equal quantities. 
 
 1. George has 1 third as many pears as apples, and the 
 number of both is 24. How many has he of each ? 
 
 Analysis. — If x represents the num- 
 ber of apples, then - will represent the 
 
 number of pears, and x + - will equal 
 
 24, the number of both. Tbe denomi- 
 nator of X is removed by multiplying 
 each term on both sides of the equation 
 by 3. (Ax. 6.) The result is 3a; + x, or 
 4a; = 72. Hence, cc=i8, the apples, 
 and 18-7-3 = 6, the pears. Hence, 
 
 3 
 
 107. When a term on either side of the equation has a 
 deiiominator, that denominator is removed hy multiplying 
 every term on hoth sides of the equation hy it. (Ax. 4.) 
 
 2. What number is that, i seventh of which is 9 ? 
 
 Ans. 6^, 
 
 3. What number is that, 2 thirds of which are 24 ? 
 
 4. A man being asked how many chickens he had, 
 answered, 3 fourths of them equal 18. How many had he ? 
 
 5. What number is that, i third and i fourth of which 
 are 21? 
 
 Analysis. — If x represent the number, then j^^^ ^ __ ^^^ 
 
 X X 
 
 will - + - = 21, by the conditions. Multiplying x X 
 
 34 -+-=21 
 
 each term on both sides by the denominators 3 3 4 
 
 and 4 separately, we have /\x+2)X=- 252. (Ax. 4.) 4^ "f" 3^ ^^ 252 
 
 Uniting the terms, 7a; = 252, and x — 36, Ans. ,' . X = 36 
 
 Proof. ^ of 36 = 12, and ^ of 36 = 9. Now, 12 + 9 = 21. 
 
 107. When a quantity on cither s'.de .of an equation has a denominator, how re- 
 move it ? 
 
MULTIPLICATION. 45 
 
 6. What number is that, 2 thirds of which exceed i half 
 of it by 8 ? 
 
 7. A general lost 840 men in battle, which equaled 
 3 sevenths of his army. Of how many men did his army 
 consist ? 
 
 8. If 3 eighths of a yacht are worth I360, what is the 
 l^hole worth ? 
 
 A.C(y 
 
 9. If -- equals 20, to what is x equal ? 
 
 10. If — is equal to 20, to what is x equal ? 
 
 4 
 
 11. If — is equal to 24, to what is x equal ? 
 
 AX 
 
 12. If — is equal to 28, to what is x equal ? 
 
 13. Henry has 30 peaches, which are 5 sixths the number 
 of his apples. How many apples has he ? 
 
 14. A farmer had 3 sevenths as many cows as sheep, and 
 his number of cows was 30. How many sheep had he ? 
 How many of both ? 
 
 15. Divide 28 pounds into two parts, such that one may 
 be 3 fourths of the other. 
 
 16. A lad having given i third of his plums to one school- 
 mate, and I fourth to another, had 10 left. How many had 
 he at first ? 
 
 17. What number is that, i third and i sixth of which 
 are 21? 
 
 18. What number is that, i fourth of which exceeds 
 
 1 sixth by 12 ? 
 
 19. Divide 36 into two parts, such that one may be 
 
 2 thirds of the other ? 
 
 20. One of my apple trees bore 3 sevenths as many apples 
 as the other, and both yielded 21 bushels. How many 
 bushels did each yield ? 
 
CHAPTER v. 
 DIVISION. 
 
 108. Division is finding how many times one quan- 
 tity is contained in another. 
 
 The Dividend is the quantity to be divided. 
 The Divisor is the quantity by which we divide. 
 The Quotient is the quantity found by division. 
 The Remainder is a part of the dividend left after 
 division. 
 
 109. Division is the reverse of multiplication, the divi- 
 dend answering to the product^ the divisor to one of the 
 factors, and the quotient to the other, 
 
 PRINCIPLES. 
 
 110. 1°. When the divisor is a quantity of the same hind 
 as the divide7id, the quotient is an abstract number. 
 
 2°. When the divisor is a number, the qiiotient is a quan- 
 tity of the same Tcind as the dividend. 
 
 3°. The product of the divisor and quotient is equal to the 
 dividend. 
 
 4°. Cancelling a factor of a quantity, divides the quantity 
 by that factor. 
 
 CASE I. 
 
 111. To Divide a Monomial by a IVIonomial. 
 
 I. What is the quotient of abed divided by cd? 
 
 Analysis.— The divisor cd is & factor of the divi- operation. 
 
 dend ; therefore, if we cancel this factor, the other cd ) abcd 
 factor ab, will be the quotient. (Prin. 4.) Ans. ab. 
 
 108. Define division. The dividend. Divisor. Quotient. Remainder. 109. Oi 
 what is division the reverse? Explain, no. Name Uie first principle. The second. 
 Third. Fourth. 
 
DIVISIOK. 4:7 
 
 2. What is the quotient of i2>al) divided by 6a ? 
 
 Analysis.— DividiDg the coefficient of the divi- opbbation. 
 
 dend by that of the divisor, and cancelling the com- 6a ) l?>ab 
 mon factor a, we have T8a6-j-6a = 3&, the quotient A^lS. $b, 
 required. (Prin. i.) Hence, the 
 
 EuLE. — Divide one coefficient by the other, and to the re^ 
 suit annex the quotient of the literal parts. 
 
 Divide the following quantities : 
 
 (3-) (4.) (5-) (6.) 
 
 2c) 4abc 4b) 2obxy 8xy ) 4ox y 16b ) S2ai 
 
 (7.) (8.) (9-) 
 
 gm ) 4sabm 2omn ) 6obcmn 24xy ) g6mnxy 
 
 SIGNS OF THE QUOTIENT. 
 
 112. The rule for the signs in division is the same as 
 ihat in multiplication. That is, 
 
 If the divisor and dividend have lihe signs, the sign of the 
 quotient will be + ; if unlike, the sign of the quotient 
 will be — . 
 
 Thus, +a X +b = +ab; hence, +ab -r- +b = +a. 
 
 —a X +b = —ab ; hence, —ab -r- +b = —a. 
 
 + a X —b = —ab; hence, —ab -. b = +a. 
 
 —a X — & = +ab; hence, +ab -i b = —a. 
 
 Divide the following quantities : 
 
 10. — ^2abc by — 4ab. Ans. Sc. 
 
 11. iSabxhj—$z. Ans. 
 
 12. 2iabc by — yab. 15. 4Sabc by — Sac. 
 
 13. — 2Sbcd by — 4cd. 16. 6$bdfx by gbx. 
 
 14. $$cdm by 7cm. 17. — 'j2acgm by Scm. 
 
 III. How divide a monomial by a monomial? 112. What is the rale for the 
 signs? 
 
48 DIVISlOITo 
 
 113. To Divide Powers of the same letter, 
 
 1 8. Let it be required to divide a^ by a\ 
 
 Analysis. — The term a^ — aaaaa, and a^ — aaa. Rejecting tho 
 factors aaa from the dividend, the result aa, or d^, is the quotient. 
 Subtracting 3, the index of the divisor, from 5, the index of the divi- 
 dend, leaves 2, the index of the quotient. That is, a^ -j- a^ _ ^2 
 (Arts. 31, no. Prin. 4.) Hence, the 
 
 EuLE. — SuUraci the index of the divisor from that of the 
 dividend. 
 
 Divide the following quantities : 
 
 19. ^ by 6^. 22. xyz^^hj xyz^. 
 
 20. x^^ by ^, 23. i6ah^ by ^ab. 
 
 21. ac^ by ac^, 24. 6xy by 3^2^ 
 
 114. The preceding principles may be summed up in 
 the following 
 
 Rule. — Divide the coefficient of the dividend hy that of the 
 divisor ; to the result annex the quotient of the literal factors^ 
 prefixing the proper sign and giving each letter its proper 
 exponent. 
 
 Proof. — Multiply the divisor and quotient together, as in 
 arithmetic. 
 
 Note. — If the letters of the divisor are not in the dividend, the 
 division is expressed by wanting the divisor under the dividend, in the 
 form of a fraction. 
 
 25. What is the quotient of ^x divided by 3?/? Ans. — • 
 
 26. — 24aWc^ -. 2>^ib. 32. 2>'^x^y^^ -^ dp?yz. 
 
 27. — 2i^x^y^^ -^ ^^y^' 33* <)6aWc-^ i2ah. 
 
 28. $aW -^ ah. 34. 84d^:i^y^ -7- jd^xy. 
 
 29. — jofiy^ -. xy. 35. loSaix^ -i- ga^a^. 
 
 30. a^b^c^ -T- aWc. 36. i^2X*y!^ -r- iix^yzK 
 
 31. 1 6aWc^ -7- Sa^^c^. 37. 121 m^n^a^ -r- 1 1 mhia^. 
 
 113. ITow divide powers of the same letter? 114. Rule for division of moiia 
 jnials ? Proof? If *^e Jitters of the divit-or are not in the dividend, what i.- done ? 
 
DIVISIO.TST, 49 
 
 CASE II. 
 115. To Divide a Polynottiial by a Monomial. 
 
 1. Divide ah ■\- ac + ad by a. 
 
 Analysis.— Since the factor a enters into oPERATioif. 
 
 each term of the dividend, it is plain that a) ab -\r dC A- ad 
 
 each term of the dividend must be divisible Ans h 4- C -i- d 
 by this factor. Hence, the 
 
 EuLE. — Divide each term of the dividend ly the divisor, 
 and connect the results hy their proper signs. 
 
 Note. — If a polynomial which contains the same factor in every 
 term, be divided by the other quantities connected by their signs, the 
 :iuotient will be that factor. 
 
 Divide the following quantities: 
 
 2. 6a^ + 10^2 __ i^a by 2a, Ans, sa^ 4- 5«5 — 7. 
 
 3. 40^ — Sa^ + i2«2 by — 2«2. Ans. — 2^2 4. 4^5 _ 6, 
 
 4. at/^ -i- a(^ -{- ad^ by a. 
 
 5. isx^y + 250:^ by s^y. 
 
 6. 6adc — 2a-i-Sah by 2a, 
 
 7. —i6by^'{-4y^hj—Sy, 
 
 8. i4X^y — yxy^ by — yx^^ 
 
 9. xy^ -f xz — xhj X, 
 
 10. 35fl5 -i- 28J— 42 by — 7. 
 
 11. i$a^ — 15^2 by 5a. 
 
 12. i6x^ac + i2acd^ — 4xa^c by — 4CC. 
 
 13. 4a* — 20^2 + Sab by 4«. 
 
 14. 3^5 -f i^a^ — 2^a^)d by 3flr5. 
 
 15. ZaV)c — xdatJ^c — 2oabc^ by ^abc, 
 
 16. 6a; {a + ^»)2 -+- 9a;2 {a -{- 5)2 by 3ir. 
 
 17. ^S{^-y)-V2>o{x — y)hYS' 
 
 18. «a;2 (5 — c) — aH {b — c) by ax. 
 
 19. i8«4 (a + by — 12«3 (^ _|. ^,)2 by 6«2 (« 4. by. 
 
 20. a"+i — «"+2 4. a«+3 by «". 
 
 115. Eow divide a polynomial by n -lonomial ? 
 
60 Divisioiy. 
 
 CASE III. 
 116. To Divide a Polynomial by a Polynomial. 
 
 I. Divide a^ + Za% + ^aV^ + ^ by a2 + 2ab + J*. 
 
 Analysis. — For conve- opbbation. 
 
 nience, we arrange tlie terms a^ + sa^b + 3«^ + Z*^ a^ -|- 2rt J 4 5^ 
 so that the first or leading a^^2a^+ aW a + b Quot 
 
 .letter of the divisor shall be ^7~~ TTTTs 
 
 the first letter of the divi- ^ + 2afr'+d 
 
 dend. The powers of this a^-{-2ai^+I)^ 
 
 letter should be arranged in 
 
 order, both in the divisor and dividend, the highest power standing 
 Jirst, the neoct highest next, and so on. The divisor may be placed on 
 the left of the dividend, or on the right, and the quotient under it, at 
 pleasure. 
 
 Proceeding as in arithmetic, we find the first term of the divisor is 
 contained in the first term of the dividend a times. Placing the a in 
 the quotient under the divisor, we multiply the whole divisor by it, 
 subtract the product, and to the remainder bring down as many other 
 terms as necessary to continue the operation. Dividing as before, a* 
 is contained in a% +b times. Multiplying the divisor by +& and 
 subtracting the product, the dividend is exhausted ; therefore a + & ia 
 the quotient. Hence, the 
 
 Rule. — I. Arrange the divisor and dividend according to 
 the powers of one of their letters ; and finding how many 
 times the first term of the divisor is contained in the first 
 term of the dividend, place the result in the quotient. 
 
 IL Multiply the whole divisor by the term placed in the 
 quotient ; subtract the product from the dividend, and tc the 
 remainder bring doivn as many terms of the dividend as the 
 case may require. 
 
 Repeat the operation till all the terms of the dividend are 
 divided. 
 
 Note. — If there is a remainder after all the terms of the dividend 
 ore brought down, vlace U over the divisor, and annex it to the quotient 
 
 ii6. How divide a polynomial by a polynomial ? Jf tba*o is a remainder, what la 
 done with it? 
 
PROBLEMS. 61 
 
 2. Divide 4^2 4- ^ab + i^ hj 2a +bo Ans. 2a + b, 
 
 3. Divide x^ + 2xy -\- y^ hj x -{- t/, 
 
 4. Divide a^ — 2ab + l^ by a — b. 
 
 5. Divide a^ -- sa^b + sab^ — i,^ hy a ^ b. 
 
 6. Divide ac ■\- be -{- ad -\- bd hy a -{• b, 
 
 7. Divide ax -{• bx — ad — bd by a + b, 
 
 8. Divide 22^ -f 'jxy 4- 6^2 by a; + 2^ 
 9/ Divide a^ — V^hya + b. 
 
 10. Divide a? — y^ by a; — y. 
 
 11. Divide a^ — l^ by o — J. 
 
 12. Divide 6^3 4. 13^5^ 4. 552 by 205 -f 35. 
 
 13. Divide «2 __ ^ __ 5 by a — 3. 
 
 14. Divide a^ -- 7,a^x + 3aic2 _ ^^js by a — a; 
 
 15. Divide 6ar* — 96 by 3a; — 6. 
 
 16. Divide x^ ■\- "jx -\- 10 by x -\- 2. 
 
 17. Divide x^ — 5^? + 6 by x — 3. 
 
 18. Divide c^ — 2cx -\- 7? hy c — a?. 
 
 19. Divide a^ + 2ab ■{- b^ hy a -{- h 
 
 20. Divide 22 (« — Z>)2 by 1 1 (a — h). 
 
 PROBLEMS. 
 
 1. A father being asked the age of his son, replied, My 
 age is 5 times that of my son, lacking 4 years; and the 
 sum of our ages is 5 6 years. How old was each ? 
 
 2. John and Frank have 60 marbles, the former having 
 
 3 times as many as the latter. How many has each ? 
 
 3. The sum of two numbers is 72, one of which is 5 times 
 the other. What are the numbers ? 
 
 4. A man divided 57 pears between two girls, giving one 
 
 4 times as many as the other, lacking 3. How many did 
 each have ? 
 
 5. Three boys counting their money, found they had 
 190 cents; the second had twice as many cents as the first, 
 and the third as many as both the others, plus 4 cents. 
 How many cents had each ? 
 
53 DIVISION 
 
 6. A farmer has 9 times as many sheepas cows, and the 
 number of both is 2oo„ How many of each ? 
 
 7. Divide 57 into two ?uch parts that the greater shall be 
 3 times the less^ plus 3. What are the numbers ? 
 
 8. Given 2X -{- 4X -\- .4; — ^ = 60, to find x, 
 
 9. A and B are 35 miles apart, and travel toward each, 
 other, A at the rate of 4 miles an hour, and B, 3 miles. In 
 bow many hours will they meet ? 
 
 lOo Given a -}- $a + 6a + 2a + y = iig^ to find a. 
 
 11. Given 85 -f- 5^ -f 7^ — 10 = 130., to find b, 
 
 12. A lad having 60 cents, bought an equal number of 
 pears, oranges^ and bananas ; the pears being 3 cents apiece, 
 the oranges 4 cents, and the bananas 5 cents= How many 
 of each did he buy ? 
 
 13. A cistern filled with water has two faucets, one of 
 which will empty it in 5 hours, the other in 20 hours. How 
 long will it take both to empty it ? 
 
 14. Given a; + - = 45^ to find a?, 
 
 15. What number is that, to ths half of which if 3 be 
 added, the sum will be 8 ? 
 
 160 Three boys have 42 marbles ; B has twice as many as 
 As and C three times as many as A. How many has each ? 
 
 17. If A has 2x dollars, and B twice as many as A, and 
 C twice as many as B, how many have all ? 
 
 r8. Divide 40 into 3 parts, so that the second shall be 
 3 times the first, and the third shall be 4 times the first. 
 
 19. A man divided 60 peaches among 3 boys, in such a 
 manner that B had twice as many as A, and as many as 
 A and B. How many did each receive ? 
 
 20. Divide 48 into 3 such parts, that the second shall be 
 equal to twice the first, and the third to the sum of the first 
 and second? 
 
 21. What number is that, to three-fourths of which if 5 
 oe added, the sum will be 23 ? 
 
OHAPTEE VI. 
 FACTORING. 
 
 117. Factors are quantities which multiplied togethei 
 produce another quantity. (Art. 86.) 
 
 118. A Composite Quantity is the product of two 
 or more integral factors, each of which is greater than a 
 unit. 
 
 Thus, 3a, 5&, also x^y^, are composite quantities. 
 
 119. Factoring is resolving a composite quantity 
 into its factors. It is the converse of multiplication. 
 
 120. An Exact Divisor of a quantity is one that will 
 divide it without a remainder. Hence, 
 
 Note. — The Factors of a quantity are always exact divisors of it, 
 and vice versa. 
 
 121. A Prime Quantity is one which has no integral 
 divisor, except itself and i. 
 
 Thus, 5 and 7, also a and 6, are prime quantities. Hence, 
 
 Note. — The least divisor of a composite quantity is a prime factor. 
 
 122. Quantities are prime to each other when they 
 have no common integral divisor, except the unit i. 
 
 Thus, II and 15, also a and 6c, are prime to each other. 
 
 123. A Multiple is a quantity which can be divided 
 by another quantity without a remainder. Hence, 
 
 A multiple is a product of two or more factors. 
 
 •117. What are factors? ii8. A composite quantity? 119. Wliat is factoring? 
 120 An exact divisor? 121. A prime quantity? 122. When prime to each other? 
 ii-3. A multioie? 
 
54 FACTORING. 
 
 PRINCIPLES. 
 
 124. 1^. If one quantity is an exact divisor of another ^ 
 the former is also an exact divisor of any multiple of the 
 latter. 
 
 Thus, 3 is a divisor of 6 ; it is also a divisor of 3 x 6, of 5 x 6, etc. 
 
 2°. If a quantity is an exact divisor of each of two other 
 quantities, it is also an exact divisor of their sum, their dif- 
 ference, or their product. 
 
 Tims, 3 is a divisor of 9 and 15, respectively ; it is also a divisor of 
 9+15, or 24 ; of 15—9, or 6 ; and of 15 x 9, or 135. 
 
 3°. A composite quantity is divisible ly each of its prime 
 factors, by the product of two or more of them, and by no 
 other quantity. 
 
 Thus, the prime factors of 30 are 2, 3, and 5. Now 30 is divisible 
 hy 2, by 3, and by 2 x 3 ; by 2 x 5 ; by 3 x 5 ; by 2x3x5, and by no 
 other number. 
 
 CASE I. 
 
 125. To Find the Prime Factors of Monomials. 
 I. What are tlie prime factors of i2«2j? 
 
 Analysis. — The coefficient 12 = 2x2x3, and a^h — adb. There- 
 fore the prime factors of i2a^6 are 2 x 2 x '^aab. Hence, the 
 
 KcTLE. — Find the prime factors of the numeral coefficients, 
 and annex to them the given letters, taking each as many 
 times as there are units in its exponent. 
 
 Note. — In monomials, each letter is a factor. Hence, the prime 
 factors of literal monomials are apparent at sight. 
 
 Resolve the following quantities into their prime factors : 
 
 2. is^f' Ans. zy^s^^y y y- 
 
 3. i2,d^b\ 7. i-jx^z. 
 
 4. lobo?]^, 8. 2$al^c3?, 
 
 5. $$aWc^, 9. Tja^c^d. 
 
 6. 2ixy^^. 10. 6$m^n^x. 
 
 124. Name Principle i. Principle 2. Principle 3. 125. How find the prime fee- 
 tors of monomials ? 
 
FACTORING. 56 
 
 CASE II. 
 126. To Factor a Polynomial. 
 
 1. Kesolve ^a^h + ^ab — Sac into two factors. 
 
 Analysis. — By inspection, we per- operation. 
 
 ceive the factor 2a is common to 2a ) /^a% + 8fl^^ — 6ac 
 
 each term ; dividing by it, the quo- 2ah + ^h — 3 c 
 
 tient 2a& + 4&-3/5 is the other factor. ^^^^ ^a izah + 4J - 3c) 
 For convenience, we enclose this fac- 
 tor in a parenthesis, and prefix to it the factor 2a, as a coejfident. 
 
 Proof. — The factor (206 + 4&— 3c) x 2a=^a^l) + %ab—6ac. Hence, the 
 
 EuLE. — Divide the polynomial ly the greatest common 
 monomial factor ; the divisor will he one factor, the quotient 
 the other, (Art. 115.) 
 
 Note. — Any common factor, or the product of any two or more 
 common factors, may be taken as a divisor ; but the result will very 
 in form according to the factors employed. (Ex. 2.) 
 
 2. Eesolve a^ + alt^ into two factors, one of which shall 
 be a monomial. Ans. db (a + ^), a (ah -{-¥), or h (a^ -f ah), 
 
 3. Factor a + ah + ac. Ans, a (i -^ h -\- c), 
 
 4. Factor by -\- he -\- t^x, 
 
 5. Factor 2ax •\- 2ay — 4az, 
 
 6. Factor ^hcx — Ghcx — ^ah(k 
 
 7. Factor Mmn — 2^dm^ 
 
 8. Factor 35^7/2 -f- 14^0:. 
 
 9. Factor 2^hdx^^^dmy, 
 
 10. Factor 6a^ -f ga^c. 
 
 11. Factor 2iao[^y -f- 35«a;y. 
 
 12. Factor 25 + 152:2 — 200^2^. 
 
 13. Factor x -\- x^ -\- ^, 
 
 14. Factor 3a; -f 6 — 9^. 
 
 15. Factor 19^52; —19^5, 
 
 126. How factor a polynomial ? 
 
56 rACTORiiir®. 
 
 CASE III. 
 127. To Resolve a Trinomial into two equal Binomial Factors. 
 
 1. Resolve x^ + 2xy -f y"^ into two equal binomial factors. 
 
 Analysis. — Since tlie square of a * operation. 
 
 quantity is the product of two equal /^xi ==: X, V^ = 2/> 
 
 factors (Art. 30), it follows tliat the ^ c^ 2 
 
 square root of a quantity is one of the " ' " '" •2' 
 
 two equal factors which produce it. (^H"^) (^ + 2/)? Ans* 
 
 (Art. 32.) Therefore the square root of 
 
 Q? is ar, that of y^ is y. And since the middle term 2Ty is twice the 
 
 product of these two terms, x^->r2xy+y'^ must be the square of the 
 
 binomial x+y. Consequently, x + y is one of the two equal binomial 
 
 factors. 
 
 2. Resolve x^ — 2xy -f- y"^ into two equal binomial factors. 
 
 ANAiiTSis. — Reasoning as before, the operation. 
 
 quantity x^—2Ty+y^ is the square of ^x^ z=x, V^ = ^ 
 
 the residual x—y. Therefore, the two ^ „ j_ 2 
 
 equal factors must be x—y and x—y. ' ' if '^ if 
 
 Hence, the (^— ^) (^ — y)> ^^S. 
 
 Rule. — Find the square root of each of the square terms, 
 and connect these roots hy the sign of the middle term. 
 
 Note. — A trinomial, in order to be resolved into equal binomial 
 factors, must have two of its terms squares, and the other term timce 
 the product of their square roots. (Art. loi.) 
 
 Resolve the following into two equal binomials: 
 
 3- 
 
 ^2 + 2a'b + J2. 
 
 9. 
 
 f+2y+i. 
 
 4. 
 
 x^—2xy-\- y\ 
 
 10. 
 
 I _ 2C2 + C^. 
 
 5. 
 
 m^ -j- 4mn + 4^2. 
 
 II. 
 
 ^2m ^ 2^myn ^ y^^ 
 
 6. 
 
 i6a^ + 8« + I. 
 
 12. 
 
 4^2" _ 4a» 4- I. 
 
 7- 
 
 49 + 70 + 25. 
 
 13. 
 
 a^ + 2aW + IK 
 
 8. 
 
 4^2 _ i2ah + 9J2. 
 
 14. 
 
 a^x^ + 2ax^y + y\ 
 
 197. How resolve a trinomial into equal binomial fectors f 
 
PACTOBING. 67 
 
 CASE IV, 
 
 2.28. To Factor a Binomial consisting of the Difference of 
 two Squares. 
 
 I. Resolve ^a^ — 9J2 into two binomial factors. 
 
 Analysis.— Both of these terms operation. 
 
 are squares; the root of the first is ^ ^0? = 2fl5 
 
 2a, that of the second is 36. But the /~w 7 
 
 difference of the squares of two quan- ^ 
 
 tities is equal to the product ot their * ' ^^ 9^ 
 
 sum and difference. (Art. 103.) Now (2«4-35) (2a— 3^), ^W5. 
 
 the sum of these two quantities is 
 
 2a + 3&, and the difference is 2a — 3& ; therefore, j\a? — 96^ = 
 
 (2a + 3&)(2a— 3&). Hence, the 
 
 Rule. — Find the square root of each term. The sum of 
 these roots will he one factor, and their difference the other. 
 
 Note. — This rule is one of the numerous applications of the for- 
 mula contained in Art. 103. 
 
 2. Resolve a^ — x^ into two binomial factors. 
 
 3. Resolve ^x^ — i6y^ into two binomial factors. 
 
 4. Resolve y^ — 4 into two binomial factors. 
 
 5. Resolve g — x^ into two binomial factors. 
 
 6. Resolve a^ — i into two binomial factors. 
 
 7. Resolve i — ^ into two binomial factors. 
 
 8. Resolve 25^^ — 16^2 into two binomial factors. 
 
 9. Resolve ^x^ — y^ into two binomial factors. 
 10. Resolve 1 — 16^2 into two binomial factors. 
 
 II. Resolve 25 — i into two binomial factors. 
 
 12. Resolve x^ — y* into two binomial factors. 
 
 13. Resolve a^x^ — b^y^ into two binomial factors. 
 
 14. Resolve m^ — # into two binomial factors. 
 
 15. Resolve a^'"" — h'^" into two binomial factors. 
 
 xaS, How fector a binomial consisting of the difference of two squares ? 
 
58 1^ AGIO KING. 
 
 CASE V. 
 
 129. Various classes of examples of higher powers may be 
 factored by means of the following 
 
 PRINCIPLES. 
 
 1°. The difference of any tivo powers of tlie same degree is 
 divisible by the difference of their roots. 
 
 Thus, (x^—f) -J- {x—y) — x+y. 
 
 {^—f) -4- {x-y) = a^+xy+^. 
 (oe*—y^) H- {x—y) — a^ + x^y+xy^+y\ 
 (a^— y5) -r- (x—y) = x^ + Q^y + x'^y'^ + xy^+y^. 
 
 2°. The differ e7ice of two even poivers of the same degree is 
 divisible by the sum of their roots. 
 
 Thus, {Q?—y^) -i- (x+y) — x—y. 
 
 (aj4_2^) ^ (x+y) = x^—x^y+xy'^—y^. 
 
 (afi—y^) 4- (x+y) = x^—x^y + a^y^—x^y^ + xy*—^, 
 
 3°. TJie Slim of two odd poicers of the same degree is divi- 
 sible by the sum of their roots. 
 
 Thus, i^+y^) -^ {x+y) = o^-xy+yK 
 
 (a^+y^) -r- (x+y) = xl^—a^y + x-y'^—xy^+y*. 
 
 (x'^+y'') ■+■ (x+y) — afi—x^y+xY—a^y^+xY—xy^+y\ etc. 
 
 Note.— The indices and signs of the quotient follow regular laws : 
 I St. The index of the first letter regularly deoreoMS by i, while that 
 of the following letter increases by i. 
 
 2d. When the difference of two powers is divided by the difference 
 of their roots, the signs of all the terms in the quotient are pltis. When 
 their sum or difference is divided by the sum of their roots, the odd 
 terms of the quotient are plus, and the even terms minus. 
 
 1^ If the principles and examples of this Case are deemed too 
 difficult for beginners, they may be deferred until the Binomial 
 Theorem is explained. (Arts. 268-270.) 
 
 129. Kecite Prin. i. Prin. 2. Prin. 3. Note. What is the index of the first let 
 ter f Of the following letter ? What is said of the si^uB ? 
 
FACTORING. 59 
 
 130. To Factor the Difference of any two Powers of the 
 same Degree. 
 
 I. Eesolve a? — y'^ into two factors. 
 
 Solution.— The binomial (a^— y^) -^ {x—y) = x^ + xi/+^. .*. x—y 
 md x^ + xy+y^ are the factors. (Prin. i.) Hence, the 
 
 'RuL^.— Divide the difference of the powers hy the differ- 
 ence of the roots; the divisor will he one factor, the quotient 
 the other, 
 
 Eesolve the following into two factors : 
 3' ^ — y^' 5. I — 
 
 131. To Factor the Difference of two even Powers of the 
 
 same Degree. 
 
 6. Eesolve a^ — Z>* into two factors. 
 
 Solution.— By Prin. 2,a^—¥ia divisible hy a + b. Thus, (a*— 6^) 
 ^ (a + b) = a^—a^b + c^—b^, the divisor being one factor, the quotient 
 the other. Hence, the 
 
 EuLE. — Divide the difference of the given powers ly the 
 sum of their roots ; the divisor ivill he one factor, the quo- 
 tient the other, (Art. 129, Prin. 2.) 
 
 Eesolve the following quantities into two factors : 
 
 7. J2 _ x\ 10. iC^ — 1. 
 
 8. d^ — ^, II. 1—^6, 
 
 9. a^ — h^, 12. a^ — I. 
 
 132. To Factor the Sum of two odd Powers of the same 
 
 Degree. 
 
 13. Eesolve a} + h^ into two factors. 
 
 Solution. — Dividing a^ + V^ by a + b, the factors are a + b and 
 a^—ab+y"* (Prin. 3.) Hence, the 
 
 EuLE. — Divide the sum of the powers hy the sum of the 
 roots; the divisor and quotient are the factors. 
 
 130. How factor the diflference of any two powers of the same degree ? 131. How 
 factor the difference of two even powers of the same degree ? 132. The sum of two 
 odd powers of the same degree t 
 
60 FACTOEIKG. 
 
 Resolve the following quantities into two factors : 
 
 14. a^-{-f. 17. i + y\ 
 
 15. a^ + I. 18. I + a\ 
 
 16. a^ + I. 19. I + W. 
 
 133. It will be observed that in the preceding examples 
 
 of this Case, binomials have been resolved into tivo factors. 
 
 These factors may or may not be prime factors. 
 
 Thus, in Ex. 6, d^-h^ = {a + 'b){a^-a'^h-\- (0)^-1)% But the /'actor 
 {a^—a?h-\-ah'^—¥') is a composite quantity = {a—b)(a^ + ¥). 
 
 134. When a binomial is to be resolved into prime fac- 
 tors, it should first be resolved into two factors, on<> of 
 which is prhne ; then the composite factor should be 
 treated in like manner. 
 
 20. Let it be required to find the prime factors of «^ - ¥. 
 
 Solution.— The ^/a^ = a\ and /y/ft^^ft^ (Art. 128.) 
 
 Now a^-¥- = (a2-&2) (^2 + jf.y But a^-h^ = («+&) {a-V). (Art. i^j.) 
 
 Therefore the prime factors of a^—¥ are {a^ + ¥){a + l){a—h). 
 
 Resolve the following quantities into their prime factors ' 
 
 21. «^— I. Arts. {dJ^ ■\-i){a-\- i){a—i). 
 
 22. i—f. Ans. (i+y^){i-{-y){i-yy 
 
 23. ofi — y^. 
 
 Ans. {x^ -xy + y^) {a^ + ccy + y^) {x •\-y){x — y). 
 
 24. x^ — 2x^y^ 4- y^. 
 
 Ans, {x^ - y^y = {x + y) {x -h y) (^ - «/) (^ ~ V)- 
 
 25. a^ — I. 
 
 Ans. (x 4- i) {x- i) {x^ -\-x^-i){xi-x-\- i). 
 
 26. a« + 2a%^ + 2'^. 
 
 ^?^5. {a -\-h){a + I) {a^ - a2> + ^) {a^ - ah -]r ^), 
 
 27. «2 _^ 9a + 18. Ans. {a + 6) (rt + 3}. 
 
 28. 4^2 ~ i2«& + 9&2. ^ws. (2a — 2fi) (20 - 3^) 
 
 (See Appendix, p. 284.) 
 
OHAPTEE YII 
 DIVISORS AND MULTIPLES. 
 
 135. A Common I>ivisor is one that will divide two 
 or more quantities without a remainder. 
 
 136. Commensurable Quantities are those which 
 haye a common divisor. 
 
 Thus, aW and dbc are commensurable by ah. 
 
 137. Incommensurable Quantities are those 
 which have no common divisor. (Art. 122 ) 
 
 Tims, ab and xyz are incommensurabla. 
 
 138. To Find a Common Divisor of two or more Quantities. 
 
 1. Find a common divisor of abx, act/, and adz. 
 Analysis. — Resolving the given quanti- operation. 
 
 ties into factors, we perceive the factor a, is abx = a xb XX 
 
 common to each quantity, and is therefore a act/ = axcXV 
 
 common divisor of tbem. (Art. 119,) Hence, ^^^i ^ axdxz 
 
 *^^ A71S. a. 
 
 Rule. — Resolve each of the given quantities into factors, 
 one of ivhich is common to all. 
 
 Find a common divisor of the following quantities : 
 
 2. -^alcd and gahm. Ans. ^ah. 
 
 3. x^yz and lahx. 6. 2ax, dlx, 14CX. 
 
 4. a% bed, ah^xy. 7. 35^^, "jm"^, 42m^x. 
 
 5. 2adc, acx% a^cy, 8. 24a^, i2al>^, 6aW. 
 
 135. What is a common divisor ? 136. Commensurable quantities ? 137. Incom- 
 mensunible quantities ? 138. How find a common divisor of two or more quantities ? 
 
0x5 DIYIS0R8. 
 
 139. The Greatest Common Divisor of two or 
 
 more quantities is the greatest quantity that will divide 
 each of them without a remainder. 
 
 Notes. — i. A common divisor of two or more quantities is always a 
 common foMor of those quantities, and the g. c. fZ.* is their greatest 
 common factor. 
 
 2. A common divisor is often called a common measure, and the 
 greatest common divisor, the greatest comm/m measure. 
 
 PRINCIPLES. 
 
 140. 1°. TJie greatest common divisor of two or more 
 quantities is the product of all their common prime factors. 
 
 2°. A commoji divisor of tioo quantities is 7iot altered ty 
 multiplying or dividing either of them hy any factor not 
 found in the other. 
 
 Thus, 3 is a common divisor of i8 and 6 ; it is also a common divisor 
 of 1 8, and of (6 x 5) or 30. 
 
 3°. The signs of a 2^olynomial may he changed hy divid- 
 ing it hy — I. 
 Thus, (— 3a + 4&— 5c)h 1 = 3a— 46 + 5c. (Art. 112.) Hence, 
 
 4°. Tlie signs of the divisor, or of the dividend, or of hoth, 
 may he changed without changing the common divisor. 
 
 141. To Find the Greatest Common Divisor of Monomials by 
 
 JPrune Factors, 
 
 I. What is the g, c, d, of 35aca;, 2Sahc, and 2iay? 
 
 Analysis. — Resolving the operation. 
 
 given quantities into their prime 3SCICX := <^XT XaXCXX 
 
 factors, 7 and a only, are com- ^SaJc = 2X2X7X«X*XC 
 mon to each ; therefore their 
 
 product 7 X a, IS the g,c,a,re- ^ -» / ./ 
 
 quired. (Prin. i.) •'• 7Xfl5 = 7«- ^ns. 
 
 139* What is the greatest common divipor of two or more quantities ? Note i. 
 What is true of a common divisor of two or more quantities? Of the g. c. d.? 
 140. Name Principle i. Principle 2. Principle 3. 
 
 * The initials g, c, d, are usod for the greatest common divisor. 
 
jivisoRS. 63 
 
 2. Find the g, c, d. of 4a^b% loaW, and i^abdx. 
 
 Analysis. — Resolving these quan- operation. 
 
 titles into their prime factors, the ^aWc = 2 x 2 X oaahhc 
 
 factor 2 is common to the coefficients; locfil^ = 2 X 5 X CLobbb 
 
 also, a and 6 are common to the lit- ^o^aMx = 2 X 7 X obdx 
 
 eral parts Now multiplying these ^^^^ 2^a^h = 2ab 
 common factors together, we have 
 ly.ay.'b — 2ab, which is the g. c, d. required. (Prin. i.) Hence, the 
 
 Rule. — Resolve the given quantities into their prime fac- 
 tors ; and the product of the factors common to all, will he 
 the greatest common divisor. (Prin. i.) 
 
 Note. — In finding the common prime factors of the literal part, 
 give each letter the least exponent it has in either of the quantities 
 
 3. Find the g, c. d, of 6aV and ^dbc, 
 
 4. Of i6a^xy and i^acx^y. 
 
 5. Of i2aWa^^ and i6a^x^zK 
 
 6. Of daV^x^^, i2a^xh^, ana i2>a^xh\ 
 
 142. To Find the Greatest Common Divisor of Quantities by 
 Continued Division, 
 
 I. Required the greatest common divisor of 30a; and 422;. 
 
 Analysis. — If we divide the greater operation. 
 
 quantity by the less, the quotient is i, 30a; ) 42:?; ( i 
 
 and 12a; remainder. Next, dividing the 302; 
 
 first divis'or 30a;, by the first remainder \ / 
 
 ^1, J.- \ ■ J XI -J I2iC)302;(2 
 
 12a;, the quotient is 2 and the remainder ^ ^ ^ 
 
 6x. Again, dividing the second divisor ^4^ 
 
 by the second remainder, the quotient 6x ) 1 2:?; ( 2 
 
 is 2 and no remainder. The last divisor, 
 
 tx, is the ff, c, d. 
 
 12a; 
 
 Demonstration. — Two points are required to be proved : 
 ist. That 6x is a common divisor of the given quantities. 
 2d. That 6a! is their greatest common divisor. 
 
 First. We are to prove that 6x is a common divisor of 30a; and 42a;. 
 By the last division, 6a; is contained in 123*, 2 times. Now as 6a; k a 
 
 141. How find the g.c. d. of monomials by prime factors? Note. In finding the 
 prime factors of the literal part, what exponents are given ? 
 
64 DIVISORS, 
 
 divisor of i2aJ, it is also a divisor of the product of i2aj into 2, or 24a;. 
 (Art. 124, Prin. i.) Next, since 6x is a divisor of itself and 24a*, it 
 must be a divisor of the sum of tx-\-2^, or 30a!, which is the smaller 
 quantity. For the same reason, since ()X is a divisor of 12a; and 30a;, it 
 must also be a divisor of the sum of 12a; + 302;, or 42a;, which is the 
 larger quantity. Hence, tx is a common divisor of 30a; and 42a;. 
 
 Second. We are to prove that 6x is the greatest common divisor of 
 30a; and 42a?. 
 
 If the greatest common divisor is not bx, it must be either greater 
 or less than bx. But we have shown that tx is a common divisor of 
 the given quantities ; therefore, no quantity less than tx can be the 
 greatest common divisor of them. The assumed quantity must there- 
 fore be greater than bx. By supposition, this assumed quantity is a 
 divisor of 30a? and 42a! ; hence, it must be a divisor of their difference, 
 42aj— 30a;, or 12a;. And as it is a divisor of 12a;, it must also divide the 
 product of 12a; into 2, or 24a;. 
 
 Again, since the assumed quantity is a divisor of 3oaJ and 24a;, it 
 must also be a divisor of their difference, which is bx ; that is, a greater 
 quantity will divide a less without a remainder, which is impossible. 
 Therefore, 6a! must be the greatest common divisor of 3oaj and 42aJ, 
 the second point to be proved. Hence, the 
 
 B,XJL^.— Divide the greater quantity by the less, then divide 
 the first divisor dy the first remainder, the second divisor by 
 the second remainder, and so on, till there is no remainder. 
 The last divisor will be the greatest common divisor. 
 
 Note. — If there are more than two quantities, fifid the g. c» d, 
 of the smaller two, then of this common divisor and a third quantity, 
 and so on with all the quantities. 
 
 2. What is the g, c, d. of 48^5, 72 a, and io8a? 
 
 Suggestion. — The </, c, d, of 48a and 72a is 24a ; and that of 24a 
 and loSa is 12a. Therefore, 12a is the (/. c, d, required. 
 
 142, «. The Greatest Common Divisor of Poly- 
 nomials is found by the above rule, as illustrated in the 
 following examples: 
 
 142. Show upon the blackboard the truth of this rule? r.-p a How find the 
 g. ^. d. of polynomials ? 
 
DIVISORS 
 
 65 
 
 3. What is the g. c. d. of /^a^ — 2ia2 4. 15a 4. 20 and 
 
 <j3 __ 6a + 8 ? 
 
 OPKBATION. 
 
 ^a^ — 2ia^ + 150^ +20 a^ — 6« + 8 istdiviBor. 
 
 \a^ — 24^2 4. 32« 
 
 4^ + 3 
 
 zst quotient. 
 
 -f 3a^ — 1 7a + 20 
 f 3^2 _ i8a -f 24 
 
 a« — 6« + 8 
 
 a^ — /^a 
 
 — 2a + 8 
 
 — 2a + 8 
 
 a — 4 'Pt remainder and 2d divisor. 
 
 a — 2 ad quotient. 
 
 Ans. a — 4. 
 
 Analysis. — Dividing the greater quantity by the less, the remain- 
 der is a— 4. Again, dividing the first divisor by the first remainder, 
 the quotient is a— 2, and no remainder. The last divisor, a— 4, is the 
 greatest common divisor. 
 
 143. It is sometimes necessary, in order to avoid frac- 
 tions, to introduce a factor into one or both the given 
 quantities, or to cancel one before finding the greatest com- 
 mon divisor. 
 
 It is also sometimes necessary to change the signs of 
 the divisor or dividend, or of both. (Art. 140, Prin. 4.) 
 
 4. What is the g, c. d,ofa^— 2xy + y^ and 01? — y^"^ 
 
 Analysis.— Dividing the 
 greater by the less, the first 
 remainder is — 2xy + 2y^. 
 Cancelling from it the com- 
 mon factor 2y, we have for 
 the second divisor —x + y. 
 Changing the signs, it be- 
 comes a;— y. (Art. 140, Prin. 4.) 
 Dividing as before, the quo- 
 tient \^ x + y, and no remainder, 
 common divisor. 
 
 OPKBATION 
 
 0^ — 2Xy-\- y^ 
 
 7? — ^2 
 
 2y) —2xy+2f' 
 
 Divisor, — ^ + y 
 
 or, ^—y 
 
 Quotient, ^-\-y 
 
 X^ — y^ Divisor, 
 I Quotient. 
 
 a? — y"^ 
 
 ■t 
 
 X?- 
 
 The last divisor, x—y, is the greatest 
 
 143. How does it affect the g. c. d. if a factor is introduced into eitlier orbotli the 
 given quantities ? How if one is cancelled ? What is true of the si^jne ? 
 
66 
 
 DIVISORS. 
 
 5. What is the g. c, d. of 4^3 — 6a;2 — 4a; + 3 and 
 
 2X^ + X^ + X—1? 
 
 dividend, 4^ — 6x^ — 4X + S 
 
 43:^ -\- 2X^ -{- 2X — 2 
 
 2d divisor, — 8,7^ — 6a; -f 5 
 2d quotient, 
 
 OPERATION. 
 
 2C(^ -^ X^ -^ X — I ist divisor. 
 2 igt quotient. 
 
 — X 
 
 3d dividend, — Sx^ — 6x -{- 5 
 — Sx^ + S ^X — 16 
 
 4th divisor, 
 4tli quotient. 
 
 — 21 ) — 42a; +21 
 , 2ir — I 
 
 — a; + 4 
 
 8a.'^ + 4X^ -{- 4X — 4 2d dividend 
 
 8'Ji^ + 6a;2 — 5ic 
 
 — 22:^ -i- 9a; — 4 3d divisor. 
 4 3d quotient. 
 
 — 23^^+93; — 4 4th dividend. 
 ~ 2X^ -{- X 
 
 2X — I is the g, c, d. 
 
 8a; — 4 
 8a; — 4 
 
 Analysis. — Dividing the greater by the less, the first term of the 
 first remainder, — Sa*-, is not contained in 2a:^> the first term of the 
 second dividend. We therefore multiply this dividend by 4, and it 
 becomes Sx^ + 4X^ + 4X— 4, and dividing this by the second divisor, the 
 second remainder is —22;'^ + 90?— 4. Dividing the preceding divisor by 
 this remainder, we see that the third remainder, —422; + 21, is not 
 contained in the next dividend. Cancelling the factor —21, the fourth 
 divisor becomes 2X — i, the greatest common divisor required. 
 
 Find the g, c. d. of the following quantities: 
 
 6. a^ — y^ and a^ — 2x2/ + ^. 
 
 7. «3 _f_ ^z an^ ^2 _j_ 2ab + ^. 
 
 8. h^ — 4 and J2 ^ 45 + 4. 
 
 9. a:^ — 9 and x^ -{- 6x -\- 9. 
 
 10. a^ — 3(1 + 2 and a^ — a — 2. 
 
 11. «^ + 3^2 -f 4flj -|- 12 and a^ -f 4^2 _{- 4^5 _{-. 3, 
 
 12. a^ -{- I and x^ + mx^ -f 772a; + 1. 
 
 13. d^ — Ifi and a^ — W, 
 
 14. fl^2 _ 2^j _j_ 4^2 and c^ — a^h + 3fl!j2 _ ^js 
 
 15. 3^ — 102;^ + 15^ + 8 and x^ — 2a;^ — 6ar^ + 4a:2 
 4- 13a; -f 6. 
 
MULTIPLES. ;37 
 
 MULTIPLES. 
 
 144. A Multiple is a quantity which can be divided 
 by another quantity without a remainder. (Art. 123.) 
 
 145. A Coimnon Multijde is a quantity which can 
 be divided by two or more quantities without a remainder. 
 
 Thus, 1 8a is a common multiple of 2, 3, 6, and 9. 
 
 146. The Least Common Multiple of two or 
 
 more quantities is the least quantity that can be divided by 
 each of them without a remainder. 
 
 Thus, 21 is the least common multiple of 3 and 7 ; 30 is the least 
 common multiple of 2, 3, and 5. 
 
 PRINCIPLES. 
 
 147. 1°. ^ multiple of a quantity must contain all the 
 prime factors of that quantity. 
 
 Thus, 18 is a multiple of 6, and contains the prime factors of 6, 
 which are 2 and 3. 
 
 2°. A common multiple of two or more quantities must 
 contain all the prime factors of each of the given quantities. 
 
 Thus, 42, a common multiple of 14 and 21, contains all the prime 
 factors of those quantities ; viz., 2, 3, and 7. 
 
 3°. The least com7non multiple of tiuo or more quantities 
 is the least quantity which contains all their prime factors^ 
 each factor heing tahen the greatest numderof times it occurs 
 in either of the given quantities. 
 
 Thus, 30 is the least common multiple of 6 and to, and contains all 
 the prime factors of these quantities ; viz., 2, 3, and 5. 
 
 144. What is a multiple ? 145, A common multiple ? 146. The least common 
 multiple ? 147. Name Principle i. Principle 2. Principle 3. 
 
68 MULTIPLES. 
 
 148. To Find the Least Common Multiple of Monomials by 
 Prime Factors, 
 
 1. Find the I, cm,* of i5«V, ^l^cx, and 9&A 
 
 Analysis. — The prime factors operation. 
 
 of the coefficients are 5, 3, and 3. l^^a'^OC^ = 3X5X«*Xa? 
 
 The prime factors of the letter a t^W-cx =1 ^xl^ XCXX 
 
 are a, a, a, a, which are denoted by qbc^z = ^X^X^X<^X^ 
 
 a*. In like manner, the prime fac- j^^^^ A^a^l^C^Z. 
 
 tors of X are denoted by a;', those of 
 
 b by b'^, and those of c by c^ ; z is prime. Taking each of these factors 
 the greatest number of times it occurs in either of the given quanti- 
 ties, the product, 45a^&-c%-2, is the I, c, in, required. (Art. 147, Prin. 2.) 
 Hence, the 
 
 Rule. — Resolve the quantities into their prime factors ; 
 multiply these factors together, tahing each the greatest num- 
 her of times it occurs in either of the given quantities. Tlie 
 product is the I, c, m, required. 
 
 Or, Find the least common multiple of the coefficieyits, and 
 annex to it all the letters, giving ecLch letter the exponent of 
 its highest poiver in either of the quantities. 
 
 Note. — Tn finding the I. c, in, of algebraic quantities, it is often 
 more expeditious to arrange them in a horizontal line, then divide, 
 «tc., as in arithmetic. 
 
 Eequired the I, c, Vfl, of the following quantities : 
 
 2. ^a\ i2a^A and 24^2:2^. Ans. yza^x^g, 
 
 3. ^ab^ 2Sbc^, and ^6a^M, 
 
 4. iSoc^'ifz, 2oy% and Sxyz^. 
 
 5. isa^l^c, ga¥c% and iSa^c^. 
 
 6. 2Sab\ i4«2J*, 35«^^, and 42a*h. 
 
 7. 2ix^yh^, 35^y^z^, and d^xyH. 
 
 8. ^mVy, i2mhiy'^, 2i>Yi^ T^mn^y^. 
 
 148. How find the I. c. tn. of monomials by prime factors ? What other method 
 * The initials I, c, tn, are used for the least common multiple. 
 
MULTIPLES. 69 
 
 149. To Find the Least Common Multiple of Polynomials. 
 
 9. Kequired the l. c. ni» of ^^ + ^ and a^ — 1^. 
 
 Analysis. — Resolving operation. 
 
 tlie quantities into their tt^— Z*^ = (« + d) X («— J) 
 
 prime factors, as in the «3+^ = (« + J) X (tt^— ffJ + ^) 
 
 margin, (« + &) is common (^^ j) >< (^_y^ ^ (cii_ah + m = 
 to both, and is their </. c. d. a dz , xq xi ^ ' 
 
 (A.rt. 139.) Now multiply- 
 ing these factors together, taking each the greatest number of times 
 it occurs in either of the given quantities, the product a*—a^b + ab^—b* 
 is the Lcm, required. (Art. 148.) 
 
 Second Method. 
 
 ^nce the g. c, d, contains all the factors common to both quan- 
 tities (Art. 147, Prin. 2), it follows if one of them is divided by the 
 g, c. d. and the quotient multiplied by the other, the product will 
 be the I, c, tn. Hence, the 
 
 EuLE. — Resolve the quantities into their prime factors 
 and multiply these factors together, taking each the greatest 
 number of times it occurs in either of the given quantities. 
 Their product is the I. e, m. reqtiired. 
 
 Or, Find the greatest common divisor of the given quanti' 
 ties^ and divide one of them hy it. The quotient, multiplied 
 ly the other, will he their I, c, m* 
 
 10. Find the I, c. m. of 2a — i and 4^2 _ i. 
 
 Solution.— The g, c. d, is 2a— 1. Now(4a2— i)-^(2a— i)=(2ffi+ 1) ; 
 and (2a + 1) X (2a— i) = 4a^— I, Ans. 
 
 Find the I, c, m, of the following quantities : 
 
 11. x^ — y"^ and x^ — 2xy + i/^. 
 
 Ans. u? — x^y — xy"^ + y\ 
 
 12. «2_j2 and a^ — tfi. 
 
 13. a? — I and x^ ■}- 2X -\- i. 
 
 14. 2^2 ^ ^a — 2 and 6a^ — a — i. 
 
 15. m^ -j- m — 2 and m^ — i. 
 
 (See AppS'Uclix, p. 284.) 
 
CHAPTER VIII. 
 FRACTIONS. 
 
 150. A Fraction is one or more of the equal parts into 
 which a unit is divided. 
 
 151. Fractions are expressed by two quantities called the 
 numerator and denominator, one of which is written below 
 the other, with a short line between them. 
 
 152; The Denominator is the quantity helow the 
 line, and shows into Jiow many equal parts the unit is 
 divided. 
 
 153. The Numerator is the quantity above the line, 
 and shows how many parts are taken. 
 
 Tlius, the expression - shows that the quantity is divided into h 
 equal parts, and that a of those parts are taken. 
 
 154. The Unit or Base of a fi-action is the quantity 
 divided into equal parts. 
 
 155. The Ter^ns of a fraction are the numerator and 
 denominator, 
 
 156. An Integer is a quantity which consists of one 
 or more entire units only ; as a, 305, 5, 7. 
 
 157. A Mixed Quantity is one which contains an 
 integer and a fraction. 
 
 Thus, a + - is a mixed quantity. 
 
 150. What is a fraction ? 151. How exprensed? 152. What does the denomina- 
 tor show? 153. The numerator? 154. What is the base of a fraction? 155. Tlia 
 terms of a fraction ? 156. An integer ? isz* ^ mixed quantity ? 
 
FKACTIOKS. 71 
 
 158. Fractions arise from division, the numerator being 
 the dividend and the denominator the divisor. Hence, 
 
 159. The Value of a fraction is the quotient of the 
 numerator divided by the denominator. 
 
 Thus, the value of 6 thirds is 6-1-3, which is 2 ; of — - is 3m. 
 
 SIGNS OF FRACTIONS. 
 
 X60. JEvery Fraction has the sign + or — , expresse(J 
 or understood, before the dividing line. 
 
 161. The Dividing Line has the force of a viiicu^ 
 luM or parenthesis, and the sign before it shows that the 
 value of the ivhole fraction is to be added or subtracted. 
 
 162. Every Numerator and Denominator is 
 
 pi-^ceded by the sign + or — , expressed or understood. 
 In this case, the sign affects only the single term to which 
 it is prefixed. 
 
 163. If the Sign before the Dividing Line is 
 
 changed from -f to — , or from — to +, the value of the 
 fraction is changed from + to — , or from — to -f . 
 
 ^, hx — ex — dx , -, . X hx — cx — dx 
 
 Thus, a -\ = a + o — c — d; but a = 
 
 x X 
 
 a — b + e + d. • 
 
 164. If all the Signs of the Numerator are 
 
 changed, the value of the fraction is changed in a corre- 
 sponding manner. 
 
 Thus, = + a + 6 ; but 
 
 158. From what do fractions arise ? 159. What is the value of a fraction • 
 160. What is prefixed to the dividing line of a fraction ? i6r. What is the force of 
 the dividing line ? 162. By what is the numerator and denominator pi'eceded ? 
 How far does the force of this sign extend? 163. If the sign hefore the dividing 
 line is changed, what is the efi"ect ? 164. If ,all the signs of the numerator are 
 changed ? 
 
72 FRACTIONS. 
 
 165. If all the Signs of the Denominator are 
 
 changed, the value is also changed in a corresponding 
 manner. 
 
 Thus, — =+a; but — = -a. Hence, 
 
 +x —X 
 
 166. If any two of these changes are made at the same 
 time, thej will balance each other, and the value of the 
 fraction wiU not be altered, 
 
 _- ab — ab —ah ab 
 
 ^-^^ -j = -ITi = - -J- = - — 6 = + «• 
 .. ab — ah ab — ab 
 
 PRINCIPLES. 
 
 167. The principles for the treatment of fractions in 
 A-lgebra are the same as those in Arithmetic. 
 
 1°. Multiplying the numerator, or \ Multiplies the 
 Dividing the denomi^iator, j fraction. 
 
 ,p, 2X2 4 2 . , 2 2 
 
 Thus, - = ^ = -. And - = - • 
 6 63 t-i-2 3 
 
 2°. Dividing the numerator, or ) r^ • • 7 .-, ^ 
 
 nr 1^' 1 • A-L 7 ' ± \ Divides the fraction. 
 Multiplying the denominator, ) -^ 
 
 »^ 2^2 I . , 2 21 
 
 6 6 6x2 12 6 • 
 
 3°. Multiplying, or dividing loth \ Does not change its 
 
 terms by the same quantity ) value. 
 
 -«, 2x2 4 2 I . , 2-T-2 I 
 
 Thus, = -t = = _. And = - 
 
 6x2 12 6 3 6-5-2 3 
 
 4°. Multiplying and dividing a \ Does not change its 
 
 fraction by the same quantity j value. • 
 
 -^ 2x2-5-2 2 
 
 Thus, = - . 
 
 6x2-7-2 o 
 
 165. If all the signs of the denominator arc changed ? i66. If both are changed? 
 167. Name Principle i. Principle 2. Principle 3. Principle 4. 
 
BEDUCTION OF FRACTIONS. 73 
 
 REDUCTION OF FRACTIONS. 
 
 168. Heduction of Fractions is changing their 
 terms without altering the value of the fractions. 
 
 CASE I. 
 
 169. To Reduce a Fraction to its Lowest Terms. 
 
 Def. — The JLoii^est Terms of a fraction are the 
 smallest terms in which its numerator and denominator 
 can be expressed. (Art. 122.) 
 
 1. Reduce - — =-^ to its lowest terms. 
 
 i$abcx 
 
 Analysis.— By inspection, we perceive the operation. 
 
 factors S, a, b, and x are common to both terms. 5^ (rax aoa ^ 
 
 Cancelling these common factors, the fraction l^abcx 3c 
 
 becomes . Now since both terms have been divided by the 
 
 same quantity, the value of the fraction is not changed. (Art. 167, 
 Prin. 3.) And since these terms have no common factor, it follows 
 
 that are the lowest terms required. (Art. 122.) 
 
 Note. — It will be observed that the factors 5, a, &, and x are prime ; 
 therefore, the product e^abx is the g, c. d. of the numerator and 
 denominator. (Art. 121.) Hence, the 
 
 Rule. — Cancel all the factors common to the numerator 
 and denominato'* 
 
 Or, Divide both terms of the fraction by their greatest 
 common divisor. (Art. 167.) 
 
 2. Reduce f- to its lowest terms. Ans. -• 
 
 i2abc 3 
 
 3. Reduce to its lowest terms. Ans. — 
 
 2,ac c • 
 
 168. What is reduction effractions? 169. What are the lowest terms of a frac- 
 tion ? How reduce fractions to the lowest terms ? 
 
14: REBTTCTIOK OP FRACTIOKS. 
 
 Reduce the following fractions to the lowest terms: 
 
 3^y . lo. 3^—3^^ 
 
 ga^y^ ' 2X^y — 2xyz 
 
 i2a^l)<? a -{-he 
 
 ^' 4abcd ' ' (a -{- be) X x 
 
 I'jh^cxy x^ — y^ 
 
 ^it^cxy ' x^ — y/^ 
 
 ^AaWc^ ax — x^ 
 
 7. — • 13. • 
 
 loSabx^y^ cfi — a^ 
 
 c^ — IP' a — I 
 
 c? 4- 2ab + 6^ c^ — 2a -\- \ 
 
 ^—y^ ^ + 1 
 
 ^* x^ — 2xy + ^' ^' a? -\^ 2xy + y^' 
 
 CASE II. 
 170. To Reduce a Fraction to a JVIiole or Mixed Quantity, 
 
 I. Reduce to a whole or mixed quantity. 
 
 Analysis.— Since the value operation. 
 
 of a fraction is the quotient of 2a -\- 4b ■{- C , c 
 
 the numerator divided by the 2 2' 
 
 denominator, it follows that per- 
 forming the division indicated will give the answer required. Now 
 2 is contained in 2a, a times ; in 46, 2& times. Placing the remainder 
 
 e over the denominator, we have a + 26 + -, the mixed quantity 
 required. Hence, the 
 
 Rule. — Divide the nunurator hy the denominator, and 
 placing the remainder over the divisor, annex it to the 
 quotient. 
 
 Note. — This rule is based upon the principle that both terms are 
 divided by the same quantity. (Art. 167, Prin. 3.) 
 
 170. How reduce a fraction to a whole or mixed quantity? Note. Upon what 
 principle is this rule baeed? 
 
KEDTJCTIOK OF FRACTIO^-S. 75 
 
 Reduce the following to whole or mixed quantities : 
 
 
 ax — x^ 
 
 
 X 
 
 3. 
 
 al--^ 
 
 a 
 
 4- 
 
 V^-^ 
 
 1) ■\-c 
 
 f 
 
 ^ + C2 
 
 a — h 
 
 a^ -{- a^ — aa^ 
 
 c? — ax 
 \2^ ■\- /i^x — zy 
 ^' h — c ^' 4X 
 
 CASE III. 
 171. To Reduce a Mixed Quantity to an Improper Fraction. 
 
 1. Reduce a -{- - to the form of a fraction. 
 
 3 
 
 Analysis.— Since in i unit there are operation. 
 three thirds, in a units there must be« b 3^^ b 
 
 times 3 thirds, or 3^; and 3? + -^=^^, "" 3~7 3 
 
 3 3 3 3 3a h 3a-{-b 
 
 the fraction required. Hence, the 1 — = — 
 
 3 3 3 
 
 Rule. — Midtiply the integer hy the denominator ; to the 
 product add the numerator, and place the sum over the 
 denominator. (Art. 65.) 
 
 Notes. — i. An integer may be reduced to the form of a fraction by- 
 making I its denominator. Thus, a = - . 
 
 2. If the sign before the dividing line is — and the denominator is 
 removed, all the signs of the numerator must be changed. (Arts. 163, 82.) 
 
 Reduce the following to improper fractions : 
 
 , cd . ahd — cd , 
 
 2. at) i- Ans, 3 = ao ^ c, 
 
 d d 
 
 xy —h , . I — ^ 
 
 3. 3X + -^— 6. a; — I + 
 
 \ •\- X 
 T ^ a — c a — t 
 
 S. a + l + ^y S. 8. + 3^. 
 

 OPEBATIOH. 
 
 
 3« = 
 
 3« 
 I 
 
 I 
 
 X 5 _ 
 
 X5 
 
 isa 
 5 
 
 Hence, the 
 
 
 76 EEDUCTION OF FKACTI0N8. 
 
 CASE IV. 
 
 172. To Reduce an Integer to a Fraction having any required 
 
 Denominator. 
 
 1. Reduce 3« to fifths. 
 
 Analysis. — Since in i« there are 5 fifths, in 
 3a there must be 3 times 5 fifths, or -^. 
 
 Or, reducing the integer 3a to the form of a 
 fraction, it becomes — ; multiplying both 
 
 terms by the required denominator, we have -^, 
 
 Rule. — Multiply the integer hy the required denominator, 
 and place the product over it. 
 
 2. Reduce 2x to a fraction having 6m for its denominator. 
 
 3. Reduce 6ax to a fraction having 4ab for its denominator. 
 
 4. Reduce ^a + 4b to a fraction having 6c^ for its 
 denominator. 
 
 5. Reduce x — y to a fraction having x -i- y for its 
 denominator. 
 
 6. Reduce 2x^y to a fraction having ^a^ — 2b for its 
 denominator. 
 
 173. To Reduce a Fraction to any Required Denominator. 
 
 I. Change - to a fraction whose denominator is 12. 
 
 Analysis. — Dividing 12, the required de- ofebatiok. 
 
 nominator, by the given denominator 3, the ^ 2 -^ 3 = 4 
 
 quotient is 4. Multiplying both terms of a X 4 4« 
 
 the given fraction by the quotient 4, the * 2X4 1 2 ' 
 
 result, — , is the fraction required. Hence, the 
 
 Rule. — Divide the required denominator hy the denomi- 
 nator of the given fraction, and multiply loth terms hy the 
 quotient. 
 
 172. How reduce an integer to a fraction having any required denominator? 
 
REDUCTION OF FRACTION'S. 77 
 
 Reduce the following to the required denominators : 
 
 2. Reduce — to thirty-fifths. 
 
 Solution. 35 -5- 7 = 5. ^o\^ = — . Ans. 
 
 7-^5 35 
 
 3. Reduce - to the denominator ac, 
 
 c 
 
 4. Reduce — to the denominator 49a. 
 
 7 
 
 5. Reduce ^ to the denominator a? — 2xy -f ^. 
 
 X — y 
 
 6. Reduce -— — to the denominator M^(x + ti)\ 
 
 x + y \ JJ 
 
 COMMON DENOMINATORS. 
 
 174. A Common Denominator is one that belongs 
 equally to two or more fractions. 
 
 PRINCIPLES. 
 
 1°. A common denominator is a multiple of each of the 
 denominators ; for every quantity is a divisor of itself and 
 of every multiple of itself (Art. 124, Prin. i.) Hence, 
 
 2°. The least common denominator is the least common 
 multiple of all the denominators, 
 
 CASE V. 
 
 175. To Reduce Fractions to Equivalent Fractions having a 
 
 Coinmon IJenoitiinaior. 
 
 fh C X 
 
 I. Reduce t> ;t> and -, to equivalent fractions having 
 a common denominator. 
 
 173. How reduce a fraction to any required denominator ? 174. What is a com- 
 jnon denominator ? Principle i ? Principle a ? 
 
78 REDUCTION OF FRACTIONS. 
 
 Solution.— Multiplying the denominators h, d, and y together, we 
 have bdy, which is a common denominator. 
 
 b X d X i/ = hdy The common denominator. 
 
 a X d X ^ = ady ) 
 
 ex i X y = hey V The new numerators. 
 
 X X l X d =z hdx ) 
 
 J a ady c hey x hdx 
 
 h~ hdy' d~~hdy' y^ hdy' 
 
 To reduce the given fractions to this denominator, we multiply- 
 each numerator by all the denominators except its own, and place the 
 results over the common denominator. Hence, the 
 
 Rule, — Multiply all the denominators together for a com- 
 mon denominator, and each numerator into all the denom- 
 inators, except its oion.for the neio numerators. 
 
 Notes. — i. It is advisable to reduce the fractions to their loicest 
 terms, before the rule is applied. (Art. 169.) 
 
 2. This rule is based on the principle, that the Dolue of a fraction is 
 not changed by multiplying both its terms by the same quantity. 
 (Art. 167, Prin. 3.) 
 
 Reduce the following to equivalent fractions having a 
 common denominator: 
 
 2. 
 
 a X ^ 
 ~c' y' 4* 
 
 
 Ans. 4^^ 4^^ ''^-. 
 4cy 4cy 4cy 
 
 3. 
 
 c h 2d 
 d' x' T 
 
 8. 
 
 2a c -\- I 
 
 4. 
 
 a h X 
 2x' ~?' ~y 
 
 9. 
 
 2 a c -\- d 
 3' ^' c-d 
 
 5. 
 
 2a X 
 
 10. 
 
 xy I 2« 
 
 Sh' a + b 
 
 z' 2' h' 
 
 6. 
 
 x — y x^y^ 
 x-\-y' x — y' 
 
 II. 
 
 2a x^ + ^^ 
 
 7. 
 
 a + h 5a— 1 
 3 ' a ' 
 
 12. 
 
 a — x a + X 
 
 a -{- x' a — x 
 
 175. How reduce fractions to equivalent fractions having a common denomina- 
 tor r JV^oie. Upon what principle is this rule based ? 
 
EEDUCTIOK OF FRACTIONS. 
 
 79 
 
 CASE VI. 
 176. To Reduce Fractions to the Least Common Denominator. 
 
 I. Eeduce - 
 
 X' xy yz 
 
 Solution. — Tlie I, c, m, of the denominators U 
 
 , and — to the I, c,d. 
 
 xyz = L c, d. 
 
 xyz -r-x = yz 
 xyz -T-xy =i z 
 
 xyz -i- yz ^x 
 
 The 
 multipliers. 
 
 a X yz 
 X X yz 
 b X z 
 xy X z 
 d X X 
 
 xyz. (Art. >j8. 
 
 _ ^?/^ 
 ~xyz 
 _bz^ 
 ~ xyz 
 dx 
 
 The 
 fractions 
 required. 
 
 yz X X xyz 
 
 To change the given fractions to others whose denominator is xyz, we 
 multiply each numerator by the quotient arising from dividing this 
 multiple by its corresponding denominator. Hence, the 
 
 Rule. — I. Find the least common multiple of all the 
 denominators for the least common denominator. 
 
 II. Divide this multiple by the denominator of each frac- 
 tion, and multiply its numerator iy the quotient. 
 
 Note. — AH the fractions must be reduced to their lowest terms 
 before the rule is applied. 
 
 Reduce the folloAving fractions to the I, c. d, : 
 
 
 a Ic y 
 
 2. 
 
 2^' a: ' 4C 
 
 
 cd 2X xy 
 
 3- 
 
 ab^ 2,a^ ac 
 
 
 a i c X 
 
 4- 
 
 ~ ' 'Z' T ' T.' 
 
 
 2 3 4 y 
 
 
 a^c 2cd x^y 
 
 S- 
 
 ah' Wc' ^U 
 
 6. 
 
 2db 3 a; I 
 
 Zac' 4' OL^c' 8 
 
 
 2« cd x^y 
 
 h 
 
 45 ' Ic' lex 
 
 8. 
 
 10. 
 
 II. — 
 
 12. -^ 
 
 a^l a — I a^+ly^ 
 a — y a + V a^-lj'' 
 2(x-\-y) a ah 
 
 zix-^tjY xy' 'ejx'-^y)' 
 d X 
 a^' Wb' 
 X m y 
 ac Wc' &d 
 X a ■\-h d 
 ' xz 
 
 y 
 
 2' 
 
 xy 
 
 13- 
 
 m-\-n m—n 
 
 m* 
 
 W 
 
 2ax^ ' ^cx 
 
 176. How reduce fractions to the least common denominator! 
 
so ADDITION OF FRACTIOKS 
 
 ADDITION OF FRACTIONS. 
 
