19268 CORNELL UNIVERSITY LIBRARY Cornell University Library oHn,an? ^^^4 031 273 174 Cornell University Library The original of tliis book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031273174 NICHOLSON'S MATHEMATICAl, SERIES. AN ELEMENTARY ALGEBRA THEORETICAL AND PRACTICAL J. W. NICHOLSON, A. M. RESIDENT ANn rttOFESSOR OF MATHEMATICS IX THE LOUISIANA STATE, UNIVEKSITY AND AGKICULTUKAL AND MECHANICAL COLLEGE. F. IIANSELL & BRO., NEW ORLEANS. PRACTICAL EDUCATIONAL SERIES, rUBLISIIED BY F. F. HilNSELL & BRO., CHAMBERS' TWENTY LESSONS IN BOOK-KEEPING. DUVAL'S STUDENTS' HISTORY OF MISSISSIPPI. HANSELL'S PRIMARY SPELLER. HANSELL'S SCHOOL HISTORY OF THE UNITED STATES. HANSELL'S HIGHER HISTORY OF THE UNITED STATES. HANSELL'S PRACTICAL PENMANSHIP, 8 Nos. HANSELL'S TRACING BOOKS, 3 Nos. HANSELL'S PRACTICAL DICTIONARY. HEMPSTEAD'S SCHOOL HISTORY OF ARKANSAS. NICHOLSON'S PRIMARY ARITHMETIC. NICHOLSON'S INTERMEDIATE ARITHMETIC. NICHOLSON'S COMPLETE ARITHMETIC. NICHOLSON'S ADVANCED ARITHMETIC. NICHOLSON'S ELEMENTARY ALGEBRA. PRACTICAL SCHOOL RECORD. PRACTICAL SCHOOL REGISTER. COPYRIGHT iSS8. F. K. HANSELL & BRO, UfMIVERSflY ^ LIBRARY/^ PRESS OF X — — : — rrM&CTROTYPltn BY L. Graham & Son, ' T. A. Slatteky ,& Bro., TJEW OKLEANS, NEW ORLEANS. PREFACE. "THIS work comprises about what is usually required for •*• admission to the best colleges. In preparing it the aim has been — 1. To secure a clear and thorough treatment of the essential subjects- in as compact a form as possible. 2. To present one difficulty at a time. For instance : (1) The several particulars in which Algebra is an extension of Arithme- tic are introduced and treated successively, thus rendering easy and natural the passage of the student from the latter to tlic former. See articles 11, 69, 203, 323, 330. (2) Each subject is treated in a similar inductive manner, thereby enabling the stu- dent to progress with" increasing interest, facility and intelli- gence. For example, see Simple Equations. 3. To embody a sufficient number of graded exercises for solution by the student in order to impress the principles on the memory, and to secure accuracy and rapidity in computation. These exercises were selected with great care; several of the examples being taken from mathema.ticaJ journals of recent date, ai;d many from the excellent works of Todhunter, Bland and other European writers.. The author feels assured that the simple and thorough man- ner in which the subject .of Factoring is treated will meet the approval of teachers, as a thbrough knowledge of this subject is essential to success in common algebraic work. Simple and Quadratic Equations have also received that broad and thorough consideration commensurate with their importance. Originality of matter in a text-book of this kind is not to be expected, but it will be found that several subjects have been treated in a manner more or less original. The reader is applications; Greatest Common 'Divisor, Fractions, Indcte minate and Impossible Problems, Inequalities, Variation, etc. In conclusion, the work is earnestly commended to the use j instructors with the hope and belief that it will be found to 1 teachable, practical and thorough. Batox Rouge, La., J. W. N. August, iSSS. SUGGESTIONS TO TEACHERS. 1st. There are about 2500 examples in this book; while th( are intended to furnish much practice, for it is only by practi( of this kind that students become proficient in algebra, yet thi are inserted rather to give the teacher a good choice from whic to select than to require any student to solve them all. 2d. In passing over the book the .first time, it is recon mended to omit the more difiicult examples and problems. 3d. Thoroughness can be attained only by frequent review and students should be drilled in factoring especially, until th( are as familiar with it as they are with the multiplication tabl CONTENTS. I. PosiTiv!: AXD Negative Numbers. Definitions 7 Addition 11 Subtraction 15 Multiplication 18 Division 21 Involution 22 Evolution 23 Compound Numbers 24 II. General or Arbitrary Numbers. Definitions 26 Numerical Value 30 Statements -31 Addition 33 Subtraction 35 Multiplication 37 Multiplication of Binomials 42 Squaring of Polynomials.... 45 Division 46 Finding the Remainder 52 Four Important Cases 64 III. Factorial Properties and Relation of Numbers. Factoring 58 Factoring of Binomials 51) Factoring of Trinomials 62 Greatest Common Divisor 68 Least" Common Multiple.... 73 IV. Fractions. Definitions and Principles. 76 Reductions 78 Addition and Subtraction. 86 Multiplication 90 Division 92 Complex Fractions 94 IV. Simple Equations. Definitions and Principles. 96 First Form 98 Second Form 99 Third Form 100 Fourth Form 101 Problems 104 Simultaneous Equations 120 Elimination by: (1) Addition and Subtrac- tion 122 (2) Substitution 123 (8) Comparison 124 Problems 129 Indeterminate Equations....l38 Incompatible Equations 143 CONTEXTS. VI. Higher Operations. Involution..' 145 Poweis of Monomials 145 Powers of Binomials 146 Evolution 148 Roots of Monomials 148 Square Roots of Polyno- mials 150 Square Roots of Aritlimct- ical Numbers 152 Cube Roots of Polynomials 154 Cube Roots of Aritiimetical Numbers 156 Fractional and Negative Exponents 159 VII. RadicAi,s. Definitions Ifi4 Reduction 165 To compare Surds 168 To add and subtract Radi- cals 169 To multiply and divide Rad- icals 170 Involution 171 Evolution 172 Square Roots of Binomial .Surds 173 Imaginary Numbers 174 Simplification of Irrational Fractions 176 Properties of Surds 177 Equations containing Rad- icals. 180 VIII. Quadratic Equations. Pure Equations 183 Affected Equations 184 Solution : (1) By Factoring 185 (2) By Completing the Square 185 (3) By a Formula 187 Formation of Equations 189 Character of the Roots 190 Higher Equations 194 Problems 197 Simultaneous Equations 201 Solution : (1) By Substitution ..201 (2; By Reduction 202 Problems 205 IX. Inequalities, Zero and Infinity. Inequalities 212 Application to: (1) Square Root 215 (2) Cube Root 216 Zero and Infinity 217 Principles 218 X. Ratio, Proportion and Variation. Ratio 221 Properties of Ratios 222 Pioportion 225 Properties of Proportionals22fi Variation 230 Principles 232 XI. Series. Series 237 I Geometrical Progression... .244 Arithmetical Progression..237 [ Harmonical Progression.. 251 Answers, 254. ALGEBRA. Chapter I. 'OSITIVE AND NEGATIVE NUMBERS. DEFINITIONS AND ELEMENTARY PRINCIPLES. Art. 1. Quantity is anything, which can be measured ; s distance, time, money. 2. To measure a quantity is to find how many times t contains another known quantity of the same kind, ailed the Unit. Thus, to measure a certain distance, we would find how many imes it contains the unit one yard; and to measure a quantity f corn, we would find how many times it contains the unit one ushel. 3. The object of measuring a quantity is to ascertain low much of it there is, and the result obtained is ailed the Yalue of the quantity. 4. The value of quantity is expressed by prefixing to he name of the unit the number which shows how lany times that unit is contained in the quantity ; as 4yd., 95gal. ALGEBRA. 5. A Known Quantity is one which has been meas- ured, or one whose value is given ; as $7, 5ft., 14da., 11. 6. An Unknown Quantity is one which has not been measured, or one whose value is to be found. 7. A quantity is often measured by means of the re- lation it sustains to other known quantities. The pro- cess of thus measuring or determining a quantity is called Calculation or Deduction. Thus, we may not know how much money D has, but if we know A has $16, B $10 more than A, C twice as much as B, and D as much as A, B and C together, we can measure or de- termine Kow much D has by calculating or deducing as follows: As B has $10 more than A he has $25, as C has twice as much as B he has $50, and as D has as much as A, B and C together he has $90 8. The Relative Values of two or more quantities are the numbers which show how many times they con- tain a common unit. Thus, if the lengths of three lines are 9 in., 11 in., and 15 in., their relative values are the numbers 9, 11, and 16, as these sus- tain the same relation to each other that the lines do. 9. The Relative Values of quantities are often called their Numerical or Abstract Values. 10. In Algebra, number and quantity are generally synonymous terms. 11. Algebra, like Arithmetic, treats of numbers, but it is an extension of Arithmetic in several particui lars. The first is this : Arithmetic treats only of positive numbers, while Algebra treats of both positive and negative numbers. t)EFINiTIONS. 12. A Posilire Number is one whose sign is plus (-f), as +12, -\-7 ; and a Negative Number is one whose sign is minus ( — ), as — 7, — 50. 13. When no sign stands before a number the sign + is understood ; thus, 7 is the same as + 7. But the sign — is never omitted. 14. Two numbers have Like signs when their signs are the same, as + 3 and + 11, or — 6 and — 13 ; and tliey have Unlike signs when their signs are different, as 7 and — 5, or — 6 and -|- 15. 15. The Absolute Value of a quantity is its value without its sign. Thus, the absolute value of -f- 12 is 12, and of — 17, 17. 16. In Algebra, as in Arithmetic, the Fundamental Operations are Addition, Subtraction, Multiplica- tion, Division, Involution and Evolution. But the introduction of negative numbers requires an extension of the meanings of some tei-ms common to arithmetic and algebra. Now, since every such extension of meaning must be consistent with, and deducible from, its arithmetical signification, it is necessary to under- stand first the relation between positive and negative numbers. 17. As symbols of nature the signs + and — denote opposite qualities, conditions, motions, directions or tendencies. It is Immaterial which quality or direction we denote by the sign -|-, provided we denote its opposite by the sign — . Thus: If gain is -|-, then loss is — ; if distance or motion up is +, then distance or motion down is — ; if warmer is +, then colder is — ; if direction to the right is +, then directionjto 10 Algebra. the left is — ; if time after a certain date is +, then time befol-e that date is — ; if time hence is +, then time since is — ; if north latitude is +, then south latitude is — ; if temperature above is +, then temperature below is — ; if more is -[-) then less is — . 18. The Algebraic "Value of a quantity is its value taken in connection with its sign of quality or direction. Thus, the algebraic values of a gain of $4 and a loss of $7, are + $4 and — $7. EXERCISE I. What are the algebraic values of the following : 1. A distance of 5 east and a distance of 8 west.? 2. 65 days after now and 50 days before now.'' 3. 75 steps forward and 22 steps backward } 4. A credit of $11 and a debit of $13? 5. A fall of 3° in temperature and a rise of 5° ? 6 I?'' north latitude and 15" south latitude ? Note. — Whatever + may mean, — means just the opposite. 7. If -|- means east, read the following: A boy went -|- 4 mi. , then — 3 mi., then — 2 rhi. 8. Read the same, calling — south. 9. If + means gain, read the following: A -|- $5, then + $4, then — $7, then — $1. 19, The relation of positive to negative numbers is shown in the following series : -6, -5, -4, -3, -2 , -1, +1, + 2, ^-3, +4, +5, +6, I r I "I "1 III I I I — ^1 — |- The positive numbers begin at and are laid o£E to the right ; and the negative numbers begin at and are laid of£ to the left. ADDITION. 11 ADDITION. 20. The Sum of two or more quantities is the result of combining them into one equivalent quantity. 21. Addition is the operation of finding the sum of two or more quantities. 22. The Parts are the quantities to be added. 23. The Sisn of addition is +. Quantities which are to be added or s-ubtracted are often en- closed in parentheses in order that the signs + and — , which denote positive and negative numbers, may not be confounded with the signs + dnd — which indicate the operations of addi- tion and subtraction. 24. The Sign of Equality is =, which is read equals, or is equal to. ILLUSTRATIONS . 1. A gain of $5 and a gain of $3 are equivalent to a total gain of $8. Hence, -^ $5 added to + $3 = + $8 ; or (+5) + (+3)=.+ 8 (1) 2. A gain of $5 and a loss of $3 are equivalent to a net gain of $2. Hence, + $5 added to — $3 = + $2 ; or (+5) + (-3)==+ 2 (2) 3. A loss of $5 and a gain of $3 are equivalent to a net loss of $2. Hence, — $5 added to + $3 = — $2 ; or (-5) + (+3)=— 2 (3) 4. A loss of $5 and a loss of $3 are equivalent to a total loss of $8. Hence, — $5 added to — $3 = — $8 ; or (- 5) ( +- 3) = - 8 (4) 12 ALGETiRA. 25. Hence, I. To add two numbers with like signs, as in (1) and (4), add their absolute values, and prefix the common sign to the surn. II. To add two numbers with unlike signs, as in (2) and {V), find the difference of their absolute values, and to it -prefix the sign of the larger. 26. Addition in Algebra does not always imply aug- mentation as it does in Arithmetic. For, when we add two numbers of opposite qualities they eliminate each other entirely or in part; and hence, the absolute value of the sum may be less than that of either of the parts. Such a result is called the algebraic su?n, when it is necessary to distinguish it from the arithmetical sum. In Algebra, the terms add and sum should always be understood in an algebraic sense. EXERCISE II. Add together: 1.-9 and +3. 5. + 7 and — 5. 2.-7 and +6. 6. + 8 and - 11. 8. + 5 and + 9. 7.-6 and — 4. 4. + 4 and - 4. 8. + 12 and - 7. 27. Since (-f-8) + (-|-2) is the same as +8+2, and (+8)+(-2) " " " " +8-2, w^e conclude : 1°. The sum of two or more numbers may be indicated by -writingthem one after the other -with their proper signs. 2°. A -plus followed by a plus, + ( + ), is eqnivalent to +, and a -plus followed by a minus, as + ( ), is eqnivalent to — . ADDITION. 13 Find the value of: 9. + 19 — 7. 12. 9 + ,+ 3. 10. — 16 — 4. 13. 6 H 4. 11. — 8 + 17. 14. 12-1 IS- IS. — 6 + 8 — f) + 9. 1°. Solution :— 64-8 = 4-2; +2 — 5= — 3; — 3 + 9 = -f 6. 2°. Solution : — 6 — 5=:— 11; +84-9 — +17; — 11 + 17 = + 6*. 16. +10+3 — 7 + 2—8—9 + 4. 17. +8 — 3 + 9 — 11 + 6 — 4 + 5. 18. — 6 + + 8 H 7 + + 12 H 4. 19. +9 + 6 — 3 + + 8-^ e -\ 3 — 5 + 7. 28. The Complemental Parts of a number are the numbers whose sum equals that number. Thus, the complemental parts of + 4 are + 3 and + 1, or + 7 and — 3, or — 5, + 11 and — 2. Find the number whose complemental parts are : 20. — 6, + 4, + 9. 24. $4, $8, + $3. 21. —7,-8,-4. 25. — $12, — $9, + $5. 22. +9, +8, —3. 26..+ 6ft., + 7ft.,— 15ft. 23. + 11, + 6,-5. 27. + + 3mi, -\ 12mi. 28. A boy made $5, then spent $3, then made $11, then spent $9 ; how much more did he then have than he had at first.? + 5 — 3 + 11—9 = + 4; hence, $4 more. 29. A man paid $8, then received $9, then received $7, then paid $20 ; how much more did he then have, than he had at first? Ans. — $12., 14 ALGEBRA. 30. B was born 5 yr. before A, C 6 yr. before B, D 7 yr. after C, and E 3 yr. after D ; how mucti older is E than A? SI. B was born 7 yr. before A, C 5 yr. after B, D 8 yr. before C, and E 12 yr. after D ; how much older is E than A ? 32. A boy went 12 mi. east, then 9 mi. west, then 17 mi. west, then 15 mi. east ; how far was he then, and in what direction, from the point of starting? 33. The temperature rose 8'^, then fell 6°, then rose 5°, then fell ll^' ; how much higher then was it than at first ? 84. A ship is in latitude 20° north ; if she sails 15° south, then 18° north, then 35° south ; in what lati- tude will she then be ? 39. A Parenthesis ( ) is used to include within it such numbers as are to be considered as forming one number. 35. James has (8 -|- 3) marbles and John (7 — 4); how many have both ? (8 + 3) = ll; 7—4 = 3; 11 + 3 = 14. 36. A ship Sailed (8 — 4 + 5) miles east, and then sailed (6 + 7 — -5) miles west; how far was. she then from the starting point? 37. Heniy had $(9— 2 — 11), then spent $(8 — 10 - 2), then earned $(11 _ 5 + 8) ; how many dollars did he then have? SUBTRACTION. 55 SUBTRACTION. 30. Subtraction is the operation of finding the miss- ing one of the complemental parts of a quantity. 31. The Subtrahend is the given part. 32. The Minuend is the quantity of which the sub- trahend is a part. 33. The Difference is the required part. 34. The Sign of Subtraction is — , which means less, and is written before the subtrahend. ILLUSTRATIONS. 1. If -f- 3 is one part of +5, the other part is + ^i since + 3 + 2 = + 5. Hence, + 5 less + 3 = + 2; or(+5)— (+ 3) = + 2, which is the same as-|-5 — 3 = + 2 (1) 2. If — 3 is one part of -j- 5, the other part is + 8, since — 3 + 8 = + 5- Hence, + 5 less — 3 = + 6 ; or (+ 5) — (— 3) = + 8, which is the same as + ^ + 3 = + 8 (2) 3. If + 3 is one part of — 5, the other part is — S, since -|- 3 — 8 = — 5. Hence, — 5 less + 3 = — 8 ; or (— 5),— (+ 3) = — 8, which is the same as — 5 — 3 := — 8 (3) O 4. If — 3 is one part of — 5, the other part is — 2, since — 3 — 2 = — 5. Hence, — 5 less — 3 = — 2; or ( — 5) — ( — 3) = 2, which is the same as — 5 -f- 3 = — 2 (4) 1 6 ALGEBfeA. 35. From (1) and (3) it is evident that a minus fol- lowed by a plus, as — ( + ), is equivalent to — ; hence, Subtracting a positive number is equivalent to adding •an equal negative number. 36. From <2) and (4) it is evident that a minus fol- lowed by a minus, as — ( — ), is equivalent to + ; hence. Subtracting a negative number is equivalent to adding an equal positive number. 37. Hence, to substract one number from another, change the sign of the subtrahend., or conceive it to be changed, and add it to the minuend. EXERCISE III. 1. One part of -(- 7 is -3, what is the cpmp. part? 2. One part of + 9 is — 5, " " " " " 3. Onepartof— ll.is+ 6, " " " " " 4. Onepartof — 10 is— 12, " " " " " 5. From — 11 take — 4. 7. From 9 take — 13. 6. From 9 take 13. S. From — 7 take + 3. Find the value of: 9. +12 — (+6) 12.'_ 6 — (-t- 8) 10. + 13 — ( — 7) 13. + 8 — (— 9) 11. - 15 - (+ 8) 14. — (_ 7) — (+ 3) How much more 15. Is 9 than 7 .? 16. Is 9 than — 12 .? 17. Is — 9 than —4.?' 18. Is — 14 than -1- 6 .? 19. Is — 160 than — 345 .? 20. Is — 151 than + 89? SUBTRACTION. 1? A man, after traveling two days, was 18 miles east of the point of starting ; how far and in what direc- tion did he go the second day : 21. If he went 10 mi. east the first day? (+18— (+10)= +8, or 8 mi. east.) 22. If he went 20 mi. east the first day ? ( + 18— (+20)= —2, or 2 mi. west.) 23. If he went 7 mi. west the first day .'' James and Henry have the same amount of money, how much more will James have than Henry : 24. If James gains $121 and Henry gains $76 ? + $121— ( + 76)= +$45, or $45 more. 25. If James gains $94 and Henry loses $17.? 26. If James loses $90 and Henry loses $100.? 27. If James loses $225 and Henry gains $148? Moses and Joshua are together; how far will it be from Joshua to Moses : 28. If Moses goes 75 ft. east and Joshua 66 ft. east? +75 ft.— (+66 ft.= +9 ft., or 9 ft. east. 29. If Moses goes 64 ft. east and Joshua 90 ft. east? 30. If Moses goes 25 ft. west and Joshua 4&ft. east? . 31. If Moses goes 37 ft. west and Joshua 72 ft. west? 38. When an expression within a parenthesis is pre- ceded by the sign — , the parenthesis may be removed if the sign of every term within the parenthesis be changed. Thus : ' (1) 8— (6 + 3 — 4)=8 — 6 — 3 + 4 = + 3 (2) _(7 — 5 + 4) +(9 — 6 + 8) = — 7+5 — 4 + 9 — 6+ 8 = + 5 M.E.A. — J IS A.LGEBRA. Find the value of: 32. 18 — 4 — (12 + 4 — 10 — 8) 33. 25 — (6 + 7) — (9 — 5 + 7 — 11). 34. — (16 — 4 -f 7 ) + 10— (8+13— 4 + 1). 35. + (16 — 4 + 11 - 5) — (18 - 13 + G — 20). 39. When parentheses are inclosed within a paren- thesis, they may be removed in succession by removing first, the innermost parenthesis ; next, the innermost of all that remain, and so on. Thus : 9 - [8 + 6-(7 -4 + 5) — 11] = 9- [8 + 6-7+4-5—11] = 9— 8 — 6 + 7 — 4+5+11 = + 14. Find the value of: 36. 16 — [9 — (10 + 4 —7) + 6] —12. 37. 8 — (7 — 2) — [11— (6 + 4 —3)] + 6. 38. —[11 _ |8 — (6 — 8 + 7) + 6i + 3] — 5. 39. 9 — (—8 + 7) — [13 — (7 — 6 + 15)] 40. 19 + 17— (21 — 3) — [16 — f 11 — (10 — 4)]] MULTIPLICATION. 40. In arithmetic, multiplication is the operation of taking one number as many times as there are units in another. 41. The Sign of multiplication is X- 42. According to the previous definition w^e have: + 3 X + 5 = + (+ 5 + 5 + 5) = + (+ 15) + 3 X - 5 = + (- 5 - 5 - 5) = + (__ 15) - 3 X + 5 = - (+ 5 + 5 + 5) = _ (+ 15) -3x-5=._C-5-5_5)=_(_i5 MULTIPLICATION. 19 Hence (35, 36), + 3X +5==+ 15..(1) — 3 X +5 = — 15..(3). + 3 X — 5=— 15..(2) _3 X — 5 = + 15..(4). 43. Hence, 1°. Algebraic Multiplication is the operation of taking one number as many times and in such a manner as is indicated by another. 2°. Like signs produce -|-, as in. (1) and (4); and unlike signs produce — , as in (2) and (3). 44. The Multiplicand is the number to be multi- plied. 45. The Multiplier is the number by which the mul- tiplicand is multiplied. 46. The Product is the result of multiplication. 47. The Factors of a product are the multiplicand and multiplier. EXERCISE LV. Multiply : 1. — 6 by + 4. 4. + 9 by + 6. 2. -(- 7 by — 3. 5. — 13 by — 34. 3. — 8 by — 7. 6. — 17 by + 42. 7. Find thevalue of — 2X+3X— 3X— 2X— 5. Solution: — 2 X+3= — 6, X — 3=+18, X — 2= — 36, X— 5:= + 180. 8. Find the value of + 3 X — 2 X— 3X — 5X + 2. Solution: -f3X— 2 = — 6, X— 3=+ 18, X — 5= — 90, X+2=— 180. 20 ALGEBRA. 48. Hence, if an even number of factors have the sign — , the product will be + ; and if an odd number of factors have the sign — , the product will be — . Find the value of: 9. +2X+2X— 3. 10. + 5 X — 3 X + 4 X — 6. 11. — 6X— 5X— 4X + 3X— 2. 12. _lx— 2X+2X+3X— 4X— 5. 13. (+ 5 + 4) X (+ 8 - 11), or (+ 9) X (— 3). 14. (6—8) X (10 — 4). 15. (7 — 5) X (9-7) x(5 — 9). 16. (6 — 4 + 7) X (3 — 8 + 12 — 20). 49. The Complemental Factors of a number are the numbers which when multiplied together will produce the given number. Thus, the complemental factors of -|- 12 are -j- 4 and + 3, or — 4 and — 3, or + 2, — 2 and — 3. 50. Multiplication may be defined as the operation of finding a number when its complemental factors are given. Find the number whose comp. factors are : 17. + 3, — 6, + 2, + 4. 18. + 2, — 3, — 4, — 5, + o. 19. — 1, — 3, — 5, — 7, — 9. 20. (6 - 2), (7 - 8), (9 — 3), (4 - 14). 21.^ + (6 - 2), -(8 - 5),- (3 - 8), + (6 - 9 + 1). 22.i- (9 - 10), - (7 + 1), _ (5 - 2), - (7 -18 + 4). DIVISION. 21 DIVISION. 51. Division is the operation of finding the missing one of the complemental factors of a number. 52. The Divisor is the given factor. 53. The Dividend is the number of which the divisor is a factor. 54. The Quotient is the required factor. 55. The Sign of division is -h, which is written be- fore the divisor, and is read divided hy or contains. Division is also indicated by writing the divisor under the dividend. + 3 = +4....(l) + 3 =-4.... (2) —3=^+4 ...(3) 56. Since+3X+4=+12, +12- Since+3X— 4=-12, —12 Since— 3X +4= -12, —12 Since— 3X —4= +12, +12 -h 3 = —4. , . .(4). Hence, in division as in multiplication, like signs produce +, and unlike signs — . EXERCISE V. One factor: 1. Of + 30 is + 5, what is the comp. factor? 2. Of — 35 is — 7, what is the comp. factor? 3. Of — 54 is'+ 6, what is the comp. factor? 4. Of + 63 is — 9, what is the comp. factor? What is the quotient : 5. If the dividend is — 156 and the divisor + 13? 6. If the dividend is — 425 and the divisor + 17? 7. It the dividen(^ -^ + 784 and the divisor — 28? 22 ALGEBRA. 8. Divide — 40 by 8. 9. Divide + 60 by — 5. 10. Divide (16 + 24) by (17 - 9) 11. Divide (24 - 60 - 12) by (11 - 17 - 6' How many times : 12. Is 4 contained in — 120.? 13. Is — 3 contained in + 243 ? 14. Is (6 — 21) contained in (7 — 112)? 15. Is — (8 — 2) contained in — (5 — 341)? What is the value of: IG. 164 -= 8.? 17. _38 -^ + 12.? 18. — 160 -^ 25 ? 19. + 110 H (17 — 2)? 20. (18 — 6 + 3) ^ (9 — 4 + 5)? 21. (25 + 6 — 7)^ — (—12.+ 9— 3)? 22. (64 4- 42 — 18) -=- ( + 9 — 21 — 32) INVOLUTION. 57. A Power is the pi-oduct of equal factors. The second power or square is the product of t-wo equal factors, and the third fower or cube is tlie product of three equal factors. Thus, the second power or square of 4 is 4 X *> O'" l^i ^"^ ^^ third power or cube of 4 is 4 X ^ X '^j o'' 64. 58. Involution is the process of finding powers 59. The Sign of involution is an Exponent or Index, which is a small figure placed at the right of, and above, a number. EVOLUTION. 23 An exponent shows the required power of a number. Thus, 5' is the second power of 5 and is equal to 5 X 6> °^ 25; 5' is the third power of 6 and is equal to 5 X S X 5) or 125. EXERCISE vr. 1. What is the fourth power of ■ — 3? (_ 3)* = — 3 X — 3 X — 3 X - - 3 = + 81. 2. What is the fifth power of — 2 .? (— 2)5 =_ 2 X — 2 X — 2 X — 2 X — 2 .= — 32. 60. Hence (48), an even power of a negative num- ber is +, and an odd power of a negative number is — . Find the value of: 3. (—6)3 6. (7— 3)* 4. (— 7)* 7. (3 — 11)3 5. (+5)s 8. (4 — 6 +5)6 9. 5 X 32 X 2 10. 7 X (- 2)3 X (- 5) 11. 4 X(— 5)2 X(— 2)3. Find the value of : 12. 2 X (9 — 3)2 X (4- 3)5. 13. 3 X (2 — 3)5 X (5— 4)''. 14. 7 X (1 — 11)' X (5 — 31 +.6)2 X (— 1)". EVOLUTION. 61. A Root is one of the equal factors of a number. The Sqtiare Root of a number is one of its t-wo equal factors, and the Cube Root of a number is one of its t/iree equal factors. Thus, the square root of 04 is 8, since 8 X 8 = 04. And the cube root of 64 is 4, since 4 X ^ X * = ''*■ 62. Evolution is the process of finding roots, 24 ALGEBRA. 63. The Sign of evolution is y", which is called the Radical Sign. A. figure, called the index, is written over the radical sign to show what root is to be taken. When no figure is written, 2 is understood. Thus, ^J^, or ^/^ means the square root of 16, = i, and J/ g means the cube root of 8, = 3. EXERCISE VII. 1. Find the 4th root of 625. Since 5x5x5x5= 625, y~625 = 5. Find the value of: 2. ]/ 36 5. {/_32. 8- ^''(60 — 12 + 1). * y 81 6. y—U. 9- ?/(— 80+19--64). 4- ^^125 7. K 256. 10. ^/( 1471 — 204 + 29). COMPOUND NUMBERS. 64. A Simple Number is one which involves no indicated operations; as 6, — 17, 325. 65. A Compound Number is one wrhich involves indicated operations ; as 8 — 3,7 — 3 + 4,8 — 5X4. 66. To Simplify a compound number is to perform the indicated operations, and thus reduce it to a simple number. 67. The Terms of a compound number are its com- plemental parts. Hence, to simplify a compound number, we must first sim- plity its terms. Thus, the terms of - 8 + CX3-18-H2— (13 — 4 + 2) + v/^ are + 8, +CX3, — 18-H2, — (13 — 4 + 2),and v^:^ Sim- plifying these, we obtain 8, + 18, — 9, — U and 7, respectively. Now adding these, we obtain + 13. COMPOUND NUMBERS. 25 68. In general, the compound number within a paren- thesis is a term, or the factor of a term ; and hence, if any, should be simplified first. Next, perform the operations indicated by X and -h, as products and quotients are generally terms. Then, perform the operations denoted by -{- and — . EXERCISE VIII. Simplify: 1. 4 X 6 + 3. 7. 12 — 6 X 3 X 5. 2. 5—3X7. 8. 15 -:- 3 + 7 + 4. 3. 2X5 — 4. 9. 25 — 4 X 3 + 6 ^ 2. 4. 3 + 8X6. 10. 25 X ( 12 — 4 + 2 ). 5. 8 -- 2 — 4. 11. ( 16 + 10 — 2 )--4+ 3. 6. 6 + 9 -H 3. 12. ( ]6 + 4)--x/(30 _5). 13. (6^2 + 7 )X [8^(5x3 — 11 ) + 3—G -^2 1. 14. [ 12 — 15 -- 5 — (9— 2X4J^Jof(]2 — l/5 X 4 - -4.) The signs X and -f- indicate opposite operations just as + and — do. Hence, when several numbers are connected by the signs X and -^, as 24 -4- 6 X 3, we may simplify the expression by dividing the product of those preceded by X by the product of those preceded by -f-. In such cases, if there is no sign before the first factor, X is understood. Simplify : 15. G -4- 3 X 2. 17. -^ 8 X 12 X 4 -^ 6. 16. 6 X 3 -^ 2 -=- 4. 18. 24 -4-3 X 5 -^ 10 X 7 ^ 14. Chapter II. GENERAL OR ARBITRARY NUMBERS. ELEMENTARY PRINCIPLES AND DEFINITIONS. 69. The second particular in which Algebra is, an extension of Arithmetic (11) is this: Arithmetic treats of particular or special numbers denoted by figures, and Algebra treats of these and also of general or arbitrary numbers denoted by letters ; as a, b, c, etc. ' 70. A General or Arbitrary Number is one which" may have any reasonable value. ILLUSTRATIONS. 1. If we say a boy is n years old, n is an arbitrary number, and may have any reasonable value like 2, 3, 6, etc. The number 70 would not be a reasonable value, since it would be inconsistent to say a boy 70 years old. 2. From a rope 100 ft. long a piece b ft. long was cut. Here, b is an arbitrary number, and may have any value between and 100, such as 5, 24, 70, etc. The number 110 would not be a reasonable vaJue, as there were only 100 ft. in all. El-EMENTARY PKINCIPLES. 27 3. James has $a, John $6 and both have $7. Here, the general numbers a and b may each have any value, algebraically, provided that their sum shall be 7 ; as, a =^3,& = 4; a = 1, 6 ^6 ; rt = 11, 6 = — 4 ; rt =^ — 13, h = 20; etc. 71. General numbers are multiplied by writing the factors side by side with no sign between them. Thus, 5 times a is written 5a ; m times n, mn; and 6 times ix + y), b (x +1J). 72. Multiplication is sometimes indicated by a dot or point (.) Thus, o. 6. c. means ay(, b y^ c, or abc, 73. A Coefflcient is the known factor of a product which shows how many times the other factor is taken. Thus, in 7a, 7 is the coefficient of d; in 7ay, 7 is the coeffi- cient of ay, or if a be known, 7a is the coefficient of y. When no numerical coefficient is written, i is understood. Thus, ay means lay. 74. A coefficient is the result of addition, and an exponent is the result of multiplication (59). Thus 5a =: a -{- a -{- a -\- a -\- a; If a = 2, 5rt = 2 + 2 -f 2 + 2 + 2 =10; and a'^ = 2 X 2 X 2 X 2 X 2 = 32. EXEIiCISE IX. What is the meaning : 1. Of 4c? Ans. c -{- c -Jf- -{- c, or 4 times C. 2. Of 6a^b^e? Ans. Gxaxaxaxbxh x c 3. Of lab^n ? 6. Of 3n*gy^ .? 4. Of 5a2c*a;2 ? 6. Of 5 (a -f b) x^y? 28 ALGEBRA. 75. Factors expressed by numbers arc called numer- ical, when expressed by letter*, literal factors. 76. In Algebra, the Symbols: 1'^, Of Numbers, are positive and negative figures and letters ; as -|- 5, — 7, -f- a, — n. 2°, Of Operations, are +, — , X, -^-, Vi exponents. 