# Full text of "Elementary Algebra / with appendix by Alfred Baker"

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jCE: 90 C^HT%. ./'.'. . . ♦ DOMATED BY m. ^. mqe & m'» ^§\mmmai ^tvu^. Elementary Algebra, J. HAMBLIN SMITH, M.A., OF GONYILLE AND CAIUS COLLEGE, AND LATE LECTUBEB AT ST. PETEE's college. CAMBEIDQB. WITH APPENDIX BY ALFRED BAKER, B.A.. MATH. TUTOR CNIV. COL. TORONTO. Sth CANADIAN COFYKIGHT EDITION. NEW BEVISED EDITION. Authorized by the Education Department, Ontario. Authorized by tlie Council of Public Ittstruction. Quebec Recotnmended by the Senate of the Univ. of Halifax. PRICE, 90 CJLIMTS. TOKONTO: ^. J. GAGE & CO. Entered according to thf Act o/ the Parliament of the Dominion of Canada, in the year one thousand eight hundred and seventy-seven, by Ada* MiLXER & Co. , in the Office of the Minister of Agriculture. PREFACE The design of this Treatise is to explain all that is commonly included in a First Part of Algebra. In the arrangement of the Chapters I have followed the advice of experienced Teachers. I have carefully abstained from making extracts from books in common use. The only work to which I am indebted for any material assistance is the Algebra of the late Dean Peacock, which I took as the model for the commencement of my Treatise. The Examples, progressive and easy, have been selected from University and College Examination Papers and from old English, French, and German works. Much care has been taken to secure accuracy in the Answers, but in a collection of more than 2.300 Examples it is to be feared that some errors have yet to be detected. I shall be grateful for having my attention called to them. I have published a book of Miscellaneous Exercises adapted to this work and arranged in a progressive order so as to supply constant practice for the student. I have to express my thanks for the encouragement and advice received by me from many correspondents ; and a special acknowledgment is due from me to Mr. E. J. Gross of Gonville and Caius College, to whom I am ndebted for assistance in many parts of this work. Tlie Treatise on Algebra by .^Ir. E. J. Gross is a continuation of this work, and is in some important points supplementary to it. J. HAMBLIXSJ^IITH. Cambridge, 1871. Digitized by tlie Internet Arcliive in 2009 witli funding from Ontario Council of University Libraries http://www.archive.org/details/elementaryalwestOOsmit CONTENTS. CHAP. PAGE I. Addition and Subtraction i II. jMultiplication 17 III. Involution 29 IV. Division 33 V. On the Resolution of Expressions into Factors . 43 VI. On Simple Equations 57 VII. Problems leading to Simple Equations . . 6i VIII. On the Method of finding the Highest Common Factor 67 IX. Fractions . 76 X. The Lowest Common Multiple .... 88 XI. On Addition and Subtraction of Fractions . 94 XII. On Fractional Equations 105 XIII. Problems in Fractional Equations . . .114 XIV. On Miscellaneous Fractions 126 XV. Simultaneous Equations of the First Degree . 142 XVI. Problems resulting in Simultaneous Equations 154 XVII. On Square Root • . . .163 XVIII. On Cube Root 169 XIX. QUADR.A.HC Equations 174 XX. On Simultaneous Equations INVOLVING Q'"*nKATics 186 XXI. On Problems resulting in Quadratic Equations . 192 XXII. Indeterminate Equations 196 XXIIi. The Theory of Indices 201 XXIV. On Surds 213 XXV. Oy Equations involving Surds . . . .229 CONTENTS. CHAP. PAGE XXVI. On the Roots of Equations .... 234 \XVII. On Ratio 243 XXVIII. On Proportion 248 XXIX. On Variation 258 XXX. On Arithmetical Progression .... 264 XXXI. Om Geometrical Progression ' . . . . 273 XXXII. On Harmonical Progression .... 282 XXXIII. Permutations 28" XXXIV. Combinations 291 XXXV. The Binomial Theorem. Positive Integrai Index 296 XXXVI. The Binomial Theorem. Fractional and Negative Indices 307 XXXVII. Scales of Notation 316 XXXVIII. On Logarithms 328 Appendix 344 \nswers . • . . . 345 1 ELEMENTARY ALGEBRA. I. ADDITION AND SUBTRACTION. 1. Algebra is the science whicli teaches the use of sym- bols to denote numbers and the operations to which numbers may be subjected. 2. The symbols employed in Algebra to denote numbers are, in addition to those of Arithmetic, the letters of some alphabet. Thus a, b, c x, y, z : a, )3, y : a', b', c' read a dash, b dash, c dash : a-^, b^, c^ read a o«e, b one, c one are used as symbols to denote numbers. 3. The number o?ie, or unity, is taken as the foundation of all numbers, and all other numbers are derived from it by the process of addition. Thus two is defined to be the number that results from adding one to one ; three is defined to be the number that results from adding one to two ; four is defined to be the number that results from adding one to three ; and so on. 4. The symbol +, read plus, is used to denote the opera- tion of Addition. Thus 1 + 1 symbolizes that which is denoted by 2, 2 + 1 3, and a + b stands for the result obtained by adding b to a. 5. The symbol = stands for the words " is e(iual to," or •' the result is." [S-A.] . ^ ADDITION AND SUBTRACTION. Thus the definitions given in Art. 3 may be presented in an algebraical form thus : 1 + 1=2, 2 + 1 = 3, 3 + 1=4. 6. Since 2 = 1 + 1, M'here unity is written twice^ 3 = 2 + 1 = 1 + 1 + 1, where unity is written thrte times, 4 = 3 + 1 = 1 + 1 + 1 + 1 pur times, it follows that a = l + l + l +1 + 1 with iinity written a times, 6 = 1 + 1 + 1 +1 + 1 with unity written h times. 7. The process of addition in Arithmetic can be presented in a shorter form by the use of the sign + . Tlius if we have to add 14, 17, and 23 together we can represent tlie process thus : 14 + 17 + 23 = 54. 8. When several numbers are added together, it is indiffe- rent in what order the numbers are taken. Thus if 14, 17, and 23 be added together, tlieir sum will be the same in w-hatever order they be set down in the common arithmetical process : 14 14 17 17 23 23 17 23 14 23 14 17 23 17 23 14 17 14 54 54 54 54 54 54 So also in Algebra, when any number of symbols are added together, the result will be tlie same in whatever order the symbols succeed each other. Thus if we have to add together the numbers symbolized by a and b, the result is represented by a + 6, and this result is the same number as that which is represented by b + a. Similarly the result obtained by adding together a, b, c might be expressed algebraically by a + b + c, or o + c + 6, or b + a + c, or b + c + a, or c + o + 6, or c + b + a. 9. When a number denoted by a is added to itself tlie result is represented algebraically by a + a, This result is for ADDITIO.y AXD SCBTRACTTO.V. the sake of brevity represented by 2a, the figure prefixed to the symbol expressing the number of times the number denoted by a is repeated. Similarly a + a + a is represented by 3a. Hence it follows that 2a + a will be represented by 3a, 3a + a by 4a. 10. The symbol — , read minus, is used to denote the ope- ration of Subtraction. Thus the operation of subtracting 15 from 26 and its con- nection with the result may be briefly expressed thus ; 26-15 = 11. 11. The result of subtracting the number h from the num- ber a is represented by a-h. Again a — h — c stands for the number obtained by taking c from a — b. Also a — b — c — d stands for the number obtained by taking d from a — b — c. Since we cannot take away a greater number from a smaller, the expression a — b, where a and b represent numbers, can denote a possible result only when a is not less than b. So also the expression a-b — c can d'enote a possible result only when the number obtained by taking b from a is not less than c. 12. A combination of symliols is termed an algebraical expression. The parts of an expression which are connected by the symbols of operation + and — are called Terms. Compound expressions are those which have more than one term. Thus a-b + c — d is a compound expression ■\nade up of four terms. When a compound expression contains hvo terms it is called a Binomial, three Trinomial, four or more Multinomial. ADDITION AND SUBTRACTION. Terms which are ju'eceded by the symbol + are called posi- tive terms. Terms which are preceded by the symb.>l — are called negative terms. When no symbol precede.s a t<'rui the symbol + is understood. Thus in the expression n -h + c-d + e -f a, c, e are called positive term.s, b, d,f negative The symbols of operation + and — are usually called posi tive and negative Signs. 13. If the number 6 be added to the number 13, and if be taken from the result, the final result will plainly be 13. So also if a number b be added to a number a, and if b he taken from the result, the final result will be a : that is, a + b-b = a. Since the operations of addition and subtraction when per- formed by the same number neutralize each other, we conclude that we may obliterate the same symbol when it presents itself as a positive term and also as a negative term in the s;ime ex- pression. Thus a-a = 0, and a-a + b = b. 14. If we have to add the numbers 54, 17, and 2? we may first add 17 and 23, and* add their sum 40 to the number 54, thus obtaining the final result 94. This process may be repre- sented algebraically by enclosing 17 and 23 in a Bracket ( ), thus : 54 + (l7 + 23) = 54 + 40 = 94. 15. If we have to subtract from 54 the .sum of 17 and 23. the process may be represented algebraically thus : 54 - (17 -H 23) = 54 - 40 = 14. 16. If we have to add to 54 the difference betweon -23 ar'A 17, the process may be represented algebraically thu!<: 54 4- (23 -17) =54-}- 6 = 60. 17. If we have to subtract from 54 the difference between 23 and 17, the process may be represented algebraically ihus : 54-(23-17) = 54-6 = 48. ADDITION AND SUBTRACTION. 18. The use of brackt . is so frequent in Algebra, that the rales for their removal and introduction must be carefully considered. We shall first tree^t of the removal of brackets in cases where symbols supply the places of numbers corresponding to the arithmetical examples considered iVts. 14, 15, 16, 17. Cd«e I. To add to a the sum of b and c. 3 is expressed thus : a + {b + c). a. irst add b to a, tlie result will be a + b. This result is too small, for we have to add to a a numV-ir /•eater than b, and greater by c. Hence our final result wili oe obtained by adding c to a + 6, and it will be a + b + c. Case II. To take from a the sum of b and»c. This is expressed thus : a — {b + c). First take b from a, the result will be a — b. 'i his result is too large, for we have to take from a a number greater than b, and greater by c. Hence our final result will be obtained by taking c from a — b, and it will be a — b — c. Case III. To add to a the difference between b and c. This is expressed thus : a + {b — c). First add b to a, the result will be a + b. This result is too large, for we have to add to a a number less than b, and less by c. Hence our final result will be ob- tJiined by taking c from a + b, and it will be a + b — c. Case IV. To take from a the difference between b and c. This is expressed thus : a — {b~ c). First take b from a, the result will be a — b. This result is too small, for we have to take from a a num- ber less than b, and less by c. Hence our final result will be obtained by adding c to a — b, and it will be a~b + c. ADDITION AND SUBTRACTION. Note. We assume that a, b, c represent such numbers that in Case II. a is not less than the sum of b and c, in Case III. b is not less than c, and in Case IV. b is not less than c, and a is not less than b. 19. Colk-cting the results obtained in Art. 18, we have a + {b + c) = a + b + c, a — (b + c) — a — b — c, a+ {b — c) = a + b-c, a — {b — c) = a — b + c. From which we obtain the following rules for the removal of a bracket. Rule I. "Wlien a bracket is preceded by the sign +, remove the bracket and leave the signs of the terms in it imchmiged. Rule II. When a bracket is preceded by the sign — , remove the bracket and change the sign of each term in it. These rules apply to cases in which any number of terms are included in the bracket. Thus a + b + {c-d + e -/) = a + b + c-d + e -f, and a + b- {c-d\-e—f) = a + b-c + d — e+f. 20. The rules given in the preceding Article for the rt- moval of brackets turuish corresponding rules for the intro- uuction of l)rackets. Thus if we enclose two or more terms of an expression in a bracket, T. The sign of each term remains the same if + pre- cedes the bracket : II. The sign of each term is changed if — precedes the bracket. Ex. a-b + c-d + e-f=a-b + {c — d) + {e-f\ a-h-tc-d + e-/=a-{b-c)-{d -e+f). ADDITION AND SUB TRA C TTON. 21. We may now proceed to give rules for the Addition and Subtraction of algebraical expressions. Suppose we have to aM. to the expression a + b — c the ex- pression d — e +f. The Sum =a + h-c + {d-e+f) = a + b-c + d-e+f (by Art. 19, Rule I.). Also, if we have to subtract from the expression a + b — c the expression d~e +f. The Difference =a + b-c-{d-e +/) = a + b-c-d + e-f{hj Art. 19, Rule II.). We might arrange tlie expressions in each case under each other as in Arithmetic : thus To a + b — c From a + b — c Add d-e+f Take d-e+f Sum a + b — c + d — e +f Difference a + b — c — d + e —f and then the rules may be thus stated. I. In Addition attach the lower line to the uyper with the signs of both lines unchanged. II. In Subtraction attach the lower line to the upper with the signs of the lower line changed, the signs of the upper line bein" unchanged. The following are examples. (1) Toa + 6 + 9 Add a-b-Q Sum a + b + Q + a-b-Q and this sum =a + a + 6-6 + 9 — 6 = 2& + 3. For it has been shown, Art. 9, that a + a = 2a, and, Art. 13, that 6-6 = 0. (2) From a + 6 + 9 Takea-6-6 Remainder a + h + 9 — a + h + Q and this remainder = 26 + 15. 8 ADDITION AND SUBTRACTION. 22. We have worked out the examples in Art. 21 at full leni.fth, hut ill practice they may he ahhreviated, by combining the symbols or digits by a mental process, thus Toc + (Z + 10 rromc + (^ + 10 Addc-d-7 Takec-rf-7 Sum 2c +3 ilemainder 2d + 17 23. We have said that instead of a + a we write 2a, a + a + a 3a, and so on. The digit thus prefixed to a symbol is called the coefficient of the term in which it appears. 24. Since 3a = a + a + a, and 5a = a + a + a + a + a, Sa + 5a = a + a + a + a + a + a + a + a = 8a. Terms which have the same symbol, whatever their coefli- cients may be, are called like terms : those which have diffe- rent symbols are called unlike terms. Like terms, when positive, may be combined into one by adding' their coefficients together and subjoining the common symbol : thus 2a; + 5ic = 7x, Zy + by + 8y = 16y. 25. If a term appears without a coefficient, unity is to be taken as its coefficient. Thus .*■ + 5x = 6a;. 26. Negative terms, when like, may be combined into one t(;rm with a negative sign prefixed to it by adding the coeffi- cients and subjoining to the result the common symbol. Thus 2x-3i/-5i/ = 2x-8i/, lor 2x~'3y-by = 2x — (Sy + 5y) = 2x-8y. So again 3x-y-iy-Gij = 3.f -lly. ADDITIOPT AMD SUBTRACTION: 27. If an expression contain two or more like terms, some being positive and others negative, we mual first collect all the positive terms into one positive term, then all the negative terms into one negative term, and finally combine the two remaining terms into one by the following process. Subtract the smaller coefficient from the greater, and set down the remainder with the sign of the greater prefixed and the com- mon symbol attached to it. Ex. 8a;-3a; = 5x, 7x — 4x + 5a; — 3a; = 1 2aj — 7a; = 5x, a-26 + 56-4& = a + 56-66 = a-6. 28. The rules for the combination of any number of like terms into one single term enable us to extend the application of the rules for Addition and Subtraction in A]g>3bra, and we proceed to give some Examples. ADDITION. (1) a -26 + 3c (2) 5a + 76-3c-4(£ 3a -46 -5c 6a-76 + 9c + 4ci 4a-66-2c 11a +6c The terms containing 6 and d in Ex. (2} destroying one another. (3) 7x-5)/+ 43 (4) 6m--13?i + 5p x + 'iy-Wz 8m+ n — ^ Zx~ y-{- bz m— n— p bx~'iy— z m+ 2n + bp IGx — ly- 3z 16m- 1 In SUBTRACTION. (1) 5a -36+ 6c (2; 3a + 76- So 2a + 56- 4c 3a -76+ 4c (3) 3a -86 + 10c 5a -66 + 2c 2a-66 + 2c (5) 3a 3a; + 7y + 122! by- 2z 146-13-5 (4) x — y + z x-y-z 2z (6) 7x-l9y-14z 6x-24j/+ 9a Zx + 2y + 14z x+ by-2Zz ib ADDITION AND SUBTRACTION. 29. We have placed the expressions in the examples given in the preceding Article under each other, as in Arithmetic, for the sake of clearness, but the same o]ierations might be ex- hibited by means of signs and brackets, thus Examples (2) of each rule might have been worked thus, in Addition, 5a + 76 - 3c - 4rf + (6a - 76 + 9c + 4fO = 5a + 76-3c-4(i+ 6a - 76 + 9c + 4i = lla + 6c; and, in Subtraction, 3a + 76-8c-(3a-76 + 4c) = 3a + 76-8c-3a + 76-4c = 146 -12c. Examples.— i. Simplify the following expressions, by combining like sym- bols in each. I. 3a + 46 + 5c + 2a + 36 + 7c. 2. 4a + 56 + 6c -3a -26 -4c. 3. 6a -36 -4c -4a + 56 + 6c. 4. 8a - 56 + 3c - 7a - 26 + 6c - 3a + 96 - 7c + 10a. 5. 5a;-3a + fe + 7 + 26-3x-4a-9. , 6. a — 6 — c + 6 + c-d + f7-a. 7. 5a + 106-3c + 26-3a + 2c-2a + 4c. EXAMPLES.— ii. ADDITION. Add together I. a + xanda-a;. 2. a + 2x and a + 3a;. 3. a - 2x and 2a - x. 4. 3x + 1y and 5x - 2i/. 5. a + 36 + 5c and 3a - 26 - 3c. a-26 + 3c and a + 26-3c. 7. 1 +a;-^ and 3-x + i/. 2x - 3i/ + 4a, hx-^y- 2;:, and 6,x + 9)/ - 82. 2a + 6 - 3x, 3a - 26 + x, a + 6 - 5x, and 4a - 76 + 6x. Examples.— iii. SUBTRACTION. I . From a + 6 take a - 6. 2 3x + i/ 2x — 1/. 3 2a + 3c + 4(i a-2c + 3i. 4. X + 2/ + 3 x-y-z. ADDITION A ND SUB TRA CTION'. 1 1 5 . From m — n + r take m — n — r. 6 a + b + c a — b — c. 7 3a + 46 + 5c 2a + 7b + 6c. 8 3x + 5ij-4z 3x + 2y-5z. 30. AVe have given examples ol' the use of a bracket. The methods of denoting a bracket are various ; thv.s, besides the marks ( ), the marks [ ], or j j, are often employed. Some- times a mark called "The Vinculum" is drawn over the symbols which are to be connected, thus « - 6 + c is used to rejiresent the same expression as that represented by a — (b + c). Often the brackets are made to enclose one another, thus a-[b+\c~{d-e-f)\]. In removing the brackets from an expression of this kind it is best to commence with the innermost, and to iemove the brackets one by one, the outermost last of all. Thu8 a-[b+\c-(d-e-f)\] = a-[b+\c-{d-e+f)\] = a-[b+ \c~d + e-f\] = a-[b + c-d + e-f] = a — b — c + d — e +/. Again 5x-(3x-7)- l4-2.c-(6x-3)f = 5x-3x + 7- |4-2ic-6x + 3j = 5x-3.o + 7-4 + 2x + 6a;-3 = 10x. Examples.— iv. beackets. Simplify the following expressions, combining all like quan- tities in each. 1. a + 6 + (3a-26). 2. a + b-{a~3b). 3. 3a + 56 -6c -(2a + 46 -2c). 4. a + 6 — c — (a — 6 — c). 5. 14x-(5x-9)- j4-3,r-(2x-3}/. 6. 4x- {3x-(2x-x-a){. 7. 15x- j7x + (3x + a^){. 12 ADDITION AND SUBTRACTlOh. 8. a.-[6+ja-(6 + a)[]. 9. 6« + [4a- j86-(2a + 46)-22&}-76]-[76 h{8a -(36 + 4a) + 86}+ 6a]. 10. 6-[6-(a + 6)-j6-(6-^6)j]. 11. 2(^-(6a-6)- }c-(5a + 26)-(a-36)}. 12. 2^- ja-(2a-[3a-(4a-[5a-(6a-x)J)])}. 13. 25a- 196 -[36 -1 4a -(56 -6c)}]. 31. We liave liitherto supposed tlie syml'ols in every ex- pression iised for illustration to represent syich numbers that 1^'A expressions symbolize results whicli wou Id be arithmetic- j,lly possible. Thus a — 6 symbolizes a possible result, so long as a is not less than 6. If, for instance, a stands for 10 and 6 for 6, a — 6 will stand for 4. But if a stands for 6 and 6 for 10, a — 6 denotes no possible result, because we cannot take the number 10 from the number 6. But though there can be no such a thini,- as a negative number, we can conceive the real existence of a negative quantity. To explain this we must consider I. What we mean by Quantity. II. How Quantities are measured. 32. A Quantity is anything which may be regarded as being made up of parts like the whole. Thus a distance is a quantity, because we may regard it as made up of parts each of themselves a distance. Again a sum of money is a quardity, because we mav regard it as made uji of parts like the whole. 33. To measure any quantity we fix upon some known quantity of the same kind for our standard, or unit, and then any quantity of that kind is measured by saying how many times it contains this unit, and this number of times is called the measure of the quantity. ADDITION AND SUBTRACTION. For example, to measure any distaiice a]ohg a road we fix lip. in a known distance, such as a mile, and express all distances by saying how many times they contain this unit. Thus 16 is the measure of a distance containing 16 miles. Again, to measure a man's income we take one pound as our unit, and thus if we said (as we often do say) that a i .an's in- con le is 50(T a year, we should mean 500 times the unit, that is, £5<'0. Unless we knew what the unit was, to say that a man's inc. -me was 500 would convey no definite meaning : all we shoald know would be that, whatever our unit was, a pound, a dollar, or a franc, the man's income would be 500 times that unit, that is, £500, 500 dollars, or 500 francs. IN.B. Since the unit contains itself once, its measure is unity, and hence its name. ;'4. Now we can conceive a quantity to be such that wheji piu to another quantity of the same kind it will entirely or in p:ii't neutralize its eff"ect. Thus, if I walk 4 miles towards a certain object and then rt turn along the same road 2 miles, I may say that the latter distance is such a quantity that it neutralizes part of my first j.iurney, so far as regards my position with respect to the point from which I started. Again, if I gain £500 in trade and then lose £400, I may say that the latter sum is such a quantity that it neutralizes liart of my first gain. If I gain £500 and then lose £700, 1 may say that the latter sum is such a quantity that it neutralizes all my first gain, and not only that, but also a quantity of which the absolute value Is £200 remains in readiness to neutralize some future gain. llegardii:g this £200 by itself we call it a quantity which will have a subtradive effect on subsequent profits. Now, since Algebra is intended to deal with such questions in a general way, and to teach us how to put quantities, alike • iT opposite in their effect, together, a convention is adopted, I ounded on the additive or subtractive effect of the quantities in question, and stated thus : "To the quantities to be added prefix the sign +, and to the quantities to be subtracted prefix the sign — , and then >vrite down all the quantities involved in such a question con- nected with these siKUS," 14 A DDITION A ND SUB TRA C TTON.^ Thus, suppose a man to trade ibr 4 years, and to gain a pounds the tirst year, to k)se 6 pounds the secoii<l year, tn ga!ii c pounds the third year, and to lose ti pounds the iuurth year. The additive quantities are here a and c, which we are to write +a and +c. The suhtractive (juantities are here h ;:nd d, whidi we are to write — h and — d, :. Eesult of trading — ■\-a — h-\-c — d. 35. Let us next take the case in which the gain for tlie first year is a pounds, and the loss lor eacli ot three subsequent years is a pounds. Eesult of trading = +a-a-a-a = - 2a. ■ Thus we arrive at an isolated quantity of a subtractivi nature. Arithmetically we interpret this result as a loss of £2a. Algebraically we call the result a negative quantity. When once we have admitted the possibility of the inde- pendent existence of such quantities as this Ave may extend the application of the rules for Addition and Subtraction, for I. A negative quantity may stand by itself, and we may then add it to or take it from some other quantity or expres- sion. II. A negative quantity may stand first in an expression which we may have to add to or subtract from any other expression. The Rules for Addition and Subtraction given in Art. 21 will be applicable to these expressions, as in the following Examples. ADDITION. (1) 5rt - 7ft = — 'la. (2) 4a-36-6a-f-76=-2rt-i-46. (3) To 4a To 5n-3S Add -3a Add -2a -26 Sum a Sum 3a -56 ADD/T/OX AND SUBTRACTION: 15 (4) 6ffl-56- 4c + 6 (5) Ix-by + Qz -5a + 76-12c-17 -ISx + 'jy-bz - a- 86 + 19c + 4 - Zx-S]j+ z -66+ 3c- 7 -l4x-42/ + 5» SUBTRACTION. (1) Fiom X Take ^^y Remainder x + y or we might represent the operation thus, (2) a + 6-(-o + 6) = a + 6 + a-6 = 2a. (3) —a -b-{a-b)= -a-b-a + h= -2a. (4) -3a+ 46- 7c + 10 5a- 9^+ 8c +19 -8a + 136-15c- 9 (5) x-y-{;3x-\-5x-(-4:y + 7x)i] = x — y~ [3x — \ —5x + 4y — Ix {■ ] =x — y — [3a; + ox - 4?/ + lx\ =x — y — Zx — bx + 4y~7x = -14x + 3i/. (6) 7a + 56+ 9c-12i -36- 12c- 8d+ 6e 7a+ 86 + 21C- 4d-6e In this example we have deviated from our previous prac- tice of placing like terms under each other. This arrange- ment is useful to facilitate the calciiiation. but is not absolutelv necessary ; for the terms which are alike can be combined independently of it. * NoTE. — The meanhig of Subtraction is liere extm !eil so that the result in Art. 18, Case iv. may he true when b is less than C. I6 ADDITION AND SUBTRACTION. Examples. — v. (I.) ADDITION. Add togethe'" 1 . 6a + 76, - 2a 45, and 3a - 5&. - 5a + 66 - 7c, - 2a + 136 + 9c, and 7a - 296 + 4r. 2;e — 3?/ + 42, - 5x + 4?/ — 72, and - 8x — 9y — 32. 4. — rt + 6 — c + (/, a — 26 - 3c + d, — 56 + 4c, and — 5c + d. 5. a + 6 - c + 7, - 2a - 36 - 4c + 9, and 3a + 26 + 5c - '-** 6. 5x - 3a - 46, 6?/ - 2«, 3a — 2i/, and 00 - 7a;. 7. a + 6 — c, c — a + 6, 26 — c + 3a, and 4a — 3c. 7a - 36 - 5c + 9*/, 26 - 3c - 5(7, and - \il + 15c. — 12a; - 5j/ + 42, 3a; + 2?/ - 32, and 9a; - 3i/ + 2. 9 (2.) SUBTRACTION. From a + 6 take —a — b. From a — 6 take - 6 + c. From a — 6 + c take — a + 6 — c. From 6x - 81/ + 3 take - 2x + 9?/ - 2. From 5a - 126 + 17c take - 2a + 46 - 3c. From 2a + 6 - 3x take 46 — 3a + 5a;. From a + 6 - c take 3c - 26 + 4a. From a + 6 + c — 7 take 8 - c — 6 + a. From l2x — 'Mj-z take 4y - 52 + x. From 8a - 56 + 7c take 2c - 46 + 2a. From 9p-4q + Zr take bq-'6p+r II. MULTIPLICATION. 36. The operation of findins .'lie sum of a numbers each equal to h is called Multiplication. The number a is called the Multiplier. h Multiijlicand. This Sum is called tbe Product of the multiplication of h by a. This Product is represented in Algebra by three distinct symbols : I. By writing the sjTubols side by side, with no sign between them, thus, ab ; II. By placing a small dot between the symbols, thus, a.h; III. By placing the sign x between tlie symbols, thus, axh ; and all these are read thus, " a into h" or " a times 6." In Arithmetic we chiefly use the third way of expressing a Product, for we cannot symbolize the product of 5 into 7 by 57, which means the sum of fifty and seven, nor can we well represent it by 5.7, because it might be confounded with the notation used for decimal fractions, as 5 -7. 37. In Arithmetic 2x7 stands for the same as 7 + 7. 3x4 4 + 4 + 4. In Algebra ab stands for ine same as + 6 + 6+ ... with 6 written a times. {a + Vjc stands for the same as c + c + 1 ... with c written a + 6 times. [s.A.] B <8 MULTIPLICATION. 38. To shew that 3 times 4 = 4 tivies 3. 3 times 4= 4 + 4-^4 = 1-rl-l + 1 ) '. I. +1+1+1^1 C 1^1 ) 4 times ii= 3 + 3 + 3 + 3 =1^1+1 \ ^'^'^' II. +1+1+1 I +1+1+1 ) Now the results obtained from I. and II. must be the same, for the horizontal colunin^: of one are identical with the verti- cal columns of the other. 39. To prove that ah = ha. ah means that the sum of a numbers each equal to h is to be taken. .'. db= 5+6+ with h written a times = h + h + to or lines = 1 + 1 + 1 + \oh terms \ + 1 + 1 + 1 + to 6 temis f J + \ to a lines. ) Again, ha= a + a + with a written 6 times = a + a + to h lines — 1 f 1 + 1 + to rt terms •\ + 1 + 1 + 1 + to « terms r y, to h lines - MULTIPLICATION. 19 Now the results obtained from I. and II. must be tlie same, for the horizontal columns of one are clearly the same as the vertical columns of the other. 40. Since the expressions ah and ha are the same in mean- ing, we may regard either a or h as the multiplier in forming «#the product of a and 6, and so we may read ah in two ways : (1) a into 6, (2) a multiplied by h. 41. The expressions ahc, ach, bac, hca, cah, cba are all the same in meaning, denoting that the three numbers symbolized by a, h, and c are to be multiplied together. It is, however, generally desirable that the alphabetical order of .the letters representing a product shoula be observed. 42. Each of the numbers a, h, c is called a Factoji of the product abc. 43. When a number expressed in figures is one of the factors of a product it always stands first in the product. Thus the product of the factors x, y, z and 9 i^ represented by 9xyz. 44. Any one or more of the factors that make up a product is called the Coefficient of the other factors. Thus in the expression 2ax, 2a is called the coefficient of x. 45. When a factor a is repeated twice the product would be represented, in accordance with Art. 36, by aa ; wlien tJiree times, by aaa. In such cases these products are, for the sake of brevity, expressed by writing the symbol with a number placed above it on the right, expressing the number of times the symbol is repeated ; thus instead of aa we write a^ aaa a^ aaaa a* These expressions a-, a^, a* are called the second, third, fourth Powers of a. The number placed over a symbol to express the power of the symbol is called the Index or Exponent. a^ is generally called the square of a. a? the cube of a. 20 MULTIPLICATION. 46. The product of a^ and o? = a^x o? = aax aaa = aaaaa = a^. Thus the index of the resulting power is the sum of the indices of the two factors. Similarly a* xa^ = aaaa x aaaaaa = aaaaaaaaaa = o.^** = a*+*. « If one of the factors be a symbol without an index, we may assume it to have an index \ that is Examples in multiplying powers of the same symbol are (1) axa'^ — a^~^' = a^. (2) 7a3 X 5a^ = 7 X 5 X a3 X a' = 35aW = 35aio. (3) a3 X a« X ^9 = a^+^^ = a^^ (4) x^y X xy^ = x-.y.x.y- = x~.x.y.y^ = x^"*"^. 1/^+2 _ ^SyS^ (5) a% X a¥ x a^V = a^+i+s. ji+s+r = ^8. jjU Examples.— vi. Multiply I. X into 3y. 2. 3x into 4y. 3, 3xy into 4xy. 4, 3a6c into ac. 5. a^ into a*. 6. a" into a. 7. 3a"5 into 4a.^62_ g, y^-^c into Sa^tc^. 9. 15a6''c^ by 12a%. 10. 7aV by 4a-6c^. 11. a^.by 3rt^ 12. 4a36a; by 5a6''?/. 13. 19x^2/3 by 4x2/ V. ^4- 17a6^2 by 3k-?/. 15. 6^y^z^ hy 8x^y-z\ 16. 3a6cby4axi/. 17. a^i'c by 8a''6''c. 18. 9m^7ip by m%-^2. 19. ay-z by 6x-z^. 20. lla%x by 3ft^"6^*m-. 47. The rules for the addition and subtraction of powers are similar to those laid down in Chap. I. for simple quantities. Thus the sum of the second and third powers of x is repre- sented by x^ + x^, and the remainder after taking the fourth power of y from the fifth power of y is represented by and these expressions cannot be abridged. MULTIPLICATION. it But when we have to add or subtract the same powers of the same quantities the terms may be combined into one : thus Sy^ + 5y^ + 7y^ = 15?/', 8x*-bx* = 3x\ 9y^-3y^-2if = 4i/. Again, whenever two or more terms are entirely the same with respect to the symbols they contain, their sum may be - abridged. Thus ad + ad = 2ad, 3a-b — 2a-b = a-b, 5a%^ + 6a%^ - ga^fes = 2^363, 1a?x— \Oa?x— 12a-x= — 15a-x. 48. From the multiplication of simple expressions we pass on to the case in which one of the quantities whose product is to be found is a compound expression. To shew that (a + b) c = ac + be. {a + b) c = c + c + c+ ... with c written a + b times, = {c + c + c+ ... with c written a times) + {c + c + c ... with c ^\Titten b times), = ac + be. 49. To sheiv that (a — b) c = ae — be. (a — b)c = c + c + e+ ... with c written a — b times, = {c + c + c+ ... with c written a times) — (c + e + c... with c written b times), = ac — be. Note. We assume that a is greater than b. 50. Similarly it may be shewn that (a + b + c) d = ad + bd + cd, (a-b — c) d = ad — bd — ed, and hence we obtain the following general rule for finding the product of a single symbol and an expression consisting of two or more terms. " Multiply each of the terms by the single symbol, and con- nect the terms of the result by the signs of the several terms of the eompound expression." i± MUL T I PLICA TtON. Examples. — vii. Multiply 1. a + 6-chya. 7. 8//!.- + 9m?i + lOn^ by mn. 2. a + 36 - 4c by 2a. 8. ^a? + 4a't6 - ZaW + 40^63 by 2a 6. 3. a^ + 3a^ + 4a by a. 9. y?\j^ — a;-?/^ + xy — 7 by a;^/. 4. Ba^^ 5a2 - 6a + 7 by 3a2. i o. m^ - 3 )»% ■\-'imv?-v? by w. 5. a2 - 2a6 + V- by ah. 11. ISa^?, _ 6a262 + oah^ by 12a263 6. o-' — 3a-62 + J3 by 3a-6. 1 2. 13.c^ - 17a;"'?/ + 5xj/2 — y^ by 8x3. 51. We next proceed to the case in which both multiplier and nuiltiplicand are comjwmid expressions. First to nnilti]ily a + b into c + d. Eepresent c + d by x. Then (a + b){c + d) = (a + b)x = ax + bx, by Art. 48, = a(c + d) + b{c + d) = ac + ad + hc + hd, by Art. 48. The same result is obtained by the following process : c + d a + b ac + ad + bc + bd ac + ad + bc + bd which may be thus described : Write a + b considered as the multiplier under c + d con- sidered as the multiplicand, as in common Arithmetic. Then •multiply each term of tlie multiplicand by a, and set down the result. Next multiply each term of the multiplicand by b, and set down the result under the result obtained before. The sum of the two results will be the product required. Note. The second result is shifted one place to the right. The object of this will be seen in Art. 56. Mi 'L TIPL ICA TIOM. 23 52. Next, to multiply a + 6 into c — i. Represent c — d by x. Then (a + 6)(c-d) = (rt + ?))x = ax + hx = a((: - (Z) + ?)(c - rZ) = ac — a*-^ + 6c — M, by Art. 49. From a comparison of this result with the factors from which it is produced it appears that if we regard the terms of the multiplicand c — (Z as independent quantities, and call them + cand —d, tlie effect of multiplying the positive terms +a and +b into the positive term +c is to produce two positive terms + ac and + he, whereas the effect of multiplying the positive terms +a and +b into the negative term —d is to produce tivo negative terms —ad and —hd. The same result is obtained hy the following process : c — d a + b ac — ad + bc-bd ac — ad + bc — bd This process may be described in a similar manner to that in Art. 51, it being assumed that a positive term multiplied into a negative term gives a negative result. Similarly we may shew that a — b into c + d gives ac + ad — be — bd. 53. Next to multiply a — b into c — d. Represent c — d by x. Then (a-b){c-d) = {a-b)x = ax — bx = a{c — d) — b{c — d) = {ac - ad) - {be - hd), by Art. 49, — ac — ad — bc + bd. When we compare this result with the factors from which it is produced, we see that The product of the positive term a into the positive term c is the positive term ac. ±4 MULTIPLICATION. The product of the positive term a into the negative term — d is the negative term — ad. The product of the negative term — h into the positive term c is the negative term — he. The product of the negative term — h into the negative term - rf is the positive term 6(7. The multiplication of c - d by a — h may be written thus : c-d. a~b ac — ad - be + bd ac — ad-bc + bd 54. The results obtained in the preceding Article enable us to state what is called the Rule of Signs in Multiplication, which is "The product of tivo positive terms or of two negative tervm is positive : the product of tivo terms, one of which is positive aiul the other negative, is negative." 55. The following more concise proof may now be given of the Rule of Signs. To shew that (a — b){c — d) = ac — ad — be + bd. First, {a - h)M= M +M^M+ ... with M written a-b times, = {M + M + M -{■ ... with M -written a times) -(M+i¥ + M + ... with M written ft times), = aM-bM. Next, let M= c-d. Then aM= a (c-d) = {c-d)a Art. 39. = ca — da. Art. 49. Similarly, bM=cb-db. .". (a — b)(c — d) = {ca — da) — (cb — db). Now to subtract (cb — db) from (ca — da), if we take away cb we take away db too mucli, and we must therefore add dh u> the result, .". we get ca - da — cb + db, which is the same as ac-ad-bc + bd. Art. 33. MUL TI PLICA T/ ON. 2$ So it appears that in multiplying {a -h) {c- d) we must multiply each term in one factor by each term in the other and prefix the sign according to this law : — When the factors viultiplied have like signs prefix +, when unlike — to theprodkct. This is the Rule of Signs 56. We shall now give some examples in ill'istration of the principles laid down in tlie last five Articles. Examples in Multiplication wwked out. (1) Multiply ic + 5 by a; + 7. (2) Multiply x - 5 by x + 7. x+ 5 x-b x+ 7 35 + 7 x^ + 5x x^ — ox + 7x + 35 +7x-35 a;2 + 12x + 35 x- + 2a;-35 The reason for shifting the second result one place to the right is that it enables us generally to place like terms under each other. (3) Multiply X + 5 by X - 7. (4) Multiply x - 5 by x - 7. x + 5 X- 5 x-7 X- 7 x^ + 5x x2_ 53. -7x-35 - 7x + 35 x2_2x-35 x'''-12x + 35 (5) Multiply x2 + ?/2 by x'-i - y'^. (6) Multiply 3ax - 5by by 7ax - 2by. 7? + if 3ax - ■ hhy x^-if lax- 2hy X* + xh/ 210^x2 - Soabxy -xV-2/* - dabxy + \Ob-y^ X* - y* 2la-x:- - Alahxi^ + lOlj^'^ 26 MUL TI PLICA TION. 57. The process in the multiplication of factors, one or both of which contains more than two terms, is similar to the processes which we have been describing, as may be seen from the following examples : Multiply ' (1) x^ + .Tt/ + 1/2 by a; — 2/. (2) a^ + 6a + 9hj a^~6a + 9. x^ + a;?/ + 2/2 a^ + 6a +9 x-y a^-6a +9 01? + X^T/ + XI/2 a* + 6a^ + 9«2 — x^i/ - xy^ — y^ ' -6a3-36(i2-54a x^-y^ + 9a2 + 54a + 81 a*-18a2 + 81 (3) Multiply 3x2 + ^^y _ ^2 by Zx^-4xy + y\ 3X2+ 43;^ _ y2 3X2- ^y.y ^ y2 9x« + 12x3y- 3xy - 12x^1/ - \Qx-y- + 4x?/^ + 3x-!/2 + 4x1/3 _ yi (4) To find the continued product of x + 3, x + 4, and x + 6. To effect this we must muUi]ily x f 3 by x + 4, and then inltiplv the result by ,(; + 6. x+ 3 x+ 4 x2+ 3x + 4x 4- I fj x2+ 7x -I- 12 x+ 6 x-''+ 7x2 + 12x + 6x2 + 42x + 72 x3 + 13x2 + 54x + 72 Note. Tlie numliers 13 and 54 are called the coefficients of x2 and X in the expression x^-" ISx^-f 54x+72, in accordance with Art. 44. MUL riPL re A T!0\ : 27 (5) Find the continued product of x + a, a + 6, and z + c. x^ + ax + bx + ab x + c 7? + aa;2 + Ix^ + ahx + ex- + acx + bcx + abc a? + {a + b + c)x^ + (ab + ac + bc)x + abc Note. The coefficients of x^ and x in the expression just obtained are a + b + c and ab + ac + be respectively. When a coefficient is expressed in letters, as in this example, it is called a literal coefficient. Examples. — viii. Multiply I. X + 3 by X -.'.). 2. a; + 15 by X — 7. 3. x - 12 by x + 10 4. X — 8byx — 7. 5. a — 3 by a — 5. 6. y — 6hyy + lS. 7. x2-4byx2 + 5. 8. x2-6x + 9 by x2-6x + 5. 9. X- + 5x - 3 by x^ - 5x - 3. 10. a^ - 3a + 2 by a^ - 3a^ + 2. II. x^ — x + 1 by X- + X— 1. 12. 3:^ + xy + y'^ hy X- — xy + y\ 13. x^ + xy + y^hyx — y. 14. a- - x^ by a* + a%^ + x*. i^ 16 17 18 19 20, 21 22 23 24. 25 26, x^ — 3x- + 3x - 1 by x- + 3x + 1. x^ + 3x^y + 9jy'- + 21y^ by x — 3?/. a^ + 2a26 + 4«l- + 86^ by a - 2b. SftS + 4a^b + ■lab'^ + ¥ by 2a - b. cr - 2a26 + Za¥ + A¥ by a- - 2ab - Zb\ a^ + Za"b - 2a6'2 + 36^ by a-^ + 2a6 - 3&2. a- — 2ax + 4x2 ]jy ^2 _l 2rta; + Ax\ 9rt- + 3ax + X- 1 ly 9a"- - 3ax + x^. X* — 2ax^ + 4a- !)}■ r^ -f 2ax2 + 4a-. a- -f 6^ -I- c^ - 06 - rf c - lie 1 ly a -i- 6 + c. x^ + 4xt/ -I- 5 J/- by x" - Zj?d - 2xy^ + 3y\ ab + cd + ac + bd bv ab • cd - ac — bd. Find the continued product ot the following expression 27. X - a, X -i- a, x^ -t- a"*, X* + a* 28. x-a, x-i-&, » — c, 28 ML Y, TIPL ICA TION. 29. 1 - a;, 1 + a;, 1 + a;2, 1 + x*. 30. % — y,x + y, x^ — xy + y^, x^ + ocy + y^. 31. a — a;, a + x, a^ + x^, a^ + sc*, a^ + x*. Find the coefficient of x in the following expansions : 32. (x-5)(x-6)(x+7). 33. (x + 8)(x + 3)(x-2). 34. (x - 2) (x - 3) (:c 4 4). 35. (x-a") (x-6)(x-c). 36. (x2 + 3x-2)(x2-3x + 2)(x^-5). 37. (X2-X + 1)(X2 + X-1)(X*-X2+1). 38. (x- - mx + 1) (x^ — mx — 1) (x* — m'x — 1). 58. Our proof of the Rule of Signs in Art? 55 is founded oil the supposition that a is greater than b and c is greater tlian d. To include cases in which the multiplier is an isolated nega- tive quantity we must extend our definition of Multiplication. For the definition given in Art. 36 does not cover this case, since we cannot say that c shall be taken — d times. We give then the following definition. " The operation of ]\[ultiplication is such that the product of the factors a — b and cv-f? tfill be equivalent to ac — ad — bc + bd, whatever may be the values of a, b, c, rf." . Now since (a — b)(c — d) = ac — ad — bc + bd, make a = and d = Q. Then (0-6) (c-0) = x c-0 x 0-6c + 6 xO. or —bxc= —be. Similarly it may be shewn that — bx —d= +bd. Examples. — ix. Multiply I . a- by — b. 2. a"^ by — a^. 3. a% by - ab-. 4. ~4a-6by — Safe^. 5. 5x^by— 6x?/2. 6. a'^ — ab + H-hy —a. 7. 3a3 + 4^2 — 5a by — 2a2. 8. —a^ — a- — ahy—a—\. 9. 3x"^/ — bxx/ + Ay'^ by — 2x — Zy. — iSiri^ — 6mn + In"^ by — m + n. 13r--17r-45 by -r-3. Tx^ - 8x%! — 92- by - x - s. — X* + x^y — x?y- by —y — x. — y^ — xy- — x-y — j^ by —x — y. III. THVOI.TJTION. 59. To this part of Algebra belongs the process called Involution. This is the operation of multiplying a quan- tity by itself any number of times. The power to which the quantity is raised is expressed by the number of times the quantity has been employed as a factor in the operation. Tlius, as has been already stated in Art. 45, a^ is called the second power of a, a? is called the third power of a. 60. When we have to raise negative quantities to certain powers we symbolize the operation liy putting the quantity in a bracket with the number denoting the incfex (Art. 45) jdaced over the bracket on the ri-ht hand. Thus ( — of denotes tlie third power of — a, ( — 2.c)* denotes the fourth jjower of — 2x. 61. The signs of all even powers of a negative quantity will be ])ositive, and the signs of the odd powers will be negative. Thus (-a)2 = (_a)x(-a) = a2, (^-af = {-a).{-a) {-a) = a-.{- a)= -a^. 62. To raise a simple quantity to any power we multiplv the index of the quantity by the number denoting the power to which it is to be raised, and prefix the proper sign. Thus the square of a^ is a^, the cube of a^ is a", the cube of - x^yz^ is - x^yh^. 36 INVOLUTIOX. 63. AVe form the second, third and fourth powers of a + 6 in the following manner : a + 6 a + 6 a^ + ab + «?) +6^ (a + 5)^ = a? + 3n-6 + 3a6'' + W a +6 a4 + 3a36 + 3a26'^ + rt63 (a + by = ft* + 4a36 + 6a%-^ + 4a¥+"b\ Here observe tlie following laws : I. The indices of (i decrease \>y unity in each term. TI. The indices of b increase by unit}' in each term. III. The numerical coefficient of the second term is always the same as the index of the power to which tiie binomial is raised. 64. We form the second, third and fourth powers of a - 6 in the following manner ; a-b a-h a^ -ab -ab +62 (a-6)2 = ^2T2a6 + 62 <7 - *: a^ - 2a~b + ab'' - a^b + -2ah--b^ (a-by = a^-:)a-l>^ Sali^-P a -b a* - 'Sa% + Ca-b- - ab^ - a^b + :Ui-b- - Sab^ + 6* (a - bf = a* - 4a-'6 + 6(t-6- - -iab^ + b*. INVOLUTION. %\ Now observe that the powers of a - 6 do not differ from the powers of a + 6 except that the terms, in which the odd, powers of 6, as 6', }?, occur have the sign - prefixed. Hence if any power of a + h be given we can write the corresponding power of a, - 6 : thus since (« + hf = a* + Sa'i?) + 1 (daW + 1 Oa-i^ + 5a6* + 6^, (a - If = a^ - 5a*6 + X^aW - lOa'^i^ + 5^54 _ y,^ 65. Since (a + 6)2 = a2 + 62 + 2a6 and (a - 6)2 = a2 + 52 _ 2a&, it appears that the square of a binomial is formed by the following process : " To the sum of the squares of each term add twice the product of the terms." Thus (a; + yf = x- + if- + 2xi/, (x-5)2 = :c2 + 25-10x, (2a; - 7i/)2 = U- + 49i/- - 28xi/. 66. To form the square of a trinomial : a + 6 + c a + 6 + c a? + 2a& + ¥ + 2ac + 26c + c-. Arranging this result thus a' + b'^ + c' + 2ab + 2ac + 2bc, we set that it is composed of two sets of quantities : I. The squares of the quantities a, b, c. II. The double products of a, b, c taken two and two. Now, if we form the square oi a-b-c, we get a-b-c a-b-c a^-ab- ac -ab + ¥ + bc -ac + bc + c^ a'^-2ab + ¥- 2ac + 2bc + c\ The law of formation is the same as before, for we have 32 INVOLVTION. I. The sqTiares of the quantities. II. The doul)le products of a, - 6, - c taken two by two : the sign of each result being + or - , according as the signs of the algebraical quantities composing it are like or unlike. 67. The same law holds good for expressions containing more than three terms, thus (a + 6 + c + (0^ = a2 + 52 + c2 + (i2 + lah + 2ac + 2arf + 26c + 26(Z + 2cd, (a-& + c-(^)2 = a2 + 62 + c2 + ,;2 - '2ah + 2ac - 2a(i - 26c + 2M - led. And generally, the square of an expression containing 2, 3, 4 or more terms will be formed l^y the following proci-.-s : " To the sum of the squares of each term add twice the product of each term into each of the terms that follow it." Examples. — x. Form the square of each of the following expressions : I. x-va. 2. v-a. 3. a^ + 2. 4. a; -3. 5. x^ + i/'. 6. x^-y\ 7. n-' + R 8. a^-W. 9. X + 1/ + 2. 10. x-y + z. II. m + w-p-?-. 12. a'" + 2x-3. 13. 3?-Qx + l. 1 4. 2x2 _ 7 J. + ()_ 1 5 _ yi + if- zK 16. X-* - -ix^y- + y*. 17. a^ + P + c^. 1 8. x-'^-y^-z^. 19. x + 2y-3z. 20. X- - '2y'^ + 5z^. Expand the following expressions : 21. (x + cf)^. 22. {x-af. 23. (x + 1)^. 24. (x-1)^ 25. (x + 2)l 26. (rt2-62)^. 27. {a + b + c)\ 28. (a-6-c)3. 29. (»i + ?i)-.(7n - ?i)-. 30. {in+ny-.{m'- — n^). 68. An algebraical product is said to be of 2, 3 dimen- sions, when tiie sum of the indices of the quantities composing the product is 2, 3 Thus ab is an expression of 2 dimensions, aWc is an exjiression of 5 dimensions. DIVISTOISr. 33 69. An algebraical expression is called homogeneous when each of its terms is of the same dimensions. Thus x'^ + xy + y- is homogeneous, for each term is of 2 dimen- sions. Also 3x^ + 4x-i/ + 5i/^ is homo<:eneous, for each term is of 3 dimensions, the numerical coefficients not affecting the dimen- sions of each term. 70. An expression is said to be arranged according to powers of some letter, when the indices of that letter occur in the order of their magnitudes, either increasing or decreasing. Thus the expression a^ + a^x + ax- + y? is arranged according to descending powers of a, and ascending powers of x. 71. One expression is said to l)P of a higher order than another wlien tlie former contains a higher power of some dis- tinguishing letter than the other. Thus a^ + a-x + rtc- + x^ is said to be of a higher order than a^ + ax + x^, with reference to the index of a. rr. DIVISION. 72. Division is the ]i!oclss liy which, when a product is given and we know one ot the factors, ihe other factor is deter- mined. The product is, vith reference to this process, called the Dividend. The given factor is called the Divisor. The factor which has to be found is called the Quotient. 73. The operation of Division is denoted by the sign -=-. Thus ab-^a signifies that ab is to be divided by a. The same operation is denoted by writing the dividend owr the divisor with a line drawn between them, thus — . a In this chapter we shall treat only of cases in which the dividend contains the divisor an exact number of times. [S.A.] Q 34 DIVISIOJV. Case I. 74. When the dividend and divisor are each included in a single term, we can usually tell by inspection the factors of which each is composed. The quotient will in this case be represented by the factors which remain in the dividend, wlien those factors which are common to the dividend and the di- visor have been removed from the dividend. Thus X"^*' Sa^ Zaa . — = = 6a, a a a^ aaaaa „ a-' aaa Thus, when one power of a number is divided by a smaller power of the same number, the quotient is that power of the number whose index is the difference between the indices of the dividend and the divisor. Thus —=a}'^~-' = a\ 15a362 _ „, '3ao 75. The quotient is ^initT/ when the dividend and the divisor are equal. Thus ^ = 1; "■^'^^l; and this will liold true wuen the dividend and the di\-isor are compound quantities. Thus ■ — r=l; -^r— S=l. Examples.— xi. Divide 1. .x" by x'. 2. x^^ by x-. 3. xhj"^ by xy. 4. x?'y^^ "hy xyh. 5. 24«6-c by 4a?). 6. 72o-6-'c^ by 9a-6-c. 7. 256«3ir(;9by USahc^. 8. 1331»i'"»'V''- b' llm-n^p\ ^. QOa^x-if by bxy. ip. 9Ga-'6-'c-3 by 126c, DIVlStOM. 35 Case II. 76. If the divisor be a single term, while the diviJeml contains two or more terms, the quotient will be found by dividing each term of the dividend separately by the divisor and connecting the results with their proper signs. m, ax + ix - Thus = a + 6, aV + a^x^ + ftx „ „ ax 12xy+16a;y-8xy2 4x1/' 2 — 3x^1/2 4- 4a;y _ 2. Examples.— xii. Divide 1 . x^ + 2x2 + a; by X. 4. m/)x* + m^p-x^ 4- m^-p^ by m'p. 2. if -y* + y^-y^ by i/^. 5.1 6a^xy - 28a^x^ + 4a''x^ by 4a'^x. 3. 8a3 + 16a26 + 24a62by8a. 6. 72xY-36xy- 18xY by 9x2^/. 7. 81m*ft'' - 54m^?i^ + 277/!,^/i2p by 3m2?i2. 8. 12xY-8xy-4xy by 4x3. 9. 169a*6 - 1 1 7a^62 + 91^25 |-,y 13^2^ • 10. 36l65c3 + 2286M-13363c5by 1962c. 77. Admitting the possibility of the independent existence of a term affected with tlie sil^mi - , we can extend the Exam- ples in Arts. 74 — 76, by taking the first term of the dividend or the divisor, or both, negative. In such cases Ave apply the Rule of Signs in Multiplication to form a Eule of Signs in Division. Thus since —axb=-ah,\ve conclude that ^[— = -a, 7 7 —ab ax -b=-ab, -.^ ^a, —ax-b = ab, — =-a; and hence the rules I. When the dividend and the divisor have the shme sign the quotient is positive. II. When the dividend and the divisor have different signs the quotient is negative. 36 D/VIS/OI\r. 78. The followinj; Examples illustrate the conclusions just obtained : (I) '^^-^.. (3) ^''=9.,. (5) _ . = -lP + ah--a-h + a^. (6) _4^^2 - - •' =3a;V-4xy + 2. Examples.— xiii. Divide 1. 72rt6by-9a6. 6. - a V — a2x2 _ ^a; by — ax. 2. - 60«8 by - 4a3. 7. - 34ft3 + 51 ^2 _ i7„_^.2 i-,y j -„ 3 -84a;Vby4ry. 8. -d,a?h^'-2A(eh'^ + 2,2a~¥hy -AaW. 4. - ISm^^i^ by Zvm. 9. - 144i';3+ 108.7-21/ -96.r?/2 ^^^ ^^x. 5. - 128ft36-'c by - 86c. 10. ¥xh- - Wx'z^ - hhfz^ by - ¥z\ Case III, 79. The third case of the operation of Division is that in which the divisor and the dividend contain more terms than one. The operation is conducted in the Ibllowing way : Arrange the divisor and dividend according to the powers of some one symbol, and ])l>ice them in the same line as in the process of Long Division in Arithmetic. Divide the first term of the dividend by the first term of the divisor. Set down the result as the first term of the quotient. Multiply all the terms of the divisor by the first term • of the quotient. Subtract the resulting product from the dividend. If there be a remainder, consider it as a new dividend, and proceed as before. DIVISION. 37 The process will best be understood by a careful study of the following Examples : (1) Divide a^ + 2«6 + 6"- by a + 6. (2) Divide a? - 2ab + b- by a-b. a + b)a^ + 2ab + b''{a + h a-b) a^ - 2ab + 6^ (a - 6 a" + ab a^ — ab ab + b'- -ab + ¥ ab + b"- -ab + b'^ (3) Divide a;^ - y^ by x^ — y^. x^-y^Jx'^-y'^^x^-hx^y^ + y* ■ , a;6 _ a;4y2 a V - T}y'^ xY - y^ x-y* - y^ (4) Divide x^ - Aa-x^ + Aa^j-^ - o" by x"^ - a^. CC2- - 0?) x^ - 4a-x* + 4a*x2 - a^ (a;* - 3a^x^ + a* x" - a V -'3ah^ + 4a*x^~a^ -3a2x* + 3a%2 aV - a° (5) Divide 3xt/ + x3 + 1/^-1 by i/ + x-l. Arranging the divisor and dividend by descending powers of x, x + y-l)x^ + 3xy + >/' - i. i^x- -xy + x + y^ + y+1 a^ + x-y - X- -x-y + x^ + Smj + y^ -1 -x^y-xy'^ + xy x^ + xy"^ + 2xy + y^-l x^ + xy-x xy^ + xy + x + y^-l • xy^ + y^-y^ xy + x + y^-1 Xy + y2-y » x + y- 1 x + y-1 38 DIVISION. 80. We must now direct the attention of the student to two points of great importance in Division. I. The dividend and divisor must be arranged accord- ing to the order of the powers of one of the symbols involved in them. This order may be ascending or descending. In the Examples given above we have taken the descending order, and in the Examples worked out in the next Article we shall take an ascending order of arrangement. II. In each remainder the terms must be arranged in. the same order, ascending or descending, as that in which the dividend is arranged at first. \ ° 81. To divide (1) 1 -x* by a;3 + x2 + a; + l, arrange the dividend and divisor by ascending powers of x, thus : 1+x + x^ + x^ -x-x^-x^-x^ -x-x^-a^-x^ (2) 48x2 + 6 - 35x5 + 58x* - 70x3 _ 333; by Gx^ - 5x + 2 - Tx^, arrange the dividend uud divisor by ascending powers of x, thus : 2-5x + 6x2 - 7^3^ 6 - 23x + 48x2 _ 70^3 + sgx* - 35x5 (^3 _ 4^ + 5^2 6-15x+18x2-21x3 - Sx-I- 30x2- 49x3 + 58x* - 8x + 20x2- 24x3 + 28x* 10x2 _ 25x3 + 30x* - 35x5 10x2 _ 25^3 + 30x* - 35x* Examples. — xiv. Divide 1. x2+15x + 50by x+10. 5. x3+ 13x2 4-54x + 72 by x-i-6. 2. x2 - 17x + 70 by x - 7. 6. x3 + x^ - x - 1 by x + 1. 3. x2 + X - 12 by X - 3. 7. x3 + 2x- + 2x 4- 1 by x + 1. 4. x2 + 13x-l-12byx + l. 8. x^ - 5x3 + 7x2 + 6x + 1 by x2 + 3x + 1. 9. X* - 4x3 ^ 2x- + 4x + 1 by x2 - 2x - 1. 10. x*-4x3 + 6x2-4x+l by x2-2x+l. ^1 Dins 10 y. 39 n. a;*-x2 + 2x-l by a;2 + x-l. 12. a;*-4x2 + 8x+ 16 l>y x + 2. 13. x^-\- -ix-y + 3xi/ + 12)/^ by x + 4y. 14. a* + 4a36 + 6a262 + 4^53 + 54 ^y ^ + 6. 15. <{5 - 5«*6 + 10a362 - I0a263 + 5^54 _ ^s ijy ^ _ 5, 16. x* - 12x' + 50x2 - 84x + 45 by x2 - 6x + 9. 17. «■• - 4a*b + 4a362 + 4^(2^,3 _ 17^54 _ i265 by a^- - 2ah - 3//. 1 8. 4a2xi - Ua^x^ + lSa*x- - Gu^x + a^ by 2(tx2 - 3a2x + a\ 1 9. X* - a;^ + 2x - 1 by x2 + X - 1. 20. X* + rt-x2 - 2a^ by x2 + 2a2. 23. x^ - y^ by x - 1/. 21. x2 - rSxy - '30y'' by x - 15j/. 24. a- - b- + ihc - c2bya - 6 + c. 22. x'' + (/' by X + y. 25. h - 'ilr + 36^ - i* by & - 1. 26. tr - 62 _ c2 + ^2 _ 2(afi - 6c) ])y a + 6 - c - rf. 27. x^ + ?/■' + ^3 - 2xyz by x + ?/ + z. 28. x^^ + ;/^" by x^ + yK 29. ^2 4.^2 4. 223}- - 2^2 + 72?- _ 3}-2 by y- q + 3r, 30. a^ + a«62 + a«6* + u-¥' + 6« by a* + a% + «-62 + a¥ + 6^ 31. x^ + x^y~ + x^y^ + x-y^' + y^ by x^ - x^y + x'-y^ - xy^ + y*. 32. 4x5 - x3 + 4,^. by 2x2 + 3^. ^ 2. 33. a'' - 243 by a -3. 34. Z;io - k by P - 1. 35- a;^ - 5x2 _ 46.5 - 40 by x + 4. 36. 48x3 - 76ax2 - 6ia-x + I05a^ by 2x - 3a. 37. ISx* - 45x' + 82x2 _ 673- + 40 by 3x2 _ 4^ ^ ^ 38. 16x* - 72a2x2 + 81a* by 2x - 3a. 39. Six-* - 256a* by 3x + 4a. 41. x^ + 2ax2 - a2x - 2a^ by x2 - a'K 40. 2«-^ + 3a26-2a62_363bya2-62. 42. a*- a262_ i264bya2 + 352 43. X* - 9x2 _ Q.j.y _ y2 by x2 + 3x + y. 44. X* - 6x^y + 9xhf - Ay^ by x2 - 3xy + 2y\ 45. X* - 8ly* by X - Sy. 47. 81a* - 166* by 3a + 26. 46. a* - 166* by a - 26. 48. 16x* - 81y* by 2x + Sy. 49. 3a2 + 8a6 + 46^ + lOac + 86c + 3c2 by a + 26 4- 3c. 50. a* + 4a2x2 + 16x* by a- + 2ax + 4x^. 51. X* + x2j/2 + y* hy X' - xy + y^. , 52. 256x* + 16x2!/2 + y* by 16x2 + 4xy + y\ 53. x^ + x*7/ - x^2/2 + x^ - 2x2/2 + y^'by xi^ + x-y. 40 DIVISION. 54. ax^ + Sa^x^ - <id?x - 2a* by a; - a. 55. a^ - ^ byx + a. 56. 2x2 + a;?/ - 3!/^ - 4i/z - X3 - 2;^ by 2x + 3y+ z. 57. 9a; + Sa;* + 14x'' + 2 by 1 + 5x + x^. 58. 12 - 38x + 82x2 - 1 12x3 + 106x* - TOx^ by Tx^ - 5x + 3. 59. x^ + 1/^ by X* - x^y + x^y^ - xy^ + y*. 