 177. When fractions have a common denominator, their 
 numerators express like parts of the same unit or base, and 
 are like quantities, (Art. 43.) 
 
 178. To Add Fractions which have a Common DenominatoPc 
 
 1. What is the sum of f , |, and | ? 
 
 Solution, f and | are V-, and f are ^i, Ans. Hence, the 
 
 EuLE.— ^flf^ the numerators, and place the sum over the 
 common denominator, 
 
 . ,, 75 45,85 . loj 
 
 2. Add — , — , and — . Ans, -^— 
 
 m m m m 
 
 3. Add ^ — , , , and ^ 
 
 2xy 2xy 2xy 2xy 
 
 . ,, 7c?a:2! i7(?a;2! iidxz , 4<7a;;? 
 
 4. Add '—r , , , — 7- , and ^^ • 
 
 $aoc ^aoc $aoc ^aoc 
 
 5. Add ?^+f to 3i-rf . 6. Add ^^±^ to 4^=^. 
 
 179. To Add Fractions which have Different Denominators. 
 
 7. What is the sum of t^ -7> and — ? 
 h d' X 
 
 bxdxx = hdx, c. d, 
 a adx c bcx 
 
 Analysis, — Since these fractions 
 
 have different denominators, their 
 
 numerators cannot be added in their 5 odx d udx 
 
 present form. (Art, 66.) We there- m hdm 
 
 fore reduce them to a common denomr ^ ~^x 
 
 inator, then add the numerators. ^^^ _^ j^^. ^ j^^ 
 
 (Art. 175.) Hence, the j^ -, Ans^ 
 
 Rule. — Reduce the fractions to a common denominator, 
 and place the sum of the nuriierators over it. 
 
 Note.— All answers should be reduced to the lowest terms. 
 
 177. When fractions have a common denominator, what is true of the numerators! 
 178. How add such fi:3,ction8 ? 179. How, whea they have different denominators ? 
 
ADDITIOI^ OF FRACTIONS. 81 
 
 Find the sum of the following fractions : 
 
 8. --f--^ + 
 V X m 
 
 2X 2y 2xy . 2X^m -\- lyhn -f 2X^y^ 
 
 mxy 
 cd Ay Ix 
 
 ^ ZX^ 2d^ S 
 
 9- 
 
 3^ + ^ + 1 
 
 
 4 5 3 
 
 lO. 
 
 ^ + J- + M. 
 
 
 3 26? 4 
 
 
 a . X 
 
 II. 
 
 
 I ■\-c ' l — c 
 
 12. 
 
 x+y x—y 
 
 
 2xy xy 
 
 I-?- 
 
 2 +x 3 -{-ax 
 
 
 y ay 
 
 ^ A. 
 
 a ah 
 
 i6. 
 
 a 271 -\- d 
 
 d sh 
 
 a d 
 17. - H 
 
 y -m 
 
 ,8. :^+ -^ 
 
 y m — n 
 
 — 4 — 16 
 
 19. — ^ + 
 
 2 7 — 3 
 
 4« 6c xm 
 
 20. -^^ — — • 
 
 X -^ y x — y b d 2>x 
 
 180. To Add Mixed Quantities. 
 
 Ji vn 
 
 1. What is the sum of « + - and d ? 
 
 c n 
 
 BOLunoiS'. — Adding the integral and fractional parts separately^ 
 
 \06 result S&a^r d ^ , tlie sum required. Hence, the 
 
 en 
 
 Efle. — Add the integral and fractional parts separately, 
 and unite the results. (Art. 179.) 
 
 Note. — Mixed quantities may be reduced to improper fractions, and 
 tlien be added by the rule. (Art. 171.) 
 
 2. What is the sum of « + - and c + - ? 
 
 2 X 
 
 3. What is the sum oi x -\- ^ and ? 
 
 m — y 
 
 4. What is the sum of 3d and a -\ ? 
 
 2 I 
 
 a ~— 1/ 
 
 5. What is the sum of 5a; + ^ and — ^ ? 
 
 2 
 
 180. How add mixed quantities ? iVofe. How else may they be added ? 
 
82 SUBTRACTION OF FRACTIONS. 
 
 181. To Incorporate an Integer with a Fraction. 
 
 c ^ 
 
 6. Incorporate the integer ah with the fraction • 
 
 Solution. — Reducing db to the denominator of tlie fraction, we 
 
 have ah = '^^^^M . and 3_«&^^ + -£z:i = 'i^x^^y^;C-d ^^^ 
 
 2,x+y 2>^+y o^x+y y^^y 
 
 Hence, the 
 
 EuLE. — Reduce the integer to the denominator of the frac- 
 tion, and place the sum of the numerators over the given 
 denominator, (Art. 172.) 
 
 za 
 
 7. Incorporate the integer 3^ with the fraction -r- 
 
 8. Incorporate — 4^ with « 
 
 9. Incorporate — a with ^ « 
 
 a — "b 
 
 10. Incorporate z^ + V with 
 
 x — y 
 
 II. Incorporate — a -\- ^h with ^• 
 
 12. Incorporate 2a; + 2y with 
 
 iC 
 
 SUBTRACTION OF FRACTIONS. 
 
 182. The numerators of fractions which have a common 
 denominator, we have seen, are like quantities, (Art. 177.) 
 Hence, they may be subtracted one from another as integers. 
 
 1. Subtract | from |. 
 
 Solution. l-l^T^^^l or -. Am, - • 
 00004 4 
 
 2. From T subtract y ^ns. t — 7 = — 7 — • 
 
 6 J III 
 
 181. How incorporate an integer with a fraction? 182. What is true of the 
 numerators of fractions having a common denominator ? How suhtract euch frac- 
 tions ? 
 
3 
 
 X 2 = 
 
 6, c. 
 
 .c?. 
 
 
 7«_ 
 3 
 
 14a 
 6 
 
 
 
 2 
 
 ga 
 6 
 
 
 I4« 
 6 
 
 9«_ 
 6 " 
 
 6' 
 
 An 
 
 SUBTRACTION OF FRACTIONS.- 83 
 
 3. From —7— subtract -^-^ 
 
 4. From — -^ subtract - — • 
 ^ a a 
 
 183. To Subtract Fractions which have Different 
 Denominators. 
 
 q. From -- subtract — • 
 ^ 3 2 
 
 Analysis. — Since these fractions 
 have different denominators, they can- 
 not be subtracted one from the other 
 in their present form. We therefore 
 reduce them to a common denominator , 
 which is 6, and place the difference of 
 the numerators over it. Hence, the 
 
 RcTLE. — Reduce the fractions to a common denominator, 
 and subtract the numerator of the subtrahend from that 
 of the minuend, placing the difference over the common 
 denominator. 
 
 Notes. — i. The integral and fractional parts of mixed quantities 
 should be subtracted separately, and the results be united. 
 
 Or, mixed quantities may be reduced to improper fractions, and then 
 be subtracted by the rule. (Art. 171.) 
 
 2. A fraction may be subtracted from an integer, or an integer 
 from a fraction, by reducing the integer to the given denominator, 
 and then applying the rule. 
 
 6. From 5«-^ take^^^. Ans, 5^:iJ£^. 
 
 X ' y xy 
 
 7. From — , take • 
 
 ' m' y 
 
 8. From ~~ , take • 
 
 m y 
 
 ^ a -\- -Kd ^ ^ m — 2d 
 
 9. From — '-^j take ^ 
 
 4 3 
 
 .•83. How when they have different denominators ? Note i. How subtract mixed 
 quantities ? Note 2. How a fraction from an integer, or an integer from a fraction T 
 
84 MULTIPLICATION^ OF FEACTIONS, 
 
 10. From — , take — - » 
 
 m y 
 
 11. From -, take m, 
 
 y 
 
 12. From 4« + -, take 3a — :^' 
 
 c d 
 
 2 3 
 
 
 14. From T-^—, take , ^ ■. 
 
 
 15. From « — -, take ^. 
 f 2 
 
 
 16. From "^ - ^ take "^-^ 
 10 ' i^ + y 
 
 
 n. From a; ^""'', take ^""^ 
 
 -a. 
 
 MULTIPLICATION OF FRACTIONS. 
 CASE I. 
 J84. To Multiply a Fraction by an Integer, 
 
 1. Multiply -7 by m. 
 
 A-{^ALysis.— Multiplying the numerator of the ofkiutioi. 
 
 fhftCtiun by the integer, the product is am. ^ ^ ^ ^^^ 
 
 (Art. i6;, Piin. i.) b ~ b ' 
 
 2. "W^at :s the product of 7- x a;? 
 
 Analysis- 'A fraction is multiplied by ^ y^ x ^ 
 
 dividing its denominator ; therefore, if we bx bx -r- X 
 
 divide hx by gc, the re.sult will be the product a a 
 
 required. (Art. 167, Knn. 1.) Hence, the ^^"ZZTJ ^^ 5* 
 
 EuLE. — Multivly i?a numerator by the integer. 
 Or, Divide the detwminaior by it. 
 
 Notes. — i. A fi'action is multiplied by a quantity equal to its 
 denominator, by cancelling the denominator. (Art. 1 10, Prin. 4.) 
 
 184. How multiply fi frdctioo by on integer? 
 
MULTIPLIOATIOK OF FRACTIONS. 85 
 
 2. A fraction is also multiplied by dJij factor in its denominator, by 
 cancelling that factor. (Art. no, Prin. 4.) 
 
 Find the product of the following quantities: 
 3- 
 
 4 
 
 5 
 6 
 
 7. 
 8. 
 
 9 
 
 10 
 
 II 
 
 ^2/ 
 
 — — T X (a — - J). -4?^5. 3a;, 
 
 —y X d. 
 
 cd 
 
 —7-^ X (3 + m). 
 
 «^ 
 
 — X 6. 
 
 24 
 
 2a; — XII , _ 
 
 v^~i X (3c + 2fZ). 14. 
 
 ax — X 6x. 
 
 SX 
 
 a + h 
 
 ; X ^x. 
 
 20X 4- 2^xy ^ 
 
 a -\- ab 
 
 he -\- c 
 
 2X-\- s 
 
 X 2ac. 
 
 X 2or2;. 
 5 a^ — z^ 
 
 
 4 «^ 
 Ans, — • 
 c 
 
 12. 
 
 a — i , 
 
 ^ X (12a: + 18). 
 
 13- 
 
 -J X (a — x), 
 
 d — x ^ ' 
 
 14. 
 
 a + b 
 
 15. 
 
 40Z — 10 ^ ^ 
 
 16. 
 
 3c — d 
 20 >^'5. 
 
 17. 
 18. 
 
 -;^ X X 12; + «). 
 
 CASE II. 
 185. To Multiply a Fraction by a Fraction^ 
 
 I. What is the product of - by — 
 
 Analysis. — Multiplying the numerator of operation. 
 
 the fraction - by d, the numerator of the axd_ad 
 
 ^ ad C c 
 
 multiplier, we have — , But the multiplier is -j -, 
 
 — ; hence the product — is JW times too lar^e. C X ffl cm 
 
 ^ d a 
 
 To correct this, multiply the denominator — X —-=:-- 
 
 by m. (Art. 167, Prin. 2.) ^ ^- ^^ 
 
 Note I. How is a fraction multiplied by a quantity equal to its denominator J 
 Note 2. How by any factor in its denominator ? 
 
e6 MULTIPLICATION OF FRACTIOITS. 
 
 2. Required the product of — -^ multiplied by — • 
 
 Analysis. — The factors 2, r, and e are opebation. 
 
 common to each term of the given frac- 2al) cm 2abcm 
 
 tions. Cancelling these common factors, 6cd ax 6acdx 
 
 the result is -^ , the product required 2aocm om . 
 
 ddcdj'jc 1(1 cr 
 (Art. 167, Prin. 3.) Hence, the ^ 
 
 Rule. — Cancel tlie common factors; then multiply the 
 numerators together for the new numerator, and the denom- 
 inators for the new denominator. 
 
 Notes. — i. Mixed quantities should be reduced to improper fractions, 
 and then be multiplied as above. 
 
 Or, i^Q fractional and integral parts may be multiplied separately, 
 And the results be united. 
 
 2. Cancelling the common factors shortens the operation, and gives 
 the answer in the lowest terms. 
 
 3. The word of in compound fractions has the force of the sign x . 
 Therefore, reducing compound fractions to simple ones is the same as 
 multiplying the fractions together. Thus, |off = fxf = ^. 
 
 Find the products of the following fractions: 
 
 2X xxii 2dy ^13 
 
 3. — X — I X — ^« 6. — ; X I- 
 
 ^ y 2d X a + 3a: 8 
 
 he X d (a + m)y.h aii 
 
 4. — X T- X — 7. ^^ — X -r-~--{ — 
 
 a by c IX {a-{-m)xc 
 
 x-^y ^x + y a±h cd 
 
 5» X ; — • o. 5 — X — • 
 
 yz y + z (? X 
 
 9. What is the product of -^— by — ^^^ ? 
 
 Solution. — Factor and cancel. Ans. —(x+y). 
 
 185. How multiply a fraction by a fraction ? Note i. How mixed quantities ! 
 Vote 2. How shorten the operation ? Note 3. What is the force of the word 0/ ia 
 compound fractiouB ? 
 
MULTIPLICATION OF FEACTI0:N^S. 87 
 
 HT li.- 1 2a , x^ — y^ . 2a (x + y) 
 
 10. Multiply by ^- Ans. — —j-^' 
 
 11. Multiply -^ by \~ * 
 
 ^ "^ 4a: "* y^ — 2xy 
 
 TIT 1 J.- 1 4^ — 2h , 2a — h 
 .2. Multiply ^—--^ by -^. 
 
 13. Multiply « + -^- by ^. 
 
 14. Multiply ^ + 1^ by -t/. 
 
 15. Multiply a; — — by - + ^. 
 
 16. Multiply « 4- -T- by — y 
 
 CASE III. 
 
 186. To Multiply an Integer by a Fraction* 
 
 dx 
 I. Multiply the integer a by — 
 
 Analysis. — Changing the integer to the 
 
 a « = 
 
 a 
 
 form of a fraction, we have - to be multiplied I 
 
 . dx ... . adx^ ^ ^. a dx _adx 
 
 by — , which equals . Hence, the 7 ^ 77; — "777 * 
 
 ^ cy* ^ cy 1 cy cy 
 
 Rule. — Reduce the integer to a fraction ; then multiply 
 the numerators 'together for the new numerator, and the 
 denominators for the new denom,inator. 
 
 Notes. — i. Multiplying an integer by b. fraction is the same as find- 
 hig B. fractional part of a quantity. Thus, a; x | is the same as finding 
 
 f of X, each being equal to — . That is, 
 4 
 2. Three times i fourth of a quantity is the same as i fourth of 
 3 times that quantity. 
 
 186. How multiply an integer by a fraction ? Note. To what is this operation 
 similar? 
 
88 MULTIPLICATION OF FRACTION'S. 
 
 Find the product of the following quantities : 
 
 9. {x'-f)x ""' 
 
 2. 
 
 , dx 
 
 abc X — 
 
 cy 
 
 3. 
 
 , b -\-c 
 ad X — ' — 
 xy 
 
 4. 
 
 ax X ' 
 
 4« 
 
 5. 
 
 (« + ^)xf 
 
 3 (^ + «/) 
 
 ^ 2 (« + /^) 
 
 II. (:z:2 + i) X ^^^ 
 
 3(^—0 
 12. 2a;^(flj — Z*) X 
 
 ;2 — Z»2 
 
 6- (3« — 2/) X ^- 13. 3a (2; — i) X ~^^ 
 
 y *. ^ V ' ^'^ _ I 
 
 7. (-^+0X^. 14. (2.. + .2)x^^. 
 
 8. (i~«^)x-^. 15. (I-^^2)x ' 
 
 I + « ^ ' n-\- \ 
 
 187. The principles developed in the preceding cases may 
 be summed up in the following 
 
 GENERAL RULE. 
 
 Reduce whole and mixed quantities to improper fractions , 
 then cancel the common factors, and place the product of the 
 numerators over the product of the denominators, 
 
 1. Multiply by 15a;. 
 
 2. Multiply ^^ by y^ — i. 
 
 3. Multiply ^ + — by —^. 
 
 ,, ,,. , 2ax ^ab , xac 
 
 4. Multiply — X — - by --^. 
 
 ^ "^ a ac '' 2ab 
 
 5. Multiply ^- by ^. 
 ^ ^ -^ loy ^ ga 
 
 6. Multiply X 7 by 
 
 a a -\- b '' a 
 
 187. What Ib the general rule for multiplying fractions ? 
 
DIVISION OF FRACTIONS. 85 
 
 7. Multiply ^— — by x + z. 
 
 8. Multiply ^ by 2y\ 
 
 if 
 
 9. Multiply by a;^ — 2xy + «/2. 
 
 a; — y 
 
 10. Multiply - — —-^ by 8;? — 2. 
 
 ^ -^ 402; — 10 -^ 
 
 11. Multiply a; — -^ by - + ^» 
 
 ^ -^ X -^ y X 
 
 2a^ 2ah 
 
 12. Multiply a + ^ by — ^ • 
 
 ., Multiply (i±l)ib,^^. 
 
 14. Multiply ^ by -5 ^' 
 
 ^ -^ 4a; y^ — 2xy 
 
 15. Multiply & + ^^^ by Z» - ^^^. 
 
 2// ^2 ^2 
 
 16. Multiply by —- 
 
 ^ -^ x — y *' ax 
 
 DIVISION OF FRACTIONS. 
 
 CASE I. 
 188. To Divide a Fraction by an Integer, 
 
 I. If 3 oranges cost — dollars, what will i cost? 
 
 71/ 
 Analysis.— One is i third of 3; therefore, opeeation. 
 
 oa 9^ . _ 3^ 
 
 I orange will cost i third of — dollars, and "^ "^ 3 — ~ 
 
 - of — dollars is — dollars. (Art. 167, Prin. 2.) 
 
 2. Divide - by m. 
 
 c ^ 
 
 Analysis. — Since we cannot divide 
 
 OPBEATION. 
 
 the numerator of the fraction by m, — -i- m rz= ^ z=. — • 
 
 we multiply the denominator by it. C ' C X m cm 
 
90 DIVISION OP FRACTIONS. 
 
 The result is — . For, in eacli of the fractions - and — , the same 
 cm c cm 
 
 number of parts is taken ; but. since the unit is divided into m times 
 
 as many parts in the loiter as in the former, it follows that each part 
 
 in the latter is only — th of each part in the former. Hence, the 
 '' m 
 
 Rule. — Divide the numerator by the integer. 
 Or, Multiply the denomhiator hy it. 
 
 Note. — If the dividend is a mixed quantity, it should be reduced 
 to an improper fraction before the rule is applied. (Ex. 3.) 
 
 Divide the following quantities 
 
 3. 
 
 a -\ by d,^ 
 
 X 
 
 
 4. 
 
 ^ + l^hjxy. 
 
 
 5. 
 
 / by s^y- 
 
 9- 
 
 6. 
 
 2«^ 1 X 
 
 — by 5. 
 
 sac -^ 
 
 10. 
 
 7. 
 
 « H hy a, 
 
 c 
 
 II. 
 
 8. 
 
 ax H by xK 
 
 12. 
 
 . ax -\- be 
 
 Ans. :: 
 
 dx 
 
 . abx 4- 11 
 Ans, — J — -• 
 abxy 
 
 c^ -\- ax , 
 
 -^ — by a-\- X, 
 
 2b '' 
 
 a^ — <^ , 
 
 ^— - — by a — c. 
 
 7? + 2xy + if . 
 
 — — — ^—^-^~ by x-\-y. 
 
 — ■ — f by « + J. 
 a — b^ 
 
 CASE II. 
 189. To Divide a Fraction by a Fraction, 
 
 This case embraces two classes of examples : 
 
 First. Those in which the fractions have a common 
 denominator. 
 
 Second. Those in which they have different denominators. 
 
 188. How divide a fraction by an integer? NoU. If the dividend is a mixed 
 quantity, how proceed ? 
 
DIVISION^ OF FRACTIONS. 91 
 
 1. At — dollars apiece, how many kites can a lad buy for 
 
 dollars ? 
 
 m 
 
 Analysis.— Since these fractions have a com- opbbation. 
 
 mon denominator, tlieir numerators are like 12a ^ ^a 
 
 quantities, and one may be divided by the ni ' m 
 
 other, as integers. (Art. 177.) Now yi is con- Ans, 4 kites. 
 
 tained in 12a, 4 times. (Art. no, Prin. i.) 
 
 2. How many times is - — contained in -^— ? 
 
 '' X X 
 
 3. What is the quotient of ^^ divided by — ^- 
 
 n c 
 
 4. It is required to divide - by -• 
 
 X y 
 
 Analysis. — Since these frac- 
 tions have different denomina- 
 tors, their numerators are unlike 
 quantities ; consequently, one 
 cannot be divided by the other 
 in this form. (Art. 114, no^e.) We 
 therefore reduce them to a com- 
 mon denominator ; then dividing 
 one numerator by the other, the 
 result is the quotient. 
 
 Or, more briefly, if we invert the divisor, and multiply the dividend 
 by it, we have the 8am£ combinations and the same result as before. 
 (Art. 185.) Hence, the 
 
 EuLE. — Multiply the dividend hy the divisor inverted. 
 Or, Reduee the fractions to a eommon denominator, and 
 divide the numerator of the dividend hy that of the divisor. 
 
 Notes. — i. A fraction is inverted, when its terms are made to 
 
 change places. Thus, ^ inverted, becomes - • 
 a 
 
 2. The object of inverting the divisor is convenience in multiplying. 
 
 3. After the divisor is inverted, the common factors should be can^ 
 celled before the multiplication is performed. 
 
 189. How divide a fraction by a fraction when they have a common denominator ? 
 When the denominators are different, how ? 
 
 
 rmST OPERATION. 
 
 a _ 
 x~ 
 
 _ay c _cx 
 ~ xy' y~~ xy' 
 
 
 ay ^ ex _ay 
 xy ' xy~ ex 
 
 
 SECOKD OPEBATIOK. 
 
 a 
 
 X 
 
 y X c ex 
 
92 DIVISION OF FBACTIONS. 
 
 Divide the following fractions : 
 
 ^ cd ^ ay ^2* 2¥c^ ^ 6bc 
 
 6. ^-r- by -^. 14. — by ^— . 
 
 taoy ''ax 4cd '' 2d 
 
 7. 3^ by a ,5. J^by^f"?. 
 
 8. -^Lby^ t6. A^by3^^-?. 
 a;— 1*^2 a -\- h '' 2y 
 
 g— I 1^ _^ a:* , «a? 
 
 ,0. 5^ by -'-5^. ,8. 36«f!i, £8aS_ 
 
 „. iil£±^) byii^±^. 10. -SLbyl^. 
 
 12. -^-r-i by 20. -—4 by -f— ^. 
 
 a + "^ a iSab *^ 36^6? 
 
 CASE III. 
 190. To Divide an Integer by a Fraction. 
 
 I. Divide the integer ydc by ^' 
 
 Analysis.— Having reduced the opbbation. 
 
 integer to a fraction, and inverted ndc -^ ^ = 
 
 the divisor, we cancel the common ^ 
 
 factor d, and proceed as in the last jdc h ']hG . 
 
 case. Hence, the ~i ^ ^ ~ 3« ' ' 
 
 Rule. — Reduce the integer to a fraction, and multiply it 
 hy the divisor inverted. 
 
 Divide the following quantities : 
 
 ^ ' ■ jx 
 
 190, How divide au integer by a fraction ? 
 
 2. 
 
 , ex 
 '^y ■■ dm 
 
 3- 
 
 ax: ^ . 
 m -{- n 
 
 4. 
 
 J/ 
 
DIVISION OF FRACTION'S. 
 
 93 
 
 191. Complex JFractions are reduced to simple ones, 
 by performing the division indicated. 
 
 8. Keduce — to a simple fraction. 
 3 
 
 Analysis. — The given fraction is equiv- a 
 
 alent to 
 
 « . 3 
 
 . Performing the division 
 
 b 
 
 , the dividend. 
 
 indicated, we have ^, the simple fraction 
 required. (Art. 189.) 
 
 -, the divisor. 
 4 
 
 ^ y 4 _ 4a . 
 
 b 3 3b 
 
 Reduce the following fractions to simple ones : 
 
 10. 
 
 II. 
 
 b 
 Id" 
 «4-i 
 
 « — I 
 
 a— I 
 
 a+ I 
 a-b 
 
 ^ + .V 
 
 x — y 
 
 a + b 
 
 Ans. 
 
 12. 
 
 13- 
 
 bed 
 
 a? — y'^ 
 
 a-b 
 
 x + y 
 x-\-y 
 
 a-^b 
 
 a + b 
 
 x — y 
 
 192. The various principles developed in the preceding 
 cases may be summed up in the following 
 
 GEN ERAL RULE. 
 
 Reduce integers and mixed quantities to improper frac- 
 tions, and complex fractions to simple ones : then multiply 
 the dividend by the divisor inverted. 
 
 191. How reduce complex fractions ? 192. What is the general rule for dividing 
 fractionB? 
 
94 
 
 DIVISION OF FKACTIONS, 
 
 1. Divide — — by sic 
 
 4xyz 
 
 2. Divide ^^ ^ by gxy. 
 
 2'jy 
 
 ^. ., i6xy , 2cd 
 -?. Divide by — 
 
 ^ fyn. '' inn. 
 
 ^. ., 42ah , 145 
 
 4. Divide -^ 5 by 
 
 5. Divide -^^^-^ by -^. 
 
 , TA. ., x'^ — 2xy + ifi , ^ — y 
 
 6. Divide / -^ by ^. 
 
 7. Divide -X ; — -„ by 
 
 Solution. — Factoring and cancelling, we have, 
 
 a^—m* _ (a® + m^) (a + m){a—m) ^ a ? -{■am _ a(a + m) ^ 
 a^—2am + m^~ {a—m)(a—m) ' a—m ~ a—m 
 
 (a^-\-m-)(a + m)(a—m) a—m a'^ + m^ m^ . 
 
 ' ^-^, — -\ X -7 ^ = , or a + — » Ans, 
 
 (a—m) {a—m) a{a + m) a a 
 
 8. Divide -^^ by -^—. 
 
 cfi — x^ ''a — X 
 
 -p. . . , 4^2 — 8c , c2 — 4 
 
 0. Divide by -* 
 
 ^ x + y '' X -\- y 
 
 10. Divide ; by ^-73-- — (• 
 
 ac -^ ax -^ A^\d -^ X) 
 
 11. Divide -=— — r bv — , — 
 
 a^ -\- & " a •\- c 
 
 12. Divide ■— by ^-^^ — -—^^ 
 
 X "^ a — x 
 
 ,3. Divide ^qr-,^jq.-^ by -^-^. 
 14. Divide ^— by ^--^. 
 
 iq. Divide by — ^^ -" (See Appendix, p. 285.) 
 
 ■^ x-^- ax -^ I ■\- a 
 
CHAPTER IX. 
 SIMPLE EQUATIONS. 
 
 193. An Equation is an expression of equality between 
 two quantities. (Art. 27.) 
 
 194. Every equation consists of two parts, called the 
 first and second members. 
 
 195. The First Member is the part on the left of the 
 sign =. 
 
 The Second Manber is the part on the right of the 
 sign =. 
 
 196. Equations are divided into degrees, according to the 
 exponent of the unknown quantity; as the first, second, 
 third, fourth, etc. 
 
 Equations are also divided into Simple, Quadratic, 
 CuMc, etc. 
 
 197. A Simple Equation is one which contains 
 only ihQ first power of the unknown quantity, and is of the 
 first degree ; as, ax = d. 
 
 198. A Quadratic Equation is one in which the 
 highest power of the unknown quantity is a square, and is 
 of the second degree ; as, ax^ -\- ex ^ d. 
 
 199. A Cubic Equation is one in which the highest 
 power of the unknown quantity is a cuie, and is of the 
 third degree ; as, aa^ + dx^ — ex = d. 
 
 193. What is an equation? 194. How many parts? 195. Which is the flri«t 
 member ? The second ? 196. How are equations divided ? What other divisions ? 
 197. What is a simple equation ? 198. A quadratic ? 199. Cubic ? 
 
96 SIMPLE EQUATIONS. 
 
 200. An Identical Equation is one in which both 
 members have the same form^ or may be reduced to the 
 same form. 
 
 Thus, ab—c = ab—c, and 8a;— 3a? = 5a;, are identicaL 
 Note. — Sucli an equation is often called an identity/. 
 
 201. The Transformation of an equation is chang- 
 ing its form without destroying the equality of its members. 
 
 Note. — The members of an equation will retain their equality, so 
 long as they are equally increased or diminished. (Ax. 2-5.) 
 
 TRANSPOSITION. 
 
 202. Transposition of Terms is changing them 
 from one side of an equation to the other without destroy- 
 ing the equahty of its members. 
 
 203. Unknown Quantities may be combined with 
 known quantities by addition, subtraction, multiplication, 
 or division. 
 
 Note. — The object of transposition is to obtain an equation in which 
 the terms containing the unknown quantity shall stand on one side, 
 and the known terms on the other. 
 
 204. To Transpose a Term from one Member of an 
 
 Equation to the other. 
 
 1. Given x -{-1) z=a, to find the value of x. 
 
 Solution.— By the problem, x+b = a 
 
 Adding —b to each side (Ax. 2), x+b—b = a—b 
 Cancelling ( + &—&) (Ax. 7), .*. x = a—b 
 
 This result is the same as changing the sign of b from + to — in 
 ihe first equation, and transposing it to the other side. 
 
 2. Given x — d z= c, to find the value of x. 
 
 Solution. — By the problem, x—d = c 
 
 Adding +dto each side (Ax. 2), x—d+d = c + d 
 Cancelling (—d + d), (Ax. 7.) .-. x = c+d 
 
 200. Identical ? 201. What is the transformation of an equation ? Note. Equality. 
 202. What is transposition of terms? 203. How combine unknown quantities T 
 Note. Object of transposition ? 
 
ONE UNKNOWN QUANTITY. 97 
 
 This result is also tlie same as changing the sign of d from — io 4- , 
 and transposing it to the other side. Hence, the 
 
 ^vJjE.— Transpose the term from one member of the equa- 
 tion to the other, and change its sign. 
 
 Note.— In the first of the preceding examples, the unknown 
 quantity is combined with one that is known by addition; in the 
 second, with one by subtraction. 
 
 3. Given b — c-{-x = a — d, to find x, 
 
 4. Given x-\-al) — c=a-\-l), to find x. 
 
 205. The Signs of all the terms of an equation may be 
 changed without destroying the equality. For, all the 
 terms on each side may be transposed to the other, by 
 changing their signs. 
 
 206. If all the terms on one side are transposed to the 
 other, each member will be equal to o. 
 
 Thus, if 05+6 = <?, it follows that x+c—d = o. 
 
 REDUCTION OF EQUATIONS. 
 
 207. Tlie Heduction of an equation consists in finding 
 the value of the unknown quantity which it contains. 
 
 208. The Value of an unknown quantity is the number 
 which, substituted for it, will satisfy the equation. Hence, 
 it is sometimes called the root of the equation. 
 
 209. The reduction of equations depends on the following 
 
 PRINCIPLE. 
 
 Both members of an equation may be increased or dimin- 
 ished by the same quantity ivithout destroying the equality. 
 
 204. How transpose a term from one member of an equation to the other? 
 205. What is the effect of changing all the signs ? 206. Of transposing all the terms f 
 207. In what does the reduction of an equation consist ? 208. What is the value of 
 an nnkno\ATi quantity ? What sometimes called ? 209. Upon what principle does 
 the reduction of equations depend ? 
 
98 
 
 SIMPLE EQUATIONS 
 
 210. This principle may be illustrated by a pair ol 
 scales. If 4 balls, each weighing i lb., are placed in each 
 scale, they balance each other. 
 
 Adding 2 lbs. to each scale, 
 
 4 + 2 = 4 + 2 
 
 Subtracting 2 lbs. from each, 
 
 4-2 = 4-2 
 Multiplying each by 2, 
 
 4x2 = 4x2 
 Dividing each by 2, 
 
 4-5-2 = 4-7-2 
 
 211. To Reduce an Equation containing One Unlcnown 
 
 Quantity by Transposition, 
 
 5. Given 2a; — 3^? -f 7 = a; -f 35, to find x. 
 
 Solution. —By the problem, 2a;— 3a + 7 = a; + 35 
 
 Transposing the terms (Art. 204), 2X—x = 35— 7 + 3a 
 Uniting the terms, (Ax. 9), Ans.x = 2S + 3a 
 
 Therefore, 28 + 3a is the value of x required. Hence, the 
 
 EuLE. — Transpose the unhnoivn quantities to one sid6, 
 and the known quantities to the other, and unite the terms. 
 
 Notes. — i. Transposing the terms is the same, in effect, as adding 
 equal quantities to, or subtracting them from each member; hence, 
 it is often called reduction of equations by addition or svbtractiovu 
 
 2. If the unknown quantity has the sign — before it, change the 
 Bigns of all the terms. (Art. 205.) 
 
 212. When the same term, having the same sign, is on 
 opposite sides of the equation, it may be cancelled. (Ax. 3.) 
 
 6. Reduce 7,x ■\- a ^ 6 = b — 4 -^ 2X. 
 
 7. Reduce x— ■^■^c=z2X'\-a — d. 
 
 8. Reduce 21/ -{- he — ad = y + 2m — 8. 
 
 9. Reduce sab — y -}- d = — 2y-f 17. 
 
 10. Reduce 4cd -f 27 — 4ic -|- J = 28 — 3a: -f 3M. 
 
 11. Reduce 5 -f c — 4a; = 32 -f & — $x + d. 
 
 12. Reduce a: -f 4 — 2a; — 3 = 3a; -f 4 -f 8 — ^x. 
 
 aio. Illustrate this principle? 211. What is the rule for reducing equations- ? 
 Vote. To what is transposition equivalent ? 212. When the same term, having the 
 fam^ sign, is on opposite sides, what may be done ? 
 
ONE unk:n^own quantity. 99 
 
 CLEARING OF FRACTIONS. 
 213. To Reduce an Equation containing Fractions. 
 
 1. Given - + -— = 27, to find the value of x. 
 
 Solution. — By the problem, ? + ??== 2? 
 
 2 6 
 Multiplying each term by 6, the I, c. ni, ) 
 
 of the denominators (Art. 148), f 3^+ 22? = 162 
 
 Uniting the terms, 52? = 162 
 
 Dividing each side by the coefficient, Ans. x = 32I 
 
 2. Given - -\- - = —, to find the value of x, 
 
 23 4 
 
 Solution.— By the problem, ? 4. ? = ^ 
 
 234 
 Mult, by 12, the I, c, in. of denominators, 6a; +42; = 90 
 
 Uniting terms, and dividing (Art. 211), Ans. x= g 
 
 Therefore, the value of x is 9. Hence, the 
 
 KuLE. — Multiply each term of the equation ly the least 
 common multiple of the denominators ; then, transposing 
 and imiting the terms, divide each member hy the coefficient 
 of the unknown quantity. 
 
 Notes. — i. An equation may also be cleared of fractions, by multi- 
 plying both sides by each denominator separately. 
 
 2. The reason that clearing an equation of fractions does not destroy 
 the equation, is because both members are fnultipUed by the same 
 quantity. (Ax, 4.) 
 
 3. A fraction is multiplied by its denominator by cancelling the 
 denominator. (Art. 184, Wote i.) 
 
 4. Removing the coefficient of a quantity divides the quantity by it. 
 
 5. If any given numerator is a multiple of its denominator, divide 
 the former by the latter before applying the rule. 
 
 6. The unknown quantity in the last two problems is combined 
 with those that are known by multiplication and division. Hence, the 
 operation is often called, reduction of equations by multiplication and 
 division. 
 
 213. Rule for clearing an equation of fractions ? Notes, i. In what other way may 
 fractions be removed ? 2. Why does not this process destroy the equation ? 3. What 
 :8 the effect of cancelling a denominator? 4. Effect of removing a coefficient' 
 5. If a numerator is a multiple of its denominator, how proceed ? 
 
100 SIMPLE EQUATIONS. 
 
 3. Eeduce — + 12 = — + i. 
 
 5 3 ; 
 
 4. Eeduce - = 6x — 66» 
 
 3 6 
 
 5. Reduce — 4--= 'zk —x. 
 
 10 5 
 
 214. When the sign — is prefixed to a fraction and the 
 denominator is removed, the sigjis of all the terms in the 
 numerator must be changed from -f to — , or — to +. 
 
 X — 2 
 
 6. Eeduce 30; = 20, 
 
 Solution.— By tlie problem, 3a; — ^^ = 20 
 
 Removing tlie denominator 5, 15a;— a; + 2 = 100 
 
 Uniting and transposing the terms, 14a; = 98 
 
 Dividing by the coeflacient, Ans. x— 7 
 
 7. Given 4^^::^^ =_ 3^, to find ^. 
 
 X d' 
 
 8. Given ^x — — = a —^ , to find x, 
 
 5 10 
 
 9. Given — a;H [-^ = -^, to find x. 
 
 3 4 24' 
 
 10. Given «-|-5 + c = --i [-^ + ^+<?-^5^ to find rzr. 
 
 2 4 
 
 215. The principles developed by the preceding illustra- 
 tions may be summed up in the following 
 
 GENERAL RULE. 
 
 I. Clear the equation of fractions. (Art. 213.) 
 
 II. Transpose and unite tlie terms. (Art. 204.) 
 
 III. Divide loth sides hy the coefficient of the unknown 
 quantity, (Art. 213, Note 4.) 
 
 Proof. — For the unknoivn quantity substitute its value, 
 and if it satisfies the equation, the worlc is riglit. 
 
 214. If sign — is prefixed to a fraction ? 315, What is the general rule for simple 
 equations ? How proved ? 
 
OlJE UNKNOWN QUANTITY, 101 
 
 EXAMPLES. 
 
 1. Given x ■] 1- - = 14, to find x, 
 
 2 4 
 
 2. Given - 4- a; = — + 40, to find x, 
 
 2 10 
 
 3. Given — -f 10 = — + 13, to find x. 
 
 ^ 5 10 ^^ 
 
 4. Given — - — \- 6 = S, to find x, 
 
 ^ X — 2 
 
 2X -t- I 
 
 5. Given x -\ =10, to find x. 
 
 6. Given 2x + ^^±1L = 18 + 9£r:29 ^^ ^^^ ^ 
 
 5 4 
 
 7. Given - H ^ - = 78, to find x. 
 
 234 
 
 8. Given ^^^-8=^-^ + 5, to find x. 
 
 4 
 
 9. Given x + ^^~^ + '^^~ = 12, to find a;. 
 
 26 
 
 10. Given 2a; — 16 = ^^ , to find x. 
 
 3 
 
 ^. 2a; — 8 , a; + 32 , ic . £ ;i 
 
 11. Given 5—^- + - = 30, to find x, 
 
 12. Given - + -^=16 + -, to find a:. 
 
 20 o 
 
 rt- 3^+1 2a; , a;— I , ^ , 
 
 13. Given — 10 = 1 — , to find x. 
 
 2 30 
 
 14. Given f- v = — — i A^ to find x. 
 
 ^ 10 6 15 ^^ 
 
 15. Given Sx + 6i — - = Si- — + •^, to find x. 
 
 -^ ^2 ^72 
 
 Note. — Sometimes there is an advantage in uniting similar terms, 
 before clearing of fractions. Tims, uniting 6^ witli 8^ ; also witli 
 
 — , we nave, 8a; = 2 + 8a; ; /. a; = 7, Ans. 
 
102 SIMPLE EQUATIONS. 
 
 i6. Given f -6 +^ = 1+ 2, to find a;. 
 
 ^ . AX ZX 
 
 17. Given — = — + 15 — 12, to find x. 
 
 5 4 
 
 18. Given 20; — 4 = - + 2, to find x, 
 
 2 
 
 19. Given ^+ ^_ i|::^3^ ^ ,^j^ j^ g^^ ^ 
 
 45 5 
 
 X 
 
 20. Given - = 5 -}- c, to find x, 
 
 (t 
 
 fix 
 
 21. Given — = ^Z, to find x. 
 
 ^. ax hx , n -, 
 
 22. Given = c, to find x. 
 
 23 
 
 ^. 2ax -{- h ex + d ^ „ ^ 
 
 2$. Given -^— = — 3_ to find x. 
 
 a c 
 
 24. Given h- = - + -, to find x. 
 
 a X 2 a 
 
 25. Given |+| + f = f + ^f-^ to find x. 
 
 26. Gi,en ^ +'-^ = '1 -^'J-^, to nnd X, 
 
 2 3 5 3 
 
 ^. 3flf + a; 6 , ^ - 
 
 27. Given - — ■ 5 =-, to find x, 
 
 X ^ x' 
 
 X — I I 
 
 28. Given — ; h i = -, to find x. 
 
 ic + I a 
 
 r^. X , X a , n -, 
 
 29. Given - H = - + «. to find x. 
 
 a c —■ a c ' 
 
 a? 
 
 30. Given x-{-h=z , to find x. 
 
 31. Given x — a = — -^— - , to find x. 
 
 X — a 
 
 32. Given 3 (^*) + p^-*) = 4 (^*), to find ^. 
 3:> Given ^-~ = x --, to find x. 
 
OKE UKKNOWK QUANTITY. 103 
 
 34. Given ~-\- x = 25, to find x. 
 
 4 2 
 
 X X 
 
 35. Given 80 = 4a; , to find x. 
 
 2 6 
 
 ,^. 2iC+I a;^^ 
 
 36. Given ! — = 2X -^-^ , to find x, 
 
 3 4 
 
 37. Given 10 -> 22; = 5^+^ - £1Z136 ^^ g^^ ^^ 
 
 3 3 
 
 38. Given a; — 3 = 15 _ ^Jli, to find x. 
 
 39. Given a; + 2 = 3a; + ^-±-? - ?-±-^, to find a?. 
 
 4 3 
 
 ^. Z^ , X — 4 a; — 10 
 
 40. Given ^ + 5 _ ^ _ 5^^ g^^ 
 
 422 
 
 ^. iia;— I ex— 11 x—i 
 
 41. Given — - — = ^ to find x. 
 
 12 4 10 
 
 42. Given ^ _ ?? = 120, to find x. 
 
 5 10 
 
 43. Given ar — 20 = — ^^+1 to find x, 
 
 5 
 
 44. Given ~^ = ^—^ ■\-t2-x, to find x, 
 
 ^ 3 
 
 45. Given ^-^ + 10 = — _ 1^=^, to find x. 
 
 46. Given — ^ - -^- = 5, to find a:. 
 
 I* -j- I Q, I 
 
 r\' X 2 + X C 
 
 47. Given ^-^ - ^_ =.____, to find x. 
 
 48. Given ^^=5+^, to find ^. 
 
 49- Given — = ^ — — , to find a?. 
 2 3 
 
 50. Given 8«= ^1^, to find x, 
 
 1 -\- X 
 
 51. Given -~^-^- = ~^yto find a;. 
 
104: SIMPLE EQUATIONS. 
 
 PROBLEMS. 
 
 216. The Solution of a problem is finding a quantity 
 which will satisfy its conditions. It consists of two parts : 
 
 First.— The Formation of an equation which will 
 express the conditions of the problem in algebraic language, 
 Second. — The deduction of this equation. 
 
 217. To Solve Problems in Simple Equations containing 
 
 one unknown Quantity. 
 
 I. A farmer divided 52 apples among 3 boys in such a 
 manner that B had i half as many as A, and C 3 fourths as 
 many as A minus 2. How many had each ? 
 
 I. Formation— Let x = A'a number. 
 
 By the conditions, - = B's ** 
 
 2 
 
 4 
 
 Therefore, by Ax. 0, x + - + — — 2= 52, the whole. 
 24 
 2c Reduction— 4a;+2a;+3«— 8 = 208 
 
 Transposing, etc., gx = 216 
 
 Removing the coefficient, x = 24, A's number. 
 
 Ans. A had 24, B had 12, and C had 18—2 = 16. 
 
 From this illustration we derive the following 
 
 GENERAL RULE. 
 
 I. Represent the unknown quantity ly a letter, then state 
 in algelraic language the operations necessary to satisfy the 
 wnditions of the prohlem. 
 
 II. Clear the equation of fractions ; then, transposing and 
 uniting the terms, divide each member by the coefficient of the 
 unhnown quantity. (Art. 213.) 
 
 Note. — A careful study of the conditions of the problem will soon 
 enable the learner to discover the quantity to be represented by the 
 letter, and the method of forming the equation. 
 
 tx6. What is the Bolution of a problem ? Of what does it coneist ? 217. What is 
 the general rale ? 
 
OKE TJKKKOWiq- QTJAI^TITY. 105 
 
 2. The bill for a coat and vest is I40 ; the value of the 
 coat is 4 times that of the vest. What is the value of each ? 
 
 3. A bankrupt had $9000 to pay A, B, and C ; he paid B 
 twice as much as A, and C as much as A and B. What 
 did each receive ? 
 
 4. The whole number of hands employed in a factory was 
 1000 ; there were twice as many boys as men, and 11 times 
 as many women as boys. How many of each ? 
 
 5. Two trains start at the same time, at opposite ends of 
 a railroad 1 20 miles long, one running twice as fast as the 
 other. How far will each have run at the time of meeting ? 
 
 6. A man bought equal quantities of two kinds of flour, 
 at $10 and $8 a barrel. How many barrels did he buy, the 
 whole cost being 1 1200 ? 
 
 7. If 96 pears are divided among 3 boys, so that the 
 second shall have 2, and the third 5, as often as the first 
 has I, how many will each receive ? 
 
 8. A post is one-fourth of its length in the mud, one- 
 third in the water, and 12 feet above water; what is its 
 whole length ? 
 
 9. After paying away I of my money, and then J of the 
 remainder, I have $72. What sum had I at first ? 
 
 10. Divide $300 between A, B, and 0, so that A may 
 have twice as much as B, and C as much as both the others. 
 
 1 1 . At the time of marriage, a man was twice as old as 
 his wife; but after they had lived together 18 years, his age 
 was to hers as 3 to 2. Eequired their ages on the wedding day. 
 
 12. A and B invest equal amounts in trade. A gains 
 1 1 260 and B loses $870; A's money is now double B's. 
 What sum did each invest ? 
 
 13. Eequired two numbers whose difference is 25, and 
 twice their sum is 114. 
 
 14. A merchant buying goods in New York, spends the 
 first day ^ of his money ; the second day, I ; the third day, 
 I; the fourth day, ^; and he then has $300 left. How 
 much had he at first ? 
 
106 SIMPLE EQUATIOKS. 
 
 15. What number is that, from the triple of which if 17 
 be subtracted the remainder is 22 ? 
 
 16. In fencing the side of a field whose length was 450 
 rods, two workmen were employed, one of whom built 9 rods 
 and the other 6 rods per day. How many days did they 
 work ? 
 
 17. Two persons, 420 miles apart, take the cars at the 
 same time to meet each other ; one travels at the rate of 40 
 miles an hour, and the other at the rate of 30 miles. What 
 distance does each go ? 
 
 18. Divide a line of 28 inches in length into two such 
 parts that one may be J of the other. 
 
 19. Charles and Henry have $200, and Charles has seven 
 times as much money as Henry. How much has each ? 
 
 20. What is the time of day, provided f of the time past 
 midnight equals the time to noon ? 
 
 21. A can plow a field in 20 days, B in 30 days, and C in 
 40 days. In what time can they together plow it ? 
 
 22. A man sold the same number of horses, cows, and 
 sheep; the horses at lioo, the cows at $45, and the sheep 
 at I5, receiving $4800. How many of each did he sell ? 
 
 23. Divide 150 oranges among 3 boys, so that as often as 
 the first has 2, the second shall have 5, and the third 3. 
 How many should each receive ? 
 
 24. Four geese, three turkeys, and ten chickens cost lio ; 
 a turkey cost twice as much as a goose, and a goose 3 times 
 as much as a chicken. What was the price of each ? 
 
 25. The head of a fish is 4 inches long ; its tail is 1 2 times 
 as long as its head, and the body is one-half the whole 
 length. How long is the fish ? 
 
 26. Divide 100 into two parts, such that one shall be 20 
 more than the other. 
 
 27. Divide a into two such parts, that the greater divided 
 by c shall be equal to the less divided by d. 
 
 28. How much money has A, if \, f , and f of it amount 
 to $1222 ? 
 
OKE UHKHOWN QUANTITY. 107 
 
 29. What number is that, |, J, J, and J of which are 
 equal to 60 ? * 
 
 30. A man bought beef at 25 cents a pound, and twice as 
 much mutton at 20 cents, to the amount of $39. How 
 many pounds of each ? 
 
 31. A says to B, "I am twice as old as you, and if I were 
 15 years older, I should be 3 times as old as you." What 
 were their ages ? 
 
 32. The sum of the ages of A, B, and C is 1 10 years ; B is 
 3 years younger than A, and 5 years older than C. What 
 are their ages ? 
 
 ^^. At an election, the successful candidate had a 
 majority of 150 votes out of 2500. What was his number 
 of votes ? 
 
 34. In a regiment containing 1200 men, there were 
 3 times as many cavalry as artillery less 20, and 92 more 
 infantry than cavalry. How many of each ? 
 
 35. Divide $2000 among A, B, and 0, giving A $100 
 more than B, and $200 less than 0. What is the share of 
 each ? 
 
 36. A prize of $150 is to be divided between two pupils, 
 and one is to have | as much as the other. What are the 
 shares ? 
 
 * When the conditions of the problem contain fractional expressions, 
 as ^, J, \, etc. , we can avoid these fractions, and greatly abridge the 
 operation, by representing the quantity sought by such a number of 
 ic's as can be divided by each of the denominators without a remainder. 
 This number is easily found by taking the least common multiple of 
 all the denominators. Thus, in problem 29, 
 
 Let \2X = the number. 
 
 Then will 6x — i half. 
 
 « " 4a; = I third. 
 
 ** " 3iC = I fourth. 
 
 " '* 225—1 sixth. 
 
 Hence, 6a; + 4a! + 3a; + 2a? = 60 
 
 .'. a; = 4 
 Finally, a;x 12 or 12a! = 48, the number required 
 
108 SIMPLE EQUATIONS. 
 
 37. Two horses cost 16 16, and 5 times the cost of one was 
 6 times the cost of the other. What was the price of each ? 
 
 38. What were the ages of three brothers, whose united 
 ages were 48 years, and their birthdays 2 years apart ? 
 
 39. A messenger travelling 50 miles a day had been gone 
 
 5 days, when another was sent to overtake him, travelling 
 65 miles a day. How many days were required ? 
 
 40. What number is that to which if 75 be added, f of 
 the sum will be 250 ? 
 
 41. It is required to divide 48 into two parts, which shall 
 be to each other as 5 to 3.* 
 
 42. What quantity is that, the half, third, and fourth of 
 which is equal to a ? 
 
 43. A and B together bought 540 acres of land, and 
 divided it so that A's share was to B's as 5 to 7. How many 
 acres had each ? 
 
 44. A cistern has 3 faucets; the first will empty it in 
 
 6 hours, the second in 10, and the third in 12 hours. How 
 long will it take to empty it, if all run together ? 
 
 45. Divide the number 39 into 4 parts, such that if the 
 first be increased by i, the second diminished by 2, the third 
 multiplied by 3, and the fourth divided by 4, the results 
 will be equal to each other. 
 
 46. Find a number which, if multiplied by 6, and 1 2 be 
 added to the product, the sum will be 66. 
 
 47. A man bought sheep for $94 ; having lost 7 of them, 
 he sold ^ of the remainder at cost, receiving $20. How 
 many did he buy ? 
 
 48. A and B have the same income ; A saves J of his, but 
 B spending $50 a year more than A, at the end of 5 years is 
 $100 in debt. What is their income ? 
 
 * When the quantities sought have a given ratio to each other, the 
 solution may be abridged by taking such a number of a's for the 
 unknown quantity, as will express the ratio of the quantities to each 
 other without fractions. Thus, taking 5a; for the first part, 2X will 
 represent the second part ; then 5 j; + 3a; = 48, etc. 
 
ONE UKKITOWK QUAKTITY. 109 
 
 49. A cistern is supplied with water by one pipe and 
 emptied by another; the former fills it in 20 minutes, the 
 latter empties it in 15 minutes. When full, and both pipes 
 run at the same time, how long will it take to empty it ? 
 
 50. What number is that, if multiplied by vi and n 
 separately, the difference of their products shall be c? ? 
 
 51. A hare is 50 leaps before a greyhound, and takes 
 4 leaps to the hound's 3 leaps ; but 2 of the greyhound's 
 equal 3 of the hare's leaps. How many leaps must the 
 hound take to catch the hare ? 
 
 52. What two numbers, whose difference is h, are to each 
 other as « to c ? 
 
 53. A fish was caught whose tail weighed 9 lbs.; his head 
 weighed as much as his tail and half his body, and his body 
 weighed as much as his head and tail together. What was 
 the weight of the fish ? 
 
 54. An express messenger travels at the rate of 13 miles 
 in 2 hours; 12 hours later, another starts to overtake him, 
 travelling at the rate of 26 miles in 3 hours. How long and 
 how far must the second travel before he overtakes the first ? 
 
 55. A father's age is twice that of his son ; but 10 years 
 ago it was 3 times as great. What is the age of each ? 
 
 56. What number is that of which the fourth exceeds the 
 seventh part by 30 ? 
 
 57. Divide $576 among 3 persons, so that • the first may 
 have three times as much as the second, and the third one- 
 third as much as the first and second together. 
 
 58. In the composition of a quantity of gunpowder, the 
 nitre was 10 lbs. more than f of the whole, the sulphur 
 4 J lbs. less than J of the whole, the charcoal 2 lbs. less than 
 \ of the nitre. What was the amount of gunpowder ? * 
 
 * The operation will be ^ortened by the following artifice : 
 
 Let 42a; + 48 = the number of pounds of powder. 
 
 Then 28a; + 42 = nitre ; jx+s^ = sulphur ; 4X + 4 = charcoal. 
 
 Hence, 390; + 49 ^ = 42a; + 48. 
 
 .'. X 
 
110 • SIMPLE EQUATIONS. 
 
 59. Divide $6 into 3 parts, sucli that J of the first, J of 
 the second, and ^ of the third are all equal to each other. 
 
 60. Divide a line 21 inches long into two parts, such that 
 one may be | of the other. 
 
 61. A milliner j)aid $5 a month foi rent, and at the end 
 of each month added to that part of 1 er money which was 
 not thus spent a sum equal to i half of this part; at the 
 end of the second month her original money was doubled. 
 How much had she at first ? 
 
 62. A man was hired for 60 days, on condition that for 
 every day he worked he should receive 75 cents, and for 
 every day he was absent he should forfeit 25 cents; at the 
 end of the time he received $12. How many days did he 
 work? 
 
 6^. Divide $4200 between two persons, so that for every 
 I3 one received, the other shall receive $5. 
 
 64. A father told his son that for every day he was perfect 
 in school he would give him 15 cents; but for every day he 
 failed he should charge him 10 cents. At the end of the 
 term of 1 2 weeks, 60 school days, the boy received $6. How 
 many days did he fail ? 
 
 65. A young man spends ^ of his annual income for 
 board, -| for clothing, -^ in charity, and saves I318. What 
 is his income ? 
 
 66. A certain sum is divided so that A has $30 less than ^, 
 B lio less than |, and $8 more than J of it. What does 
 each receive, and what is the sum divided ? 
 
 67. The ages of two brothers are as 2 to 3 ; four years 
 hence they will be as 5 to 7. What are their ages ? * 
 
 Note. — To change a proportion into an equation, it is necessary to 
 assume the truth of the following well established principle : 
 
 If four quantities are proportional, the product of the extremes is 
 
 * A strict conformity to system would require that this and- similar 
 problems should be placed after the subject of proportion ; but it is 
 convenient for the learner to be able to convert a proportion into an 
 equation at this stage of his progress. 
 
ONE UNKNOWN^ QUANTITY-. Ill 
 
 equal to the product of fhe means. Hence, in such cases, we have only 
 to make the product of the extremes one side of the equation, and the 
 product of the means the other. 
 
 Thus, let 2X and 3a; be equal to their respective ages. 
 
 Then 2a;+4 : 3a' + 4 - 5 : 7- 
 
 Making the product of the extremes equal to the product of the 
 
 mean^ we have, 
 
 140;+ 28 = 150;+ 20. 
 
 Transposing, uniting terms, etc., a; = 8. 
 
 .'. 2X= 16, the younger ; and 3a; = 24, the older. 
 
 68. What two numbers are as 3 to 4, to each of which if 
 4 be added, the sums will be as 5 to 6 ? 
 
 69. The sum of two numbers is 5760, and their difference 
 is equal to J of the greater. What are the numbers ? 
 
 70. It takes a college crew which in still water can pulLat 
 the rate of 9 miles an hour, twice as long to come up the 
 river as to go down. At what rate does the river flow ? 
 
 71. One-tenth of a rod is colored red, ^ orange, -^ yellow, 
 -}^ green, -f^ blue, ^^ indigo, and the remainder, 302 inches, 
 violet. What is its length ? 
 
 72. Of a certain dynasty, | of the kings were of the same 
 name, J of another, J of another, ^^ of another, and there 
 were 5 kings besides. How many were there of each name ? 
 
 73. The difference of the squares of two consecutive 
 numbers is 15. What are the numbers? 
 
 74. A deer is 80 of her own leaps before a greyhound ; 
 she takes 3 leaps for every 2 that he takes, but he covers as 
 much ground in one leap as she does in two. How many 
 leaps will the deer have taken before she is caught ? 
 
 75. Two steamers sailing from New York to Liverpool, a 
 distance of 3000 miles, start from the former at the same 
 time, one making a round trip in 20 days, the other in 
 25 days. How long before they will meet in New York, 
 and how far will each have sailed ? 
 
 ("See Appendix, p. 286.) 
 
CHAPTEE X. 
 SIMULTANEOUS EQUATIONS. 
 
 TWO UNKNOWN QUANTITIES. 
 
 218. Simultaneous * Equations consist of two oi 
 more equations, each containing two or more unk7iown 
 quantities. They are so called because they are satisfied by 
 the same values. 
 
 Thus, x + y^'j and 5a?— 4y = 8 are simultaneous equations, for in 
 each aj = 4 and y = 3. 
 
 219. Indepe^ident Equations are those which 
 express different conditions, so that one cannot be reduced 
 to the same form as the other. 
 
 Thus, 6aj— 4^=14 and 23^ + 3^=: 22 are independent equations. 
 But the equations ic + y = 5 and 3a; + 3^ = 15 are not independent, for 
 one is directly obtained from the other. Such equations are termed 
 dependent. 
 
 Note. — Simultaneous equations are usuaWj independent ; but inde- 
 pendent equations may not be simultaneous ; for the letters employed 
 may have the same or different values in the respective equations. 
 
 Thus, the equations a?+y=7 and 2a?— 2^=14 are Independent, 
 but not simultaneous ; for in one x = 7—y, in the other x= 7+y, etc. 
 
 220. Problems containing more than one unknown quan- 
 tity must have as mani/ simultaneous equations as there are 
 unknown quantities. 
 
 If there are more equations than unknown quantities, 
 some of them will be superfluous or contradictory. 
 
 218. What are Bimultaneous equations ? 219. Independent equations ? 220. Ho^ 
 many equations must each problem have ? 
 
 * From the Latin ^i'n^id, at the same time. 
 
TWO UNKN^OWN QUAl^TITIES. 113 
 
 If the number of equations be less than the number of 
 unknown quantities, the problem will not admit of a definite 
 answer, and is said to be indeterminate. 
 
 221. JSllmination* is combining two equations 
 which contain two unknown quantities into a single equa- 
 tion, having but one unknown quantity. There are three 
 methods of elimination, viz. : by Comparison, by Substitution. 
 and by Addition or Subtraction, 
 
 CASE I. 
 222. To Eliminate an Unknown Quantity by Comparison, 
 
 I. Given x -\- y =z i6, and a; — «/ = 4, to find x and y. 
 
 Solution.— By the problem, 
 « ** 
 
 Transposing the ^ in (i), 
 " the 7/ in (2), 
 
 By Axiom i. 
 Transposing and uniting terms. 
 
 Substituting the value of y in (4), 
 
 x+p= 16 
 
 (I) 
 
 x-y= 4 
 
 (2) 
 
 X = it-y 
 
 (3) 
 
 x= A+y 
 
 (4) 
 
 4+y= it—y 
 
 (5) 
 
 2y= 12 
 
 (6) 
 
 .-. y= 6 
 
 
 aj= 10 
 
 
 In (5) it will be seen we have a new equation which contains 
 only one unknown quantity. This equation is reduced in the usual 
 way. Hence, the 
 
 EuLE. — I. From each equation find the value of the quan- 
 tity to he eliminated in terms of the other quantities. 
 
 II. Form a new equation from these equal values, and 
 reduce it ly the preceding rules. 
 
 Note. — This rule depends upon the axiom, that things which are 
 equal to the same thing are equal to each other, (Ax. i.) 
 
 For convenience of reference, the equations are numbered (i), 
 
 (2), (3), (4), etc. 
 
 • 
 
 221. What is elimination? Name the methods. 222. How eliminate an unknown 
 ttuantity by comparison ? Note. Upon what principle does this rule depend ? 
 
 * From the Latin eliminare, to cast out. 
 
114 SIMPLE EQUATIONS. 
 
 2. Given x -\- y = 12, and x—y-ir^ z= 8, to find x and y 
 
 3. Given 3X-]-2y = 48, and 2x—$y = 6, to find x and y. 
 
 4. Given x-\-y = 20, and 2:^+3?/ = 42, to find x and y. 
 
 5. Given 4^+32/ = i3> and 3^^ + 21/ = 9, to find x and ?/. 
 
 6. Given 30; +2?/ =118, and a; +5?/= 191, to find a; and ^^'. 
 
 7. Given 4x-\-^y = 22, and 72:4-32^=27, to find x and 2/. 
 
 CASE II. 
 223. To Eliminate an Unknown Quantity by Substitution. 
 
 8. Given x-{-2y = 10, and ^x-^2y z= 18, to find x and y. 
 
 Solution. — B7 the problem, x+2i/=io (i) 
 
 " 3a;+22/ = i8 (2) 
 
 Transposing 2y in (i), . x = 10— 2y (3) 
 
 Substituting tlie value of x in (2), 30—4^ = 18 (4) 
 
 Transposing and uniting terms (Art. 211), 42/ = 12 (5) 
 
 '- y= 3 
 Substituting the value of 2^ in (i), a; = 4 
 
 5^" For convenience, we first find the value of the letter which is 
 least involved. Hence, the 
 
 Rule. — I. From one of the equations find the value of the 
 unknown quantity to he eliminated, in terms of the other 
 quantities. 
 
 II. Substitute this value for the same quantity in the 
 other equation, and reduce it as before. 
 
 Notes. — i. This method of elimination depends on Ax. i. 
 2. The given equations should be cleared of fractions before com- 
 mencing the elimination. 
 
 9. Given x + ^y = 19, and $x — 2y = 10, to find x and y. 
 
 10. Given - + ^ = 7, and - -|- ^ = 8, to find x and 11. 
 
 23' 3^2' ^ 
 
 11. Given 2x-{-^y =z 28, and 30;+ 2^ = 27, to find x and y. 
 
 12. Given 4X-{-y = 43, and 5^+2^ = 56, to find x and y. 
 
 13. Given s^+^ = 7^> and 5?/ + 32 = 'jx, to find x and y. 
 
 14. Given 4X-]-sy= 22, and yx+^y = 2^, to find x and y. 
 
 223. How eliminate an unknown quantity by substitution? If^ote Upon what 
 principle docs this method depend? 
 
TWO UNKNOWN QUANTITIES. 115 
 
 CASE III. 
 
 224. To Eliminate an Unknown Quantity by Addition or 
 Subtraction, 
 
 15. Given 4a; + 3^ =18, and 5a:— 2?/=: 11, to find x and y. 
 
 Solution. — By tlie problem, 4a; + 3^=18 (i) 
 
 " " e,x—2y~ II (2} 
 
 Multiplying (i) by 2, the coef. of y in (2), 8a5+ 6y = 36 (3) 
 
 Multiplying (2) by 3, the coef. of y in (i), isx—6y = 33 (4) 
 
 Adding (3) and (4) cancels 6y, 23a; = 69 (5) 
 
 ,', X =3 
 Substituting the value of a; in (i), 12 + 3^ = 18 
 
 y= 2 
 
 ^° In the preceding solution, y is eliminated by addition, 
 
 16. Given 6a: + 5?/ =2 8, and 8a; + 3?/— 30, to find x and y. 
 
 Solution. — By the problem, 6a; + SV = 28 (1) 
 
 " '* 8a; + 3y= 30 (2) 
 
 Multiplying (i) by 8, the coef. of a; in (2), 48a; + 402/ = 224 (3) 
 
 Multiplying (2) by 6, the coef. of x in (i), 48a; + i8y = 180 (4) 
 
 Subtracting (4) from (3), 22^ = 44 
 
 .'. y= 2 
 
 Substituting the value of y in (2) 8a;+6 = 30 
 
 .'. x= 3 
 
 B^ In this solution, x is eliminated by subtraction. Hence, the 
 
 EuLE. — I. Select the letter to he eliminated; then multiply 
 or divide one or doth equations hy such a number as ivill 
 make the coefficients of this letter the same in loth, (Ax. 4, 5.) 
 
 II. If the signs of these coefficients are alilce, subtract one 
 equation from the other ; if unlihe, add the two equations 
 together. (Ax. 2, 3.) 
 
 Notes. — i. The object of multiplying or dividing the equations is to 
 equalize the coefficients of the letter to be eliminated. 
 
 2. If the coefficients of the letter to be eliminated are prime num- 
 bers, or ^rime to each other, multiply each equation by the coefficient 
 of this letter in the other equation, as in Ex. 15. 
 
 224. What is the rule for elimination by addition or snbtraction ? Notes. — i. The 
 object of multiplying or dividing the equalion ? 2. If coefficients are prime ? 
 
116 SIMPLE EQUATIONS. 
 
 3. If not prime, divide the I. c, in* of the coefficients of the lett^i 
 to be eliminated by each of these coefficients, and the respective 
 quotients will be the multipliers of the corresponding equations. 
 Thus, the I, c, in. of 6 and 8, the coefficients of a; in Ex. 16, is 24; 
 hence, the multipliers would be 3 and 4. 
 
 4. If the coefficients of the letter to be eliminated have common 
 factors, the operation is shortened by cancelling these factors before 
 the multiplication is performed. Thus, by cancelling the common 
 factor 2 from 6 and 8, the coefficients of x in the last example, they 
 become 3 and 4, and the labor of finding the I, c, in. is avoided. 
 
 17. Given 3a; +4?/ =2 9, and 'jx-\-iiy=j6,to find a: and y. 
 
 18. Given 9a;— 4^=8, and 13^-1- 72/== loi, to find ir and ^. 
 
 19. Given ^x—'jy='jy and 12a: + 5^/^94, to find x and y. 
 
 20. Given 3^^+21/= 118, and 0:4-5^=191, to find xandy. 
 
 21. Given 4a; -{-52/= 22, and 7a; + 32/= 27, to find x and y. 
 
 Rem. — The preceding methods of elimination are applicable to all 
 simultaneous simple equations containing two unknown quantities, 
 and either may be employed at the pleasure of the learner. 
 
 The first method has the merit of clearness, but often gives rise to 
 frax^tions. 
 
 The second is convenient when the coefficient of one of the unknown 
 quantities is i ; if more than i, it is liable to produce /7•ac^^07^«. 
 
 The third never gives rise to fractions, and, in general, is the most 
 simple and expeditious. 
 
 EXAMPLES. 
 
 Find the values of x and y in the following equations : 
 
 1. 2a; + 3^= 23, 5. 5a; + 7^= 43, 
 5a; — 2^= 10. \ix -\- 9?/= 69. 
 
 2. 4a; + y— Z^, 6. 8a; — 2iy= n, 
 4y+ x= 16. 6a: + 357/= 177. 
 
 3. 30a; + 4oy = 270, 7. 21?^ + 20a; = 165, 
 50a; 4- 30^ = 340. 772^ — 30^ = 295. 
 
 4. 2a; + jyzzz 34, 8. I la; — loy = 14, 
 SX+ gy= 51. s^+ iy= 41. 
 
 Notes. — 3. If not prime, how proceed ? 4. If the coefficients have comuion feo- 
 tors, how shorten the operation ? 
 
TWO UNKNOWK QUANTITIES. ll? 
 
 lO. 
 
 II. 
 
 12. 
 
 13- 
 
 14. 
 
 6y~ 
 
 2a; = 208, 
 
 15. 
 
 Sx + 
 
 y = 
 
 '42, 
 
 loa; — 
 
 42/ = 156. 
 
 
 2X + 
 
 42/ = 
 
 18. 
 
 4X + 
 
 Sy= 22, 
 
 16. 
 
 2X + 
 
 42/ = 
 
 20, 
 
 5^- 
 
 72/= 6. 
 
 
 4X + 
 
 52/ = 
 
 28. 
 
 SX- 
 
 5«/= i3> 
 
 17. 
 
 4X-\- 
 
 32/ = 
 
 50, 
 
 2X + 
 
 7^= 81. 
 
 
 3«- 
 
 32/ = 
 
 6c 
 
 5^- 
 
 72/= 33j 
 
 18. 
 
 3^ + 
 
 52/ = 
 
 57, 
 
 iia: + 
 
 121/ = 100. 
 
 
 5« + 
 
 32/ = 
 
 47. 
 
 i* 
 
 |=.8, 
 
 19. 
 
 ; + 
 
 2( _ 
 3 
 
 7> 
 
 re 
 2 
 
 ^ = 21. 
 
 4 
 
 
 i- 
 
 4 
 
 5. 
 
 162; + 
 
 17?/ = 500, 
 
 20. 
 