3°, Of Aggregation, are the bar, | ; the vinculum, ; the parenthesis, ( ) ; the brackets, [ ] ; and the brace, j \ . Thus, each of these expressions . signifies that a-\-h is to be treated as one number. [d+i*]; Sa + 6|. 77. An Algebraic Number is any quantity expressed in algebraic symbols of numbers, operations and aggre- gations. -Thus the following are algebraic numbers: 1°, 5a, which is 5 times the number denoted by a. 2°, 4a6 + 3 (6-c)2, which is 4 times the product of the num- bers denoted by a and 6, increased by 3 times the square of the difference between the numbers denoted by 6 and c. 78. Whec an algebraic number is made up of parts connected by the signs + and — , each of these parts is called a Term (67). Thus, in the number « -f- 26c — 4w3, there are three terms, rt, 26c and — 4w3. 79. A Monomial is an algebraic number of one term ; as ia^b, — a^ft, -|- la^c^x. 80. A Binomial is an algebraic number of two terms ; a + 6^ 5a; — 3c«, ELEMENTARY PRINCIPLES. 2& S.li. A Trinomial is an algebraic number of three terms; as a — b -{- c, a^ — Ibc^ -f- mn^x. 82. A Polynomial is an algebraic number of two or more terms •; as a^ — 5qx + 7c* + ^^• A polynomial may be regarded as a compound num- ber (65). 83. Like or Similar Terms are those containing the same powers of the same literal factors ; as a^C, and 7a^c, oa^ (6 — cy and 8a^ (b — c)2. 84. The Degree of a term is the number of its literal factors, or the sum of the exponents of its literal factors. Thus, 3a6 is of the seconddegree, ba^b, 7abc, and 8c* are of the third degree. 85. A Homogeneous Polynomial has all its terms of the same degi-ee ; as 5x^ — 3xy ; ia'c — Bx* + Ix^y^. 86. A Polynomial is arranged according to the pow- ers of some letter, when the exponents of that letter either descend or ascend in order of magnitude. Thus, ax^ -f bx^ — 5cx -}- d is arranged according to the descending powers of a? ; and 5y ^ &y^ + 7y8, according to the ascending powers of y. 87. Two numbers are KeciprocalS of each other when their product is unity; as | and |, a and., The reciprocal of a number is that number inverted. ^0 ALGEBRA. NUMERICAL VALUE. 88. The Numerical Value of an algebraic number is the number obtained by giving a particular value to each ktter, and then performing the operations indicated (68). EXERCISE X. If «=5,6 = 4, c = 3,d = 2,e = l, ^=0, find the numerical values of the following algebraic numbers : 1. 7a + 6& — 3d. e. aa — (&2 _j_ gz _ ad). •2. 86 — 6jj -I- 7c — oa. 7. (abc + bed) -^ cde. 3. ab + ce~ ad +cx. 8. lOc* — 6a^x + Sgs. 4. 0,26 — 62c -I- d^it!. 9. ft3— 63 c2_d2 5. fl3_^ d3_ z,2^e2. ^^3:-^ +7ird" 10. aA—_d^h3j^ _ a3 _ 8. (a _d) _ (a_|_ ft_p) T^ 12. (a^—C") X (il-d— cd)H-(da_ta)_ 89. In Algebra, the Symbols : 1°, Of Eelation, are =, > and <, which stand for IS equal to," "is greater than," and " is less than," respectively. Thus, 5-f3=8;4x 5>18;12--3<5. 20, Of Deduction, are .-. and •.-, which stand for hence and " because," respectively. Thus, 2 + 3==5, .-. 3 = 5-2. 6-3-2 •• 2 + az=6. ''' • ^ STATEMENTS. 31 3°, Of Continuation, are dots, , or dashes, , which are read, " and so on." Thus, 2+3+4 7 ; means 2 + 3 + 4 + 5 + 6 + 7, or 27. 90. The product 1x2x3 10, is indicated by writing the last factor | io . That is, |_6^ = 1x2x3x4x5x6 = 720. STATEMENTS. 91. A Statement is the expression of some relation between numbers. 92. A statement is expressed in Common Language, when the relation and numbers are written in words; and in Algebraic Language, when the relation and numbers ai-e expressed in algebraic symbols. Thus, " eight and seven are fifteen " is a statement expressed in common language, and " 8 + 7 = 15 " is an expression of the same statement in algebraic language. 93. Case I. — To translate, a statement from com- mon, to algebraic language. EXERCISE XI. 1. The sum of two positive numbers is greater than their difference. Solution: To translate this statement, we first as- sume letters to denote the numbers. Let a and h be the letters, then the statement becomes (a+6)> (a -6). 94. Rule. — For the words., substitute the letters and signs which denote the numbers, the relations and the operations to be performed. 32 ALGEBRA. 2. The sum of any two numbers increased by their "difference is equal to two times the greater number. 3. One-half of the sum of any two numbers, dimin- ished by one-half of their difference, is equal to the smaller of the two numbers. 4. The sum of any two numbers multiplied by their difference, is equal to the difference of their squares. 6. The square of the sum of two numbers, is equal to the sum of their squares increased by two times their product. 6. The square of the difference of two numbers, is equal to the sum of their squares diminished by two times their product. 95. Case II. — To translate a statement from algebraic to common language. EXERCISE XII. 1. i(a+&)+i(a — 6)=a. Solution : One-half of the sum of any two numbers, increased by one-half of their difference, is equal to the larger of the two numbers. . 96. Rule. — For the letters representing numbers^ and the signs indicating relations and operations, substitute words. 2. (a +1) fa — 1) = rt2 — 1. 3. (a + 3) {a — 3) = rt2 — 9. 4. (a + &)2 + (a _ t)2 = 2 (a^ + fta). 5. {a + 6)2 _ (a _ &)2 ^ 4^6. ADDITION. 33 0. {a + by =a^ + 3a2b -(- Saft^ _^ js. 7. 4 (a + 6) = 4 a -f 4 6. g a — 6 a 6 3 3 3 9- i/a X 6x c = i/~o' X i/T X i/TT 10. (a X fe X c)» = a» X &^ X c3. ADDITION. 97. Principles. — 1°, TAe sum of two or more num- bers is the same in -whatever order they are added. 1°, Similar terms only may be united into one. EXERCISE Xiri. 1. Add 5a2 .f 3a6 _ 6d; — a^ _ hah + 2^ ; 2a2 -\-M — hxMj\, — 2ab — 3x''y. Solution. For conven- Ba^ + 3rt& — 6d ience, we write similar — a^ — 5a6 + 2d terms in the same column, 2a'' + 3d — 5x^y. and then add the coeffi- — 2ab — Sxhj. dents (25.) Ga^-iab- d - 8x^y. 98. Rule. I. Unite similar terms by -prefixing the sum of the coefficients to the common letters. II. Connect dissimilar terms by their signs. 2. Add 4a — 3& ; 6a + 9& ; — 2a — 6c ; — 86 + 10c. 3. Add 7a2— 66c; Sa^ + 76c; + 66c — 5; -fa^— 7. 4. Add 8a»— 6a?^;— 8a2 + 5a;ll» + 6;— 3a3— 5. 5. Add 5a* + 3a^ — Sa^ ; — Za^ + Sa^ — 11a; + 3a2 — 9a + 7. e. Add 4a;y2 + Zx^y, 5x^y — Ixxj^ ; IxHf- — &x^y. M.E.A. — '3 34 ALGEBRA. 7. Add 9a»b, — 8a^b^ , + Qa^b, — 7ab^, + 9a2i2, — 5a»&, + 4a&3. 8. Add as + 3aH, — SaH — b^, 263 ^ 4^26, — a3 — 2a2ft. 9. Add 3x^y — ixy^ +8x^, + 2xy^ —bx^ + Ix^y, — bx — 8fl?2 + 70,3,2 _ 41/ + eajay — 5ar + iia,2 + 8?/, 10a;2 — 2a? — ix^y^, _ Tir^y _(- 2ic + 3. 10. Add bV ab and 7l/a6^ Ans. 12\/ub 11. Add 6l/«i(c, — 4l/a6c, + Sj/o/i^. 12. Add b{a — &), 8(a — 6 ), 3(a — 6). Add: 13. 61/ ac + Vi/wT ^ bx; i-\/~ac — 4i/"~m -)- Ix. 14. 7l/ a — /> — 81/ « + c; — 5l/a — 6 — 6f' a + c. 15. 3a2 _ 6 (a — m)2 ; — 9«2 + 11 (a _ w)2. 16. 5a;y+ Ax^z — 3.r.?2, 7a7j/ + Qx^z — xz^, and — lOxy -\- 20x^z — 15a!»2. 17. 3a + b ~ e, 9a ~ 66 — ' lOc + /, 8a + 6J — 8c + 3/, 4ffl — 6 -f e + (7, and 6a — 36 — 8c. 18. (a — c)2 + 8« V~6~— icsx; 5(a — c)2 _ lOai/f — 3e3a5. 19. Find the sum of 8a2 _ 3ab + bac — 3(a + I) + 4|/^_ 6ae + 7a6 _ 3V~2^ and - iab — 6a2 + 5(a + i) — i/fT SUBTRACTION. 35 SUBTRACTION. 99. To Reverse a number is to change its sign. 100. Principle . — Subtracting a number is equivalent to adding the reverse of that number (37). 101. Rule. — Change the signs of the numbers to be subtracted, or conceive them, to be changed, and unite the terms as in addition. EXERCISE Xrv. 1. From + 5a2 — 6a6 — U^ take + 3a2 + bob — Solution . — For convenience we write the subtrahend under the + Ba^ — 6a6 — 76*. minuend, placing similar terms + 3a' + S'*'' — ^^^ — S. in the same column, and proceed -\- 2a^ — lla6 + 26* + 6. according to the rule. 2. From x^ -- bx — Q take Zx^ — 5a; + 3. 3. From 36 + 7a — 6c take 3c — 46 + 5a. 4. From 6an — lax + 8 take — 6 — 1 an -\- 5ax. From : 5. 6o6 take iab, 9. m take n. 6. 9x^y take — Sx'y. 10. m take — n. 7. — 7a6^ take 5ab^. 11. — in take n.' 8. — 8xz take — 3xz 12. — m take — n. How much more : 13. Is 5a2x than — Sa'^x i 14. Is — 9xy than 5a;y ? 15. Is 7l/3 than 4l/ 3 .? 16. Is 9 1/ a than— 3l/a? 17. Is a + 6 than a — 6 ? 18. Is 5a — 3a! + 2y than — 5a — 5y + 7aj? 3g ALGEBRA. 19. From 7c take — 3c ? 20. From 7l/ m take — 3l What number must be added : 21. To 5a;2 to make 8a;» ? 22. To a to make & ? 23. To — c to make — a ? 24. To — 5a to make 3a? 25. To a2 _)_ 6 to make a^ —b? 2G. To a? — y to make a? + 2/ ? 27. To x^ —6x + 17 to make — 5x' + 7a5 + 15 .? 28. To 3l/"o" — 6l/^ + 21/5 to make b\/~a _ 81/T— 3l/T.' Reduce to simplest terms (38) : 29. 2a + 3& — (4a — 56 + c). 30. 3a2 — 26a; -\- x^ — (56a; + 4a2 —^x^}, 31. (a + 6) + (6 + c) — (a + c). 32. (3a — 26 — c) — (« — 26 + 3c). 33. (2a;2 +y) — (21/ — 36) — (56 — 3x^). 34. 1 _ (1 _ a;) + il—x + a;2) — (1 — a; + a;* — iB3). 85. 3a — [26 — (5a — 36) — a]. 86. 5aj — [4a? — (3a; — 2a;) — a;]. 37. 7a —•[3a — { 4a — (5a — 2a) | J. 38. 2a; — [7a; — (6a; + 1)] + [8a; — (6a; — 1)]. _ 39. 3V^"2"— [51/2"— (4l/'2~ — 2l/"2~) + 7^2 ]■ 102. If we wish to enclose any number of the terms »i a polynomial within a parenthesis, we may place : 1°. The sign -\- before the wliole ; 2°. The sign — before the whole, provided the sign of every term within the parenthesis be changed. MULTIPLICATION. 37 Thus, a — b -\- c — d = a — (6 — c+ ^)) or (a _6) +(c_d). Change the polynomial a — 6 + 2o — 3d + 4e — 5/ + a; — bmn, into a binomial by enclosing within parentheses : 40. The first two terms, and the last six terms. 41. The first three terms, and the last five terms. 42. The first four terms, and the last four terms. 43. The first five terms, and the last three terms. 44. The positive terms, and the negative terms. 45. From the sum of a and h take their difference. 46. From the difference of a and h take their sum. 47. From the sum oi X? — Zx -{- b and 2a? -{■ Ix — 4, take their difference. • 48. From the difference of a= + 3a6— bW and ¥ + 3a6 — oft", take their sum. MULTIPLICATION. 103. Principles. — 1°. The product of several fac- tors is the safne in whatever order they are multiplied. 2°. Like signs -produce -[-, unlike signs — . 104. Multiplication of monomials. EXERCISE XV. 1. Multiply + Ga^lr't/ by — Wb. Solution. + Ga^Vy X — 4a''6 =: + Gaaabby X — 4aa6 _ _|_ 6 X — 4 aaaadbbby = — 24:a^b^y — — , .2ia^ + ^b" + ^y. 38 ALGEBRA. 8. a2 hya^. 9. a" by a™. 10. as'hy a^\ 11. «"+" by a°-". 12. — 7a2 by 3a". 13. — 5tt2c'' by — 3a"" c6. 105. Rule.— 7b zf^e product of the numerical co- efficients annex the literal factors, giving each an ex- ponent equal to the sum of all its exponents in the several factors. Multiply: 2. 6a* by Sa^. 3. 7a3 by ha'^h. 4. 8a26 by — 3a5,T. 6. — 9a63 by — hh'^xK 6. 8e2n2a;by Ta^a;*. 7. + 9.r2y3jj, by — ^x^z^. Find the value of : 14. + 9o36*c2a; X SaSftc*^^^, 16. — yVft" c X -+ 6a'"6'"'c\ 16. — 6(a + 6)2a;s X — 10(a + lyx^. 17. + n(a + 6)3(a _ 6)4 x — 7(a + ft)2(a — 6)5. 18. — 16(a + 2)3(a + l)na+ Z*) X — 2(tt + 2)* (a+ l)3(a — 6)2. 19. + 7(a — 6 + c)3 X — 5(a — 6 + c)2. 20. + 3a3& X — 2a262c x — SoSa;" x ^hHj. 106. Multiplication of polynomials hy mono- mials. 21. Multiply (8 + 6) by 3. (8 + 5) X 3 = (8 + 5) + (8 + ,5) + (8 + 5) = 8 + 8 + S + 5 +5+5 = 3x8 + 3 X 5 z= 39. 107. Principle.—^ number is multiflied bv multi- plying its complcmental parts. 108. Hence, Kule.— Multiply each term of the poly- nomial by the monomial^ and add the partial products. MULTIPLICATION. 39 22. Multiply Sa^ft _ 5ab^c — 2ax by 4a62. Solution: 3a^b — 5ab^c ~ 2ax. 12a3ft3 _ 20a^b*o — Sa'b^x. Multiply : 23. 5as — 3a2 -|- 4a — 2 by ia^. 24. 7a;* + 5x^ — Sx^ — 8a! by — Sa;^. 25. «»« — 3a2a;2 + 3ax — 1 by ia^x. 26. 3(13 _ Qa'b + lab^ — 2b^ by — Sa^b^. 27. c5 — 6c* 4- &2c3 — b^c^ by a^fisc. 28. — 7a5 _(- 6a*a; — 5a3x'i + ia^x^ by — 2a^x3y. 29. 5(a + 6)3 + 6(a+ 6)2 by 3(a + 6)2. 30. 6a2(a — 6)3 — 7a3(a— 6)* by — 5a*(a— 6)3. 31. 7(a; + y)3(aj_y)2 _ 9{x + y)(x — 2/)5 by 4(a; +y)*( a;— y)5. 82. 50" — Ga"" + 7a^ by 4a2. 33. — 5a» + 12a'' — Sa" by — 3a". 34. _ 6a362 + 10a56° — 7a°65 + 3a^"63° ^^y 9a»b'°. 109. Multiplication of polynomials by poly- nomials. 35. Multiply ( 8 + 5)by (7 — 3). (8 + 5) X C7 - 3) = 13 X (7 - 3), (Art. 107.) = 7 X 13 - 3'X 13 = 7 X ( 8 + 5) - 3 X ( 8 + 6-* = 7X8 + 7X3-3X8-3X3- 110. Principle.— yl number is muitiflied by multi- flying each of its comp. parts by each of the comp. parts of the multiplier. 111. Hence, '^\x\s^.— Multiply each term of one factor by each term of the other, and add the partial products . 40 ALGEBRA. 36. Multiply 6 — 4 + 5 by 8 — 3. 1" Solution. 2° Solution. 6 — 4 + 6=7; 6—4 + 6 8 — 3= 5; 8—3 7 X 6 = 35, Ans. 48 — 32 + 40 — 18+ •12— 15 48 — 50 + 62 — 16 = 35, Ans. Let the next five examples be solved and verified in the same manner. Multiply : 37. 9 +7 — 5 by 7— 2. 38. 5 — 6 +3 — rbyS — 10. 39. 10 — 4 + 3 by 6 — 5 — 4. 40. 2 — 4 + 3 — 7 by 5 — 2 + 3. 41. 22 + 32 — 22 by 32 —21. 42. Multiply — 3a; + 6 — 23;" by 'ii? — 3x. Solution — It is best always to arrange both factors according -|- 6 — 3a; — 2a;' to the ascending or descending — 3a; + 2x^ powers of some one of the — 18a; + 9a;2 -f- 6x^ letters. In this instance they -f- 12a;'' 6x' 4a;' are arranged according lo the — 18a; + 21x' — 4a;*, Ans. ascending power of x. Multiply : 43. a + 6 by a + 6. 44. a~bhy a — i. 45. a + & by a — b. 46. 5a + 3& by 5a + 36. 47. 7c — 6a; by 7c — ex. 48. 8a2 _|_ 7j by 8a2 — 76. 49. ffl2 + 2a6 + 62 by a^ + 2o6 + 6». 50. a;2 _ 4x1/ -j- 4y2 by 002 _ ^^^ _j_ ^^^^ MULTIPLICATION. 41 Multiply : 51. a2 + 6oc + 9c2 by a^ — 6ac + 9c^. 52. x^ -{- xy -{- y^ hy X — y. 53. o2 — Say + 9y^ by a + 3y. 54. as -f- o6 + a* + a2 _|_ 1 by a^ — i. 55. a» — 3a26 + 3a&2 — h^ by a^ — 2a& + 62. 56. a^ -\- ah -\- b' by a^ + ao + c^. 57. aj2 -j- y2 -f 2;3 _ 2a;y — 2xz -\- 2yz hy x ~ y — z. 58. a^ + 2ab + b' + a + b + l.by a + 6 — 1. 59. Ax* — 12x'y -\- 9y^ —2x^+3y + l by 2x^ — 3y + 1. Find the products of: 60. X ~3, X — 2, a; + 1, and a? + 3. 61. X — i, X — 2,x -\- 2, and a? + 4, 62. a; + 2, a; + 3, and a;2 — 6a; + 6. 63. a; — 4, a; + 4, and x^ + 16. 64. a;* — a; + 1, ar" + a; + 1, and a;^ — a;' + 1. 65. a; + a, a; + b, and x -\- c. 66. 5a! — ea;" + a;3 — 24 and x^ + 7 — 4a;. 67. a' + Sab" — fis — Sa^ft and ¥ + a' + 3a¥ + 3a'b. Simplify : 68. (a + by, or (a + 6)(a + 6). 69. (a + 6)3, or (a + 6)(a + 6)(a + 6). 70. (a + 6)*, 01 (a + 6)(a + 6)(a + &)(« + &). 71. (a!2+ 2 xy. 74. (0 a + 6c)3. 72. (ajs— 3a;2)2. 75. (a^ — 2)*. 73. (2 a; + 3)*. 76. (2a^ — 5a^by. Show that each of the following is equal to 0: 77. {n + 6)2 — (a — 6)2— 4a6. 78. (a; + l)^— {x — 1)^ _ 3(ar + 1)2 — 3(0;— 1)2+4. 79. (a+ 1)*— 4(rt + 1)»+ 4(a+ 1)2— 4(a— 1)2 — 4(a_l)3 _(a_i)4. 42 ALGEBRA. MULTIPLICATION OF BINOMIALS WHOSE FIRST TERMS ARE EQUAL. 112. I. — When the second terms are positive. EXERCISE XVI. 1. Multiply X -\- ahy X -^ b. Ans. x^ -\- {a + b) X -\- ab . 113. Hence, Rule. 1°, Square the first term; 2°, multiply the sum of the second terms^ by the first; 3°, multiply the second terms ; 4°, add the three products. 2. Multiply £» + 5 by a; -f 6. 1°, a;X!c=x2;2°, (5 + 6) X a; = + Ux; 3°, 5 X 6 = + 30; 4°, «2 ^ iij. _Y 30, Ans. Multiply : 3. a? + 3 by a? + 7. 8. y + 7 by y + 20. 4. a; + 6 by a; +- 8. 9. aj^ + 11 bya;^ + 5. 6. a; + 7 by a; + 13. 10. x^ + 3rt by x^ + ha. 6. a? + 12 by a? + 5. 11. x^ + 66 by x^ + 76. 7. a; + 9 by a? + 10. 12. a;* + 3» by a;* + low. 114. II. — When the second terms are negative. 13. Multiply (x — 5) by (x ~ d) 1°, xXx = X'; 2°, (- 5 -9) X a: = - lix; 3°, - 5 X - 9 = +46; 4:°,x^ _ 14k + 45. y ~ Shyy— 11. a?^ — 11 by 372— 9_ ■ x^ — 6c by a;2 — 8c. x^ — 10» by a;3 — 7»i. a;4 _ 6a2c by *■*— lia^c. 14. X — - 8 by a: — 10. 19. 15. ■ X — - 12 by a; — 20. 20. 16. X — 13 by X — 7. 21. 17. X — - 30 by a; — 20. 22. 18. X — - 14 by X — 7. 23. MULTIPLICATION. 43 115. III. — When the second terms have unliJee signs, the positive term being the greater. 24. Multiply a? + 8 by a? — 3. 1°, xy,x—x^;2°, (+ 8 — 3) a: = H- 5 a;; 3°, + 8 X — ii = — 24; 4°, x2 _|_ 5a; _ 24, Ans. 25. a? + 9 by a? — 2. 29. x — 11 by a? + 20. 26. a; + 20 by a; — 5. 30. x"^ — 8« by x^ -\- 12rt. 27. a; + 15 by a; — 6. 81. as^ + 10a by x'^ — la. 28. a; — 7 by a; + 11. 32. x^ -\- 126'by x^ — \0h. 116. IV. — When the second tertnn have unliJce signs, the positive term being the less. 33. Multiply X — 11 by aj + 4. 1°, xXx=^x^; 2°, (—11 + 4) a; = — 7 a;; 3°, — 11 X * = — 44; 4°, a;2 — 7 a; — 44, Ans. 34. a? — 9 by a? + 2. 38. a; + 11 by a? — 20. 35. a; — 20 by a; + 5. 39. x^ + 8a by x^ — 12a. 36. ar — 15 by a; + 6. 40. a?2 — 10a by x^ + 7a. 37. a; + 7 by a; — 12. 41. ajs — 126 by a;^ -|- 106. THREE SPECIAL CASES. 117. I. — When, the second terms are equal and positive. 42. Multiply a; -j- a by a; + «. As before, we find x^ + 2aa; + a', Ans.; hence, (a; + a) 2 = a;2 + 2ax + a'. That is, 118. TAe square of the sum of t-wo numbers is equaj. to the sum of their squares -(- t-wice their product. 44 ALGEBRA. 119. II. — When the second terms are equal and negative. 43. Multiply x — ahy x — a. As before we find x^ — 2ax -\- a', hence, (a; — a) 2 = a;2 — 2ax + a=. That is, 120. The square of the difference of two numbers is equal to the sufn of their squares — twice their product. 121. III. — When the second terms are equal and have unlike signs. 44. Multiply X -\- ahy X — a. As before, we find x^ —a^, Ans. ; hence, (x -f a) (z — a) = x2 — a2. That is, 122. The product of the sum and difference of two numbers is equal to the difference of their squares. 123. A Formula is a general truth expressed by symbols. The following examples should be solved by the pre- ceding formulas and rules. Simplify: 46. (5 + 3)2. 53. (a + 9)2. 46. (5 — 3)2. 54. (a — 8)2. 47. (5 4- 3) (5 — 3). 55. ya + 9) (a _ 9). 48. {X + 6)2. 56. (aa _^ qy . 49. (a; — C)2. 57. („= _ 5)2. 60. (x + 6) (a? — 6). 58. (a2 _j_ x\) (a^ — U) 61. (2a? +1)2. 59. (3,/ -1)2. 52,1 (y + 4a) 2. go. (z — 3a) 2. MULTIPI.ICATION. 45 Simplify : 61. (y2 + 6a)2. 68. («2 — 5a) 2. 62. (a;2 + Iby. 69. (a;2 — 96)2. 63. (3fl52 + 2xy. 70. (4a!2 — 3xy. 64. (5a;2 + 4aa;)i'. 71. (5a;2 — 66ir)2. 65. Isax + 6a^y. 72. (3ma! — 7n2)2. 66. (oasft + lab^y. 73. (6M3a; — 10n2a!2)2. 67. (9 + 07) (9 — X) 74. (8a6 + 262) (8a6 — 262). 75. (5a;2 — 3y2) (5aj2 +3y2). 76. (7a3 + 4a6) (Ta^ — 4a6). 77. 6 by 2a^ — 2y^. 66. fflS — 3a*62 _(_ 3a2i,4 _ ^s by as — So^ft + Sab" — 67. a;* + a;^^^ + y* by x^ + xy + y'. 68. 16a864 + 4a666 + a*b^ by 4«*&2 — 2as&s ^ a^i**. 69. 16ai2ft8c4 _ a4&8ci2 by 2a362e 4 o&^cs. 60. aje — 2a;3 + 1 by a;2 — 2a; + 1. 61. a* — 2*262 _^ &4 by «« + 2a6 + 62. 62. ajs + 3/3 + a3 — 3a;7/2( by .r + 2/ + 2. 63. zi — 24»3 + 206^2 — 744a + 945 by g^ — Sz+ 15. 64. 5a7 — 22a66 + 12a562 _ 6a*6' — 4a»64 + 8a265 by a» _ 4a2& + 263. 65. a;* + a;* + a!2 + a; + 1 by a;2 _ 2ar +1. Divide (without simplifying terms): 66. ix + a)2 + 7 (« + a) + 12 by ix + a) + 3. 67- {X — c)2 — 9 (a; — c) + 20 by {x — c) — 4. 68. (a; + n)2 — 3 (a; + «) — 28 by ( a; + ») — 7. 69. (a; + n)2 _ 25a2 by (ar + w) + 5a. 70. (ar + a)3 — 64 by (a; + a) — 4. 71. (a; + a — c)2 + 5 (a; + a — c) — 24 by (a; + a — c) — 3. 72. {X + c)* - (* + nY by (a; + c)" + {x -\- ny. 52 ALGEBRA. ' FINDING THE REMAINDER WITHOUT DIVIDING. 136. In certain cases the remainder may be found without performing the operation of division. The two following cases are the bases of so many important for- mulas and principles, that they are here presented un- der the forms of Propositions. 137. Prop. I. — If a quantity containing y, can be divided by y — a so as to leave a remainder which does not contain y, the remainder will be the same as the quantity itself with a written in the place of y. Thus, if we divide y — a into y^ -\. Ay -\- B, the remainder will be a" -\- Aa -}- B. Proof. Let y^ •+ my'^ -i- ny + r represent the quan- tity to be divided by y — a. Denote the quotient by Q, and the remainder, which does not contain y, by B. Then, y» +my' +ny + r=Q{y — a) + B. Now, writing a in the place of y, we have a^ +ma^ +na + r=Qx + B, or a* + TOflZ -{- na + r = B. 138. Prop. II. — Jf a quantity containing y, can be divided by y + a so as to leave a remainder which does not contain y, the remainder will be the same as the quantity itself with — a written in the f lace ofy. Thus, if we divide y -{- a into y^ + Ay -j- B, the remainder will be a' Aa + B. Proof. Let y^ + my^ + ny + r represent the quan- tity to be divided hyy + a. Denote the quotient by Q, and the remainder, which does not contain y, by B, DIVISION. 53 Then 3,3 _|_ tny^ ^ny+r = Q{y + «) + ig. Writing — a in the place of y, we have {-ay-\-m{—ayJrK—«') + i'=Qi—a'-\-a) + R; or, — a^ -{- ma^ — na -\- r = It. EXERCISE XIX. 139. In solving the following examples, the student should prove the work by division, until the principle is understood. 1. If y* — 5y2 ,|_ 9^ _ 7 be divided hy y — 2, what Will be the remainder.? Writing 2 in the place of y, we have (137) (2)3 — 5(2)2 _^ 9(2) _ 7 = — 1, Ans. 2. If the same quantity be divided hy y -\- 2, what will be the remainder ? Writing — 2 in the place of y, we have (138) (— 2)s - 5(— 2)2 + 9(— 2) — 7 = — 53, Ans. 3. If 3/* -\- a^ be divided hy y — ff, what will be the remainder.? Ans. (137), (ay -\- a^, or 2a^. 4. If y^ -{- a^ be divided hy y -\- a, what will be the remainder.? Ans. (138), (— ay + a», orO. Find the remainder 5. When y^ — 6y + 5 is divided by y — 1. 6. y2^ey + o " " 2/+1. 7. y2_2y — 3o " " y — 5. 8. » '* y2—2y — S6 " " y + 5. 9. ojs _ ex^ _|- iLu _ 6 is divided by X ~ 2. 10. ajs _j_ 6a;2 + lla? + 6 " " a? + 2, 11. a;5 -\- 32 is divided by on — 2. 12. a;s _)- 32 " " a? -)- 2. 54 Algebra. 16. jf2 — a2 17. x^ -\- a^ 18. x^ + as 19. ar* + a* 20. a?* — a* 21. yB 4- as 22. y5 flS 23. 2/6 + ae 24. ^6 — a^ Find the remainder: 13. When y^ — St/^ + 2y is divided by y — 3, 14. " ys — 3y2 +21/ " " 2/ + 3. One quantity is said to be divisible' by another when the remainder is equal to 0. 15. Is 3/2 — a2 divisible by 2/ — a? y + a? " X — a? " a; + a? " X -\- a, or a; — a? " a; + a, or a; — a ? " y-^a, or y — a? " 3/ + a, ory — a.? " y + a, or y — a? " 2/ + a, or 2/ — a.? FOUR IMPORTANT CASES. 140. There are some cases in Division which occui so often in algebraic operations that they should be carefully noticed and remembered. In considering these cases, we shall employ 2» to de- note an even number, and 2re + 1 to denote an odd number. Let it be observed that (— «)^° = a'° ( 48 ), and (— a)2" + ' = (— ay° K —a = — a''' + '. 141. Case I. — The difference of two equal odd powers of any two numbers is divisible by the dif- ference of the nutnbers. § That is, 3/'° +^ — a^" + ? is divisible by 2/ — «• For, if we divide the former by the latter, and con- tinue the operation until we obtain a remainder which does not contain ^,rthat remainder will be (137) t)ivisi6}4. 55 a»" + i _ a2»+i ; which = 0. a5 _ &6 "^^^S' ^_^ = «* + «'& + a*&* + ai^ +^*' We may write the quotient by inspection by observ- ing that: 1°. The number of terms in the quotient is equal to the exponent of the powers, and the signs of all the terms are plus . 2°. The first term of the quotient is obtained by di- viding the first term of the dividend by the first term of the divisor. 3°. Each succeeding term of the quotient may be found by dividing the preceding term by the first term of the divisor, and multiplying the result by the second term of the divisor (disregarding the sign). EXERCISE XX. Write by inspection the results of the following : 1. (a»— 1) ^(a — 1). 5. (n*- 16)^(w - 2). 2. (63_ 8) -=- (6 — 2). 6. (8 — 27a;3) ^ (2 - 3a;). S. (a'— 6'') -f- (a — 6). 7. (x^—y^) -^ (so — y). 4. (&4— a;4) -4- (6 — x). 8. (32a6_ i) -^ (2a — 1). 9. (25a*aj2— 9) -^ (^5a^x— 3). 10. (1 — 64a6c9) H- ( 1 — 4a2cs). 11. (216 — 64) H- (6 — 4). 12. (1331 — 343c*) -r- (11 — 7c). 142. Cash II. The sum of two equal odd powers of two numbers is divisible by the sum, of the numbers. That is, y*° + ^ -{- a"" + ' is divisible by y -|- * 56 ALGEBRA. For (138), the remaincier after division is ( — «)'° + ' + a'" + \ or (140) — ft'" + ' -[- a"" + \ which = 0. Thus, "' + ^^ = a* — «=>& + ft^&a _ «&3 _|_ J4. a + b The quotient may be found as in Case I, but the signs are alternately plus and minus. Write by inspection the results of the following; 13. («3 +a;3)_=-(rt4-a;). 17. (U + c«) ^ (4 + c^). 14. (x^ + l)^(a?+l). 18. (a8-|-i25) -r- {a^ -\- 5). 15. ( 8 + «3) -J- (2+a). 19. (aS + 3265) --(ft+26). 16. {a'' + V'') -i- {a + b). 20. (343+ a^) -=-(7+ a). 2i. (64a3 + lOOOd") h- (4a -|- 10^2 ). 22. (1331 + 216) -- (11 + 6). 23. (1024a6 _j- 24366) ^ (4^ _^ 3^), ^4. [(a + &)3 + ajs] -f- [(a + &) + a;]. 143. Case III. — Th? difference of two eqttal even pow rs of two numbers is divisible by the sum and also by the difference of the numbers. That is, ^■' ; — a^ is divisible by j^ + a and y — a. • For (138), the remainder after dividing hy y -\- a is (— ay- — a?'\ which — 0. and (137), the remainder after dividing by (y — a) is («)'° — («)'% which = 0. When the divisor is the difference of the numbers we find the quotient as in Case I ; and when the divisor is the sum of the numbers, we find the quotient as in Case II, biVisioN. 57 Write by inspection the results of the following : 25. (a* — 1) -f- (ft _ 1). 30. (a?* — 81) -=- (a? — 3). 26. (a* — 1) -=- (a 4- l). gl. (a^x'S— 1)^ {ax^ — 1). 27. (a* — 68) -^ (a + 6). 32. (64 ~ c^) h- (2 + e). 28. (a6—6« )--(«— 6). 33. (64 — c°) -h (2 — f). 29. (16a!* — l)-f-(2a7+l). 34. (1— r( + y). = (a — 2/) (6 + 2')- 18. Find the factors of x^ + 9x + 20. a;2 + 9a; + 20 = (x^ + ix) + (5a; + 4 X 6) = a;(a; + 4) + S(x + i) = (x + 5) (a; + 4). Express the following in terms of their factors : 19. x'' + ax + ex + ac. 27. x" -{- (a -\- h)x + ab. 20. y" + % + ny + bn. 28. x^ -\- Ix -{- 12. 21. ab -\-bm — an — mn. 29. y^ _ (m _|_ n)y -\- mn. 22. a& — bm — a« + mm. 30. y^ — 8y -|- 15. 23. y2 — ay — by -\- ab. 31. a?2 -|- (a — b)x — aft. 24. «2 j^cz — bz — be. 32. x^ + Bx — 28. 25. a2j2 _^ b^n—a^m—mn. 33. i/^ — (m — n)y — m». 26. oo2 — c^i/a _a6 + by^. 34. y^ — 3y — 10. FACTORING OF BINOMIALS. 148. Wftcw the binomial is in the form of two squares with a minus sign between tJtem. 1. Find the factors of a;2 — 25. The square root of x^ is x, and the square root of 25 is 6, Hence (122), a;2 _ 25 = (.» + 5) (a? — 5). '30 AI.GEBIIA. Rule. — Find the square root of the first, and also of the second term. Take the sum of these roots for one factor^ and the difference of the roots for the other factor. EXERCISE XXII. Find the factors of : 2. flj2 — 81. 9. fflZjz _ 4_ 3. a;2 _49. 10. IGa^ — m*. 4. 4i/2 — 25. 11. 9a262 _ i. 5. 81a*&2 _ 144. j2. looajs^z — 121a2&2 6. 121a«c2_4-t;2. 13 64c6ar2 _ 25^8. 7. 36a;8 — 169. 14., 49a2y6 _ 9a6y2. 8. a* — 40062 15. i _ 225a864c2. 149. A binomial may often be resolved into three or more factors. Thus : aS _ ftS ^ (^4 _|_ J4) (^4 _ J4) = (a4 _|_ J,) (^2 _j^ ^,2) (a2 _ j2) = (a* + &*) (a2 + &2) (a _^ jj (a— 6). Find all the factoi-s of: 16. x^ — 1. 18. c* —625. 17. 1— 16a8. 19. ai6_i,i6. . 20. Find the factors of {a + 6)2 — (c + <7)2. (a4.i)2_ (c+rf) = {(a + i) + (<: + rf)||(.«+^)_(<.^^| = {« + * + c + =. 21. x-^z^ + 38.r.s6 _|_ sei^c. 12. u^-\- 10rt3c2 + 25e*. 22. 64«s _)_ iGOdft^ + 100«2. 154. When the. first term of the triuomial is not a perfect square, EXERCISE XXV. 1. Find the factors of 6x^ + 5x — 4. The first terms of the two factors might be 6* and .V, or 2x and Sx, since 6x X * = ^"^t ^"'^ ^-^ + * 2* X 3* = Gx^-. 2x — 1 Likewise the last terms of the two factors 8.v might be 4 and 1, or 2 and 2 (disregarding — 3v the signs). ^ 5x From these it is necessary to select such as will produce the middle term of the trinomial, and they are found by trial to be 3x and 2x, and + 4 and — 1. .-. 6*2 4. 5« _ 4 = (3.V 4- 4) (2x — 1). 2. Find the factors of 18a;2 _ 9j? — 5. 155. The following method of solution is often pref- erable, which, for the present, is given without demon- stration. Dividing the first term and multiplying the last terra by 18, the coefficient of x-, x^ — dx — 90. Factoring (152), (x — 15) (x + 6). 66 ALGEBIIA. Multiplying x in each factor hy 18, the coefficient of a;2, (18x — 15) (t8a; + 6). Dividing by 18, or 6 X 3, (18a; ;^ 15) ( 1 8a: + 6), 3 6 or (6a; — 5) (3a; + 1). Find the factors of: 3. 6a;2 + 11a? + 3. 4. 12a;2 _ nx — AO. 5. 10a!2 — l\x + 3. e. 20a!2 _ 41a; + 20. 7. 12a!2 4- iTar — 7. 8. b%x^ — 11a; — 15. 9. 18a;2 — 3a; — 10. 10. 24a;2 — 14a; — 5. 11. 12a;2 + 23.'i; + 10. 12. 24a;2 + a; — 3. 13. 6a;2 + 13a;3 — qx^ —py^ + qy^. 68 ALGKBUA. 37. 7a;* — 49d?2 — 420. 38. x^ — x"^ -{- x — \. 39. ox^ — 6aa;2 — 40a. 40. x^ — ^x^ — a;^ -f 3,r. 41. 24:X^ + Mx^ —80a;. 42. 4a2 -f 8o& — 3ac — 66c. 43. 24a3 — 34a2 — 80a. 44. (a;^ + 2a;)2 — (2a; + 1)2. 45. a*a;2 — 5a2a; — 24. 46. x^ + aa;2 — a'^x — a». 47. a' + «6 4- a3. 48. 4a262 — (a2 + 62 — c2)2. 49. 3x^y^ — xy^ — 2y^. 50. (a?2 — y2 —^2)2 _4!/222. GREATEST COMMON DIVISOR. 159. The Factors of a number are sometimes called its Divisors, since it maybe divided by any one of them. 160. A Common Divisor of two or more numbers IS any factor common to those numbers. 161. The Greatest Common Divisor of two or more numbers, denoted by G. C. D., is the greatest factor common to those numbers. 162. When the num bers can be resolved into factors by inspection. EXERCISE XXVIII. 1. Find the G. C. D. of: 28a36*, 42a5ft3^ and 70a266c. 28a3&* = 22 X 7 X a' X &* 42o563 = 2 X 3 X 7 X «** X 6' 70a265c = 2 X 5 X 7 X a^ X 6" X c. .. theG. C. D. = 2X 7X«^ X 6^ = l'*«^6^ 2. Find the G. CD. of: 6a(«;2 4- 8^ + 15), Sa(^x^ — 2x ~ 15), A:a^{x^ — 9). 6a(x2 + 8a; + 15) == 2 X 3 X « X (^ + 3) (^ + 5) 8a(a;2 — 2x — 15; =2^ x a X (a;+ 3) (a — 5) ia^{x^— 9) = 22 X a2 X (^ + 3) {x - 3) . . the G.t:. D. = 2 X a X (.'■ + 3) = 2ax + 6a. GREATEST COMMON DIVISOR. 69 Hence, Rule: 1°, Resolve each number into its lowest factors. 2**, Select from these the lowest power of each common factor, and find the product of these powers. FindtheG. C. D. of:' 3. 4asj and Qa^h^x. i. a^b^c and b^c^x. 5. Ub^xy and Sab^x'^. 6. 5d^n^ and la^bn. 7. ISc^n^x and 12b''o*x. 8. 52a'^b^c and lllbv^x. 9. 63c*a;32;2 and ^la^x^z^. 10.' a{a + 6)3 and a^{a + ft)2. 11. Sasft^c*. 16a363c3, and 20a*b*c^ 12. 40j;*2/3c2^ 64ic33/5c*, and 120fl!22/c3. 13. (jf 4- 2/)2 and a;^ —y^. 