60. (a-x^ + h\j'^ - (a?h- + x'^y'^) by ax + by + ab + xy. 61. a& (x- + 1/2) + xj/(a2 + 62) by ax + by. 62. X* + (262 - a^)x2 + ¥ by x2 + ax + b\ 82. The process may in smne cases be shortened by the use of brackets, as in the following Exaui2:)le. x + 6^x^ + (a + 6 + c) x2 + (a6 + ac + 6c) x + a6c(x2 + (a + c) x + ac x^ + 6x2 (a + c) x2 + (a6 + ac + bc) x (a + c) x2 + (a6 + 6t) x acx + abc acx + abc x — l)x^- mx* + na^ — ?ix2 + mx — 1 (:'•■* - (m - 1) x^ x^-x* -(//i.-7i-l)x2-(?n,-l)x+l. - (m - 1) X* 4- nx^ -(m-l)x* + (m-l) x3 — (rn — n—l)x^ — nx^ -{m-n-1) a^+{vi-n-l) X- -(m- 1) x2 + mx - (m - 1) x2 + (in -1) X x-1 x-1 Examples.— XV. Divide 1. X* - (a2 _ 6 _ c) x2 - (6 - c) ax + be by x2 - ax + c. 2. y^-(l + m + n) y^ + {hn + In + mn) y - Imn by y-n. 3. x^ - {m - c) x'* + {n - cm + d)oi^ + {r + en - dm) x^ + {cr + dn) x + dr by x^ - mx- + ')ix + r. 4. X* 4- (5 + a) x3 - (4 - 5a + 6) x2 - (4a 4- 56) x 4- 46 by x2 4- 5x - 4. 5. x*-(a4-64-c4-rf) x3 4-(a64-ac4-a<i4-6c4-6(i4-cd) x^ - (a6c 4- a6(/ 4- acd + bed) x 4- abed by x* - (a 4- c) x 4- ac. division: 41 83. Tlie following Exainples in Division are of great importance. Divisor. Dividend. Quotient. x + y x2-2/2 x-y x-y x^-y^ x + y x + y x^ + y^ x^ - xy + y^ x-y oc^-y^ x^ + xy + y^ 84. Again, if vre an'ange two series of binomials consisting respectively of the sum and the ditference of ascending powers of x and y, thus x + y, X" + y-, y? + y^, xf^ + ?/■*, x'' + y^, a;" + 7/'', and so on, x-y, X-- y-, sr - y^, x* - y^, af" - y^, x'^ - 1/", and so on, x + y will divide the odd terms in the upper line, and the even in the lower x-y will divide all the terms in the lower, but none in the upper. Or we may put it thus : If n stand for any whole number, x^ + jj" is divisible by x + y Avhen n is odd, by x-y never ; x^-y" is divisible hy x + y when n is even, hy x-y alv.jys. Also, it is to be observed that when the divisor is a;-y all the terms of the quotient are positive, and when the divisor is x + y, the terms of the quotient are alternately positive and negative. x^ — v^ Thus^-^ — -^ = a^ + x'^y + xy^ + y^, x-y xJ + 'lf~ ^ = x^- afy + xhf - oi?y^ + x^y^ - xy^ + y^, J- =x^- xhf + x^y^ - x^y^ + xy* - y^. 45 Tilvisroy: 85. These properties may bf easily remembered by taking the four simplest cases, thus, x + j/, x-y, x- + y-, x^-yp, of which the first is divisible by x + y, second x-y, third neither, fourtli both. Again, since these properties are true for all values of x and y, suppose y = \, then we shall have x^-\ , x^-l a;+l X - 1 3? + l , , x3-l , - = x--x+l, '=x- + x+l. X+l X- I Also x^ + l , , ., , ;- = X* - X/ + .X- - X + 1, x-i- 1 x^- 1 = x^ + X-* + x^ + X- + X + 1. X- 1 Examples. — xvi. "Without going through the process of Division write down the quotients in the following cases : 1. When the divisor is m + n, and the dividends are respectively m^ - n^, m^ + n^,'ni^ + n^, m^ - n^, m^ + ?i^. 2. When the divisor is m - n, and the dividends are respectively vi^ - n~, m^ - n^, m* - 7i*, m^ - n^, vi' — iiJ. 3. AVhen the divisor is a+1, and the dividends are respectively (r-1, r»^+ 1, a^ + l, a" + l, a*-l. 4. When the divisor is y-\, and the dividends are respectively 2/2-1, !/5- 1,2/5-1,7/7 _ 1^2/9 _1. V. ON THE RESOLUTION OF EXPRES- SIONS INTO FACTORS. 86. We shall discuss in this Chapter an operation which is the opposite of that which we call Multiplication. In Mul- tiplication we determine the product of two given factors : in the operation of which we have now to treat the product u given and the factors have to be found. 87. For the resolution, as it is called, of a product into its component factors no rule can be given which shall be applic- able to all cases, but it is not difficult to explain the process in certain simple cases. We shall take these cases separately. 88. Case I. The simplest case tor resolution is that in which all the terms of an expression have one common factor. This factor can be seen by inspection in most cases, and there- fore the other factor may be at once determined. Thus a^ + ah = a(a + b), 2a3 + 4a2 + Sa = 2a {a? + 2a + 4), 23?y - 1 %xhf + bAxij = 9xy (x- - 2xy + 6). EXAMPLPS.— xvii. # Resolve into factors : 1. 5a;2-15x. 5. a^-ax^ + hx'^ + cx. 2. 3rc" + 18x2-6a;. 6. 3afy^-2lxY + ^'^x^y*- 3. 49y--Uy + 7. 7. 54a%'i + 108a%'* - 24Sa^b\ 4. 4x^y-\2x-y2 + Sxy\ 8. 45a;"(/i^ - 90^5^7 - 360xV^. 44 RESOL UTTOJSr INTO FA CTORS. 89. Case 41. The next case in point of simplicity is that in which four terms can be so arranged, that the first two have a common factor and the last two have a common factor. Thus a;^ + ax + 6a; + a6 = {^ + ax) + (ix + a6) = .r (.r 4- a) 4- & (x + a) = (x + 6) (x + a). Again ac - ad - be + bd = (ac - ad) - (he - hd) = a{c-d)-h{c-d) = (a-h) (c-d). Examples. — xvi ii. Resolve into factors : 1 . x^ -ax-bx + ab. 5. ahx^ - axy + hry - y\ 2. ab + ax — hx — x"^. 6. abx — ahy + cdx — cdy. 3. bc + hy -cy - y^. 7. cdx^ + dmxy - cnxy - m n y'. 4. hm + mn + ab + an. 8. abcx-b^dx-acdy + bd'y. 90. Before reading the Articles that follow the student is advised to turn back to Art. 56, and to observe the manner in which the operation of multiplying a binomial by a binomial produces a trinomial in the Examples there given. He will then be prepared to expect that in certain cases a trinomial can be resolved into two binomial factors, examples of which we shall now give. 91. Case III. To find the factors of x- + 7x+12. Our object is to find two numbers whose product is 12, and whose sum is 7. These will evidently be 4 ai^d 3, * .-. x^ + 7x + 12 = (x 4- 4) (x + 3). Again, to find the factors of x2 + 56x + 662. Our object is t4 find two numbers whose product is 65*, and whose sum is 56. These mil clearly be 36 and 26, .•. X- + 56x + 66- = (x + 36) (x + 26). ffF.SOLUTION- INTO FACTORS. Examples.— xix. Resolve into factors : a:2+llx + 30. X-+ 17x + 60. 2/2+13^ + 12. i/ + 2l!/ + 110. ?7i2 + zbvi + 300. m? + 23»i- + 102. a2 + 9«6 + 862, a;- + 13ma; + 36??;-^. 9. j/2 + 19?!?/ + 48n2. 10. ^- + 2^ + 1002)2. 1 1 . a-* + 5x- + 6. 12. u:'' + 4x3 + 3. 13. a:2i/2 + 18a:i/ + 32. 14. xh/-\-1x^y'+l2. i;. m'o + 10?rt=+16. 93. 16. ?i' + 27?i2+ 14032. Case IV. To find the factors of a;— 9x + 20. Our object is to find two negative terms -whose product is 20, aud whose sum is — 9. These? will clearly he - 5 and - 4, .-. x2 - 9.C + 20 = (,.; - 5) (x - 4). Exam ples. — xx. Resolve into factors : x--7x+ 10. X- - 29x + 190. 2/2 - 237/ + 132. y- - 30y +- 200. 7r-43?i+.460. 6. 'n^-57n + 56. 7. 3:^-7x^ + 12. 8. a262-27a6 + 26. 9. Mc''-ll62c3 + 30. 10. x-yh--l3xijz + 22. 92. Case V. To find the factors of x2 + 5x - 84. Our object is to find two terms, one positive and one negative, w liose product is - 84, and whose sum is 5. These are clearly 12 and - 7, .-. x2 + 5x - 8^= (x + 12) (x - 7). 46 RESOLUTION INTO FACTORS. Examples. — xxi. Resolve into factors : I. x^^'lx-m. 6. ■ 62 + 256-150. 2. a:2 + 12a;-45. 7- a;8 + 3ar*-4. 3. rt2+n(j_i2. 8. a;V + 3xi/-154. 4. a2+13a-140. 9- 7/i'0+15rr65- 100. 5. 6- +13?^ -300. 10. 7(2 +17,1 -390. 94. Case VI. To find the factors of ;«- - 3.C - 28. Our object is to find two terms, one positive and one negative, whose product is - 28, and whose sum is - 3. These will clearly be 4 and - 7, .-. a;2-3a;-28 = (2; + 4)(j;-7). Examples.— xxii. Resolve into factors : I. ■j?-hx- 66. 2. x^ - Ix - 18. 3- ni^ - 9??i - 36. 4- ?i2_ 11,^-60. 5- 1/-13J/-14. 2- -152 -100. .a-io _ 9.,.5 _ 10. c-d-'-24ctZ-180. in^'n- - mhi - 2. 95. The results of the four jnvcvding articles may be tims stated in general terms : a trinomial of one of the forms X" + ax + h, x^ - ax + h, x- + ax - b, x--ax- h, nuiy be resolved into two simple factore, when b can be re- solved into two factors, such that their sum, in the fii-st two iorms, or tlieir difference, in the last two forms, is equal to a. 96. We shall now give a set of Miscellaneous Examples oh the U'solution into factors of expressions which come under one or other of the cases already explained. RESOLUTION INTO FACTORS. 47 Examples. — xxiii. Kesolve into factor.s : 1. a;--15x + 36. 8. a;" + TOx + ?ia; + mu.. 2. a;^ + 4.o-45. ^ 9. 1/^- 41/^ + 3. 3. a^W' - 1 Ga6 - 36. i o. a;^ - «&x - cxy + a6c. 4. a:^- 3?;.'x*- IOjh^. II. a;2 4. (^fj _ j^ ^. _ g5_ 5. T/^ + 1/^-90. 12. a;- - (c - rf) X - c(i. 6. x*-a:--110. 13. ab^ - bd + cd - abc. 7. ar^ + 3«x- + 4ff2x. 14. 4^^ - 28^!/ + 48i/2. . 97. We have said, Art. 45, that when a number is mnlti- plied by itself the result is called the Square of the number, and that the figure 2 placed over a number on the right hand indicates that the number is multiplied by itsell'. Thus a^ is called the square of a, (x - yf is called the square oi x-y. Tlie Square Root of a given number is that number whose square is equal to the given number. Thus the square root of 49 is 7, because the square of 7 is 49. So also the square root of a^ is a, because the square of a is a^ : and the square root of {x - y)- is x-y, because the square of x-y is (x- yy. The symbol ^1 placed before a number denotes that the square root of that number is to be taken : thus ,j2b is read " the square root of 25." Note. The square root of a positive quantity may be either positive or negative. For since a multiplied by a gives as a result a', and - a multiplied by - a gives as a result a^, it follows, from our definition of a Square Root, that either a or - a may be regarded as the square root of a^. But throughout this chapter we shall take only the positive value of the square root. '48 RESOLUTION INTO FACTORS. . 98. We may now take the case of Trinomials which are verfect squares, which are really included in the cases dis- cussed in Arts. 91, 92, but which, from the importance they nssume in a later part of our suliject, demand a s^eparate con- .-ii deration. 99. Case VII. To find the factors of Seeking for the factors according to the hints given in Art 9i, we find them to be a; + 6 and x+ 6. That is a;2 + 12x + 36 = (x + 6)2. Examples. — xxiv. Resolve into factors : 1. a;2+18a; + 81. 6. a;* + 14a;2 + 49. 2. a;2 + 26a; + 169. 7. a;2+ 10xi/ + 25t/2. 3. a;2 + 34x4-289. 8. tn!^ + \(5mhi- + QAn*. 4. y' + 'iy+l. 9. x« + 24.1-3 + 144. 5. 22 + 2002+10000. 10. x-?/2 + 162x«/ + 6561. 100. Case VIII. To find the factors of .r-^12x + 36. Seeking for the factors according to the hints given in Art. 92, we find them to be x - 6 and x - 6. That is, x2 - 12x + 36 = (x - 6)2. Examples.— XXV. Resolve into factors : I. x2-8x + 16. 2. x2-28x + 196. 3. x2-36x + 324. 4. 2/2 _ 40j/ + 400. 5 . c2 - lOUs + 2500. 6. .i"* - 22x2 + 1 2 1 . 7. x2 - 30x1/ + 225?/2 8. 7H^-32?/i.2„2 + 256«*. n. x«- 38x3 + 361. RESOLUTION INTO FACTORS. 49 101. Case IX. We now proceed to the most important case of Resolution into Factors, namely, that in -which the ex- pression to be resolved can be put in the form of two squares with a negative sign between them. Since m^ - n^ = (m + n) (m - n), we can express the difference between the squares of tw<i quantities by the product of two factors, determined by tlie following method : Take the square root of the first quantity, and thasquaie root of the second quantity. The sum of the results will form the first factor. The difference of the results will form the second factoi'. For example, let a^ - Ir be the given expression. Tlie square root of a- is a. The square root of h- is h. The sum of the results is a + 6. The difference of the results is a — b. The factors will therefore be a + 6 and a - 6, that is, a^-¥ = {a + b) (a - b). 102. The same method holds good with resjDect to com- pound quantities. Thus, let a^ - (6 - c)- be the given expression. The square root of the first term is a. The square root of the second term is 6 - c. The sum of the results is a + b-c. The difference of the results is a — b + c. .. a^- {b-cy = {a + b-c){a-b + c). Again, let (a - by - (c - d)- be the given expression. The square root of the first term is a - 6. The square root of the second term is c - d. The sum of tlie results is a-b + c-d. The difference of the results is a-b-c + d. :. {a-b/- {c-df=(a-b + c-d^ (a-b-c + d). [6.A.] • D 50 kESOLUTION INTO FACTORS. 103. The terms of an expression may often be arranged 30 as to form two squares with the negative sign between them, and then tlie expression can be resolved into factors. Thus a2 + 52_c2_f^2 + 2a6 + 2cd = a2 4.2a6 + 62_c2 + 2cd-d2 = {o? + 2a6 + 62) - (c2 - 2cd + d?) = (a + 6)2-(c-d)2 = (a + 6 + c - c?) (a + 6 - c + (Z). Examples. — xxvi. Resolve into two or more factors : I. x^-y^. 2. x2-9. 3. 4a;2-25. 4. a* -a;*. 5. x^-,!. 6. a;''_i. 7. x^-l. 8. m'^-lQ. 9. 36!/2-4922. 10. Slxhf - 121a262. J I. (a _ 5)2 _ ^2, ■ 12. x^ - (m - nf. 13. (a + 6)- - (c + (Z)2. 24. 2a;2/-a;2-i/2^1_ 14. {x + yf-{x-yf. 25. x^-2yz-y--z^. 15. x2-2xt/ + i/-2!2. 26. a2-462-9c2 4-12ic. 16. (a-^6)2-(m + Ji,)2. 27. a*- 1661 17. a2_2ac + c2-62_26ji_(^2_ ^g. l-49c2. 18. 2bc-b^-c^ + d\ 29. a2 + 62_c2_rf2_2a5_2crf 19. 2xy + x'^-\-y'^-z^. 30. a- - 6- + ^2 - ^2 _ 2ac + 266?. 20. 2mn - m2 - %2 ^ ^2 + 52 _ 2^6. 3 1 . Sa'^.r'^ - 27ax, 2 1 . (ax + byy - 1. 32. a''6'' - c*. 22. (ax + %/ - (ax - 6?/)2. 33. (5x - 2)- - (x - 4)2. 23. l-a2-62 + 2a6. 34. {7x + 4yy--{2x + 3y)l 35. (753)2 -(247)2. 104. Case X. Since =x2-«a: + a2, and -^ = x2 + ax + a2 (Art. 83), X ~r d- X — CV we know the following important fj^^jts ,• RESOLUTION INTO FACTORS. 51 (1) The suTfi of the cuhes of two numbers is divisible by the swm of the numbers : (2) The difference, between the^ cvhes of two numbers is divisible by the difference between the numbers. Hence we may resolve into i'actors expressions in the form of the sum or difierence of the cubes of two numbers. Thus ic3 + 2 7 = :<? + 33= (./ + 3) (x- - 3x + 9) 3/3_64 = 2/3-43=(2/-4)0/ + 42/+16). Examples.— xxvii. Express in factors the following expressions : I. a3 + 63. 1. a? ^W. 3. a3_8. 4. a;3 + 343. 5. 63 _ 125. 6. x3 + 64;/3. 7. a3-216. 8. '6y? + Tif. 9. 64a3- 100063. 10. 729x3 + 512t/3. Express vcijour factors each of the following expressions : II. x^-i/^ 12. x^-1. 13. 0*^-64. 14. 729 -j/S. 105. Before we proceed to describe other processes in Algebra, we shall give a series of examples in illustration of the principles already laid down. The student will find it of advantage to work every example in the following series, and to accustom himself to read and to explain with facility those examples, in which illustrations are given of what may be called tlu short-hand method of expressing Arithmetical calculations by the symbols of Algebra. Examples. — xxviii. 1 . Express the sum of a and b. 2. Interpret the expression a-b + c. 3. How do you express the double of x ? 4. By how much is a greater than 5 ? 5. If a; be a whole number, what is the number next above it ? 6. Write five numbers in order of magnitude, so that x phall be the tliird of the five, RESOLUTION INTO FACTORS. 7. If a be multiplied into zero, what is the result ? 8. If zero be divided by x, what is the result ? 9. What is the sum of a + a + a . . . written d times ? 10. if the product be ac and the multiplier «, what is th« iiiulli[ilicand? 1 1 . What number taken from % gives 1/ as a remainder ? 12. J. is a; years old, and B is y years old ; how old was A when B was born ? 13. A man works every day on week-days for x weeks in llie year, and during the remaining weeks in the year he does not work at all. During how many days does he rest ? 14. There are x boats in a race. Five are bumped. How many row over the course ? 15. A merchant begins trading with a capital of x pounds. He gains a pounds each year. How much capital has he at the end of 5 years 1 •' _ « 16. A and B sit down to play at cards. A has x shillings and B )j shillings at first. A wins 5 shillings. How much has each wiieu they cease to play ? 17. There are 5 brothers in a family. The age of the eldest is X years. Each brother is 2 years younger than the one next above him in age. How old is the youngest ? 18. I travel x hours at the rate of y miles an hour. How- many miles do I travel ? 19. From a rod 12 inches long I cut off x inches, and then I cut off y inches of the remainder. How many inches are left ? 20. If n men can dig a piece of ground in q hours, how many hours will one man take to dig it ? 21. By how much does 25 exceed xl 22. By how much does y exceed 25 ? 23. If a ]iroduct has 2m repeated 8 times as'a factor, how do you express the product ? 24. By how much does a + 2h exceed a-2bl 25. A girl is X years of age, how old was she 5 ye.irs since ? RESOL UTTON INTO FA C TO RS. 53 26. A boy is y years of age, how old will he he 7 years hence 1 27. Express the difference between the squares of two numbers. 28. Express the product arising from the multiplication of the snui of two numbers into the difference Ijetween the same numbers. What value of x will make 8a; equal to 16 ? What value of x will make 28a; equal to 56 ? X What value of x will make ^ equal to 4 ? What value of x will make a; + 2 equal to 9 ? What value of x will make a; - 7 ecpial to 16 ? What value of x will make a;- + 9 equal to 34 ? What value of x will make a.-- 8 equal to 92 ] Examples. — xxix. Explain the operations symbolized in the following expres- sions : 1. + 6. 2. (I- -IP-. 3. 4(t2 + fc3_ 4_ 4(f(2^^2)_ 5. a2_26 + 3c. 6. a + m-Kh-c. 7. ((( + ?h)(6- c). 8. s]^. 9. ^x- + t/. 10. a + 2(3-c). II. (a + 2)(3-c), 12. . -,-. 13. -^^^ -^ . 14. — -■- — ^- . 4ab ■" x-y ^ ^f^ + y Examples. — xxx. If a stands for 6, b for 5, x for 4, and y for 3, find the val ue of the following expressions : I. a + x-b~y. 2. a + y-b-x. 3. 3a + 4y-b-2x. 4. 3(a + 6) - 2(x - y). 5 . (a + x){b- y). 6. 2a + 3 r y). 7. (2a + 3)(x + y). 8. 2a + Zx + y. 9. ^-^ti'. 10. abx. II. ab{x + y). 12. ay{b-\-xf. 54 * kEsdLUTION INTO FACTOkS. 13- ah{x- y)^. 14. v/56. 1 6. isfW- 17. {J^+b)\ 19. ^2axy. 20. a^ + ¥ + y 15- Jy^- 18. J5bx. 21. 3a + (2x-i/)2. 22. |«-(&-i/)f !a-(a;-i/)f. 24. 3(« + 5 -?/)H4(/i4-x)*. 23. (a-6-2/)2 + (a-a; + 2/)2. 25. 3 (tt - 6)^ + (4x - 1/2)2. EXAMPLES. — XXXi. 1. Find the value of Sabc -a^ + P + c^, when a = 3, /> = 2, c=l. 2. Find the value of 3? + y^ -z^ + 3xyz, when ic = 3, y = 2, g = 5. 3. Subtract «'- + c^ from (a + c)-. 4. Subtract (x - 1/)- from x- + y'^. 5. Find the coefficient of x in the expression {a + byx-{a + bxf. 6. Find the continued product of 2x - VI, 2x + n, x + 2m, x - 2n. 7. Divide acr^ + {be + ad) r^ + {bd + ae)r + be by ar + h; and test your result by putting a = b = c = d = e=l, and r = 10. 8. Obtain the product of the four factors (a + h + c), (b + c-a), {c + a- b), {o. + b-c). What does this become when c is zero ; when 6 + c = a; when a = b = c'{ 9. Find the value of (a + 6)(6 + c) -{c + d)(d + a)-{a + c) (b-d), ' where b is equal to d. 10. Find the value of 3a + (26 - c2) + ! (;2 - (2a + 5b) ! + { 3c - (2a + 36)|2, wh(;i a = 0, 6 = 2, c = 4. RESOLUTION INTO FACTORS. 11. If a = l. // = 2, c = 3, d=A, shew that the numerical values are ec^ual of j(Z-(c-& + a)f{(fZ + c)-(6 + o)i, and of d^ - (c^ + 6-) + a- + 2 (6c - arf). 12. Bracket together "the different powers of x in the follow- in g expressions : (a) ax^ + bx'^ + ex + dx. . (/?) ax^ - h:i? - ex"- - dx} + 2x2. (7) A'j? - ax^ - 3x- - hi? - 5x — ex. (S) {n + x;'-(h-xy. (e) {mx- + qx. + 1 )■' - (?!x2 + qx+l )2. 13. Multiply the three factors x-a, x-b, x-c together, and arrange the product according to descending powers of x. 14. Find the continued product of (x + a) {x + b){x + c). 15. Find the cube of a + h + c; thence without further multiplication the cubes of a + 'j-c; b + c -a; c + a-b; and subtract the sum of these three c\\ >e< from the first. 16. Find the product of (3a + 2b) (3a + 2c - 3b). and test the result by making a = 1, 6=:c = 3. 17. Find the continued product of a-x, a + Xj a- + x'-, a'' + a;'*, a* + x^. 1 8. Subtract (b - a) (c - d) from {a - 6) (c - d). What is the value of the -result when a = 26 and tZ = 2c ? 19. Add together J) + y) (a + x), x-y, ax - by, and a(x + y). 20. What vaiue of x will make the difference between (x + 1) (x + 2) and (x - 1) (x - 2) equal to 54 ? 21. Add together ax -by, x- y, x(x - ?/), and (a - x) {h - y). 22. ^T:iat value of x will make the difference between (2x + 4) (3x + 4) and (3x - 2) (2x - 8) equal to 96 ? 23. Add together 2mx - 3ny, x + y, 4(m + n)(x- y), and mx + vy. 24. Prove that {x + y + z)^ + x'^ + y^ + z- = {x + ij)'^ + {y + z)^ + {x + zy. 56 RESOUmON INTO FACTORS. 25. Find the product of (2a + 36) (2a + 3c - 20;, and test the result by making a = \, 6 = 4, c = 2. 26. If a, b, c, d, e ... denote 9, 7, 5. 3, I, find the values of ab - cd ., - . 62 _ ^2 ___; (6c-«f?)(W-c.); ^^-; andrt«-c^ ::/. Find the value of 3a6c - a^ + Ir + c^ when a = 0, 6 = 2, c = l. ;3. Find the value of .^ , , 2«62 • c^ , , I, 1 n 3a- H ^:, Avhen a = 4, 6 = 1, c = 2. 29. Find the value of (a-6-c)2 + (6-«-r)-' + (c-a-6)2'when a=l, 6 = 2, c = 3. 30. Find the value ol' (rt + 6 - c)2 + (a - 6 + c)2 + (6+ c-fl)2 when a=;l, 6 = 2, c=4. 31. Find the value of (a + 6)2 + (6 + c-)'- + (e + a)2 when a= - 1, 6 = 2, c= -3. 32. Shew that if the sum of any two nnmhers divide the ditt'erence of their squares, the quotient is equal to the differ- ence of the two numbers. 33. Shew that the product of the sum and difference of anv t\Mi numbers is equal to the difference of their squares. 34. Shew that the square of the sum of any two consecu- ti\e integers is always greater by one than four times their jjriMluct. 35. Shew that the square of the sum of any two consecutive even whole numbers is four times the square of the odd number between them. . 36. If the number 2 be divided into any two parts, the ditlerence of their squares will always be equal to twice liie difference of the parts. 37. If the number 50 be divided into any two parts, tli- difference of their squares will always be equal to 50 timdi tli-.- difference of the parts. 38. If a number n be di^dded into any two parts, the difference of their squares will always be equal to n times the difference of the parts. ON SIMPLE EQUA TIONS. 57 39. If tw'o numbers differ by a imit, their product, together with the sum of their squares, is equal to the difference of the cubes of the numbers. 40. Shew tliat the sum of the cubes of any three consecu- tive whole numbers is divisible by three times the middle number. VI. ON SIMPLE EQUATIONS. 106. An Equation is a statement that two expressions are equal. 107. An Identical Equation is a statement that two ex- pressions are equal for all numerical values that can be given to the letters involved in them, provided that the same value be given to the same letter in every jiart of the eciuation. Thus, 0-<; + a)2=a;2-!-2ax + a2 is an Identical Equation. 108. An Equation of Condition is a statement that two expressions are equal for some particular numerical value or values that can be given to the letters involved. Thus, a;+l = 6 is an Equation of Condition, the only number which x can represent consistently with this equation being 5. It is of such equations tliat we have to treat. 109. The Root of an Equation is that number which, wlien ])at in the place of the unknown quantity, makes both sides of the equation identical. 110. The Solution of an Equation is the process of find- ing what number an unknown letter must stand for that the eipiation may be true : in other words, it is the method of fuuling the Eoot. The letters that stand for imknown numbers are usually X, y, z, but the student must observe tliat any letter may stand for an unknown number. 111. A Simple Equation is one which contains the first jiower only of an unknown quantity. This is also called ^n Equation of the First Decree. 58 ON SIMPLE EQUATIONS. 112. The following Axioms form the grounANVork of the solution of all equations. Ax. I. If equal quantities be added to equal quantities, the sums will be equal. Thus, if a = &, Ax. II. If equal quantities be taken from eaual quantities, the remainders will be equal. Thus, if x = y, ^ x-z = y -z. Ax. III. If equal quantities be multiplied b^ equal quan- tities, the products will be equal. Thus, it a = h. Ax. IV. If equal quantities be divided uy equal quantities, the quotients will be equal. Thus, if xy = xz, y=z. 113. On Axioms I. and II. is founded a process of great ntilitv in the solution of equations, called The Traksposition OF Terms from one side of the equation "c the other, which may be tlius stated : " Any term of an equation may be transferred from one side of the equation to the other if its sign be changed." For let x-a = h. Then, bv Ax. I., if we add a to both udes, the sides remain equal : therefore x-a + a = b + a, that is, x = b + a. Again, let x + c = (l. • Then, by Ax. II., if we subtract c liom ^u,•ih. side^ the sides remain equal : therefore iC + c~c = d~c, l.uat is, x=d-c, ON SIMPLE EQUATIONS. 59 114. We may change all the signs of each side of an equa- tion without altering the equalit}'. Thus, if a-x — h-c, x-a = c-b. 115. We may change the position of the two sides of the e(|nation, leaving the signs unchanged. Thus the equation a - b = x - c, may be written thus, X- c = a -b. 116. We may now proceed to our first rule tor the solution of a Simple Equation. Rule I. Transpose the known terms to the right hand side (>f the equation and the unknown terms to the other, and com- I'ine all the terms on each side as far as possible. Then divide both sides of the equation by the coefficient of the unknown quantity. This rule we shall now illustrate by examples, in which x stands for the unknown quantity. Ex. 1. To solve the equation, 5x - 6 = 3x + 2. Transposing the terms, we get 5x - 3x = 2 + 6. Combining like terms, we get 2x = 8. Dividing both sides of this equation by 2. we get x = 4, and the value of x is determined. Kx. 2. To solve the equation, 7x + 4 = 25x - 32. Transposing the terms, we get 7x~25x= -32-4. Combining like terms, we get -18x=-36. Changing the signs on each side, we get 18x = 3t). Dividing both sides V)y 18, we get x = 2, and the value of x is determined. 6o OX SIMPLE EQUATIONS. Ex. 3. To solve the equation, 2a; - 3a: + 120 = 4j; - 6x+ 132. that is, 2x - 3x - 4x + 6a; = 13z - liv/, or, • 8x-7x=I2, therefore, x=12. U.X. 4. To solve the equation, 3a; + 5-8(13-a;) = 0, that is, 3x + 5-104 + 8x==0. or. 3a; + 8x=104-5, ' or. llx=99, therefore, a;=9. Ex. 5. To f.olve the equation, ()a;-2(4-3x) = 7-3(17 -■<, that is, 6a;- 8 + 6x= 7 -51 + :).£, or, or, 6x4-6:c-3a;=7-51+8, 12x-3a;=15-51, or, 9x= -36, therefore. x= -4. EXAMPLES.— :?cxxii. 9. 26-8x = 80-1435. 10. 133-3x = x-83. 11. 13-3x = 5x- o. 12. 127 + 9x= 12x4-100. 13. 15-5x=6-4x. 14. 3./ -22 = 7x-J-6. 15. 8 + 4x=12.c-16. 1 6. 5.r - (3x - 7) = 4x - ^6x - 3o). 17. 6x - 2(9 - 4x) + 3 (5x - 7'i = lOx - (4 + 16x) + 35. 18. 9x-3(5x-6) + 30 = O 19. 12x - 5 (9x + 3) + 6(7 - 8x) + 783 = 0. 20. X - 7(4x - 11) = 14(x - 5) - 19(8 - x^ - € :. 21. ^T + 7)(x-3) = (x-5)(x-15). 1. 7x + 5 = 5x+ll. 2. ]2x + 7 = 8x + 15. , 3. 236.c+425 = 97x + 564 4. 5x - 7 = 3x + 7. 5. 12x-9 = 8x-l, 6. 124x+19 = 112x + 43. 7. 18- 2^=27 -5x. 8. 125-7x=145-12.''. PROBLEMS LEADING TO SIMPLE EQUATIONS. 6i 22. (x-8)(x + 12)=(ic+l)(a;-6). 23. {x - 2)(7 - x) + (x - 5) (x + 3) - 2(x - 1) + 12 =^ 0. 24. (2a! - 7) (x + 5) = (9 - 2.r) (4 - x) + 229. 25. (7 - 6x) (3 - 2x) = (4.C - 3) (3j; - 2). 26. 14 - a; - 5 (x - 3) (x + 2) + (5 - x) (4 - Sx'i = 45.r - 76. 27. (x + 5)2-(4-x)-=21x. 28. 5(x - 2)2 + 7(x - 3)2 = (3x - 7)(4.t - 19, + 42. 29. (3x - 17)2 + (4x - 25)2 _ (5_^ _ 09)2 = ] . 30. (x + 5) (x - 9) + (x + 10)(x - 8) = (2x + 3) (x - 7) - 1 13. VII. PROBLEMS LEADING TO SIMPLE EQUATIONS. 117. When we have a question to resolve by means 01 Algebra, we represent the number sought by an unknown symbol, and then consider in what manner the conditions of the question enable us to assert thot tv:o exjjressiotis are equal. Thus we obtain an equation, and by resolving it we determine the value of the number sought. The wliole difficulty connected with the solution of Alge- braical Problems lies in the determination from the conditions of the question of tiro different expressions having the same numerical value. To explain this let us take the following Problem : Find a number sucli that if 15 be added to it, twice the sum will be equal to 44. Let X represent the number. Then x + 15 will represent the number increased by 15, fu I 2(x + 15) will represent twice the sum. But 44 will represent twice the sum, therefore 2 (x + 15) = 44. Hence 2x4-30 = 44, tliatis, *2x=14, or, x=7, and therefore the number sought is 7. 62 PROBLEMS LEADING TO SIMPLE EQUATIONS. 118. We shall now give a series of Easy Problems, in which the conditions by which an equality between two expres- sions can be asserted may be readily seen. The student should be thorouffhly familiar with the Exanijdes in set xxviii, the use of which he will now find. We shall insert some notes to explain the method of repre- senting quantities by algebraic symbols in cases where some difficulty may arise. Examples. — xxxiii. 1. To the double of a certain number I add 14 and obtain as a result 154. What is the number ? 2. To four times a certain number I add 16 and obtain as a result 188. What is the number ? 3. By adding 46 to a certain number I obtain as a result a number three times as large as the original number. Find the original number. 4. One number is three times as large as another. If I take tlie smaller from 16 and the greater i'rom 30, the remaiii- deis are equal. What are the numbers % ;. Divide the number 92 into four parts, such that the first is greater than the second by 10. greater than the third by 18, and greater than the fourth by 24. 6. Tlie sum of two numbers is 20, and if three times the' smaller number be added to five times the greater, the sum is 84. What are the numbers ? 7. Tlie joint ages of a father and his son are 80 years. If the nge of the son were douliled he would be 10 years older than his father. What is the age of each? 8. A man has six sons, each 4 years older than the one next to Jiim. The eldest is three times as old as the youngest. Wiiat is the age of each ? 9. Add .£24 to a certain sum, and the amount ^dll be as much above ^80 as the sum is below ^80. What is the sum \ 10. Thirty yards of cloth and lorty yards of silk together cost £66, and the silk is twice as valuable as the cloth. Find the cost of a vard of each. PROBLEMS LEADING TO SIMPLE EQUATIONS. 63 11. Find the number, the double of which being added to 24 the result is as much above 80 as the number itself is below 100. 12. The sum of ^500 is divided between A, B, C and D. A and B have together ^280, A and G X260, A and D ^'220. How much does each receive ? 13. In a company of 266 persons, composed of men, women, and children, there are twice as many men as there are women, and twice as many women as there are children. How many are there of each ? 14. Divide i'1520 between A, B and C, so that A has ^100 less than B, and B i;'270 less than C. 15. Find two numbers, differing by 8, such that four time? the less may exceed twice the greater by 10. 16. A and B began to play with equal sums. A won £0, and then three times ^-I's money was equal to eleven times B'a money. What had each at first ? 17. A is 58 years older than B, and ^'s age is as much above 60 as B's age is below 50. Find the age of each. 18. yl is 34 years older than B, and A is as much above 50 as B is below 40. Find the age of each. 19. A man leaves his property, amounting to J7500, to be divided between his wife, his two sons and his three daughters, as follows : a son is to have twice as much as a daughter, and the wife .£500 more than all the five children together. How much did each get ? 20. A vessel containing some water was filled up by pour- ing in 42 gallons, and there was then in the vessel 7 times as much as at first. How many gallons did the vessel hold 1 21. Three persons. A, B, C, have .£76. B has .£10 more ihan A, and C has as much as A and B together. How much lias each ? 22. Wliat two numbers are those whose difference is 14, and their sum 48 ? 23. A and B play at cards. A has £72 and B has £52 when they begin. When they cease playing, A has three times as much as B. How much did A win ? 64 PROBLEMS LEADING TO SIMPLE EQUATIONS. Note I. If we have to express algebraically two parts into which a yiven number, suppose 50, is divided, and we repre- sent one of the parts by x, the other will be represented by :.() - X. Ex. Divide 50 into two such parts that the double of one \rAxt may be three times as great us the other part. Let X represent one of the parts. Then 50 - x will represent the other part. Now the double of the first part will be represented bv 2x, and three times the second part will be represented by 3 (50 - x). Hence 2a; = 3 (50 -x), or, 2x=150-3x, or, 5a; = 150; .-. x = 30. Hence the parts are 30 and 20. 24. Divide 84 into two such parts tliat three times one part may be equal to four times the other. 25. Divide 90 into two such parts that four times one part may lie equal to five times the other. 26. Divide CO into two such parts that one part is greater tlian llie other by 24. 27. Divide 84 into two such parts that one part is less than t'.ie (idler by 36. 28. Diviile 20 intn two sucli parts that if three times one 1 art be added to five times the other part the sum may be S4. Note 1 1. When we have to compare the ages of two per- sons at one time and also some years alter or before, we must lie caretul to remember that hoih will be so many years older or younger. Thus if X be the age of .4 at the present time, and 2* be the age of B at the present time, The age of .4 5 years hence will be a; + 5, an<l the age of B 5 years hence \\ iil be 2j + 5. PROBLEMS LEADING TO SIMPLE EQUATIONS. 65 Ex. ^ is 5 times as old as B. and 5 years hence A will only be three times as old as B. What are the ages of A and B at the present time ? Let X represent the age of B. Then bx will represent the age of A. Now a; + 5 will represent £'s age 5 years hence, and 6x + 5 will represent ^'s age 5 years hence. Hence 5x + 5 = 3 (x + 5), or 5x + 5 = 3x+15, or 2x=10; .'. x = 5. Hence A is 25 and 5 is 5 years old. 29. A is twice as old as B, and 22 years ago he was tliree times as old as B. What is yl's age ? 30. A father is 30 ; his son is 6 years old. In how many years will the age of the father be just twice that of the son \ 31. A\% twice as old as B, and 20 years since he was three times as old. What is £'s age ? 32. A is three times as old as B, and 19 years hence he will be only twice as old as B. What is the age of each ? 33. A man has three nephews. His age is 50, and the joint ages of the nephews are 42. How long will it be before tlie joint ages of the nephews will be ec^ual to the age of the uncle \ Note III. In problems involving weights and measures, after assuming a symbol to represent one of the unknown quantities, ^ve must be careful to express the other quantities in the same terms. Thus, if x represent a number of pence, all the sums involved in the problem 7nust be reduced to pence. Ex. A sum of money consists of fourpenny pieces and si.x- pences, and it amounts to £1. IBs. 8d. The iiuniber of coins is 78. How many are there of each sort ? [s.A.] S 66 PROBLEMS LEADING TO SIMPLE EQUATIONS. Let X be the number of l'ouT2)enny pieces. Then Aj-y is their value in fence. Also 78 — X is the number of sixpences. And 6 (78 — X) is their value in 'pence. Also £\.. 16s. 8(Z. is eqtiivalent to 440 pence. Hence 4a; + 6 (78 - a) = 440, or 4a; + 468- Gx = 440, from which we find x= 14. Hence there are 14 fourpenny pieces, and 64 sixpences. 34. A bill of ^100 was paid with guineas and half-crowns, and 48 more hulf-ciowus than guineas were used. How many of eacli were paid ? 35. A person paid a bill of £3. 14s. w'ith shillings and hall-crowns, and gave 41 pieces of money altogether. How many of each were paid ? 36. A man has a sum of money amounting to £11. 13s. 4d., consisting only of shillings and fourpenny pieces. He has in all 300 pieces of money. How many has he of each sort ? 37. A bill of .£50 is paid with sovereigns and moidores of 27 shillings each, and 3 more sovereigns than moidores are given. How many of each are used ? 38. A sum of money amounting to £42. 8s. is made up of shillings and half-crowns, and there are six times as many half-crowns as there are shilling-. How many are there of each sort ? 39. I have £5. 1 1*-. 3(/. in sovereigns, shillings and pence. I have twice as many shillings and three times as many pence as I have sovereigns. How manv have I of each sort J YIII. ON THE METHOD OF FINDING THE HIGHEST COMMON FACTOR. 119. An expression is said to be a Factor of another expression wiieii the latter is divisible by the former. Thus 3a is a factor of 12a, 5xy of lox^y\ 120. An expression is said to be a Common Factor of two or more other expressions, when each of the latter is divisil)le by the former. Thus 3a is a common factor of 12a and 15a, 3xy of Ibx^y^ and 2\x^y^, 4z of 82, 12^2 and I6z^ 121. The Highest Common Factor of two or more expres- sions is the expression of highest dimensions by which each of the former is divisible. Thus 6a2 is the Highest Common Factor of 12a2 and 18a^, bx^y of 10x^1/, 15x^2/2 and 25x*2/^ Note. That which we call the Highest Common Factor is named by others the Greatest Common Measure or the Highest Common Divisor. Our reasons lor rejecting these names will be given at the end of the chapter. 122. The words Highest Common Factor are abbreviated thus, H.C.F. 123. To take a simple example in Arithmetic, it will readily be admitted that the highest number which will divide 12, 18, and 30 is 6. Now, 12 = 2x3x2, 18 = 2x3x3, 30=-2x3x5. 68 MET HOP Oh h IN DING THE Having thus reduced the numbers to their sirajylht factor?, it appears that we may determine the Highest Common Factoi- in the i'ollowing way. Set down tlie factors of one of the numbers in any order. Place beneath theiii tlie factors of the second number, in !-uch order tliat fact(jrs like, any of those of the first number shall stand under those factors. Do the same for the third number. Then the number of vertical columns in which the numbers are alike Avill be the number of factors in the h.c.f., and if we multiply the figures at the head of those columns together the result will be the h.c.f. required. Thus in the example given above two vertical columns are alike, and therefore there are two factors in the h.c.f. And the numbers 2 and 3 which stand at the heads of those columns being multiplied together will give the H.C.F. of 12, 18, ana 30. 124. Ex. 1. To find the h.c.f. of aW-x and a%^X'. aWx = aaa .bb .x, aWx^ = aa . bbb . xx ; :. 'E.c.F.=aabbx = a'b^x. Ex. 2. To find the h.c.f. of 34a26»c* and 51a%*c*. Ma%^c* = 2 X 17 xaa . bhbbbb . cccc, 5la%*c- = 3 X 17 X aaa . bbbb . cc ; .*. B..c.F. = 17 aabbbbcc = 17a--'6V. Examples.— xxxiv. Find the Highest Common Factor of \. a*h iuu\ a'b^. 3. 14x-_ir' and 24.r^. 3. a^yh and x'-y-z-. 4. -ibm-ny and mmhip*. HIGHEST COMAfOA' FACTOR. 69 5. 18rt7)'-c'd and Z^a^hcd?. 8. 17 pq~, 'i4p^q and 5\p^q^. 6. «3?)-, arb^ and a^6*. 9. Sx^j/Sg*, Ux^z^ and 20x*3/V. 7. 4a7), lOac and 305c. 10. 3(teY, QOx^ and 120x3t/4 125. The student must he urged to commit to memory the following Table of furms which can or cannot be resolved into factors. Where a blank occurs after the sign = it signifies tliat the form on the left hand cannot be resolved into simpler factors. x^-y^ = {x + y){x-tj) x^-l=(x+l)(x-l) x^ + y^= a:- + 1 = x^ — y^ = {x — y){x'^ + xy + y^) x?-l = {x—l)(x^ + x + l) 3^ + y^ = (x + y){x--xy + y^ a^+ l = (x+ 1) (a;^ — x + l) a^-y* = {x'^ + y^-){x'^-y^) x*-l = {x^+l: {x^-1) x* + y^= a;* + 1 = x^+2xy + y' = {x + yy x'^-h2x+l={x+iy x^-2xy + y^ = {x-yf x--2x+l = {x-iy 05^ + 'Ax^y + Sxy^ + y^ = {x + yY x? +.3x2 + 3x + 1 = (x + 1)3 X? - 3x2?/ + 33.^^2 _yZ^(^^_ yy x^ - 3x2 + 3x - 1 = (x - 1)3 The left-hand side of the table gives the general forms, the right-hand side the particular cases in which 1/= 1. 126. Ex. To find the h.c.f. of x^-l, x2-2x-H, and a;2 + 2x-3. X2-I=(x-I)(x+1), x2-2x + l = (x-l)(x-l), x2-i-2a;-3 = (x-l)(x-l-3), .". H.C.F. =X-1. Examples. — xxxv. 1. a2 - 52 and a^ - b\ 4. a^ + a^ and (a -I- x)'. 2. a- — h- and a* — b*. 5. 9x2 _ i ^j^^j ^^x + 1)2. 3. a2_x2 and (a — x)2. 6. 1 -25a^ and (1 — bay. 7. x2 - 1/2, (x + yY and x2 4- Zxy + 2y'. 8. x2 — y^, x^ — y^ and x2 — Ixy + 6?/2. 9. x2 — 1, x3 — 1 and x2 + x - 2. 10. 1 — a"^, \ + a^ and a2 + 5^ -1- 4. METHOD OF FINDING THE 127. In large numbers the factors cannot often be deter- mined by inspection, and if we have to find the h.c.f. of two such numbers we have recourse to the following Arithmetical Rule : " Divide the greater of the two numbers by the less, and the divisor by the remainder, repeating the process until no rr mainder is left : the last divisor is the h.c.f. required." Thus, to find the h.c.f. of 689 and 1573. 689; 1573(2 1378 T95;689(3 585 Tb4; 195(1 104 9i; 104(1 91 13; 91 (7 91 .-. 13 is the H.i F. of 689 and 1573. Examples.— xxxvi. Find the h.c.f. of 1. 6906 and 10359. 4. 126025 and 40115. 2. 1908 and 2736. 5. 1581227 and 16758766. 3. 49608 and 169416. 6. 35175 and 236845. 128. The Arithmetical Rule is founded on the following )peration in Algebra, which is called the Proof of the Rule foi finding the Highest Common Factor of two expressions. Let a and h be two expressions, arranged according to de- scending powers of some common letter, of which a is not of lower dimensions than h. Let h divide a with -p as quotient and remainder c, c h g A. d c r with no remainder. HIGHEST COMMOX J-ACTOR. 71 The form of the operMtiun may be shewn thus : fb d) c (r rd Then we can shew !. That rf is a common factor of a and 6. II. That any other common factor of a and 6 is a factor of rf, and that therefore d is the Higliest Common Factor of a and b. For (I.) to shew that rf is a factor of a and b : b = qc + d = qrd + d = {qr + I) d, and .'. d is a factor ot b ; and a=|j6 + c —P (3<^ + d) + c = pqc+pd + c =pqrd+pd + rd = {pqr+p + r) d, and .•. d is a factor of a. And (11.) to shew that any common factor of a and b is a factor of d. Let 8 be any common Factor of a and b, such that a = viS and b = n8. Then we can shew that 8 is a factor of d. For d = b-qc = b~q(a-pb) = b - qa + pqb =n8 - qm8 + pqnS = {n-qm +pqn) 8, and .". 8 is a factor of d. Now no expression higher than d can be a factor of d ; :. d is the Highest Common Factor of a and b. 72 METHOD OF FINDING THE 1 29. Ex. To find tlie h.c.f. of x'- + 2a; + 1 and x-"' + 2x- + 2a- -^ I. a;2 + 2x+l_;a;3 + 2x2 + 2x4-l(x x3 + 2x2 + X x+l^x2 + 2x+ 1 (,x+ 1 x^ + x x+1 x+ 1 Hence x+1 being the last divisor is the h.c.f. required. 130. In tlie algelnaical process four devices are frecjuently useful. Tliese we shall now state, and exemplify each iu the next Article. I. If the sign of the first term of a remainder be negative, we may change the signs of all the terms. II. If a remainder contain a factor which is clearly not a common factor of the given expressions it may l)e removed. III. We may'nmliiply or divide either of the given expres- sions by any number which does not introduce or remove a common factor. IV. If the given expressions have a common factor which can be seen by inspection, we may remove it from both, and find the Highest Common Factor of the parts which remain. If we inultiply this result by the ejected factor, we shall obtain the Highest Com- mon Factor of the given expressions. 131. Ex. I. To find the h.c.f. of 2x2 - x - 1 and 6x2 -4x- 2. 2x«-x-i;6x2-4x-2(3 6x2 -3x- 3 - x + 1 HIGHEST COMMON FACTOk. ^% Change tlie signs of the remainder, and it becomes x— 1. • a!-i;2!fc2-x_ 1(2x4-1 2x2 _ 2x x-1 x-1 The H.c.F. required is x — 1. Ex. 1 1 . To find the h.c.f. of x^ + 3x + 2 and x^ + 5x + 6. x2 + 3x4-2;x2 + 5x-l-6(l x2 + 3x + 2 2x + 4 Divide the remainder by 2, and it becomes x + 2. x + 2;x2 + 3x + 2(x+l x2 + 2x x + 2 x + 2 The H.c.F. required is x + 2. Ex. III. TofindtheH.c.F.of 12x2 4.x-land 15x"+8x + l. Multiply 15x2 + 8x + l by 4 12x2 + a; - 1> 60x2 + 32a; + 4 (5 60x2+ 5a;_5 27x + 9 Divide the remainder by 9, and the result is Zx-k-l. 3x+i;i2x2 + x-l(4x-l 12x2+ 4x ^x^ -3x-l The H.c.F. is therefore 3x + l. Ex. IV. To find the h.c.f, of x^ — 5x2 + 6x and x--Kte2 + 21x. Remove and reserve the factor x, which is cotuiiKJii tu both expressions. U METHOD OF FINDING TfTK Then we have remaining x^ — 5x + 6 and x- — lOx + 21. The H.c.F. of these expressions is x — 3. The H.c.F. of the original expressions is therefoie x^ — 3x Examples.— xxxvii. Find the h.c.f. of the following expressions : 1. x2 + 7x + 12 and x2 + 9x + 20. 2. x2 + 12x + 20 and x^ + 14x + 40. 3. x2 - 17x + 70 and x2- 13x4-42. 4. x2 + 5x-84andx2 + 21x+108. 5. X- + X— 12 and x^ — 2x-3. 6. x^ + 5x2/ + ^y^ ^^^ ^^ + ^^V + 9j/^- 7. x^ - 6x?/ + 8j/- and x^ — Sxy + 16?/^. 8. x2 - 13x1/ - 30?/2 and x^ - 18xt/ + 45i/2. 9. x^ — y^ and x- — Ixy + 1/-. 10. x^ + \f and x^ + 3x-i/ + 3x?/2 + 1/^. 11. X* — 1/* and X- — 2x1/ + ^2. 12. x^ + 1/^ and x^ + y^. 13. X* — 2/* and 3? + 2x1/ 4 y"^. 14. a^ _ 52 ^ 26c - c- and a- 4- 2a?> 4- 6- - 2ac - 26c 4- c-. 1 5. 12x2 4- Ixy 4- 2/" and 28.C- 4- 3x!/ — y-. 1 6. 6x- 4- xxj - 1/2 and 39x2 _ 22x1/ 4- 3 j/*. 17. 1 5x2 — 8x1/ 4- 1/2 and 40x2 — 3x1/ — 1/2. 18. x*-5x3 4-5x2-l and x*4-x3-4x2 4-x4-l. 19. X* + 4x2 + 1 6 an,| x5 + X* - 2x3 4- 1 7^.2 _ 1 Ox 4- 20. 20. X* 4- x2(/2 4- 1/* and x* 4- 2x''i/ 4- 3x2)/2 4- 2xi/' 4- y*. 21. x" - Gx^ 4- 9x2 - 4 ^mi x^4-x^-2x*4-3x2-x-2. ftlGHEST COMMON FACTOR. 75 22. 1 5a* + lOa^ft + ^aW + Gai^ - 36* and Ga^ + IQa^i + 8a62 _ 563. 23. ISa:^ - Hcc^?/ + 24a;y2 - Ti/^ and 27x3 + SSx^j/ - 20x1/2 + gi/^. 24. 21x2 - 83xy - 27x + 22!/2 + 99;,' and 12x2 _ 353.^ _ ^y. -33t/2 + 227/ 25. 3a3-12a2-a26 + 10rt6-262and %a^ -\~ia?-h^M}P--W. 26. 1 8a3 _ I8a2x + 6ax'' - Gx^ and 60a2 - 75ax + 15x2. 27. 21x3-26x2 + 8xand 6x2-x-2. 28. 6x* + 29n2x2 + 9rt* and Sx^ - 15ax2 + a^x - ou\ 29. x^ + x^i/2 + x^y + T/3 and x* — ?/*. 30. 2x3 + 103-2 + 14a; + 6 and x^ + x2 + 7x + 39. 3 1 . 45a3a; + 3a2x2 - 9ax3 + 6x* and 1 8a2x - Sx\ 132. It is sometimes easier to find the h.c.f. hj reversing the order in which the expressions are given. Thus to find the h.c.f. of 21x2 + 38x + 5 and 129x2 + 221x+ 10 the easier course is to reverse the expressions, so that thev stand thus, 5 + 38x + 21x2 and 10 + 22 lx+ 129x2, and jjjgj^ j-^ proceed by the ordinary process. The h.c.f. is 3x + 5. Other examples are (1) 187x3 - 84x2 + 31x - 6 and 253x3 - 14x2 ^ 29x - 12, (2) 371^/3 + 262/2-50?/ + 3 and 469i/ + 7oi/ - 103?/ - 21, of which the h.c.f. are respectively llx — 3 and 7?/ + 3. 133. If the Highest Common Factor of three expressions a, b, c be reciuired, find first the h.c.f. of a and b. If d be the h.c.f. of a and b, then the h.c.f. of d and c will be the h.c.f. of a, b, e. i«%. Ex. To find the h.c.f. of x'3 + 7x2 - X - 7^ ^.3 ^. 5^.2 _ 2; _ 5^ and x2 - 2x + 1. The H.C.F. of x3 + 7x2 - X - 7 ^nd x3 + 5x2 - x - 5 will be found to be x2 — 1. The H.C.F. of x2-l and x2-2x+l will be found to be x-1. Pence x— 1 is the h.C-f. of the three expressiona. 76 FRACTIONS. Examples. — xxxviii. Find the Highest Common Factor of 1. a;2 + 5x + 6, x2 + 7x+10, and a;2 + i^^20- 2. x3 + 4x2-5, a;3-3x + 2, andx3 + 4x2-8x + 3. 3. 2x2 + x-l, x2 + 5x + 4, and x^-i-l. 4- V^-'f-V^^-, 3l/2-2i/-l, and 1/3 -2/2 + 2/ -1. 5. 3?- 4x- + 9x - 10, x^ + 2x2 _ 3x + 20, ami x^* + Sx'" - v»x -»- S."). 6. x3 _ 7a,2 + i6x - 12, 3x3 - 143.2 + igj.^ ^^^,1 5x3-10x- + 7x-14, 7. ■j/3 — 5?/2 + ii^_ 15j y3_y2^3y^.5 aj^Q 2i/3-72/-+ieT/- 15. XoTE. We use the name Highest Common acrm n..«Tead of Greatest Common Measure or Highest Common Divisor lor the following reasons : (1) We liave used the word " Measure '• in An. a different .*ense, that is, to denote the number of tiW^~ any quantity contains the ^lnit of measurement (2) Divisor does not necessarily imply a quanntv wuich is contained in another an exact number of times. '1 nus in performing the operation of dividing 333 tiy 13, we can 13 divisor, but we do not mean that 333 coutains 13 au eicact number of times. IX. FRACTIONt 135. A QDANTITY a is called an Exact i)ivisoK 01 m civwn- tity b, when h contains a an exact number :)i mnes. A quantity a is called a Multiple of a (juanniy 0, M^acu a contains b an exact number of times. ■FJHACTIOXS. 77 136. Hithori-o we have treated of quantities wliicli coutniii the unit of »»^«^«iiremeut in each case an exact ninube:- of times. We have p'^w to treat of quantities which contain some exact divisor of a pnmary unit an exact number of times. 137. We must first explain what we mean by a primary unit. We said in Art. 33 that to measure any quantity we take a known standard or unit of the same kind. Our choice as to the quantity to be taken as the unit is at first unrestricted, but when once made we must adliere to it, or at least we must give distinct notice of any change which we make with re.spect to it. To such a unit we give the name of Primary Unit. 138. Next, to explain what we mean by an exact divisor of a primary unit. Keeping our Primary Unit as our main standard of mea- surement, we may conceive it to be divided into a number of parts of equal magnitude, any one of which we may take as a Subordinate Unit. Thus we may take a pound as the unit by which we mea- sure sums of money, and retaining this steadily as the primary unit, we may still conceive it to be subdivided into 20 equal parts. We call each of the subordinate units in this case a shilling, and we say that one of these equal subordinate units is one-twentieth part of the primary unit, that is, of a pound. These subordinate units, then, are exact divisors of the primary unit. 139. Keei^ing the primary unit still clearly in view, we represent one of the subordinate units by the followinf' nota- tion. We agree to represent the words one-third, one-fifth, and one-twentieth by the symbols ^, -, — , and we say that if the Primary Unit be divided into three equal parts, - -will represent one of these parts. 78 FRACTIONS. If we have to represent two of these subordinate units, we 2 3 do so by the symbol - ; if three, by the symbol - ; if four, by o o " 4 the symbol -, and so on. And, generally, if the Primary Unit be divided into h equal parts, we represent a of those parts by the symbol ■ . 140. The symbol t we call the Fraction Symbol, or, more briefly, a Fraction. The number helow the line is called the Denominator, because it denominates the number of equal parts into which the Primary Unit is divided. The numbi-r above the line is called the Numerator, because it enumerates how many of these equal parts, or Subordinate Units, are taken. 141. The term number may be correctly applied to Frac- tions, since they are measured by units, but w'e must be careful to observe the following distinction : An Integer or Whole Number is a multiple of the Primary Unit. A Fractional Number is a multiple of the Subordinate Unit. 142. The Denominator of a Fraction shews what multiple the Primary Unit is of the Subordinate Unit. The Numerator of a Fraction shews what multiple the Fraction is of the Subordinate Unit. 143. The Numerator and Denominator of a fraction are called the Terms of the Fraction. 144. Having thus explained the nature of Fractions, we next proceed to treat of the operations to which they are sub- jected in Algebra. 145. Def. If the quantity x be divided into b equal parts, and a of those parts be taken, the result is said to be the fraction ,- of x. Jfxhe the unit, this is called tlie fraction j-. FRACTIONS. 79 146. If the unit be divided into b equal parts, y will represent one of the parts. r two T three And generally, T will represent a of the parts. 147. Next let us suppose that each of the b parts is sub- divided into c equal parts : then the unit has been divided into be equal parts, and T- will represent one of the subdivisions. -=— two DC And generally, a — a be 148. To shew that r = t- be b Let the unit be divided into b equal parts. Then j- will represent a of these parts (1). Next let each of the b parts be subdivided into c equal parts. Then the primary unit has been divided into be equal parts, and -J— will represent ae of these subdivisions (2). Now one of the parts in (1) is equal to c of the subdivisions in (2), .'. a parts are equal to ac subdivisions ; , a ac "'b^'k' 8o FRACTIONS. Cor. We draw Irom this proof two inferences : I. If tlie numerator and denominator of a fraction be multiplied by llie same number, the vahie of the frac- tion is not altered. II. If the numerator and denominator of a fraction be divided by the same number, the value of the fraction is not altered. 149. To make the important Theorem established in the preceding Article more clear, we shall give the following proof that K = o7x, ^y taking a straight line as the unit of length. I I I I I I I I I I M I I I I I I I I I A E D F B C Let the line AG be divided into 5 equal parts. Then, if B be the point of division nearest to C, AB is I of AC. (1). Next, let each of the parts be subdivided into 4 equal parts Then AG contains 20 of these subdivisions, and AB 16 :. ABi^^^oiAJ. (2). Comparing (1) and (2), we conclude that 4^1^ 5~20" 150. From the Theorem established in Art. 148 we derive the following rule for reducing a fraction to its lowest terms : Find the Highest Covimon Factor of the numerator and denomi- nator and divide both by it. The res^ilting fraction vnll be one equivalent to the original fraction expressed tJi the simplest terms. FRACTIONS. 8i 151. When the numerator and denominator each consist of a single term the h.c.f. may be determined by inspection, or we may proceed as in the following Example : , To reduce the fraction , ^ „,, ., to its lowest terms, 10a^6-c* _ 2 X 5 X aaabbcccc 12a-6V^ 2 X 6 X aabbbcc ' We may then remove factors common to the numerator and denominator, and we shall have remaining -— — j- : " 6x0 .'. the required result will be -^^ 152. Two cases are especially to be noticed. (1) If every one of the factors of the numerator be removed, the number 1 (being always a factor of every algebraical expression) will still remain to form a numerator. 3a'C Zaac 1 Thus I2ah'^ 3 X 4 X aaacc 4ac' (2) If every one of the factors of the denominator be removed, the result will be a whole number. „, I2ah- 3 X 4 X oMacc Thus .^ , = — ^ = 4ac. 3a-c o X aac This is, in fact, a case of exact division, such as we have explained in Art. 74. Examples. — xxxix. Reduce to equivalent fractions in their simplest terms the following fractions : 4a2 12a3' 8x3 ^' 36x2- IQx^yh^ 45x^2*' 7o567c« 5' 21a36V 6. tT- 3abc blay-z 8xYz^ •iia-yz^' 9- 6a^y»Z^' [8.A.] F §2 FRACTIONS. 2\0mVp a? 14m*x lO. -. II. . 12. ■• 42m%2jp2" • d--\-ab' 21m^p -7mx xy Aax + 2x^ mi + w- 3x2/2 — 5x2^z" 8ax^ — 2x2' 3- abc + bcy' 4a^x + 6ah j 12 aF- - 6ah „ c^-4a^ ' ■ 8x2-18?/' ^-^^ 86-C-2C ■ ^ ' c2 + 4ac + 4a2 3x* + 3x2|/2 ^ labhfi - 7abY ^9' 5x* + 5x27/' ^'^ 14a%x» - 14a%2/2' lOx-lOy 5x9 4, 45t^2 "°' 4x2 -8x1/ + 4?/^ ^5' lOcx" + 90crfx2' ax + by , 10a2 + 20a6+106* ^ 26. 7a2x2 - 7b-y^' ' 5a^ + 5a% 6ab + Scd 4x2 _ g^-y ^ 4^2 27a262x - 48c2cZV ^" 48(x-i/)2 ' xy-xyz - 3mx + 5?ix2 2«3 — 2rt32' ■ 3to?