 2a; + 
 
 2/ = 
 
 50, 
 
 17a; — 
 
 3«/= no. 
 
 
 i* 
 
 ^ _ 
 
 7 
 
 5. 
 
 
 PROBLEMS. 
 
 
 
 
 1. Required two numbers whose sum is 70, and whose 
 difference is 16. 
 
 2. A boy buys 8 lemons and 4 oranges for 56 cents; and", 
 afterwards 3 lemons and 8 oranges for 60 cents. What did 
 he pay for each ? 
 
 3. At a certain election, 375 persons voted for two candi- 
 dates, and the candidate chosen had a majority of 91. How 
 many voted for each ? 
 
 4. Divide the number 75 into two such parts that three 
 times the greater may exceed seven times the less by 15. 
 
 5. A farmer sells nine horses and seven cows for I1200; 
 and six horses and thirteen cows for an equal amount. 
 What was the price of each ? 
 
 6. From a company of ladies and gentlemen, 15 ladies 
 retire; there are then left two gentlemen to each lady. 
 After which 45 gentlemen depart, when there are left five 
 ladies to each gentleman. How many were there of each at 
 first? 
 
118 SIMPLE EQUATIONS. 
 
 7. Find two numbers, such that the sum of five times the 
 first and twice the second is 19; and the difference between 
 seven times the first and six times the second is 9. 
 
 8. Two opposing armies number together 21,110 men; 
 and twice the number of the greater army added to three 
 times that of the less is 52,219. How many men in each 
 army? 
 
 9. A certain number is expressed by two digits. The 
 sum of these digits is 11 ; and if 13 be added to the first 
 digit, the sum will be three times the second. What is the 
 number ? 
 
 10. A and B possess together I570. If A's share were 
 three times and B's five times as great as each really is, then 
 both would have I2350. How much has each ? 
 
 11. If I be added to the numerator of a fraction, its value 
 is I ; and if i be added to the denominator, its value is ^. 
 What is the fraction. 
 
 12. A owes $1200; B, $2550. But neither has enough to 
 pay his debts. Said A to B, Lend me | of your money, 
 and I shall be enabled to pay my debts. B answered, I 
 can discharge my debts, if j^^ou lend me J of yours. What 
 sum has each ? 
 
 13. Find two numbers whose difference is 14, and whose 
 sum is 48. 
 
 14. A house and garden cost I8500, and the price of the 
 garden is ^ the price of the house. Find the price of each. 
 
 15. Divide 50 into two such parts that f of one part, 
 added to f of the other, shall be 40. 
 
 16. Divide I1280 between A and B, so that seven times 
 A's share shall equal nine times B's share. 
 
 17. The ages of two men differ by 10 years ; 15 years ago, 
 the elder was twice as old as the younger. Find the age of 
 each. 
 
 18. A man owns two horses and a saddle. If the saddle, 
 worth $50, be put on the first horse, the value of the two 
 is double that of the second liorse ; but if the saddle be put 
 
TWO UNKNOWN QUANTITIES. 119 
 
 on the second horse, the value of the two is $15 less than 
 that of the first horse. Kequired the value of each horse. 
 
 19. A war-steamer in chase of a ship 20 miles distant, 
 goes 8 miles while the ship sails 7. How far will each go 
 before the steamer overtakes the ship ? 
 
 20. There are two numbers, such that ^ the greater added 
 to -J the less is 13 ; and if I the less be taken from | the 
 greater, the remainder is nothing. Find the numbers. 
 
 21. The mast of a ship is broken in a gale. One-third of 
 the part left, added to ^ of the part carried away, equals 
 28 feet; and five times the former part diminished by 
 6 times the latter equals 12 feet. What was the height of 
 the mast ? 
 
 22. A lady writes a poem of half as many verses less two 
 as she is years old ; and if to the number of her years that 
 of her verses be added, the sum is 43. How old is she ? 
 How many verses in the poem ? 
 
 23. What numbers are those whose difference is 20, and 
 the quotient of the greater divided by the less is 3 ? 
 
 24. A man buys oxen at $65 and colts at $25 per head, 
 and spends $720 ; if he had bought as many oxen as colts, 
 and vice versa, he would have spent 1 1440. How many of 
 each did he purchase ? 
 
 25. There is a certain number, to the sum of whose 
 digits if you add 7, the result will be 3 times the left-hand 
 digit ; and if from the number itself you subtract 18, the 
 digits will be inverted. Find the number. 
 
 26. A and B have jointly $9800. A invests the sixth part 
 of his property in business, and B the fifth part of his, and 
 each has then the same sum remaining. What is the entire 
 capital of each ? 
 
 27. A purse holds six guineas and nineteen silver dollars. 
 Now five guineas and four dollars fill |-J of it. How many 
 will it hold of each ? 
 
 28. The sum of two numbers is a, and the greater is n 
 times the less. What are the numbers ? 
 
120 SIMPLE EQUATION". 
 
 THREE OR MORE UNKNOWN QUANTITIES. 
 
 225. The preceding methods of elimination of two 
 unknown quantities are applicable to equations containing 
 three or more unknown quantities. (Arts. 222-224.) 
 
 226. To Solve Equations containing three or more Unknown 
 
 Quantities. 
 
 I. Given 3a; + 2^ — 52? = 8, 2X -{- $y -\- 4Z = 16, and 
 $x — 6y ■}- ^z = 6, to find x, y, and z. 
 
 -.UTlON.— By the problem. 
 
 30!+ 2y- sz = 
 
 8 
 
 (I) 
 
 K ft 
 
 2X+ 3y+ 4^ = 
 
 16 
 
 (2) 
 
 tt U 
 
 SX— 6y+ 3Z = 
 
 6 
 
 (3) 
 
 Multiplying (i) by 2, 
 
 6x+ 4y—ioz = 
 
 16 
 
 (4) 
 
 (2) by 3, 
 
 6x+ gy+i2Z = 
 
 48 
 
 (5) 
 
 Subtracting (4) from (5), 
 
 5.?/ + 22S = 
 
 32 
 
 (6) 
 
 Multiplying (2) by 5, 
 
 ioa; + 1 5^ + 202 = 
 
 80 
 
 (7) 
 
 (3) by 2, 
 
 1005— i2y+ 6s = 
 
 12 
 
 (8) 
 
 Subtracting (8) from (7), 
 
 27y + i4Z = 
 
 68 
 
 (9) 
 
 Multiplying (6) by 27, 
 
 1352^ + 5942 = 
 
 864 
 
 (10) 
 
 (9) by 5, 
 
 1352/ + 70s = 
 
 340 
 
 (II) 
 
 Subtracting (11) from (10), 
 
 5242 = 
 
 524 
 
 T 
 
 (12) 
 
 Substituting the value of 2 in (6), y = 
 
 1 
 
 2 
 
 
 Substituting the value of y 
 
 and z in (2), x = 
 
 3 
 
 
 Ana. x = 3, y = 2, 2=1. 
 
 Hence, the 
 
 
 
 Rule. — I. From thegiveti equations eliminate one unknown 
 quantity, by combining one equation with another, 
 
 II. From the resulting equations eli?ninate another unknown 
 quantity in a similar manner. Continue the operation ufitil 
 a single equation is obtained, with but one unknown quan- 
 tity, and reduce this by the preceding rules. 
 
 Note. — The letter having the smallest coefiScients should be elimi- 
 nated first ; and if each letter is not found in all the given equations, 
 begin witli that which is in the least number of the equations. 
 
 226. What iB the rale for solving equation? having three or more unknown quan- 
 tities ? Note. Which letter should be eliminated flret ? 
 
THEEE OR MORE UNKNOWN QUANTITIES. 121 
 
 Eeduce the following equations : 
 
 2. 5^ — 3«/ + 2^ = 28, 5. 5ic + 2?/ 4- 42; = 46, 
 SX + 2y — 4Z = 15, 32; + 2?/ + z = 23, 
 3^ + 42; — a; = 24. loa; + 5?/ + 4^ = 75. 
 
 3. 2a; 4- 5«/ — 32^ = 4, 6. a; + ?/ + 2; z= 53, 
 4^ — 3y + 22; = 9, a; + 2y 4- 32; = 105, 
 5x + 6y—2z=ziS. ic + 32^ + 40 = 134. 
 
 4. 2a; -\-sy — 4Z = 20, 7. 33: + 42; = 57, 
 
 ^—2?/ + 3^= 6, 2y— z = ii, 
 
 SX — 2y-}-sz = 26. . 5a: + 3?/ =: 65. 
 
 234 345 450 
 
 9. Required the value of w, x, y, and z in the following 
 
 equations : 
 
 w+x-\-y-\-z=i4 (i) 
 
 2W 4- a; + 2/ — 2? = 6 (2) 
 
 2w -\- z^ — y + z = 14 (3) 
 
 «^ — ^ + 32/ + 4^ = 31 (4) 
 
 SOLUTION. 
 
 Adding (i) and (2), 2)'^+ 2X-\- 2y = 20 (5) 
 
 " (2) and (3), 410+ 4X =20 (6) 
 
 Multiply (3) by 4, Sw+i2X— 4^+42= 56 (7) 
 
 Subtract (4) from (7), 7z^+i3a;— 7^ = 25 (8) 
 Multiply (5) by 7, 2it^+i4a; + i4y =140 (9) 
 
 " (8) by 2, 14^^ + 26a!— i4y = 50 (10) 
 
 Add (9) and (10), 35?5 + 4ac =190 (11) 
 
 Multiply (6) by 10, 40^0 + 401? =200 (12) 
 
 Subtract (11) from (12), 5«/J = 10 
 
 .'. w = 2 
 Substituting value of w in (6), 8 + 4aj = 20 
 
 .'. «— 3 
 Substituting value of w and a? in (5), etc., 
 y = 4, and 2 = 5. 
 6 
 
L;i2 
 
 SIMPLE EQUATIONS. 
 
 227. The solution of equations containing many unkno\;\'t] 
 quantities may often be shortened by substituting a single 
 letter for several. 
 
 lo. Eequired the value of w, x, y, 
 and z in the adjoining equations. 
 
 io-^x-\-y= 13 
 w-\-x-\-z = 17 
 iv-\-y-i-z = 18 
 x+y-{-z = 2i 
 
 (I) 
 (2) 
 
 IS) 
 (4) 
 
 Note. — Substituting s for the sum of the four quantities, we have, 
 
 8 = w + x+y + z. 
 
 Equatien (i) contains all the letters but z, s—z = 13 (5) 
 
 (2) « « " p, s-y = 17 (6) 
 
 (3) " " '* X, s-x =18 (7) 
 
 (4) " " " w, s-w = 21 (8) 
 
 Adding the last four equa- ) 
 
 tions together. 
 
 or 4S—{z + y-{-x + w) j^ = 69 
 or 4s— 8 
 
 That is. 
 
 3s = 6g 
 
 .-. 8= 23 
 
 Substituting 23 for s in each of the four equations, we have, 
 w = 2, x= s, y = (>, z= 10. 
 
 II. Eequired the value of v, 
 Wf X, y, and z, in the adjoining 
 equations. 
 
 (9) 
 (10) 
 
 V -\-w-\-x-\-y := 10 (1) 
 
 V -\-W-\-X-\-Z = II (2) 
 
 V -\-w-\-y-\-z=i2 (3? 
 
 V +x -\-yi-z = is (4) 
 
 lw + x+y-{-z = i4 (5) 
 
 Note. — Adding these equations, 4v + 4w + 4X + 4y+4Z = 60 (6) 
 
 Dividing (6) by 4, v+ w+ x+ y+ z = 15 (7; 
 Subtracting each equation from (7), we have, 
 
 s = 5, y = 4, x = 3, w = 2, and v = 1. 
 
 I£. W + X -\- Z z= 10, 
 
 X -}- y + z = 12, 
 w +x -\-y= 9, 
 w -\- y -\- z = II. 
 
 13- 
 
 I I 
 
 X y 
 
 5 
 6' 
 
 y z 12' 
 
 X z 4 
 
 (See Appendix, p. 286.) 
 
THREE OR MORE UNKNOWN QUANTITIES. 123 
 
 PROBLEMS. 
 
 1. A man has 3 sons ; the sum of the ages of the first 
 and second is 27, that of the first and third is 29, and of 
 the second and third is 32. What is the age of each ? 
 
 2. A butcher bought of one man 7 calves and 13 sheep 
 for $205 ; of a second, 14 calves and 5 lambs for I300; and 
 of a third, 12 sheep and 20 lambs for $140, at the same 
 rates. What was the price of each ? 
 
 3. The sum of the first and second of three numbers is 
 13, that of the first and third 16, ana that of the second 
 and third 19. What are the numbers ? 
 
 4. In three battalions there are 1905 men: J the first 
 with J in the second, is 60 less than in the third ; | of the 
 third with | the first, is 165 less than the second. How 
 many are in each ? 
 
 5. A grocer has three kinds of tea: 12 lbs. of the firsts 
 13 lbs. of the second, and 14 lbs. of the third are together 
 worth $25 ; 10 lbs. of the first, 17 lbs. of the second, and 
 II lbs. of the third are together worth $24; 6 lbs. of the 
 first, 12 lbs. of the second, and 6 lbs. of the third are together 
 worth 1 1 5. What is the value of a pound of each ? 
 
 6. Two pipes, A and B, will fill a cistern in 70 minutes, 
 A and will fill it in 84 minutes, and B and C in 140 min. 
 How long will it take each to fill the cistern ? 
 
 7. Divide $90 into 4 such parts, that the first increased 
 by 2, the second diminished by 2, the third multiplied by 2, 
 and the fourth divided by 2, shall all be equal. 
 
 8. The sum of the distances which A, B, and C have 
 traveled is 62 miles; As distance is equal to 4 times C's, 
 added to twice B's; and twice A's added to 3 times B's, is 
 equal to 17 times C's. What are the respective distances ? 
 
 9. A, B, and C purchase a horse for $100. The payment 
 would require the whole of A's money, with half of B's ; or 
 the whole of B's with i of C's; or the whole of C's with J 
 of A's. How much money has each ? 
 
OHAPTEE XI. 
 GENERALIZATION. 
 
 228. Generalisation is the process of finding a 
 formula, or general rule, by which all the problems x)f a 
 class may be solved. 
 
 229. A Problem is generalized when stated in 
 general terms which embrace all examples of its class. 
 
 230. In all General I^roMems the quantities are 
 expressed by letters, 
 
 I. A marketman has 75 turkeys; if his turkeys are mul- 
 tiplied by the number of his chickens, the result is 225. 
 How many chickens has he ? 
 
 Note. — This problem maybe stated in the following general terms : 
 
 231. The Product of two Factors and one of the Factors 
 
 being given, to Find the other Factor.* 
 
 SuGGESTiON.^If & product of two factors is divided by one of them, 
 it is evident the quotient must be the other factor. Hence, substituting 
 a for the product, h for the given factor, we have the following 
 
 General Solution. — ^Let x = the required factor. 
 
 By the conditions, xxb,OThx = a,the product. Hence, the 
 
 FOKMULA. X = ^' 
 
 h 
 
 Translating this Formula into common language, we have the 
 following 
 
 EuLE. — Divide the product ly the given factor ; the quo- 
 tient is the factor required. 
 
 228. What is generalization ? 129. When is a problem generalized ? 230. Hov» 
 are quantities expressed in general problems ? 
 
 * New Practical Arithmetic, Article 93. 
 
GENERALIZATIOIS'. 125 
 
 Generalize the next two problems : 
 
 2. A rectangular field contains 480 square rods, and the 
 length of one side is 16 rods. What is the length of the 
 other side ? 
 
 3. Divide 576 into two such factors that one shall be 48. 
 
 4. The product of A, B, and C's ages is 61,320 years; A 
 Is 30 years, B 40. What is the age of C ? 
 
 Note. — The items here given may be generalized as follows : 
 
 232. The Product of three Factors and two of them being 
 given, to Find the other Factor. 
 
 Suggestion. —Substituting a for the product, h for one factor, and 
 e for the other, we have the 
 
 General Solution. — Let x = the required factor. 
 
 By the conditions, xxbxc, or bcx = a, the product. 
 
 Removing the coeflBcient, we have the 
 
 FOKMULA. X = i — 
 
 be 
 
 Rule. — Divide the given product hy the product of the 
 given factors ; the quotient is the required factor, 
 
 5. The contents of a rectangular block of marble are 504 
 cubic feet ; its length is 9 feet, and its breadth 8 feet. What 
 is its height ? 
 
 6. The product of 3 numbers is 62,730, and two of its 
 factors are 41 and 45. Required the other factor. 
 
 7. The amount paid for two horses was $392, and the 
 difference in their prices was 1 1 8. What was the price of 
 each ? 
 
 Note. — From the items given, this problem may be generalized 
 as follows : 
 
 231. When the product of two factors and one of the factors are given, how find 
 the other factor ? 232. When the product of three fiactors and two of them are 
 given, how And the other foctor ? 
 
126 ge:n^eralization". 
 
 233. The Sum and Difference of two Quantities being given, 
 to Find the Quantities. 
 
 Suggestion. — Since the sum of two quantities equals the greater 
 filns the less ; and the less plus the difference equals the greater ; 
 it follows that the sum plus the difference equals twice the greater. 
 Substituting .s for the sum, d for the difference, g for the greater, 
 9xid I for the less, we have the following 
 
 General Solution. — Let ^ = the greater number, 
 
 and 1= '* less " 
 
 Adding, gr-f-? = «, the sum. 
 
 Subtracting, g—l — d, the difference. 
 
 Adding sum and difference, 2g = s+d 
 
 8A-d 
 
 Removing coefficient, g = , greater. 
 
 Subtracting difference from sum, 2I = s—d 
 
 g ^ 
 
 Removing coefficient, I = — — , less. Hence, the 
 
 Formulas. 
 
 This problem may be solved by one unknown quantity. 
 
 7 — ^ — ^ 
 
 2 
 
 FORMATION OF RULES. 
 
 234. Many of the more important rules of Arithmetic 
 are formed by translating Algehraic Formulas into common 
 language. Thus, from the translation of the two preceding 
 formulas into common language, we have, for all problems 
 of this class, the following general 
 
 Rule. — I. To find the greater, add half the sum to half 
 the difference. 
 
 II. To find the less, suUract half the difference from half 
 the sum. 
 
 8. Divide I1575 between A and B in such a manner that 
 A may have $347 more than B. What will each receive ? 
 
 233. When the stun and difference of two quantities are given, how find the 
 quantities ? 234. Give the rule derived from the last two formulas. 
 
GENERALIZATION". 137 
 
 9. At an election there were 2150 vot^s cast for two 
 persons ; the majority of the successful candidate was 346. 
 How many votes did each receive ? 
 
 10. If B can do a piece of work in 8 days, and m 12 
 days, how long will it take both to do it ? 
 
 Note. — Regarding the work to be done as a unit or i, the problem 
 may be thus generalized : 
 
 235. The Time being given in which each of two Forces can 
 produce a given Result, to Find the Time required by the united 
 Forces to produce it. 
 
 Suggestion. — Since B can do the work in 8 days, he can do i eighth 
 of it in I day, and C can do i twelfth of it in i day. Substituting a 
 for 8 days, and h for 12 days, we have the 
 
 General Solution. — Let x = the time required. 
 
 Dividing x by a, we have - = part done by B, 
 
 a 
 
 X 
 
 ** X by 6, we have, h~ ** ** ^' 
 
 By Axiom q, - + - = i, the work done. 
 
 a J) 
 
 Clearing of fractions, bx+ax = db 
 
 Uniting the terms, (a + b)x = ab,B and C*s time. 
 
 Removing tlie coefficient, we have the 
 
 Formula. x = i-- 
 
 HvLiE.— Divide the product of the numbers denoting the 
 time required ly each force, hy the su7n of these numbers ; 
 the quotient is the time required by the united forces. 
 
 11. A cistern has two pipes; the first will fill it in 
 9 hours, the second in 15 hours. In what time will both 
 fill it, running together ? 
 
 235. The time beiug given in which two or more forces can produce a result, how 
 find the time reauired for the united forces to produce it ? 
 
128 GEN^EKALIZ ATIOl!f. 
 
 1 2. A can plant a field in 40 hours, and B can plant it in 
 50 hours. How long will it take both to plant it, if they 
 work together ? 
 
 236. In Generalizing Problems relating to Per- 
 centage, there is an advantage in representing the quantities, 
 whether known or unknown, by the initials of the elements 
 or factors which enter into the calculations. 
 
 Note. — The elements 0Yfact(y,'s in percentage are, 
 ist. The Base, or number on wliicli percentage is calculated. 
 2d. The Rate per cent, which shows how many hundredths of 
 the base are taken. 
 
 3d. The Per^centage, or portion of the base indicated by the rate. 
 4th. The Amount, or the hsiseplus or minus the percentage. Thus, 
 Let b = the base. p = the percentage. 
 
 r = the rate per cent, a = the amount. 
 
 13. A man bought a lot of goods for $748, and sold them 
 at 9 per cent above cost. How much did he make ? 
 
 Note. — The items in this problem may be generalized as follows : 
 
 237. The Base and Rate being given, to Find the 
 
 JPerceutage,* 
 
 Suggestion. — Per cent signifies hundredths; hence, any given 
 per cent of a quantity denotes so many hundredths of that quantity. 
 But finding a fractional part is the same as multiplying the quantity 
 by the given fraction. Substituting h for the cost or base, and /• for 
 the number denoting the rate per cent, we have the 
 
 General Solution. — Let p = the percentage. 
 
 Multiplying the base by the rate, br = p. Hence, the 
 
 FoKMUiA. p = hr. 
 
 Rule. — Multiply the base hy the rate per cent ; the product 
 is the percentage. Hence, 
 
 Note. — Percentage is a product, the factors of which are the base 
 and rate. 
 
 236. Note. What are the elements or factors in percentage? 237. When base 
 and rate are given, how find the percentage ? 
 
 ♦ New Practical Arithmetic, Arts. 336 — 340. 
 
GENERALIZATION^. 129 
 
 14. The population of a certain city in 1870 was 45,385 ; 
 in 1875 it was found to have increased 20 per cent. What 
 was the percentage of increase ? 
 
 15. Find 37^ per cent on $2763. 
 
 16. A western farmer raised 1587 bushels of wheat, and 
 sold 37 per cent of it. How many bushels did he sell ? 
 
 17. A teacher's salary of $2700 a year was increased $336. 
 What per cent was the increase ? 
 
 Note. — The data of this problem may be generalized as follows : 
 
 238. The Base and Percentage being given, to Find the Rate, 
 
 Suggestion. — Percentage, we have seen, is a product, and the hase 
 is one of its factors (Art. 237, note) ; tlierefore, we have the product 
 and one factor given, to find the other factor, (Art. 231.) Substituting 
 2> for the product or percentage, and b for the salary or haae, we 
 have the 
 
 General Solution.— Let r = the required rate. 
 Then (Art. 231), p-r-h = r. Hence, the 
 
 Formula. r = ^' 
 o 
 
 Rule. — Divide the percentage ly the lase; the quotient 
 is the rate. 
 
 18. From a hogshead of molasses, 25.2 gallons leaked out. 
 What per cent was the leakage ? 
 
 19. A steamship having 485 passengers was wrecked, and 
 291 of them lost. What per cent were lost? 
 
 20. A man gained $750 by a speculation, which was* 
 25 per cent, of the money invested. What sum did hb 
 invest ? 
 
 Note. — The particular statement of this problem may be tran». 
 formed into the following general proposition : 
 
 238. When the hase and percentage are given, how find the rate? 
 
130 GENERALIZ ATIOIT. 
 
 239. The PeVcentage and Rate being given, to Find the Base, 
 
 Suggestion. — We have the product and one of its factors given, 
 to find the otJier factor. (Art. 237, note.) Substituting p for the 
 percentage, and r for the rate per cent gained, we have the 
 
 General Solution. — Let 6 = the base. 
 
 Then (Art. 237), _p-r-7' = 6. Hence, the 
 
 Formula. b = —' 
 r 
 
 KuLE. — Divide the percentage by the rate, and the quotient 
 
 is the base. 
 
 21. A paid a tax of I750, which was 2 per cent of his 
 property. How much was he worth ? 
 
 22. A merchant saves 8 per cent of his net income, and 
 lays up $2500 a year. What is his income ? 
 
 23. xit the commencement of business, B and C were 
 each worth $2500. The first year B added 8 per cent to 
 his capital, and C lost 8 per cent of his. What amount 
 was each then worth ? 
 
 Note. — The items of this problem may be generalized thus : 
 
 240. The Base and Rate being given, to Find the Amount, 
 
 Suggestion. — Since B laid up 8 per cent., he was worth his origi- 
 nal stock plus 8 per cent of it. But his stock was 100 per cent, or once 
 itself; and 100 per cent, + .08 = 108 per cent or 1.08 times his stock. 
 
 Again, since C lost 8 per cent, he was worth his original stock 
 minus 8 per cent of it. Now 100 per cent minus 8 per cent equals 
 100 per cent — .08 — 92 per cent, or .92 times his capital. Substituting 
 b for the base or capital of each, and r for the number denoting the 
 rate per cent of the gain or loss, we have the 
 
 General Solution.— Let a = the amount. 
 
 Then will b(i+r) = a, B's amount. 
 
 And b (i—r) = a, C's amount. 
 
 Combining these two results, we have the 
 
 Formula. a = b{i ± /•)• 
 KuLE. — Multiply the base by i i the rate, as the case may 
 require, and the result ivill be the amount. 
 
 239. When percentage and rate are given, bow find the base ? 240, How find the 
 amount when the base and rate are given ? 
 
GENEKALIZATION. 13i 
 
 Note. — When, from the nature of the problem, the amount is to he 
 greater than the base, the multiplier is i plus the rate ; when less, the 
 multiplier is i .ninus the rate. 
 
 24. A man bought a flock of sheep for $4500, and sold it 
 25 per cent above the cost. What amount did he get for it ? 
 
 25. A man owned 2750 acres of land, and sold 33 per 
 cent of it. What amount did he have left ? 
 
 241. The elements or factors which enter into computa- 
 tions of interest are the principal^ rate, time, interest, and 
 amount. Thus, 
 
 Let p = the principal, or money lent. 
 ** r = the interest of $1 for i year, at the given rate. 
 ** t = the time in years. 
 " i = the interest, or the percentage. 
 " a ~ the amount, or the sum of principal and interest. 
 
 26. What IS the interest of $465 for 2 years, at 6 per cent? 
 Note, — The data of this problem may be stated in the following 
 
 general proposition : 
 
 242. The Principal, the Rate, and Time being given, to Find 
 
 the Interest, 
 
 General Solution.— Since r is the interest of $1 for i year, 
 pxr must be the interest of p dollars for i year ; therefore, pr x t 
 must be the interest of p dollars for t years. Hence, the 
 
 Formula. i = prt. 
 
 EuLE. — Multiply the principal hy the interest of %i for 
 the given time, and the result is the interest. 
 
 27. What is the interest of I1586 for i yr. and 6 m., at 
 8 per cent ? 
 
 28. What is the int. of $3580 for 5 years, at 7 per cent ? 
 
 29. What is the amt. of $364 for 3 years, at 5 per cent ? 
 
 Note.— This problem may be stated in the following general terms '. 
 
 Note. When the amount is j^reater or less than the base, what is the multiplier? 
 
 241. What are the elements or factors which enter into computations of interest? 
 
 242. When the principal, rate, and time are given, how find the interest ? 
 
132 GEN^ERALIZATION. 
 
 243. The Principal, Rate, and Time being given, to Find the 
 
 Amount, 
 
 General Solution. — Reasoning as in the preceding article, 
 
 the interest = prt. 
 But the amount is the sum of the principal and interest. 
 /. p+prt = a. Hence, the 
 
 FoBMULA. a = p-{- prt. 
 
 Rule. — Add the interest to the principal, and the result is 
 the amount. 
 
 30. Find the amount of $4375 for 2 years and 6 months, 
 at 8 per cent. 
 
 31. Find the amt. of $2863.60 for 5 years, at 7 per cent. 
 
 244. The delation between the four elements in the 
 
 Formula, a = p ^ prt, 
 is such, that if any three of them are given, the fourth may 
 be readily found. (Art. 243.) 
 
 245. The Amount, the Rate, and Time being given, to Find 
 
 the Principal. 
 
 Transposing the members and factoring, we have the 
 
 Formula. p 
 
 1 -\- rt 
 
 32. What principal will amount to I1500 in 2 years, at 
 6 per cent ? 
 
 2,3' What sum must be invested at 7 per cent to amount 
 to $300 in 5 years ? 
 
 246. The Amount, the Principal, and the Rate being given, 
 to Find the Time. 
 
 Transposing 2> and dividing hy pi', (Art. 244), we have the 
 
 Formula. t = ^' 
 
 pr 
 
 34. In what time will $3500, at 6 per cent, yield $525 
 
 interest ? 
 
 243. The amount f 244. What is the relation between the four elements in the 
 preceding formula. 245. When the amount, rate, and time are given, state the 
 formula, 246. When the amount, principal, and rate* are given, state the formula. 
 
GENERALIZATIOK. 
 
 133 
 
 247. The Hour and Minute Hands of a Clock being together 
 at I2!VI., to Find the Time of their Cottjuncfioii between any 
 two Subsequent Hours. 
 
 35. The hour and minute hands of a clock are exactly 
 together at 12 o'clock. It is required to find how long 
 before they will be together again. 
 
 Analysis. — The distance around the dial of a clock is 12 hour 
 spaces. When the hour-hand arrives at I, the minute-hand has passed 
 12 hour spaces, and made an entire circuit. But since the hour-hand 
 has moved over one space, the minute-hand has gained only 11 spaces. 
 Now, if it takes the minute-hand * hour to gain 1 1 spaces, to gain 
 I space will take -^j of an hour, and to gain 12 spaces it will take 12 
 times as long, and 12 times -^j hr. = ^f hr. = Tj\ hour. Or, 
 
 Let X = the time of their conjunction. 
 
 Then 11 spaces : 12 spaces :: i hour : a; hours. 
 
 Multiplying extremes, etc., iia; = 12 
 
 Removing coefficient, x = i jJy hr., or i hr. 5y\ min. 
 
 36. When will the hour and 
 minute hand be in conjunction 
 next after 3 o'clock ? 
 
 Suggestion. — Substituting a for i^ 
 hr., the time it takes the minute-hand to 
 gain 12 spaces, h for the given number 
 of hours past 12 o'clock, t for the time of 
 conjunction, we have the following 
 
 General Solution, a x h = t, the 
 time required. Hence, the 
 
 Formula. t = ah. 
 
 Rule. — Multiply the time required to gain 12 spaces by 
 the given hour past 1 2 o^dock ; the product will de the time 
 of conjunction. 
 
 37. At what time after 6 o'clock will the hour and minute 
 hand be in conjunction ? 
 
 38. At what time between 9 and 10 o'clock ? 
 
 (See Appendix, p. 287.) 
 
 247. What is the formula for finding when the hands of a clock will be in 
 conjunction ? Translate this into a rule. 
 
CHAPTER XII. 
 INVOLUTION * 
 
 248. Involution is finding a power of a quantity. 
 
 249. A Power is the product of two or more equal 
 factors. 
 
 Thus, 3x3 = 9; axaxa = a^', 9 and a^ are powers. 
 
 250. Powers are divided into differe^it degrees ; as first, 
 second, third, fourth, etc., the name corresponding with the 
 number of times the quantity is taken as 2i factor to produce 
 the power. 
 
 251. The First Power is the quantity itself. 
 
 The Second Poiver is the product of two equal factors, 
 and is called a square. 
 
 The Third Power is the product of three equal factors, 
 and is called a cuhe, etc. 
 
 Note. — The quantity caUed the first 'power is, strictly speaking, not 
 a power, but a root. Thus, <2^ or a, is not the product of any two equal 
 factors, but is a quantity or root from which its powers arise. 
 
 252. The Index or Exponent f of a power is a, figure 
 or letter placed at the right, above the quantity. Its object 
 is to show hoto many times the quantity is taken as a factor 
 to produce the power. 
 
 Thus, «' = a, and is called the first power. 
 
 a'^ = axa, the second power, or square, 
 a^ = axaxa, the third power, or cube, 
 <35* = ax ax ax a, the fourth power, etc. 
 
 248. What is involution? 249. A power? 250. How divided? 251. The first 
 power ? Second power 1 Third ? 252. What is the index or exponent ? Its object ? 
 
 * Involution, from the Latin involvere, to roll up. 
 f Index (plural, indices), Latin indicare, to indicate. 
 Exponent, from the Latin exponere, to set forth. 
 
IKVOLUTIOK. 135 
 
 Notes. — i. The index of the first power being i, is commonly 
 omitted. 
 
 2. The expression «* ig read " a fourth," "the fourth power of a" 
 or " a raised to the fourth power ;" x^ is read, " x nt\v" or " the 
 wth power of x." 
 
 Read the following: a\ d^, x\ y^% %^, IT, cC". 
 
 253. Powers are also divided into direct and reciprocal, 
 
 254. Direct Powers are those which arise from the 
 continued multiplication of a quantity into itself. 
 
 Thus, the continued multiplication of a into itself gives the series, 
 a, a^, a', «*, «^ a^, etc. 
 
 255. Reciprocal Powers are those which arise from 
 the continued division of a unit by the direct powers of that 
 4uantity. (Art. 55.) 
 
 Thus, the continued division of a unit by the direct powers of a 
 ^ves the series, 
 
 I I I I I I 
 
 a' ~a^* ^3' "^' ¥5' ¥' 
 
 256. Reciprocal Powers are commonly denoted by 
 prefixing the sign — to the exponents of direct powers of 
 the same degree. 
 
 Thus, -^a~\ 4; = «-^ — , = a-', -^ = a"^, etc. 
 
 257. The difference in the notation of direct and recip- 
 rocal powers may be seen from the following series : 
 
 (I.) a\ a\ a\ a\ a\ i, \, -^, ^, -^„ ^, etc. 
 
 (2.) a^, aS a% a\ a\ a^, a-\ a'"^, a"^, aS a~^ etc. 
 
 Note. — The first half of each of the above expressions is a series 
 of direct powers ; the loM half, a series of reciprocal powers. 
 
 258. Negative Exponents are the same as the 
 exponents of direct powers, with the sign — prefixed to 
 them. 
 
 Note. The index i ? 253. How else are powers divided? 254. Direct powers? 
 255. Reciprocal? 256. How is a reciprocal power denoted ? 
 
136 INVOLUTION. 
 
 N0TES.-~i. This notation of reciprocal powers is derived from the 
 continued division of a series of direct powers by their root; that is, by' 
 subtracting i from the successive exponents. (Art. 113.) 
 
 2. The use of negative exponents in expressing reciprocal powers 
 avoids fractions, and therefore is convenient in calculations. 
 
 3. Direct poicers are often called positive, and reciprocal powers, 
 negative. But the student must not confound the quantities whose 
 exponents have the sign + or -, with those whose coeJicientshaiYe the 
 sign + or — . This ambiguity will be avoided, by applying the term 
 direct, to powers with, positive exponents, and reciprocal, to those with 
 negative exponents. 
 
 259. The Zero I^oiver of a quantity is one whose 
 exponent is o ; as, a° ; read, *^ the zero power of a." 
 Every quantity with the index o, is equal to a unit or i. 
 
 For, — r= «"-» = ao (Art. 113) ; but — = i ; hence, «« = i. 
 
 SIGNS OF POWERS. 
 
 260. When a quantity is positive, all its poivers are 
 positive. 
 
 Thus, axa = a"^', axaxa=: a^^ etc. 
 
 When a quantity is negative, its even powers are positive, 
 and its odd powers negative. 
 
 Thus, —a X —a = a^ ; —a x —a x —a = —a^, etc. 
 
 FORMATION OF POWERS. 
 
 261. All JPowers of a quantity may be formed by 
 multiplying the quantity into itself. (Art. 249.) 
 
 262. To Raise a Monomial to any Required Power. 
 
 The process of involving a quantity which consists of 
 several factors depends upon the following 
 
 259, What is the zero power ? To what is a quantity of the zero power equal ? 
 a6o. Rule for the signs ? 
 
INVOLUTION". 137 
 
 PRINCIPLES. 
 
 1°. The power of the jwoduct of two or more factors is 
 equal to the product of their powers. 
 
 2°. The product is the same, in whatever order the factors 
 are tahen. (Art. 87, Prin. 3.) 
 
 1. Given 3^^ to be raised to the third power. 
 
 SOLUTION. 
 
 {UhJ = 3a¥ X 3«^ X sah^ (Art. 261), 
 or, sxsxsxaxaxaxb^xb^xIP (Prin. 2), 
 
 .-. (3ab^y = 27«3&6, Ans. 
 Involving each of tliese factors separately, we have, (3)3 = 27 ; 
 (af = a^ ; and {¥f = ¥ ; and 27xa^x¥ = 2']a^¥, Ans. Hence, the 
 
 EuLE. — Eaise the coefficient to the poiuer required, and 
 multiply the index of each letter by the index of the power, 
 prefixing the proper sign to the result, (Art. 90.) 
 
 Notes.— T. A single letter is involved by giving it the index of the 
 required power. 
 
 2. A quantity which is already a power is involved by multiplying 
 its index by the index of the required power. 
 
 3. The learner must observe the distinction between an index 
 and a coefficient. The latter is simply a multiplier, the former shows 
 how many times the quantity is taken as a, factor. 
 
 4. This rule is applicable both to positive and negative exponents. 
 
 2. What is the square of abc ? 
 
 3. What is the square of — abc ? 
 
 4. What is the cube of xyz ? 
 
 5. What is the fifth power of abc ? 
 
 6. What is the fourth power of 2x^y ? 
 
 7. What is the third power of 6a^^ ? 
 
 8. What is the fourth power of $aM^c? 
 
 9. What is the sixth power of 2a^c^'i 
 
 10. What is the eighth power of abcd'i 
 
 11. What is the nih power of xyz ? 
 
 262. How raise a monoinial to any power? Note. A pingle letter ? A" quantity 
 already a power ? Distinction between index and coefficient ? 
 
138 INVOLUTION. 
 
 12. Find the fifth power of {a + hy. 
 
 13. Find the second power of {a -h hY", 
 
 14. Find the nth power of {x — «/)*". 
 
 15. Find the 7it\\ power of {x + yy. 
 
 16. Find the second power of (a^ + If). 
 
 17. Find the third power of {aW¥). 
 
 263. To Involve a Fraction to any required Power* 
 
 18. What is the square of — ^? 
 
 EuLE. — Raise both the numerator and denominator to the 
 required power. 
 
 'i(it^ 
 10. Find the cube of ^ — • 
 2a 
 
 20. Find the fourth power of ^ 
 
 21. Find the square of ^— gTn* 
 
 2 
 
 22. Find the wth power of -• 
 
 23. Find the wth power of — ^ • 
 
 <y 
 
 264. A compound quantity consisting of two or more 
 terms, connected by + or — , is involved by actual multi- 
 plication of its several parts. 
 
 24. Find the square of 3a + Z>2. Ans. ^a^-^-SaU^-^hK 
 
 25. What is the square of « + 5 + c ? 
 
 Ans. 0? + 2ab + 2ac + 5^4- 2bc + &; 
 
 26. What is the cube of a; + 2?/ + 2 ? 
 
 265. It is sometimes sufficient to express the power of a 
 compound quantity by exponents. 
 
 Thus, the square of a + & = (« + &)2 ; the nth power of a& + c + 3«?^ - 
 
 363. How involve a fraction? 264. How involve a compound quantity « 
 
INVOLUTION- 
 
 139 
 
 
 
 
 
 L 
 
 — 
 
 :i 
 
 
 - 
 
 
 ' 
 
 
 FORMATION OF BINOMIAL SQUARES. 
 
 266. To Find the Square of a Binomial in the Terms of 
 its Parts. 
 
 1. Given two numbers, 3 and 2, to find the square of 
 their sum in the terms of its parts. 
 
 Illustration.— Let the shaded part of the diagram represent the 
 square of 3 ; — each side being divided into 
 3 inches, its contents are equal to 3 x 3, or 
 9 sq. in. 
 
 To preserve the form of the square, it is 
 plain equal additions must be made to two 
 adjacent sides ; for, if made on one side, or on 
 opposite sides, the figure will no longer be a 
 square. 
 
 Since 5 is 2 more than 3, it follows that 
 two rows of 3 squares each must be added at 
 
 the top, and 2 rows on one of the adjacent sides, to make its length 
 and breadth each equal to 5. Now 2'into 3 plus 2 into 3 are 12 squares, 
 or tvyice the product of the two parts 2 and 3. 
 
 But the diagram wants two times 2 small squares, to fill the comer 
 on the right, and 2 times 2, or 4, is the square of the second part. We 
 have then 9 (the square of the first part), 12 (twice the product of the 
 two parts 3 and 2), and 4 (the square of the second part). Therefore, 
 
 (3 + 2)2 = 32 + 2 X (3 X 2) + 2^. 
 
 2. Required the square oix-^y, Ans. x^-{-2 xxy-\-y\ 
 
 Hence, universally. 
 
 The square of the sum of two quantities is equal to the 
 square of the first, plus twice their product, plus the square 
 of the second. 
 
 Note. — The square of a Unomial always has three terms, and con- 
 sequently is a trinomial. Hence, 
 
 No binomial can be a perfect square. (Art. loi.) 
 
 266. To what is the square of the sum of two quantities equal ? How illustrate 
 the square of the sum of two quautities in the terms of its parts ? 
 
140 BII^OMIAL THEOREM. 
 
 267. All Binomials may be raised to any required 
 power by continued multiplication. But when the expo- 
 nent of the power is large, the operation is greatly abridged 
 by means of the Bi7iomial Theorem.* 
 
 268. The Binomial Hieorem is a general formula 
 by which any power of a binomial may be found without 
 recourse to continued multiplication. 
 
 To illustrate this tlieorem, let us raise the binomials « + & and a—h 
 to the second, third, fourth, and fifth powers, by continued multipli- 
 cations : 
 
 {a + hf = a^ + 2(0) + ¥. 
 
 (a + hf = a^ + sa^b + saf^ + ¥. 
 
 {a + Vf = a* + ^a% + ta%'^ + 40*3 + j^^ 
 
 {a + yf = a^ + 5a*& + loaW + loarW + 506* + }fi. 
 
 (a - bf = a? - 2ab + b^. 
 
 {a - bf ^ «3 _ 3^2^ + 2>ab'' — b\ 
 
 (a - bf = a'^ - 4a^b + 6a?b^ - ^ab^ + b\ 
 
 {a - bf -oJ> - sa^ft + \oaW - loa^&a + 5^54 _ y,^ 
 
 269. Analyzing these operations, the learner will discover 
 the following laws which govern the expansion of Unomials : 
 
 1. The number of terms in any power is one more than the 
 index of the power. 
 
 2. The i7idex of the first term or leading letter is the 
 index of the required power, which decreases regularly by i 
 through the other terms. 
 
 The index of the following letter begins with i in the 
 second term, and increases by i through the other terms. 
 
 3. The sum o/the indices is the same in each term, and is 
 equal to the index of the power. 
 
 268. What is the Binomial Theorem ? 269. What is the law respecting the num- 
 ber of terms in a power ? The indices of each quantity ? The sum of the indice* 
 iu each term ? 
 
 * This method was discovered bj Sir Isaac Newton, in 1666. 
 
BINOMIAL THEOEEM. 141 
 
 4. The coefficient oi the first and last term of every power 
 is I ; of the second and next to the last, it is the tJidex of 
 the power; and, universally, the coefficieuts of any two 
 terms equidistant from the extremes, are equal to each other. 
 
 Again, the coefficients regularly increase in the first half 
 of the terms, and decrease at the same rate in the last half. 
 
 5. The signs follow the same rule as in multiplication 
 
 270. The preceding principles may be summed up in the 
 following 
 
 GENERAL RULE. 
 
 I. In'DICES. — Give the first term or leading letter the index 
 if the required power, and diminish it regularly hy i through 
 the other terms. 
 
 TJie index of the following letter in the second term is i, 
 and increases regularly hy i through the other terms. 
 
 II. Coefficients. — The coefficient of the first term is i. 
 To the second term give the index of the poiuer ; and, 
 
 universally, multiplying the coefficient of any term hy the 
 index of the leading letter in that term, and dividing the 
 product by the index of the following letter increased hy i, 
 the result will he the coefficient of the succeeding term. 
 
 III. Signs. — If hoth terms are positive, mahe all the terms 
 positive; if the second term is negative, make all the odd 
 terms, counting from the left, positive, a?id all the even terms 
 
 the binomial formula. 
 
 n—- I 
 (a + hy = a'' -\-n x a""-^ h + n x a^'-^h^ etc. 
 
 Note. — The preceding rule is based upon the supposition that the 
 index is a positive whole number ; but it is equally true when the index 
 is e\ih.Qr positi've or negative, integral or fractional. 
 
 The coeflacients of the first and last terms ? The law of the signs ? 270. What 
 is the general rule ? 
 
142 BINOMIAL THEOKEM. 
 
 Expand the following binomials : 
 
 1. {a + by. 6. (^-\-zy^. 
 
 2. (a — by. 7. (a — by. 
 
 3. {c + ay, 8. (m + 7iy\ 
 
 4. {x + yy, g. (x—y)^. 
 
 5- {^—yy* lo. {a+by. 
 
 2T1.. When the terms of a binomial have coefficients or 
 exponents, the operation may be shortened by substituting 
 for them single letters of the first power. After the opera- 
 tion is completed, the value of the terms must be restored. 
 
 11. Required the fifth power of i^ + 3^2 
 Solution.— Substitute a for x'^, and h for 32^2 . then 
 
 (a + If = a^ + 5a^& + loaW + ioa?l^ + soft* + 6^. 
 Restoring the values of a and &, 
 
 (a;2 + 3^2)5 _ ajio + 1 53,8^2 ^ goa?^^ + 2-joxh^ + 4052;^ + 2432/^®. 
 
 12. Expand (x^ — 2,by. 
 
 A71S. 7^ — i2^b + s^b^ — io8ic2^»3 4. 8iJ4. 
 
 272. Every power of i is i, and when a factor it has no 
 effect upon the quantity with whicli it is connected. (Art. 94, 
 note.) Hence, when one of the terms of a binomial is i, it 
 is commonly omitted in the required power, except in the 
 first and last terms. 
 
 Note. — In finding the exponents of such binomials, it is only 
 necessary to observe that the mm of the two exponents in each term 
 is equal to the index of the power. 
 
 13. Expand (ic + i)^. 15. Expand (\ ~ ay. 
 
 14. Expand \b — i)*. 16. Expand (i -f of, 
 
 (See Appendix, p. 287.) 
 271. When the terms have coefficients or exponents, how proceed ? 
 
POWERS or POLYNOMIALS. 143 
 
 273. A Polynomial may be raised to any power by actual 
 multiplication, taking the given quantity as a factor as many 
 times as indicated by the exyonent of the required power. 
 But the operation may often be shortened by reducing the 
 several terms to two, by substitution, arid then applying the 
 Binomial Formula. 
 
 17. Kequired the cube of x -^ y -\- z. 
 
 Solution. — Substituting a for (y + e), we have aj + (y +2) = aj + a. 
 By formula, {x-vaf = q^-\- yi?a + ym^ + a^ 
 
 Restoring the value of a, 
 
 274. To Square 2i Polynomial without Recourse to 
 Multiplication. 
 
 18. Eequired the square oi a -\- h -{• c. 
 
 Solution. — By actual multiplication, we have, 
 
 {a + h^-cf = a'^ + 2ab+2ac + ¥ + 2bc+c^. 
 Or, changing the order of terms, 
 
 a'^ + ¥ + c^ + 2<zb + 2ac+2bc. 
 Or, factoring, we have, a'^ + 2aib + c) + ¥ + 2bc + cK 
 
 19. Eequired the square ot a -{- b -\- c + d. 
 
 Solution. — By actual multiphcation, we have, 
 
 a^ + 1^ + d^ + d^ + 2db + 2ac + 2ad + 2bc + 2bd + 2cd. 
 Or, changing the order of the terms, and factoring, we have, 
 a? + 2a(b+c-\-d)->r ¥ 4- 2& (c + cZ) -4- c^ + 2cd + d^. Hence, the 
 
 KuLE. — To the sum of the squares of the terms add twice 
 the product of each pair of terms. 
 
 Or, To the square of each term add twice its product into 
 the sum of all the terms which folloio it. 
 
 20. Eequired the square oix -\- y -{- z. 
 
 21. Eequired the square of a — h -{- c. 
 
 22. Eequired the square oi a -\- x -\- y ■\- z. 
 
 273. How may a polynomial be raised to any required power ? 274. What is the 
 rule foi- squaring a polynomial ?^ _ - 
 
 a 
 
 OFT..E -^X 
 
144 ADDITION OF POWERS. 
 
 275. When one of the terms of a binomial is 2^ fraction, 
 it may be involved by actual multiplication, or by reducing 
 the mixed quantity to an improper fraction, and then 
 involving the fraction. (Art. 171.) 
 
 22t. Kequired the square of x -{• ^\ and x — J. 
 
 « + i a; — i 
 
 a- + i a!- i 
 
 + ja; + i -ia; + i 
 
 aj'+ X + \ a^— X + \ 
 
 Or, reduce the mixed quantities to improper fractions. Thus, 
 
 a; + - = -= — ; and x = , (Art. 171.) 
 
 22 22^'' 
 
 /2a!+i\2 4a;-+4a;+i . /2aj— iV 
 
 4a;2— 4a;+i 
 
 Expand the following mixed quantities : 
 
 24. (fl^ + J)^. 26. (— f + 2al)cf. 
 
 25 
 
 276. Powers are added and subtracted hke other 
 quantities. (Arts. 67, 77.) For, the same powers of the 
 same letters are like quantities; while powers of different 
 letters and different powers of the same letter are unlike 
 quantities, and are treated accordingly. (Arts. 43, 44,) 
 
 28. To 7^2 + 5 (« 4- J)3 _ 6a; + 3^ + «' 
 
 Add — 30^3 + 4 (^ + ^f -f 4a; 4- 43^2 — fl< 
 
 Ans. 4(f + g (a -^ by — 2X + $0^ -{- 43^ -^ a^ — a^ 
 
 29. From 3^3 + 5 J2 — 4(^8 4. ^^^a _ ^5 
 
 Take — 40^^ + 3^ + 3^^ — 5^ + a* 
 
 Ans. 7^8 + 22>2 _ 7^8 4- 5^:3 4. 4^2 _ ^^s _ ^4 
 
 275. How involve a binomial, when one term is a fraction ? 276. How are power* 
 added and subtracted ? Why ? 
 
DIVISION OF POWERS. 145 
 
 MULTIPLICATION OF POWERS. 
 277. To Multiply Powers of the Same Moot. 
 
 1. What is the product of 2>tt'^b^ multiplied by aWi 
 
 Solution. — Adding the exponents of each letter, we have yi^ 
 and 6^ Now 3«^ x &^ = 3a66% Ans. (Art. 94.) 
 
 2. Multiply ^a^¥ by a~^l)~\ 
 
 Solution. — Adding the exponents of each letter, as before, we 
 have 3a'^62, Ans. Hence, the 
 
 EuLE. — Add the exponents of the given quantities, and the 
 result will be the product. (Art. 94.) 
 
 Notes. — i. This rule is applicable to positive and negative exponents. 
 
 2. Powers of different roots are multiplied hy writing them one 
 after another. 
 
 Multiply the following powers: 
 
 3. fl^^ by a^. 7. a~*b by a~^bK 
 
 4. x~^ by ar\ 8. a~*cd by a'^c^d^. 
 
 5. b-^ hj bK 9. ¥ c-^ y-^ hj b-^ (^y\ 
 
 6. a"* by a\ 10. a^y^z^ by a~^y-^^. 
 
 DIVISION OF POWERS. 
 278. To Divide Powers of the Same Root, 
 
 II. Divide a^ by a\ 
 
 Solution. — Subtracting one exponent from the other, we have 
 a^-T-a^ = a?, the quotient sought. (Art. 113.) Hence, the 
 
 Rule. — Subtract the exponent of the divisor from that of 
 the dividend ; the result is the quotient. (Art. 113.) 
 
 Note. — This rule is applicable to positive and negative exponents. 
 
 ••"•t. How multiply powers of the same root ? Note. Of different roots 1 278. What 
 is tne rule for dividing powers of the same root ? 
 
146 TRAKSFEREING FACTORS. 
 
 Divide the following powers : 
 
 12. a^ by ar\ i6. oc^yz^ by ar^y^z~\ 
 
 13. x~^ by ic^. 17. i2aPh~^c by 3flf2Z'"^c^ 
 
 14. ¥ by 52. 18. 6a;4^V ^y 22^-2 ?/;22. 
 
 15. c~^ by c~s. 19. 6oa%^(^ by 5«~2Z>%-A 
 
 279. The Method of denoting Reciprocal Powers shows 
 that any factor may be transferred from the numerator of a 
 fraction to the denominator, and vice versa, by changing the 
 5i^;i of its exponent from + to — , or — to +. (Art. 256.) 
 
 20. Transfer the denominator of —3 to the numerator. 
 
 q6 I 
 
 Solution. ^ = «^ ^ "T^ia ~ ^^ ^ ^~^ = a^a;-^, Ans. 
 
 21. Transfer the denominator of --^ to the numerator. 
 
 x~^ 
 
 Solution. — -=— = 61^-^ — - = axaj^ = aofi, Ans. 
 
 ■^ 
 » ^5 
 
 22. Transfer the numerator of — to the denominator. 
 
 01 a^ I . I I I . I . 
 
 Solution. — =- x a5__^___i.^-5_ — ^ ^;^5^ 
 
 2^ 2^ y o^ y or^y 
 
 2^. Transfer the numerator of — to the denominator. 
 
 y 
 
 Solution. — = — ^ x - = -^ , Ans. 
 y a^ y a^y 
 
 ax~^ 
 
 24. Transfer ar'^ to the denominator of • 
 
 y 
 
 25. Transfer y^ to the numerator of j—^' 
 
 26. Transfer d~^ to the denominator of —q— 
 
 27. Transfer af* to the numerator of — -• - 
 
 279. What inference may be drawn from the method of denoting rec;pri>4.Al 
 powers ? How transfer a factor ? 
 
OHAPTEE XIIL 
 
 EVOLUTION * 
 
 280. Evolution is finding a root of a quantity. It is 
 often called the Extraction of roots. 
 
 281. A Hoot is one of the equal factors of a quantity. 
 
 Notes. — i. Powers and roois are correlative terms. If one quantity 
 is a power of another, tlie latter is a root of the former. 
 Thus, a^ is the cube of a, and a is the cube root of a^. 
 
 2. The learner should observe the following distinctions : 
 ist. By involution a product of equal factors is found. 
 2d. By evolution a quantity is resolved into equal factors. It is the 
 reverse of involution. 
 
 3d. By division a quantity is resolved into two factors. 
 4th By subtraction a quantity is separated into two parts. 
 
 282. Boots, like powers, are divided into degrees ; as, the 
 square, or second root; the cube, or third root; the fourth 
 root, etc. 
 
 283. The Square Hoot is one of the two equal factors 
 of a quantity. 
 
 Thus, 5 X 5 = 25, and axa = a^ ; therefore 5 is the cquare root of 
 25, and a the square root of a^. 
 
 284. The Cube Hoot is one of the three equal factors 
 of a quantity. 
 
 Thus, 3 X 3 X 3 = 27, and axaxa = a^; therefore, 3 js the cube 
 root of 27, and a is the cube root of a^. 
 
 280, What is evolution? 281. A root? Note. Of what is evolution the reverse? 
 283. What is the square root ? 284. Cube root ? 
 
 * From t^ie Latin evolvere, to unfold. 
 
148 EVOLUTION. 
 
 285. Eoots are denoted in two ways : 
 
 ist. By prefixing the radical sign ^ to the quantity.* 
 2d. By placing a fractional exponent on the right of the 
 quantity. 
 
 Thus, ^/a and a* denote the square root of a. 
 
 y\/a and a^ denote the cube root of a, etc. 
 Notes.— I. The figure placed over the radical sign, is called the 
 Index of the Root, because it denotes the name of the root. 
 Thus, ^/a, and \^a, denote the square and cube root of a. 
 
 2. In expressing the square root, it is customary to use simply the 
 radical sign ^ , the 2 being understood. 
 
 Thus, the expression /Y/25 = 5, is read, ''the square root of 25 = 5." 
 
 3. The method of expressing roots hj fractional exponents is derived 
 irom. the manner of denoting powers by integral indices. 
 
 Thus, a^=axaxaxa; hence, if a'^ is divided into four equal 
 factors, one of these equal factors may properly be expressed by a. 
 
 286. The numerator of a fractional exponent denotes the 
 power, and the denominator the root. 
 
 Thus, a^ denotes the cube root of the first power of a; and a* denotes 
 the fourth root of the third power of a, or the third power of the 
 fourth root, etc. 
 
 Read the following expressions : 
 
 1. «i 4. bk 7. dh 10. af. 
 
 2. «1 5. c^. 8. m"^. II. «»*. 
 
 3. a^, 6. x^. 9. n^. 12. x§, 
 
 13. Write the third root of the fourth power of a. 
 
 14. Write the fifth power of the fourth root of x. 
 
 15. Write the eighth root of the twelfth power of y. 
 
 287. A JPerfect Power is one whose exact root can 
 be found. This root is called a rational quantity. 
 
 285. How are roots denoted? 286. What docs the numerator of a fractional 
 exponent denote ? The denominator? 287. What is a perfect power ? 
 
 * From the Latin radix, a root. 
 
 The si^n /v/" is a corruption of the letter r, the initial of ra4i^* 
 
EVOLUTION". 149 
 
 288. An Imperfect 'Power is a quantity whose exact 
 
 root cannot be found. 
 
 289. A Surd is the root of an imperfect power. It is 
 often called an irrational quantity. 
 
 Thus, 5 is an imperfect power, and its square root, 2.23 + , is a surd. 
 
 Note. — All roots as well as poicers of i, are i. For, a root is a 
 factor, wliich multiplied into itself produces a power ; but no number 
 except I multiplied into itself can produce i. (Art. 272.) 
 
 Thus, r, i^ I", and y^i, /y/i, /y/i, etc., are all equal. 
 
 290. Negative Exponents are used in expressing roots as 
 well as j902(;er5. (Arts. 255, 257.) 
 
 Thus, — T = a-i ; -T = a-i ; -r = a-n ^ 
 
 a* a* ^t 
 
 291. The value of a quantity is not altered if the index 
 of the power or root is exchanged for any other index of 
 the same value. 
 
 Thus, instead of x^, we may employ x^, etc, Hence, 
 
 'i92. A fractional exponent maybe expressed in decimals. 
 'fhus, a^ = a^ = a°*^ . That is, the square root of a is equal to the 
 fifth power of the tenth root of a. 
 
 Express the following exponents in decimals : 
 
 16. Write a^ in decimals. 19. Write h'^ in decimals. 
 
 1 . 
 
 17. Write a^ in decimals. 20. Write x^ in decimals. 
 
 18. Write a^ in decimals. 21. Write y^ in decimals. 
 
 22. Express a^ in decimals. Ans. a^ = ^0.333333+ ^ 
 
 23. Express x^ in decimals. Ans. x^ = 2^0.66666+ ^ 
 
 24. Express y^ in decimals. Ans. y^ = y-^. 
 
 «5. Write a^ in decimals. Ans. a~s = a^'^. 
 
 Note. — In many cases, & fractional: exponent can only be expressed 
 approximately by decimals. 
 
 288. An imperfecl power ? 289. A rard ? 290. Are negative exponents used in 
 expressing roots 1 292. How are fractional exDonents sometimes expressed ? 
 
150 EVOLUTION. 
 
 293. The Signs of Hoots are gc remed by the following 
 
 PRINCIPLES. 
 
 1°, An odd root of a quantity has the same sign as the 
 quantity. 
 
 ■ 2°. A71 even root of a positive quantity is either positive 
 or negative, and has the double sigti, ±. 
 
 Thus, the square of +a\aa^, and the square of —a is a^ ; therefore 
 the square root of a^ may be either +a or —a\ that is, ^/a^ = ± a. 
 
 3°. The root of the product of several factors is equal to 
 the product of their roots. 
 
 Notes. — i. The ambiguity of an even root is removed, when it is 
 knqwn whether the power arises from a positive or a negative quantity. 
 
 2. It should also be observed that the two square roots of a positive 
 quantity are numerically equal, but have contrary signs. 
 
 294. An Even Hoot of a 7iegative quantity cannot be 
 found. It is therefore said to be impossible. 
 
 Thus, the square root of —a^ is neither +a nor —a. For, +ax +a 
 = +a^ ; and —a x —a = +a^. Hence, 
 
 295. An even root of a negative quantity is called an 
 Imaginary Quantity. 
 
 Thus, \/— 4, V'— a^ 'V^— a^ are imaginary quantities. 
 
 296. To Find the B^oot of a Monomial. 
 
 I. What is the square root of a^? 
 
 Analysis. — Since a'^ — axa, it follows that one opeeation. 
 
 of the equal factors of «? is a; therefore, a is its ^/a? = a 
 
 square root. (Art. 283.) 
 
 Again, since multiplying the index of a quantity by a number 
 raises the quantity to a corresponding power, it follows that dividing 
 the index by the same number resolves the quantity into a correspond- 
 ing root. Thus, dividing the index of a^ by 2, we have a' or a, which 
 is the square root of a^. 
 
 293. What principles eovem the sign?, of roots ? When is the doable sisu used ? 
 Illustrate this. JVote. When is the ambiguity removed ? 294. What is an even root 
 of a negative quantity ? Illustrate. 295. What is it called ? 
 
EVOLUTION. 151 
 
 2. What is the square root of 9^^^ ? 
 
 Analysis. — Since 9 = 3x3, the index of operation. 
 
 jj* = 2 X 2, and the index of &2 = i x 2, it ^/oaf^ := 2t^^b 
 
 follows ^shat the square root of 9 is 3, that of 
 a^ is a-.; and that of ¥ is 6^ or &. Therefore, ^^a?¥ = ^a^h. Hence, the 
 
 Rule. — Divide the index of each letter hy the index of 
 the required root; to the result prefix the root of the 
 coefficient with the proper sign. (Art. 293.) 
 
 Note. — This rule is based upon Principle 3. If a quantity is an 
 imperfect power, its root can only be indicated. 
 
 296, a. Tlie root of a Fraction is found hy extracting the 
 root of ^ach of its terms. 
 
 Find the required roots of the following quantities: 
 
 3. Va^. 10. 'v/36«^Z>2. 
 
 "s/a^ or a. 11. ^/^^y\ 
 
 3 /- 
 
 ^/ AfXy. 12. v64a% 
 
 ^2>a%\ 13. ^i^xy. 
 
 V^^ahc, 14. A/49^y. 
 
 V^i6^. 15. ^/r^. 
 
 297. To Extract the Square Moot of ihe Square of a 
 Binomial. 
 
 I. Required tb€ square root of a^ + 2a5 4- l\ 
 
 Analysis. — Arrange the terms operation. 
 
 according to the powers of the letter O^ ■\- 2ab + Z>2 ( fl^ -f d 
 
 a ; the square root of the first term ^2 
 
 is a, which is the first term of the \_ 'h\ h A- Jvi 
 
 root. Next, subtracting its square ' / ■, j^ 
 
 from the given quantity, bring down 2^0 + (r 
 the remainder, 2ab + W. 
 
 296. How find the root of a monomial ? iVb/«.— Upon what principle is this rulf 
 based ? 296 a. How find the root of a fraction ? 
 
152 EVOLUTION. 
 
 Divide the ist term of this rem. by 2a, double the root Oms found, the 
 quotient h is the other term of the root. Place & both in the root and on 
 the right of the divisor. Finally, multiply the divisor, thus increased, 
 by the second term of the root, and subtracting the product from 
 2ab + h^, there is no remainder. Therefore, a + & is the root required. 
 
 The square root of a:-—2db+l^ is found in the same manner, the 
 terms of the root being connected by the sign — . Hence, the 
 
 HuLE. — Find the square roots of the first and third ter^n^ 
 and connect them hy the sign of the middle term, 
 
 2. What is the square root of a^ -{- 4X -\- 4? 
 
 3. What is the square root of «2 _ 2^5 4- i ? 
 
 4. What is the square root of i + 2X -\- a^? 
 
 5. What is the square root oi x^ -\- ^x -\- ^ ? 
 
 6. What is the square root of «^ — a + } ? 
 
 7. What is the square root of x^ -{- ix -] ? 
 
 4 
 
 298. To Extract the Square Root of a Polynomial. 
 
 8. Required the square root of 4«* — i2a^-\-s^^+6a-{-i, 
 
 OPBRATION. 
 
 40* — i2fl^3 + 5^2 _f- 6« 4- I ( 2a^ — $a — i 
 
 4a* 
 
 ^2- 
 
 3«) - 
 
 I2fl3 
 
 + 9«2 
 
 + 6a+i 
 
 4^2 
 
 -6a- 
 
 -0 
 
 -4«2 
 -4«^ 
 
 + 6«+ I 
 + 6a + I 
 
 Analysis. — The square root of the first term is 2a', which is the 
 first term of the root. Subtract its square from the term used 
 md bring down the remaining terms. Divide the remainder by 
 louble the root thus found ; the quotient —3a is the next term of the 
 root, and is placed both in the root and on the right of the partial 
 divisor. Multiply the divisor thus increased by the term last placed 
 in the root, and subtract the product as before. 
 
 Next, divide the remainder by twice the part of the root already 
 found, and the quotient is —i, which is placed both in the root and 
 on the right of the divisor. 
 
 297. How extract the square root of the square of a binomial? 
 
EVOLUTIOK. 153 
 
 Finally, multiply the divisor, thus increased, by the term last 
 placed in the root, and subtracting the product, as before, there is no 
 remainder. Therefore, the required root is 2a"^—3«—i. Hence, the 
 
 KuLE. — I. A rra?ige the terms according to the jjowers of 
 some letter, beginning ivith the highest, find the square root 
 of the first term for the first term of the root, and subtract 
 its square from, the given quantity. 
 
 IL Divide the first term of the ''remainder by double the 
 root already found, and place the quotient both i7i the root 
 and on the right of the divisor, 
 
 III. Multiply the divisor thus increased by the term last 
 placed in the root, and subtract the product from the last 
 dividend. If there is a remainder, proceed with it as before^ 
 till the root of all the quantities is found. 
 
 Proof. — Multiply the root by itself, as in arithmetic. 
 
 Note. — This rule is essentially the same as that used for extracting 
 the square root of numbers. 
 
 Extract the square root of the following quantities : 
 
 9. x^ + 2xy + y"^ -{- 2XZ + 2yz + z\ 
 
 10. 
 
 a^ — 4ab -\- 2a -{- 4J2 _ 4J _}. i. 
 
 II. 
 
 a^ + ^a^ + 4^2 _ 4^2 _ 85 -f 4. 
 
 12. 
 
 I — 4^2 _|_ 4^4 4- 22; — 4^2^; + x\ 
 
 13. 
 
 4a^ — i6a^ + 24^2 _ i6a + 4. 
 
 14. 
 
 a^-ab+ ibK 
 
 
 X^ 1/2 
 
 15- 
 
 2-1- — • 
 
 299. The fourth root of a quantity may be found by 
 extracting the square root twice j that is, by extracting the 
 square root of the square root. 
 
 Thus, y^iea* = 4a^, and /^4a^ = 2a. Therefore, 2a is the fourth 
 root of i6a\ 
 
 Proof. 2ax2a = 4a^ ; 4a^ x 40^ = i6a\ 
 
 The eighth, the sixteenth, etc., roots may be found in like 
 manner. 
 
 298. Of a polynomial ? 299. How find the fourth root of a quantity ? The eighth ? 
 
CHAPTER XIV. 
 RADICAL QUANTITIES. 
 
 300. A Madical is the root of a quantity indicated by 
 the radical sign or fractional exponent. 
 
 Notes.— I. The figures or letters placed before radicals are coefficients. 
 
 2. In the following investigations, all quantities placed under tlie 
 radical sign, or having a fractional exponent, whether perfect or 
 imperfect powers, are treated as radicals, unless otherwise mentioned. 
 
 301. The Degree of a radical is denoted by its index, 
 or by the denominator of its fractional exponent. (Arts. 
 285, 286.) 
 
 Thus, '^ax, a^, and {a + Vf, are radicals of the same degree. 
 
 302. Lihe Madicals are those which express the 
 same root of the same quantity. Hence, like radicals are 
 like quantities. (Art. 43.) 
 
 Thus, s^y/a'^— & and 3/y/«^— &, etc., are like radicals. 
 
 REDUCTION OF RADICALS. 
 
 303. Reduction of Madicals is changing their 
 form without altering their value. 
 
 304. The Simjylcst Form of radicals is that which 
 contains no factor whose indicated root can be extracted. 
 Hence, in reducing them to their simplest form, all exact 
 poivers of the same name a« the root must be removed from 
 under the radical sign. 
 
 300. What is a radical ? 301. How is tlie decree of a radical denoted? 302. What 
 are like radicals ? 303. Define reduction of radicals. 304. What is the simplest form 
 }t radicals ? 
 
EEDUCTIOK OF RADICALS. 155 
 
 CASE I. 
 305. To Reduce a Radical to its Simplest Form. 
 
 I. Reduce ^/\Wx to its simplest form. 
 
 Analysis. — By inspection, we ofebation. 
 
 perceive that tlie given radical is ^ \Wx = ^ ^0? X 2X 
 
 composed of two factors, <^a? and ^^ / — g / — 
 
 IX, the first being a perfect square " 
 
 and the second a surd. (Art. 289.) .'. ^/ l^a^X = ^aV2X 
 Removing ga^ from under the rad- 
 ical sign and extracting its square root, we have 3a, which prefixed 
 to the other factor gives s^y'z^, the simplest form required. 
 
 2. Reduce 4^/a^ — a^x to its simplest form. 
 
 Analysis. — Factoring operation. 
 
 the radical part, wejiave 4^^.^^ = 4^a^x{a-x) 
 
 the two factors, \/a^ and 3,-— », 
 
 ^a—x, the first being a 
 
 perfect cube, and the sec- .'. ^^ a^—a^X = /^a\^a—X 
 
 ond a surd. Remove a^ 
 
 from under the radical sign, and its root is a, which multiplied by 
 the coeflBcient 4, and prefixed to the radical part, gives ^a^a—x, the 
 simplest form. Hence, the 
 
 Rule. — I. Resolve ilie radical into two factors, one of 
 which is the greatest power of the same name as the root. 
 