14. 4^2 — 1 and (2x— 1)2. 15. (rt — by and a2 _ J2. jg. „* _ j* and a^ —h^ 17. ajs — c3 and a? — c. 18. 4(a; — i/)* and 6(0? — j/)^. 19. x^ -\-%x -\- 20 and a;2 — 16. 20. a;2 _ 9a! + 20 and a:2 — 25. 21. a!2 + 3a; — 54 and a;2 — 81. 22. a!2 — 4a; — 60 and a;2 — 36. 23. a;2 + 7a; — 18 and a;2 + 12a; + 27. 24. a;2 + 6a; — 16 and a;2 — 6a; + 8. 26. a;2 — 2a; — 35, x^ —2x + 14, and a;2 — 49. 26. a;2 + 11a; — 26, a;2 + 9a; —52, and a;2 + 15a; -f 26. 163. When one of the numbers can be factored and the other cannot, we may ascertain by trial which of the factors of the first are contained in the second. 27. a;2 — 7a; + 10 and a;' — 6a;2 + 11a; — 6. x^ 7a; + 10 =(x — 2) {x — 5). Now by trial we find that x — 2 will divide a' — 6a;2 -\- llx — 6, andx— 5 will not. Hence, X — 2 is the G. C. D. 70 ALGEBRA. 28. x^ — y^ and x^ + y^. 29. x^ -\- x~& and 2x^ — 12x^ + 21a! — 10. 30. a2 _ 7(ij _^ 1262 and ba^ — ISa^h -\- llab^ — 663. 164. When the numbers can not be readily re- solved into their factors by inspection. 31. Find the G. C D. of 70p and IBSj), where p may be regarded as representing a polynomial. The G. C. D. in this case is evidently 5p, which we obtain by inspection; but we shall find this G, Ct D. by the two follow- ing methods, for the purpose of illustrating three important principles. 1°. We may divide the greater number by the less, then divide the lebs number by the remainder, and continue to divide the last divisor by the last remainder until nothing re- mains ; the last divisor, as 5/, will be the G. C. D. required. FIRST SOLUTION. 70iS,)l«5iiS(2 140/> 26;*) 70;* (2 50/ 20;*) 25/1(1 20/ 6/) 20/ (4 2°. We may divide both num- bers by any common factor as 5, and set that factor aside for one of the factors of the G. C D. 3°. We may multiply or divide either dividend or divisor by any factor which is not a factor of the other. Thus, we divide lip by 2, be- cause 2 is not a factor of Sip ; again we divide S3p by 3, because 3 is not a factor of Up. ABBREVIATED METHOD. 5 I 70p, W5p . lip, 33p. I4P ^ 2 = 7p; SSp-i-S — llp 1p) Up{l Jp_ ip) rp (1 ip 3i)4i)(l p)Sp(S Sp greatest commo.v divisor. 71 165. Demonstration of the 1° principle. Evidently, 166p — 2(702?) = 25p. That is, l&5p, and — 2(70^;) are the c6mp. parts of 25/) ; hence, any number which will divide each of the two former will divide the latter (130). In general^ any number which will divide the divi- dend and divisor will also divide the divisor and re- mainder. Hence, it follows that any number: which will divide 16flp and 70p, will divide 70p and 25p; \vhich will divide lOp and 25p, will divide 252? and lOp; which will divide 25j) and lOp, will divide lOp and S^). Now 5p is the G. C. D. of lOp and bp, hence it is the G. C. D. of 165i> and 70?. The same reasoning may be applied to any two' numbeis whatever. 32. Find the G- C. D. of x^ + 7a;2 + llx + 14 and x^ — x — 6. x^ + 7x2 _|. i7j; _|_ 14 I a:2 _ x —6. a;3 _ a:2 _ 6a; a; + 8 8x2 _j. 23x + 14 8x2 — ga _ 48 31x + 62 = 31(x + 2). Rejecting 31 by Princ. 3°, x + 2 becomes our next divisor. x"— X— 6 X + 2 x2 +2x X — 3 — 3x — 6 — 3x — 6 Hence, x + 2 is the G. C. D. I. Find the G. C. D. of a?^ + y^ and x^ —y^. (j;3 _|_ j,3) ^ ^a;2 — V^ ) = X (quo.) and xxj'^ + »/= or y^ (X + 2/) (rem.) Rejecting rj"-, Princ. 3°, (x» — y^) ^ (x + V) =x — y. Hence, x + y is the G. C D. 72 ALGEBRA. All the principles involved in finding the G. C. D. are illustrated in the solution of the next example. 34. Find the G. C. D. of 4xS + 9x* +2xS — 2x^ — 4x and 6x* + 10*3 _ 2*2 -|- Ax Since the factor x is common to both, we divide out by x, and retain the a; as a factor of the G. ''. D. Again, since 2 is a factor of the second number and not of the first, we divide the second by 2 and throw it away: The two numbers thus become ' ix* + 9x3 _|_ 2a;2 _ 2x — 4 and 3x^ + Sx^ — x + 2. Now, in order that the first may be divisible by the second, we multiply it by 3, and then proceed thus: 12x« + 27x3 _j_ ^2 _ 6x — 12 |_3x3_-)-_5x2 —x + 2. V2.r* + 20x3 — 4x2 _|_ gj; 4;,; ^ 7 ' 7x3 ^ 10x2 — 14x — 12. We multiply this remainder by'3 to make it divisible by the divisor and continue thus.: 21x3 + 30x2 _ 423: _ 3G 21x3 _|_ 353-2 _ 73; _(_ 14 — 5x2 _ 35J. _ 50 or — 5(,/j2 -\- 7x + 10) Now rejecting the factor — 5, we continue thus : 3x3 _|_ 5j;2 _ 3; -)- 2 I'j-^ + Tx + 10 3x3 _|_ 21x2 _[_ 3o .r 3x — 16 ' — 16x2 — Six + 2 — 16x2 _ ii2j; __ 160 + 81x + 162 or 81 (x + 2) Rejecting the factor SI, we proceed thus: x2 -|- 7x + 10 |x 4- 2 x2 + 2x x + 5 5x + 10 5x + 10 Hence, x(x + 2) or x2 -f- 2x is the G. C. I>, LEAST COMMON MUI.TU'LE. 73 Find theG.C. D. of: 35. a?2 + 20a7 + 91 and x^ — 10a; — 119. 36. a;2 + 28a? + 187 and a;^ — 2,» — 143. 37. 3a;2 _ 6aj + 3 and 6x^ + 6,» — 12. 38. 10x2 -|- ar — 60 and 6a!2 + a; — 35. 39. 28a;2 _ 19a; + 3 and 20a;2 + 7a; — 3. 40. a» + 3a2& + 3ab^ + &» and «» + h«. 41. 3a;3 -j- Gx' — 24a; and 6a;3 — 96a;. 42. x^ — a^ + 2xy + 3^2 and x^ + «2 + iax — 2/2. 43. 2/3 — ?)y^z — Ays^ -f 2s» and 7^2 _|_ lOf/^ + 3^2. 44. 7a2 — 23a6 + 662 and ha^ — 18a2& + llfl62 _ efts. 45. a;* — Sx* + 21a;2 — 20a; + 4 and 2,«3 — 12a;2 + 21a;— 10. 46 2a;4 — 12a;S + 19a;2 _ 6.r + 9 and 4x3 _ 18,052 _(- 19a; — 3. 47. 6a;* -|- a;^ — a; and 4a;3 — 6a;2 — 4.» + 3. 48. 2a;* + lla;3, — 13a;2 — 99a; ^45 and %v^ — 7a;2 — 46a; — 21. 40. 3a;5 — \0x^ + 15a; + 8 and x^ — 2.«* — 6a;3 + 4a;2 + 13a; + 6. LEAST COMMON MULTIPLE. 166. A Multiple of a numbei- is a number of which the given number is a factor. 167. A Common Multiple of two or more num- bers is a number of which each of the given numbers is a factor. 168. The Least Common Multiple of two or more numbers, denoted by L. C, M., is the least number of which each of the given numbers is a factor. 74 ALGEBRA. 169. Principle.' — The L. C. M. of two or more nutn- bers is Ihe least number that contains the comflemental factors of those numbers. 170. When the nwmhers can he resolved into their factors hy inspection. EXERCISE XXIX. 1. Find the L. C. M. of loa2&3 and 20a6*c. 15«2^3 ^ 3 ^ 5 -y; ^2 -^ J3 2Cnb^c = 22 X 5 X « X ** X c. .-. L. C. M. = 2= X 3 X 5 X «' X ** X c = GOaH'c. 2. Find the L. C. M. of 8(a?2 _ 9), 9(a;2 + 6a? + 9), and 5(i»2 — 6a; + 9). 8(j;2 — 9) =2^x + 3) (x — 3). 9(^2 + 6x + 9) = 32(x + 3)2; 5(x' _ 6x + 9) = 5 (x-3)2. .-. L. C. M. = 23 X S^X 5X (« + 3)2 (X — 3)2 = 360 (x' — 18x2 + 81)'. Hence, Rule. 1°, Separate each number into its lowest Jactors. 2°, Select the highest power of each factor and find the -product of these powers. Find the L. C. M. of : 3. 6a36 and Sa^b^c. 4. a^b^c and b^c^x. 6. 12c2a;3 and ox^y^. 6. a^^c^ and a^'b^c*. 7. lOfes^jj^ and 1562a;3,,. g. ^^(aj ^ j)3 gnj a(^_|_ j)3 9. 16a*62c, 24a363c2, and 12a2&c5. 10. 25n^mx, dn'^x*!/, and mSa;^. 11. a;2 + 3a? -f- 2 and x^ — 4. 12. a2 — 62 and (a —by. 13. aj2 — 5* + 6 and x^ - 9. 14. 9 — 4*2 and (3 + 2x)^. LEAST COMMON MULTIPLE. 75 15. x'^ + Sjj + 16 and x^ — 16. 16. 6(0;— 1)2 and 9(a;— 1)3. 17. x2 — 9, j;2 ^ 2x — 15 and a;^ — ix — 21. 18. x^ -{- X— 30, a?2 + 12a! + 36 and x^ — 10* + 25. 19. o(a + by (a — b), 6(ffl + by (a — by and 3(a + &) (a— 6)3. 20. 4(a + & 4- c)2(rt + 6), and 6a(a + b+ o){a + &^3 («-&). 171, When the numbers cannot be easily factored by inspection. 21. Find the L. CM. of a^b^c and a'bc^x. The L. C. M. in this case is a^b^c^x, which we obtain by in- spection. For the purpose of illustrating a principle let us solve the problem thus : The product of the numbers is a^h^c*x. The G. C. D. of the numbers is a^bc. a'-'b^c*x -=- a^bc = n'b'c'x, Ans, 172. Principle. T&e L. C. M. o/ iwo numbers is equal to their product divided by their G. C D. Hence, Rule. i°. Find the G. C. D. of the two num bers. 2°. Divide one of the numbers by their G. C. D., and multiply the other -by the quotient. Find the L. C. M. of : 22. a;2 — 23a; + 112 and x^ ^ x — 56. 23. aj2 .^ 28a! + 187 and a;2 — 2a; — 143. 24. a* — lab -f- fts and a* — h'^. 25. x^ +*6a;2 + 11a; + 6 and x^.-\- Ix^ + 14a; + 8. 26. a;* -f 13a;2 + 36 and a;* — 13.r2 + 36. 27. a;3 + 2x^y — xy^ — 2y^ and x^ — 2x^y — xy^ -f 2yK Chapter IV. FRACTIONS. DEFINITIONS. 173. A Common Fraction is one number written over another with a line between them. Thus, A, ^, *!+_?; which are read 3 lOths, a 6ths, (a + b) 10 b a — c (a — c)ths. 174. The Terms of a fraction are the two numbers by which the fraction is expressed ; the Numerator is the term above the line, the Denominator is the term below the line. 175. A Proper Fraction is one whose numerator is less than the denominator ; and an Improper Fraction is one whose numerator is equal to or exceeds the denominator. 176. An Entire Number or Integer is a number which does not contam a fraction ; and a Mixed Number is one which contains both an entire and a fractional part. 177. A Complex Fraction is one having a fraction in either of its terms ; and a Compound Fraction is a fraction of a fraction. FUNDAMENTAL PRINCIPLES. 178. Every number, whether entire or fractional, may be regarded as representing a single thing, or the coefficient of a single thing. "FRACTIONS. 77 Thus, as a single thing, 5 means one 6; and %, one %. As a coefficient of a single thing, 5 means 5 oues; and %, % of one. CASE I. 179. A fraction, regarded as a single thing, is a quotient, the numerator being the dividend.^ and the denominator the divisor. Hence, 1°. The value of a fraction is the quotient obtained by dividing the numerator by the denominator. Thus, the value of is 4; of z. _ a — x. 3 - a + X ' 2°. The dividing line has the force of a vinculum or parenthesis, and the sign before it is the sign of the fraction. 3°. The 1°, 2° and 3° principles of Art. 126, are applicable to fractions, if numerator be substituted for dividend, denominator for divisor, and fraction for quotient. Thus, lo. l^ = 2;but^-?, or J1-^ = 2X3 = 6. 16 16 X* 16 -=-4 5°. If the signs of the terms are alike the fraction is plus ; if unlike, minus. Thus, ±J or Zli = + 2 and ±J or ^-^ = - 2. '4.4. —4 — 4 + i 6°. Changing the sign of either term changes the sign of the fraction, but changing the signs of both tejrms does not change the sign of the fraction. '78 ALGEBRA. CASE II. 180. A fraction, regarded as a CO-efflcient, rep- resents one or more of the equal parts of unity. For (179)» -^ = « -f- ?; = « times _, Hence, = — of one = a times 1 /Ah of one b Hence, the denominator shows into how many equal parts the unit is divided, and the numerator shows how many of these parts are taken. NOTE. — For practical and business purposes, it is perhaps better to consider a fraction as expressing one or more of the equal parts of unity; hence it is so regarded and treated in the Arithmetics of this series; but for the more general purposes of algebra, it is better to treat fractions as quotients. REDUCTION OF FRACTIONS. 181. Reduction of fractions is changing their terms without altering the value of the fractions. 182. Principle. — Multiplying a quantity by a num- ber and then dividing the result by the same number., does not change the value of the quantity. Thus, i X 12 = 9; 9 — 12 = ll .-. 1 = A 4 12 i 12 Hence, the value of a fraction is not changed by multiplying or dividing both of its terms by the same number (179). To employ the preceding principle in the reduction of fractions, it is necessary first to ascertain by what, and how, to multiply a fraction in order that the prod- uct may be an entire number. FRACTIONS. 79 183. If a fraction be multiplied by a multiple of the denominator, the product will be an integer. d ahn Thus, ^ multiplied by ftw (126),=: — i— = an. To perform the multiplication, divide the multi^^lier by the denominator and multiply the quotient by the nunterator. EXERCISE XXX. 5 1. Multiply g- by 18. 18H-9 =2; 2X 5 = 10, Ans. -J. |[ 2. Multiply — p-5 by x"^ + 5j; + 6. fxz + 6a; + 6) -H- Cx + 3) = a; + 2 ; (x -f 2) X (a:— 1) :^ a;M i — 2, Ans. Simplify: 8 7 X 28. 4. ^^ X C^'' — 4). 11 5(1 — &) «• 1^X45. 6.. A_^-^x3(«+&;3. 7. I X («^ + «). 8. ^^^ X «<«=> _ 1). 9. jY X 22a;. 10. 1^3 X (aj^ — a; — 6). 1^- It^ X (^ + «)^- "• ^ X ("' + *')• *^- "6^i^X30a2«='a;^ 16. ^.3 1 L X (a^« - y"). 80 ALGEBKA. Simplify : 17. 51 X 9. 2 , 5X9 = *5; g-X9=6; 45 + 6 = 61 Ans. 18. 7| X 8. 19. [x+l + g^) X 12a;. 20. 5f X 14. / 5x^ + 6x\ 21- (^ + 3 + -^TT-) x(^^-i). 22. 9f X 20. 23. (2a;^ 7 + ^^3^ ) X (^^ + a;- 6.) 24. (-1 + 52^ — 2|) X 12. Art. 107. (38 + 5L _ 23.) X 12 = 8+66 — 33=41. 25. (3|- + I _ ^) X 12. 26. (5:1 _ 4f + I-) X 24. /«+l 3j? — 7 5\ ^*- i~5~ + Ans. 29- (f X i-) X 40. Art. 103, (f xj)X40 = |x (JX>0) =fX 36 = 31, Ans^ FRACTIONS. 81 Simplify : 80- (-1 X -7) X 63. 3J. (I X ^) X &«. 32. (2^- X 3^) X 12. Zx 5aj \ , fX X + 2 X — 3\ „^ „ „„. 184. To reduce a fraction to its lowest terms. 185. A fraction is reduced to lower terms by divid- ing its terms by any common factor. Hence, Rule.^ — Divide the numerator and denominator by all the conivion factors, or by the G. C. D. oj the terms. EXERCISE XXXI. Reduce to lowest terms : ba^l^o (Divide both terms by Sa^B^). 16a;sy*g (a + ^Y (^ + «')^ 20a^y^z^' ' (^a + hy (a + xy' 36a*b?j; 5a^ (1 + xy (1 — a;)« *■ 63a*c2a;2' 16a^ (1 + '»)=' (1 — <») 39msn^x bla'^e^ (a + xy (a — xY *• Qba^m^x '• 85a6c (a + aj)2 (a — a?)' ' a;8 4- 4a; — 21 _ (a? — 3) (a? + 7) ^ «? — 3 - ®- a!2 — a; — 66 ~ (a; — 8) (a? + 7) 07 — 8. U. R. A. — 6. 82 ALGEBRA. Reduce to lowest terms : a;2 — 64 x"^ ■\- X — 72 10 y. x2 _ 12*' + 32 11. a;2 + 12.^; + 27 aj3 .- 81 13. a;2 + 5aj — 36 x"^ + 2a7 — 24 15. (a; + 1)2 12. 14. 16. a;2 — iT — 90 a;2 — 49 4i» ( a; 4- 7)" a2 — 2a — 3 ffl2 _ 10a -f- 21' ajs _ 6aj2 + 11a?— 6 a;3 + 3 a!2 — ai — 3 ^"" aj^ — 6a; + 8 Whem common factors cannot be determined by inspection, the 6. C. D. of the terms must be found by Art. 164, and the terms divided by it. as _ 3rt2 _|. 4rt _ 2 8,t;2 — 6a; — 35 '"aS— o2— 2«+2 12a;2 — 23a; — 77 2a;2 -—xy—Xhy"^ 6a3 — 23a2 + 16ffl — 3 19- 2a;2 — Zxy — 93 a;3 _|_ 2a;2 _ 63a; ' a+b ' ' r^ ~2x — 35 a* + 6* 12.);3 -f 4a-2— 27a!— 9 ®" a + 6 ■ ■ 2iM^5jqr3 187. To reduce an integer or fraction to a re- quired denominator. Rule. Multiply by the required denominator, and ■write the product over that denominator. This does not change the value, as it is equivalent to multiply- ing and then dividing by the same number. EXERCISE XXXIII. Reduce : 20 1. 5 to 4ths. 5X4=20, -^• 3 3 6 2. g- to lOths. g X 10 = 6, Yo"- a « , <*** 3. ^ to ?)wths. ^ X o» = <"*; i^' 4. Y^ to36ths. 6. ^YZn^o (a^ — l)ths. 6. - — to n^xHhs. 7 . ^ , ., to (a-^ — 9)ths. nx X -{- 6 8. |^%o 126a;ths. 9. J^ to (a'^ + x — 12)ths. 10. ■^'+_y+^ to (0!== + a; - 30)ths. 84 XLGEisftX. 188. To reduce a mixed number to an improper fraction. EXERCISE XXXIV. n 1. Reduce a -\- -^ to dths f« + -|] X 'i'= ("i + «, ad ■\- n d Rule. Multiply the integral pizrt by the denom- inator, to the product add the numerator, and place the sum over the denominator. Reduce to the fractional forms : 3a7 + 1 (x — 2')2 2. 2a; + 1 + ^ 3. x + 2 4- - — r^r- ' ' X ' ' X -\-2 Bx + 4 9-^ 11 , ' 1 -t-^^^ ' X X — a^fta C. ab + ; ' ab -'-^+m 8. a;2 + 3a; + 2 + x^ — 3a;2 + 2ir + 4 X— 2 9. a2 — 2a + 3 — a^ — 3a2 +4a— a 189. To reduce fractions to a com,mon denomina- tor, or quotients to a common divisor. EXERCISE XXXV. 1. Reduce g, ^ and g- to I2ths. Multiply each fraction by 12, we get 8, 9, 10. 8 9 10 Divide each of these by 12, we get r^-. ^r-. -r^- M.A \.z 12 Irfe ACTIONS. 8^ kule. 1°, Find the L. C M. of the denominators. 2°, Multiply each fraction by this L. C. M. 3°, Write the L. C M. under each product. Reduce to a common denominator : 3 oa; J 7c a — , — , and — . The L. C. M. of 4, e« and 8«2 is 24«2. ■ 3 5a; 7c ,, ^, , ,x24ai=18c!2 20aa;, 21c; (18a^20ax, 21c,) -^ 24a ^ = ^J^, ???^, ii^ . s 24a^ 24,(2 ' 24a2 ' -^ns. 3. !, !land I^. 4. !^izi_ and ^J^ilZ. 3 6 10 8 12 5. Z^, «and_£. 6. ^t^' and ?!Llzi^ 3 ft 6a 3a 66 7. m,^and^. 8. «-±^ and i^izi. m 3y a — b a -{- b 9. ^, A and-?-. 10. !^±^and?^Zziy. 2a 3aa Sa? 2 x — 3^ 2x -f- dy 11. 12. 13. 3a; w^ 8(1 +a?) 4(l + a;)2 2(1 + xy' x^ a? -f 2 a; — 3 a!2 — 9 ' a!2 + 3a? ' a;2 — 3aj * X -{- 3 X + 2 x+1 x? + 3x+ 2' a?2 + 4aj+ 3 x^ +.5x + 6' 86 Algebra. ADDITION AND SUBTRACTION. EXERCISE XXXVI. 2 5 l.Add -5- and 77- o b The L. C. M. of 3 and C is G. f- _^ I'j multiplied by 6 = (4 + 5) = 9. 9 divided by 6 = ^ = 1^, Ans. 6 X + 1 3 2. Simplify ;^—j + 2(^^2)- The L. C. M. of re^ _ 4 and 2(v^ — 2) is 2(a;2 — 4) Multiply " + ^ by 2(i«2— 4), 2«+2 a;2 ^4 Multiply ?_!_ by 2(x2 — 4), 3 a; — 6 2(a; 4- a) Add, 5 a; — 4 Divide this sum by 2(«2 — 4), 5« — 4 2C*2 —4)- 190. To add fractions : Multiply each fraction by the L. C. M. of the denominators , add the products, and divide the sum by that L. C. M. 5 , , 3 3. From s take . 6 4 The L. C. M. of 6 and 4 is 12. (^1 — _?^ multiplied by 12 = CIO — 8) = 1. 1 divided by 12 = L Ans. ADDITION AND SUBTRACTION. « +4 b 4. Simplify - g^ — — fi 6• (a + 6) (a + c) + (6 + c) (6 + a) + c2 — «6 (c + a) (c + 6) ■ a2 — 6c Z*^ + «c , «• (a _ 6) (a — c) + (6 + c) (&—«)'+' ca 4- fl6 (e — a) (0 + &) ■ 90 ALGEBRA. MULTIPLICATION . EXERCISE XXXVII. 3 7 1. Multiply F by j- I g- X .i) X (5 X 4)) Art. 188, Exs. 29 to 34, = 3 X 7, (3 X T) ^ (5 X 4) = -fxi' '^"'■ 2. Multiply ^ by ^• (^ X I) XU'X^)=aXc; (rt X c) -^ ('' X <«) = )^5T '^"'• 192. Rule. Cancel com.7non factors in the numer- ators and denominators, then multiply numerators together for a new numerator, and denominators together for a new denominator. Mixed nufnbers must be reduced to itnprofer frac- tions.. Simplify : 8 15(a; — a) [ ^ a J i 7. ?^X ^''. 8.fa-f")x '^ 42a 30a; V a a —X 9. ^(^ + ■^) X ^("~'^) 10. ^_i' X ^''il^^^ 3(«— a-) 8(a + a:)' ' 2 — y 2(j.' — 3; 11. ^ X — 12 1^ X -i X -^^-^ 4a» 9a;4' ' 4« 7// 4a ^a.'^ MULTIPLICATION., 91 , , J I Simplify : j3 5(« + xy Ujx + y y 7(0? + yy 16(a + x) ' ^. x^ —9 ,, a;2 — 25 14. X — X -\- b X — Z When the terms are polynomials, it is best to resolve them into their factors, in order to deti'ct common factors in the numerators and denominators. ^^ x^ -]- bx -\- & ^ a;2 + 9ir -f- 20 15. — ;; ; — = ; — T7; X a;2 + Vaj + 10 x^ — ^x ~ 28 (X + 2) (a -f 3) (X + 4) (» 4- 5 ) _ X + 3 (X + 2) (x + 5) (x + 4) (K — 7) ■" X — 7' a;2 — 1 ^^ ica — x — & 16. -T X a;a _ 4 x^ -\- 2j? + 1 052— 4a;+3 a'2 + 5j; + 6 1 7 • — :: — : — : : — "... ^^ a!2 + 4a; + 3 x^ — 7j3 + 12 a" — 9 (.g + 4h ,r2 — 7,g + 12 'a,'2 — 16 (« — 3)2 ^ *2 -i- Ix + 12' tc2 + 8a; + 15 ^ a;2_7a;-f-10 -r" + 8.x — 28 ^ a;2+5a;— 14 ^ a;^ — a? — 12 a'^ — 25 3aa; o^ — a;^ fee + ix <■ — a; ^''* 4&y -^ e2 — a'2 ^ a2 + «a; ^ a — a;' l_a,2 1 — yg / a; • 21- r+"l/ ^ iT^ X 1^1 + i_^ a* — 6* , , a — b 22. 7 TTTT X (a — 6)2 «2 -f- <,6 (x^ X \ ( .«2 a; , , ^«^ (P - ^ + 1 ) X (T2 + ^ + ^ 24, (.z;2— a;+l) X (^+J + :i:? /f|3— 6»\ /a + t\ / a2 _ <,;> -I- 6 26- ( ^Fqi'js ) (^a — 6J U^ -f a* 4- &^, 92 ALGEBRA. Simplify: 6a;2 —ic—1 10a;2 + 9a; -|- 2 ^^- 5x^ + 23x + 12 ^ 3ir2 — lla;-^' a;2 — 2x — 15 ,, ac^ 4- 6ac — 7 a?2 — « — 72 a;2 —2x — (i3 x^ + Bx — iO ^ x^ + ix + 3 DIVISION. EXERCISE XXXVIII. 1. Divide =- by -=. ■' d — multiplied hy bd = ad; — multiplied by bd = be. Hence (126), ^ -h — = ad-i-bc = ^r— > Ans. b d be 193» Rule I. — Multiply both dividend and divisor by the L. C. M. of their denominators^ and divide the fortner product by the latter ; or, II. — Invert the divisor and -proceed as in multiplica- tion. • Divide : 2. ?f by "§-. 3. -^l- by -??_. ^ 4cX—2 . 2x ^ a — h , a^ ~ ab 4. -3— by^- 5. ^^-^-^by^^— ^. x^ -^1 X + 1 a3 — aj3 (a _ aj)2 *■ 5 °y 10 • ■ aJ~+l^ '^y (a + xy • a!!2 — 9 , x — 3 m^ — n^ n — m 8- ^2 _ 16 by ^qri" ®- cs 4- 8 ''y a?3 — X x — 10- ^TT^by^^ ^ X + 4: "• c3 + 8 ".y c + 2 a?3 — X X — 1 DIVISION. 93 Divide : 11. ^ — ^hy ar2 — 7a! x"^ — 13a; + 42 Simplify : a)g + a;y -\- y^ a* + ?/ a;2 — a!y 2a / 1*- rT-7.^(l 15. 13 a;2 — a!y + j/^ " x ^-y 2a I a — & ' a+6 ■ \^^ ~ a + 6 (a;2 — 8)2 a;^ +' 2a.-2 — 80 (a;2 + 8)2 ■ a;* — 2*2 — 80 a;2 — 14a! -1-49 a!2 — 15a! -f 56 16. [«''■ - ^tI^(-^ + a!2 — 12a! +36 ' a!^ — 11a! + 30 1 \ / 1\ 17. 19. ^^. + 2+4^)-.(a!+-^ 20. (.3+1+ _Lj ^ (—1+4 21. f^_4+^\ ^ IX _^ \^2«2 a!2 y \^ 2a a! 22. fa._ft._e.2+26c) ^ ^-Iz^. \ / a + 6 + c 23. (a2 _ 6. _ ,.2 _ 2&.) ^ ^JL±^. V I a + ft — e 24. (x^ — Sax — 2a2 + -^^ ] ~- (zx — 6a - ^"^^ X -\- Saj \ a! + 3ai( 94 ALGEBRA. COMPLEX FRACTIONS. EXERCISE XXXIX. 194. 1. Simplify -. 4 3 4- li 2. Simplify j^X 4i _ 2i ^ T i> V 3 4 2-12 2 ^ 25 _ 64 25' It is often shorter to multiply both terms of the fraction by the L. C. M. of the denominators of the fractions contained in the terms. Thus, in (1) multiply both terms by 12, and in (2) multiply both terms by 12. Simplify : 3 3 . T + y !v-i + 2 r *• "3" 2" '"+ 1 + X + 1 X 2as , 3a; 27a7 -R- + -^ 3a7 + 5 8 ^ 2a;+ 3 COMPLEX FRACTIONS. 95 Simplify : a-\- X a — X a?-|- 1 1 X — 1 7. a — X « + X a; f 1 a; — 1 a + x a — ■X a — X a-\- X X — 1 x-\-l ^ I 1 10. x2 — n.v + 72 9. x-\-y ~'~ X' -y a;2 4- 22a; + 120 X J r x^ — 21* + 108 X — y x-\ a;2 + 18ar + 80 11. a+ b a2 a_6 + a2 + 62 62 63' "65 12. c— b c3 — 63 c + 6 ~ c'S + 6? a — b «» a + b ~ a^ .+ c + b ^ c2 -(- 62 c — 6 + e2 — 62 13. x^ + ?/2 a,.2 _ y2 X H- y x^ x"- X _y2 — y . 14. a -{- b a — 6 a — b "*" « -I 6 tt2 + ^2 , a^ —b^ x — y X + y a^ ^b'^~ a^ -\- 62 1.1 1 16. 1 x + 1 « + 1 X + 1 6+1 X c 17 a 18. 1 b + aj-+ 1 d + e l+a!-h 1 / 3— a; Chapter V. SIMPLE EQUATIONS. DEFINITIONS AND PRINCIPLES. 195. An Equation is an expression of the equality of two quantities. Thus, 12 — 7 = 5, is an equation, as it expresses tlie equality of the compound number 12 — 7 and the simple number 5. 196. The First Member or Side of an equation is the quantity on the left of the sign = ; and the Second Member or Side is the quantity on the right of the sign =. 197. To Yerify an equation is to reduce both sides to their simplest forms, and thus show that they are equal. EXERCISE XI Verify : 1.6 + 5 — 2=8+1. 2.10 — 7+3 = 11—5. 3. 14 — (5 + 3)=4 — (— 2). 4. 10 — (11 — 3) = — (3 — 5). 5. (6 — 2) (8 — 5) = — 3 (—4). C. 18 — (11 — 2) (10 — 8) = 0. 7— 1 _ 7^- 3 _ 7 — 3 2 ~^ 2 SIMPLE EQUATIONS. Verify : 10. 4—2^5—2 3(4 —2) ' a — la — 1 5(ffi — 1) 3~ + 2 6 3a; — 4 2(x + 1) a; — 20 4 3 12 Let the student ascertain by trial what number put in the place of, or substituted for a;, will satisfy the fol- lowing : 11. 9 —a; = 5. 12. 5 X 4 — Sa? = 14. 13. 3ar— 7 = 2a;+ 8. 14. 5a; — (2a; — 1) z= 10. XX Ap ^ — la; — 3 x—l. 15. 2 + 3 = 10. ^*'- -2~ 4^-=~r 198. Known Numbers (5) are usually represented by figures, or by the first letters of the alphabet, as a, 6; c, &C. 199. Unknown Numbers (6) are usually represented by the last letters of the alphabet, as a;, y, z. 200. An Identical Equation is one in which the two members are equal, whatever numbers the letters stahd for. As {% + o)» =x^ -\- 2ax + a". 201. A Conditional Equation is one which is true only on condition that the letters representing unknown numbers stand for particular values. As, ex's H to 16, Art. 197. V.E.A.— 7. ALGEBRA. 202. The Yalne of the unknown number (or a Root of the equation) is the number which substituted for it will satisfy the equation. 203. A Simple Equation, or an equation of the first degree, is one which contains only the first power of the unknown number. 204. To Solve an equation is to find its root, or the value of the unknown number. 205. The third particular in w^hich Algebra is an ex- tension of Arithmetic(ll, 69) is the solution of prob- lems by means of equations and their transformations. 206. To Transform an equation is to change its form without destroying the equality. 207. The transformations of equations are based on the following : Principle. If equal changes be made in both mem- bers of an equation, the results ivill be equal. EXERCISE XLI. First Form : ax = h. 1. Find the value of ic in 5a; = 20. Dividing both members by 6 6 | 53; = 20 is making equal changes in a; = 4. both members. Hence, SIMPLE EQUATIONS. 99 208. Rule. Divide both members by the coefficient of 00. Find the value of a; in: 2. 6 a; = 18. ^. ax = b. 10. (^a — i)a) = d. 3. 7x =: 35. 7, mx = c. 11. (a -\- n) x z= m. 4. 12a; = 27. %.rx = ar. 12. («+5)i» = rt2 _25. 5. 9aj = 42. 9.TO2a! = cTO3. I3.(re — 3)a; = 7j3 _27. Second Form : «« + bac — cx= in — n. 14. Solve 7a; — 3a; + 5ir = 30 — 5 + 11. Reducing by uniting similar terms, we have 9x= 36, .-. x = i. « 309> Rule. Reduce each member to a simple num- ber, and •proceed as with the First Form.. 15. 9aj + 11a; — 5a7 = 20 — 3 + 13. 16. 11a; — hx — 2a; = 13 + 7 + 12. 17. 6a; — 8ar + 7a; — 2a; = 5 + 19 — 6 + 9. 18. 5a; + 4a; + 3a; — 2.2; = 17 — 7 + 28 — 8 + 5. 19. 6a! — (3a; — 7a; + 8a;) = 3 + 7 — (14 — 6 + 8). 20. ax — &ai -(- ca; = m ■ — n. (a — & -|- c)^ = "' — '"•• •'• '" — " *= a—b + c' 21. aa; + 6a; — nx = m -{- r. 22. ma; — nx ^= n -if- m — a. 23. ax -\- bx = a'^ — 62_ 34, j^a; — M.r = jm^ — n^. 25. 6a; 4- ca; = Z»3 -(- c^. 26. ca; + a; = c^ -f: 1. 27. ax -\- bx ^^ m(a -\- b) -\- n{a -\- b). 28. ax — cx = c{a^ — c^) + a{a — c)2. 29. a'^x + aa; + a; = a* + a2 + 1. 100 ALGEBRA. Third Form : ax + in — cae => a — bx + n. 210. To Transpose a term is to carry it from one side to the other, which is done by subtracting it from both members (207). Thus, take the equation k -|- 4 ^ 8. Subtract -j- 4 from both members, x =8 — 4. Again, take the equation x — 4 = 8. Subtract — 4 from both members, x =8+4. Hence^ a term may be transposed by changing its sign and -writing it in the other member. 30. Solve hx — 7,» + 12 = —%x + 28. Transpose, hx — 7a; + 6a; = — 12 + 28. Reduce, 4a; == IG. . ^ . Divide by 4, x^i. 211. Rule. — Transpose all the terms' containing the unknown number to the first member, and all the others to the second m,ember, and then proceed as with the Second Form. Find the values oi x'vs\: 31. 7a; — 18 + 5a7 = + 18. 35. cx—h -^n — mx = o. 32. 8ic + 19 — 5 = j;. 36. ex + n^ = «,r + c'S. 33. 11a; — 3a; — 37 = 5. 37. ca; — w^ = c^ — nx. 34. ax -\- m = n — bx. 38. 5a; — 20 = 50 — 2a^. 39. 5a; + 15 — 4a; + 12 = 10a; — 50 — 40. 40. 3x + 18 — (16 - 3a-) = 50. 41. 6a; + 6 + (4a; + 8) = 192 — (15a; + 3). 42. 3(3a; — 2) + 5(.r — 6) = 18a; — 4. 43. 6(,r + 1) + 4(a; + 2) — 3(5 — a;) = 12 X 14. 44. p(x —p)= q(x — q). 45. (.r + fl) (x — a) = (a- — «,) (a; -b)+ab + b^. SIMPLE EQUATIONS. 101 T- ^1 T-. X a 6 ax Foui'th Form: — 4- — = — — • n ~ 111 r in 46. Solve 7a; 4- 5 5.r — 6 8 — 5a! 6 "~ 4 = 12 Multiply both members by 12, the L. C. M. of the denomina- tors, and we have : 14a; + 10 — 16a; + 18 = 8 — 5a;. Transpose, 14a; — 15a; + 5a; = 8 — 18 — 10 Reduce, 4a; = — 20 Divide by 4 x = — 5. 212. Rule. — Multiply each term by the h. CM. of the denominators, and froceed as with the Third Form. If a fraction is preceded by the sign — ^, the sign of every term of the numerator must be changed when the denominator is rennoved. The sign of all the terms of both members may be changed, as it is equivalent to multiplying both mem- bers by — 1. "■ " = *-!■ 48. X X 2+3 = 9, 3 a; 3a; *^- 2 5 5 5 2* ^^ X X . ^. X ^ 1 99 51. ^+_V2=^T-' + '^-=l^- 4 o + 1). 5 4 9 55. —2 ^- - 14 g— . 5a;_4 7.T+6 2a;— 1 Sa;— 2 56 12 3 4 57. -^(28 — 9a;)— ^ (3a;— 11) = Uj—4x. 58. y (5a; — 1) — -i (9a; — 7) — ^ (5 — 9*) =^ 0. 1 1 1 59. -J- a; — — (2a; — 9) ^ (5a? + 8) = 0. „. 3a; — 7 , 25 — ix 5x — 14 60. — 3~ + — g— = — 3 61. 4a; + 17— — — (7a; + 2) = 19a;. «^-T — ir- + ^ = ^-(77 + ^ a; + 2 a; + 2 6a; + 7 _ 2a;— 2 , 2a; + 1 i5 ~ 7a; — 6 "^ 5 2 6 1 6S. 2a; — 5 3a; — 1 a — 3 6a; + 1 2a; — 4 _ 2a; — 1 lo 7a; — 16 5 67. [x + ^^{x -^ -^ ^^ (^x -^ b) {X - 3). 68. (^_|)(*-+ i)-^ = (.-._o)(^ + 3). 69.^^ ^— = ^ 1_. X — 2 a; — 4 a; — 6 a; — 8 SIMPLE EQUATIONS. 103 X — 1 X — 2 X — 5 X — 6 ' X — 2 X — 3 X — 6 X —^' 71 '''i^ — X x^ — c 72. ax ax x' -\- a X + c X -\- c X -\- a' 3a — 6a? 1 78. ax 3 2 _, - ax — b 74. 6a — ^x. 3 76. ^_ — & r=. ^ a -\- 1 a — 1 yg c ^ «! _ l+je ■ a2 — 62 a ~ b a+ b' 77. (a + a;) (6 + ic) - a(6 + c) = ^ + a;*. X +a X + b 2(a -\- 6) (a — 6) X -\- b X -\- a {x -{- b) {x -{- a) 79 25-^0^ , 16a; + 4^ _ _23_ 5 'a7+13a;+2 ar+l 80. -M^- + -f^^ + 1!1L±A)^ = Sex + ^^ a + 6 (a4-6)3 ai^a-\-by a 78. 104 ALGEBRA. PROBLEMS. 213. A Problem is a question proposed for solution. 314:. A Condition is a relation existing between the known and unknown quantities of a problem. In a problem the conditions are expressed in common language. 215. The solution of a problem by Algebra consists of two parts : i^, The Statement: that is, expressing the condi- tions by means of equations. 2'-', The Solution of the equations. No general rule can be given for the statement of a problem, but by adopting the following method the student will soon acquire the ability to make statements intelligently and readily: To the student. Take or assume some number, and by trial ascertain whether or not it is the answer. If you can do this, it shows that you understand the conditions. If you can not do this, ask the teacher to show you how to ascertain whether or not the assumed number is the answer: this will help you to understand the conditions. Now take x as the representative of the unknown number, and proceed with it just as you did with the assumed number, and where it ought to be equal to so and so, make it equal to it, and you have the statement. EXERCISE XLII. 1. Two boys have 24 marbles and the second boy has twice as many as the first; how many has each.? Trial Solution. Suppose the first boy has 6 marbles Then the second boy has 2 times 6, or 12 " and both bo^'s have 18 " But both boys have 24 " ••• 18 ought to be equal to 24. gut, as this is not the case, 6 is not the correct answer. PROBI-EMS. 105 Solution. Suppose the first boy has x marbles. Then the second boy has 2 times x, or 2a; " and both boys have x -\- 2x, or Sx " But both boys have 24 " 3a; = 24. Whence a; = 8, the number the first has, and 2a; =. IG, the number the second has. Comparing the two solutions, we see that "a;" takes the place of ''6," the supposed number, and "=:" the place of "ought to be equal to." 2. James paid $12 for a hat and coat, and the coat cost 3 times as much as the hat; what did each cost.' 3. A farmer paid $17.50 for a hog and cow, paying 6 times as much for the cow as the hog ; what did each cost ? 