/ + 5)!X1/' 153. We shall now give a set of Examples, some of which may be worked by Resolution into Factors. In others the H.C.F. of the numerator and denominator must be found by the usual process. As an example of the latter sort let us take the following : To reduce the fraction „-, — ^-„ — .so ^, to its lowest terms. •' 2x3-9x2-38x + 21 Proceeding by the usual rule for finding the H.C.F. of the numerator and denominator we find it to be x - 7. Now if we divide x^ — 4x2 — 19x— 14 by x — 7, the result is x2 + 3a; + 2, and if we divide 2x^-9x2-38x4-21 byx-7, the result is 2x2 + 5x — 3. x2 + 3x + 2 Hence the fraction „ (. . "q i^ equivalent to the proposed fraction and is in its lowest terms. Examples. -xl. a2+7a + 10 ^ x2-9x + 20 x« -2x-3 a2 + 5a + 6' '' x2-7x + 12" ^' x^-\0x^2l' FRACTIONS. 83 x2-18xj/ + 452/^ x^ + x^ + l x^ + 2x3?/^ + 2/' x^-8x?/-105i/2' 5- a;- + x + l* ' a;^-i/6 ''■ x3 + 2x2"-3x + 20' ^"^^ m3-7m + 6 ' „ x^-5x-+ llx — 15_ a^ + 1 a^_x2 + 3x+5 ^' a3 + 2a- + 2a+r a:3-8x2 + 21x-18 , 3ax2-13ax4-14a ^" 3x3-16x'-^ + 21x ■ ^ ■ 7x3- 17x2 + 6x • x3-7x2 + 1 6x-12 14x2-34x + 12 '°" 3x3- 14x2 +T6x • 17- 9ax2-3'9ax + 42a' x* + x3j/ + X1/3 - 1/^ 10a -24a2 + 14^3 x* — x^?/- x]/'* — ?/'' * 15 — 24a + 3a2 + 6a3" a3 4 4a- - 5 2a63 + a6- - 8a6 + 5a ^^' a3-3a + 2* ^^' 763-1262 + 56 * 63 + 462-56 a3_3^2 + 3^_2 3x2 + 2x-l a^-a-%) x^-3x^ + 4x-2 x3 + x2-x-l "' a- + a-12' "^ x3-x2-2x + 2 (x + y + g)2 + ( g - j/)2 + (x - g)2 + (y - a;)2 X2 + ?/2 + ^2 2x* - x3 - 9x2 + 1 3x - 5_ 1 5fj3 ^ab-2b'^ ^5" 7x3-19x2 + 17x-5 ' 3^" 9a2 + 3a6 -262' 16x*-53x- + 45x + 6 x2-7x + 10 8x^-30x3 + 31x2^1 2" ^■^" 2x2 - X - 6 " 4x2 - 1 2_ax + 9a2 x3 + 3x2 + 4,^ + 1 2 ^'^' 8x3-27a3 • 35- x3T4x2T4x + 3* 6.< r^-23x2 + 16x -3 x*-x2_-2x + 2 6x3-17x2+llx-2' ^ 2x3 -x^l"' x3-6x 2 + nx- 6 x3 - 2x2 - 1 5x + 36 ^9" x3-2x2-x + 2 ■ ^^' 3x2^x^- 15 ' 7n3 + m2 + m — 3 3x3 + x2-5x + 21 ■^ ' '»i3 _• Q^-v.2 _i_ P^«v» 1^ Q* -3 ' 3)762 + 5m + 3' :)"• 6x3 + 29x2 + 26x-2r x^ + 5x* — x2 — 5x X* — x3 - 4x2 — X + 1 ^'' x4 + 3x3-x-"3 ' ^^: 4x3-3x2-8x-l " a2 - 62 - 26c - c2 a3-7a2+I6(x-12 ?^* a2 + 2a6 + 62-?" ^O- 3^3 ::T4a2 + i6a ' FRACTIONS. 154. The fraction t is said to be a proper fraction, ■when a is less than h. The fraction t is said to be an improper fraction, when a is greater tlian h. 155. A whole number x may be written as a fractional number by writing 1 beneath it as a denominator, thus -. 156. To prove that 5- of j = r3- a ocL \ Divide the unit into bd parts. ^^'^n'^^d = 6«^^ (Art. 148) = r of be of these parts (Art. 147) = T- of 6c of these parts (Art. 148) = ac of these parts (Art. 147). But yj = ac of these parts; a ^ c _ac •'■bd~bd' This is an important Theorem, for from it is derived the Rule for what is called Multiplication of Fractions. We extend the meaning of the sign x and define , x t (which according to our definition in Art. 36 has no meaning) to mean r of -„ and we conclude that y >< -i = t^. which in words trives b d b d bd ^ us this rule — " Take the product of the numerators to form the numerator of the resulting fraction, and the product of the denominators to form the denominator." The same rule holds good for the multiplication of three or more fractions, FRACTIONS. 85 157. To shew that r^-7= t- • a be The quotient, x, of r divided by -5 is such a number that x multiplied by the divisor 3 will give as a result the dividend t- . arc _ a •• li~b'' d p xc d . a .-. - of -r = - of |- ; c a c xcd _ ad " cd be ' ad ■' ^=k- Hence we obtain a rule for what is called Division of Fractions. _,. a c ad Since r-r- j = T~) d be a c _a d b^d'h^'c' Hence we reduce the process of division to that of multiph' cation by inverting the divisor. 158. The following are examples of the Multiplication an 1 Division of Fractions. 2x o _ ^■'^ 3a _ 6ax 2x I. 3„2>^'^"-3„2>^-l- = 3^ = -- 3x_^ _3x^3a_3x 1 _ .3x _ x ^' 26* *~26 • T~26 ^3a~6a6~2a6* 4a^ 3c _ 3 X 4 X a-c _ 2a ^' 9c^ ^ 2a~ 2 X 9x ac- ~ 3c' 14x2^ '7x_ 14x2 9.v_ 9xl4xa:2? / 2x ^ 27y^'' 9y''27y^^7x~7x27xxy^~^' 2a 96 5c _ 2a x 96 x 5c _3 5' 36 '^ TOc ^ 4a ~ 36 x~10or4a " 4* 86 PR ACTIONS. x^ — 4x x^ + 7.x_ x(x — 4) x(x + 7) x^ + 7x^ x-4 x-(x + l) x — 4 _x(x-4)x{x + 7)_ ~x2(x + 7)(x-4)~ a'i _ 52 ^ 4(a2_ ab) _ «2- ?)2 a2 ^j, '' a^ + 2ab + b- ' a- + ab ~ a^ + 2ab + b"^ 4{a'^ - ab) _{a + b){a — b) a{a + b) ~{a + b){a + b) 4a{a-b) _(a + b){a-b)a{a + b) _1 ~{a + b)\a + b)4a(a-b)'~4' Examples.— xli. Simplify the following expressions : 3x 7x 3a 26 4x^ 3x 4y^9y' ^' 4b^3a' ■ ^' df^^V' 80253 I5xy2 Q^y2^ 20a%k 2a 46 5f 45x2?/ ^ 24a42- 5- ioa^^c'' mxij-z' 56 "" 3c ^ 6a" Sx^y 5yh I2xz Ici^b* 20cM- 4ac 4x^ ^ 6x2/ ^ 20x^" 5"c2d3 ^ 42^463 "" shd' 9vihi^ hifiq 24x2w2 25A;3m2 "On^q 3pm ^ -^ ^ X ^ TO X — X — ^ 82?3g3 2x2/ 90mn" ' 14712^2 'jop'm 4k^n Examples. — xlii. Reduce to simple fractions in their lowest terms : a- 6 a2-62 x2 + x-2 ■t 2 - 13x + 42 a2 + a6 a2 - a 6' ^' x^ - 7x x2 ^ gx x^ + 4x 4x2j^l2x x2-llx + 30 x2-3x X2"^X ^ 3x2Tl 2X 5 • 2-2 _ y^ + y ^ a;2 _ 53;" x2 + 3x + 2 x2-7x + 12 , x2-4 x2_25 O. -r, i X ./-^ 5x + 6 x2 + x ■ ■ x^ + S^u x^ + 2x' g-i _ 4 a + 3 a2-9a + 20 a2 - 7 a '' a^ — ba + 4 a- - 1 Oa + 21 «-' - 5a' 62-76 + 6 62+106 + 24 6^ - 862 8- 62:^36-4 "" 62 - 1 46 + 48 "" 62 + 66 • t^R ACTIONS. 87 ■y^ xy- 27/2 ^x^-xy II. 13- x^ - 3xy + 2y^ '" x^ + xy {x - yy^ {a + by~c^ c2 - (g - b)- d' - {h - cY"" c^ - {a + hf {x - m)- — n- X- - (n - nt)^ (x - n)- - m^ X- - {m - n)'^' ( a + hy-(c + dY {a-hf-{d- cf (a + cy -{b + df ""[a- cf -{d- bf X? — 2xy + y- -z^ x + y-z x^ + 2xy + y^-z- x-y + z' Examples.— xliii. Simplify tlie following expressions : 2a 36 X ~5c" Aa , rix ' rSab. bx . -2. 2 l£M^^^ . 8x*^_^2x3 14s ■ 7z' ■^' loab^ ' SOoi^ ^ 2^; - 2 > - 1 • 5x 11 11 X' - a.c + 2 ■ X - r ^' x- - 17x + 30 ■ X - 15' 158. We are now able to justify the use of the Fraction Symbol as one of the Division Symbols in Art. 73, that is, we can shew that j is a proj^er representation of the quotient resulting from the division of a by b. For let X be this quotient. Then, by the definition of a quotient. Art. 72, b xx = a. But, from the nature of fractions, , a X y = a; a :.-r=x. THE LOWEST CO^TmO^ :■ i'.^. 159. Hi-re we may state an important : neorciu, whicn -«>' shall require in the next chapter. If ad = be' to shew that , = -,. b a Since ad = bc, ad be bd~ 'bd a c 'b~ 'cC X. THE LOWEST COMMOri iviui-TIPLE. 160. An expression is a Common Multh uk of two or more other expressions when the former is esLnunv divisible by each of the hitter. Thus 24x^ is a common multiple of 6, 8x^ and 12a^. 161. The Lowest Common Multiple of two or more expressions is the expression of loivest dimenbi^ns which is exactly divisible by each of then, , Thus ISx* is the Lowest Couimou iuun-ipie of 6j;*, ^x^, and 3x. The words Lowest Common Multiple are abbreviated into L.c.M. 162. Two numbers are said to be prime to each other which have no common factor but unity. Thus 2 and 3 are prime to each other. 163. If a and b be prime to each other the fraction is in its lowest terms. Hence if a and b be prime to each other, uud i=^, «J>tl if m be the h.c.f. of c and d, ^ 1 1. ^ o = — and = —. 7/1 m THE L O IVES T COMMOX MUL TIPL E. Sg 164. In finding the Lowest Common Multiple of two or more 'expressions, each consisting of a single term, we may proceed as in Arithmetic, thus : <1) To hnd the l.c.m. of ^a?x and 18ax'^, 2 4a%, 18ax3 a 2a\ 9acc3 X 2a\ 9x3 2a\ 9x2 L.C.M. = 2 X a X .7- X 2a2 X 9x2 = 36^83*^ (2) To find the l.c.m. of ab, ac, be, a ab, ac, 6c b b, c, be c 1, c, c 1, 1, 1 L.C.M. = a X 6 X c = a6c. (3) To find the l.c.m. of 12«2c, 146c2 and SGoJ*, 2 12a2c, 14&c2, 36a¥ 6 a 6a\ a-c, 7bc% ' nc\ 18a62 ~3a62 h ac, lbc\ 362 c ac. 7c2, 36 a. 7c, 36 L.C.M. = 2 X 6 X a X 6 X c X « X 7c X 36 = ^biaWc^. Examples Find the L.C.M. of I. 4a^x and 6<(-x-. Zxhj and 12.i:y-. -xliv. 4a36 and 8*262. ax, a"x and rt2x2. 2ax, 4ax2 and x^. 6. ab, a-c and 6-'c^ 7. a'^x, a^y and x-y-. 8. blaH^, 34ax3 and ax*. 9. 52)'q, lOq^r and 20pqr. I p. 18ax2, 72ai/2 and 12x?/. 90 THE LOWEST COMMON MULTIPLE. 165. The method of finding tlie l.c.m., given in tlie pre- ceding article, may be extended to the case of compound expressions, when one or more of their factors can be readily determinea. Thus we may take the following Examples : (1) To find the l.c.m. of a-x, a^ — x^, and a^ + ax, a — x, a^ — x^, a^ + ax 1, a + x, a^ + ax a — x a +x 1, 1, a L.C.M. = (« — x){a + x)a = (a^ — .r^) ar=a^ — ax^. (2) To find the l.c.m. of .t^- i, x*-l, and 4x''-4.r:», a;2-l I x^~l,x*-l,4x^-4x* I 1, x^+l, 4x* L.C.M. = (a;2 - 1) (x2 + 1) 43;^ = (x* - 1) 4x* = 4x8 _ 43.4, 166. The student who is familiar with the methods of resolving simple expressions into factors, especially those given la Art. 125, may obtain the L.C.M. of such expressions by a process which may be best explained by the following Ex amples : Ex. 1. To find the l.c.m. of a--x^ and a^-x\ a^ - x^ = (« - x) (a + x), o3 - a;3 = (rt — x) (a^ + ax + x'^ Now the l.c.m. must contain in itself each of the factors in each of these products, and no others. .•. L.C.M. is (a - x) {a + x) (a- + ax + x"^, the factor a-x occurring once in each product, and therefore once onlv in the L.c.Jr. Ex. 2. To find the l.c.m. of a~ — b-, a^ — 2ab + b'', and a^ 2ab + 1 a2-52 = (rt + 6)(a-6), a^-2ab + b^-=(a-b){a-b), a^ + 2ab + b^ = la + b)(a + b); t.C.M. is (a + b){a- h) (a - b) {a + b). THE LOWEST COMMOX MULTIPLE. 91 the factor a — h occurrinrj txoice in one of the products, and a + 6 occurring twice, in another of the products, and therefore each of these factors must occur lunce in the l.c.m. Examples. — xlv. Find the L.C.M. of the following expressions : 1. x^ and ax + x^. 10. x^ - 1, a;^ + 1 and x^ - 1. 2. a.-2 — 1 and a;2 — X. ii. x^-x, x^— 1 and x^ + 1. 3. a-^ — 52 and a^ + aJ, 12. x^- 1, x^-x and x^- 1. 4. 2x-l and 4x--l. 13. 2a + 1, 4a^- 1 and 8a^ + l, 5. a + 6 and a^^-W". 14. x + ?/ and 2x2 + 2x?/. 6. x+ 1, X- 1 and x^— 1. 15. (a + 6)- and a2_52_ 7. x+ l,x^— 1 andx2 + x+ 1. 16. a + 6, a- 6 and a^ — 62_ 8. x+1, x2+l andx^+l. 17. 4(1 +x), 4(1 -x)and 2(1 -x^). 9. X— 1, x^- 1 and x^— 1. 18. x— 1, x- + x + 1 and x^— 1. 19. (a — 6) (a — c) and (a — c) (6 - c). 20. (^x + l)(x + 2), (x + 2)(x + 3) and (x+l)(x + 3). 21. x^ - ?/^, (x + 1/) 2 and (x -ijf- 22. (a + 3) (a + 1), (a + 3) (a - 1) and a^ - 1. 23. '3?{x — 'ijy, x{x^ — y-) and x + y. 24. (x+l)(x+3), (x + 2)(x + 3)(x + 4) and (x + l)(x + 2). 25. x^ — y"^, 2{x — yY and 12 (x^ + i/^). 26. 6(x2 + x?/), ^(xy-y-) and 10(x2-i/2). 167. The chief use of the rule for iinding the l.c.m. is for the reduction of fractions to common denominators, and in the simple examples, which we shall have to put before the student in a subsequent chapter, the rules which we have already given will be found generally sufficient. But as we may have to find the L.C.M. of two or more expressions in which the elementary factors cannot be determined by inspection, we must now pro- ceed to discuss a Rule for finding the l.c.m. of tv, o expressions which is applicable to every case-. 92 THE LOWEST COMMON MULTIPLE. 168. The rule for finding the l.c.m. of two expressions o and h is this. Find d the higliest common factor of a and 6. Then the l.c.m. of a and i = , x h, a b or. = 3 X o. ♦ a In words, the l.c.m. of two expressions is found by the fol- lowing process : Divide one of the expressions by (he h.c.f. and multiply the quotient by the other expression. The result is the L.C.M. The proof of this rule we shall now give. 169. To find the l.c.m. of two algebraical expressions. Let a and h be the two algebraical expressions. Let d be their h.c.f., X the required L.C.M. Now since x is a multiple of a and 6, we may say that X = ma, X = 7i6 ; .". ma = nb ; fii b / J . , --v .-.- = - (Art. 159). n a Now since x is the Loxcest Common Multiple of a and h. m and n can have no common factor ; ;. the fraction ~ must be in its lowest terms ; n :. m = T and n = -, lArt. 163). d d - Hence, since x = ma, b x = -,xa. d Also, since x—nb, " J, x = -,x 0. a THE LOWEST COMMON MULTIPLE. 93 170. Ex. Find the l.c.m. of x2 - 13x + 42 and x^ - 19x-+ 84. First we find the h.c.f. of the two expressions to be x — 7. „, (x2-13x + 42)x(x2-19x + 84) Then l.c.m. = ^ ' -\ '. x-1 Now each of the factors composing the numerator is divisible by X — 7. Divide x- — 13x + 42 by x — 7,'and the quotient is x - 6. Hence l.c.m. = (x - 6) (x^ - 19x + 84) = x^ - 25x2 _,. iggx - 504. Examples. — xlvi. Find the l.c.m. of the following expressions : 1 . X- + 5x + 6 and .x- + 6x + 8. 2. ft' -a-20 and n^ + a- 12. 3. x^ + 3x + 2 and x- + 4x + 3. 4. x2+llx + 30 andx2+12x + 35. 5. x2-9x-22 andx2-13x4-22. 6. 2x2 + 3x + 1 and x^ - x - 2. 7. x^ + x^y + xy + y^ and x* - y*. 8. x^ - 8x + 15 and x- + 2x- 15. 9. 21x2 _ 2Gx + 8 and 7x'' - 4x''* - 21x + 12. 10. x^ + x^y + x?/2 4- y^ and x^ - x-ij + xy^ - y^. 11. a^ + 2a-b - ab^ - 2P and a^ - 2a'^b - ab- + 2¥. 171. To find the l.c.m. of three expressions, denoted by a, b, c, we find m the l.c.m. of a and b, and then find M the L.C.M. of m and c. M is the l.c.m. of a, b and c. The proof of this rule may be thus stated : Every common multiple of a and 6 is a multiple of m, and every multiple of m is a multiple of a and b, therefore every common multiple of m and c is a common multiple of a, b and c, and every common multiple of a, b and c is a common multiple of m and c, and therefore the L.C.M. of m and c is the l.c.m. of a, b and c. 94 OM ADDITION AND SUBTRACTION Examples. — xlvii. Find the l.c.m. of the following expressions : x- - 3,/; + 2, x- - 4x + 3 and a;^ - 5x + 4. x- + 5x + 4, a- + 4x + 3 and x- + 7x+ 12. X- - 9x + 20, x^ - 1 2x + 35 and x^ - 1 Ix + 28. 4. 6x2 - X - 2, 21x2 - 17x4- 2 and 14x2 + 5^ - i. 5. x^ - 1, x- + 2x - 3 and 6x2 _ 3; _ 2. x3 - 27, x2 - 15x + 36 and x^ - 3x2 _ ^x + 6. XL ON ADDITION AND SUBTRACTION OF FRACTIONS. 172. Having established the Rules for finding the Lowest Common Mi;ltiple of given expressions, we may now proceed to treat of the method by which Fractions are combined by the processes of Addition and Subtraction. 173. Two Fractions may be replaced by two equivalent fractions with a Common Denominator by the following rule : Find the l.c.m. of the denominators of the given fractions. Divide the'L.c.M. by the Denominator of each fraction. Multiply the first Numerator by the first Quotient. Multiply the second Numerator by the second Quotient. The two Products Avill be the Numerators of the equivalent fractions whose common denominator is the L.C.M. of the original denominators. The same rule holds for three, four, or more fractions. 174. Ex. 1. Reduce to equivalent fractions with the lowest common denominator, 2x + 5 , 4x-7 -3- and -^. of FRACTIONS. 9$ Denominators 3, 4. Lowest Common Multiple 12. Quotients 4, 3. New Numerators 8a; + 20, 12x-21. 8a; + 20 12x-21 Equivalent Fractions 12 ' 12 Ex. 2. Reduce to equivalent fractions with the loweet common denominator, 56 + 4c 6a -2c 3a -56 a6 ' ac '' be ' Denominators a6, ac, be. Lowest Common ^Multiple abc. Quotients c, b, a. New Numerators 56c + 4c', 6ab - 26c, 3a- - 5a6. „ , ^ „ . 56c + 4c'- 6a6 - 26c 3a- - 5a6 lljqiuvaient 1" Tactions — -, , = , ; . a6c abc abc Examples. — xlviii. % Reduce to equivalent fractions with the lowest common denominator : 3x , 4x ^ a -b , d^-ab 1. —-and -• 6. -— - and r^-. 4 5 a'^6 a6'' 3x-7 , 4x-9 „ 3 ,3 2. — TT- and ^ —. 7. ^ and . 6 18 1-rX 1-z 2x-4y , Sx-Sy „ 2 ,2 3. - ■■ ^ and ^rp-^. 8. , — and , -. •^ Sx'' lOx 1-2/- 1+2/'' 4a + 56 , 3a — 46 5 1 6 4. — r-v- and -- — . 9. and ^ „ -la- oa ^ 1 — X 1 - x'' 4a — 5c 1 3a — 2c a , 6 and ^ ^ ., -. 10. - and 5ac 12a-c ' ' c c(6 + a!)' II. , ,.—,, .and (a-6)(6-c) (a-6)(a-c)' 12. -r^ TT-^ , and a6(a — 6)(a — c) ac (a - c) (6 — c)' o6 ON ADD/TW.V AND SUBTRACTION ,.Tr m 1 *.!, 4. * c ad + bc 175. To sliew that v + ?= — I'l" ■ Suppose the unit to be divided into bd equal parts. 7 Then j-j will represent ad of these parts, 6c Id Now ^ = g, by Art. 148, , c 6c and J = r^. d bd Hence t + -. will represent ad + be of the parts. But — i," will represent aa + be of the pares. bd _, „ a c ad + be .nerefore^ + -^=-^^. By a similar process it may be shewn that a e _ ad — be h~d^~~bdr' ,„„ _,. a c ad + bo 176. Since J 4-^=-^, our Rule for Addition of Fractions will run thus : "Reduce the fractions to equivalent fractions having the Lowest Common Denominator. Then add the Numerators of the equivalent fractions and place the result as the Numerator of a fraction, whose Denominator is the Common Denominator of the equivalent fractions. The fraction will be equal to the sum of the original frac- tions." The beginner should, however, generally take two fractions at a time, and then combine a third with the resulting fraction, as will be shewn in subsequent Examples. . , . a e ad — be Also, since ^-5=--^^-, the Rule for Subtracting one fraction from another will be, OF FRACTIONS. gy •' Reduce the fractions to eijuivalent fractions having the Lowest Common Denominator. Then suhtract the Numerator of the second of the equivalent fractions from the Numerator of the first of the equivalent fractions, and place the result as the Numerator of a fractijjn, Avhose Denominator is the Common Denominator of the equivalent fractions. This fraction will be equal to the difference of the original fractions." These rules we shall illustrate by examples of various degrees of difficulty. Note. When a negative sign precedes a fraction, it is best to place the numerator of that fraction in a bracket, before combin'ing it with the numerators of other fractions. 177. Ex. 1. To simplify 4x - 3|/ 3x + 7|/ _ 5a; - 2)/ 9x + 2y 7 ^ ^ ~I4 2l " "*" ~42~' Lowest Common Multiple of denominators is 42. Multiplying the numerators by 6, 3, 2, 1 respectively, 24x-a % ^ii^l y_ _ lOx-4?/ 9x + i'y ~~ 42 "*" 42 42"~^'''~42~ 24x - 1 8;/ + 9x + 2 1 7/ - ( 1 (\c - 4y) + 9x + 2y ~ 42 _2 4x-18?/ + 9x + 21?/-10x + 47/ + 9x + 2;/ 42 _ 32x 4- 9 ;/ 42~'' » Tv o 'p • ^•r 2x+l 4x + 2 1 Ex. 2. To simplify — h =. ^ 3x ox 7 Lowest Common Multiple of denominators is 105x. Multiplying the numerators by 35, 21, 15x, respectively. 70.C + 35 _ 84x + 42 1 5x lU5x l()5x 105x _ 70x 4- 35 - (84x + 42) + 15a; ~ lOSx 70x4-35-84x-42-H5x x-7 ^ 105x ~T05x' 98 ON ADDITION AND SUBTRACTION Examples.— xlix. 4x + 7 3a; -4 3a -46 la-h + c 13a -4c I. ^— +— ,-v— • 2. ^ + .,— . 5 15 / .J 12 4x - 3i/ 3a; + 7i/ bx — 2?/ 9x + ly 3- ~y~"'^~l4 21^'^~42~^" 3a;-2i/ 5x-7y 8x + 2y ^' ~5x '*' 10a; "^ ~ 25"* cv 4a;2 - 7i/2 3a; - 81/ 5-2 j/ 5' "3x2" + ^6x~ "'' ~~[2~' . 4a2 + 56_2 3a^26 7^2a W^^ 56 "^ 9 • 4x + 5 3x - 7 9 ^' ""3 5x~''"l2x2" 5a + 26 _ 4c - 36 6a6 - 76c 3c 2a 14ac 2a + 5c 4oc - Zc^ bac - 2c^ 3X1/-4 51/2 + 7 6x2-11 10. 5—5 ' 6 T - \ jc^-' xy* x-^y a - 6 4a - 56 3a - 76 a?h a^bc b^c^ 178. Ex. To simplify a-6 a+b a + b a-b' L.C.M . of denominators is a2 - 62. Multiplying the numerators by a-6 anri a +6 respectively, we get a'^ - 2ab + b- a- + -lab + 6^ a2-62 "^""^a— ftl _a2- 2a6 + 62 + a^ + 2ab + b- a2 - 62 _2a2^+^62 ~ fi2_j2-- or FRACTIONS. 99 EXAMPLES.- -1. 1 1 a; - 6 a; + 5' 1 1 ^' x-7 x-3' 3- 1 1 1 +X l-X ?±1 _ ?J"_1 x-y x + y' 1 2 5' l-X 1-X2* 6. a (ad - be) x c c{c + dx) XX o 7. + . 8. X+y x-y 1 X- — + X (x-yy 2 3a j^ "* x + a {x + ay 2a 1 + 1 . (a + x) 2a (a - a) 179. Ex. 1. To simplify 3 5 6 1 + y 1-1/ 1 + 1/2" Taking the first two fractions 3 5 1+y l~y 1 3-3?/ b + by ~1-/ ' l-t/2 8 + 2y ^ -1-7/' we can now combine with this result the third of the original fractions, and we have 3 5 1 + 7/^1-7/ 1 6 - + 7/2 8 + 27/ 6 1-7/2 1+7/2 ^ 8 + 27/ + 87/2 ^27/3 _ 6 - 6?/2 ~ ' 1-jr* "l^p _ 8 + 27/ + 87 /2 + 27/ 3 - 6 + 6y2 1^7/* _ 27/3+147/2-|.27/ + 2 1-y* too ON ADDITION AND SUBTRACTION Ex. 2. To simplify 2 2 2 (a-6)(6-c)"'"(a-6)(c-a)^(6-c)(c-a)' I..C.M. of first two (lenominatoi-s being (a - 6) (6 - c) (c - a) _ 2c -2a '26 - 2c 2 ~ (a - 6y (6 - cH^ a) ^ (a-b) (6-"^H7-T) ■*■ (6 - c) (c - a) 26-2CT _ 2 ~ (^Ujlh~=^cj~{^^ ^ (6^ r-) (c - «)• L.C.M. of the two denominators being (a -h) {b- c) (c - a) 26-2ffl + 2a-26 {a -b)(b- c) (c -a) (a- b) (6 - c) (c - a) =0. Examples.— li. 1 Jl_ 2a _J^ 1 26 4¥ '• H-ft''"l-a''"l-a2" "^^ «-6 a + b a'^ + b^ a*-^b*' 1 1 2a; X y x^ 1-x 1 + x l+X' y x + y x' + xy X x^ X ^ x + Z x-4 x + 5 ^ x-l x-1 x-3 a;-2 x-3 x-4 8 -^ _^_ _ 5a^ X - a (x - a)"-^ (x - a)3* 1 1 3 ^' x-l x + 2 (x+l)(x + 2)' 1 3 lo. (r + 1 ) (x + 2) (x + 1) (x + 2) (x + 3)* X X - 1 "*'x^'T'^x+r 1 1 £2. (d + c) (a + d) (tt + c) (a +• e)' a-6 6-c c-a '^" (ftTcncTo) (c + a) (aT6) (a + 6) (fTc)* OF FRACTIONS. X — a x — b {a - b)' x-b x-a (x - a){x-b)' x + y 2:c x-y-x^ '■ y x + y y{-''--y')' 1 6. 17- a + 6 6 + c c + ii (6 -c){c- a) (c -a) (a- b) (a — b){b- c)' x 2xy x^ + xy + y^ x^ — y^' i8 2,2^2 ^( a-br- + {b-cy + {c-af ^ a-b b-c c — a {a — b){b — c){c-a) a + b 2a a-b - a^ '9- 'b^~^+b^a^b^¥' 1 1 1 ^°' {n+l){n + 2) ()i+l)(?H-2)(?i + 3) (?i + l)(7i + 3)* a^ — be b- - ac c^ — ab 2 1 1 + (a + 6)(a + c) {b + a){b-\-c) {c + b){c + a) „ .-,• O'b -, — ab . ►,« 180. bmce X~"' ^ ~T'~^' ^'^' ' ab _—ab 'b~~^T' Fioiu this we learn that we may change the sign of the (lenoiiiinat(ir of a traction it we also change the sign of the numerator. Hence if the numerator or denominator, or both, be expres- sions Avith more than one term, we may change ibe sign of every term in the denominator ii we also change the sign of every term in the numerator „ a — b —(a-b) c-d -{c-a) ~ - r+i ' or, writing the terms of the new i'raction so that the positive terms may stand first, _b — a d-c' I02 ON ADDITION- AND SCB TRACTION 181. t-X. To simplify — ^-- . Changing the signs of the numerator and tleiiominator of tlu^ second fraction, X (a + x) - bax + x- a-x a—x _a!}c + x^-( — 5ax + x-) ax + x^ + bax -x^ _ 6ax a — x a — x a-x' 182. Again, since —ab= the product of -a and h, and «6= tlie proiUict of +a and h, the sign of a product will he cliani,'ed hy changing the sic;ns of one of the factors composing the product. Hence (a — b)(b- c) will giv. a set of terms, . and (6 -a) {b- c) will give the same set of terms icith dif- ferent signs ' This may be seen hy actual multiplication : (a - h) {b -c) — (ib-ac-b^ + he, (b- a) (b - c)= — ah + ac + Ir — he. Consequently if we have a fraction 1^ (a - h) {h - c)' and we change the factor n-h into h-a, Ave shall in effect change the sign of every term of the expression which Avould result from the multiplication of (a - b) into {b - c). Now we may change the signs of the denominator if we also change the signs of the numerator (Art. 180) ; 1 -J " (a - 6) (6 - c) ~ (b-a) [b- c)' If we change the signs of two factors in a ilenominator. the sign of the numerator will remain unaltered, thus 1 1 ia-b)(b-c)~(h-n)(c'-by I I OF FRA C TICNS. io3 183. £x. Simplify 1 1 (a-6)(6-c) {h-a){a-c) (c-a){c-b)' First change the signs of the factor (b-a) in the second fraction, changing also the sign of the numerator ; and change the sigTis of the factor (c - a) in the third fraction, changing also the sign of the numerator, , . 1 -1 -1 the result is , rr-r, r + ; Tx / \ ~ 7 w i\- (a -b) {b- c) (a - o) (a - c) [a -c) [c- b) Next, change the signs of the factor (c - b) in the third, changing also the sign of the numerator, .11.- 1 - 1 1 the result is ^ tttt ^ + -. jv-? ; - 7 ttt — ;. (a -b) [0- c) [a -o) {a- c) {a - c) (6 - c) L.c.M. of the three denominators is (a -b) {b- c) {a - c), _ a-c -b+c a-b ~{a-b){b- c) (a - c) (a- b) {a -c) (b- c) (a -b)(a- c) (6 - c) a-c-b->rc- {a -b) (a -b)(b- c) {a - c) (a - h) (6 -c){fl- c) " .0. Examples.— lii. X x-y 3 + 2x2 — 3a; 16a! — a;^ ' x-y y-x' ' 2 — x 2-f-x x^ — 4' jc x x^ 114 ^- x+l~l-x'''x2-l* ^' 61/ + 6 ~ 2?/ - 2 "•■ 3^7?" 5- 1 2 1 (m-2)(m-3) (m-l)(3-m) (m - 1) (to - 2)" ^- (a-6)(x + 6)"*"(6-a)(x + a)* ''" a2-62 a^ - fe3 "•" a^ + F 1 1 1 °' 4(l+x) 4(x-l) 2(l + x2)' 1 ^ 1 • , 1 ' 1 _-.__„ 1 a(a-b)(a — c) b{b-a)ib-c) cic-a)(c-b)' I04 ADDITION AND SUBTRACTION OF FRACTIONS. 184. Ex. To simplify 1 1 a;2-llx + 30 a;--12x + 35' Here the denominators may Le exprcssi.il in lacinrs, and "Wo have 1 1 (x — 5)(x — G) (re — 5) (x — 7)" The L.C.M. of the denominators is (x — 5) (x — 6) (x — 7), and we have X— 7 X— 6 + - (a;-5)(x-6)(x-7) (x-5) (x-G) (x-7) _ 2x-13 "~(x-5)(x-6)(x-7)' Examples. — liii. 1 : + -^ x''' + 9x + 20 x2 + 12x + 35* 1 1 x2-13x + 42 x2^15x + 54' , __i + i ^' x2 + 7x-44 x2-2x-143" 1 2x 1 ^' x2 + 3x + 2'*'x- + 4x + 3"^x^ + 5x + 6* m 2ni 2?7),?i 15. — h — ^ n m + n {m + nf 1+x 1-x 2 l+x + x^ 1— x + x- 1+x^H-x** 5 2 7x 7x ^* 3^( 1^^) ~ T+x "^ 3x2T3 " 3x- - 3' 1 1 J j 8(x-l)^4(3-x) 8(x-5) (l-x)(x-3)(x-5) X* Q. 1 - X + X- - X^ + - — — . ^ 1+a; XII. ON FRACTIONAL EQUATIONS. 185. We shall explain in this Chapter the method of solving, first, Equations in which fractional terms occur, and secondly, Problems leading to such Equations. 186. An Equation involving fractional terms may be reduced to an equivalent Equation without fractions by mul- tiplying every term of the equation by the Lowest Common Multiple of the denominators of the fractional terw^. This process is in accordance with the principle laid down in Ax. III. page 58 ; for if both sides of an equation be multi- plied by the same expression, the resulting products will, by that Axiom, be equal to each other. 187. The following examples will illustrate the process of clearing an Equation of Fractions. EX. 1. 1 + ^.8. The L.c.M. of the denominators is 6. Multiplying both sides by 6, we get 6.5 6x ,„ T+6=^^' or, 3x + x = 48, or. 4x = 48; .-. cc = ]2. Ex. 2. X x + \ _ 2 + -7-"-'- The L.c.M. of tie denominators is 14. Multiplying both sides by 14, we get 14x 14X + 14 ,^ -— + 14x-2a^ (o6 ON FRACTIONAL EQUATrON"^. or, • .7a; + 2a; + 2 = 14x-28, or, 7a; + 2x-14x= -28-2, or, -5x=-30. Changing the signs of both sides, we get 5x = 30; .-. a; = 6. 188. The process may be shortened from tlie foUowin.i,' considerations. If we have to multiply a fraclion by a multi])le of its denominator, we may first divide the multiplier by the denominator, and then multiply the numerator by the quotient. The result will be a whole number. Thus, - X 12 = a;x 4 = 4x, ^^x56=(x-l)x8 = 8x-8. EX. 1. M + 1-39. The L.c.M. of the denominators being 12, if we multiply the numerators of the fractions by 6, 4, and 3 respectively, and the other side of the equation by 12, we get 6x + 4x + 3x = 468, or, 13x = 468; ,-. x = 36. Ex, 2. §-^ + ^ = ^. " X 2x 3x 12' The L.c.M. of the denominators is 12x. Hence, if Ave mul- tiply the numerators by 12, 6, 4, and x respectively, we get 96 -90 + 28= 17x, or, 34=17x, or, 17x = 34; /. x=2. O.y FRA C no A' A L EQ UA TIONS. 107 EXAMPLES.— liv. ,. 1 = 8. . f = ». , ,M = 8. X X „ .-,„ 4x ^ , 2a; 176 -4a; 4. --- = 3. 5- 36--g- = 8. 6. -=-^— 2a: , 7x ^ a; + 2 X - 1 a- - 2 o 2x 4.7; a; x_ 3 X ^- "3" + ^^- 5+^- '^- 2 + 3-^4-4- 3x , 5x ^ x + 9 2x 3x-6 „ 9. — + 5 = — + 2. 19. -—4-^ = —^ — + 3. '^4 6 ^4(0 7x , 9x „ 17 -3x 29-llx 28x+14 ,0. _-5 = ---8. 20. _^- - = _-^_+_^_ 5x _ ^. 7a; 2x-10 . II. -r--8 = 74-T7i. • 21. — = — = 0. 9 12* " 7 X 6 X , ,, X 3x + 4 4x-51 12. ,- — 4 = 24--. 22. 7 + 47 - = u 23- ^-3 = -l-L X X 24. 12+x ^ 6 X X 25. 1 1 1 4"+10^ + 20^ = = 40. 13. 56-| = 48-^|. 3x 180 -5x _- ^4- -4- + — 6— = '^- 3x , , X - 8 ^5- T-^^ = -2- , X X X 13 , -1 3-x _5 ._,! '6- 2 + 3 + 4=12- ^6- 24^ + -2-=V-% .-,3 31 325 ^'^^ ""4 x~x 100' ^ „1 18 -X ,1 1 3-2x 2 ^^- 22 + -3-=¥ + 3 + -10- + 5- X X 5x _- ,2 ^„ ^9- 3+4-6-- 12= ¥-^^- 7x+2 ,- 3x 3x + 13 17x 3°- -10—12-^=-^-^. 108 ON FRA CTION.A L EQUA T!Oy<!. 189. it must next be observed tbat in clearing an equation !)[■ fractions, whenever a fraction is precedeii by a negative sign, u e must place the result obtained by multiplying that nume- : ilor in a hraclcet, alter the removal of the denominator. For example, we ought to proceed thus : — Ex. 1. ^±! = ^^Z^_^ 5 2 7 Multiply by 70, the l.c.m. of the denonii:)ators, and we get I4x + 28 = 35.7; - 70 - (10.c - 10), or 14.c + 28 = 35a;-70-10x+10, I'.om which we shall find a; = 8. Ex.2. 12-2^_.4J? + 2^1. bx 'ix Multiplying by 15a;, the L.C.M. of the denominators, we get 51-6x-(20a; + 10) = 15x, or 51-6jj-20x-10 = 15x, from which we shall find a;= 1. Note. It is from want of attention to this way of treating fractions preceded by a negative sign that beginners make so many mistakes in the solution of equations. Examples.— Iv. x + 2 „, 5x 5a; 9 3-x 1. 5x— 2- = 71. 4- T-T = 4— 2-- 3-x ,2 „ 5x-4 „ l-2x ^- ^—3- = '3- 5- 2..- -—=.—— . 5-2.7; „ 6x-8 , x + 2 14 3 4 5x 3. -— - + 2 = x —. 6. -2- = -9- ^ . 5x + 3 3 - 4x x_31 9-5x 7' ~8~~ 3 ^l~^ "~6~" „x + 5x-2.(; + 9 „x + 2x ^- -l—~^-^U- '°- ^-='— 8- = 3 x+1 x-4_x + 4 x + 5_x + -2 X — 2 GN FRACTIOXAL EQUATIONS. tog x + 2 x-2 'a;-! ^ 2x x + 3 ., _. ID. -i; ^— =3x-21. 5 2 7' 7 +-7_a 11 x + 9 3x-6 „ 2x 2i; + 7 9x - 8 x-11 H- — , r- = 3--. 17. ^ 7x-31 8 + 15x_7x-8 ■ 4 26~~~22~" 8x-15 llx-1 7x + 2 19. 3 7 13 ■ 7x + 9 3x+l_9x-13 249-9X- 20. ~^- 7— -4 T4- • X -^ XX X 10 -X -„3 190. Literal e(|nations are those in wliicii known quantilifs are represented by letters, usually the first in the alphabet. The following are examples : — Elx. 1. To solve the equation ax + bc = bx + ac. that is, ox -bx — ac- be, or, {a-b)x = {a — b)c. therefore, x = c. Ex. 2. To solve the equation a-x + bx- c = h-x + cx- d, that is. (i^x + bx- b'-x -cx = c-d, or. (a^ + b- b- - c)x = c — d, therefoi-e, c-d " a^ + b-b'-c Examples.— Ivi. 1. ax+bx = c. 4. dm - ox = bc- ox. 2. 2a — ex = 3c — 56x. 5. abc- a-x = ax — a-b. 3. bc + ax — d = a^b-fx. 6. 3acx — 6bcd=l2cdx + abc. ox FRACTIOXAL EQUATIONS. 7. A;- + ?><xckx 4- 3A; = ^x + Zahh — li^ - ackx. 8. — ac^ + b'^c + obex = abc + cmx — ac-x + h-c — mc. 9. {a + X + b) (a + b - x) = {a + x) {b — x) — ab.* 10. (a — x){a + x) = 2a^ + 2ax — x'. 11. (a2 + a;)2 = x2 + 4a2 + a*. 1 2. (o" - a;) (a- + x) = a* + 2ax — x-. ax-b x + ac m (p-x + x^) mx^ 13. l-a = . 17. — ^ ' = mqx-\ . •^ c c px p 3a-bx 1 „ X , c TA. ax 7; — = ^. lo. — o = -, — x. ^ 2 z ad Aax - 26 x'^ — aa-x2xa 15. 6a 3— = x. 19. -fc^— 2,- = y--- , 6x4-1 a{x^-\) 3 ab-x- 4x-ac 16. ax = ~^ -. 20. , = . XX c ox ex ab + x b' -x x — b ab-x 22. ¥ a'^b a^ 6- 3ax — 2b ax — a ax 2 36 26 , ax 05 - 2"?. am — 6 — -^H = 0. ■^ 6 m ^a263^ 62^ 3a2c- _ 3acx _ ^-^^ab^x (a + 6) a(a4-6) a + 6~ 6 (a + 6) " ax^ ax ^ a6 , , 1 25. r + a + — = 0. 27. — = 6c + rf+ . o~cx C X X , a(d- + x-) ax _ m(a — x) 26. ^— J ^ = ac + -j-. 28. c = a+~' -^. ax a 3a + x 29. (a + x) (6 + x) - ffl (6 + c) = ^ + x'i ace (a + by.x , „, 30. — T-— ^^ ox = ae-36x. - d a 191. In the examples already given the L.C.M. of the denominators can 'generally In- deterniined by inspection. When compound expressions appear in the denominators, it is sometimes desirable to collect the fractions into two, one ON FRA CTTONA L EQUA TIONS 1 1 1 on each side of the eqiiution. When tliis has been done, we can clear the equation of fractions liy multiplying the nu- merator on the hft by the denominator on the right, and the numerator on the right by the denominator on the left, and making the produ ts equal. For, if ^ = -j, it is evident that ad = bc. ' a F 4x + 5_13x-6_2.'c-3_ 10 7x-l-4 ~ 5 ' 4a: + 5 2x-3_13a:-6^ ■■ To 5~~ 7.C + 4 ' 4x + 5 - (4a; - 6) _ 1 3x - 6 ^ ' 10 7x + 4 ' ll_13x-6^ •■ 10~ 7x + 4 ' .-. ll(7x + 4) = 10(l;ix-6); whence we find a; = -— r-. 06 Examples.— Ivii. 3x + 7 3X-1-5 ,2 5 ^ 4X-I-5 4x + 3" x-l-6 X 2x + 5 2x- 5' 2x + 7 4x- 1 x + 2 2x- 1' 5x-l 5x- 3 2x + 3 2x- 3* 1 2 4-^ — 1 - 5x 1 - 2x 1 1 3 7- —- + 8 x-1 x+1 X--1' 4x + 3 8x+19 7x-29 9 18 5x-12" X .'■- - ox _ 2 3 ox — 7 3* 3x + 2 2x - 4 ^ '' 3^^-^^-"3="' '"• ^:rr+^T2-=^- II. l(x + 3)-^(ll-x) = |(x-4)-l(x-3). (x+_lM2x + 2)_.^_ J, x+_l_^l_ (x-3)(x + 6) " ■ ^ x + 1 x-l~l-x2- (2x + 3)x 1 , 2 8 4.5 •^ 2.(- + i 3x ^ 1 - X 1 + X 1 - x^ ON FRA C TIONA L EQUA TIONS. , 4_ _3^ iA = _.? ■ x-8''2j;-16 24 3a;-24' :c*-(4a;2-20a; + 24) , „ a;2 - 2.C + 4 „ 2rc* + 2x3 -23x2 + 31a; 18. ., —r ; = 2X'' - 4x - 3. X- + 3x - 4 (A 2\ 1 3x-(4-5x) 192. Equations into which Decimal Fractions enter do not present any serious difficulty, as may be seen from the follow- ing Examples : — Elx. 1. To solve, the equation •5x = -03x + l-41. Turning the decimals into the form of Vulgar Fractious, we get 5x_2x_ 141 10 ~ loo "^ loo* Then multiplying both sides by 100, we get 50x = 3x+141; therefore 47x=141; therefore x = 3. Ex. 2. 1 •2x - i^^^ = -ix + 8-9. ■5 First clear the fraction of decimals by multiplying its numerator and denominator by 100, and we get i-2x-— |=^ = -4x + 8-9; DO ^, , 12x 18x-5 4x 89 therefore -^-^y ___ = _ + _; therefore 60x - 1 8x + 5 = 20x + 445 ; therefore 22x = 440; therefon- x = 20. ON' FRA CTIONAL EQ UA TIOMS. 1 1 3 Examples. — Iviii. 1. •5.-c-2 = -25x + -2x-l. 2. 3-25X-5-1 H-x — ■75x = 3-9 4-"5x. 3. •125x + -01x=13--2x-+-4. 4. -S.c + 1 -SOS.t + -Sx = 22 -95 - • 1 95x. 5. •2x--01x + -005x=ll-7. 6. 2-4x-:^^^;:-^^ = -8x + 8-9. 7. 2-4x- 10-75 = -25x. 8. •5x + 2- •75a;=-4.r.- 11. 9. ^^ + 3-875 = 4-025. 10. 2-5x ^=^{ i~^)~"^ 2 + .'.•/! ^\ . 5x4-3 8 "' 8-5 -2 J 1--1X -48x 3-4x .„__ "• Y-x=^4 -.^' ■ ''■ ^—^-=^^^3. 2-3x 5x _2x-3_x-2 ^^7 14. ?i^ + --. ■04(x + -9) = 24I-2. •45X--75 1-2 -Sx-'e 15...5X + ___ = _-___. , , 3-5x 24 -3x .^, 16. -5 ^ „ =-3/5x. X — 2 8 •135X--225 -36 -09x--18 17. •15X + = - ^^. 193. To shew that a simple equation can only have one root. Let x = a be the equation, a form to which all equations of the first degree may be reduced. Now suppose a and /3 to be two roots of the equation. Then, by Art. 109, a = a, and /? = «, , therefore a = P\ in other Avords, the two supposed roots are identicaL XS.A.1 H XIII. PROBLEMS IN FRACTIONAL EQUATIONS. 194. We shall now give a series of Easy Problems resulting for the most part in Fractional Equations. Take the following as an example of the form in which such Protjlems should be set out by a beginner. "Find a number such tliat the sum of its third and fourth parts shall be equal to 7." Suppose X to represent the number. Then - will represent the third part of the number, o and - will represent the fourth part of the number. X X Hence ^ + t "^^'i^^ represent the sum of the two parts. But 7 will represent the sum of the two parts. Therefore ^ "^ 4 ^ "^^ Hence 4a; + 3x = 84, that is, 7x = 84, that is, a; =12, and therefore the number sought is 12. Examples. — lix. 1. What is the number of which the half, the fourth, and the tilth jiaits added together give as a result 95 ? 2. ^^"llat is the number of which the twelfth, twentieth, and fortieth parts Ridded together give as a result 38 ? 3. What is the number of which the fourth part exceeds the Ulth part by 4 1 PR OR f. EMS IX FRACIIOXAL EQUATIONS. 115 4. Wiiat is the iniiuber of wliicli the twenty-fifth part exceeds the thirty-tifth jjart by 8 \ 5. Divide GO into two such parts that a seventh part of one may be ec^ual to an eighth part of the other. 6. Divide 50 into two such parts that one-fourth of one parr being added to five-sixths of the other part the sum may be 40. 7. Divide KK) into two such parts that if a tliird part of tlie one be subtracted from a fourth part of the otlier the remainder may be 11. 8. \Yhat is the number which is greater than the sum of its third, tenth, and twelfth parts by 58 ? 9. "When I have taken away from 33 the fourth, fifth, and tenth pai'ts of a certain number, the remainder is zero. Wliat is the number ? 10. What is the number of which the fourth, fifth, and sixth parts added together exceed the lialf of the number by 112? 11. If to the sum of the half, the third, the fourth, and the twelfth parts of a certain number I add 30, the sum is twice as large as the original number. Find the number. 12. The difference between two numbers is 8, and the quotient resulting from the division of the greater by the less is 3. What are the numbers ? 1 3. The seventh part of a man's property is equal to his whole property diminished by £1626. What is his property ? 14. The difference between two numbers is 504, and the quotient resulting from the division of the greater by the less is 15. What are the numbers ? 15. The sum of two numbers is 5760, and their difference is equal to one-third of the greater. What are the numbers ? 16. To a certain number I add its half, and the result is as much above 60 as llie number itself is below 65. Find the number. ii6 PROBLEMS IN FRACTIONAL EQUATIONS. 17. The difference between two numbers is 20, and one- seventb of the one is equal to oue-third of the other. What are the numbers ? 18. The sum of two muubers is 31207. On dividing one by the other the (^uolient is fouud to be 15 and the remainder 1335. What are tlie numbers ? 19. Tlie ages of two brothers amount to 27 yeai-s. On dividing the age of the elder by that of ihe younger the quo- tient is 3i. What is the age of each ? 20. Divide 237 into two sucb parts that one is four-fifths of the other. 21. Divide £1800 between A and B, so that 5's share may be two-sevenths of ^'s share. 22. Divide 46 into two such parts that the sum of the quotients obtained by dividing one part by 7 and the other by 3 may be equal to 10. 23. Divide the number a into two such ]iarts that the sum of the quotients obtained bv dividing one part by 7?i and the other by n may 1*6 equal to h. 24. The sum of two numbers is a, and their difference is h. Find the numbers. 25. On multiplying a certain number by 4 and dividing the product by 3, I obtain 24. Wiiat is the number ? 5 26. Divide £864 between A, B, and G, so that A gets — of what B gets, and C"s share is equal to the sum of the shares of A and B. 27. A man leaves the half of his property to his wife, a sixth part to each of his two children, a twelfth part to his brotlier, and the rest, amounting to £600, to charitable uses. What was the amount of his property ? 28. Find two numbers, of which the sum is 70, such that the first divided by the second gives 2 as a quotient and 1 as a remainder. 29. Find two niimbers of Avliich the difference is 25, such that the second divided by tlie tiist ^jives 4 as a quotient and 4 as a renjainde:. PROBLEMS IN FA' ACTIONAL EQUATIONS. W; 30. Divide tlie number ^08 into two parts snch that the sum of tlie fourtli of the tjreater and the tliird of the less is less bv 4 tliau four times the difference between the two parts. 31. There are thirteen days between division of term and the end of the first two-thirds of the term. How many days are there in the term ? 32. Out of a cask of wine of which a fifth part had leaked away 10 i^allons were drawn, and then the cask was two-thirds full. How much did it hold 1 '^3. The sum of the ages of a f;ither and son is half what it will be in 25 years : the difference is one-third what the sum will be in 20 years. Find the respective a.ges. 34. A mother is 70 years old, her daughter is e.xactly half that age. How many years have passed since the mother was 3J times the age of the daughter ? 35. A is 72. and B is two-thirds of that age. How long is it since A was 5 times as old as B ? Note I. If a man can do a ]iiece of work in x hours, the part of the work which he can do in one hour will be repre- sented by -. •' X Thus if A can reap a field in 12 hours, he will reap in one hour — of the field. Ex. A can do a piece of work in 5 days, and B can do it 12 days. I do the work ] in 12 days. How long will A ami B working together take to Let X represent the number of days A and B will take. Then - will represent the part of the work they do daily Now - represents the part A does daily, and Yg represents the pai-t B does daily. irS J'ROBLEMS IN FRACTIONAL EQUATIONS. Hence - + -- will represent the part A and B do daily. .1 1 1 1 Consequently ^4-^^ = -. Hence 12x + 5x = 60, or 17x = 60; 60 •■• ^ = 17- 9 That is, they will do the work in 3r— days. 36. A can do a piece of work in 2 days. B can do it in 3 days. In what time will they do it if they work together ? 37. A can do a piece of work in 50 days, B in 60 days, and G in 75 days. In what time will they do it all working together ] 38. A and B together finish a work in 12 days ; A and G in 15 days ; B and G in 20 days. In what time will they finish it all working together ? 39. A and B can do a piece of work in 4 hours ; A and G in 3- hours ; B and C in 5= hours. In what time can A do it alone ? , 40. A can do a piece of work in 2;^ days, B in 3.^ days, and G ill ?> days. In what time will they do it all working together ? 41. A does - of a piece of work in 10 days. He then calls in B, and they finish the work in 3 days. How long would B take to do one-third of the work l>y liimself ? Note II. If a tiip can fill a vessel in x hours, the part of the vessi'l llllcd 1>\ it in om- Innir will be represented by . Ex, Three taps running separately will fill a vessel in 20, 30, and 40 minutes respectively. In what time will they fill it when thev all run at the same time \ PROBLEMS IN FRACTIONAL EQUATIONS. 119 Let X represent the number of minutes they will take. Then - will represent the part of the vessel filled in > minute. Now - represents the part filled by the first tap in 1 minute, 1 30 J_ 40 second . third.. 1111 Hence 20 + 30 + 40 = ? or, multiplying both sides by 120ic, 6a; + 4x + 3.x = 120, that is, 13a; = 120; 120 ••• ^=-iy 3 Hence they will take 9 ^^ minutes to fill the vessel. 42. A vessel can be filled by two pipes, runnincr separately, in 3 hours and 4 hours respectively. In what time will it be filled when both run at the same time ? 43. A vessel may be filled by three different pipes : by the first in I5 hours, by the second in 3- hours, and by the third iu 5 hours. In what time will the vessel be filled when all three pipes are opened at once ? 4i|. A bath is filled by a pipe in 40 minutes. It is emptied by a waste-pipe in an hour. In what time will the bath be full if both pipes are opened at once ? 45. If three pipes fill a vessel in a, 6, c minutes running separately, in wliat time will the vessel be filled when all three are opened at once ? I20 PROBLEMS IN FRACTIONAL EQUATIONS. 46. A vessel containing 755 "allons can be filled bv three pipes. The first let<? in 12 gallons in Z- minutes, the second 15- gallons in 2r minutes, tlie third 17 gallons in 3 minutes : in what time will the vessel })e filled by the three pipes all running together? 47. A vessel can be filled in 15 minutes by three pipes, one of which lets in 10 gallons more and the other 4 gallons less than the third each iiiiniite. The cistern holds 2400 gallons. How much comes throug'n each pipe in a minute ? Note III. In questions involving distance travelled over in a certain time at a certain rate, it is to be observed that Distance ^t^. — .is = linie. Rate That is, if I travel 20 miles at the rate of 5 miles an hour, number of hours I take = -^. 5 Ex. A and B set out, one from Newmarket and the other from Cambridge, at the same time. The distance between the towns is 13 miles. A walks 4 miles an hour, and B 3 miles an hour. Where will they meet ? Let X represent their distance from Cambridge when they nu^et. Then 13 -a: will represent their distance from Newmarket. X Then - = time in hours that B has been walking. 13- 4 X A And since both have been walking the same time, X 13- - X 3" 4 » or 4.x = = 39- -3x, or 7x = .'. x = = 39; 39 " 7' PROBLEMS IN FRACTIONAL EQUATIONS. 121 4 That is, they meet at a distance of 5- miles from Cam- bridge. 48. A person starts from Ely to walk to Cambridge (wliich 4 is distant 16 miles) at the rate of 4- miles an hour, at the y same time that another person leaves Cambridge for Ely walking at the rate of a mile in 18 minutes. Where will they meet ? 49. A person walked to the top of a mountain at the rate of 2- miles an hour, and down the same way at the rate of o 3^ miles an hour, and was out 5 hours. How far did he walk altogether ? 50. A man walks a miles in 6 hours. "Write down (1) The number of miles he will walk in c hours. (2) The number of hours he will be walking d, miles. 51. A steamer which started from a certain place is fol- lowed after 2 days by another steamer on the same line. The first goes 244 miles a day, and the second 286 miles a day. In how many days will the second overtake the first ? 52. A messenger who goes 31 ^ miles in 5 hours is followed after 8 hours by another who goes 22- miles in 3 hours. When will the second overtake the first ? 53. Two men set out to walk, one from Cambridge to London, the other from London to Cambridge, a distance of 60 miles. The Ibrmer walks at the rate of 4 miles, the latter 3 at the rate of 3- miles an hour. At what distance from Cam- 4 bridge will they meet ? 54. A sets out and travels at the rate of 7 miles in 5 hours. Eight hours afterwards B sets out frrnu the same place, and travels along the same road at the rate of 5 miles in 3 hours After what time will B overtake A. ? 122 PROBLEMS TN FRACTIONAL EQUATIONS. Note IV. In problems relatincj to clocks the chief point to be noticed is that the minute-hand moves 12 times as i'ast as the hour-hand. The following examples should be carefully studied. Find the time between 3 and 4 o'clock when the hands of a clock are (1) Opposite to each other. (2) At right angles to each other. (3^ Coincident. «g3 (1) Let ON represent the position of the rainiite-hand in Fig. I. OD represents the position of the hoiu-luind in Fig. I. M marks the 12 o'clock point. T 3 o'clock The lines OM, OT represent the position of the hands at 3 o'clock. Now suppose the time to be x minutes past 3, Then the minute-hand has since 3 o'clock moved over the urc MDN. And the hour-hand has since 3 o'clock moved over the arc TD. Hence arc MDN= tvelve times arc TJX If then we represent MDN by x, we shall represent TD by . Also we shall represent MT by 15, and DX in- 30. PROBLEMS IN FRACTIOiVAL EQUATIONS. T?3 Now MDN = MT ^TD-\- UN, that is, x=15 + — +30, or 12a; = 180 + x + 360 or llx = 540; 540 .•.x=— . Hence the time is 49-- niimites past 3. (2) In Fig. II. the description given of the state of the clock in Fig. I. applies, except that DN will he represented hy 15 instead of 30. Now suppose the time to he x minutes past 3. Then since MDN= MT+TD + DN, x=15 + ^ + 15. from which we get 360 8 ^ that is, the time is 32— minutes past 3. (3) In Fig. III. the hands are both in the position ON. Now suppose the time to be x minutes past 3. Then since MN=MT+TN, IK ^ ^=15 + ^2, or 12x=180 + x, 180 or x = --, 4 that is, the time is 16 — minutes past 3. 55. At what time are the hands of a watch opposite to each other, (1) Between 1 and 2, (2) Between 4 and 5, (3) Between 8 and 9 ] 124 PROBLEMS IX FRACTIONAL EQUATIO.VS. 56. At what time are the hands of a vatch at light angles to each other, (1) Between 2 and 3. (2) Between 4 and 5, (3) Between 7 and 8 \ 57. At what time are the liands of a watch together, (1) Between 3 and 4, (2) Between 6 and 7, (3) Between 9 and 10 ? 58. A person buys a certain number of apples at the rate of five for twopence. He sells half of them at two a j)enny, and the remaining half at three a penny, and clears a penny by the transaction. How many does he buy ? 59. A man gives away half a sovereign more than half as many sovereigns as he has : and again half a sovereign more than half the sovereigns then remaining to him, and now has notliing left. How much hud he at first ? 60. ^Miat must be the value of 71 in order that may be equal to -— wlien a is - ? 3u + 69a 61. A body of troops retreating before the enemy, from which it is at a certain time 25 miles distant, marches 18 miles a day. The enemy jairsues it at the rate of 23 miles a day, but is fiist a day later in starting, then after 2 days is forced to halt for one day to repair a bridge, and this they have to do again after two days' more marching. After how many days from the beginning of the retreat will the retreating force be overtaken ? 62. A person, after ]iaying an income-tax of sixpence in the pound, gave away one-tliirteentli of his remaining income, and had .£540 left. What was his original income ? 63. From a sum of money I take away £bO more than the half, then from the remainder £.10 more than the filth, then fiom the seconil remainder ;£20 more than the fourth part : and it last onlv i;iO remains. W!;at was the original sum ' PROBLEMS IN FRACTIONAL EQUATIONS. 125 64. I bou;4lit a certain number of eggs at 2 a penisy, and the same nuuiher at 3 a penny. T sold tlieni ut 5 for twopence. and lost a petiny. How man}' eg;,'S aid I Luy ? 65. A cistern, liolding 1200 gallons, is tilled by 3 pipes A, B, C in 24 minutes. The pipe A re'Uiires 30 minutes more than C to fill tlie ci-stern, and U) gallons le.~s run tl.rough C per minute than through .4 and B togellier. What time would each pipe take to till the cistern by itstdf ? 66. A, B, and (-' drink a barrel of beer in 24 days. A and 4 B drink „rds of what C does, and B drinks twice as much as A. o In what time would each separately drink the cask ] 67. A and B shoot by turns at a tari;et. A puts 7 bullets out of 12 into the centre, and B puts in 9 out of 1-. Between them they put in 32 bullets. How many shots did each fire? 68. A farmer sold at market 100 head of stock, horses, oxen, and sheep, selling two o.xen for every horse. He obtained on the sale £2, 7s. a head, li he sold the horsgs, oxen, and sheep at the respective prices .£22, £12, lOs., and £1, 10s., how many horsesi^oxen, and sheep respectivir-ly did he sell ? 69. In a Euclid paper A gets 160 marks, and i> just passes. A gets full marks for book-work, and twice as many marks for riders as B gets altogether. Also B, sending answers to all the questions, gets no marks for riders and half marks for book-work. Supposing it necessary to get - of full marks in order to pass, find the number of marks which the paper carries. 70. It is between 2 and 3 o'clock, but a person looking at the clock and mistaking the hour-hand lor the minute-hand, fancies that the time of day is 55 minutes earlier than the reality. What is the true time ? 71. An army in a defeat loses one-sixth of its number in killed and wounded, and 4(X)0 prisoners. It is reintbrced by 3000 men, but retreats, losing a fourth of its nundjer in doing so. There remain 18000 men. What was the original force / 72. The national debt of a country was increased by one- fourth in a time of war. During t\\ enty years of peace widen ii6 Oy MISCELLANEOUS FRACTrON^. followed £25,000,000 was paid off, and at the end of that time the interest! was reduced from 4J to 4 per cent. It was then found that the interest was the same in amount as before the war. What was the amount of the debt before the war ? 73. An artesian well supplies a brewery. The consump- tion of water goes on each week-day from 3 a.m. to 6 p.m. at double the rate at which the water flows into the well. If the well contained 2250 gallons when the consumption began on Monday morning, and it was just emptied when the con- sumption ceased in the evening of the next Thursday but one, what is the rate of the influx of water into the well in gallons per hour ? XIV. ON MISCELLANEOUS FRACTIONS. 195. In this Chapter we shall treat of various matters con- nected with Fractions, so as to exhibit the mode of applying the elementary rules to the simplification of expressions of a more complicated kind than those which have hitherto been discussed. 196. Tlie attention of the student must first be directed to a point in which the notation of Algebra difiers from that of Arithmetic, namely wktn a whole number and a fraction stand side by side vdth no sign between them. • 3 3 Thus in Arithmetic 2'- stands for the sum of 2 and -. / 7 But in Algebra x- stands for the product of x and ". So in Algebra 3— — stands for the product of 3 and ; ° c c . „a + b 2a + Zb that 18, 3 = — - — c c ON MISCELLANEOUS FRACTIONS. 127 Examples. — Ix. Simplify the following fractions : 1, a + x + 3-. 3. ^ + 2—*^. X ^ X x-y a- + ax jx-a .a + b ^a'^ — b^ 2. • s- -2 — -, 4. 4 ,~2--. — jT,. x^ X a-h a^-\-¥ 197. A fraction of which the Numerator or Denominator is itself a fraction, is called a Complex Fraction. y X Thus -, ■% and — are complex fractions. a a m b n A Fraction whose terms are whole numbers is called a Simple Fraction. All Complex Fractions may be reduced to Simple Fractions by the ]nocesses already described. We may take the follow- ing Examples : a b_am_a n _an ^ m~b ' n b m bm n b___d_/a c\ /m _p\_ad-bc , mq-np ^~' m p \b dJ \n q/ bd ' nq n q _ad — bc nq _ nq (ad - be) bd mq - np bd {mq — np)' ,„, 1+x ,, . /, 1\ ,, , x-t-1 (3) _ = (i+x)^(^l + ^j = (l+x)--— — _l+a; X _x(l +x' 1 + - X 1 "x+1 1+x * t28 OM MISCELLA.VF.OUS FRACTTOyS. 1 1 ^^ X . 1 Vl-a; 1 + x/ ■ Vl-x 1 + x' 1+x-l+x cc + x^+l-. 1-x 1+x (5) 1_3;2 • l_a;2 _ 2x l-x^_ 2j ~ 1 - a;2 1 + x=^ ~ 1+ x"-^" 3 3 3 3 3,3, 3(1 -.c) , 3-,ix 3 l-x+3 l-x+3 4-x 1 — X 1 - X 3 3 (4 - x) _ 12 -3x '^ 4-x + 3-3x ~4^x + 3-3x~ 7-4x' 4 — X Examples.— IxL Simplify the following expressions : 4 « « 5 X 7/ X • 1 -g* ^' 7 ~ 1' ^' x-u" ^' r 3— ^ 1+- 23 ^^a 0-D 2-x + -„ 1+i 1 XX a^ -x + rtx-a 2x 7. — ^- 8. 2I • 9- r a x^-a'' 1-rx^ x-u x + V .X , 1 x+y X -y 7+1 pj. II. s. 1 x + 1 x-y x + y 1 + X+y x-v ON MISCELLANEOUS FRACTIONS. 129 ., 2771 - 3 + — 1 + a: + X- m ■ '^- -T-T- ^4. 2m- 1 1 + - -I .,- X x' m a + b _b_ J_ L i. b a + h ah ac be a 6 aft 198. Any fraction may Tie split np into a number of trac- tions equal to the number ol tenns iu its numerator. Tiius a^ + x^ + x + l x^ I!? X 1 X* X* x'* :f^ X* 1111 X X- X^ X* Examples.— Ixii. Split up into four fractions, cacli in its lowest terms, the following fractions : a* + 3a3 + 2a- + 5a 9«3 - 1 2^2 + 6a - 3 '• 2a* ■ ^ 108 ■ a^bc + alri + abc^ + feed' 18;?-+ 12y^-36r2 + 72sg abed, ' 'Spqrts x^-3x2y + 3x?/--?/ 10x3 - 25x2 + 75x- 125 -^ 5' x'-y ' ■ 1000 ■ 199. The quotient obtained by dividing the unit by any fraction of that unit is called The Reciprocal of that fraction. Thus -, that is, -, is the Reciprocal of ?-. a a ^ 6 b 200. "VVe have shewn in Art. 158, that the fraction symbol r is a proper representative of the Division of a by b. In r.s.A.] 1 no ox MISCELLANEOUS LR ACTIONS. Chapter IV. we treated of C3=es of division in which the divisor is contained an exact number of times in the dividend. We now proceed to treat of cases in which the divisor is not con- tained exactly in the dividend, and to shew the proper method of representing the Quotient in such cases. Suppose we have to divide 1 by \-a. We may at once represent the result by the fraction . But we may actually perform the operation of division in the following ■way. \-a) 1 (1 +a + a2 + a3-t-... \-a a i3-a4 The (^lotient in this case is interminable. We may carry on the operation to any extent, but an exact and terminable Quotient we sliall never find. It is clear, liowever, that the terms of the Quotient are formed by a certain law, and such a succession of terms is called a Series. If, as in the case before us, the .scries may be indefinitely extended, it is called an Infinite Serie.s. If we wish to express in a concise i^tiu the result of the operation, we may sto|) at any term of the quotient and write the result in the following way. _!__ _a_ l-a~^'l-a' 1 - a 1 - a' 1 , ., «^ ;; = 1 + a + a- + :; , \-a \-a = 1 + a T ((- + a^ + :; , I -a 1-a' ON MrsCELLANEOUS FRACTIONS. 13' always bein^ careful to attach to that term of tlie quotient, at which we intend to stop, the remainder at that point of the division, placed as the numerator of a fraction of which the divisor is the denominator. Examples. — Ixiii. Carry on each of the following divisions to 5 terms in the quotient. 1. 2 by \+a. 7. 2. m by m + 2. 8. 3. a - 6 by a + 6. 9. 4. a^ + X- by a^ - x^. 10. 5. ax by a- X. 11, 1 by 1 + 2x- - 2x2. 