 II. Extract the root of this power, and multiplying it ly 
 the coefficient, prefix the result to the other factor, with the 
 radical sign letween them. 
 
 Notes. — i. This rule is based upon the principle that the root of 
 the product of two or more factors is equal to the product of their 
 roots. 
 
 2. When the radicals are small, the greatest exact power they 
 contain may be readily found by inspection. 
 
 3. Reduce 3V5o«^^^ to its simplest form. Ans. isav^x. 
 
 4. Reduce 6\/s4^l/ to its simplest form. Ans. iSxV 2y. 
 
 305. Becite the rule. Ifote. Upon what based ? 
 
156 EEDUCTIOK OF EADICALS. 
 
 306. To Find in large Radicals the Greatest Power 
 corresponding to the indicated Root. 
 
 5. Eeduce ViSya to its simplest form. 
 
 OPERATION. 
 
 4 ) 1872 ^1872 = ^4x4x9 X V13 
 
 4 ) 468 " = A/144 X V13 
 
 9)117(13 V1872 = 12V135 ^'ns. 
 
 Analysis. — Divide the radical by the smallest power of the same 
 degree that is a factor of it ; the quotient is 468. Divide this quan- 
 tity by 4 ; the second quotient is 117. Th% smallest power of the same 
 degree that will divide 117, is 9. The quotient is 13, which is not 
 divisible by any power of the same degree. The product of the 
 divisors, 4x4x9= 144, is the greatest square of the given radical. 
 Extracting the square root of 144, we have i2/y/i3, the simplest form 
 required. Hence, the 
 
 'Rv'LE,— Divide the radical by the smallest power of the 
 same degree ivhich is a factor of the given radical. 
 
 Divide this quotient as before; and thus 'proceed till a 
 quotient is obtained which is not divisible by any power of 
 the same degree. The continued product of the divisors will 
 be the greatest poiver required. 
 
 Note. — This rule is founded on the principle that the 'product of 
 any two or more square numbers is a square, the product of any two 
 or more cuhic numbers is a cuhe, etc. 
 
 Thus, 2' X 32 z= 36 = 62 ; and 28x38 = 216 = e^. 
 
 Eeduce the following radicals to their simplest form : 
 
 \^54a^c. 
 
 yVga^ — 2'ja^b, 
 '\/64a^y. 
 </8i^. 
 V46Sa^. 
 
 306. How find the greatest power corresponding to the indicated root, in large 
 radicals ? Jffote. On what principle is this rule founded ? 
 
 6. 
 
 Va^. 
 
 12. 
 
 7- 
 
 A/8a2^. 
 
 13- 
 
 8. 
 
 2 Vgxy. 
 
 14. 
 
 9- 
 
 3^^. 
 
 15. 
 
 10. 
 
 SV^iSS- 
 
 16. 
 
 II. 
 
 6\/252«2J. 
 
 17. 
 
EEDUCTION OF RADICALS. 157 
 
 CASE II. 
 
 307. To Reduce a Rational Quantity to the Form of a 
 Radical. 
 
 I. Keduce ^a^ to the form of the cube root. 
 
 Analysis.— The cube root of a quantity, we operation. 
 
 have seen, is one of its three equal factors. (3^^)^ = 2'ja^ 
 
 (Art. 284.) Now 3a'' raised to the third power . ^ „2 \/^2T^cfi 
 
 is 27a*. Therefore 3^^ = ^i']a^. Hence, the 
 
 Rule. — Raise the quantity to tlie power denoted hy the 
 given root, and to the result prefix the corresponding radical 
 sign. 
 
 Note. — The coeflBcient of a radical, or any factor of it, may be 
 placed under the sign, by raising it to the corresponding power, and 
 placing it as a factor under the radical sign. 
 
 2. Reduce 2a^ to the form of the cube root. 
 
 3. Reduce (2a + d) to the form of the square root. 
 
 4. Reduce {a — 2b) to the form of the square root. 
 
 5. Place the coefficient of saVb under the radical sign. 
 
 6. Place the coefficient of 2a^\^ab under the radical sign. 
 
 7. Reduce 2x^y^z^ to the form of the fourth root. 
 
 8. Reduce ^abc to the form of the cube root. 
 
 9. Reduce ^{a — b) to the form of the cube root. 
 10. Reduce a^ to the form of the cube root. 
 
 Note. — When a power is to be raised to the form of a required 
 root, it is not the given letter that is to be raised, but the power of the 
 letter. * 
 
 I I. Reduce ah^ to the form of the fourth root. 
 
 12. Reduce a — b to the form of the square root. 
 
 13. Reduce a'^ to the form of the nth root. 
 
 307. How reduce a rational quantity to the form of a radical? Note. How place 
 a coefficient under the radical sign ? Note. How raise a power to the form of * 
 required root ? 
 
158 REDUCTIOK OF EADICALS, 
 
 CASE III. 
 
 308. To Reduce Radicals of different Degrees to others of 
 
 equal Value, having a Common Index. 
 
 1. Eeduce a^ and b^ to equivalent radicals of a common 
 index. 
 
 Analysis. — The fractional opebation. 
 
 indices i and i, reduced to a J = -^ and i = -rj 
 
 common denominator become ^.^ a^ = a^ and b^ = ^t\ 
 
 j\ and f But af' = {a^)^, 4 , ,. i , , 3 ,„, 1 
 
 and 5^^ = (63p. (Art. 174.) ^" "= ^^ ^^ ^^^ ^" ^ ^^^^ 
 Therefore (a^)^'^ and (ft^)" are the radicals required. Hence, the 
 
 EuLE. — I. Eeduce the indices to a common denominator. 
 
 II. Raise each quantity to the power expressed hy the 
 numerator of the new index, and indicate the root expressed 
 ly the common denominator. (Art. 174.) 
 
 Eeduce the following radicals to a common index : 
 
 2. a^ and (hc)^. 7. V4S and \^2aK 
 
 3. 3^ and 5^. 8. a^ and 5«. 
 
 4. a^ and 6^. 9. l^ and c». 
 
 5' Vs^, V^3, and v^. 10. (« + h)^ and (a- <^)i 
 6. '\/2x^ and Vs^^. 11. (^ — ^)^ and (a? 4 «/)i 
 
 CASE IV. 
 
 309. To Reduce a Quantity to any Hequired Index^ 
 
 I. Eeduce a^ to the index ^. 
 
 Analysis. — Divide the index | byl; operation. 
 
 wo have | or ^. Place this index over a; l""~^"f^=i^XT^^I 
 
 it becomes <z', and setting the required i" "^ 1 ^^^ F ^^ 7 
 
 index over this, the result, (a*)^, is the /, (^v^j -<4ws. 
 answer. Hence, the 
 
 30^. 5ow reduce radicals to a coininon iude? ? 
 
ADDITION" OF EADICALS, 159 
 
 Rule. — Divide the index of the given quantity hy the 
 required index, and 'placing the quotie7it over the quantity, 
 set the required index over the whole. 
 
 Note. — This operation is the same as resolving the original index 
 into two factors, one of which is the required index. (Art. 126, note.) 
 
 2. Eeduce a^ and ifi to the index J. 
 
 Solution. J-^i = |xf = |, tlie first index. 
 
 |-i-i = f X f = f , the second index. 
 Therefore, (a^)^ and (&")* are the quantities required. 
 
 3. Reduce 3^ and 43 to the common index ^, 
 
 4. Reduce a^ and i^ to the common index \, 
 
 5. Reduce a^ and h^ to the common index J. 
 
 6. Reduce a^ and ifi to the common index -f. 
 
 7. Reduce w^ and 5^ to the common index J-. 
 
 ADDITION OF RADICALS. 
 310. To Find the Sutn of two or more Radicals. 
 
 1. Wliat is the sum of 3 V» and 5 Va ? 
 
 Analysis. — Since these radicals are opekation. 
 
 of the same degree and have the same ^^/a -f- S^^ = SV^ 
 
 radical part, they are like quantities. 
 
 (Art. 43.) Therefore their coefficients may be added in the same 
 manner as rational quantities. (Art. 67.) 
 
 2. What is the sum of 3^8 and 4^/18 ? 
 Analysis.— These radicals operation. ^ 
 
 are of the same degree, but 3^8 = 3V4 X ^2 ■=. 6^/2 
 
 the radical parts are unlike ; /—x r~ r~ /~ 
 
 ^, . XI . -, 4Vi8 = 4VQ X V2 = I2V2 
 
 therefore, they cannot be ~t ^ y » "^ 
 
 united in their present form. /. 6y2 -j- 12 V 2 = 18 y 2 
 
 Reducing them to their sim- 
 plest form, we have 3V^8 = 6-^/2, and 4y^i8 = 12-^/2, which are 
 like radicals. (Arts. 302, 305.) Now 6y'2 and 12-^/2 = 18-^/2, Ans. 
 
 309. How reduce quantities to any required index? Note. To what is this oper- 
 ation simiJar ? 
 
160 SUBTRACTION OF RADICALS. 
 
 3. What is the sum of 3A/1S and 4'V^24. 
 Analysis. — Reducing the operation. 
 
 radicals to tlieir simplest form, 3\/l8 = '^'V/o X a/z = qV^ 
 
 we have 3 V^ = 9^2, and ^^- ^ ^^g x 'v/3 = S^v^S 
 
 4/^24 = 8/^3, whicli are un- ,— 3 ,- 
 
 like quantities, and can only ^^^- 9V 2 + Ws 
 
 be added by writing tliem one after the other, with their proper signs. 
 (Arts. 43, 67.) Hence, the 
 
 Rule. — I. Reduce the radicals to their simplest form. 
 
 II. If the radical parts are aliJcCf add the coefficieiits, and 
 to the sum annex the common radical. 
 
 If the radicals are unliTce, write them one after another, 
 with their proper signs. 
 
 Note. — To determine whether radicals are alike, it is generally 
 necessary to reduce them to their simplest form. (Art. 305.) 
 
 Find the sura of the following radicals : 
 
 4. '\/i2 and \/27. 9. 3V^ and 4V^i28. 
 
 5. V20 and V48. 10. 7^/243 and 5 A/363. 
 
 6. 2\^¥ and ^\^a^. 11. «V8i^ and 3«'v/49^. 
 
 7. a^/2,a% and cA/276/A 12. dV^S^.smd Vs^^c, 
 
 8. 3Vi8«a;2 and 2^/32^^. j^, 41^/^ and 5^/^. 
 
 SUBTRACTION OF RADICALS. 
 
 311. To Find the Difference between two Radicals. 
 
 I. From 3V45 subtract 2 A/20. 
 Analysis. — Reducing to the operation. 
 
 simplest form, we have gy^s 3'\/45 = 3^9 X V^ = gVs 
 
 and 4 a/ 5. which are like /— r r r 
 
 .^ ^^ r r 2V20Z11 2V4 X V5 =4V5 
 quantities. Now 9 y 5 — 4^/5 ■ 
 
 = 5a/5, the difference re- .•.3^45-2^20 = 5^/5 
 
 quired. Hence, the 
 
 310. How ad4 radioals ? iVpfe. How determine whether tUey are like quantities I 
 
MULTIPLICATION OF RADICALS. 161 
 
 EuLE. — Reduce the radicals to their simplest form ; change 
 the sign of the subtrahend, and proceed as in addition of 
 radicals, (Art. 310.) 
 
 (2.) (30_ (4.)_ 
 
 From 4 a/772 V480 4 A/320 
 
 Take V44^ 4 A/63 — 5-A/80 
 
 5. From 3a/49«S take 2A/25fit^. 
 
 6. From ^^/a + ^ take 3V^« + 5. 
 
 7. From 3a/^ take — 4a/^. 
 
 8. From 3V^25o^% take 2'V^54J%. 
 
 9. From — a~^ take — 2^"^. 
 10. From 5a/J take 2a/J. 
 
 MULTIPLICATION OF RADICALS. 
 312. To Multiply Radical Quantities. 
 
 1. What is the product of 3a/« by 2 a/^. 
 Analysis. — Since these radicals are operation. 
 
 of the same degree, we multiply the 3a/^ X 2\^b ■= 6\^ab 
 
 radical parts together, like rational 
 
 quantities, and to the result prefix the product of the coefficients. 
 
 2. Multiply 3a/^ by 2'\/c. 
 
 AiiTALYSTS. — As these radicals are of differ- opbbation. 
 
 ent degrees, they cannot he multiplied together 3 ^/a =z 3 (ai) 
 
 in their present form. We therefore reduce 3/— / 2\ 
 
 2 A/ C '~~ 2 ( C^ / 
 
 them to a common index, and then, multiply- ^ ^ 
 
 ing as before, we have 6^a^c\ ^^«- ^ {a^cP)^ 
 
 3. Multiply «3 by a^. 
 
 Analysis. — These radicals are of different degrees, but of the same 
 radical part or root ; we therefore multiply them by adding their 
 fractional exponents. ^ + 1 = f . Therefore, a^ x a^ = a^. Hence, the 
 
 311. How subtract radicals f 
 
162 MULTIPLICATION OF RADICALS. 
 
 Rule. — I. Reduce the radicals to a common index, 
 II. — Multiply the radical parts together as rational quan- 
 tities, and placing the result under the common index, prefix 
 to it the product of the coefficients. 
 
 Notes. — i. Roots of like quantities are multiplied together by 
 adding their fractional exponents. (Art, 94.) 
 
 2. This rule is based upon the principle that the product of the 
 roots of two or more quantities is the same as the root of the product. 
 (Art. 293, Prin. 3.) 
 
 3. The product of radicals becomes rational^ whenever the numer- 
 ator of the index can be divided by its denominator without a 
 remainder. 
 
 4. If rational quantities are connected with radicals by the signs + 
 or — , each term in the multiplicand must be multiplied by each term 
 in the multiplier. (Art. 98.) 
 
 Multiply the folkwing radicals: 
 
 4. 5a/i8 by 3V20. 10. OP- by q>, 
 
 5. a\/x by h^fx, 11. 7V^4 by 3^4. 
 
 6. ^/a + i by V« — ^. 12. V9« by Vi6a. 
 
 7. ^fax by ^fcy, 13. \/i8 by ^/ 2. 
 
 8. c^ by (^, 14. "^Zax by ^fwx, 
 
 (9-) (X5-) 
 
 Multiply a -f- V? Mult. « + \/x 
 
 By g-f- V5 By i + l\/x 
 
 ac + cVi a + Va; 
 
 + aVd + v^ -t- ahVx -f Ja: 
 
 J/i5. «c + cA/^-f «A/^-f V^ Ans. a-{-V^-\-ahVx-^bx 
 
 16. 2\/f by 2\/J. 18. (m-]-n)^hj{m-\'n)^ 
 
 17. 4V|by3V|. ^^^ J^^^J~^_. 
 
 313. How multiply radicals? iVbfe*. How arc roots of like quantities multi- 
 plied? Upon what principle is this rule based? When does the product of 
 radicalB become r'vtional? If radicals are connected with rational quantities, how 
 multiply them? 
 
DIVISION" OF RADICALS. 163 
 
 DIVISION OF RADICALS. 
 313. To Divide Radical Quantities. 
 
 1. Diyide 4^/^400 by 2\/Sa. 
 
 Analysis. — Since the given radicals are operation. 
 
 of the same degree, one may be divided by 4\/T4ac / — 
 
 the other, like rational quantities, the quo- /^— ^^ 2V3C 
 
 2 V o<^ 
 
 tient being Y 3c. (Art. iii.) To this result 
 
 prefixing the quotient of one coeflBcient divided by the other, we have 
 2 -v/sc, the quotient required. 
 
 2. Divide 4 Vac by 2 Vet. 
 
 Analysis. — Since these radicals are aVoc a (ac)^ 
 
 of different degrees, they cannot be — rp- = ^ 
 
 divided in their present form. We 2\/a 2 [a)^ 
 
 therefore reduce them to a common 4\/ttC 4 (aW^ 
 
 index, then divide one by the other, and •*• s/~~ ^^ , „. 1 
 
 2 A/ (l 2 \ (I I ^ 
 
 to the result prefix the quotient of the ^ ^ ' 
 
 coefficients. Tlie answer is 2^0^. or 2 (a(?)^, Ans. 
 
 . 3. Divide a^ by a^ 
 
 Analysis. — These radicals are of different opbeation. 
 
 degrees, but have the same radical part or root. ^h — ^1 
 
 We therefore divide them by subtracting the 1. 2 
 
 fractional exponent of the divisor from that of 
 
 the dividend. (Art. 113.) Reducing the expo- a^ — a^ z=: a^ 
 nents to a common denominator, a^ = a^, and . ^^ . /^^ — ■ rA 
 as— a«. Now a^-i-a^=a^, Ans. Hence, the 
 
 Rule.— I. Reduce the radical parts to a common index. 
 
 11. Divide one radical part hy the other, and placing the 
 quotient under the common index, prefix to the result the 
 quotient of their coefficients. 
 
 Note. — Boots of like quantities arfe divided Tyy subtracting thefrat- 
 tional exponent of the divisor from that of the dividend. (Art. T13.) 
 
 313. How divide radicals ? 2^ote. How divide roots of like quantities ? 
 
164 IKVOLUTIOK OF KADICALS. 
 
 Divide the following radicals : 
 4. ^/\2a^c by ^/ ^c. 10. 14a V^ by 7a/^. 
 
 9 
 
 d^Mx^ by 2^/dx. 11. (« + 5)t by (a + Z>)«. 
 
 (a^ -f fl^ic)^ by a^, 12. 3^500;^ by v^. 
 
 12 (ay)T by («y)i 13. ^/x^ — f- by ^/x-\-y, 
 
 2^1^/ ax by S-v/flf. 14. 16^/32 by 2^/4. 
 
 iS^cs/^rc by 2cV^. 15. 8^512 by 4^/2. 
 
 INVOLUTION OF RADICALS. 
 314. To Involve a Radical to any required Power. 
 
 I. Find the square of a^, 
 
 OPERATION. 
 
 Analysis. — As a square is the product of two 1 i 2 
 
 Ct^ X d^ "^^^ (X^ 
 equal factors, we multiply the given index by 
 
 the index of the required power. Hence, the .*. 0^, Ans. 
 
 EuLE. — Multiply the index of tlie root 'by the index of tlie 
 required power, and to the result prefix the required poivei 
 of the coefficient. 
 
 Note. — A root is raised to a power of the same name by removing 
 the radical sign or fractional exponent. (Ex. 2.) 
 
 2. Find the cube of ^/a + J. Ans. a ■\-b. 
 
 3. Find the cube of a^, 
 
 4. What is the square of 3a/2^. 
 
 5. What is the cube of 2\^. 
 
 X ■ 
 
 6. Eequired the cube of -V2X, 
 
 7. Required the cube of 4\ / — • 
 
 V 4 
 
 8. Find the fourth power of 3A / — 
 
 9. What is the square of « + \/y ? 
 
 314. How Involve radicals to any required power ? Note. How raise a root to a 
 yower of the same name? 
 
BVOLUTIOK OF RADICALS. 165 
 
 EVOLUTION OF RADICALS. 
 315. To Extract the Boot of a RadicaL 
 
 I. Find the cube root of a^'i/W. 
 
 Analysis. — Finding the root of a radi- ope ratiok. 
 
 cal is the same in principle as finding the V^a^V^^ = v ^3^1 
 
 root of a rational quantity. (Art. 296.) 3 
 
 Reducing the index of the radical to an V^^^t =: ab^, Ans. 
 
 equivalent fractional exponent, we extract 
 
 the cube root by dividing it by 3. The result is a&^. Hence, the 
 
 Rule. — Divide the fractional exponent of the radical by 
 the number denoting the required root, and to the result 
 prefix the root of the coefficient. 
 
 Notes. — i. Multiplying the index of a radical by any number is the 
 same as dividing the fractional exponent by that number. 
 
 Thus, /^a = aK Multiplying the former by 2, and dividing the 
 latter by 2, we have ^a = a". 
 
 2. If the coefficient is not a perfect power, it should be placed under 
 the radical sign and be reduced to its simplest form. (Art. 305.) 
 
 2. Required the square root of gV^, 
 
 3. Required the square root of 4\^^. 
 
 4. Find the cube root of 3a/^. 
 
 5. Find the cube root of 2bV2b. 
 
 6. What is the cube root of a (bc)^ ? 
 
 7. What is the fourth root of ^^f ? 
 
 8. What is the fourth root of Va^ V^^ ? 
 
 9. Find the seventh root of i28a/«. 
 10. Find the fourth root of Vayby, 
 
 II. Find the fifth root of 4a^V2a. 
 12. Find the nth root of aVbc. 
 
 315. How extract the root of a radical ? Notes. To what is multiplyinff the index 
 Of a radical equivalent ? If the coefficient is not a perfect power, what is dope ? 
 
166 CHAI^GIKG RADICALS TO llATIOKALS. 
 
 CHANGING A 
 
 RADICAL TO 
 QUANTITY. 
 
 A RATIONAL 
 
 CASE I. 
 316. To Change a Radical Monomial to a Rational Quantity 
 
 1. Change V^ ^^ ^ rational quantity. 
 
 Analysis. — Since multiplying a root of a 
 quantity into itself produces the quantity, it 
 follows that /y/a x ^ a = a, which is a rational 
 quantity. (Art. 287.) 
 
 2. It is required to rationalize c^. 
 
 OPERATION. 
 
 OPERATION. 
 
 Analysis. — A root is multiplied by another 
 root of the same quantity by adding the expo- ^i y^ ^1 -^^ ^ 
 
 nents ; therefore we add to the index \ such a 
 fraction as will make it equal to i. (Art. 94.) 
 
 Thus, as X as = as+ s = fl^i = «^ the rational quantity required. 
 
 3. It is required to rationalize ^. 
 
 Solution.— Multiplying a^ by ic«, the result is 
 a?, which is a rational quantity. Hence, the 
 
 OPERATION. 
 
 Rule. — Multiply the radical hy the same quantity having 
 such a fractional exponent as, when added to the given 
 expo7ient, the sum shall be equal to a unit, or i. 
 
 4. Required a factor which will rationalize a^. 
 
 5. What factor will rational 
 
 6. What factor will rational 
 
 7. What factor will rational 
 
 8. What factor will rational 
 
 9. What factor will rational 
 xo. What factor will rational 
 
 ze Va^c? 
 ze '^(a + by. 
 ze Va^? 
 ze ^^+jY? 
 ze \/(a+ by? 
 ze V{a + b + c)? 
 
 316. How reduce ft rftdicftl monomifll to » r^Uonal quantity? 
 
CHANGING RADICALS TO RATIONALS. 167 
 
 CASE II. 
 317. To Change a Radical Binomial to a Rational Quantity. 
 
 1. Ifc is required to rationalize ^s/a + ^/}). 
 Analysis, — The product of the sum and opbratiok. 
 
 difference of two quantities is equal to the /y/^ _i- ^J}) 
 
 difference of their squares (Art. 103); there- r- ,j- 
 
 fore (/;/«+ \/&) multiplied by {'s/a—^h) 
 
 = a—b, which is a rational quantity. a + ^/ob 
 
 Therefore, the factor to employ as a multi- — yai — b 
 
 plier is V«- V^- a — b, AnsT 
 
 2. What factor will rationalize Vx — Vy ? 
 
 Analysis.— If the binomial ^x — y'y is operation.^ 
 
 multiplied by the same terms with the sign of Vcc — Wy 
 
 the latter changed to + , we have w'~ . ^T. 
 
 (\/x - Y^y) X {^^x+ ^y) = x-y. ~~~ ~ 
 
 _ _ JO *""• y 
 
 (Art. 103.) Therefore, y\/x + ^/y is the fac- ,- .- 
 
 tor required. Hence, the ^^^- V^+Vy 
 
 Rule. — Multiply the binomial radical by the correspond- 
 ing binomial with its connecting sign changed, 
 
 3. What factor will rationalize x -\- 4 V9 ? 
 
 4. Rationalize A/9 — V^. 
 
 5. What factor will rationalize VT + Vet ? 
 
 6. Rationalize 6 — V^. 
 
 7. What factor will rationalize Vsa — V^b ? 
 
 8. Rationalize Vci' — Vs. 
 
 9. What factor will rationalize 3 's/a + a/S ? 
 10. What factor will rationalize 4 V^a — 5 V^? 
 
 317. How reduce a radical binomial to a rational quantity ? 
 
168 RADICAL FRACTIONS. 
 
 CASE III. 
 
 318. To Change a Radical Fraction to one whose Numerator 
 or Denominator is a Rational Quantity. 
 
 1. Change-^ to a rational denominator. 
 
 Analysis. — Multiply both terms of the operation. 
 
 fraction by the denominator A^h, and the a y^ ^^ aVb 
 
 result is —x— , whose denominator is V ^ X yb ^ 
 
 rational. (Art. 167, Prin. 3, note.) Hence, the 
 
 Rule. — Multiply both terms of the fraction ly such a 
 factor as will malce the required term ratio7ial. 
 
 Note — Since the product of the sum and difference of two quan- 
 tities is equal to the difference of their squares, when the radical 
 fraction is of the form — =, if we multiply the terms by 
 
 (y'a + -y/ft), we have a — & for the denominator. (Art. 103.) 
 
 2. Rationalize the denominator of -~z» Ans. - — ^. 
 
 Va ^ ^ a 
 
 3. Rationalize the numerator of ---• Ans, 
 
 ^/x Vox 
 
 4. Rationalize the denominator of -■ 
 
 x 
 
 5. Rationalize the denominator of — -• 
 
 6. Rationalize the denominator of —^ —• 
 
 ^x — Vy 
 
 Of 
 
 7. Rationalize the denominator of —j= —* 
 
 I 
 
 I + V3 
 
 3 — V3 
 
 8. Rationalize the denominator of 
 
 9. Rationalize the denominator of 
 
 318. How reduce a radical fraction to one whose numerator or denominator is a 
 rational quantity? \^'^len the fractions contain compound quantities, what prin- 
 ciple enters into their reduction? 
 
BADICAL EQUATIOI^S. 169 
 
 RADICAL EQUATIONS. 
 
 319. A Madical Bquation is one in whiCh the 
 unknown quantity is under the radical sign. 
 
 320. To Solve a Radical Equation. 
 I. Given a/^ 4-2 = 7, to find x. 
 
 Analysis. — Transposing 2, we have, y5 + 2 = 7 
 
 /\/x = 5. Since 5 is equal to the >\/x, it /— 
 
 follows that the square of 5, or 25, must be V — 7 — — 5 
 the square of y\/x. Therefore, x = 25. 
 
 a; = 52 = 25 
 
 2. Given 2a -{- \/x = ()a, to find x. 
 
 Solution. — ^By the problem, la + ^Jx — t^n 
 
 By transposing, ^x = ya 
 
 By involution, x = 49a' 
 
 3. Given 5V^a; + i = 35, to find x. 
 
 Solution.— By the problem, 5/y/a; + i = 35 
 
 Removing coefficient, ^^x + i = 7 
 Involving, a? + i = 343 
 
 Transposing, X = 342. Hence, the 
 
 Rule. — Involve both sides to a power of the same name as 
 the root denoted by the radical sign. 
 
 Note. — Before invohing the quantities, it is generally best to clear 
 of fractions, and transpose the terms, so that the quantities under tJ.a 
 radical sign shall stand alone on one side of the equation. 
 
 Eeduce the following radical equations: 
 
 4. « H- v^ -\- c = d. 
 
 8. v^22; + 3 — 6 = 13. 
 
 5. ^/X+ 2z=S. 
 
 9. v"^ — 4 = 3- 
 
 6. sVa; — 4 4-S = 7i• 
 
 10. 2^a; — 5=4. 
 
 7• 3a/^=24. 
 
 II. 5A^ = 30. 
 
 319. ^ hat is a radical equation ? 320. flow solved ? Note. What should be done 
 before involving the quantities ? 
 
 8 
 
170 RADICAL EQUATIONS, 
 
 12. Given ^ "^^ — -i— = J2, to find iR 
 
 13. Reduce a/cl^ +Vx=z ^/"^^^ . 
 
 Analysis. — By removing tlie opkkation. 
 
 denominator the first member is /~Z 7=^ 3 + ^ 
 
 squared. But x is still under y -r V ^ . / v 
 the radical sign. This is re- 
 
 moved by involving both mem- ^ + v ^ = 3 + ^ 
 
 bers again. Atis. X = ($ -\- C — fl') 
 
 14. Given ^^^ = :^, to find ST. 
 
 2^2 
 
 15. Given x + a/«^ 4- ic^ = _______ to find a;. 
 
 Note. — If the equation has two radical expressions, connected with 
 other terms by the signs + or — , it is advisable to transpose the terms 
 so that one of the radicals shall stand alone on one side of the equation. 
 By involving both members, one of the radicals becomes rational ; and 
 by repeating the operation, the other will also disappear. 
 
 16. Given V4 + 5.'^^ — ^z^ =-- 2, to find x. 
 Transposing, 'Y/4 + S-c = ^/3x + 2 
 
 Involving, 4 + 5« = 4 + 4\/3'« + 3« 
 
 / — ^ 
 Transiiosing and dividing by 4, ysx — - 
 
 Involving, ^x = — 
 
 4 
 
 Transposing and multiplying by 4, x'^ = I2j; 
 Hence, x = 12, Ana. 
 
 17. Given a/^ -f 12 r=: 2 + v^, to find x. 
 
 18. Given Vs x Vx 4-2 = 2 + Vs^, to find Xo 
 
 ^. Vx X — ax , „ , 
 10. Given = — — , to find x, 
 
 (See Appendix, p. 289.) 
 
CHAPTER XV. 
 QUADRATIC EQUATIONS. 
 
 321. Equations are divided into different degrees, as the 
 fii-st, second, third, etc., according to the powers of the 
 unknown quantity contained in them. 
 
 An equation of the First Degree is called a Simple 
 Equation^ and contains only the /r*^ power of the unknown 
 quantity. 
 
 An equation of the Second Degree is called a Quad- 
 vatic Equation, and the highest power of the unknown 
 quantity it contains is a square. 
 
 An equation of the Third Degree is called a Cuhic 
 EquoMon, and the highest power of the unknown quantity 
 it contains is a cul:)e. 
 
 An equation of the Fourth Degree is called a 
 Biquadratic, etc. 
 
 322. Quadratic Equations ai-e divided into pure 
 
 and affected. 
 
 323. A Pure Quadratic contains the square only of 
 the unknown quantity ; as, x^ = 5. 
 
 324. An Affected Quadratic contains both the first 
 and second powers of t h e unkn own quantity ; as, x^-\-ax= cd. 
 
 Notes. — i. Pure quadratics are sometimes called incomplete equa- 
 tions ; and affected quadratics, complete equations. 
 
 321. How are equations divided ? What is an equation of the first de^^ree ? The 
 second? Third? Fourth? 322. How are quadratic equations divided ? 325. What 
 I? h pure quadratic ? 324. An afl;"ected quadratic ? Note. What are they sometimes 
 called ? 
 
172 PURE QUADRATICS. 
 
 2. A Complete Equation contains every integral power of tlie un- 
 known quantity from that which denotes its degree down to the zero 
 power. 
 
 An Incomplete Equation is one which lacks one or more of these 
 powers. 
 
 PURE QUADRATICS. 
 
 325. Every pure quadratic may be reduced to the form 
 
 a;3 = «. 
 
 For, by transposition, etc., all the terms containing x^ can be 
 reduced to one term, as lyx^ ; and all the known quantities to one 
 term, as c. Then will 
 
 W = c. 
 
 Dividing both members by &, and substituting a for the quotient of 
 e-r-b, the result is the form, 
 
 x^ = a. 
 
 326. Pure quadratic equations have two roots, which are 
 the same numerically, but have opposite signs. (Art. 293, 71.) 
 
 Thus, the square of +a and of —a is equally a^. Hence, 
 
 V^= ±a, 
 
 327. To Solve a Pure Quadratic Equation. 
 
 I, Find the value of a; in 6 = h 2, 
 
 9 3 
 
 Solution.— Given 6 = — + 2 
 
 9 3 
 
 Clearing of fractions, 5a^ — 54 = 3^^ + l8 
 Transposing, etc., 2x'^ = 72 
 
 Removing coefficient, a^ = 36 
 
 Extracting sq. root, x = ±6, Ans, 
 
 Substituting b for 2, and c for 72, in the third equation, we have 
 the form, bx^ = c. 
 
 Removing coefficient, etc., x^ = a. Hence, the 
 
 Rule. — Reduce tlie given equation to the form x^ = a, and 
 extract the square root of holh members. (Art. 296.) 
 
 336. Hqw manjT roots nas a pure ooa^raticf 
 
PURE QUADRATICS. 173 
 
 Find the value of x in the following equations : 
 
 2. 3^5^ — 5 = 70. 10. 20^ + 12 ^ 3a;2_ 27. 
 
 3. 92?^ + 8 = 3a;2 -f- 62. 11. 72^2 — 7 = 32^2 + 9. 
 
 4. 5a;2 -I- 9 = 2a;2 ^ 27. 12. ahy^ = a*. 
 
 5. 6a« + 5 = 40?^ + 55. 13. {x + 2)2 = 4a; + 5. 
 
 6. ^ + 35 = 3^+7. 14. ii^-i=^^^. - 
 4 4 
 
 £^±8_^-6 ^(2^+9) __ 3^-1-6 
 
 8. - = ". 16. —5 1 f- = -. 
 
 42a; 4 — a;4H-a;3 
 
 :r . 2 a; . ^ ax^ia — 2) 
 
 9. - + - = - + -• 17. — ~T = I — ic. 
 
 328. Radical equations, when cleared of radicals, often 
 become pure quadratics. 
 
 18. Given Vx^ + n = \/2x^ — 5, to find a:. 
 
 Solution.— Clearing of radicals, aj2 + ii = 2aJ' — 5 
 
 Transposing and extracting root, jc = ± 4 
 
 4a; 
 
 19. Given 2V^ — 5 = — , to find ar. 
 
 20. Given 2V^--4 = 4V^---i, to find a?. 
 
 21. Given ^/^"-M = , to find a?. 
 
 V a; — ^ 
 
 22. Given \/- — ^^^ = Vx, to find a; 
 
 23. Given = 'v/iTT^, to find x. 
 
 24. Given ■ = \/x — lo, to find x, 
 
 vx + 10 
 
 327, What is the rule for the solution of pure quadratics? 328. What may 
 radical equations become ? 
 
174 PUEB QUADKATIOS. 
 
 PROBLEMS 
 
 1. The product of one-third of a number multiplied by 
 one-fourth of it is io8. What is the number ? 
 
 2. What number is that, the fourth part of whose square 
 being subtracted from 25, leaves 9 ? 
 
 3. How many rods on one side of a square field whose 
 area is 10 acres ? 
 
 4. A gentleman exchanges a rectangular piece of land 
 50 rods long and 18 wide, for one of equal area in a square 
 form. Kequired the length of one side of the square. 
 
 5. Find two numbers that are to each other as 2 to 5, and 
 whose product is 360. 
 
 6. If the number of dollars which a man has be squared 
 and 7 be subtracted, the remainder is 29. How much 
 money has he ? 
 
 7. Find a number whose eighth part multiplied by its 
 fifth part and the product divided by 16, will give a quotient 
 of 10. 
 
 8. The product of two numbers is 900, and the quotient 
 of the greater divided by the less is 4. What are tlie 
 numbers ? 
 
 9. A merchant buys a piece of silk for I40.50, and the 
 price per yard is to the number of yards as 3 to 54. 
 Required the number of yards and the price of each. 
 
 10. Find a number such that if 3 times the square be 
 divided by 4 and the quotient be diminished by 12, the 
 remainder will be 180. 
 
 11. A reservoir whose sides are vertical holds 266,112 
 gallons of water, is 6 feet deep, and square on the bottom. 
 Required the length of one side, allowing 231 cubic inches 
 to the gallon. 
 
 12. What number is that, to which if 10 be added, and 
 from which if 10 be subtracted, the product of the sum and 
 difference will be 156 ? 
 
AFFECIEB QUADBAIICS 175 
 
 AFFECTED QUADRATICS* 
 
 329. An Affected Quadratic Equation is one 
 
 which contains the^rs^ and second powers of the unknown 
 quantity ; as, aoi? -^ Ix z=i c, 
 
 330. Every affected quadratic may be reduced to the form, 
 
 in which a, b, and x may denote any quantity, either 
 positive or negative, integral ov fractional. 
 
 For, by transposition, etc., all the terms containing x^ can be 
 reduced to one term, as cx"^ ; also, those containing x can be reduced 
 to one term, as dx ; and all containing the known quantities can be 
 reduced to one term, as g. Then, cx^ + dx = g. 
 
 Dividing both members by c, and substituting a for the quotient of 
 d-r-c, and b for the quotient of g -i- c, we have, 
 a^ + ax = b. 
 
 Take any numerical quadratic, as-^ S = x^ + — — 4, 
 
 Clearing of fractions, Saj^ — 4a; — 48 = 6aJ* + 4a; — 24 
 
 Transposing, etc., 2af^ — Sx = 24 
 
 Removing the coefficient, a^ — 4X = 12 
 
 Substituting a for 4, and 6 for 12 in the last equation, we have, 
 x^ — ax = b. Hence, 
 
 All affected quadratics mag be reduced to the general form, 
 %^ ±^ax = b, 
 
 331. The First Member of the general form of an 
 affected quadratic equation, it will be seen, is a Bi?iomial, 
 but not a Complete Square. One term is tvanting to make 
 the square complete. (Art. 266, note.) The equation, 
 therefore, cannot be solved in its present state. 
 
 329. What is an affected quadratic equation ? 330. To what general form may 
 every affected quadratic be reduced? 331. What is true of the first member of the 
 general form of an affected quadratic ? 
 
 * Quadratic, ivGm the Latin qtiadrare^ to make square. 
 Affected, made up of different powers ; from the Latin ad and/acw>, 
 to make or join ta 
 
176 AFFECTED Q tJ ADHATICS. 
 
 332. There are three methods of completing the square 
 and solving the equation. 
 
 FIRST METHOD. 
 
 I, Given a:' + 2aa; = J, to find the value of x. 
 
 Analysis. — The first opeeation. 
 
 and third terms of the 01^ -{- 2ax =. b 
 
 Bquare of a binomial are a? -j- 2ax •\- d^ =. a^ -{■ h 
 complete powers, and the x 4- a = -h a/^2 i a 
 
 second term is twice the 
 
 product of their roots; 
 
 or the product of one of the roots into twice the other. (Art. loi.) 
 
 In the expression, x'^ + 2ax, the first term is a perfect square, and 
 the second term 2ax consists of the factors 2a and x. But x is the 
 root of the first term x'^ ; therefore, the other factor 2a must be tmce 
 the root of the third term which is required to complete the square. 
 Hence, half of 2a, or a, must be the root of the third term, and a^ the 
 term itself. Therefore, x- + 2ax + a^ is the square of the first member 
 completed. 
 
 But since we have added a^ to the first member of the equation, 
 we must also add it to the second, to preserve the equalitj. 
 Extracting the square root of both members, and transposing a, we 
 have x = ~a± ^/a'^ + b, the value sought. (Art. 297.) 
 
 2. What is the value of a; in 20^ + x = 64 — yx? 
 
 Analysis. — 'transposing — 70? operation. 
 
 and removing the coefficient of a^, 2X + x =z 64. — 'jx 
 
 we have the form x^ + 4X = 32. But 23^ -}- 8x = 64 
 
 the first member, x'^ + 4X, is an incom- a^ -{- 4X =z 22 
 
 plete square of a binomial. x^ 4- ax 4- A ^= '16 
 
 In order to complete the square, 
 we add to it the square of half the 
 coefficient of x. (Art. 266.) Now, 
 
 having added 4 to one member of *• €., X = 4 OT — 8 
 the equation, we must also add 4 to 
 
 the other, to preserve their equality. Extracting the root of both, 
 
 and transposing, we have a; = 4, or — 8. (Art. 297.) 
 
 333. How many methods of completing the square ? 
 
 a?+ 2 = ± 6 
 
 X= — 2 ±6 
 
AFFECTED QUADRATICS. 177 
 
 Notes.— I. Adding the square of half the coefficient of the second 
 term to both membei-s of the equation is called completing the square. 
 
 2. The first member of the fourth equation is the square of a bino- 
 mial ; therefore, its root is found by taking the roots of the first and 
 third terms, which are perfect powers. (Art. 297.) From the process 
 of squaring a binomial, it is obvious that the middle term (4a;) forms 
 no part of the root. (Art. 266.) 
 
 333. From these illustrations we derive the following 
 
 EuLE. — I. Reduce the equation to the form, x^ ±ax = b. 
 
 II. Add to each member the square of half the coefficient ofx 
 
 III. Extract the square root of each, and reduce the result 
 ing equation. 
 
 3. Find the value of a; in — 2:?? -f 2>ax = — 65. 
 
 Solution. — By the problem, —2a;' + ^ax = —6b 
 
 Removing coefficient of x^, —x'^ + ^ax = — 3& 
 
 Making x^ positive (Art, 140, Prin. 3), x^—4ax = 3& 
 Completing square, a^—^ax + ^a^ — 4a' + 3& 
 
 Extracting the root, x—2a = ± Y4a'^ + 3ft 
 
 .*. x = 2a± ^\a? + 38 
 4. Given ^ ■\- ax -\- bx zzz d, to find x, 
 
 OPEBATION. 
 
 ix? ■{■ ax -\- bx =z d 
 a^+ (a -\-b)x = d 
 
 Analysis. — Factoring the terms which contain the first power of x, 
 we have ax + hx = {a + b)x ; hence, (a + h) maybe considered a com* 
 pound coefficient of x. By adding the square of half this coefficient to 
 both members, and extracting the root, the value of x is found. 
 
 333. What is the rule for the first method of solving affecj^d quadratics » 
 
178 AFFECTED QUADRATICS, 
 
 5. Given 32; — 2ic2 _ _ ^^ ^0 ^^^ ^.^ 
 Solution. — By the problem, 3X — 2X^ = — g 
 
 Making x^ positive, etc., 
 
 
 x^- 
 
 3X_ 
 2 
 
 .9 
 2 
 
 
 Completing square. 
 
 X'- 
 
 f- 
 
 9 
 16 
 
 2 16 
 
 81 
 ~ 16 
 
 Extracting root, 
 
 
 X- 
 
 _3 ^ 
 4 
 
 ±9 
 4 
 
 
 /. X = 
 
 f±f. 
 
 i. e. 
 
 , X = 
 
 : 3 or — 
 
 li 
 
 6. Given $x^ — 24X = — 36, to find x. 
 
 Ans. + 6 or +2. 
 
 Note, — The two Toots of an affected quadratic may have the same 
 or different signs. Thus, in the 6th and 12th examples they are the 
 same; in the ist, 2d, 3d, 4th, and 5th, they are different. 
 
 7. Given ^x^ — 40a; = 45. to find x. 
 
 8. Given x^ — 6ax = d, to find x. 
 
 9. Given 2X^ + 2ax = 2 (J -f c), to find x. 
 
 Solution. — Completing the square, x"^ + ax -^ — = b b + c, 
 
 4 4 
 
 Extracting root, x+- = ±-i/— + b + c 
 2 'A 
 
 Transposing, 3?= ± y — \- b + c 
 
 10. Given 2X^ — 22a; = 120, to find x. 
 
 11. Given x^ — 140 z= 13a;, to find x. 
 
 12. Find the value of a; in' x^ — ^x ■{• 1 = k^x — 15. 
 
 Solution. — By the problem, a;^— 3a; + i = 5a;— 15 
 Transposing, aj*— 8a; = — 16 
 
 Completing square, a;'— 8a5+i6 = o 
 
 Extracting root, «— 4 = o 
 
 .*. a; = 4 
 
 Note. — In this equation, both the signs and the numerical values of 
 the two roots are alike. Such equations are said to have equal roots. 
 
 i\ro<«,— What signs have the roots of an affected quadratic f 
 
AFFECTED QUADRATICS. 179 
 
 SECOND METHOD. 
 
 334. When an affected quadratic equation has been 
 reduced to the general form, 
 
 x^ -\- ax = h, 
 its root may be obtained without recourse to completing the 
 square. 
 
 I. Given x^ -{-Zx=. 65, to find x. 
 
 Analysis. — After the square operation. 
 
 of an affected quadratic is com- /y3 a_ Q/y (\c 
 
 pleted and the root extracted, 
 the root of the third term is 
 
 a; = — 4 ± a/65T^6 
 transposed to the second mem- •*• 2; = 4 i 9 
 
 her, by changing its sign. (Art. i,e.y X = ^ or — 1 3 
 
 204.) 
 
 Now, if we prefix half the coeflScient of x, with its sign changed, to 
 plus or minus the square root of the second member increased by the 
 square of half the coefficient of x, the second member of the equation 
 will contain the saiTie combinations of the same terms, as when the 
 square is completed in the ordinary way. Hence, the 
 
 Rule. — Prefix half the coefficient of x, with the opposite 
 sign, to plus or miuus the square root of the second member, 
 increased hy the square of half the coefficietit of x. 
 
 Solve the following equations : 
 
 2. 
 
 3^^ — 9^ — 3 = 207- 
 
 8. 
 
 a« -f- 4ax — b. 
 
 3- 
 
 42^2+ 120; + 5=45. 
 
 9- 
 
 32)2 — 74 = 6a; + 31. 
 
 4. 
 
 ix^— 14a: -f- 15=0. 
 
 10. 
 
 a^ -\- i^ = 6x. 
 
 5. 
 
 4a;2 _ pa; __ 28. 
 
 II. 
 
 {X — 2) {X— l)= 20c 
 
 6. 
 
 2X X + 2 
 
 12. 
 
 x+ 1 X _is 
 X ^ x-\-\ 6 ' 
 
 7. 
 
 a^ + ^--ah = d, 
 I? 
 
 13. 
 
 ^^^-j^ch = bd. 
 c 
 
 334. What is the second method of solving affected quadratics ¥ 
 
180 AFFECTED QUADRATICS. 
 
 THIRD METHOD. 
 
 335. A third method of reducing an affected quadratic 
 equation may be illustrated in the following manner: 
 
 1. Given a'3^ -^-Ixzzl c, to find x. 
 Analysis. — Multiply- operation. 
 
 ing the given equation by CL^ -\- hx ^z c 
 
 a, the coefBcient of a;^ and /^a^a^ + /[ahx = ^ac 
 
 by 4, the smallest square ^^Z^s _j_ ^al)X-\- W = 4«c -f J3 
 
 number, we have ^^«. i z> i . / i — m 
 
 4«2a;2 + ^ahx = ^ac, _L 
 
 the first term of which is . /„ _ — ^±V4^g + ^ 
 
 an exact square, whose 2a 
 
 root is lax. Factoring 
 the second term, we have /s^abx = 2 (2aajx5). (Art. 119.) 
 
 As the factor 2.ax is the square root of 4a^a;-, it is evident that ^a^^ 
 may be regarded as the first term, and /^cibx the middle term of the 
 square of a binomial. Since ^ahx is twice the product of this root lax 
 into &, it follows that & is the second term of the binomial ; conse- 
 quently, 6' added to both members will make the first a complete 
 square, and preserve the equality. (Axiom 2.) Extracting the square 
 root, transposing, etc., we have, 
 
 X = — — , the value of x required. 
 
 2. Given 2:1^ 4- 3a; = 27, to find x. 
 
 Solution. — By the problem, 2X^ + 3a; = 27 
 
 Multiplying by 4 times coef. of a;*, i6ar'^ + 24a; = 216 
 Adding square of 3, coef. of a?, i6«2 + 24a; + 9 = 225 
 Extracting root, 4a;+3 = ± 15 
 
 Transposing, 425 = —3 ± 15 
 
 .\ flj = 3 or — 4|. 
 
 336. From the preceding illustrations, we derive the 
 KuLE. — I. Reduce the equalion to the form, aa^ ±_bx = c. 
 
 II. Multiply both members hy 4 tiines the coefficient of a^. 
 
 III. Add the square of the coefficient of x to each member, 
 extract the root, and reduce the resulting equation. 
 
 336. What is the rale for the third method of reduciug affected qoadrhtics f 
 
AFFECTED QUADRATICS. 181 
 
 Notes. — i. Wlien the coeflBcient of x is an even numbei it is 
 sufficient to multiply both members by the coefl&cient of x^, and add 
 to each the square of half tlie coeflScient of x. 
 
 2. The object of multiplying the equation by the coefficient of x^ is 
 to make the first term a perfect square without removing the coefficient. 
 (Art. 251.) 
 
 3. The reason for multiplying by 4, is that it avoids fractions in 
 completing the square, when the coefficient of a; is an odd number. 
 For, multiplying both members by 4, and adding the square of the 
 entire coefficient of x to each, is the same in effect as adding the square 
 of half the coefficient of x to each, and then clearing the equation of 
 fractions by multiplying it by the denominator 4. 
 
 4. This method of completing the square is ascribed to the Hindo<^ 
 
 3. Given ^a^ -\- 4X = 39, to find x, Ans. 3 or — 4^. 
 
 Reduce the following equations : 
 
 4. x^ — $0=: —X. 8. 2aj2 — 6a; = 8. 
 
 5. 5^ -f 32)2 _ 2. 9. 32^2 4- 5a; = 42. 
 
 6. 4X^ — 'jx — 2 = o. 10. a^ — 15a; = — 54. 
 
 7. 50^2^ 2a; = 88. II, 92^— 7a; =116. 
 
 337. The preceding methods are equally applicable to all 
 classes of affected quadratics, but each has its advantages in 
 particular problems. 
 
 The first is perhaps the most natural, being derived from 
 the square of a binomial ; but it necessarily involves frac- 
 tio7iSy when the coefficient of x is an odd number. 
 
 The second is the shortest, and is therefore 2i favorite with 
 experts in algebra. 
 
 The advantage of the third is, that it always avoids 
 fractions in completing the square. 
 
 The student should exercise his judgment as to the method 
 best adapted to his purpose. 
 
 Notes. When the coefficient of x is an even number, how proceed ? Object oJ 
 multiplying by coefficient Qix'^'i By 4 ? 
 
182 AFFECTED QUADRATICS. 
 
 EXAMPLES. 
 
 Find the yalue of x in the following equations: 
 
 1. x^ — ^x^—i, 17. 3:^2 _ 7;;c _ 20 = o. 
 
 2. a;2_^^_,_^^ jS. ^x^ -- ido ^ IX, 
 
 3. 2a;2 __ y^ _. _ 2. 19. 2X^— 2X=\\. 
 
 4. a;2 -{- loic = 24. 20. (ir — 2) (a; — i) = 6. 
 
 5. 6:^2— 130: + 6 = 0. 21. 4(i»2__ i) _4^__ I, 
 
 6. 14a; — a;2 = 33. 22. (2X — 3)2 := 8i?;. 
 
 7. iz;2— 3 = — _^. 23. 3ic— 2 = 
 
 6 •'^' ^- --a;-i 
 
 8. ^^ = ^-^. 24. 4a; = 14. 
 
 16 100 — ga? „ t 40? 
 
 to. - + - = -. 26. a;« 4- - = -. 
 
 X a a 22 
 
 11. a^ + 2/?za; = i^. 27. x^ — 2nx = m^ — 7A 
 
 12. a^ + f=ii. .28. 9£.:=£)^£Z13«. 
 
 '3- ^2_6^-+y-^-3- 
 
 14 
 
 4^ a; — I 9a? + 7 
 
 14 — ic 3ic ~~ X * 
 
 15. 2A/rc2 — 4^ — J __ _ ^^ 
 
 16. V^^S + 6 = a; + 5. 
 
 338. An Equation which contains but two powers of the 
 unknown quantity, the mdex of one power being kvice that 
 of the other, is said to have the Quadratic Form. 
 
 The indices of these powers may be either integral or 
 fractional. 
 
 Thus, a^— a;2 = 12 ; aj^^ + ar* = A ; and ^x — ^x = c, are equations 
 of the quadratic form. 
 
 Note. — Equations of this character are sometimes called trinomial 
 equations. 
 
 338. When has an equation the ouadratic forinf 2fo(e. What are such equatioDB 
 called? 
 
AFFECTED QUADRATICS. 183 
 
 339. Equations of the quadratic form may be solved by 
 the rules for affected quadratics. 
 
 I. Given a^ — 2X^ = 8, to find x, 
 
 Solution — By the problem, ic* — 2a;' = 8 
 Completing square, x^ — 2X^ + i = g 
 
 Extracting square root, aj^ — i = i 3 
 
 Transposing, a:^ = 4 or — 2 
 
 Extracting square root again, a; = ± 2, or ± ^y/— 2 
 
 . 2. Given 0^ — 40^ = ^2, to find x. 
 
 Solution. — By the problem, afi — ^ = 32 
 
 Completing square, afi — 4iB^ + 4 = 36 
 Extracting square root, ar* — 2 = ± 6 
 
 Transposing, etc., a;' =1 8 or —4 
 
 Extracting cube root, . a? = 2 or ^y^— 4 
 
 3. Given x^" — 4bx^ = a, to find x. 
 
 Solution.— By the problem, x^* — 45a?* = a 
 
 Completing square, a;-^'* — 4hx^ + 45* = a + 45* 
 
 Extracting square root, a;» — 26 = ± ya+4ljr' 
 
 Transposing, a;" = 2& ± ya+^ 
 
 Extracting the Tith root, x = V 26 ± ya+4b^ 
 
 4. Given x^ + S = 6x^, to find x. 
 
 5. Given x* — 2x^ = 3, to find ar. 
 %. Given afi — ju^ = o, to find x. 
 
 x^ X 1 
 /. Given {- - = -^, to find ar. 
 
 2 ^ 4 32' 
 
 8. Given v^ + ^\/x — i, to find x. 
 
 9. Given 4a; + 4\/x +2 = 7, to find x. 
 
 10. Given ^, ' _ = - — -=--, to find «. 
 4 + V a; VaJ 
 
184 AFFECTED QUADRATICS. 
 
 PROBLEMS. 
 
 1. Find two numbers such that their sum is 12 and theii 
 product is 32. 
 
 2. A gentleman sold a picture for I24, and the per cent 
 lost was expressed by the cost of the picture. Find the cost 
 
 Note. — Let ic^tlie cost. 
 
 Then = the per cent. 
 
 100 ^ 
 
 X 
 
 We now have x — x y. — = 24, to find the value of 05. - 
 100 
 
 3. The sum of two numbers is 10 and their product is 24. 
 What are the numbers ? 
 
 4. A person bought a flock of sheep for |8o ; if he had 
 purchased 4 more for the same sum, each sheep would have 
 cost %i less. Find the number of sheep and the price of 
 each. 
 
 5. Twice the square of a certain number is equal to 65 
 diminished by triple the number itself. Required the 
 number. 
 
 6. A teacher divides 144 oranges equally among her 
 scholars ; if there had been 2 more pupils, each would have 
 received one orange less. Required the number in the 
 school. 
 
 7. A father divides $50 between his two daughters, in 
 such a proportion that the product of their shares is $600. 
 What did each receive? 
 
 8. Find two numbers whose sum is 100 and their product 
 2400. 
 
 9. The fence enclosing a rectangular field is 128 rods 
 long, and the area of the field is 1008 square rods. What 
 are its length and breadth ? 
 
 10. A colonel arranges his regiment of 1600 men in a 
 solid body, so that each rank exceeds the file by 60 soldiers. 
 IIow many does he place in rank and file ? 
 
AFFECTED QUADRATICS. 185 
 
 11. A drover buys a number of lambs for $50 and sells 
 them at I5.50 each, and thus gains the cost of one lamb. 
 ^Required the number of lambs. 
 
 12. The sura of two numbers is 4 and the sum of their 
 reciprocals is i. What are the numbers ? 
 
 13. The sum of two numbers is 5 and the sum of their 
 cubes 65. What are the numbers? 
 
 14. The length of a lot is i yard longer than the width 
 and the area is 3 acres. Find the length of the sides. 
 
 15. A and B start together for a place 300 miles distant; 
 A goes I mile an hour faster than B, and arrives at his 
 journey's end 10 hours before him. Find the rate per hour 
 at which each travels. 
 
 16. A and B distribute $1200 each among a certain 
 number of persons. A relieves 40 persons more than B, and 
 B' gives to each person $5 more than A. Required the 
 number relieved by each. 
 
 17. Divide 48 into two such parts that their product may 
 be 252. 
 
 18. Two girls, A and B, bought 10 lemons for 24 ceiL% 
 each spending 12 cents ; A paid i cent more apiece than B: 
 how many lemons did each buy ? 
 
 19. Find the length and breadth of a room the perimeter 
 of which is 48 feet, the area of the floor being as many 
 square feet as 35 times the difference between the length 
 and breadth. 
 
 20. In a peach orchard of 180 trees there are three more 
 In a row than there are rows. How many rows are there, 
 and how many trees in each ? 
 
 21. Find the number consisting of two digits whose sum 
 is 7, and the sum of their squares is 29. 
 
 22. The expenses of a picnic amount to $10, and this sum 
 could be raised if each person in the party should give 30 cts. 
 more than the number in the party. How many compose 
 the party ? 
 
186 AFFECTED QUADKATICS. 
 
 23. Find two numbers the product of which is 120, and 
 if 2 be added to the less and 3 subtracted from the greater, 
 the product of the sum and remainder will also be 120. 
 
 24. Divide 36 into two such parts that their product shall 
 be 80 times their difference. 
 
 25. The sum of two numbers is 75 and their product is 
 to the sum of their squares as 2 to 5. Find the numbers. 
 
 26. Divide 146 into two such parts that the difference of 
 their square roots may be 6. 
 
 27. The fore- wheel of a carriage makes sixty revolutions 
 more than the hind-wheel in going 3600 feet ; but if the 
 circumference of each wheel were increased by three feet, it 
 would make only forty revolutions more than the hind- 
 wheel in passing over the same distance. What is the 
 circumference of each wheel ? 
 
 28. Find two numbers whose difference is 16 and their 
 product $6. 
 
 29. What two numbers are those whose sum is i J and the 
 sum of their reciprocals si ? 
 
 30. Find two numbers whose difference is 15, and half 
 their product is equal to the cube of the less number. 
 
 31. xi lady being asked her age, said, If you add the 
 square root of my age to half of it, and subtract 12, the 
 remainder is nothing. What is her age ? 
 
 32. The perimeter of a field is 96 rods, and its area is 
 equal to 70 times the difference of its length and breadth. 
 What are its dimensions ? 
 
 33. The product of the ages of A and B is 120 years. If 
 A were 3 years younger and B 2 years older, the product of 
 their ages would still be 120. How old is each ? 
 
 34. A man bought 80 pounds of pepper, and ^6 pounds 
 of saffron, so that for 8 crowns he had 14 pounds of pepper 
 more than of saffron for 26 crowns; and the amount he 
 laid out was 188 crowns. How many pounds of pepper did 
 he buy for 8 crowns ? 
 
SIMULTAKEOUS QUADRATICS. 187 
 
 SIMULTANEOUS QUADRATIC EQUATIONS. 
 TWO UNKNOWN QUANTITIES. 
 
 340. A Moniogeneotis Equation is one in which 
 the sum of the exponents of the unknown quantities is the 
 same in every term which contains them. 
 
 Thus, x'^—y'^ = 7, and x^—xy + y^ = 13, are each homogeneous. 
 
 341. A Symmetrical Equation is one in which 
 the unknown quantities are involved to the same degree. 
 
 Thus, x''+y^ = 34, and x^y—xy^ = 34, are each symmetrical. 
 
 342. Simultaneous Quadratic Equations con- 
 taining two unknown quantities, in general involve the 
 principles of Biquadratic equations, which belong to the 
 higher departments of Algebra. 
 
 There are three classes of examples, however, which may 
 be solved by the rules of quadratics. 
 
 ist. When one equation is quadratic, and the other simple, 
 2d. When both equations are quadratic and homogeneous, 
 3d. When each equation is symmetrical. 
 
 343. To Solve Simultaneous Equations consisting of, a 
 
 Quadratic and a Simple Equation. 
 
 I. Given x^ -{- y'^=z 13, and x -{- y =z 5, to find x and y. 
 
 Solution.— By the problem, x^+y^ = 13 
 
 (i) 
 
 x+y= 5 . 
 
 (2) 
 
 By transposition, x = s—y 
 
 (3) 
 
 Squaring each side of (3) (Art. 102), x'^ = 2S — ioy+y^ 
 
 (4) 
 
 Substituting (4) in (i), 2S—ioy + y'^+y'^ — 13 
 
 (5) 
 
 Uniting and transposing, 2y'^—ioy = — 12 
 
 (6) 
 
 Comp. sq. (Art. 336,710(6), 4y^—2oy + 2S = —24 + 25 
 
 (7) 
 
 Extracting root, 2^—5 = ± i 
 
 
 .-. y=:3 or 2. 
 
 
 Substituting value of y in (3), a; = 2 or 3. Hence, the 
 
 
 340. What is a homogeneoua equation ? 341. A symmetrical equation ? 
 
188 ' SIMULTAN^EOUS QUADRATICS. 
 
 Rule. — Find the value of one of the unhnoivn quantities 
 in the simple equation ly transposition, and substitute this 
 value in the quadratic equation. (Arts. 221, 223.) 
 
 ^olye the following equations: 
 
 2. 0:2+^2:^25, 5. a:2 + ^2:=244, 
 
 y —X = 2. 
 6. 3^:2 _ ^2 —25 1, 
 
 ^ + Ay = 38. 
 
 4. a^J_^2_28, 7. 8a;2_j_ ^^2_ 728, 
 
 6y — x=z 15. 
 
 344. To Solve Simultaneous Equations which are both 
 Quadratic and Homogeneous. 
 
 8. Given x^-\-xy = 40, and y'^-^xy = 24, to find x and y. 
 
 X +?/ -- 
 
 = 7- 
 
 x^-^-f-- 
 
 = 74, 
 
 x + y=i 
 
 : 12. 
 
 x^-f-. 
 
 = 28, 
 
 x-y = 
 
 : 2. 
 
 jUTIon. — By the problem. 
 
 x^ + a^ = 4o 
 
 (I) 
 
 »< «c 
 
 y^ + xy = 24 
 
 (2) 
 
 Let 
 
 x = py 
 
 (3) 
 
 Substituting ^3^ in (i). 
 
 pY +pf - 40 
 
 (4) 
 
 " (2), 
 
 /+i>2/^ = 24 
 
 (5) 
 
 Factoring, etc., (4), 
 
 ^ p'+p 
 
 (6) 
 
 *' (5), 
 
 ^ i+P 
 
 (7) 
 
 Equating (6) and (7), 
 
 40 24 
 p'+p i+p 
 
 (8) 
 
 Clearing of fractions. 
 
 S + 5P = 3P' + 3P 
 
 (9) 
 
 Transposing, etc., 
 
 3P'-2p = 5 
 
 (ID) 
 
 Comp. sq., 3d meth. (Art. 336), 
 
 gp'^—6p+i = 15 + 1 = 16 
 
 (II) 
 
 Extracting root, 
 
 3P-1 = ± 4 
 
 (12) 
 
 Transposing, 3^ = i ± 4 
 
 Dropping the negative value, p = | 
 
 Substituting value of ^ in (7), y^ = 24-^(1 +|)=9 
 Extracting root, y = ± 3 
 
 Substituting value of p and y in (3), a? = fx ±3 = ±5. 
 
 Hence, the 
 
 343. Rule for solution of equations coneieting of a quadratic and simple equ^ 
 tion? 
 
SIMULTANEOUS QUADRATICS. 189 
 
 Rule. — I, For one of the unknown quantities substitute 
 the product of the other into an auxiliary quantity, and then 
 find the value of this auxiliary quantity. 
 
 II. Find the values of the unknown quantities by substi- 
 tuting the value of the auxiliary quantity in one of the 
 equations least involved. 
 
 Note. — ^An auxiliary quantity is one introduced to aid in the sola 
 >,ion of d problem, as p in the above operation. 
 
 9. Given 
 A.nd 
 
 It^^"" o^' 1 to find a; and y. 
 
 10. Given x ^y = 2, ].«, , 
 
 1 1. Gireu 3^ - 72/' - - I, I t„ g„3 ^ ,„^ 
 And 4^y= 24, ) ^ 
 
 1 2. Given ^ - ^y + 3^ = '9, 1 to fi^a ^ ^„3 
 And xy=i5,) ^ 
 
 345. To Solve Simultaneous Quadratic Equations when each 
 Equation is Symmetrical. 
 
 13. Given x + y =zg, and xy = 20, to find x and y. 
 
 .UTiON.— By the problem, x+y= 9 
 
 (I) 
 
 xy = 20 
 
 (2) 
 
 Squaring (i), aj» + 2a'2^ + / = 8 1 
 
 (3) 
 
 Multiplying (2) by 4. 40;^ = 80 
 
 (4) 
 
 Subtracting (4) from (3), a?—2xy+y^ - I 
 
 (5) 
 
 Extracting sq. root of (5), x—y — ±1 
 
 (6) 
 
 Bringing down (i), x + p = 9 
 
 
 Adding (i) and (6), 2X = 10 or 8 
 
 (7) 
 
 Removing coefficient, a; = 5 or 4 
 
 
 Substituting value of a; in (i), y = 4 or 5 
 
 
 Notes. — i. The values of x and y in these equations are not equal, 
 but interchangeable ; thus, when x = s, y=4 ; and when aj = 4, y = 5 
 
 344. How solve equations which are both quadratic and homogeneous ? I^ote, 
 What is an auxiliary quantity ? 
 
190 SIMULTANEOUS QUADRATICS. 
 
 2. The solution of this class of problems 'caries according to the 
 given equations. Consequently, no specific rules can be givren that 
 will meet every case. But judgment and practice will readily supply 
 expedients. Thus, 
 
 I. When the sum and product are given. (Ex. 13, 15.) 
 Find the difference and combine it with the sum, (Art. 2 24.) 
 
 II. When the difference and product are given. (Ex. 1 6.) 
 Find the sum and combine it ivith the difference, 
 
 III. When the sum and difference of the same powers are 
 given. (Ex. 14, 17.) 
 
 Combine the two equations by addition a7id subtraction, 
 
 rV*. When the members of one equation are multiples of 
 the other. (Ex. 18.) 
 
 Divide o?ie by the other, and then reduce the resulting 
 equation, 
 
 14. Given a;^ -f ^^ = 5, (i) ) . « , , 
 
 ^ ^ , 1 ^1 X ^ to find a; and V. 
 
 And x» —y^ =. i, (2) ) 
 
 Solution, — Adding (i) and (2), and dividing, a;^ = 3 
 
 Involving, a? = 27 
 
 Subtracting (2) from (i), etc y* = 2 
 
 Involving, y = 8 
 
 15. Given 2^ + ^=27, ],«, , 
 
 ^ . , ^^ „" ^ to findrraudy. 
 And xy = 180, \ ^ 
 
 16. Given a? — v= 14, )./>;, , 
 
 . , ^ > to find X and y. 
 
 And xy = 147, ) 
 
 ,"*"'\'^^'ltofinda:andy. 
 * — ^^ = 3? ' 
 
 17. Given x^ + y^ = 7, 
 And X 
 
 18. Given o^y^ + x^f =z 12,] , ^ ^ , 
 A :i 00 ^ r to fin<i ^ and y. 
 And a?y -{- xy^ = 6, ) ^ 
 
 Note,— What is true of the solution of Bimultaneous quadratics? When the tmm 
 and product are ffiven,how proceed? When the difference and product? When 
 the sum and difference of the same powers are given ? When the members of one 
 tquation are multiples of the other ? 
 
SIMULTANEOUS QUADBATIC8. 193 
 
 PROBLEMS. 
 
 1. The difference of two numbers is 4, and the difference 
 of their cubes 448. What are the numbers ? 
 
 2. A man is one year older than his wife, and the product 
 of their respective ages is 930. What is the age of each ? 
 
 3. Required two numbers whose sum multiplied by the 
 greater is 180, and whose difference multiplied by the less 
 is 16. 
 
 4. In an orchard of 1000 trees, the number of rows 
 exceeds the number of trees in each row by 15. Required 
 the number of rows and the number of trees in each row. 
 
 5. The area of a rectangular garden is 960 square yards, 
 and the length exceeds the breadth by 16 yards. Required 
 the dimensions. 
 
 6. Subtract the sum of two numbers from the sum of 
 their squares, and the remainder is 78 ; the product of the 
 numbers increased by their sum is 39. What are the 
 numbers ? 
 
 7. Find two numbers whose sum added to the sum of 
 their squares is 188, and whose product is 77. 
 
 8. A surveyor lays out a piece of land in a rectangular 
 form, so that its perimeter is 100 rods, and its area 589 
 square rods. Find the length and breadth. 
 
 9. Required two numbers whose product is 28, and the 
 sum of their squares 65. 
 
 10. A regiment of soldiers consisting of 1154 men is 
 formed into two squares, one of which has 2 more men on a 
 side than the other. How many men are on a side of each 
 of the squares ? 
 
 11. Required two numbers whose product is 3 times their 
 sum, and the sum of their squares 1 60. 
 
 12. What two numbers are those whose product is 6 times 
 their difference, and the sum of their squares 13 ? 
 
 CSee A'^r)cnclix, p. 290.) 
 
CHAPTER XVII. 
 RATIO AND PROPORTION. 
 
 346. Ratio is the relation which one quantity bearb k 
 pother with respect to magnitude. 
 
 347. The Ter^ns of a Ratio are the quantities 
 compared. The first is called the Antecedent, the second 
 the Consequent y^ and the two together, a Couplet 
 
 348. The HUjn of ratio is a colon : \ placed between the 
 two quantities compared. 
 
 Ratio is also denoted by placing the consequent under the 
 antecedent, in the form of 2^ fraction. 
 
 Thus, the ratio of a to 6 is written, a : &, or ^ • 
 
 349. The Measure or Value of a ratio is the quotient 
 of the antecedent divided by the consequent, and is equal 
 to the value of the fraction by which it is expressed. 
 
 Thus, the measure or value of 8 : 4 is 8-7-4 = 2. 
 