4. A man and his son earned $6.40; what did each earn if the man earned 4 times as much as the. son. 5. General Problem. The sum of two numbers is «, and the greater is t times the less ; what are the numbers ? is Ans. Less = ; greater 1 -\- t 1 + < 216. A General I'roblem is one in which the given numbers are expressed by letters. The answer to a general problem is a formula, by which all similar questions may be solved. Thus, to solve problem 2, we substitute 12 for s and 3 for t in the preceding formulas and obtain Less = _il^=3, greater =4^ = 9. Problems 3 and 4 may be solved similarly. 106 ALGEBRA. 6. William and Charles have 37 nuts and Charles has 11 more than William ; how many has each. Let X = the number William has. Then K + 11 := the number Charles has, and X -\- X -\- 11 = the number both have. But 37 = the number both have. .-. X +a;+ 11 = 37. Reducing, etc., 2x =26. Whence, x = 13, William's number, and 13 4- 11 := 24, Charles' number. 7. A horse and cow cost $153, and the hor^e cost $75 more than the cow ; what did each cost ? 8. James and John have $12, and John has $3.70 less than James ; how much has each ? 9. Two boys were 173 feet apart, and traveled towards each other until they met; how far did each go if one of them went 21 feet further than the other.? 10. The sum of two numbers is S and their difference is d ; what are the numbers .'' 11. Two boys, A and B, have 75 apples, and B has 7 more than 3 times as many as A ; how many has each .'' Let X = the number A has. Then 3a; + 7 = " " B " and a; + 3x + 7 = " " both have. But 75 = " " " " .•.a; + 8a; + 7= 75. Transposing, etc. 4x = 68. Whence, x = 17, A's number, and 75 — 17 = 58, B's " 12. A father wishes to divide 147 peaches between two sons so that one of them shall have 13 less than 7 times as many as the other. How many must he give to each ? PROBLEMS. 107 13. Henry lacks 27 years of being 3 times as old as Moses, and the sum of their ages is 33 years. What are their ages ? 14. C and D together paid a debt of $279 thus : I) first paid- $24, and then paid 4 times as much of the remainder as C. What did each pay ? 15. The sum of two numbers is s, and one of them is d more than t times the other ; what are the num- bers ? 16. The sum of the third and fourth parts of a num- ber is 35 ; what is the number.'' Let X = the number. Then X S == " third of the number and X i = " fourth " " and - + 3 ^ X T = " sum of the two parts. But 35 (I (( tt t( (1 (( X ~3 -+ ■ X T = 35 Multiplying both sides by 12, ix + 3x = 420 Reducing, 7x = 420 Whence, x = GO, the number. 17. Find the number whose fourth is 18 more than its tenth ? 18. Find the number whose ninth is 15 less than its sixth. 19. Find the number whose ninth and sixth to- gether make 15. 20. The third of a number exceeds its tenth by 28 ; what is the number.' 108 ALGEBRA. 21. Find the number whose mth part is d more than its nth part. 22. Find the number whose with and «th parts are together equal to S. 23. Gordon and John have together $66, and one- sevenih of Gordon's money is equal to one-fifth of John's; how much has each.? Let * =; the am't Gordon has. Then 60 — x = the am't John has ; and '— == one-seventh of the am't Gordon has, 7 ' , 60 — X = one-fifth the am't John has. 6 X 60 — * 7 5 Whence, ^ = 35, the am't Gordon has, and 60 — 35 =25, the am't John has. 24. Divide 100 into two parts such that if one part be divided by 8, and the other by 12, the quotients shall be equal.? 25. What time of day is it if one-fourth of the time past noon is equal to one-fifth of the. time till midnight.? 26. Divide 46 into two such parts that if one part be divided by 7, and the other by 3, the sum of theparts shall be 10. 27. A house and garden cost $850, and five times the price of the house was equal to twelve times the price of the garden. What is the price of each .? 28. A lot of 15 hogs cost $118; if the larger hogs cost $10 each and the smaller $6 each, how many of each kind were there? 29. The sum of two numbers is s, and the mth part of the first is equal to the »th part of the second. What are the numbers .? PROBLEMS. 109 30. Divide s into two parts such that if one part be divided by m, and the other by n, the sum of the quo- tients shall be a. 31. Divide 25 into two, parts such that the second divided by the first gives 2 as a quotient and 4 as a re- mainder. Let X = the first part. Then 25 — K = the second part . Now, according to the condition, if 25 — x were less by i, it would contain X, 2 times Hence, 25 — x — 4 „ X Whence, x = 7, and 25 — x= 18. 32. Divide 60 into two parts such that the second divided by the first part gives 5 as a quotient and 6 as a remainder. 33. Divide 20 into two parts such that three times the second divided by two times the first gives 1 as a quo- tient and 11 as a remainder. ■34. Divide s into two parts such that the larger divided by the smaller gives j' as a quotient and ^ as a remainder. 35. A certain sum is to be divided among A, B and C. A is to have £30 less than the half, B is to have JEIO less than the third, and C is to have £8 more than the fourth. What does each receive } 36. Divide 108 into three parts such that J of the first, ^ of the second and i of the third shall be equal to each other. 37. Divide 180 into four parts such that the first increased by 5, the second decreased by 5, the third 110 ALGEBRA. multiplied by 6 and the fourth divided by 5, shall be equal to each other. 38. A and B play at a game and commence with equal sums. After A has won $20, one-fifth of the amount he has is equal to one-third of what B has ; how much had each at first? 89. A man bought a buggy at a certain price and paid twice as much for a horse ; if the buggy had cost $10 more and the horse $55 less, then the cost of the horse would have been one-fourth more than that of the buggy; what did each cost.'' 40. A lady having been asked her age replied: "Three-fifths of my age lacks 29 years of being 1 year more than twice my age." What was her age? Note I. x, x -\- I, x -j- 2, etc., are consecutive numbers. 41. The difference of the squares of two consecutive numbers is 37 ; what are the mimbers.? / 42. Find three consecutive numbers such that 10 times the first increased by 6 times the second shall be equal to 15 times the third. 43. In a basket is a number of apples. Henry takes out 10 apples and ^ of what remains ; then John takes out 20 apples and J of what remains ; and then they find that there are 20 apples remaining ; how many were in the basket .' Note II.— If a; = a person's age now, '^^" a; — « = his age n years since, ^""^ X + n = liis age n years hence. 44. In 9 years a boy will be 4 times as old as he was 6 years ago. How old is he.? PROBLEMS. Ill Let X = the number of years in his age. Then x — 6 = the number of years of his age 6 yrs. ago, and X -\- 9 = the number of years of his age 9 3'ears hence. .-. a; + 9 = 4(a; — 6). Whence, x = ll. 45. A man is 3 times as old as his son, and 5 years hence he will be 5 times as old as the son was 5 years since. What is the age of each ? 46. A is twice as old as B, and 5 years ago he was 2J^. times as old ; what is the age of each.? 47. Eleven years ago the age of A was 44 yfears, and of B 13 years ; in how many years will A be twice as old as B .'' 48. A is 54 years old ; B's age is ^ of A's, and C's age is 4 of B's ; how long is it since A's age was equal to the sum'pf B's and C's.? 49. In m years a man will be n times as old as he was r years ago. How old is he? Note III. — If a man can do a work in n days, (1) In one day he can do _ part of the work, (2) In two days he can do _ part of the work, n (3) In X days he can do _ part of the work, (4) In n days he can do - (= 1) the whole work. n 50. A can do a piece of work in 4 days, and B can do it in 6 days. How long will it take A and B to- gether to do the work ? Let X = the number'of days it will take A and B together. Now, -r = the part A can do in one day. 112 ALGEBRA. X Hence, -r = the part A can do in x days. Again, v.- = the part B can do in one day. Hence, -w = the part B can do in x days. Now as A and B do the whole work in x days, we have X X T + ^ = 1- Whence, x = 2f. 51. A can do a work in 8 days, B in 10 days, and C in 20 days ; in what time will they do it, all working to- gether? 52. A can do a work in 6^ days, B in 7j^ days and- C in 11^ days ; in what time will they do it, all work- ing together.' 53. A can do a work in 1.0 days, but with, the help of B he can do it in 6 days ; how long would it take B to do it alone .'' 51. A can dig a trench in one-half of the time that B can ; B can dig it in two-thirds of the time that C can ; all together they can dig it in 6 days ; find the time it would take each of them alone .'' 55. Eight boxes or 24 barrels will fill a certain room ; what equal number of boxes and barrels will fill the same room .'' 56. A box will hold 20 oranges ; after putting 7 oranges and 2 peaches in the box, I observe that it is ■^ full. How many peaches will the box hold ? 671 A cistern can be filled by one pipe in 10 hours, by another in 15 hours and emptied by another in 12 hours ; supposing it at first empty, in what time will it be filled when all the pipes are running.'' PROBLEMS. 113 58. Two pei'sons, A and B, could finish a work in 12 days ; they worked together 7 days when A was called off and B finished it in 8 days.- In what time could each do it .' 59. A and B together can do a piece of worlt in 12 days, A and C in 15 days, B and C in 20 days. In what time can they do it, all working together ? 60. If three pipes can fill a cistern in «, b and c min- utes, respectively, in what time will it be filled by all three running together ? Note IV. — In questions involving distance, time and late, we have: (1) Distance = Time X Rate. Distance (2) Time (3) Rate Rate Distance Time 61. A traveler who goes at the rate of 22 miles in 4 hours, is followed after 5 hours by another, who goes at the rate of 38 miles in 6 hou-s. In how many hours will the second overtake the first. Since the first goes 22 miles in 4 hours, his rate per hour is 5% miles (3;. Since the second goes 38 miles in 6 hours, his rate per hour is 6)4 miles (3). Let X = the number of hours the first is traveling, Then« — 5 = " " " ■' "secondis " Then 5'» = " " " miles the first travels, andeu'* — 6)= " " " " "second " They both travel the same distance, ... 5^-« = (i-(x^ 5). 2 3 Whence, x = 38, Hence, 38 -^ 5 = 33, Ans. N. E. A —8. 114 ALGEBRA. 62. A and B depart from the same place to go in the same direction ; B travels at the rate of 9 miles in 4J hours, and A 10 miles in 3^ hours, but B has the start of A by 5 hours ; in how many hours will A over- take B ? 63. A and B depart at the same time from the same place, to travel in the same direction around an island 36 miles in circumference ; A travels 3 miles an hour and B 2 J ; after how many hours shall they come together .' 64. One horse (A) runs at the rate of 1 mile in 2J minutes, and another (B) at the rate of 1 mile in 3^ minutes ; if B has the start of A f of a mile how long must A run first, before he overtakes B? Second, be- fore he will be i of a mile in advance of B ? 65. A man went to town in 6J houi's ; in going he traveled at the rate of 6 miles per hour and in returning at the rate of 4 miles per hour. Hdw far was it to town .' 66. Two persons set out from the same place in op- posite directions. The rate of one of them per hour is 3 miles less than double that of the other, and in 5 hours they are 45 miles apart. Find their rates. 67. A left a certain town at the rate of a miles per hour, and in n hours was followed by B at the rate of b miles per hour. How far did A travel before he was overtaken by B ? 68. A hare makes three leaps to the dog's 2, but 3 of the dog's leaps are equal to 7 of the hare's. If the hare has a start of 60 of her own leaps, how many leaps must the hound make to overtake the hare.'' PROBLEMS. Ua Since the hare has the start of 60 of her own leaps we Let a = the length of a leap of the hare's, then !2 = " " " hound. 8 Let X = the number of leaps of the hound, then — = " " " hare. 2 Then, a'OO + —\ = the whole distance. I 2 ) Tax Jeo + ^*' And Divide by a, 60 + / Whence, = Tax 3 3x 2 = 7x 3 x = 72. Ans, 69 A hare has 30 leaps the start of a hound, and makes 6 leaps to the hound's 5, but 7 of the hound's leaps are equal to 9 of the hare's. How many leaps will she make before she is caught? 70. A hound makes 6 leaps while a hare makes 8, but 2 of the hound's leaps are equivalent to 3 of the hare's. The hare has a start of 50 of the hound's leaps. How many leaps does each make before the hare is caught .'' In this case, since the distance between the hound and hare is expressed in hound leaps, we let a = the length of a leap of the hound. 71. A hound makes 5 leaps while a hare makes 7, but 8 leaps of the hound are equal to 13 of the hare's. The hare has a start of 72 of the hound's leaps. How many leaps will the hare take before she is caught ? Note V.— In problems relating to clocks, it is to be observed: lie ALGEBRA. 1°. The minute or long hand moves 12 times as fast as the hour or short hand. Hence, if » := the number of minute- bpaces traveled over by the long hand, which will be the case in the following problems, then — ^ the corresponding number traveled over by the short hand . 2°, The difference between x and — expresses the distance in minute-spaces that the long hand will gain on the short hand in X minutes. Hence, X — ^ = Distance gained in x minutes. 72. Find the time between 3 and 4 o'clock when the hands of a clock: 1". Are together. 2". Are at right angles to each other. 3°. Are opposite to each other. 1'. At 3 o'clock the short hand is 16 minute-spaces ahead of the long hand, and when, the latter gains this distance on the former, they will be together. .-. a — £ = 15. 12 Whence, x = l&-^^. Hence the time is 16j^ min. past 3 o'clock. 2°. The hands are at right angles to each other when one is 15 minute-spaces ahead of the other. Hence, they are at right angles at 3 o'clock and will be so again between 3 and 4 o'clock. For, when the long hand gains 15 minute-spaces the hands will be together and when it gains 15 more they will be at right angles. Hence it must gain 30 in all. .-. X— ^ = 30. 12 Whence, x = 32-[fif . Hence, the time is 32^inin. past 3 o'clock. PROBLEMS. 117 8°- The hands are opposite when one is 30 minute-spaces ahead of the other. At 3 o'clock the short hand is 15 minutes ahead of the long hand. Hence, it must first gain 15 and then 30, or 45 in all. V. 12 X = 49;jV. Hence, the time is i9-^ min. past 3 o'clock. 73. At what time are the hands of a watch together: 1°. Between 5 and 6 ? 2°. Between 8 and 9 ? 3°. Between 11 and 12.' 74. At what time are the hands of a watch at right angles: 1°. Between 1 and 2 ? 2°. Between 4 and 5 ? 3°. Between 7 and 8 .' 75. At what time are the hands of a clock opposite to each other : 1°. Between 1 and 2? 2°. Between 4 and 5 ? 3°. Between 8 and 9 ? 76. At what time between 7 aod 8 : I'', Is the minute hand 5 minntes ahead of the hour hand? 2*^, Is the minute hand 10 minutes behind the hour hand. 77. A number of boys have a basket which contains a number — y x+y _ 2 + 10 — •'• 3£ y^ 1 _ ^ J^ 10 ~ 15 ~ 9 -~ 12 ~ 18' 8 a; ^ 11 ^'^ ~ 3 = 12 " 15 + 10' 126 ALGEBRA. 33. 2ic + .5y = 1.8, .5* — .83^ = .08. 34. 2x — -^ = 8, 8—x 2v + 1 35. x-\-y = s, 36. 3a; + 2y = a, X — y=zzd. Ax — 3yz=b. x—2 10 — a? y— 10 37. -^- - —3— ^ -^, 2y + 4 2a! + y ar + 13 3 8 = ¥~' 88. aa!+ by =m, (1) ex -\- dy = n (2) (1) X Ci OCX 4- 6cj = cm (3) (2) X «i acx + adij = an (4) (3) — (4), (6c — ad)y = cm — an „,. cm — an Whence, y = -; -• "bc — ad To find X: (1) X <^. adx + bdy = dm (5) (2) X 6. bcx + bdy = bn (6) (5)— (6), {ad — bc)x = dm — bn ,„, dm — bn Whence, a; = — ^ r- • ad — be i9.mx—ny = r, 40. ax-\-l)y = e, px~qy zz= s. mx — ny = 0. .1^,1 a b --- = 6 ^ ^_w a? 3/ ' X ~ y — "" X V X — V *^- T+b + db = 2a, -/ = 1. EQUATIONS. 127 226. If there are three equations, (1), (2), (3), in- volving three unknown quantities, 1°. Combine two of the equations, as (1) and (2), so as to eliminate one of the unknown quantities, and call the resulting equation (6). 2°. Combine the remaining equation, (3), with either of the other two, as (1) or (2), so as to eliminate the same unknown quantity, and call the resulting equa- tion (9). 3°. Combine (6) and (9) so as to find the unknown quantities which they involve. 4°. Substitute the values thus found in (1). (2) or (3), and thus find the remaining unknown quantity. EXERCISE XLIV. 1. Find the values 2x + 3y + 4z = 16 (1) oi x,y andz'in: 3x + 2ij — 5z = 8 (2) 5x — 6y+3z= C....(3) Note. — It is immaterial which unknown quantity we elimi- nate first, but there will be less work in eliminating the one which requires the least number of multiplications or the smallest multipliers to make its coefficients equal. For in- stance, it is best to eliminate y first in the preceding example. (1)X2 4x-\-6y+ 8z= 32 (4) (3) is Bx — ey+ Sz= 6 (5) (4) + (5) to + 112= 38 (C) (2) X 3 9x + 6y— 15z = 24 (7) (3) is 5x — Gy+ 3z = 6 (8) (7) + (3) Ux — 122= 30 (U) (6)X14 12Ga; +1640 = 532 (10) (9)X9 126x —1083 = 270 (11) (10) — (11) 2622 = 262 2=1 Substituting 1 tor z in (11), we find x= 3 Substituting 1 for z and 3' tor a; in (1), we have 6-|-Sy + 4 = 16; whence, 2/ = 2. 128 ALGEBRA. 227. If four or more equations are given, involving four or more unknown quantities, we proceed in a similar manner. Thus : Eliminate one of the unknown quantities between three or more pairs ot the equa- tions ; then a second between the different pairs of the resulting equations ; and so on. Solve the following : 2. 2x— y -{■ 2; = 9, 8. 4a; — 3y + 22 = 40, X — 2y + 32 = 14, 5ar + 9y — ■ 72 = 47, Sflj 4- Ay— 2z= 7. 9a; + 82; — 32 = 97. 4. 2a; — 32^ + oz = 15, 5. 5x — 6y + Az= 15, 3a! + 2y — 2=8, 7a; + 4?/ — 32 = 19, -a; + 5y + 22 = 21. 2a; + 2/ + 62 = 46. 6. 2a; — 4a/ -{- 9z = 28, 7. 2x + 3y — iz ^ 10, 7a; + 3y— 52= 3, — 43/ + 02 + 3a; = 14, 9a; + lOy — II2 =4. 62 + 4a; — 5y = IB- 8. 3x — 3y + 2=0, 9. a; + jf + 2 = 29, 2a; — 72/ + 42 = 0, x + 2y + 3z = 62, 9a; + 5j^ + 3» = 28. ix + y + iz ^ 10. 10. 7a;— 3y = i, 11. a; + 3/+ 2+ m = 14, II2 — 7m = 1, 2x + 3y + 4z + 5u = 54, 42 — 7y = 1, 4a; — 5y — 72 + 9m = 10, 19a; — 3m = 1. 3x + 4y+2z—3u= 11. 12. ay + 6a; = c, 13. aa; -\-by + e z = n, ex -\- az = h, a^x -\- h^y -\- c^z = n^, iz -{- cy = a. a^x -\- l>^y + c^z = n^. 14. hx -\- ay = db, Ih. x -{- y -\- z = a -\- I -\- c, ex + «2 = ac, hx-\- cy -\-az = n^ -{-b^ -\-e^, cy -\-lz = be. ex + ay -\-bz = a^ -\- b"^ -\-c'^. fftOBLEMS. 129 PROBLEMS. EXERCISE XLV. 228. — 1. Find two numbers such that if tne first be added to 4 times the second, the sum is 29 ; and if the second be added to 6 times the first, the sum is 36. Let X = the first number, and y = the second. Then x + 4?/ = 29 (1) and 6a; + 2/= 36 •. (2) (1)X6, GK + 242/ = m (3) (S) — (2), 23?/ = 138 2/ = . 6. Substituting 6 for rj in (2), we have 6a; + 6 = 36. Whence, a; = 5. 2 Find two numbers such that 2 times the first plus the second is 17, and 2 times the second plus the first is 19. 3. Find two numbers such that 4 times the first less 1 is equal to 3 times the second plus 2 ; and 3 times the first plus 12 is equal to 4 times the second less 12. 4. Divide 100 into two parts such that 10 shall be the difference between the first part and 5 times the second part. 5. A and B together have $570. If A had 3 times as much as he has, and B 5 times as much as he has, the sum would be $2350. How much has each } 6. If A's money were increased by $36 he would have 3 times as much as B ; but if B's money were diminished by $5 he would have ^ as much as A. How much has each .' N. E. A: — 9. 130 ALGEBRA. 7. Find two numbers such that the sum of 4 times the first and ^ of the second is 53 ; and the sum of 3 times the second and J of the first is 48. Let a; = the first and y = the second. Then, ix + iy = ' 53 (1) ix+Sy= 48 (2) a)X9. 36a: + 3j/= 477 (3) (3) — (2), 35|a! = 429 (4) (4) X *, liSx = 1716 X = 12. Substituting 12 for Kin (2), we find y = 16. 8. Find two numbers such that 7 times the first in- creased by ^ of the second shall be 144; and 10 times the second increased by ^ of the first shall be 324. 9. Find two numbers such that 3 times the first diminished by ^ of the gecond shall be 121 ; and 2 times the second diminished by ^ of the fii'st shall be 76. 10. Divide 100 into two parts such that 3 times the first part diminished by J of the second shall be JIO. 11. If A had 6 times as much money as he has he would lack 4^ of what B has of having $1100; and if B had 7 times as much as he has he would lack J of what A has of having $1000. How much has each.? 12. Find two numbers such that ^ of the first and I of the second together may be equal to the difference of 3 times the first and the second, and this difference equal to 11. 13. Divide 9 into throe parts such that the sum of the first, 2 times the second, and 3 times the third shall be 22 ; and the sum of the first, 4 times the second, and 9 times the third, 58. PROBLEMS. 131 14. A certain fraction becomes equal to ^ when 1 is added to its numerator, a«d equal to ^ when 1 is added to its denominator. What is the fraction ? X Let — = the required fraction. x+ 1 1 Then, -^— = 2- and 1 y + 1 3 From the solution of these equations it is found that x = 3, and ^ = 8. Therefore the fraction is |. 15. If the numerator of a fraction is decreased by 2 its value becomes J ; if the denominator is increased by 3 its value becomes ^. What is the fraction ? 16. What fraction is that whose numerator being doubled, and the denominator increased by 7, becomes f; but, the denominator being doubled, and the nu- merator increased by 2, becomes f.' 17. A fraction which is equal to J, becomes f when a certain number is subti'acted from both of its terms ; and 4, when the same number is added to its numerator and 5 to its denominator. What is the fraction ? 18. The numerator of one fraction is 1, and the de- nominator of another is 3 ; what are the fractions if their sum is |, and their difference, ^? 19. Find two fractions whose numerators are 1 and 2, whose sum is ^, and whose sum will be f if the de- nominators are interchanged .'' Note. — If x and y represent the tens' and units' digits of a number respectively, then the number itself is represented by IOk 4- y. Likewise, a number consisting of three digits may 132 ALGEBRA. be represented by lOOa; -\- lOy -\-z, in which x, y and 2 represent the digits of the number. Thus, 753 means 700 +50 + 3, or 100 times 7 + 10 times 5 + 3. Here, 7, 5 and 3 are the digits of the number 753, 20. The sum of the two digits of a number is 6, and if 18 be added to the number the digits will be in- terchanged. What is the number } Let X = the digit in the tens' place, and y = the digit in the units' place. Then lOz + !/ = the number. By the conditions, a; + ?/ = 6 (1) and lOx + 2/ + 18 = 10?/ + a; (2) From (2) to— 9?/=— 18 (3) (3)-h9 X— y = — 2 (4) (1) + W, 2x = i,.-.x = 2 CI) — l4), 2!/= 8, .■.y = i. Hence, the number is 24. 21. A number expressed by two digits is 4 times the sum of the digits, and if 27 be added to the number the digits will be interchanged ; find the number.? 22. A number of two digits is 4 times the sum of its digits, and 3 times the number is equal to the square of the sum of its digits. What is the number.? 28. A number is expressed by three digits. The middle digit is twice the left-hand digit, and one less than the right-hand digit. If 297 be added to the num- ber, the order of the digits will be reversed ; find the number. 24. A number is expressed by three digits. The sum of the digits is 18 ; the number is equal to 99 times the sum of the first and third digits, and if 693 be sub- tracted from the number the order of the digits will be reversed. Find the number. PROBLEMS. 133 25. If a certain number be divided by tlie sum of its two digits the quotient is 3, and the remainder 3 ; if the digits be interchanged, and the resulting number be di- vided by the sum of the digits, the quotient is 7, and the remainder 9. What is the number.? 26. The sum of the three digits of a number is 16. If the number be divided by the sum of its hundreds' and units' digits the quotient is 77 and the remainder 6 ; and if it be divided by the number expressed by its two right-hand digits, the quotient is 16 and the remainder 5. Find the number. •27. The sum of the two digits of a number is «, and if 9d be added to the number, the order of the digits will be reversed. What is the number.? 28. A person has two horses, and a carriage which is worth $150. The value of the first horse and carriage is 2 times that of the second horse, and the value of the second horse and carriage is 3 times that of the first horse. What is the value of each horse ? 29. A person has two horses, and a carriage which is worth $c. The firs-t horse and the carriage are worth TO times the second horse, and the second horse and carriage are worth n times the first horse. What is the value of each horse ? 30. Five years ago the age of a father was 7 times that of his son ; five years hence the age of the father will be 3 times that of the son. What are their ages? 31. Two years hence A will be twice as old as B was 6 years since, and 8 years hence B will be twice as old as A was 15 years since. What are their ages } 134 ALGEBRA. 32. I£ A give B $5, they will have equal sums of money; but if B give A $5, then 7 times A's money will be equal to 11 times B's. How much has each.' 33. If A give B $a, A will have twice as much as B ; and if B give A. $6, A will lack $C of having 4 times as much as B. How much has each ? 34. A bought 5 hogs and 7 sheep for $47i, and B bought of the same lot 9 hogs aind 4 sheep for $48J ; what was the price of the hogs and sheep ? 35. A pound of tea and 3 pounds of sugar cost 6 shillings, but if sugar were to rise 50 per cent, and tea 10 per cent, they would cost 7 shillings. Find the price of tea and sugar. 36. A grocer has two kinds of tea which are worth 40 cents and 72 cents a pound. How much of each must he take to make a mixture of 50 pounds worth 60 cents a pound ? 37. A -wine-merchant has two kinds of wine which are worth 90 cents and 28 cents a quart, out of which he forms a mixture which is worth $61.20. He sells the mixture at 50 cents a quart and gains $3.80. How many quarts of each did he put into the mixture.? 38. A and B can do a certain work in 24 days, B and C in 40 days, A and C in 30 days. In how many days can each do the work? Let X = the number of days it will take A to do the work. y — << l( II g II Z = " <( (I Q II 111 Then, — > — i — > respectively, will denote the part each can do in one day. PROBLEMS. 135 1115 ... Hence, - + - = ^j = J20 *-^^ i , i_l_-i- (2) !/ ^ z — 40 — 120 L , L_l_-J_ (3) K ^ z ~ 30 ~ 120 2 2 2 12 fA^ (1) + (2) + C3), - + - + - = J20 (*^ 2 2 10 f^s (1)X2, - + - =120 ^'^ W - (5), ^ = 120' whence, z = 120. (4) - 2 X (2), X = 120' '"'"^ence, s = 40. C4) -2X (3) i^ = 120' '"thence, 2/= 60. 39. A and B can do a certain work in 4| days, A and C in 6 days, and B and C in 8 days. In how many days can each do the work? 40. A box will hold 20 spellers and 15 readers ; and 8 spellers and 12 readers will fill f of it. How many of each will it hold ? 41. A had enough money to buy 10 hats and 12 caps ; after buying 8 hats and 4 caps he observed that he had spent I of his money. How many of each could he have bought with all his money.' 42. A, with the help of B for 4 days, can do a cer- tain work in 15 days ; B, with the help of C for 6 days, in 20 days ; C, with the help of A for 7 days, in 22 days. How long will it take each alone to do the work .? 43. A cistern has three pipes A, B, and C. A and B will fill it in rt minutes; A and C in 6 mmutes ; B and C in c minutes. How long will it take each alone to fill it.? 136 ALGEBRA. 44. A can row a boat down stream at the rate of 9 miles per hour, and up stream at the rate of 3 miles per hour. At what rate does the stream flow, and at what rate could A row the boat in still water ? Let X == A's rate per hour in still water, and »/ = the rate per hour of the stream. Then x + y = A's rate per hour down stream, and x — y = (( (( tt « up " .-. X + y = s, X — 2/ = 3, Whence, x = G, y = 3. A crew which can pull at the rate of 15 miles an hour down stream, finds that it takes 3 times as long to come up the river as to go down. At what rate does the stream flow ? 46. A man sculls down a stream, which runs at the rate of 3 iniles an hour, for a certain distance iti 3 hours, and finds that it takes him 12 hours to i-eturn. Find the distance he pulled down the stream and the rate of his pulling. 47. A boatman can row down stream a distance of a miles and back again in 6 hours ; and he finds that he can row d miles against the current in the same time he rows miles with it. Required the time in going and returning, also the rate of the current. 48. Find three numbers, so that the first with half the other two, the second with one-third the other two, aijd the third with one-fourth the other two, shall each be equal to 34. 49. A says to B, " give me -J- of your money and I shall lack $52 of having twice as much as you ; " B says to A, " give me ^ of your money and I shall lack PROBLEMS. 137 $40 of having thrice as much as you." How much has each ? 50. If the length and width of a rectangular field were increased by 5 rods and 4 rods respectively, its area would be increased by 240 square rods ; but if the length and width were diminished by 4 rods and 5 rods respectively, its area would be diminished by 210 square rods. Required, its length, width and area. 51. A and B run a mile. First A gives B a start of 44 yards and beats him by 51 seconds ; at the second heat A gives B a start of 1 minute and 15 seconds, and is beaten by 88 yards. Find the times in which A and B can run a mile separately. 52. The fore-wheel of a carriage makes six revolu- tions more than the hind-wheel in going 120 yards ; if the circumference of the fore-wheel be increased by J of its present size, and the circumference of the hind- wheel by ^ of its present size, the six will be changed to four. Required the circumference of each wheel. 