1 + X by 1 - X + x^ 1 + h by 1 - 2&. x^ — 6^ by X + 6. a^ by x-h. b bv a + x. 1 2. a^ by (a + x)^ 13. If the divisor be x-a, the quotient x--2ax. and the remainder 4a^, what is the dividend ? 14. If the divisor be m - 5, the quotient m^ + 5m^ + Ibm + 34, and the remainder 75, what is the dividend ? 201. If we are required to multiply such an expression as x^ X 1 , X 1 ¥ + 3 + 4^^^2-3' we may multiply each term of the former by each term of the latter, and combine the results by the ordinary methods of addition and subtraction of fractious, thus a;2 X 1 X 1 2 3 X^ X^ X 4+-6- + 8 X^ X 1 6 9 12 «* X 1 t ' 72 LX t32 Or^ MISCELLAiWEOUS ER ACTIONS. Or we may first reduce tlie mulliplicaiid and the multiplier to single I'ractioiis und proceed in the loUowing way : (-2+3 + 4)42-3) _ 6.x2 + 4x + 3 3a:-2_ 18ar^4-x- 6 12 ^ 6 ~ 72 "72+72 72~ 4 +72 12 This latter process will be louud the simpler ty a beginner. Examples.— ixiv. Multiply a- a \ , a \ 11,11 '• y-6 + 3^'-^'4-5- 5- ^ + 6^by^-p-. , 11, 1 ,111,111 xa;-^ X a c ' a c 7. 1 + - + -r by 1 - - + -^. 8. l+-a:f-.r-byl--x + -x--x3. 5^ 37 2 1_1 9' 2x2 + x'3 ^x-^"x 2" 10. pr + -5- + 2 by j:r - -T - 2. 2b2. If we have to divide such an expression as ^ o 3 1 X x^ by X + -, we may proceed as in the division of whole numbers, carefully observing that the order of descending powers of x is *^' ^' *' t' X2 ' X3 ON MISCELLANEOUS ER ACTIO. \S. 133 Any isolated digits, uo 1, 2, .j ... will stand between x , 1' and -. X Tims the expression ■! 1 r. O y ^ 5 arranged according to descendinfj powers of x, will stand thus, 5 3 1 a^ + 3x2 + 5x + 4 + - + _ + The reason for this arrangement will be given in the Chapter on the Theorv of Indices. Ex. x + l ]x3 + 3x + ^ + -,l x2 + 2 + 4 x/ a. x^ ^ X'* x^+ X 2x 3 X 2x 2 + - X 1 - + X 1 1 - + X 1 X3 Or we may proceed in the follov/ing way, which will be found simpler by the beginner. (x3 + 3x + ^ + l3)-(x+^) 3 1 )-^lx+ , x/ x^ -f 3x^ + 3x- + 1 , x2 + 1 x^^ ■ X x« + 3x* + 3x2 + 1 3. x^ x- + 1 X* + 2x' +1 X* 2x^ 1 , ^ i = = —, + -^ + - = x- + 2 + -.. X- X^ X' X- X' 134 ON MISCELLANEOUS FRACTIONS. Examples.— Ixv. Divide : 2 1 V, 1 fill,' 1. %'■ — nDya; + -. 4. c° — t- bv c — -5. 1 1 X V^ X 1/ 2. a--j-„hy a--.. 5. -5 + 2 + % by - + ^. b^ •' b y^ x^ -^ y X 3. m-' + -3bym + -. 6. -4 + -wg + ri bv — ,--r 4 rs- a-' w'^ X y . X y 7. -.-^,-3-4-3'- by --4 1/-* x-* 1/ X y X _ 3x5 , , 77 , 43 „ 33 „- , a;^ 8. -r - 4x* 4- — x3 - —X- - ^x 4- 27 by — - X 4- 3. a^ ¥, a b 1113,111 9. i:i + ^byT + - 10. -, + M + ~i — jrDy- + rH — • ^ ¥ a? ■' a a^ ¥ c^ abc ■' a b c 203. In dealing with expressions involving Decimal Frac- tions two methods may be adopted, as will be seen from the following example. Multiply -Ix - -21/ by •03x 4- -4?/. We may proceeil thus, applying the Eules for Multiplication. Addition, and Subtraction ot Decimals. •\x—-2y •03x 4- -Ay •003x2 -Ouexy 4-04 xy--08y» •003x"2T-03l--,v--0%2' Or thus, _ x-2y 3x4^40y ~ 10 ^ lOOT" ^ 3x2 4-3 4x?/-80y3 ~ 1060 = -003x2 4. .034x2/ - -083/2. The latter method will be found the simpler for a be^nner. ON MISCELLANE O US ERA C TIONS. 1 35 Examples.— ixvi. Multiply : (. -Ix- -3 by -53; +07, 2. •05x + 7by-2 -3, 3. -Sx - -2!/ by •4x + -ly, 4. 4-3x + b-2y by •()4x - -06?/. 5. Find the value of a^ - 6^ + c^ + Zabc when a = -03, h=-\, and c = -07. 6. Find the value of 01? - 3ax2 + 3a^x - n^ when x = '7 and (i = 'OS. , 204. When any expression E is put in a form of which /is E a factor, then -^ is the other factor. Thus a + h = a{ \ c 7 T , ab-\-ac-¥hc So (U) + ac-^oc = aoc. -, and a?+2an/ + 2/2 = x2.(?l±^^±^') EXAMPLES.— Ixvii. 1. Write in factors, one of which is a^x, the series a^x + a^ocr + a^x^ + a4X'* + . . . 2. Write in factors, one of which is xyz, the expression xij -XZ + yz. 3. Write in factors, one of which is x^, the expression X- + x!/ + y'. 4. Write in factors, one of which is a + 6, the expression (a + 6)3-c(a + 6)2-d(a + 6) + e. 136 O.V MISCELLANEOUS FRACTIONS. 205. We s!i;ill now give two examples of a process by which, when certain Iractiuns are known to be equal, otlier relations between the q^uautities involved in them may be. deteruiineJ. This i)rocess will be found of great use in a later part of the suliject. and the student is advised to pay particular attention to it. (1) If ^= J, shew that 0, a — b c — ^' T ^ a Let r = X- Then 3 = X ; a .■. a = \b, and c = \d. Now a + 6^X6+^&^6(X + l)_X + l a'-b \b-b~bl\-l)~\-l' c + d_\d + d d{X+l) X + 1 and c-d Xd-d (Z(X-l) X-T TT C'-^b , c + fZ - . , , ;^+i tience — _-^ and ~-_-j being each equal to — — are equal to one another. (2) If =- = = .shew that m + u + r = 0. a — 0- c c - a Let a-b~^' o-c r = X, c-a then m = Xa-X6. n = \b- Xc, r='Kc - Xa ; .•. m + n + r = X(T -X^-L \6-Xc + Xc-Xff = 0. ON MISCELLANEOUS FRACTIONS. 137 Examples. — Ixviii. O. C 1. If r = -7 prove the following relations : . 'lZ^ — —A ^r\ 8a + & _ 8c + d , . rt _ _c , c2-6-_a6 , N 3ft _ 3c . ll« + 6_13a + 6 ^^^ 4a - 56 "" 4c - 5rZ" ^^^ TlcTrf ~ 13c+"5" / ^ "' + ^^_ C' + tf - «2-rt6 + 6-_c-'-ff(' + d* W a2_p-^rr"ci2- ^^) a2 + a6 + 62-c^ + cf/ + f;2- Tr ^ m 71 , , 2. 11 r = T = , then i + 7/1-}- ?i = 0. a — n — c c — a ^ jfO c e ^, ^ « la + vic + ne 3. ii r = -, = 7, T>rove that y = .-7 1 > a J' ' h Ib + md + nf a+h b+c_c+t c a 4- -11 — 7;— = = . prove that a = 6 = c. '■ r, ■ i- 3- 11 1- =r = r> sliew that fJ= ^/ ^- . •*. 6. II T, -J. J he in descending order of magnitude, shew xi-fi + c-l-e., , a , , e i_j7T— ->is less than ^- and greater tlian ■^. 7. If -'=^ shew that ^^A^4x, + 5y,^ 2/1 2/1. '^i + y^/i '^2 + 92/a T4-^ <^ 1 ^1 , rt--t-a6 ab-b- 8. Il5 = ^,shewthat-^^ = ^^_^, 9. If^ = .%hewthat7«-+.^, = I^,. o a 6a + ob 3c-f5(i 138 Oy MTSCELLANEOUS FRACTIOf^S. lo. If r be a f roper fraction, shew that -r is greater than r, c being a positive quantity 6 + II. If r bt- an improper fraction, shew that t — : is less than r, c being a positive quantity. 206. We shall now give a series of examples in the svorking of which most of the processes connected with fractions will be introduced. Examples.— ixix. I. Find the value of Sa^ H y^ when a = 4, b = ^, c=l. ^- Sin^pMy 7x^-12x + 5 -'^"'^ a^'-H4a-45 - 3. Simplifv(^t^_^--^)^(«_tP + ^-n -^ ^ ' \a-j9 a+p/ \a-^ a+p/ 4. Add together a;2 t/2 2^ ^2 j,2 3.2 j;2 j-i y2 4" e'^S' 4""6"*" 8 ^""^ 4"^ 6"^8' and subtract 2- - x- + ^ from the result. 5. Find the value of -5 — .,~ „ — :r^ wheii a=4, 6 = -, c = l. 6. Multiply |x2 + 3ax - \a^ by 2x2 -ax-%, 01 xi ^ a^-ft- „, 36- 7. Shew that -. tw = « + 26 -t- r- ' (a - 6)-^ a-h OM MTSCELLANEOUS FRACTIONS. 130 8. Simplify ?^ + ^^ + 4-^!. ^ X x-y x^ — xy- 01 ^, , 60x3 -17x2- 4x + l ,- „, ^ 49 Q. Shew tnat , ., . ^ = 12x-25h ^. ^ 5x- + 9x - 2 .j; + 2 „. ,.„ x*-9.r'' + 7x2 + 9x-8 10. fei«^Pl'fy^4 + 7^3Z9^2_7^V8' 11. Simplify ^^+ j— . 1-.-- J 1 2. Simplify a+'ah + 6^ f a + «6 + ^V^tt- )• 13. Multiply together U + 1)\}' + j>)y ~ \ 14. Add to!::'ether -, -, — -, -, and shew that if their ^ ° a+V h+V c+l sum be equal to 1, then ahc = a + 6 + + 2. ^. ., X b h^ h b^ , 15. Divide--! 5 + - + --, by x-a. a b c . r-^cH '-a-\ =-0 16. Simplify r , and shew that it is equal --rC + i-^aH ^6 a c to -^^ ' — T^ — if 2s = a + 6 + c. be 17. Shew that -^ + p+ _- = --— ^. a-^x a~x a^-T-x^ 1 8. Simpllly r + r - 2-:^ — r-,. ^ •'a-b a + b a^ + b- _. ,.. 6 a + 6 a2 + 62 19. Simplify--^ --2 - + 2^^^ZTy „. ,.,. a--ab + ¥ (i--ly^ I40 ON MISCELLANEOUS FRACTIONS. 2 2 1 . Simplify r-j — ,-Tr, — (x2_i)-' 2x--4x + 2 l-x2 ... (^^-V'-vlah-c^ a->rh->rc 22. Sunpuly-n T. — ,., — ^TT-^i— -• '^ '' c^ - a^ — b' + 2ab b + c — a 23. Simplify /-^■.l--j-\-^-^^-x-^^y ^ X X . /x-rt\3 x-2« + 6 , a + 6 24. rind the value of I r I ^j, when x = — s"~' \x-6/ x + a-26 2 a" - (6 - c)2 Zy- - (fi - c)- c- - (a - 6)^ , „. ,.„ (x2-4x)(x2-4)a 26. Simplify^— ^,-^3-i-. 27. Simplify ^^^^,^^,---^ 28. Simplify ^ + — -5 r-r, + -s r---rr-^ rr^ •^ - ar X- X (x^ + l)^ x2+l 2--(x- + l)' T,,. . , x^ X a a^ , X a 29. Divide -, - - + b by . a'^axx^-'ax 30. Simplify |2—_-_^^-^^4-^-^.}^^-. 31. Simplify ^^ " ^ + ^^'^ ^ (^ : '^l:^ ( V ")'^ ^ ^" - ^^\ „ , l-x-3x2 - l+3a2 + 2x3 3^- ^^^^ (3-2.c-7x^)3 ^^""^ (-332^^x-^v 33. Sim^lih(i±f,-i--t)^(^J-'^^). *'-' -^ • \X'-y- X' + y/ \x-y x + y/ 34. si.p,ir,Q-:->)(-!^-i).(^-OC-.|^e-0 35. Simplify a--(<6 g- 4- a6 + 6- /_ 2a3_ _ , \ / , _ ^gft \ OM MISCELLANEOUS FRACTLOXS. 141 36. Simplify 1 _ 1^+ 2(a;-l)- 4(x-l) 4(a;+l) (x - 1)-' (x + 1)" 37. Prove that 1 1 — + s—as-h&-c /111 \ 4- + ... =s(- +r+ -+ ... )-". \a be I ab.c a {a - h) {x - a) h{b — a) {x - h) x {x - u) {x - 1)' 38. Tf s = a4 ?) + c+ ... to 71 term?, slie-\v that -b s- T- -, — + a c 39. MuUith-(^,-^^.>y -^$^/?-_. jy i- •■ \ X' - y- X- + y-/ X- — y-y + {x- + ij-)- , a-x , a- — x^ 1 + 1 + -, r. ,.r « + •'<; rt-4X' 4.0. Simpluv — •.. ^ , ^ •' a-x U- — X- a + x a- + X' 41 . Divide x^ + 3 - s( ^ - x- ) + 4f x + - ) liy x+ -. 42. If s = rt + 6 + c + ...tow terms, shew th;it s-a s — b .« - r , + + +...=71-1. 5 S S 43. Divide (-"- — ^- - I l.y ( ..^^-. + X0- ^•^ \x-|/ x + 2// \x- + y X'-y-y 1--^. /I 44. bimphfy ^—^ -T 45- If r_-a6 =^;j:ri' P^«^'^ that --p^-^ = a6.i a c a 46. Simplify p* + 4p^q + 6p-q' + 4p(f + 2^ ^ 1^ + ^P'1 -*- •^P^ '^ * 'f pi - 4p^q + Gjy-q' - 4pq^ + q* ' p^ - 3p-q + 3pg- - ^ 142 SIMULTAXEOUS F.QUATIOXS 48. Simplify 1 1 y(xi/3 + x + a) y+~ ^ z 1 1 1 a: y ^ iniBlifr * " ■'' " ~ '^ ^"^ ~ ^ • ' ^"^ " ^^' 49. (a - 1/) (a - x)2 (a - 1/;2 (« _ x) 3 50. c.:..-„i:f.. f'^'^ 3-a-6-c i'"--' 1 1 1 a^ 0-c ' T)c ca ah Simpliiy -l{a'^-b'^. a — - — 51- ♦ b XV. SIMULTANEOUS EQUATIONS OF THE FIRST DEGREE. 207. To determine several unknown quantities we must have as many independent equations as there are unknown quantities. Thus if we had this equation given, x + y = 6, we could determine no definite values of x and y, for a; = 2) x = 4) x = 3) or other values miglit he given to x and y, consistently with the equation. In fact we can find as many pairs of values of X and 1/ as we please, which will satisfy the equation. OF THE FIRST DEGREE. 143 "We must have a second equation independent of tlie first, and then we may tind a pair of values of x and y wliicii will satis/ u both equations. Thus, if besides the equation x + y = Q, we had anotlier equation x-y = 2, it is evident that the values of x and y which will satisfy both equations are x = 4 y = 2 since 4 + 2 = 6, and 4-2 = 2. Also, of all the pairs of values of x and y which will satisfy one of the equations, there is but one pair which will satisfy the other equation. We proceed to shew how this pair of values may be found. 208. Let the proposed equations be 2x + 7i/ = 34 5x + 9i/ = 51. Multiply the first equation by 5 and the second equation by 2, we then get 10x + 35j/ = 170 10.c+18?/ = 102. The coefficients of x are thus made alike in both equations. If we now subtract each member of the second equation from the corresponding member of the first equation, we shaU. get (Ax. II. page 58 j 35-j/-18i/=170-102, or 17(/ = 68; -■• 2/ = 4. We have thus obtained the value of one of the unknown symbols. The value 01 the other may be found thus : Take one of the original equations, thus 2x + 7j/ = a4. Now, since y — 4,7y = 28; .: 2x + 28 = 34; .-. x = 3. Hence the pair of values of x and y which satisfy the ecjuations is 3 and 4. 144 SIMULTANEOUS EQUATIONS Note. The process of thus obtaining from two or more equalioiis an equation, Ironi wliicli one ol the unknown quanti- ties has disappeared, is called Elimination. 209. ^Ye worked out tlie steps fully in the example given in the last article. We shall now work au example in the lorm in which the process is usually given. Ex. To solve the equations 5a; + 4)/ = 58. Multiplying the first equation hy 5 and the second by 3, 15x4-35^ = 335 15x4- 12// = 174. Subtracting, 23^=161, and therefore 2/ = 7. Now, since 3x4- 7;/ = 67, 3x4-49 = 67, .-. 3x = 18, .-. x = 6. Hence x = Q and y=*7 are the values required. 210. In the examples given in the two preceding articles we made the coetticients of x alike. Sometimes it is more con-" venient to make the coefficients ol" y alike. Thus if we have to solve the equations 29x4-27/ = 64 13x4- 2/ = 29, we leave the first equation as it stands, and multiply the second equation by 2, thus 29x4- 2?/ = 64 2Gx + 2!/ = 58. Subtracting, 3x = 6, and therefore x = 2. Now, since 13x4-y = 29, 264-?/ = 29, .-. 2/ = 3. Hence x = 2 and ?/ = 3 are the values required. OF THE FIRST DEGREE. 145 Examples — Ixx. I. 2a; + 7i/ = 41 2. 5.<; + 8i/=101 3. 13.r+ l??/^ 189 3a; + 4i/ = 42. 9x + 2y = 95. 2a;+ i/ = 21. 4. 14x + 9?/ = 15G 5. a; + 152/ = 49 6. 15;(;- 19?/= 132 7x + 2)/ = 58. 3x+ 7)/ = 71. Sax + 17?/ = 226. 7. 6a; + 4?/ = 236 8. 39.j; + 277/ = 10o 9. 72a; + 14?/ = 330 3a; +15?/ = 573. 52a; + 297/ =133. G3x+ 7?/ = 273. 211. "We .shall now give some examples in which negative signs occur attached to tlie coelUcieut ol y in one or both of the equations. Ex. To solve the equations: 6a; + .^o?/ = l77 8a;- 217/ = 33. Multiply the first equation by 4 and the second by 3. 24x+140?/ = 7()3 24x- 63?/ = 99. Subtracting, 2037/ = G09, and therefore y = ^- The value of x may then be found. Examples.— Ixxi. I. 2x + 7y = 52 2. 7x- 47/ = 55 3. x + 7/ = 9G 3x-5// = 16. 15x-13// = 109. x-7/ = 2. 4. 4x+ 9?/ = 79 5. a; + 197/ = 97 6. 29x-14?/ = 175 7x-17?/ = 40. 7x-53?/ = 121. S7.:;-5G?,'-4y7. 7. 171x-213?/ = 642 8. 43x+ 2?/ = 26G 9. 5x + 9?/ = lfi8 114x-326?/ = 244. 12x-17?/ = 4. 13x-27/ = 57. fs.A.l - 146 . STMUL TANEOUS EQUA TIONS 212. We have hitherto taken examples in ■which the coetiicients of x are both positive. Let us now take the lolljv . ing equations : 5x -7y = 6 9y-2x = 10. Change all the signs of tJie second equation, so that we get 5a; — 7i/ == 6 2x-9y= -10. Multiplying by 2 and 5, I0x-Uy = l2 10x-45j/=-50. Subtracting, -141, f 45?/ = 12 + 50, or, Sly = 62, or, y = 2. The value of x may then be found. Examples.— ixxii. I. 4x~7y = 22 2. 9.c-5?/ = 52 3. 17x + 3i/ = 57 7y-'Sx=l. 8y-'3x = 8. 16i/-3x = 23. 4. 7y + 3x = 7S 5. 5.c-3i/ = 4 6. 3x + 2i/ = 39 19i/-7x = 136. 12!/-7x=10. 32/-2a;=13. 7. 5y-2x = 21 8. 9?/-7x=13 9. 12a;+ 7y=176 13x-4y=l-20. 15x-7t/ = 9. 32/-19a; = 3. 213. In the preceding examples the values of a; and y have been jiositive. We shall now give some equations in which x or y or both have negative values. Ex. To solve the equations: 2x-9y = U 3x-4y = 7. Multiplying the equations by 3 and 2 respectiveiy, we get 6x-27y = :yi 6x- 8}f=14. OF THE FIRST DEGREE. i^'j Subtracting, -19*/ = 19, or, 192,'=-19, or, y=-l. Now since 9y= - 9, 2x - 9y will be et[uivalent to 2x - ( - 9) or, 2x + 9. Hence, from the first equation, 2x + 9 = ll, .-. x = l. Examples. — Ixxiii I. 2x + 3i/ = 8 2. 5x~2y = bi 3. 3x-5y = ol 3x + 7y = 7. 19a; -3?/ = 180. 2x + 7i/ = 3. 4. 7i/-3x=139 5. 4x+ 9?/= 106 6. 2x-7i/ = S 2x + by = i)l. 8a; + 172/=198. 4!/-9x=19. 7. 17x+122/ = 59 8. 8x + 3y = :i 9. 69i/-17x=103 19x- 4?/ = 153. 12x + 9(/ = 3. 14.c- 13^/= -41. 214. We shall now take the case of Fractional Equations involving two unknown quantities. Ex. To solve the equations, 2x-^:-^=4 5 3y = 9— 3-. First, clearing the equations of fractions, we get 10x-y + 3 = 2O 9y = 27-x + 2, from which we obtain, 10x-y = 17 x + 9y = 29, and hence we may find x = 2, i/ = 3. 148 SIMULTANEOUS EQUATIONS Examples.— Ixxiv. I. 1 + 1 = 7 2. iOx + | = 210 3. ^ + 7j/ = 251 1 + 1 = 8. 10!/-? = 29(). | + 7x = 299. 4. —,-^ + 5 = 10 5. 7a; + Tr = '413 6. — --^ = 10-^ 3 2 5 3 ^--1 + 7 = 9^ 39x = 142/-l609 ^^=| + 1. 7. x-'^-^ = 5 10. ^;^ + 8, = 31 4u :^— = 3. =i-— +10x = 192. 3 4 8. | + 8 = |-12 II. ?^J^ + 3x = 2y-6 x + ?/ w 2x-'V „^ 1/ + 3 1/-X „ ^^ 3x-5;/ „ 2x + y x-2 10-x 1/-IO 2 o 3 4 x-2y/_x 7/ 2>/ + 4_4 x + y+13 S ____ + _ _- ^ . 13. — ---^ + 3x = 47/-2 ■^ 13 5x + 6i/ 3x-2v_5, _ .^ 5x-3 3.7;-19 , 37/-x .4. -2 -o" = ^-^3- 2x + y _ 9x - 7 _ % t? _ f^ + ^ ~2 8 "4 16 ■ 4x + 5?/ ^5 -40-" = ^-^ 2x - 1/ „ 1 -3-^"^' = 2- OF THE FIRST DEGREE. 149 215. We have now to explain the nietliod of solving Literal Equations involving two xniknowu quantities. Ex. To solve the equations, ^ ax-\-by = c px + q>j = r. \. Multiplying the first equation Ly p and the second by a, we get apx + bpy = cp apx + aqxj = ar. Subtracting, bjnj — aqij = cp — ar, or, {bp -aq)y = cp - ar ; _ cp - ar bp — aq' We might then find x by substituting this value of y in one of the original equations, but usually the safest cour.se is to begin afresh ami make the coefficients of y alike in the original equations, multiplying the first by q and tlie second by b, which gives aqx + bqy = cq bpx + bqy = br. Subtracting, aqx — bpx = cq- br, or, {aq — bp)x = cq-br; _ cq- br aq-bp' Examples.— ixxv. I. 'mx + ny = e, 2. ax + by = c 3. ax-by = m px + qy =f. dx - cy =/. ex + ey = n. 4. ex =dy 5. mx-ny = r 6. x + y = a x + y = e. m'x + n'y = r x-y = b. 7. ax + by = c 8. abx + cdy = 2 o. , = ^; ' -^ •' ^ h + y 3a + x dx + fy = c-. ax-cy= -rj-. ax + 2by = d. 156 ^TMtJLTAYF.OTJS F.QUATIOI^S \o. bcxi-2b -aj = II. {b + c)(;x + c — b) + a(7j + a) = 2a^ ^, a(c^-b^) 2h^ , ay (b + cy ^ be c (b-c)x a' (8b-2m)bm 12. 3x + 5i/ = — , ij „ — 0- - r/r , 6-a; - , t-(b + c + vi) my = ■ni^x + (b + 2m) &m. 216. We now proceed to the solution of a particular class of Simultaneous Equations in which the unknown symbols appear as the denominators of fractions, of which the following are examples. Ex. 1. To solve the equations, a b - + -=c X y m TO J =(L X y Multiplying the first by m and the second by a, we get am bm + ^ =cm X y am an , X y ~ ' bm an — H = cm- y y -ad. Subtracting, bm + an . or, =cm-ad, y or, bm + an = (cm - ad) y, _^bm + an ^ cm -ad' Then the value of x may be found by substituting this value of y in one of the original etjuations, or by making the terms containing y alike, as iu the example given ia Art. 215. 6P THE FIRST DEGREE. t^t ElX. 2. To solve the equations: X 3y~27 Ax'^y 72" Multiplying the second equation 1->y 8, we cret 2__5 _-i_ a~3i/~27 2 8_11 X y~ 9' Subtracting, 5 8_ 4 11 •3y y~27 9" Changing signs, 5 8 11 4 Sy'^y~9 27' or. 5 + 24 33-4 dy 27 ' whence we find y=9, and then tlie vahie of x may be found by substituting 9 lor y in one of the original equations. * Examples. — Ixxvi. X y 2. 1 2 - + - = a a; y 3- a b - + - = c X y i + =^ = 20. X y 3 4 . - + - = ?). X y b a - + - = d. X y a b - + - = m X y 5- X y 6. 5 2 „ Sx by a b X y 7 1 ^ X y~ 6x 102/ ""■ 2 ax 3 :5 8. m n 1 = m + n nx my 5 ax 2 by" 3. n - + X TO ., ., — =TO- + ?l-. I.<2 SIMULTANEOUS EQUATIOh'S 217. There are two other methotls of solvin.n Simultaneoua El [nations of which we have hitherto made no mention, because they are not generally so convenient and simple as the method which we have explained. They are I. The metliod of Substitution. If we have to solve the equations a; + 3?/= 7 2x + 4?/ = 12 we may find the value of x in terms of y from the first equa- tion, thus a; = 7-3(/, and substitute this value for x in the second equation, thus 2 (7 -37/) + 47/= 12, from which we find i/ = l. We may then find the value of x from one of the original equations. II. The method of Comparison. If we have to solve the equations 5x + 2j/ = 16 7x-3!/= 5 we may find the values of x in terras of y from each equation, thus x = — - — -, from the first equation. x = — — -, from the second equation. Hence, equatini; these values of x, we get 16^27/ _5+_3y 5 '~ 7" ' an equation involving only one unknown symbol, from which we obtain !/ = 3, and tlun the value of x may be found fr<ini one of the ori-inal e(iuati'>'>s. OF TFJE FIRST DEGREE. 153 218. If tliere be ihrRe, unknown symbols, their values may be found from tliree independent eijuations. For from two of the equations a third, which involves only tioo of the unknown symbols, may be found. And from the remaining equation and one of the others a fourth, containing only the same two uukuown symbols, may be found. So from these two equations, which involve only two un- known symbols, tlie value of these symliols may be found, and by substituting these values in one of the original equations the value of the third unknown symbol may be found. Ex. 5x-6y + 4z=15 7.'c + 47/-3.-;=19 2x+ 7/ + 6.v = 46. Multiplying the first by 7 and the .second by 5, we get 35.<;-42i/ + 283=105 3Jx + 20?/-15s = 95. Subtracting, -62)/ + 433 = 10 (1). Again, multiplying tlie first of the original equations by 2 and the third by 5, we get 10.c-12r/ + 83 = 30, lOx + by + 30z = 2Z0. Subtracting, - 17?/ -222= -200 (2). Then, from (1) and (2) we have 62>j-43z= -10 17(/ + 222 = 200, from which we can find ij = 4 and s = 6. Then substituting these values for tj and z in the first equa- tion we find the value of x to be 3. Examples. — Ixxvil. 1. 5x + 7y- 22 = 13 3. bx-3y + 2z = 21 8x + 3!/+ 2 = 17 8x- y-3z= 3 x-4?/ + 103 = 23. 2x + 3i/ + 2z = 39. 2. 5x + 3)/-6.i; = 4 4. 4x-5y + 2z= 6 3x- y + 2z = S 2x + 3?/- 2 = 20 x-2y + 2z = ± 7x-4.v + 32 = 35, 154 PROBLEMS RESULTING TiV 5 x+ y+ z= 6 8. 4x-3j/4- 2= 9 5x + 47/ + 3;2 = 22 9x + 1/ - 62 = 1 6 15a; + 10)/ + 62 = 53. x-4i/ + 3z= 2. 6. 8x + 47/-3.v = 6 9. 12a; + 5?/ -42 = 29 a; + 32/— z = 7 13x- 2^4-52 = 5b 4x-52/ + 4z = 8. 17a;- ?/- z = 15. 7. x+ y+ 2 = 30 10. y-x + z=- 5 8a; + 4?/ + 22 = 51) z - ?/ - x = - 25 27a; + 9;/ + 32 = 64. x + ?/ + 2 = 35. XVI. PROBLEMS RESULTING IN SIMUL- TANEOUS EQUATIONS. 219. In the Solution of Problems in which we represent two of the numbers sought by unknown symbols, usually x and y, we must obtain two independent equations from the condi- tions of the question, and then we may obtain the values of the two unknown symbols by one of the processes described in Chapter XV. Ex. If one of two numbers be multiplied by 3 and the other by 4, the sum of the products is 43 ; and if the former be multiplied by 7 and the latter by 3, the difference between the results is 14. Find the numbers. Let X and y represent the numbers. Then 3a; + 4*/ =43, and 7x — 'iy = 14. From these equations we have 21x + 28!/ = 301, 21a;- 9?/ = 42. Subtracting, 37j/ = 259. Therefore J/ = 7. and tlu'U tlie value of a; may be found to be o. Hence the numbers are 5 and 7. A. SIMULTANEOUS EQUATIONS. 155 Examples.— Ixxviii. The snm of two numbers is 28, and tlieir difference is 4, find the numbers. ill 2. The sum of two numbers is 256, and their difference is 10, find the numbers. 3. Tlie sum of two numbers is 13'5, and tlieir difference is 1, find the numbers. ■6 ^4. Find two numbers such that the sum of 7 times the "^greater and 5 times the less may be 332, and the product of their difference into 51 may be 408. .jjl 5. Seven years ago the age of a father was four times that •^of his son, and seven years hence the age of the father will be double that of the son. Find their ages. ^^6. Find three numbers such that the sum of the first and •second shall be 70, of the first and third 80, and of the secoud and third 90. 7. Three persons A, B, and G make a joint contribution which in the whole amounts to ^400. Of this sum B contri- butes twice as much as A and £20 more ; and G as much as A and B together. What sum did each contribute? 8. If A gives B ten shillings, B will have three times as much money as A. If B gives A ten shillings, A will have twice as much money as B. What lias each ? 9. Tlie sum of £760 is divided between A, B, G. The shares of A and B together exceed the share of G by £240, and the shares of B and C together exceed the share of A by £360. Wliut is the share of each ? ^(^ 10. The sum of two numbers divided by 2, gives as a quo- "^tient 24, and the difference between them divided by 2, gives as a quotient 17. What are the numbers? ^^ II. Fiml two numbers such that when the greater is divided by the less the quotient is 4 and the remainder 3, and when the sum of the two numbers is increased by 38 and the result divided by the greater of the two numbers, the quotient is 2 and the remainder 2. /A 12. Divide tlie number 144 into three such parts, that (/'when the first is diviiled by the second the quotient is 3 and the remainder 2, and when the third is divided by the sum of the other two parts, the quotient is 2 and the remainder 6. IS6 PROBLEMS RESULTING IN H 13. A and B buy a horse for £120. A can pay for it if B will advance half the money he has in his pocket. B can pay for it if A will advance two-thirds of the money he lias in his pocket. How much has each ? .^14. "How old are you?" said a son to his father. The father replied, "Twelve years hence you will be as old as 1 was twelve years ap;o, and I shall be three times as old as you were twelve years ago." Find the age of each. 1,^5. Eequired two numbers such that three times the greater exceeds twice the less by 10, and twice the greater together with three times the less is 24. ^7' li6. The sum of the ages of a father and son is half what it "^vill be in 25 years. The difference is one-third what the sum will be in 20 years. Find their ages. , / ' 17. If I divide the smaller of two numbers by the greater, • / the quotient is '21 and the remainder "OIjT. H" I divide the greater lunuber by the smaller, the quotient is 4 and the remainder '742. Find the numbers. 18. The cost of 6 barrels of beer and 10 of porter is £51 ; the cost of 3 barrels of beer and 7 of porter is £32, 2s. How much beer can be bought for £30? 19J The cost of 7 lbs. of tea and 5 lbs. of coffee is £1, 9s. 4il. : the cost of 4 lbs. of tea and 9 lbs. of coffee is £1, 7s. : what is the cost of 1 lb. of each ? 20. The cost of 12 horses and 14 cows is £3S0 : the cost of 5 horses and 3 cows is £130 : what is the cost of a horse and a cow respectivel}' 1 21. The cost of 8 yards of silk and 19 yards of cloth is £18, 4s. 2d.: the cost of 20 yards of silk and IG yards of doth, each of the same quality as the former, is £25, 16s. Sti. How much does a yard of each cost ? 22. Ten men and six women earn £18, 18s. in 6 days, and four men and eight women earn £(>, Cs. in 3 days. What are the earnings of a man and a woman daily ? 1 1)23. A farmer bought 100 acres of laud for £4220, part at ^^£37 an acre and part at £45 an acre. How many acres had lie of each kind? SIMULTAi^EOUS EQUATIONS. 157 Note I. A number consisting of two digits may be repre- sented algebraically by lOx + y, where x and y represent the significant digits. For consider such a number as 76. Here the significant digits are 7 and 6, of which the former has in consequence of ils position a local value ten times as gre'it as its natural value, and the number represented by 76 is equivalent to ten times 7, increased by 6. So also a number of which x and y are the significant digits will be represented Ijy ten times- x, increased by y. If the digits composing a number lOx + y be inverted, the resulting number will be lOy + x. Thus if we invert the digits composing the number 76, we get 67, that is, ten times 6, in- creased by 7. If a number be represented by lOx + y, the sum of the digits will be represented by x + y. A number consisting of three digits may be represented algebraically by 100x+ lOy + z. Ex, The sum of the digits composing a certain number is 5, and if 9 be added to the number the digits will be inverted. Find the number. Let lOx + y represent the number. Then x + y will represent the sum of the digits, and lOy + x will represent the number with the digits inverted. Then our equations will be x + y = 5, 10x + y + 9 = l0y + x, from which we may find x = 2 and ?/ = 3 ; .". 23 is the number required. ^'^ 24. The sum of two digits composing a number is 8, and if 36 be added to the number the digits will be inverted. Find the number. jvi^25. The sum of the two digits composing a number is 10, and if 54 be added to the number the digits will be inverted. What is the number ? 1S8 PROBLEMS RESUL TING IN 26. The sum of the digits of a munber less than 100 is 9, and if 9 be added to the number the digits will be inverted. What is the nundjer? 27. The sum of the two digits composing a number is 6, and if the number be divided by the sum of the digits the quotient is 4. '\Vliat is the number ? 28. The sum of the two digits composing a number is 9, and if the number be divided bv the sum of the digits the quotient is 5. What is the number ? 29. If I divide a certain number by the sum of the two digits of wliich it is composed the quotient is 7. If I invert the order of the digits and then divide the resulting nund)er dinnnished by 12 by tlie difference of the digits of the original number the quotient is 9. What is the number ? A 30. If I divide a certain number by the sum of its two digits the quotient is 6 and the remainder 3. If I invert the digits and divide the resulting number by the sum of the digits the quotient is 4 and the remainder 9. Find the number. 31. If I divide a certain number by the sum of its two digits diminished by 2 tlie quotient is 5 and the remainder 1. If I invert the digits and divide the resulting number by the sum of the digits increased by 2 the quotient is 5 and the re- mainder 8. Find the number. ^i 32. Two digits which form a number change places on the addition of 9, and the sum of these two numbers is 33. Find the numbers. 33. A number consisting of three digits, the absolute value of eacli digit being the same, is" 37 times the square of any digit. Find the number. 34. Of the three digits composing a number the second is double of the third : the sum of the first and third is 9 : the sum of all the digits is 17. Find the number. .1 (35. A number is composed of three digits. The sum of the digits is 21 : the sum of tlie fust and second is greater than the third by 3; and if 198 be added to the number the digits will be inverted. Find the number. SIMUL TANEOUS EQ UA TIONS. I J9 Note II. A fraction of which the terms are unkno\vn may be represented by -. Elx. A certain fraction becomes ^ when 7 is added to its denominator, and 2 when 13 is added to its numerator. Find the fraction. Let - represent the fraction a; + 13_ are the equations ; from which we may find a; = 9 and i/=ll. 9 That is, the fraction is yy. 36. A certain fraction becomes 2 when 7 is added to its numerator, and 1 when 1 is subtracted from its denominator. What is the fraction ? 37. Find such a fraction that when 1 is added to its 1 3' numerator its value becomes -, and when 1 is added to the denominator the value is -. 4 38. What fraction is that to the numerator of which if 1 be 1 ^2 added the value will be ^ : but if 1 be added to the denominator. the value will be ;^ ? 39. The numerator of a fraction is made equal to its denominator by the addition of 1, and is half of the deno- minator increased by 1. Find the fraction. 40. A certain fraction becomes - when 3 is taken from the numerator and the denominator, and it becomes - when 5 i6o PROBLEMS RESUL TING IN is added to the numerator and the denominator. "What is the fraction ? 7 41. A certain fraction hecomes ^ when the denominator is 20 increased hv 4, and — ^ when the numerator is diminished by 15 : determine the fraction. 42. What fraction i.< that to the numerator of which if 1 be added it becomes , and to the denominator of which if 17 be added it becomes - ? o Note III. In questions relating to money put out at simple interest we are to observe that T Principal x Rate x Time Interest = , where Eate means the number of pounds paid for the use of £100 for one year, and Time means the number of years for which the money is lent. 43. A man puts out £2000 in two investments. For the first he gets 5 per cent., for the second 4 per cent, on the sum in\ested, and by the first investment he has an income of £10 more than on the second. Find how much he invests in each case. 44. A sum of money, put out at simple interest, amounted in 10 months to £5250, and in 18 months to £5450. "What was the sum and the rate of interest ? 45. A sum of money, put out. at simpie interest, amounted in 6 years to £52(10, and in 10 years to £6000. Find the sum and the rate of interest. Note IV. "When tea, spirits, wine, beer, and such com- modities are mixed, it must be observed that quantity of ingredients = quantity of mixture, cost of ingredients = cost of mi.\ture. Ex. I mix wine which cost 10 shillings a gallon with another sort which cost 6 shillings a gallon, to make 100 SIMULTANEOUS EQUATIONS. i&i gallons, -which I may sell at 7 shillings a gallon vithout profit or loss. How much of each do I take ? Let X represent the number of gallons at 10 shillings a gallon, and \j 6 Then a; + 2/=100, and 10x + 6t/ = 700, are the two equations from which we may find the values of X and y to be 25 and 75 respectively. 46. A wine-merchant has two kinds of wine, the one costs 36 pence a quart, the other 20 pence. How niucli of eacli must he put in a mixture of 50 quarts, so that the cost price of it may be 30 pence a quart ? 47. A grocer mixes tea which cost him Is. 2fZ. per lb. with tea that cost him Is. %d. per lb. He lias 30 lbs. of the mi.vture, and by selling it at the rate of Is. 8(Z. per lb. he gained as much as 10 lbs. of the cheaper tea cost him. How many lbs. of each did he put in the mixture? Note V. If a man can row at the rate of x miles an hour in still water, and if he be rowing on a stream that runs at the rate of 1/ miles an hour, then X + 1/ will represent his rate down the stream, X — ?/ wp 48. A crew which can pull at the rate of twelve miles an hour down the stream, finds that it takes twice as long to come up a river as to go down. At what rate does the stream How ? 49. A man sculls down a stream, which runs at the rate of 4 miles an' hour, for a certain distance in 1 hour and 40 minutes. In returning it takes him 4 hours and 15 minutes to arrive at a point 3 miles short of his starting- place. Find the distance he pulled down the stream, and the rate of his pulling. 50. A dog pursues a hare. The hare gets a start of 50 of her own leaps. The hare makes six leaps while the dog makes 5, and 7 of the dog's leaps are equal to 9 of the hare's. How many leajJS will the hare take before she is caught ? l62 PROBLEMS RESUL TIXG IN 51. A grevhoimd starts in pursuit of a hare, at the distance of 50 of liis own leaps Irom ber. He makes 3 leaps while the bare makes 4, and he covers as much ground in two leaps as the hare does in three. How many leaps does each make before the hare is caught ? i;2. I lay out half-a-crown in apples and pears, buying the apples at 4 a penny and the pears at 5 a jtenny. I then sell half the apples and a third of the pears for thirteen pence, •which was the price at which I bought them. How many of each did I buy ? 53. A company at a tavern found, when they came to pay their reckoning, that if there had been 3 more persons, each would have paid a shilling less, but had there been 2 less, each would have paid a shilling more. Find the number of the company, and each man's share of the reckoning. 54. At a contested election there are two members to be returned and three candidates. A, B, and C. A obtains 1056 votes, B, 9S7, C, 933. Now 85 voted for B and C, 744 for B only, 98 ibr C only. How many voted for A and C, for A and B, and for A only ? 55. A man walks a certain distance : had his rate been half a mile an hour faster, he would have been H hours less on the road; and had it been half a mile an hour slower, he ■would have been 2h hours more on the road- Find the distance and rate. 56. A certain crew pull 9 strokes to 8 of a certain other crew, but 79 of the latter are equal to 90 of the former. Which is the faster crew ? Also, if the faster crew start at a distance equivalent to four of their own strokes behind the other, how many strokes will they take before they bump them ? 57. A person, sculling in a thick fog, meets one barge and overtakes another which is going at the same rate as the former ; shew that if a be the greatest distance to which he can see, and b, b' the distances that he sculls between the times of his first seeing and passing the barges, 2^1 I a h h'' STMUL TA NEOUS EQUA TTONS. 1 63 58. Two trains, 92 feet long and 84 feet long respectively, are moving with uniform velocities on parallel rails in opposite directions, and are observed to pass each other in one second and a half ; but when they are moving in the same direction, their velocities being the same as before, the faster train is observed to pass the other in six seconds; find the rate in miles pei» hour at which each train moves. 59. The fore-wheel of a carriage makes six revolutions more than the hind-wheel in 120 yards ; but only four revolu- tions more when the circumference of the fore-wheel is increased one-fourth, and that of the hind-wheel one-fiith. Find the circumference of each wheel. 60. A person rows from Cambridge to Ely (a distance of 20 miles) and back again in 10 hours, and fihds he can row 2 miles against the stream in the same time that he rows 3 miles with it. Find the rate of the stream, and the time of his going and returning. 61. A number consists of 6 digits, of which the last to the left hand is 1. If this numl>er is altered by removing the 1 and putting it in the unit's place, the new number is three times as great as the original one. Find the number. XVII. ON SQUARE ROOT. 220. In Art. 97 we defined the Square Root, and explained the method of taking the square root of expressions consisting of a single term. The square root of a positive quantity may be, as we explained in Art. 97, either positive or negative. Thus the square root of 4a'- is 2a or - 2a, and this ambiguity is expressed thus, J4a^=±2a. In our examples in tnis chapter we shall in all cases regard the square root of a single term as a positive quantity. l64 OKT SQUARE ROOT. 221. The sfjuare root of a product may be found by taking the square root of each factor, and multiplying the roots, so taken, together. Thus y/^' = ab, 222. The square root of a fraction may be found by taking the square root of the numerator and the square root of the denominator, and making them the numerator and denominator of a new fraction, thus V4a^_2a 8lP" 9b 4 96' 2bx-y^ _ 5ory^ 492'^ ^~72^' Examples. — Ixxix. Find the Square Root of each of the following expressions ; 2. Slants. 3. 121mio?ii2,.u. 5. 11289a*b^z^. 6. lG9a^%^c^^. 1 25a^6<' I. 4x-y'^. 4- Ma'^b^^cl 9a2 y- 1G62- 256x^2 289/" 4a2c** ^* 121x«i/">' 625«2 "• 3246'-i* 223. We may now proceed to investigate a Rule for the extraction of the square root of a compound algebraical expression. "We know that the square of a + 6 is a'^ + '2nb + b'^, and there- fore a + 6 is the square root of a- + 2ab + b'. If we can devise an operation by which we can derive a + b from a^ + 2ab + b', we shall be able to give a rule for tlie extraction of the square root. Now the first term of tlie root is the square root of the first term of the square, i.e. a is the square root of a^. Hence our rule begins : "Arrange tlie terms in the order of magnitude of the indices of one of the quantities involved, then take the square root of the ON SQUARE ROOT. idg jirst term and net down tlie result as the first term of the root: subtract its square from the given expression, and bring down the remainder :'' thus d^ + 2ab + b- (a a- 2ab + b'^ Now this remainder may he represented thus &(2a + 6^: hence if we divide iab + b"^ by 2a + b we shall obtain rh the second term of the I'oot. Hence our rule proceeds : '" Double the first term of the root and set 'fowr the result as the first term of a divisor:'' thus our process up to this point will stand thus : a^ + 2ab + b^ [a a? 2a , 2a6 + &2 Now if we divide 2ab by 2a the --eRult is b, and hence we obtain the second term of the root, and if we add this to 2a we obtain the full divisor 2a + b. Hence our rule proceeds thus : '• Divide the first term of the remainder by this first term of the divisor, and add the result to the first term of the root and also to the first term of the divisor:" thus our process up to this point will stand thus : a^ + 2ab + b-{^a + b a2 2a+b 2ab + 62 If now we midtiply 2a + 6 by 6 we obtain 2ab + b^, which we subtract from the first remainder. Hence our rule proceeds thus : ^'Multiply the divisor by the second term of the root and sub- tract the result from the first remainder :' tiius our process will stand thus : i66 ON' SQUARE KOOT. a2 + 2a6+6%a + fe o2 2tH-6 2a6 + 62 2a6 + 62 If there is now no remainder, the root has been found. If there he a remainder, consider the two terms of the root already found as one, and proceed as before. 224. The following examples worked out will make the process more clear. (1) o2-2a6 + 62(^a-6 ft2 2a- 6 I -2a6 + 62 ■ -2a6 + 62 Here the second term of the root, and consequently the second term of the divisor, will have a negative sign prefixed, because ->, — = -o. 2a (2) (3) 6p + 42 101-6 9p^ + 24pq+l6q-(^3p + 4q 9p2 24pq + 16^2 24pq + IQq- 25x2-60x + 36(5x-6 25x'- - 60z + 36 -60X + 36 Next take a case in which the root contains three terma. a- + 2ab + b- — 2ac - 2bc + c-{^a + b — e a2 2a + 6 2ab + b--2ac-2bc + c^ 2ab + 6"- 2a + 26 - c - 2ac - 26c + c^ - 2ac - 26c + c* ON SQUARE ROOT. 167 When we obtained the second remainder, we took the double of + 6, consiflereJ as a single term, and set down the result as the first part of the second divisor. We tlien divided the first term of the remainder, — 2ac, by the first term of' the new- divisor, 2a, and set down the result, - c, attached to the part of the root already found and also to the new divisor, and then multiplied the completed divisor by -c. Similarly we may proceed when the root contains 4, 6 or more terms. Examples.— Ixxx. Extract the Square Eoot of the following expressions : 1. 4a- + V2ab + 9b\ 6. x^ - 6x^ + I9x' - 30x + 2b. 2. lG¥^-24kH^ + 9l^ 7. 9x^+12x3+ 10x2 + 4x+l. 3. a-b-+l62ab + 65Gl. 8. 4r*- 12)-3+ 13?---Gr+ 1. 4. /-38?/3 + 361. 9. 4)i* + 4)i3-7n2_47i + 4. 5. 9a26V - 102a6c + 289. 10. l-6x+ 13x2-12x3 + 4x* 11. x8- 4x5 + 1 Ox* -12x3 + 9x2. 1 2. 4y* - 12yh + 2oyh^ - 24yz^ + 16a*. 13. a^ + 4ah + 4¥ + 9c' + 6ac + '[2bc. 1 4. a^ + 2a'6 + 3a^b- + 4a"¥ + 3a-b* + 2ab^ + W. 15. x8-4x5 + 6x3 + 8x-' + 4x+l. 1 6. 4x* + 8ax3 + 4a2x2 + 1 662x2 + \<oab-x + \ 66*. [7. 9 - 24x + 58x2 - 116x3 + 129x* - 14.0x5 + ioOx«. f 8. 1 6a* - 4Qa?b + 2ba?b' - 80a62x + 646'x2 + 64a26a;. 1 9. 9a* - 24a^p^ - •^OaH + 1 da-f + 40apH + 25^2, 20. 4?/*x2 - 1 2 y^x^ + 1 7i/2x* -\2yx^ + 4x^. 2 1 . 25x*2/2 _ 30x37/3 + 29x2?/* - 1 2xif + 4y^. 22. 16x* - 24x3?/ + 25x2y2 _ 12x»/3 + 4y\ 23. 9a2-12a6 + 24ac-166c + 452 + i6c2. 24. x* + 9x2 + 25-6x3+10x2-30x. 25. 25x2 _ 20x2/ + 4 )/2 + 9^2 _ 1 2i/3 + SCtea. 26. 4x2 (a;2 _ ^) + ^^3 (j, _ 2) + y2 (43.2 4. 1). i68 ON SQUARE ROOT. 225. "When .any fractional terms are in tlie expression of which we have to liiid the Scjuare Root, we may pmceed as in the Examples just given, taking care to treat the fractional terms in accordance witli the rules relatiun to fractionsi 8 16 Thus to find the square root of '•''^ — K^ + o^• 9 81 .,8 16/ 4 X" — -.c + ■ 9 81 {^-\ 2x- 8 16 9^ + 81 8 16 ■9^-*-8l Since 8 c_8_2_8 1_4 9^ 9 • 1~9''2~9- 8 16 Or we niis^ht reduce x^--^z + -^ to a single fraction, whicli 9 oi would he 81.T--72X4-16 8l ' and then take the square root of each of the terms of the fraction, with the I'oUowin result : 9x-4 , . , . ^, 4 ' — - — , which IS the same as x - -. Examples.— ixxxi. I. 4a'' + -~ - a^b-. lb 9 ^ a-' a- 9 4- TT + 2 + -T. 5. x*-2x^^ + 2x--x + - 3. «*-2 + -7. 6. X* + 2x3-.T+ ,. 4 353 54 7. 4a- - 12a& + ah"- + 9)h- - t + tt-- ' 2 lo ON CUBE ROOT. '^9 16 32 8. x^ + 8x- + 24 + ^ + --,-. X* x^ 9 ^ . 16 p ., .^ , „ , ,16 , lb y i 1 4 9 4 6 12 a;^ y'' z- xy xz yz 71 5 n^ 25 .)». 12. a^J^ - Gffkrf + " ■'- + 9c'ch + -Jq- ~. 4x2 j,2 9!/2 • ^ 6?/ 12xw ^-^ X- 3- X 2^ 4??i2 9,i,2 16^ 247i 14. -—+-^ + 4 + — . n- nf n iii cfi ^ h' c- d? ah 2ac ad be bd cd^ ^ 5 ■ ¥ "^ iTi "^ 2^ '*' T ~ 6 "*" Ts" ~ y ~ To "^ T " y * 1 6. 49x4 - 28x3 - 1 7x2 + 6x + ?. 4 1 7. 9x* - 3ax^ + 66x3 + abx'^ + b^x^. 4 1 8. 9x* - 2x3 - l^lx'' + 2x + 9. XVIII. ON CUBE ROOT. 226. The Cube Root of any expression is that expression whose cube or third power gives the proposed expression. Thus a is the cube root of a^, 3b is the cube root of 276^. Tlie cube root of a negative expression will be negative, for since (-a)3=— ax —ax -a=-a^, the cube root of - a^ is —a. 170 ON CUBE ROOT. So also - 3a; is the cube root of - 27:c-'', and — 40^6 is the cube root of - %\a^}?. The sA^mliol I] is used i-o denote tJae operation of extracting the cube root. Examples. — Ixxxii. Find the Cube Roots of the following expressions : I, Sal 2. 27xY- 3- -125m3n3. 4. -216ai263. 5. 34361^8. 6. - lOOOa^iVa. 7. -1728m2in24. 3. 133100/^13/ 227. We now proceed to investigate a Rule for finding the cube root of a compound algebraical expression. We know that the cube of a + & is a^ + 3a-5 + 3a6- + 5^, and thereiore a + 6 is the cube root of a'' + 3a-6 + 3a62 + ft'. We observe that the first term of the root is the cube root of the first term ol the cube. Hence our rule begins: "Arrange the terms in the orrler of magnitude of the indices of one of flic quantities involved, then take the cube root of the first term and set down the result as the first term of the root: subtract its cube from the given expression, and bring down the remainder;" thus a^ + 3a-b + 3ab^ + ¥{^a a3 3a-b + -3ab' + P Now this remainder may be rejiresented thus, b {3a- + 3ab + b-) ; hence if we divi<1e 3a-b + 3ab- + P by 3a- + 3ab + b^, v:e shall obtain +b, the second term of the root. Hence onr rule proceeds : " Multiply the sqiiare of the first term, of the root by 3, and xet doivn the result as the firU term of a divisor:" thus our process up to this point will stand thus : OiV CUBE ROOT. 171 a3 + 7,orh + 3a&- + h^ (a a 3a2 I 3a26+3a62 + 63 Now if we divide 3a-6 by 3a2 the result is 6, and so we obtain the second term oi the root, and if wc add to 3a2 the expression 3a6 + 6'- we obtain the full divisor 3ti- + 3a6 + 6^. Hence our rule proceeds tlius : " Divide the first term of the remainder by the first term of the divisor, and add the result to the first term of the mot. Then take three times the product of the first and second terms of the root, and also the square of the second term, and add these results to the first term of the divisor." Thus our jsrocess up to this point will stand thus : a^ + 3a^6 + ZalP' + 6^ (^a + 6 «3 3a2 + 3a6 + 62 Za'-h + ZaV' + l^ If we now multiply the divisor by h, we obtain 3a26 + 3a6- + 6^ which we subtract from the first remainder. Hence our rule proceeds thus : "Multiply the divisor by the second term of the root, and sub- tract the result from the first remainder :" thus our process will stand thus : a3 + 3a26 + 3aZ)2 + &3(,a + 6 a 3a2 + 3a6 + 62 3a- 6 + 3a 6- + 1^ Za-b + Zalfi + h^ K tliere is now no remainder, the root has been found. If there be a remainder, consider the two terms of the root already found as one, and proceed as before. 228. The following Examples may render the process more clear: 172 ON CUBE ROOT. Ex. 1. 3a2-12o + 16 rt3_i2a2 + 48ft-64(a-4 «3 12a'- + 48a -64 12a2 + 48a-64 Here observe that the second term of the divisor is formed thus : r- 3 times the product of a and — 4 = 3xax -4= — 12a. Ex. 2. a;« - 6r^ + 15a;* -20x3 + 15x2 -6x+l(x2-2x+l 3x-* - 6x3 + 4_j2 _ (jj.-, + X5^.4 _ 20x3 + 15x2 - 6x + 1 -6x-'+ 12x^-8x3 3x'*-12x3 + 15x2-6x+l 3x*- 12x3 + 15x2 -6x + l 3x* - 12x3 -f 15x2- 6x + l Here the formation of the Urst divisor is similar to that in the preceding Examples. The formation of tlie second divisor may be explained thus: Regarding x2 — 2x as one terra 3 (:c2 - 2x)2 = 3 (x* - 4x3 + 4^2) = 3_^4 _ I2x3 + 12x2 3x(x2-2x^xl = 3x2-6x 12 = i and adding these results we obtain as the second divisor 3x* -12x3 + 15x2 -6x + l. Examples. — Ixxxiii. Find the Cube Root of each of the following expressions: 1. a^-2>d^h-vZa\r-h^. 2. 8a3 + 12a2 + 6a + 1. 3. o3 + 24a26 + 192a62 + 51263. 4. a3 ^ 3(^2^ + 3rt^2 + 53 + 3„2(. 4. 6a5g ^ 352(5 + 3^^^ + "ihc- + cl 5. JC3 - 3x2)/ + 3j;j^2 _ y^ + 3j.2~ _ g_j.y.. ^ 3j^2~ + 3jv2!i _ 3y~2 + j^^ 6. 27x" - 54x° + 63x* - 44.t3 + 2 1x2 - 6x + 1. ON CUBE ROOT. 173 7. 1 - 3a 4- 6a- - 7a3 + 6a* - 3a'' + a". 8. x^ - 3x-i/ + 3x1/2 _ ^3 + 833 + 6x^2 - Vixxjz + G?/-2 + 12x32 _ i2y;:2_ 9. a" - 12a-^ + 54a* - 1 12a3 + 108a2 - 48a-+ 8. 10. 8»^'' - 367/t= + 66»i* - %Zm^ + 33??i2 - 9m + 1. 11. x^ + 6x-.v + 1 2x.i/2 + 8?/3 _ 3x-2 - 1 2xw2 - 1 2 v^s + 3a;~2 + ^y^i _^z_ 12. %m? - 36?7i-?i + 547JiJi,2 — 2Tu' — Mm-r + 36m?(?- — 27'7i-r 4- 67?ir2 — 9?i7"2 — r'. 1 3. ?7i3 + 3??i2 - 5 H 5 =. "^ ?/(." 7?l'* 229. The fourth root of an expression is found by taking tlie sejuare root of the s(|iiare root of the expression. Thus 4/16aS6* = ^l^a^h" = 2a26. The sixih root of an expression is found by taking the cube root of the scj^uare root of the expression. Thus 4/64ai266 = .^8a663 = 2a26. Examples.— Ixxxiv. Find the fourth roots of 1. 16u*-96a3x + 216a-x2-2T6ax3 + 81cc*. 2. l + 24a2 + 16a<-8a-32al 3. 625 + 2000X + 2400x2 + 1280x3 + 256a;^. Find the sixth roots of 4. a« - ^w'h + 15a*62 - ma?\? + ISa^t* - Gai^ + ^6. 5. x6 + 6x5 + 15x'- + 20x3+ 15x2 + 6x + l. 6. m" - 12771^ + 60771* - 160?jr + 2407?i2 - 192m + 64 XIX. QUADRATIC EQUATIONS. 230. A Quadratic Equation, or an equation of two dimen- sions, is one into which the square of an unknown symbol enters, without or with the tirst power ol' the symboL Tims a;2=16 and x--i-6x = 27 are Quadratic Equations. 231. A Pure Quadratic Equation is one into which the square of an unknown symbol enters, the fii-st power of the symbol not appearing. Thus, x-=16 is a. pure Quadratic Equation. 232. An Adfected Quadratic Equation is one into which the square of an unknown symbol enters, and also the lii-st power of the symbol. Thus, x^ + 6x = 21 is an adfected Quadratic Equation. Pure Quadratic Equations. 233. When the terms of an equation involve the square of the unknown symbol oiibj, the value of this square is either given or can be found by tlie pi-ocesses described in Ciiapter XVII. If we then e.xtract the square root of each side of the equation, the value of the unknown symbol will be determined 234. The following are examples of the solution of Pure Quadratic Equations. QUA DRA TIC EQUA TIONS. I7S Ex. 1. x^=\%. Taking the square root of each side x=±4. We prefix the sign ± to the number on the right-hand side of the etjuation, for the reason given in Ait. 220. Every pure quadratic equation will therefore have two roots, equal in magnitude, but with different signs Ex.2. 4a;2 + 6 = 22. Here 4x'- = 22-6, or 4x^=16, ' or x- = 4 ; .-. x=±2. That is, the values of x which satisfy the equation are 2 and - 2. Ex. 3. '^^ -^^ Here 128 (5x2-6) = 216 (3x2-4). or 640x2-768 = 648x2-864, or x2= 12 ; .-. X=±V12. Examples.— ixxxv. I. x'^=Qi. 2. x2 = a252_ 3. x2- 10000 = 0. 4, x2-3 = 46. 5. 5x2-9 = 2x2 + 24. 6. 3ax2=192a5c6. x2 — 12 x2-4 o 7. - — - — = — -. — . II. mx- + n=q. 3 4 ^ 8. (500 +x) (500- x) = 233359. 12. x2-ax + 6 = ax(x- 1) 8112 45 57 9- — =3x. 13. 2^J-3 = 4^:r5- rl ■> ,„ r.. /^ OW 42 35 10. 5-x--18x + 6o = (3x-3)2. 14. ^^32 = ^733- 176 QUADRA TIC EQUA TTO.VS. Adfeded Quadratic Equations. 235. Adfectecl Qnailratic Equations are solved by adding a certain term to both sides of tiie e(.|uatiou so as to make the left-hand side a perl'ect sq^uare. Having arranged the equation so that the first term on tlie left-hand side is tlie square of tlie uui<no\vn symbol, and the second term the one containing the lirst power of tiie unknown, quantity (the known symbols being on the right of the equa- tion), we add to both ddes of the equation the square of half th-e coefficient of the second term. The left-hand side of the equa- tion then becomes a perfect square. If we then take the square root of both sides of tiie equation, we shall obtain two simple equations, fiom which the values of the unknown symbol may be determined. 236. The process in the solution of Adfected Quadratic Equations will be learnt by tlie examples which we shall give in this chapter, but before we proceed to them, it is desirable that the student should be satisfied as to the way in wliich an expression of the form x^ + ax is made a perfect square. Our rule, as given in the preceding Article, is this : add the square of half the coefficient of the second term, that is, the square of 5, that is, -^. We have to shew then that 4 is a perfect square, whatever a may be. This we may do by actually performing the operation of extracting the square root of x'^ + ax + —, and obtaining the result X + A with no remainder. QUADRATIC EQirATIOiVS. 1 77 237. Let us examine this process by the aid of numerical coefficients. Take one or two examples from the perfect squares given in page 48. We there have x^+ 18x+ 81 which is the square of x+ 9, a;2 + 34x + 289 x+ll. ic- — 8x + 16 X— 4, a;2-36u; + 324 cc-18. In all these cases the thinl term is the square of half the coefficient of x. For 81= (9)^ = (\^)', 289 = (17)^ = (=^y, 324 = (18)2 = (''|y. 238. Now put the question in this shape. What must we add to X' + ax to make it a perfect square i Suppose b to represent the quantity to be added. Then x'^ + ax + 6 is a perfect square. Now if we perform the operation of extracting the square root of x- + ax + b, our process is x^ + ax + hi x + - X 2x + H ax + 6 2 a" ax + --r- 4 '-T faA.] M 178 Q VADRA TIC EQUA TlONS. Hence in order that x^ + ax + 6 may ba a perfect square we must have t 4 i-?-o, or t=-, (ly That is, 6 is equivalent to the square of half the coefficient ofx. 239. Before completing the square we must be careful (1) That the square of the unknown symbol has no coeffi- cient but unity, (2) That the square of the unkno^vn symbol has a positive These points will be more fully considered in Arts. 245 and 246. 240. We shall first take the case in Avhich the coefficient of the second term is an even number and its sign positive. Ex. a;--l 6x = 40. Here we make the left-hand side of the equation a perfect square by the following i)rocess. Take the coetficient of the second term, that is, 6. Take the half of this coefficient, that is, 3. Square the result, which gives 9. Add 9 to both sides of the equation, and we get x2-|-6x + 9 = 49. Now taking the square root of both sides, we get x + 3=±7. QUADRATIC EQUATIONS. 179 Hence we liave two simple equations, a; + 3=+7 (1), and 35 + 3= -7 .'. (2). From these we find the values of x, thus: froui (1) x = 7-3, that is, x = 4, from (2) x= — 7 - 3, that is, a;= - 10 Thus the roots of tlie equation are 4 and - 10. EXAMPLES. — IXXXVi. I. x- + 6x = 72. 2. a;-+12x = 64. 3. a;2 + 14x = 15. 4. x2 + 46x = 96. 5. x-+128x = 393. 6. x- + 8x-65 = 7. x2+18x-243 = 0. 8. x'-^ + 16x- 420 = 0. 241. We next take the case in which tlie coethcient of the second term is an even number and its sign negative. Ex. x^-8x = 9. The term to be added to both sides is (8-7-2)^, that is, (4)-, that is, 16. Completing the square x2-8x+ 16 = 25. Taking the square root of both sides z-4=±5. This gives two simple equations, a;-4=+5 (1), a;-4=-5 (2), From (1) x=5+4, .-. x = 9; from (2) x=-5 4-4, .-. x=-l. Thus the roots of the e'j;iation are 9 and - 1. I 1 80 Q UADRA TIC EQ UA TIONS. EXAMPLES. — IXXXVii. I. a;2-6a: = 7. 2. x--Ax = ^. 3. a;2-20x = 21. 4. a;2-2x = 63. 5. a;2- 12x+ 32 = 0. 6. x2-14x + 45 = () 7. x'' - 234x + 13688 = 0. 8. (x - 3) (x - 2) = 3 (5x + 14). 9. x(3x-17)-x(2x + 5) + 120 = 0. 10. (x - 5)- 4- (x - "7)- = X (x - 8) + 46. 