 Note. — That quantities may have a ratio to each other, they must 
 be so far of the same nature, that one can properly be said to be equal 
 to, or greater t or les8 than the other. 
 
 Thus, a foot has a ratio to a yard, but not to an hour, or a pound 
 
 350. A Simple Itatio is one which has but two terms *, 
 
 fts, a\h, 8 : 4. 
 
 346. What is ratio ? 347. What are the terms of a ratio ? 348. The pign ? How 
 »lso is ratio denoted ? 349. The measure or value ? Note. What quantities have a 
 ratio to each other ? 350. What is a simple ratio ? 
 
 * Ardecedent, Latin ante, before, and cedere, to go, to preceoe. 
 Consequent, Latin con, and sequi, to follow. 
 
 f Tlie sign of ratio : is derived from the sign of division -f-, the 
 horizontal line being dropped. 
 
RATIO. 193 
 
 351. A Compound Matio is the product of two or 
 more siinple ratios. 
 
 Thus, 4 • 2 [ are eacli simple But 4x9: 2x3 
 
 9:3$ ratios. is a compound ratio. 
 
 Note. — The nature of compound ratios is the same as that of sim- 
 ple ratios. They are so called to denote their origin, and are usually- 
 •expressed by writing the corresponding terms of the simple ratios one 
 •mder another, as above. 
 
 352. A Direct Ratio arises from dividing the ante- 
 cedent by the consequent. 
 
 353. An Inverse^ or Heciprocal Matio arises 
 from dividing the consequent by the antecedent, and is the 
 same as the ratio of the reciprocals of the two numbers 
 compared. 
 
 Thus, the direct ratio of a to a& = -r, or v, and that of 4 to 
 
 4 I 
 
 12 = — , or - • 
 12 3 . 
 
 The inverse ratio of a to a& = — , or 6 ; of 4 to 12 = — , or 3. 
 
 tt 4 
 
 It is the same as the ratio of the reciprocals, - to -^ , and - to — • 
 
 a ab 4 12 
 
 Note. — A reciprocal ratio is expressed by inverting the fraction 
 which expresses the direct ratio. When the colon is used, it is 
 expressed by inverting the m^der of the terms. 
 
 354. The ratio between two fractions which have a 
 common denominator, is the same as the ratio of their 
 numerators. 
 
 Thus, the ratio of f : f is the same as 6:3. 
 
 Note. — When the fractions have different denominators, reduce 
 them to a common denominator; then compare their numerators. 
 (Art. 175.) 
 
 355. A Ratio of Equality is one in which the quan- 
 tities compared are equal, and its value is a unit or i. 
 
 351. What is a compound ratio? l<!ote. Why so called? 352. What is a direct 
 ratio ? 353. A reciprocal ? 355. What is a ratio of equality ? 
 
 * Inverse, from the Latin in and verto, to turn upside down, to invert. 
 
 9 
 
194 RATIO. 
 
 356. A !Ratio of Greater Inequality is one whose 
 antecedent is greater than its consequent, and its value is 
 greater than i. 
 
 357. A IRatio of Less Inequality is one whose 
 antecedent is less than its consequent, and its value is less 
 than I. 
 
 358. A Duplicate Hatio is the square of a simple 
 ratio. It arises from multiplying a simple ratio into itself, 
 or into another equal ratio. 
 
 359. A Triplicate Matio is the cule of a simple ratio, 
 and is the product of three equal ratios. 
 
 Thus, the duplicate ratio of a to 5 is a* : 6*. 
 The triplicate ratio of a to & is a^ : 5^. 
 
 360. A Suhduplicate Hatio is the square root of a 
 simple ratio. 
 
 361. A Subtriplicate Matio is the cuhe root of a 
 simple ratio. 
 
 Thus, the svbduplicate ratio of a; to y is ^\/x : ^/y. 
 The sicbtriplicate ratio of a; to y is y^x : ^y, etc. 
 
 362. Since ratio may be expressed in the form of a 
 fraction, it follows that changes made in its terms have the 
 same effect on its value, as like cUamjes in the terms of a 
 fraction. (Art. 167.) Hence, the following 
 
 PRINCIPLES. 
 
 I®. Multiplying the antecedent, or ) ,^ ... ,. ^- 
 
 T^' 'f' J.1 ^ \ MuUivhes tho ratio. 
 
 Dividing the consequent, j 
 
 2**. Dividing the antecedent, or ) r.- -t ^t 
 
 !•#- 7,. 7 . ,1 ± \ Divides the ratio. 
 
 Multiplying the consequent, ) 
 
 3**. Multiplying or dividing loth \ Does 7iot alter the value 
 
 terms hy the same quantity, f of the ratio. 
 
 356. Of greater inequality? 357. Of lef? inequality? 358. A duplicate ratio? 
 359. Triplicate? 360. Subduplicate ? 361. Subtriplicate? 363. Name Principle i. 
 Principle a Principle a> 
 
EATia Idd 
 
 EXAMPLES. 
 
 1. What is the ratio of 4 yards to 4 feet? 
 
 Solution. 4 yards = 12 feet ; and the ratio of 12 ft. to 4 ft. ia y 
 
 2. What is the ratio of 6a^ to 2X ? Ans, 3a:, 
 
 3. What is the ratio of 40 square rods to an acre ? 
 
 4. What is the ratio of i pint to a gallon ? 
 
 5. What is the ratio of 64 rods to a mile ? 
 
 6. What is the ratio of Sa^ to 4a ? 
 
 7. What is the ratio of isaic to ^ab? 
 
 8. What is the ratio of $5 to 50 cents ? 
 
 9. What is the ratio of 75 cents to 16 ? 
 
 10. What is the ratio of 35 quarts to 35 gallons P 
 
 11. What is the ratio of 20,^ to 4a ? 
 
 12. What is the ratio otx^ — y^tox-^-r/? 
 
 13. What is the compound ratio of 9:12 and 8 : 15 ? 
 Solution. 9 x 8 = 72, and 12 x 15= i8a Now 72-5-180 = t^, Ana 
 Or, 9 : 12 = ■^, and 8 : 15 = ^V Now j\ x yV = t¥o = tu» ^^■ 
 
 14. What is the compound ratio of 8 : 15 and 25 : 30? 
 
 15. What is the compound ratio of a:b and 2h : ^ax? 
 
 16. Eeduce the ratio of 9 to 45 to the lowest terms. 
 Solution. 9 : 45 = t\» and ^\ = |, Ans. 
 
 17. Reduce the ratio of 24 to 96 to the lowest terms. 
 
 18. Reduce the ratio of 144 to 1728 to the lowest terms. 
 
 19. What kind of ratio is 25 to 25 ? 
 
 20. What kind of a ratio is ab: ab? 
 
 21. What kind of ratio is 35 to 7 ? 
 
 22. What kind of ratio is 6 to 48 ? 
 
 23. Which is the grearter, the ratio of 15 : 9, or 38 : 19? 
 
 24. Which is the greater, the ratio of 8 : 25, or V4 '- V25, 
 
 25. If the antecedent of a couplet is 56, and the ratio 8 
 what is the consequent? 
 
 26. If the consequent of a couplet is 7, and the ratio 14, 
 what is the antecedent ? 
 
196 pBOPOBiioifir. 
 
 PROPORTION. 
 
 363. Proportion is an equality of ratios. 
 
 Thus, the ratio 8 ; 4 = 6 : 3, is a proportion. That is. 
 Four quantities are in proportion, when the first is the same mvlti- 
 pie or part of the second that the third is of the fourth. 
 
 364. The Sign of Proportion is a double colon : :,♦ 
 or the sign =. Thus, 
 
 The equality between the ratio of a to 5 and c to (? is expressed by 
 
 a : 6 : : c : <f , or by T = 3 
 d 
 
 The former is read, " a is to & as c is to (f ;" the latter, *'& is contained 
 
 In a as many times as d is contained in c.** 
 
 Note. —Each ratio is called a couplet , and each term a proportional. 
 
 365. The Terms of a proportion are the quantities 
 compared. ^\iq first and fourth are called the extremes, the 
 second and third the means, 
 
 366. In every proportion there must be at least four 
 terms ; for the equality is between two or more ratios, and 
 each ratio has two terms. 
 
 367. A proportion may, however, be formed from three 
 quantities, for one of the quantities may be repeatedy so as 
 to form two terms ; as, a\h : : J : c. 
 
 Note. — Care should be taken not to confound propoHion with ratio. 
 In common discourse, these terms are often used indiscriminately. 
 Thus, it is said, " The income of one man bears a greater proportion 
 to his capital than that of another,'* etc. But these are loose expressions 
 
 In a simple ratio there are but two terms, an antecedent and a 
 cjonsequent ; whereas, in a proportion there must at least be four 
 terms. (Arts. 350, 366.) 
 
 363. What is proportion? 364. The sign of proportion? Note. What is each 
 ratio called ? 365. What are the terms of a proportion ? 366. How many terms in 
 every proportion ? 367. How form a proportion from three quantities ? 
 
 * The sign : : is derived from the sign of equality =, the four 
 
 points being the terminations of the lines. 
 
PBOPOETION. 197 
 
 Again, one ratio may be greater or less than another, but one 
 proportion is neither greater nor less than another. For equality does 
 not admit of degrees. In scientific investigations, this distinction 
 should be carefully observed. 
 
 368. A Mean Proportional between two quantities 
 is the middle term or quantity repeated, in a proportion 
 formed from three quantities. 
 
 369. A Third Projjortional is the last term of a 
 proportion having three quantities. 
 
 Thus, in the proportion a:h :: b:e,b\ask mean proportional, and 
 e a third proportional. 
 
 370. A Direct Proportion is an equality between 
 
 two direct ratios ; as, a:b : : c:d, $16 : : 4:8. 
 
 371. An Inverse or Heciprocal Proportion is an 
 
 equality between a direct and reciprocal ratio ; as, 
 8:4 :: i'.h 
 
 372. Analogous Terms are the antecedent and con- 
 sequent of the same couplet. 
 
 373. Homologous Terms are either two antecedents 
 
 or two consequents. 
 
 PROPOSITIONS. 
 
 374. A Proposition is the statement of a truth to be 
 proved, or of an operation to be performed. 
 
 Propositions are of two kinds, theorems and proUems, 
 
 375. A TJieorem is something to be proved. 
 
 376. A Problem is something to be done. 
 
 377. A Corollary is a principle inferred from d 
 preceding proposition. 
 
 368. What is a mean proportional ? 369. Wliat is a third proportional ? 370. A 
 direct proportion? 371. An inverse or reciprocal proportion? 372. What ar« 
 analop:ou8 terms ? 373. Homologoni?? 374. What is a proposition ? How divided? 
 375. What is a theorem ? 376. A problem? 377. A .^roUary? 
 
198 PEOPORTIOK, 
 
 378. Tho more important theorems in proportion are the 
 following: 
 
 Theorem L 
 
 If four quantities are 'proportional^ the product of the 
 extremes is equal to the product of the means. 
 
 Left a : d :: e : d 
 
 By An. 363, |=| 
 
 dearing of fractions, a<2 = && 
 
 Verification by NuMBERa 
 Given, a: 4:: 8: 16; and 2x16 = 4x8. 
 
 Cor. — ^The relation of the four terms of a proportion to 
 each other is such, that if any three of them are given, the 
 
 fourth may be found. 
 
 Thus, since ad — be, it follows that 
 
 a = bc+dt b = ad-T-c, e = ad-r-b, and d = bc-t-a, (Ax. 5.) 
 
 Notes. — i. The rule of Simple Proportion in Arithmetic is founded 
 upon this principle, and its operations are easily proved by it. 
 
 2. This theorem furnishes a very simple test for determining 
 whether any four quantities are proportional. W© have only to 
 multiply the extremes together, and the means. 
 
 Theorem IL 
 
 If three quantifies are proportional, the product of the 
 extremes is equal to the square of the mean. 
 
 Let a : 6 : : b • t 
 By Art. 363, |=:- 
 
 Oearing of fractions, ac = b*. . 
 
 Again, 9:6:: 6:4, and 4x9 = 6'. 
 
 Cor. — A mean proportional between two quantities is 
 equal to the square root of their product 
 
PEOPOETIO]Sr. 199 
 
 Theorem HL 
 
 ff the product of two quantities is equal to the product of 
 two others, the four quantities are proportional ; the factors 
 of either product being taken for the extremes, and the factors 
 9fthe other for the means. 
 
 Let adr=lfe 
 
 DividingbjM, 1 = 1 
 
 Or, by Art. 363, a:b :: e : d, 
 
 Agoia, 4x6 = 3x8, and 4:3:: 8 :& 
 
 Theoeem IV. 
 
 If four quantities are proportional, they are proportional 
 when the means are inverted. 
 
 Let aih :i ei d, then a i e 11 h: d 
 
 a e 
 
 For, hj Art. 363, 
 
 h~d 
 
 Multiplying by -. * s= j 
 
 Or, aie II hi d. 
 
 Again. 3 : 6 : • 4 : 8, and 3 : 4 : : 6 : 8. (Th. i.) 
 
 Note. — This change in the order of the means \a called 
 ** AlterTuUian,** 
 
 Theorem V. 
 
 If four quantities are proportional, they are proportional 
 when the terms of each couplet are inverted. 
 
 Le. a ih M e \ dt then h ', a .: d \ t 
 By Theorem i, ' ad = he 
 
 By Theorem 3, h : a :: d : e. 
 
 Again, 6 : 2 :: 15 : 5, then 2 : 6 :: 5 : 15. (Th. i.) 
 
 Cor. — ^If the extremes are inverted, or the order of the 
 terms, the quantities will be proportional. 
 
200 PROPORTIOK. 
 
 Notes. — i. If the terms of onlj one of the couplets are inverted, 
 ihe proportion becomes reciprocal, 
 
 2. The change in the order of the terms of each couplet Ls called 
 ^Inversion** 
 
 3. This proposition supposes the quantities compared to be of the 
 same kind. Thus, a line has no relation to weight. (Art 349, note.) 
 
 Theorem VL 
 
 If four quantities are proportional , ttvo analogous or twt^ 
 homologous terms may be multiplied or divided by the sam4 
 quantity without destroying the proportion. 
 
 Let a:h :: e I d 
 
 Multiplying analogous terms, am : bm :: e : d 
 and a : & 11 em i dm 
 
 ^' h^d 
 
 Hence. (Art 362. Prin. 3). ^ = |. and | = ^ 
 
 Multiplying homologous terms, am : 5 : : cm i d 
 
 And a '. hm '.: c : dm. 
 
 .„. , . ^ am em 3 a e 
 
 Hence. (Ax 4. 5), X=T' ""^ i^=^ 
 
 Dividing analogous terms, — : — : r e : d, 
 
 and « : 6 ; • — : — 
 
 m m 
 
 Dividing homologous terms, — : b :: — : d 
 
 b d 
 
 and ^^ • ;r • ' ^ • ;r 
 
 m m 
 
 Clearing of fractions (Th. i), ad=:be 
 
 Cor. — All the terms of a proportion maybe multiplied 
 01 divided by the same quantity without destroying the 
 proportion. 
 
 Notes. — i. When the homologous terms are multiplied or divided, 
 both ratios are equally increased or diminished. 
 
 2. When the analogoiLs terms are multiplied or divided, the ratios 
 are not altered. 
 
PEOPOHTION. 201 
 
 Theokem VIL 
 
 If four quantities are proportional, the sum of the first 
 and second is to the second, as the sum of the third and fourth 
 is to the fourth. 
 
 Let a h II ei d, then fl+6 : 6 :: e-\-d : d 
 _ a c 
 
 *^°'' »=5 
 
 Adding I to each member, ^ + i = -% + 1. (Ax. a.) 
 
 Incorporating i, — r— = —-^ 
 
 Therefore (Art. 363), a+h il :: c + d : d 
 
 Again, 4:2 :: 6:3, then 4+2 : 2 :: 6 + 3 : 3 
 KoTE. — This combination is sometimes called ** Composition.'* 
 
 Theorem VIII. 
 
 // four quantities are proportional, the difference of the 
 first and the second is to the second, as the difference of the 
 third and fourth is to the fourth. 
 
 Let a : 6 : : c : d, then a— & : h :: c—d : d 
 
 Subtracting i from each member, r- — i = - — x 
 
 d 
 
 - ,. a— 6 c—d 
 
 Incorporating —1, — v— = ^ - 
 
 Therefore, a— 6 : 6 : : c—d : <2 
 
 Again, 4:2 : : 6:3, then 4—2 12:: 6—3 : 3 
 
 Note. — This comparison is sometimes called *' Division" * 
 
 * The technical terms. Composition and Division, are calculated 
 rather to perplex than to aid the learner, and are properly falling into 
 disuse. The objection to the former is, that it is liable to be mistaken 
 for the composition or compounding of ratios, whereas the two 
 
202 
 
 PEOPORTION. 
 
 Theoeem IX. 
 
 If two ratios are respectively equal to a third, they are 
 equal to each other. 
 
 Let a : b :: m : n, 
 a _m 
 l~"n' 
 
 Then 
 
 and 
 and 
 
 d :: m I n 
 d ~" » 
 
 
 That is, a : b = e : d 
 
 Again, 12 : 4 = 6 : 2, and 9:3 = 6:3 
 A 12 : 4 = 9 : 3 
 
 Theoeem X. 
 
 When any numler of quantities are proportional, any 
 antecedent is to its consequent, as the sum of all the ante- 
 cedents is to the sum of all the consequents. 
 
 Let a I b :: e : d ::e:/, etc. 
 
 Then a : b :i a+c+e : b+d+f, etc. 
 
 For (Th. i), ad — be 
 
 And, ** af — be 
 
 Also, ab = ba 
 
 Adding (Ax, 2), 
 Factoring, 
 
 ab+ad+af = ba+bc+be 
 a(b+d+f) = b{a+c+e) 
 
 Hence, (Th. 3), a :b i: {a+e+e, etc.) : (J)+di-f, etc.) 
 
 » ■ — — — — — — — — — - — ■ — ~— 
 
 operations are entirely different. In one the terms are added, in the 
 other they are multiplied together. (Art. 351,) 
 
 The objection to the latter is, that the change to which the term 
 division is here applied, is effected by subtraction, and has no 
 reference to division, in the sense the word is used in Arithmetic and 
 Algebra. Moreover, the alteration in the terms of Theorem 6 is 
 produced by actual division. Usage, however ancient, can no longer 
 justify the employment of the same word in two different senses, in 
 explaining the same subject. 
 
PEOPOETI02!r. 203 
 
 Theorem XL 
 
 If the corresponding terms of two or more proportions are 
 multiplied together, the products will he proportional. 
 
 Let a ih :: ci d, and e i f i: g : h 
 
 Then ae :hf :: eg : dh 
 
 For. -=5 '«'<1 7=f 
 
 Halt ratios together (Ax. 4), h^~^ 
 
 Hence, (Th. 3), ae : bf z: eg : dh. 
 
 Theoeem xn. 
 
 If four quantities are proportional, like powers or roots 
 of these quantities are proportional. 
 
 Let a : b : : e : d, then t = 3 
 
 a 
 
 ■DA a« c* 
 
 Hence (Th. 3), «» : 6» ; ; c» : d» 
 
 Extracting sq. root, a* : 6* : : c* : d* 
 Again, 2:3 : : 4:6, then 2« : 3' : : 4" : 6* 
 
 In like manner, /y^ : ^/g : : -y/iS : -y/is. 
 
 Note.— The index » may be either integral or fractional. 
 
 Theorem XIIL 
 
 Equimultiples of two quantities are proportional to the 
 ruantities themselves. 
 
 Smce 1=3, hy Art. 362, Pnn. 3, r— = ^ 
 fid" '^ bm d 
 
 Hence, am : im ii c : d. 
 
204 PEOPORTIOK 
 
 PROBLEMS. 
 
 1. The first three terms of a proportion are 6, 8, and 3. 
 What is the fourth ? 
 
 Let x = the fourth term. 
 
 Then 6:8 : : 3 : x 
 
 /. 6x = 24, and x = 4. 
 
 2. The last three terms of a proportion are 8, 6, and 12. 
 What is the first? 
 
 3. Eequired a third proportional to 25 and 400. 
 
 4. Required a mean proportional between 9 and 16. 
 
 5. Find two numbers, the greater of which shall be to 
 the less, as their sum to 42 ; and as their difference to 6. 
 
 6. Divide the number 18 into two such parts, that the 
 squares of those parts may be in the ratio of 25 to 16. 
 
 7. Divide the number 28 into two such parts, that the 
 quotient of the greater divided by the less shall be to the 
 quotient of the less divided by the greater as 32 to 18. 
 
 8. What two numbers are those whose product is 24, and 
 the difference of their cubes is to the cube of their difference 
 as 19 to I ? 
 
 9. Find two numbers whose sum is to their difference as 
 9 is to 6, and whose difference is to their product as i to 12. 
 
 10. A rectangular farm contains 860 acres, and its length 
 is to its breadth as 43 to 32. What are the length and 
 breadth ? 
 
 11. There are two square fields; a side of one is 10 rods 
 longer than a side of the other, and the areas are as 9 
 to 4. What is the length of their sides ? 
 
 12. What two numbers are those whose product is 135, 
 and the difference of their squares is to the square of their 
 difference as 4 to i ? 
 
 13. Find two numbers whose product is 320 ; and the 
 difference of their cubes is to the cube of their difference 
 as 61 to I. 
 
CHAPTER XVIII. 
 PROGRESSION. 
 
 379. A Progression is a series of quantities which 
 increase or decrease according to a fixed law. 
 
 380. The Terms of a Progression are the quan- 
 tities which form the series. The first and last terms are 
 the extremes ; the others, the means, 
 
 381. Progressions are of three kinds: arithmetical, 
 geometrical, and Tiarmonicah 
 
 ARITHMETICAL PROGRESSION. 
 
 382. An Arithmetical Progression is a series 
 which increases or decreases by a constant quantity called 
 the common difference. 
 
 383. In an ascending series, each term is found ly adding 
 the common difference to the preceding term. 
 
 If the first term is «, and the common difEerence d, the series is 
 
 a, a+d, a + 2d, a + ^d, etc. 
 If a = 2, and d = 2, the series is 2, 5, 8, 11, 14, etc. 
 
 384. In a descending series, each term is found by 
 sultr acting the common difference from the preceding term. 
 
 If a is the first term, and d the common difference, the series is 
 a, a—d, a— 2d, a— 3d, etc. 
 
 In this case, the common difference may be considered —d. Hence, 
 the common difference may be either positive or negative. And, since 
 adding a negative quantity is equivalent to subtracting an equal 
 
 379. What is a progression? 380. The terms? 381. How many kinds of 
 progrepsion? 382. An arithmetical i)rogre8Pion ? What is this constant quantity 
 •ailed ? 383. An ascending series ? 384. A descending series ? 
 
306 AEITHMETICAL PROGKESSION. 
 
 positive one, it may therefore properly be said that each successive 
 term of the series is derived from the preceding by the addition of tlie 
 common difference. (Art. 75, Prin. 3.) 
 
 Notes.— I. The common difference was formerly called arithmetical 
 ratio; but this term is passing out of use. 
 
 2. An Arithmetical Progression is sometimes called an Equidifferent 
 Series, or a Progression by Difference. In every progression there may 
 be an infinite number of terms. 
 
 385. If four quantities are in arithmetical progression, 
 the sum of the extremes is equal to the sum of the means. 
 
 Let a, a + d, a + 2d, a + 3d, be the series. 
 Adding extremes, etc., 2a + 3d = 2a+sd. 
 
 Or, let 2, 2 + 3, 2 + 6, 2 + 9, be the series. 
 Then 2 + 2 + 9 = 2 + 3 + 2 + 6. 
 
 386. If three quayitities are in arithinetical progression, 
 the sum of the extremes is equal to double the mean. 
 
 Let a, a + d, a + 2d, be the series. 
 Then 2a + 2d = 2{a + d), 
 
 Again, let 2,2 + 4,2 + 8, be the series. 
 Then 2 + 2 + 8 = 2(2 + 4). 
 
 Cor. — An Arithmetical Mean between two quantities 
 may be found by taking half their sum. 
 
 387. In Arithmetical I^rogression there are five 
 elements to be considered: the first term, the common 
 difference, the last term, the number of terms, and the sum 
 of the terms. 
 
 Let a = the first term. 
 
 d = the common difierence, 
 I = the last term. 
 n = the number of terms. 
 8 = the sum of the terms. 
 
 The relation of these five quantities to each other is such 
 that if any three of them are given, the other two can be 
 found. 
 
 385. What is true of four quantities in arithmetical progression ? 386. Of three 
 quantities? 387. Name the elements in arithmetical progression? What relation 
 have they to each other { 
 
ARITHMETICAL PKOGRESSION. 207 
 
 CASE I. 
 
 388. The First Term, the Common Difference, and Number 
 of Terms being given, to Find the Last Term. 
 
 Each succeeding term of a progression is found by adding the 
 common difference to the preceding term. (Art. 384.) Therefore the 
 terms of an ascending series are 
 
 a, a+d, a+2d, a+sd, etc. 
 The terms of a descending series are 
 
 a, a—d, a — 2d, a— 3d, etc. 
 It will be seen that the coefficient of d in each term of both series 
 is one less than the number of that term in the series. Therefore, 
 putting I for the last or nth. term, we have 
 
 Formula I. I = a ±{n— i)d. 
 
 EuLE. — I. Multiply the common difference hy the number 
 of terms less one. 
 
 II. When the series is ascending, add this product to the 
 first term ; when descending, subtract it from the first term, 
 
 1. Given « = 3, d= 2, and n-=i, to find I. 
 
 l = a± (n—i)d = 3 + (7—1)2 = 15, Ans. 
 
 2. Given « = 25, ^ = — 2, and w = 9, to find I 
 
 3. Given az= 12, d = 4, and n = 15, to find L 
 
 4. Given a = 1, d= —^, and n =13, to find t 
 
 5. Given « = |, d = \, and n = g, to find I. 
 
 6. Given « = i, d=^ — .01, and n = 10, to find h 
 
 7. Find the 12th term of the series 3, 5, 7, 9, 11, etc 
 Note. — In this problem, a = 3, d — 2, n = 12. Ans. 25. 
 
 8. Find the 15th term of i, 4, 7, 10, etc. 
 
 9. Find the 9th term of 31, 29, 27, 25, etc. 
 
 10. What is the 30th term of the series i, 2 J, 4, 5I, etc. 
 
 11. Find the 25th term of the series x + ^x + ^x-^-yx, etc. 
 
 12. Find the nth term of the series 2a, sa, Sa, iia, etc. 
 
 388. What is the rule for finding the last term ? 
 
208 AKITHMETICAL PROGEESSIOIJ", 
 
 CASE II. 
 
 389. The Extremes and Number of Terms being given, tc 
 Find the Sum of the Series. 
 
 Let a, a + d, a + 2d, a + 3d . . . I, be an arithmetical progression, 
 the sum of which is required. 
 
 Since the sum of two or more quantities is the same in whatever 
 order they are added (Art. 63, Prin, 2), we have 
 
 8 = a+ {a+ d) + {a+ 2d) + (a+ 3d) + , , . +1 
 Inverting, 8 = 1 + {I— d) + {I — 2d) + {I — 3d) + . . . +a 
 Adding, 28 = a + I + {a + I) + {a+l) + {a + l) + . . . +a+l 
 
 .'. 28 = (a + T) taken n times, or as many times as there are 
 terms in the series. 
 That is, 28 = {a + l)n. Hence, the 
 
 Formula II. s = i — ^t_Z x n. 
 
 2 
 
 EuLE. — Multiply half the sum of the extremes ly the 
 number of terms. 
 
 Cor. — From the preceding illustration it follows that the 
 sum of the extremes is equal to the sum of any two terms 
 equally distant from the extremes. 
 
 Thus, in the series, 3, 5, 7, 9. 11, 13, the sum of the first and last 
 terms, of the second and fifth, etc., is the same, viz., 16. 
 
 1. Given « = 4, / = 148, and n = 15, to find 5. 
 Solution. 4+148 = 152, and (152-5-2) x 15 = 1140, Arts, 
 
 2. Given a = ^, I = 30, and n = 50, to find s. 
 
 3. Given a = 6, Z = 42, and n = 9, to find 5. 
 
 4. Given « = 5, ? = 75, and n = 35, to find s, 
 
 5. Given a = 2, I = i, and w= 17, to find s. 
 
 6. Find the sum of the series 2, 5, 8, 1 1, etc., to 20 terms. 
 
 7. Find the sum of the series i, i^, 2, 2^, etc., to 25 terms. 
 
 8. Find the sum of the series 75, 72, 69, 66, 63, etc., 
 to 15 terms. 
 
ARITHMETICAL PRO G R E SSI OIT. 209 
 
 390. The two preceding formulas are fundamental, and 
 furnish the means for solving all the problems in Arith- 
 metical Progression. From them may be derived eighteen 
 other formulas. 
 
 By Formula L 
 
 391. This formula contains /owrtfiyere/i^ quantities; the 
 first term, the common difference, the last term, and the 
 number of terms. If any three of these quantities are given, 
 the other may be found. (Art. 388.) 
 
 I. 1 = a ±, (n — i) ^ / a,cl, and 71 being given. 
 
 3. Given d, Z, and n, to find a, the first term.* 
 
 Transposing (n—i) d in (i), 
 
 a = l± {n—i)d. 
 
 4. Given a, I, and n, to find d, the common difference. 
 
 Transposing in (i), and dividing by {n—i), 
 
 , I — a 
 
 a = • 
 
 n — I 
 
 5. Given a, d, and ?, to find n, the number of terms. 
 
 Clearing of fractions and reducing (4), 
 
 I — a 
 
 n = —-=— + I. 
 
 d 
 
 1. Given a = 2$, d=i 3, and n = 12, to find I 
 
 2. Given a = ^S, d = 5, and n = 45, to find I 
 
 3. Given d=: 3, Z = 35, and 71 = 9, to find a. 
 
 4. Given I = ^y, d = 5, and 71 = 21, to find a, 
 
 5. Given « = 15, 1 = 85, and n=z ^1, to find d, 
 
 6. Given a = 28, 1= 7, and ?^ = 26, to find ^. 
 
 7. Given a=z 2^, d = 5, and Z = 5^38, to find w. 
 
 8. Given a= 6, t?= 6, and Z=ii52, to find w. 
 
 * For Formula 2' see Art. 389. 
 
21(/ AEITHMETICAL PROGRESSION. 
 
 By Formula II. 
 
 392. In this formula there are four different quantities: 
 the first term, the last term, the number of terms, and the 
 sum of the terms. If any three of these quantities are 
 given, the other may be found. (Art. 389.) 
 
 2. s = X iif a, I, and^^^ being given. 
 
 Note. — For Formulas 3-5, see Article 391. 
 
 6. Given I, n, and s, to find a, the first term. 
 Clearing (2) of fractions, dividing and transposing, 
 
 a = 1. 
 
 n 
 
 7. Given a, n, and s, to find I, the last term. 
 
 Transposing in (6), we have 
 
 I = a. 
 
 n 
 
 8. Given a, I, and 5, to find n, the number of terms. 
 Clearing (7) of fractions, transposing, factoring, and dividing, 
 
 2.S 
 
 w = ', 
 
 a ^-l 
 
 1. Given «= 9, 7 = 41, and w= 7, to find 5. 
 
 2. Given « = \, ? = 45, and n = 50, to find s. 
 
 3. Given ^ = 50, d:= 4, and w = 12, to find a. 
 
 4. Given a= 9, Z = 41, and 5 = 150, to find 71. 
 
 5. Given d=z 7, ?=2i, and ?^ = 35, to find 05. 
 
 6. Given a = 46, 1= 24, and 5 = 455, to find n. 
 
 7. Given a = 27, w = 9, and 5 = 72, to find I 
 
 8. Given a = 72, n= 8, and s = 288, to find I 
 
 9. Find the sum of the series 3, 5, 7, 9, etc., to 15 terms. 
 
 10. Find the twentieth term of 5, 8, 11, 14, 17, etc. 
 
 11. If the first term of an ascending series is 5, and the 
 common difference 4, what is the 15th term? 
 
ARITHMETICAL PROGRESSION. 
 
 211 
 
 393. The remaining twelve fonnulas are derived by 
 combining the preceding ones in such a manner as to 
 eliminate the quantity whose value is not sought. They 
 are contained in the following 
 
 TABLE. 
 
 No. 
 
 GlVBN. 
 
 Required. 
 
 FOBUUXAS. 
 
 9- 
 lo. 
 II. 
 
 12. 
 
 14. 
 
 15- 
 
 16. 
 
 17. 
 18. 
 19. 
 20. 
 
 d n, s 
 
 d, I, s 
 
 a, I, s 
 
 I, n, s 
 
 a, n, 8 
 
 d, n, s 
 
 a, d, s 
 
 a, df s 
 
 d, I, s 
 
 a, d, n 
 
 a, d, I 
 
 d, I, n 
 
 a = 
 
 25 — dn^ + dn 
 
 271 
 
 "^^f^yO+i/"'^' 
 
 _ 2(nl — s) 
 w (w— i) 
 
 25 — 2an 
 
 d = 
 d=z 
 d = 
 
 n^ — n 
 
 ±^(2^—^)24- Sds—2a-{-d 
 
 71=1 
 
 n=z 
 
 2d 
 
 2l+d± V(2l ^df—Ms 
 
 2d 
 
 n 
 
 5 = - [2« + (^^ — i) ^ 
 
 Ij^ a Z2 _ ^2 
 
 5 = Y i— 
 
 2 2d 
 
 5 = - [2?— (7^— i)cZ] 
 
 ._ Of the twenty fonnulas in Arithmetical Progression, the jirsA 
 two are indispensable, and should be thoroughly committed to memory ; 
 the next six are important in the solution of particular problems. The 
 remaining twelve are of less consequence, but will be fountJ 
 interesting to the inquisitive student. 
 
212 ARITHMETICAL PROGRESSION. 
 
 394. By the fourth formula in x\rt. 391, any number of 
 arithmetical means may be inserted between two given 
 terms of an arithmetical progression. For, the number of 
 terms consists of the two extremes and aU the intermediate 
 terms. 
 
 Let m = the number of means to be inserted. 
 
 Then m + 2 = n, the whole number of terms. 
 
 Substituting m + 2 f or n in the fourth formula, we have 
 
 d = . Hence, 
 
 m+i 
 
 27ie required number of means is found by the continued 
 addition of the common difference to the successive terms. 
 
 1. Find 4 arithmetical means between i and 31. 
 
 2. Find 9 arithmetical means between 3 and 48. 
 
 PROBLEMS. 
 
 1. If the first term of an ascending series is 5, the common 
 difference 3, and the number of terms 15, what is the last 
 term ? 
 
 2. If the first term of a descending series is 27, the 
 common difference 3, and the number of terms 12, what is 
 the last term ? 
 
 3. If the first term of an ascending series be 7, and the 
 common difference 5, what will the 20th term be ? 
 
 4. Find 5 arithmetical means between 2 and 60. 
 
 5. What is the sum of 100 terms of the series -J-, f, i, f, 
 h 2, i, h 3» etc. 
 
 6. If the sum of an arithmetical series is 18750, the least 
 term 5, and the number of terms 20, what is the common 
 difference ? 
 
 7. Required the sum of bhe odd numbers i, 3, 5, 7, 9, 11, 
 etc., continued to 76 terms? 
 
 8. Required the sum of 100 terms of the series of even 
 numbers 2,^4, 6, 8, 10, etc. 
 
ARITHMETICAL PRO GRESSI 0:N^. 213 
 
 9. The extremes of a series are 2 and 47, and the number 
 of terms is 10. What is the common difference ? 
 
 10. Insert 8 means between 6 and 72. 
 
 11. Insert 9 means between 12 and 108. 
 
 12. The first term of a descending series is 100, the 
 common difference 5, and the number of terms 15. What 
 is the sum of the terms ? 
 
 Note. — i. In Arithmetical Progression, problems often occur in 
 which the terms are not directly given, but are implied in the 
 conditions. Such problems may be solved by stating the conditions 
 algebraically, and reducing the equations. 
 
 13. Find four numbers in arithmetical progression, whose 
 ium shall be 48, and the sum of their squares 656. 
 
 Let X = the second of the four numbers. 
 
 And 1/ = their common difference. 
 
 By the conditions, (a^— y) + x + {x + i/) + (x + 2y) = 48 (i) 
 
 And (ps—yf + x^ + {x+ yf + {x+ 2y* = 656 (2) 
 
 Uniting terms in (i), 42; + 2^= 48 (3) 
 
 " "(2), 4»' + 4^+62/« = 656 (4) 
 
 Transposing and dividing in (3), y = 2^ — 2x (5) 
 
 Dividing (4) by 2, 27l^ + 2xy+2f = 328 (6) 
 
 Substituting value of y, 2X^ + 20(24—20;) + 3(24 — 2xY = 328 
 Reducing, ic*— 2405 = —140 
 
 Completing square, etc., a; = 14 or 10 
 
 Substituting in (5) y = — 4 or 4 
 
 Hence the required numbers are 6, 10, 14, and 18. 
 
 Note. — 2. The first two values of x and y produce a descending series ; 
 the other two an ascending series. In both the numbers are the same. 
 
 14. Find three numbers in arithmetical progression whose 
 Bum is 15, and the sum of their cubes is 495. 
 
 15. If 100 marbles are placed in a straight line a yard 
 apart, how far must a person travel to bring them one by 
 one to a box a yard from the first marble ? 
 
214 AKITHMETICAL PROGRESSION-. 
 
 1 6. How many strokes does a common clock strike in 
 ?4 hours? 
 
 17. A student bought 25 books, and gave 10 cents for the 
 first, 30 cents for the second, 50 cents for the third, etc. 
 What did he pay for the whole ? 
 
 18. A boy puts into his bank a cent the first day of the 
 year, 2 cents the second day, 3 cents the third day, and so 
 on to the end of the year. What sum does he thus lay up 
 in 365 days? 
 
 19. The clocks of Venice go on to 24 o'clock. How many 
 strokes does one of them strike in a day ? 
 
 20. What will be the amount of li, at 6 per cent simple 
 interest, in 20 years ? 
 
 21. What three numbers are those whose sum is 120, and 
 the sum of whose squares is 5600 ? 
 
 22. A traveller goes 10 miles a day ; three days after, 
 another follows him, who goes 4 miles the first day, 5 the 
 second, 6 the third, and so on. When will he overtake the 
 first? 
 
 23. Find four numbers, such that the sum of the squares 
 of the extremes is 4500, and the sum of the squares of the 
 means is 4100. 
 
 24. A sets out from a certain place and goes i mile the 
 first day, 3 miles the second day, 5 the third, etc. After he 
 has been gone 3 days, he is followed by B, who goes 1 1 miles 
 the first day, 12 the second, etc. When will B overtake A ? 
 
 25. The first term of a decreasing arithmetical progression 
 is 10, the common difference |, and the number of terms 21. 
 Required the sum of the series. 
 
 26. A debt can be discharged in 60 days by paying $1 the 
 first day, $4 the second, $7 the third, etc. Required the 
 amount of the debt and of the last payment 
 
GEOMETEICAL PEO G EES SIGN . 215 
 
 GEOMETRICAL PROGRESSION. 
 
 395. A Geometrical Progj^ession is a series of 
 quantities which increase or decrease by a constant multiplier 
 called the ratio. Hence, 
 
 The ratio may be an integer or o^ fraction. 
 
 Note, — When the ratio 'ib fractional, the series will decrease. For 
 multiplying by a fraction is taking a certain part of the multiplicand 
 as many times as there are like parts of a unit in the multiplier. 
 
 396. In a geometrical series, each succeeding term is 
 found by multiplying the preceding one by the ratio. 
 
 Thus, if o is the first term, and r the ratio, the series is 
 
 a, or, ar^ ar*, or*, ar^y ar^, etc. 
 If the ratio is 3, the series is 
 
 a, ax 3, «x3*, ax3«, etc. 
 If the ratio is |, the series is 
 
 a, ax^, axjx^, oxjx^x^^, etc. 
 
 397. An Ascending Series is one which increases 
 by an integral ratio ; as, 2, 4, 8, 16, 32, etc. 
 
 398. A Descending Series is one which decreases 
 by a fractional ratio; as, 64, 32, 16, 8, etc. 
 
 399. When the ratio is a positive quantity, aU the terms 
 of the progression are positive; when it is negative, the 
 terms are alternately positive and negative. 
 
 Thus, if the first term is a, and the ratio —3, the series is 
 «» — 3«» +9«» —2705, +Sia, etc. 
 
 395. Wliat l8 a geometrical progression ? 397. What Is an ascending serlM f 
 J98. Descending? 399. Wlxat law governs the signs? 
 
216 GEOMETBICAL PROGRESSION. 
 
 400. In geometrical progression there are five elements, 
 the first term, the last term, the number of terms, the 
 common ratio, and the sum of the terms. 
 
 Let a = the first term, 
 / = the last term, 
 n = the number of terms, 
 r = the ratio, 
 8 = the sum of the terms. 
 
 The relation of these five quantities to each other is such 
 that if any three of them are given, the other two can be 
 found. 
 
 CASE I. 
 
 401. The First Term, the Number of Terms, and the Ratio 
 
 being given, to Find the Last Term. 
 
 In this problem, a, w, and r are given, to find I, the last term. 
 
 The successive terms of the series are 
 
 a, ar, ar^, ar^, ar^^ etc, to ar»-^ (Art. 397.) 
 
 By inspection, it will be seen that the ratio r consists of a regular 
 series of powers, and in each term the index of the power is one less 
 than the number of the terms. Therefore, the last or nth term of the 
 series is ar''-'^. Hence, we have 
 
 Formula I. I =. ar"^^. 
 
 Rule. — Multiply the first term ly that 'power of the ratio 
 vjhose index is one less than the number of terms. 
 
 CoR. — Any term in a series may be found by the preceding 
 rule ; for the series may be supposed to stop at that term. 
 
 1. Given a= ^, n ^ 6, and r = 2, to find I 
 
 2. Given a= 2, w = 8, and r = s, to find I. 
 
 3. Given a = 72, 71 = 5, and r = |, to find I, 
 
 4. Given a= 5, 7^ = 4, and r = 4, to find I, 
 
 5. Given a= 7, ?i = 5, and r = 2, to find I. 
 
 6. Given a = 10, n = 6, and r = — 5, to find I 
 
 400. Name the elements In jceometrical prog^Bsion. 401. How find the last 
 term? 
 
GEOMETRICAL P ROGRESSIOlif 217 
 
 CASE II. 
 
 402. The First Term, the Last Term, and the Ratio being 
 given, to Find the Sum of the Terms. 
 
 In this problem, a, I, and r are given, to find 8, 
 
 Since 8 = the sum of the terms, we have 
 
 8 = a-\-a/r + ar'^ + a7^-{- .... +a7^-2 + ar»-\ (i) 
 
 Multiplying (i) by r, 
 
 rs = ar + ar^ + a7^ + ar^+ . . . . +aT^^ + ar*. (2) 
 
 Subtracting (i) from (2), ra—s = ar*—a. (3) 
 
 Factoring and dividing, a = • (4) 
 
 In equation (4), ar^ is the last term of (2), and is therefore the 
 product of the ratio by the last term in the given series. 
 Substituting Ir for ar^, we have 
 
 Ir -~a 
 
 Formula II. 
 
 r— I 
 
 EuLE. — Multiply the last term hy the ratio, from the 
 product subtract the first term, and divide the remainder by 
 the ratio less one. 
 
 PW' For the method of finding the sum of an infinite descending 
 aeries, see Art. 435. 
 
 1. Given a = 2, ? = 500, and r = 3, to find the sum. 
 
 _ Ir—a 500 X 3 — 2 . 
 
 Solution. « = = = 749, Ans, 
 
 2. Griven « = 3, 1=^ 9375? and r = 5, to find s, 
 
 3. Given a = 9, l^ 9000, and r = 10, to find s. 
 
 4. Given a = s^ ^ = 20480, and r ^ 4, to find s. 
 
 5. Given a =: 15, 1= 3240, and r = 6, to find s. 
 
 6. Given a = 2^, 1= 6400, and r =z 4, to find 5. 
 
 402. How find the sum of the terms ? 
 10 
 
218 GEOMETRICAL PE0GRESSI02T. 
 
 403. The two preceding formulas furnish the means loi 
 solving all problems in geometrical progression. Thej may 
 be varied so as to form eighteen other formulas. 
 
 By Formula I. 
 
 404. The first formula contains /owr different quantities: 
 the first term, the last term, the ratiOy and the number of 
 terms. If any three of these quantities are given, the oiuer 
 may be found. By the first formula, 
 
 I. I = ar**^^; a, n, and r being given. (Art. 401.) 
 For formula 2, see Article 402. 
 
 3. Given Z, n, and r, to find or, the first term. 
 
 Factoring (i), and dividing by y-^ 
 / 
 
 4. Given a, ?, and 7^, to find r, the ratio. 
 
 Dividing (i) by «, and extracting the root denoted by the index, 
 
 5* Given a, ?, and r, to find n, the number of terms. 
 
 I 
 
 a 
 
 Dividing (i) by a, r""^ 
 
 By logarithms, log r (n— i) = log ? — log a 
 
 -. , log I — log a 
 
 Dividing, etc., n = —-i^-^— + ^' 
 
 Note.— Sin«e this formula contains logarithms, it may be deferred 
 till that subject is explained. 
 
 1. Given « = 3, n = $, and r = 10, to find I 
 
 2. Given a = 5, n = 6y and r = 5, to find /. - 
 
 3. Given Z =1 256, n = S, and r = 2, to find a. 
 
 4. Given I = 243, ^ = 5, and r = 3, to find a. 
 
 5. Given a = 2^ Z = 2592, and /i = 5, to find n 
 
 6. Given a = 4, / := 2500, and w = 5, to find n 
 
GBOMETKICAL PRO GRESSIOl?'. 219 
 
 By Formula IL 
 
 405. This formula contains four different quantities : the 
 fird term, the last term, the ratio, and the sum of ths 
 terms. If a7iy three of them are given, the other may be 
 found. By the second formula, 
 
 1/p ft 
 
 2, s = — , a, I, and /• being given. (Art. 402.) 
 
 For formulas 3-5, see Article 404. 
 
 6. Given I, r, and s, to find a, the first term. 
 
 Clearing (2) of fractions, etc., 
 
 a = Ir — s {r — i) 
 
 7. Given a, r, and 5, to find Z, the last term. 
 
 Transposing in (6), 
 
 ;r = a + s (r — i). 
 Dividing by r, 
 
 / — « + g(r~»i) 
 
 8. Given a, 7, and s, to find r, the ratio. 
 
 Clearing (2) of fractions, 
 
 sr — s-= Ir — a. 
 Transposing in the last equation, 
 
 sr — Ir = s — a. 
 
 Factoring, etc., 
 
 8 — a 
 r = 
 
 8-1 
 
 1. Given a =^ 2, I z= 162, and r = 3, to find s, 
 
 2. Given / =1 54, r = 3, and s =z 80, to find a. 
 
 3. Given o^ = 4, r = 5, and s z^ 624, to find I. 
 
 4. Given a = 4, I = 12500, and s = 15624, to find n 
 
 5. Given a = 5, I = 180, and r ^ 6, to find s. 
 
 6. Given a = t, r = ^, and s = 847, to find I, 
 
220 
 
 GEOMETRICAL PROGRESSION?-, 
 
 406. The remaining twelve formulas are derived by 
 combining the preceding ones in such a manner as to 
 eliminate the quantity whose value is not sought. 
 
 TABLE. 
 
 No 
 
 Given. 
 
 Requered. 
 
 FOKMULAS. 
 
 9 
 
 lO 
 
 II 
 
 12 
 
 13 
 
 14 
 
 15 
 
 16 
 
 17 
 
 18 
 19 
 20 
 
 n, r, s 
 
 I, n, s 
 
 a, n, s 
 
 n, r, s 
 
 a, I, s 
 
 a, r, s 
 
 h r, s 
 
 tty 71, S 
 
 I, 71, S 
 
 a, n, r 
 
 I, n, r 
 
 a, I, Tfi 
 
 a = 
 
 {r—\)s 
 
 /•" — I 
 
 a{s — ay-^ = / (5 — iy-\ 
 
 l{s — ly-^ = a{s — ay-K 
 
 r" — I 
 
 71 = log ^ - log Q^ ^ 
 
 log(s—a)—log(s—iy 
 ]og[a + {r—i)s] — log a 
 log r 
 
 _log|— log [Ir— (r— 1)5] 
 
 ~" log r 
 
 n 
 
 + 1 
 
 s s 
 
 r" r = 1 
 
 a a 
 
 /•" + 
 
 %-i 
 
 I— S 1 — 8 
 
 a(r'' — i) 
 
 r - 
 
 - 1 
 
 
 
 ^/i — " v« 
 
 Of the twenty formulas in Geometrical Progression, the fird 
 two are fundamental, and should be thoroughly committed to memory ; 
 the next six are important in the solution of particular prohlems. The 
 remainder are less practical. 
 
GEOMETRICAL PK OGRE SSI ON. 221 
 
 407. By the fourth, formula (Art. 404), any number of 
 geometrical means may be found between two given 
 quantities. 
 
 Let m = the number of means required. 
 
 Then m + 2 = n. 
 
 Substituting m + 2 for n in the formula, we have 
 
 '• = ©-^ 
 
 The ratio being found, the means required are obtained by continued 
 multiplication. 
 
 1. Find two geometrical means between 3 and 192. 
 
 Solution, r = Q* = j/i^? = ^ej = 4. 
 
 The ratio being 4, the first mean is 3 x 4 = 12 ; the second is 
 12x4 = 48. 
 
 2. Find three geometrical means between | and 128. 
 
 PROBLEMS. 
 
 1. In a geometrical progression, the first term is 6, the 
 last term 2916, and the ratio 3. What is the sum of all the 
 terms ? 
 
 2. In a decreasing geometrical series, the first term is |, 
 the ratio i, and the number of terms 8. What is the sum 
 of the series ? 
 
 3. What is the sum of the series i, 3, 9, 27, etc., to 15 
 terms? 
 
 4. Find the sum of 12 terms of the series, i, |, f, -jy, etc. 
 
 5. If the first term of a series is 2, the ratio 3 and the 
 number of terms 15, what is the last tenn ? 
 
 6. What is the i6th term of a series, the first term of 
 which is 3, and the ratio 3 ? 
 
 Note. — When the terms of the series are not stated directly, they 
 may be represented algebraically. 
 
222 GEOMETEICAL PKOGRESSIOK. 
 
 7. Find three numbers in geometrical progression, such 
 that their sum shall be 2 1, and the sum of their squares 189. 
 
 Let the three numbers be x, ^/xy, and y. 
 
 By the conditions, a;+ ^/m-\-y = 21 (i) 
 
 And a;2 + a;y+y2 = 189 (2) 
 
 Transposing and sq. (i), as^ + 20;^ + ^^2 = 441 _ 42 /y/^ + xy (3) 
 
 Subtracting (2) from (3), iry = 25 2 - 42 'v/^+ xy (4) 
 
 Transposing, etc., ^/xy = 6 (5) 
 
 Involving, ^ icg^ = 36 (6) 
 
 And 32^ = 108 (7) 
 Subtracting (7) from (2), c^—2xy+y^ = 81 
 
 Extracting root, x—y = 9 (8) 
 
 Substituting (5) in (i), x+y = 15 (9) 
 Combining (8) and (9), a? = 12 
 
 y= 3 
 Hence the numbers, 12, 6, and 3, Ans. 
 
 8. A father gives his daughter $1 on New Year's day 
 t(>vrards her portion, and doubles it On the first day of every 
 month through the year. What is her portion ? 
 
 9. A dairyman bought 10 cows, on the condition that he 
 should pay i cent for the first, 3 for the second, 9 for the 
 third, and so on to the last. What did he pay for the last 
 cow and for the ten cows? 
 
 10. A man buys an umbrella, giving i cent for the first 
 brace, 2 cents for the second brace, 4 for the third, and so 
 on, there being 10 braces. What is the cost of the 
 umbrella ? 
 
 11. The sum of three numbers in geometrical progression 
 is 26, and the sum of their squares 364. Find the numbers. 
 
 12. What would be the price of a horse, if he were to be 
 sold for the 32 nails in his shoes, paying i mill for the first, 
 2 mills for the second, 4 for the third, and so on ? 
 
 13. Find four numbers in geometrical progression, such 
 that the sum of tlie first three is 130, and that of the last 
 three is 390. 
 
GEOMETRICAL PROGRESSION^. 223 
 
 14. A man divides $210 in geometrical progression among 
 three persons; the first had $90 more than the last. How 
 much did each receive ? 
 
 15. There are five numbers in geometrical progression. 
 The sum of the first four is 468, and that of the last four is 
 2340. What are the numbers? 
 
 16. The sum of $700 is divided among 4 persons, whose 
 shares are in geometrical progression ; and the difference 
 between the extremes is to the difference between the means 
 as 37 to 12. What are the respective shares? 
 
 17. The population of a town increases annually in 
 geometrical progression, rising in four years from loooo to 
 1 464 1. What is the ratio of annual increase ? 
 
 18. The sum of four numbers in geometrical progression 
 Is 15, and the sum of their squares 85. What are the 
 numbers ? 
 
 HARMONICAL PROGRESSION.* 
 
 408. An Harmonical I^rogression is such, that 
 of any three consecutive terms, the first is to the third as the 
 difference of the first and second is to the difference of the 
 second and third. 
 
 Thus, 10, 12, 15, 20, 30, 60, 
 
 are in harmonic progression ; for 
 
 ID ; 15 :: 12—10 : 15—12 
 12 : 20 :: 15—12 : 20—15 
 15 : 30 : : 20—15 '• 30—20 
 20 : 60 : : 30—20 : 60—30 
 Let a, 6, c, d, e, /, g, be an harmonical progression, then 
 
 a : c : a—b : b—c, etc. 
 Note.— When three quantities are such, that the first is to the 
 tJiird as the difference of the first and second is to the difference of the 
 second and tliird, they are said to be in Harmonical Proportion. 
 Thus, i, 3, and 6, are in harmonical proportion, 
 
 408. What is an harmonical progression ? 
 "•' If a musical string be divide.] in harmonical proportion, the 
 /ifferent parts will vibrate in harmony. Hence, the name. 
 
224 HARMONICAL PROGRESSIOK. 
 
 409. To Find the Third Term of an Harmonical 
 Progression, the First Two being given. 
 
 Let a and h be the first two terms, and x the third term. 
 
 Then a : x :: a-l : l-x 
 
 Multiplying extremes, etc., ab—ax = ax—hx 
 
 Transposing, etc., 2ax—bx = db 
 
 Factoring, and dividing by 2a— h, we have the 
 
 ab 
 
 FORMtJLA. X 
 
 2a — h 
 
 Rule. — Divide the product of the first two tenns hy twice 
 the first mi?ms the second term; the quotient will le tht 
 third term. 
 
 Note. — This rule furnishes the means for extending an harmonic 
 progression, by adding one term at a time to the two preceding terms. 
 
 1. Find the third term in the harmonic series of which 
 12 and 8 are the first two terms. Ans. 6. 
 
 2. Find the third term in the harmonic series of which 
 12 and 1 8 are the first two terms. Ans. 2,^. 
 
 3. If the first two terms of an harmonic progression are 
 15 and 20, what is the third term? Ans. 30. 
 
 4. Continue the series 12, 15, 20, for two terms. 
 
 Ans. 30 and 60. 
 
 5. Continue the series 7^^, 9, 12, for two terms. 
 
 A71S. 18 and 2,^, 
 
 410. To Find a Mean or IVIiddle Term between Two Terms 
 of an Harmonic Progression. 
 
 Let a and c be the first and third of three consecutive terms of an 
 harmonic progression, and m the mean. 
 
 Then a i c :: a—m . m—e 
 
 Mult, extremes and means, am—ac = ac—cm 
 Transposing and uniting, am + cm = 2ac 
 
 Factoring and dividing by a + c, we have the 
 
 „ 2ac 
 
 Formula. m = • 
 
 a + c 
 
 409. How find the third term of an harmonical progression, the first two being 
 tfiven? 
 
HARMOtsTICAL PROGRESSION". 225 
 
 Rule. — Divide twice the product of the first and third 
 terms hy their sum; the quotient will he the mean or middle 
 term. 
 
 6. The first and third of three consecutive terms of an 
 harmonic progression are 9 and 1 8. Required the mean or 
 middle terra. 
 
 Solution. 2x9x18 = 324, and 9 + 18 = 27, 
 Now 324-J-27 = 12, Ans. 
 
 7. rind an harmonic mean between 12 and 20. Ans, 15. 
 
 8. Eind an harmonic mean between 15 and 30. Ans. 20. 
 
 411. The Reciprocals of the terms of an harmonic 
 progression form an arithmetical progression. 
 
 Thus, the reciprocals of 10, 12, 15, 20, etc., viz., 
 
 tV» tV» tV» "eV* ■^' ^^^•» 
 are an arithmetical progression, whose common difference is ^V 
 
 Again, let a, b, c be in harmonic progression. 
 
 Then a : c :: a—b : b—c 
 
 Mult, extremes and means, ah—ac = ac—bc 
 
 Dividing hj abc, ^ ~ ^ (^^- 3^4') 
 
 Conversely, the reciprocals of an arithmetical progression 
 form an harmonic progression. Thus, 
 
 The reciprocals of the arithmetical progression i, 2, 3, 4, 5, etc., 
 viz., \, ^, \i\, i, etc., are in harmonic progression. 
 
 412. If the lengths of six musical strings of equal weight 
 and tension, are in the ratio of the numbers 
 
 i> h h h h h etc., 
 
 the second will sound an octave above the first ; the third 
 will sound the twelfth ; the fourth the double octave, etc. 
 
 410. Kow find a mean between two terms of an harmonic progression? 
 411. Whai do the reciprocals of an harmonical prog; ession fonL . 
 
INFIl^ITE SERIES, 
 
 INFINITE SERIES. 
 
 413. An Infinite Series is one in which the successive 
 terms are formed by some regular law, and the numher of 
 terms is unlimited. 
 
 414. A Converging Series is one the sura of whose 
 terms, however great the number, cannot numerically 
 exceed definite quantity. 
 
 415. A Diverging Series is one the sum of whose 
 terms is numerically greater than 2ii\y finite quantity. 
 
 416. To Expand a Fraction into an Infinite Series, 
 
 Remark. — Any common fraction whose exact value cannot be 
 expressed by decimals, may be expanded into an infinite series. 
 
 1. Expand the fraction J into an infinite series. 
 
 Solution. 1-5-3 = .333333, and so on, to infinity. 
 
 Or, I ^3 = ^3^ + ^ + ^^ + ^^, etc. Hence, the 
 
 EuLE. — Divide the numerator hy the denominator* 
 
 2. Reduce to an infinite series. 
 
 1 —X 
 
 I— aj)i (i + aj+a;2 + a^+a^, etc., the quotient. (Art. 1 70. ) 
 +x 
 
 +ic*, etc. 
 Therefore, = i+x+x'^ + q? + v^-{-q?, etc., to infinity. 
 
 413. What is an infinite series? 414. A converging series? 415. Diverging? 
 416. How expand a fraction into an infinite fcrie?? 
 
INFINITE SERIES. 227 
 
 Let x = ^; then will = = 2 ; and tlie series will be 
 
 ^ ' i — xi—i 
 
 I + i+i + i + iV + aV* ®tc., tlie sum of which = 2. 
 
 If a? = 4, then will = = |, and the series will become 
 
 ** I — a; I — i ^ 
 
 i+i + i+^T+^T + ^5. etc. = |. 
 
 Notes. — i. If a? is less than i, the series will be convergent . 
 
 For, when x is Icm than i, the remainder must continually decrease ; 
 therefore, the further the division is carried, the less will be the 
 quantity to be added to the last term of the quotient in order to 
 express the exact value of the fraction. 
 
 2. If X is greater than i, the series will be divergent. 
 
 For, when x is greater than i, the remainder must constantly 
 increase ; therefore, the farther the division is csLrried, the greater will 
 be the quantity either positive or negative to be added to the quotient. 
 
 3. Eeduce the fraction to an infinite series. 
 
 Solution. i-7-(i+a;) = i—x+^—a^+x*—x^ + , etc. 
 
 This series is the same as that in Ex. 2, except the odd powers 
 of X are negatice. 
 
 Let a? = 2- ; then will = f ; which is equal to the series 
 
 I -i + 1 -i + iV- aV + , etc. 
 
 4. Reduce the fraction to an infinite series. 
 
 I —X 
 
 Ans. I + 20; + 2x^ -f 2x^ -f- 2X^, etc. 
 
 417. A fraction whose denominator has more than two 
 terms, may also be expanded into an infinite series. 
 
 5. Expand 5 into an infinite series. 
 
 ^ I — X -\- x^ 
 
 I— a?+«^) I {i+x—Q^—x^+x^, etc., Ans, 
 
 x—x^ 
 x—x^ + a ? 
 
 —a?, etc 
 
228 INFINITE SERIES. 
 
 418. To Expand a Compound Surd into an Infinite Series, 
 
 6. Keduce v i + x to an infinite series. 
 
 OPERATION. 
 
 
 2+-| +x 
 
 a? 
 
 + « + — 
 4 
 
 X^ I 25"^ 
 
 4 8 "^64 
 
 2 + a? + -Z +"5 — z~» ®tc. Hence, the 
 
 4 lo I o 04 
 
 EuLE. — Extract the square root of the given sutfl 
 (Art. 298.) 
 
 7. Expand ^/^^2 Ans.x-t^^^^, etc. 
 
 8. Expand \/2, or Vi + i. ^ws. i + -f — ^ + -jV? ^^c* 
 
 419. The Binomial Theorem applied to the Formation of 
 Infinite Series. 
 
 The Binomial Theorem may often be employed with 
 advantage, in finding the roots of binomials. For a root is 
 expressed like a power, except the exponent of one is an 
 integer, and that of the other is q> fraction. 
 
 9. Expand {x 4- yY into an infinite series. 
 
 Solution. — The terms without coefficients are 
 
 «*» aj~5y, a;~«y% a;~»^, a;~»2^, etc. 
 
 1 X —4 
 The coefficient of the second term is + ^ ; of the 3d, = — J ; 
 
 — 1 X — J 
 
 of the 4th term, — = + ^\, etc. 
 
 The series is «* + ^x'^y — lx~^y^ + ^^x~^y^» etc. 
 418. How expand a surd into an infinite eeriei f 
 
INFINITE SERIES. 229 
 
 420. When the index of the required power of a binomial 
 is a posiiivG integer, the series will termiriate. For, the 
 index of the leading quantity continually decreases by i ; 
 and soon becomes o; then the series must stop. (Art, 269.) 
 
 421. When the index of the required power is negaiive, 
 the series will never terminate. For, by the successive 
 subtractions of a unit from the index, it will never become 
 o ; and the series may be continued indefinitely. 
 
 10. Expand {(^+1/)^ into an infinite series, keeping the 
 factors of the coefiicients distinct. 
 
 ^ ^ 2X 2.^ 2.4.6aj* 2.4.6.8a;' 
 
 It. Expand VJ, or (i + i)^, keeping the factors of th€> 
 coefficients distinct. 
 
 Ans. i+i— i-+_3 3.-J 3- 5- 7 t,. 
 
 2 2.4 2.4.6 2.4.6.8 2.4.6.8.10 
 
 422. An Infinite Series must not be confounded with an 
 
 Ivfinite Quantity, 
 
 423. An Infinite Quantity is a quantity so great 
 that nothing can be added to it. 
 
 424. An Infinite Series is a series m which the 
 number of terms is unlimited, 
 
 425. The magnitude of the former admits of no increase; 
 while in the latter the number of terms admits of no 
 increase, and yet the sum of all the terms may be a small 
 quantity. 
 
 Thus, if the series i + l +i + TV+"s'*> ®*^'» ^^ which each succeeding 
 term is half the preceding, is continued to infinity, the sum of all the 
 terms cannot exceed a unit. 
 
 426. When one quantity continually approximates 
 another without reaching it, the latter is called the Litnit 
 of the former. 
 
330 INFIiflTE SERIES. 
 
 427. An Infinitesimal is a quantity whose value is 
 less than any assignable quantity/. 
 
 428. The Sign of Infinity^ or of an infinite 
 quantity, is a character resembling an horizontal figure 
 eight ( CO ). 
 
 The Sign of an Infinitesimal is zero ( o ). 
 
 429. One infinite series may be greater or less than 
 another. 
 
 Thus, the series i + i + i + ^ + A* etc., whose limit is 2, is greater 
 than the series i + i + g + t^ + sV* 6*^., whose limit is i. 
 
 430. Since an infinitesimal is less than any assignable 
 quantity, and in its limit approaches zero, when connected 
 with finite quantities by the sign -f or — , it is of so little 
 value that it may be rejected without any appreciable error. 
 
 431. An infinite series may be multiplied by a finite 
 quantity. 
 
 Thus, if the series 222222, etc., is multiplied by 3, 
 
 the product 666666, etc., is three times the multiplicand. 
 
 432. An infinite series may also be divided by a finite 
 quantity. 
 
 Thus, if the series 888888, etc., is divided by 2, 
 the quotient 444444, etc., is hali the dividend. 
 
 433. If a. finite quantity is multiplied by an infinitesimal, 
 the product will be an infinitesimal. For, with a given 
 multiplicand, the less the multiplier, the less will be tlie 
 product. Thus, xxo =z o. 
 
 434. If a fi7iite quantity is divided by an infinitesimal^ 
 the quotient will be infinite. Thus, ic -^ o = 00 . 
 
 If a finite quantity is divided by an infinite quantity, the 
 quotient will be an infinitesimal. Thus, a: -^ 00 = o. 
 
 If an infinitesimal is divided by a finite quantity, the 
 quotient is an infinitesimal. Thus, o -j- 2; = o. 
 
 Note.— In higher mathematics, the expression o -f- o admits of 
 various interpretations. 
 
IKFIl^ITE SEEIES. 231 
 
 435. To Find the Sum of a Converging Infinite Series, the 
 First Term and Ratio being given. 
 
 By the second formula in geometrical progression, we liave for an 
 
 increasing series (Art. 402), 
 
 Ir — a ar^ — a 
 
 8 — , or • 
 
 r—i r— I 
 
 In a decreasing series, tlie ratio r is less tlian i ; therefore. I or a/"»-i 
 is less than a. (Art. 398.) 
 
 Tliat both terms of the fraction , or may be positive, 
 
 T — I r — I 
 
 we change the signs of both (Art. 166), and 
 
 a — Ir 
 
 8 = • 
 
 i — r 
 
 But, in a decreasing infinite series, I becomes an infinitesimal, or o; 
 therefore, Ir = o. (Art. 427.) Hence, rejecting the iuHnitesimal from 
 
 f = , we have the 
 
 Formula. s = • 
 
 I — 1* 
 
 EULE. — Divide the first term ly i minus tlie ratio, 
 
 1. Find the sum of the infinite series 
 
 I + i + i + ^7 + A^ etc. 
 
 2. Required the sum of the infinite series 
 
 I — } + i — i+, etc. 
 
 3. Find the sum of the series \ + J + J +? etc, 
 
 4. Find the sum of the infinite series i + I + I +* eta 
 
 5. Find the sum of the series f + f + 27 +> etc. 
 
 6. Find the sum of the series 3 + 2 + | -f , etc. 
 
 7. Find the sum of the series 4 + ¥ + f| +> etc. 
 
 8. Find the sum of the series .ZZZZ^ etc. 
 
 9. Find the sum of the series .ddddd, etc. 
 
 10. Find the sum of the series - H -^■\ 5 + etc. 
 
 11. Suppose a ball to be put in motion by a force which 
 impels it 10 rods the first second, 8 rods the next, and so on, 
 decreasing by a ratio of ^ each second to infinity. Through 
 what space would it move ? 
 
OHAPTEE XIX. 
 
 LOGARITHMS* 
 
 436. The LogarithTn of a number is the exponent ol 
 the power to which a given fixed number must be raised to 
 produce that number. 
 
 437. This Fixed Numler is called the ^ase of the 
 
 system. 
 
 Tlius, if 3 is the base, then 2 is the logarithm of 9, because 3^ = 9 ; 
 and 3 is the logarithm of 27, because 3^ = 27, and so on. 
 
 Again, if 4 is the base, then 2 is the logarithm of 16, because 
 4^^ = 16 ; and 3 is the logarithm of 64, because 4^ = 64, and so on. 
 
 438. In forming a system of logarithms, any number, 
 except I, may be taken as the base, and when the base is 
 selected, all other numbers are considered as some power or 
 root of this base. Hence, there may be an unlimited 
 number of systems. 
 
 Note. — Since all powers and roots of i are i, it is obvious that other 
 numbers cannot be represented by its powers or roots. (Art. 289.) 
 
 439. There are tv/o systems of logarithms in use, the 
 Napierian system,! the base of which is 2.718281828, and 
 the Common System, whose base is 10. J 
 
 The abbreviation log stands for the term logarithm. 
 
 436. Wliat are logarithms ? 437. What is this fixed number called ? 439. Name 
 the systems in use. The base of each. 
 
 * The term logarithm is derived from two Greek words, meaning 
 the relation of nwrnhers. 
 
 f So called fi-om Baron Napier, of Scotland, who invented log- 
 arithms in 1614. 
 
 X The common system was invented by Henry Briggs, an English 
 mathematician, in 1624. 
 
LOGARITHMS. 233 
 
 440. The Sase of common logarithms being lo, all 
 other numbers are considered as powers or roots of lo. 
 
 Thus, the log. of i is o ; for lo" equals i (Art. 259) ; 
 ** *' 10 is I ; for 10^ " 10 ; 
 
 ** *• icx) is 2 ; for lo^ " 100 ; 
 
 ** " 1000 is 3 ; for lo' " 1000, etc Hence, 
 
 The logarithm of any number between i and 10 is a 
 fraction; for any number between 10 and 100, the logarithm 
 is I plus a fraction ; and for any number between 100 and 
 1000, the logarithm is 2 plus a fraction, and so on. 
 