53. A railway train after traveling for one hour meets with an accident which delays it one hour, after which it proceeds at ■§■ of its former rate, and arrives at the terminus 3 hours behind time ; had the accident oc- curred 50 miles further on, the train would have ar- rived 1 hour and 20 minutes sooner. Required the length of the line. 54. A's capital is invested at a certain per cent. B's capital is $1000 more, and he receives one per cent, more than A ; C's capital is $1500 more, and he re- ceives two per cent, more than A. B's income is $80 greater than A's, and C's income is $150 greater than 138 ALGEBRA. A's. Find the capital of each, and the rate at which it is invested. 55. If A give B as much money as B has, then B give A as much as A has, then A give B as much as B has, then B give A as much as A has, each would have $16. How much has each? 56. A merchant has three casks. A, B and C, each lontaining wine. If he pour from A into B and C each as much as it contains, then pour from B nito A and C each as much as it contains, then pour from C into A and B each as much as it contains, each would contain 16 gallons. How many gallons in each ? INDETERMINATE EQUATIONS. 229. An Indeterminate Equation is one in which the unknown numbers have an indefinite number of values. 230. I. — One equation containing two or more un- known ttumbcrs is indeterminate. Thus, take the equation dx — iy = 12 (1) 12 + 42/ By reducing, we get x = g Let 3/ = 1, 2, 3, i, 5, etc. — Then x = 5^, 6f, 8, 9j, 11^, etc. Any two corresponding values of x and y will satisfy (1). 231. II. — Simultaneous equations are indeterminate when the number of tmknown numbers involved ex- ceeds the num.ber of independent equations. For by eliminating, the equations may be reduced to INDETERMINATB EQUATIONS. 139 one equation involving two or more unknown numbers, which is indeterminate, as before shown. Thus, take these equations: 8* -f 4j, — 6a = 40 (1) 5* — 3)/ + 2« = 25 (2) Here we have txvo equations involving three unknown num- bers. Multiply (.2) by 3, add the resulting equation to (1), and we obtain 18* — hy — 115, which is indeterminate. 232. Generally, the object in solving indeterminate equations and problems, is to find only the positive integral values of the unknown quantities. EXERCISE XLVI. 1. Solve in positive integers 4a; -|- 9y = 115 (1) 115—92/ 3 —a ,„, Transpose, etc., % = 1 : = 28 — 2y + — i '^ -^ x-2S + 2y=-~ (3) Since the values of x and y are to be integral, x — 28 + 2!/ 3—2/ will be integral, and hence, — j — will be integral, otherwise we should have an integer equal to a fraction. Hence, we may write — T — = n, an integer. Then, 3 — y =i in, or y = 3 — 4n (4) Substitute in (2) , a; = 22 + 9)i (5) In (4) and (5) n may have any integral value, except such as produce negative values of x or y. Hence, in (4), m can have no positive value greater than 0, but may have any negative value. But we see from (5) that If the absolute value of n exceeds 2 (n being negative), x will be negative. 140 ALGEBRA. Therefore, n can have but three values: 0, — 1, — 2. Substituting these values in (4) and ^^5), we have 2/= 3, 7, 11; X = 22, 13, 4. 233. In solving equations like the preceding it is desirable that the coefficient of y should be one more than some multiple of the coefficient of x, as in (1). If this relation does not exist, the equation should be multiplied by such a number as will make the new coefficient of y one more than some multiple of the original coefficient of a;, and then proceed as in the solution of the next example. 2. Solve in positive integers hx — 14y = 11. . . .(1) Here, 14 -h 6 = 2 quo. and 4 rem., which is not in the desired form. Multiply (1) by 4, and we have 4X5* — 562/ = 44 (2J Now, 56 -H 5 = 11 quo. and 1 rem., which is in the desired form. It is useless to consider 4, as it is a factor of all the terms. Transpose, divide by 6, 44 + 56i/ 4 + » *+ 2/ Make — 7 — = ra, whence, ^= 5m — 4 (4) Substitute in (3) a; = 14ii — 9 (5) Here it is evident that n may have any positive value. Making n = 1, 2, 3, etc.. We have 2/ = 1, 6, 11, etc., and a; = 5, 19 23, etc. Solve in positive integers : 3. 3a; 4- 1y = 29. 4. 4ar — 7y = 13. 6. dx -\-\\y--^%\. e. arc incompatible. ( x — 6y=ld,) 337. Incompatible Conditions are tliosc which may be represented by incompatible equations. 238. Impossible Problems are those which involve incompatible conditions. Note.— Let the student show that the following are impossi- ble problems. To do so, it will be sufficient to show that the equations representing the conditions are incompatible. 2. Divide $75 between A and B so that if B give A $5, A will then have 2 times as much as B ; but if A give B $5, they will have equal sums. 3. A line 96 feet long is cut into two parts whose lengths are proportional to 5 and 3. What are the lengths of the parts, if one part is 10 ft. longer than the other.? 4. A and B buy 200 acres of land at $2 per acre, each paying $200. If A takes the better portion of the land at $2^ per acre and B the remainder at $lf per acre, how much of the land will each receive ? Chapter VI. HIGHER OPERATIONS. INVOLUTION. 239. Any power of a number may be expressed by giving it the exponent of the required power. Thus, the third power of 5a'b is (5a^by ; 2ab f2ab\* the fourth powerof -5 — is -;;— i the nth power of 2a + 6 is (2a + 6)" . POWERS OF MONOMIALS. 240. It follows from Art. 59, that Hence, to find any power of a monomial, multiply the exponent of each factor by the exponent of the re- quired power. The coefficient, if any, may then be simplified as in the above example. 241. It follows from Art. 192, that, /2n2j\s_2rt2ft 2rt2& 2rt^6 Sa^fta Hence, to find any power of a fraction, raise the terms separately to the required power, N.E.A; — 10 146 ALGEBRA 242. It is evident that (60), 1°. All even powers of a number ai'e positive. 2°. All odd powers of a nuTnber have the same sign as the number itself. EXERCISE XLVIII. Write the second members of the following equa- tions : 1. (a2)2 = 2. (3a2c)5 = 3. (— la^bz^y = 4. (a*)3 = 6. (— 2x^yy = 6. (— ib^c" )* = 7. (a»)*= 8. (—Sx^zy = 9. (— 3«»a;3y)5 := 10. (a26»)s = 11. (5a*&2)2 = 12. (2rt"'&»c)6 = .„ /a^&N* /2.«2y\3 / 5a2&\* 16. (-f j ^ 17. _(- ^-j =. 18. (- -^) = POWERS OF BINOMIALS. 243. By actual multiplication we find that: (a + 6)i=a + 6. (a + by =a2 + 2a6 + b^. (a + 6)3 = a» 4- 3a^b + Safi^ -f fta. (a + by = a* + 4a3& + 6a^b^ + 4a63 _|_ 64. From these results it will be observed that: I. The number of terms is greater by one than the exponent, of the power to which the binomial is raised. II. In the first term, the exponent of a is the same as the exponent of the power to which the binomial is lN'V6l,lJti6>J. 147 raised ; and it decreases - by one in each succeeding term. III. 6 appears in the second term with 1 for an expo- nent, and its exponent increases by one in each suc- ceeding term. IV. The coefficient of the first term is 1. V. The coefficient of the second term is the same as the exponent of the power to which the binomial is raised. VI. The coefficient of each succeeding term is found from the next preceding term by multiplying its coeffi- cient by the exponent of a, and dividing the product by a number greater by one than the exponent of h. 244. If 6 be negative, the terms in which the odd powers of 6 occur are negative. Thus : (a — 6)4=a*— 4a3&+6a262— 4afts +6*. EXERCISE XLIX. Write by inspection the results : 1. (rt + xy = 2. (a + xy =- 8. (a; + yy = 4. (a — a;)s= 5. (ar— «)*= 6. (ar— 3/)Sz= 7. (a + ly = 8. {x + ly = 9. ix+ ly = 10. (.T — 1)« rrz 11. (a? — 1)*= 12. (y — 1)6 = 345. The same method may be employed when the terms of a binomial have coefficients or exponents. Let it be observed that the following results are in accordance with the preceding rule, provided, that a be called " the first term of the binomial," and 6, "the second term of the binomial." 148 ALGEBRA. (3a; — y^y = (Sxy — 3(3a;)2(y=') + 3(3.r)(2,2)2 _ (2,2)3 = 27x2 — 27x'y^ + 9xy* — y'. (c2 — ia;)« = (c^)' - 4(0^)3 ('x) + e(c^ya=^y — 4(c2) (ia:)^ + (ix}* = c8 — 2c»a; + Ic^K^ — {o^x^ + J^k*. 246. Similarly, a polynomial of three or more terms may be raised to any power by first writing it under the form of a binomial. Thus : (x' + 2x + sy = j «^ + (2x + 3) j ' = (x^y + 3(a;2)2(2x + 3) + 3(x=)(2a; + 3)= + (2x + sy = x' + 6x^ + 21a;* + iix^ + CSa;^ + S4a; + 27. Simplify, or expand the following: 13. {x+ sy. 14. (a;2— 2a)3. 15. (a^+a + iy. 16. (x— 2)3. 17. (a2_3«)4. ig. (.^2_a;_i)2. 10. (a + 4)*. 20. (fts + 4c)3. 21. (1 — 2a + a2)*. 22. (l + 2a)s. 23. (2a2— 3j/)*. 24. (5 — 2£P — a!2)3. 25. (2a— 1)B. 26. (2«3— Jit,')5. 27. (ipa — 5aj + 2)^. EVOLUTION. ROOTS OF MONOMIALS. 247. Since the third power of 2aH is 2^ a^b^, the , cube root of 2^a^h^ is 2a^b. Since the fourth power of + Sa^jz q^ of — Sa^i^ is + 3*ai2&8^ the fourth root of + S^a^H^ is ± Sa^h^. Hence, to find the root of a monomial, divide the exponent of each factor by the index of the root, and take the product of the resulting factors. 248. Since the third poAver of — is — 5' the cube root of —X is -— - teVOLUTlOIf. 14& Hence, to find the root of a fraction, extract the re- quired root of the terms separately. 249. It is evident that (242), 1°. Any even root of a positive number will have the double sign^ ±. 2°. There can be ?io even root of a negative' number. 3°. Any odd root of a number ■will have the same Hgn as the number. Thus, l/l6a< = ± 4«2 ; VZZ'^v. = _ 2«2. Bull/ — «^ is neither -1" a nor ^^^ rt, for (-j- a)^ = -|- a^, and 250. The indicated even root of a negative number is called an Imaginary or Impossible Number. EXERCISE L. Simplify : 4. Vofi. 5- v/'si^sy^ 6. J/216a3a!9s6. '• V'^x^. 8. g,''27ai263. 9. J/i21aioft4^, 10. t-^l6tt8. 11. ^^_l25ce3/i5.12. ^^_rt2ii!)i2c9*». l/!^. 14. i/l?^^. 15. i/: jg ^ . ^^ .. _,.„„.„ .. _., 8ffl6C»Xl2 9«a V 64c« "^ 27b^y^'^ 16- 1/16"+ 1/257 17- i/25"+ l/l6"+ V^^'. 18- l/81a2 — 1/49^37 19. i/l00a*P"+ Kl25a6FT 160 ALGfiBRA. 02 2a + 6 + 2ab + b^ + 2ab + b' SQUARE ROOTS OF POLYNOMIALS 251. Since the square root of a^ -\- 2ab -{- b^ isa -\-b, we may determine a general rule for extracting the square root of a polynomial by observing the manner in which a -\- b may be derived from a^ -\- 2ab -)- b^. a2 _(. 2ab + l>^{a + 6 The first term is a", and its square root is a, which is the first term of the root. Subtract its square, a', from the polynomial, and bring down the remainder + 2ab -\- b^ ■ Now we may obtain the sec- ond term of the root, + 6, by dividing the first term of the re- mainder, -|- 2ab, by the double of the part of the root already found, 2a. Next, we add this new term, 6, to twice the part of the root already found, 2a, for a complete divisor, 2a -{- b. Multiply this divisor by the new term, b, and subtract the prod- uct from the remainder. EXERCISE LI, 1. Extract the square root of 25a;* — iOx^y -f IQy^. 26x* — iOx^y + 16!/2 {5x^ — iy 25»* lOx'' — iy — iOx'^y + 162^2 — iOx^y + 162/2 The square root of the first term is Bx^, which is the first term of the root. The second term of the root, — iy, is ob- tained by dividing —40j:2s/ by 2 times Ba:^ lOa:^, and this new term is also added to the trial divisor, 10*, to complete the divisor, lOx' — iy, 252. The same method will apply to polynomials of more than three terms, if, after each subtraction, the part of the root already found be doubled for a trial EVOLUTION. 151 divisor, and the new term of tlie root be added to the trial divisor for a complete divisor. Before beginning the work, always arrange the poly- nomial in reference to a certain letter. 2. Extract the square root of 5aj2 — 12a;3 + 6x + ix^ + 1. First arrange the polynomial in reference to x, and then pro- ceed thus : ix* — 12a;!' ^6x2 + 6a; -f- 1 (^x" — 3a; — 1 ix* ix^ — 3x 12x3 4- Ba:2 + 6a; + 1 ■ 12a;» 4- ^"^^ ix^ — 6x — 1 — ix' + 6a; + 1 — ix^ 4- 6a: -I- 1 Extract the square roots of : 8. 9ffl»6* + 12a62 ^ 4, 4. 64a;6 — 80a!*3; -f 25x'^y'i. 5. 100 a^b*c^ + 20a3&2c -|- 1. 6. X* — 2x^ -\- Sx^ — 2a; + 1. 7. X* — ix^ +8x + 4. 8. 4a;* + 12a;'' + 5x^ — 6a; + 1. 9. 4a;* — 4a;3 + bx' — 2a; -f 1. 10. 4a;* — 12rta;3 + 2oa^x^ — 24a»a; + 16a*. 11. 25x* — 30aa;» + ida'^x'^ — 24aSa; + 16a*. 12. a;B — 6ax^-\-15a^x*—20a^x^+15a*x''—&n^x-{-a«. a;2 4 IS. x*—x^ + -^ + ix — 2+ ^■ a* , a^ a^ „ x^ 14. -. — 4- s — +6xy + x^ 8y3 + Uxy' sy^ + 6x 'y + x=\_ 2!/ + » 12y + 12xy^ + I2xy' + tix + 6* 'y +=^'' ^y + x3 The cube root of the first term is 2y, which is the first term of the root. The second term of the root, + x, is obtained by dividing + 12xy^ by 3X (2j/)^, or 122/2. Now, to the trial divi- sor 12!/", we add Sy^2yx -\- *", and obtain the complete divisor, 12|/2 + 6xy + a;2 256. The same method may be applied to polyno- mials of a greater number of terms, provided that, after each subtraction, the part of the root already found be regarded and treated as the first term of the root. 2. Extract the cube root of 64a;8 — liix^ + 156a7* - 99a;3 + 39x^ — 9a; + 1. I 4a:2 — 33? + 1 64a;» — iHx^ + 156x« —d9x^+Z9x^—9x + l 48»« Six' ^ — 36a;3 -|- 9a;2 I — lUx'^ + 156a;« — 99x^ + 3S)a:2 — 9a; + 1 48a:* — SGx^ + Sa" ( — Uix<' + lOSa:^ — 27a» S(,ix^—3xy= 48JC*— 72x34- 27a;2 + 12a;2 — 9» -f 1 48a;* — 72a;3 + 39a;2 — 9a; + 1 48a;* — 72z3 + 39a;2 — 9a; -f 1 48x* — 72a;3 + 39x2 _ 9a; -f 1 For convenience of arrangement the root is placed above the polynomial. The first term of the root, ix^, is found by taking the cube root of the first term of the polynomial, and the first trial di- visor, 48»*, is obtained by taking 3 times the square of the first term of the root. The first complete divisor is found by adding to the trial divisor 3 X i^^^) X (— ^^) + (— 3a=)'- 156 ALGEBRA. The part of the root already found is ix' — Zx, which wo now call the first term of the root. Hence, 3(4«^ — Sx)^ = iSx* — 3C«' -|- 27»^ is the trial divisor, to which we add Sx (ix^ — 3*) X 1 + 1) or 12«2 — 9* -j- 1, to obtain the complete divisor'. Find the cube roots of : 3. a3 — 9a2 + 27a — 27. 4. 1/3 4- 12y2 + 48y + 64. 5. 8a3 — S6«26 + 54a6a — 276s. C. x6 + Gx^ — 40J73 + 96a? — 64. 7. x" — 12a;5 -f- o4a!« — 112ii!S + 108a;2 — 48ir + 8. 8. 8x« — 36x6 _|_ 66j;* — 6Bx^ + 33x^ — 9x + 1. 9. Sx« + 48ca;5 + 60c^x* — 80c^x^ — 90c^a;2 + 108 c^x — 27c6. 10. x» + 6a;8 _ 64a;6 — 96a;6 + 192ir* + 512a!3 — 768 a; — 512. 11. a» + 3a2(a? + 5) + 3a(a;2 + lOa; + 25) + x^ + lox^ + 75a; + 125. 12. as _ j3 _|_ c3 _ 3(a2ft _ a2c — ab" —ac^ — hH + ftc2) — 6a&e. CUBE ROOTS OF ARITHMETICAL NUMBERS. 357. The method of extracting the cube roots of polynomials, when considered in connection with the following principle, enables us to devise a rule for ex- tracting the cube roots of Arithmetical numbers. 258. Principle. T'he cube of^ a number contains three times as many Jigurcs as the number^ or three tim.es as many less one or two. For, the cube of any number: 1°. Between 1 and 10, is a number between 1 and 1000. EVOLUTION. 157 2°. Between 10 and 100, is a number between 1,000 and 1,000,000. 3°. Between 100 and 1000, is a number between (100) ' and (1000)3. Hence, if a number be pointed off into periods of three figures each, beginning at the right, there will be as many figures in the integral part of the root as there are periods in the number. EXERCISE LIV. 1. Find the cube root of 15,625. 16,625 I 20 + 6 o3 = 8,000 3a2 = 3 X (20)2 = 3 X 2= X 100 = 1200 3(i& = 3 X 20 X 5 = 300 62 = 52 = 25 7,625 7,626 1525 Separate the number into 3-figure periods. Let a denote the value of the figure in the tens' place of the root, and b that of the figure in the units' place. Then a must be the greatest multiple of 10, which has its root less than 15,000; this is 20. Subtract a^, or 8000, from the given num- ber. Divide the remainder by the trial divisor, Sa^, which ie 1200, and the quotient is b, that is 5. Next, complete the divisor by adding to the trial divisor Zab -\- b'^, or 300 + 25 = 326, which gives 1526. Multiplying this by 6, or 5, and sub- tracting the product from 7C26, we find the remainder to be 0; hence, 20 -|- 5, or 25, is the required cube root. If the root contains three figures, proceed as in square root. See bottom of page 152. 158 ALGEBRA. 2. Find the cube root of 12,812,904. 12,812,904 [230 + 4 2= = 8 3 X 2' X 100 = 1,200 3X2X3X10= 180 3= = 9 4,812 1,389 4,167 3X (23)2 X 100 = 168,700 8X23X*X10= 2,760 i' = 16 645,904 645,904 161,476 After separating the number into 3-figure periods, we find 23, the cube root of ttie greatest cube contained in 12,812, as before. We then consider 230 as the new value of a, and pro- ceed to find the new value of 6, 4, as in the last example- NoTE. — The notes under Art. 254 are applicable to cube root with the change of " square " to " cube." Find the cube roots of: 3. 13824. 5. 421875. 7. .019683. 9. 34.965783. 4. 77308776. 6. 109.215352. 8. .726572699. 10. 122615327232. M V A.i, u . ^ 1331. ^ 59319 11. Find the cube root of j^^ , of ^^j^. 12. Find the cube root of 2 to three decimal places. 13. Find the cube root of 3 to three decimal places. 14. Find the cube root of 5 to three decimal places. FRACTIONAL, AND 'NEGATIVE EJJPONENTS. 159 FRACTIONAL AND NEGATIVE EXPO- NENTS. SIGNIFICATION AND PRINCIPLES. 259. To find tlie ineaning of a fractional expo- nent. T he sixth power of a is a" . The cube root of the sixth power of rt is a •* ^ a''. In general, the mth power of a is a". The )ith root of the mth power of a is « » . Hence,, whether m be divisible by « or not, a" may denote v^a", or t/ie nth root of the mth power of a. Thus : c^ = VW; x^ = Vx^. 260. To find the meaning of a negative eacpontnt. By Art. 128. we have «" =1 • (1) \_ (1) --«3 (127), '*'' ^o"' In general, (1) -^ a» , a-» = -_. Hence, a~" is the reciprocal of a"- Thus, a- -^; \j) =[-) ' ^FT = «*• Hence, a factor may be changed fi-om the numerator to the denominator, or vice versa, if the sign of its ex- ponent be changed. Thus, ^_-i^ = -3- = a^hcr-n-^ = -.-^i^" 261. It has been shown that, when in and. M are in- tegers, i- a" X &" = («&)" • ^II- «"'"^ «" = <<" II. a™ X a" = «"+"• IV- («" )" = «"*"• /"— ». 'IC© ALGEBKA. It is now to be shown that these formulas are true, whether m snd n be integral or fractional, positive or negative. Notes. — 1. In addition to the notation under Art. 222, we shall employ the following: (1)» is read, taking the nth power of both sides of equation 1; (2)" is read, taking the Jith root of both sides of equation 2; and similarly tor other numbers. 2. For the present, p, q and r will denote only positive num- bers; and s and < will represent positive, entire or fractional numbers. 262. Case I. — To prove that Formula I is true: 1° When m is a fraction. We have a X'' — ab (1) ay , Art. 240, fflp X '"' = (ab)p (2) — _p p p (2) ' , Arts. 247, 259, cr X & "" = (.ah) ' (3) 2°. When m is negative. Wel^a^^ ¥Xy = ^ (1) (1)'' «vX^=-py7- (2) Or, Art. 260, a-' X ''- = («'') - (3) 263. Case II. — Toprove that Formula Ills true: 1°- When m and n are fractions. We have ap X "'=«''■*' ' (1) (l)*- Arts. 247, 259, o'Xa' =«' •• (2) 2°. When m and n are negative, 1 . . 1 _!_ ' + ' Or, Art. 260, o- • X o~ ' = o- ' Thus: a^X «" =o^; o^ X «^ = «^ "^ ' =«"■ We have — V — _. — : o' ^ a\ a' ■ FIIACTIONAL ANt) NEGAT'IVE EXPOiNENTS. 161 264. The process of proving the generality of For- mula III, is similar to that of Case II. 265. Case IV. — To prove that Formula IV is true: V. When n is a fraction. We have, Art. 240, («•')''= «"""'■ (1) (1)' Art. 259, («»')'■ =n""^- (2) 2° When n i-i tugative. 1 1 We have, ^^„,y ^ ^„. Or, (n"')-- " = a-"" = a""- '. Thus: (aO^=i»; (J)'* = a~ ' ; (oT *)~^ =a. EXERCISE LV. 266. Express with fractional exponents : Express with radical signs : Express with positive exponents : 4. o-S; 2a;-22/-S; 5a6-* ; x-^ -\- 2ar^y — ^oc-^y^. Express as entire numbers : a _ ab _ liix^ . b . c^ . 5a^cx ^- b' c^x' n^' Sac' '^' b'dy" Simplify : C. a^xa^; b*xb^; c^Xo^; fl?^xA 7. a*Xffl ^; «^x« ^; « xa ^; "^ xw^ M.E.A. II 162 Al.GEIJKA. Simplify : •8. a*X ^/a ; i/rx c"^ ; ^/^X V^ ; V^XV^ 9. aH^cxab'^c^; fl^Pc^Xa^&^c"^. 10. a^Xa^Xa~^ ; V^ c* X c~ * r^ x ^" ^^~ ^ . 11. a*-^a^; e^-=-c^; x^—x'^ ; if'^—i/K 12. (M*)^-(«2)=; ,r*--(a;-2)i; 10x^yi--^5y^i/^^ 13. (mt)% yr; (3,-4)1; (.-!)- 1; (ah-i)-\ U. (2x^yiz~^) % (— 3a-26-^c^j>^)~^. 16. 16^. lel = ({/i^)' = (±2)' = ± 8. "■ (^r. (5" = (?/ = (»)' = (I)' = f 18. (32)*; (64)-*; (64)-«; {MyK 19. Multiply «4 — rt^fii + aifti _ ;,! by o^ -f i*. a 11 11 a a* — a'^h" -\- a'^b? — h* 1 1 a* + 6« i 1 2~1 1 3 a — a«6« + 0*62 — a*b* 3 1 2 1 13 + g^fc" — a<62 -f a*b* — b a _ 6 FRACTIONAL AND NEGATIVE EXPONENTS. 163 Multiply : 20. a-2 — 2a-i6 + Ja — a&s by'a-s + 2a-2&. 21. x-^y^—x-^y — 2x-^hy2x^y-'^ -}-2x^y~^ —ix*y-«. 22. a^ + h^ + a~h by ab'^ — a^ + h^' 23. ar^ — asT/^ + x^y — y^ by x -\- x^y^ + V- 24. a ^ + a*6* + 6 by a^ — a^fi^ + 6. 25. 2x^ _ 307^ _ 4 + a;"^ by 3a;^ + a? —2x^. 26. a?" ^H-Sa;" ^+3ar ^+1 by oT —3a; ^+3a; ^—1. 27. Divide a;^ — a;^ _ 6 by a;^ — 3. ^f _ a;i _ 6 I z^ — 3 K^ — 3a:^ a^ + a + 2a!^ — 6 2a;^ — 6 b2 — xv^ 4- a;2^« — 1/2 bv x^ Divide : 28. x^ — xy^ + a5% — y^ by a;^ — y^. I 1 29. a; — y by a;* — y*. 30. a + 6 by a^ + 6^. 31. x^ + x^a^ + a^ by a;^ + x^a^ + a^. 32. 2a;6y-8 _ 5a;^y-2 -|- 7a;3y-i — bx'^ + 2.rj/ by a;^?/"* — x'^y~^ + *'2/'^- Extract the squaie root of : 33. 9a!-4— 18a!-sy^+15a;-2y— 6a;-ii/^+y2. 34. 4o— 12aH^+96^+16a*c^— 24&^ci+16c* Chapter VII. RADICALS. DEFINITIONS. 267. A Radical is a root of a number, indicated lij a radical sign or a fractional exponent. As, J/il^ ai. Va + 6, (a — a;)^. When the root can be exactly obtained, it is called a Rational Number; and when it cannot be exactly obtained, it is called an Irrational Number or Surd. 268. The Degree of a radical is denoted by the in- dex of the radical sign, or by the denominator of the fractional exponent. Thus : 1°, 1/19, or a*, is a radical of the second degree. 2°, 1/57 or 6*, is a radical of the third degree. 3°, V 3, or x*, is a radical of the fourth degree. 269. A Quadratic Radical is one of the second de- gree. 270. li an indicated root is multiplied by a number, the former is called the Kadical Factor, and the latter the Coefficient of the radical factor. Thus, in 5l/ 2 and ((« + hyc, /"F and (« -|- Z/)^ are the radi- cal factors, and 5 and e are the coefficients. RADICALS. 165 271. Similar Radicals are those whose radical fac- tors are the same. As, 5l/y and aVY; mVWc and nVh'^c. 372. A surd is-in its simplest form when the number under the radical sign is a whole number and as small as possible. REDUCTION. EXERCISE LYI. 273. To reduce a rational number to a radical. 1. Reduce 5 to a radical of the 3d degree. 5 = 5* = (63)^ = e/ 125: 2. Reduce 2a^h to a radical of the 4th degree. Complete the second members of : 3. ha = ■^/ . 4. ±4ffl36ac = J/ . 5. — 2a2 = i/ , 6. 2a; + 1 = |/~ 7. 3c2ar = J/ . 8. (a;— 2)=K" — 3aa;2 ^ y . lo. (aj^ + as)* == t^ 11. ^ = K—. 12. -^ = K—. 274, To carry the coefficient of a radical under the radical sign. 13. In 5a l/26, carry 5a under the radical. BaV2b = l/(5a)^ V2b == V25a^ X 26 = T/aOa^ft. 14. Complete this : 2x^ Vhx = ?/ . 166 AT.GEBRA. Complete the following : 15. ei/V= V . 16. (a + 6) }/a'= V 17. sV¥=i/ . 18. (a; — 2)l/S+2=l/" 19. 3a2i> i/2a = f/ • 20. (a — b) t^c(a — 6)-3 = y . 21. —2a^b^y3aH = f/ ■ 22. a / c , a ._ _3/4 23. T 17' 3 = K" 24. ^,/i_^+^ = 3^ 275. To remove a fastor from under the radical sign to the coefficient. The reduction Is pertormed by reversing the process of Art. 274. 25. Complete this : 5t/ 12a^c = ( ) ;/ 3c. 12a'c -4- 3c = ia^. *•. Sl/TIo^c = Sl/io^ VSc = 5 X 2a l/ic = 10a VSc. Complete the following : 26. V8 = ( ) 1/ 2. 27.- 2e/i6,= ( ) e/l. 28. i/27 = ( ) i/3. 29. 5^600 = ( )l/l. 30. i/l8b = ( )t/6. 31. 3e/80 = ( ) ^/g. 32, t/567 = ( )i/7. 33. 4t/l250=( )e/2. 34. l/25a3 6 ^ ( ) y^ gg l/r92a6ft?e"^( )i/-'3^. 36. V432^aW=( )Vi^. 37. 7l/62o«*64c=( )W. 38. ^l/| = ( )v^21. 39. 6?/^=( )e/i8. RADICALS. 167 Reduce to simplest forms : 40. 3l/a*F. 41. 5l/8pPT 42. 2\/98a^b*x». 43. Bl/Tox^y.. 44. VT92aW&r 45. 6aVT80a^¥c^. 46. Vsia^b. 47. ^^160i«^. 48. 2j/108a:62. 49. VbOa^c. 50. t/l62a563c^ 51. 4aF64aiop"^ 62. i/l8a?sy* — 27aj*ys. 53. i/ax^ — 6ax +9a. 54. i/(a;a— y2) (OJ + j/). 55. Vl92a^b^ + S20a^b*. 66. i/f: 1/1 = ■/fxl = v^^xe = iv/e: 57. l/|7 58. l/|7 59. t/^ 60. l/^ 61. |l/>. 62. 11/ 1 63. |l/^. 64. f J/^ 65. VJ. 66. f/^ 67. al'"'^. 68. I""^^ 376. To change the degree of a radical. 69. Complete this : V2a^=v^ '. 12 ^ 3 = 4; (2a2c)« = 16a8c' . .-. V2a^= ^/16o»c*. 70. Complete this : l^d^ = V T 2 H- 6 = J ; (8a«)^ = 2a^. .-. v^te* = j^2a^. Complete the following: 71. V^ia^^ —V . 72. V^^ =V 73. l/l6a*b^c^^=V • 74. V2ab^^'^V' 75. V2ba^b^c^ =V ■ 76. Vbax^ = V 77. Va^b^c^ =V . 78. V—ia^s= ^V~ 168 ALGEBRA. = ( )^- 80. (ayS)* = ( )*; (a^fte)* = ( )*; (asfc*)* 377. Tio reduce radicals to the same degree. 81. Reduce Vi and f/3 to the same degree. The L, C. M. of 4 and 6 is 12. Reduce to a common index : 82. 1/ 2, K3 and v^5. 83. 3*, 2^, and 5* 84. 1/3, V^ and F 7. 85. 4^, 2^, and 7^ 86. H, K2 and ij/T 87. 3^, 7^, and 4^. 88. ig/^, e/p and f/^. 89. ?/^, e.''^ and ^t^^ 278. To compare surds. EXERCISE LVII. 1. Which is the greater, 9l/"2 or 7|/'3.? 91/2 = 1/81x2 = 1/1627 t/I = 1/49 X 3 = i/UTT .■ . 91/2 is greater than iVz. 2. Which is the greater, V^ or V^ ? l/l = V~i; l/~3 = v/"y, ?/"3 is greater than ^'"g, RADICALS- 169 Which is the greater : S. 3l/7 or 2t/15? 4. 5l/6 or 6l/5? 5. 6l/n or 5T/T5f ? 6. lll/"6 or 5l/29? ?• e/2ore/l? 8. t/4 ore/5? 9. l/f or ?/||? 10. fl/"7 or fl/lO? 279. To atM and sitbtract radicals. EXERCISE LVIir. 1. Find the sum of aVlSa^x and ■t/32a^x. VZia^x = Vina* X.^aa: = iaWiax. 3a'i/2a3c + 4aV2ax = (3ar2 + ia^)V2ai, = laWiax. Simplify : 2. i/50 + l/8. 3. 3i/l2 + 2i/75— 4i/48. 4. ,/27— 1/12. 5. 4i/l8 — 3i/50+ 5i/98". 6. 5i/aac + av/DeT 7. ?/320-2?/i35 + Sf/ia 8, 1/320862 + i/iSosFTe, 2i/243rt«C + 5^^19206^. 10. 5^24 + t/54. 11. 4^/500079^2 _3e/108a;9^ 12. l/SeSa^ajSy — \/2^a^x^ + 2aa?ST/3ay. 13. V2t>6a^x^ + aFl08a3a;* — 3a'>xV32x. 14. •!: + i/f 15. vj—i/i. ^ t, « X^ f /T 16. Ji/F+ |i/^ *^- 6 1/ IT <^ 1/ ^i 18. I ri + m: 19' IX -2j- -«T/ IF' 170 ALGEBHA. 280. To multiply and divide radicals. EXERCISE LIX. 1. Multiply h\/2aH by 3ai/6a2. 2. Divide 5^^ by 3i/27 6^T = 65/167 3l/2" = 35^87 5^16 ^S^/s (5 -^ 3) S/l6-=-8 = lfv''2: Simplify: 3. 4i/6a X Si/Soc! 15ai/l2a=6 5. 7i/l0a2 X 2i/5fflft2. 7. 5i/14aB X 2«i/7a2c. 9. 6i/l2 X 3v/37 11. .|i/^ X Av/^ 13. 4e/|' X se^i: 15. s^'T X 41X3: i. 15i/6a8a; ^ 5i/2a. 6. 9a;i/5ia2 ^ SoJi/sT 8. 12a^\/d6a^x -h ^v^2aF. 10. 61/20- - 3i/l5. 12. fv^- -Kf 14. iKl- -i^T 16. 2?/3 - -3i/T 17. X f 18. X Note. — The wth power of the »ith root of a number is equal to that number. Thus, (t/T) = 7, {yr+ X + 3;-]^'= 1 + a: + a;2. 19. Multiply i/T + 4i/T by v"^ ~ 7i/Y. VT + 4t/T ■/¥ — 7i/T 2+41/6 — 7l/T 28 X 3 _ 3l/ 6 — 8i . 82 — Sv' 6. HADICALfe. l7l * Multiply : 20. i/T + i/T by i/T — 1/5^ 21. i/T + i/y by i/T + i/87 22. 2i/T — 5i/2" by 3i/3" + iv^Y. 23. 5 + 3i/T by 9 — 4:t/T. 24. 1X2" + i/T + 1/5" by i/T — 1/^ + i/T 26. 3v^ — 2i/"6' + i/T by 6v/T + 4i/T + 2i/7: 26. i/^ + i/T + t/c~ by i/T — i/TT 4- i/'cT 27. 1/1 + a; + 2i/l — a? by 1/I + 00 — 5i/l — a;. 28. i/2+3a! + 71/3+2S by 3i/2+35 — 4i/3H-2a;. 29. 1/1 +ar + 2i/T+2x — 3i/l— Saj by 1/1"+^ — 21/1+2* -I- 3i/l—'6x. INVOLUTION OP RADICALS. 281.— EXERCISE LX 1. Raise l^ 2 to the 9th power. (v/"2)» = (2^)9 =2^= 23 = 8. 2. Raise 2^^ to the 5th power. (2v^) = = 2^ (3) ^ = 2* (3') ^ = 321/2437 Find the : ' 3. 4 th power of 2i/«. 4. 3d power of Bal^x. 6. 2d power of Sv^wi^ 6. 4th power of 2a;2 v^gT 7. 3d power of — f/a^x. 8. 9th power of — xVa^. 9. 6th power of f 1/37 10. 10th power of —J1/2T 11. 2d power of a?!/ y- 12. nth power of t/^. 172 AI-GEDRA. Simplify (245) : 13. (l + i/"!)! 14. (l — l/li)? 15.(2 + 1/5)! 16. (1/^ + 3)? 17. (1/^-1)* 18. (l/F+v'^). 19. (2 + e/^)!. 20. (3--J/1)! 21. (v''2 + e/l)! 22. («*+&i)! 23. („l+ „^)! 24. (v/"^+e/i;)t 25. (1 +.r + V14- x). 26. (1/1 + *• + 1/1 — *■)• EVOLUTION OF RADICALS. 283. EXERCISE LXI. 1. Extract the cube root of 2i/l) (2i/!6')3 = 23(6^)^ = 2h°- = VY v-^ 2. Extract the fourth root of ft* ?/T(5 (a^Vli)* = (o*)* (l6=)» = a(l6»)^ = 0(2)3 ^ aV2'. Find the: 3. Cube root of 8i/a^. 4. Square root of 5^^16. 5. Fourth root of dVa^. 6. Cube root of 27v/87 7. Sixth root of x^^-i/a^. 8. Tenth root of a^Vb^'o. 9. Cuberootof y ]/ —' *®' Fourth root of^ |/-- 11. Square root of x -\- 2\/xy -\- y. 12. 'Square root of v^ — 2t/ub -\- v'^. 13. Square root oiV~a-{- 2Vab + f^ 14. Square root of ^Va^ — 20\/ab + 2bVb^'. 15. Cube root of a — sVa^ -f sV ax" — x. 16. Square root of a -f 2i/a6 -(- i — 2i/ac — 2i/ bo -{-c. ■EVOI.DTIOM OF KADICA-LS. 1T3 283. To extract tJie square root of binotnlal surds. 17. Extract the square root of 8 + 4i/37 Problems of this kind may' often be solved by in- spection, thus : since the square of i/^ -\- i/TT = (a+6) -t-2v/«6, the square root of (a+6)+2T/«6 is v^+v^b. Hence, if the given expression be written in the form in -{- 2i/ji^ we have only to find two numbers whose sum is m and whose product is n, and connect their square roots by the intermediate sign of the given ex- pression. Thus, 8 + 4i/"3 = 8 -1- 2 X ^^^ = 8 -f- 2/127 Now, we find by inspection that 6 -|- 2 = 8 and G X 2 = 12. Hence, the required root is i' 6 -|- V^ 18. Find the square root of 12 — 8i/2^ 12 — 8i/'2 = 12 ^ 2v/32r 8 -|- 4 = 12 and 8 X * = 32. Hence, . ^12—81/2 =1/4—1/8 = 2 — 21/2. Find the square root of : 19. 4 -f 21/37 20. 7 — 4i/3r 21. 7 + 2]/To. 22. 18 + 8v/5T 23. 17 + 4i/l57 24. 13 — 2i/30. 25. 10 + 2\/n. 26. 11— 6t/2T 27. 38 — 12i/ia 28. 16 — 21/55". 29. 35 + lOi/lO. 30. 24 — i/252. 284. General Solution. Let us suppose 1/ »» -|- v' n ^ /a -}- /ft. 174 ALGEBRA. Then, from what precedes, (( -\- l> = m ( I ) and iab = n (2) Square (1;, a^ + 2ab + b^ = m^...{3} Subtract (2) from (3), a= — 2ab + b^ = in' — n ,/z Extract the square root a — b ^ v m'' — h (4) Add (1) and (4), reduce, a = \{in -\- V^m'^ — n). Subtract (4) from (1), b = \{m — v^H' — n). 1/ OT + i/?i— 1/ .J(ot + v^m' — )i) + "|/ i(«s — -H). 31. Find the square root of 17 + A\/).a. 17 _(_ 4l/l5 = 17 + 1/240. Here, 9» = 17and )j = 240. Sub- stituting in the above we have j/l7 + i/240 = 1/ i (17 + v-'4'J) + l/hO-'i — y'i'-^) = i/Ii + 1/57 Ans. The preceding examples may be solved in a similar manner. IMAGINARY NUMBERS. 