242. We now take the case in wliich the coefficient of the second term is an oiii number. Ex. 1. x2-7x = 8. The term to be added to both sides is Completing the square , ,. 49 ^ 49 o ^ 49 81 or, x'^ - 7x + -r = -r- 4 4 Taking the square root of both sides 7 .9 •"-2=±2- This gives two simple equations, 7 9 ^-2=+2 <!>• 7_ 9 ^^ ^~2~~2 ^''^• From (1) ^"^9+2' or!^=9-) •■•x = 8; 9 7 —2 from (2) x= - - + -, or, x = — -, .-. x= -1. Thus the roots of the equation are 8 and - 1. QUA DRA TIC EQUA TIONS. l8l Ex. 2, a;2-x = 42. The coefficient of the second term is 1 The term to be added to both sides is /. a;' - x + :: = 42 + - 4 4 1 169 or, x--x + ^= — ; 1 ^13 2 - 2 Hence the roots of the equation are 7 and —6. Examples. — Ixxxviii. I. a;2+7a; = 30. 2. x2-llx=12. 3. x2 + 9x = 43-. 4. x2-13x=140. 5. x2 + x = — . 6. x2-x = 72. 7. x2 + 37x = 3690. 8. x2 = 56 + x. 9. x(5-x)(-2x(x-7)-10(x-6) = 0. 10. (5x-21)(7x-33)-(17x+15)(2x-3) = 448. 243. Our next case is that in which the coefficient of the second term is a fraction 0/ which the numerator is an even number. Ex. i?-jx = 2\. 5 The term to be added to both sides is 4 4 4 5 2o 2o „ 4 4 529 r82 Q UADRA TIC EQUA TIONS. 2 ^23 5 - 5 21 Hence the values of x are 5 and - -=-. Examples. — Ixxxix. „ 2 35 ,4 3 „ 28x 1 ^ I. x2--x = -g-. 2. ^^ + 5^= -25- 3. ^''-9- + 3 = f^- ,83_ ,43 -„16 16 4- ^ -n^-ll = ^- 5- ^^ + 35^ = 7- 6. x==-y-'; = y. 7. x2-|^x + ^| = 0. 8. a;2_4 ^45_ 244. "We now take the case in which the coefficient of the second term is a fraction u7i-ose numerator is an odd number. Ex. ^-W^- The term to be added to both sides is 2 7 49_13G 49 ''•'' ~3'' + 36~ 3 "^36' „ 7 49 1681 °' ^-3-^-^36 = -36-' 6-6 17 Hence the values of x are 8 and — ^. Examples.— xc. I. X2-2T=8. 2. x2-j2; = 98. 5 3. .x2 + _a. = 39. 4. x^ + ^x=76. g 5. x2--x=16. 6. x--^x + 6 = 0. 7. x2-— X- -34 = 0. 8. ^-3 3 / 4 QUADRATIC EQUATIONS. 183 245. The square of the unknown symbol tnust not be pre- ceded by a negative sign. Hence, if we have to solve the equation • 6x — X- = 9, we change the sign of every temi, and we get x2-6x= -9. Completing the square a;2-6x + 9 = 9-9, or x^ - 6x + 9 = 0. Hence a; - 3 = 0, or x = 3. Note. We are not to be surprised at finding only one vaJue for x. The iuterpretatiuu to be j>laced on such a result is, that the two roots of the equation are equal in value and' alike in sign. 24(5. The square of the unknown symbol must have no coefficient but wiity. Hence, if we have to solve the equation 5x2-3x = 2, we must divide all the terms by 5, and we ^.et , 3x 2 X" — — = -. o o 2 From which we get x = l and x= — -. 247. In solving Quadratic Equations involving literal co- efficic-nts of the unkiiown symbol, the same rules will apply as in tlie cases of numerical coefficients. Thus,' to solve the equation ?^-?-2 = 0. X a Clearing the equation of fractions, we get 2a2-x2-2ax = 0; therefore -x^-2ax= -2a\ or x^ + 2ax = 2a^. r84 Q UADRA Tld EQUA TIONS. Completing the sqiiare X- + 2ax + a^ = Sa^, whence* x + a=^±. ^J'i • a ; therefore a; = - o + ^3 . a, or x = — a - ^^3 . a. The following are Examples of Literal Quadratic Equations. EXAMPLES.— XCi. 7m- I. x2 + 2ax=a2. 2. x^ — 4ax = 7a2. 3. x^ + Zmx = -^ . , 5n ._372^ _rt2 fe2 4. ^--Y^-~2~- ''■ (x + a)2 (x-«)2-*- 5. x^ + (a-l)x = a. 8. adx-acx- = bcx-bd. 6. x'^+ {a-h)x = ab. 9. cx-{ — -y = (a + b)3:?^. a"x^ 2ax b- ^ 10. -To +-2 = 0- ^ , Sa^x 6a2 + a6-262 JZx 11. abx--{ = „ . . c c^ c 12. (4a2 - 9cd-) X- + (4a2c2 + 4abd'-) x + (ac^ + bd^^ = 0. 248. If both sides of an equation can be divided by the nnknowTi symbol, di\ide by it, and observe that is in that case one root of the equation. Thus in solving the equation x^-2x2 = 3x, we may divide by x, and reduce the equation to the form x2-2x = 3, from which we get x = 3 or .r= - 1. Then the three roots of the original equation are 0, 3 and - 1. We shall now give some Miscellaneous Examples of Quad- ratic Equations. Q UADRA TIC EQUA TIONS. 185 Examples.-— xcii. I. x2-7a; + 2 = 10. 2. x--5x + 3 = 9. 3. a;2-llx-7 = 5. 4. x2-13x-(j = 8. 5. x'- + 7x-18 = 0. 6. 4x - -"---^ = 22. x-3 7. x--9x + 20 = 0. 8. 5x-3— ~4 = — J,— ■ x-3 2 9. x--Gx-14 = 2. 10. —^ -—:!?- = 2. ^ x-^3 2x + 5 4x X- 7 x + 7~2xT3' 14- 2^^""~3^+~8 = ^- ^5- '^•^ - = 26. 16. 2x- = 18x-40. 4 + 3x 15 — X 7x — 14 „ , o 3x-5 ^f__l 7_2x-5_3x-7 '^' 9x 3x-25~3' ^°' 4~^+y""~27~' 4X-10 7-3x 7 , ^,., , ,, 21. ; =-. 22. (x-3,- + 4x = 44. x + o X 2 ^ -^ x+11 „ 9 + 4x ^ , „ ,11 21. —^='-—^—' 24. 6x- + x = 2. 25. x--^x = ^. 26. x2-x = 210. 27. — % + - = 3. 28. ^-11=5. ' X 4- 1 X 3 3 X 3 x-1 = 15. 1 2 x + 2' 3 "5' 10 3- ¥" 14 -2x X- 22 " 9" 12 8 32 30. 15x'- — 7x = 46. 32. 4x 20 - 4x 5-x X 34- X 7 x + 60 3x-5' — + -- = 2-- ' 5-x'4-x x-f2' •'"' 7-x X 10' JV ^- + 7-^ = -,-o- 36. ;^— ; + — — = 2 37. x-+(a + 6)x + rt6 = 0. 38. x2-(6-a)x-a6 = 0. 39. x^ - 2ax 4- rt' — t- = 0. 40. X- - (rt- -a^)x — a^ = 0. , a 2a- „ a- + 62 41. x2 + ^x--^ = <-. 42. x---^x + l=0. XX. ON SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS. 249. For the solution of Simultaneovis Equations of a de- gree hi^'Iier than the first no lixed rules can be laid down. We shall ])oiiit out the methods of solution which may be adopted with advantage in particular cases. 25('. If the simple power of one of the unknown symbols can be expressed in terms of the other symbol by means of one of the given equations, the Method of Substitution, explained in Art. 217, may be employed, thus: Ex. To solve the equations x + y = 50 xy = 600. . From the first equation x = 50-y. Substitute this value for x in the second equation, and we get (50 - y) . y = 600. This gives 50-/- i/^ = 600. From which we find the values of y to be 30 and 20. And we may then find the corresponding values of x to be 20 and 30. 251. But it is better that the student should accustom himself to work such equations symmetrically, thus : To solve the equations x + y = 50 (1), x!/ = 600 (2), From ( 1 ) x'^ + 2xy + y- = 2500. From (2) 4x7/ = 2400. O^ SIMULTANEOUS EOUArrONS. &^c-. 187 Subtracting, x^ — 2xy + y- = 100, :. x-y=± 10. Then from this equation and (1) we find X = 30 or 20 and y = 20 or 30. Examples. — xciii. I. x + y — 40 2. x+v = 13 3. a; + i/ = 29 xj/ = 300. xy = m. xy=\ 00. 4. .c — 1/=19 5. x-y = 45 6. x-?/ = 99 •c?/ = 66. xy = 250. i^l/ = 1 00. 252. To solve the equations x-y=\2 (1), a;2+!/2 = 74 (2). From (1) '*x2-2.r;/ + i/2 = 144 V^)- Subtract this t'roui (2), then 2x1/= -70, .• 4x1/= -140. Add this to (3), then x^ + 2xy + y- = A, :. X + 1/ = ± 2. Then from this equation and (1) we get X = 7 or 5 and y= — 5 or — 7. Examples. — xciv. I. x-y = A 2. x-y=l() 3. x-«=14 x2 + 2/2 = 4o. x2 + 2/2 = 178. x2 + i/2 = 436. 4. x-t-i/ = 8 5. x + y = l-2 * 6. x + i/ = 49 x2 + i/2 = 32. x2 + i/2=104. ' x2^y2=,ie81. i88 ON SIMUL TANEOUS EQUA TJONS 253. To solve the equations a^ + 2/3 = 35 (1), a; + 2/ = 5 (2). Divide (1) by (2), then we get x^-xy + f^l (3), From (2) a;2 + 2xi/ f i/^=25 (4), Subtracting (3) from (4), 3a;2/ = 18, .". 4^2/ = 24. Then I'rum this equation and (4) we get x''^-2x?/ + i/2= 1, :.x-y=±\; andl'rom this etiuation and (2) we find x = 3 or 2 and ^ = 2 or 3. Examples.— xcv I. ftr' + i/3 = 91 2. a;^ + i/ = 341 ' 3. x3 + i/3 = 1008 x + y = l. x^y = \\. x + i/=12. 4. ic3_y3 = 5(J 5, 3.3_^3 = 98 6. x3-i/3 = 2T9 x-y = % x-y = 2. x-y = S. '254, To solve the equations s+r6 ^'^' 1 1 13 ,-, F + ?=3«-' ^')- From (1), by squaring it, we get 1 2 1 25 ,.,, -^ + — + -2=^ (3). x^ xy y' 3d From this subtract (2), and we liave ^_'12 xi/ 36 ' . ^_24 xn 36' IXVOLVING QrADA\4 7/CS. 189 Now subtract this from (3), and avo gi't x^ xy y'^ 36' X y - o • and from this equation anil (1) we find x = 2 or 3 and i/ = 3 or 2. Examples. — xcvi. I. 1 1 9 x'^y~20' 2. 1 1_3 X y~A' 3- 1 1 r. X 1/ 1 1 41 1 1 5 x2"^/~16' -^ + -4 = 13. X- 1/- 4. 1 1_ 1 X y~\-2 5- 1-1=2^ X y 2 6. 1-1 = 3. X ?/ I 1 7 x^ 1/- 144' x^ ?/^ 4" X^ i/^ 255. To solve the equations x2+3xj/=7 a), ^ + 4y'-=18 (2} If we add the equations we get x- + 4xi/ + 4!/- = 25. Taking the square root of each side, and taking only the positive root of the riglit-hand side into account, X + 2?/ = 5 ; .•. x = 5- ly. Substituting this value for x in (2) we get (5-22/)2/ + 4!/2=18, an equation by which y may be determined. Note. In some examples we mu-t subtract the second equation from the first in order to get a perfect square. 190 ON SIMULTANEOUS EQUATIONS 256. To solve the equations x^-f = i^ (1), a;2 + xi/ + i/2=p (2). Dividing (1) by (2) we get x-i/ = 2 '3), squaring, x'--'±xy-\-y- = \ (4}. Subtract this from (2), and we have 3^2/ = 9; .'. 4x?/ = i2. Adding this to (4), we get a;2 + 2.rj/ + ?/2= 16 ; .-. X + ?/ = ± 4. Then from this equation and (3) we lind a; = 3 or — 1 , and ;/ = 1 or — 3. 257. To solve the equations a;2 + y2 = (;5 ^Y\ 01/ = 28 (2,. Multiplying (2) by 2, we have ! + i/2 = 65) 2a:?/ = 56) ' .-. X2 + 2X!/ + J/2=121^_ a;2-2x2/ + i/= 9) ' .-. x + i/=±ll (A), x-xj=± 3 (B). The equations A and B furnTsh four pairs of siiii])lr equations, x + i/=ll, .r + ?/=ll, a- + !/=-ll, a: + )/=-ll, x-i/ = 3, x-2/=-3, x-2/ = 3, x-?/=-3. from whicli we find the values of x to be 7, 4, -7 aixl -4. and the corresponding values of ?/ to be 4, 7. —4 and - 7. 258. The aititiee, l\v wliich the solution of the equation.^ eiven in this article is eti'ected, is a])plicable to cases in wliich the equations are homogeneonx mid <>/ the savie orilcr. INVOLVING QUADT?ATICS. 19^ To solve the equations x2 + xy = 15, Suppose y = mx. Then x'^ + mx^=l^. from the first eonation. and mx'^ -m'^x^ = z, from the secona equation. Dividing one of these equations by the other, x^ + mx^ _15 mx^ - m^x~ 2 ' x^n+m) 15 or ^^ — = — x^ {m - m^) 2 ' 1 + m .15 o^ 2= o- m — m^ 2 From tliis equation we can determine the values of m. 2 One of these values is ^, and putting this for m in the o 2 equation x- + 7nx-= 15, we get x'^ + -x''=l5. From which we find a;= ±3, and then we can find y from one of the original equations. 259. The examples which we shall now give are intended as an exercise on the methods of solution explained in the four preceding articles. Examples.— xcvii. I. a;^ — y^ = 37 2. x- + 6x?/ = 144 3. x'^ + xy = 2l0 x- + xy + y- = 37. 6xy + 36?/ = 432. y- + xy = 23 1. 4. a;2 + 7/2 = 68 5. x^ + y^=l52 6. 4x- + 9xy=l90. xy=l(J. x^-xy + y-=l9. 4x-5y = l0. 7. x^ + xy + y- = 39 8. x^ + xy = 6Q 9. 3x- + 4x!/ = 20. 3y'--5xy = 2o. xy — y' = b. 5xy + 2y- = l2. \o. x^-xy + y- = 7 11. x'^ — xy = 35 12. 3x^ + 4xy + 5ij- = 7l. :ix^ + l3xy + 8y- = liy2. xy + y^ = 18. ox + 7y = 29. i2,.'X^ + y-- = 212S 14. u;2 + 9xy = 340 15. x^ + y-==225 x^-xy + y-=l24:. 7xy -y'^=\7l. xy=]()8. XXI. ON PROBLEMS RESULTING IN QUADRATIC EQUATIONS. 260. The method of stating problems resiiltiug in Quad- ratic Equations does not require any general explanation. Some of the Examples which we shall give involve ane unknown symbol, others involve tivo. Ex. 1. What number is that whose square exceeds the number by 42 ? Let X represent the number. Then x^ = x + 42, ^ or, a;'''-x = 42; thereiore x- — x + - = —j- ; 4 4 whence x - ^ = ± -^. And we find the values of .'• to be 7 or — 6. Ex. 2. The sum of two numbers is 14 and the sum of their squares is 100. Find the numbers. Let X and y represent the numbers. Then x-f-!/ = l'i, and .x2 4-r = i00. Proceeding as in Art. 252, we find x = 8 ur 6, y = 6 or 8. Hence the numbers are 8 and 6. ON PROBLEMS RESULTING, dj'c. 193 Examples. — xcviii. *<j \ I. What number is that whose half multiplied by its third part gives 864? 2. What is the number of which the seventh and eiglith parts being multiplied together and the product divided bj' 2 3 the quotient is 298^ ? ^ 3. I take a certain number from 94. I then add the number to 94. I multiply the two results together, and the result is 8512. What is the number ? {.4. What are the numbers whose product is 750 and the quotient of one by the other 3- ? 5. The sum of the squares of two numbers is 13001, and the difference of the same squares is 1449. Find the numbers. * 6. The product of two numbers, one of which is as much •^ above 21 as the other is below 21, is 377. Find the numbers. A \ 7. The half, the third, the fourth and the fifth parts of a ^certain number being multiplied together the product is 6750. Find the number. 8. By what number must 11500 be divided, so that the quotient may be the same as the divisor, and the re- mainder 51 .' g. Find a number to which 20 being added, and from which 10 being subtracted, the square of the first result added to twice the square of the second result gives 17475. 10. The sum of two numbers is 2G, and the siim of their squares is 436. Find the numbers. 11. Tlie difference between two numbers is 17, and the sum of their squares is 325. What are the numbers ? 1 2. What two numbers are they whose product is 255 and the sum of whose squares is 514 ] a-/ '3- Divide 16 into two parts such that their product ' added to the sum of their squares may be 208. [S.A.] N 194 ON PROBLEMS RESULTING \ 14. What number added to its square root gives 'as a result 1:332 ] 3 15. What number exceeds its square root by 48^? 16. What number exceeds its square root by 2550 ? ^ 17. The product of two numbers is 24, and their sum niultiiilied by their difference is 20. Find the number.*. 18. What two numbers are those whose sum multiplied (; by the greater is 204, and whose difference multiplied by the less is 35 ? f\ 19. What two numbers are those whose I'.ifference is 5 'and their sum multiplied by the greater 228 ? ; 20. Find three consecutive numbers whose product is V equal to 3 times the middle number. ^ 21. The difference between the .squares of two consecutive niimbers is 15. Find the numbers. 3 2. The sum of the squares of two consecutive numbers is 481. Find the numbers. 23. The sum of the squares of three consecutive numbers is 365. Find the numbers. ' Note. If 1 buy x apples for y pence, - will represent the cost of an apple in pence. If I buy X sheep for z pounds, - will represent the cost of a sheep in pounds. Ex. A boy bought a number of oranges lor 16(f. Had he bought 4 more for the same money, he would have paid one-third of a penny less for each orange. How many di<l he buy ? Let as represent the number of oranges. Then — will represent the cost of an orange in pence. „ 16 16 1 Hence — = - — , + ^, X a; + 4 3 or 16(3x + 12) = 48x + x2 + 4x, or x2 + 4.i- = 192, from which we find the values of x to be 12 or — 16. Therefore he bouglit 12 oranges. IN Q UA DRA TIC EQCA TIOXS. 19$ 24. T buy a number of handkerchiefs for £\l. Had I bought 3 more for tlie ^^anle money, they would have cost one shilling each less. How many did I buy '{ 25. A dealer bought a number of calves for £80. Had he bought 4 more for the same money, each calf would liave cost £\ less. How many did he buy ? 26. A man lx)ught some pieces of cloth for £33. 15s., which he sold again for £1. 8.?. the piece, and gaiueil as much as one piece cost him. What did he give for each piece ? 27. A merchant bought some pieces of silk for £180. Had he bought 3 pieces more, he would have paid £3 less for each piece. How many did he buy ? 28. For a journey of 108 miles 6 hours less would have sufficed had one gone 3 miles an hour faster. How many miles an hour did one go ? 29. A grazier bought as many sheep* as cost him £60. Out of these he kept 15, and selling the remainder for £54, gained 2 shillings a head by them. How many sheep did be buy ? 30. A cistern can be filled by two pipes running together in 2 hours, 55 minutes. The larger pipe by itself will fill it sooner than the smaller by 2 hours. What time will each pipe take separately to fill it ? 31. The length of a rectangular field exceeds its breadtli by one yard, and the area contains ten thousand and one hundred square yards. Find the length of the sides. 32. A certain number consists of two digits. The left- hand digit is double of the right-hand digit, and if the digits be inverted the product of the number thus formed and the original number is 2268. Find the number. 33. A ladder, whose foot rests in a given position, just reaches a window on one side of a street, and when turned about its foot, just reaches a window on the other side. If tlie two positions of the ladder be at right angles to each other, and the heights of the windows be 36 and 27 feet respectively, find the width of the street and the length of the ladder. tg6 ON PROBLEMS RESULTING, dr-r. 34. ('lot]), bein<,r wetted, shrinks up - in its length and o ~- in its width. If the surface of a piece of cloth is di- 3 minished by 5- square jards, and the length of the 4 sides by 4- yards, what was the length and width of the cloth % 35. A certain number, less than 50, consists of two digits whose difference is 4. If the digits be inverted, the difference between the squares of the number thus formed and of the original number is 3960. Find the number. 36. A plantation in rows consists of 10000 trees. If there had been 20 less rovvs, there would have heen 25 more trees in a row. How many rows are there ? 37. A colonel wished to form a solid square of his men. The first time he had 39 men over: the second time he in- creased the side of the square by one man, and then he found that he wanted 50 men to complete it. How many men were there in the regiment ? XXII. INDETERMINATE EQUATIONS. 261. WHEisr tlie number of unknown symbols exceeds that of the independent equations, the number of simultaneous values of the symbols will be indefinite. We propose to ex- plain in this Chapter how a certain number of these values may be found in the case of Simultaneous Equations involving two unknown quantities. Ex. To find the integral values of x and y which will satisfy the equation 3x + 7y=lO. Here 3a;=10-7|/; .-. x=3-2j/ + ^^. Now if X and y are integers, -— must also be an integer. INDE TERMINA TE EQUA TIOXS. 197 1 —1/ Let — ;— = in, then 1 — ^ = 3?7i ; .". i/ = 1 — 3m, and a; = 3 — 2i/ + m = 3 — 2 + 6m + ?^^ = 1 + 77w ; or the general solution of the equation in whole numbers is x = l + 1 m and y = \ — 3?7i, where rii may be 0, 1, 2 or any integer, positive or negative. If m = 0, x= 1, 1/= 1 ; if m=l, x= 8, 3/= -2; if m = 2, x= 15, 1/= -5; and so on , from which it appears that the only positive inte- gral values of x and y which satisfy the equation are 1 and 1 . 262. It is next to be observed that it is desirable to divide both sides of the equation by the smaller of the two coefficients of the unknown symbols. Ex. To find integral solutions of the equation lx->rby = Z\. Here by = ^\-lx: 1 - 2x 1 — 2e Let — --^ = m, an integer. Then 1 -2x = 5m, whence 2x=l -5fli; 1 — m 2 -2"^- T . 1 — m- . , Let =n, an integer. Then 1 -m = 2n, whence m = l -2n. Hence x = n-27?i = ?i — 2 + 4n = 5n — 2 ; y = 6-x + )n = 6-5?i-l-2 + l-2«, = 9-7». Now if n = (K x= -2, 2/=: 9; if n = l.x= 3,2/= 2; if n=2, X— 8,ys:~-5. and 80 on. 198 INDE TERMLVA TE EQUA TIO.VS. 263. In how many ways can a person pay a bill of £13 with crowns and guineas? Let X and y denote the number of crowns and guineas. Then 5a; + 21?/ --=260; .-. 5a; = 260-211/; x = 52-4v-|. ^ 5 Let ^ = m, an integer. Then y = 5m, and x = 52-4y — m = 52 -21m. If 771 = 0, 2 = 52, y= 0; m=l, x = 31, y= 5; m = 2, a;=10, y = 10; and higher values of m will give negative values of x. Thus the number of ways is three. 264. To find a number which when divided by 7 and 5 will give remainders 2 and 3 respectively. Let X be the number. x-'2 Then — „— =an integer, suppose m; I and =an integer, suppose n. Then x — 7-ni + 2 and x = 5?i + 3; .-. 7?rt + 2 = 5?i + 3; 2m-l :. 5n = 7m - 1, whence n = m + - 5 Let — ' — =p, an integer. Then 2m = 5p+l, whence m=2p -»-— g— v Let ~n— — 9.} *^ integer. Then j9 = 2g-l, m = 2|) + (7 = 4(7-2 + 7 = 5g-2, x=7m + 2 = 357- 12. INDE TERM IS' A TE EQ L 'A T/ONS. I99 Henct if » q = 0,x=-l'2; if Q = \,x= 23; if q=2,x= 58; and so on. Examples. — xcix. Find positive integral solutions of I. 5x + 72/ = 29. 2. 7x+192/ = 92. 3. 13x+19!/=ll70. 4. 3x + 5i/ = 26. 5. \Ax-by = l. 6. llx+15?/ = 1031. 7. llx + 7i/ = 308. 8. 4x-19?/ = 23. 9. 20x-9!/ = 683. 10. 3x + 77/ = 383. II. 27x + 4i/ = 54. 12. 7x + 9^ = 653. 13. Find two fractions with denoiuinators 7 and 9 and their sum -^~. DO 14. Find two proper fractions with denominators 11 and 82 13 and their difference -77:. 14.3 15. In how many ways can a debt of £\. 9s. be paid in florins and half-crowns ? 16. In how many ways can £20 be paid in half-guineas an(f half-crowns ? 17. What number divided by 5 gives a remainder 2 and by 9 a remainder 3 ? 18. In how many different ways may £11. 15a-. be paid in. guineais and crowns ? 19. In how many different ways may £4. lis. Qd. be paid with half-guineas and lialf-crowns .' 20. Shew that 323x- 527?/ =1000 cannot be satisfied by integral values of x and y. INDETERMINATE EQUATIONS. 21. A farmer buys oxen, sheep, and hens. The whole number bought was 100, and the whole price .£100. If the oxen cost .£5, the sheep ;£1, and the hens Is. each, how many of each had he? Of how many solutions does this Problem admit ? 22. A owes B 4s. lOd.; if A has only sixpences in his pocket and B only fourpenny pieces, how can they best settle the matter ? 23. A person has £12. 4s. in half-crowns, florins, and shil- lings ; the number of half-crowns and florins together is four times the number of shillings, and the number of coins is the greatest possible. Find the number of coins of each kind. 24. In how many ways can the sum of £h be paid in exactly 50 coins, consisting of half-crowns, florins, and four- penny pieces \ 25. A owes B a shilling. A has onlj' sovereigns, and B has only dollars worth 4s. 3d. each. How can A most easily pay Bl 26. Divide 25 into two parts such that one of them is / divisible by 2 and the other by 3. 27. In how many ways can I pay a debt of £-1. 9s. with crowns and florins ? :b 28. Divide 100 into two parts such that one is a multiple of 7 and tlie other of 11. 29. The sum of two numbers is 100. The first divided by 3 5 gives 2 as a remainder, and if we divide the second by 7 the remainder is 4. Find the numbers * ^ 30. Find a number less than 400 which is a multii^li^ <•» 7^ 'f and which when divided by 'z, o, *, 5. 6. gives as a iciiiiuuder in each case 1. XXIII. THE THEORY OF INDICES. 265. The number placed over a symbol to express tlie power of the symbol is called the Index. Up to this point our indices have in all cases been Positive Whole Numbers. We have now to treat of Fractional and Negative indices ; and to put this part of the subject in a clearer light, we shall commence from the elementary principles laid down in Arts. 45, 46. 266. First, we must carefully observe the following results : (a3)2=a6. For a^ X a^ = a . a . a . a . a = a^, and (a^y = a^.a^ = a.a.a.a.a .a=a^. These are examples of the Two Rules which govern all combinations of Indices. The general proof of these Rules we shall now proceed to give. 267. Def. "When m is a positive integer, a" means a, a. a with a written m times as a factor. 268. There are two rules for the combination o^ndices. Rule I. a'"xa" = a''^. Rule II. {'(*")••=«-. 269. To prove RvLE 1. a"^ = a.a .a to m factors, a'-'a^a.a to /i factors. 402 THE THEORY OF INDICES. Therefore ^" X a" = (rt . o . a to m factors) x (a . a . a to ?i. factors) = a .a. a io -^m + n) factors. = a'"+", by the Definition. To prove Rule 11. (a'")" = a'' .oT' .a" tc n lactors, = (a.a. a m m taei-ors) (a . a . a ... to m factors) . . . repeated n times, = a.a .a to mn factors, = 0"", by the Definition. 270. We have deduced immediatehj from the Definition that when m and n are positive integers a" x a'' = a'*+ . When m and n are not positive integers, tlie Definition has no mean- ing. We therefore extend the Definition by saying that a" and a", whatever m and n may lie, shall be such Uiat a" x a' = a"*+", and we shall now proceed to shew what meanings we assign to a" in consequence of this definition, in the following cases. p 271. Case I. To find the meaiiing of a', p and q being positive integers. ? p p,p a''xa'> = a'' ', P P T fj.? ? ?+?+? «» X «' X o» = a» 1 xa'' = a'' » »; and by continuii^g this process, xa'x to (/ lactors = a» « « But by the nature of the symbol 4/ i^a^ >^ ^a'' X to q factor8 = a'; p p :. a^xa'' X to q factors = ^/a' x ^o' x . . . to ^ factora ; p THE THEORY OF INDICES. 203 272. Case II. To find the meaning of a~\ s beirig a po.si.- tive number, ivhole or fractional. We must first find the meaning of a". We have Now a "xa" .•. </," — i. a' X a~' = 1; :. a~' _ 1 ~ a'' 273. Thus the interpretation of a*" has been deduced Irom Rule I. It remains to be proved that this interpretation agrees with Rule II. This we shall do by shewing that Rule II. follows from Rule I., whatever m and n ma\' be. 274. To shew that {a"')" = a'"" for all values of m and ?i. (1) Let n be a positive integer : then, whatever m may be, (ft"*)" = a" . rt"* . o"" to n factors ™m-f-"i-i-"»+ ... to n t«rm3 (2) Let n be a positive fraction, and equal to ~,p andV being positive integers ; then, whatever be the value of m, ^ - ^ + ^+...109 terms (a"*)' X («")« X to 5' factors = (a")' ' = «"•", by (1). But a' xa' x to 5' factors = a ' ' that is, Ca''y = a'^. 204 THE THEOR V OF INDICES. (3) Let n— —s., s being a positive number, whole or frac- tional : then, whatever m may be, («")- = - — _ bv Art. 272, (a*")" ^ = -;j, by (1) and (2) of this Article ; that is, (0°*)"= ^— - 275. We shall now orive some examples of the mode in which the Theorems established in the preceding articles are applied to particular cases. We shall commence with exam- ples of the combination of the indices of two .single terms. 276. Since x'" x x" = x'''+", (1) x" X af— = yf^-' = X*. (2) X' X X = x'+i. (4) ft^—'.fc" ''xa"-'".6''-".c = a'"-"+"-"'.6"-''+^-".C = 1.1. c = c. Since (x"')" = x"", (1) (.c6)3 = x6^3 = a.ij. (2) (x^)^ = x*'^=.o'. (3) {a^^ = a"'^=^(iK 278. Since x' = 4/x-% (1) x^= Vr*. (2) x^=^i2; THE THEORY OF INDICES. 205 Note. When Examples are given of actual numbers raised to fractional powers, they may often be put in a form more fit for easy solution, thus : (1) 144-^ = (144)3 = (V141)'=12-'=1728. (2) 125^ = (125^)''i = ( 4/125)''^ = 5- = 25. 279. Since (ic"*)" = a;™", (1) j(a;"')"j'' = (x""')'' = x""'P. (2) {(«-"*)-"}'■ = («'""/ = «"•"'. (3) I (x-")" i*" = (x-'"") " = X-"""'. 280. Since x-" = — , X" we may replace an expression raised to a negative power by the reciprocal (Art. 199) of the expression raised to tlic. same positive power : thus (1) a-i = -. (2) a-^= \. (3) a~^= -\. Examples. — c. (1) Express with fractional indices : 1 . ^x5 4- 4/x2 + ( Jxy. 3. 4/^^ A ( ^'af + a J^. (2) Express with negative indices so as to remove all p<n\ er.s trom the denominators : 1 a 6^ 3 o(? 5x^ X X x^ x'^ X* '^ 42/-3'' Tt/s-* yz x^ 3x 4 ocy I z y^ y^ y*' ^^^ bx^y'^ x?y^' (3) Express with negative indices s<- as to remove all powers from the numerators ; 206 THE THEORY OF INDICES. 1 x X? x* 4o-6- 3f( l]x t t f M^ ih^^ ^/(T'ofi (4) Express with root-symbols and positive indices : -i o -» -I 2 12 X^SX-X* y-^ y-i 3y-i • -2 -i -f I o X X X 2. x"3 + (/~S + 2-3. 4- "X "^ ir^ "*" ~^' 281. Since x"-^x" = — = x'" .x~" = x"-", X" (1) X«-=-X^ = X«-3 = X*. (2) x3H-xS = x3-8 = X-5 = -1 (3) x"-^x"-" = x"-''"-"'^x"'-'^" = x". (4) a'-ra'^ = a'-<^' = a'-»^=a--=— . (5) x*-^x^ = x5 ^ = x* (6) X^^X^ = X^"6=X^"^ = X~^ = X~^=^- x^ 282. Ex. Multiply a^ -a-' + a'-\ by a' + 1, a^ — a^ + a'-l o'+l a*' - a*' + a*'-a' ay-1 EXAMPLES.— Ci. Multiply 1. x'' + x'y' + T/*' by .r*' - x'j/' + y*. 2. a'" + 3a^y' + 9a-t/*" + 27y'" by a" - Sy". 3. x**- 2ax*' + 4a« by x*^ + 2ax='' + 4a\ THE THEORY OF INDICES 207 a"* + 5" + c' by a"' - 6" + c". a" + 6" - 2c' by Sa"* - 6 + c''. x*" — x"i/" + 2/-" by x-^" + ar''^/" + 1/**. ap*+p _ 5?° 4. cp by a''"-' + 6*-'' + c^"*. Form the square of ar''' + x' + 1. Form the square of x^ — x^ + 1. 283. Ex. Divide, x*" - 1 by x" - 1. x" - 1) x*" - 1 {x'" + x'*" + X' f 1 x^p . -1 a?p. -x^ 7?" -1 T?"- -xJf xf- -1 af- -1 Examples.— cii. Divide I . X*" — y*"* by X"' - y". 3, x*' - y'' by x' - y. 10 x"* + 1/'" by x" + y\ 4. o"'' + 6'°' by a^*" + 6' x'" - 243 by x" - 3. a*" + 4a-"'x^" + 16x*" by a*" + 2a"'x" + 4x^" Qx" + 3x*'' + Ux'" + 2 by 1 + Sx' + x^''. 14&*"'c'" - ISi'^c'"* - 56''" + 4b-"'c'"' by 6»~ + 6"c^'" - 26^'"c" Find the square root of a*" + Ga'"" + IQa*"" + 20a^ + 15a-'» + Ga" + 1. Find the square root of 2o8 THE THEOR V OF INDrCES I Fractional Indices. 284. Ex. Multiply J - ah^ + b^ by J + 6*. J-ah^ + b^ a^ + b^ a - a%* + 0*6* + ah^-ah^ + b a +b Examples. — ciii. ' Multiply 1. x^-2x^+lhj x^-l. 2. 2/* + 2/^ + ?/^+ 1 by J/*- 1. 3* a* - x^ by a^ -t a^x^ + x-^. 4. a^ + b^ + c^ - a*b^ - a^c^ - b^c^ by a* +b^ + c^. 5. 5x^ + 2x^y^ + 3x^1/2 + 7y^ by 2xi - ??/*. 4 31 sa 12 4, a 1 6. to"' + TO-'7i' + 7*i,-^?i-^ + 7?i"?i'^ + "" by TO" - n", 7. m^ - 2dhn^ + 4d- by m^ + 2dhn-> + -ld~. 8. 8 J + 4ah^ + 5ah^ + 96^ by 2«^ - 36*. Foi-m the square of each of tlie following expressions : 9. x^ + a^. 10. x^-a^. II. x'^ + y^'. 12. a + ti 13. x2-2x* + 3. 14. 2x' + 3x'+4. 11;. x^-y^ + zK 16. x* + 2i/J-a* THE THEoky OF INDICES. 2c 9 285. Ex. Bividt a-bby ija- i/b. 1 1 Putting a^ for ^'a, and b^ for i/b, we jjroceed thus ; J -b^)a-b{J + ah^ + aihi + },i 3 1 a-a^b'^ ah^-b ah^-a^b^ ah^-b a^b^-ah^ ah^ - h ah^-b EXAMPLES.— Civ. Divide 1. x-yhy x~--y^ 2. a — bhy o^ + 6* 3. x--y \)y x-^ -y' 4. a + b by a^ + b^ 1 I 5. x + yhyx^+y'' 1 1 6. m — n by ?)i^ — 7i''. .s _ a,/i. 7.x- Sly by x^ - 3y^. 8. 81a-166by 3«5^-2Ai. 9. a-x by a;-^ +a~. 1 10. 1*1 — 243 by m" — 3. 1 1 11. a;+17x-- + 70 by a:2 + 7. 12. x^ + x^ - 12 bv X* — 3. 13. 63 _ 3t 5 + 36 _ 5I by b^ - 1. 14. x + y+z- 3.'?:3y3;.'3 ^y x* + 1/* + 2^. g 1 1 15. X - 5x3 - 46x3 - 40 by x^ + 4. 1 .!_ 1 1 i 1 16. m + m^n^ + n by m2 -m*ri* + ?i2, 17. ^ - 4^)* + 6p2 •_ 4pi + 1 by ^- - 2pi + 1. 18. 2x + x^y^ ~3y- 4i/^3^ - xh^ - 2 by 2x^ + 3^/2 + z^. 4 31 2g la 4. 19. x + 1/ by x-"^ -x^y^ +x^y" -x-'^y^ +y". Sio THE THEORY*OF INDICES. Negative Indices. 28(5. Ex. Multiply x~^ + x~-y~'^ + x~'^y~^ + y~^ by x"' — y~^. x^^ + x~~y~^ + a:~^T/~^ + y~^ - x~^i/~^ — x~^y~^ — x~'i/~^ - y" x"*-i/~* Examples. — cv. Multiply I. «-i + 6-1 by (i-i - 6-1. 2. x-3 + 6-2 by xr^ - 6-2. 3. x^ + x + x-i + x-^ by x-x-i. 4. X-- I4-X-2 by x2+ l+x~2. 5 . a-2 + 6-2 by a-2 - 6-2. 6. a"! - 6-1 + f-i by a-i + ft-i + c-i. 7.1 + «6-i + a-6-2 by 1 - a6-i + a'^b-^. 8. a26-2 + 2 + a-262 by a26-2 - 2 - a'-b-. 9. 4x-3 + 3x-2 + 2x-i + 1 by x-2 - x-i + 1. r o. ^x-2 + 3x-i - 1 by 2x-2 - x~i - J. 2 3 -^ 2 287. Ex. Divide x^ + l+x'- by x-l+ x~\ X-l+X-lJ x2+l +X-2 (^X+1+X~l X2 - X + 1 X + X-2 X - 1 + x-^ l-x-i + x-3 l-X-l + X-2 Note. The order of the powers of a is a', a^, (1(1, a", a~^, a~'-, a'^'.. u serii^s which may be written thus 3 2 1 1 1 1 a a-" a** THE THEOR Y OF INDICES. EXAMPLES.— CVi. Divide I. a;2 - X"- by *; t i ' j. > 6~^bya — 6~^ 3. 771^ + ?i~^ by 7?i + 7^~*. 4. c^ - tZ~^ by c - d~^. 5. x^~^ + 2 + x~-y'^ by x?/~i + x~^y. 6. a-* + a-'-^t-s + h-* by a-^ _ ^-i^-i + 5-2. 7. x^y~^ - x~^y^ — Sxy~^ + '3x~^y by xy~^ - x~hj. ^ 3x-5 . . 77x-3 43X-2 33x-i „, g. — -4x-4--g J-+27 a;-2 by 7:;^ — x~'^ + 3. ^ 2 g. a^6~^ + «~^6^ by «6~^ + rt"'6. 10. a~^ + 6"^ + c~^ - 3a~^6~^c'i by a~^ + ft-^ ^-c~^ 288. To shew that (rt5)'' = a". J", (at)" = a6.a6. a?)... to 71 factors = (a . a . a . . . to n iactors) x (6 . 6 . 6 . . . to ti factors} = a" . 6". We shall now give a series of Examples to introduce the various forms of combination of indices explained in this Chapter. Examples. — cvii. 1 . Divi de x^ - 4x!/ + Ax^y + Ay^ by x* + 2x~y^ + 2y. _i_ _i_ 2. Simplify )(x»"*)3.(x6)-'j3-». 3. Simplify (.r^o* . xi«^)^-l j _i i_U ,.- ) 1 1 x + a x-a 4. Simpliiv < -TT — 5 — ^ 2 r: — 3- ^ ^ - jx''-a^ x^ + a'' x- + a- \ a THE THEORY OF INDICES. 5. Multiply |x-2 + 4x-i - 1 by -isr'^ - 2xr^ - \. 01 it af^' i""* x*~^ 6. Simplify ' — ^ . 7. Divide x^ - 2/"" by x" + ?/". 8. Multijjly (a^ + 6^)-' by a^ - 1^. 9. Divide a — 6 by 4^rt - 4/^. 10. Prove that (a^)" = (a")-. 11. If a"'" = (a'")", find 7?i iu terms of n. 1 2. Simplify cc''+*+' . a;*+'~' . x'*~'+' . x*^^. 13. Simplify(^--j-^(^-,^) . 14. Divide 4^' by —. 15. Simplify [j (a-")- }^]-[ j (a"')" I"']- 1 6. Multiply iC + 1"- 2c" by 2a"' - 36. 17. Multiply a'"-"})"-" by a"^6'>-"c. 18. Shew that --+<^^")^:("^^^^ = ^^^. 19. Multiply x^ + x^ + 1 by x^ - x^ + 1 and their product by x^ - x^ + 1. 20. Multiply a" - 6a"— ^ x + ca"— ^ a;2 by a" + 6ti"-' x - ca'-'h^. 2 1 . Divide x^*"*-^' - i/2«<^-ii by x*^*"" + !/«<'-". 22. Simplify j (a")" "•i'»+i. 23. Multiply x^"" + x'^yf" + x'y^ + y"'' hy af—y'. 24. Write down the values of 625^ and 12~^. 25. Multiply .•-•'"•-'"• - 2/'— 1)" by x" - y". 26. xM u] tiply x^ + 3.C- - 1 by x^ - :>x"i. XXIV. ON SURDS. 289. All numbers which we cannot exactly determine, because they are not multiples of a Primary or Subordinate Unit, are called SurdS. 290. We shall confine our attention to those Surds which originate in the Extraction of roots where the results cannot be exhibited as whole or fractional numbers. For example, if we perform the operation of extracting the square root of 2. we obtain 1-4142..., and though we may carry on the process to any required extent, we shall never be able to stop at any particular point and to say that we have found the exact number which is equivalent to the Square Root of 2. 291. We can approximate to the real value of a surd by finding two numbers between which it lies, differing from each other by a fraction as small as we please. Thus, since V2 = 1-4 142 14 15 1 a/2 lies between :— and -—, which differ by :r- ; 10 10 ■' 10 also between -—- and -— -, which differ by tt^k, 100 100' ■^ 100 also between ^ and ^^^, ,vhich differ.by -^-. And, generally, if we find the square root of 2 to n places of decimals, we shall find two numbers Ix'twec^; wliich ^2 lies. differinLT ironi eucli other by the fraction ,^- . 214 ON^ SURDS. 292. Next, we can alwaj's find a fraction differing from the real value of a surd by less than any assigned quantity. For example, suppose it required to find a fraction differ- ing from ^'2 by less than ^o- Now 2(12)''^, that is 288, lies between (16)- and (17)2, .'. 2 lies between ( t;^) and (rs) ; .•. ^2 lies between -^ and j^ ; .-. J2 differs from r— by less than r-^. 12 -^ 12 293. Surds, though they cannot be expressed by whole or fractional numbers, are nevertlieless nuinlx-rsof which we mav form an approximate idea, and we may make three assertions respecting them. (1) Surds may be compared so far as asserting that one is greater or less than another. Thus ^^^3 is clearly greater than ^'2, and 4^9 is greater than ^fS. (2) Surds may be multiples of other surds : thus 2 ^^2 is the double of J2. (3) Surds, when multiplied together, may produce as a result a whole or fractional number: thus V2x ^2 = 2, 294. The symbols ^^a, ^a, ^/a, i^a, in cases where the second, third, fcurth, and n*^ roots respectively of a ainnot be exhibited as wliole or fractional numbers, will represent surds of the second, third, fourth, and Jt"" order. These symbols we may, in accordance with tlie principles laid down in Chapter XXIII., replace by a*, a^, a*, a". ON SURDS. 215 295. Surds of the same order are those for which the root- symljol or surd-iudex is the same. 1 Tliiis ^a, 3 >Ji^h), 4 i^l(inn), r^ are surds of the same order. Like surds are those in which the same root-symbol or surd- index appears over the same quantity. Thus 2 sja, 3 Ja, 4a^ are like surds. 296. A whole or fractional number may be expressed in the form of a surd, by raising the number to the power denoted by the order of the surd, and placing the result under the symbol of evolution that corresponds to the surd-index. Thus 0= Mja\ b ' Ib^ 297. Surds of different orders may be transformed into surds of the same order by reducing the surd-indices to fractions with the same denominator. Thus we may transform ^fx and ^y into surds of the same order, for and. ^y = y^^y^ = ^l/y\ and thus both surds are transformed into surds of the twelfth order. Examples.— cviii. TransforTn into Surds of the same order : I. Va;and ^y. 2. 4/4 and ^2. 3. ^(18) and 4/(50). 4. 'J^2 und ;i/2. 5. ^/rt and ;;/6. 6. 4^(a + 6) and i^{a-b). 298. If a whole or fractional number be multiplied into a surd, the product will be represented by plaqing.the multiplier and the multiplicand side by side with no sign, or with a dot (.) between them. Thus the product of 3 and ^f2 is represented by 3 ^^2, of 4 and 5 v'2 by 20^2, of rr and Jc by a ^/c. 2i6 ON SURDS. 299. Like surds may he combined by the ordinary pro- cesses of addition and subtraction, that is, by adding the coefficients of the surd and placing the result as a coefficient of the surd. Thus ,v/« + «/« = 2 V**? X Jc- ^/c = (.C - 1) i^C 300. We now proceed to prove a Theorem of great ini- ]>ortance, which may be thus stated. The root of any expression is the saw,e as the product of the roots of the separate factors of the expression, that is sj(ah) = ^la . ^h, ^{xyz)=^x.^y. »/z, ;:/(pqr)= ;'2). s^q.^fr. We have in fact to shew from the Theory of Indices that 1 11 (aby =0" . h". Now \(ahy>r = (abf = ab, 11 11 '^ 1 and Irt". ?)" j" = (a")". (6")" = rt". 6" = rt.6; ^ 111 .". (ab)" =rt". b". 301. We can eometimes reduce an expression in the form of a surd to an equivalent expression with a whole or frac- tional niimber as one factor. Thus v'("2) = V(-fi X 2) = ^/CM) . ^/2 = 6 ^/2, 4/(128) = ^(64 X 2) = ^(64) . ^72 = 4 ^f'2, !j{a'x) = a^a" . Zfr = a . ^J/x. O.V SURDS. Examples.— cix. Reduce to equivalent expressions with a whole or fractional number as one factor : I. V(24). 2. ^/(50). 3. V(4a3). 4. s'{l2fiaH^). 5. v/(32?/s3). 6. ^/(lOOOa). 7. V(720c2). 8. 7.V(396x) 9. 18.J(^x3). -T-. II' \^('t^ + 2a-x + ax-). 12. V(a^-2x2|/ + XJ/2). 13. v'(5Oa2_iO0a64-5O&-). 14. V(63c*?/-42cy + 7y3). 15. 4/(54rt662), 16. 4/(1 60xV). 17. 4/(108m9ni»y. 18. 4/(1372ai65i6). ig. 4/(3;* + 3x31/ + 3xV- -^ xr)- 2a 4/(rt*-3a36 + 3a262-a&3). 302. An expression containing two factors, one a surd, the other a whole or fractional number, as 3 »J2, a ^x, may be transformed into a complete surd. Thus 3 v'2 = (32)i V2 = V9 . V2 = ^/(18), a^fx = {a^)K ^x= 4/a3. ^x= ^{a^x). Examples.— ex. Reduce to complete Surds : I. 4V3. 2. 3^/7 4- 24/6. 5. 3/^''^ 7. 4«V(3x). 9. (»^+^)-^G~-3- 3. 54/9. 6. 3 V«- ^■•W(£> '«+«(,7y- \x + v/ ' \x- - 2xv + v'-' 2i8 ON SURDS. 303. Surds may be compared by transforming them into surds of the same order. Tlius if it be required to determine whether s/^ be greater or less than 4^3, we proceed thus : V2 = 2^ = 26= 4/23= ^8, 4/3 = 3^ = 3^=4/32=4/9. And since 4^9 is greater than ,^8, ^3 is greater than ^'2. Examples. — cxi. Arrange in order of magnitude the lullowing Surds : 1. J3 and 4/4. 6. 2 ^87 and 3 ^33. 2. VlO and 4/15. 7. 2 4/22, 3 4'7 and 4 V2. 3. 2 V3 and 3 ^'2. 8. 3 ^/19, 5 4/I8 and 3 4/82. 4. ^J'l^rld^{~). 9- 2 4^14, 5 4/2 and 3 4'3. 5. 3^7 and 4^3. 10. ^ ^72, | ^3 and ^ v'-i. 304. The following are examples in the application of the rules of Addition, Subtraction, Multiplication, and Division to Surds of the same order. 1. Find the sum of ^'18, ^a28, ami ^'32. v/(18)4- v/(128)+ V(32)= ^'(9x2)+ ^/(64x2)+ v'(16 x 2) = 3V2 + 8s'2^4^'2 = 15 ^'2. 2. From 3 ^/(75) take 4 ^/(12). 3 ^/(75) - 4 x/(12) = 3 x/(25 x 3) - 4 ^'f4 x 3) = 3.5.^'3-4.::. x'3 = 15 ^3 - 8 J3 = 7^3. ON SURDS. 3. Multiply v/R ^y V(12). ^/8x v/(12)= v/(8xl2) = V(96) = V(16 X 6) = 4^6. 4. Divide ^/32 by ^18. x /(32) _ >v/(16x 2) ^ 4^2 ^ 4 V(18) V(9x2) 3V2 3" Examples. — cxii. Simplify 1. V(27)+ 2^(48) + 3^(108). 11. ^6 x */8, 2. 3^(1000) +4^(50) + 12^(288). 12. ^(14) x ^(20). 3. a VC^^a;) + & sj(t»^x) + c sj{c^x). 1 3. ^/(50) x V(200). 4. ^(128) + 4/(686) + 4/(16). 14. 4/(3rt26) X 4/(9a6''!). 5. 7 4/(54) +3 4/(16) +4/(432). 15. 4/(12a6) x 4/(8a^6S). 6. V(96)- V(54). 16. ^/(12)- V3. 7. V(243)-V(48). 17. x/(18)-vio'V 8. 12 ^(72) -3^(128). 18. 4/(rt-^)-^ 4/(«?'-). 9. 5 4/(16) -2 4/(54). 19. 4/(a36)-^ 4/(a63). 10. 7 4/(81) - 3 4/(1029). 20. V(^2 + ^,3y) ^ ^/^^ + 2a;2y + a;^)/^). 305. We now proceed to treat of the Multiplication of Compound Surds, an operation which will be frequently ?e- quired in a later part of the subject. The Student must bear in mind the two following Rules ; Rule I. sjax Jb= ^/(ab), Rule II. ^ax ^a = a, which will be true for all values of a and b. ON SURDS. EXAMPLES.— cxiii. Multiply s]x by ^y. V(3;-2/)i>y Vy- 6 ,^x by 3 sjx. 7V(* + l)by 8V('X+1) lO^a^by 9V(a;-l). 9- \'35 ^y - 'J^- 10. V(^-l) by - sl{^.-l). 11. 3 ^/.c by - 4 ^x. 12. - 2 ^a by - 3 ,^a. 13- \/(a;-7)by - ^x. 14. -2 V(a; + 7) by -3 ^a:- 15. -4Vra2-l)by -2^(a2-l). V(3x) by ^/(4x-) . 16. 2 V(a^ - 2a + 3) by - 3 ^{a^ - 2a + 0). 306. The following Examples will illustnite tlie wny of proceeding in forming the products of Compound Surds. Ex. 1 . 1 o multiply ^x + 3 by ^a: + 2. ^/x + 3 Vx + 2 « + 3^x + 2v'a; + 6 X + 5 v'a; + 6 Ex- ?. To multiply A^x + Zjy by 4 ^r - 3 ^hj. AJx + ZsJy 4 Vx - 3 y/j/ 16x + 12v'(x?/) -12V(xj/)-9?/ 16x - 9?/ Ex. 3. To form tlu' scpiare ofV(a;-7)- ^^x. V(x-7)- Vx ^(x-7)-Va; x-7 - ^/(x2-7x) - ^/(x2-7x)+a; 2x-7-2\/(x''2'-7x) ON SURDS. at Ex AMPLES.— CXiv. Multiply I. ^x + 7 hj ,^'o: + 2. 2. v'-^-5 by Vx+3. 3. J(a + 9) + 3 by ^f{a + 9) - 3. 4. V(a-4)-7byV(a-4) + 7. 5. S^x-1 hj i^x + 4. 6. 2^/{x-i>) + 4hy3J{x-5)-6. 7. ^(6 + x) + ^fx by ^/(6 + x) - v'x. 8. V(3-;;+l)+ v/(2x-l)byV3x- V(2a;-1). 9. s^a + J{a - x) by .Jx - J(a - x). I o. V (3 + x) + Jx by ^/(3 + x). 11. sjx+ ijy+ ijz hy Jx- Jy+ Jz. 12. Ja+ J(a — x)+ Jx hy Ja- J(a-x)+ Jx. Form the squares of the following expressions : 13. 21+ ^/(x2-9). 17. 2V^--3. 14. J{x + Z)+ J(:x + 8). 18. J{x + y)- J{x-y). 15. JX+ J{x-A). 19. Jx.J{x+l)-J{x-l). 1 6. J{x - 6) + v'a;. 20. ^/(.c + 1) + V^ . V(-^' - 1 • • 307. We may now extend the Theorem explaineil in Art. 101. We there shewed how to resolve expression^ df the form a2-6« into factors, restricting our observations to the case of perfect squares. The Theorem extends to the difference between any tivo quantities. Thus a-b={Ja+ Jh){Ja- Jb). » x^-y = {x+ Jy) {x- Jy). l-x==(l+ Jx) (1- Jx\ 222 ON SURDS. 308. Hence we can always find a multiplier which will fVie tVoiii surds an expression of any of the /oitr forms I. a+ s/b or 2. Ja+ Jb, 3. a- s,fh or 4. Ju- Jb. j.'Oi- since the first laid third of these expressions give as a product a'^~b, which is free from surds, and since the second and fourth give as a product a-b, which is free from surds, it follows that the required multiplier may be in all cases found. Ex. 1. To find the multiplier which will free from surds each of the following expressions: I. 5+V3. 2. ^6+^5. 3. 2- ^o. 4. x/7- ^'2. The multipliers will be ^ I. 5-^3. 2. V6-V5. 3. 2+^5. 4. V7+V2. The products will be I. 25-3. 2. 6-5. 3. 4-5. 4. 7-2. That is, 22, 1, -1, and 5. ct Ex. 2. To reduce the fraction ^_ ^^ to an equivalent. fraction with a denominator free from surds. Multiply both terms of the fraction by 6+ ,^c, and it be- comes ab + atjc b^-c ' which is in the required form. Examples.— cxv. Express in factors : I. c-d. 2. c2-d. 3- c-d^. 4. 1-1/. 5. 1-Sx\ 6. 5m- -\. ■ 7. 4a2-3x. 8. 9-8?i. 9- 11«--16. 10. p'^ - 4r. 1 1 . jj - 83^ 12. rt*" - b\ ON SVRDS. 223 Reduce the following fractions to equivalent fractinus Avitli denominators free from surds, ,, 1 „ N^L ,- 4 + 3V2 16 ^ 17 V3 2-V2 V«+Vx V(™'+*1)- V('h2-1) ^" Va - V*' ■ \/(»i2 + 1) + ^{m^ - i) * 1- Vrc' ~^' a~ s/(a2-l)- V(a + x) + v'(a - a;) ^ a+ sj{a^-x-) ^{a + x)— sjia — x)' ~ ' a- sj{a? - x^)' 309. The squares of all numbers, negative as well as posi- tive, are positive. Since there is no assignable niamber the square of which would l)e a negative quantity, we conclude that an expression which appears under the form sfi - 'i^) represents an impossible quantity. 310. All impossilile square roots may be reduced to one common form, thus V(-«2)=Vla-x(-l)f=>2.N/(-l) = a.V(-l) ^(-a;)=VI^ x(-l)\=Jx .^'■-1). Where, since a and sjx are possible numbers, the whole impossibility of the expressions is reduced to the appearance of ^( - 1) as a factor. 311. Def. By ,^/(-l) we understand an expression which ivhen multiplied l»y itself produces - 1. Therefore }n/(-i)P=U'(-i)!--v^(-i)=(-i)-v^(-i)=- V(-i), *U/(-i)l*=U'(-i)!MV(-i)P=(-i)-(-i)=i, «V-d so on. :524 ON SURDS. Examples.— cxvi. ^lultiply, oLservini,' tluit ^ - ax ^1 -h= - ^fab. 1. 4+ ^/(-:3)l_.y4- V(-3). 2. V3-2V(-2)l,y ^'3 + 2 ^/( - 2). 3. 4V(-2)-2V2l.y^^/(-2)-3V2. 4. V(-2)+ V(-3j+ x/(-4)by V(-2)- V(-3)- ^/(-4). 5. 3 V( - «) + x^( - b) by 4 V( - «) - 2 v/( - 6). 6. a + s,f( - a) by a - ^f( - a). 7. a^{-a) + b^'{-h) by a .yA; - a) - 6 V( - &)■ 8. a+/5v'(-l) bya-/ix/(-l). " 9. 1- V(l-e') by 1+^/(1-62). I o. t''^'-" + e"''^ '-" by e""^ '-'' - e"^^'-". 312. "We sliall now gi\ e a few Miscellaneous Examples to illustrate the principles explained in this Chapter. Examples. — cxvii. 1. bnnphly ^^^^^J-^^-^A 2. Prove that |1+ ^(-1)^+11- v/(-l)j2 = 0. 4. Prove that 11+ ^f{-\)'r- \1- ^{-l)\^= ^'(-16). ; . D i \- i 1 1 e .r^ + (I M ly x- + ^/2ax + a^. 6. Divide 7a'' 4-?c' by m-— ^^f2mn + n-. 7. Siniplil'y ,^f {x^ + 2x-y + xy-) + ^' {x^ - 2x-y + xy^). 8. Simplify- , „ , yr, ami verify by puttniL: „ -. ;) and i = 4. ox sunDS. 225 9. l^iiul the square of « >» /r - sf{cd). 10. Find the square of aV'^ — -j^r 11. Siniplit'y 12. Smipuiy — — — ^ i. x/(l-^'-') „. ,.,. ic-l ( a;-l \-x ) 13. Simphtv { — , r- + T- > , 14. Form the square of . /( \v + •' ) - /i / ( ! - 3 ). 15. Form the square of i^(^x + a) - sj{x — a). 16. :\Iiiltiply J/(a^'»-"6''"'+V-"') by xy(a''&"^*c— ^0- 17. Raise to the 5"" power —\ — a^l{- 1). 1 8. Simplify 4/(81 ) - ^l{ -512)+ 4/(192). 19. Simplify ^-^y( 3-3 ). 20. Simplify ~~„ j '4/(32:'-a;^ - GSjjV + 441^^^: ._ 1029^52) j . X— I 21. Simplify 2('h -\)^ ( - _-, -1 5— ,- \ •' - ^^V 2/i*-6/i3 + 6?t^-2?i/ 22. Simplify 2(?i - 1) ^(63) + \ v/(112) - ^'(^j!^ 2 -../!l75(n-l)2c^!xA_2 /(^;j 23. Wliat is the difference between s/jl7- v/OW)!x VI17+ v/(33)J and 4/ ! ( i") + ^/( 1 :!! )) I X 4/ ) f;5 - V( 129)1 [S.A.] 226 ON SURDS. 313. We have now to treat of the method of finding the Square Root of a Binomial Surd, that is, of an expression of one of the following forms : m+ s]n^ m— /y/n, where m stands for a whole or fractional number, and tjn for a surd of the second order. 314. We have first to prove two Theorems. Theorem I. If Ja = m+ ^n, m must fee zero. Squaring both sides, a='mP'-\- 2m ^n + n ; .". 2ni ^11 = a — m- — n ; , a — Tn? — n that is, V*i, a surd, is c(iual to a -whole or fractional number, which is impossible. Hence the assumed equality can never hold unless in =0, in which case ijn= s,hi. Theorem II. 7/"fe+ ^'a=^m^ Jn, then must fe=7?i, and For, if not, let b — m, + x. Then m + x+ ^a=m+ ^/n, or x+ ^fa= x^i ; which, by Theorem I., is inipossiblii unless a; = 0, in which case h = vi and ^'(f= i>^^n. 315. To find the Square Boot of : + ^fb. Assume V(*+ V^)= >/•*+ \'v- Then a+ ,Jb = x + 2 VC-r.V) + y ; ••• x + y = a (1). 2n'(-'-."V v7, ^i), froni which we have to tiud x auu ij. ON SURDS. 227 Now from (1) »2 ^ 2x1/ + 1/^ = a-, and from (2) 4xy = h ; .•. x"-2x2/+ (/- = a2 — 6; Also, x + y = a. From these equations we find and y-- 2 " i2 ' Similarly we may show that ^'(» - ^'») = ^l " "4""- *' } - ^^^^f^l . 316. The practical use of this method will be more clearly seen from the following example. Find the Square Root of 18 + 2 VC^T). Assume V{ 18 + 2 ^(77) | = V« + Vy. Then 18 + 2 V(77) = a; + 2 ^(xy) + y ■ .-. a; + 2/ = 18 ) 2V(a^) = 2V(77)r Hence x^ + 2xy + 1/- = 324 ) 4a-j/ = 308J"' .'. x^ - 2xy + y"^ = \^ ; :.x-y=±A; also, x + y=l8. Hence a = ll or 7, and y = l or 11. That is, the square root required is ^^(11)+ ^^7. 228 ON SURDS. Examples. — cxviii. Find the square roots of tlie following Binomial Surds: I. 10 + 2^/(2^. 2. 16^2^(55). 3- 9-2^(14). 4. 94-42V5- 3- 1-3-2^/(30). 6. 38-12^(10). 7. 14-4V6. 8. 103-12^/(11). 9. 7.^> - 12 ^/(21). 10. 87-12v'(42). IT. 3_^-v/(10). 12. .57-12^/(1.5). 317. It is often easy to determine the square roots of expressions such as those given iu the preceding set ot Examples hxj insjiedion. Take for instance the expression 18 + 2 \/(77). What we want is to find two numbers wliose sum is 18 and whose product is 77 : these are evidently 11 and 7. Then 18 + 2 V(7V) = 11 + 7 + 2 ^(11 x 7) = U/(ll)+^/7p. That is v/(ll)+ \'' is the .s(iuare root of 18 + 2 ^/(77). To effect this resolution by inspection it is necessary that the coefficient of the surd should be 2, and this we can always ensure. For example, if the proposed expression be 4+ /v/(15), we proceed thus : 8 + 2V(15) 5 + 3 + 2^(5x3) 4+ V(15) = 2 V2 ~\ J2 J ' :. — 75^ is the square root of 4+ \/(15). Again, to find the Square Root of 28 - 10 is/3. 28-10^/3 = 28-2^/(75) = 25 + 3-2v/(2.-)'x.3) = :5- V3)2; :. 5 - ^3 's f '16 sipuire root required. XXV. ON EQUATIONS INVOLVING SURDS. 318. Any equation may be cleared of a single surd, by transposing all the other terms to the contrary side of the equation, and then raising each side to the power correspond- ing to the order of the surd. The process will be explained by the following Examples. Ex. 1. ^'.r = 4. Raising both sides to the second power, a; =16. Ex. 2. 4/x = 3. Raising both sides to the third jjower, a; = 27. Ex. 3. Via;2 + 7)-x=l. Transposing the second term, J(a;2 + 7) = r+a;. Raising both sides to the second power, X- + 7 = 1 + 2x + a;2, .-. x = 3. Examples.— cxix. I. Jx = 1. 2. v/-c = 9. 3. x^ = b. 4. 4/a; = 2. 5. x- = Z. 6. 4/x = 4. 7. v/(x + 9) = 6. 8. ,./(x-7)-7, 9. V(.r-15) = 8. 10. (x-9)^=12. II. ^(4x-16) = 2, 12. 2()-3Va; = 9. 230 ON EQUATIONS INVOLVING SURDS. 13. 4/(2a; + 3) + 4 = 7. 17. ^/(4x2 + 5x-2) = 2x + l. 14. h-\-CsJx = a. 18. x/(9x2-12a;-51) + 3 = 3x. 15. V0^"-9) + x = 9. 19. v^(^''-"^ + '^)-«=-K- 16. ^(x^- 11) = x- 1. 20. ^'iLoy? — '^inx-Vii)-hx = m. 319. When ^iro surds are involved in an equation, one at least may be made to disappear Ly disposing the tenns in such a way, tliat one of the surds stands by itself on one side of the equation, and then raising each side to the power cor- responding to the order of the surd. If a surd be still left, il can be made to stand by itself, and removed by raising each side to a certain power. Ex. 1. ^(x-16)+ v'-c = 8. Transposing the second term, we get ' ^/(x-16) = 8- ^Ix. Then, squaring both sides (Art. 306), 3;-16 = 64-16V« + a;; therefore 1 6 ^/.c = 6 i + 1 6, or 16Va: = 80, or /y/x = 5 ; x = 25. Ex. 2. V(^ - 5) + sK^ + T) = 6. Transposing the second term, V(-c-5) = 6- ^'Crr';). Squaring both sides, x - 5 = 36 - 1 2 sj{x + 7) + x + 7 , therefore 12 ^'(x + 7) = 36 + x + 7-x + 5. or 12V(x + 7) = 48, or V(x + 7) — 4. Squaring both sides, x + 7 = 1 6 ; therefore x = 9- r V EQUA TIONS LWOLVIXG SURDS. 23 1 Examples. — cxx. 1. v''(16 + x)+ Jx = 8. 6. 1+ v/(3a; + l)= x/(4x + 4). 2. ^f{.C-\(3) = 6- s,tx. 7. l~ ^J{l-■6x) = •ls/(^■-'•c)■ 2,. s/{x + 15) + ^.0= 15. 8. a - ^/(x - a) = ^x. 4. ^'{x -21)= ^'x - 1. 9. V^ + v/(x - 7?l) = y. 5. v'(-c-l) = 3- v/(u; + 4). 10. V(x-1)+ ^'(:c-4)-3 = 0. 320. When surds appear in the denominiitors of fractions in equations, tlie equations may be cleared of fractional terms by the process described in Art. 186, care being taken to follow the Laws of Combiualiou of Surd Factors given in Art. 305. Examples.— cxxi. 36 28 2. Vx+,/(.-21) = ^^. 4. V(x-15)+V^ = -^i*^-^. 9a ^I{ax) + h^ b-a jjx+l6 _ s'-'- + S2 '' x + 6 ~h- ^{ax)' 9- ^/x + :r~^Zr+12' o /I , / N /I / \ 4+ ,^/x v/:c-8 ./;/;- 4 8. (1 + Vx) (2 - Va;) = — ^- . 10. \_- ,. = >, — -. 321. The following are examples of Surd Equations result- ing in quadratics. Ex.1. 2^x^^^-'5. r'learing the equation of fractions, 2a; + 2 = 5 ^jz. 232 ON EQUA TIONS INVOL VING SURDS. Squaring both sides, we get 4x2 4-8x + 4=25x; whence we find re = 4 or -. 4 Ex.2. V(-'' + 9) = 2V^-3. Squaring both sides, a; + 9 = 4x - 1 2 ^/x + 9 ; therefore \1 sjx = 3a;, or A: ,Jx=x. Squaring both sides, 16x = x2. Divide by x, and we get 16 = a;. Hence tlie values of x which satisfy the equation are 16 and (Art. 248). Ex.3. v/(2x+l)+2^x = ^^-^-j^. Clearing the equation of fractions, 2a; + l + 2v'(2x2 + x) = 21; therefore 2 ^(2x2 + x) = 20 - 2x, or V(2x2 + ;i;) = 10-x. Squaring both sides, 2x"^ + x = 1 00 - 20x + x*, whence x = 4 or -25 322. We sliall now give a set of examples of Surd Equa- tions some of which are reducible to Simple and others to Quadratic Equations. Examples.— cxxii. I. 4x - 12 ^/x = 16. 4. V(6x -11)= V(249 - fix^). 2." 45-14Vx=-x. 5. >/(6-x) = 2- ^/(2x-l). 3. 3V(7 + 2.c2) = 5^/(4x-3). 6. x-2 ^',4-3x) + 12 = 0. 7. v/(2x + 7) + V(3x -18)= v\7x + 1). 8. 2 V(204 - 5x) = 20 - ^'(3x - 68). ON EQUATIONS INVOLVING SURDS. 233 9. Vx-4 = -^^. 14. V(x + 4)+ V(2x-1) = 6. 10. V:c+ll=^^?^. 15. V(13x-1)- v/(2x-l) = 5. >y X — 11 11. V(.c + 5). V(a; + 12) = 12. 16. V(7x+1)- V(3x+1) = 2. 1 2. V(a; + 3) + V(a; + 8) = 5 ^x. 17. VC-l + x) + V^ = 3. 525 13- v'(25 + x)4- V(25-x) = 8. 18. v/x+ V(a! + 9975)=-7=. 20. V(x2-l) + 6 = -^^^ . 21. V(('^-«)" + 2«/) + 6-S=a;-a+i. 22. Vl(^ + «)' + 2aft + 6-J=6-a-a;. 23. V(x + 4)- V^=J(,r + |). a; — 1 5 24- ^;/^I^=^ + 4- 26. V(a; + 4)+ V(a; + 5) = 9. V(a;-4)- 25 . V(4 + .r) - v'3 = ^x. 27. ^fx + ^{x - 4) = -j^ 28. x2 = 21+ ^(^2-9). 29. V(50 + a;)- V(50-x)=2, 30. V(2xr4)- J(|+6) = l. 31. V^3 + .r)+^/x= ^ V(3 + x)' 1 _J ^1^ ^ 3^- V(a; + 1) ■*" \./{x ~i)~ ^/{x' - !)• 3x -r ■^f(4x — x^ XXYI. ON THE ROOTS OF EQUATIONS. 323. We have already proved that a Simple E(iuation can have only one root (Art. 193) : Ave have now to prove that a Quadratic Equation can have only two roots. 324. We must first call attention to the following fact : If m7i = 0, either m = 0, or n = 0. Thus there is an ambiguitv : but if we know that m cannot be equal to 0, then we know for certain that n = 0, and if we know that w cannot be equal to 0, then we know for certiiin that m = 0. Further, if lmn = 0, then either 1 = 0, or 7?i = 0, or n = 0, and so on for any number of factors. Ex. 1 . Solve the equation (x - 3) (x + 4) = 0. Here we must have x-3 = 0, or x + 4 = 0, that is, X = 3, or X — — 4. Ex. 2. (x - 3a) (5x - 26) = 0. Here \m must have x-3a = 0, or 5x — 26=0, , . 26 that IS, «=3a, or x = — . o OM THE ROOTS OF EQUATIONS. 23$ Examples. — cxxiii. I. (a;-2)(a;-5)=0. 2. (x-3) (x + 7) = 0. 3. (a; + 9)(x + 2)=0, 4. (x-5a)(a;-6?*) = 0. 6. (19x-227) (14a; + 83)=0. 5. (2a; + 7)(3.c-5) = 7. (5x-4m)(6x- lln) = 0. 8. (a;2 + hax + Sa^) (x^ - Tax + 1 2a2) = 0. 9. (x^ - 4) (x- - 2«x + cfi) = 0. 10. X (x^ - 5x) = 0. 11. (acre - 2ffi + 6) {bcx + 3a - 6) = 0. 12. (ex - (f ) (ex - e) = 0. 325. The general form of a quadratic equation is ax^ + bx + c = 0. Hence aix^ + -x + -) = 0. \ a a/ Now a cannot =0, .-. x^ + -x + - = 0. a a ... b e Wnting x> for - and q for -, we may take the following as the type of a quadratic equation of which the coeflBcient of the first term is unity, x'^-irfx + q — O. 326. To show that a quadratic equation has only two roots. Let x^ +px + 5' = he the equation. Suppose it to have three different roots, a, b, c. Then a'^ + ap + q = (1), ¥+bp + q = i..-(2), c2 + cp + q = (3). Subtracting (2) from (1), a^-b^+(a-b)p = 0, or, {a-b){a + b-\-p) = 0. 236 ON THE ROOTS OF EQUATIONS. Now a-b does not equal 0, since a and 6 are not alike, :. a + h+p = (4). Again, subtracting (3) from (1), a^ — c^ + (a — c) p = 0, or, {a — c){a + c+p)=0. Now a — c does not equal 0, since a and c are not alike, .