 441. By means of negative exponents, this principle may 
 be applied to fractions. 
 
 Thus (Art. 256), the log. of .1 is —i ; for 10-^ equals .1 ; 
 " '* ,01 is —2 ; for 10-2 " .01; 
 *' " .001 is —3; for 10-^ ** .001. 
 
 Therefore, the logarithms for all numbers between i and 
 0.1 lie between o and — i, and are respectively equal to — i 
 plus a fraction; for any number between o.i and 0.0 1, the 
 logarithm is —2 plus a fraction ; and for any number 
 between o.oi and o.ooi, the logarithm is — 3 plus a fraction, 
 and so on. 
 
 Hence, the logarithms of all numbers greater than 10 or 
 less than i, and not exact powers of 10, are composed of 
 two parts, an mteger and o. fraction. 
 
 Thus, the logarithm of 28 is 1.44716; 
 
 and of .28 is 1.44716. 
 
 442. The integral part of a logarithm is called the 
 Characteristic ; the decimal ^^rt, the llantissa, 
 
 443. The Characteristic of the logarithm of a whole 
 number is one less than the number of integral figures in 
 the given number. 
 
 Thus, the characteristic of the logarithm of 49 is i ; that of 495 is 
 2 ; that of 4956 is 3 ; that of 6256.414 is also 3, etc. 
 
 440. What is the logarithm of any number from i to 10? From 10 to 100? From 
 100 to 1000? 442. What is Ihe integral part of a logarithm called? The decirna.] 
 part ? 443. What is the characteristic 01 the logarithm of a whole numlier ? 
 
234 LOGARITHMS. 
 
 444. The Characteristic of the logarithm of a 
 decimal is negative, and is one greater than the number of 
 ciphers before the first significant figure of the fraction. 
 
 Thus, the characteristic of the logarithm of ^^ or .i is — i ; that of 
 3^^o' or .01, is — 2 ; that of ^i^^, or .001, is —3, etc. (Art. 256.) 
 
 The logarithm of .2 is — i with a decimal added to it ; that of 05 
 !s — 2 with a decimal added to it, etc. 
 
 Note. — It should be observed that the characteristic only is negative, 
 while the mantissa, or decimal part, is always podtive. To indicate 
 this, the sign — is placed over the characteristic, instead of before it. 
 
 Thus, the logarithm of .2 is i. 30103, 
 
 " " *' .05 is 2.69897, eta 
 
 445. The Decimal Part of the logarithm of any 
 number is the same as the logarithm of the number 
 multiplied or divided by 10, 100, 1000, etc. 
 
 Thus, the logarithm of 1876 is 3.27325 ; of 18760 is 4.27325, etc. 
 
 TABLES OF LOGARITHMS. 
 
 446. A Table of Loffaritlmis is one which contains 
 the logarithms of all numbers between given limits. 
 
 447. The Table found on the following pages gives the 
 mantissas of common logarithms to five decimal places for 
 all numbers from i to 1000, inclusive. 
 
 The characteristics are omitted, and must be supplied by 
 inspection. (Arts. 443, 444.) 
 
 Notes. — i. The first decimal figure in column is often the same 
 for several successive numbers, but is printed only once, and is 
 understood to belong to each of the blank places below it. 
 
 2. The character ( ♦ ) shows that the figure belonging to the place 
 it occupies has changed from 9 to o, and through the rest of this line 
 the first figure of the mantissa stands in the next line below. 
 
 444. What Is the characteristic of the loj]jarithm of a decimal ? 445. What is the 
 effect upon the decimal part of the lo<?. of a number, if the number is multiplied or 
 divided by 10, 100, ioop, etc. <^6, What is a table of logarithms f 
 
LOGAKITHMS. 235 
 
 448. To Find the Logarithm of any Number from I to 10. 
 
 Rule. — Look for the given numher m the first line of the 
 table ; its logarithrn will he found directly below it. 
 
 1. Find the logarithm of 7. Ans. 0.84510. 
 
 2. Find the logarithm of 9. A7is, 0.95424. 
 
 449. To Find the Logarithm of any Number from 10 
 
 to 1000, inclusive. 
 
 Rule. — Looh in the column marhed N" for the first two 
 figures of the given numher , and for the third at the head 
 of one of the other columns. 
 
 Under this third fiqure* and opposite the first tioo, will 
 he found the last decimal figures of the logarithm. The first 
 one is found in the column marked 0. 
 
 To this decimal prefix the proper characteristic. (Art. 443.) 
 
 Note. — If the number contains 4 or more figures, multiply the 
 tabular diflference by the remaining figures, and rejecting from the 
 right of the product as many figures as you multiply by, add the rest 
 to the log. of the first 3 figures. 
 
 3. Find the logarithm of 108. Ans, 2.03342. 
 
 4. Find the logarithm of 176. Ans. 2.2 ^^^\. 
 
 5. Find the logarithm of 1999. Ans. 3.30085. 
 
 450. To Find the Logarithm of a Decimal Fraction. 
 Rule. — Take out the logarithm of a ivhole number consist- 
 
 ing of the same figures^ and prefix to it the 'proper negative 
 characteristic. (Art. 444.) 
 
 Note. — If the number consists of an integer and a decimal, find the 
 logarithm in the same manner as if all the figures were integers, and 
 prefix the characteristic which belongs to the integral part. (Art. 443 ) 
 
 6. What is the log. of 0.95 ? Ans. 1.97772. 
 
 7. What is the log. of 0.0125? Ans. 2.09691. 
 
 8. What is the log. of 0.0075 ? ^^^^- 3'375o6. 
 
 9. What is the log. of 16.45 ? Ans. 1.21616. 
 10. What is the log. of 185.3 ? Ans. 2.26787. 
 
 448. How find the logarithm of a numher from i to 10? 449. From 10 to 1000 ? 
 450. How find the log. of a decimal ? Nots. Of fta integer and a decime^l ? 
 
236 LOGARITHMS. 
 
 451. To Find the Number belonging to a given Logarithm. 
 
 Rule. — Look for the decimal figures of the given logarilhin 
 in the table under the column marked ; a7id if all of them 
 are 7iot found in that column, look in the other colu?nns on 
 the right till you fi7id them exactly, or very nearly ; directly 
 opposite, in the column marked N, will he found the first 
 iwo figures, and at the top, over the logarithm, the third 
 figure of the given number. 
 
 Make this number correspond to the characteristic of the 
 given logarithm, by pointing off decimals, or by adding 
 ciphers, if necessary, and it will be the number require L 
 
 Note. — If the characteristic of a l(9garithin is negative^ the number 
 belonging to it is 2, fraction, and as many ciphers must be prefixed to 
 the number found in the table, as there are units in the characteristic 
 le%s I. (Art. 444.) 
 
 452. When the Decimal Part of the given Logarithm is 
 not exactly, or very nearly, found in the Table. 
 
 Rule. — From the given logarithm subtract the next less 
 logarithm found in the tables ; annex ciphers to the remain- 
 der, and divide it by the tabular difference {marked D) 
 as far as necessary. 
 
 To the number belonging to the less logarithm annex the 
 quotient, and make the number thus produced correspond to 
 the characteristic of the given logarithm, as above. 
 
 Note. — For every cipher annexed to the remainder, either a sig- 
 nificant figure or a cipher must be put in the quotient. 
 
 11. What number belongs to 2. 1 7231 ? Ajis. 148.7, 
 
 12. What number belongs to 1.25 261 ? A71S. 17.89. 
 
 13. What number belongs to 3.27715 ? Jns, 1893. 
 
 14. What number belongs to 2.30963 ? Ans. 204. 
 
 15. What number belongs to 4.29797 ? Ans. 19858.29. 
 
 16. What number belongs to 1. 14488 ? A?is. 0.1396. 
 
 17. What number belongs to 2.29136 ? Ans. 0.01956. 
 
 18. What number belongs to 3.30928 ? Ans. 0.002038. 
 
 451. How find the number belonging to a logarithm? 
 
LOGARITHMS. 23? 
 
 453. Computations by logarithms are based upon the 
 following principles : 
 
 1°. TJie sum of the logarithms of two numbers is equal to 
 the logarithm of their product. 
 
 Let a and c denote any two numbers, m and n their logarithms, 
 and b the base. 
 
 Then b^ = a 
 
 And ft» = (J 
 
 Multiplying, 6*+» = og, 
 
 2°. TJie logarithm of the dividend diminished hy the 
 logarithm of the divisor is equal to the logarithm of the 
 quotient of the two nn7nlers. 
 
 Let a and c denote any two numbers, m and n their logarithms, 
 
 and b the base. 
 
 
 Then 
 
 fe"* = a 
 
 And 
 
 &• = <? 
 
 Dividing, 
 
 6«-« = a-i-c 
 
 454. To Multiply by Logarithms. 
 
 1. Required the product of 35 by 23. 
 
 Solution. — The log. of 35 = 1.54407 
 
 « «' » 23 = 1. 36173 
 
 Adding, 2.90580. (Art. 453, Prin. i.) 
 
 The number belonging is 805, Ans. Hence, the 
 
 Rule — Add the logarithms of the factors; the sum ivill 
 he the logarithm of the product. 
 
 Notes. — i. If the sum of tlie decimal parts exceeds 9, add the tens 
 figure to the characteristic. 
 
 2. If either or all the characteristics are negative, they must be 
 added according to Art. 65. But as the mantissa is always positue, 
 that which is carried from the mantissa to the characteristic must be 
 considered positive. 
 
 2. What is the product of 109.3 by 14.17 ? 
 
 3. What is the product of 1.465 by 1.347 ? 
 
 4. What is the product of .074 by 1500 ? 
 
 453. Upon what, two principles are computations by logarithms based? 454. How 
 multiply by logarithms f 
 
238 LOGARITHMS. 
 
 455. To Divide by Logarithms. 
 
 5. Kequired the quotient of 120 by 15. 
 
 Solution.— The log. of 120 = 2.07918 
 «« 4i « jg _ 1. 1 7609 
 
 " ** '* quotient = 0.90309. Ans. 8. Hence, the 
 
 Rule.— ^rom the logarithm of the dividend subtract the 
 logarithm of the divisor ; the difference will he the logarithm 
 of the quotient. (Art. 453, Prin. 2.) 
 
 Notes. — i. When either of the characteristics is negative, or when 
 the lower one is greater than the one above it, change the sign of the 
 subtrahend, and proceed as in addition. 
 
 2. When I is carried from the mantissa to the characteristic, it 
 must be considered positive, and be added to the characteristic before 
 the sign is changed. 
 
 6. What is the quotient of 12.48 by 0.16 ? 
 
 7. What is the quotient of .045 by 1.20? 
 
 8. What is the quotient of 1.381 by .096 ? 
 
 456. Negative quantities are divided in the same manner 
 as positive quantities. 
 
 If the sign of the divisor is the same as that of tlie 
 dividend, prefix the sign + to the quotient ; but if different, 
 prefix the sign — . 
 
 9. Divide —128 by — 47. 
 
 10. Divide — 186 by — 0.064. 
 
 11. Divide — 0.156 by —0.86. 
 
 12. Divide — 0.194 by 0.042. 
 
 457. To Involve a Number by Logarithms. 
 
 Multiplication by logarithms is performed by addition. (Art. 453.) 
 Therefore, if the logarithm of a quantity is added to itself once, the 
 result will be the logarithm of the second power of that quantity ; if 
 Added to itself twice, the result will be the third power of that 
 quantity, and so on. Hence, the 
 
 Rule. — Multiply the logarithm of the number by the 
 exponent of the required power, 
 
 455. How divide by them ? 457 How involve a number by logarithms ? 
 
LOGARITHMS. 23b 
 
 Notes. — i. This rule depends upon the principle that logarithms 
 are the exponents of powers and roots, and a power or root is involved 
 by multiplying its index into the index of the power required. 
 
 2. In this rule, whatever is carried from the mantissa to the 
 characteristic is positive, whether the index itself is positive or negative. 
 
 13. What is the cube of 1.246. 
 
 Solution. — The log. of 1.246 is 0.09551 
 
 Index of the required power is 3 
 
 Log. of power is o.28b53. Ans. 1.93435. 
 
 14. What is the fourth power of .135 ? 
 
 15. What is the tenth power of 1.42 ? 
 
 16. What is the twenty-fifth power of 1.234? 
 
 458. To Extract the Hoot of a Number by Logarithms. 
 
 A quantity is resolved into any number of equal factors, by dividing 
 its index into as many equal parts. (Art. 281.) Hence, the 
 
 Rule. — Divide the logarithm of the number ly the index 
 oj the required root. 
 
 Note. — This rule depends upon the principle that the root of a 
 quantity is found by dividing the exponent by the number expressing 
 the required root. (Art. 296.) 
 
 17. What is the square root of 1.69? 
 
 Solution.— The log. of 1.69 is 0.22789 
 The index is 2, 2 ) .22789 
 
 Logarithm of root, 0.11394. Ana. 13. 
 
 18. What is the cube root of 143.2 ? 
 
 19. What is the sixth root of 1.62 ? 
 
 20. What is the eighth root of 1549 ? 
 :?!. What is the tenth root of 1876 ? 
 
 459. If the characteristic of the logarithm is negative, 
 and cannot be divided by the index of the required root 
 without a remainder, make it positive by adding to the 
 characteristic such a negative number as will make it 
 exactly divisible by the divisor, and prefix an equal positive 
 number to the decimal part of the logarithm. 
 
 4s8 ■^Tow extract the root ? 
 
240 LOGARITHMS. 
 
 22. It is required to find the cube root of .0164* 
 Solution.— The log. of .0164 is 2.21484. 
 
 Preparing the log., 3)3 + 1.21484 
 
 1.40494. Ans. 0.25406 +. 
 
 23. What is the sixth root of .001624 ? 
 
 24. What is the seventh root of .01449 ? 
 
 25. What is the eighth root of .0001236 ? 
 
 460. To Calculate Compound Interest by Logarithms. 
 
 Rule. — Find the amount of i dollar for i year ; multiply 
 its logarithm ly the numler of years, and to the product add 
 the logarithm of the principaL The sum will be the logarithm 
 of the amount for the given time. 
 
 From the amount subtract the principal, and the remain^ 
 der will be the interest. 
 
 Notes. — i. If the interest becomes due half yearly or quarterly, 
 find the amount of one dollar for the half year or quarter, and multiply 
 the logarithm by the number of half years or quarters in the time. 
 
 2. This rule is based upon the principle that the several amounts in 
 compound interest form a geometrical series, of which the principal is 
 the first term, the amount of $1 for i year the ratio, and the number 
 of years + i the number of terms 
 
 26. "What is the amount of 1 15 65 for 40 years, at 6 per 
 cent compound interest ? 
 
 Solution.— The amt. of |i for i year is $1.06 ; its log., 0.02531 
 
 The number of years, 40 
 
 Product, 1. 01 240 
 
 The given principal, $1565 ; its log., 3-1945 3 
 
 Ans. I16103.78. 4.20693 
 
 27. Wliat is the amount of $1500, at 7 per cent compound 
 interest, for 4 years ? Ans. $1966.05. 
 
 28. What is the amount of I370, at 5 per cent compound 
 interest for 33 years ? ^/? 5. $1851.274-. 
 
 460. How calculate compound interest by logarithms ? 
 
LOGARITHMS. 
 
 241 
 
 TABLE OF COMMON LOGARITHMS. 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D 
 
 
 
 .00000 
 
 .3oio3 
 
 .47712 
 
 .60206 
 
 .69897 
 
 .77815 
 
 .84510 
 
 .90309 
 
 .95424 
 
 
 10 
 
 
 0432 
 
 0860 
 
 1284 
 
 1703 
 
 2119 
 
 253 1 
 
 2938 
 
 3342 
 
 3743 
 
 4.6 
 
 
 4i39 
 
 4532 
 
 4922 
 8636 
 
 53o8 
 
 569. 
 
 6070 
 
 6446 
 
 6819 
 
 7.88 
 
 7555 
 
 379 
 
 12 
 
 7918 
 
 8279 
 
 ini 
 
 9342 
 
 & 
 
 ♦037 
 
 ♦38o 
 
 ♦721 
 
 1059 
 
 349 
 
 i3 
 
 .11394 
 
 1727 
 
 2057 
 
 2711 
 
 3354 
 
 3672 
 6732 
 
 3988 
 
 43o2 
 
 322 
 
 14 
 
 46i3 
 
 4022 
 7898 
 
 5229 
 
 5534 
 
 5836 
 
 6137 
 
 6435 
 
 7026 
 
 7319 
 
 3oi 
 
 i5 
 
 7609 
 
 8184 
 
 8469 
 
 8752 
 
 9033 
 
 93i3 
 
 9590 
 
 9866 
 
 ♦ 140 
 
 281 
 
 i6 
 
 .20412 
 
 o683 
 
 ^ll 
 
 1219 
 
 38o5 
 
 1484 
 
 1748 
 43o4 
 
 2011 
 
 2272 
 
 253 1 
 
 lin 
 
 264 
 
 \l 
 
 3045 
 
 33oo 
 
 4o55 
 
 455 1 
 
 4797 
 7184 
 
 5o42 
 
 249 
 
 7875 
 
 5768 
 
 6007 
 
 6245 
 
 6482 
 
 67.7 
 
 6951 
 
 7416 
 
 7646 
 
 235 
 
 19 
 
 8io3 1 833o 
 
 8556 
 
 8780 
 
 9003 
 
 9226 
 
 9447 
 
 9667 
 
 9885 
 
 223 
 
 20 
 
 .3oio3 
 
 0320 
 
 o535 
 
 0750 
 
 0963 
 
 1175 
 
 1 387 
 3445 
 
 1597 
 
 1806 
 
 2oi5 
 
 212 
 
 21 
 
 2222 
 
 2428 
 
 2634 
 
 2838 
 
 3041 
 
 3244 
 
 3646 
 
 3846 
 
 4044 
 
 203 
 
 22 
 
 4242 
 
 4439 
 
 4635 
 
 483 1 
 
 5o25 
 
 5218 
 
 541 1 
 
 56o3 
 
 5704 
 7658 
 
 5984 
 7840 
 
 \tl 
 
 23 
 
 6173 
 
 636 1 
 
 6549 
 
 6736 
 856i 
 
 6922 
 
 7107 
 8917 
 
 7291 
 
 7475 
 
 24 
 
 8021 
 
 8202 
 
 8382 
 
 !IS 
 
 9094 
 
 9270 
 
 9445 
 
 9620 
 
 178 
 
 25 
 
 9794 
 
 9967 
 
 ♦ 140 
 
 ♦3l2 
 
 ♦654 
 
 ♦824 
 
 IX 
 
 1162 
 
 i33o 
 
 171 
 
 i65 
 
 26 
 
 .41497 
 
 1664 
 
 i83o 
 
 1996 
 
 2160 
 
 2325 
 
 2488 
 
 2814 
 
 2975 
 456o 
 
 27 
 
 3i36 
 
 3297 
 
 3457 
 
 36i6 
 
 3775 
 
 3933 
 
 tS] 
 
 4248 
 
 44o5 
 
 1 58 
 
 28 
 
 4716 
 
 4871 
 
 5o25 
 
 5179 
 
 5332 
 
 5485 
 
 5788 
 
 5939 
 
 6090 
 
 1 53 
 
 29 
 
 6240 
 
 6389 
 
 6538 
 
 6687 
 
 6835 
 
 6982 
 
 7129 
 
 7276 
 
 7422 
 
 7567 
 
 '47 
 
 3o 
 
 .47712 
 
 7857 
 
 8001 
 
 8144 
 
 8287 
 
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 8572 
 
 8714 
 
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 143 
 
 3i 
 
 9i36 
 
 9276 
 
 9416 
 
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 io55 
 
 9831 
 
 9969 
 
 l322 
 
 ♦ 106 
 
 ♦243 
 
 i38 
 
 32 
 
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 0786 
 
 0920 
 
 1 188 
 
 1455 
 
 1 587 
 
 1720 
 
 1 33 
 
 33 
 
 i85i 
 
 1983 
 
 2114 
 
 2244 
 
 2375 
 3656 
 
 25o5 
 
 2634 
 
 2763 
 
 2892 
 
 3020 
 
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 34 
 
 3i48 
 
 3275 
 
 34o3 
 
 3529 
 
 4778 
 
 3782 
 
 3908 
 
 4o33 
 
 4i58 
 
 4283 
 
 126 
 
 35 
 
 4407 
 
 453 1 
 
 4654 
 
 4900 
 
 5o23 
 
 5i45 
 
 5267 
 
 5388 
 
 6703 
 
 123 
 
 36 
 
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 5751 
 
 5871 
 
 5991 
 
 6110 
 
 6229 
 
 6348 
 
 6467 
 
 6585 
 
 116 
 
 u 
 
 6820 
 
 6937 
 
 7054 
 
 7171 
 
 7287 
 8433 
 
 74o3 
 
 7519 
 
 7634 
 
 iin 
 
 7864 
 8995 
 
 7978 
 
 8093 
 
 8206 
 
 8320 
 
 8546 
 
 8659 
 
 8771 
 
 n3 
 
 39 
 
 9107 
 
 9218 
 
 9329 
 
 9439 
 
 9550 
 
 9660 
 
 9770 
 
 9879 
 
 9988 
 
 ♦097 
 
 no 
 
 40 
 
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 o3i4 
 
 0423 
 
 o53i 
 
 o638 
 
 Vstt 
 
 o853 
 
 0959 
 
 1066 
 
 1172 
 
 108 
 
 41 
 
 1278 
 
 i384 
 
 1490 
 
 1595 
 
 1700 
 
 1909 
 
 2014 
 
 2118 
 
 2221 
 
 io5 
 
 42 
 
 2325 
 
 2428 
 
 253 1 
 
 2634 
 
 2737 
 
 2839 
 
 2941 
 
 3o43 
 
 3i44 
 
 3246 
 
 102 
 
 43 
 
 3347 
 
 3448 
 
 3548 
 
 3649 
 
 3749 
 
 3849 
 
 3949 
 
 4048 
 
 4147 4247 
 
 100 
 
 44 
 
 4345 
 
 4444 
 
 4542 
 
 4640 
 
 4738 
 
 4836 
 
 
 5o3i 
 
 5128 
 
 5225 
 
 98 
 
 45 
 
 5321 
 
 5418 
 
 55i4 
 
 56io 
 
 5706 
 
 58oi 
 
 5992 
 
 6087 
 
 6I8I 
 
 95 
 
 46 
 
 6276 
 
 6370 
 
 6464 
 
 6558 
 
 6652 
 
 6745 
 
 6839 
 
 6932 
 
 7025 
 
 7II7 
 
 
 47 
 
 7210 
 
 73o2 
 
 7394 
 
 7486 
 
 7578 
 
 7669 
 
 7761 
 
 7852 
 
 7943 
 
 8o34 
 
 1 
 
 48 
 
 8124 
 
 8215 
 
 83o5 
 
 8395 
 
 8485 
 
 8574 
 
 8664 
 
 8753 
 
 8842 
 
 8931 
 9810 
 
 49 
 
 9020 
 
 9108 
 
 9197 
 
 9285 
 
 9373 
 
 9461 
 
 9548 
 
 9636 
 
 9723 
 
 88 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
u^ 
 
 LOGARITHMS. 
 
 N. 
 
 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 B. 
 
 5o 
 
 .69897 
 
 9984 
 0842 
 
 ♦070 
 
 ♦i57 
 
 ♦243 
 
 ♦329 
 
 ♦41 5 
 
 ♦5oi 
 
 ♦586 
 
 ♦672 
 
 86 
 
 5i 
 
 .70757 
 
 0927 
 
 1012 
 
 I933 
 
 1181 
 
 1265 
 
 1 349 
 
 1433 
 
 i5i7 
 
 85 
 
 52 
 
 1600 
 
 1684 
 
 1767 
 
 i85o 
 
 2016 
 
 2099 
 
 2181 
 
 2263 
 
 2346 
 
 83 
 
 53 
 
 2428 
 
 25l0 
 
 2591 
 
 2673 
 
 2754 
 
 2835 
 
 2917 
 
 2997 
 
 3078 
 
 3i59 
 
 8! 
 
 54 
 
 3289 
 
 3320 
 
 3400 
 
 3480 
 
 3560 
 
 1 3640 
 
 IVot 
 
 3799 
 4586 
 
 3878 
 
 3957 
 
 80 
 
 55 
 
 ■74036 
 
 4ii5 
 
 4194 
 
 4273 
 
 435i 
 
 4429 
 52o5 
 
 4663 
 
 4741 
 
 78 
 
 56 
 
 4819 
 
 4896 
 
 4974 
 
 5o5i 
 
 5i28 
 
 5282 
 
 5358 
 
 5435 
 
 55ii 
 
 'll 
 
 ^7 
 
 5588 
 
 5664 
 
 5740 
 
 58i5 
 
 5891 
 
 1 5967 
 
 6042 
 
 6118 
 
 6193 
 6938 
 7670 
 
 6268 
 
 58 
 
 6343 
 
 6418 
 
 6492 
 
 ',fj, 
 
 6641 
 
 6716 
 
 6790 
 
 6864 
 
 7012 
 
 75 
 
 59 
 
 7085 
 
 7159 
 
 7232 
 
 7379 
 
 7452 
 
 7525 
 
 7597 
 
 7743 
 
 73 
 
 6o 
 
 .77815 
 
 7887 
 
 7960 
 
 8o32 
 
 8104 
 
 8176 
 
 8247 
 
 83i9 
 
 8390 
 
 8462 
 
 72 
 
 6i 
 
 8533 
 
 8604 
 
 8675 
 
 8746 
 
 8817 
 
 8888 
 
 8958 
 
 9029 
 
 9099 
 
 9169 
 9865 
 
 
 62 
 
 9239 
 
 9309 
 
 9379 
 
 9449 
 
 9519 
 
 9588 
 
 9657 
 
 9727 
 
 9796 
 
 63 
 
 9934 
 
 ♦oo3 
 
 ♦072 
 
 ♦ 140 
 
 ♦209 
 
 ♦277 
 0956 
 1624 
 
 ♦346 
 
 ♦414 
 
 ♦482 
 
 ♦55o 
 
 64 
 
 .80618 
 
 0686 
 
 0754 
 
 0821 
 
 0889 
 
 1023 
 
 n?7 
 
 n58 
 
 1225 
 
 ll 
 
 65 
 
 I2gi 
 
 i358 
 
 1425 
 
 1491 
 
 2l5l 
 
 1 558 
 
 1690 
 
 1823 
 
 1889 
 2543 
 
 66 
 
 1954 
 
 2020 
 
 2086 
 
 2866 
 
 2282 
 
 2347 
 
 24i3 
 
 2478 
 
 65 
 
 tl 
 
 2608 
 
 2672 
 
 2737 
 
 2802 
 
 2930 
 
 2995 
 
 3o59 
 
 3i23 
 
 3187 
 
 64 
 
 325i 
 
 33i5 
 
 3378 
 
 3442 
 
 35o6 
 
 3569 
 
 3632 
 
 3696 
 
 3759 
 
 3822 
 
 63 
 
 69 
 
 3885 
 
 3948 
 
 4011 
 
 4073 
 
 4i36 
 
 4199 
 
 4261 
 
 4323 
 
 4386 
 
 4448 
 
 63 
 
 70 
 
 .84510 
 
 4572 
 
 4634 
 
 4696 
 
 4757 
 
 4819 
 
 4881 
 
 4942 
 
 5oo3 
 
 5o65 
 
 62 
 
 71 
 
 5i26 
 
 5.87 
 
 5248 
 
 5309 
 
 5370 
 
 543 1 
 
 5491 
 
 5552 
 
 56i2 
 
 5673 
 
 61 
 
 72 
 
 5733 
 
 5794 
 
 5854 
 
 5914 
 
 5974 
 
 6o34 
 
 6094 
 
 6i53 
 
 6213 
 
 6273 
 
 60 
 
 73 
 
 6332 
 
 6392 
 
 645i 
 
 65io 
 
 6570 
 7157 
 
 6629 
 
 6688 
 
 6747 
 
 6806 
 
 6864 
 
 59 
 
 74 
 
 6923 
 
 6982 
 
 7040 
 
 ]6T 
 8253 
 
 7216 
 
 7274 
 
 7332 
 
 7390 
 
 7448 
 
 5o 
 
 75 
 
 7306 
 8081 
 
 7564 
 
 7622 
 8196 
 
 7737 
 83o9 
 
 ml 
 
 7852 
 8423 
 
 7910 
 
 Itl 
 
 8024 
 
 58 
 
 76 
 
 8139 
 8705 
 
 8480 
 
 8503 
 9154 
 
 57 
 
 77 
 
 8649 
 
 8762 
 
 88r8 
 
 8874 
 
 8930 
 
 8986 
 
 9042 
 
 9098 
 9653 
 
 56 
 
 78 
 
 9210 
 
 9265 
 
 9321 
 
 9376 
 
 9432 
 
 9487 
 
 9542 
 
 9598 
 
 9708 
 
 55 
 
 79 
 
 9763 
 
 9818 
 
 9873 
 
 9927 
 
 9982 
 
 ♦037 
 
 ♦091 
 
 ♦ 146 
 
 ♦200 
 
 ♦255 
 
 55 
 
 80 
 
 .90309 
 
 o363 
 
 0417 
 
 0472 
 
 o526 
 
 o58o 
 
 o634 
 
 0687 
 
 074; 
 
 0795 
 
 54 
 
 81 
 
 0849 
 
 0902 
 
 0956 
 
 1009 
 
 1062 
 
 1116 
 
 1 698 
 
 1222 
 
 1275 
 
 i328 
 
 54 
 
 82 
 
 i38i 
 
 1434 
 
 1487 
 
 1 540 
 
 1593 
 
 1645 
 
 1751 
 
 i8o3 
 
 1 856 
 
 52 
 
 83 
 
 1908 
 
 ig6o 
 
 2012 
 
 2o65 
 
 2117 
 
 2169 
 
 2221 
 
 2273 
 
 2324 
 
 2376 
 
 52 
 
 84 
 
 2428 
 
 2480 
 
 253 1 
 
 2583 
 
 2634 
 
 2686 
 
 2737 
 
 2788 
 
 2840 
 
 2891 
 
 52 
 
 85 
 
 2942 
 
 2993 
 
 3o44 
 
 3095 
 
 3i46 
 
 3.97 
 
 3247 
 
 3298 
 
 3349 
 
 3399 
 
 5i 
 
 86 
 
 345o 
 
 3doo 
 
 355i 
 
 36oi 
 
 365i 
 
 3702 
 
 375i 
 
 38o2 
 
 3852 
 
 3902 
 
 5i 
 
 ll 
 
 3952 
 
 4002 
 
 4o52 
 
 41 01 
 
 4i5i 
 
 4201 
 
 425o 
 
 43oo 
 
 435o 
 
 4399 
 
 5o 
 
 4448 
 
 4498 
 
 4547 
 
 4596 
 
 4645 
 
 4694 
 
 4743 
 
 4792 
 
 4841 
 
 4890 
 
 8 
 
 89 
 
 4939 
 
 4988 
 
 5o37 
 
 5o85 
 
 5i34 
 
 5i82 
 
 523 1 
 
 5279 
 
 5328 
 
 5376 
 
 90 
 
 .95424 
 
 5473 
 
 5521 
 
 5569 
 
 56i7 
 6095 
 
 5665 
 
 57.3 
 
 5761 
 
 5809 
 
 5856 
 
 48 
 
 91 
 
 5904 
 
 5952 
 
 6000 
 
 6047 
 
 6142 
 
 6190 
 
 6237 
 
 6284 
 
 6332 
 
 47 
 
 92 
 
 6379 
 
 6426 
 
 6473 
 
 6520 
 
 6567 
 
 6614 
 
 6661 
 
 6708 
 
 6755 
 
 6802 
 
 tl 
 
 93 
 
 6848 
 
 6895 
 7359 
 
 6942 
 
 6988 
 
 7035 
 
 7081 
 
 7128 
 
 7'74 
 
 7220 
 
 7267 
 
 94 
 
 73.3 
 
 74o5 
 
 745i 
 
 '411 
 
 7543 
 
 7589 
 
 ^', 
 
 7681 
 8137 
 8588 
 
 7727 
 8182 
 
 46 
 
 95 
 
 7772 
 8227 
 
 78i8 
 8272 
 
 7864 
 83i8 
 
 ITe^ 
 
 8000 
 
 8046 
 
 45 
 
 96 
 
 8408 
 
 8453 
 
 8498 
 
 8543 
 
 8632 
 
 45 
 
 97 
 
 8677 
 
 8722 
 
 8767 
 
 881 1 
 
 8856 
 
 8901 
 9344 
 
 8945 
 9388 
 
 8990 
 
 9034 
 
 9078 
 
 45 
 
 98 
 
 9123 
 
 9167 
 
 9211 9255 1 
 
 9300 
 
 9432 
 
 9476 
 
 9520 
 
 44 
 
 99 
 
 9564 
 
 9607 
 
 965 1 
 
 9695 
 
 9739 
 4 
 
 9782 
 
 9826 
 
 9870 
 
 9913 
 
 9957 
 
 43 
 
 K. 
 
 
 
 1 
 
 2 
 
 3 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 D. 
 
CHAPTER XX. 
 
 MATHEMATICAL INDUCTION AND 
 BUSINESS FORMULAS. 
 
 461. 3Iathematical Induction consists in proving 
 by trial that a proposition is true in a certain case ; and, 
 finding it true in the next case, then in the third, and so 
 on, we conclude it must be true in all similar cases. 
 
 462. Many of the principles and formulas of Arithmetic 
 and Algebra are established by this mode of reasoning. 
 
 463. Take the familiar principle in Arithmetic: 
 
 The product of any two or more numlers is the same, in 
 whatever order the factors are taken. 
 
 To prove this principle of two numbers, as 5 and 3, the pupil 
 represents the number 5 by as many unit marks in a 
 
 41, 4P 4t, 4C; ^ 
 
 horizontal row, and under this places two similar rows. 
 
 # # # # # 
 He sees that the number of marks in the horizontal 
 
 # # # # # 
 row taken 3 times is equal to the number of marks in 
 
 a perpendicular row taken 5 times ; that is, 3 times 5 = 5 times 3. 
 
 He then takes three factors and finds the proposition true, and so 
 
 on. Hence, he concludes the principle is universally true. 
 
 464. Next, suppose it be asserted : 
 
 The product of the sum and difference of two quantities is 
 equal to the difference of their squares. (Art. 103.) 
 Taking two quantities, as 4 + 3 and 4—3, or a + & and a—T). 
 Multiply 4 + 3 Or a + 6 
 
 By 4 — 3 By <g — & 
 
 42 + 4 X 3 a^ + ah 
 
 — 4x3 — 3* — ab — W 
 
 Product, 4^ ^~3^ cf~ -¥ 
 
 461. In what does Mathematical induction consist ? Illustration. 
 
244 MATHEMATICAL INDUCTION^. 
 
 He takes another example of two quantities, and finds the 
 statement true ; then another, and so on. Hence, he concludes the 
 proposition is a universal truth. 
 
 465. Suppose this proposition were enunciated : 
 
 Tlie sum of any numher of terms of the arithmetical series 
 I, 3, 5, 7, etc, to n terms^ is equal to n^. 
 
 We see by inspection that the sum of the first two terms, 1 + 3 — 4, 
 or 2- ; that the sum of the first three terms, 1 + 3 + 5 = 9> or 3^ ; that 
 the sum of four terms, i + 3 + 5 + 7 = i6, or 4-, and so on. Hence, we 
 may conclude that the proposition is true if the series be extended 
 indefinitely. Or, 
 
 Since we know the proposition is true when n denotes a small num- 
 ber of teims, and that the value of any term in the series, as the 5th, 
 7th, 9th, etc., is equal to 211—1, we may suppose for this value of n, 
 that 1 + 3 + 5 + 7 +(2/i— I) = w^. (i) 
 
 Adding 2n + 1 to both members, we have 
 
 1 + 3 + 5 + 7 •••• +(2ri— i) + (2?i + i) = 7i2 + 2?i + i. (2) 
 
 Factoring second member of (2), 7)? + 2n+i = {n+ if. 
 
 Therefore, since the sum of n terms of the series = 71^, it follows 
 that the sum of w + i terms = {n+ if, and so on. Hence, the prop- 
 osition must be universalis/ true. 
 
 466. In Geometry, we have the proposition : 
 
 The sum of the three angles of a triangle is equal to two 
 right ajigles. 
 
 We find it to be true in one case; then in another, etc. 
 Hence, we conclude the proposition is universally true. 
 
 Notes. — I. It is sometimes objected that this method of f>roof\B less 
 satisfactory to the learner than a more rigorous process of reasoning. 
 But when it is fully understood, it is believed that it will produce the 
 fullest conviction of the truths designed to be established. 
 
 2. In metaphysics and the natural sciences, the term induction is 
 applied to the assumption that certain laws are general which by 
 experiment have been proved to be true in certain cases. But we 
 cannot be sure that these laws hold for any cases except those which 
 have been examined, and can never arrive at the conclusion that they 
 are necessary truths. 
 
BUSINESS FOEMULAS. 245 
 
 BUSINESS FORMULAS. 
 
 467. The principles of Algebra are not confined to the 
 demonstration of theorems and the solution of abstruse 
 equations. They are equally applicable to the development 
 of formulas and for business calculations. , 
 
 Note. — In reciting formulas, the student should first state the 
 proposition, then write the formula upon the blackboard, explaining 
 the several steps by which it is derived as he proceeds. He should 
 then translate the formula from algebraic into common langv/ige. 
 
 PROFIT AND LOSS. 
 
 468. Profit and Loss are computed by the principles 
 of Fercenlage. 
 
 469. To Find the Profit or Loss, the Cost and the per 
 
 cent Profit or Loss being given. 
 
 Let c denote the cost, r the per cent profit or loss, and p th* 
 percentage, or sum gained or lost. 
 
 Since per cent means hundredths, r per cent of a number must be r 
 hundredths of that number. (Art. 237.) Therefore, 
 
 c dollars x r = p, the sum gained. Hence, the 
 
 Formula. p = cr. 
 Rule. — MiiUipI^ the cost hy the rate per cent, and the 
 product will he the jjrqfit or loss, as the case may require. 
 (Art. 237.) 
 
 1. Suppose c = $3560, and r = 12 per cent. Required 
 the profit. 
 
 Solution. $3560 x .12 = $427.20, Ans. 
 
 2. If a house costing $4370 were sold at 8 per cent lese 
 than cost, what would be the loss ? 
 
 470. To Find the per cent Profit or Loss, the Cost and 
 
 the Sum Gained or Lost being given. 
 
 Let c denote the cost ; p the percentage, or sum gainea or lost ; and 
 r the per cent. 
 
246 BUSINESS FOKMTJLAS. 
 
 Since the cost multiplied by tlic rate gives ^, the given profit or 
 loss, it follows thatp-f-c must be the rate. (Art. 469.) Hence, the 
 
 Formula. r = —* 
 c 
 
 Rule. — Divide the gain or loss hy the cost, and the quo- 
 tient will he the per cent profit or loss, 
 
 3. A farm costing $2500 was sold for $500 advance. 
 Required the per cent profit. Ans, 20 per cent. 
 
 4. A teacher's salary being $i8oo a year, was raised $300. 
 What per cent was the increase ? 
 
 471. To Find the Cost, the Profit or Loss and the per cent 
 
 Profit or Loss being given. 
 
 Let p denote the sum gained or lost, r the per cent, and c the cost. 
 
 Since the sum gained is equal to the cost multiplied by the rate per 
 cent (Art. 469), it follows that p dollars the sum gained, divided by r 
 the rate, will be the cost. Hence, the 
 
 Formula. c = — • 
 r 
 
 Rule. — Divide the profit or loss by the rate per cent. 
 
 5. The gain on a bill of goods was $67.48. At 25 per 
 cent profit, what was the cost ? Ans. $269.92. 
 
 6. An operator in stocks lost $1575, which was 12I per 
 cent of his investment. What was the investment? 
 
 472. To Find the Selling Price, the Cost and per cent 
 
 Profit or Loss being given. 
 
 Let c denote the cost, r the per cent of profit or loss, and s the 
 selling price. 
 
 When there is a gain, the amount of %i = i+r; when there is a 
 loss, the amount of $1 = i— r ; and the amount of c dollars = c{i± r). 
 Therefore, we have the general 
 
 Formula. s = c{i ±r). 
 Rule. — Multiply the cost hy i plus or minus the rate, as 
 the case may require. (Art. 240.) 
 
 7. A man paid $750 for a piano. For what must he sell 
 it to gain 15 per cent? Ans. $862.50. 
 
 8. A man bought a carriage for $960, and sold it at a loss 
 of 12 J per cent. What did he receive for it ? 
 
BUSINESS FORMULAS. 247 
 
 473. To Find the Cost, the Selling Price and the per cent 
 
 Profit or Loss being given. 
 
 Let 8 denote the selling price, r the per cent profit or loss, and c the 
 cost required. 
 
 When there is a gain, the selling price equals the cost plus r per 
 cent of itself, that is, i+r times the cost; when there is a ^o««, the 
 selling price equals the cost minus r per cent of itself, that is, i— r 
 times the cost ; therefore the cost equals the selling price divided bj 
 I ± r. Hence, we have the general 
 
 Formula. c = — ; — • 
 
 EuLE. — Divide the selling price hy i plus or minus the 
 rate, as the case way require ; the quotient will he the cost. 
 
 9. A goldsmith sold a watch for $175 and made 20 per 
 cent profit. What was the cost ? 
 
 Solution. 1 + 20 per cent = 1.20 ; and 
 
 $175 -r- 1.20 = $i45.83i» ^ns, 
 
 10. A jockey sold a horse for $540, and thereby lost 
 10 per cent. What did the horse cost him ? 
 
 SIMPLE INTEREST. 
 
 474. The Elements or Factors involved in calcula- 
 tions of interest are the same as those in percentage, with the 
 addition of time* 
 
 475. Interest is of two kinds, simple and compound. By 
 the former, interest is derived from the principal only ; by 
 the latter, it is derived both from the principal and the 
 interest itself, as soon as it becomes due. 
 
 476. To Find the Time in which any Sum at Simple Interest 
 win Double itself, at any given Rate Per Cent. 
 
 Let p denote the principal, r the rate per cent, i the given interest, 
 and t the ti jt e in years. 
 
 Then i = prt, (Art. 242.) 
 
 Making i equal to j?, p = prt. 
 
 Dividing by pr, - ^ t. Hence, the 
 
248 BUSINESS FORMULAS. 
 
 Formula. t — —. 
 r 
 
 'Rule,— Divide i by the rate, and the q^lotient will he the 
 time required. 
 
 11. How long will it take $1500, at 5 per cent, to double 
 itself ? 
 
 Solution. t = - = — = 20. Ans. 20 years. 
 r .05 "^ 
 
 12. How long will it take $680, at 6 per cent, to double 
 itself? 
 
 13. How long will it take I8475, at 10 per cent, to double 
 itself ? 
 
 477. To Find the Rate at which any Principal, at Simple 
 
 Interest, will Double itself in a Given time. 
 
 By the preceding formula, t = -» 
 
 T 
 
 T 
 
 Multiplying by - , we have the 
 t 
 
 Formula. !• = -. 
 
 Rule. — Divide i hy the time ; the quotient will he the rate 
 per cent required. 
 
 [For other formulas in Simple Interest, see Arts. 242-246.] 
 
 14. If $1700 doubles itself in 8 years, what is the rate? 
 
 Ans. 12J per cent 
 
 15. At what rate per cent will $5000 double itself in 
 40 years ? 
 
 COMPOUND INTEREST. 
 
 478. Interest may be compounded annually, semi- 
 annually, quarterly, etc. It is understood to be com- 
 pounded annually, unless otherwise mentioned. 
 
 479. To Find the Atnoiitit of a given Principal at Compound 
 
 Interest, the Rate and Time being given* 
 
 Let p denote the principal, r the rate, n the number of years, ana 
 a the amount. 
 
BUSINESS FORMULAS. 249 
 
 Since the amount equals the principal plus the interest, it follows 
 that the amount of $i fori year equals i+r; therefore, p(i+r) 
 equals the amount of p dollars for i year, which is the principal for 
 the second year. 
 
 Again, the amount of this new principal p{i+r) for i year = 
 p{i+r){i+r) = p{i+ rf, which is the amount of p dollars for two 
 years. 
 
 In like manner, /) {\-\-rf is the amount of p dollars for three years 
 i,nd so on, forming a geometrical series, of which the principal p is 
 the first term, i+r the ratio, and the number of years + i, the num- 
 ber of terms. The terms of the series are 
 
 p, p(i + r), i)(i + r)2, p(n.7.)8^ p(i + r)*, . . . . i7(i+r>». 
 
 The last term, p (i + Tf» is the amount of p dollars for n years. 
 Hence; the 
 
 Formula. a = p (i + ry. 
 
 Rule. — Multiply the principal by the amount of %i for 
 I year, raised to a power denoted by the number of years ; 
 the product will he the amount, 
 
 480. To Find the Compound Interest for the given Time 
 
 and Rate. 
 
 SuUract the principal from, the amount, and the remainder 
 will he the compound interest. 
 
 Note. — When the number of years or periods is large, the oper- 
 ations are shortened by using logarithms. 
 
 1 6. What is the amount of $842, at 6 per cent compound 
 interest, for 4 years ? 
 
 Solution. $842 x (1.06)^ = $1063, Am. 
 
 17. What is the amount of $1500, at 5 per cent for 6 yrs., 
 compound interest ? 
 
 r 
 
 481. If the interest is compounded semi-annually, - will 
 
 denote the interest of $1 for a half year. Then, at com- 
 pound interest, the amount of p dollars for n years is 
 
 ^\ 2n 
 
 '{ 
 
 •+-,) 
 
'('*f- 
 
 250 BUSINESS FORMULAS. 
 
 482. If the interest is compounded qiiarte7'ly, then - 
 
 4 
 will denote the interest of $i for a quarter. Then, at 
 
 compound interest, the amount of jj dollars for n years is 
 
 471 
 
 etc. 
 
 4/ 
 
 1 8. What is the amount of $2000, for 3 years at 6 per 
 fent, compounded semi-annually ? Ans, I2388.05. 
 
 19. What is the amount of I5000 for 2 years, at 4 per ct., 
 compounded quarterly ? 
 
 483. By transposing, factoring, etc., the formula in 
 A.rt. 479, we have. 
 
 The first term, « = -, • 
 
 The number of terms, n = -^ — ;- — ^~ • 
 
 log. (I + 7-) 
 
 The ratio, r = l-r — l. 
 
 DISCOUNT. 
 
 484. Discount is an allowance made for the payment 
 of money before it is due. 
 
 485. The JPresent Worth of a debt payable at a future 
 time is the sum which, if put at legal interest, will amount 
 to the debt in the given time. 
 
 486. To Find the Present Worth of z Sum at Simple Interest, 
 
 the Time of Payment and the Rate being given. 
 
 Let 8 denote the sum due, n the number of years, and r the interest 
 of $1 for I year. 
 
 Since r is the interest of $1 for i year, nr must be the interest for n 
 years, and i +7?r the amount of $1 for n years. Therefore, s-5-(i + nr) 
 is the present worth of the given sum. 
 
 Putting P for present worth, we have the 
 
 Formula. B = — ; • 
 
 I 4- fir 
 
 Rule. — Divide the sum due hy the amount of $1 for the 
 
 given time and rate; the quotient loill he the present worth. 
 
BUSINESS FORMULAS. 251 
 
 487. To Find the Discount, the Present Worth being given. 
 
 SuUract the present worth from the debt. 
 
 20. What is the present worth of $2500 payable in 4 yrs., 
 interest being 7 per cent ? What the discount ? 
 
 Solution. P = — ?— = 1^ = $1953.125, pres. worth ,, , 
 
 i + nr 1.28 'a' V3J D, I' M ^^^^ 
 
 $2500 — $1953.125 = $546,875, discount. 
 
 1 
 
 21. What is the present worth of $3600 due in 5 years, at 
 6 per cent ? WTiat is the discount ? 
 
 22. Find the present worth of $7800 due in 6 years, 
 interest 5 per cent ? What the discount ? 
 
 COMPOUND DISCOUNT. 
 
 488. To Find the Present Worth of a Sum at Compound 
 Interest, the Time and the Rate being given. 
 
 Let s denote the sum due, n the number of years, r the rate per ct. 
 
 Since r is the rate, i + r is the amount of $1 for i year ; then the 
 amount for n years compound interest is (n-r)«. (Art. 479.) That is, 
 $1 is tlie present worth of (1+ r)« due in n years. Therefore, «-t-(i + r)n 
 must be the present worth of the given sum. 
 
 Putting F for the present worth, we have the 
 
 FOKMULA. JP = V 
 
 (I + rf 
 
 Rule. — Divide the sum due hy the amount of li at com- 
 potmd interest, for the given time and rate; the quotient 
 will he the present worth. 
 
 23. What is the present worth of $1000 due in 4 years, at 
 
 5 per cent compound interest ? 
 
 Solution, P = — ?-- = f^, = $822.71, Ans. 
 (i+r)" (i.o5)'» 
 
 24. What is the present worth of $2300 due in 5 years, at 
 
 6 per cent compound interest ? 
 
252 BUSINESS FORMULAS. 
 
 COMMERCIAL DISCOUNT. 
 
 489. Commercial Discount is a per cent taken 
 from the face of bills, the marked price of goods, etc., 
 without regard to time. 
 
 490. To Find the Cofmnercial Discount on a Bill of Goods, 
 the Face of the Bill and the Per Cent Discoant being given. 
 
 Let 6 denote the base or face of the bill, r the rate, and d the 
 discount or percentage. 
 
 Then bxr will be the discount. Hence, the ' 
 
 Formula. d = br. 
 Rule. — Multiply the face of the Mil hy the given rate, and 
 the product will be the cornmercial discount, 
 
 491. To Find the Cash Value or Net Proceeds of a Bill. 
 
 Subtract the commercial discount from the face of the hill. 
 
 25. Required discount and net value of a bill of goods 
 amounting to $960, on 90 days, at 12^ per cent off for cash ? 
 
 Solution. $q6o x .125 = $120, discount ; $960 — $120=1840, Ans, 
 
 26. Required the cash value of a bill amounting to I2500, 
 the discount being 10 per cent, and 5 per cent off for cash. 
 
 492. To Mark Goods so as to allow a Discount, and make 
 
 any proposed per cent Profit. 
 
 Let c denote the cost, r the per cent profit, and d the per cent disc. 
 
 Since ris the per cent profit, 1 +r is the selling price of $1 cost, and 
 c{i+t) the selling price of c dollars cost. 
 
 Again, since d is the per cent discount from the marked price, ano 
 the marked price is 100 per cent of itself, i—d must be the net value 
 of $1 marked price. Therefore, 
 
 c(i +7") -5- (i— <?) = the marked price. 
 
 Putting m for the marked price, we have the 
 
 Formula. m = -- — -—• 
 I — d 
 
 Rule. — Multiply the cost hy i plus the per cent gain, a7id 
 
 divide the product hy i minus the proposed discount. 
 
BUSINESS FORMULAS. 25,3 
 
 27. A trader paid I25 for a package of goods ; at what 
 price must it be marked that he may deduct 5 per cent, and 
 yet make a profit of 10 per cent ? 
 
 _ c(i+r) $25x1.10 ^ . . 
 
 Solution, m = --'^ — ~ = -^-^ = $28.94 + , Ans. 
 
 i-d .95 
 
 28. A merchant buys a case of silks at I1.75 a yard; 
 what must he mark them that he may deduct 10 per cent, 
 and yet make 20 per cent ? 
 
 29. A grocer bought flour at $6^ a barrel; what price 
 must he mark it that he may fall 3 per cent, and leave a 
 profit of 25 per cent ? 
 
 INVESTMENTS. 
 
 493. The Value of an Investment in National and 
 State securities, Railroad Bonds, etc., depends upon their 
 market value, the rate of interest they bear, and the cer- 
 tainty of payment. 
 
 494. The Dividends of stocks and bonds are reckoned 
 at a certain per cent on the par value of their shares, which 
 is commonly $100. 
 
 495. To Find the Per Cent which an Investment will pay, the 
 Cost of a Share and the Rate of Dividend being given. 
 
 Let c denote the cost or market value of i sliare of stock, p \is par 
 value^ and r the annual rate of dividend. 
 
 Since r is the rate of dividend and p the par value, pr must be the 
 dividend on i sliare for i year. Therefore, pr -i- c will be the per cent 
 received on the cost of i share. (Art. 238.) 
 
 Putting JR for the per cent received on the cost of i share, we have 
 the 
 
 Formula. M = -— 
 c 
 
 KuLE. — Find the dividend on the given shares at the given 
 rate, and divide this hy the cost ; the quotient will he the per 
 cent received on the invostment* 
 
254 BUSINESS FOKMULAS. 
 
 Note. — When the stock is above or lelow par, the premium or 
 discount must be added to or subtracted from its par value to give the 
 cost. 
 
 30. What per cent interest does a man receive on an 
 investment of $5000 in the Bank of Commerce, its dividends 
 being 10 per cent, and the shares 5 per cent above par? 
 
 Solution. — The premium on the stock = $5000 x .05 = $2500 
 Therefore, the cost = $5000 + 1250 = $5250. 
 
 Again, the dividend on stock = $5000 x. 10 = $5oa Therefore, 
 $5oo-f-$525o = 9II per cent, Ans. 
 
 31. A invested $6000 in New York 6 per cent bonds, at 
 3 per cent premium. What per cent did he receive on his 
 investment ? 
 
 32. A man lays out 1 1000 in Alabama 10 per cents, at a 
 discount of 20 per cent. What per cent did he receive on 
 his investment ? 
 
 33. What per cent will a man receive on 50 shares of 
 Pennsylvania Eailroad stock, the premium being 4 per ct., 
 and the dividend 10 per cent? 
 
 34. Which are preferable, Massachusetts 6 per cent bonds 
 at par, or Ohio 8 per cent bonds at 2 per cent premium? 
 
 496. To Find the Amount of a given Remittance which a 
 Factor can Invest, and Reserve a Specified Per Cent for his 
 Commission. 
 
 Let s denote the sum remitted, and r the per cent commission. 
 
 The sum remitted includes both the sum invested and tlie commis- 
 sion. Now $1 remitted is 100 per cent, or once itself ; and adding the 
 per cent to it, we have i+r, the cost of $1 invested. Therefore, 
 « -4- (i+r) must be the amount invested. 
 
 Putting a for the amount invested, we have the 
 
 FOEMULA. a = 
 
 I+r 
 
 Rule. — Divide the remittance ly 1 plus the per cent 
 commission ; the quotient will he the amount to invest. 
 
BUSINESS FOEMULAS. 255 
 
 35. A clergyman remitted to his agent $2500 to purchase 
 books. After deducting 4 per cent commission, how much 
 does he lay out in books ? 
 
 Solution, a = — ^ = ^^^^ = $2403.85, Ans. 
 i+r 1.04 
 
 ;^6. A gentleman remitted $25000 to a broker, to be 
 invested in stocks. After deducting i J per cent, how much 
 did he invest, and what was his commission ? 
 
 SINKING FUNDS. 
 
 497. Slnkinf/ Funds are sums of money set apart or 
 deposited annually, for the payment of public debts, and for 
 other purposes. 
 
 CASE I 
 
 498. To Find the A^nouitt of an Annual Deposit at Compound 
 
 Interest, the Rate and Time being given* 
 
 Let s denote the annual deposit or sum set apart, r the rate per 
 cent, n the number of years, and a the amount required. 
 
 Since the sam^ sum is deposited at the end of each year, and put 
 at compound interest, it follows that the deposit at the end of the 
 ist year = s 
 2d " =s + «(i+r) 
 3d " = s + s{i + r) + 8{i+rf 
 nth. " — 8 -\- s{i+r) + s{i+rf . , . . + 8{i ^-r)«-^ 
 
 forming a geometrical series ; the annual deposit being the first term, 
 the amount of $1 for i year the ratio, the number of years the num- 
 ber of terms, and the annual deposit multiplied by the amount of $1 
 for I year, raised to that power whose index is i less than the number 
 of years, the last term ; and the amount is equal to the sum of the 
 series. (Art. 402.) Hence, we have the 
 
 (i + ^^r— I 
 FOEMULA. a = S. 
 
 r 
 
 Rule. — Multiply the amount of $i annual deposit for the 
 given time and rate hy the given annual deposit ; the product 
 tvill he the amount required. 
 
256 BUSINESS FORMULAS. 
 
 37. A clerk annually deposited $150 in a savings bank 
 which pays 6 per cent compound interest. What amount 
 will be due him in 5 years ? 
 
 Solution, a = ^ -^— ~ a = ' ^ $150 = $845.75. Ans. 
 
 38. A man agrees to give $300 annually to build a church 
 What will his subscription amount to in 4 years, at 7 pei 
 cent compound interest? 
 
 39. If a teacher lays up $500 annually, and puts it at 
 5 per cent compound interest for 10 years, how much will 
 he be worth ? 
 
 CASE II. 
 
 499. To Find the Ammal Dejyosil required to produce a 
 given Amount at Compound Interest, the Rate and Time being given. 
 
 By the formula in the preceding article, we have 
 
 (r + r)« — I 
 
 ^^ s = a. 
 
 r 
 
 Dividing by coefficient of .s, we have the 
 
 FOKMULA. S=za^ (i + vY —J_ 
 
 r 
 
 Rule. — Divide the amount to he raised hy the amount of 
 $1 annual deposit for the given time and rate ; the quotient 
 will he the annual devosit rec 
 
 Note. — To cancel the debt at maturity, the sum set apart as a 
 sinking fund is supposed to be put at compound interest for the giver 
 time and rate. 
 
 40. A father promises to gives his daughter $5000 as a 
 wedding present. Suppose the event to occur in 5 years, 
 what sum must he annually deposit in a Trust Company, at 
 5 per cent compound interest, to meet his engagement ? 
 
 (I + ^)« _ I (105)^ — I $250 
 
 Solution. » = a -*- L— !— -^ = $5000 -5- i — ^ = — ^ = 
 
 r .05 .276 
 
 $905.80, An8. 
 
BUSINESS FORMULAS. 257 
 
 41, A man having lost his patrimony of $20000, wishes 
 to know how much must he annually deposited at 10 per 
 cent, to recover it in 5 years ? 
 
 42. A county borrows $30000, at 6 per cent compound 
 interest, to build a court-house ; what sum must be set 
 apart annually as a sinking fund to cancel the debt in 
 10 years? 
 
 ANNUITIES. 
 
 500. Annuities are sums of money payable annually, 
 or at regular intervals of time. They are computed accord- 
 ing to the principles of compound interest. 
 
 CASE I. 
 
 501. To Find the Amount of an Unpaid Annuity at Compound 
 
 Interest, the Time and Rate per cent being given. 
 
 Let a be the annuity, 1 + r tlie amount of $1 for i year, and n the 
 number of years. 
 
 The amount due at the end of the 
 
 ist year = a, 
 
 2d *' = a + a(i+r; 
 
 3d " = a + a{i+r) + a{i-\-rf, 
 
 4th " = a + a{i + r) + aii^rf + a{i+rf, 
 
 nth. year = a + a{i +r) + a{i+rf + a{i-\rf . . . a{i +r)"-i. 
 
 forming a geometrical progression, the annuity being the first term, the 
 amount of $1 for i year the ratio, the number of years the number of 
 terms, and the annuity multiplied by the amount of $1 for i year 
 raised to that power whose index is i less than the number of years, 
 the last term. Therefore, the amount is equal to the sum of the 
 series. 
 
 Putting 8 for the amount (Art. 498), we have the 
 
 Formula S = (^ + ^)" — i ^ 
 
 r 
 
 Rule. — Multiply the amount of %i annuity, for the given 
 time and rate, hy the given annuity. 
 
258 BUSINESS FORMULAS. 
 
 Note. — Logarithms may be used to advantage in some of the 
 following examples. 
 
 43. What is due on an annuity of $650, unpaid for 
 4 years, at 7 per cent compound interest ? 
 
 Solution. 8 = (L±j£_Ill « ^ (L^^ll $550 = $2886, Ans. 
 r .07 
 
 44. An annual pension of |88o was unpaid for 6 years ', 
 what did it amount to at 6 per cent compound interest ? 
 
 45. An annual tax of I340 was unpaid for 7 years; what 
 was due on it at 5 per cent compound interest? 
 
 CASE II. 
 
 502. To Find the Present Worth of an Annuity atCompound 
 Interest, the Time of Continuance and the Rate being given. 
 
 Let P denote the present worth ; then the amount of P in ?i years 
 will be equal to tlie amount of the annuity for the same 'time. 
 Therefore^ 
 
 r 
 Dividing each member by (i + r)" (Art. 279), we have the 
 
 Formula. 
 
 Note. — In applying the formula, the negative exponent may be 
 made positive by transferring the quantity which it affects from the 
 numerator to the denominator (Art. 279). 
 
 1 
 
 r r 
 
 EuLE. — Multiply the present worth of an annuity of $1 
 for the given time by the given annuity. 
 
 46. What is the present worth of an annuity of $375 for 
 6 years, at 7 per cent compound interest ? 
 
 Solution. P = l^Jll^ « = i-( i.o7H ^ = l=Jt ^^^^ 
 r .07 .07 
 
 r= $1785.71, Ans. 
 
 47. What is the present worth of an annual pension of 
 $525 for 5 years, at 4 per cent compound interest ? 
 
BUSINESS FORMULAS. 259 
 
 CASE III. 
 
 503. To Find the Present Worth of a Perpetual Annuity, the 
 
 Rate being given. 
 
 Let n denote infinity, then reducing the formula in Art. 502, we 
 have this 
 
 Formula. -P = v C^^*- 435-) 
 
 Rule. — Divide the annuity hy the interest of %i for 
 I year, at the given rate, 
 
 48. What is the present worth of a perpetual scholarship 
 that pays $150 annually, at 7 per cent compound interest? 
 
 Solution. P = - = i^ = $2142.86, Ans. 
 r .07 
 
 49. "What is the present worth of a perpetual ground rent 
 of $850 a year, at 6 per cent ? 
 
 CASE IV. 
 
 504. To Find the Present Worth of an Annuity, commencing 
 in a given Number of Years, the Rate and Time of Continuance 
 
 being given. 
 
 Let n be the number of years before it will cottimence, and N the 
 number of years it is to continue. Then, 
 
 P — I — (i + r)-^"+-^) __ I — (i + r)-" 
 ~ r r ' 
 
 Performing the subtraction indicated, we have the 
 
 Formula. J» = - [(i + /»)-« — (i + r)"^^. 
 
 Rule. — Find the present worth of the given annuity to 
 the time it terminates ; from this subtract its present worth 
 to the time it commences. 
 
 50. What is the present worth of an annuity of I600, to 
 commence in 4 years and to continue 12 years, at 7 per 
 cent interest ? 
 
 SoLxrriON. P = ? [(i + r)-« — (i + r)-«--»], 
 
 P=?^[(i.o7H-(i.o7)-^«], 
 p ^ $600 X .4242^^^^^^^^ 
 
260 BUSINESS FORMULAS. 
 
 51. A father left an annual rent of I2500 to his son for 
 6 years, and the reversion of it to his daughter for 12 years. 
 What is the present worth of her legacy at 6 per cent 
 interest ? 
 
 CASE V. 
 
 505. To Find the Annuity, the Present Worth, the Time, 
 and Rate being given. 
 
 By the formula in Article 502, 
 
 p-'- 
 
 (I +r)-«^ 
 r 
 
 Dividing by the coefficient of a 
 
 , we have the 
 
 Formula. a 
 
 JPr 
 
 I — (i + r)- 
 
 EtiLE. — Divide the present worth hy the present worth of 
 an annuity of%i for the given time and rate. 
 
 52. The present worth of a pension, to continue 20 years 
 
 at 6 per cent interest, is $668. Eequired the pension. 
 
 Pr $668 X. 06 $668 X. 06 ^ ^ . 
 
 Solution, a = — = ^^^-7 — -:—r. = ^ ,_, — ■ = $58.23,47^. 
 
 53. The present worth of an annuity, to continue 30 years 
 at 5 per cent interest, is $3840. What is the annuity? 
 
 Note. — The process of constructing formulas or rules, it will he 
 seen, is based upon the principles of generalization combined with 
 those of algebraic notation. The student will find it a profitable 
 exercise to form others applicable to different classes of problems. 
 
CHAPTEE XXI. 
 
 DISCUSSION OF PROBLEMS. 
 
 506. The Discussion of a Problem consists in assign- 
 ing all tfie different values possible to the arbitrary quanti- 
 ties which it contains, and interpreting the results. 
 
 507. An Arbitrary Quantity is one to which any 
 value may be assigned at pleasure. 
 
 Problem. — If h is subtracted from a, by what number 
 must the remainder be multiplied that the product may be 
 equal to c ? 
 
 Let X = the number. 
 
 Then (a — ft) a? = c. 
 
 Therefore, x = -. 
 
 a — o 
 
 508. The result thus obtained may have five different 
 forms, depending on the relative values of a, h, and 6*. To 
 represent these forms, let m denote the multiplier. 
 
 I. Suppose a is greater than h. In this case a — 6 is positive, 
 and c being positive, the quotient is positive. (Art. 112.) Consequently, 
 the required multiplier must be positive, and the value of x will be of 
 
 the form of + m. 
 
 « 
 
 II. Suppose a is less than 5. In this case ^ — & is negative^ 
 and c divided by a —6 is negative. (Art. 112.) Hence, the required 
 multiplier must be negative, and the value of x is of the form of — m. 
 
 III. Suppose a is equal to h. In this case a — 5 = o. There 
 fore, the value of x is of the form of — , or a; = - = 00. (Art. 434.) 
 
 506. In what does the discaeeiou of problems consist ? 507. What is an arbitrary 
 quantity! 
 
262 DISCUSSION OF PROBLEMS. 
 
 IV. Suppose c is o, and a is either greater or less than h. 
 
 In this case the value of x has the form — , or a; = o. 
 
 m 
 
 V. Suppose c equals o, and a equals J. In this case the 
 
 value of a; has the form -• 
 o 
 
 Note. — The student can easily test these principles by substituting 
 lumbers for <z, &, and c. 
 
 509. The Discussion of Problems may be further illus- 
 trated by the solution of the celebrated 
 
 PROBLEM OF THE COURIERS.* 
 
 Two couriers A and B, were traveling along the same 
 road in the same direction, from C toward Q ; A going at the 
 rate of m miles an hour, and B ^ miles an hour. At 
 12 o'clock A was at a certain point P \ and B d miles in 
 advance of A, in the direction of Q. When and where were 
 they together ? 
 
 P d Q 
 
 This problem is general ; we do not know from the statement 
 whether the couriers were together before or after 12 o'clock, nor 
 whether the place of meeting was on the right or the left of P. 
 
 Suppose the required time to be after 12 o'clock. Then the time after 
 12 is positive, and the time before 12 is negative ; also, the distance 
 reckoned from P toward Q is positive, and from P toward C is negntive. 
 
 Let t — time of meeting in hours after 12 o'clock ; then mt = dis- 
 tance from P to the point of meeting. 
 
 Since A traveled at the rate of m miles an hour, and B n miles an 
 
 hour, we have 
 
 mt = the distance A traveled. 
 
 And nt- " '* B 
 
 Again, since A and B were d miles apart at 12 o'clock, 
 
 mt — nt = d. 
 
 Factoring and dividing we have the 
 
 * Originally proposed by Clairaut, an eminent French mathemati- 
 cian, born in 1713. 
 
DISCUSSION OF PROBLEMS. 263 
 
 Formula. t = • 
 
 7n — u 
 
 The problem may now be discussed in relation to tlie 
 time t, and the distance 7nf, the two unknown elements. 
 
 I. Suppose m > n. 
 
 Upon this supposition the values of t and mt will both be positive; 
 because their denominator m — n is positive. Now since t is positive, 
 it is evident the two couriers came together after 12 o'clock ; and as 
 mt is positive, the point of meeting was somewhere on the right of P. 
 
 These conclusions agree with each other, and correspond to the 
 conditions of the problem. For, the supposition that m>n implies 
 that A was traveling faster than B. A would therefore gain upon B, 
 and overtake him some time after 12 o'clock, and at a point in the 
 direction of Q. 
 
 Let cZ =: 24 miles, m = S miles, and n = 6 miles. 
 
 By the formula, t = = ^ ^ = 12 hours. 
 
 m—n 8—6 
 
 mt = % y. 12 — ()6 miles A traveled. 
 
 71? = 6 X 12 = 72 " B *' 
 
 Now, 96 — 72 = 24 m. their distance apart at noon, as given above. 
 
 These values show that the couriers were together in 12 hours past 
 
 noon, or at midnight, and at a point Q, 96 miles from P and 72 miles 
 
 from d. 
 
 II. Suppose m <in. 
 
 Then in the formula, the denominator m — n\^ negative, therefore 
 both t and mt are negative. 
 
 Hence, both t and mt must be taken in a sense contrary to that 
 which they had in supposition (I), where they were positive ; that is, 
 the time the couriers were together was before 12 o'clock, and the 
 place of meeting on the left of P. 
 
 This interpretation is also in accordance with the conditions of the 
 problem under the present supposition. For, if w < n, then B was 
 traveling faster than A ; and as B was in advance of A at 12 o'clock, 
 he must have passed A before that time, somewhere on the left of P, 
 in the direction of C. 
 
 Let (Z = 24 miles, m = 5 miles, and ti = 8 miles. 
 
 By the formula, t = = — — = — 8 hours. 
 
 m—n 5—8 
 
 . And mt = 5 X — 8 = — 40 miles A traveled 
 
264 DISCUSSIOIS" OF PKOBLEMS. 
 
 These values show that the couriers were together 8 hours before 
 noon, or at 4 o'clock a. m., and at a point C, 40 miles from P and 
 64 miles from d. 
 
 III. Suppose m =zn. 
 
 Upon this supposition we have m — n = o, and 
 
 ^ d T . md 
 
 t = - = CO, also mt = — = 00. 
 o o 
 
 According to these results, t the time to elapse before the couriers are 
 together, is infinity (Art. 434) : consequently they can never be together. 
 In like manner mty the distance from P of the supposed point of 
 meeting, is infinity ; hence, there can be no such point. 
 