285. An Imaginary Number is an indicated even root of a negative quantity (^50) ; as 1/ — a, t/ — rt^. In contradistinction, all other quantities, rational or irra- tional, are called 7-eal quantities. 286. Imaginary numbers jnay be reduced to one form. Thus, V—m'^ = v''i)i-( _ 1) = mi/;^ \^ — n = v^llc"^^!)^ = i/ jT \/—T. IMAGINARY NUMBERS. 175 287. Since 1/ — 1 is the square root of — 1, v^~^ I X 1/^^ 1 = — 1. Therefore, (v/^-rr^-i; and so on. Hence the successive powers of v'^^ 1 form the repeating series, + 1/— 1, —1, — 1/— l, +!• EXERCISE LXII. 1. Multiply 2 + iA=^ by 2 — i/^oT 2+1/^1^9 = 2+3v/— 1, and2-i/ — 9 = 2-3iA-^l (2+3iA=n) (2-31/:^) = 4-9(-l) = 13, Ans. 2. Divide 1/— 185 by 5i/— 1. ^IIl85 = i/T85 v/— 1, and Si/^^H! = i/25 v/ — 1 ^185v/=:i fT85 1^ li/lSS: Simplify : 5. 1/12 X 1/— 8. 6. 1/15 -- 1/— 5. 7. 4i/^"3 X 2i/=^'3: 8. 81/— T -- 21/^—^0: 9. (2v/=:4)3, 10. (5i/=^)2. 11. (3i/=^)*. 12- ( 2i/= 2)5. 13. (2 + 5i/^^) X (2 — 5v/— ^)- 14. (3 + 7i/— T) X (3 — TV— 2)^ 15. (i^ + 4v/=^) X (t/3 — 4i/- 5), 176 AI.GEBRA. 19 16. (oi/— 3 + 4i/^"2) X (5i/^ 3 — 4i/— 2). 17. (1 + 2v/— "1)3. 18. (1 — 3i/:=n')2. Verify the following : SIMPLIFICATION OF IRRATIONAL FRACTIONS. 288. To simplify a fraction having an irrational denominator is to reduce it to an equivalent fraction whose denominator is i-ational. The process consists in multiplying both terms by such a number as shall produce the desired result EXERCISE LXIII. ' 289. When tJie denominator is a monomial. Reduce the following to equivalent fractions having rational denominators : 2 -\-x '■ w Multiply both terms by 5^25, (2 -\-x) V25 _ (2 +x) v'lE (2 4- a:) VIE VlXVIl = i-'m = 5 - 3 „ 5 ^6 1/^ 1X8 1/I8 2 f ./^ 6 ^^ — ^•^ 7 3 + i/"5 i/l ' ' 2V\ i/=~2 ' SIMPLIFICATION OF IRUATIONAL FRACTIONS. 177 8 Uli^ 9 6 + 3i/^ j^ na + ai/— "2 290. Wlien the denominator is a binomial, cott- taining only radicals of the second degree. Find equivalent fractions with rational denomina- tors, for the following : Multiply both terms by i/T -\- \/Y (the denominator with its intermediate sign changed), y/ 5" + v^ (i/y + i/2 ") (i/T + 1/2") _ 7 + 2/In v/y- v/2 = (,/y _ y2')(v^-5-+ v^) - 3 12. ^=^^. 18. '-±^. u. 1^±^. ts 5+1/-7 v/-"8+4- i/"6"-3t/-^ -3 15. ;=■• I"' — 7 ■■■'• / — ■;==■ 5— i/— 7 1/ — 8—4 1/ 6 H-3i/— 3 18 ?±i^S3- 19. "~'^— Z^ 20. ^+i^^-4 v^ffl+a; + ]/a—x ^^ i/g^— 1 — i/V +1 PROPERTIES OF IRRATIONAL NUMBERS. 291. I. — -A/b irrational number can he expressed by a fraction. For, if possible, let v/"=^•• ....(1) Then, (■!)», 6» a = ^ •••• (2; Now, if — is in its lowest terras, — is in its lowest, terms, and. c x = 21. 61. 3a;2 + 25a; = — 8. 62, hx^ — 46a; = — 9. 63. lOa;^ — 23a; = — 12. „^ ic^ , a? 1 4a;2 17 x «*■ T + 8 =-24- fiS- -3- - T - 3 =«• 66. 6a; + '—^^ = 44. 67. 4a;— --^'=14. 7 3 22 6a; + 35 — 3x X -^' X 5 — X 15 — X X — i 8a; « _20_ 68- s — x; — — r— -= -1- 69. a'^ — 4 a; + 2 5 7n --- fi -- „, 3a;— 7 4a;— 10 „1 70- ^+2 - 6 = 3^- 71. -^- + -^-5- = 3^. 79 ^° 10 27 ^^ 2a; 2a; — 5 1 '2-^^5-13=--. 73. ^—^+--- = 83. 74. '>^ + ^ ^ ^ 2_ yg_ x — i/x-j-i ^ _5 a; — i/a; 3 iP + i/i"+l H' QUADRATIC EQUATIONS. 189 2a; — 3 Sa? — 5 5 77 3aj— 5~^2a;— 3~ 2 3x —2 2a; — 5 8 2a? — 5 3iC — 2 ~ 3 a;+3 2.r; —3 x — 3 '^* ar + 2 ~ a; — 1 + x^^ = ^^ 79. lO(2a;+3) (a;— 3) + (7a;+3)2 = 20(a;+3) (a;— 1). 80. x" — 2aar + «« — 6^ = 0. 81. OCX'' — (6c — a one of the roots, as r, is positive and the other (?■') negative. Again, since r — »■' = — j>, the negative root is numerically the greater. Thus, the roots of x^ _|_ 5j; _ 14 = 0, are + 2 and — 7. In the second form, since r X '"' = — 9) one of the roots, as )•, is positive and the other (}•') is negative. Again, since )' — ?•' = +p, the positive root is numerically the greater. Thus, the roots of x^ — 3x — 40 = 0, are + 8 and — 5. In the third form, since )■ X '"' = -\- I1 tbe roots have like signs, and since — r — r' = — p, both roots are negative. Thus, the roots of x^ -j- 5x + 6, are — 2 and — 3. In the fourth form, since r X ''' = + 4> the roots have like signs, and since + J" -f- r' = -^-p, both roots are positive. Thus, the roots of x^ — llx + 28, are + 4 and + 7. 307. — VALUES OF THE ROOTS. In the first form. .ri) '•'=-^-1/9 +^^- 2 1/ ■" ^ 4 192 ALGEBRA. In the second form 4 + ^/^T? .(2) In the third form .(3) In the fourth form ■ 2 ^ l^ 4 •W '= 1-— i/^ — 2 ^ i 308. UNEQUAL ROOTS. The roots of an equation of the first or of the second form are unequal ; for, in either of these r—r' -2i/7+^^. In an equation of the third or of the fourth form •-»•' =2i/^-g; that is, the roots are unequal if -j- is greater or less than q. 309. EQUAL ROOTS. The roots of an equation of the P^ third or of the fourth form are equal if -7- is equal to q. 310. REAL ROOTS. The roots of an equation of the first or of the second form are real; for, in these forms, ^ + q is positive. The roots of an equation of the third or of the fourth form pi > are real if -7- is not less than q. CHARACTER OF THE ROOTS. 193 311. IMAGINARY ROOTS. The roots cf an equation of the third or of the fourLh form are imaginary if -j^ is less than g; for in this case, -7- — } is neg.T.ive (250). 312. Imaginary roots indicate' incompatible condi- tions. For, (324, K.), (^^^)'> ah .... (1) That is, the square of one-half of the sum of two numbers is greater than their product. Now, in (3) and (4) the roots are imaginary when (^yj of a; and y in - x -\- y = 8 (2) (1) H- (2), x-y = 2 (3) Combine (2) and (3) x = 5, y = 3. 29. Find the values a-s '-|- ?/' = 1008, (1) of a; and 2/ in x -\- y =12 (2) (1)h-(2), a;2- xy + y^^ 84 (3) (2)2 a;2 + 2X2/ + 2/2 = 144 (4) (4) — (3), , Saii/ = 60, or Ki/ = 20 (5) Combine (2) and (6), Art. 318. Find the values of x and y in : 30. a;2 — y2 = 33, 81. x^ — y» = 973, X —y —^. X —y =1. 82. a;*— y*=3471, 83. x^—y^=Zl, a;2 4- 2/2 = 89. x — y =1. 34. a;* + x^y^ + y^ = 133, aja — j??/ + y2 = 7. 85. X* + a;22/2 + 3/* — 931, x^ -{- xy -\- y^ = 49. 86. a; + \/xy + y = 14, a;^ + /rj/ + 1/2 = 84. 87. a;* + 2/2 +0-2/ = 133, X -\- y —Vxy — 7. 38. x« + y« = 35, 89. aj2 — aij/ = 4.5, a'^(a' + y) = 30. l/'i" + i/y = 5. 40. a; + y = 9, 41. x^ — ys ■= 19, as' + y* = 3, 372 _|_ a;y _|. j,a __ jg^ PROBLEMS. 205 330. III. — When the equations are homogeneous and of the second degree. 42. Find the values 2g^ — 4,ri/ + S.r^ =17 (1) of X and y in y^ — a?^ _. jg ^2) In (1) and (2) make y = v», x'{2v^ — iv + 3) = 17 (3) (6) 17 - 16 From (3), «" = g^a —4^-1-3 ; ^^°^ (*">. «' = ^JIT' 17 16 S 13 ••• 2v^ - 4« + 3 - ^-=a- Whence, ,; = y or y 5 In (5) make » = — ; whence, « = ±3, j'=±5. 13 6 13 In (6) make* = -r't whence, w = d_ „ ^ = ± -»"• Find the values of x and ]/ in : 43. a?2 + 3xy = 54, ;ry + 4i/2 = 115. 44. a!2 + a;^ + 4y^ = 6. 3a;2 -)_ 8y^ = 14. 45. 6a;2 — oxy + 2y^ = 12, Soj^ + 2xy — 3y' ——B. 46. 2a!a _ 7^;^ _ 2^2 = 5, Sjjy _ x^ + 6y* = 44. PROBLEMS. EXERCISE LXXII. 321. The difference of two numbers is 5, and the sum of their squares is 193. Find the numbers. Let X and y be the numbers. Then, x —y = 6 (1) and a;2+2/» = l93 (2) From (1), z= 5 -\-y Substitute in (2), (5 + J/)^ + j/^ = 193 Whence, j/ = + 7 or — 12. Substitute in (i;, x = + 12 or — 7. 206 ALGEBRA. 2. Divide 40 into two such parts that the sum of their squares shall be 818. 3. What two numbers are those whose difference multiplied by the less produces 42, and by their sum. 133.? 4. What two numbers are those whose sum multi- plied by the greater gives 144, and whose difference multiplied by the less gives 14? 5. The difference of the cubes of two numbers is 19, and their product multiplied by their difference is 6 ; find the numbers. 6. What number is that which is equal to twice the product of its two digits, and if 27 be added to it, the order of the digits will be reversed ? Let X and y be the digits Then 10a; + y = the number. Hence, 10a; -f- J/ = 2xy. (1) and 10a; + !/ + 27 = 10.V + a; (2) Reduce (2), y = a; + 3 (3) Substitute this value of y in (1), 10a; + (x -I--. 3) = 2x(x + 3) Or, lla; + 3 =jJx2 +6a; Whence, x ^ 3, or — \. Substitute in (3), y = 6, or -f 2^. Therefore the number is 36. 7. What number is that which, when multiplied by its units" digit, the product is 69 ; and when multiplied by its tens' digit, the product is 46 ? 8. There are four consecutive numbers, of which if the first two be taken for the digits of a number, that number is the product of the other two. Find the four numbers. PROBLEMS. 207 9. The fore wheel of a carriage makes 6 revolutions more than the hind wheel in going 120 yards ; but if the circumference of each wheel be inci'eased one yard, it will make only 4 revolutions more than the hind wheel, in the same distance ; required the circumfer- ence of each wheel. Let X = circum. of hind wheel in yards, and y = circum. of fore wheel in yards. 120 Then, = number of revolutions of hind wheel. 120 and — — = number of revolutions of fore wheel. Then, by the conditions: 120 120 T- = ^ + « « 120 120 ljTpt = r+i + -^ (2) mXxy, 120*= I20y + 6xy... (3) (2)X(J'+1)(«+1), 120*4-120= 120^4-120+4«y+i*+4j/+4.. (4,1 Or, ' xy = 29x — 31y—l (5) From (3), xy = 20» — 20y (6) (5) — (6), = dx — ny — y (7) WK ny + l Whence, k = — s (o) llj/2 + y 220y + 20 Substitute in (6), ^ = — ^ — — 20y 5 Whence, y =. i, or — jj-* Substitute in (8), * = 5, or — -g-- Therefore the two circumferences are 4 and 5 yards. 10. Two men, A and B, traveled from C to D, a distance of 60 miles. They started at the same time, and A reached D 5 hours in advance of B ; but if B's rate per hour had been 2 miles more and A's rate 1 208 ALGEBUA. mile less, then B would have reached D 2 hours in ad- vance of A. Required the rates at which A and B traveled per hour. 11. It requires 200 rods of fence to enclose two square lots whose contents are 1300 square rods. Re- quired one side of each lot. 12. A person bought a number of pigs for $35 ; after losing two of them he sold the rest at $J a head more than he gave for them, and by so doing gained $1 by the transaction. Find the number of pigs he bought. 13. A gentleman sends a lad into the market to buy 12 cents worth of oranges. The lad having eaten a couple, the gentlemen pays at the rate of a cent for fifteen more than the market price. How many did the gentleman receive.? 14. There are three numbers, the difference of their differences is 8 ; their sum is 41 ; and the sum of their squares 699. Find the numbers. Let X, y and z be the numbers. Then (jc — y) — (.y—z')=S or, x — 2y + 2= 8 (1) x+ y-]-z=il (2) ~ ' + 3'^ + »' = 699 (3) (2)-(l), 3y=33, .-. 2/ =11. In (2) substitute U for 2/, X -|- 2 = 30 (4) In (3J substitute 11 for y, a;' + e^' = 578 (S) ' * Combine (4) and (5), and we find a= 23, g= 7. 15. There are three numbers, the difference of their differences is 1 ; the difference of the differences of their squares is 25 ; and the square of the intermediate PROBLEMS. 209 number is equal to the product of the other two. Find the numbers. 16. The sum of two numbers multiplied by their difference is 21, and the sum of their squares multiplied by their product is 290. Find the numbers. Let * and y be the numbers. Then, (« + S*) (« — S>) = «" — 2/' = 21 (1) and («2 -{- y^)!xy = x^y + xy^ = 2!I0 (2) (l)2X»'3''i x'y^ — ix^y* +a;2s^» = 441*22/2 (3) (2)2, x'>y'' + 2x*y* -\-x^y^ — Uim (i) (4) _ (3), ix*y* = 84100 — 441»2i/2 Whence, taking the positive value, «j/ = 10 (5) Substituting in (2) , «i2 -f 2/2 _ 29 (C) Combining (5) and (6), * = 6, y = 2. 17. A person bought two cubical stacks of hay for $41, each of which cost as many nickels per solid yard as there were yards in a f^ide of the other, and the greater stood on more ground than the less by 9 square yards. What was the price of each ? 18. A grocer sold 80 pounds of tea 'and 100 pounds of coffee for $65 ; and he sold 60 pounds more of coffee for $20 than he did of tea for $10. What was the price of each 1 19. A and B together can do a certain work in 12 days; if A could do the work in 8 days less time, and B in 6 days less time, both together could do it in 8 days. How long will it take each alone to do the work? 20. The height of the walls of a rooin is 11 feet, and the length of the ceiling is 2 feet more than its width. The cost of plastering the walls and ceiling at %\ per N.E.A:— 14 210 ALGEBllA. square yard, is $38^^, no allowance being made for doors and windows. Find the length and width of the 21. A sets off from C to D, and B at the same time from D to C, and they travel uniformly; A reaches D 16 hours, and B reaches C 36 hours, after they have met on the road. Find in what time each has performed the journey. 22. A certain number of workmen can move a heap of stones in 8 hours from one place to anoth-er. If there had been 8 more workmen, and each workman tiad carried 5 lbs. less at a time, the whole work would have been completed in 7 houi's. If, however, there had been 8 less workmen, and each had carried 11 lbs. more at a time, the work would have occupied 9 hours. Find the number of workmen and the weight which each carried at a time. Note. — The longer side of a right-triangle is called the hypotenuse (H), and the other two sides, the base (B) and/ej-- pendicular (P). Formula; H' = B^ -|- P^. 23. A cord 100 feet long is stretched over the limb of a tree in such a manner that while one of its ends reaches the ground directly under the limb, the other touches the ground at a distance of 40 feet from that point. Find the height of the limb. 24. A ladder 50 feet long is leaning against a wall so as to reach a window. Now if the bottom of the ladder be pulled out 16 feet farther from the wall, it will throw the top of it 8 feet below the window. Required the height of the window from the ground. PROBLE^iife. Sll 26. A and B are two towns on the bank of a stream, and are 42 miles apart. A man rows from A to B and back again, and finds, first, that he is 3 hrs. 12 min. longer upon the water than he would hav^been had there been no current; second, that he was 8 hours longer going from A to B than B to A. Find his rate of rowing, and the rate of the current. 26. Find the price of eggs per dozen when two more for 12 cents lowers the price one cent per dozen. 27. Two farmers drove to market 100 sheep between them, and returned with equal sums. If each of them had sold his sheep at the same price that the other did, the one would have returned with $180 and the other with $80. • At what price per sheep did they sell, and how many sheep had each ? 28. A ladder just reaches a window on one side of a street, and when turned about its foot, just reacljies a window on the other side. If the two positions of the ladder are at right angles to each other, and the heights of the windows are 45 and 28 feet respectively, find the width of the street and the length of the ladder. 29. Find two numbers whose product equals the dif- ference of -their squares, and the sum of whose squares equals the difference of their cubes. Chapter IX. INEQUALITIES, ZERO AND INFINITY. INEQUALITIES. 322. An Inequality is a statement in algebraic lan- guage that one quantity is greater or less than another. Thus, a. >> 6 is an inequality and indicates that a is greater than 6, or that a — 6 is positive. Again, a <^ 6 is an inequality and expresses that a is less than b, or that a — 6 is negative. The form a >• 6 > c indicates that h is less than a but greater than c, or that & is intermediate in value between a and c. 323. The fourth particular in which Algebra is an extension of Arithmetic (205) is the use of inequali- ties in deducing certain relations and properties of quantities. 324. The principles (207) applied to the solutions of equations may be applied to inequalities, except that if each side of an inequality have its sign changed, the sign > will be reversed. Thus, 3> 2 (1) Multiply (1) by — 1, — 3 < — 2 ...(2; Square (.2) 9 > 4. EXERCISE LXXIII. 1. If 5.17 — 17 > 3a; + 7, show that x > 12. Transpose, 5a: — 3a; > 17 + 7. Reduce, 2a; > 24. Pivide by 2, a; > 12. INEQUALITIES. 21S Show that : 2. If 3a; — 5 > 13, then a; > 6. 7x ox X 8. If y — -g- > y — 3, then x < 5. 4. If 2a; — 5 > 25, and 3a; — 7 < 2a; + 13, then a; > 15 and x < 20. 6. If 3a; + 1 > 13 — x, and 4a; — 7 < 2a; + 3, then 5 > a; > 3. 6. If 2a; + 4y > 30, and 3x + 2y = 31, then a; < 8, y > 3.i. 7. If 5a; + Sy >121, and 7a; + 4y = 168, then a;< 20, y>7. Note. — In the following exercises the letters a, b, c, etc., represent any positive, real and unequal numbers, unless other- wise stated. 8. Show that a^ -\- h" > 2ab (A) Evidently, (a — 6) ^ is positive. Or, ' (a — &)2 >0. Expand, a^ — 2ab + Z>2 > 0. Transpose, n^ _j_ js ~> 2ai. 9. Show that the sum of any fraction and its recipro- cal is greater than 2. Divide both sides of (A) by ab. 10. Show that «3 -j- &3 > a26 + ai»2 (B) (A) X «. a3+fl!62>2n»6 (1) (A) X 6, a^b + b^>2ab- (2) f(l) + (2)] a' +63 >o26-f a62. 11. Show that ffl2 -f &2 + c2 > ab + ac + bc (C) By (A), «2+6-'>2a6 fl) a^' + c^ > 2ac (2) 62 +c2 >2hc (3) L(l) -^ (2) + (3)] H- 2, a2 + fc2 _^ (.2 ;> ab + nc -(- 6c. 514 ALGEBftA. 12. Show that a^ +b^ + c« > 3abc •••••• --(P) Multiply (C) hy a + b + c, and reduce. 13. Show that — y^ > V^r? (E) In (A) make r = a', r' = &^ and reduce. U. Showthat 'jtl^Jl^' > lA^i^ (F) In (D) make r — a', »" = &% »" = c' , and reduce. 15. Showthat — -^^>Vn (G) n In (E) make r = a, r" = — , and reduce. 16. Show that —3^— > e/Tt • • --(H) re In (F) make r = r' = a, r" = rr^ and reduce. 17. Showthat/-^— > «&. (K) y _L »•' 4. J-" J_ j'" 18. Show that — ■ g ■ > t/''rr'r"r'" . (L) r -4- r' r" -i- r"' 111 (K) make a -. —^ — and 6 = ^ , and reduce. 19. If a-|- 6-j- c = wi, ah ~\- ac -{- he = ti , and ahc—r, show that (1) ma>3n; (3),....k3 > 27r2; (2), .. m='>27r; (4), . . .{m^—2ny :> 21r^. 325. The principles of inequalities afford elegant methods of deducing many .useful formulas. We pur- pose now to show how they may be employed in ob- taining an approximate square or cube root of a num- ber. 326. An Approximate Koot is a number which dif- fers but little from the true root. Thus, an approximate square root of 7 is 2, or (more ap- proximate) 2.6, or (still more approximate) 2.64. 327, To find the approximate square root of a number. _ rt2 + n If a > |/ n, then — 5- — is a more approximate value of l/ M than a is. n For, since a > Vn, a^ > n, and — < o (1) a^ -\-n . Add a to both sides, reduce, — gj- — <; a (2) a^ + n _ But, (.G), ^^>v^» (2) a^ + m Hence, a > ^u > V n. That is, *" ~'~" is intermediate in value between a and ■j/'^ 2a 388. Hence, to find a more approximate value of l/~«, substitute anQi affroximate value of it for a in — ^ , and reduce the result to its simf lest form. EXERCISE LXXIV. 1. Find the approximate square root of 24. Take a = 5 for 1' approx. root. Now substitute 5 for a in ^ — and we get the 2° approx. in root viz. : tq = 4.9, 216 Algebra. a^ -L 24 Again, substitute 4.9 for a in ^ and we get the 3° ap- 4801 prox. root, viz. : -ggy- = 4.89896 +. The correct answer to eight decimal places is 4.89897949. Hence, the 3° approx. root is true to four decimals. Now the 4° approx. root, which may be found by substituting a' +24 the 3° approx. root for a in — ^ — , is still more nearly equal to the true answer. Continuing thus, the approximation may be carried to any degree of accuracy. Find the 3° approximate square root: 2. Of 2, taking 1 J for the 1° approx. root. 3. Of 26, taking 5 for the 1^ approx. root. 4. Of 10, taking 3 for the 1° approx. root. 5. Find the approx. value of l/l -J- x. Take 1 for the 1° approx. value. 1' + 1 + X 2 +x Then, g = — ^ — = 2° approx. value, \{1±JLY m n M.^]^ on ^ ,,_ 8 + 8^+ ^ and.^--^j +(l + «)J^2a + .):- «^,^ r2 1 + "2" — "8^ + fe ^''^" ~ *^® ^° ^PP''0''- value, which is true to four terms. 329, To find the approximate cube root of a number. It a i> ^ „^ then — g^ — is a more approximate value of n than o is. For, since a > yli, a^ > n and -^ < o. Add 2a to both sides, reduce, ?^1±1* ^ „ ' 3a^ ^ But (H), ^I'-iJ? . , ,_ 2E116 And Infinity. Si? 2a^ +w Hence, a > — g^ — > J/^. See Art. 328. Vn. 2a= + » That IS, — g-j — is intermediate in value between a and EXERCISE LXXV. 1. Find the approximate cube root of 9. Take a =-- 2 for 1° approx. root. 2a3 + 9 Now substitute 2 for a in — s— j — and we 25 get the 2° approx. root, viz.: t- = 2.083 -)-. 25 2rt3 + 9 Again, substitute jg for a in — n^ and we 23i01 get the 3° approx. root, viz. : rfogn = 2.C8008 -f, which is the correct answer to five decimals. Find the 3° approx. cube root: 2. Of 3, taking ] J for the 1° approx. root. 3. Of 2, taking IJ for the 1° approx. root.' 4. Of 11, taking 2^ for the 1° approx. root. ZERO AND INFINfTY. 330. The fifth particular in which Algebra is an ex- tension of Arithmetic (333) is the extended significa- tion and use of zero, or 0. 331. In Arithmetic, all zeros are regarded as being equal to each other, since each denotes none or nollifng in comparison with a given finite quantity. But it does not follow from this that all zeros are equal in compar- ison with each other. §1S AtGfifeftA. 332. Finite Quantities are the ordinary quantities denoted by figures and letters. 333. Zero, denoted by 0, is tlie result of subtracting a finite quantity from itself. As, a — a '= 0, nx — nx = 0. 334. Infinity, denoted by oo , is the reciprocal of 0. Thus,-——- = TT = cc. PRINCIPLES AND PROPERTIES. (See Art. 345.) 335. I. — Zero stands for a quantity numerically less than any finite quantity. For, however small a finite quantity may be, it is more than no quantity, or nothing. 336. II. — Infinity stands for a quantity numerically greater than any finite quantity. For, let — = g. Now q expresses the number of times that x is contained in 1. Hence, 1°, q is larger in proportion as x is smaller ; 2°, so long as x, however small, has a finite value, q will also have a finite value. Therefore if x be less than any finite number, or 0, q is greater than any finite number. Hence, y = oo ; in which oo stands for a quantity greater than any finite quantity. 337. \\\.- A finite quantity is not changed by the addition or subtraction of zero. "Por, since a — « = 0, a + = a, and a — = a .(A) ZERO AND INFINITY. 219 338. IV. — The product of zero by a finite quantity is zero . For, m X (« — a) = ma — ma or, mXO = (B) 339. V. — Tke quotient of zero by a finite quantity is zero. For, rB) 0=toXO (1) (l)--"*. ¥j = 0- (C) 340. VI. — Any finite quantity divided by zero is equal to infinity. For, taking the reciprocal of (C), m 1 -Q- = -jf = 00 (D) 341. VII. — A finite quantity divided by infinity is equal to zero. For, multiplying (D) by _, !!L = (E) 342. VIII. — Infinity multiplied by a finite quantity is infinity. 1 m For, '" X o" = T or, m X <» = 00 (F) 343. IX — Infinity is not changed by the addition or subtraction of a finite quantity. m For in (A) substitute — forO (E), 00 m m a + — = a, and a — — = a . 00 00 220 ALGEBKA. Multiply by oo , aX°° + in = aX '^ > and rtX<» — »ll = aX<»• ButoX<» = 00 (^)' lience, 00 + «i = oo , and co — m = oo . (G) 344. X. — Each oj the followinfr may have any Jinitc 00 value.vtz- : -^, y_ (x> , — , oo — oo . " 00 For, 1°, Since (B) m X = 0. "o" = '» (H) m 2°, Since (D) -y- =- oo , X ■» = '«---(K) 3°, Since (F) m X <» = oo ' ^ = «» (L) 4°, Since (G) oo = oo + m, oo — oo =m-.(M) GEOMETRICAL ILLUSTRATIONS. 345. These illustrations are intended to g've the student a clearer idea of the ineaning of a few of the preceding formulas. It we denote the length of a line by a, then since a point terms no appreciable part of a line, the length of a point is 0. The meaning then, 1°. Of n + = a, and a — ^ o, is this : a -point, -when added to or subtracted from a line, does not change its value, a 2°. Of Q- = 00 I « point is contained in a line an infinite number of times. 3°- Of 00 X = o!; "» infinite number of points make up a line. 4°- Of 55" = "i if " ^""^ ^^ diz'ided into an infinite number of parts, each part is a point. Chapter X. RATIO, PROPORTION AND VARIATION. RATIO. 346. Ratio is the relation between two like quanti- ties exprebsed by their quotient Thus, the ratio of m to » is — , which is often written vi : n. n When the colon is thus employed it is called the .sign of ratio, and is equivalent to -,-. If A and B are lines whose lengths are 6 and 2 inches, re- spectively, then 5 in. ^ = " = nr. = 2i-. which is read, the ratio of A to B is 2J. 347. Commensurable Quantities are those which contain a common unit an exact number of times. Thus, if the lengths of the lines A and B are 1| and 2J feet, . A and B are commensurable. For, A = 1| ft. = li'j- ft. = 21 twelfths ft.; and, B = 2i ft. = 2-1^^- ft. = 28 twelfths ft. In this case, A : B = |J = 5- 348. Incommensurable Quantities are those which do not contain a common unit an exact number of times. Thus, if the side of a square (S) is 1 foot, the length of the diagonal (D) is v/ 2 feet. Now V 2 and 1 are incommensur- able, as l/ 2 can not be expressed by any rational part of 1. In this case, D : S ^ l/ 2 , which is called an incommensur- able ratio. 222 ALGEBRA. 349. The ratio of two incommensurable quantities may always be expressed as near the true value as we please by means of a fraction, if we only make the common unit of measure sufficiently small. Thus (348). iV of a ft. is contained in S 10 times, and in D a small fraction over 14 times. Hence, we have very nearly D : S = H = l-4- Again, making the unit smaller, -j^ of a ft. is contained in S 100 times, and in D a little over 141 times. Thisgives a nearer approximation, viz.: D : S = j^J- :=.1.41. 350. The Terms of a ratio are the quantities com- pared ; the Antecedent is the first term or dividend ; the Consequent is the second term or divisor, and the two terms together form a Couplet. 351. A Simple Ratio is one whose terms are entire. 352. A Compound Katio is the ratio of the products of the corresponding terms of two or more ratios. 363. A Duplicate Ratio of two quantities is the ratio of their squares. Thus, x^ : j/2 is the duplicate ratio of x : y. 354. The Triplicate Ratio of two quantities is the ratio of their cubes. Thus, x^ : 2/3 is the triplicate ratio of a; : y. 355. The Reciprocal of a ratio is the result of inter- changing the places of its terms. Thus, the reciprocal of m : n is n : m. PROPERTIES OF RATIOS. 356. 1.— If both terms of a ratio be multiplied or divided by the same number, the ratio is not altered. Or, a : 6 = a X « : 6 X « = o -=- « : 6 -^ ». T?ATIO. 223 357. II. — // both terms of a ratio be increased by the same quantity^ the ratio will be made more nearly equal to unity; that is, if the ratio is a proper frac- tion, it -will be increased ; and if it is an improper fraction, it will be decreased. a ' Let a : 6, or t-> be the ratio. a -\-n Add n to both terms, a -\- n : h -\- n, or t .■ • The difference between: a a b — a 1°. ldnd^isl-^ = — ^ a 4- n a + re 6 — a 6 — a 6 — a Now, numerically, — ^ — > ^ _|_ ,j • Hence, t-J — is more nearly equal to unity than r-i for the difference between the latter and 1 is greater than the differ- ence between the former and 1. EXERCISE LXXVI. 1. Compound 3 : 4 with the duplicate of 2 : 5. 2' * . „ . 8 . The duplicate of 2 : 5 is gj = 35' 3 : 4 = -j- > — V — - — . Ans. 25^ i - 25 2. Which is the greater, the reciprocal of 5 : 2 or the triplicate of 2 : 3. 2 23 8.2 54 The former = -g' the latter = g-y = g^i 5 ^^ 135 and ^ — jgg- Hence, the formei is the greater. 224 ALGEBkA. 3. Which is the greater 3 : 4 or 5 : 6 ? 4. How much less is 3 : 4 than (3 + 1) : (4 + 1) ? 6. How much greater is 6 : 5 than (6 + 2) : (5 + 2) ? 6. If a; : 3 added X : 4 = 14, how much is x} 7. If 2* : 5 is 35 more than 3x : 11, what is the value of X ? Find the value of x in the following equations : 8. a; : 3, -(- the reciprocal of a; : 3, = 3J. 9. X : 5, + the duplicate of a; : 5, = ||. , 10. The duplicate of 3a; : 4 = the triplicate of a; : 2. 11. Two numbers are in the ratio 3 : 4, and if 4 be added to each, they are in the ratio 4 : 5. Find the numbers. Suggestion. Let 3.r and 4a; represent the numbers. 12. Two numbers are in the ratio 5:6, and if 5 be added to each, they are in the ratio 11 : 13. Find the numbers. 13. The ratio of the ages of A and B is 3 : 7, and 15 years ago it was 1 : 4. What are their ages? 14. If 3 pounds of tea are worth as much as 5 pounds of coffee, and 7 pounds of coffee as much as 11 pounds of sugar, find the ratio of the value of tea to that of sugar. 15. One-seventh of the diameter of a circle is con- tained nearly 22 times in the circumference. Find the approximate ratio of the circumference to the diameter of a circle. 16. Two numbers are in the ratio of 2 to 3, and if 1 be subtracted from 7 times the first and 6 be added to PROPORTION. 22B 5 times the second, the results are in the ratio of 5 to 6. Find the numbers. 17. There are two roads from A to B, one of them 14 miles longer than the other, and two roads from B to C, one of them 8 miles longer than the other. The dis- tances from A to B and from B to C along the shorter roads are in the ratio of 1 to 2, and the distances along the longer roads are in the ratio of 2 to 3. Determine the distances. PROPORTION. 358. A Proportion is an equality of ratios. 359. The equality of two ratios may be indicated by the sign =, or by the sign :: . The equality of 12 : 3 and 8 : 2 may be expressed thus, 12 : 3 = 8 : 2, or 12 : 3 :: 8 : 2. • 360. The Proportionals are the four numbers com- pared ; the Extremes are the first and fourth terms ; and the Means are the second and third terms. 361. The algebraic test of a proportion is that the two fractions which represent the ratios shall be equal. 362. If three quantities a, b, c, are so related that a : b = b : c, b is said to be a Mean Proportional be- tween a and c. 363. If four quantities a, b, c, d are so related that 1 1 a:b = — : -ri c a they are said to be Inversely or Reciprocally propor- tional. M.E.A. — 15 ^26 ALGEBRA. 364. A Continued Proportion is several equal ra- tios connected by the sign =. As, 4 : 2 = 6 : 3 == 14 : 7 = 22 : 11. PROPERTIES OF PROPORTIONALS. 365. I. — The product of the extremes is equal to the ■product of the means. For, if a : 6 :: c : d (1) .