-. a + c+p = (5). Then subtracting (5) from (4), we get 6-c = 0, and therefore h = c. Hence tliere are not more than hvo distinct roots. 327. We now procet-d to show the relations existing be- tween the Roots of a quadratic equation and the Coefficients of the terms of tlie equation. 328. x'^^-px + q=0 is tlie general form of a qtiadratic equation, in which the co- efficient of the first term is unity. Heni'e x'^+px= —q x'^ + 'px+^-^=^-q, Now if a and /? be the roots of the equation, «=-i-V('i-') '"• ^--i-V(t-') <''• Adding (1) and (2), we '.y\ a + j3= -p (.3). v ON THE ROOTS OF EQUATIONS. Multiplying (1) and (2), we get or a/3=^--^-+2, or ay8 = (2 (4^. From (3) we learn that tiiM, sum of the roots is equal to the coefficient of the second term with its sign changed. From (4) we leam that the product of the roots is equal to the last term. 329. The equation x'^ + px + q = has its roots real and different, real and equal, or impossible and different, according as 'p- is > = or < Aq. For the roots are 2 -i-V(?-')'"" and _2 _/(»?. A „,r-ti4£!zil). i-V(?-'> First, let p~ be greater than Aq, then >J{p^ - Aq) is a possible quantity, and the roots are different in value and Ijoth real. Next, let2'^ = 4g', then each of the roots is equal to the real quantity -^. Lastly, let ^^ be less than Aq, then \f{p- — Aq) is an impos- sible quantity and the roots are different and both impossible . Examples.— cxxiv. I. If the equations ax- + bx + c = 0, and a'x^ + h'x + c' = 0, have respectively two roots, one of which is the reciprocal of the other, prove that (aa' - cc')^ = {aV - he') {a'b - b'c). 238 ON THE ROOTS OF EQCA 2. If a, /? be the roots of the equation ax- + 6a; + c = (), prove that .) no &^ — 2ac ' a- 3. If a, ^ be the roots of tlie equation ay? + 6x + c = 0, prove that ac'j? -i- {2ac ~ b'^) x + ac = ac \-^~ r,)\^— )- 4. Prove that, if tlie roots of the equation ax- + bx + c = i) be equal, nx- + bx + c is a perfect square witli respect to x. 5. If a, y8 represent the two roots of the equation x^-{l + a) a; + ^(l + a + «") =0, show that a- + /3'- = a. 33O. If a and /3 be the roots of the equation x^+px + q=Oy th en x'^ + jjx + 3 = (a; - a) (x - yS). For since ^= - (a + /?) and q = afB, 3? + px + q = x- ~ {a + (B) X + a/B = {x~a){x-(3). Hence we may form a quadratic equation of which the roots are given. Ex. 1. Form the equation whose roots are 4 and 5. Here x-a = x — 4andx-/3 = x-5; .•. the equation is (x - 4) (x — 5) = ; or, x--9x + 20 = 0. Ex. 2. Form the equation whose roots are ^ and - 3. 2 .111(1 r— /?=)•-!- 3 ■ Here x - a = x -- and x - ^ = x + 3 ; th« equation is f x - ^ j (x + 3) = ; er, (2x-l)(x + 3)=0; or, £3:- + 5x-3 = 0. UN THE ROOTS OF EQUATIONS. 239 Examples.— cxxv. Form the equations whose roots are I. 5 and 6. 2. 4 and -5. 3. -2 and -7. 12 5 4. 2 and-. 5. 7aud-- 6. v/3 and - ^3. 7. m + 7i and. m — n. 8. - and . 9. -7^ and-. a jd pa 331. Any expression containing x is said to be a Function of X. An expression containing any symbol x is said to Ll; a positive integral function of a; when all the powers of x con- tained in it liave positive integral indices. 3 1 For example, bx^ + 2r^ + ^x* + j~x^ + 3 is a positive integral ■ 1 • function of :r, but Qx^ + Scc^ + 1 and 5a-" - 2x~^ + 3x- + 1 are 1 not, because the first contains x^, of which the index is not integral, and the second contains a;"^, of which the index is not positive. 332. The expression 5x^ + 40;' + 2 is said to be the expres- sion corresponding to the equation 5x^ + 4x^ + 2 = 0, and the latter is the equation corresponding to the former. 333. If a be a root of an equation, then x-a is a factor of tlie corresponding expression, provided the equation and expression contain only positive integral powers of x. This principle is useful in resolving such an expression into factors. We have already proved it to be true in the case of a quadratic equation. The general proof of it is not suitable for the stage at which tlie learner is now supposed to be arrived, but we ■will illustrate it by some Examples. 240 ox TUF. ROOTS OF EQUATTOA^S. Ex. 1 . Rrsul ve 2oc2 - 5x + 3 into factors. If we solve the equation 2x^-5.15 + 3 = 0. we shall find thai its roots five 1 and -. Now divide 2/--5.r + 3 by x-1 ; the quotient is 2j6-3 that is o(.,:- I); .'. the L;i\'eii eA])ression = 2 (a; - 1) ( x - ^ I. Ex. 2. Eesolve 2x^ + a;-— ll.f- 10 jnto factor.?. By trial we find that this expre.ssion vanishes if we put x= - 1 ; tliat is, — 1 is a root of the e(jiiation Sx^ + x^-ll.t- 10 = 0. Divide the expression l>y x4- 1 : the quotient is Sx^-x- 10 ; .'. the expre.ssion = (2x- - x - 10) (x + 1) = 2(x^-|-5)(. + l). We must now resol\-e x- - -b into factor.s, by solving the corresponding equation x^ ~'- — b=0. The roots of this equation are - 2 and g; .-. 2x3 + x2 - 1 Ix -10 = 2 (.»• + 2) (x - ^) (x + 1) = (x + 2)(2./;-5)(x + l). Examples.— cxxvi. Resolve into simijle factors the following expressions : I. .t3-11x2 + 36x-36. 2. x^-7.c2 + l4x-8. 3. x-"* - 5.1-2 - 4(i.-- - 40. 4_ 4x3 + 6.1-2 + X-1. 5. 6.r3+ll.»;2-9x-14. 6. 3?^y^ -^^-Zxyz. 7. a^-P~c^-2ahc. 8. 3x3-x2-23x + 21. 9. 2.f3 - 5x2 _ i7.r + 20. ■ 10. 15.1-3 + 41.r2 + 5.r - 21. ON THE ROOTS OF RQClATlONS. 241 334. Tf we can find one root of such an equation as 2a;3 + a;2-llx-10 = 0, we can find all the roots. One root of the equation is - 1 ; .-. (x + l)(2x2-x-10) = 0; .-. x+l = 0, or2a;2-a;-10 = 0; .. x= - 1, or —2, or -. Similarly, if we can find one root of an equation involving the 4"" power of x, we can derive from it an equation involving the 3'* and lower powers of x, from which we may find the other roots. And if again we can find one root of this, the other two roots can be found from a (quadratic equation. 335. Any equation into which an unknown symbol or ex- pression enters in two terms onl3', having its index in one of the terms double of its index in the other, may be solved as a ([uadratic equation. Ex. Solve the equation x^ — Qx^ = l. Regarding x^ as the quantity to be obtained by the solution III the equation, we get therefore x^-3=±4; • therefore x^=7, or x^= — 1. Hence x= ^'7 or x= ^ -1^ and one value of ^^ - 1 is - 1. 336. In some cases by adding a certain quantity lo both sides of an equation we can bring it into a form capable of solution, thus, to solve the equation x2 + 5.>; + 4 = 5 ^'(x^ + 5,/; + 28), add 24 to each side. Then x^ + 5x + 28 = 5 s'{x'^ + 5x + 28) + 24 ; or, a;2 + 5a; + 28-5 V(a'-"^ + 5x + 28) = 24. This is now in the form of a quadratic e([uation, the un- known quantity being ^f{x- + 5x + 28), and completing the square we have fs-A-l Q 242 0\ THE kOOTS OF EQUATIONS. 95 191 .-. V(a;2 + 5.r + 28)-|=±^; whence »J{x^ + 5x + 28) = 8 or - 3 ; .-. .r'' + 5x + 28 = 64 or 9; from which we may find four values of x, viz. 4, - 9, ani 5+ V(-51) Examples.— cxxvii. Find roots of the following equations : I. x-*- 12x2=13. 2. x6+14x3 + 24=0. 3. x»-t- 22x^ + 21=0. 4. x-'" + 3x"' = 4. 5. x--3X-» = ^. 6. ^=-20;-=-. 7. x-2 + 3x-i = ^. 8. x-'"'-x-'' = 20. 9. x2-2x + 6(x2-2x + 5)2 = ll. 10. x^-x + S V(2x2-5x + 6) = — ^ — . 11. x2-2V(3x2-2ax + 4) + 4 = |*(x + ^ + l). 12. ax + 'i >J{x--ax + a'^=x^ + '2,a. 337. Every equation has as many roots as it has dimen- sions, and no more. This we have proved in the case of simple and quadratic equations (Arts. 193, 323). The general proof is not suited to this work, but Ave may illustrate it by the following Examples. Ex. 1. To solve the equation x-^- 1=0. One root is clearly 1. Dividing b}- x — 1, we obtain x- + x + 1 = 0, of which the roots -1+ V-3^^.. -l-x/-3 are ^r^'— and ^ . 2 « ON RA no. 243 Hence the three roots are 1, ~ and ^^ . Ex. 2. To solve tht equation x^-\=0. Two of the roots are evidently + 1 and - 1. Hence, dividing by (x- l)(x + 1), that is by a;^- 1, we obtain a;2 + 1 = 0, of which the roots are v^— 1 and - v''— I- Hence the /our roots are 1, - 1, ^^- 1, and — \'— 1. The equation x^ — 6x^ = 7 will in lii<;e luauner have six roots, for it may be reduced, as in Art. 335, to two cuT)ic equations, x^ - 7 = and x^ + 1 = 0, each of which has three roots, which may be found as in Ex. 1. XXVII. ON RATIO. 338. If a and B stand for two unequal quantities of the same kind, we may consider their inequality in two ways. We may ask. (1) By ichat quantity one is greater than the other ? -The answer to this is made by stating the difference be- tween the two quantities. Now since quantities are represented in Algebra by their measures (Art. 33), if a and b be the measures of A and B, the difference between A and B is represented algeljraically by a-b. (2) By how many times one is greater than tlie other? The answer to this question is made by stating the number of times the one contains the other. Note. The quantities must be of the same kind. We can- not compare inches with hours, nor lines with surfaces. 339. The second method of comparing ^4 and B is called finding the Eatio of A to B, and we give the following ileti- nition. Def. Eatio is the relation which one quantity bears to another of the same kind with respect to the number of t,ime: the one contains the other. 244 ON RATIO. 340. The ratio of A io B is expressed thus, A : P>. A and B are called the Terms of the ratio. A is called the Antecedent and B the Conskquent. 341. Now since quantities are represented in Algebra hy their measures, we must represent the ratio between two (quantities by tlie ratio Ijetween their measures. Our next step then must be to sliovv how to estimate tlie ratio between two numbers. This ratio is determined by finding how many times one contains the other, that is, by obtaining the quotient resulting from the division of one by the other. If a and 6, then, be any two numbers, the fraction j- will express the ratio of a to b. (Art 136.) 342. Thus if a and b be the measures of A and B respec- tively, the ratio of A to JB is represented algebraically by the fraction r. 343. If a or b or both are surd numbers, the fraction ^ may also be a surd, and its approximate value can be found In' Art. 291. Suppose this value to be ' , where m and n are whole numbers : then we sliould say that the ratio A : B is aj (proximately re])resented by — . 344. Ratius may be compared witli each other, by com- paring ihe fractions by wliich they arc denoted. Thus the ratios 3 : 4 ami 4 : 5 may be compared by com- 3 4 paring the fractions - and -. These are equivalent to — and ^ resi)ectlvely ; and since gx is greater than 7,—, the ratio 4 : 5 is greater than the latio 3:4 ON RATIO. Ms Examples. — cxxviii. 1. Place in order of magnitude the ratios 2 : 3, 6 : 7. 7 : 9. 2. Compare the ratios x + 3y : x + 2y and x + 2ij : x + //. 3. Compare the ratios x-5y : x — 4y and x-Zy : x - -ly. 4. What number must be added to each of the terms of th« latio a : h. that it may become the ratio c : d? 5. The sum of the squares of the Antecedent and Conse- quent of a Eatio is 181, and tlie product of the Antecedent and Consequent is 90. What is the ratio? 345. A ratio of greater inequality is one whose antecedent is greater than its consequent. A ratio of less inequality is one whose antecedent is less than its consequent. This is the same as saying a ratio of greater inequality is represented by an Improper Fraction, and a ratio of less in- equality by a Proper Fraction. 346. A Ratio of greater inequality is diminished by adding the same nuwher to both its terms. Thus if 1 be added to both terms of the ratio 5 : 2 it becomes 6 : 3, which is less than the former ratio, since g, that is, 2, is less than -. And, in general, if x be added to both terms of the ratio a : h, where a is greater than 6, we may compare the twu ratios thus, ratio a + x : 6 + a- is less than ratio a : b, if -i be less than -y, b + x V .„ ' ab+bx . ■, ,, ab + ax it ,-i5 — r- be less than -=- — ;— , 62 + bx V + bx^ if ab + bx be less than ab + ax, if 6x be less than ax, if b be less than a. Now b is less than o ; :. a +x -.b + x \?, less than a : h. 246 CyV l^A no. 347. We may observe that Art. 346 is iiieiely a repetition of that which we proposed as an Example at the end of the chapter on Miscellaneous Fractions. There is not indeed any necessity for us to -vvearj' the reader with examples on Ratio: for since we exjiress a ratio by a fraction, nearly all that we mi.t,'ht have had to say about Ratios has been anticipated in our remarks on Fractions. 348. The student may, however, work tlie following Theo- rems as Examples. (1) If fl : 6 be a ratio of greater inequality, and x a positive quantity, the ratio a — o:: b — x is greater than the ratio a : b. (2) If (/ : h he a ratio of less inequality, and x a positive quantity, llie ratio a + x : b +x is greater than the ratio a : b. (3) If a : i be a ratio of less inequality, and x a positive quantity, the ratio a — x: b — x is less than the ratio a : b. 349. In some cases we may from a single equation involv- ing two unknown symbols determine the ratio between the two symbols. In other words we may be ahle to determine the relative values of the two symbols, though we cannot determine their absolute values. Thus from the equation 4x = 3?/, X 3 we get - = -. y 4 A^ain, from the equation 3x2 = 2?/-, ■we"et'., = ^; and therefore =-\t. " 2/- 3 y x/.3 Examples. — cxxix. Find the ratio of x to y from the following equations : 1. 9.1 = 6)/. 2. ax = by. 3. ax-by = cx + dy. 4. x-4-2a-?/ = 5?/-. 5. a;2- 12,r)/= 13;r. 6. x' + mxy = n^y-. 7. Find two numbers in the ratio of 3 : 4. of \\ hich the sum is to the sum of their scjuares :: 7 : 50. 8. Two numbers are in the ratio of 6 : 7, and when 12 is addid to each ihe resulting numbers are in the ratio 1 I 12 : 13. Find the nimibers. OK RATIO. 247 9. The sum ol' two iiujuLers is 100, and the nunriieis are in the ratio of 7 : 13. Find them. 10. The ditt'erence of the squares of two numbers is 48, and the sum of the nvimber^ is to the difierence of the num- bers in the ratio 12:1. Find the numbers. 11. If 5 gold coins and 4 silver ones are worth as much as 3 gold coins and 12 silver ones, find the ratio of the value of a gold coin to that of a sih'er one. 12. If 8 gold coins and 9 silver ones are Avorth as much as 6 gold coins and 19 silver ones, find the ratio of the Aaliie of a silver coin to that of a srold one. 350. Ratios are compounded by multipljing together the fractions by wliich they are denoted. Thus the ratio compounded of a : 6 and c : fZ is ac : hd. Examples. — cxxx. Write the ratios compounded of the ratios 1. 2:3 and 4:5. 2. 3 : 7, 14 : 9 and 4 : 3. 3. a;- — y- : x^ + y^ and x- - xy + y- : x + y. 4. a^ — b^ + 2bc - c^ : a^ - 6- - 2hc - c^ and a + b-rC : a + h - r. 5. m^ + n^ : vi^ - n^ and m — n : m + n. 6. x^ + 5x + 6 : y'-' — ly + 12, and y^ - 3)/ : x- + 3x. 351. The ratio a^ : b'^ is called the Duplicate Ratio of a ; 6. Thus 100 : 64 is the duplicate ratio of 10 : 8, and 36a;2 : 2oy^ is the duplicate ratio of 6x : by. The ratio a^ : ¥ is called the Triplicate Ratio of a : &. Thus 64 : 27 is the triplicate ratio of 4 : 3, .and 343x^ : 1331)/^ is the triplicnte ratio of fx : lly. 248 ox PROPORTION. 352. The definition of Ratio given in Euclid is the sanu! jf^ in Algebra, and so also is the expression for the ratio that one quantity bears to another, that is, A : B. But Euclid cannot employ fractions, and hence he cannot represent the value of a ratio as we do in Altjebra. XXVIII ON PROPORTION. 353. Proportion consists in the equality of two ratios. The algebraic test of Proportion is tlud the two fractions representing the ratios must he equal. Thus the ratio a : b will be equal to the ratio c : d, and the/o?(?- numbers a, b, c, d are in such a case said to be in proportion. ^ 354. If the ratios a ■ b and c ; d form a proportion, we express the fact thus : a : b = c : d. This is the clearest manner of expressing the equality of the ratios a : b and c : d, but there is another way of expressing the same fact, thus a : b :: c : d, which is read thus, a is to 6 as c is to d. The two terms a and d are called the Extremes. , b and c the Means. 355. When four numbers are in proportion, product of extremes = product of means. Let a, b, c, d lie in jiroportion. ON PROPORTION. 249 Multiplying both sides of the equation by M, we get ad = he. Conversely, if ad = bc we can show that a : b=c ', d. For since ad = be, dividing both sides by bd, we get ad_hc hrVd' that is, h^ d' ^'^' " '•^ = '^ '• ^' 356. liad = bc, Dividing by cd, we get - = j, i-fe- a : c = b : d; d c Dividing by ab, we get r = -, i-t^- '^ : 6 = c : a ; Dividing by ac, we get - = -, i.e. d : c = b -.a. 357. From this it follows that if any 4 numbers be so related that the product of two is equal to the product of the other two, we can express the 4 numbers in the form of a pro- portion. The factors of one of the j^roducts must form the extremes. The factors of the other product must form the means. 358. Three, quantities are said to be in Continued Pro- portion when the ratio of the first to the second is equal to the ratio of the second to the third. Thus a, b, c are in continued projjortion if a : b = b : c. % The quantitj' b is called a Mean Proportional lietween a and c. 25© ON proportion: Four quantities are said to be in Continued Proportion when the ratios of the first to the second, of the second to tlie third, and of the tliird to the fourth are all equal. Tlius a, b, c, d are in continued proportion when a : b = b : c = c : d. 359. We showed in Art. 20.5 the process by wliicli when two or more fractions are known to be equal, otlier relation? between the numbers involved in them may be determined That process is of course applicable to Examples in Ratio and Proportion, as we shall now show by particular instances. Ex. 1. li a : b = c : d, prove that a^ + b'^ : a^- -¥ = <:'- + d^:c^- d^. Smce a : o = c : d, t= j. a Let r=X. ThenT = \; d :. a = \b, and c = \d. Now and ft^ + 6' _ X^b- + ¥ _ />-(\^+l) _ X2+J c'^ + d^- _ \\l- + d'^ _ d- (X^+l)_X^ + l a^ + b'^_c- + d- ^t^b^~^~d^'' Hence ti^at is, a2 + 6« : a2 _ 52 = c2 + ^^2 . ^2 _ ^2^ Ex. 2. If « ; fe :: c : d, prove that a:c:: ^{a* + ¥): i/{c* + d^), LetJ = X. Then^ = X; d ' ,: a = \b, and c = \d. ON PROPORTION. 251 a _yb _h c y^d d' i/ic'+d^) ~ ^(Md* +'d') ■" ;^d^. */^\* 4-1) ^¥~'d- Hence that is, a:c:: ^{a^ + b*) : ij{c* + d^^ . Ex. 3. ] f a : 6 = c : f? = e : /, prove that each of these ratios is equal to the ratio a + c + e: b + d +f. Let - | = \, | = X, ^ = X. Then a = \b, c = Xrf, e = X/. ^ a + c + e 'Kb + \d + \f_H b + d+f) _^ °^'^ b + 'd+f~~b + d+f " 'b + d+f TT a + c-re a c e ^"^^^ b^drrb=d=f' that is, a + c + e : b + d +f— a : b = c : d = e :/. Ex. 4. If a, b, c are in continued jiroportion, show that a"^ + b'^ : b'^ + c^ ~ a : c. Let ~ = X.- Then- = X. c Hence a='\b and 6 = Xc:. a^ + b'^ _X-b- + b-_b\ \^+l )_b-(\^' + l)_b'^ _ac_a b'^ + c'^ ~ T- + C-' ~ T-c^^Tc- ~ f^X'-4-l)~ c''^ ~ c^ ~ c" Ex. 5. If Uxi + b : 15t- + d=l2a + b: 12c + d, ])rove that a :b = c : d. Since 15a + 6 : 15c + c?=12a + ?* : 12c + ci, and since product of extremes = product of means. 252 ON PROPORTION. (15CH-6) (12c + i) = (15c + d) (12a +6), or, 180ac+ 126c + 15ad + 6d = 180ac + 12a<Z+ 156c + 6rZ, or, 126c + Ybad = 12ad + 156c, or, 3ad = 36c, or, atZ = 6c. Whence, by Art. 355, a : b = c : d. Additional Examples will be found in page 137, to which we may add the following. EXAMPLES. — CXXXi. 1 . li a : b = c : d, show that a + b: a = c + d :e. 2. U a : b = c : d. show tliat a^ - 1- : b^ = c- - d"^ : d^. "?. It a, : Oi = a2 : 6,, show that — ^ ,- ==-. 4. If a : 6 :: c : c?, show that 3a- + ab + 26'' : 3a- - 26^ : : 3o- + cd + 2d^ : 3c2 - 2d-. 5. If ffl : 6 = c : rf, show that ft2 + 3ab + ¥ : c^ + 3tY/ + d- = 2«6 + 362 . 2cd + 3d-. 6. Ifa:6 = c:rf = e :/ then a : b — mc — ne : md-nf. 7. If — a, —6, any parts of a, b, be taken from a and 6 n n respectively, show that a, b, anil the remainders form a propor- tion. 8. If a : 6 = c : d = e :/, show that ac : bd = la^ + mc- + ne- : lb- + ind- + nf-. 9. If (/, : 6i = aj : 63 = 03 : 63, show that (/,- 4-^,2 + ^^2 . 5^2 ^5^2 + 5^2 .. „^-.- . i,i^ av PROPORTION. 253 10. If ai : 6i = rt2 : 62 = a3 : h-i, show that a^a.2^ + a^Og + 03(11 : h^.^ + 6063 + 6361 = a-^ : 6,2. -rpa2-a6 + 62 c--cd + c?2 ,,,.,, a c a d 11. It „- — , --, „ = -.,— — — „, snow that either r = 3 or t = -• 12. If a2 + 6- : rt^ - /*- = c2 + c?2 . cii - rf2^ phow that a: b — c : (^. 13. If rt : 6 = c : (Z, show that (rt - c) (a^ - r-') (6 - d) (¥- d^y 14. If rtj : h^ — a., : /*.,, show that On the Geometrical Treatment of Proportion. 360. The definition of Proportion (viz. the equality of ratios) is the same in Euclid as in Algebra. (Eucl. Book v. Def. 6 and 8.) But the ways of testing whether two ratios are equal are quite different in Euclid and in Algebra. The algebraic test is, as we have said, that the two fractions representing the ratios must be equal. Euclid's test is given in Book v. Def. 5, where it stands thus : " The first of four magnitudes is said to have the same ratio to the second which the third has to the fourth, when any equimultiples whatsoever of the first and third being taKen and any equimultiples whatsoever of the second and fourth : " If the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth : " If the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth : or. 254 ON PROPORTION. " If the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth." We shall now show, first, how to deduce Euclid's test of the equality of ratios from the algebraic test, and secondly, how to. deduce the algebraic test from that employed by Euclid. 361. I. To show that if quantities be proportional accord- ing to the algebraical test they will also be proportionai according to the geometrical test. If a, 6, c, d be proportional according to the algebraical test, a _c Multiply each side by — , and we get ma _mc nb ruT Now, from the nature of fractions, if ma be less than nb, mc will also be less than nd, and if ma be equal to nb, mc will also be equal to nd, and if ma be greater than nb, mc will also be greater than nd. Since then of the four quantities a, b, c, d equimultiples have been taken of the first and third, and equimultiples of the second and fourth, and it appears that when the multiple of the first is greater than, equal to, or less than the multiple of the second, the multiple of the third is also greater than, equal to, or less than tlie multiple of the fourth, it follows that a, b, c, d are proportionals according to the geometrical test. 362. II. To dediioe the algebraic test of proportionality from that given by Euclid. Let a, h, c, d be proportional according to Euclid. Then if s- is not equal to -3, let , be equal to , (1). EXAMPLES ON RA TIO. ±%^ Take to and n such that via, is greater than nh, but less than n (i + x) (2). Then, by Euclid's definition, TOC is greater than nd (3). But since, by il), -77-,— ^ = —7? and, by (2), wa is less than i!(6 + x), it follows that 7/i.c is less than nd (4). The results (3) and (4) therefore contradict each other. Hence (1) cannot be true. Therefore -7 is equal to -^. We shall conclude this chapter with a mixed collection of Examples on Ratio and Proportion. EXAMPLES. — CXXXii. 1. VL a-h -.h-c •.-.h : c, show that i is a mean proportional between a and c. 2. If a : 6 : : c : rf, show that a^^W : ^\ = c'' + d^:-^-. a+b c+d and a : b :: ^/{ma* + nc*) : i/(^mb^+ iid*). 3. li a : b :: c : d, prove that ma — nb _ mc - nd ma + nb mc + nd' 4. If ba + 2b': 7a + 36 : : 56 + 3c : 76 + 3c, 6 is a mean proportional between a and c. 5. If 4 quantities be proportional, and the first be the greatest, the fourth is the least. If a + 6, TO 4- n, m-n,a — b be four such quantiti««j show that h is greater than n. 25^ k^AMPLES ON RA TlO. 6. Solve the equation x-\ : a;-2 = 2x + l : x + 2. 7. If — , — = — 5—, show that the ratios a : b and c : d are b a also equal. 8. In a mile race hetween a bicycle and a tricycle, their rates were proportional to 5 and 4. The tricycle had half-a- niinute start, but was beaten by 176 yards. Find the rates of each. 9. li a : b :: c : d and a is the greatest of the four quanti- ties, show that a- + d- is greater than b~ + c^. 01 .1, .-plOa + fe 12a + 6 , , , 10. bhow that it vt; i=rEi 3) i^hen a : :: c : a. lOc + d 12c + d' 11. U X : y :: Z : 2 and x : 25 : : 24 : 1/, find x and y. 12. If a, b, c be in continued proportion, then (1) a : a + b :: a-b : a-c; (2) (a2 + f^) (62 + C-) = {ab + bcf. 13. If a : : : c : a, show that — j— = — t— ; and hence solve the equation ah — bc — dx_a — h — c bc + dx b + c ' 14. If a, b, c are in continued proportion, show that a y- nib : a - mb :: b + vie : b - mc. 15. li a : h :: ."> : 4, find the value ot the ratio 1 3 16. The sides of a triangle are as 2- : 3- : 4, and the peri- 2 4 '^ meter is 205 yards: tiiid the sides. 17. The sides of a triangle are as 3 : 4 : 5, and the peri- meter is 480 y^rds : find tlie sides. AND PROPORTION. 257 1 8. Assuming a + 6 :^ + 9 '■'-p — i • a-b, prove that the sum of the greatest and least terras of any proportion is greater than the sum of the other two. '^'1 19. A waterman rows 30 miles and back in 12 hours, ■md he finds that he can row 5 miles with the stream in the same time as 3 against it. Find the rate of the stream. A,^ 20. There are three equal vessels A, B, C ; the first con- tains water, the second brandy, the third brandy and water. If the contents of B and G be put together, it is found that the mixture is nine times as strong as if the contents of A and G had been put together. Find the ratio of the brandy to the water in the vessel G. 21. A factor buys a certain quantity of wheat which he sells again so as to gain 5 per cent, on his outlay, and thus clears £16. Had he sold it at a gain of 5s. a quarter lie would have cleared as many pounds as each quarter cost shillings. How many quarters did he buy, and what did each quarter cost him ? 22. A man buys a horse and sells it for £144, gaining as much per cent, as the horse cost him. What was the price of the horse 1 23. I buy goods and sell them again for £96, gaining as much per cent, as the goods cost. "What is the cost price ? 24. A man bought some sheep and sold them again for £24, gaining as much per cent, as the sheup cost him. What did he give for them ? ^^ 25. A certain crew, who row 40 strokes per minute, start 'at a distance equivalent to four of their own strokes behind another crew, who row 45 strokes to the minute. In 8 minutes the former succeed in bumping the latter. Find the ratio between the lengths of the strokes of the two boats. 26. The time which an express train takes to travel a journey of 180 miles is to that taken by an ordinary train a»s 9 : 14. The ordinary train loses as much time from stoppnges as it would take to travel 30 miles without stoppini;. The express train only loses half as mucli time as the o:hfi- in this 25S OA' VARIATION. manner, and it also travels 15 miles an hour quicker. Sup- posing the rates of travelling uniform, what are they in miles per hour ] , 27. An article is sold at a loss of as much per cent, a? it /lis, worth in pounds. Show that it cannot be sold for more than ^25. XXIX. ON VARIATION. 363. If a sum of money is put out at interest at 5 per cent, the principal is 20 times as great as the annual interest, what- ever the sum may be. Hence if x be the principal, and y the interest, x = 2()i/. Now if we change x we must change w in ike same propor- tion, for so long as tlie rate of interest remains the same, x will always be 20 times . as great as y, and hence if a: be doubled or trebled, y will also be doubled or trebled. This is an instance of what is called Direct Vari.\tion, of which we may give the I'ullowing definition. Def. One quantity y is said to vary directly as another quantity x, wlien y depends on x in such a manner tiiat any increase or decrease made in the value of x produces a propor- tional increase or decrease in the value of 1/. 364. If x = my, where m is a constant quantity, that is. a quantity which is not altL-rcd by any change in the values of j, and y, y will vary directly as x. For any increase made in the value of x must produce u proportional increase in the value of y. Thus if x be doubled, y must also be doubled, to prt-serve the e(|uality of x and my, since m cannot be changed. ON VAI^IATION. 259 365. Suppose a man can reap an acre of corn in a day. Then 10 men can reap 60 acres in 6 days, and 20 men can reap 60 acres in 3 dayss So that to do the same amount of work if we double the number of men we must halve the number of days. This is an instance of what is called Inverse Variation, of which we may give the following definition. Def. One quantity y is said to vary inversely as another quantity x, when y depends on x in such a manner that any increase or decrease made iu the value of x produces a propor- tional decrease or increase in the value of y. 366. If a; = — , where m is constant, y y will vary inversely as x. For any increase made in the value of x must produce a pro- portional DECREASE in the value of y. Thus if x be doul)led, y must be halved, to preserve the equality of x and — . ■ For 2x = = — . y y 2 367. If 1 man can reap 1 acre in 1 day, 5 men can reap 20 acres in 4 days, and 10 men can reap 80 acres in 8 days. That is, the number of acres reaped will depend on the product of the number of men into the numl^er of days. This is an example oi joint variation, of which we may give the following definition. Def. One quantity x is said to vary jointly as tw^o others ;/ and j-, when any change made in x produces a proportional change in the product of y and z. 368. One quantity x is said to vary directly as y and inverseh' as z when x varies as -. z 2bo O.y \ AklAT.'OS'. 369. Theorem, ll x varies as y when % is constant, and «s z when '/ is eonstiiiit, then wlien y and z are both variable, % varies as yz. Let x — tth. yz. Then we have to show thnt in is constant. Now when z is constant, X varies as y ; .". mz is constant. Now z cannot involve y, since z is constant when y changes, and therefore m cannot involve y. Similarly it may be shown that m cannot involve z ; :. m is constant, and X varies as yz. 370. The symbol oc is used to express variation; thus xocy stands for the words x varies as y. 371. Variation is only an abbreviated form of expressing proportion. Thus when we say that x varies as y, we mean that x bears to y the same ratio that any given value of x beiirs to the corresponding value of y, or x : J/ = a given value of x : the corresponding value of y. And similarly for the other kinds of variation, as will be .seen from our examples. Ex. 1. If xoc y and i/oc,t, to show that xocj. Let x=my, and y = iiz. Then substituting this value of y in the first equation. x = m}U ; •Old therefore, since mn is constant, • OCS. ON VARIATION. 2C1 Ex. 2. If a;cci/ and xocz, then will xcc ^/(i/js). Let x = mi/, and x = ns. Then x^ = mnyz\ ;. x= v/(mTO) . V(l/«)- Now J(mn) is constant ; .-. re ex: VM- Ex. 3. If y vary as x, and when x=l, i/ = 2, what wiU be the value of y when x = 2 ? Here 1/ : x= a given value of y : corresponding value ot x; :.y:x = 2:l: .-. y = 2x. Hence, when x = 2, y = 4. Ex. 4. If A vary inversely as B, and when A = 2, B=12, what will B become Avhen A =91 Here yl : -j, = a given value of A corresponding value ofS' A 1 12" 9 12" - ^ . 2 *^ 2 1 \2' 1 B 24 9 8 ~3' -1 Hence, when ^ = 9, whence Ex. 5. If A vary jointly as B and G, and when yl = 6, J5 = 6 and (7= 15. find the value of A when 5= 10 and C=3. Here A : BG= a given value of A : corresponding value of BC\ :. A :BC='6 : 6x15; .-. 90.4 =6B0. 262 Uy iARIAlJOW Henee, when J5= 10 and C=3, 90^ = 6 X 10 X 3 ; ••"*-90-^- Ex. 6. If z vary as x directly and y inversely, and if when 2 = 2, x = 3 and i/ = 4, what is the value of z when x=15 and TT a; -. ■ 1 r corresponding value of x Here % : - =*'a civen value of z : ^ — ^.-^ = jr- ; y " corresponding value oi y X \ z :- y -:!. . 32 ■■ 4 _2j; nd j/ = 8, 32 4 30 8' .'. «-= 120 ^ ^T4- = ^- Examples. — cxxxlii, II 1. If ^oc-^ and Boc — then will ^ocC. 2. If .-loc^then Avill^oc^. 3. U Acr.B and Coc D then will ^ C<x 5D. 4. If xccj/, and when x = 7, J/ = 5, find the value of z whew y = 12. 5. If xec- , and when x» lO, y = t, find the value of y when :c = 4. ON VARIATION. 263 6. \i xazyz, and when x = l, i/ = 2, a = 3, find the value of y when X = 4 and ^ = 2. 7. If xoc^, and when x = 6, ]/ = 4, and 2 = 3, find the value of X when 1/ = 5 and a = 7. 8. If 3x + 5?/ oc 5a; + 3y, and when a; = 2, 1/ = 5, find the value . X of -. y 9. If ^cci> and B^ozC^, express how J. varies in respect of a 10. If z vaiy conjointly as x and y, and 2=4 when x=l and 2/ = 2, what will be the value of x when s = 30 and y = Si 11. If ^ocB, and when A is 8, 5 is 12; express A in terms of B. 12. If tlie square of x vary as the cube of y, and x = 3 when 7/ = 4, find the equation between x and y. 13. If the square of x vary inversely as the cube of y, and « = 2 when i/ = 3, find the equation between x and y. 14. If the cube of x vary as the square of y and x = 3 when i/ = 2, find the equation between x and 3/. I?. If XOC5J and i/oc-, show that xoz-. ^ z' y 16. Show that in triangles of equal area the altitudes vary iiurersely as the bases. 17. Show that in parallelograms of equal area the altitudes vary inversely as the bases. 18. H y=p + q + r, where p is invariable, q varies as x, and r varies as x^, find the relation between y and x, supposing that when a;=l, y = 6; when x = 2, y=ll ; and when x = 3, 2/ = 18. 19. The volume of a pyramid varies jointly as the area of its base and its altitude. A pyramid* the base of which is 9 264 ON ARITHMETICAL PROGRESSION. feet square and the height of which is 10 feet, is found to con- tain 10 cubic yards. What must be the height of a pyramid upon a base 3 feet square in order that it may contain 2 cubic yards ? 20. The amount of glass in a window, the panes of which are in every respect equal, varies as the number, length, and breadth of the panes jointly. Show that if their number varies as the square of tlieir breadth inversely, and their length varies as their breadth inversely, the whole area of glass varies as the square of the length of the panes. XXX. ON ARITHMETICAL PROGRESSION. 372. An Arithmetical Progression is a series of numbers wliich increase or decrease hxj a constant difference. Thus, tlie following series are Arithmetical Progressions: 2, 4, 6, 8, 10; 9, 7, 5, 3," 1. The Constant Difference being 2 in the first series and - 2 in the second. 373. In Algebra we express an Arithmetical Progression thus : taking a to represent the first term and d to represent the constant ditt'erence, we shall have as a series of numbers in Arithmetical Progression o, a + d, a + -Id, a + ;j(/, and so on. We observe that the terms of the series differ only in the coefficient of d, and that each coefficient of d is always less by 1 than the number of the term in which that particular coefficient stands. Thus the coefficient of d in the 3rd term is 2, in the 4th 3, ? in the 5th 4. OM ARITHMETICAL PROGRESSION. 265 Consequently the coefficient of d in the m"" term will be Therefore the v!^ term of the series will be a -i- (n - 1; d.. 374. If the series be a, a + d, « + 2(i, 'anil % the last term, the term next before z will clearly be 2 - d. and the term next before it will be s - 2d, and so on. Hence, the series written backwards will be 2, z - fZ, 3 - 2rf, a + 2d, a + rf, a. 375. To find the s^im of a series of numbers in Arithvietical Progression. Let a denote the iirst term. ... d the constant difference. ... z the last term. ... n the number of terms. ... s the sum of the 7i terms. Then s = a+(a + d) + (rt + 2(f)+ +{z~2d) + (z-d) +z. Also s = z + {z-d)+ {z-2d)+ +(« + 2d) + (a + rf) + a, the series in the second case being the same as in the lirst, Vmt written in the reverse order. Therefore, by adding the two series together, we get •2s={a + z) + (a + z) + (a + z)+ + {a + z) + (a + z) + (a + z) ; and since on the right-hand side of this equation we have a series of n numbers each equal to a + z, we get 2s = n{a + z)', This result may be put in another form, because in the place of z we may put a + {n—l)d, by Article 373. Hence s = ~\a + a + {n- i.)d\, thatio, =|{2a + (n-l)di. / 266 ON ARITHM£:TICAL PROGRESSION. 376. We have now obtained the following results : a(-=a + (M-l)rf (A), «=|(« + ^') (B), < = |)2« + (n-l)cf( fC). From one or more of these equations we have in Examples to determine the values of a, d, n, s or z. We shall now jjro- (•ee<l to j^ive instances of such Examples. Ex. 1. Find the LAST TERM of the series 7, 10, 13, ...... to 20 terms. Taking the equation z = a+ {n — \)i, for a put 7 and for n put 20, and we get « = 7 + (20-l)i, or, !J! = 7+19d. Now d is always found by taking the first term from ike second^ and in this case, r^=10-7 = 3; .-. 2 = 7 + 19x3 = 7 + 57 = 64. Ex. 2. Find the last term of th - series 12, 8, 4, to 11 term.=^. In the equation z = a-^ {n—\)d, ]iut a = 12 and n=ll. Then z=\<i + \Od. Now d = 8-12=-4. Hence, a = 12 -40 =-28. Examples. — cxxxiv. Find the last term of each of the following siiries • 1. 2, 5, 8 to 17 terms. 2. 4, 8. 12 to 50 terms. On arithmetical PROGRESSIOiV. Z'oJ 3- ^ 29 15 . -.^ ^ 7,-r,-Tr to 16 terms. ' 4 2 4. 1 5 ^,—1, -- to 23 terms. r.. 5 11 . n. 6' 2' 6 to 12 terms. 6. -12, -8, -4 to 14 terms. 7- -3, 5, 13 to 16 terms. 8. w-ln-2n-3 , , to ?i terms. n n n 9 (x -^yf-^x^ + y-, {x-ijf to n terms. lO. a- fe 4a- 36 7a — 56 ., ^, — -^— to 71 terms. a + 6' a + 6' a + 6 377. Ex. 1. Find the sum of the series 3, 5, 7 to 12 terms. In the equation s = -{2a+ (« - 1) (l\ put 3 for a and 12 for n, and we get 19 Now d = 5 - 3 = 2, and so s = ^{6 + 22; =6x28 = 168. Ex. 2. Find the sum of the series 10, 7, 4 to 10 terms. s = |!2a + (»-l)d{: put 10 for a and 10 for n, then ,=12j20 + 9d(. 26S ON ARITHMETICAL PROGRESSION. Now rf«=7 - 10= - 3, and therefore a = l?)2<v-27!=5xf- 7)= -35. EXAMPLES. — CXXXV . Find the sum of the following series : 1. 1, 2. 3 to 100 tdins. 2. 2, 4, () to 50 t(M-m3. 3. 3, 7, 11 to 20 terms. 4. -, , -7 to 15 terms. 4' 2 4 5. -9, -7, -5 to 12 term 6. -. -, - to 17 terms. 6' 2' 6 7. 1, 2, 3 TO n terms. 8. 1, 4, 7 to ?i terms. g. 1, 8, 15 TO n terms. n - 1 ?! - 2 ?i - 3 . , 10. , , to 7? terms. n n n 378. Ex. What is tlie Constant Differekci!; when the first term is 24 and the tenth teiiii is - 12? Takintf tlie equation (A), z = a + (n - l)d, ajid re^'ardinij tlie tenth as the last term, we get -12 = 24 + (10-l)rf. or - 36 = Off, whence v,m obtain d= — 4. ON ARITHMETICAL PROGRESSION. t j Examples. — cxxxvi. What is the Constant Difference in the following cases % I. When the first term is lOo and the twentieth is — 14. 2 c fifty-first is - x. 3 —-= forty-ninth is 5-. 3 3 4 — ^ twenty-fifth is -21-. 5 -10 sixth is -20. 6 150 ninety -first is 0. 379. Ex. What is the First Term when the 4()th term is 28 and the 43rd term is 32 ? Taking equation (A), 2 = a + (n- l)c?, and regarding the last term to be the 40th, we get 28 = a + 39(7 (1). Again, regarding the last term to be the 43rd, we get 32 = a-f42tf (-:) From equations (1) and (2) we may find the value of a to be -24. Examples. — cxxxvii. I. What is the first term when (i) The 59th terra is 70 and the 66th term is 84; (2) The 20th term is 93 - 356 and the 21st is 98 - 376 ; (3) The second term is ^ and the 55th is 5-8 ; (4) The second term is 4 and the 87th ir; - SO ? fijro ON ARITHMETICAL PROGRESSION. 2. The Sinn of the 3rd and 8th terms of a series is 31, and the sum of the 5th and 10th terms is 43. Find the sum of 10 terms. 3. The sum of the 1st and 3rd terms of a series is 0, and the sum of the 2nd and 7th terms is 40. Find the sum of 7 terms. 4. If 24 and 33 Ije the fourth and fifth terms of a series, what is the 100th term ] 5. Of how many terms does an Arithmetical Progression consist, whose difference is 3, fir.st term 5 and last term 302 ? 6. Supposing that a body falls through a space of 16^^ feet in the first second of its fall, and in each succeeding second 32- feet more than in the next preceding one, how fir will a body fall in 20 seconds? 7. What debt can be discharged in a year by weekly pay- ments in arithmetical progression ; the first pajTiient being 1 shilling and the last ^5. 3*'. ? 8. Find the 41st term and the sum oi 4i lernis in each of the following series : (1) -0,4,13 (2) 4a2, 0, -40.2 (3) 1 + a-, 5 + 3.r, 9 + 5x (4) -4 -1'4 V.5.) 4> 20 9. To how many terms do the following series extend, and what is the sum of all the terms ? (1) 1002 10,2. (2) -0, :: ,186. ON- ARITHMETICAL PROGRESSION. 271 (3) 22X, -Sa; -72-3je. / X 1 1 (4) 2' 4 -^ (5) m-\ 137(1 -m), 135>a -m). (6) a; + 254, x + 2, x-2. 380. To insert 3 arithmetic niMyis between 2 and j.O. The number of terms will be 5. Taking the equation z = a + {n- i) d, we have 10 = 2 + (b-l)d. Whence 8 = 'id; :. d=2. Hence the series will be . 2. 4, 6, 8, 10. Examples. — cxxxviii. 1. Insert 4 arithmetic means between 3 and 18. 2. Insert 5 arithmetic means between 2 and —2. 2 ■*. Insert 3 arithmetic means between 3 and -. ^ 3 4. Insert 4 arithmetic means between - and -. 381. To insert 3 arithmetic menus between a and b. The number of tenu^ in the series will be 5. since the;, are to be 3 terms in addition to the iirst term a and the last term b. Taking the equation 2 = a + (n — iy a, we have to find d, having giveii a, z = b and n — b. 272 UN ARl'l HAfETICAL PROGRESSION. Hence h = a-\-(^-\)d. or, 4<i=6-a, .". d=— p . Hence the series will be 6 — a h — a 3(6 — a) . «, a + —4—, « + -2"' '^ + ~ 4 — "' °' that IS, a, -^p-, — g— , —J—, 6. Examples. — cxxxix. 1. Insert 3 arithmetic means between rfi and n. 2. Insert 4 arithmetic means between m + 1 and in-\. 3. Insert 4 arithmetic means between 11^ and ?!- + 1. 4. Insert 3 arithmetic means between x^ + y- and a;'-^ — y'-. 382. We shall now give the general form of the proposition " To iiisert m arithmetic means beticeen a aiid b." The number of terms in the series will be 7?i + 2 Then taking the equation z = a + (n-'\)d, we have in this case b = a + {m + 2 - 1 ) f/, or, b=a+{m + l)d. Hence d= ,, m + V and the form of the series will be , 26 - 2a , b — a . m+1 m+V ' bm-b + ia bm + a , m+1 ' m + l' ' a b-a ' m+1' 26 -2a a + — , m+1 ' that is, am + b avi - a + 26 '^' »r+T' m + 1 ' XXXI. ON GEOMETRICAL PROGRESSION. 383. A Geometrical Progression is a series of mnuliers which increase or decrease by a constant factor. Thus the following series are Geometrical Progressions. 2, 4, 8. 16, 32, 64; 12 3 2 A ^• ^^' "*' 4' 16' 64' _1 ^ __! Jl 2' 16' 128' 1024' The Constant Factors being 2 in the tirst series, - in tin- 4 second, and — - in the third. 8 Note. That which we shall call the Constant Factor is usually called the Common Ratio. 384. In Aljj;eV)ra we express a Geometrical Progression thus : taking a to represent the jfirst term and / to represent the Constant Factor, we shall have as a series of numbers in (ieonii'trical Progression a, of, af'^, af^, and so on. We observe that the terms of the series differ only in the index of/, and that each index of/ is always less by 1 than the number of the term in which that particular index stands. Thus the index of/ in the 3rd term is 2, in the 4th 3, in the 5th 4 Consequently the index of/ in the nth term will be n - 1. Therefore the ?ith term of the series will be a/"~'. [s.A,] 8 274 ON GEOMETRICAL PkOGKESSlON. Hence if z be the last term, 385. If the series contain 7! terms, a being the first term and / tlie Constant Factor, the last term will be a/""', the last term but one will be a/"~*, the last term but two will be a/*~*. Now a/"-' x/=a/'-i x/i = o/'-i+' = a/", a/"-^ X /= «/--^ X /' = a/"-'+' = a/-\ a/^s X /= a/'-=' X /I = a/"-='+' = a/-*. 386. We may now proceed to Jind the tmm of a teries of numbers in Geometrical rrogression. Let a denote the first term, / the constant factor, 71 the number of terms, s the sum of the n terms. Then s = a + af+ af+...+ af-^ + of-'' + af-K Now multiply both sides of this equation by/, then fs = af+ af^ + af+ ... + af"-^ + «/"-' + af". Hence, subtracting the first equation from the second, fs-s=^af"-a. ■•• «(/-!)=« (/"-I); •■'- f-i • Note. The proposition just proved presents a difficult}' to a beginner, which we shall endeavour to explain. When we multiply the series of ?! terms a + af+af-i- + af'-^ + af-^ + c^f^ OiV GEOMETRICAL PROGRESSION. 275 by/, we shall obtain another series af+af- + af + + a/'-' + «/--» + a/", which also contains n terms. Though we cannot fill up the gap in each series completely, we see that the terms in the two series must be the same, except the first term in the former series, and the last term in the latter. Hence, when we subtract, all the terms will dis- appear except these two. 387. From the formulae : 2 = a/"-' (A), .."-^' (B, prove the following : (a) sJj^. (y) a=fz-{f-\)s. f s — a z' 388. Ex. Find the last term of the series 3, 6, 12 to 9 terms. The Constant Factor is -, that is, 2. In the formula 3 = a/— », putting 3 for a, 2 for/, and 9 for n, we get 3 = 3x25 = 3x256 = 768. Examples.— cxl. Filid the last term of the following series 1. 1, 2, 4 to 7 terms. 2. 4, 12, 36 to 10 terms. 3. 5, 20, 80 tu 9 terms. 276 ON GEOMETRICAL PROGRESSION. 4. 8, 4, 2 to 15 terms. 5. 2, 6, 18 to 9 terms. 6- ^' 1^' 4. to 11 term.. 2 1 1 ^ n, 7- -3' 3' -6 to 7 term*. 389. Ex. Find thf sum of the series 3 2 3 6, 3, ^ to 8 terms. Generally, s= — 7— j — anti here a = 6,/=^, « = 8, 2 2 6__ 6_ 256 256 _ 766 " _1 1 ~~"6T* 2 2 EXAMPLES.— CXli. Find the .sum of the following series : 1. 2, 4, 8 to 15 terms. 2. 1 , 3, 9 to 6 terms. 3. a, ax^, ax* to 13 terms. 4.. a, -, -., to 5) terms ,r ./- 0^--x-, a- X, — ; — to 7 terms. ' a + x osr Geometrical pkockESSioM. 277 6. 2, 6, 18 to n terms. 7. 7, 14, 28 to ?i terms. 8. 5, -10, 20 to 8 terms. 2 1 1 ^ r, ^ 9- -3J 35 -g to 7 terms. 390. To find the sum of an Infinite Series in Geometrical Progression, when the Constant Factor is a proper fraction. If/ be a proper fraction and n very large, /" is a very small number. Hence if the number of terms be infinite, f" is so small that we may neglect it in the exjiression ,_«(/"- 1) /-I, ' and we get -a "I-/' 391. Ex.1. Find the sum of the series 5 + 1 + 7 + to infinity. Here /=1-| = !' 4 _ 0^ 3 16_-1 •'•*~i-7~7~3~T~^3- ^-4 3 2 8 Ex. 2. Sum to infinity the series g ~ o + 07 ~ Here /=-|^|=-!; 3 3 a 2 V 2 27 / 4\ , 4 26" -(-9) ^+- 9 27^ ox GEOMETRICAL PROGRESSION, Examples. — cxlii. Find the sum of tht I'olldwiug infiniie series; I. 1, i \ 9- 4^ 2*. .. 2' 4' 2. 1. -. ~ lo. 2z^, - -SSa;^ -i in 3- 3, -, - II. o, 6, o Z t 2 11 11 4- o. o> 5, 12. 3' 3' 6' 10' 10^' 13. X, -I/, . 3 1 )• 4' 4' 11 ^ 86 2" ~3 ''^' 100' 10000 •7- 8, I, 15. -54444 _ 3' 8. l|, -5, 16. -83636, 392. To inst'ti 3 geometric means heticeen 10 and 160. Taking the equation z^af"^^, we put 10 for a, 160 for z, an.l 5 for ?i, and we obtain 160=10./'-': .-. 16=/*. Now 16 = 2x2x2x2 = 2*; • 2* =/*. Hence /=i;. and the serie.s will be 10, 20, 40, 80, 160. ON GEOMETRICAL PROGRESSlOX. if^ Examples.— cxliii. 1. Insert 3 geometric means between 3 and 243. 2. Insert 4 geometric means between 1 and 1024. 3. Insert 3 geometric means between 1 and 16. 4. Insert 4 geometric means between - and — -. 393. To insert m geometric riieans between a (ind b. The number of terms in the series will be m + 2. In the formula z = af'''''^, putting b for z, and to + 2 for n, we get or, 6 = f^"'+l; •••' ~a' or, /=~t:. Hence the series will be, 1 _i_ _}_ 1 rt, a X — p , a X 6-^— r-, 6-^— T-, t, that is, II J 1 a, (rr . &)-+', (ft"-i.62)m+i^ ^ (a^S—y+i, (-; . Z/"-)"^!, />. 394. AVe shall now give some mixed Examples ou Aiitl:- raetical and Geometrical Progression. Examples. — cxliv. I. Sum the following series : (i) 8 + 15 + 22+ tol2terms. (2) 116 + 108 + 100+ to 10 terms. 28o ON GEOMETRICAL PROGRE.SSTOM. (3) 3 + 2'^12"^ to infinity. '4) 2 - - + — - to infinity. 4 oz 1 2 11 (5) 2~3~y ^"^"^ terms. 112 (6) 9~o+q— to 6 terms. 1 5 (7) g-1-^- to 29 terms. (8) s + l + l?+ toSterms. (9) 3 + 9 + 27"^ ••••■■• to infinity. , , 3 14 Ol i ^/^i . (10) V — T7^-r-- to 10 terms. 5 10 lo ('0 /v/?- v'6 + 2V(l-''')- to 8 terms. V 5 , , 7 7 35 ^ - i (12) -^ + s — r-+ to .5 terms. o 2 4 2. If the continued product of 5 terms in Geometrical Progression be 32, show that the middle term is 2. 3. If a, h, c are in arithmetic jirogression, and a, ?/, (• :i;v .1 • 1 *i 4. ^ <* + <' in geometrical jirogressioii, show that 17 = 5 — 77 — r. 4. Show that the arithmetical mean between a and h i- i:reater than tlie geometrical mean. 5. The sum of the first three terms of an arithmetic series is 12, and the si.xth term is 12 also. Find the sum of the first 6 terms. 6. What is necessary that «, 6, c may be in geometric pro- grescsion ? ON GEOMETRICAL PROGRESSION. 281 7. If 271, X and -^r- are in cfeometric protrression, what is x? 8. If 2n, ?/ and — are in aritlimetic progression, what is 1/? 9. The sum of a geometric progression whose firet term is 1, const nit factor 3, and number of terras 4, is equal to the sum of an arithmetic progression, whose tirst term is 4 and constant difference 4 ; how many terms are there in the arithmetic pro- gression? 10. The tirst (7 + ?i) natural numbers when added together make 153. Find n. 11. Prove that the sum of any number of terms of the series 1, 3, 5, is the square of the number of terms. 12. If the sum of a series of 5 terms in arithmetic progres- sion be 95, show that the middle term is 19. 13. There is an arithmetical progression whose first term is 1 4 3„, the constant difference is 1,;. and tlie sum of the terms is 22. Required the number of terras. 14. The 3 digits of a certain number are in arithmetical progression ; if the number be divided liy the sum of the digits in the units' and tens' place, the quotient is 107. If 396 be subtracted from the number, its digits will be inverted. Required the number. 15. If the {'p-Vfjf' term of a geometric progression be'm, and the {p — qf" term be n, show that the 2^"" term is >^f{mn). 16. The ditt'erence between two numbers is 48, and tlie arithmetic mean exceeds tlie geometric by 18. Find the numbers. 17. Place three aiithmctic means between 1 and 11. 18. The first term of an increasing arithmetic series is "034, the constant difference •0004, and the sum 2-748. Find the number of terms. 19. Place nine arithmetic jueans between 1 ami - 1, 282 ON HARMONIC AL PROGRESSION. 20. Prove that every term of the series 1, 2, 4, is greater by unity than the sum of all that precede it. 21. Show that if a series of my) tenns forming a geometrical progression whose constant factor is r be divided'into sets of p consecutive terms, the sums of the sets will foim a geometrical progression whose constant factor is r'. 22. Find five numbers in arithmetical progression, such that their sum is 55, and the sum of their squares 765. 23. In a geometrical progression of 5 terms the difference of the extremes is to the difference of the 2nd and 4th terms as 10 to 3, and the sum of the 2nd and 4tli terms equals twice the product of the 1st and 2nd. Find the series. 24. Show that the amounts of a sum of money put out at Compound Interest form a series in geometrical progression. 25. A certain number consists of three digits in geometrical progression. The sum of the digits is 13, and if 792 lie added to the number, the digits will be inverted. Find the number. 26. Tlie population of a county increases iu 4 years from 10000 to 146 il ; what is the rate of increase ? XXXII. ON HARMONICAL PROGRESSION. 395. A Harmonicai Progression is a series of numbeis of which the reciprocals form an Arithmetical Progression. Tims the series of numl)ers «, h, c, <l, is a Harmon ical • .•1 .1111. .-.,., Progression, 11 the series , r> -> "7? ^s an Anthmetical a bed Progression. If a, b, e be in Harmonicai Progression, b is called the Harmonicai Mean between </ and c. Note, There is no way of finding a general expression for the sum of a Harmonicai Series, but manv problems with OA' HARMONICAL PROGRESSTON. 283 reference to sncb a series maybe sobbed by inverting tbe terms and treating the reciprocals as an Arithmetical Series. 396. J/a,,b, c he in Harmonical Progression, to show that a : c :: a — b : b — c. Since -, -,) ' are in Arithmetical Protrression, c 6 5 a' b-c a—b DC ao ab a-b a a — b or - = T^ — . c b — c 397. To insert m harmonic means between a and b. First to insert m arithmetic means between - and t- a h Proceeding as in Art 357, we have a ' or a = 6 + (m + l).a6(£ , a-b ab (?«.+ 1) Hence the aritlimetic series will be 11^ a-b 1 , 2 (ffl - 6) 1 m{a-b ) 1 a' rt'a6(ni+l)' a ab{m+iy a a6(m+l)' b' 1 6?M + a 6//1 + 2a-b am + b 1 a' a6(»i-rl)' ab{m+l)' ah m + iyb' Therefore the Harmonic Series is ab(m + \) abjm+^l) ab{m + l) ' 6m -r a ' hm + 2a-b' am + b ' 284 ON HARMONICAL PROGRESSION. 398. Given a and h the first two terms of a series in Har- monical Progression, to find the n*'' term. -, T are the first two terms of an Arithmetical Series of o which the common difference is t — . 6 a The w"' term of this Arithmetical Series is 1 (n — 1) (o - 6) _ 5 + Tia - a — n6 + ft a ah ah * (Tta - g) - inh - 2b) _ ( n - 1) a -{n-2) h ah ~ ah .'. the n** term of the Harmonical Series is (rr^)a-(n-2)6" 399. Let a and c be any two numbei-s, 6 the Harmonical Mean between tliem. 1111 Then t — = --i-> b a c 2 a + c or T= ; ac ,_ 2ac ~a + c' 400. The following results should be remembered. Arithmetical Mean between a and c = —^ — . Geometrical Mean between a and c= ^ac. 2ac Harmonical Mean between a and c = — — . , a + G ON HARMONICAL PROGRESSION. 285 Hence if we denote the Means by the letters A, G, H respectively, A X 11=——- X = ac that is, (? is a mean proportional between A and H. 401. To show that A, G, H are in descending order of magnitude. Since ( ^'a - aJc)- must be a positive quantity. ( V« - */c)^ is greater than 0, or a — 2 ,Jac + c greater than 0, or a + c greater than 2 i^fac, a + c . , — or -— greater than ^ac ; that is, A is greater than G. Also, since a + c is greater than 2 ^ac, Jac (a + c) is greater than 2ae ; ,— . , 2ac :. Jac IS greater than — ; — ; ^ * a + c i.e. G is greater than H. Examples.— cxlv. I. Insert two harmonic means between 6 and 24. 2 four 2 and 3. 3 three - and -. 4- foiir -and—. 286 ON HARMONICAL PROGkESSlON. 5. Insert five harmonic means between — 1 and 2~^. 6 five ^and--. 7 SIX 3 and — . 8 n 2x and By. 9. Tlie sum of three terms of a harmonical series is Yg> *"*! the first term is - : find the series, and continue it both ways. 10. The arithmetical mean between two numbers exceeds the geometrical by 13, and tlie geometrical exceeds the har- monical by 12. What are the numbers? 11. There are four numbers a, 6, c, d, the first three in arithmetical, the last three in harmonical progression ; show that (t : 6 = c : rf. 12. If X is the harmonic mean between m and n, show that _1_ _1_ = J_ 1 x-tii x-n m n 13. The sum of three terms of a harmonic series is 11, and the sum of their squares is 49 ; find the numbers. 14. If X, y, z be the //"", 5"*, and r* terms of a h.p., show that {r-q)y- + {P -r)xz + {q- p) xy = 0. 