 This interpretation agrees with the supposed conditions of the 
 problem. For, at 12 o'clock the two couriers were d miles apart, and 
 it m = n they were traveling at equal rates, and therefore could 
 never meet. 
 
 IV. Suppose d = o, and m either greater or less than n. 
 
 We then have t = = o, and mt = o. 
 
 m — n 
 
 That is, both the time and distance are nothing. These results show 
 that the couriers were together at 12 o'clock at the point P, and at no 
 other time or place. 
 
 This interpretation is confirmed by the conditions of the problem. 
 For, if d = 0, then at 12 o'clock B must have been with A at the point 
 P. And if m is greater than n, or m is less than n, the couriers were 
 traveling at different rates, and must either approach or recede from 
 each other at all times, except at the moment of passing ; therefore 
 they can be together only at a single point. 
 
 V. Suppose d = Of and m = n. 
 
 Then we have < = - » and mt = -' 
 
 o o 
 
 These results must be interpreted to mean that the time and the 
 distance may be anything whatever, and that the couriers must be 
 together at all times, and at any distance from P. 
 
 This conclusion also corresponds to the conditions of the problem. 
 For, if d5 = o, the couriers were together at 12 o'clock, and if m = n, 
 they were traveling at equal rates, and therefore would never part. 
 
IMAGIJS^ARY QUANTITIES. 265 
 
 IMAGINARY QUANTITIESo 
 
 610. An Imaginary Quantity is an indicated even 
 root of a ne^a^m quantity ; as, V— i, V— «, ^—7. 
 
 Notes. — i. Imaginary quantities are a species of radicals, and are 
 called imaginary, because tliey denote operations wliicli it is impossi- 
 ble to perform. (Art. 294.) 
 
 2. Though the operations indicated are in themselves impossible , 
 these imaginary expressions are often useful in mathematical analyses, 
 and when subjected to certain modifications, lead to important results. 
 
 Sllr Imaginary quantities are added and subtracted like 
 other radicals. (Arts. 310, 311.) 
 
 But to multiply and divide them, some modifications in 
 the rules of radicals are required. (Arts. 312, 313.) 
 
 512. To Prepare an Imaginary Quantity for Multiplication 
 and Division. 
 
 Rule. — Resolve tlie given quantity into two f actor s, one 
 of which is a real quantity, and the other the imaginary 
 expression V — 1. 
 
 Notes.— I. This modification is based upon the principle that any 
 negative quantity may be regarded as the product of two quantities, 
 one of which is — i. Thus, —a = a x — i ; —W = ¥ x — i. 
 
 2. The real factor is often called the coefficient of the imaginary 
 expression, -y^— i. 
 
 I. Multiply V^^ by V—'b, 
 
 Solution. \^—a = ^a x -v/— i, and ^y/— 6 = '^b x /y/— 1« 
 Now y'a X -y/^ X y^ X y^/^ = -y/oS x — i = —y\/ab. Ana, 
 
 2. Multiply + V--^ by — V— y. 
 
 3. Multiply V— 9 by V— ^. 
 
 51a What are Imagimary quantities? 511. How added and subtracted I 
 51a How prepare them for mxiltiplicatiou aud divisiou 1 
 
 1% 
 
266 IMAGIN^ARY QUANTITIES. 
 
 513. It will be seen from the preceding examples: 
 
 First. That the product of two imaginary quantities is 3 
 real quantity. 
 
 Second. That the sign before the product is the opposite 
 of that required by the common rule for signs. (Art. 92.) 
 
 For, while the sign to be prefixed to an even root is ambiguous, 
 this ambiguity is removed when we know whether the quantity whose 
 root is to be taken has been produced from positive or negative 
 quantities. (Art. 293,) 
 
 4. Multiply V— 2 by ViS, 
 
 5. Multiply a/— ^ by \/y. 
 
 Note, — i. From these examples it will be seen that the product of 
 a reoyl quantity and an imaginary expression, is itself imaginary. 
 
 6. Divide V— ^ by V— ?/. 
 
 
 Solution.— '^ 1 = X^-I^v^^ = V -, Ans. 
 
 7. Divide V —x by V~—x, 
 
 Note. — 2. Hence, the quotient of one imaginary quantity divided by 
 another, is a real quantity ; and the sign before the radical is the same 
 as that prescribed by the rule. (Art. 92.) 
 
 8. Di^*ide ^/^x by a/.V> 
 
 9. Divide Vx by V—y, 
 
 Note. — 3. Hence, the quotient of an imaginary quantity divided by 
 a real one, is itself imaginary, and vice versa. 
 
 10. Divide 10 a/— 14 by 2 a/— 7, 
 
 1 1. Divide c V'— i by d V — i, 
 
 514. The development of the different powers of V — i. 
 
 12. (a/^)2 = -I. 15. (V^)5 = +V^. 
 
 13. (V^^)^ = -V-i. 16. {V-if - -I. 
 
 14. {^y~lY=: +1. 17. {^~~i)l= -V'-^. 
 
 Hence, tJie even powers are alternately — i and +1, and the odd 
 powers — /y/— I and +y'— i. 
 
INDETERMIITATE PROBLEMS, 267 
 
 INDETERMINATE PROBLEMS. 
 
 515. An Indeterminate Prohlem is one which does 
 not admit of a definite answer. (Art. 220.) 
 
 Note. — Among the more common indeterminate problems, are 
 
 I St. Those whose conditions are satisfied by different values of the 
 
 same unknown quantity. (Art 220.) 
 
 2d. Those which produce identical equations. (Art. 200.) 
 
 3d. Those which have a less number of independent simultaneous 
 
 equations than there are unknown quantities to be determined. 
 4th. Those whose conditions are inconsistent with each other. 
 
 1. Given the equation a; 4- y = 9, to find the value of x. 
 
 Solution. — Transposing, x = g — y, Ans. This result can be 
 •verified by assigning any values to x or y. 
 
 2. What number is that, J of wliich minus i half of itself 
 is equal to its 12th part plus its sixth part ? 
 
 Let 
 
 
 X = 
 
 the number. 
 
 Then 
 
 3« 
 4 
 
 X _ 
 2 ~~ 
 
 12^6 
 
 Clearing of fractions, etc.. 
 
 
 gx = 
 
 gx 
 
 Transposing and factoring. 
 
 (9- 
 
 ■Ci)X = 
 
 
 
 
 •• 
 
 . X = 
 
 
 
 
 IMPOSSIBLE PROBLEMS. 
 
 616. An Impossible I^roblemh one, the conditions 
 of which are contradictory or impossible. 
 
 I. Given a; -f y = 10, x — y = 2, and xy = 38. 
 
 OPERATION. 
 
 Solution. — By combining equations (i) x -^ y = 10 (i) 
 
 and (2), we find x = 6 and y = 4. Again, x y = 2 (2) 
 
 xxy = 6x4 = 24. But the third condition " 
 
 2X :i^ I 2 
 requires the product of x and y to be 38, 
 
 which is impossible. • *• ^ = o 
 
 1=^ 4. 
 
 515. Wb«t is au in4eterifti»ate problem ? 516. Wsat is an impossible problem ? 
 
268 NEGATIVE SOLUTIONS. 
 
 2. What number is that whose 5th part exceeds its 4th 
 part by 15? 
 
 3. Divide 8 into two such parts that their product shall 
 be 18. 
 
 NEGATIVE SOLUTIONS. 
 
 517. A Neffative Solution is one whose result i& a 
 minus quantity. 
 
 518. An odd root of a quantity has the same sign as the 
 quantity. An even root of a positive quantity is either 
 positive or negative, both being numerically the same. 
 (Art. 293.) 
 
 But the results of problems in Simple Equations, it is 
 understood, are positive; when otherwise it is presumed 
 there is an error in the data, which being corrected, the 
 result will be positive. 
 
 1. A school-room is 30 feet long and 20 feet wide. How 
 many feet must be added to its width that the room may 
 contain 510 square feet ? 
 
 Solution. — Let x = tlie number of feet, 
 
 Then (20 + a;) 30 = area. 
 
 By conditions, 600 + 3oaj =510 
 Transposing, 30a; = — 90 
 
 .-. X — — 2) ft., Ans. 
 
 Notes. — i. It will be observed that this is a problem in Simple 
 Equations. The steps in the solution are legitimate and the result 
 satisfies the conditions of the problem algebraically, but not arith- 
 metically. Hence, the negatke result indicates some mistake or 
 inconsistency in the conditions of the problem. 
 
 If we subtract 3 ft. from its width, the result will be a positive 
 quantity. 
 
 2. Were it asked how much must be added to the width that the 
 room may contain 690 square feet, the result would be + 3 feet. 
 
 517. What is a ueeative eolution? 
 
hor'N-er's method. 269 
 
 3. In suc« cases, by changing some of the data, a similar problem 
 may be easily found whose conditions are consistent with a possible 
 result. 
 
 2. What number must be subtracted from 5 that the 
 remainder may be 8 ? 
 
 Solution.— Let x = the number. 
 
 Then 5 - a? = 8 
 
 Transposing, a; = — 3, Ans. 
 
 3. A man at the time of his marriage was 36 years old 
 and liis wife 20 years. How many years before he was twice 
 as old as his wife ? A71S, — 4 years. 
 
 HORNER'S METHOD OF APPROXIMATION.^ 
 
 519. This method consists in transforming the given 
 equation into another whose root shall be less than that of 
 the given equation by the first figure of the root, and 
 repeating the operation till the desired approximation is 
 found. 
 
 The process may be illustrated in the following manner: 
 
 Let it be required to find the approximate value of x in the general 
 equation, 
 
 A^ + Bx^ + Cx = D. (i) 
 
 Having found the first figure of the root by trial, let it be denoted 
 by a, the second figure by 6, the third by c, and so on. 
 Substituting a for x in equation (i), we have, 
 
 Aa^ + Ba? + Ca = B, nearly. 
 
 Factoring and dividing, 
 
 * So called from the name of its author, an English mathematician, 
 who communicated it to the Royal Society in 18 19. 
 
270 hoenek's method. 
 
 By putting y for tlie sum of all the figures of the root except the 
 first, we have x = a^y, and substituting this value for x in equation 
 (i), we have, 
 
 Aia^yJ + B^a + yy + C{a+y) = J); 
 
 or A {a^ + 3aV + 3«/ +2^) + ^ («^ + 2ay+y'^) + C(a+y) = D. 
 
 or Aa^ + sAal^y + sAay^ + ^^^ + Ba'^ + 2Bay + 5y- -^-Ca-^Gy — B. 
 
 Factoring and arranging the terms according to the powers of ^. 
 we obtain 
 
 Ay^ + (J5 + 3-4%2 + ((7+ 25a + ^Aa})y = B-{Ga + Ba? + Aa^. (3) 
 
 To simplify this equation, let us denote the coefiicient of y'^ by B', 
 that of ^ by C", and the second member by D' ; then, 
 
 Ay^ + B'f + C'2^ = D\ (4) 
 
 It will be seen that equation (4) has the same form as (i). It is the 
 first transformed equation, and its root is less by a than the root of 
 equation (i). 
 
 By repeating the operation, a second transformed equation may be 
 obtained. Denoting the second figure in the root by &, and reducing 
 as before, we find, 
 
 , _ D' . 
 
 C' + B'h + AJ^' ^5^ 
 
 Putting z for the sum of all the remaining figures in the root, we 
 have y = !)-{■ z; and substituting this value in equation (4), we obtain 
 a new equation of the same general form, which may be written, 
 
 As^ + B"z'' + C"z = D", (63 
 
 This process should be continued till the desired accuracy is attained. 
 The first figure of the root is found by trial, the second figure from 
 equation (5), and the remaining figures can be found from similar 
 equations. 
 
 But it may be observed that the second member of equation {5) 
 involves the quantity 5, v.'hose value is sought. That is, the vahie of 
 6 is given in terms of h, and that of c would be given in terms of c, and 
 so on. For this reason, equations such as (5) might appear at first 
 «ight to be of little use in practice. This, however, is not the case ; 
 for after the root has been found to several decimal places, the value 
 of the second and third terms, as B'b + AH^ and B"c + Ac^ in the 
 denominators, will be very small compared with C and G ', conse- 
 
horiter's method. 271 
 
 quently as & is very nearly equal to D' divided by C , tliey may be 
 neglected. Therefore the successive figures in the root may be 
 approximately found by dividing D' by C, D" by (7', and so on, 
 regarding C , C , etc., as approximate divisors. 
 
 In transfonning equation (i) into (4), the second member D' and 
 the coefficients G and B' of the transformed equation may be thus 
 obtained. 
 
 Multiplying the first coeflBcient A by a, the first figure of the root, 
 and adding the product to B, the second coeflBcient, we have, 
 
 B + Aa (7) 
 
 Again, multiplying this expression by a, and adding the product to 
 C, the third coeflicient, we have, 
 
 C + Ba + AaK (8) 
 
 Finally, multiplying these terms by a, and subtracting the product 
 from B, we have 
 
 l)-iCa + Ba? + Aa^) = D', 
 
 which is the same as the expression for B' in equation (4). 
 
 Now to obtain C, we return to the first coeflScient, multiply it by a, 
 add the product to expression (7), and thus have the sum 
 
 B + 2Aa, (9), 
 
 which we multiply by a, and adding the product to expression (8) 
 obtain, 
 
 C + 2Ba + 2,Aa^ = C\ 
 
 which is the desired coefficient of y in equation (4). 
 
 Finally, to obtain B' , we multiply the first coeflBcient by a, and add 
 the product to expression (9), and thus obtain, 
 
 B + sAa = B'. 
 
 In this way the coeflBcients of the first transformed equation are 
 discovered ; and by a similar process the coeflBcients of the second, 
 third, and of all subsequent transformed equations may be found. 
 
 520. This metliod of approximation is applicable to 
 equations of every degree. For the solution of cubic equa- 
 tions, it may be summed up in the following 
 
 KuLE. — 1. Detach the coefficients of the given equation, 
 and denote them hy A; B, (7, and the second member hj D. 
 Find the first figure of the root iy trial, and represent it by a. 
 
272 HORNER'S METHOD. 
 
 Multiply A ly a, and add the product to B. Multiply 
 the sum hy a and add the product to C. Multiply this sum 
 by a and subtract the product from D. The remainder is 
 the first divideyid, or D', 
 
 II. Multiply A by a and add the product to the last sum 
 under B. Multiply this sum hy a and add the product to 
 the last sum under G, The result thus obtained is the first 
 divisor, or C. 
 
 III. Multiply A by a and add the product to the last sum 
 U7ider B. The result is the second coefficient, or B'. 
 
 IV. Divide the first dividend by the first divisor, TJie 
 quotient is the second figure of the root, or b. 
 
 Y. Proceed in like manner to find the subsequent figures 
 of the root. 
 
 Note. — i. In finding the second figure of the root, some allowance 
 should be made for the terms in the divisor which are disregarded ; 
 otherwise the quotient will furnish a result too large to be subtracted 
 from ly. 
 
 
 EXAMPLES. 
 
 I. Given a^ + 2x^ 
 
 -{- Z^ = 24, 
 
 to find X, 
 
 
 SOLXJTION. 
 
 
 A B 
 
 C 
 
 
 D a he 
 
 I +2 
 
 + 3 
 
 = 
 
 24 a? = ( 2 . 8, An& 
 
 2 
 
 4 
 
 J 
 II 
 
 
 22 
 
 2 = jy 
 
 3 
 
 6 
 
 12 
 
 23 = (7' 
 
 
 1.891712 
 
 .108288 = D" 
 
 2 
 
 .6464 
 
 
 
 8 = J5' 
 
 23.6464 
 
 
 ' 
 
 .08 
 
 .6528 
 
 
 
 8.08 
 
 24.2992 = 
 
 C" 
 
 
 ..08 
 
 
 
 • 
 
 8.16 
 
 
 
 
 .08 
 
 
 
 
 9.24 = B" 
 
 
 
273 
 
 Note. — 2. In the following example, tlie last figures of tlie root are 
 found by the contracted method of division of decimals, an expedient 
 which may always be used to advantage after a few places of decimals 
 have been obtained. (See Higher Arithmetic.) 
 
 2. Given ic^ + izx^ — i2>x = 216, to find x. 
 
 
 
 
 SOLUTION. 
 
 
 A 
 
 B 
 
 
 
 
 D abe 
 
 I 
 
 + 12 
 
 
 -18 
 
 = 216 (4.24264+. 
 
 
 _4 
 
 
 +64 
 
 184 
 
 
 x6 
 
 
 +46 
 
 32 = iy 
 
 
 A 
 
 
 Jo 
 
 26.168 
 
 
 20 
 
 
 126 = 0" 
 
 5.832 = D" 
 
 
 j4 
 
 
 4.84 
 
 5.468224 
 
 
 24 = 
 
 B 
 
 130.84 
 
 .363776 = 2)"' 
 
 
 24.2 
 
 
 4.88 
 
 275385 
 
 
 24.4 
 24.6 = 
 24.64 
 24.68 
 
 = B" 
 
 135.7 2 = G' 
 .9856 
 
 T36.7 056 
 .9872 
 I37.0|9|2j8 = 
 
 88391 
 82615 
 
 5776 
 5508 
 
 X = 4.24264+, Ans. 
 
 3. Given a:* + 32:2 + 5a; = 178, ^o find x. 
 
 a; = 4.5388, Ans, 
 
 4. Given $3? + 92:2 — 7^ __ 2200, to find x, 
 
 X = 7.1073536, Ans. 
 
 5. Given a^ + a^ + x= 100, to find x, 
 
 a? = 4.264429+, Ans. 
 
274 TEST EXAMPLES FOR REVIEW. 
 
 TEST EXAMPLES FOR REVIEW. 
 
 1. Required the value of 
 
 6a + 4^ X 5 + 8«j -^ 2 — 3a + 12a X 4. 
 
 2. Required the value of 
 
 (8a; + 3^) 5 + 4^ + 7 - (53J + 9^) "^ 7- 
 
 3. Required the value of 
 
 sax — ab + 4cd — {2ax — 4«5 + 2cd), 
 
 4. Required the value of 
 
 4bc + [s^d — {2xy — mn) 5 + ^dc], 
 
 5. Show that subtracting anegative quantity is equivalent 
 to adding a positive one. 
 
 6. Explain by an example why a positive quantity 
 multiplied by a negative one produces a negative quantity ? 
 
 7. Explain why a minus quantity multiplied by a minus 
 quantity produces a positive quantity. 
 
 8. Given ^_(a; + 8)=^ + — - i7f, to find a;. . 
 
 3 9 7 
 
 g. Given ~ — i h 2a; = ^^ X ^ to find x, 
 
 ^ 5 5 32 
 
 10. Resolve ^d^c — 6Ir^(^ — c^d into two factors. 
 
 11. Resolve $(^y — - gxh — iSx^yz into two factors. 
 
 1 2. Resolve a^' — if^' into two factors. 
 
 13. Resolve 8a — 4 into prime factors. 
 
 14. Resolve a^ — i into prime factors. 
 
 15. Divide 31 into two such parts that 5 times one of them 
 shall exceed 9 times the other by i. 
 
 16. Make an algebraic formula by which any two numbers 
 may be found, their sum and difference being given. 
 
 17. Two sportsmen at Creedmoor shoot alternately at a 
 target; A hits the bull's-eye 2 out of 3 shots, and B 3 out 
 of 4 shots; both together hit it 34 times. How many shots 
 did each fire ? 
 
TEST EXAMPLES FOE REYIEW. 275 
 
 1 8. Find two quantities the product of which is a and the 
 quotient b, 
 
 10. Reduce -7 7 to its lowest terms. 
 
 ^ 00 — 
 
 20. Reduce -^ 75 to its lowest terms. 
 
 c? — W' 
 
 21. Resolve 90:^ _|. \2xyz + 42:2 jnto two factors. 
 
 22. Resolve ()W- — die + c^ into two factors. 
 
 23. Make a formula by which the width of a rectangular 
 surface may be found, the area and length being given ? 
 
 24. A square tract of land contains J as many acres as 
 there are rods in the fence inclosing it. What is the length 
 of the fence ? 
 
 25. A student walked to the top of Mt. Washington at 
 the rate of \\ miles an hour, and returned the same day at 
 the rate of 4^ miles an hour ; the time occupied in traveling 
 being 13 hours. How far did he walk? 
 
 26. Given l ^^— = o, to find x. 
 
 1 —X 
 
 27. Prove that the product of the sum and difference of 
 two quantities, is equal to the difference of their squares. 
 
 28. Prove that the product of the sum of two quantities 
 into a third quantity, is equal to the sum of their products. 
 
 29. Reduce 7-7 , \ "~ ^ —^^ — r to its lowest terms. 
 
 fj^ J4 
 
 30. Reduce j-^ r"^— low-o-ri^v *o i*s lowest terms. 
 
 31. Reduce ;; ^ to a single fraction havirisj 
 
 I — a* I -\- a?' ° *^ 
 
 the least common denominator. 
 
 32. Find a number to which if its fourth and fifth part 
 be added, the sum will exceed its sixth part by 154. 
 
 33. Two persons had equal sums of money ; the first 
 spent $30, the second I40: the former then had twice as 
 much as the latter. What sum did each have at first ? 
 
276 TEST EXAMPLES FOR REVIEW. 
 
 34. A French privateer discovers a ship 24 kilometers 
 distant, sailing at the rate of 8 kilometers an hour, and 
 pursues her at the rate of 12 kilometers an hour. How 
 long will the chase last ? 
 
 35. Given '^ ^ = 7 and 7^ — 3^ _ ^ ^^^ 
 
 7 2 *^ 
 
 9: and y, 
 
 36. Given x = ^"7 f 5 and 4^ ^!^^ = 3, to find 
 
 a; and y, 
 
 37. Make a rule to find when any two bodies moving 
 toward each other, will meet, the distance between them and 
 the rate each moves being given ? 
 
 38. A steamer whose speed in still water is 12 miles an 
 hour, descended a river whose velocity is 4 miles an hour, 
 and was gone 8 hours. How far did she go in the trip ? 
 
 39. Find a fraction from which if 6 be subtracted from 
 both its terms it becomes }, and if 6 be added to both, it 
 becomes J. 
 
 40. Required two numbers whose sum is to the less as 8 
 is to 3, and the difference of whose squares is 49. 
 
 41. Given iox-\-6y =. 76, 4^—20 = 8, and 6x-\-Zz=: 88, 
 to find X, y, and z, 
 
 42. Given 20; + 3^ -f 2; = 24, ^x ■\- y •\- 2Z z=z 26, and 
 a; -f 2y + 32; = 34, to find x, y, and z. 
 
 43. Three persons, A, B, and 0, counting their money, 
 found they had 1 180. B said if his money were taken from 
 the sum of the other two, the remainder would be |6o; 
 C said if his were taken from the sum of the other two, the 
 remainder would be \ of his money. How much money 
 had each ? 
 
 44. The fore-wheel of a steam-engine makes 40 revolutions 
 more than the hind-wheel in going 240 meters, and the 
 circumference of the latter is 3 meters greater than that of 
 the former. What is the circumference of each ? 
 
 45. A man has two cubical piles of wood; the side of one 
 
TEST EXAMPLES FOR REVIEW. 277 
 
 is two feet longer than the side of the other, and the differ- 
 ence of their contents is 488 cubic feet. Required the side 
 of each. 
 
 46. Required a formula by which the height of a rectan- 
 gular solid may be found, the contents and base being given. 
 
 47. Divide 126 into two such parts that one shall be a 
 multiple of 7, the other a multiple of 11. 
 
 48. A tailor paid $1 20 for French cloths ; if he had bought 
 8 meters less for the same money, each meter would have 
 cost 50 cents more. How many meters did he buy ? 
 
 49. A shopkeeper paid 1 175 for 89 meters of silk. At 
 what must he sell it a meter to make 25 per cent ? 
 
 50. Make a formula to find the commercial discount, the 
 marked price and the rate of discount being given. 
 
 51. A man pays $100 more for his carriage than for his 
 horse, and the price of the former is to that of the latter as 
 the price of the latter is to 50. What is the price of each ? 
 
 52. Make a formula to find at what time the hour and 
 minute hands of a watch are together between any two 
 consecutive hours? 
 
 53. A father bequeathed 165 hektars of land to his two 
 sons, so that the elder had 35 hektars more than the younger. 
 How many hektars did each receive ? 
 
 54. What number is that, the triple of which exceeds 40 
 by as much as its half is less than 5 1 ? 
 
 55. A butcher buys 6 sheep and 7 lambs for $71 ; and, at 
 the same price, 4 sheep and 8 lambs for I64. What was the 
 price of each ? 
 
 56. At a certain election, 1425 persons voted, and the 
 successful candidate had a majority of 271 votes. How 
 many voted for each ? 
 
 57. A's age is double B's, and B's is three times C's; the 
 sum of all their ages is 150. What is the age of each ? 
 
 58. Reduce the V243 to its simplest form. 
 
 59. Reduce Vy^ 4- ay^ to its simplest form. 
 
278 TEST EXAMPLES FOR REVIEW. 
 
 60. Reduce x^ and y^ to the common index |^. 
 
 61. Reduce ${a — h) to the form of the cube root. 
 
 62. A farmer sold 13 bushels of corn at a certain price ; 
 and afterward 17 bushels at the same rate, when he received 
 I3.60 more than at the first sale. What was the price per 
 bushel ? 
 
 63. A sold two stoves. On the first he lost $8 more than 
 on the second; and his whole loss was I2 less than triple 
 the amount lost on the second. How much did he lose on 
 each? 
 
 64. A number of men had done J of a piece of work in 
 6 days, when 12 more men were added, and the job was 
 completed in 10 days. How many men were at first 
 employed ? 
 
 65. A company discharged their bill at a hotel by paying 
 $8 each; if there had been 4 more to share in the payment, 
 they would only have paid $7 apiece. How many were 
 there in the party ? 
 
 66. In one factory 8 women and 6 boys work for $72 a 
 week; and in another, at the same rates, 6 women and 
 1 1 boys work for $80 a week. How much does each receive 
 per week ? 
 
 67. What factor can be removed from ViSZX? 
 
 68. Given Vx + 12 = \/a + 12, to find x. 
 
 69. Given — - = ^ ~- , to find y. 
 
 y Vy 
 
 70. Given Vz^ — ^ah = a — h, to find x. 
 
 71. From a cask of molasses \ of which had leaked out, 
 40 liters were drawn, leaving the cask half full. How many 
 liters did it hold ? 
 
 72. Make a formula to find the per cent commission a 
 factor receives, the amount invested and the commission 
 being given. 
 
 73. Divide 20 into two parts, the squares of which shall 
 be in the ratio of 4 to 9. 
 
TEST EXAMPLES FOR REVIEW. 279 
 
 74. After paying out ^ of my money and then J of the 
 remainder, I had I140 left. How much had I at first? 
 
 75. If I be added to both terms of a fraction, its yalue 
 will be J; and if the denominator be doubled and then 
 increased by 2, the value of the fraction will be ^. Kequired 
 the fraction. 
 
 76. Tiffany & Co. sold a gold watch for $171, and the per 
 cent gained was equal to the number of dollars the watch 
 cost. Kequired the cost of the watch. 
 
 77. Two Chinamen receive the same sum for their labor; 
 but if one had received I15 more and the other $9 less, then 
 one would have had 3 times as much as the other. What 
 did each receive ? 
 
 78. A drover bought a flock of sheep for $120, and if he 
 had bought 6 more for the same sum, the price per head 
 would have been |i less. Required the number of sheep 
 and the price of each. 
 
 79. A certain number which has two digits is equal to 
 9 times the sum of its digits, and if 63 be subtracted from 
 the number, its digits will be inverted. What is the 
 number ? 
 
 80. Two river-boatmen at the distance of 150 miles apart, 
 start to meet each other ; one rows 3 miles while the other 
 rows 7. How far does each go ? 
 
 81. A and B buy farms, each paying $2800. A pays I5 an 
 acre less than B, and so gets 10 acres more land. How 
 many acres does '^ach purchase ? 
 
 82. Find a factor that will rationalize ^/x -\- V7. 
 
 83. Find a factor that will rationalize V^ — V^. 
 
 84. Given ^W- + Vx z= -_^-i— ^ to find x. 
 
 V{b^ + Vx) 
 
 85. The salaries of a mayor and his clerk amount to 
 I13200; the former receives 10 times as much as the latter. 
 Kequired the pay 01 eacn. 
 
 S6. What two numbers are those whose sum is to their 
 
280 TEST EXAMPLES ¥0E REVIEW. 
 
 difference as 8 to 6, and whose difference is to their product 
 as I to 36 ? 
 
 87. What two numbers are those whose product is 48, and 
 the difference of their cubes is to the cube of their difference 
 as 37 to I ? 
 
 S8. Find the price of apples per dozen, when 2 less for 
 12 cents raises the price i cent per dozen. 
 
 89. Two pedestrians set out at the same time from Troy 
 and New York, whose distance apart is 150 miles ; one goes 
 at the rate of 24 m. in 3 days, and the other 14 m. in 2 aays. 
 When will they meet ? 
 
 90. The income of A and B for one month was 1 187 6, 
 and B's income was 3 times A's. Required that of each ? 
 
 91. A farmer bought a cow and a horse for $250, paying 
 4 times as much for the horse as for the cow. Find the 
 cost of each. 
 
 92. A man rode 24 miles, going at a certain rate ; he then 
 walked back at the rate of 3 miles per hour and consumed 
 12 hours in making the trip. At what rate did he ride ? 
 
 93. It costs $6000 to furnish a church, or |i for every 
 square foot in its floor. How large is the building, pro- 
 vided the perimeter be 320 feet? 
 
 94. Find 5 arithmetical means between 3 and 31. 
 
 95. Find the sum of 50 terms of the series -J-, i, if, 2, 2 J, 
 S, Zh 4, 4i etc. 
 
 96. A dealer bought a box of shoes for $100. He sold all 
 but 5 pair for $135, at a profit of li a pair. How many 
 pair were there in the box ? 
 
 97. Two numbers are to each other as 7 to 9, and the 
 difference of their squares is 128. Eequired the numbers. 
 
 98. In a pile of scantling there are 2400 pieces, and the 
 number in the length of the pile exceeds that in the height 
 by 43 : required the number in its height and length. 
 
 99. Bertha is J as old as her mother, but in 20 years she 
 will be f as old. What is the age of each ? 
 
 100. Fifteen persons engage a car for an excursion ; but 
 
TEST EXAMPLES FOE REVIEW. 281 
 
 before starting 3 of the company decline going, by which 
 the expense of each is increased by $1.75. What do they 
 pay for the car ? 
 
 loi. When the hour and minute hands of a clock are 
 together between 8 and 9 o'clock, what is the time of day ? 
 
 102. A and B wrote a book of 570 pages; if A had 
 written 3 times and B 5 times as much as each actually 
 did write, they would together have written 2350 pages. 
 How many pages did each write ? 
 
 103. A man and his wife drink a pound of tea in 12 days. 
 When the man is absent, it lasts the woman 30 days. How 
 long will it last the man alone ? 
 
 104. Find the time in which any sum of money will 
 double itself at 7 per cent simple interest. 
 
 105. A purse contains a certain sum, in the proportion of 
 I3 of gold to $2 of silver ; if $24 in gold be added, there will 
 then be $7 of gold for every $2 of silver. Eequired the sum 
 in the purse. 
 
 106. A and B in partnership gain $3000. A owns f of 
 the stock, lacking $200, and gains $1600. Required the 
 whole stock and each man's share of it. 
 
 107. In the choice of a Chief Magistrate, 369 electoral 
 votes were cast for two men. The successful candidate 
 received a majority of one over his rival : how many votes 
 were cast for each ? 
 
 108. Two ladies can do a piece of sewing in 16 days; after 
 working together 4 days, one leaves, and the other finishes 
 the work alone in 36 days more. How long would it take 
 sach to do the work ? 
 
 1 09. If a certain number be divided by the product of its 
 two digits, the quotient is 2 J ; and if 9 be added to the 
 number, the digits will be inverted : what is the number ? 
 
 no. Find 4 geometrical means between 2 and 486. 
 
 III. A trader bought a number of hats for $80 ; if he had 
 bought 4 more for the same amount, he would have paid |i 
 less for each : liow many did he buy ? 
 
282 TEST EXAMPLES FOB KEVIEW. 
 
 112. If the first term of a geometrical series is 2, the ratio 
 5, and the number of terms 12, what is the last term? 
 
 113. A tree 90 feet high, in falling broke into three 
 unequal parts ; the longest piece was 5 times the shortest, 
 and the other was 3 times the shortest : find the length of 
 each piece. 
 
 1 14. The sum of 3 numbers is 219 ; the first equals twice 
 the second increased by 11, and the second equals | of the 
 remainder of the third diminished by 19: required the 
 numbers. 
 
 1 15. Required 3 numbers in geometrical progression, such 
 that their sum shall be 14 and the sum of their squares 84. 
 
 1 1 6. A pound of coffee lasts a man and wife 3 weeks, and 
 the man alone 4 weeks : how long will it last the wife ? 
 
 117. Two purses contain together I300. If you take $30 
 from the first and put into the second, each will then 
 contain the same amount : required the sum in each purse. 
 
 118. A clothier sells a piece of cloth for I39 and in so 
 doing gains a per cent equal to the cost. What did he 
 pay for it? 
 
 119. A settler buys 100 acres of land for $2450; for a 
 part of the farm he pays $20 and for the other part $30 an 
 acre. How many acres were there in each part ? 
 
 120. What is the sum of the geometrical series 2, 6, 18, 
 54, etc., to 15 terms? . 
 
 121. There are 300 pine and hemlock logs in a mill-pond, 
 and the square of the number of pines is to the square of the 
 number of hemlocks as 25 to 49 : required the number of 
 each kind. 
 
 122. A ship of war, on entering a foreign port, had 
 sufficient bread to last 10 weeks, allowing each man 2 kilo- 
 grams a week. But 150 of the crew deserted the first night, 
 and it was found that each man could now receive 3I kilo- 
 grams a week for the remainder of the cruise. What was 
 the original number of men ? 
 
APPENDIX. 
 
 621. To Extract the Cube Root of Polynomials* 
 
 I. Required the cube root of a^ + 3^5 — 3a* — i la' -4 
 6^8 4. 12a — 8. 
 
 OPERATION. 
 
 a^'+Sa"— a^"*— iia^+6a"2 + i2a— 8 ( a*+a— 2, Roots 
 
 a^, the first subtrahend, 
 ist Trial Divisor, 3a'' ) 3a^— 3a**— iia^ etc., first remainder. 
 
 Com. D., 2,(1^ + yi^ + a- ) 3^*^ + 3^^+ d^ 
 
 2d Tr. D., 3a^ + 6«3 4-3a^ ) — 6a*— i2a3 + 6«2 + i2a— 8, 2d remainder. 
 Complete Divisor, 
 
 2,a^ + da^ — 3a'^ — 6a + 4 ) — 6a*— I2a^ + 6a^ + i2g— 8 . Hence, the 
 
 Rule. — I. Arrange the terms according to the powers of 
 one of the letters^ take the cube root of the first term for the 
 first term of the root, and subtract its cube from the given 
 polynomial. 
 
 II. Divide the first term of the remainder hy three times 
 the square of the first term of the root as a trial divisor, and 
 the quotient ivill be the next term of the root. 
 
 III. Complete the divisor by adding to it three times the 
 product of the first term by the second, also the square of the 
 second. Multiply the complete divisor by the second term of 
 the root, and subtract the product from the remainder, 
 
 IV. If there are more than tioo terms in the root, for the 
 second trial divisor, take three times the square of the part 
 of the root already found, and completing the divisor as 
 before^ continue the operation until the root of all the terms 
 is found. ( See Key) 
 
 * For Horner's Method of Approximation, see p. 269. 
 
I. 
 
 x^ ^ gx + 20. 
 
 2. 
 
 a^ ■}- ya— 18. 
 
 3- 
 
 a^ — isa + 40. 
 
 4. 
 
 2aho^ — i4aJc - 
 
 5- 
 
 icy _2xy + I 
 
 6. 
 
 8x^ — 32?/2. 
 
 7. 
 
 a;2 + yh + ?y?7?2;. 
 
 8. 
 
 i2«2a; — Sa^y + 4^2;. 
 
 9. 
 
 a^ — Sci^x 4- 3a!a;2 — a^. 
 
 10. 
 
 I -a^. 
 
 II. 
 
 I + Sa\ 
 
 284 APPENDIX. 
 
 2. Required the cube root of a^ + ^a^ + 30^52 + b^. 
 
 3. Find the cube root of x^ -\- 6x^ + 122; + 8. 
 
 4. Find the cube root oi a^ — 6aPy + izxy^ — 8y^. 
 
 5. What is the cube root of Sa^ — 48«r2 _}. gSa — 64. 
 
 6. Find the cube root of 27^3 — S4a^x + :^6ax^ — S.'?;^^ 
 
 7. What is the cube root of a^ — 6a^ -[- 15^* — 20^3 _j. 
 15^2 _ 6« + I. 
 
 8. The cube root of a;^ — 3ic8 ^ Sx^ — 6a^ — 6x^ i- Sa:^- 
 3^+ I. 
 
 522. Factor the following Polynomials:* 
 
 6odb, 
 
 12. a^ — h^x^. 
 
 523. Find the </. c. c?. of the following Polynomials: 
 
 1. 4^2 — 4ax — 15^ and 6a^ + 'jax — 3:r2. 
 
 2. ^ax^yh^, \2y?^, and i6«3a;3;2;2. 
 
 3. i6a;2 — 2/2, and 16:^2 _ <^xy 4- ?/2. 
 
 4. 6^2 ■\- wax +3a;2, and 6^2 +7«a; — '^'^^ 
 
 5. «* — 5^, and a^ — ^2flr3. 
 
 6. jr^ — a^, and a;* — «*. 
 
 524. Required the h c. m. of the following Polynomials. 
 
 1. 6fl2 _ 4fl5, 4058 _^ 2«, and 60^2 _|_ 4^, 
 
 2. 4 (i + a^), 8(1— «), 4(1— «^); and 8 (i + «)• 
 
 3. c? — la -{• I, «'* — I, and a^ -\- 2a ■\- i. 
 
 4. 12 («J2 _ J3)^ 4 (^2 _^ ^j)^ and 18 (a2 _ J2). 
 
 5. 4(^2 — I, 2a — I, and 4^2 -|- i. 
 
 6. 4 (i 4- a2), 8(1 + a), 4 (i — a% and 8 (i — «). 
 
 * The following problems are classified and may be studied in con- 
 nection with the subjects to which they refer, or be omitted till the 
 other parts of the book are finished, at the option of the teacher. 
 
APPENDIX. 285 
 
 525. Unite the following Fractions i 
 
 X — 2 - 3iP — 3 
 
 I 2ab 
 
 - ^3 - ^ ' ^4 _ ^^ 
 
 ab be ac 
 
 5. From « + 3^ take z(^ — h-\ ^^« 
 
 2 3 
 
 6. From 5a; + t take 2:?; 
 
 c 
 
 ' X I 
 
 7. From « + ic 4- -^ -^ take a — a; + 
 
 0^5 — 2^2 a; 4- ^ 
 
 8. From take 
 
 a a — I 
 
 I ■ , 2 
 
 0. From take -„• 
 
 ^ I — X I — x^ 
 
 526. IWultipIy the following Polynomials: 
 
 a — b , 2b a^ x^ — ifi 
 
 1. I —7 X 2 H ^' 5. — ; — X T^ . 
 
 a -\-b a — b X -^ y ab 
 
 4a ^x 2b , s^ ^ o 5^ 
 
 2. — +^x h— • 6, a^ — I X — ^ 
 
 3a; 2^ 3a; 4a a — I 
 
 3. a;2 __ 2xy + y^ x —^-^- 7. -^ ^ x *^ 
 
 X — y 2a 5a — 10 
 
 75 2^2 — 4<5j3 ^y ^^. 
 
 527. Divide the following Quantities: 
 
 , 2a 2a 3V 2y 
 
 «— 3 «— 3 ^ 2y — 2 y — i 
 
 II .1 ab + b^ b 
 
 X xy^ ' -^ y ^' a^ — b^ ' a — b 
 
 ^* ' "^ 2; • ' x^' ^' \i+a "^ i-a/ • (i^a^ 
 
286 APPEN-DIX. 
 
 528. Simplify the following Fractions: 
 
 3^ x^ 
 
 2a — 
 
 - 2 
 
 2a 
 
 a — 
 
 X + 
 
 I 
 
 X 
 
 3 
 
 y + 
 
 4 
 
 .-c^- 
 
 -f 
 
 f . . 
 
 I 
 a 
 
 -f ' 
 '^ a¥ 
 
 l- 
 
 -^ + 1 
 
 lO. 
 
 529. Solve the following Equations: 
 
 1. A house and barn cost $850, and 5 times the price of 
 the house was equal to 12 times the price of the barn. 
 Required the price of each. 
 
 2. A, B, and C, together have 145 acres of land ; A owns 
 two-thirds and B three-fourths as much as C. How many- 
 acres has each ? 
 
 3. From a cask \ full of water^ 21 liters leaked out, when 
 \ the water was left. Eequired the capacity of the cask. 
 
 X — 4 s^ -j- 14 I 
 
 4. IX — 4 = - — — - — — 
 
 ^ 4 3 12 
 
 ^3^ + 9 7^ + 5 16 + 40; 
 
 2 3 5 
 
 a; + 8 a; — 6 
 
 Y X ^=^ X ■\- 2. 
 
 4 3 
 
 a; + 8 X — 6 
 
 X — 2 ^=z X A, • 
 
 4 3 
 
 2a; -h I / — X — 3^ 
 
 + 6. 
 
 ^-3 \ 
 4 / 
 
 3 
 
 9. The hour and minute hands of a watc^ are together 
 at 12 M. At what time between the hours of 7 and 8 p. m. 
 will they again be in conjunction ? 
 
 10. A merchant supported himself 3 yrs. for £50 a year, 
 and at the end of each year added to that part of his stock 
 which was not thus expended, a sum equal to \ of this part. 
 At the end of the third year his original stock was doubled. 
 What was that stock ? 
 
APPENDIX. 287 
 
 530. Solve the following Simultaneous Equations,* 
 
 I. ^^±-'^=26. 3. ^-^ =9. 
 
 3 •'42^ 
 
 6x 6y ^ 2x — 2 y 
 
 -23 4 
 
 a. - + - = 8. 4. - + ^ = d. 
 
 32 23 
 
 X y ^ . y 
 
 2^3 3 4 
 
 5. A man bought a horse, buggy, and harness for I400 ; 
 he paid 4 times as much for the horse as for the harness, 
 and one-third as much for the harness as for the buggy ; 
 how much did he pay for each? 
 
 6. What number consisting of two figures is that to 
 which, if the number formed by changing the place of the 
 figures be added, the sum will be 121 ; but if subtracted, 
 the remainder will be 9 ? 
 
 X + y — z = 
 
 0; 
 
 10. 
 
 xy = 600 ; 
 
 x+ z—y = 
 
 2; 
 
 
 xz = 300 ; 
 
 y -hz-'X = 
 
 4. 
 
 
 yz = 200. 
 
 
 
 II. 
 
 . y z 
 
 X . z 
 
 
 
 « 3 4 
 
 c d 
 
 
 
 a; , , z 
 -+y + - = 33. 
 
 w -\- X -^ y = 
 
 :6; 
 
 12 
 
 ?^ -f 50 — ic; 
 
 W -{■ X -{- z = 
 
 9; 
 
 
 X + 120 = 3y; 
 
 IV + y -h z = 
 
 8; 
 
 
 y -{• 120 = 2Z', 
 
 X +y + z = 
 
 7- 
 
 
 ^ + 195 = Z^' 
 
 13. A's age added to 3 times B's and C's, is 470 jts. ; 
 B's added to 4 times A's and C's, is 580 yrs. ; and C's added 
 to 5 times A's and B's, is 630 yrs. What age is each ? 
 
 14. What 3 numbers are those whose sum is 59 ; half the 
 difference of the first and second is 5, and half the differ- 
 ence of the first and third is 9 ? 
 
288 APPEl^DIX. 
 
 531. Generalize the following Problems and translate the For- 
 mulas into Rules* 
 
 1. A dishonest clerk absconded, traveling 5 miles an 
 hour ; after 6 hours, a policeman pursued him, traveling 
 8 miles an hour. How long did it take the latter to over- 
 take the former ? 
 
 Note, — Substitute c for clerk's rate, p for policeman's rate, n foi 
 flumber of hours between starting, and x for the time required. 
 
 Formula. x = • 
 
 p — c 
 
 2. A can do a piece of work in 2 days, B in 5 days, and 
 in 10 days ; how long will it take all working together 
 to doit? 
 
 Note. — Let «, &, and c represent the numbers. Then x = dbe -*• 
 {ctb + ac + be). 
 
 -r, abc 
 
 Formula. x = — ; ^-« 
 
 ab + ac + be 
 
 3. Divide $4400 among A, B, and C, in proportion to the 
 
 numbers 5, 7, and 10. 
 
 Note.— Put a, b, and c, for the proportions, s for the sum of the 
 proportions, and n for the number to be divided. 
 
 4. A father is now 9 times as old as his son ; 9 years 
 hence he will be only 3 times as old : what is the age 
 of each? 
 
 632. Expand the following by the Binomial Theorem : 
 
 1. {2a--2fiY. 4. (^' + /)'. 
 
 2. {zx + 2y)\ 5. {a + a-^)\ 
 
 3. (i + 3«)'. 6. (fl2 _ 2a)\ 
 533. Find the Product of the following Powers : 
 
 1. dbor^ by a^. 3. x~'^ by x~^, 
 
 2. a^h-^x-^}ijarWx-\ 4. y-'^hjy\ 
 
 634. Divide the following Powers: 
 
 1. 6a~» by 3«-2. 3. 1 2x-^ by 4X~^. 
 
 2. Sa-*bc-^ by 4a^¥c^, 4. {a + a;)-» by {a + a;)-»». 
 
APPEi^^DIX. 289 
 
 535. Transfer Denominators to Numerators, thus forming 
 entire Quantities. 
 
 J ^, ^, -^ — • 
 
 536. Unite the following Radicals : 
 
 1. V48 + V27 + '\/243. 4. X A/25^ + "^/z^xf^c, 
 
 2. A/54^ — '\/9(>x 4- A/24a;. 5. V^ofi^^ — '\/2oa^x. 
 
 3. 8 V^^^jJ + 2 Vi^. 6. 3'V^i28iz:3z/';2 —• ^x \^i6yz. 
 
 537. Find the Product of the following Madicals: 
 
 1. (« 4- ?/)i X (^ + 70^. 3- (^' + y)^ X (x H- 2/)i 
 
 2. 4 + 2^/2x2 — \/2. 4- 3^ V^^ + ^ X 4 a/^. 
 
 538. I>ividing one Radical by Another. 
 
 1. (fl^%)- -^ («a;)^. 4. (^ + yf -J- (^ + 2^)". 
 
 2. 24a; ^ay -^ 6 V^. 5. 4« a/«^ -i- 2 V«c- 
 
 3. "s/ i6a^ — \2aH -^ 2a. 6. yoyp-i-yViS. 
 
 539. Required the Factors which will Rationalize the fol- 
 lowing Radicals: 
 
 1. 2 ^/a + V7. 4. \/5 — \/5. 
 
 2. ic + Vy. 5- 4 V2S — 5 Vy. 
 
 3. Thedenom. of-^. 6. The d. of -— -^-= 
 
 2V3 V3+V2+I 
 
 540. Solve the following Madical Equations: 
 
 1. Given Vx + i = Vn + ^j to find x. 
 
 2. Given \/x + 18 — ^5 = V^ — 75 to find x, 
 
 3. Given Vx^ —11=5. 5. (13 + ^23 + y"^)^ = 5. 
 
 6 _ 
 
 4- /--— =^ = V 3 + ^- 6. 2 V^ = V iz^ + 3«. 
 
290 APPEKDIX. 
 
 541. Solve the following Quadratics, 
 
 2. — 
 
 0-^ - 
 
 ^~3 
 
 "~~ ' 
 
 .. J ^ 
 
 i6 
 
 X 
 
 lOO — 
 
 4^2 
 
 gx ^ 
 
 = 3. 
 
 ^ 
 
 a^ 
 
 I 
 
 
 2 
 
 4 
 
 32 
 
 
 Vi^ 
 
 L±_? _ 
 
 4 — 
 
 V5 
 
 4 -(- Va; \^x 
 
 5. Find two numbers whose differenco is 12, and the suni 
 of their squares 1424. 
 
 6. Eequired two numbers whose sum is 6, and the sum 
 of their cubes 7 2. 
 
 7. Divide the number 56 into two such parts, that their 
 product shall be 640. 
 
 8. A and B started together for a place 150 miles distant. 
 A's hourly progress was 3 miles more than B's, and he 
 arrived at his journey's end 8 hrs. 20 min. before B. What 
 was the hourly progress of each ? 
 
 9. The diiference of two numbers is 6 ; and if 47 be 
 added to twice the square of the less, it will be equal to the 
 square of the greater. What are the numbers ? 
 
 10. The length added to the breadth of a rectangular 
 room makes 42 feet, and the room contains 432 square feet. 
 Required the length and breadth. 
 
 11. A says to B, the product of our years is 120 ; if I 
 were 3 yrs. younger and you were 2 jrrs. older, the product 
 of our ages would still be 120? How old is each ? 
 
 12. Vx^ + a/^ = 6 Vx. 
 
 13. X 4- '\/x + 6 = 2 + 3 a/^ + 6. 
 
 14. A man bought 80 lbs. of pepper and 100 lbs. of 
 ginger for £65, at such prices that he obtained 60 lbs. more 
 of ginger for £20 than he did of pepper for £10. Whai 
 did he pay per pound for each ? 
 
COLLEGE EXAMINATION PROBLEMS. 
 
 542. I. Divide i^x^ — ^~ hjx 
 
 5<? c 
 
 2. Divide a* — ¥ hy {a — b), 
 
 3. Solve tne equation a; + = 12 — -^ — • 
 
 4. Multiply 3 V45 — 7 V7 ^y Vif + 2 Vpf 
 
 5. Divide a^b^ by «^5i 6. Divide xy~^ by a;%~^, 
 
 7. Given ^cd^ -[- 2X — 9 = 76, to find x. 
 
 8. Given ia;^ _ t^; + 7I = 8, to find x. 
 
 9. Find two numbers, the greater of which shall bw to 
 the less as their sum to 42, and as their difference to 6. 
 
 10. Find the value of i + | + ^f + ^f + ^^^* ^^ infinity. 
 
 11. Find the third term of {a + by^. 
 
 12. Expand to four terms (i + x^)~k 
 
 543. I. Divide . ^ + - by — ' — ^ ; 
 
 2. Find the product of a^, a\ a^ and a ^^. 
 
 2a 
 
 Va^ + ^ 
 
 3. Solve the equation x + V«^ + ^^ 
 
 4. Solve ^^ ^1 -^— = o. 
 
 a; X -\- I X -{■ 2 
 
 5. Solve V^^ I = a? — !• 
 
 6. Find the value of f + i + J +, etc., to infinity. 
 
 7. Given x-{-'\/x : x—Vx :: 3 ^^ + 6 : 2 V^, to find x, 
 
 8. Expand to four terms (a^ + x)'^. 
 
 9. Expand to four terms {x^ — y^)~^. 
 
 544. I. Find the g. c. d. and the I. c. m. of (243aioj5 _^ ^ j 
 and {^ia%^ — i) by factoring. 
 
 T^. ., 6 V^ 1 20c Vl^ 
 
 2. Divide ^— r by ^— • 
 
 25 v«^ 2iaJ wd^ 
 
 3. Solve the equations 2X — y — 21, 20^2 + ^/^ = 153. 
 
292 APPENDIX. 
 
 4. A person buys cloth for $90. If lie had got two yards 
 more for the same sum, the price would have been 50 cts. 
 per yd. less. How much did he buy, and at what price ? 
 
 5. Expand (a — b)^ by the binomial theorem. 
 
 6. Factor 4^^ — ^y\ 
 
 7. Multiply 3 Y ? by 2 W |- 
 
 8. Given x + 2y = y and 22:4-3^ = 12, to find z and y, 
 
 9. Eeduce a V4Sa^d and Vj|- to their simplest forms. 
 
 10. Given 4- - = 12 , to find x. 
 
 32 3 ^ 
 
 X or — — rt' 
 
 645. I. From ^x ■{ — = subtract x — 
 
 2h c 
 
 2. Multiply together -^^ — , ~~~2> ^^^ ^ "^ ~ — * 
 
 I -J- y X -f- X I — X 
 
 3. Extract the square root of Sal)^ + a* — 4a^ -f- 4 J*. 
 
 4. From 2 V3I0 take 3 'V^4o. 
 
 5. Divide a"^^^ by a^b~^. 6. Solve a;* + 40^2 — 12. 
 
 7. Solve a;2 — X Vs = x — ^ \/^. 
 
 8. What two numbers are those whose sum is 2a, and the 
 sum of their squares is 2^* ? 
 
 9. What two numbers are those whose difference, sum, 
 and product are as the numbers 2, 3, and 5 respectively ? 
 
 10. Find three geometrical means between 2 and 162. 
 
 11. Expand to four terms 
 
 546. T. Divide 120;^ — 192 by ^x — 6. 
 
 2. Divide a;* H 7 by 7 — m, 
 
 3. Solve the equation 21 + ^-^^ = 5^ + 2Zzi2?. 
 
 ^ 16 82 
 
 4. Find the product of a^, «^, «^, and «~^. 
 
 5. Solve the equation ~ — g— = 2. 
 
 3/ 2X 
 
 6. Find 6 arithmetical means between i and 50. 
 
COLLEGE PEOBLEMS. 293 
 
 7. How many different combinations may be formed of 
 eight letters taken four at a time ? 
 
 8. Expand {a — l)~^ to four terms. 
 
 9. Divide 150 into two such parts that the smaller may 
 be to the greater, as 7 to 8. 
 
 10. Given 5a; + 2?/ = 29 and 2^ — a; = — i, to find 
 X and y, 
 
 2 T 
 
 547. I. Solve the equation {- 4 = a 
 
 2. What is the relation between a, ap, and ar^ ? 
 
 3. Find two numbers such that the sum of | the first 
 and \ of the second equals 1 1, and also equals three times 
 the first diminished by the second. 
 
 4. Give the first three and last three terms of (2« ) • 
 
 5. Find the r/. c. d, of a^ — W and a^ — lab + V^, 
 
 6. Find the I, c. in, of (p? — ^), and 4 (a — x\ and 
 
 7. Add -=^-^ f , — ^ =:r, and ^—, tto* 
 
 5 (a — ^ 5 (« — *) 5_(« — ^)^ 
 
 8. Find the value of x in "^x + a = V^ + a. 
 
 648. I. Eeduce r 5 to its lowest terms. 
 
 c? — x^ 
 
 2. Multiply ar^m by -^ ; and divide ar^W by -^ 
 
 o 1 J.1 . • nx — 6 X — 5 <^ 
 
 3. Solve the equation 7 ^^— = — 
 
 ^ 35 6;r — loi s 
 
 4.Give.Z£±J_(._£^) = ,. 
 
 a^ X 
 
 5. Solve the equation - -^ [- 7I = 8. 
 
 6. It is required to find three numbers such that the 
 product of the first and second may be 15, the product of 
 the first and third 21, and the sum of the squares of the 
 second and third 74. 
 
29":^ COLLEGE PROBLEMS. 
 
 7. Find tlie sum of n terms of the series i, 2, 3, 4, 
 5^ 6, etc. 
 
 8. Expand to five terms {cfi — ^3)-^. 
 
 9. Find the sum of the radicals A/300 and a/tJ. 
 10. Solve the quadratic - -| = 2^, 
 
 549. I. Find the sum and difference of \/i8«^and 
 
 2. Multiply 2 Vi - a/3^ by 4 V3 - 2 a/^. 
 
 3. Solve the equation """ - 4- ^^ ~~ ""^ z=z n ^ ^ -r ^ ^ 
 
 7 5^4 
 
 4. Solve the equation ~~ ^ ^~^ = — . 
 
 a; — 2 X — I 20 
 
 5. The sum of an arithmetical progression is 198 ; its 
 first term is 2 and last term 42 ; find the common differ- 
 ence and the number of terms. 
 
 6. Expand to four terms {a^ — ]/)^, 
 
 7. Simplify the radical (cfi — 20^!) -\- aW)^, 
 
 8. A and B together can do a piece of work in 3I days, 
 B and in 4f days, and and A in 6 days. Required the 
 time in which either can do it alone, and all together. 
 
 9. Find 3 numbers such that the prod, of the first and 
 second may be 15, the prod, of the first and third 21, and 
 the sum of the squares of the second and third 74. 
 
 550. I. Given - + -==2, - + - = 3, --{-- = 3; find 
 X, y, and z. 
 
 2. Given ^ =: , to find x. 
 
 S — X 3 12 
 
 3. Find the I. c. m. and g. c. d. oi x^ ■\' ^ — 21 and 
 x^ — X — $6, 
 
 4. It takes A 10 days longer to do a piece of work than 
 it takes B, and both together can do it in 12 days. In how 
 many days can each do it alone ? 
 
 5. Substitute ?/ + 3 f or a; in >* — a? -{■ 2X^ — t, \ simplify 
 and arrange the result. 
 
ANSWERS. 
 
 Page 15. 
 
 I, 2. Given. 
 
 3. 2 cts. A, 6 cts. 0. 
 
 4. $8, h. ; I32, c. 
 
 5. 9 and 27. 
 
 6. 4^,0; 8^, B; 16^?, A 
 
 7. 121/, son ; z^y> father. 
 
 Pagre 16* 
 
 8. $20, B's ; I80, A's. 
 
 9. 15. 30. 45- 
 
 10. $7, calf; $56, cow. 
 
 II. $5.25, bridle; 
 $10.50, saddle; 
 $110.25, horse. 
 
 12. $3000, daughter; 
 
 I6000, son ; 
 
 $27000, wife. 
 13- 234, 702, 936. 
 
 Page 19. 
 
 1-3. Given. 
 
 4. 98^. 
 
 5. 18. 
 
 INTRODUCTION. 
 
 6. 10. 
 7- 34- 
 
 8. 17. 
 
 Page 21c 
 
 1. 60. 
 
 2. 40. 
 
 3. ac + 85. 
 
 4. 55 — 2t7. 
 
 5- 35- 
 
 6. 24. 
 
 7. 3^ + 21/ + «5. 
 
 8. 6b — jcx + sa, 
 
 9. Z>a;^ + ca:?/. 
 
 Page 22, 
 
 10. -^-^ + a. 
 
 2Z 
 
 b — a , 
 
 II. f- 2;^. 
 
 12. 3^ + i?^«/ + %2;. 
 
 ^3- d 
 
 ' 14. 92. 
 15. 120. 
 
 ax — Qty — Zia; + by 
 
29G 
 
 S U B T K A C T I If . 
 
 ADDITION. 
 
 Page 24, 
 
 l^wgre 26. 
 
 1, 2. Given. 
 
 I. 240^ + 25—36?. 
 
 3. 2iah. 
 
 2. iGmn—xy-^-hc 
 
 4. 11 xy. 
 
 3. i6hc -\- xy -—mn 
 
 5. isa\ 
 
 4. 4fl!5— 3w/^ + 2^ 
 
 6. — 22,hcd. 
 
 5. i52:?/-i-«5 + 5. 
 
 7. — i6:r3|/2. 
 
 6. Given. 
 
 8. 45a^2. 
 
 7. 21 {a + h). 
 
 9. -39«^^y. 
 
 8. igc{x—y). 
 
 10. 2gWd7n\ 
 
 9. 7«V^y. 
 
 II. Given. 
 
 10. 6^/^?. 
 
 12. 4. 
 
 II. loVa; — y. 
 
 13. 5. 
 
 
 
 Page 27. 
 
 
 12. Given. 
 
 J*asre ;^5. 
 
 13. a{T~6h + ^d 
 
 14, 15- Griven. 
 
 —3>m). 
 
 16. 82;. 
 
 14. 2/(^^ + 3 — 2C 
 
 17. «^c. 
 
 -5m). 
 
 18. — 12J. 
 
 15. m{g + ab--jc 
 
 19. — 12?/. 
 
 + 3^). 
 
 20. — 2m. 
 
 16. i?;(l3«r_3^_|_^ 
 
 21. I. 
 
 -3^+?^). 
 
 22. 75- 
 
 17. a;?/(a4-5-c). 
 
 23. Given. 
 
 I. Given. 
 
 4. 
 
 5- 
 6. 
 
 7. 
 8. 
 
 9- 
 10. 
 II. 
 12. 
 
 13. 
 
 14. 
 
 15- 
 16. 
 
 17. 
 
 18. 
 19. 
 20. 
 
 . 16 cts., b ; 
 30 cts., k. 
 
 Page 28, 
 
 26 peaches ; 
 49 pears. 
 15 and 70. 
 15 K 25 g. 
 I. 
 
 7. 
 
 7. 
 
 356, A; 94, B. 
 
 36 and 141. 
 
 8. 
 
 9 cts., top; 
 
 23 cts., ball. 
 
 $9? b; I31, s. 
 
 26 cts., A. M.; 
 
 74 cts., p. M. 
 
 Given. 
 14. 
 
 7. 
 12. 
 60. 
 20. 
 
 I'cffire 31, 
 
 I, 2. Given. 
 
 3. i4^yz. 
 
 4. — 62«J. 
 
 5. I gab. 
 
 i^UBTRACTION. 
 
 6. 2 7ip?/. 
 
 7. 43(ic. 
 
 8. 37fl;:?:l 
 
 9. 5i«2J. 
 10. — 44.7:^^2. 
 
 Page 32, 
 
 11. z^a^b. 
 
 12. o. 
 
 1 3- —lim^x. 
 14. 53^^«/- 
 
MULTIPLICATION 
 
 297 
 
 15- 
 
 $150. 
 
 28. 
 
 9(«— ^-f-a;). 
 
 38. 
 
 xy{%—ab-^c 
 
 i6. 
 
 25°. 
 
 29. 
 
 5(« + J). 
 
 
 -d). 
 
 17. 
 
 $420. 
 
 30. 
 
 -7ix'-yy 
 
 39- 
 
 c(2a + bm + d) 
 
 18. 
 
 4xy—6a. 
 
 31. 
 
 $600, A's. 
 
 
 
 19. 
 
 iS^+i6am. 
 
 32. 
 
 60°. 
 
 
 
 20. 
 
 i8a;2 + ?/2-f6«. 
 
 
 
 
 rage 34, 
 
 21. 
 
 13^^ + ^— iC 
 
 
 
 I- 
 
 ■3. Given. 
 
 
 —5^ + 3^^- 
 
 
 JP«rflre 33, 
 
 4- 
 
 b—c + d—m. 
 
 22. 
 
 gcd—ab—2m 
 
 33. 
 
 Given. 
 
 5. 
 
 Sx-\-y—ab 
 
 
 + 3^ + 4^. 
 
 34. 
 
 {2h — C-i-d)3^. 
 
 
 + 4^. 
 
 23. 
 
 iSm— 23. 
 
 35- 
 
 (ab—c—d 
 
 6. 
 
 2a—b—c-{-x 
 
 24. 
 
 12a;' — 13a;. 
 
 
 +x)y' 
 
 
 +y+d. 
 
 25. 
 
 i6«Z'+ i^c+d. 
 
 36^ 
 
 a^d-b + c). 
 
 7. 
 
 a—b-{-c—a 
 
 26. 
 
 a—b + c. 
 
 37. 
 
 x(ab—2iC—d 
 
 
 •\-c-^c—a-\-b. 
 
 27. 
 
 6(a + b). 
 
 
 -m). 
 
 
 
 MULTIPLICATION. 
 
 rage 36, 
 
 1-4. Given. 
 
 5. 42abc. 
 
 6. ^^abcxy. 
 
 7. 2>dmxy. 
 
 8. dibcdxyz. 
 
 9. ^6abxy. 
 
 10. 42acdx. 
 
 11. $4bcdm. 
 
 12. d^adfxyz. 
 
 Page 37. 
 
 13. Given. 
 
 14. — 45rt5a:^. 
 
 15. 42abcd. 
 
 16. i^2abcxy. 
 
 17. — 41 4abcxy. 
 
 18. g4^bcdxy. 
 
 Page 38. 
 
 19-21. Given. 
 
 22. i$x^y^. 
 
 23. 24aW. 
 
 24. a^x^y^. 
 
 25. «2j'«+«. 
 
 26. dx^yH. 
 
 27. i8«5^V. 
 
 28. 18. 
 
 29. 240. 
 
 30. 6a^^. 
 
 31. — i8a'*J%. 
 
 32. 4^^1/2^ 
 
 33. 2iaW. 
 
 34. 4oc^xy. 
 
 35. — 28a4^3. 
 
 36. —6x^y^. 
 
 37. 2jaWc^, 
 
 38. — 28a3c*. 
 
 39. a:2^V. 
 
 I, 2. Given. 
 
 3. 6fljca;2 + 8c2d 
 
 4. i^aWx—6acdx 
 
 + 3«^- 
 
 5. —8^25^ 
 -\-6ab^d—2bdm> 
 
 6. — 15«% 
 
 + 2oa2^cH-ioaV 
 
 7. 8. Given. 
 
 I. 6aa; + 3Ja; 
 + 2ay + Jy. 
 
298 
 
 DIVISION-. 
 
 
 Page 40, 
 
 24. 24a^x^—6a^yK 
 
 16. a^b^+2adcd 
 
 2. 2,a^+4ay 
 
 25. 6/2_|_^^^_j_^^^ 
 
 + c^d^. 
 
 —lbx—\ly. 
 
 + ^ca;2. 
 
 17. 9a^—4y^* 
 
 3. \2M—2>cd 
 
 26. Given. 
 
 18. rz;4__^2. 
 
 —4ad + ac, 
 
 27. ^2_^2. 
 
 19. x^—2xy'^-\-y^. 
 
 4. 6ixt/—2ab 
 
 28. «^ + a3_|.^_|.j^ 
 
 20. 4«^— ic2. 
 
 '\-6cxy—2ac, 
 
 29. x^-\-^x^y 
 
 
 5. z^x-\-4'bx—cx 
 
 +3^y'+f' 
 
 Pwgre 45. 
 
 —Z(iy—Aby-\-cy. 
 
 30. a2'^ + 2a'^Z»« 
 
 6. ^ax-^^ay + az 
 
 4-52-. 
 
 I, 2. Given. 
 
 •\-ibx-\-zhy-\-lz. 
 
 31. a;2 4-2:ry + 2a;2? 
 
 3. 36. 
 
 7. i^cdmx—Galm 
 
 + ^2+2y;2; + ^2. 
 
 4. 24 chickens. 
 
 — 2icdnx-{-gah7i. 
 
 
 5. Given. 
 
 8. 24abcx-\-i2mx 
 
 
 6. 48. 
 
 —Z2ahcy~\(imy, 
 
 JPagre 45. 
 
 7. i960. 
 
 II. labd^xyz"^. 
 
 I. a^-{-2a+i. 
 
 8. I960. 
 
 12. \iaUH'''^\ 
 
 2. 4a^ + 4a+i, 
 
 9- 25. 
 
 13. aV+^ 
 
 3. 4«2— 4c^^ + ^l 
 
 10. i6w~ 
 
 14. c:z;(«4-^)^. 
 
 4. 2:2+2a;y + ?/2. 
 
 II. 56. 
 
 15. 5^(«_^)5. 
 
 5. x^—2xy\-y\ 
 
 12. 77. 
 
 16. abc(x-{-yY'^''. 
 
 6. I— ;rl 
 
 13. 36 apples. 
 
 1^, ~3^(« + ^)^. 
 
 7. 49^^-14^^+^^ 
 
 14. 70 sheep ; 
 
 
 8. i6m^—gn^. 
 
 100, both. 
 
 
 9. ic4_^2. 
 
 15. 16 and 12 
 
 P«9re 41, 
 
 10. I — 4gx^. 
 
 16. 24 plums. 
 
 20, a^-{-h^. 
 
 II. i6:z;2— 80:4-1. 
 
 17. 42. 
 
 21. a^^^s^H^**. 
 
 12. 25^2^105+1. 
 
 18. 144. 
 
 22. a:^ + a;2-j-i. 
 
 13. I — 2a; + 2:1 
 
 19- 2if; i4f 
 
 23- 3^ + 4^^i/ 
 
 14. i+4^ + 4''i'^- 
 
 20. i4yV bu., one; 
 
 — I3z2— 4:z;2^2 
 
 15. 64b^—48ab 
 
 6^ bu., other, 
 
 + 222:^-30. 
 
 + 9«^. 
 DIVISION. 
 
 
 Page 47. 
 
 4. 5^^. 
 
 7. sab' 
 
 I, 2. Given. 
 
 5. 5- 
 
 8. z^c. 
 
 3. 2^5. 
 
 6. 2/7i 
 
 9. 4mn, 
 
DIVISION" 
 
 299 
 
 10, II. Given. 
 
 12. — 3C. 
 
 £3. 7J. 
 
 14. 5^- 
 
 15. -65. 
 t6, ydf. 
 17. —gag. 
 
 Page 48, 
 
 r8. Given. 
 
 20. a:®, 
 
 ^i. c^. 
 
 22. 2:^. 
 
 23. 4^» 
 
 24. — • 
 
 y 
 
 25. Given. 
 
 26. 8a^»c2. 
 
 27. — 6^2f. 
 
 28. 5a5. 
 
 29. 72;2?/. 
 
 30. «Jc. 
 
 31. 2a5c. 
 
 32. Sx^y^z, 
 S3. Sa^c. 
 34. i2d^x^y. 
 
 I2iC2 
 
 a 
 
 36. 12X^z\ 
 
 37. iim^n. 
 
 35- 
 
 Pagre 49. 
 
 1-3. Given. 
 4. I^+(^-\-d*. 
 
 5- 3^+5- 
 
 6. 35^—14-45. 
 
 8. — 2a;+y. 
 
 9. y^+z—i, 
 
 10. — 5a— 45 + 6. 
 
 11. sab—sa. 
 
 12. — 4a;2 — scP 
 ■{■ax, 
 
 13- a^— 5^ + 25. 
 
 14. i+5«— 9«^. 
 