- o c Then y = ^ (2) (2) X ^^, lA = be. 366. II. — //" the product of two numbers be equal to the- product of two others^ the four numbers are pro- portionals. Let ad = 6c (1) (D^ftd | = -|- (2) Or a ; 6 :: c : d. 367. Ill- — Four proportionals are in proportion: 1°, By Inversion. That is, The second : the first :: the fourth : the third. Let a : 6 : : c : (2. Then (365), he = ad, (1) (1) ^ ac, - = -, Or, h : a :: c : d. 368. 2°, By Alternation. That is, The first : the third :: the second : the fourth. Let a : & ;; c : d. Then ad, — he (1) ,-i\ ,1 ah ^^^^'^' ^ = ¥' Or a : c v. h : A. PROPORTION. 227 369. 3°, By Composiiion. That is, The first -f- the second : the second :: the third -\- the fourth : the fourth. Let a : 6 :: c : d a c then T~"d W h d Evidently, T"" d" '-^^ a + b c + d a) + (2). -r-=-i- C3) or, a + 6 : 6 : : c + d : rf. 370. 40, By Division. That is, The first — the second : the second :: the third — - the fourth : the fourth Let o : 6 :: c : ^- 13. \i u cc X when y and z axe constant, and M oc Jf when X and 3 are constant, and M oc « when « and y are constant, how does m vary when a;, y and « are variable. 14. If one quantity vary directly as another, and the former be | when the latter is f , what will the latter be when the former is 9? 15. Given that the area of a circle varies as the square of its radius, prove that the area of a circle whose radius is 5 inches is equal to the sum of the areas of two circles whose radii are 3 and 4 inches. VARIATIOUr. 235 16. Given that the volume of a sphere varies as the cube of its radius, prove that the volume of a sphere v,-hose radius is 6 inches is equal to the sum of the vol- umes of three spheres whose radii are 3, 4 and 5 inchea. Application to Problems in Proportion. 17. If 4 hats cost $15, how much will 7 hats cost.? Suggestion. Let x denote hats and y the value of hats, then so (X. y, whence x = my. 18. If $40 draw $8 interest in a certain time, what interest will $90 draw in the same time f 19. If a pole m ft. high cast a shadow n ft. long, what is the height of a tree that casts a shadow r ft. long? 20. If 4 men in 9 days can do a certain work, how long will it take 10 men to do the same work ? I Suggestion. Let a; denote men and 2/ time, then x OC — m whence x = —• 21. If 25 men can do a piece of work in 24 days, how many men can do the same in 30 days ? 22. If 15 persons consume 6 barrels of flour in 9 months, how long will it take 20 persons to consume lOf barrels? Let X denote number of persons, y time and z number of bar- rels of flour; then a; oc ^' that is a; qc « when y is constant, and X varies inversely as y when z is constant. Hence, a;= — (1) y Now when a; = 15, ?/ = 9 and » == 6. Substituting in (1) we find«j= i^. Therefore, a: = — (2) 2 'iy 236 atlgebrA. Now we wish to know the value of y when x = 20 and z = lOJ. Making these substitutions in (2) we find y = 12, Ans. 28. If 6 persons consume 240 pounds of sugar in 8 months, how much will 9 persons consume in 11 months ? 24. If 12 men can do as much work as 25 women, and 5 women do as much as 6 boys, how many men would it take to do the work of 75 boys ? 25. If 12 oxen eat 3^ acres of grass in 4 weeks, and if 21 oxen eat 10 acres in 9 weeks, how many oxen will eat 24 acres in 18 weeks, the grass growing uniformly.? Let X denote oxen, t weeks, y acres of grass, z growth of grass. 1°- The amount of growth varies as the number of acres mul- tiplied by the time. Hence, e cc ty, or z = nty (1) 2°. The number of oxen varies as the total amount of grass divided by the time. Hence, oc oc — , or tx = my -{-mz (2) Combine (1) and (2), making ??ire = r, tx = my + rty (3j Now, when a; = 12, ?/ = 3^ and t = 4. Hence, 10m + iOr =U4 (4) Again, when a; = 21, y ^= 10 and t ^ 9. Hence, 10 m + 90r =189 (5) Combining (_i) and (5), »• = ^%, m = ^. Substitute in (3), reduce, 10 tx = WSy +9ty (6) Now, in (6) make J = 18, J/ = 24, and we get s = 36; hence, 36 oxen. Ans. 26. A cistern has a stream of water running into it ; it has 10 pipes ; all together will empty it in 2^ hr., and G pipes will empty it in 6 J hr. In what time will 3 pipes empty it? Chapter XI. SERIES. 390. A Serifs is a succession of quantities, so related that each may be derived from one or more preceding ones, in accordance with some fixed law. 391. The Terms of a series are the quantities of which it is formed; 392. The Extremes of a series are the first and last terms, and the Means are the terms between the ex. tremes. ARITHMETICAL PROGRESSION. 393. An Arithmetical Progression is a series in which each term may be derived from the preceding term by the addition of a common difference. 394. An Increasing Arithmetical Progression is one in which the common difference is positve. Thus, 3, 7, 11, 15, 19, 23, etc., is an increasing arithmetical progression in which the common difference is + 4. 395. A Decreasing Arithmetical Progression is one in which the common difference is negative. Thus, 13, 8, 3, — 2, — 7, — 12, etc., is a decreasing arithmeti- cal progression in which the common difference is — 6. 396. There are five parts or elements in an arith- metical progression, viz. : first term, /a^^term, num.ber of terms, common difference^ and sum of all the terms. 238 A'LGE'BRA. These parts are so related that any two of them may be found when the other three are given. 397. To find the last term when the first term, the common difference and the numbet' of terms are given. Let a denote the first term, d! the common difference, n the number of terms, and I the last term; then the progression will be a, (a + d), (a + 2d), (a + Zd), (a + M), etc. That is, the coefficient of d in any term is one less than the number of that term in the series; consequently, the mth or last term will be « -|- ()i — V)d. Wheiice, putting I for the »th term, we have Z = n + ()8 — l)(i, in which d is either positive or negative, according as the series is. an increasing or a decreasing one. Hence, Rule. To the first term add the froduct of the common difference by the number of terms less one. EXERCISE LXXIX. 1. If the first term is 4, the common difference 5, and ihe number of terms 30, what is the last term.' 2. If the first term is 7, the common difference 3, and the number of terms 20, what is the last term? 3. What is the tenth term if the first term is 23 and the common diffeience — 2? 4. When the first term is 31 and the common differ- ence — 3, what is the fifteenth term.? ^ 5. When the first term is 5| and the common differ- ence 3J-, what is the twelfth term .? 6. Find the 13th term of the series 11, 16, 21, etc. 7 What is the 21st term of the series 103, lOOj, 98, J»5^ etc? ARITHMETICAL PROGRESSION. 239 398. To find the sum of the terms tvhen tlie first term, common difference and number of terms are given. Let a denote the first term, d the common difierence, n the number of terms, I the last term, and 8 the sum of the terms. Then, s= a+ (a + d) + (a + 2d) + + 1, or, writing the terms in the reverse order, s = I -\- (I —d) + {l — 2d) + + a. Therefore, by adding these equations, term by term, 2s= (« + + (a + + (« + + + (« + 0- Here (a + V) is taken as many times as there are terms, or n times;" whence, 2s = n^a -\- 1), sor = \n(a -\- 1) . Hence, Rule I. — Find the last term (397). II. — Multiply the sum of the first and last terms by half the number of terms. 8. If the first term is 5, the last term 62, and the num- ber of terms 20, what is the sum of the terms .'' 9. If the least term of a series of numbers in arithmeti- cal progression be 4, the greatest 100, and the number of terms 17, what is the sum of the terms? 10. Find the sum of the natural series of numbers 1, 2, 3, 4, etc., carried to 1000 terms. 11. Find the sum of n terms of the same series. 12. Required the sum of the series of odd numbers 1 , 3, 0, 7, etc., carried to 101 terms. 13. Find the sum of n terms of the same seriee. 14. How many times does the hammer of a cornmpa. clock strike in a week.' 240 Ai,GEHK.\. Find the sum 16. Of 20 terms of the series 2, 6, 10, 14. etc. 16. Of 32 terms of the series 4, 3|, 3^, 3J, etc. 17. Of 12 terms of the series 1, If, 2J, etc. 18. Of 50 terms of the series ^, f , 1, etc. 19. Of n terms of the series 9, 11, 13, etc. 20. Of n terms of the series 1, f , f , etc. 21. Of 30 terms of the series — 27, — 20, — 13, etc. 22. Of n terms of the series J, — §, — ^, etc. n — 1 n — 2 n — 3 23. Of n terms of the series — -; — ? — - — i ; etc. ti n 71 399. To insert a given nuinbei- of arithmetical means betiveen two given terms. Let it be required to insert in arithmetical means be- tween a and I. Since there are to be iti terms between a and Z, there will be m -|- 2 terms in all, a being the first and I the last. Now to find d, the common difference, substitute to + 2 for n in the formula I — a I = a -\-(n — l)d, which gives, after reducing, d = — ^^■ Hence, to find the common difference, divide the differxnce of the two numbers by the number of means plus one. 24. Insert 4 arithmetical means between 3 and 18. 18 —3 Common difference = —. — r-r = 3, 4+1 ' Hence, the means are 3 + 3 = G, 6 -|- 3 = 9, 9 -|- 3 = 12, 12 + 3 = 15. 25. Insert 1 arithmetical mean between 5 and 13. 26. Insert 5 arithmetical means between 2 and 26. 27. Insert 7 arithmetical means between 5 and 33. 28. Insert 9 arithmetical means between 3 and 4. 29. Insert 15 arithmetical means between 4|. and 5^. ARITHMETICAL PROGRESSION. 241 400. The formulas I = a + {n — l)d (A) s=in(a+l) (B) form two independent equations, and contain together all the five elements of an arithmetical progression. Hence, when either three of these elements are given, the o.'^her two may be determined either from one of the equations or from combining the equations. 30. Required the first term of an arithmetical pro- gression, when the last term is 62, the common difference 3, and the number of terms 20. The required term (a) and the three given terms (I, d and n) are all contained in (A.) ; hence, the problem may be solved by substituting 62 for I, 3 for d and 20 for n in (A), and reducing. Thus, 62 = a + (20 — 1)3 ; whence, a = 5. 31. Find the common difference, when the last term is 149, the first term 4, and the number of terms 30. 32. Find the number of terms, when the last term is 59, the first term 3, and the common difference 4. 33. Find the first term, when the sum of the terms is 670, the number of terms 20, and the last term 62. 34. When the first term is 7, the number of terms 6, and the sum of the terms — 18 ; what is the last term ? 35. The extremes are 2 and 52, and the sum of the terms 297 ; what is the number of terms ?' 36. When the last term is 1, the first term — 6, and the number of terms 15 ; what is the common differ- ence ? 37. The extremes are 3 and 45, and the common "difference 2 ; what is the sum of the series? N.e.a:— i^ 242 ALGEBRA. Since neither (A) nor (B) contains all of the given parts and the required part, it is necessary to use both formulas. Substituting 3 for a, 45 for I; and 2 for <^ in (A) and (B), we obtain 45= 3 + 2()i — 1) (1) 8=^n(S + i5J (2) Now, combining (1) and (2) so as to eliminate the super- fluous element n, we obtain S = 528, Ans. 38. The sum of a certain number of terms of the series 19, 17, 15, etc., is 91 ; find the number of terms.? Here, a = 19, d = — 2, and 8= 91. Substituting in (A) and (B), we have 1= 19 — 2(ji — ]) (1) dl = in(l9 + 1) (2) Combining (1) and (2) so as to eliminate the superfluous element I, we have n^ — 20)1= —91. Whence, n = 13, or 7. 39. What is the common difference when the first term is 1 , the last 50, and the sum of the terms 204. 40. The first term of a decreasing arithmetical series is 10, the number of terms 10, and the sum of the terms So ; find the last term and the common difference. 41. The last term of an arithmetical progression is 52, the common difference 5, and the sum of the series 297 ; find the first term and the number of terms. 42. Insert 4 arithmetical means between 5 and 7. 43. The sum of the first two terms of an arithmetical progression is 4, and the fifth term is 9 ; find the series. 44. What is the series of which ^n(M — 1) is the nth term ? %bstit«te I, 2, 3, 4, etc., successively for n. ARITHMETICAL PROGRESSION. 243 45. The Mth term of an arithmetical progression is |(3re — 1) ; find the first term, the common difference, and the sum of n terms. 46. How many terms of 16 + 24 + 32 + 40 + etc. amount to 1840 ? 47. How many terms of the series 5, 4, 3, etc., must be taken to make 14? 48. Find the sum of n terms ofl — 3 + 5 — 7 + etc. 49. Find the sum of n terms of 1 — 2 + 3 — 4 + etc. 50. Find four numbers in arithmetical progression, such that the product of the extremes shall be 27, of the means 35. Let X — Srj,x — y,x + y, x + Sy, be the numbers. 51. There are 4 numbers in arithmetical progression, their sum is 32, and their product 3465; find "the num- bers. 52 The sum of 3 numbers in arithmetical progression is 30, and the sum of their squares 308 ; find the num- bers. Let X — y, x,x-\-y, be the numbers. 53. The sum of the 4th powers of three consecutive numbers is 353 ; find the numbers. 64. The product of 4 consecutive numbers is 1680 ; find the numbers. 55. The sum of 7 numbers in arithmetical progres- sion is 35, and the sum of their cubes 1295 ; find the numbers. 56. Two travelers (A and B) set out from the same place at the same time ; A travels uniformly at the rate 244 algebrX. of 3 miles an hour, but B's rate is 4 miles the first hour, 3J the second, 3 the third, and so on. In how many hours will A overtake B ? 57. The sum of 6 terms of an arithmetical progres- sion is 93, and the 6th term is 28 ; what is the series.? 58. A traveler (A) sets out for a certain place, and goes 1 mile the first day, 2 the second, 3 the third, and so on. In 5 days afterward another (B) sets out, and travels 12 miles a day. How long and how far must B travel before he and A wjll be together.? 59. If a falling body descends 16^ feet the first sec- ond, three times this distance the next, five times the next, and so on, how far will it fall the 30th second, and how far altogether in half a minute ? 60. Two hundred stones being placed on the ground in a straight line, at the distance of 2 feet from each other; how far will a person travel who shall bring them separately to a basket, which is placed 20 yards fi-om the first stone, if he starts from the spot where the basket stands ? 61. Prove that the sum of any 2n -\- 1 consecutive in- tegers is divisible by 2» -{- 1. GEOMETRICAL PROGRESSION. 401. A Geometrical Progression is a series in which each term may be derived from the preceding term by multiplying it by a common ratio. 402. An Increasing Oeometrioal Progression is one in which the ratio is greater than unity. Thus, 3, 6, 12, 24, 48, etc., is an increasing geometrical pro- gression in which the ratio is 2. GKOMKTRICAL PROGRESSION. 24;i 403. A Decreasing Geometrical Progression is one in which the ratio is less than unity. Thus, 27, 9, 3, 1, J, etc., is a decreasing geometrical pro- gression in which the ratio is ^. 404. There are five parts or elements m a geometri- cal progression, viz. : j^r5^ term, lasi term, number oi terms, ratio, and sum of all the terms. These parts are so related that any two of them may be found when the other three are given. 405. To find the last term when the first term, the ratio, and the number of terms are given. Let a denote the first term, r the ratio, and n the number of terms; then the successive terms of the series will be a, ar, ar^, ar^, ar* ar"-'. That is, the given first term is a factor of each of the terms, and the exponent of )■ in the second term is 1, in the i/iird term 2, in the /our/ i term 3, and so on, to the nth term, in which it is n— 1. Therefore, if I denote the last term, we shall have I = ar"-'-. Hence, Rule. — Multiply the first term, by the ratio raised to a ■power whose exponent is one less than the number of terms. EXERCISE LXXX. 1. Find the last term of a geometrical progression whose first term is 5, ratio 3, and number of terms 7. I = a»-»-^ = 5 X 3'"^ = 5X3" = 3G45. 2. Find the 9th term of a geometrical progression whose first term is 5, and ratio 4. 3. Find the 8th term of a geometrical progression whose first term is 28672, and ratio \. 4. Find the 7th term of the series 5, 10, 20, etc. 246 ALGEBRA. 5. Find the 6th term of the series 3|, 2^, 1 J, etc. 6. Find the 8th term of the series —21, 14, —9 J, etc. 406. To find the sum of the terms when th". fi,rst term,, the ratio, and the number of terms are given. Let a be the first term, r the ratio, n the number of terms, and s the sum of terms. Thus g = a + ar -\- ar' + ar"-i (1) (1) X n «»■ = ar + ar^+ aj-'>-i+ ar" . ..(2) (2) — (1), sr — s = oc" — a. Whence g^aQ-" — 1) ^gv^ r — 1 If I denote the last term, we have I = o?'»-i . Hence, substituting in (3), s = 'l^-^ (4) r — 1 Equation (3) gives the value of s in terms of the given parts, but (4) is often a more convenient form. Hence, "RxXe..— Multiply the last term by the ratio, subtract the first term, and divide the remainder by the ratio less one. 1. Find the sum of a geometrical series whose first term is 1, ratio 2, and last term 1024. 8. Find the sum of a geometrical sei-ies whose first term is IJ, ratio 4, and number of terms 9. Find the sum: 9. Of 9 terms of the series 1 + 3 + 9 + etc. 10. Of 7 terms of the series 8 + 20 + 50 + etc. 11. Of 6 terms of the series 1| -|- 2f + 4f + etc. 12. Of 10 terms of the series 2 — 2^ -)-23 — etc. 13. Of n terms of the series 3 + 2 + | -f etc , 14. Of n terms of the series \, \, J, etc. GEOMETRICAL PROGRESSION. 247 407. To find, tlie sum of the terms when the pro- gression is decreasing and the number of terms in- finite. The value of s may be written thus, a(l— r") Now suppose (■ less than unity; then the larger n is the smaller will »•" be, and by taking n large enough r" can be made as small as we please. If then n be taken so large that r" may be neglected in comparison with unity, the value of s reduces a to :j This is the case when n is infinitely great. Hence, Rule. — Divide the first term by one less the ratio. 15. Find the sum of 4, 2, 1, J, etc., to infinity. n 4 s = nry. = nZT = 8, Ans. 16. Find the sum of 9, 3, 1, ^, etc., to infinity. 17. Find the sum of §, J, \, ^, etc., to infinity. 18. Sum to infinity 5 — J + ^V — -z^ + ^*<^' 19. Sum to infinity | + 1 + ^\ + etc. 20. Sum to infinity f — § + ^\ — etc. 21. Sum to infinity 1 + i + t^j + etc. 22. Sum to infinity 1 — i + t^ — etc. 23. Sum to infinity 1 + "^ + ^ 4" etc. 24. Sum to infinity ^ ""^ IT + ^ — etc. 408. To insert a given number of geovnetrical means between two term,8. Let it be required to insert m geometrical means be- tween, a and {. 2.18 AI.GEnilX. Since there are to be m terms between a and I, counting a and I there will be m + 2 terms in all. Now, to find r, the ra- tio, substitute )» + 2 for H in the formula 1= «r"-',andwe obtain r = "ii/ — • [/ a Hence, to find the ratio. Divide the last term by the firsts and extract that root of the quotient whose indeo is 07ie more than the number of means. 25. Insert 4 geometrical means between 7 and 224. Ratio = ■'+^/a|i = V¥i = 2. Hence, the means are 7 X 2 = I't, 1* X 2 = 28, 28 X 2 = 5G, 5C X 2 = 112. Find: 26. The geometrical mean between 4 and 9. 27. The geometrical inean between i and f . 28. The geometrical mean between -\- 1 and — 1. Insert : 29. Two geometrical means between 5 and 320. 30. Three geometrical means between 6 and 486. 31. Seven geometrical means between 2 and 13122. 32. Three geometrical means between -\- 1 and — 1. 33. Five geometrical means between + 1 and — 1. 409. The formulas I = ar»-i (C) s = !i^l^ (D) r— 1 ^ ^ form two independent equations, and contain together the five elements of a geometrical progression. Hence, in general, when either three of these elements are given the other two may be determined either from one of the eq^uations, or from coinbining the equations. GEOMETRICAL PROGRESSION. 24& 34. Find the first term of a geometrical progression when the Jast term is 1536, the ratio's, and tihe niiiiiii- bar of terms 10. 35. The last term of a geometrical progression is 12500, the first term 4, and the number of terms 6 ; find the ratio and sum of the terms. 86. The last term of a geometrical progression is 1536, the first term 3, and the sum of the terms 3069 ; find the ratio. 37. The last term of a geometrical progression is 885735, the ratio 3, and the sum of the terms 1328600 ; find the first term. 38. The first term of a geometrical progression is l,the ratio 3, and the sum of the terms 9841 ; find the last term and the number of terms. 89. The first two terms of a series in geometrical progression are f and J ; what are the next two terms? 40. The first and third terms of a geometrical series are 4 and 16 ; what is the fourth term ? Note. — Numbers in geometrical progression may be denoted by a;, xy, xy^, etc., in which x is the first term and y the ratio. Again, three terms may be expressed by x^, xy, y' ; four terms by —> x, y, —\ five terms by —i x', xy, y', — ; etc. y Jj ~ y •*' In all these cases the square of any term is equal to the prod- uct of the two adjacent terms. 41. The sum of three numbers in geometrical pro- gression is 7 ; and the sum of their reciprocals is J ; find them. 42. There are four numbers in geometrical progres- 250 ALGEBRA. sion, the sum of the first and third is 36, and the sum of the second and fourth is 70 ; find them. 43. There are four numbers in geometrical progres- sion, the sum of the extremes is 28, and the sum of the means 12 ; find them. 44. There are three numbers in geometrical progres- sion whose product is 27, and the sum of their cubes 757 ; find them. 45. There are four numbers in geometrical progres- sion, which, being inci-eased by 5, 7, 7, and 3 respect- ively, the sums are in arithmetical progression ; find them. 46. There are four numbers in arithmetical progres- sion, which, being increased by 2, 4, 8, and 15 respect- ively, the sums are in geometrical progression ; find them. 47. Suppose a body to move 20 miles the first minute, 19 miles the second, 18^ the third, and so on forever; required the utmost distance it can reach. 48. A father divided $147 among three sons, the shares being in geometrical progression ; and the first had $63 more than the last. How much had each ? 49. Fmd the geometrical series in which the sum of the first two terms is 36, and of the next two 324. 50. A person who saved every year half as much again as he saved the previous year had /n 7 years saved ^£"102 19s. How much did he save the first yeai ? 51. There are four numbers, the first three of which are in geometrical progression, and the last three in HARMONICAL PROGRESSION. 251 arithmetical progression ; the sum of the first and last is 14, and the sum of the second and third is 12 ; find the numbers. HARMONICAL PROGRESSION. 410. Quantities are in Harmouical Progression when their reciprocals form an arithmetical progression. Thus, 1, i, i, ^, etc., are in harmonical grogression, because their reciprocals 1, 3, 5 , 7, etc., form an arithmetical progression. 411. To find the relation between any three con- secutive terms. Let a, b, c be the three terms, then their reciprocals _, _, -L .„ . a b c will be in arithmetical progresssion . Hence, l. — L = L — L c b b a f^^ (1) X «6c, ab — ac = ac — be or> a(6 — c) = c(a — 6) (2) c 6 — c Hence, of three consecutive terms of a harmonical progression, the first is to the third as the first minus the second is to the second minus the third. EXERCISE LXXXI. 1. The first two terms of a harmonical progression are 30 and 10 ; find the third term. Let X = the third term, 30 30 — 16 then — = .„ __ , ; whence a; == 6, ^as,.. 252 ALGEBRA. 2. The first two terms of a haimonical progression ab ■ are a and 6 ; prove that the third term is „ -r- 3. Find the harmonical mean between 21 and 28. Let X = the harmonical mean. 21 21 — a; then So =^ ng ' whence, x =; 24, Ans. 4. Prove that the .harmonical mean between a and 412. All problems relating to quantities in harmoni. cal progression, which are susceptible of solution, may be solved by inverting the terms and applying the rules ol arithmetical progression. It should be remembered, however, that there' is no general rule for finding the sum of the terms. 5. Find the 26th term of the series |,f,¥>etc (1). Inverting the terms and we have 4. f. iS>etc (2) which is an arithmetical progression, in which the common difference is § — | = — j'j . Now (397), the 26th term of (2) is f-(26-l),\=-i Hence, the 26th term of (1) is — |, Ans. 6. Insert 3 harmonical means between 2 and 10. Inverting, we have to insert 3 arithmetical means between i and i^o. Art. 399, (Z = (iij - i) - 4 = - A. Hence, the arithmetical means are J, i\, J, and the required harmonical means 2^, 3J, 5. 7. Insert 2 harmonical means between 3 and 12, 8. Find the 23d term^ of f, f , etc. HARMONICA!. PROGKESSIOX. 253 9. Insert 6 harmonical means between 3 and — 1. 10. The first term of a harmonic series is^, and the 6th ^ ; find the intermediate terms. 11. Find the next two terms of the series 3, f , f . 12. The arithmetical mean of two numbers is 5, and the harmonical mean is 3^, find the numberis. If a, b and C are in : 13. Arithmetical progression, prove that b^ > ac. 14. Geometrical progression, prove that b^ = ac. 15. Harmonical progression, prove that b^ < ae. 16. If a, b, are in arithmetical progression, and b, c, d are in harmonical progression, prove that a, b, c, d are in geometrical progression. 17. If any three quantities are in harmonical pro- gression, prove that the three quotients obtained by dividing each of the quantities by the sum of the other two, are also in harmonical progression. ANSWERS. Page 10. 1, + 5, — 8 ; 2, + 65, — 50 ; 3, + 75, — 22; 4, + $11, — $13; 5, -3°, +5°; 6, + 17o, — 15°; 7, A boywent east 4 mi., then west 3 mi., then west 2 mi. ; 8, A boy went north 4 mi., then south 3 mi., then south 2 mi. ; 9, A gam of $5, then- a gain of $4, then a loss of $7, then a loss of $1. Page 12. 1, — 6 ; 2, — 1 ; 3, + i4 ; 4, ; 6, + 2 ; 6,-3; 7,-10; 8, + 5. Page 13. 9, + 12; 10, —20; 11, + 9; 12, + 12; 13, + 2 ; 14, — 3 ; 16, — 5 ; 17, + 10 ; 18, + 3 ; 19, + 13. Page 14. 30, 1 yr. ; 31, 2 yr. younger; 32, 1 mi. east; 33, 4° lower; 34, 12° south; 36, 1 mi. east; 37, $9. Page 16. 1, + 4 ; 2, + 14 ; 3, — 17 ; 4, + 2 ; 5, — 7 ; 6, — 4 ; 7, + 22 ; 8, — 10 ; 9, + 6 ; 10, + 20 ; 11, — 23 ; 12, — 14 ; 13, + 17 ; 14, + 4 ; 15, + 2 ; 16, + 21 ; 17, — 5 ; 18, — 20 ; 19, + 185 ; 20, — 240. Page 17. 23, 25 mi. east; 25, $111; 26, $10; 27, 373 ; 29, — 26 ft. or 26 ft. west; 30, — 71 ft. or 71 ft. west ; 31, + 35 ft. or 35 it. east. Page 18. 32, + 16 ; 33, + 12 ; 34, — 27 ; 35, + 27; 36, - 4; 37, + 5 ; 38, ^ 10; 40, + 13; 41, Page 19, 1,-24; 2,-21; 3, + 56 ; 4, + 54: 5., + 442 ;. 6, — 714., ANSWERS. 255 Page 20. 9, — 12 ; 10, 4- 360 ; 11, + 720 ; 12, + 240; 13, —27; 14, —12; 15, —16; 10,-117; 17, — 120 ; 18, — 720 ; 19, — 945 ; 20, + 240 ; 21, + 120 ; 22, + 168. Page 21. 1,4-6; 2, + 5; 3.-9; 4,-7; 5, — 12; 6,-25; 7, — 28. Page 22. 8, — 5 ; 9, — 12 ; 10, 4- 5 ; 11, + 4 ; 12, — 30 ; 13, — 81 ; 14, + 7 ; 15, — 56 ; 16, — 20J ; 17, -3i; 18, 4-6|; 19,-7^; 20, + 1J; 21, +4; 22, — 2. Page 23. 3, - 216 ; 4, 4- 2401 ; 5^+ 15,625 ; 6, 4- 256 ; 7, -512 ; 8, + 243 ; 9, + 90 ; 10, 4- 280 ; 11, — 800; 12, 4-72; IS, — 3 ; 14, + 2,800,000. Page 24. 2, ± 6 ; 3, ± 3 ; 4, 4- 5 ; 5, — 2 ; 6, — 4 ; 7, ± 4 ; 8, ± 7 ; 9, — 5 ; 10 ± 6. Page 25. l, 4- 27 ; 2, — 16 ; 3, + 6 ; 4, + 51 ; 5, 0; 6,4-9; 7, —78; 8, + 16 ; 9,4" 16; 10, + 250 ; 11, + 9; 12, + 4; 13,4-20; 14, + 2; 15,4; 16, 2J; 17, 1; 18, 2. Page 27.' 3, 7 XaXbXbX & X«; *, 5 X «Xa XcXcXcXcXaJX*; 5, 3XmXmXmX« xgxyxyxy; ^ bx{a + b)xooxoi!xy- Page 30. 1,53; 2,28-; 3,13; 4,52; 5,118; 6,10; 7, 14 ; 8, 805 ; 9, 62 ; 10, — 5 ; 11, 4^ ; 12, ^f Page 32, 2, (a + b)+ia — b) =2a; 3, K« + ^) _ |(a _ 6) = & ; 4, (a + i*) X (« — ^) = «^ — *' ; 5, (a + by = a2 4- ^-2 4- 2ctb ; 6, (a— by = a^ + &« — 2ab. Page 33, 2, 8tt - 26 + 4c ; 3, 16a^ 4- 76c - 12 ; 4, 5a3 - 14a2 + I6a + 1; 5, 5a* — 20a + 7; 6, 2xy^ 4- 2x^y + Ix^y"^. 256 AI^GEBRA. Page 34. 7, Wa^b + aH^ — 3ab^ ; 8, 2a''b + b^ ; 9, 9x^y -f 5a;i/2 + 11*2 — 10a; + 4y — Zx^y'^ + 3; 11, lOVabe ; 12, 16( ffi — & ) ; 13, lOV ac + 3|/m + 2a;; 14, 2i/a — ft — 141/ a + c ; 15, — 6a2-(-5(a — n)^ ; 16i 2xy + 33a;2« — li)xz^; 17, 30a— 36— 26e + 4/+ gr; 18, 6(a — p)2— 2«l/"6— 7c3a;; 19, 2;i2— ac^- 2(a +6). Page 35. 2, — 2a;2— 9 ; 3, 76 + 2a — 9c ; 4, 12a» — 12ax-\-U; 5, 2a6;6, 12.2;22/; 7, — 12a.62 ;8, — 5a;2:;9,JM— ■n; 10, m-\-n; 11, — m — n ; 12,— m-fM; 13, llffl^j;;!^^ —Uxy ; 15, 3V^, 16, 12l/a7 17, 26 ; 18, 10a— 10a; + 7y. Page 36. 19, lOc ; 20, lOl/m"; 21 , 3a;2 ; 22, 6 — a ; 23, — a + c;24,8rt;25,— 26; 26,21/; 27, — 6a;2+ 13a;— 2 ; 28, 21/' a — 21/' 6 — 5l/5"; 29, — 2rt + 86 — c ; 30, —a* — 76a; + 3a;2 ; 31,26; 32, 2a — 4c; 33, 5a;2— y— 26; 34, X + x^ ; 35, 9a — 56 ; 36, 3a; ; 37, 5a ; 38, 3a; + 2 ; 39, — 7l/'27 Page 37. 40, (a — 6) + (2c — 3(i + 4e - 5/+ a; — 5jnn);41,(a — 6-)-2c) — (3d— 4e + 5/— a; + 5tom) ; 42, (a — 6 + 2c — 3(1) + (4e — 5/ + a; — 5mn) ; 43, (a _ 6 + 2c — 31^ + 4e) — (5/— a; + 5 mn) ; 44, (a + 2e + 4e + a;) _ (6 + 3d + 5/ + 5mw) ; 45, (a + 6) — (a — 6) or 26 ; 46, (a — 6) — (a + 6) or — 26 ; 47, 2(2.x2+ 7a; — 4) ; 48, — 2(6^+ 3a6 — 562). Page 38. 2, 18a^; 3, 35a5&; 4,— 24a''6a;; 5, + idab^x^; 6, 56a3c2w2a;5 ; 7, — SGx^y^z^ ; 8, aS ; 9, a^t"; 10, a^^: 11, a2c. 12, — 21a»+2; 13, 15a"'+2c"+5 ; 14, 72a866cew2 a;2 i 15. — 42 a3»6*'»c'+i ; 16, 60 (a + b)^x'' ; 17, — 77(a -(- 6)4 (a — 6)8 ; 18, 32 (a + 2)' (a + 1)5 (a + 6) (a — 6)« ; 19, — 35 (a — 6 + c) 5 ; 20, bia^b^c^x^y. Page 39. 23, 20a5 — I2a4 _|. leaS _ Sa^ ; 24, — 35a;7 — 25a;e + 15a;5 -f- 403;" ; 2oJ 4a5x2 — 12a*a;8 ANSWERS. 257 + 12asa;2 _ 4^235; 26, — Qasfia + 18 a*6» — 21 a^b* + 6 a^fts . 27, ffl2j3e6 _ a^fyics _|_ azfisc* — a'^b^c^ ; 28, lia'^ajsj/ — 12a«x*y + lOaSipSy — Sa**^!/; 29, 15(a + 6)6 + 18(a + by ; 30, — 30a6(« — ^')® + 35 a''(a—by ; 31, 28(a;+3/)''(a7— i/)' — 36(a;+y)5 (a; _ 2^)10 ; 32, 20a»+2 — 24a'"+^2 _^ 28a'' ; 33, ISa-'+s — sea"*' + 9a2''; 34, — 54a"+362»+2 + 90a"+663" — 63a2"&2"+6 _|_ 27«s''66". Page 40. 