15. If the H.M. between each pair of the numbers, a, b, c he in a. P., then b'-, a-, c'^ will be in H.P. : and if the h.m. be in H.P., b, a, c will be in H.P. 16. Show that ^ — ~+ =4, >7, or >10, according as c — c — a c is the A., G. or H. mean between a and b. XXXIIi. PERMUTATIONS. 402. The different arrangements m respect of order of suc- cession wliich can be made of a given number of things are called Permutations. Thus if from a box of letters I select two, P and Q, I can make two permutations of tliem, placing P first on the left and then on the right of Q, thus : P, Q and Q, P. If I now take three letters, P, Q and R, I can make six per- mutations of them, thus : P, Q, B ; P, R, Q, two in which P stands first. Q,P,R; Q,R,P, Q R,P,Q; R,Q,P, R 403. In tlie Examples just given all the things in each case are taken together ; but we may be required to find how many permutations can be made out of a number of things, when a certain number only of them are taken at a tinie. Thus the permutations that can be formed out of the letters P, Q, and R taken tivo at a time are six in number, thus: P,Q; P,R; Q,P; Q,R; R,P; R, Q. 404. To find the nuvdicr of jJermutations of n different things taken t at a time. Let a,h, c, d ... stand for n difi'erent things. First to find the number of permutations of the n things taken two at a time. , If a be placed before each of the other things 6, c, d ... of which the number is n— 1, we shall have n—\ permutations in which a stands first, thus ah, ac, ad, 2^8 PERMUTA TIONS. If I be placed before each of the other thiiifjs, a, c, d ... we shall have « - 1 permutations in which b stands first, thus : ba, be, bd, Similarly there will be n- 1 permutations in which c stands first: and so of the rest. In this way we get every possible permutation of the 71 things taken two at a time. Hence there will be n . (n - 1) permutations of n things taken two at a time. Next to find the number of permutations of the n things taken three at a time. Leaving a out, we can form (n- 1) . (n — 2) permutations of the remaining (n - 1) things taken tivo at a time, and if we place a before each of these permutations we shall have («- 1) . (?i- 2) permutations of the n things taken three at a time in which a stands first. Similarly there will be (n - 1) . (n — 2) permutations of the n things taken three at a time in which b stands first : and so for the rest. Hence the whole number of permutations of the n things taken three at a time will be n.(n-l). {n-2), the factors of the formula decreasing each by 1, and the figure in the last facto? being 1 les^s than the niuiiber taken at a tinu. We now assume that the formula holds good for the number of permutations of n things taken r—1 at a time, and we shall proceed to show that it will hold good for the number of per- mutations of n things taken r at a time. The number of permutations of the n things taken r—1 at a time w iU be n.{n-l).(n-2) [„- } (r- 1) - I [], tliat is ?i..(?i-l). («.-2) (n-r + 2). 'Leaving a out we can form {n - 1) . (n - 2) (« - 1 — r + 2) permutations of the (n-l) remaining things takrn r — 1 at a time. Putting a before each of these, we shall have (n-l). {n-2) (n-r+l) periiuitatiniis of the n things taken r at a time in which a stands fir>l. PERMUTA TIONS. 289 So again we shall have (to — l).(n — 2) (?i-r + l) per- mutations of the n things taken r at a time in whicli h stands first ; and so on. Hence the whole numtier of permutations of the n things taken r at a time will be n.(«-l).(?i-2) (7i-r+l). If then the formula holds good when the n things are taken r- 1 at a time, it ■will hold 'good when they are taken r at a time. But we have shown it to hold when they are taken 3 at a time ; hence it will hold when they are taken 4 at a time, and so on : therefore it is true for all integral values of r* 405. If the 71 things be taken all together, r = n, and the formula gives n. (n— 1) . (?i-2) (n — n-l- 1) ; that is, n.(n-l).(7i-2) 1 as the number of permutations that can be formed of n dif- ferent things taken all together. For brevity the formula TO. (71- 1). (71-2) 1, which is the same as 1.2.3 to, is written 1 77. This symbol is called /a ciorwZ n. Similarly \r is put for 1 . 2. 3 r ; [r-1 for 1.2.3 {r-\\ Ohs. i 7i = n . 1 71 - 1 = n . (ti — 1) . ?! — 2 = &c. 406. To find the numbei- of jpermutations of n things taken all together ivhen certain of the things are alike. Let the n things be represented by the letters a, b, c, d and suppose that a recurs p times, b q times, c r times, and so on. * Another proof of this Theorem may be seen in Art. 475. £s.A.l ^ 290 PERMUTA TIONS. Let P represent the whole number of permutations. Then if all the p letters a were changed into f other letters, different from each other and from all the rest of the n letters, the places of these -p letters in any om permutation could now be interchanged, each interchange giving rise to a new permu- tation, and thus from each single permutation we could form 1.2 p permutations in all, and the whole nutnber of per- mutations would be (1 . 2 ...^) P, that is [p . P. Similarly if in addition the g letters h were changed into 5 letters different from each other and from all the rest of the 7i letters, the whole number of permutations would be k.l^.P; and if the r letters c were also similarly changed, the whole number of permutations would be ind so on, if more were alike. But when the^, g, and r, &c., letters have thus been changed, we shall have n letters all different, and the number of permu- tations that can be formed of them is \ n (Art. 405). Hence P .\p . \q .\r = ?i ; \p.\q. [r Ex AMPLES. — CXivi. 1. How many permutations can be formed out of 12 things taken 2 at a time ? 2. How many permutations can be formed out of 16 things taken 3 at a time ? 3. How many permutations can be formed out of 20 things taken 4 at a time 1 \ 4. How many changes can be rung with 5 bells out of 8 ? 5. How many permutations can be made of the letters in the word Examination taken all together \ y,^. In how many ways can 8 men be placed side by side ? CO MB IN A no MS. igt 7. In how many ways can 10 men be placed side by side ? 8. Three flags are required to make a signal. How many signals can be given by 20 flags of 5 different colours, there being 4 of each colour ? 9. How many different permutations can be formed out of the letters in Algebra taken all together ? I o. The number of things : number of permutations of the things taken 3 at a time = 1 : 20. How many things are there? 11. The number of permutations of in things taken 3 at a time : the number of permutations of j?i + 2 things taken 3 at a time = 1:5. Find m. 12. In the permutations of a, b, c, d, e, f, g taken all together, find how many begin with cd. 13. Find the number of permutations of the letters of the product a^b^c* written at full length. 14. Find the number of permutations that can be formed out of the letters in each of the following words : Conceit, Talavera, Calcutta, Proposition, Mississippi. XXXIV. COMBINATIONS. 407. The Combinations of a number of things are the diflerent collections that can be formed out of them by taking a certain number at a time, without regard to the order in which the things stand in each collection. Thus the comliinations of a, b, c, d taken tu-o at a time are ab, ac, ad, be, bd, cd. Here from each combination we could make tico permuta- tions : thus ab, ba ; ac, ca ; and so on : for ab, ha are the same combination, and so are ac, ca. < Similarly the combinations of a, b, c, d taken three at a time are abc, abd, acd, bed. Here from each combination we could make six permuta lions ; thus abc, acb, bac, bca, cab, cba : and so on. igi COMBIXATIONS. And, generally, in accordance with Art. 405, any combina- tion of n things niuy he made into 1 , 2 . 3 ... n permutations. 408. To fuul tJie number of combinations of n different things taken x at a time. Let C, denote the nnmher of combinations required. Since each conibinatinn contains r things it can be made into I r permntations (Art. 405) ; .•. the Avliole number of permutations = : r . (7,. But also (from Art. 404) the wliole number of permutations of n tilings taken r at a time — n{n—\) (n-r + 1); .-, I r . C, = n (?i - 1) (?i - r + 1) ; . ^ _ n{n-\) (?t-r + l) 409. To show that the number of combinations of n things taken t at a time is tlis same as the number taken n — r at a time. _, n. (n- 1) (7i-r+l) ^'" 1.2.3 r ' and c n,(n-l) \n-in-Hl\ 1.2.3 (n-r) _ n.(n-l) (r+1) ~ 1.2.3 {n-r) ' Hence C, _ n.(n-l) (n-r+l) 1.2. 3 (n-r ) C^~ 1.2.3 r ^n.{n-l) (r+1) n.(?i-l) (n-r+l). (n-r) 3.2.1 "^ 1.2.3 r. (r+1) (n-l).n \n = 1. That is. O.-'O,^ COMBINATIONS. 293 410. Making r=], 2, 3 r- 1. r, r+ 1 in order, --, _ p _ " ''^ — 1 /-I _ 'I 71-1 71-2 ^ ^^71.01-1) (»-r+2) 1.2 (r-l) „ n.(n-X) (7i-r + 2). Oi -r+1) 1 .2 (r-1). '• 71 . (n — 1) (7? — r + 1 ) . (71 — r) 1.2 r.(r+l) c;.=i. Hence the general expression for the factor connecting Cv, one of the set of numbers Cj, Cj, C^i C',, with C^i, that which stands next before it, is , that is, ^^^7^-r + l r With regard to this factor , we observe r (1) It is always positive, because 71 + 1 is greater than r. (2) Its value continually decreases, for 7! - r + 1 71+1 r -1' which decreases as r increases. 11 J- -^ 1^ (3) Though continually decreases, yet for several •» successive values of r it is greater than unity, and therefore each of the corresponding terms is greater than the preceding. (4) When r is such that '^— is less than unity the cor- responding term is less than the preceding. 294 . COMB IN A T/OXS. 71 — 7* -*- 1 (5) If 11 and r be such that '■ — = 1, C, and C^, are a pair of equal terms, each greater than any preceding or suLse- quent term. Hence up to a certain term (or pair of terms) tlie terms in- crease, and after that decrease : this term (or pair of terms) is the greatest of the series, and it is the object of the next Article to determine what value of r gives this greatest term fur )iair of terms). 411. To find the value of t for which the number of combina- tions of n thincjs taken r together is the greatest. n.(n-l ) ( n -rH-2) ^r-,- - jf_2 (r-i) ^ _ n. (n-1) fw-7- + 2 ) (n-r + 1) ' 1.2 [r-[) * r „ _n.(7i-l) (n-r+l) n-r ^^^ 1.2 r r+l ■ Hence, if 0, denote the number of combinations required, C C j^- and -^ must neither of them be less than 1. a n-r+l -But Jt— = -y Cr r+l and rr~ = — • C'^i n-r vt r+l. T* + l. Hence is not less than 1 and is not less than 1, r n — r or, n — r+ 1 is not less than r and r+l not less than n -r, or, n + 1 is not less than 2r and 2r not less than n — l; :. 2r is not greater than ?! + 1 and not le.ss than n—l. Hence 2r can have only three values, 7i — 1, n, n + 1. Now 2r must be an even number, and therefore (1) If n be odd, ?! - 1 and 7i + 1 being both even numbers, 2r may be equal to 7i - 1 or ?» + 1 ; COMBTNA rroNs. 295 n— 1 w+ 1 (2) If n be even, n-\ and n + 1 being both odd numbers, 2r can only be equal to n ; n ■■' = 2- Ex. 1. Of eight things how many must be taken together that the number of combinations may be the greatest pos- sible ? Here « = 8, an even number, therefore the number to be taken is 4, which will give = — - — - — - or 70 combinations. 1x2x3x4 KXi 2. If tlie number of things be 9, then the numlier 9 _ 1 9 -I- 1 to be taken is — v— or — g— , that is 4 or 5, which will givf respectively 9x8x7x6 1x2x3x4 9x8x7x6x5 , or 126 combinations, and or 126 combinations. 1x2x3x4x5 Examples. — cxlvii. /^ I. Out of 100 soldiers how many different parties of 4 can be chosen ] ( 2. How many combinations can be made of 6 things taken ' 5 at a time / A 3. Of the combinations of the first 10 letters of the alphabet / 'taken 5 together, in how many will a occur ? ^ /\ 4. How many words can be formed, consisting of 3 cnn- sonants and one vowel, in a language containing 19 consonants and 5 vowels ? 5. The number of combinations of n things taken 4 at a time : the number taken 2 at a time =15 : 2. Find n. 6. The number of combinations of n things, taken 5 at 296 COMBINA TIONS. 3 a time, is 3_ times the number of combinations taken 3 at a time. Find n. . 7. Out of 17 consouants and 5 vowels, how many words \ r can be formed, each containing 2 vowels and 3 consonants ? . Q 8. Out of 12 consonants and 5 vowels how many words can be formed, each containing 6 consonants und 3 vowels ? 9. The number of permutations of n things, 3 at a time, is 6 times the number of combinations, 4 at a time. Find n. 10. How many different sums may be formed with a guinea, a half-guinea, a crown, a half-crown, a shilling, and a sixpence ? ; -^ II. At a game of cards, 3 being dealt to each person, any one can have 425 times as many hands as there are cards in the pack. How many cards are there ? I 12. There are 12 soldiers and 16 sailors. How many dif- / ferent parties of 6 can be made, each party consisting of S soldiers and 3 sailors ? //, 13. On how many nights can a different patrol of 5 men be drfiughte<l from a corps of 36 ? On how many of these would any one man be taken \ XXXV. THE BINOMIAL THEOREM. POSITIVE INTEGRAL INDEX. 412. The Binomial Theorem, first explained by Newton, is a method of raising a binomial expression to any ])ower without going through the process of actual multipli- cation. 413. To investigate the Binomial Theorem for a Positive Integral Index. THE BINOMIAL THEOREM. 29? By actual multiplication we can show that (x + aj (z + a,) = x^ + («! + a^) X + Oja, (x + ai) (x + Oa) (x + Og) =x3 + (a^ + Oj + 03) x^ (x + aj) (x 4- a^) (x + 03) (x + a^) = X* + (oj +02 + 013 + 04) i? . + (ttitta + ajCTj + c^a^ + a^a^ + a.ja4 + 0304) x* + (cfi<*2% + ffliffl2«4 + d-fl'^di + a2a3a4) x + a^ajCtja^. In these results we observe the following laws : I. Each product is composed of a descending series of powers of x. The index of x in the first term is the same as the number of factors, and the indices of x decrease by unity in each succeeding tenu. II. The number of terms is greater by 1 than the number of factors. III. The coefficient of the _^rs< term is unity. of the second the sum of a^, a.^, tij ... of the third the sum of the products of %, rtj, rt3 ... taken two at a time. of the/o?t?-;/i the sum of the products of Oj, «2, ^3 ... taken three at a time. And the last term is the product of all the quantities «1, «2> «3 Suppose now this law to hold for 7i — 1 factors, so that (x + tti) (x + aj) (x + «3) (x + a„_i) = x"-i + S'l . rc"-2 + S^ . x"-^ + ,^8 . x"^+ + S„_i, where .S\ = a^ + aj + 03 + . . . + a„_i, that is, the sum of aj, a.,, 03 ... a„_i, Sf=aja.2 + a^a.^ + a^Oj + . . . + aia„_i + a„a„_i + ... that is, the sum of the products of dj, a^, a^ ... a.»_i, taken two at a time. 298 THE BT.VOMIAL THEOREM. S3 = a^a^a-i + aiCiM^ + . . . + a^aM^^i + aia^a„_y + ... that is, the sum of the products of Oj, aj...a^„ taken three at a time, that is, the product of a^, a^, 0.3 ... dn-i- Now multiply both sides hy x + a„. Then {x + ai)(x + a.,) ... (.T + a„_i) {x + a„) =x" + Si X"-' + ,S', X"-- + S3 x"-^ + ... + a„ x"~^ + a„Sj x"'^ + a„S.^ x"^^ + . . . + a,S„ _i =x'' + {Si + a„) x"-i + (S3 + a„Si) x"'"^ + {S3 + a„S.,) x"-^ +... + a„S„_i. Now Si + a„ = ai + a„ + a3 + ... +a„_i + a„ that is, the sum of Oj, a.^, 0-3... a„, /Sj + a„Si = S^, + rr„ (f?! + Oj + . . . + a._i), that is, the sum of the products of a^, aj,..a,„ taken two at a time, Sg + a„S.2 = S3 + a„ {aia.2 + a^a^ +...), that is, the sum of the products of a^, aj...a,, taken three at a time, that is, the product of Oi, a,, ^3 ... o,. If then the law holds good for n-l factors, it will hold good for n factors : and as we have shown that it holds good up to 4 factors it will hold for 5 factors : and hence for 6 factors : and so on for any number. THE BINOMIAL THEOREM. ig^ Now let each of the n quantities a^, a^, a^... a„he equal to a, and let us write our result thus : {x + a^) {x + a.^) ...{x + a„) ^x' + Ai . x"~* + ^2 . x"-'+ ... +A^. The left-hand side becomes {x + a) {x + a)...{x + a) to n factors, that is, {x + a)'. And on the right-hand side Ai = a + a + a+ ...to n terms = ?;«, A^ = a^ + a^ + a^+ ...to as many terms as are equal to the number of combinations of n things taken two at a time, that . n .(n-l) . _ n.{n-l) .. A^- ^^ .a, A3 = a^ + a^ + a^+ ...to as many terms as are equal to the number of combinations of n things taken three at a time, that . n.{n-l) .{n-2) 1.2.3 _^. (n-l). (n-2 ) ^' 17273 •"'' A„ = a . a . a ...to n factors = a". Hence we obtain as our final result / N„ - - 1 n . (n - I') „ . , {x + a)" = x" -t- Tiaa;""' -| ^^ — ^-■' a-x*-^ n.(n-\) . (n-2) , .^ 1.2.3 -r...-rt* 414. Ex. Expand (x + a)6. Here the number of terms will be seven, and we have ^6.5.4.3 ,2,6.5.4.3.2 , . + 1727^74 "^^17273:476 " ^^^" — x^ + Qaufi + 15aV -I- 20aV -i- 15a*x*-i- 6a^x + afi. 300 THE BINOMIAL THEOREM. Note. The coefficients of terms equidistant from the end and from the beginning are the same. The general proof of this will be given in Art. 420. Hence in the Example just given when the coefficients of font terms had been found those of the other three might have beeu written down at once. Examples.— cxlviii. Expand the following expressions : I. (a + x)*. 2. (6 + c)8. 3. (a + 6)^ 4. (x + i/)8. 5. (5 + 4a)*. 6. {a^^hcf. 415. Since 1 n . (« - 1) , , (,-c + a) " = a:" + naT^"'^ + -^ — „— ' . aV'* 4- ... + a", if we put x= 1, we shall have (1 +a)" = l +na + — Y~a~~ -^ "^ ••• +<* • 416. Every binomial may be reduced to such a form that the part to be expanded may have 1 for its first term. Thus since x + a = x(l+-Y (x + a)- = x"(l+^); and we may then expand (l + - j and multiply each term of the result by x". Ex. Expand (2.c + 3y)\ (2x + 32/)6=(2x)''.(l+||y , 5-4.3.2 /3i/y /3yy| "^1.2.3.4-\2x/ ■^V2x/ I THE BINOMIAL THEOREM. -^oi = 32x5 + 240x*y + 720x3i/2 + loSOxy + SlOxi/^ + 243?/. 417. The expansion of (x — a)" will be precisely the same as that of {x + a)", except that the sign of terms in -which the odd powers of a enter, that is the second, fourth, sixth, and other even terms, will be negative. Thus [X - a)" = X" — 7utx"~' + — ^ — - — . ah:"'* TO ■ (w - 1) . (?i - 2) 1.2.3 for (x — a)'=\x + (-a)\' ^x' + ni-a) x'-' + ^_lC^_rLl) ( _ afx'-' + &c. Ex. Expand (a - c)*. / ^^ ^ r4 5-432 5.4.3,, 5.4.3.2^ ^ (a - cf = a» - 5a*c -- j— g ~ r~2"~3 "^ 12 3 4 ~ = a^ - 5a*c + 10u\'2 - lOa^c^ + 5ac* - c^ Examples. — cxlix. Expand the following expressions : I. (a-x)«. 2. (b-cy. 3. (2x-3j/)». 4. (l-2x)5. 5. (l-x)io. 6. (a^-by. 418. A trinomial, as a + b + c, may be raised to any power by the Binomial Theorem, if we regard two terms as one, thus : +^4^*-(«-»)-'-'*-^ 302 THE BINOMIAL THEOREM. Ex. Expand (l+x + a;2)3. (l+x + x-)^ = (l+x)3 + ;3(l+a;)'''.a;2 + ^-|(l+x).x* + x« = (1 + 3ic + 3a;''^ + x3) + 3 (1 + 2x + a;2) o-? + 3(l+u,).c-' + x« = 1 + 3a; + 3a;2 + ic^ + 3:c2 + 6x3 + 3x^ + 3a;'» + 3XS + .7-'' = 1 + 3x4-6x2 + 7x3 + 6x* + 3xS + x«, Examples.— cl. Expand the following expressions : I. (ft + 26-c)-'. 2. (l-2x + 3x2)3. 3. (x3_a.2 + j.-^3_ 4. (3x^ + 2x« + l)3. 5. ^x + 1--). 6. (a^ + 6^-c^); 419. To jind the r"" or general term of the expansion of (z + a)". We have to determine three things to enable us to write down the r"" term of the expansion of (x + a)". 1. The index of x in that term. 2. The index of a in that term. 3. The coefficient of that term. Now the index of x, decreasing by 1 in each term, is in the r* term ?i — 7-+ 1 ; and the index of a, increasing by 1 in each term, is in the r"" term r— 1. For example, in the tliird term the index of x is n — 3 + 1, that is, n-2 : the index of a is 3 - 1, that is, 2. m assigning its proper coefficient to the ;•"' term we have to determine tlie last factor in the denominator and also in the numerator of the fraction n.{n-l).{n-2).(n-Z) 1.2.3.4 TffE BINOMIAL THEOREM. 303 Now the last factor of the denominator is less by 1 than the number of the term to which it belongs. Tlius in the 3'* term the last factor of the denominator is 2, and in the ?•"■ term the last factor of the denominator is r— 1. The last factor of the numerator is formed by subtracting from 7i«the number of the term to which it belongs and adding 2 to the result. Thus in the S"* term the last factor of the numerator is 71-3 + 2, that is ?i— 1 ; in the 4* 71- 4 + 2, that is 9i-2 ; and so in the j-"" ?i - r + 2. Observe also that the factors of the numerator decrease by unity, and the factors of tlie denominator increase by unity, so that the coetticient of the r"" term is n.(n-l). {n - 2) {n - r + 2) 1.2.3 (r-1) ~"' Collecting our results, we write the r* term of the expansion of (x + a)" thus : n.(7i-l).(n-2) in-r + 2) ' 1.2. 3 (r-1) •" ••" • Obs. The index of a is the same as the last factor in the denominator. The sum of the indices of a and x is n. Find Examples. — cli. The 8*'term of (1+x)". The 5'" term of (a^ - ft^)". The 4'" tf rm of (« - 6)i«>. The 9"" term of (2a6-cd)». The middle term of (a — 6)^^. The middle term of (a^ + h°)^. The two middle terms of (a - by^. The two middle terms of (a + x)^. 304 THE BINOMIAL THEOREM. 9. Show that the coefficient of the middle term of 1.3.5 (4n-l) (a + as)*" is 2"" x 1.2 .3 2n 10. Show that the coefficient of the middle term of (a + xr- is 2-^ X ^-2!^) (^" + ^) (^"- 1) (^^-^ 1) 1.2 420. To &how that the coefficient of the r"" temi from the btyianing of the expansion of (x + a)" is identical with the coeffi- cient of the r"^ term from the end. Since the number of terms in the expansion is n+ 1, there are n+l—r terms before the r"" term from the end, and there- fore the r^'term from the end is the (n — r + 2)'^ term from the beginning. Thus in the expansion of (x + a)*, that is, X* + 5ax* + lOa^x^ + lOa^x^ + 5a*x + a°, the 3rd term from the end is the (5 - 3 + 2)"', that is the 4"" term from the beginning. Now if we denote the coefficient of the r*^ term by (7„ and the coefficient of tbe (?) -r + 2)"' term by C«_^2, we have n.{n-l) (n-r + 2) C,= C'.-H-l — 1.2 (r-1) _ n.(n-l) {n-(n-r + 2) + 2{ , 1.2 (71-7- + 2-1) n. (n- 1) r 1 .2 (n-r+l) Hence C, n.(n-l) (n-r + 2) 1. 2 (n-r+l) CZ^,~ 1.2 (r-l) "" n.(n^) r n.(Ti-l) (n- r + 2 ).(n- r+ 1) 2.1 - 1.2 (r- 1) .r (n - 1). n In , . , , = 4^:= = 1, which proves the proposition, n THE BINOMIAL THEOREM 305 421. To find the greatest term in the expansion of (x + a)% n being a positive integer. Tlie r* term of the expansion {x + a)' is n.{n-l) {n-r + 2) , 1.2 (r-1) The (r + 1)"" term of the expansion (x + a)" is n.(n-l) (n-r + 2). jn-r + l) 1.2 (r-l).r Hence it follows that. we obtain the (r + l)* term by multi- plying the r"" term by n - r + I a r ' x' When this multiplier is first less than 1, the r"" term is the greatest in the expansion. Now . - is first less than 1 r X when na-ra + a is first less than rx, or na + a first less than rx + ra, or r (x + a) first greater than a {n + 1), r first greater than — ^^ -. or x + a re \, 1 *. <^ ('"' + ^) i.u n-r+1 a . , ,, If r be equal to , then .- = 1, and the x+a « r X (r + l)"" term is er^ual to the r*, and each is greater than anv other term. Ex. Find the greatest term in the expansion of {4 + ay, when « = „. Here -±±^^£11.)^^^^^.^^. X + a ^3 11 11 ^' ^ + 2 T The first whole number greater than 2^ is 3, therefore th« greatest term of the expansion is the .3rd. [s.A.j 17 2o6 THE BINOMTAL THEOREM. 422. To find the sum of all the coefficients in the expansion of{l+x)\ a- /^ \, 1 n . (71— 1) „ Since (1 +x)" = l +71X+ — - — ^— V + n.(?z — 1) , -<-— ^-g- V-' + ruf^ + af putting x = l, we get _. , n.(n-i) n.(n-l) 2"=l + n + — 1 2~ '*' — 1 "2" ' or, 2" = the siiiu of all the coefficients. 423. To shon: that the sum of the coefficients of the odd term in the expansion of (1 + a;)" is equal to the smn of the coeffi^yients of the even terms. Since ,, , , 7i,(n-l) , n. (n- l).(n — 2) , (l+x)-=l + T?x + — j-^'^ + 123 ^"^ putting x= - 1, we get (1-1) _l-n + -j--2~ j-2-3 + or. ,|„^ .,.(,-,.(.-., ^ } = sum of coefficients of odd terms - sum of co- efficients of even terms ; .". sum of coefficients of odd terms = sum of coefficients of even terms. Hence, by the preceding Article, 2" sum of coefficients of odd terms = — = 2»~^; 2" sum of coefficients of even terms = g- = 2"~*. XXXVI. THE BINOMIAL THEOREM. FRACTIONAL AND NEGATIVE INDICES. 42-L We have shown that when m is a positive integer, N_ T m.(m-\) , (l+x)" = H-mz+ l^-r — ^ X-+ We have now to show that this equation holds good when . . , . 3 . . TO is a positive fraction, as -, a negative integer, as - 3, or a 3 negative fraction, as - --. We shall give the proof de-vised by Euler. 425. If m be a positive integer we know that ,, ,_ , m.(m.— 1), m . Cm-1) . lm-2) , (l+a;)"=i-r7nj;+ p-^ — -3^ + ^ , ^ g ^x^+ Let us agree to represent a series of the form m . (m — 1) „ 1+77!X+ p2 ^ + •' by the symbol /(m\ irhatever the valw of m maii be. Then we know that when m is a positive integer (l+x)"'=/(m) ; and we have to show that, also, when m is fractional or negative ■ ■' (l+.r;;-=/(»0. o- J-/ \ -, - m.(m-l) , Sine* f{m) = l + 'mx-\ ;-- ■ z-+ /(??) = 1 + nx + — ~-^ — X-+ 3o8 THE BIAOMTAL THEOREM. If we multiply together the two series, we shall obtain an. expression of the form 1 + aic + &x2 + cx^ + dx*+ that is, a series of ascending powers of x in which the coeffi- cients a, 6, c are formed by various combinations of m and n. To determine the mode in which a and h are formed, let ua commence the multiplication of the two series and continue it as far as terms involving a;^, thus ,, , _ TO . (m - 1) /(m) = l+mx4-- — ^^—^ — --x'+ /(n) = l +«a; + f-^" f{m) xf(n) = l+mz+ — pg"" ■*" + nx + mnx^+ n.(n-l) , ^ 1.2 / s (m.(m — 1) l + (m + n).i + | ^-g-^ n . (u — 1) , , 1.2 ( Comparing this product with the assumed expression l+ax + bx- + cz^ + dx* + we see that a = m + n, . , to.(to-I) n.(n—l) and b = — i"^ +mn + — ] o m^ -m + 2mn + 7i- — n "" T72 (m + n) . (m + n— 1) " 172 • P/^ACIYOXAL AND NEGATIVE INDICES. 309 Similarly we could show hy actual viultiplication that (m + n) . (m + n — l). {m + n-2) *" 1T273 ' ,_ (m + 7i) . (m + n — 1) . (m + n — 2) . (m + n - 3) TTaTsTi • Thus we might determine the successive coefficients to any extent, but we may ascertain the law of their formation by the following considerations. Tlnj forms of the coefficients, that is, the way in which m and n are involved in them, do not depend in any way on the values of m and n, but will be precisely the same whether m and 71 be positive integers or any numbers whatsoever. If then we can determine the law of their formation when m and n are positive integers, we shall know the law of their formation for all values of m and n. Now when m and n are positive integern, /(m) = (l+x)", /(u) = (l+x)"; " /(™) x/(w) = (1 + x)" X (1 + x)- = (H-a;)"'+- , , , (m + n) .{m + n -\) , = l + {m + n)x + ^ -Y-a — V+ ... =/(m + n). Hence we conclude that whatever he the values of m and n f(m)x f{n)=f{m + n). Hence f{m + n+p)=f{m).f{n+p) =f(m).fin).f{p), and »o generally /{m + n+p + ...)=f{m).f{n)./{p)... 3 to THE BLVOM/AL 'IHEOREM. Xuw let m = n=p= ... =j-, h ami k being positive integ(;rs, then ^h h h ^fh h h ^ , \ /i^^ + j + ^+.-. tot terms j =/a)./(J)./©...to. facto. h (h \ , h k-\k V, k 1.2 which proves the theorem for a positive fractional index. Again, since f{m).f{n)=f{m + n) for all values of m and n, let 71= -VI, then /(m)./(-m) =/(«!-?») =/(0). , . , mJm-l) ., Now the series l + mx+ -^—h ^'+--' becomes 1 when vi = 0, that is,/(0) = l ; .-. /(m)./(-770=l; •■■•^(-™^=/(7^=(TT^-=^^ + ^>'^' .-. (l+x)-=/(-7n) ■■ / \ - 771 ( - 771 - 1) 2 , = l+(-77l)x+ j—g -X2+ ... which proves the theorem for a negative index, integral or fractional. FRACTIONAL AND NEGATIVE INDICES. 3:1 426. Ex. Expand (a + x)2 to four tenna. .a'' .J? ... 1 1.2.3 3 111 4. -T ft 5 z z o = a2+ g + — ^ 2a 2 8a2 16«- Or we might pmceed thus, as is explained in Art. 416. a-0 .= ia-')a-^) ^ 1 /I = aMl + ^ ^^ 2 a 1.2 o^ = a^ -'1+ „H , ... ^ ( 2a 8a2 16a3" j i , X X2 X2 2a2 8a^ IGa^ 1.2.3 Examples. — clii. Expand the following expres-sions : I. (1 + x)2 to five terms. 7. (1 -x-)^ to five terms. 2. (1 +a)-^ to four terms. 3. (a -f- x)^ to five terras. 4 (1 + 2x)2 to five terms. 5. ( a + —^ H to four terms 1 i i 6. (o^ ^x*;-' to four terms. 8. (1 - a^)^ to four terms. 9. (1 — 3x)^ to four terms. 10. (x- — -^y^to four terms. 1 1. (1 — x)* to four terms. 12. ( o" "" g ) ^^ three terms. 312 THE BINOMIAL THEOREM. 427. To exjxnid (1 +x)~'. (1 +x)- = l + (-«). x + ^1^^"^^ X* -n.(-n-l).{-n-2) 1.2.3 = 1 na; + -j--2~x ^ ^ ^ .x-+ the terms being alternately positive and negative. Ex. Expand (1 +x)~^ to five terms. /, , x, , o 3.4, 3.4.5, 3.4.5.6 . = 1 - S.c + 6x2 _ 10x3 4- 15x* - ... 428. To expand (1 - a;)—. _ -n{ -n-l)(-n--2) . 1.2.3 n.(n + l) „ n. (?i + l)(n + 2) , the tenns being all positive. Ex. Expand (1 -x)~^ to five terms. „ ,, , „ 3.4, 3.4.5, 3.4.5.6, (l-x)-3=l+3x + — .2+^_^_^^+____.^+... = l + 3x + 6x2 + 10x3+ 15x*+ ___ Examples.— cliii. Expand 1. (1 +a)~2 to five terms. 4. ( 1 - 5) to five terms. 2. (1 - 3x)-i to five terms. 5. (a2-2.r)~^ to five term*. 3. ( 1 — T ) to four terms. 6. (a^ — x^)""" to lour tenus. THE BINOMIAL THEOREM. 313 429. To ex'pand{\+x)-'n, 11 \ n / n 2 "*" _l(_l_l)(_I_2) 1.2.3 x'^^ ... n 2?i^ fan-* Examples. — cliv. Expand I. (1 + x2)~2 to five terms. 4. (1 + 2a;)~2 to five terms. 3. (1 — c'^)~2 to five terma. 5. (ft^ + x*)'^ to four terms. 2 _i 3 (a^ + is*) ^ to four terms. 6. (o' + x^) ' to four terms. 430. Observations on the general expression for the term involving X' in the exjjayisions (1 + x)" and (1 - x)". The general expression for the term involving x', that is the (r+ l)"" term, in the expansion of (1 +x)'' is n.(n-l)...(7i-r+l) 1.2 r ' •'■'■ From this we must deduce the form in all cases. Thus the (r+ l)* term of the expansion of (1 -x)" ifl found liy changing x into ( — x), and therefore it is n.(n-l)...(7i-r+l) r72~;:~r -^""^^ *"' V^^ 1.2 r ' 314 THE BINOMIAL THEOREI.I. If n be negative and = — m, the (r+ 1)* term of the expan- sion of (1 +x)" is ( — m)(-m— 1)... -m — r + 1) , 1.2 r *' (-1)'. j?n..(m+l)...(m + r-l)jx'. °'"' 1727::::^.:::::::^^ If n be negative and = -m, the (r+ 1)* term of the expan- sion of (1 + x)" is (-l)-. )m.(m+l )...(m + r-l) { _ 1.2 r "^ ^' m. (m+ 1) ... Cwi + r- 1) , 1.2 ....r -•^• Examples. — civ. Fiuil the r"' terms of the following expansions : 1. (l+a-y. 2. (l-x)i2. 3. {a-x)\ 4. (5x + 2!/)». 5. (1+x)-^ 6. (l-3x)-^. 7. (l-a;)"-. 9. (l-2x)~^. 10. (a2-x2)"5. 1 1. Find the (r + 1)"" term of (1 -x)-^. '* S. (a + x)^'. 12. Find the (r+ 1)* term of (1 -4s;) -. 13. Find the (r + 1)'*" term of (1 + x)''. 14. Show that the coefficient of x'+* in (1 +x)''+^ is the sum of the coefficients of x' and^ j;"^* in (1 +x)", I ;. What is tlie fom th term of ( « — ) - ? • 16. "What is the tiftli term of (a^-i'-)^ ? 17. Wliat is the ninth term of (a2 + 2x-)- ? 18. What is the tenth term of (a + 6)"" ? ig. What IS the seventh term of ya-^b)" \ THE BINOMIAL THEOREM. 315 431. The following are examples of the application of the Binomial Theorem to the approximation to roots of numbers. (1) To approximate to the square root of 104. V104 = ^/(lOO + 4) = 10 ( 1 + ~-f =io|i+i.-i-+tk:'J.(Ay ( 2 100^ 1.2 VlOO/ ^2V2 /V2 /.(J \% ia-')a- ^) = 10|l4-A__?__ + _J . 1 ( 100 10000 1000000 [ = 10-19804 nearly. (2) To approximate to the fifth root of 2. 4/2 = (1 + 1)^ = 1 + 1 + 1. l(l-l) + l.l.Cl-l)(l-2U... 5^2 5\5 / 6 5 \5 / \5 / ^j^l_^ _3 21_ 5 25 "^250 2500"^'" = 1 + -^ + nearly 25 2500 = 1-1236 nearly. (3) To approximate to the cube root of 25. 4/25=4/(27-2) = 3il-.-U. Here we take the cube next ahove 25, so as to make the second term of the binomial as small as possible, and then proceed as before. Examples.— clvi. Approximate to the following roots : I. ;/31. 2. ^108. 3. 4^260. 4. 4^31. XXXVII. SCALED OF NOTATION. 432. The sjinbols employed in our common system of Arithmetical Notation are the nine digits and zero. These digits when written consecutively acquire local values from their positions ■s\-ith respect to the place of imits, the value of every digit increasing ten-fold as we advance towards the left hand, and hence the number ten is called the Radix ot the Scale. If we agree to represent the number ten by the letter t, a number, expressed according to the conventions of Arithmetical Notation by 3245, would assume the form 3<3 + 2?2 + 4« + 5 if expressed according to the conventions of Algebra. 433. Let us now suppose that some other number, as^re, is the radix of a scale of notation, then a number expressed in this scale arithmetically by 2341*will, if five be represented by /, assume the form 2/3 + 3/2-r4/+l if expressed algebraically. And, generally, if r be the radix of a .scale of notation, a number expressed arithmetically in that scale by 6789 will, when expressed algebraically, since the value of each digit increases r-fold as we advance towards the left hand, be repre- sented by 67-' + 7r* + 8r + 9. 434. The number which denotes the radix of any scale will be represented in that scale by 10. Thus in the scale whose radix is five, the number fire will be represented by 10, Scales of xo ta tion. 3 17 In the same scale seven, being equal to five + two, ■will therefore be represented by 12. Hence the series of natural numbers as far as tv:enty-j\,vt will be represented in the scale whose radix is five thus : 1, 2, 3, 4, 10, 11, 12, 13, 14, 20, 21, 22. 23, 24, 30, 31, 32, 33, 34, 40, 41, 42. 43, 44, 100. 435. In the scale whose radix is eleven we shall require a new symbol to express the number ten, for in that scale the number eleven is represented by 10. If we agree to express ten in this scale by the symbol t, the series of natural numbers as far as twenty-three will be represented in this scale thus : 1, 2, 3, 4, 5, 6, 7, 8, 9, t, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, It, 20, 21 436. In the scale whose radix is tioelve we shall require another new symbol to express the number eleven. If we agree to express this number by the symbol e, the natural numbers from nine to thirteen wdll be represented in the scale whose radix is twelve thus : 9, t, e, 10, 11. Again, the natural numbers from twenty to twenty-five will be represented thus : IS, 19, If, le. 20, 21. 437. The scale of notation of which the radix is two, is called the Binary Scale. The names given to the scales, up to tliat of which the radix is twelve, are Ternary, Quaternary, Quinarj-, Senarv, Septenary, Octonary, Nonary, Denary, Undenary and Duo- denary. 438. To perform the operations of Addition, Subtraction, Multiplication, and Division in a scale of notation whose index is r, we proceed in the same way as we do for numbers ex- pressed in the common scale, with this difference onlv, that r must be used where ten would be used in the common scale : which -will be understood better by the following examples. 3 1 8 SCALlLS OF NOTA TION. Ex. 1. Find the sum of 4325 and 5234 in the senary scale. 4325 5234 the sum -14003 which is ohtained by adding the numbers in vertical lines, carrying 1 fur every six contained in the several results, and set' ing do%vu tlie excesses above it. Thus 4 units and 5 units make nine units, that is, six units together with 3 units, so we set down 3 and carry 1 to the next column. Ex. 2. Find the difference between 62345 and 53466 in the septenarv scale. 62345 53466 the difference = 5546 which is obtained by the following process. We cannot take six units from five units, we therefore add seven units to the five units, making 12 units, and take six units from twelve units, and then we add 1 to the lower figure in the second column, and so ou. Ex. 3. Mul iply 2471 by 358 in the duodenary scale. 247 1 358 17088 e < e 5 7 193 8333 18 Ex. 4. Divide 367286 by 8 in the nonary scale. 8 ; 367286 ~42033 The following is the process. We ask how often 8 is contained in 36, which in the nonary scale represents thirty-three units ; the answer is 4 and 1 over. We then ask how often 8 is con- tained in 17, which in the nonary scale represents sixteen units; the answer is 2 nnd no remainder. And so for the other digits. SCALES OF NOTAtiON. M$ Ex. 5. Divide 1184323 hy 589 in the duodenary scale. 589; 1184323 ('2483 e5fi 22f3 KeO 3e32 39i!0 1523 1523 Ex. 6. Extract the square root of 10534521 in Vhe senary scale. 10534521 ( 2345 4 43 253 213 504 4045 3224 5125 42121 42121 I 2 3 4 5 6 7 8 9 lo scale. Examples. — clvii. Add 23561, 42513, 645325 in the septenary scale. Add ,3074852, 4635628, 1247653 in the nonary scale. Subtract 267862 i'roni 358423 in the. nonary scale. Subtract 124321 from 211010 in the quinary scale. Multiply 57264 by 675 in the octonary scale. Multiply 1456 by 6541 in the septenary scale. Divide 243012 by 5 in the senary scale. Divide 3756025 by 6 in the octonary scale. Extract the square root of 25400544 in the senary scale. Extract the square root of 56898(1 in the duodenary ^io SCALES 01' NorATiohr. 439. To transform a given integral number from one scale to another. Let N be the given, integer expressed in the first scale, r the radix of the irw scale in which the number is to be expressed, a, b, c in,}:!, q tlie digits, n + l in number, expressing the number in the Jiew scale ; so that the number in the new scale will be expressed thus : ar" + br"""^ + cr"~- + + mf- + jn- + q. We have now from the equation J\r= ar" + 6r"~^ + cr""2 ^ ^ ^,ij-2 ^ pj. ^ q^ to determine the values of a, 6, c m, p, q. Divide N by r, the remainder is q. Let A be the quotient : Uien A = ar"-^ + br"-- + cr"^^ + +mr+p. Divide A by r, the remainder is p. Let B be the quotient ; then B = ar''-^ + br"-^ + cr"-^+ +m. Hence the iirst digit to the right of the number expressed in the new scale is q, the first remainder ; second p, the second remainder ; third m, the third remainder ; and thus all the digits may be determined. Ex. 1. Transform 235791 from the common scale to the scale whose radix is 6. 6 235791 6 39298 remainder 3 6 6549 remainder 4 6 1091 remainder 3 6 181 remainder 5 6 30 remainder 1 6 5 remainder I remainder 5 The number required is therefore 5015343. SCALES OF NOTATIOiY. 32i The digits by which a number can be expressed in a scale whose radix is r will bel, 2, 3 t- 1, because these, with 0, are tiie only remainders which can arise from a division in which the divisor is r. Ex. 2. Express 3598 in the scale whose radix is 12. 12 12 12 12 3598 299 remainder t 24 remainder e 2 remainder remainder 2 .•. the number required is l^&i. 440. The method of transforming a given integer from one scale to another is of course applicable to cases in which both scales are other than the common scale. We must, however, be careful to perform the operation of division in accordance with the principles explained in Art. 438, Ex. 4. Ex. Transform 142532 from the scale whose radix is 6 to llie scale whose radix is 5. 5 142532 5 20330 remainder 2 5 2303 remainder 3 5 300 remainder 3 5 33 remainder 3 5 4 remainder 1 remainder 4 The required number is therefore 413332. Examples. — clviii. Express 1. 1828 in the septenary scale. 2. 1820 in the senary scale. 3. 43751 in the dnodenarv scal(». rs.A.i SCALES OF NOTATIOK. 4. 3700 in the quinary scale. 5. 7631 in the binary scale. 6. 215855 in the duodenaxy scale. 7. 790158 in the septenary scale. Transform 8. 34002 from the quinary to the quaternary scale. 9. 8978 from the undenary to the duodenary scale. 10. 3256 from the septenary to the duodenary scale. 1 1. 37704 from the nonary to the octonary scale. 12. 5056 from the septenary to the quaternary scale. 13. 654321 from the duodenary to the septenary scale. 14. 2304 from the quinary to the undenary scale. 441. In any scale the positive integral powers of the num- bei which denotes the radix of the scale are expressed by 10,100, 1000 Thus twenty-five, which is the sqiiare of five, is e.xpressed in the scale whose radix is live by 100: one hundred and twenty- five will be expressed by 1000, and so on. Generally, the ?;"' power of the number denoting the radix in any scale is exjjressed by 1 followed by n cyphers. The highest number that can be expressed byj:^ digits in a scale whose radix is r is expressed by ;•'' - 1. Thus the highest nuni\)er that can be exjjressed by 4 digits in the scale whose radix is five is 10^ - 1, or 10000 - 1, that is 4444. The least number that can be expressed by 2' digits in a scale whose radix is r is expressed by j-^-^ Thus the least numlier that can be expressed by 4 digits in the scale whose radix is five is JO*-' or 103, tiij^t is 1000, SCALES OF NO TA TION. 323 442. In a scale whose radix is r, the sum of the digits of an integer divided by (»■- 1) will leave the same remainder as the integer leaA^es when divided by r — 1. Let iV be the number, and suppose Then JV=a(r''-l) + 5(r''-i-l) + c(r"-2-l)4- ... +m(r2- l)+p(r- 1) + jrt + 6 + c+ +m+j:> + 7{. Now all the expressions r" - 1, r""^ — 1 r-- 1, r— 1 are divisible by r - 1 ; N . ^ a + b + c+ m + p + q .. ;-= an integer +- :; — - ; r— 1 ° r— 1 which proves the proposition, for since the quotients differ by an integer, their fractional parts must be the same, that is, the remaindt^ra after division are the same. Note. From this proposition is derived the test of the accuracy of the result of Multiplication in Arithmetic by cast- ing out the nines. For let A = Qm + a, and B = ^n + h ', then AB=Q{Qmn + an + 6m) + ah ; that is, AB and ah wlien divided by 9 will leave the same remainder. Radical Fraction?. 443. As the local value of each digit in a scale whose radix is r increases 7'-fold as we advance from right to left, so does the local value of each decrease in the same proportion as we advance from left to right. If then we affix a line of digits to the right of the units' place, each one of these having from its laosition a value one-r"" part of the value it would have if it Avere one place further to the left, we shall have on the right hand of the units' place a series of Fractions of which the denominators 324 SCALES Of- NOTAlIOAr. are successively r, r''-, r^, , while the immerators may be any numbers between r— 1 and zero. These are called Eadical Fractions. In our common sy.^tem of notation the word Radical is replaced by Decimal, because ten is the radix of the scale. Now adopting the ordinary sj'stem of notation, and markins^ the place of units by putting a dot ' to the right of it, we have the following results : 246-4789 = 2 X 10^4 4x 104-6 + A + _^"^^ + _|_^_9^. In the denary scale 246-4789 = 2: in the quinary scale 324-4213 = 3x 10^+2. 104-44-A + _L+_l_ + J_, remembering that in this scale 10 stands ioifive and not for teii (Ai-t. 434). 444. To shoio that in any scale a radical fraction is a proper fraction. Suppose the fraction to contain n digits, a, b, c Then, since r - 1 is the highest value that eacli of the digits can have, - + -5 + ... is not greater than (r- 1)^-+ -t, + ... to n terms) r r^ \r r- / than(r-l) 'y~ not greater --1 r ( r" — \ ") not greater tlian (r - 1) ■; — I ; (?-'\r-l)j not greater than ; not greater than I - — r" SCALES OF iVOTA TTON: 325 Hence the criveu Iraction is less than 1, and is therefore a proper fraction. 445. To transform a fraction expressed in a given scale into a radical fraction in any other scale. Lut F be tlie given fraction expressed in the first scale, r the radix of the new scale in which the fraction is to be expressed, a, b, c.the digits expressing the fraction in the nev/ scale, so that r r- r^ from which eqnation the values of a, 6, c.are to be deter- mined. Multiplying both sides of the equation by r, TP be r r* b c Now - + ^+ ••• is a proper fraction by Art. 444. Hence the integral part of Fr will =a, the first digit of the new fraction, and the fractional part of Fr will b c = - + -,+ ... r r- Giving to this fractional part of Fr the symbol F-^^ we have Alultiplying both sides of the equation by r, F,r = b + -+ ... r _ Hence the integral part of F^r=^b, the second digit of the new Taction, and thus, by a similar process, all the digits of the lew fraction may be found. \26 SCALES OF NOTATION. 3 Ex. 1. Express = as a radical fraction in the quinary Kcale : 7 7 7' 1-5^5 5 , 25 ., 4 7 / 7 4 . 20 ^ 6 i?x5 = — = 2 + -, 7 / 7' 6 , 30 , 2 7 7 7' 2 , 10 , 3 ^xo = -=l+-; therefore fraction is •203241 recurring. Ex. 2. Express •84375 in the octonary scale : •84375 8 6-75000 8 6^00000 The fraction required is •66. Ex. 3. Transform ■ 42765 from the nonary to the senary scale. • •42765 6 2-78133 6 5 •23820 6 155430 6 365800 TTie fraction required is •2513... SCALES OF NOTATION. 327 Ex. 4. Transfonn 6l24-i275 from the duodenary to the quaternary scale : •^275 4 3-4«58 4 1-75 i8 4 2-5e68 4 l-e(28 Number required is 10223230-3121 4 el24 4 2937- - remainder 4 834- -remainder 3 4 20e- - remainder 2 4 62- - remainder 3 4 16- - remainder 2 4 4- - remainder 2 4 1- - remainder 0- - remainder 1 Examples. — clix. 25 1. Express :^^ in the senary scale. 3 2. Express — in the septenary scale. 3. Express 23' 125 in. the nonary scale. 4. Express 1820"3375 in the senarj' scale. 5. In what scale is 17486 written 212542 ? 6. In what scale is 511173 written 1746305 ? 7. Show that a number in the Common scale is divisible (1) by 3 if the sum of its digits is divisible by 3. (2) by 4 if the last two digits be divisible by 4. (3) by 8 if the last three digits be divisible by 8, (4) by 5 if the number ends with 5 or 0. 32^ ON LOGARITHMS. (5) by 11 if the difference between the sum of the digit* in the odd places and the sum of those in the even places be divisible by 11. 8. If iV be a number in the scale whose radix is r, and n be the number resultinsr when the digits of N are reversed, show that iV- 7i is divisible by r- 1. XXXVIII. ON LOGARITHMS. 446. Def. The Logarithm of a number to a given base is the index of the power to which the base must be raised to give the number. '^ Thus if m = a', x is called the logarithiu of m to the base a. For instance, if the base of a system of Logarithms be 2, 3 is the logarithm of the number 8, because 8 = 2^: and if the base be 5, then 3 is the logarithm of the number 125, because 125 = 5^ 447. The logarithm of a number in to the base a is written thus, logaWi ; and so, if vi = a', X = log^m. Hence it follows that m = a'"^"". 448. Since 1 = 0", the logarithm of unity to any base is zero. Since a = a}, the logarithm of the base of any .system i3 unity. 449. We now proceed to describe th;it wliish is called the Common System of logarithms. The ba,8e of the system is 10. ON LOGARITHMS. %n By a system of logarithms to tlie base 10, we mean a succes- sion of values of x whicli satisfy the equation 771=10" for all positire values of m, integi-al or fractional. Such a system is formed by the series of logarithms of the natural numbers from 1 to 100000, which constitute the logarithms registered iu oiir ordinary tables, and which are therefore called tabular logarithms. 450. Now 1 = 100, 10 = 101, 100 = 102, 1000 = lO-'', and so on. Hence the logarithm of 1 is 0, of 10 is 1, of 100 is 2, of 1000 is 3. and so on. Hence for all numbers between 1 and 10 the logarithm is a decimal less than 1, between 10 and 100 the logarithm is a decimal between 1 and 2, between 100 and 1000 a decimal between 2 and 3, and so on. 451. The logarithms of the natural numbers from 1 to 12 stand thus in the tables : No. Log 1 0-0000000 2 0-3010300 3 0-4771213 4 0-6020600 5 0-6989700 6 0-7731513 No. Log 7 0-8450980 8 0-9030900 9 0-9542425 10 1-0000000 11 1-0413927 12 1-0791812 The logarithms are calculated to seven places of decimals 33<5 ON LOGARITHMS. 452. The integral parts of the logarithms of numbers higher than 10 are called the characteristics of those logarithms, and the decimal parts of the logarithms are called the mantisscB. Thus ■ 1 is the characteristic, •0791812 the mantissa, of the logarithm of 12. 453. The logarithms for 100 and the numbers that succeed it (and in some tables those that jirecede lOOj have no charac- teristic prefixed, becfiuse it can be supplied by the reader, l)eing 2 for all numbers between 100 and 1000, 3 for all between 1000 and 10000, and so on. Thus in the Tables we shall tind No. Log 100 0000000 101 0043214 102 0086002 103 0128372 104 0170333 105 0211893 which we read thus : the logarithm of 100 is 2, of 101 is 2-0043214. of 102 is 2-0086002; and so on. 454. Logarithms are of great use in making arithmetical computations more easy, for by means of a Table of Logarithms the operation of j\Iultiplication is changed into that of Addition, . . . Division Subtraction, ... Involution Multiplication, ...Evolution Division, as we shall show in the next four Articles. 455. The logarithm of a product is equal tc. the sum of the logarithms of its factors. Oy LOGARITHMS. 33» Let m = a', and n = a". Then mn = a"'^' ; ••• log, mn = x + y = log„m + ^O'^ji. Hence it follows that log^mnp = log^TO + \og^n + log^^'j and similarly it may be shown that the Theorem holds good for any number of factors. Thus the operation of Multiplication is changed into that of Addition. Suppose, for instance, we want to find the product of 246 and 357, we add the logarithms of the factors, and the sum is the logarithm of the product : thus log 246 = 2-3909351 log 357 = 2-5526682 their sum = 4-9436033 whicli is the logarithm of 87822, the product required. Note. We do not write logio246, for so long as we are treating of logarithms to the particular base 10, we may omit the suffix. 456. 77ie logarithm of a quotient is equal to the logarithm of the dividend diminished by the logarithm of the divisor. Let m = a", and n = a^. Then -^a'-"; n , m .: los„——x — y = log^7)i — log„n.. Thus the operation of Division is changed into that of Sub- traction. 532 ON LOGARITHMS. If, for example, we are required to divide 371 "49 by 52-376, we proceed thus, log 371 -49 = 2-5699471 lo" 52-376 = 1-7191323 their difference = '8508148 which is the logarithm of 7-092752, the quotient required. 457. The, logarithm, of any jwiver of a number is equal to the product of the logarithm of the number and the index denoting the poiver. Let 771 = a". Then TO' = a"; .•. logjn' = rx = r . lo" m. Thus the operation of Involution is changed into Multipli- cation. Suppose, for instance, we have to find the fourth power of 13, we nauy proceed thus, log 13 = 1-1139434 4 4-4557736 which is the logarithm of 28561, the number required. 458. The logarithm of any root of a number is equal to the quotient arising from the division of the logarithm of the number by the nurnher denoting the root. Let m = a*. Then 1 X m' = a" ; 1 -' a; _1 r ' log^TO. Thus the operation of Evoluiion is changed into Division. ON LOGARITIJMS. y^. If, for example, we have to find the fifth root of 16807 we proceed thus, ' 5 |_4|2254902, the log of 16807 •8450980 which is the logarithm of 7, the root required. 459 The common system of Logarithms has this advanta-e overall otliers for numerical calculations, that its base is th" same as tlie radix of the common scale of notation. Hence it is that the same mantissa serves for all numl.ers which have tlie same significant digits and difler only in the position of the place of units relatively to those digits. For, since log 60 = log 10 + log 6=1+ log 6, log 600 = log 100 + log6 = 2 + log6,' log 6000 = log 1000 + l(3g 6 = 3 + log 6, t is clear that if we know the logarithm of any number as 6 ve also J.novv tlie logarithms of the numbers resulting' from nultiplymg that number by the powers of 10. So again, if we know that log 1-7692 is -247783, TO also know that log 17-692 is 1-247783, log 176-92 is 2-247783, log 1769-2 is 3-247783, log 17692 is 4-247783, log 176920 is 5-247783. .1^ unit^^'' ™"'^ ""'^'^ *^'^' ""^ '^' logarithm., ot numbers less Since \ = \Q\ •^=^o=l«"^ 334 ON LOG/.RITRMS the logarifhm of a numl^er between land "1 lies between and -1, between -land '01 -land -2, between •()! and '001 —2 and -3, and so on. Hence the logarithms of all numbers less than unity are negative. "We do not require a separate table for these logarithms, for w^e can deduce them from the logarithms of numbers greater Lhan unity by the following process : log-6 =log j^ =log6-loglO =log6-l, log-06 =logi^ =log6-loglOO =log6-2, log -006 = log ^^- = log 6 - log 1000 = log 6 - 3. Now the logarithm of G is •7781513. Hence log-6 = - 1 + -7781513, which is written 1-7781513, log -06 = - 2 + -7781513, which is written 2-7781513, log -006= - 3 + -7781513, which is Avritten 3-7781513, i the characteristics only being negative and the mantissse positive. I 461. Thus the same mantissce serve for the logarithms of all numbers, whether greater or less than unity, which have the same significant digits, and differ only in tlie position of the place of units relatively to those digits. It is best to regard the Table as a register of the logarithms of numbers which have one significant digit before the decimal point. ox LOGARITHMS. 335 No. ! Log For instance, when we read in the tables 144 | 1583625, we interpret the entry thus log 1-44 is -1583625. We then obtain the following rules for the characteristic to be attached in each case. I. If the decimal point be shifted one, two, three ...n places to the right, prclix as a characteristic 1, 2, 3 ... n. II. If the decimal point be shifted one, two, tlu'ee...TO places to the left, prefix as a characteristic f, 2, 3 ... w. Thus log 1-44 is -1583625, .-. log 14-4 is 1-1583625, log 144 is 2-1583625, log 1440 is 3-1583625, log -144 is 1-1583625, log -0144 is 2-1583625, log -00144 is 3-1583625. 462. In calculations with negative characteristics we follow lae rules of algebra. Thus, (1) If we have to add the logarithms 3-64628 and 2-42367, M-3 first add the mantissoe, and the result is 1-06995, and then add the characteristics, and this result is 1. The final result is 1 + 1-06995, that is, -06995. (2) To subtract 5-6249372 from 3-2456973, we may arrange the numbers thus, - 3 + -2456973 -5 + -6249372 1 + -6207601 the 1 carried on from the last sul)traction in the decimal places changing — 5 into — 4, and then — 4 subtracted from - 3 giving 1 as a result. Heuce the resulting logarithm is 1-6207601. 336 ON LOGARITHMS. (3) To multiply 3-7482569 Ly 5. 3-74825C9 5 12-7412845 the 3 carried on from tlie last multiplication of the decimal l>laces being added to — 15, and thus giving — 12 as a result. (4) To divide 14-2456736 Ly 4. Increase the negative characteristic so that it may be exactly divisible by 4, making a proper compensation, thus, 14-2456736 = 16 -i- 2-2456736. 14-2456736 16 + 2-2456736 - Then ^ = ^ =4 + -5614184; and so the result is 4-5614184, Examples. — clx. 1. Add 3-1651553, 4-7505855, 6-6879746, 2-6150026. 2. Add 4-6843785, 5-6650657, 3-8905196, 3-4675284. 3. Add 2-5324716, 3-6650657, 5-8905196, -3156215. 4. From 2-483269 take 3-742891. 5. From 2-352678 take 5 4286 19. 6. From 5-349162 take 3-624329. 7. Multiply 2-4596721 by 3. 8. Multi])ly 7-4296S3 by 6. 9. Multiply 9-2843617 by 7. 10. Divide 6-3725409 by 3. 1 1. Divide 14-432962 by 6. 1 2. Divide 4-53627188 by 9. 463. "We shall now explain how a system of logarithms calculated to a base a may be transformed into another system of which the base is 6. ON LOGARITHMS. Let m be a number of which the logarithm in the first system is x and in the second y. Then m = a*', and 771=6". Hence h^ = a', ■•^=iog.