 15. 2a—4h—sc. 
 
 16. 2(^ + 5)2 
 
 + 3^(«4-2')2. 
 
 17. 9i?r-9^. 
 
 18. a;(5— c) 
 
 19. 3^2 — 2a. 
 
 20. a— a^+tt^. 
 
 I, 2. Given. 
 
 3. i»+y. 
 
 4. «— 5. 
 
 5. «2_2a5 + 52, 
 
 6. c-^-d. 
 
 7. a;— d 
 
 8. 20; + 32/. 
 
 9. «— &. 
 
 10. ic+y. 
 
 11. a^+ah-^-l^. 
 
 12. 3<Z + 26. 
 
 13. a4-2._ 
 
 14. c^—2ax + a?. 
 
 15. 2a.-3+4a:2 + 8a; 
 + 16. 
 
 16. rc + 5. 
 
 17. a;— 2. 
 
 18. c — X. 
 
 19. a + J. 
 
 20. 2 (a — Z>). 
 
 1. 10 yrs., son ; 
 46 yrs., father. 
 
 2. 15, F.'s m. ; 
 45, J.'s m. 
 
 3. 12 and 60. 
 
 4. 12 and 45 p. 
 
 5. 31 cts., ist; 
 62 cts., 2d; 
 97 cts., 3d. 
 
 Page 52, 
 
 6. 20 cows ; 
 180 sheep. 
 
 7. 13I, less; 
 43 J, greater. 
 
 8. 9. 
 
 9. 5 hours. 
 
 10. 8. 
 
 11. 7. 
 
 12. 5 of each. 
 
 13. 4 hours. 
 
 14. 32>%' 
 
 15. 10. 
 
 16. 7 m., A's No. ; 
 14 m., B's ; 
 
 21 m., C's. 
 
300 
 
 FACTOEIiq^G 
 
 17. ^2x, A'sm.; 
 ^x, B's m. ; 
 $Sx, C's m. ; 
 $14^, all. 
 
 ^^' 5> 15^ and 20. 
 19- 10, A's; 
 20, B's; 
 
 30, C's. 
 
 20. 8, 16, 24 
 
 21. 24. 
 
 FACTORING, 
 
 T'af/e 54, 
 
 I, 2. Given. 
 
 2, 3j 3j cf^^h, 
 
 2, 2, $hxxxyy. 
 5, 7» aaabicc. 
 
 3, 7. ^?/«/;2;;a;;?. 
 iixxyyyz. 
 
 S* 5^ 5^ abhcxxx, 
 9' 1, uaahccd. 
 io» 5> ^Ziifnmnnnx, 
 
 Page 55, 
 
 1-3. Given. 
 
 4. ^ (y + <? + 3^). 
 
 5. 2a (a; + ?/ - 2^;). 
 2,hc{x — 2x—'a). 
 Mm {n — 3). 
 -ja {sm + 2:r). 
 2']cl{bx ~ 2my). 
 
 ^axy (3.T + 5). 
 
 J2. 5(5 +3^^ — 4^W 
 
 13. x{i -\-x + x^). 
 
 14- 3 (^ + 2 - 3?/). 
 
 15. i9«5(a;— i). 
 
 I, 2. Given. 
 
 3- (« + J) (« + J). 
 
 6. 
 
 7. 
 8. 
 
 9- 
 10. 
 II. 
 
 4. {x^y)(x-y). 
 
 5. (m 4- 27^) (m + 2n) 
 
 6. (4a + i) (405 + j). 
 
 7- (7 + 5) (7 + 5). 
 
 8. (2« — 3Z») (2« — 3J) 
 
 9- (y + i) (2/ + i). 
 
 10. (l — c2) (l — c2). 
 
 11. C:?;"* + ^») (a;»^ 4- 3^«). 
 
 12. (2a«-- i) (2a»— i). 
 
 13. {a^ -h h') (a^ + i^), 
 
 14. («^ + y) («2;8 -f- y). 
 
 1. Given. 
 
 2. (« + re) (« _- x). 
 
 3. (3^ + 4«/)(3^-4y) 
 4- (^+2)(y-.2). 
 
 5. (3 + ^O (3 -- x). 
 
 6. («5 + i) (« - i). 
 
 7. (i + ^) (i - Q. 
 (5« + 4^) (5«5 - 4^). 
 {2X + y){2X^y), 
 (i +4«)(i — 4«). 
 (5 4- I) (5 -I). 
 
 {X^ i- tf) (x^ - y2), 
 
 (ax + %) (ax — %). 
 («'« 4- b") (a^ — 5«). 
 
 8 
 
 9 
 
 10, 
 II. 
 
 T2. 
 
 13. 
 14. 
 
MULTIPLES, 
 
 301 
 
 12. (a -f- i)(fl' — 0^ + a^ 
 
 Page 59. 
 
 
 -a^ 
 
 4. a3 — a^-\. a — i). 
 
 I. Given. 
 
 13. Given. 
 
 2. (x^i)(a^ + x+ i). 
 
 
 3- (^ - 2^) (^ + ^y + ^y 
 
 Page 60. 
 
 4- a;y + a;^ + y^). 
 
 14. (ar + «/) (a:* - a:3y + a;y 
 
 4. (:?;— i)(a;+ i). 
 
 - ^if + ^). 
 
 5. (I -6^) (I +6^). 
 
 15- (a+i)(a2_-a4. i). 
 
 6. Given. 
 
 16. {a+ i)(a« — a8 + a2_.^ 
 
 7. (b^x)(b + x). 
 
 + 1). 
 
 8. (^ + ;^)(^_^;2 + 6Z22-;23). 
 
 17. (I +^)(i-y + ^^)- 
 
 9. (« 4- b) (a^ - a*b + a^^ 
 
 18. (i +a)(i _« + a2_«? 
 
 --aW + «54 _ j5). 
 
 + A 
 
 lO. (.T + i) {t^ — X^ -\-X— l). 
 
 19. (i + J)(i_J + J2_63 
 
 II. (i + a)\i ^a-\-a^ — a^ 
 
 4.54_^5_|_ J6). 
 
 4- a* - a^). 
 
 20-28. Given. 
 
 DIVISORS. 
 
 Page 61. 
 
 Page 63. 
 
 7. a + J. 
 
 I, 2. Given. 
 
 I, 2. Given. 
 
 8. J 4- 2. 
 
 3. ^' 
 
 3. 3«^- 
 
 9. iP + 3- 
 
 4. J. 
 
 4. 2«a:y. 
 
 10. a — 2. 
 
 5. «<7. 
 
 5. 4a^T^z\ 
 
 II. « + 3. 
 
 6. 20;. 
 
 6. 6aa:2;2;2. 
 
 12. a; 4- I. 
 
 7. 7^i- 
 
 
 13. a — h. 
 
 8. 6al). 
 
 Paflre «(>. 
 
 14. a — 5. 
 
 
 1-5. Given. 
 
 15. a;3 4. 3a;2 4. 32: 
 
 
 6. a; — 2/. 
 
 + 1. 
 
 MULTIPLES. 
 
 Page 68. 
 
 7. si5x^f2^. 
 
 13. Q^ + a?'— a;— I. 
 
 I, 2. Given. 
 
 8. Z^mhiY* 
 
 14. 6a^ 4- iia* 
 
 3. 56fl4^,2cs^. 
 
 Page 69. 
 
 -3«-2. 
 
 4. 2>oxY^' 
 
 5. 9oa3^%'5. 
 
 6. 42oa^J*. 
 
 9-11. G 
 12. a* -f 
 
 _ 7,; 
 
 iven. 
 
 — 2. 
 
302 
 
 BEDUCTIOK OF FBACTION^S. 
 
 REDUCTION 
 Page 74, 
 
 1-3. Given. 
 I 
 
 d' 
 
 I 
 
 a — h 
 x — y 
 
 zy — 3^, 
 
 2X — 2Z 
 
 I 
 X 
 
 I 
 
 OF FRACTIONS. 
 
 X' 
 
 '-\-f 
 
 
 X 
 
 a 
 
 + x 
 
 
 I 
 
 a 
 
 — I 
 
 
 I 
 
 x + y 
 
 3- h 
 
 Page 75. 
 
 1. Given. 
 
 2. a — x. 
 
 a 
 4. h ^ c. 
 
 5.5 + . + ^--. 
 
 6. a — h. 
 
 7. 5 + 7 
 
 b — a 
 
 8. a + a; + 
 
 9. 3a; + I - 
 
 « 
 
 fl^ — X 
 
 I. 2. Given. 
 ^ A^y — h 
 
 , X^ — X 
 o. • 
 
 a; 4- I 
 
 i2«c — a + 5 
 
 '• — 3^ — 
 
 5^ 
 
 J'asre 76. 
 
 I. Given. 
 i2ywa; 
 
 2. 
 
 6m 
 
 24a%x 
 
 iSac^ 4- 24b(^ 
 
 6^ 
 a^ — y^ 
 
 66?V?/ — 45a;2y 
 3«2 — 2b 
 
 Page 77* 
 I, 2. Given. 
 ab 
 
 2I«2 
 
 49« 
 
 ^ — y^ 
 
 ^' Q? — 2a;?/ + y^ 
 
 6. 
 
 32 Q;3(a:4-y) 
 
REDUCTION or FRACTIONS. 
 
 303 
 
 Page 78, 
 I, 2. Given. 
 
 3 
 
 2CX 2ld CpX 
 
 2dx' 2dx^ 2dx 
 
 6. 
 
 lO. 
 
 II. 
 
 12, 
 
 ac^y 2hxy 2(^x^ 
 zc^xy' 2(?xy^ 2c^xy 
 2c? 4- 2db 2>^x 
 
 x^ — 2rry + y'^ 
 
 a::^ -[- 2xy + ^2 
 
 «2 -{- «5 15^ — 3 
 3« ' 3«^ * 
 
 2Wc—2T)^d :^ac—T,ad 
 
 2it^c^l6^d 
 zWc—^d 
 2'bxy hz 4az 
 
 2bz ' 2bz' 2bz 
 
 ax + ay 6x -\- 6y 
 
 2X + 2y 2X + 2y 
 
 2X^ 
 
 ^y 
 
 2X + 2y 
 c? — 2ax 4- a^ 
 
 a^ + 2ax -\- a^ 
 a^ — Q? 
 
 2. 
 
 Page 79. 
 
 I. Given. 
 
 2acx 4i^c^ hxy 
 4dcx' 4bcx' 4bcx 
 
 2,(?d 2'bcx 2>bxy 
 ^ay 4by zcy 
 
 5- 
 
 \2X 
 
 12?/' 12?/' 12?/' \2y 
 
 4abc^ Sod b x^y 
 i6a^ jSa^c 24a; 
 
 6. —- ^ , — 
 
 24a^c' 24a^c' 240^0' 
 
 24a^c 
 
 ac 2cd 2xy 
 
 2bc' 2bc' 2bc 
 
 8. 
 
 10. 
 
 II. 
 
 13. 
 
 (a-^by (a-bf a^^W- 
 a^ — b^' a^ — W' a2_j2 
 
 4xy{x^ y) 6a{x-hy) 
 6xy(x-\-y)' 6xy{x + y)' 
 
 abxy 
 6xy{x + y)' 
 ad bx 
 aW' aW 
 b'^cdx acdm 
 
 aWchV aWcH' aWM 
 '^^ ayz + byz dy^ 
 
 x^z 
 xy\ 
 
 xyH 
 
 ' xyH 
 
 4cmx^ -\- 4cnx^ 
 
 6acm — 6acn :^a^m^x 
 i2a^cx^ ' i2a^cx^ 
 
304 
 
 ADDITION OF FRACTIONS. 
 
 ADDITION 
 
 Page 80. 
 
 I, 2. Given. 
 
 2xy 
 
 Sgdxz 
 
 Sabc 
 
 X 
 
 ya + 2b 
 
 7. Given. 
 
 rage 81, 
 
 8. Given. 
 
 10. 
 
 II. 
 
 12. 
 
 13. 
 
 4$a + 12:?: + 20^ 
 
 60 
 
 Sax -]- 6 -\- <)ay 
 
 12a 
 
 ah — ac + hx -\- ex 
 
 z^ — y 
 
 2xy 
 2a + 2ax 4- 3 
 ay 
 
 OF FRACTIONS. 
 am — dy 
 
 ax — a y + ahx + «5^ 
 
 5g<^ + 3Q^^ + zbda? 
 ^' iSdx 
 
 ^ah — 2dn — d^ 
 
 16. 
 
 17. 
 
 18. 
 
 my 
 nx — mx 
 
 my — 7iy 
 19. — 6. 
 
 4adx + 6hcx — Mm 
 
 20. 
 
 I. Given. 
 
 bdx 
 
 2. a + c + 
 
 hx -f 2d 
 
 2X 
 
 3' x + 
 
 am — ay — dx -\-bd 
 
 bm — by 
 
 4. $d +a -\- b — c 
 
 xy + z 
 
 Sdh 
 
 5- 5^ + 
 
 2flj — by 
 2b~^' 
 
 Page 82. 
 
 6. Given. 
 
 ^bd 4- 2« 
 
 7. ^-^— . 
 
 8 ^ + ^ — 4gy . 
 
 c 
 
 X -\- y ^ a^ 
 
 a 
 
 3^2 _ 2a;y — y^ 4- a — b 
 
 x-^y 
 
 X — y — a^ -i- Sab — 5^ 
 
 II. ^ 7 "^ — 
 
 a — b 
 
 2x^-\-2xy— 2x—2y4-a-\-h 
 
 X— I 
 
 10 
 
 12. 
 
MULTIPLICATIOK Of FRACTIOi^S. 
 
 305 
 
 SUBTRACTION OF FRACTIONS. 
 
 rage 84. 
 
 rage 83 
 
 I, 2. Given. 
 
 gahc 
 
 3- 
 
 4- 
 
 d 
 
 5, 6. Given. 
 
 ay — dm ■\- hm 
 
 ^' " my 
 
 8. 
 
 hy — f/y -f- J?;i 
 12 
 
 /i?/ 4- 7im 4- dm 
 
 10. -^— ^= ~ 
 
 my 
 
 h — my 
 II. ^. 
 
 y 
 
 , M + c7t 
 12. a A 9 — • 
 
 cd 
 
 6« + 55 — 2^/ 
 
 13- 
 
 14. 
 
 «<? + 
 
 5<? + 
 
 ex 
 
 15. flf 
 
 bd—dx-^ly — xy 
 2X + 3<Z«/ 
 
 2V 
 
 , a^ — y^ — loa + lob 
 
 16. -^^ -^ 
 
 10:?; -\- loy 
 
 MULTIPLICATION OF FRACTIONS. 
 
 rage 85. 
 1-4. Given. 
 
 5. h + ^d. 
 
 . ab 
 
 6. — • 
 
 4 
 
 6ca; — 9c?/ + 4dx — 6dy 
 
 ISC + 4d 
 
 8. 2abc, 
 a + b 
 
 10. 
 
 4 + 5^ 
 
 2«2 _f- 2^25 
 
 Z> + I 
 
 11. 8a;2 + 122;. 
 
 12. 2ax — 2bx + 3« — 35. 
 
 13. abc, 
 a + b 
 
 5 
 
 5 
 
 9c — 3^ 
 
 • 
 
 4 
 17. 3^y + 3^- 
 
 18 ^' 
 
 rz; — j5 
 
 Page 87. 
 
 I, 2. Given. 
 3. 6xy. 
 
306 
 
 MULTIPLICATIOK OF FRACTIOKS 
 
 4. 
 
 5- 
 6. 
 
 7. 
 
 8. 
 
 9> 
 
 II. 
 
 12. 
 
 13- 
 
 14. 
 
 15- 
 
 16. 
 
 I. 
 
 2. 
 
 3- 
 
 4. 
 
 5- 
 6. 
 
 'ay 
 
 0? — y^ 
 yh + «/;2;2* 
 
 3 
 8« -1- 24a; 
 
 ^ (o? + ^) 
 
 10. Given. 
 
 y — 3^ 
 
 2xy 
 h — 2a 
 
 2,ab 
 
 ly 
 xy -\- 2X -\- y'^ -\- 2y 
 
 xy 
 xS — y^ 
 ~^ 
 2& + 4. 
 
 Page 88. 
 
 Given. 
 
 abdx 
 
 dbd 4- acd 
 xy 
 
 mx + nx 
 
 — — — • 
 
 4 
 4«c + 4cA 
 
 sr— I 
 
 8. 7a; — 7«:r. 
 aco; — acy 
 
 10. 
 
 12. 
 
 13 
 
 14 
 
 Zac 
 
 2aa^ 4- 2«:p 
 
 3^ — 3 
 Sx^y 
 
 a-\-b 
 6am 
 
 X ■{- I 
 2dbxy 4- y^xy 
 
 4« 4- 6 
 15. I — w. 
 
 ^cx — 3^a? 
 
 1, • 
 
 4 
 
 2. 3a;(?/4- i). 
 
 xy -\- 2X -\- y^ -\- 2^ 
 ^' ■" ^^^ 
 
 4. 9a:. 
 
 S-6- 
 
 6. X, 
 
 x — z 
 
 8. 6ay. 
 
 9. a?-5^3 
 
 2i?7 4- y 
 10. -^- 
 
 5 
 
 X/^ — IP 
 
 12. 254-4. 
 
 13. 2a (c 4- d). 
 
 y — 3^ 
 2xy 
 
 15. y"' 
 
 14 
 
DlVISiei^ OF FRACTIONS. 
 
 307 
 
 I 
 
 5- 
 6. 
 
 7- 
 8. 
 
 9- 
 lo. 
 
 II. 
 
 12. 
 
 -4. Given. 
 
 2« 
 
 Wo' 
 
 a; 
 
 Vb 
 
 DIVISION OF FRACTION 
 
 4. Given. 
 
 7.^. 
 21/ 
 
 8. 
 
 a; — I 
 a^ — a 
 
 10. 
 
 II. 
 
 12. 
 
 h -\-c 
 
 Given. 
 3 times. 
 5- 
 
 9. Given, 
 a^ + 20? + I 
 a?' — 20^+1 
 
 oc^ — y'^ 
 
 ^ — xy^ + ^y — ?/3 
 
 2x^y 
 
 3 
 
 11. 6. 
 
 12. - 
 
 2a + 2^ 
 
 ^ z^j^zy^ 
 
 abcxy 
 X — a 
 
 15. 
 
 3<2:2; + 3^^ 
 
 a — h 
 
 13- 
 
 
 16. -4^ 
 
 3«2J + ilz 
 
 18. — . 
 
 ' 4^y° 
 
 I. Given. 
 abdmy 
 
 ex 
 mx -f wa; 
 
 ax + a;2 
 
 253^2 — ^xy 
 
 y 
 
 6 50^^ + 5« 
 
 rr + I 
 7. 32; — 3fl5:r. 
 
 jPasre 9fl. 
 
 5^ 
 
 243^J/^' 
 24^^ 
 
 x-\-y 
 
 22,XZ 
 
308 SIMPLE EQUATIONS. 
 
 U (x — y) 
 
 6. 
 
 c 
 
 7. Given. 
 a 
 
 8. 
 
 £0. 
 
 « 4- a; 
 c + 2 
 3« (c2 __ ^) ' 
 
 2C 
 
 11 
 
 a2 _ «c 4- c2 
 12 4 (q^^ — zaa; 4 a^) 
 IX 
 
 13- 
 
 («^ 4 hf 
 
 X 
 
 a^ -\- 2X •\- I 
 c 
 
 X 
 
 SIMPLE EQUATIONS, 
 
 Page 97. 
 
 5. 24f 
 
 
 12. 24. 
 
 I, 2. Given. 
 
 6. Given. 
 
 
 13. 14. 
 
 3. a — J 4 c — ^. 
 
 
 4ad 
 
 14. -i^. 
 
 4. a 4 ^ —ab + c. 
 
 
 15. Given. 
 
 
 8. '^^- 
 
 2h — c 
 
 
 
 ] 
 
 t6 
 
 
 Page 98, 
 
 9' 7i* 
 
 
 
 5. Given. 
 
 i6c 
 
 
 Page 102, 
 
 6. 2 — a 4 5. 
 
 10. 
 
 15 
 
 
 16. 12. 
 
 7. ^>-|-c — « — 3. 
 
 
 
 17. 60. 
 
 8. «6Z — ^c 4 2m 
 
 
 
 18. 4. 
 
 — 8. 
 
 Page . 
 
 101. 
 
 19. 20. 
 
 9. 17 — sai — d. 
 
 I. 8. 
 
 
 20. aJ 4 ac. 
 
 10. 4^6? 4 6? — 3M 
 
 2. 50. 
 
 
 dn 
 21. — • 
 
 — I. 
 
 3. 30- 
 
 
 a 
 
 II. 32 — c 4 6?. 
 
 4. 9. 
 
 
 6c 
 
 12. II. 
 
 5- 7. 
 
 
 22. T' 
 
 3^4 2b 
 
 
 6.9. 
 
 
 ad — be 
 23. • 
 
 
 7. 72. 
 
 
 ^ ac 
 
 Pagre 100. 
 
 8. 3a 
 
 
 2ab 
 
 I, 2. Given. 
 
 9. 5- 
 
 
 24. 
 
 ac— 2C 
 
 3. 15. 
 
 10. 28. 
 
 
 5« 4 13-^ 
 
 4. 12. 
 
 II* 12. 
 
 
 ^* 24 
 
ONE UKKKOWK QUANTITY. 
 
 309 
 
 
 24^ + I. 
 
 50c — 2005 
 
 52 
 
 
 b^ 
 
 26. 
 
 ^r 
 
 3« — 6 
 
 45 
 
 
 a— I 
 
 
 2 
 
 27. 
 
 4 
 
 
 
 ^'- 2 
 
 
 
 28. 
 
 I 
 
 
 
 32. 7*- 
 
 
 
 2a — I 
 
 a^(c — a ■}- en- 
 
 -ac) 
 
 3Z' 34. 
 
 
 
 29. 
 
 
 C2 
 
 
 
 
 rage 103, 
 
 Page 105. 
 
 
 Page 106. 
 
 34. 
 
 20. 
 
 
 I. Given. 
 
 15. 
 
 13- 
 
 35. 
 
 24. 
 
 
 2. $8, vest ; 
 
 16. 
 
 30. days. 
 
 S^' 
 
 I. 
 
 
 $32, coat. 
 
 17. 
 
 240 m., one; 
 
 37- 
 
 If 
 
 
 3. $1500, A; 
 
 
 180 m., other. 
 
 38. 
 
 i6f 
 
 
 $3000, B ; 
 $4500, C. 
 
 18. 
 
 12 in., one; 
 
 39- 
 
 lA- 
 
 
 4. 40 men; 
 
 
 16 in., other. 
 
 40. 
 
 36. 
 
 
 80 boys; 
 
 19. 
 
 $25, H. ; 
 
 41. 
 
 II. 
 
 
 880 women. 
 
 
 $175, 0. 
 
 42. 
 
 1200. 
 
 
 5. 40 miles ; 
 
 20. 
 
 8h. 24m. A. M. 
 
 43- 
 
 i4f 
 
 
 80 miles. 
 
 21. 
 
 9 A days. 
 
 44. 
 
 5- 
 
 
 6. 1 33 J barrels. 
 
 22. 
 
 32 of each. 
 
 45- 
 
 4f. 
 
 
 7. 12 p., ist; 
 
 1 
 
 23- 
 
 30, 75, and 45. 
 
 46. 
 
 !(.- 
 
 ^). 
 
 24 p., 2d ; 
 60 p., 3d. 
 
 24. 
 
 25 cts., ch. ; 
 
 47. 
 
 2a — 25 
 
 4-c 
 
 8. 28f feet. 
 
 
 75 cts., goose; 
 
 26 
 
 
 9. $120. 
 
 
 $1.50, turkey. 
 
 48. 
 
 3a — 6 
 
 
 10. $50, B's sh. ; 
 
 25. 
 
 8 ft. 8 in. 
 
 4 
 
 
 $100, A's sli.; 
 
 26. 
 
 40 and 60. 
 
 
 66? 
 
 
 $150, C's sh. 
 
 
 ad , 
 
 49. 
 
 5^* 
 
 
 II. 18 yrs., w. ; 
 
 27. 
 
 — . — -7? less. 
 c + d' 
 
 50. 
 
 I — 8« 
 I + 8a 
 
 
 7,6 yrs., m. 
 12. $3000. 
 
 
 ac , 
 c + d' ^''''^'■ 
 
 
 a 
 
 
 13. 16 and 41. 
 
 28. 
 
 $1128. 
 
 ^i- 
 
 2. 
 
 4 
 
 
 14. $6000, 
 
 
 
310 
 
 SIMPLE EQUATIONS, 
 
 Page 107. 
 
 29. Given. 
 
 30. 60 lbs., b.; 
 
 1 20 lbs., m. 
 
 31. isyrs., B's; 
 
 30 '' A's. 
 
 32. 32|yrs., C's; 
 37l " B's; 
 
 40J " A's. 
 
 33. 1 1 75 votes d.; 
 1325 " s. 
 
 34. 164 artillery; 
 472 cavalry; 
 564 infantry. 
 
 35- ^533i B's; 
 
 $633^, A's; 
 
 Ussh O's. 
 36. $56.25, one; 
 
 ^93-75> other. 
 
 rage 108. 
 
 37- h3^> V^- one. 
 
 I280, " other 
 SS, i4yrs.,y'ngest; 
 
 16 " next; 
 
 18 " eldest. 
 
 39. i6|days. 
 
 40. 550. 
 
 41. 30 and 18. 
 
 42. 
 
 13 
 43. 225 acres, A; 
 
 315 " B. 
 
 44. 2f hrs. 
 
 45. 5, istpart; 
 8, 2d " 
 
 2, 3d " 
 24, 4th '* 
 
 46. 9. 
 
 47. 47 sheep. 
 
 48. $120. 
 
 rage 109, 
 
 49. 60 min. 
 
 d 
 
 50. 
 
 m — n 
 
 51. 300 leaps. 
 
 52. 
 
 a — c 
 he 
 
 one. 
 
 , other. 
 
 53. 72 lbs. 
 
 54. 36 hours; 
 312 miles. 
 
 55. 2oyrs., s.; 
 40 yrs., f. 
 
 56. 280. 
 
 57. $324, ist; 
 I108, 2d ; 
 $144, 3d. 
 
 58. Given. 
 
 Page 110, 
 
 59. 8, ist part; 
 12, 2d '' 
 16, 3d ** 
 
 60. 9 in. and 1 2 in 
 
 61. $75. 
 
 62. 27 days. 
 
 63- ^J575> one; 
 $2625, other. 
 
 64. 12 days. 
 
 65. $720. 
 
 dd, $384, sum ; 
 I162, A'ssh.; 
 $118, B's *' 
 $104, C's " 
 
 67. Given. 
 
 Pasre m. 
 
 (i%, 6 and 8. 
 69- 3456, one; 
 2304, other. 
 
 70. 3 m. an hour, 
 
 71. 400 in., 
 or 33i ft. 
 
 72. 8 k. of one name 
 6 k. of another; 
 3 k. « " 
 
 2 k. " " 
 
 73. 7 and 8. 
 
 74. 240 leaps of d. 
 
 75. 100 days; 
 30000 m., ist; 
 24000 m., 2d ; 
 
SIMPLE EQUATIOifS. 
 
 311 
 
 Page 114:. 
 
 1. Given. 
 
 2. x^% y=4. 
 
 3. ir=i2, ^=6. 
 
 4. a;=i8, «/=2. 
 
 5. ic=i, 2^=3. 
 
 6. rr=i6, «/=35- 
 
 7. tr=3, ?/=2. 
 
 8. Given. 
 
 9. .T=4, 2/=5- 
 
 10. x—df y=i2. 
 
 11. a;=5, y=6. 
 
 12. ic=io, ?/=3. 
 
 13. a:=ii, 2^=9. 
 
 14. a;=3, 2^=2. 
 
 Page 116, 
 
 15. 16. Given. 
 
 17- ^-3. 2/=5- 
 
 18. a:=:4, y=T, 
 
 19. a;=7, «/=2. 
 
 20. x=i6, 2/=35- 
 
 21. a:=3, ^=2. 
 
 1. a:=4, «/=5. 
 
 2. ir=8, y=2, 
 
 3. a;=5, ?/=3. 
 4- ^=Zy y=4' 
 
 5. ir=3, ?/=4. 
 
 6. 0^=12, y=3. 
 
 7. ^=3. y=5' 
 
 8. t?;=4j 5^=3- 
 
 TWO UNKNOWN QUANTITIES. 
 Page 117. 
 
 9- x=S4> y=4^' 
 
 10. a;=4, ^=2. 
 
 11. x=i6, ^=7. 
 
 12. a;=8, y=i» 
 
 13. a;=6o, 2/=36. 
 
 14. x=io, y=z20. 
 
 15- ^=5? y=2. 
 
 16. x=2, y=4. 
 
 17. a.=8, y=6, 
 
 18. a;=4, ^=9. 
 
 19. ic = 6, 
 
 «/=I2. 
 
 20. ir=i8, y=i4. 
 
 1. a;=43, ^=27. 
 
 2. 4 cts., lemons ; 
 6 cts., oranges. 
 
 3- 233 v.; 142 v. 
 
 4. 21 and 54. 
 
 5. $48, cow; 
 $96, horse. 
 
 6. 40 1.; 50 g. 
 
 7. 3 and 2. 
 
 8. mil m., one; 
 9999 m., other. 
 
 9. 56. 
 
 10. $320, B's; 
 $250, A's. 
 
 12. I900, A's; 
 $2400, B's. 
 
 13. 31 and 17. 
 
 14. $6000 h. ; 
 $2500 g. 
 
 1 5. 30 and 20. 
 
 16. $560, B's ; 
 $720, A's. 
 
 17. 25 y. and 35 y. 
 
 18. I180, ist; 
 $115, 2d. 
 
 Page 119. 
 
 19. 140 m., ship; 
 
 1 60 m., steamer 
 
 20. 12 and 18. 
 
 21. 108 ft. 
 
 22. 3oyrs.; 
 13 verses. 
 
 23. 10 1. ; 30 g. 
 
 24. 3 oxen ; 
 21 colts. 
 
 25. 53- 
 
 26. $5000, B's cap.; 
 
 $4800, A's " 
 
 27. $21 or 63 g. 
 an 
 
 28. x = 
 
 y = 
 
 n+i* 
 
 a 
 n-\-i 
 
3VZ 
 
 GEKEBALIZATIOK. 
 
 THREE OR MORE UNKNOWN QUANTITIES, 
 
 rage 121, 
 
 
 6. ir=24, y 
 
 = 6, ^=23. 
 
 I. Given. 
 
 
 7. ^=7i 
 
 y 
 
 = 10, z=^. 
 
 2. x=7, y=s, 
 
 Z=4* 
 
 8. ic=24, y 
 
 = 60, ;z=i20. 
 
 3. x=2, y=s, 
 
 2=5- 
 
 9-1 1. Given. 
 
 4. x=S, y=4, 
 
 Z = 2. 
 
 12. W=2 
 
 , X-. 
 
 =3» y=A, ^=5 
 
 5. x=4, «/=3, 
 
 z=S' 
 
 13. X=2 
 
 > y-- 
 
 =3, ^=4. 
 
 
 4. 630 men, ist; 
 
 7. 
 
 18 = ist; 
 
 rage 123. 
 
 675 " 2d; 
 
 
 22 = 2d; 
 
 I. 12 yrs., ist; 
 
 600 « 3d. 
 
 
 io==3d; 
 
 15 " 2d; 
 17 " 3d. 
 
 2. I5, s. ; $4, 1- ; 
 $20, c. 
 
 5. 50 cts., ist; 
 60 " 2d; 
 80 " 3d. 
 
 8. 
 
 40 = 4th. 
 46 m., A's; 
 9 " B's; 
 
 7 '' C's. 
 
 
 6. 105 min.. A; 
 
 9- 
 
 $64, A's; 
 
 3. 5, 8, and 11. 
 
 210 " B; 
 
 
 I72. B's; 
 
 
 420 " C. 
 
 
 I84. C's. 
 
 G 
 
 ENERALIZATIO 
 
 N. 
 
 
 rage 125. 
 
 12. 22|-hrs. 
 
 24. 
 
 I5625. 
 
 I. 3 chickens. 
 
 13. I67.32. 
 
 25. 
 
 1842I A. 
 
 2. 30 rods. 
 
 Page 129. 
 
 26. 
 
 $55.80. 
 
 3. 12. 
 
 
 27. 
 
 $190,32. 
 
 4- S^TTrFS- 
 
 14. 9077. 
 
 28. 
 
 I1253. 
 
 5. 7 ft 
 
 6. 34. 
 
 15. 1x036.12^. 
 
 29. 
 
 $418.60. 
 
 16. 587.19 bu. 
 
 30- 
 
 $5250. 
 
 7. I205, $187. 
 
 17. i2f per cent. 
 
 1 8. 40 per cent. 
 
 31- 
 
 32. 
 
 $3865.86. 
 $1339.29. 
 
 Pagres 127, 128. 
 
 8. $961, A's; 
 $614, B's. 
 
 19. 60 per cent. 
 
 20. $3000. 
 
 Pages 130-133. 
 
 33- 
 34. 
 
 35. 
 36. 
 
 $222.22+. 
 
 2i yrs. 
 Given. 
 3h. i6T^m.p.M. 
 
 9. 1248, g.; 
 
 21. 137500- 
 
 37- 
 
 6h.32y8ym. P.M. 
 
 902, 1. 
 10. 4f days. 
 
 22. I31250. 
 
 23. $2700, B's; 
 
 38. 
 
 9h.49-^m.P.M. 
 
 II. 5|lirs. 
 
 $230C 
 
 . C'8. 
 
 
 
INV0LUTI02«" 
 
 313 
 
 Page 137. 
 
 2. aW(^' 
 
 3. aWc^, 
 
 4. .T^^/V. 
 
 5. a^b^c^, 
 
 6. i6x^y*. 
 
 7. 2\6a^h^. 
 
 8. 62sa>W(^. 
 
 9. 64«i2^,6^i2. 
 10. aW(^(P. 
 
 II. a;y5f. 
 
 INVOLUTION. 
 rage 138 
 
 12. (a+^y^. 
 
 13. (« + ^')^ 
 
 14. (a: — 2^)'««. 
 
 15. (^ + ^r- 
 
 16. («3 4-J3)2. 
 
 17. aWii2. 
 2'ja^^ 
 
 19 
 
 20. 
 
 8^3 
 
 21. - 
 
 22. 
 
 23 
 
 49^ 
 
 2»» 
 
 rtvnn'lMy.n 
 
 26. ic3_j_6/p2^ _j_ 62^3 
 4-120:^2^242:^ 
 + i2X-{-2>y^ 
 
 + 243/2 + 24^ 
 + 8. 
 
 1. «4 + 4fl3J + 6aW + 4a5s + J*. 
 
 2. i«5 _ 5^4§ 4. loaW — IOa2^,3 _{. 50,^4 _ J5. 
 
 3. c^ + 1(^d + 2ic56?2 + 356^^ + z$(^d*^\-2lc^d^-{■^cd^+d^, 
 
 4. ic^ -f 6a:5y + i5.T^?/2 ^ 20:2:^2/^ 4- isa:^?/^ + 6xy^ 4- ?/^. 
 
 5. x"^ — 7a:^y 4- 2 la^y^ — zs^y^ 4- 35^^^ — 2 1 a:^^^ 4- ^xy^- -y'^. 
 
 6. ^^® 4- loyh 4- 45«/^;2;2 4- \2oy*7? 4- 2io?/V 4- 2<^2y^^ 
 4- 2\oif^ 4- 120?/%^ 4- 45?/22« 4. io?/;2;3 4- 2;io. 
 
 7. a^—ga% 4- 36a'^^2 _ 34^6^ 4- i26«5J4 _ 126^4^,5 ^84^8^ 
 
 8. m^^ 4- iim}^n 4- $$m^n^ 4- 165WW 4- 33om'';i* 4- 46 2772^^2^ 
 4- 462771^7^^ 4-33om%''4-I65m3/^8 4 55^27294-11^7^*0 4- w'^. 
 
 9. a:'^ — i2x^^y 4- 66a:io^2 _ 22ox^y^ 4- 4952:^7/* — 7920:'^^ 
 4- 924a::6/ — 'jg2X^y'^ -\- 4gsc(^y^ — 22o:x?y^ 4- 66:z^yo 
 
 — 122:7/1* -f 7/*2. 
 
 10. a*4-wa"~^J-f- w 
 — I 
 
 I „_o 70 , n—i n 
 - fl^** 2 52 _j_ ^ ^ _ 
 
 (?"-3 5s 
 
 >»- 
 
 n — 2 7i — 
 
 X X 
 
 3 4 
 
 #4- etc. 
 
314 
 
 EVOLUTION, 
 
 13. 0^ -{. ^x^ -\- ^x + I, 
 
 .14. h^ — 4b^ + 6b^ — 45 + 1. 
 15- I — 505 + loa^ — io«3 -I- 5^4 
 
 16. I -j- ^« 4- 
 
 n — I 
 
 a^ + n 
 
 71—1 n — 2 
 
 X 
 
 a^ 4- etc. 
 
 2 23 
 
 J*a</es 143, 144. 
 
 17. a:34_3^2^4.3^22;4.3^^2_|_6^^2; + 3:6-;2;2_f_^3_^2^2^_j_3^^2^^3 
 
 18, 19. Given. 
 
 2o. x^ + 2X (y -\- z) -\- y^ + 2yz + z\ 
 ii. a^ — 2a\b — c) -\- If^ — 2bc + c\ 
 22. a^ ^ 2a{x -{-J/ -{- z) + x^ -\- 2x(y + z) + y^ + 2yz + 2;2 
 
 23. Given. 
 
 ga^ + 12a + 4 
 
 24 
 
 25. 
 
 4^2 — 4ac + 6^ 
 
 26. 
 
 27. 
 
 36 — iSSabc + i()6aWc^ 
 
 49 
 ^(2 — 6hmxy -f gm^x^y^ 
 
 m^ 
 
 28, 2g. Given. 
 
 MULTIPLICATION AND DIVISION OF POWERS. 
 
 Pages 145, 
 
 I, 2. Given. 
 
 146, 
 
 10. a^y-h'^. 
 
 11. Given. 
 
 12. all. 
 
 20- 
 24. 
 
 23. Given, 
 a 
 
 3. ««. 
 
 4. ars. 
 
 
 13. a;-ii. 
 
 14. &2. 
 
 25. 
 
 ay-^ 
 b 
 
 5. b-\ 
 
 6. «"*+«. 
 
 
 15. C-2. 
 
 16. x'''y~^zK 
 
 26. 
 
 a 
 
 7. «-7^. 
 
 8. a^c^cR 
 
 
 17. 4«&c~*. 
 
 18. 32^2/. 
 
 27. 
 
 bx-"" 
 a 
 
 9. ^6^. 
 
 
 19. I2a^C^. 
 
 EVOLUTION. 
 
 
 
 Pages 148, 
 
 149. 
 
 15. #. 
 
 16. «-25. 
 
 19. 
 
 20. 
 
 b'\ 
 
 x^\ 
 
 13. ai 
 
 
 17. «-2. 
 
 21. 
 
 ^.75; 
 
 14. tci 
 
 
 j8. a-6. 
 
 22- 
 
 ■25. Given. 
 
RADICAL QUANTITIES 
 
 315 
 
 t*a(/es 151, 152, 
 
 I, 2. Given. 
 
 3- a^' 
 
 4. a^. 
 
 5. 4*a;%i 
 
 6. 20^1^. 
 
 8. 2a'". 
 
 9. 35a52;8, 
 
 10. 6^2^. 
 
 11. 2 3 a; 3^ 3, 
 
 12. 8«t^>3, 
 
 13- (i3)^^V- 
 
 14. ya^y. 
 
 15. 30^3^7. 
 
 16. 1^. 
 
 1. Given. 
 
 2. ic 4- 2. 
 
 3. a— I. 
 
 4. I -[-X, 
 
 5. ^ + 1. 
 
 6. « — J. 
 
 7. ^ + -• 
 
 8. Given. 
 
 9. a; + «y 4- 2;. 
 
 10. a — 2(3> -f I. 
 
 11. a^ -\- 2b — 2. 
 
 12. I — 2^2 4- ir. 
 
 13. 20^ — 4a 4- 2. 
 
 14. a 
 
 X y 
 15. ~* 
 
 y » 
 
 Page 156 
 
 1-5. Given. 
 6. aVb- 
 
 7. 2a v^. 
 
 8. 6\/xy. 
 
 , 3 /- 
 
 9. 6v3. 
 
 10. i5V^5. 
 
 11. ^SaV^b. 
 
 12. 3a\^2^. 
 
 13. 2m\/i — 3^> 
 
 14. 4^V^2^- 
 
 15. sa\^b. 
 
 16. 6a'v/i3C. 
 
 17. I2a'\/ll. 
 
 rage 157» 
 
 I. Given. 
 
 RADICAL QUANTITIES. 
 
 4. («3^(i296)i 
 
 5. v^i5625, 
 v/87, ^8. 
 
 6. ^^4^8^ '^T^. 
 
 7. v^«~^ v^i^". 
 
 8. ::;7^, "^i-. 
 10. v^c^+Ty, 
 
 II. ^{x — y)\ 
 
 2. ^/8^. 
 
 3. ^/(zfl^ 4- yf- 
 
 4. V(« — 2^)2. 
 
 5. a/9«^^- 
 
 6. ^8^7^. 
 
 7. yi6x^y^z. 
 
 8. l/"^. 
 
 27 
 
 9. ^27 (« — J)' 
 10. 
 
 11. V^rti2^. 
 
 12. \/(« — bf, 
 
 13. a/«^. 
 
 J><«</es J5«, 159, 
 
 1. Given. 
 
 2. (^3)i^ (^^)i. 
 
 3. 9^, (i25)i 
 
 V^l^ 4- yf. 
 
 I, 2. Given. 
 
 3- (3*)S (4^)i 
 
 4. (a'»)i (J")i 
 
 5. (a*)J, (J^)i. 
 
 6. (ai)i, (JtI)I. 
 
 7. (asr, (^^3", 
 
316 
 
 DIVISIOK OF RADICALS. 
 
 ADDITION OF RADICALS. 
 
 Page 160. 
 
 1-3. Given. 
 
 4. 5V3. 
 
 5- 2^/5+4^/3. 
 
 6. (2^ + 3a) V^. 
 
 7. (a2 4- 3c)V'3«^. 
 
 8. (90; 4- 8a) a/2«. 
 
 9. 25^/2. 
 10. 118V3. 
 
 11. 30a v^. 
 
 12. (5^a;-|-6ic2) ^/c^ 
 
 13. 40!^^/?/ 
 
 SUBTRACTION OF RADICALS, 
 
 Page 161. 
 
 1. Given. 
 
 2. 81/7. 
 
 3. 4V30 — i2a/7- 
 
 4. 52^/5- _ 
 
 5. {2ix—io)^/ax 
 
 6. 2'V^« + ^. 
 
 7. 7^Z>. 
 
 8. 95\/2&a;. 
 
 9. a~2. 
 10. ilVs- 
 
 MULTIPLICATION OF RADICALS. 
 
 Page 162. 
 
 1-3. Given. 
 
 4. 90A/10. 
 
 5. aJic. 
 
 6. a/«^ 
 
 ^. 
 
 7. \acxy. 
 
 ' 8. a\/«^. 
 9. Given. 
 
 10. v«^ « 
 
 11. 42A/2. 
 
 12. 12a. 
 
 13. 6. 
 
 14. ^ax. 
 
 15. Given. 
 
 16. Vs- 
 
 17. 2^/5- 
 
 18. (w + w)'V^//z4-yi 
 
 19 
 
 V" 
 
 4. Vs^^^ or 
 
 «\/3«^- 
 
 5. 3\/^. 
 
 6. (a^ta:)i 
 
 DIVISION OF RADICALS 
 
 7. i2(a«/)i 
 
 8. 35\/^. 
 
 9. -^V&. 
 
 10. 2a 
 
 V^f 
 
 11. (« + 5)«. 
 
 12. i5a:V^. 
 
 13. 'v/a; — y. 
 
 14. l6\/2. 
 
 15. 32- 
 
KADICAL EQUATIONS. 
 
 317 
 
 INVOLUTION OF RADICALS. 
 
 Page 164, 
 t, 2. Given. 
 3. at 
 
 4. 18a;. 
 
 5. Sa. 
 
 xi 
 
 6. — V2X. 
 
 4 
 
 7. Saa^Vct. 
 
 8. 9&2. 
 
 9. a2_j_2ay^-}-^. 
 
 EVOLUTION OF RADICALS. 
 
 rage 165, 
 
 1. Given. 
 
 2. 3V^a. 
 
 3. 2V^3:r. 
 
 4. ^9^«/- 
 
 5. 'V/SR^ 
 
 6. \/^bc, 
 
 '• ^r 
 
 8. aci 
 
 9. 2«^V 
 
 10. fl^^J^*^. 
 
 11. Vza. 
 
 12. fl«^6w^«. 
 
 1-3. Given. 
 
 4. ak 
 
 1 a 
 
 5. a^c^, 
 
 6. (« + h)l 
 
 7. Vttc. 
 
 8. \^x-{-y, 
 
 9. 'V^« + J. 
 
 10. a/« + 5 + c. 
 
 Pai/e i67. 
 
 I, 2. Given. 
 
 3. a; — 4^/9. 
 
 4. 3- 
 
 5. A/7 — v«- 
 
 6. 31. 
 
 7- Vs^ + ^3^. 
 
 8: a — 5. 
 9. 3'v/a — Vs. 
 10. 4^2^ + sVb. 
 
 Page 168, 
 1-3. Given. 
 
 xys/ci -\- Vc) 
 
 A/3 — I 
 
 A/3 4- 
 
 RADICAL EQUATIONS 
 
 Pages 169, 170, 
 
 1-3. Given. 
 
 4. (d — a— cy. 
 
 5- 25. 
 
 6. 4f|. 
 
 7. 256. 
 
 8. 1 100. ' I — fl 
 
 9- 3^- 
 
 10, 21. 
 
 11. 252. 
 
 12. 
 
 2« 
 
 13. Given. 
 
 I 
 14. 
 
 IS- «a/}. 
 
 16. Given. 
 
 17. 4. 
 
 18. ^. 
 
 I 
 
 19, 
 
 1 —a 
 
318 
 
 AFfECTED QUADRATICS. 
 
 PURE QUADRATICS, 
 
 
 14. 
 
 ^= ± 2. 
 
 2. a? = ± 8. 
 
 Page 173. 
 
 15. 
 
 X=:±S, 
 
 3. 40 rods. 
 
 I. Given. 
 
 16. 
 
 X=: ± 1. 
 
 4. 30 rods. 
 
 2. X=±^. 
 
 17* 
 
 ^=± ' . 
 
 5. 12, one; 
 
 3- ^ = ± 3- 
 4. ar = ± 4. 
 
 18. 
 
 a— I 
 Given. 
 
 30, other. 
 6. $6. 
 
 5- a: =±5. 
 
 19. 
 
 x=±s. 
 
 7. 80. 
 
 6. x= ±4, 
 
 20. 
 
 a; = ± 2«. 
 
 8. 15, less. 
 
 7. a;==±6. 
 
 21. 
 
 x = ±Vc^ + (P 
 
 60, greater. 
 
 8. a; = ± 4. 
 
 22. 
 
 x=±\. 
 
 9. 27 yds.; 
 
 9. ic = i V6, 
 
 23- 
 
 X=i±^/w^^l^ 
 
 I1.50, price 
 
 10. x= ±: 7. 
 
 24. 
 
 x= ±26. 
 
 10. x=z ±: 16. 
 
 II. a; = ± 2. 
 
 
 
 II. 77 ft. 
 
 12. 0;= Jt «• 
 
 
 Page 174=, 
 
 12. x= ± 16. 
 
 13. a;=±i. 
 
 I. 
 
 X^±Z6, 
 
 
 AFFE 
 
 CTED QUADRA 
 
 TICS. 
 
 rages 178, 179, 
 
 7. 
 
 a 
 
 Page 181. 
 
 1-5. Given. 
 
 '^Tb 
 
 I, 2. Given. 
 
 6. a; = 6 or 2. 
 
 ^/ah-{-d-{- f,. 
 
 3. 3 or - 4*. 
 
 7. a; = 9 or — I. 
 
 
 V 4^^ 
 
 4. 5 or — 6. 
 
 8. a: = 3a 
 
 8. 
 
 — 205 
 
 5. J or - 2. 
 
 9. Given. 
 
 6. 2 or — J. 
 
 7. 4 or — 4|. 
 
 ±Vb'i- Aa\ 
 
 10. 15 or —4. 
 
 9- 
 
 7 or — 5. 
 
 8. 4 or — I. 
 
 II. 20 or — 7. 
 
 10. 
 
 3 ± 2V— I. 
 
 9. 3 or — 4-|. 
 
 12. Given. 
 
 II. 
 
 6 or — 3. 
 
 10. 9 or 6, 
 
 I. Given. 
 
 
 
 II. 4 or - 3f 
 
 
 12. 
 
 2 or — 3. 
 
 
 2. 10 or — 7. 
 
 
 
 
 
 3. 2 or — 5. 
 
 13- 
 
 ^ 
 
 Page 182* 
 
 4. 3 or i|. 
 
 ± 
 
 2(; 
 
 I. 3 or I. 
 
 5. 4 or - I J. 
 
 / W 
 \ hd—ch-\ — z. 
 
 2. 4 or I. 
 
 6. a; = 2. 
 
 
 V 4^ 
 
 3- 3ori. 
 
AFFECTED QUADRATICS. 
 
 319 
 
 4. 2 or — 12. 
 
 5- liorf. 
 
 6. 1 1 or 3. 
 
 7. I J or — li- 
 
 8. - J or - iM. 
 
 9. 4 or 2^. 
 
 10. i±V—a^+i' 
 IT — m 
 
 12. I or — if. 
 
 13. I or — 28. 
 
 14. 10 or — ^. 
 15- -ior-i 
 
 16. 4 or — I. 
 
 17. 4 or — if. 
 
 18. 5 or — 4f. 
 
 19. I J or — |. 
 
 20. 4 or — I. 
 
 21. if or — f. 
 
 22. 4ior 4. 
 
 23. 3 or — if 
 
 24. 4 or — i|. 
 
 25- |orf 
 
 26. f or — I. 
 
 27. n ±m. 
 
 28. 3^ or 3a— 3 J. 
 
 1-3. Given. 
 
 ± 2 or ± ^2. 
 ± A/3 or 
 
 J or — J. 
 
 8. I or - 
 
 9. 4iorJ. 
 10. 4 or — 
 
 2lf 
 
 Pagres 184, 185. 
 
 1. 8 or 4, one ; 
 4 or 8, other. 
 
 2. $60 or $40. 
 
 3. 6 or 4, one ; 
 
 4 or 6, other. 
 
 4. 1 6s., $5 each. 
 
 5. 5 or — 6|. 
 
 6. 16 scholars. 
 
 7. $30 or $20; 
 $20 or $30. 
 
 8. 60 or 40, one ; 
 40 or 60, other. 
 
 9. 36 rds. length ; 
 28 " breadth 
 
 10. 20 in file; 
 80 in rank. 
 
 11. 10 lambs. 
 
 12. 2 and 2. 
 
 13. 4 and I. 
 
 14. 121 yds. long; 
 120 ** wide. 
 
 15. 6 m., A's rate; 
 
 5 m., B's rate. 
 
 16. 120, A; 
 80, B. 
 
 17. 42 and 6. 
 
 18. 4 lemons, A; 
 b " B. 
 
 19. 14 ft., length ; 
 10 " breadth. 
 
 20. 12 rows; 
 
 15 trees in each 
 
 21. 52. 
 
 22. 20 persons. 
 
 rage 186. 
 
 23. 8or— 10, less; 
 15 or— 12, gr. 
 
 24. 16 and 20. 
 
 25. 50 and 25. 
 
 26. 121 and 25. 
 
 27. 12 ft., fore-w. ; 
 15 ft., hind-w. 
 
 28. 2 or —18, one; 
 18 or —2, oth. 
 
 29. I and i, 
 
 30. 3, less; 
 
 18, greater. 
 
 31. 16 or 36 yrs. 
 
 32. 28 rods, length; 
 20 " breadth. 
 
 33- 15 yrs., A's; 
 
 8 yrs., B's. 
 34. 20 lbs. pepper. 
 
 rages 188, 189. 
 
 1. Given. 
 
 2. a; = 4 or 3; 
 
 2^ = 3 or 4. 
 
 3. a; = 7 or 5 ; 
 
 ^ = 5 or 7. 
 
 4. x = S, y =z 6. 
 
 5. a:=io or — 12; 
 y=i2 or — 10. 
 
 6. tz; = 10; 
 y=i2^or7. 
 
>j3iO 
 
 ARITHMETICAL PROGRESSION?'. 
 
 7. iC = 9; 
 
 y = 4or Iff. 
 9. a; = 5 or 4; 
 
 ^ = 4 or 5. 
 ro. a; = 5 or —3; 
 
 «^ = 3 or —5. 
 ti. a; = 3; 2^=2. 
 ^2. x=±s; 
 
 15. 2?= 15 or 12; 
 y= 12 or 15. 
 
 3. J. 
 
 4. |. 
 
 5. f 
 
 6. 2fl^. 
 
 7. 3^- 
 
 8. 10. 
 
 9- i- 
 
 Piagre 204, 
 
 2. 4. 
 
 3. 6400. 
 
 4. 12. 
 
 16. a; = 21 or — 7, 
 ?/ = 7 or — 21. 
 
 17. xz=62s; 
 y=i6. 
 
 18. a;= 2 or i; 
 2/ = I or 2. 
 
 1. 8 or -4, gr.; 
 4 or —8, less. 
 
 2. 30 yrs., wife ; 
 3 1 " man. 
 
 3. 0^ = 9^2, gr.; 
 ^=±^2, less. 
 
 RATIO. 
 
 10. J. 
 
 a 
 
 11. — 
 2 
 
 12. X — y. 
 
 14. f 
 
 2 
 
 15. — 
 •^ 3a: 
 
 
 PROPORTION 
 
 5. 32 and 24. 
 
 6. 10 and 3. 
 
 7. 16 and 12. 
 
 8. 6 and 4. 
 
 9. 48 and of. 
 
 \. 40 rows ; 
 
 25 trees in each 
 ;. 40 yds, length; 
 
 24 " breadth. 
 9 and 3. 
 
 1 1 and 7. 
 31 rds., length; 
 19 " width. 
 ± 7 and i: 4. 
 
 25 m. and 23 m. 
 
 1 2 and 4. 
 3 or — 2, one ; 
 2 or —3, other. 
 
 19. Equality. 
 
 20. Equality. 
 
 21. Gr. inequality. 
 
 22. Less inequal'y. 
 
 23. f I > ¥• 
 
 25. 7. 
 
 26. 98. 
 
 10. 
 
 TI. 
 12. 
 13- 
 
 430 r., length ; 
 320 r., breadth, 
 20 r. ; 30 r. 
 9 and 15. 
 20 and 16. 
 
 2. 9. 
 
 3. (>^' 
 
 ARITHMETICAL 
 
 4. — 5- 
 
 5. If 
 
 6. .91. 
 
 8. 43- 
 
 PROGRESSION 
 
 Page 207* 
 
 9- 
 
 15- 
 
 10. 
 
 44i. 
 
 II. 
 
 49^. 
 
 12. 
 
 3^'' 
 
 a. 
 
GEOMETKICAL PROGRESSION". 
 
 321 
 
 Page 208, 
 
 2. 762J. 
 
 3. 216. 
 
 4. 1400. 
 
 5- 25i 
 
 6. 610. 
 
 7. 175- 
 
 8. 810. 
 
 rage 209. 
 
 1. 58. 
 
 2. 278. 
 
 3. II. 
 
 4. —43- 
 
 5. 2}. 
 
 6. -fj. 
 
 7. 1024. 
 
 8. 192. 
 
 Page 210. 
 
 1. 175- 
 
 2. 1 130. 
 
 3. 6. 
 
 4. 6. 
 
 5. 259. 
 
 6. 13. 
 
 7. —II. 
 
 8. o. 
 
 9- 255- 
 10. 62. 
 
 II 61. 
 
 1. I, 7, 13. 19.25, 
 31- 
 
 2. 3. 7 J, 12, i6|, 
 21,25^,30,34^, 
 39> 43|. 48. 
 
 1. 47. 
 
 2. —6. 
 
 3. 102. 
 
 4. 2, 1 1 1, 2I§, 31, 
 
 4of , 5oJ» 60. 
 
 5. i683f 
 
 6. 98A. 
 
 7. 5776- 
 
 8. foioo. 
 
 J'aflre 213. 
 
 9. 5- 
 
 10. 6, 13I 2o|, 28, 
 35i 421, 50. 
 Slh 64I, 72. 
 
 11. 12, 21.6, 31.2, 
 40.8, 50.4, 60, 
 
 69.6, 79.2, 88.8. 
 98.4, 108. 
 
 12. 975- 
 
 14. 3, 5, and 7. 
 
 15. loioo yards, 01 
 5|: mi., nearly. 
 
 Page 2 14. 
 
 16. 156. 
 
 17. $62.50. 
 
 18. $667.95. 
 
 19. 30c, 
 
 20. $1.20, int.; 
 $2.20, ami 
 
 2f. 20, 40, and 60. 
 
 22. 16.61 + days. 
 
 23. 30. 40, 50. 60. 
 
 24. 3 days. 
 
 25. 140. 
 
 26. $i78,]astpay't; 
 I5370, debt. 
 
 GEOMETRICAL PROGRESSION 
 
 Page 216. 
 
 1. 160. 
 
 2. 4374. 
 
 3. 4i. 
 
 4. 320. 
 
 5. 112. 
 
 6. —31250. 
 
 P^ 
 
 iges 217-221. 
 
 3- 2. 
 
 2. 
 
 1 17 18. 
 
 4. 3- 
 
 3- 
 
 9999. 
 
 5. 6. 
 
 4- 
 
 27305- 
 
 6. 5. 
 
 5- 
 
 3885. 
 
 I. 242. 
 
 6. 
 
 8525. 
 
 2. 2. 
 
 I. 
 
 30000. 
 
 3- 500 
 
 2. 
 
 15625. 
 
 4. 5. 
 
322 
 
 BUSINESS FORMULAS. 
 
 5- 215- 
 
 6. 567. 
 
 2. i, 2, 8, 32, 128. 
 
 I. 4371. 
 
 3- 7174453- 
 
 4. 2imu. 
 
 5- 9565938. 
 
 6. 43046721. 
 
 9. $196.83, 1. c. ; 
 I295.24, wh. c. 
 
 10. $10.23. 
 
 11. 2, 6, 18. 
 
 12. I4294967.295. 
 
 13. 10, 30, 90, 270. 
 
 Page 223, 
 
 14. $120, $60, $30 
 15- 3. 15. 75. 375^ 
 1875. 
 
 16. $108, I144, 
 $192, I256. 
 
 17. 1^,01 I.I. 
 
 18. 8, 4, 2, I. 
 
 INFINITE SERIES. 
 
 Page 231. 
 
 1. li. 
 
 2. J. 
 
 3. I. 
 
 4. if 
 
 Pagre 257. 
 
 2. 1548.86. 
 
 3- I-973- 
 4. Ill 
 
 6. 78. 
 
 7. .0375- 
 
 8. 14.38. 
 
 9. 2.723. 
 10. 2906.3. 
 
 5. 2. 
 
 6. 9. 
 
 7. 10. 
 
 8. }. 
 
 LOGARITHMS. 
 
 11. .1814. 
 
 12. — 4.619. 
 
 Page 239. 
 
 14. .0003321. 
 
 15- 33-335- 
 
 16. 191.77. 
 
 18. 5.23. 
 
 19. 1.0836. 
 
 9- I- 
 
 10. • 
 
 a — I 
 ir. 50 rods. 
 
 20. 2.504. 
 
 21. 2.124. 
 
 Page 240* 
 
 23- -342 + . 
 
 24. .546+. 
 
 25. .324-f. 
 
 26. Given. 
 
 BUSINESS 
 Pages 245-260 
 
 2. $349.60. 
 
 4. i6| per cent. 
 
 6. $12600. 
 
 FORMULAS, 
 
 8. $840. 
 
 IP. $600. 
 
 1 2. 1 6f years. 
 
 13. 10 years. 
 ^5* 2| per cent. 
 
 17. $2010.14. 
 19. $5414.28. 
 21. $2769.23, pr.w.; 
 
 P-77: 
 
 disc. 
 
 22. $6000, pr. w. ; 
 $1800, disc. 
 
 24. $1718.75- 
 26. $2125. 
 28. $2.33j. 
 
 29. $8.83+. 
 31- 5MI^ per ceiki. 
 32. 12^ per cent 
 33' 9ts per cent. 
 34. 0. 8 per cents. 
 ^6. $24630.54, in.; 
 
 $369.46, com. 
 38. $1332. 
 3?- $6290.15. 
 
TEST EXAMPLES, 
 
 323 
 
 40. $905.80. 
 
 41. $3278.69. 
 
 42. $2278.48. 
 
 44. $6130.67. 
 
 45. $2767.60. 
 47. $2336.25. 
 49. $14166.67. 
 
 51. $14775- 
 53. $249.77. 
 
 Pages 265-268, 
 
 -6. 
 
 4. 6v — I. 
 
 5. V—xy, 
 7. I- 
 
 V? 
 
 y 
 
 a/2. 
 
 10. 5 
 
 c 
 a 
 
 2. Impossible. 
 
 3. Impossible. 
 
 I. 
 2. 
 3- 
 
 4- 
 
 5- 
 6. 
 
 7- 
 8. 
 
 9- 
 
 10. 
 
 II. 
 
 12. 
 13- 
 14. 
 
 15- 
 16. 
 
 Pagre 274, 
 
 75«. 
 
 57^+ 7- 
 
 3ff2; + 3fl5j-f2C(^ 
 
 7^6? + 3Cf? 
 
 — io:c?/+ 5rym. 
 
 31. 
 
 c (3^ _ 6Z»2c 
 
 — ccl). 
 
 3^ (2/ — 3^ 
 
 2x2 (2« — i). 
 
 X (a— l). 
 20, II. 
 
 17. 24 shots. 
 
 EST EXAMPLES. 
 rage 275, 
 
 18. ±^/ab 
 
 ± 
 « + I 
 
 4 
 
 19. 
 
 20 
 
 a — h 
 
 x{zxy + 2z), 
 
 22. (3^_c)(3^ — c). 
 23- • 
 
 24. 640 rodSc 
 
 25. 29 J miles. 
 b-i 
 
 26. 
 
 27. 
 28. 
 29. 
 
 30- 
 
 ^ I — «^ 
 
 32. 120. 
 
 33. $50. 
 
 b+ I 
 
 I. 
 
 a — b 
 3^2 
 
 «^ 
 
 34. 
 35- 
 36. 
 37. 
 38. 
 39- 
 40. 
 41. 
 
 42. 
 43- 
 
 44. 
 
 45- 
 46. 
 
 47- 
 48. 
 49. 
 
 50- 
 51- 
 
 Page 276. 
 
 6 hours. 
 
 x = s; y 
 
 2. 
 
 85 f miles. 
 
 8ig.;5il. 
 x = 4', y = 6\ 
 
 x=2) y = ^', 
 
 X r= $40 A, 
 
 y = $60 B, 
 z = $80 0. 
 3 meters f. w. 
 6 " h. w. 
 8 ft. one; 10 ft 
 
 49 ana 77. 
 48 meters. 
 $2.46 a meter. 
 
 $100 horse, 
 $200 carriage. 
 
324 
 
 TEST EXAMPLES. 
 
 52. . 
 
 53. 65 hectares y. 
 100 " elder. 
 
 54. 26. 
 
 55. I5 1.; $6s. 
 
 56. 577 v.; 848 V. 
 
 57. 9oyrs.A;45y. 
 B; yy. C. 
 
 58- 9 Vsj 
 
 59. 2/ a/i + «. 
 
 Page 278, 
 
 60. {:c^)^ (f)i. 
 
 61. V27 (a — h)\ 
 
 62. 90 cts. 
 
 63. $10; $18. 
 
 64. 15 men. 
 
 65. 28 persons. 
 
 66. I4 b. %(k w. 
 
 67. 9. 
 
 68. «. 
 
 69.-!-. 
 
 ^ I — « 
 
 70. « -f J. 
 
 71. 240 liters. 
 
 72. . 
 
 73. 12 and 8. 
 
 74. ^180. 
 
 75- A- 
 
 76. $90. 
 
 77. ^321. 
 
 78. 24 s. $5 pr. 
 
 79. 81. 
 
 80. 45 m. ; 105 m. 
 
 81. 80 A; 70 B. 
 
 82. Vx — ^7. 
 
 83. V3^ + vsy- 
 
 84. ^2 ^ 6t? — 2^2^/ 
 
 + 9 — 6^^+^^. 
 
 85. 1 1 2000 Mayor; 
 $1200 Clerk. 
 
 S6. 216 g. ; 3of 1, 
 
 l^agre 280. 
 
 87. 8 and 6. 
 ^^. 8 cts. 
 
 89. 10 days. 
 
 90. I1407 B; 
 $469 A. 
 
 91. $50 cow; 
 I200 h. 
 
 92. 6 miles. 
 
 93. 100 ft. x6oft. 
 
 94. lh 12^ 17, 
 
 2 if, 26f 
 
 95- 637!.^ 
 
 96. 50 pair. 
 
 97. 18 and 14. 
 
 98. 32 h.; 75 1. 
 
 99. 10 y. B; 
 30 y. M. 
 
 100. $105. 
 
 Page 281, 
 
 1 01. 8:43^7yo'cik 
 
 102. 250 pp. A; 
 320 " B. 
 
 103. 20 days. 
 
 104. i4f yrs. 
 T05. I30. 
 
 106. $9000 whole i 
 $4800 A; 
 $4200 B. 
 
 107. 184 V. and 
 185 V. 
 
 108. 24 d.; 48 d. 
 
 109. 45. 
 
 no. 6, 18, 54, 162, 
 
 111. 16 hats. 
 
 Page 282, 
 
 112. 97656250.^^ 
 
 113. 10 ft.; 30 ft.; 
 50 ft. 
 
 114. 95; 42; 82. 
 
 115. 2, 4, 8. 
 
 116. 12 weeks. 
 
 117. |i8o; $120. 
 
 118. |!30. 
 
 119. 45 A.; 55 A 
 
 120. 14348906. 
 
 121. 125 p. 175 b 
 
 122. 375 men. 
 
 " OF THE 
 
 DIVERSITY 
 

 ^D 21-100^-8/34 
 
 Is and 
 
 *rooklyn 
 
 been to 
 
 e lower 
 
 itiaustive 
 
 ic which 
 
 but that 
 
 enabling 
 
 e author 
 
 n posses- 
 
 ience into 
 
 products, 
 
 lies are fol- 
 
 bff is made 
 
 to take the 
 ' Sentential 
 
 d through to 
 lar and High 
 .-. This work 
 ae of analyz- 
 e Btudent to 
 that is being 
 unfolded, to 
 '. G. 6. Albfe, 
 ool, Oshkosh, 
 
 York. 
 
AText-Book on English Literature, 
 
 Witli copious extracts from the leading authors, English and Ameri- 
 can. With full Instructions as to the Method in which these are 
 to be studied. Adapted for use in Colleges, High Schools, 
 Academies, etc. By Brainerd Kellogg, A.M., Professor of 
 the English Language and Literature in the Brooklyn Collegiate 
 and Polytechnic Institute, Author of a " Text-Book on Rhet- 
 oric," and one of the Authors of Reed & Kellogg's " Graded 
 Lessons in English," and ** Higher J^sons in English." 
 Handsomely printed. 12mo, 478 p] 
 
 The Book is divided into the following Te\ 
 
 Period I. — Before the Norman Con 
 Prom the Conquest to Chaucer's dej 
 From Chaucer's death to ElizabetT 
 beth's reign, 1558-1603. Period 
 Restoration, 1603-3660. Period 
 death, 1660-1745. Period VII 
 Revolution, 1745-1789. Pe: 
 1789, onwards. 
 
 Each Period is preced< 
 great historical events tl 
 ing the literature of th 
 
 The author aims ' ' 
 help himself to. 
 their relations to 
 the authors' ti 
 helping- to 
 asshonlrl' 
 
 allow^ 
 
 8el( 
 
 t 
 
 . Period II.— 
 0. Period III.— 
 eriod IV. — Eliza- 
 th's death to the 
 storationto Swift's 
 th to the French 
 rench Revolution, 
 
 brief resum^ of the 
 . shaping or m color* 
 
 that which he cannot 
 
 places in the line and 
 
 pil; it throws light upon 
 
 great influences at work, 
 
 it points out such of these 
 
 limits of a text-hook would 
 
 writers of each Period. Such are 
 
 c traits of their authors, both in 
 
 of these extracts have ever seen the 
 
 them have been worn threadbare by 
 
 iiie pupil's familiarity with them in the^ 
 
 selections are to be studied, soliciting and 
 exactmg tii^ jm.«.i.^..u a.h ^y^.cj step of the way which leads from the 
 author's diction up through his style and thought to the author himself, 
 and in many other ways it places the pupil on the best possible footing with 
 the authors whose acquaintance it is his business, as well as his pleasure, to 
 make. 
 
 Short estimates of the leading authors, made by the best English and 
 American critics, have been inserted, most of them contemporary with us. 
 
 The author has endeavored to make a practical, common-sense text- 
 book : one that would so educate the student that he wot;dd know and 
 enjoy good literature. 
 
 '• I find the book in its treatment of English literature superior to any other I 
 have examined. Its main feature, which should be the leading one of all similar 
 books, is that it is a means to an end, simply a guide-book to the study of Enj;li!?h 
 literature. Too many studants in the past 'have studied, not the literature of the 
 English language, but some author's opinion of that literature. I know from ex- 
 perience that your method of treatment will prove an eminently successful one." — 
 Jarms H. Shults, Prin. qf the West High School, Cleveland^ 0. 
 
 Effingham maynard & Co., Publishers, New York.