37, 55; 38, 35;' 39, —27; 40, —36; 41, 63 ; 43, a' + 2ab + b^ ; 44, a^ — 2db + 6^ ; 45, a2 — 62 ; 4G, 25ffl2 -j- 30a6 + 96^ ; 47, 49c2 — 84ca; + BGx^ ; 48, 64a* — 4962 ; 49, a* + 4a36 + 6a262 + 4a6s + 6* ; 50, x* — 8x^y + 24a;2y2— 32xy^ + 16y*. Page 41. 51, a* — 36a'^c'^ + 81c* ; 52, x^ —ij^; 53, a3 + 27y^ ; 54, a" -^ 1 ; 55, a^ — 5a*6 + 10a362 — 10 a263 + 5a6* — 6^ ; 56, a* + a86 + a^c + a262 + a^c^ + a26c +a62c + ac26 + 6^02 . 57,^53 — y^— z^—3x^y —Zx^e-\-3xy^ + 3xs^ — Zz^y— Zy'^z-^- 6xyz; 58, as + 3a26 + 3a62 + b^ — 1 ; 59, 8x« — 3&x*y-{- Mx^y^ — 27^3 + 1; 60, cc* — a;^ — llx^ + 9a; + 18 ; 61, «* — 20a;2 + 64 ; 62* go^ — x^ — ISx'^ — 6x + 36 ; 63, 07* — 256 ; U, x» + x^ + 1 ; 65, x^ + ax^ + 6a;i'+ cx'+ abx + acx + J>cx + abc; 66, x^+ S6x^ — lOa?* — 86a;2 + 131a; — 168 ; 67, a^ —66 + 3^26* — 3ft*62 ; 68, o2 + 2a6 + &« ; 69, «« + 3a26 + 3a62 + 63 ; 70, a* + 4a36 + 6a263 + 4a63 + 6* ; 71, x^ + ix»+ 4x^ ; 72, x«—6x^+ 9a!* ; 73, 16a;*+ 96a!S _|_ 216a;2 + 216ar+81 ; 74, 125a3 + 450a2c + 540ac2 + 216c3 ; 75, aS— 8a6 + 24a4 — 32a2 _,_ le ; 76, 4a6 — 20a56 + 25«*62. Page 42 . 3, a;2 4- 10a; + 21 ; 4, a?2+ 14a? + 48 ; 14, xi- 18a; + 80 ; 15, x^ — 32a; + 240. M.E.A. — 17 258 ALGEBRA. Page 43. 26, a;^-{- Ix — 18; 32, x^+ 2bxS— 12062 . 34, a.2 — 707 — 18 ; 35, x^ — Ux — 100. Page M. 45, 64 ; 47, 16 ; 49, x^ — 12a; + 36 ; 50, a;2 — 36 ; 51, Ax^ -j- 40? + 1 ; 53, a^ + 18a -j- 81 ; 55, a2 _ 81 ; 56, a* + 14a2 + 49 ; 58, a* — 121 ; 60, z^ — &az + 9a2. Page 45. 61, y* + 12ay + 36a2 ; 63, 9a?* + 12x^ + 4a;2 ; 65, 'da'^x^ + 36a3a; + 36rt* ; 67, 81 — a;^ ; 69, a;* — imx^ + 8162 ; 71^ 25it;4 — 606a;3 + 36 62a?2 ; 73, 36»8a?2 _ 120 n^x^ + lOOrt^iT* ; 75, 25a;* — 9^/* ; 77, a*66 _rt2j8. 78^ 1 _ i44«8. 79^ SSaea!*^^ — 1; 80; a8_81. Page 46. 3, 16 ; 4, 100; 5, 16 ; 6, a;*— 4a;S + 4ar2 ; 7, a!4-(- 4a;3 — 2a!2 — 12a; + 9 ; 8, x^— &x^ -\- 9a;* — 10a;3 + 30a;2 + 25 ; 9, 4rt* — 12a36 + 25a2ft2 _ 24068 + 166* ; 10, o2 — 2a6 + 2ac + 0^— 2hc + 62; 11, a^ _j- &2_j_c2_2a6— 2ac +26c; 12, a* + 2a^ -f az _ 8a2a;— 8aa; + 16a;2; 13, 4a'e— 12a;S + 25a;* — 28a;3 -f 22a;2 — 8a; + 1 ; 14, 4 — 4« + 9^2 _ Sa^ + 6a* — 4a6 + a8; 15, 0* — 4aS6 + 16a262 _ 24a63 + 366*; 16, a6— 6a«+ 15a*- 20 i^ + 15a2— 6 1+ 1. Page 47. 1, 3ffl36; 2, — 3a;; 3, 5as ; 4, — 3a*c; 5, 5a2«3 ; 6, 263^3 ; 7, — 3ms ; 8, — 6a6c ; 9, 2a36c3 ; 10, ao^x; 12,60=1; 13, c6=l; 14, ( a + 6)0= 1 ; 15, a!0=l. Page 48. 17, 4 ; 18, - 1 ; 19, — 2a + 4«26 — 5a3c; 20, 2ay + &a^xy^ — l ; 21, — 5 + 7a62y — Sa^b^nx; 22, 5ae»w + 7 — 10a2c6 ; 23, 2a2— 36*+ 4c8 ; 24, 4(a -f 6)3— 2(a + 6) + m ; 25, — 4a(a — x) — 3c(a — a;)* + 5 ; 26, — 6(a - a;)2(6 + y) + c{h + yy. ANSWERS. 259 Page 50. 30, 5 ; 31, — 2 ; 32, 15 ; 33, — 2 ; 34, 39; 35, J; + 6 ; 86, a?— 7 ; 37, a; — 5 ; 38, a; + 4; 39, x — 7; 40,37— 9a; 41, a? — 2a26; 42, a;2+3a;+ 9;43,aj2_4; 44, ay^+ Sbx^ ; 45, a'>+ ab + b^ ; 46,- a3+ a^b + ab^ + 63 ; 47, aj*+ 5a52+ 13a; + 14 ; 48, a^— oa? + aj2 ; 49, aj3— 2x'^y + 23/8. Page 51. 50, 2a2+ 5aa! + as^ ; 51, a;*— aj^j/ + xy^ — ys ; 52, 24a;2— 2aa! — SSa^ ; 53, 20a^— 80ac + loc^ ; 54, a — b; 55, 3a* + 3a^y''+ 3y* ; 56, ««+ ^a^fe + 3a&24-6s; 57^a!2— a!y + 2/2; 58, 4a*62_|. 20363+ a^fi* ; 39,8a966es_4a76ec5+ 2a5&8c7_ a^b^c^ ; 60, a?*+ 2a;3 + 3a;2+ 2a; + 1 ; 61, a^— 2a6 + b^ ; 62, x''+ y^+ z^ — xy - xz — yz; 63, z^— 16a + 63 ; 64, 5a*— 2a36 5(2aJ — 1) „„ + 4a2&2 ; 65, x''+ 3x + 6 + a;^_2a; + l ' ««' ^ + " + 4; 67, a; — c — 5;68, a; + M + 4;69, aj + n— 5a; 70, (ar + a)2 + 4(a; + a) +16; 71, a; + a — c + 8; 72, (aj+ c)2— (ar + n)2. Page 53. 6,0; 6,12; 7,-20; 8,0; 9,0; 10, — 12 ; 11, 64 ; 12, 0. Page 54. 13, 3; U, — 60; 15, yes; 16, yes; 17, no; 18, yes. Page 55. 1, a^ + a + 1; 3, a^ + a^b + a*&=' _|_ asfts _^ ^,2?^4 + a65 + &8 ; 6, 4 _(- 6a; + 9a;2 ; 9, .5a2a; + 3. Pac-^ 56. 13, a2 — fla; + a;2 ; 15, 4 — 2a + a^ ; 18, a6 _ 5a3 + 25 ; 23, 256a* - 192a3& + Uia^b^ — 108ab» + 81&*. Page 57. 25, a3 + a^+a+l; 26, as^a^ + a-l; 29 8^-40.=' + 2.; - 1 ; 31, a3a.e + «<=«.* + a^.^ +1 ; 260 ALGEBRA. 36, 27a;3 — ISx^y + 12xy^ — 8y»; 39, a? —2; 40, (a + 2)3 + (a + 2)2 2a? + (a + 2) (4a;2) + Sx^ ; 41, a;2 + 5a; + 1 ; 42, «» — a^ (3aa — 2a + 1) + a (Sa^ — 2a +1)2 — (3a2 — 2a + 1)8. Page 58. 2, x(a + x); 3, xy^x—y), 8, m(a + &); 9^ J(a2 _|_ c2 — M); 10, ac(6a — 3& + 7ac); 15, (a + a?) X (a + &)• Page 59. 19, (x + a)(ar + c); 21, (a + »»)(&— »); 22, (a — m) X (6 — w); 25, (a2 + w)(&^ — »«) ; 28, (a; + 3) (« + 4) ; 31, (a? + a) X (a? — 6); 34, (2/ + 2)(3/ - 5). Page 60. 2, (a? + 9)(a? — 9); 5, (9a26 + 12) (9az6 — 12) ; 8, (a2 + 206)(a2 — 20&) ; 15, (1 + 15 a^&2c)(i_ioa3&2c); 16, (a;2 + l)x(a7 + l)(a;— 1) ; 17, (1 + 4a*)(l + 2a2)(l — 2a2); 19, (aS + js) (a* + hi'){a'^+b^){a + b)x(a — b); 21, (x-+a + n) (a; + a — m) ; 22, (x — b + m){x —b —m); 23, (a; + 7)(a; — 3); 24, 2/(12 —y); 26, (2a;)(2a) ; 27, (a;2 + 4a: + l)(fl;2 — 1) ; 30, (a; — a + 36)(a; — a — 36). Page 61. 32, (a — 6)(a2 + a& + 62 ) . 33^ (a- _ 2) (a!2 4. 2a! + 4) ; 34, (x — 3)(a;3 + 3a!2 + 9a! + 27) ; 35, [y — 4:){y' + iy + 16) ; 39, (2a! — Sy^ix^ + 6xy + 92/2); 41, [(a;+ 1) — 2] [(a!+ 1)3 + 2(a; + 1)2 + iix + 1) + 8] ; 45, (a! + 2) X {^^ — 2a; + 4) ; 46, (a; + 3)(a!* — 3a!3 .^^^2 _27a! + 81) ; 47, (a! + 4)(a!2 — 4a; +16); 50, (46+ 5)(1662_206 + 25); 62, lix + 1) + 3] [(a; + 1)2 _ 8(a7 + 1) + 9]. Page 63. 3, (x + 3)(a; + 6) ; 4, (a; + 2)(a; + 7) ; 8, (a; + 4a)(a! + 8a) ; 11, (2/2 + 8a2)(2/2 + 8a2). Page 63. 20, (a;— l)(a!— 10); 21, (a!— 2)(a! — 9) ; 28, {X — 7«)(a; — 9a) ; 32, {y» — 56)(2/3 - 76) ; 37, ANSWERS. 261 (a? — 2)(a? + 7) ; 38, (a? — 3)(a; + 8) ; 45, {x — 8a) {X + 15a) ; 47, (a;2 — 15c)(a;2 -f 20c). Page 64. 54, [x + 2)(a! — 7) ; 55, {x + 3)(aj' — 8) ; 63, {X + 8»)(a; — 13n) ; 69, (j^s^* + \.Q){y^z*' — 11). Page 65. 3, ix-\-iy -, 4, (a;— 8)^ ; 13, (^l—Mh^y ; 20, (18a2c — ac3)2. Page 66. 3, (3a? + 1) (2a? + 3) ; 4, (3a; — 8) (4a; + 5) ; 5, (2a; — 1) (5a; — 3) ; 6, (4a; — 5) (5a; — 4) ; 9, (6a; — 5) (3a! + 2) ; 10, (6a; — 5) (4a; + 1). Page 67. 2, (a^ + « + l) (a^ — a + l) ; 3, (Sa^ + 3ac + 5c2) (3a2 — 3ac + oc^) ; 5, (7a2 + 13a6 + 1162) (7ffl2 _ i3a& -I- 11&2) ; 7, (4a;2 + ^xy — 7y2) X (4a;2 —^xy —ly^); 9, (Sa^ -\- xy — Ay^) {hx^ — xy — 4y2) ; 13, (12 -\- x — 9a;2) X (12 — a; — 9a;2). Art. 158. 1, (a;— l)(a;+ 1); 2, (a; — 7) (a; — 8) ; 3, Alt. 151 ; 5, Art. 150 ; 6, (a; — 11) (a; + 4) ; 10, Art. 154; 12, (3.i; + 4) (2a; —'1) ; 15, a;(a; + a) (a: — aa; + a2); 16, (3a — 2)(2a — 5); 17, a^{a — J}) (a2 + a6 + 62) ; is, Art. 156 ; 19, 5(1 + 2a;) (1 — 2a; + 4a!2); 20, (2a;2 + a; + 1) (2a;2 — a; + 1) ; 21, 7(2a; — 3) (4*2 -f 6a; + 9) ; 22, Art. 156 ; 23, a;(8a;3 4- 1) (4a;2 + 2a; + 1) (2a; — 1) ; 26, (5a; — 1) (a;2 + 1) ; 27, 3(a;2 — 5) (a!^ _ 8) ; 28, Art. 154 ; 29, a;(a; — 3) (a; + 5) ; 30, Art. 154 ; 33, 5(a;* + 7) (a;* — 12) ; 36, (p — 2)(a; + 2/)(a; — y). Page 68. 37, l{x^ + 5)(a;2-12) ; 38, (a;^ + l)(a:- 1); 40, a;(a;+ l)(a;— l)(a; — 3); 41, a;(8a; — 10) (3a; + 8) ; 42, Art. 147 ; 43, Art. 154; 44, Art. 148 ; 46, Art. 147 ; 47, a3(«' + « + 1)(«* — a + 1) ; *8, (a 4- & 4- r)(a + 6 — c)(c — a + 6)(c + a — l)-, 49, y^i^x + 2y){x — y) ; 50, (a; + y — ^X^^ — ^ + «) (a; + y + z)ix — y — A. 2 262 ALGEBRA. rage 69. 3, 2a^b ; 5, ib'^x ; 8, 13hc ; 10, a{a + h)' ; lS,x + y; U,2x—1; 17,0! — c; 18, 2(x — y)» ; 19, a? + 4 ; 20, a? — 5 ; 21, a? + 9 ; 23, a; + 9 ; 25, x — 7. Page 70. 28,x + y; 29, a; — 2 ; 30, a — 36. Page 73. 35, x+7 ; 36, a; + 11 ; 37, 3{x — 1) ; 38, 2x + 5; 39, 4.B — 1 ; *0, rt + 6 ; 41, Sx^ + I2x ; 42, a-\- X ^y; i^,y -\- z; 44, a — 3& ; 45, a? — 2 ; 46, a; — 3 ; 47, 2a; — 1 ; 48, 2x^ + 7a; + 3 ; 49, (a; + I)*. Page 74. 3, 24n3i»2c; 4. a^ftsc^as; 5, eOe^ajayz ; 8, ft2(a + hy ; 9, 48.1*63^5 ; n^ (,x, _|_ i)(a; _|_ 2)(a; ^2); 12, (a + &) (a— i;)2; 13, (j;^ _ 5a; + 6) (a; + 3) ; 14, (3 + 2a;)2(3 — 2a;). Page 75. 17, (a; — 3)(a; + 3)(a; + 5)(fl; — 7) ; 18, ( X— 5)2(a; + 6)2 ; 19, 30(a + fc)H« — ^Y \ 20, 12a (a + 6 + c)2(a + 6)8(a — Z») ; 22, [x^— 2Sx + 112) (a; + 8) ; 23, (_x^+ 28a; + 187)( a; — 13) ; 24, (a — by (a2+ 62)(a + h) ; 25, (a;3+ 6^^^ lla; + 6)(a; + 4) ; 26, (a;4— 16)(a;*— 81); 27, (x^~ y'^Xx'^— iy''). Page 79. 3, 20 : 4, a;2 + a; — 6 ; 5, 33 ; 6, 15(a2— 62); 7, 5(a+l); 8, 7c2(a+l); 9,26a;; 10, (a; + 2)2 ; 11, Sa^ftca;; 12, 2/2 - 25 ; 13, a;2 — a2 ; 14, fts _ 2a26 -I- 2a62 — 63; 15, 18ffl662cm; 16, x^ + 2a;*y3 + 2a;2y8 + y^, Page 80. 18, 62 ; 19, 12a;2 -|- 18a; + 10 ; 20, 78 ; 21, 6a;s+ 4a;2 —7a; — 3 ; 22, 192 ; 23, ox»~ 18a;2-)- 32 ; 25, 46; 26, 43; 27, 3a!— 17; 28, x^ — 47a;2 + 224a! — 48. Page 81. 30, 15; 31, a;«; 32, 91; 33, 15a;2 ; 34, a; -x -6a;, Art. I80. 1, ^^, 2, ^^,^; 3, "^qij-; ANSWERS. 263 x+8 ^„ a? — 8 ^^ a?+3 Page 82. », ^Hi? l®' ^Hio' "' ^=9' x—^ _ a;+9 .. a + 1 ,, a; + 1 . 12' TF'' 1^' S+6 5 "' F:^' 1^' ^2 + 2^-3' a;2_4a;_|_3 ^„ gz — 2ffl + 2 2.^ — 5 16i ^ZTi ! "' a2 — 2 ' ^^'Sjj— 11' S„h3 _464 289 12 6,1- -F+T- 20 Page83. 7, «''-«& +&^8>^+ 4-^rp-5' ». «" a -f- ji -^ i- Art. 187. 4,gg; 5, ^, _ ^ > 6> -i^.^' , 2a;2 — 5a;— 3 « 10^". a a;" — 8a; + 15 '' — ;^2-zr9 — ' *' I2&^' ' so^+ ^ - 12 10, ic2— 9 a;8— 223; — 15 x^+ x— 30 * 2a?«+ 8. Page 84. 2, f:^:i2:^r-_* ; a, ^^g" ' ^^ 2 2^M:i^J:L?; 3, 264 ALGEBRA. . 2aj2 — 20? — 1 , 12x X . x'-+ x — 2 „ 2a:(aj2 — x — 1) _ a ^' 0,-2 ' *' J^Tl- Page 85. 3, !2, H^, ^if 4,6^-^ 6^M; ^ ' 30' 30' 30' ' 24 ' 24 ' g 35a 6a; ISa' 3&c . » 2«r6 + 46^ 2ffla— Sfl-ft . ' Ibab' 15ab* I5a6' ' Gab ' 6a6 ' y 3wgy 3wy wta; . „ {a + by {a—by_ ' 3my ' Zmy' 3my ' ' a^— b^' a^—b^ ' g GOcKC 56a; 27a2 . 4a;2+ 12xy +Jy^ ' 24a2a;' 24a%'' 24a5^' ' Ix^'^^dy^ ' 4a;a— 12a;y+9yz . ^^ 9(1 + a;)^ Ca;(l + a^) 4o;2— 93/2 ' ' 8(l+a;)3' 8(1 + a;)»' 4a;2 _ jg a;8 x"— x — 6 a;* — 9 8(1 4- a;)*' ' a!3— 9a;' x^— 9x ' a^ — 9a; ' 13, a;2+ 6a; + 9 a;2+ 4a; + 4 (a; + l)(a;+ 2)(a; + 3)' a;3+ 6a;2+ lla; + 6' aja-f 2a; + 1 x^+ 6x^+ 11a; + 6' 17a; 2a; + 8 _ x o 3a; ~ 8 Page 87. 5, ^; 6,?^+ «; 7, ^. 8, 12 ' 'a; — a;8' '24' x^ —9' 9 ^^-±J-10 a' — 9 . 11 41— a; „ 2«2 13 ^^'^^ 14— 2a* . IK 3&a; — 2fla;+3a 16, ?(«!±i!). 1- lla. ,,, 4a6 ANSWERS. 265 ■Pa^e 88. 19, ^^^ ,- 20, l^i±i^±i; a, 71a; — 513/ . 80.153 + 64*2 4- 84.i; + 45 ' 50F~' ^^' 60iS ' 23 2x^+710+1 2^ lla?+64 ' a;3+12a;a+47a;+60' ' (a;+4)(a?— o)(a.'+6) ' a; + 6 ;r + 7 ^^' x^ + ^- + 3 ' ^®' a:2 + 6*' + 8 ' 2 • 2 2^ (a,_4)(a;_5)(a!_6)' ^*' 4*-3 — a?' 2a;g + 8a; — 1 1 ^®' (1 — a!)(a! + 2)2 ' ®**' a; + 2 ' ^*' °- 2a; — 3 ax — b^ Page 89. 32, ^^, _ j^^^^. _^ g^ ; 33, ^5^3^; 81a — ib 2ab^ 4« 34, gj ; 35, ^4 _ j,4 ; 36,j:p-^; 37,0; 88, — 1; 39, 0; 40, 0; 41, 0. llfls . 3(x + a) Page 90. 8, YsftS *' i^; S, io(a;-a)5 6, f(a2+a;2); 7, ||; 8, a + a;; 9, J; 10, IJ; 2a2 ^ "' 3^5 *^' 2aa;' Page 91. 13, |^^-^y¥ ; 1*. ^^ - 2^ - i^ ; a;2_4a;+3 ^, a;^ + a; — 2 16. ^.2 _ ip _ 2 5 "' -^2 _ 3.r - 4 ' ^*'^' 19, 1; 20, 4^; 21, -^; 22, - , 266 ALGEBRA. ^bTS + S + i; 24, .^ + i + ^;25, 1. •"(10ir2 + 9ag + 2 ) (2a! — 1) Page 92. 26, p^r:p23x + 12)Ca? — 4) ' «— 1 9 _! A ^°^~^ 5, — p-; 6, 2(a; — 1); ', (^S^r;^^ + a;2)|,a — a;) ' Page 93. ll, ^ ; i''' 6 "~ a ' IB, iSqrys-; l*- l; l^' a; us' ( 3a; + 18)(2ir — 3 ) *' (2ir+ 6)(2a; + 3); a2 ^ a,2 ^ _*i+J; o 1 Page 95. 7.— g^^^? »' 2^ ' "''^^ a;2 —64 _ gi+ a^^ +6^ . 10. i^ZIiii' *^> ab\a—Ly _ j,c(b- cy . ANSWERS. 267 fee + 1 adf + ae 4 ^®' ffl&c + ffl + c' *'' bdf+Te + cf' **' 3(aj'+l)' Page 97. 11, 4; 12, 2; 13, 10; 14, 3; 15, 12; 16, 7. Page 99. 2,3; 3,5; 4, 2^; 5, 4§; 6,-; 7,^; 8,«;9,cm; 10, ^^:i:^ ; 11, -q^; 12,«-5 ; 18, m^ + 3M + 9; 15,2; 16,8; 17,9; 18, 3J; 19,-3; 21, J+fc^; 22,^zr-„-; 23,0-6; 24,«.+«; 25^ ft2 _ 6c + c« ; 26, c2 — c + 1 ; 27, m + w ; 28, «2 + c2 ; 29, a^ —a + 1. n — m Page 100. 31, 3; 32,-2; 33, SJ; 84,-^-5^; 35, ^^ : 36, c2 + c» + m2 ; 37, c2 — CM + n2 ; 38,10; 39,13; 40,8; 41,7; 42,-8; 43,13; 44, y -\-q; 45, a -\- b. Page 101. 47,3; 48, 10|; 49,5; 50,24; 51,13; 52, 11 ; 53, 7. Page 102. 54, 20; 55, 13; 56, 4; 57, 4; 58, 3 ; 59, IV: 60,4; 61, f^; 62,7; 63,1; 64,3; 65, ^f; 66, — 2 ; 67, 12 ; 68, 12 ; 69, 6. P.ige 103. 70, 4j; 71, ^-J-^; 72, — K« + "); ■3 6 ^ + 3 ■ U^^^; 75,4(1 -'»='); '*' 6a + 2/>' »*' a+ 3. ' '2^ 268 Algebra. 80. a H- 6 Page 105. 2, $3, $9; 8, $2.50, $15; 4, $1.28, $5.12. Page 106. 7, $39, $114; 8, $4.15, $7.85; 9, 76 ft., 97 ft. ; 10, J(« + <^)> K«—^) ; 12, 20, 127. Page 107. 13, 15 jr., 18 yr. ; 14, $51, $228; >i — d ts + d 15, ^-77^' 7473- ; 17, 120; 18, 270; 19, 54; 20, 120. „ ,„„ dmn mns Page 108. 21,5^-; 22,,-^^^p^; 24,40,60; 25, 20 min. past 5; 26, 28, 18; 27, $250, $600; ms ns 28,7,8; 29, m -f- TO m-{-n Page 109. 80. ""'" - '^, "»-"" !!'; 32, 9, 51; n — m n — m 33, 9i, lol; 34, tizl, ^l+S- 35, A i£:i62, B o 5 q-\- 1 g-\- 1 £\1S, C ^"104; 36, 24, 36, 48; 37, 20, 30, 5, 125. Page 110. 38, $80; 39, $90, $180; 40, 20 yr. ; 41, 18, 19 ; 42, 24. 25, 26 ; 43, 100. Page 111. 45, 15 yr., 45 yr. ; 46, 20 yr., 40 yr. ; 47, 7 yr. ; 48, 24 yr. ; 49, '^'' +— . Page 112. 51, 3,^ da. ; 62, 2|f da. ; 53, 15 da. ; 54. 11 da., 22 da., 33 da. ; 56, 6; 66, 30; 67, 12 hr. ANSWERS. ^69 Page 113. 68, A 32 da., B 19^ da. ; 59, 10 da. ; 60, -^h—-. ab -\- ao -\- be Page 114. 62, 10; 63, 72; 64, 3f min., 6} min. ; 66, 15 mi.; 66, 4 and 5 mi. per hour; 67, ^ Page 116. 69, 420; 70, 600, 450; 71, 728. Page 117. 73, 1°, 27^1 min. past 5, 2°, 43^ min. past 8, 3°, no time; 74, 1°, 21^^ min. past 1, 2°, 5^ min. and 38^ min. past 4, 3°, 21^ min., and 54^^ min. past 7; 76, 1°, 38^^ min. past 1, 2°, 54^^ min. past 4, 3°, 10^ min. past 8 ; 76, 1°, 43,'t min. past 7, 2°, 27,^ min. past 7; 77,93. Page 118. 78, ^; 79, 70; 80, $150; 81, $20; 82, 6 miles; 83, 17; 84, 60; 85, 12, 7. Page 119. 86, $126; 87, 32 min. 18^ sec. past 5 ; 88, A in 10, B in 12, C in 15, all in 4 days ; 89, 15 mi. per day; 90, ^; 91, hare _^!!i_, hound ^^-. 92, 1°, 7t^ sec, 2°, 46f|f sec, 3°, 29^| sec. Page 123. 2, a; = 2, y = 3 ; S, x = U, y =4. 4,a! = 20,y = 10; 6,a; = 6,y=12; 6,«=16,i/=7 l,x = 7,y = U; 8, x = 8, y = 4; 9, a; = 5, y = 2 10, a; = 60, y = 30. Page 124. 12,a; = 1; y = 3; 13, * =60, y = 36; 14, ar = 5, y=4; 15, a? = 12, y = 20; 16, a; = 5, y = 7; 17,ar=— 21^, y=38i; 18, a; = — 6, y= 12; 19, a; = 18, y = 6. Page 125. 21, a> = 2, y = 6 ; 22, a; = 21, y = 15 ; 270 ALGEBRA. 23,a? = 3,y = 4; 24, ii?=12, y = 9; 26, a? =4, 2^ = 3 ; 2&, X = 3, y = 5 ; 27, a? = 7, y = 11 ; 2S, x — 13, 3/ = 11 ; 29, ar = 144, ?/ z= 216 ; 80, a? = 4, y = 6 ; 31, 07 = 2, 2/ = 4 ; 32, a? z= 2, 3/ = — 1. Page 126. 33, a? = .8, 2/ = .4 ; Si, x==5, y = 5; 3a + 26 35, xz=i(s + d), y = K« — ^) ; 36, a? = ~— , 4ns — 3& _ ffr — ns y = —iT- ■^Sl,x = 7,y = 10; S9,x = ^ii^~, pr — ms en cm V = ; 40, X := ; — T — ) V = : — T — ) " mq — np ' an -^ om " an -\- bin 2 2 2a «. «> = iqn,' y =^r^h > 42, a. = ^^^, 26 y=-^I-d' 43, a? = (a + 6)2, y = (a — 6)2. Page 128. 2, a? = 3, y = 2, 2; = 5 ; 3, a; = 10 y = 2, = 3; i, x = 2, y :^3, z = 4:; 5, a; = 3, y=^i,z = e; 6, a7=2, y = 3, 2;=4; 7,a; = 5,y=4, 2 = 3 ; 8, a? = 1, 3/ = 2, 2: 3= 3 ; 9, a7= 8, 3/ = 9, 2 = 12 ; 10, X — 4:,y = 9,2 = 16, « ^ 25; 11, 07 = 2,3/ = 3,2 = 4, 62 1 c2 _ a2 «2 _L. c2 — 62 « = 5; 12, a7 = 26H , 3,= ^^^ , Qg + 62 — c2 w(w — 6)(w — c) ^ — 2a6 ' 13, a? _ ^^^^ _ ^^^^ _ ^ ^ ; a 6 c 14, as = -g-j 3/ = -g-, 2 := -g- ; 15, a; = 6 + c — a. Page 129. 2, 5, 7 ; 3, 12, 15 ; 4, 85, 15 ; 5, $250, $320; 6, A $42, B $26. Page 130. 8,20,32; 9,42,45; 10,40,60; 11, A $180; B $140; 12, 7, 10; 13, 1, 3, 5. ANSWERS. 271 . Page 131. U,Z^; 16,1; 17, if; 18, §,i; 19, Page 132. 21, 36 ; 22, 48 ; 23, 245 ;. 24, 891. Page 133. 25,39; 26,853; il , ^(lls — 9d) ; 28, (m + l)e (n + 1)0 $90, $120; 29, ^^ ^ , , ^ ^ , ; 30, lOyr., ^ ' ^ ' ' inn — 1 mn — 1 > .? ' 40 yr.; 31, 30 yr., 22 yr. Page 134. 32, $50, $40; S3, A 6a + 56 + c^ 3a +56 + 6 ^ , ^ , . ,, B -^ — —> 34, $3 J, $44; 35, tea 5s., sugar 4d. ; 36, 18| lb., 31i lb. ; 37, 40, 90. Page 135. 39, 8, 12, 24 days; 40, 40 spel., 30 read.; 41, 16 h., 32 c. ; 42, 18, 24, 36 days; 43, 2a6c 2a&c A in ao + bc- ab '"'"•' ^ ^" ab + &c -ae ""'"•' ^ 2abe C in -r — j — — j— mm. oo -|- ac — be Page 136. 45, 5 mi. per hour; 46, Dist. 24mi., be hd a{c^ — d^) . rate 5 mi. per hour; «, ^qj^' ^qrd' 2M ' 48, 10, 22, 26 ; 49, $60, $80. Page 137. 50, 3a, 20 rd., 600 sq. rd. ; 51, A in 5 min., B in 6 min. ; 52, 4 and 5 yd. ; 53, 100 mi. ; 54, Principals, $3000, $4000, $4500 ; rates 4, 5, 6 per cent. Page 138. 55, A $21, B $11 ; 56, 26, 14, 8 gal. Page 140. 3, X = b, y =2; 4, a; = 5, 12, etc., y = l,5, etc. ; 5, a;=10, y=l; ,6, a? = 5, 10, etc., y = 2, 8, etc. ; 7, a? = 4, y =- 6 ; 8, a; = 10, 19, etc., 272 ALGEBRA. y — 3, 16, etc. ; 9, a? = 15 or 2, y = 1 or 2 ; 10, a? = 8, 21, etc., 3^ = 3, 8, etc. ; 11, no values ; 12, x — 2, 37, etc., 2/ = 3, 46, etc. Page 142. 15, 41, 59; 16, it cannot; 17, 23; 18, two, viz: 4, f and V- *; 19, M, H and ||, |f ; 20, 40, 110, 180, etc. ; 21, ^, |f , etc. ; 22, 58 ; 23, 6, 56, 8, or 12, 42, 16, or 18, 28, 24, or 24, 14, 32 ; 24, 61. Page 143. 25, 34, 13, l,or 37, 8, 3, or 40, 3, 5 ; 26, 4, 11, 25, or 8, 2, 30; 27, 4 ox., 6 h., 90 hens; 28, 5 m., 3 w., 33 c. ; 29, A 1, B 15, C 29— each rec'd 65 ct. Piige 146. 1, a*; 2, 243ai<'c5; 3, — SiSa^b^z^ ; 4, rtl2 . 5^ iQx^iyi ; 6, 25668c4» ; 7, ffl4" ; 8, — ^Ix^z^ ; 9, — 243«s»a;i52/5. iq, aioftiS; n^ iba^l^ • 12, a86* 8aj6«3 625a8ft* 64a6»i-c- 13, -g^; 14,1251,; 1^, ^g^ ; 243a;B3^iB ,^21 ffl6»fti2 . ^®' 1024 ' 1^' — rei4a;T ' *^' cS" ' Page 147. 1, a»-\- 3a^x + 3aa;2+ a;3 . 2, a*+ 4a3a; -f 6a^x^ + 4aa53 + a?* ; 3, a;B _)_ 5a;4 3^ -j_ 103733,2 _|_ iq a2j/3 _(_ 5a;y4 + 2/s ; 4, aS _ 3^2^, .^ 3aa;2 _ 3,3 . 5^ a;* — 4aa!3 + 6a^x^—4:a^x-\- a* ; 6, ajS— ox^y-^ 10a;3 2/2 — 10aj22/3 ^ 5a;3/* — i/S ; 7, as + Sa^ + 3a + 1 ; 8, «* + 4a;3 + 6a?2 + 4a! + 1 ; 9, a;6 + 5a!* + 10a!3 + 10a!2 + 5a! + 1 ; 10, a!3 — 3a!2 _(- 3^! _ 1 ; n, x* — 4a:3 -f 6a;2 _ 4ai + 1 ; 12, j/s — 5y* + lOy^ — lOj/^ -|- oy— 1. Page 148. 13, x» + 9x^ + 27* + 27; 14, a!^ — 60a!* + 12a2a!2 — 8a3 ; 15, a^ + 3a^ + ea* + 7as + 6a2 + 3a+l; 16, ai^ — ea!^ + I2a! — 8 ; 17, a« — 12oT 4- 54a" — 108a« + 81a* ; 18, a!* — 2a!3 — a!^ ANSWERS. 273 -|- 2x + 1 ; 19, a* + 16a3 + dGa" + 256a + 256 ; 20, al + Ua^c + 48a3c2 + 64c8 ; 21, 1 — 8a + 28a2 ^ a^as + 70a* — 56a5 + 28a6 _ 8a'' + a» ; 22, 1 4- 10a + 40a2 + SOas + 80a* + 32a5 . gg, lOaS — ^a^y + 216a*y2— 2iea^y3+ Sly* ; 24, 125— 150a; — XSiX^ + 52a;8 _(- Sx* — 6x^ — x« ; 26, 32a6 — 80a* 4_ apd*— 40a2+ 10a— 1 ; 26, 32ai5_40ai2a7+20a9x2 _5^?a;3_|_£- a^x^—^; 27,x«—15x^+81x*—l86x« 4- iq2a;2 — 60a; 4- 8. Pagf! 149. 1, ±a2 ; 2, — 2a26; 8, ± 12ai63c; 4, a^ ; 8, ± 8x3y ; 6, 60x3^^ ; 7, 4a!2 . g, 3a4& ; 9, + HaSft^^s ; 10, ± 2a2 ; 11, — 5c2y« ; 12, — a^fiic^a;; 13, ± --; 14, + !^^-^; 15, -?^^; 16, 9; 17, 7; 18, 2a; 19, 15a26i 20, 10a; 21, —a^x^. Page 151. 3, 3a&2+ 2 ;'4, SajS- 5a;y ; 5, lOa^b^c + 1 ; 6, «a— a; + 1 ; 7, aj^— 2a; — 2 ; 8, 2a;2+ 3a; — 1 ; 9, 2a;2_ a; + 1 ; 10, 2x2— saa, + 4a2 ; 11, 5a;a— 3aa; + 4a2 ; 18, x^- Bax' + Ba^x — a^ ; 13, a;^ — -| + - ; , a^ a X ''' T + ¥ - ¥■ rage 153. 3, 95 ; 4, 2625 ; 5, 7.82 ; 6, 125 ; 7, 1168 ; 8, 31.28 ; 9, 292 ; 10, 1404 ; 11, 280.9 ; 12, 234 ; 13, 1564 ; 14, .0733. -D ir^ ir 12. 1ft 25. jy ^. 18 11.19 37.97 Page 154. 15,—; 16,--, i^'^^- ^»' 2 ' .)-• 20 291^; 21, 1.4142 +; 22, .4714 +; 23,11.157 + ; 24,' 1 7320 ;'25, .'433 + ; 20, 15.832 + ; 27, 2.23606 +; 28, .37267 + ; 29, 10.068 +. N.E.A. — 18 274 AI.GEBRA. Page 166. 3, « — 3 ; 4, y + 4 ; 5, 2a — 3& ; 6, a;2 4- 20? — 4 ; 7, x^— 4a; + 2 ; 8, 2x^— 3x + 1; 9, 2as2-i-4cx — 3c2; 10, a;3+ 2aj2— 4a! — 8 ; 11, a + a; + 5: 12, a — 6 + c. Page 158. 3, 24 ; 4, 426 ; 5, 75 ; 6, 4.78 ; 7, .27 ; 8, .899; 9, 3.27; 10,4968; 11, i±, g; 12, 1.259; 13, 1.442; 14, 1.709. Page 161. 1, x^, a^; a^, a^, a^", ah^ ; 2, x^y^, 1 2 5a 1 2y iVx'^y^ 5^V^ir^, 4, ^, ^^, P' ^ + ^ -^— ; 5, «&-!, a&c-2a!-i, mn-^r-iaj^, S-ia-ifte-^, a;3 4-i&~2^c^, 5a3&-2cd-ia;j^-i ; 6, a"^, 6^*, c^, as^t; 7, a^, n"^, a"i, rii^. . Page 162. 8, a, c^, ^•'cl^ ^v^a; -' ; 9, ah^c^, ab^c^; 10, a^^,lic~ix^; 11, a^, c^^, a;^^, i/"*; 12, Hi*, a;iJ, 2a!^jf*; 13, m^, a;"^, y'i, z, a-^b^ ; 14, |a!-6y-2st, ^y/^a^.i^o-ix~i; 15, 729a364c*, ^- ia-ia;V^ ; !»' «' *' ^' ym~^' Page 163. 20, a-^ —Sa'h^ + a-^^ — 2a-'b* ; 21, 2a!-iy — lOxy-'- + 8x^y-^ ; 22, a^ft-^ + aH^ + arh^ ; 23, x^ + a;^y — xy^ — y^ ; 24, a + a^i + b^ ; 25, 6a!2 — 7a;* — 19a;^ + 5a; + 9a;^ — 2a;^; 26, x-^ ~~ Aj^swafts. §75 30, a*— fflSft^+'ft^; 31, x^ -- x^a^ + a^ ; 32, 2x^ — 3xy + 2y2 ; 33, 3ar-2 — Sx-^y^ +y; 34, 2a^ — 36* ■ Page 165. Fill the blanks with: 3, 25a'; i, 16a^b*c' ; 6, ix' _(- 4a; +1 ; 7, 27g^x^ ; 8, ips _ ea?^ + 12a? — 8 ; 9, 81a*ir8 ; 10, a;* + 2x3 + x' ; U,-^; 125aec* ^^' ~ 2166S ■ Page 166. Fill the blanks with: 15, 180; 16, (a + bya ; 17, 243 ; 18, (a!^ — 4)(a; — 2) ; 20, c ; 21, - 24a"67 . 22, -^ + ^ ; 23, A; 2i,x^ — a+^; 26, 2 ; 27, 4 ; ^8, 3 ; 29, 25 ; 31, 6 ; 33, 20 ; 35, Sa^&^c* ; 38,3^; 39,2. Page 167. 40, U^VT ; 41, \0ahV2a; 42, Ua^' !iin/2ax; 43, ISaj^l/l^; 44, Sa'^h^c^VSac; 45, 36a2i!»2c|/6«6 ; 46, 3ffl^^3a6; 47, 2xyV20x' ; 48, 6a;2|/4«; 49, 2aK5c; 50, 3act/2abH» ; 51, 8a3 6K2c; 52, 3.rv/l/2a'i/2 — 3a;2y ; 53, (a? — 3)T/"a ; 54, (a? + j/) l/S^^; 55, iabVSab'^ + 56 ; 57,^1/10; 68, ^l/30; 59,^1/14; 60,^1/165; 61, ^^Vl/^T; 62, Jl/T; J3, Jt/15; 64,4i/«S; 65, jJ/^; 66, ^e/lO ; 67,x\^^96; 6 6 68, -j-Va'b. 276 ALGfeBttA. Art. 276. Fill ih'^ blanks with: 71, 806; 72, 27aOx^ ; 73, 2fl&c3; 74, 32«5&io^i5. 75, 5fl&c2. . Page 168. Fill the blanks with: 79, a, aeft*, a^^fca* ; 80, x,ab', ah. Alt. 277. 82 ^yU, ^VS^, 'K125; 83, 3^ 2^ 5*; 84, e/27, VT, VT; 86, 4*, 2"^, 7^; 88, 'K^, 'v/P", 'Fc8 ; 89, d^, x^i, z^^. Page 169. 3, Sl/T; 4, 6t/T; 5, ei/Il; 7, f^: 9, e/||; 10, fl/lO. Art. 279. 2, 7l/T; 3, ; 4, l/T ; 5, 32l/¥; 6, 8fll/T; 7, 4!/^; 8, lab\/2a; ,9, 6a2v''^+20a2^^3c; 10, if/T+sVY ; 11, llx^Viy'^ ; 12, 2 -a73(a+ 1)1/302/; IB, a^xVii; 14, Jl/'T; 15; \v 3 ; 16,|v^6 ; 17, —[;xz;zr-^^'^\ 18' i\^18; 19, ^Via^b^x. ' Page 170. 3, 48ffll/3cj" 4, SaVSx^ 5, 70a&i/2«; C, 9aT/2T 7, 70a3a!v'2M"; 8, 12a*l/37 9, 108 ; 10, *\/3] 11, ]^jl/37 12, 1; 13, 4^'15"; 14, iV3&; 15, I^VaM'; IG, 1^/727 17, ^?/^ 18, ^^^^J~ Psige 171. 20, 2 ; 21, 14 + 81' 37 22, — 22 — IVT; 23, — 39 + 71/77 24, 4 + 21/Tol' to, 5G + 12l/li5"; 2G, « + c — 6 + 21/ (77; 27, n.r — 9 — 3l/l— ic^; 28, — 78 — 47j; + 17i/6 + 13a; + 6x2;- 29, 20j? — 32 -f 121/1 — aT— 6a^ ANSWERS. 277 Art. 281. 3, 16o2 ; 4, 126a»x; 5, 9l/m; 6, 96x^1/6'; 7, — V^Fi 8, — a-'x«Va~, 9, f^; 10, ^^ ; H- a^^'i/^ 12, "i^ Page 172. 13, 3 + 2V¥] 14, 10 — ei/sj is, 38 4- 17l/'57 16, 11 + 61/2] 17, 56 — 24i/57 18, 91/T+ 11 1/27 19, 4 + 4i/3"-f e/97 20, 25 — 27?/2" + 9^T; 21, 21/T+ 6i/r+ 6t/2^''2~+ 4; 22, rt + 2g/a362 + ^7*27 23, rt2 + 3ai/a"+ 3a + l/oT 24, a^ + 4a ya»62+ 6aF62+ 46i/a~+ &v/&7 25, 2 + 3a; + ay^ + 2(1 + a;)l/l + a; ; 26, 2 + 2l/l — a;a. 3, 2l/a7 4, l/Tl/T; 5, + VJVa\ 6, 3t^ 7, a;2t/iil 8,i/^ 9, ■(/ -^ ; 10, e/|T 11, Vx + 1/2/; 12, e^- Vb; 13, f/a"+ ^^ 14, iVa — hVb- 15, J''^— ^-^ 16, l/a"+ i/T- Page 173. 19, 1 + 1/3] 20, 2 — l/SJ 21, i/'2 + \/~b; 22, 2l/"2 + l/lO; 23, l/'S + 2l/"3; 24, l/lO — l/"3; 25, l/"3 + l/"7; 26, 3 — l/"2 ; 27, 2l/"5 — 3l/'2; 28, l/TI — l/T ; 29, 5 + l/lO; 30j.l/21 — l/"3. Page 175. 3, — 3l/'2; 4, 2 ; 5, 4l/="6 ; 6, l/=^; 7,-24; 8,2; 9, — 64l/=l; 10,-75; 11,324; 12, 1281/^=^; 13, 29; 14, 107; 15, 83. Page 176. 16, — 43; 17, — 11 — 2l/=n:; 18, -8 - 6l/=n. 2, |l/T ; 3, |l/'2 ; 4, i/'^ ; 5, ?^ - - ' •2?S ALGEBRA. „ 2 — i/lO , 3i/II2 + t/— 10 b, , ,, __ Page 177. 8, ^^ + ^ ; 9, 2^^ + e/72. IQ. wg/ a + Fli i/^=T; 12, — jg ; 13, — ^^^ ' M,-5-2^; 15,J!+1^; 1«, _ i±!^; 17, _ 7 + 6i/="2 . ^e^ _ 5 + « + 4i/a + 1 . jg^ 2a2 _ 1 _ 2«v^«-:ri ; ^20, ^^-2 + ^i/^ i^; 2 ^ + ^^'-^% 22, ^^J-=T 21 Page 180. 2, a? = 3, y = 4 ; 3, a? = 4J, y = 6 ; 4, a; = 4, y = 4 ; 5, a? = 9, 3/ = 12 ; 6, a? = 3, 2/ = 2 ; 7, a;=7, y = 5. Page 181. 4, 3 ; 5, 5 ; 6, 5 ; 7, — 3^ ; 8, 9 ; 9, 4 ; 10, 4 ; 11, 12 ; 12, 81 ; 13, 5 ; 14, 2 5 16, 4 ; 16, 25 ; 17,0; 18, 4; 19,0. Page 182. 20, 4 ; 21, 6 ; 22, — ^^ ! 23, cCi/w^-a) ; 962 24, 25; 25, 56i; 26, 6; 27, -^; 28, 6; 29, %; SO, &8 — 4aa 4a ~' Page 184. 3, ± 8 ; 4, ± 7 ; 6, ± 6 ; 6, dz i/Tl ; 7, ±4; 8, ±9; 9, ±i; 10, ± | ; 11, h= 1 ; 12, ± J ; 13, ±i/ll; 14, ±|l/-2; 15, + 1 ; 16, ± | ; 24a «(62_c2) 17, ±-7-; 18, ±-^—: ANSWfeRS. S79 Page 185. 4, — 2, — 9 ; 6, — 3, — 7 ; 6, 3, 12 ; 7, 2, 11 ; 8, 3, — 8 ; 9, 4, —11 ; 10, 8, —2 ; 11, 9, —4 ; 12,-6,-6; 13,7,-12; 14, §, - ^ ; 15, §, — 2 ; 16, —3, —J; 17, J., — 3. Page 186. 21, + 2, — 8 ; 22, + l , — 9 ; 23, + 10, - 2 ; 24, + 12, — 2 ; 25, + 1, — 13 ; 26, 7 + i/l70, 7 — i/l70 ; 27, 1, — 4 ; 28, 2, — 7. Page 187. 29, 10, — 3 ; 30, 20, — 5 ; 31, —1,— 9 ; 32, — 1, — 17 ; 33, 2, 20 ; 34, i(n+^/lS), J(ll— i/l3) ; 36, — 6, — 7 ; 36, 6, 13 ; 37, 7, — \9 ; gg, 4, — | ; 39, 5, — 6J; 40, 5, — 5|; 41,3,3^; 42, 9, — 8f; 43,5, - 5f ; 44, V, y ; 45, 4, - 4 ; 46, 24, - C ; 47, 4, - |; 48, 3, ^ ; 49, 7J, 4 ; 60, 8, 2 ; 51, b, — b — 2a; 62, 3», 4c — 3m; 63,4, — ! 54, a, — 1. a a Page 188. 58, 3, — ^ ; 59, V' ^3 ; 60, ^, — 3 ; 61,-^,-8; 62, 9, i; 63, f, |; 64, — J, - ^ ; 65, 1 + 1/409T gg^ 7^ ^. gj^ 4 _ J. gg^ 4^ _ ^,. gg^ _3^ 2^^, 70^10,-1; 71, 7,- V; 72, 9, H; 73, 6, J-J; 74, i/^ = — 5; 76, 8,— f. Page 189. 76, f, 1;77, V, 1 ; 78, 0, 4; 79, 1, — f ; 80, « ± & ; 81, A, - i; 82, 0, 2a6- rrc -j^. ^^ a c a -\- b — 2g ah + be + oe ± i/a^b'^-\- b^c^+ a^c'-^—ul > f{-+ b+ c); a -{- b -\- e. 84, 86, 2a 2b a +6' a + b' 86, iJi/^zt gi/7^. i> n" ■m -(bn ± v/«^»»M-~''^'"^ — a^n^)< 280 ALGEBRA. Page 190. 4, aj2 _ 14^1; + 45 = ; 6, a;^ —2ax + a2— &2=0; 6, a;2 — 5^ — 84 =0; 7,0!^— 2bx — a2 + 62 = ; 8, a;2 + 7£t! — 30 =r ; 9, x^ — 6c + 9c2 — 1 =0; 10, a?2+ 12a; + 3of = 0; 11, aj^ — 6aj + 7 &2 _ rt2 a2_/;)2 = ; 12, 072 + --^^ — ,r_i=o ; 13, x^—ax+ ab ax a 0; 14,072 — — _ 4 a2+62 ^^ — ^ — 0; \^, x"^ — ax -{- ^ = ; 16, a?2 — aa; — & = ; 17, fla72 _j_ ^ + e = 0. Note. In quadratic equations all the answers to each will not always be given. Page 196. 7, ± 2, ± l/TO; 8, 4, :i; 9, 1,-2; 10, 8,Vi^-; 11, 9, (-41)^; 12,1; 13, 16, i|f|i; 14,4, -9; 15, t, fe<^; 16, 0, -5, I, -V-; ", i, i; ,„ , , — 3 ± 1/109 18, 1, — 4, g— ; 19, 1, ^ ; 20, — 1 ± v'^'d , -1+1/2 ;21, 2,-2,3, 7; 22, 3, i, — =^-. Page 197. 23, 1, — 1, 5, 7; 24, 4,1, 3, 2; 25, —1, — 3 ± 1/ 33 — 3 ± 1/^ — 3 ± i/^TY — ^, o ; 26, -. ' ~ • 3, 3, 17 ; 4, 7, 13; 5, 10 + 1/1^5, 10 _ i/_ 5. Page 198, 7,3; 8,4; 9, " - ^ =J^ ; 12, $4 • 13, 4 mi. Page 199. 14, 10, 11, 15, 14, 15; 16, 6, 7, 8, 9; 17, 14rd., 16rd. ; 18, 7 ch., 14 ch. ; 19,3 ft.; 20,10 in;; 21, 12, 18; 22, 3, — 10; 23, ^^-. Page 200. 24, 6 hi-;, 8 hi-.; 25, 40 ft.; 26, 6,18; 27, 2 ft, 5 ft.; 28, 30s. per week; 29, A $200, B $300; 30, 76 mi, ' ANSWfeks. m Page 201. 2, (900)2 acres ; 3, a; = 3, i/ = 4 ; 4, a;=4, 2/ = 3; 5, a; = 7, y=4; 6, a- = 3, f i/T, 2/ = 2, ^i/T; 7,a; = fi/2T, y = ^i/21; 8, a? = 3, 2/ = — 4, Page 203. 12, a? = 2, 5, 1/ = 5, 2 ; 13, a; = 8, — 2, y = 2, — 8; 14,37 =7, 11, y = 11, 7 ; 15, Jt = 12, — 7, y = 7, — 12 ; 16, a; = 8, 6, y = C, 8 ; 17, a? = 15, — 10, y = 10, — 13 ; 18, x = 24, 10, y = 10, 24 ; 19, x ^ 12, _ 5, 2/ = 5, — 12 ; 20, a; = 2, 3, 2/ = 3. 2 ; 2^, a; = 2, 5, 3^ = 5, 2; 22, ar = |, J, 2/ = *, i; 23, a; = ^, _ ^, 2, = ^, - ^ ; 24, = ± 2, 2/ = ± 1 ; 25, x = 3, y = 2i 26, ar = 5, 2/ = 2 ; 27, a; = 3. 2/ = 4. Page 204. 30, a; = 7, 2/ = 4 ; 31, a; = 10, y = 3 ; 32, a; = 8, 3/ = 5 ; 33, a; = 2, 1/ = 1 ; 34, a; = ± 3, ± 2, 2/ = ± 2, ± 3 ; 35, a; = ± 0, ± 3, 3/ = ± 3, ± 5 ; 36, a; = 8, 2, 2/ = 2, 8 ; 37, a: = 9, 4, 2/ = 4, 9 ; 38, a; = 3,2/ = 2; 39, a; = 9, 2/ = 4; 40, a;= 8, 1/ = 1; 41, x=B,y = 2. Page 205. 43, x = 3, 3G, 2/ = 5, — V ; 44, .r = 2, i/|72/ = i, - 2v/|"; 45, ar = 2, - 2, -^. 2/ = 3, - 3, — -;= ; 46, a; ± 1, ± 14f , 2/ = + 3, ± 3J. 1/31 Page 206. 2, 23, 17 ; 3, 13, 6: 4, 9, 7 ; 5, 3, 2 ; 7, 23; 8, 5, 6, 7, 8, or 1, 2, 3, 4. PagJ 207. 10, A 6 mi., B 4 mi. ! Page 208. n, 30 rd., 20 rd. ; 12, 14; 13, 18 ; 15, 9, 0,4. Page 209. 17, $25, $10; 18, 50 ct., 25 ct ; 19, 20 da., 30 da.; 20, 18 ft., 16 ft. S6S ALGkftftA. Page 210. 21, A 40 hr., B 60 hr. ; 22, 36 men, 77 lb., or 28 men, 45 lb. ; 23, 42 ft ; 24, 48 ft. Pilge 211. 25, 5 mi., 2 mi. per hour; 26, 9 ct. ; 27 $2, $3, and 60, 40 sheep ; 28, street 73 ft., ladder 53 ft; 29, JI/^ i(5 + v/5). TJ oifl o 577 „ 5201 . 721 Page 216. 2, __; 3, ; 4, -—. " 408 1020 228 Page 217. 2, ^!1; 3, |Z£2i!; 4, 2.223 + * 4563 297675' ^ Page 224. 3, 5 : 6 ; 4, ^V; 5, ^\; 6, 24; 7, 275; 8, I, 9; 9, 2, — 7; 10, 4^; 11, 12, 16; 12, 50, 60; 13, A 27, B 63 ; 14, |f ; 15, 3.14 + ; 16, 8, 12. Page 225. 17, short road from A to.B 26 mi., from B to C 52 mi. Page 229. 8, 12; 9, 35; 10, 4|; 11, 13; 12, ^, 3; j3^ £(a4^c^)2 1^^ ^ J. 15^ 18^ 27; 16, 18, 14; 17, V" — '^J 15yd., 25yd; 18, 6, 9; 19, o^g, 4f^ ; 20, 28, 21. Page 230. 23, $25f ; 24, 2, 4 ; 25, 171J lb.; 26, 3, 6. Page 233. 7, 2/ = fr; 8, 20. Page 234. 9, «■ ^ S(x + y); 10, y ^ II, z = Sxy, 13, u cc a;i/»; 14, 16. Page 235. 17, $26^; 18, $18; 19, — ft. ; 20,. 3^ da.; 21, 20. Page 236. 23, 495 lb.; 24, 30 ; 26, 55 hr. x" Y Page 238. 1, 149 , 2, 64; 3, 5 ; 4, — 11 ; 5, 42; 6, 71 J 7,53. ANSWERS. 283 Page 239. 8, 670 ; ft, 884 ; 10, 600500 ; 11, in(n + 1)> 12, 10201: 13, n^; 14, 1092. Page 240. 15,800; 16,4; 17, 61 J; 18,425; 19, «(8 + «); 20, !lIi^=L^; 21, 2235; 22,^ (13 -7m); 23, ?iZ^; 25, 9; 26, 6, 10, 14,18,22; 27, 8], 12, etc., to 29J; 28, 3.1, 3.2, etc., .... 3.9 ; 29, 4^, 4^, 4j%, etc., to O^j. Page 241. 31, 5 ; 32, 15 ; 33, 5 ; 34, — 13 ; 35, 11 ; 36, |. Page 242. 39, 7 ; 40, i = 7, (7 = — ^ ; 41, a = 2, » = 11 ; 42, 5.4, 5.8, 0.2, 6.6 ; 43, 1, 3, 5, etc. ; 44, 0, 1, 3, 6, 10, etc. n Page 243. 45, a =|, d = ^, s = ^(3n + 1) ; 46, 20; 47, 4 or 7; 48, — n( — 1)"; 49, |[1 — (2» + 1) (— 1)»] ; 50, 3, 5, 7, 9 ; 51, 5, 7, 9, 11 ; 52, 8, 10, 12 ; 63, 2, 3, 4 ; 54, 5, 6, 7, 8 , 55, 2, 3, etc. to 8 ; 56, 5 hr. Page 244. 57, 3, 8, 13, etc. ; 58, 3 or 10 da. and 36 or 120 mi. ; 59, 948|J and 14475 ft. ; 60, 19 mi. 160 rd. 640 ft. Page 245. 2, 327680; 3, If; 4, 320. Page 246, 5, f; 6,f2f; 7, 2047; 8, 1310711 ; », 9841; 10, 3249|; H, V" [(5)" — IJ 5 12, — |(2i»- 1) ; 18,9[1-(|)"]; 14, J7^ -I- 284 ALGEBRA. Page 247. 16, 13^; 17, IJ ; 18, f?; 19, 2; 20, f J; 21,1; 22,1; 23,^-; 24, -|^- Page 248. 26, 6; 27, |; 28, l/— T; 29,20.80; 30, 18, 34, 162; 31, 6, 18, 54, etc.; 32, (— 1)^, (- 1)*, (- 1)^; 33, (- 1)*, (- l)i. (_ 1)^, etc. Page 249. 34, 3; 35, »• = 5, « = 15624 ; 36, 2, 37, 5 ; 38, « = 9 ; 39, f , ^% ; 40, 32 ; 41, 1, 2,4; 42, 7, 14, 28, 56. Page 250. 43,1,8,9,27; 44,1,3,9; 45,2,4,8, 16; 46, 6, 8, 10, 12; 47, 400 mi. ; 48, $21, $42, $84; 49, 9, 27, 81, etc. ; 50, £3 4s ; 51, 2, 4, S, 12. Page 252. 7, 4, 6; 8, ^S-. Page 253. 9,7, -21, -V, -h - U, -U; 10. h h A. A; 11, A, A; 12, 2, 8.