^; ■ X logj) ' \ ■y-io^y^- Hence if we multiply the loyiirithm of any number in the system of which the base is a by = we shall obtain the logarithm of the same number in the system of which the 'base is h. This constant mulliiilier -. — , is called The Modulus of the log„o ■' system of which the base is b with reference to the system of which the base is a. 464. The common system of logarithms is used in all numerical calculations, but there is another system, which we must notice, emj^loyed by the discoverer of logarithms, Napier, and hence called The Napierian System. The base of this system, denoted by the symbol e, is the number which is the sum of the series of which sum the first eight digits are 2-7182818. 465. Our common logarithms are formed from the Loga.' rithms of the Napierian System by multiplying each of tha [s.A.j Y 338 ON LOGARITH.\fS. ■latter by a common multi])lier culled Tlie Modulus of the Couimon System Tliis modulus is, in accordance with the conclusion of Art. 463, i— ^. log, 10 That is, if I and iV V)e the logarithms of the same number in the common and Napierian systems respectively. Now log, 10 is 2-30258509 ; 1 . ] or -43429448, ■ ■ log, 10 2-30258509 and so the modulus of tlie common system is -43429448. 466. To prove that log,6 x log(,rt = 1. Let ft; = log A Then 6 = a'; .-. - = log,a. X Thus loga6 X logjft = .-; x - = 1. 467. The following are simple examples of the method ot applying the princi})les explained in this Chapter. Ex. 1. Given log 2 = -3()l()3()0, log 3 = -4771213 and log 7 = -8450980, find log 42. Since 42 = 2x3x7 log 42 = log 2 + log 3 + log 7 = -3010300 4- -4771213 + -8450980 =r 1-6232493. ON LOGARITHMS. 339 Ex. 2. Giveu log 2 = -3010300 and log 3 = -4771213, find the logarithms of 64, 81 and 96. log 64:* log 26 = 6 log 2 log 2 = -3010300 6 log 64 = 1-8061800 log 81 = log 3* = 4 log 3 log 3 = -4771213 4 log 81 = 1-9084852 log 96 = log (32 X 3) = log 32 + log 3, and log 32 = log 2-^ = 5 log 2; .-. log 96 = 5 log 2 + log 3 = 1-5051500 + -4771213 = 1-9822713. Ex. 3. Given log 5 = -6989700, find the logarithm of ^(6-25). log (6-25)^ = i log 6-25 = ^ log ~g = J (]og625-log 100) = ^(Iog5''-2) = i(4log5-2) = i (2-7958800 -2) = -1136657. Examples.— clxi. 1. Given log 2 = -3010300, find log 128, log 125 and log 2500. 2. Given log 2 = -3010300 and log 7 = -8450980, find the logarithms of 50, -005 and 196. 3. Given log 2 = -3010300, and log 3 = -4771213, find the logarithms of 6, 27, 54 and 576. 4. Given log 2 = -3010300, log 3 = -4771213, log 7 = -8450980, find log 60, log -03, log 1-05, and log -0000432. 340 ON LOGARITHMS. 5. Given log 2 = -SOinS'OO, log 18 = 1 -2552725 and log 21 = 1-3222193, find log -00075 and log 31-5. 6. Given log 5 = -6989700, find the logarithms of 2, -064, J, and (5,0; • 7. Given log 2 = -3010300, find the logarithms of 5, -125, 8. What are the logarithms of -01, 1 and 100 to the base 10? What to the base -or? 9. What is the characteristic of log 1593, (1) to base 10, (2) to base 12 ? 10. Given —^ = 8, and x = 3i/, find x and y. 11. Given log 4 = -6020600, log 1-04 = -0170333 : (a) Find the logarithms of 2, 25, 83-2, (-625)"~. (6) How many digits are there in the integral part ot (1-04)0000? 12. Given log 25 = 1-3979400, log r03 = -0128372 : (a) Find the logaritlims of 5, 4, 5r5, (•064)"'~. (6) How many digits are there in the integral part of (1-03)000? 13. Having given log 3 = -4771213, log 7= -8450980, log 11 = 1-0413927: find the logarithms of 7623, - -^ and ^^r^. 14. Solve the equations : (i> 4096'=-^. (4) a-?)-=<;. (2) (:4y = 6-25. (5) a^.\^-^-^c^-\ (^) a^.]f = m. (6) a'lr =&~\ ON LOGARITHMS. 34! 468. We have explained in Arts. 459 — 461 the advantages of the Common System of Logarithms, wliich may be stated in a more general form thus : Let A be any sequence of figures (such as 2-35916), having one, digit in the integral part. Then any niiniber iV having the same sequence of figures (such as 235-916 or -00235916) is of the form A x 10", where n is an integer, positive or negative. Therefore logjjxY= logjo( J. x 10") = log^,^ + n. Now A lies between lO*' and 10\ and therefore log ^ lies between and 1, and is therefore a proper fraction. But logjjiV and logjo.4 differ only by the integer 71 ; .". logjp^4 is the fractional part of logu,iV. Hence the logarithyns of all numbers having THE same SEQUENCE OF FIGURES have the same mantissa. Therefore one register serves for the m.antissa of logarithms of all such numbers. This renders the tables more comprehensive. Again, considering all numbers which have the same sequence of figures, the number containing t'co digits in the integral part =10. J., and therefore tlie characteristic of its logarithm is 1. Similarly the niimber containing m digits in the integral part = 10". A, and therefore the characteristic of its logarithm is m. Also numbers which have no digit in the integral part and one cypher after the decimal point are equal to A . 10~' and A . 10~^ respectively, and therefore the characteristics of their logarithms are - 1 and — 2 respectively. Similarly the number having m cyphers following the decimal point = ^ . 10-<™+"; .'. the characteristic of its logarithia is ~{m + 1). Hence we see that the characteristics of the logarithms of all nuvibers can be determined b)j inspection and therefore need not be itj'istered. This renders the tables less bulky. 34:2 ON LOGARITHMS. 469. The method of using Tables of Logarithms does not fall within the scope of this treatise, but an account of it may be found in the Author's work on Elementaky Trigonometry. 470. We proceed to give a short explanation of the way in which Logarithms are applied to the .solution of questions relating to Compound Interest. 471. Suppose r to represent the interest on .£1 for a year, then the interest on P pounds for a j^ear is represented by Fr, and the amount of P pounds for a year is represented, by P + Pr. 472. To find the amount of a (jiven sum for any time at conifpound interest. Let P be the original principal, r the interest on £\ for a year, n the number of years. Then if P^, P„, P.^...P„ be the amounts at (he end ol 1, 2, 3 . . . n years, Pi = P +Pr = P (1 + r, P2 = Pi + Pir-P,(l+r)=P(l+7f P3 = P„ + P.,?- = P., (1 + 7-) = P (1 + if, P,. = P(l+r)". 473. Now suppose P„, P and r to be given : then by the aid of Logarithms we can find n, for logP„ = log !P(l + r)"| = log P + nlog(l+r) ; _ log 7'„-l()gP log(i+r) I ON- LOGARITHMS. . 343 474. If the interest be payable at intervals other than a year, the fornmla P^ = P(1 +r)" is applicable to the solution of tlie question, it being observed that /• represents the interest on £\ for the perio'l on wliich the interest is calculated, half- yearly, quarterly, or for a*iy other period, and n represents the number of such periods. For example, to find the interest on P pounds for 4 years at compound interest, reckoned quarterly, at 5 per cezit. per annum. Here r=l of A = l^ = .0i25, n = 4x4 = 16; .-. P„ = P(1 + -0125)16. Examples.— clxii. N.B. — The Logarithms required may l)e found from the extracts from the Tables given in pages 329, 330. 1. In how many years will a sum of money double itself at 4 per cent, compound interest ? 2. In Iiow many years will a sum of money double itself at 3 per cent, compound interest \ 3. In how many years will a sum of money double itself at 10 per cent, compound interest ? 4. In how many years will a sum of money treble itself at 5 per cent, compound interest ? 5. If £F at compound interest, rate ?•, double itself in n years, and at rate 2r in m years : show that in : n is greater than 1 : 2. 6. In how many years will £1000 amount to £1800 at 5 per cent, compound interest ? 7. In how mnny years will £P double itself at 6 per cent, per ann. compound interest payable half-yearly 1 APPENDIX. 475. The following is another method of proving the prin- cipal theorem in Permutations, to which reference is made in the note on page 289. To prove that the number of pernjfiitatioHs of n things taken r at a time is n . (n - 1) (n — r + 1). Let there be n things a, h, c, d If n things be taken 1 at a time, the number of permutations is of course n. Now take any one of them, as a, then n - 1 are left, and any one of these may be put after a to form a permutation, 2 at a time, in which a stands first: and hence since there are n things which may begin and each of these n may have n - 1 put after it, there are altogether n (n — 1) permutations of n things taken 2 at a time. Take any one of these, as ab, then there are n-2 left, and any one of these may be put after ab, to form a permutation, 3 at a time, in which ab stands first : and hence since there are n{n — 1) things which may begin, and each of these n{n - 1) may have n-2 put after it, there are altogether n(n — 1) (ti - 2) permutations of n things taken 3 at a time. If we take any one of these as abc, there are ?i - 3 left, and so the number of permutations of n things taken 4 at a time is n.(n-l){n-2){n-3). So we see that to find the number of permutations, taken r at a time, we must multiply the nvimber of permutations, taken r— 1 at a time, by the niimber formed by subtracting r— 1 from n, since this will be the number of endings any one of these permutations may have. Hence the number of permi;tations of n things taken 5 at a time is n(n-l)(7j-2) (n-3) x (n-4), orn(n- 1) (h -2) (n-3) (n-4); and since each time we multiply by an additional factor the number of factors is equal to the number of things taken at a time, it follows that the number of permutations of n thinga taken r at a time is the product of the factors n.(n-l)(n-2) (n-r+1). A :^ S W E R s. i. (Page 10.) I. 5a+ 76-f 12c. "7 a + 3b + 2c. 3- 2a + 26 + 2a 4- 6a + 2h + 2c. 5- '2x-7a + 3b-2. 6. 0. 7- 126 + 3c. ii. (Page 10.) I. 2a. 2. 2a + 5A 3. 3a — 3x. 4. Sx + Sr/. 5. 4a + 6 + 2c. 6. 2ti. 7. 4. S. 13x-y-6z. 9. 10a— 76- X. iii. (Page 10.) I. 26. 2. a: + 2]/. 3. a + 5c + d. 4. 2y-{-2z. 5. 2r. 6. 26 + 2c. 7. o-36-c. 8. By + z. iv. (Page 11.) I. 4a -6, 2. 46. 3. a 4-6 -4c. 4. 26. 5. 14x + 2. 6. 2x + a. 7. 6x — a. 8. a. 9. 2a -6. 10. 2a. 11. c. 12. x + 3o. 13. 29a -276 + 6c. ^r. (Page 16.) Addition. I. 7a-26. 2. -106 + 6c. 3. -llx-Sy-6z. 4. -66-5c + 3d. 5. 2a. 6. -2x-2a + b + 4y. 7. 7a + 46 - 4c. 8. 7a — 6 + 7c. 9. — 6?/ + £<;. [S.A.] ^* 34& AxYSlV£RS. I. 4- 7- lo. Subtraction. 2a + 26. 8x-l7j/ + 5. - 3a + 36 — 4c. 6rt - 6 + 5c. 2. a - c. 3. 2a - 26 + 2c. 5. 7a -166 + 20c. 6. 5a-36-8x. 8. 26 + 2c -15. g. llx-7i/ + 45;. II. 12^-95 + 2r. I. 2xy. 5- a?. 9- 180a^65c*. 13- 76x*i/%3. 16. 12a-6can/. 19. ahx^yh^. Vi. (Page 20.) 2. \2xy. 3. 12a;2i/2. 6. a*. 7. 12^561 10. 28a"6r'". 11. Ba' 14. 51a6*c-2/2;. 17. 8a"6-V. 20. 33a206i6m2x. 4. 3a26c2 8. 35a66c*. 12. 20a*b^xy. 15. 48x8t/i<'2«. 18. ^mhi^p^. Vii. (Page 22.) I. a2 + o6-ac. 2. 2rt- + 6a6 — Sac. 3. a^ + Za^ + Aa^. 4. 9a5-15a'*-18a^ + 21a2. 5. a'j _ 2a252 + ^js^ 6. 3a56-9a*63 + 3a264. 7. 8/)i% + 9m2ji2+ lOmiiA 8. 18a66 + 8a562-6a*63 + 8a36*. 9. a; Y - ary + x^ - 7xi/. to. m^n - 2m-n^ + Smn^ — 71'*. 11. 1 44a^6* - 72a'*6^ + 6()a^6^ 1 2. 104a;*i/ - 136.c'!/2 + 4t)x-)/'' - ^xyK I. :c- + 12x + 27. 4- x^- 15.r + 56. 7. jc* + a;- - 20. 9- .T*-31x2 + 9. II. .v;-* - .X- + 2x - 14. a« - .r«. 16, a;*-81y*, viii. (Page 27.) 2. x2 + 8x - 105. 3. X' - 2x - 1 20. 5. «2 — 8a+15. 6. i/' + 7j/ — 7S. 8 . X* - 1 2.c3 + 50x2 _ 84x + 45. 10. a" — 3a^ — 3a* + 1 3a' - 6a- — Ga + 4. 12. .x* + .>;2v2 + (/■». 13. x^-y^ 15. a-^-5.i;3 + 5x-- 1. 17. a* -166*. 18. 16a* -6*. ANSWERS. 347 19. a^ - Aa*h + 4aW + Aa^h^ - llah* - 1265, 20. a= + ba*h + aW - lOa-1? + 12a6^ - 2}y>. 21. a* + 4a-x- + J6j;*. 22. Sla^ + Qa^x^ + x*. 23. x8 + 4a-x*+16a*, 24. a^ + 6^ + c3 _ 3a6c. 25. x^ + x*y - 9x3?/2 _ 20x2^/3 + 2x7/< + 15?/*. 26. a^fi- + c-d- - rt-c- - 6-cZ-. 27. s? - a\ 28. x^ - ax- + 6.';- - cx^ — abx + acx - hex + abc. 29. 1 c*. 30. x^ — y^. 31. a^S-x^^ 32. -47. 33. 2. 34. -14. 35. ab + ac + hc. 36. -60. 37. 2. 38. m^. ix. (Page 28.) I. -a%. 2. -a*. 3. _a363_ ^ l^aW. 5. -30xV. 6. -a3 + a26-a62. 7, -6a5-8a*+ lOal 8. a* + 2a3 + 2a2 + «. 9. - 6x3y + x^y^ + y^j/S _ 1 9y4_ lo. 5m3 + 7/i,2/i- 137H7i2 + 77i3. II. - IS/'^ - 22?-2 + 96r + 135. 12. - 7X* + X^Z + 8x2^2 + 9x^2 + 923 13. x« + xy. 14. x* + 2x3?/ +2x22/2 + 2X2/3 + 1/4. X. (Page 32.) I. x'^ + ^ax + a^. 2. x2-2ax + a2. 3. a:- + 4x + 4. 4. x2_gj.^.9_ g_ x* + 2.r2y2 + ,y. 6 x4-2x2?/2 + y4. 7. a6 + 2a363^56_ g_ a6-2rt363 + 56 9. x^ + j'2 + ^2 + 2x?/ + 2x2 + 2yz. 10. x2 + 1/2 + ^2 _ 2x?/ + 2x3 - 2yz. 11. m2 + n2 + 2)- + 7-2 + 2 m n - 2mp -27nr~2iip-2nr+ -Jj r. 12. x* + 4x3-2x2-12x + 9. 13. X* - 12x3 + 50x2 -84x + 4y. 14. 4x* - 28x3 + 85x2 -126x + 81. 15. x* + i/ + ^+2xY-2x-z^-2y^'^, 34^ ANSWERS. 1 6. cc8 - 8x«2/2 + 1 8x*i/* - 8a;2i/8 + f, 17. a6 + 66 + c« + 2a3i3 + 2a3c3 + 26V. 18. x^ + 2/8 + 2^ - 2x^2/3 - 2x^»^ + 2yh^. 19. x^ + 4t/2 + Qz^ + 4x1/ - 6x2 — 1 '2,yz. 20. X* + 4?/* + 252* - 4x^2/2 + 10x2g2 _ 2O2/V. 21. x^ + 3ax2 + Sa^x + a^. 22. x^ - Sax^ + Sa'x - o*. 23. x3 + 3x2 + 3x+l. 24. x3-3x2 + 3x-l. 25. x3 + 6x2+12x + 8. 26. a6-3a<62 + 3a26*-6fi. 27. a' + 3a26 + 2,a¥ + 6^ -|. c' + 3a«c + 6a6c + Zhh + 3ac2 + 36c2. 28. a3 - 3a26 + ZaV^ -h^-c^- Zah + Gahc - W-c + Sat- - Zhc-. 29. m* - 2?>i-?i2 + n*. 30. m* + 2m^n - 2mn^ - n*. xl. (Page 34.) I. a^. 2. x*. 3. x^?/. 4. x*y^. 5. 66c. 6. 8c*. 7. 16a266c8. 8. 121m«ri«2>^ 9- 12a3xy*. 10. 8a*6c^. xii. (Page 35.) I. x' + 2x + l. 2. 2/' -1/2 + 2/-!. 3. a* + 2rt6 + 36-. 4. X* + m2?x- + m^p*. 5. Aay -Ix + x^. 6. 8x^1/^ — 4.r-!/2 _ 2y. 7. 27m%*-18m%'* + 97ny. 8. 3xy - 2x!/^ - y*. 9. 13u26-9a62 + 76. 10. 196V + 12&V- 76c*. Xiii. (Page 36.) I. -8. 2. 15a^ 3. -21x't/'. 4. -6m2rj. 5. 16a^6. 6. a-x-Jrax + l. 7. -2a2 + 3a-x*. 8. 2 + 6a=6 - 8a*66. 9. — 1 2x2 4. 9_^. j^ _ 8y2_ 10. - x^ + i^x V + fry*. Xiv. (Page 38.) I. x + 5. 2. x-10. 3. x + 4. 4. x+12. ^. x2+7x+12. 6. a;--l. 7- x^ + x+l. ANSIVERS. 349 8. x3-3x2 + 33;+l. 9. X--2X-1. 10. x^-'ix+l. II. x^-x + l. 12. x3-2x2 + 8. 13. x- + 3y'^. [4. a^ + ^a^ + Zah' + y^. 15. a* - 4a35 + Ga^fcs _ 4a{,3 + j4 16. x2-6x + 5. 17. a^ — ^a~h + Zah^ + A¥. 1 8. 2rtx^ - 3a-x + a'. 19. x^ - x + 1 . 20. x^ — a^. 21. x + 2i/. 22. X* - x^y + x^i/^ - X1/3 + 2/*. 23. x^ + x*2/ + x-'ff^ + x-y' + xj/* + y^. 24. « + 6 — c. 25. -6 + 2&--61 26. a-6 + c-d. 27. x^ — xy — x.: + y^ — i/z+2^ 28. x*' — x^2/- + x^!/* - x^j/" + y*. 29. 2J + 29-r. 30. a* - 0^6 + a'6'^ - a6^ + 6*. 31. X* + x% + X"2/2 + xt/' + 7/*. 32. 2x' - Sx''^ + 2x. 33. a^ + 3a3 + 9a2 + 27a + 81. 34. ^-7 + ^* + ^. 35. x2-9x-10. 36. 24x^-2ax-35a2. 37. 6x2-7x + 8. 38. 8x3+12ax2-18a2x-27a». 39. 27x3 - 36ax2 4- 48a2x - 64a3. 40. 2a + 36. 41. x + 2a. 42. a^-Alfi. 43. x'^-3x-y. 44. x--3xy-2y-. 45. x^ + Sx^y + 9xi/2 + 27i/3. .46. a^ + 2a% + 4ab^ + 8¥ 47. 27a3-18a26+ I2a62-8R 48. 8x3-12x2i/ + 18x?/2_27i/». 49. 3« + 26 + c. 50. a2-2ax + 4x2. 51. x^ + xy + y" 52. I6x--4xy + y'. 53. x^ + xy-y-. 54. flx2 + 4«-x + 2flA 55. a-x. 56. x-y-z. 57. 3X--X + 2. 58. 4-6x + 8x'-'-10x'. 59. x + y. 60. ax + by-ab-xy. 61. bx + ay. 62. x^ - ax + 6-. XV. (Page 40.) I. x2 + ax + 6. 2. 2/2 - (^ + to) 1/ + Zm. 3. «;'4-cx + (/. 4. x^ + ax-b. 5. x2 - (6 + 0?) X + 6ci. xvi. (Page 42.) I. m~n, m^ — mn + v?, m* — in?n + ni^n- - mn^ + n*, vv" - mhi + &c., m* - m'n + <Ssc. 556 ANSWERS. 2. 'm-^n,w?^- mn + n-, w? + mhi + &c., m* + mhi + &c., m® + m*n + &c. 3. a - I, a^ - a + I, a* - a^ + &c., a^-a^ + &c., a^ - a^ + &c. 4. y + l,y'^ + y+l, y* + y^ + &.c., y^ + y^ + &c., y^ + y'' + &,c. xvii. (Page 43.) I. 5a; (x- 3). 2. 3x{x^ + Gx-2). 3. 7(7i/2-2i/ + 1). 4. 4a;y (a;2 - 3x1/ + 2?/2), 5. a;(x^ — ax^ + 6x + c). 6. 3xY (x^i/ - 7x + V). 7. 27a%^{2 + 4a%'^-9a^b3). 8. 45xy(xV-2x-8i/). xviii. (Page 44.) I. (x-a)(x-6). 2. (a-x)(6-!-x). 3. (b-y)(c + y). 4. (a + m) (6 + n). 5. (ax + y) (bx - y). 6. (a6 + cd) (x - j/). 7. {ex + my) {dx - nyy 8. (ac - bd) (bx - dy). xix. (Page 45.) I. (x + 5)(x + 6). 2. (x + 5)(x + 12). 3. {y + U){y + l). 4. (!/ + ll)(i/+10). 5. (»i- + 20)(?n + 15). 6. (m + 6)(m + 17). 7. (a. + 86) (a + 6). 8. (x + 4?)i)(x + 9m). 9. {y + 3n)(y + l6n). 10. (s; + 4^j) (2 + 25;?). II. (x^ + 2) (x2 + 3). 12. (x^+l)(x3 + 3), 13. {xy + 2){xy+l6). 14. (xY- + 3) (xy + 4), 15. (m5 + 8)(7rt5 + 2). 16. {n + 20q){n + 7q). XX. (Page 45.) V (x-5)(x-2). 2. (x-19)(x-10). 3. {y-U)iy-l-2). 4. iy-20)(y-10). 5. (n- 23) (71 -20). 6. (7i-56)(ji-l). 7. (.x3-4)(x3-3). 8. (ab - 26} {ab- I). 9. (6'-c»-5)(6V-6). 10. (xi/~-ll)(xy«-2). Al^SlVEHS. 35» xxi. (Page 46.) I. (a; 4- 12) (a; -5). 2. (x + 15)(x-3). 3. (a+12)(a-l). 4. (a + 20) (a -7). 5. (& + 25) (6 - 12). 6. (6 + 30) (6 -5). 7. (x* + 4)(x*-l). 8. (x!/+14)(x2/-ll). 9. (m5 + 20)(m5-5). 10. (7i + 30) (ji- 13). xxii. (Page 46.) I. (x-ll)(a; + 6). 2. (x-9)(.r; + 2). 3. (m- 12) (/?!, + 3). 4. (7i-15)(n + 4). 5. (2/-14)(i/ + l). 6. (3- 20) (2 + 5). 7. (x5_i0)(x5 + i). 8. (cd-30)(cd + 6). 9. (m% - 2) (m% + 1). 10. (;>Y - 12) (i^V + '^)' xxiii. (Page 47.) I. (x-3)(x-12). 2, (x + 9)(a;-5). 3- (a6-18)(a6 + 2). 4. (x* - 5m) (x* + 2m). 5- (l/3+10)(j/3-9). 6. (x2+10)(x2-ll). 7- z (.r^ + Zax + 4a2). 8. (x + to) (x + n). 9- (2/3-3)(r/3-l). 10. (xy — ab) (x-c). II. {x + a) (x - 6). 12. (x - c) (x + d). 13- (a6 - d) (6 - c). 14. 4.(x-47/)(x-32/). xxiv. (Page 48.) I. (x + 9)2. 2. (x + 13)2. 3. (x + 17)2. 4. (2/ + 1)2. 5- (2+100)2. 6. (X2 + ■7)2. 7. (x + 52/)2. 8. (m2 + 87*2)2, 9. (x3 + 12)2. 10. (X7/ + 81)2. XXV. (Page 48.) I. (x-4)2. 2. (x-14)2. 3. (x-18)2 4. (7/ -20)2. 5. (3-50)2. 6. (X2-11)2. 7. (x-157/)2. 8. (77^2 - 1 67*2) ». 9. (it'- 19)2. 3S2 ANSH'EH^. xxvi. (Page 50.) 1. {x + y){x-y). 2. (x + 3)(x-3). 3. (2x + 5) (2x - 5). 4. (a2 + x2)(a-'-x-^). 5. (a; + l)(a;-l). 6. (x3 + 1) (x^- 1). 7. (:c* + 1) (x* - 1 ). 8. (m2 + 4) {m^ - 4). 9. (61/ + Tz) (6?/ - 72). 10. (9xr/ + lla6) (9xi/-lla6). II. {a-h + c) {a-h-c). 12. (x + m-n) (x-m + n). 13. (a + b + c + cO (« + ^-<^'~^)- H- 2xx2y. 15. (x-i/ + z)(x-i/-z). 16. {a-h + m + n) {a-h-m-n). 17. (ffl-c + 6 + (0 (^-c-^-c^)- 18. (a + 6-c) (a-6 + c). 19. (:c + t/ + z) (x + y-a). 20. (a-6 + m-n) (a-6-m + n). 21. {ax + h]i+l){ax + hy-\). 22. 2axx2by. 23. (H-a-6) (l-a+'O- 24. (l+x-i/)(l-x + 2/). 25. (X + 2/ + 2) (X-1/-2). 26, (a + 26 -3c) (a -26 + 3c). 27. (rt2 + 46)(rt2-46). 28. (1 + 7c) (1 - 7c). 29. {a-b + c + d){a-b-c- d). 30. (a + 6 - c - rf) (a - 6 - c + d). 31. 3ax(ax + 3)(ax-3). 32. (a^t^ + c*) (a-^t^ - c*). 33. 12(x-l)(2x + l). 34. {9x + ly){5x + y). 35. 1000x506. xxvii. (Page 51.) I . ((( + /;; (cr -nh + b"^). 2. (a - b) (a« + a6 + i^). 3. (« - 2) (rt2 + 2a + 4). 4. (x + 7) (.r--7x + 49). 5. (6-5) {b- + 56 + 25). 6. (x + 4?/) (x^ - -ixy + 16?/2). 7. (a-6)(rt2 + 6rt + 36). 8. (2x + 3^) (4x2 - 6xj/ + 9i/»). 9. (4a - 106) (Ifia^ + 40a6 + 10062). ' 10. (9x + Sy) (8 lx-2 - 72xy + 64 j/2). II. {x + y) {jc- - xy + y-) {x - y) {x- + xy + y-)- ANSWERS. 353 20. n^. 21. 25 -z. 5. x-5. 26. 1/ + 7. 29. 2. 30. 2. 34. 5. 35. 10. 12. (x+l)(x2-x + l)(x-l)(a;2 + x+l). 13. (a + 2) (a2 - 2a + 4) (a - 2) (a2 + 2a + 4). 14. (3 + 2/)(9~3y + 2/2)(3-2/)(9 + 37/ + 7/2). xxviii. (Page 51.) I. a + 6. 2. Take 6 from a and add c to the result. 3. 22/., 4. a -5. 5. x + l. 6. x— 2, x-1, x, x+l, x + 2. 7. 0. 8. 0. 9. da. 10. c. II. x-i/. 12. x-y. 13. 365 -6x. 14. x-10. 15. x + 5a. 16. A has X + 5 shillings, B has 1/ - 5 shillings. 17. x-8. 18. xy. 19. 12-X-2/. 22. y — 25. 23. 256r/i*. 24. 4b. 27. x2_^2 28^ (x + 2/)(x-?/). 31. 28. 32. 7. 33. 23. XXiX. (Page 53.) 1. To a add b. 2. From the square of a take the square of h. 3. To four times the square of a add the cube of b. 4. Take four times the sum of the squares of a and b. 5. From the square of a take twice b, and add to the result three times c. 6. To a add the product of m and b, and take c from the result. 7. To a add m. From b take c. Multiply the results together. 8. Take the square root of the cube of x. 9. Take the square root of the sum of the squares of x and y. 10. Add to a twice the excess of 3 above c. 1 1. Multiply the sum of a and 2 by the excess of 3 ab^ve c. [S.A.] g 354 ANSWERS. I. 2. 2. 0. 3- 17. 4- 31. 7- 105. 8. 27. 9- 14. 10. 120. •3- 30. 14. 5. 15- 3. 16. 4. 12. Divide the sum of the squares of a and h by four times the product of a and h. 13. From the square of x subtract the square of y, and take the square root of the result. Then divide tliis result by the e.xcess of x above y. 14. To the square of % add the square of ?/, and take the square root of the result. Then divide this result by the square root of the sum of x and y. XXX. (Page 53.) 5. 20. 6. 33. II. 210. 12. 1458. 17. 49. 18. 10. 19. 12. 20. 4. 21. 43. 22. 20. 23. 29. 24. 41536. 25. 52. xxxi. (Page 64.) I. 0. 2. 0. 3. 2ac. 4. Ixy. 5. a^-^h"-. 6. 4x* + (6m - 6?i) x' - (4m ^ + 9??in + 4?r) x'- + (6™^?i — 6m?i-) X + 4m^n^. 7. cr^ + dr + e. 8. - a* - 6* - c* + 2*262 + 2tt2c2 + 2i-c2. When c = 0, this becohies - a* - 6* + 2*262. When 6 + c = «, the product becomes 0. When a = h = c, it becomes 3a*. 9. 0. 10. 34. 12. (a) (a + 6)x2+(c + rf)x. (/S) (a-6)x3-(c + (Z-2)x2. (7) (4-a)x3-(3 + ?))x2-(5 + c)x. (5) a^ - 62 + (2a + 26) x. (e) (7)1.2 _ ^2^ a^s ^ ^271! 2 — 2?((;) x^ + (2wi — 2?i) x2. 1 3. .x^ _ ((,, 4. 6 + c) x2 + (((6 + rtc + 6c) X - a6c. 14. x^ + (« + 6 + c) x2 + (rt6 + ac + 6c) X + a6c. 15. (a + 6 + c)3 = a3 + 3a-6 + 3rt62 + 63 + c3 + 3rt2c + 6a6c + 36'-c + Zac^ + 36c2. (d + 6 _ c)3 = a^ + 3a26 + 3<»6'- + 6^ - c^ - 3rt2c - ^a\)c - 36-c + 3(a-2 + Zhc", ANSCVERS. (6 + c-a)3=-a3 + 3a26-3rt62 + 63 + c3 + 3a2c -6a&c + 362c-3ac2 + 35c2. (c + a - 6)3 = «3 _ 3f^25 + 2aV' -h'^ + c^ + ^ah - 6rt6c + 362c + 3rtc2 - 36c^. The sum of the hi?t three subtracted from the first gives 24a6c. 1 6. 9a2 + 6ac-3«6 + 46'--662. 17. a^^-x^^. 1 8. 2ac - 26c — 2«fZ + 26c/. The value of the result is — 26c. 19. a6 + a:i/ + (6+ l+2a)a; + (2a-6- 1)2/. 20. 9. 21. 06 + 2:- + (a -6+1) a; -(a + 6 + 1)7/. 22. 2. 23. (7m + 4?i + l)a;+ (1 -6>i — 477i)?/. 25. 4a2 + 6ac + 2a6 + 96c-662. 26. 3; 128; 3; 118. 27. 9. 28. 44. 29. 20. 30. 35. 31. 18. xxxii. (Page 60.) I. 3. 2. 2. 3- 1- 4- 7. 5- 2. 6. 2. 7- 3. 8. 4. 9. 9. 10. A ?!s. 54. II. 2. 12. 9. 13- 9. 14. -7. 15. 3. 16. 7 17- 2. 18. 8. 19. 10. 20. 6. 21. 4. 22. lit. 23- 3. 24. 15. 25. 1. 26. 2. 27. 3. 28. 4. 29. 6. 30. -1. xxxiii (Pa^e 62.) I. 70. 2. 43. 3.' 23. 4. 7,21. 5. 36,26,18.1:2. 6. 12, 8. 7. 50, 30. 8. 10, 14, 18, 22, 26, 30. 9. .iC 10. 12 shillings, 24 shillings. 11. 52. 12. A has £130, B il50, C jElSO, D £90. 13. 152 men, 76 women, 38 children. 14, £350, £450, £720. 15. 21, 13. 16. £8. 15s. 17. 84, 26. 18. 62, 28. 19. The wife £4000, each son, £1000, each daughter £5no. 20. 49 gallons. 21. £14. £24, £38. 22.31,17 356 AX.sirhRS. 23. £21. 24. 48, 36. 25. 50, 40. 26. 42, 18. 27. 60, 24. 2.S. 8, 12. 29. 88. 30. 18. 31. 4a 32. 57, 19. 32,- -4. 34- SO. 128.. 35. 19, 22. 36. 200, 100. 37. 23, 20. 38. 53. 318. 39. 5, 10, 15. xxxiv. (Page 68.) I. a%. 2. x-y-z. 3. 2x-y. 4. 15m2?ijs. 5. 18a&c(f. 6. a2j2_ 7_ 2. 8. 172)2. 9- 4a;2j/222. 10. SOxV- XXXV. (Page G9.) I. a-h. 2. a'^-fcl 3. a — x. 4. a + x. 5. 3x + l. 6, l-5a. J. x + y. 8. x-y. 9. x-1. 10. 1+a. XXXVi.' (Page 70.) I. 3453, 2. 36. 3. 936. 4. 355. 5. 23. 6. 2345. xxxvii. (Page 74.) I. x + 4. 2. x+10. 3. x-7. 4. x + 12. 5. x-3. 6. x + 2y. 7. x-4!/. 8. x-l5y. g. x-y. 10. x + y. 11. x-y. 12. x + y. 13. x + y. 14. a + 6- c. 15. -ix + y. 16. 3x-!/. 17. bx-y. 18. x* + x^- 4X- + X + 1. 19. x--2x + 4. 20. x^ + xy + y"^. 21. x^ + x"- — x-1. 22. 3a^ + 2a6-6-. 23. Zx — y. 24. 3x-lli/. 25. 3a-6. 26. 3(a-x). 27. 3x-2. 25. 3x2 + al 29. x2 + 2/l 30. x + 3. 31. (3a + 2x)a-. xxxviii. (Page 76.) I. x-f2. . 2. x-1. 3. x + 1. 4. y-1. 5. x2-2x + 5. 6. x-2. 7. J/- -2!/ + 6. ANSWEfiS. 357 xxxix. (Page 81.) J^ 2a; _56 2x2 aW'c^ , 4xy 3y _ 5h-c 5" ~3~' 36c* 7* 2aa* * 4a^' 4 5 m a 2??ix 3x^2/^' .P ' " a + 6* ■ Sm^p — x 1 2a + x jr^ o2 3?/ -5x3* 4ax- — x' -*' 6c' ' 2x-3y' 3ab . - c-2a 3 17. sr . 18. ^. 19. -. ' 2bc + c c + 2a ^5 5 1 2 20. ^ — -. 21. s srr- 22. 2x-2y* ' 7ax-7by' ' 9abx—12cdx' xy 62 1 , 2a + 26 23. .r-^. 24. ^.-. 25. TT- 26. ;;— . 1 - X -7. 12- 28. -. Xl. (Page 82.) a + 5 x-5 a + l a + 3* * x-3' '■ 03-7* . ^JL^y x^-x+l 6 — +^' ^ x + 7y ^ oi?-y^ x-2 X--3 x2_5a; + 6 ^* x+"4* ■ x + r ^" 3x2 -7x ■ x2 - 5X + 6 x2 + XT/ - y' 3x2 _ 8x ' * x2 - xy - 2/2* a2 + 5a + 5 6^ + 56 m2 + 4m '"• a2 + a-2* ^^' 6^ + 6-5* ^^ m2 + m-6' a2-a + l 3ax-7a 14x-6 "5* a2 + a+l' ^ ■ 7x2 -3x' ^7- g^x - 21a' g 10fl -14a2 2a62 + .3a6-5a ' ' 15-9a-6a2' '9" 762^I"56 ^* 358 ANSWERS. a^-a+l 3x-l a-b 20. , „ ;,. 21. „ ,, 22. 5. a;- - 2x + 2 - 2x- + 'ix-b 23. -^^32-- 24. 3. 25. ^^_5 • 4x- + 9x + l 2x-3a „ x-3 2x--3x-2" ^'^* 4x2 4.6ax + 9a2- ^^- x-^' m — 1 x^ + 5x 2Q. T- 30- 5- 3'- 6a + 26 x2 + 4 , X3 + X--2 ^5' X- + X+1' ^ 2x^ + 2x + r ^'^' x^-2x + 3 x3-2x^-2x + l ^ 2x2 + 5x-3" ^^' 4x2-7x-l " '^°* 3a2-8a x + 3 ■ x-5 2j; + 3' a:2 + x-i2 3x + 5 ■ a2_5a + 6 Xli. (Page 86.) I. \2f 5. ax. 1 ^- 2- 2x' 2- 3y3- ^•1- 3 7- 8- bkm? 10. -j 4pq 4- by 9ax' 8. Sa^c' 9d2' 4- 3mnxi/ 423g2 xlii. (Page 86.) a-h 4 (x 4- 2) (x - 4) ~W ^' 3' ^' x(x-2) • (x-l)(x- 6) x-6 g (x-2)(x-5) x''' * ^* X — 3' * a^ ' 1 01 n V c- a+b 7. 1. 8. 0. 9. — ^— . 10. f. x—y c-a —0 x — m + n - x — y-z* II. . 12. 1. 13. ~ . x + m-n "' x + y + z JJVSIV£J?S. 359 xliii. (Page 87.) lOae ^- 2bx- 3 3. ^. 4. 4 36?ix" 3 5- 4- , hx ^- 4i' 5x 7- I4- ^- x-2- 9- 1 x-2' Xliv. (Page 89.) I. 12a3x2. 2. 12x2?/2. 3. Sa^ftz 4. a^x*. 5. 4ax3. 6. aW(^. 7. a3x22/2. 8. 102a2xi 9. 20p222r. 10. l^ax^y^. Xlv. (Page 91.) I. x2^(j^a.)_ 2. x^-x. 3. a{a'^-}p). 4. 4x2-1. 5. a3 + 63. 6. x2-l. 7. (x5-l)(x + l), 8. (x2 + l)(x3+l). 9. (X + I)(x3-1). 10. X*-l. II. x(x3-l) (xHl). 12. X (x + 1) (x^ - 1). 13. (2a-l)(8a3 + l). 14. 2.(;2 + 2x2/. 15. (a + 6)2 (a -6). 16. a2_ft2_ 17. 4(l-x2). 18. x3-l. 19. (a - 6) (a - c) (6 - c). 20. (x + 1) (x + 2) (x + 3'). 21. (x + 2,')2(x-i/)2. 22. (a + 3)(a"''-l). 23. x2(x2-i/). 24. (x + l)(x + 2)(x + 3)(x + 4). 25. 12(x-?/)2(x3 + 2/3). 26. 120x1/ (x2- 1/2). Xlvi. (Page 93.) I. (x+2)(x + 3)(x + 4). 2. (a-5)(a + 4)(a-3). 3^ (x+l)(xH-2)(x + 3). 4- (x + 5)(x + 6)(x + 7). 5. (x-ll)(x + 2)(x-2). 6. (2-,: + 1) (x+1) (x-2). 36o ANSWERS. 7. (x2 + y)(x + y)(x2 + i/2)(x-2/). 8. (x-5)(x-3)(x + 5). 9. (7z-4)(3x-2)(x2-3). 10. (a;2 + j/2)(x + i/)(x-2/). II. (a2- 62) (a + 26) (a -26). xlvii. (Page 94.) I. (a;-2)(a;-l)(x-3)(a;-4). 2. (a; + 4) (x + 1) (a; + 3). 3. (a; -4) (a; -5) (a; -7). 4. (3x - 2) (2x + 1) (7x - 1). 5. (x+l)(x-l)(a; + 3)(3x-2)(2x+l). 6. (x-3)(x2 + 3x + 9)(x-12)(x'--2). xlviii. (Page 95.) 15x 16x 9x-21 4x-9 ^' W "20"* ^' 18 ' ~T8~' 4x-8?/ 3x2 -Bxy 20a + 256 Q>a'-%ah lOx- ' 10x2 • 4- iOrt-^~' 10a2 • 48a--60ac 15a- 10c , ah-W a^-a% 3 - 3x 3 + 3a; 2j^2!/2 2-2y2 1-X-" 1-X2' • l_2/4' i_y 5 + 5x 6 a6 + ax b lO. l-x2' l-x2' • c(6 + x)' c{b + x)' a—c b—c {a'^'b)(h~c)(a-cy (a-b) (b-c) (a-c)' c{b-c) 6( a-6) o6c(a-6) (a-c) (6 — c)' abc (a - 6) (o - c) (6 - c}' xlix. (Page 98.) 15X+17 71a -206 -56c 32x + 9tf I . 2. . 1. . 15 84 -^ 42 16x» + 55 x'+ 4x1/ - 55!/ , 27x2 - 2x2;/ _ ^ gj-y _ 28^2 ^ "■ 50x ■ '■ li:.-'- AI^SWERS. 361 ISOffl^ + 54ffl6 + 3316^ - 20a62 S Ox^ + 64 x2 + 84a; + 4 5 9062 • 7- gQ^2 35rt2 + 23a6 + 2l6c-42c 2 Aa?c - Zac^ - 3ac + 7c» 2 lac ■ ^" a-c^ lly2-8xY-4xy-7x' 3a* - 7a^b + 4a%c - 5ahh + ahc^ - ¥c* aWc^ 1. (Page 99.) 2x-l 4 ^ 2 (x-6)(x + 5y - (x-7)(x-3y •"■ (i+x)(i-xy 4- 4x1/ - 1 y. a + bx 1+x" 'c + dx' (x + i/)(x-3/y 7- 2x2 2x-i/ 2x4- 5a (x-t/)2- 9- (^^^jr {x + y){x-yy 10. 1 (a + x) (a-x)" U. (Page 100.) I. 2 4x 2x 86' 1-x*" ^" 1-x*" "^^ a8~l> 5- x + y y , 3x3 + 20x2 _ 32a; _ 235 • (x + 4)(x-3)(x+7)" 7. 3x3- 24x2 + 60x -46 3x2-2ax-6a- (x-2)(x-3)('x -4)- ^- (x-a)3 • 9- 6 X (x-l)(x + 2)(x+l)' — (x+l)(x + 2)(x + 3)" 1 1 3x2 x2-r e-d ■ {a + c){a + d){a + ey ^^' ' 14- 2. 15. y . 16. 0. 17. ";^-^^ 362 ANSWERS. i8. 0. 19- -A^, 20. 0. 21. 0. a + 6 lii. (Page 103.) ?/ 1 3x2 y + 6 ^ ^ 3(1-^ ^' x-1/' ^" 2 + « ^* x-'-r "^ 3(1-2/2) 5. 0. 6. , r-^ rr. 7 ^ (x + a) (x + 6) c* -</ 1 2 1 1-x* ^ (x-2)(y-8;) aoc liii. (Page 110.) 2x + ll 2(x-8) (x + 4)(x + 5)(x + 7)* ■ (x-6)(x-7)(x-9)' 2x - 1 7 2 7/1^ + 4m2n + m?!^ 4- rr^- 5- ■'■ (x-4)(x + ll)(x-13)' ^ x + 3" ^" n(m + 7i)2 , - Ilx3-x2 + 25x-l „ - 1 6. 0. 7. ;r-7^ ,, . 8. 0. 9. :, ■. ' 3(l-x*) ^ l + x liV. (Page 107.) I. 16. 2. 12. 3- 15. 4. 28. 5- 63. 6. 24. 7- 60. 8. 45. 9- 36. 10. 120. II. 72. 12. 96. 13- 64. 14. 12. 15- 28. 16. 1. 17- 8. 18. 9. 19. 7. 20. 4. 21. 5. 22. 1. 23- 1. 24. 3 2' 25. 100. 26. 24. 27. 2 28. 6. 29. 24. 30. 4. IV. (Page 108.) 16. 2. 5. 3. \. 4. 1- S- 8. AA'SU'EI^S. 363 ^•4 12. 12. 18. 9. 7. 9. 13- 8. 19. 9. 8. 2. 9. 11. 14. 7. 15. 9. 20. 9. 21. 10, 10. 6. 16. 7. II. 2. 17. 7. 14- 25. c a + 6' 6c — rfm a — 5 3a&-2Jfe-3 4ac-l a 10. -2- 3a +1 „ ahd + ac 18. — J — T. aa + a 22. 1.^ Ivi. (Page 109.) 3c -2a ^" 56^T' 6 (a + c) ^ 1 + a 15 2. 18a + 2& 4a + 3 ■ 19. 6-1. 23. 6m. 6c c2-6" o a (m - 3c + 3o) c- a + m 26. 29. 12. 0. 16. ^, — . 0^6 — bc + d 6bd + ah 3a^l2cl {a + hf b — a' 13- _6_ a-r -f- 21. 2a^ F-"i- 3a36c + 2a%^ + «6^ ^"^ 63 + 3a3cT3a26c + 2a2p- c ac T' 27. 30- a6-l 6c + d' a'-e (c — d) Xofi'+Wjd' I. 2. 6. 1 7* I. 9. 16. 12. Ivii. ^Tage 111.) 2. 15. 7- 5 2' 3- 1- 4- 13" 8. 6. 9. -7. 12. 19. 13. 1. 14. 4. 1 2" *^- 8* 17. 2. 18. \. 19. i 7 5- To- 10. 6. 15. -- 20. 3. 35 364 AmiVERS. Iviii. (Pacre 11:1) 4f)Q I. 20. 2. 3. 3. 40. 4. ~. 5. 60. J ^ 46 ^ 6. 10. 7. 5. 8. 20. q. 3. 10. -^. II. 8. 12. 100. 13. 0. 14. -1. 15. 5. 16. -. 17. 5. liX. (Page 114.) I. 100. 2. 240. 3. 80. 4. 700. 5. 28,32. 6. A.-\ 7. 24, 76, 8. 120. 9, 60. 10. 960. II. 36. 12. 12,4. 13. £1897. 14. 540, 36. 15. 3456, 2304. 16. 50. 17. 35, 15. 18. 29340, 1867 19. 21, 6. 20. IO5I, 13l| 21. X has £1400, B has £400. 22. 28, 18. m (nb - a) n (mb -a) a + b a — b 23. — !^ ■% -5^ . 24. -^r-, -5-. 25. 18. ■^ n-m m-n ^2' 2 ^ 26. £135, £297, £432. 27. £7200. 28. 47, 23. 29. 7,32. 30. 112,96. 31. 78. 32. 75 gallons. 33. 40, 10. 34. 20. 35. 42 years. 36. 1^ days. 37. 20 days. 38. 10 days. 39. 6 hours. 40. I53 days. 41. 4- days. 42. 1.:, hours. 43- 48'. 44. 2 hours. 45- abc , , minutes. ab + ac + oc 46. 48|. 47. 51;r, 6I.3, 47.J gallons. 48. 9_ miles from Ely. 000 i ANSWERS. 365 , , -1 ac Id ^13 49. 14 miles. 50. -J, — . 51. 11—. 30 52. 42 hours. 53. 30.-- miles. 54. 50 houi's. 55. (1) 38^ past 1. (2) 54^- past 4. (3j 10-- past'S. 56. (1) 27-- past 2. (2) 5^ and also 38— past 4. 9' 6' (3) Slj- past 7, and also 54— past 7. 57. (1) 16^ past 3. (2) 32^ past 6. (3) 49^ past 9. 58. 60. 59. £3. 60. ^. 61. ISidays- 62. .£600. 63. ^£275. 64. 60. 65. 90', 72', eC. 66. 126, 63, 56 days. 67. 24 68. 2, 4, 94. 69. 200. 70. 2*, 5—. 71. 30000. 72. X200000000. ' 73. 50. Ix. (Page 127.) z* + ax + 3a I. . X a2 + 3ax-2x' ^" x{x-yy 2a3 + 6a26 + 6a62 + 26» "^ (o-6)(a2 + 62) • Ixi. (Page 128.) 8-13X xy V y 4 j^_^_ x^ + ox'^ + l 5- 2a;2-x3 + r , x^-x + l a^ + a + 1 D- - — -. • 7- • X a 3«6 ANSWERS. 8. 1 X. Q. -. lO X X. II, - -2x]/ a(a2 + 2a6 + 262) (a + 6)2 14. «t-L I- 1 ^3- '^' c(a-6-c)' Ixii. (Page 129.) I. 13 15 2. fi 6 c (i -1 + - + J + -. a c a a 3- ^_3 + ?_y. y'^ y X 3?' 4- i2~¥"^18~3(j' 5- 6p 4} 12r 24s grs jjrs fqs fqr 6. 100 40^^40 !5 Ixiii. (Page 131.) 1. 2-2a + 2a2_2a3 + 2a* , 2 4 8 16 2. 1 + — 3 + —4 m m" 771"* m* , 26 262 265 26* -' a a- a"^ a* 2x2 2x* 2x« 2x» x2 x^ X* X^ 5. X + — +-2 + -3 + -4 -' a a^ a^ a* ^ 6 6x 6x-' hs? 6x* a a^ a^ a* a° 7. 1 - 2x + 6x2- 16x3 + 44x* 8. l + 2x + x2-x3-2x* 9. 1+36 + 66= + 126' + 246* , ^ ,, 263 26* 10. x--6.c + o^ + -^ X X^ ANSWERS. 367 a2 a26 0^52 a^ a^ X x^ x^ x* x^ ' , 2x 3x2 4a;3 5a4 12. 1- — + .. - , +—r.... n. n- n9 n* a a- a'" a* 13. x^-3ax- + 2a^x + 4a\ 14. m< - lOm^ - 41to - 95. Ixiv. (Page 132.) J x^ x2 23x 1 , a^ _49a^ la 1 9^4'''l20'''20' ^' 20 "600''" 60" 15- 3- ^*-^- 4. x*+l + - X* 5 ---^ 6. 12 11 a^ ac b'^ c^' 7- l+a2 + a^- «■ '4'-f x« '64 9- 5 7 107 5 7 X*"^2'i3-i2x2 + 6x'^6- ^°- ¥ a* 0*^ ^• Ixv. (Page 134.) 1 ,1 , m 1 I. «--. 2. a + T. 3. m2-- + - X b n n^ , c' c2 c 1 XV ^ a d^ d^ d^ •' y X 6. -^ + -7 + 75. 7. -0-2 + ^2. 8. -x5-5x2+--x + 9. 62 ""^o2- ^°- a2 ab ac"*" 62 6c "^"?- Ixvi. (Page 135.) I. -05x2 -•143x- -021, 2. •01x2+l-25x-21. 3. -12x2 + -13x2/ --141/2. 4 -172x2- -05x2/- -3 12?/2. 5. 0. 6. -300763. 368 ANSWERS. Ixvii. (Page 135.) /, «o «, „ a4 , \ 1. OiXl 1+— a: + — x2 + — x3+ ... I. ^ aj ttj ttj / 2. a;i/2( + -). 3. x2(l+^ + ^). " \z y xf \ X xV 4- (a + 6) I (a + 6)2-c(a + 6)-d + -^l. Ixix. (Page 138.) 2x2 + 3x-5 ,a2 + 5(j_i4 2«}7 I. 46. 2. — = r — and — — r — . 3. -v- - , 7x-5 a + 9 -^ a^+f'^ 37x2-71/2-1922 11 4- 24 • 5- -9- , 60x* + 42ax'-]07a2x2+10aSx + 14o* 6. j2 • „ x' + xhi + 2w^ X - 8 x2 a 8- tY^ Sr-- lO- —,-5- II- ^i 4- 12. :; — , x{T/ — y^) X + 8 1 - X* 1-6 „ 1 ah + ac + hc + 2 a + 26 + 2c + 3 ^' ~F a6c + a6 + ac + ic + a + 6 + c + 1' 1 6 62 52 8a252 6(a2 + 6») 15. 2 5- 18. 4 ,4. 19. -7-5 — roi. ■' a ttx a^x ax- a* — o* ^ a {a^ - 6^) a^ + ¥ 1 a + b- c 22. ^ (a-6)2.(a2 + 6-')" 2(x + l)2- "" a-6 + c A 1 ^ (x-4)(x + 2)2 23. X. 24. 0. 25. 1. 26. ^^ -. 27. X ,2 29- ^2 + i 30. 1. W X' -2 + 5x + 17x2-ll.r '-21x* 3^- (3^2x'^7x?V* 28. X3(x2 4 31- ^1 3. 33- r-' + y^ 34. 2. i ANSIVERS. 369 35- 2a-6 . r. r- 36. 0. ^9- :^,(;--i + 2/2)- X 40, -, 41. x2 + 3x + 3-- + -,. X X- (.^ + 2/2)2 ^^- x« + 3/* 44- 1- 46. ^ + ^ A7 1 48. 1. p-q ^'' (x2 + l)(a;3 + l)- 49- 2a2 - ax - ay. 50. a + 6 + c (a3-63)2. Ixx. (Page lib.) I. x = 10 2. x=9 3. x = 8 2/ = 3. y = 7. i/ = 5. 4. x = 6 5. x=19 6. x = 5 l/ = 8. i/ = 2. 2/ = 3. 7. x = 16 8. x=2 9. x = 4 y = 35. y=l. i/ = 3. Ixxi. (Page 145.) I. x=12 2. x = 9 3. x = 49 4. x = 13 y = A. 2/ = 2. 2/ = 47. t/ = 3. 5- x = 40 6. x = 7 7. x = 5 8. x = 6 !/ = 3. l/ = 2. i/=l. y = 4. 9- x = 7 2/ = 17. Ixxii. (Page 146.) I. x = 23 2. x = 8 3. x = 3 4. x = 5 2/ = 10. 2/ = 4. !/ = 2. t/ = 9. 5- x = 2 6. x = 7 7. x=12 8. x = 2 t/ = 2. y = 9. y=9. 2/ = 3. 9- x = 3 |/ = 20. rs.A.] sa 370 1 ANSWERS. Ixxiii. (Page 147.) I. x = 7 2. x=9 3- x = 12 4- x= -2 2/= -2. J/=-3. 1/=-3. i/ = 19. 5- x= -5 6. x=-3 7. x = 7 8. 1 ^ = 2 2/ = 14. y=-2. 3/= -5. ■^ 3 9- x=-2 y=i. Ixxiv. (Page 148.) I. x = 6 2. x = 20 3- x = 42 4- x = 10 2/ = 12. 2/ = 30. 2/ = 35. y = 5. 5- x = 9 6. x = 4 7- x = 5 • 8. x = 40. 2/= 140. 2/ = 9. y = 2. 2/ = 60. 9- 13- x=12 2/ = 6. x = 6 lO. 14. x=19 1/ = 3. x=19^ II. 15. x = 6 2/ = 12. 1 12. 3201 ^~ 708 278 ^=59- y= -17 y=-^' 5* Ixxv. (Page 149.) eg -nf ce + bf em + bn x^ * 2 X= '^ "^ x = mq — np ' bd + ae ^' ae + bc _ mf— ep _cd-af _ an — cm mq - np' ^~ bd + ae ^ ae + bc' de n'r + n/ , a + b x = — r^ c. x= — , 7- 6. x = — ^— c + d ■' mn +mn 2 _ ce _ to/ — mV _ a — 6 ^~c + d' ^~mn' + m'n' ^~ 2~ " c(/-6c) „ 1 2b^-6a^ + d x= ^;; , / 8. x = -T- 9. x= 5 a/-6(i aft ^ 3a _c(ac-rf) _J_ 3a'-fe^ + rf ^~ af-bd- '-''cd- y~ 36 • ANSWERS. 371 a ofi bm 10. x = T- II. x = i — 12. x = r be b + e b — m _a + 2b _ ¥-c^ _ bm ^ c ■ ^~~a ■ ^~b + m' Lxxvi. (Page 151.) _ 1 _ ^_ _ 6- - g'' '• ^~2 ^- ''~b-2a 3- -"'bd-ac 1 2 _ ¥-a^ ^4- '^~S^^- y~bc-a<r 2a 61 , 1 26 61 1 103' ^ 5' x = - 8. x = - a n 1 1 Ixxvii. (Page 153.) I. x=\ 2. x = 2 y = 2 y = 2 2 = 3. 2 = 2. 5. x = l 6. x = l 2/ = 2 2/ = 4 2 = 3. 2 = 6. 9. x = 2 10. x = 20 t/ = 9 2/ = 10 3. x = 4 4. x = 5 2/ = 5 y = 6 2=8. 2 = 8. 2 8. x = 5 7- ^ = 3 , = 6 y=-7 2!=7. 2 = 36^. 2=10. Ixxviii. (Page 15'.) I. 16, 12. 2. 133, 123. 3. 7-25, 6-25. 4. 31, 23, 5. 35, 14. 6. 30, 40, 50, 372 ANSWERS. 7. £60, £140, £200. 8. 22s., 26s. 9. £200, £300, £260. 10. 41, 7. II. 47, 11. 12. 35, 11, 98. 13. £90, £60 14.60,36. 15.6,4. 16.40,10. 17.503,1072 18. 10 barrels. 19. ^s.,\s. Sd. 20. £20, £10. 21. 15s. \Qd., 12s. 6(f. 22. 4s. 6d., 3s. 23. 35, 65 24. 26. 25. 28. 26. 45. 27. 24. 28. 45. 29. 84. 30. 75. 31. 36. 32. 12. 33. 333. 34. 584. 35. 759. 36. I 37. A 38. I 2 7 35 19 39.3- 40.^9. 41.41. 42.40- 43. £1000. 44. £5000, 6 per cent. 45. £4000, 5 per cent. 46. 31^, 18| 47. 20, U). 48. 3 miles an hour. 49. 20 miles, 8 miles an hour. 50. 700. 51. 450,600. 52. 72, 60. 53. 12, 5s. 54. 750, 158, 148. 55. 15 and 2 miles. 56. The second, 320 strokes. 58. 50,30. 5 59. 4 yd. and 5 yd. 60. -, 6, 4 miles an hour respectively. 61. 142857. IxxiX. (Page 164.) I. '2rt]. 2. 9af6*. 3. IIw^hV. 4, Sa?lf>c. 5. 267a26x». 6. ISa'S^c^. 7. ~. 8. — V ^ ' Ah 2a? 5a-63 16x6 25a ^- llxV '°- 171/2- ^'- "1S6- Ixxx. (Page 167.) I. 2a + 36. 2. 4fr^-3P. 3. a6 + 81. 4. 1/8-19. 5. 3a6c-17. 6. x--Zx + b. 7. 3a;- + 2x + l. ^N^IVERS. 373 8. 2r2-3r+l. 9. 2u2 + 7i-2. 10. l-3x + 2x2. II. x3-2x2 + 3x. 12. 2^2-32/3 + 42-. 13. a + 26 + 3c. 14. a^ + arb + aki' + h\ 15. x^-2x- — 2x-l. 16. 2j;2 + 2ax + 46-. 17. 3 - 4x + 7x- - lOx^. 18. 4a2_5o6 + 86x. 19. Za^-Aap^-bt. 20. 2i/2a; - 3yx2 + 2x^. 2 1 . 5x^2/ - 3x?/2 + 2 y^. 22. 4x2 - 3x2/ + 2i/2. 23. 3a -26 + 4c. 24. x^ — Zx + b. 25 . 5x - 2i/ + 3^. 26. 2x2 - y + i/2. IXXXi. (Page 168.) 3 a - 1 5. x2_a;+ 6. x2 + x--. 8. x2 + 4 + ^. 9. -i^a^x + ia---. x' 6 4 II. 6m --+^. n 5 2x 3i/ z 13. — --^ + -. "^ z z X a b e _d ^5- 3"4''"5 2" « o ftX , 17. 3x2- — + OX. Ixxxii. (Page 170.) I. 4 4- a b b^a- 7- 2a-36 + ^. 4 10. 1 2 3 + -. X y z 12. ab - 3cd + Y- 14. 2m 3» 16. 7x2-2x-| 18. 3x2-1-3. I. 2a. 2. 3x2t/2. • 3- - binn. 4- - 6o<6. 5. 7V>c<'. 6. -lOafc^c*. 7- - I'lm'n^. 8. llo'6«. %1A AJ^SWMRS. Ixxxiii. (Page 172.) I. a-h. 2 !. 2a + 1. 3. a + 86. 4. a + 6 -r c. 5- x-y + z. 6. 3x2-2x+l. 7. 1 - a + o2 8. x-y + 2z. 9. a2-4a + 2. 10. 2m2-3m+l. II. x + 2y — z. 12. 2m-3;i-r. 13. IXXXiv. (Page 173.) 1 TO 4- 1 . TO I. 2a - 3x. 2. 1 - 2a. 3. 5 + 4x. 4- a-h. 5. x+1. IXXXV. (Page 175.) 6. TO - 2. I. ±8. 2. ±ah. 3. ±100. 4- ±7. 5- ± v'(ll). 6. ±8a2cl 7. ±6. 8. ±129 9- ±52. 10. ±4. II. ± J(' 7?l / 12. W(.- I> 13. ± x/6. 14. ±2v^2 Ixxxvi. (Page 179.) I. 6, -12. 2. 4, -1(5. 3. 1, -15. 4. 2, -48. 5- 3, -131. 6. 5, -13. 7. 9, -27. Ixxxvii. (Page 180.) 8. 14, -30. I. 7,-1. 2. 5,-1. 3. 21, -1. 4. 9, - 7. 5- 8,4. 6. 9, 5. 7. 118, 116. 8. 10±2v'34. 9- 12, 10. lO. 14,2. Ixxxviii. (Page 181.) I. 3, -10. ,^ ■■ 7 25 2. 12, -1. 3- 2' -y 4. 20, -7. 5- 1 5 4' 4" 6. 9, -8. 7. 45, -82. 8. 8, -7. 9. 4. 15. TO. 290, 1. ANSWERS. 37S Ixxxix. (Page 182.) I. 7 3' 5 3' 2. 1 3 5' 5" 3- 3.^- 4- 1, 3 11* 5- 3 5 5' f 6. ^.-1 7. 8, 2 3" 8. xc. (Page 182.) I. 3, 8 3' 2. u.,-|. 3- ^. -¥■ 4- 8, 19 2* 5- ^.-^- ^- ^. 1 7. 8, 17 4" 8. 7 3 2' 14" xci. (Page 184.) I. -a±V2-a- 2. 2rt±^/ll.a. 3. 2'~"2* , ^ , cfi + ah a^ — ah 5. 1, -a. 6. 6, -a. 7. — • •^ a-b ' a + b c+ J{c'^ + 4ac) c- ^ (c^ + 4ac) ^" 2 (a + 6) ' 2 (oTi) ' 4- Zn, n ~2' 8 d h c' a 0. 62 ¥ ac' ac 2a -b 3a + 26 II. , , ac be ac^ + bd^ ac- + bd^ 2a + 3d Vc' ~2a^3f?v'c' XCii. (Page 185.) I. 8, -1. 2. 6, -1. 3. 12, -1. 4. 14, -1. 9 4' 5. 2, - 9. 6. 6, 5. 7- 5, 4. 8. 4, - 1. 9. 8, - 2. 376 ANSWERS. lo. 3, -^. II. 7,^. 12. 12, -1. 13. 14, -1. 14. \-\ 15- 13, -y. 16. 5,4. 17. 36,12. o ^ c 25 5 „ 10' „ 10 18.6,2. 19.18,-3. 2°-7,-y. 31. .,-y. ^ r o 1 12 2 1 22.7,-5. 23.3,-2. 24.2,-3. "5.3.-,. 26. 15,-14. 27. 2, -|. 28. 3, -^. 29. 2,|. o 23 o 14 .5 o 21 30. 2, --. 31. 3, -— . 32. 4, -„. 33. 3 15' -^ ■ ' 3' -^- ' 3' •'■^" ' ir 58 13" 58 34. 14,-10. 35. 2,--. 36. 5,2. 37. -a, -6. 38. -a,h. , , o , a 2a. ft 6 39. a + 6, a -6. 40. a-, -a'. 41. ^, -y. 42. p -. xciii. (Page 187.) I. a; = 30 or 10 2. a; = 9or4 3. 2; = 25 or 4 y = 10 or 30. y = 4 or 9. j/ = 4 or 25. 4. 2 = 22 or -3. 5. x = 50 or - 5 6. a;=100 or - ] j/ = 3or-22. i/ = 5or-50. i/=lor-l()0 XCiv. (Page 187.) I. x = 6or-2 2. a;==13or-3 3. x = 20or-6 y = 2 or — 6. i/ = 3or — 13. y = Gor-20. 4. x = 4 5. a;=10or2 6. x = 40 or 9 y = 4. 1/ = 2 or 10. j/ = 9or40. XCV. (Page 188.) I. a; = 4 or 3 2. a: = 5 or 6 3. a- = 10 or 2 i/ = 3or4. J/ = 6 or 5. t/ = 2 or 10. 4. a;==4or-2 5.a; = 5or-3. 6. x=7or-4 y = 2 or — 4, 1/ = 3 or - 5. y = 4 or - 7. i AA'SIVERS. 377 XCVi. (Page 189.) 1 . 2 = 5 or 4 2. a; = 4 or 2 y ==-4 or 5. y = 2 or 4. 1 4 a; = 3 5. x = ^ y = 4. y = 2. y = xcvii. (Page 191.) 3- 1 1 x = — or — 3 2 1 1 ^ = 2°^ 3 6. 1 I I. j:=^-i or- -3 2. x=±6 3- x=±10 y — 'i 01- -4. 2/= ±3. y=±n. 4- x=±8 5- 2 = 5 or 3 6. 95 x = 5or-^ 33 !/ = 2or-y. y=±2. y = 3 or 5. 7- x=±2 y=±5. 8. x=6 y=5. 9- z=±2 2/=±l. 10. x=±2 II. 2=±7 12. 11 2=3 or - !/ = 2or|. y=±3. i/=±2. '3- a- = 10 or 12 14, x = 4 or 85 "8" 19 8" 15- 2=±9 or ±1-2 ?/ = 12 or 10. t/ = 9 or y=±l2 or ±9 xcviii. (Page 193.) I. 72. 2. 224. 3. 18. 4. 50, 15, 5. 85, 76. 6. 29, 13. 7. 30. 8. 107. 9. 75. 10. 20, G. II. 18,1. 12. 17,15. 13. 12,4. 14. 1296. 15. 56^ 16. 2601. 17. 6, 4. 18. 12, 5. 19. 12, 7. 20. 1, 2, 3. 21. 7,8. 22. 15,16. 23. 10,11,12. 24. 12. 25. 16. 26. £2, 5s. 27. 12. 28. 6. 29. 75. 30. 5 and 7 liours. 31. 101 yds. and 100 yds. 32. 63. 33. 63 It., 45 It. 34. 16 yds., 2 yds. 35. 37. 36. 100. ^j. 1975, 378 ANSWERS. XCix. (Page 199.) 1.35 = 3 2. x = b 3. 05 = 90, 71, 52. ..down to 14 i/ = 2. 2/ = 3. |/ = 0,13,26 upto52. 4. a: = 7, 2 5. x = 3,8,13... 6. x = 91, 76, 61 ...down to 1. y=\,A. i/ = 7,21,35... i/ = 2,13,24 up to 68. 7. a; = 0, 7, 14,21,28 8. a; = 20,39... 9. a; = 40,49... i/ = 44,33, 22, 11,0. i/ = 3,7... i/=13,,33... 10. a- = 4, ll...uptol23 II. x = 2 12. x = 92,83....2 1/ = 53, 50... down to 2. y = 0. y=l, 8... 71. 4 3 8 2 13. i^ and-. 14. yyand— . 15. 3ways, viz. 12,7,2; 2,6,10. 16.7. 17.12,57,102... 18.3. 19. '2. 21. 19 oxen, 1 sheep and 80 hens. There is but one other solution, that is, in the case where he bought no oxen, and no hens, and 100 sheep. 22. A gives 5 11 sixpences, and B gives A 2 fourpenny pieces. 23. 2, 106, 27. 24. 3. 25. A gives 6 sovereigns and receives 28 dollars. 26. 22, 3 ; 16, 9; 10, 15; 4, 21. 27. 5. 28. 56, 44. 29. 82, 18 ; 47, 53 ; 12, 88. 30. 301. C. (Page 205.) ^27 2 2 3 jf (1) I. x"2+x^ + x2. 2. x^y + x^y^+x'y. 4 5-fi. 121312 3. o5 + a=i+a2. 4. x-^yz^ + a^yH + a-''yz-'- . (2) I. x-' + ax-^+b^x-' + 3x-*. 2. x^y-* + 3xy-* + 4y-*. x^y~h~^ 5xhrh~^ , , 3. ^^ +7 — +a-r*2"'- 4. ^^ + ^g'- + x-''y-*z. 11 1_ ^ 1_ _1_ J^ ANSWERS. 379 4 3 1 a ^b~c^ X ^y 1 1 1 (4) :. 24/.^.34/(x,^).i,. 2. ^-^^ + i ci. (Page 206.) I. x*'' + x'^''y^'' + y*''. 2. a*"-81i/*". 3. x*'' + 4aV + 16a'», 4. tr" + Sa^c' - 6=" + c^ 5. 2a- " + 2a'"6" - 4a"'c'' - a'"b - b"+^ + 26c' + crc- + b"c^ - 2c'^"-. 6. x^' + x""'"". 2/""'"" — ^''-j/""- 7/"'"""+"'. 7. x*'' + ar"y-'' + y*\ 8. a='^ - a'''-'' b"^ + a^^-^ c" + a''°+'' . S^-^* - 6 + S'-'" c" + a''^+'' c'-" - b^^c^-" + c. 9. x*' + 2x«'' + 3x2'' + 2x''H-l. 10. x*''-2x'"' + 3x-''-2x''+l. cii. (Page 207.) 1. x^" + ar'^y'" + x'"if"' + y^"'. 2. x^" — x'"!/" + x^'y-" - x^'y^" + y*". 3 . r'' + x^'i/' + x^'y^ + x-'y^' + x'y*' + ]f'. 4. a'^" - a"6^ + a*''6*» - a^''&*'' + h^. 5. x'-* + Sx** + Qx-" + 27x'' + 8] . 6. a-" - 2a"'x" + 4x-". 7. 2-x'' + 3x*^ 8. 46"'c"'-552"'. 9. a'"' + 3«="' + 3a'" + l. lo. a"' + 6'' + c', 38o ANSWERS. ciii. (Page 208.) I. x-3a;^ + 3x^-l. 2. y-\. 3. a^-x^. 4. a + h + c-ZaH^c^. 5, 10a;-lla;*i/* + 5xi?/*-21?/. 4 12 6. vw — w. 7. m-^ + 4d^m3 + 16rf. 8. 16a + 8a"^6^ + \Qah^ + 18a^6^ - 2Aah^ - 12a^6^ - 15a^ b? -276. fi 1 2 a 1 1 A y. -. +2a^x^ + a^. 10. x3-2a3x3+a3. 4 3 2 A '^ i II. x^ + 2x^y^ +y^. 12. a- + 2a65' + 62^ 13. x-4x* + 10x2-12x* + 9. 1 4. 4x* + 1 2x' + 25x'^ + 24x'^ + 16. 15. x^ - 2x^2/^ + 2x^z* + 2/* — 22/*2* + 3*. 16. x2 + 4x%4 - 2x*a* + 41/2 - Ay^z^ + a^^ Civ. (Page 209.) I. x^ + j/* 2. a^ — 6^, 3. yfl -{-T^y* + y'\ 2112 ^ZlZS^XSik ^. a^-a^h^ + o^. 5. x^ -x^y -\-x^y^ -x^y ^y. 6. m 8 + m* w* + m2 71,3 4. m* «, 2 ^ jnByj,^ + n^. 7. X* + 3x^1/^ + 9x*i/2 + 27t/^. 8. 27a^ + 18a^6i + 12ai6^ + 86^. 9. a^-xi 10. ?>i"+3m5 + 9m5 + 27m5 + 81. 11. x2 + 10. 12. x^ + 4. 13. -h + 2h^-h^. a 11 1152 11 14. x^ -x^y^ —x^z^ +y^ + z^ —y^z-^. 15. x3-9x3-10. 16. m^ + 1111^11^ + 11^. 11 111 11 J7. 'p~-2p^ + \. 18. x- -(/--;:-. 19. x'^+y^. ANSIVERS. 381 CV. (Page 210.) I. a-2-&-2 2. x-«-6-*. 3. a;4-x-4. 4. iC' + l+oj-*. 5. a-'*_j-4_ 5_ a-2 + 2a-ic-i-6-2 + c-2. 7. l + a26-2 + a*6-4. 8. a^J"* - a-*6* - 4a-262 _ 4. 9. 4a;-5-x-* + 3a;~' + 2a;-2 + x-^-Vl. cvi. (Page 211.) I. x-x-^. 2. a + 6~^ 3. m2-mn-i + w,-2. 4. c* + c3ci-i + c2i-2 + cd-3 + d-4. 5, xy-i + x-Y 6. a-'^ + a-'^h-^ + h~\ 7. a;2?/-2 _ 2 + a;-2,^2_ 8. |x-3-5x-2 + lx-i + 9. 9. a26-2-l + a-262 I o. a-2 - a-ift-i - a-ic-i + 6-2 - &-ic-i + c-2. cvii. (Page 211.) 2 11 y?*+I2 I. x*-2x%2 + 2y. 2. X '"-' • 108+18a 3. X 3i-2 . A 2a (x*-a*)^' ^ , 22 _, 421 _, 10 . 1 5. 7x-*+ g-x 3-^^a; «-yx-i + -. 6. x". 7. x"-i/". 8. a2 + 2a26'5-2a^65-6^ 9. a^ + a3 63+63. u. 771 = ™"^'. 12. x^+2H-2=, 13. x^*. 14. I6a'^. 15. a'^'-p. 16. 20^"" + 2a" 6'' - 4(( c"-3a"6-3/>''+i + 66c". 17. c. 382 AJ\rslVE/?S. 19. x^ + x^-hl. 20. a"^ + 2a'"+»-* . bcu^ - a""^-^ b-x- - a"^-^ c V. 21. x'^'-"-i/^'^". 22. a"~^ 23. x^'-tf". ~^' ''' 144 ^^' ic'"''-xV~''"--c""""""i/" + r"- 26. x + Zx^-2x^-7x^ + 2x~^. cviii. (Page 215.) I. 4'x3, 4'j/'. 2. '4^(1024), jys. 3. 4/(5832), 4/(2500). 4. •";'2", "■;/2'-. 5. ';/a', 'V^/h". 6. 4/(a2 + 2a6 + 6'-0, 4/(a3-3a26 + 3a52-63). cix. (Page 217.) I. 2v'6. 2. 5^'2. 3. 2a Ja. 4. oa-d ^(5d). 5. 4zV(2y2). 6. 10x/(10a). 7. 12c ^'5. V5x „ ,a II. (a + x).,Ja. 12. {x-y)i^lx. • 13. 5(a-6).^'2. 14- (3c2-t/).V(7l/). 15- 3u^4'(26-). 16. 2x?/2 . 4/(20a»7\ 17. 3m3„3^/(4„). 18. va^fe^ 4/(46). 19. (x + y).^x. 20. (o-t).4/a. ex. (Page 217.) I. V(48). 2. V(63). 3- 4^(1125). 4- v.96). 5. I~. 6. V(9a). 7. >v'(48a^x). 8. ^'{Zah . AmWERS. 383 CXi. (Page 218.) The numbers are here arranged in order, the highest on the left hand. I. x^3, 4/4. 2. J\0, 4/15. 3. 3^/2,2^/3. 6. 2 ^/87, 3 s^33. 7. 3 4'7, 4 ^'2, 2;'22. 8. 5 4/I8, 3 ^'19. 3 s"^:^. 9- 5 ^'2, 2 ^^14, 3 4/3. 10. |x'2,^,^3,i,'4. cxii. (Page 219.) I. 29 ^'3. 2. 30 ^ao+ 164^/2. 3. {a^ + h^ + c^) ^'x. 4. 134/2. 5. 33 4'2. 6. V6. 7. 5^'3. 8. 48^2. 9. 44/2. 10. 0. II. 4v'3. 12. 2^'(70). 13. 100. 14. 3a6. 15. 2a6 4/(126). ,6. 2. ,7. I .8. 4/? ,9. J V. X 20 .+X1/ cxiii. (Page 220.) I. ^{xy). 2. s'{xy-y^). 3. x + !/. 4. s'{v^-y^). 5. ISx. 6. 56(.f+l). 7. 90v'(-r--x). 8. 2x^3. 9. -X. 10. 1-x. II. -12x. 12. 6rt. 1 3. - s'{^ - 7x). 1 4. 6 v^(x2 + 7x). 15. 8 (a2 - 1 ). 16. -6a2+12a-18. CXiv. (Page 221.) 1. x + 9^'.r+14. 2. x-2^/x-15. 3. a. 4. a-53. 5. 3x + 5^/x-28. 6. 6.v-54. 7. 6. 8. V(9x^ + 3x) + ^'(6x2 - 3x) - ^'(6x- - x - 1 ) - 2x + 1 . 3^4 AmWE}?S. s/iax) + sj{ax — x^) - js/{a^ -ax)-a + x. 3 + X+ ^{Sx + x"^). 11. x-y + z + 2s,/xz. 2x + 2j{ax). 13. 4Z2 + 42j{x^-9)+x^. 2x+ll+2V(a;2+lla; + 24). 15. 2x - 4 + 2 ^(a;''' - 4x). 2a;-6 + 2V(a;2-6x). 17. 4x + 9-U^x. 2x-2s/{x--y-). 19. x- + 2x-l-2y/{x?-x). x2 + l + 2V(a^-x). cxv. (Page 222.) I. {^c+ ,^d)(Jc- ^fd). 2. {c+ ^d){c- ^d). 3. ( s'c + d){Jc-d). 4. (1 + -Jy) (1 - Vy). 5. {\+ ^•i.o-:){\~ >JZ.x). 6. (V5.m + l)(V5.m-l). 7. |2a+ v/(3x)n2a- V(3x){. 8. |3 + 2V(2n)} j3-2v/(2n){. 9. U'(ll).« + 4nV(ll).«-4i. 10. {p + 2^r){p-2Jr). II. (v'p+ V3.2)(ViJ- V3.2). 12. {a" + 6^'na"-6^}. «W6^ oW^. 15. 24+17^2. ■^ a- -b a — o 16. 2+^2. 17. 3 + 2^3. 18. 3-2^/2. a + x + 2 yj{ax) 1 + x + 2 ^x IQ. • 20. ^ . ^ a-x 1-x 0+ V(a2-x2) 22. TO-- v'(w*-l)- 2a2-x2 4-2aV(a2-x2) 23. 2a2 - 1 + 2a V(a2 - 1). 24- CXVi. (Page 224.) I. 19. 2. 11. 3- 8-26v/(-l). 4- 5+4^/3. 5. 2h-\-2 ^'{ah)-\2a. 6. a- + a. 7. i^-a'. I AA'SW^/^S. 385 cxvii. (Page 224.) I. x+y ^ x+y SJixy)- 3. 2^{xy)' 5. x^- v'2.ax + a-. 6. vi^+ ^f2.vin + n^. 7. 2x i^x. 2« v''^ - 26 Va 0. - , . a-b 9- ah J „ fd -^+cd-2ac^-^. 10. -'^'-^ + aU II. x-l '3- ,,.• 14. l-^\/{^-^y ^5. 2x-2V(x2-a2). 16. a'^¥c. 17. -l4-5a2(2-a2) + a(10a2 a4_5)^(_l) 18. 8 + 74/3. 19. 4^/(3cx). 20. x^^CS^?'"). 21. -4/(-47i0. 22. (9?i-10).^7. 23. 0. t cxviii. (Page 228.) "l. v/7+V3. 2.^/11+^5. 3. ^/7-v/2. 4.7-3^5. 5. v/10- V3. 6. 2^5-3^2. 7. 2^3- v/2. 8. 3^11-2. 9. 3 V" - 2 -v/3. 10. 3 V7 - 2 v/6. 11. h ^10 - 2). 12. 3 ^/5 - 2 ^3. I cxix. (Page 229.) •I. 49, 2. 81. 3. 25. 4. 8. 5. 27. 6. 256. 7. 27. 8. 56. 9. 79. 10. 153. 11. 6. 12. 36. 13. 12. 14- '^ — 2^' 15- 5- 16. 6. 17. 3. 18. 10. 19. _. 20. ^3^ .rs,A.i 2 p :86 ANSWERS. cxx. (Page 231.) I. 9. 2. 25. 3. 49. 4. 121. 5. 1^. 7.0,-8. 8. (-2-)- 9. (-^). 6. 8,0. 10. 5. cxxi. (Page 231.) I. 26. 2. 25. 3. 9. 4. 64, 5 ^^ o 6. -^ 7. a. 8. \ or 0. 9. 64. 10. 100. 5 4 cxxii. (Page 232.) I. 16, 1. 2. 81, 25. 3. 3, 2^. 4. 10, - 13. 5. 5.? 6. -4.-32. 7.9,-3? 8. 28.1||f 9. 49. 10. 729. II. 4, -21. 12. 1 or—. 13. ±24. 145 25 14. 5 or 221, 15. 5or— r-. 16. 5 or 0. 17. i^. 18. 25. ^ ^ 121 ' 36 19. ±9^/2. 20. ± ^/65 or ± *y5. 21. 2a. 22. -2a. 23. gOJ^-^e- '^- 4- "=5- 1-^. 26. ^1^-. 27. ^. 28. ±5 or ±3 ^'2. 29. ±14. 1,30. 6or-y-. 31. 1. 32. ^. 2>2,. 2or0. 34. or ^j^. cxxiii. (Page 23.'i.) I. 2,5. 2. 3, -7. 3. -9,-2. 4. ba,Gh. _7 5 A ?^ _§^ ^ Yk^l 5- 2' 3' l9"' 14" ^' 5 ' 6 ■ A.VSlVERS. 387 8. -2a, - 3a and 3a, 4a, 9. ±2, a. 10. 0,5. 2a - & & - 3a d e , — r — . 12. -, -. ac be c c CXXV. (Page 239.) I. a;2-llx + 30 = 0. 2. .x^ + x- 20 = 0. 3. x2 + 9x + 14 = 0. 4. 6x2-7x + 2 = 0. 5. 9x2 -58a; -35 = 0. 5 a;2-3 = 0. 7. x^ — 2mx + m--n- = 0. 8. x^ i^-c + — ^ = 0. a/3 a/3 2 a^ - /3- a/3 • cxxvi. (Page 240.) I. (x-2)(x-3)(x-6). 2. (x-l)(x-2)(x-4). 3. (x-io)(x+i)(x+4). 4.4(x+i)(x+l::^)(x+^-:!^). 5. (x + 2)(x+l)(6x-7). 6. (x + 1/ + 2) (.(;2 + y^ + z^ — xy — xz- yz). 7. (a-fe-c) (a2 + 6^ + c2 + a6 + ac-6c), 8. (x-l)(x + 3)(3x-7). 9. (x-l)(x-4)(2x + 5). 10. (x+1) (3x + 7)(5x-3). cxxvii. (Page 242.) I. Vl3orV-l. 2. 4/ -2 or 4/- 12. 3. ^V _ 1 or 4/- ?1 . 4. lorV-4. S.^aor^--. 6 25.,,.; 1 L 7- ::9F^7- ^- (D'*^^- (-D- 9- 1 or 1 ±2 VI- „ 1 5± V1329 10. 3 or — - or ~ . 3 4 A.VSWERS. „ a + 6 a±2J(a2-3o) 11. a + 2, or - --— , or - ^\ \ o o 1 2. 0, or a, or ^'- — '. cxxviii. (Page 245.) I. 6 : 7, 7 : 9, 2 : 3. 2. The second is the greater. 3. The second is the greater. 4. ^iz^, 5. I0:9or9:10. « CXXix. (Page 246.) I. 2:3. 2. h:a. 3. 6 + (Z:a-c. 4. ±^6-1:1. 5. 13 : 1, or, - 1 : 1. 6. ± s,l{rn^ + 4m^ - in : 2. 7. 6, 8. 8. 12,14. 9. 35,65. 10. 13,11. 11. 4:1. 12. 1:5. CXXX. (Page 247.) 8 8 x-v a-h+c 1. TT- 2. pr. 3. -. 4. , 15 9 -^ x + y ^ a-o-c m^ - mn + n^ , (x + 2) y ^' !«?■■¥ wn + n^' ' (y — 4)x' cxxxii. (Page 255.) 6. X = 4 or 0. 8. 440 yds. and 352 yds. per minute. II. x = 30, 2/ = 20. 62 ^3- T 9 15- 41. 16. 50, 75 and 80 yards. 17- 120, 160, 200 yards. 19. I5 miles per hour. 20. 1 : 7. 21. 160 quarters, ^2. 22. £80. 23. £60. 24. £20, 25. 90:79. 26. 45 miles and 30 miles. ANSWEI^S. 389 cxxxiii. (Page 262.) 4. 16|. 5. 5. 6. 12. 7. 3^. 8. I 9. Aoz(P. 10. 5. II. A=Ib. 12. 64x2 = 91/3. 13. x2 = — 3-. 14. 4x3=271/2. 18. i/ = 3 + 2x + x2. 19. 18ft. CXXXiv. (Page 266.) I. 50. 2. 200. 3. lo|. 4. -32.^ 5. -2| 6. 40. 7. 117. 8. 0. 9,20/ o\ 3an - 26?^ - 2a + 6 9. x^ + y^-2{n-'2)xy. 10. r . CXXXV. (Page 268.) I. 5050. 2. 2550. 3. 820. 4. 30. 5. 24. 6. -34. 7. ^^^^l 8. ?^^^l b 2 2 7n2 - 5» w - 1 10. -2-. CXXXvi. (Page 269.) I. -6. 5. -2. 2. X "25" 1 3- 8- 4- 7 8* 6. -4 cxxxvii. (Page 269.) 1. (I) -46. (2) 36-2. (3) ?. (4) 4-4. 2. 155. 3, 112. 4. 888. 5. 100. 390 ANSWERS. 6. 6433|. 7. £135. 4s. 8. (i) 355,7175. (2) - ISGa^, -3116«- (3) 161 + 81a;. 3321 + 1681X. (4) 119^, 2357^. (5) 8^, I74I. 9. (i) 126, 63252. (2) 25, 2250. (3) 45, -1570-5X. (4) 99, -1163^. 4 (5) 71, 4899(1 -m). (6) 65, 65x + 8190. cxxxviii. (Page 271.) I. 6, 9, 12, 15. 2. \\, |, 0, -| -1^. „_5^ 5 1 ^ 1^ ? 11 CXXXiX. (Page 272.) 3m + TO ?n. + n m + 3?i 4 ' 2 ' 4 ■ 5m 4- 3 5mH-l 5m — 1 5to-3 2. 5 ' 5 ' 5 ' 5 • 6n2 + l 5n2 + 2 5?i2 + 3 5n2 + 4 J- 5 ' 5 ' 5 ' 5 ■ 4- 2x2 + 1/2 9 ' ^2 2x2-^2 CXl. (Page 275.) I. 64. 2. 78732. 3. 327680. 4. -J-. ^ ^ 2048 5. 13122. 6. 16384. 7. -— . ' ' 96 ANSWERS. 391 CXli. (Page 276.) I. 65534 2. 364. 3- ;^-i- 4- x^ {x- ■1) 1)' (a-x)jl- 5" (a + u;)». (1 (^ + ^)^t 6. 3--1, — a-x) 7- 7(2»- ■!)• 8. -425. CXlii. (Page 43 9- -96- 278.) I. 2. 4 ^- 3- 27 3- 8-- ^ 1- 5. ll. 6. •3. 7- 4 3.4 ^-1 16x5 9- ^^3- ^°- 8x2+1- II. a2 a-V 12 1 • 9- ^3. x2 86 x + y '■^- 99" 15- 49 90" . 46 ^6- 55- cxliii. (Page 279.) I. 9, 27, 81. 2. A, , 16, 64; ,256. 3- 2,4, 3 9 27 81 ^ 4' 8' 16- 32' cxliv. (Page 279.) I. (i) 558. (2) 800. / ^ 18- (3) T (4) f (5) -2-- (^) 486- (7) 1189 2 ■ (8) 13^. (9) 1- (10) -84. (II) - 9999 V3 (Vio + i).V5' , , 3157 5. 42. 6. ac=b'^. 7- ±1. 8. n+i-. 471 392 ANSWERS. 9- 4. 10, . 10. 13- 4. 14. 642. 1 6. 49, 1. 17- 3| 6, 4 18. 60. 19. 4 3 5' 5' 2 5' 1 5' 0, 1 5' 2 6' 3 ~5' 4 "5" 22. 3, 7, 11, 15, 19. 23- 5, 15, 45, 135, 405. 25. 139. 26. 10 per cent. cxlv. (Page 285.) 1111 „ „ , 2 ^ 6' 9' 12' 15* >• -^, ^, ^, L, 3* 6. 3 3 4' 2' 0°, 3 3 2' 4' 7- 6 3 6 3 5' 4' 11' 7' 6 17' 3 10' 8. 6a5?/ (n + 1) 3?i2/ + 2a; ' 6x1/ (% + l) 3?ii/ + Ax — 'iy » 6x1/ (n + l) ■■' 2nx + Sy ' 9- 1 4' 1 2' 1 1 1 '*' 2' 4' 6' 5 ^"^31' 5 5 15 24' 17' 2' 3' 5 4* 10. 104, 234. cxlvi. 13- (Page 2, 3, 6. 290.) I. 132. 2. 3360. 3- 116280. 4- 6720. 5. Ill 8' 6. 40320. 7. 3628800. 8. 125. 9- 2520. 10. 6. II. 4. 12. 120. 13- 1260. 14. 2520, 6720, 5040, 1663200, 34650. cxlvii. (Page 295.) I. 3921225. 2. 6. 3. 126. 4. Ii628a 5. 12. 6. 12. 7. 816000. S. 3353011200. 9. 7. 10. 63. II. 62. 12. 123200. 13. ;376992 ; 52360 ANSWERS. 393 cxlviii. (Page 300.) a* + 4^x + %d?x^ + Aax^ + x*. 66 + 66=c + 156^c2 + m?(? + 1552c* + Gtc^ + c«. a" + 7a66 + 21a562 + 35^453 + 35^35* + 'iXaP-lP + TaS^ + h\ 7? + 8x^y + 283fiy^ + 563^y^ + '70x*y* + 563^y^ + 28x2y« + 8xy-'-{y\ 625 + 2000a + 2400a2 + 1280a3 -i- 256a*. cxlix. (Page 301.) 1 . a^ - Ga^x + 15a'*x2 - 20a V + 1 5a2x* - 6ax^ + x^. 2. V - Wc + 216^02 - 356*c3 + 356V - 216-V + Ihc^ - c\ 3. 32x5 _ 240x*i/ + 720x31/2 - lO^Ox-y"^ + 810xi/* - 243i/5. 4. 1 - 1 Ox + 40x2 _ 80x3 + SOx* - 32x5. 5. l-10x + 45x2- 120x3 + 210x*- 252x5 + 210x6 -120x7 + 45x8- 10x9 + xi». 6. a24 _ 8a2i62 + 28ai86* - 56tti566 + 70ai268 - 56a96^o + 28a66i2_8o36u + 5i6^ 9 Cl. (Page 302.) 1 . a3 + 6a26 - 3a«c + 1 2o62 _ 1 2a6c + 3ac2 + 863 _ 1 262c + 66c2 - c^. 2. 1 - 6x + 21x2 - 443^ + 63x* - 54x5 + 27x6. 3. x9-3x« + 6x"-7x6 + 6x5-3x* + x3. 4. 27x4-54x^ + 63x3 + 44x2 + 21x3+6x^ + 1. 5. x3+3x2-5+?3--,. -' X- x^ 6i o^ + 6^ - c^ + Zah^ + 3a«6^ - 3a^ci - 36^ci + 2a^c^ i 1 111 + 365c^-6a*Z)*c*. 594 ANSWERS. Cli. (Page 303.) r. 330a;7. 2. ^^ha}%^. 3. - IGlTOOa^^^s^ 4. 192192a666c8d8. 5. 12870a86s. 6. TOa^ii 7. -92378ai069and92378a96i«. 8, I7l6a7a;6 aud iheaSx^. Clii. (Page 311.) 1 1 2 _L 3_ ^ 4 I. , l+^aJ-gX +^ga; ^28** , la a^ 4a^ i X a;2 5J.3 loa;* 3. «^ + -| — 5+ — I n- 3aS 9a5 Sla^ 243a^ 4. 1 + X - -X2 + -X^ - -X*. I 1 1 _& „ 5 -a , •^ 6 54 1 4 _i 1 2 -- i 4 6. «-^+5-«"-^*-25-*"^'-125' x^ X* x^ 5x^ 7- 2~8~16~ l28" _ , 7 „ 14 , 14 - 8. l-3a^ + ^«*-,-ia«. 9x 27x2 135 9- ■^-T~~32' 128-'^- 10. a=^-^y + 6^+5i^- ANSWERS. 395 ,5 __5^ , 35 6'^ 72"^ 1£96 /2\2 2 /3\i _i 3/3U 4 , Cliii. (Page 312.) 1 . 1 - 2a + 3a2 - 4a'' + 5rt^ 2. 1 + 3x + Qx^ + Tls? + 81:c* 5 „ 5 , , Zr? x^ 5x^ 5 . a''" + 10a-'' c + 60a-'*x- + 280a-'«./;3 + 1 1 20a-' V. , 1 , 6x^ 21.x^ 56x a'^ I I a^ Cliv. (Page 313.) , x2 3a;* 5a:6 35»8 2 8 16 128 3x2 15^ 353.6 315^ ^- ^^ 2""^ 8 "*" 16 "^"128 • 2 7 98 3. X ^X Z +^^X Z -—X . . j_ 3x2_5x3 35^ . 1 «' 3x* 5x6 "^^ . 2 2 ■*■ 8 • '• a 2^3--8,j5-i6^^r , 1 x^^ 2x6 143.9 ■ a"3a5"''9a^~81ai<>- Clv. (Page 3 U.) I. 7^;9-;) ,^1. 3. (-i)-.12^L^(M::r) ,- 1-^ ;••(»•-!) ^ ^ 1.2...(r-l) • • 3. (_i)^i. 8.7...( 10-r) ., 396 ANSJFEJiS. 4- i.V;;.y_iy • (5^)"- • (22/)-^ 5- ( - 1)--^ »• • x-^ 6. r.(r+l).(7- + 2) „ 1.3.5...(2r-3) /xX-^. 6 •^^''^ • 7. i.2.3...(r-l)-W 8. 1.2.5...(3r-7) / x\-\\ 1.2.3...(7--1) "V 3a/ • 9- 7.9.11...(2r + 3) 1.2.3... (r-1) •* • lO. a"2 3.7.11... (4r- 5) /^Y^-^^ 4'-^* 1.2. 3. ..(r-1) *\cJ • II. (r+l)(r+2) 1.3.5...(2r-l) 2 •''• '- 1.2.3..:r -^^ ^• 13- 1.3.5...(2r-l) 5 1 1.2.3...r •^^''^' '5. le-^i^. 1 6. 3 .,„ 429 xi« 128 •«^'- »7 -128-a^- 1 8. 1.2 9 •'' •^• 19. (1 - 5?7i) (1 - 4m) (1-m) l-« 1.2 6m6 •" clvi. (Page 315.) I. 3-14137.... 2. 1-95204.... 3. 3-04084.... 4. 1-98734.... Clvii. (Page 319.) I. 1045032. 2. 10070344. 3. 80451. 4. 31134. 5. 51117344. 6. 14332216. 7. 31450 and remainder 2, 8. 522256 and reinainder 1. 9. 4112. 10. 2437. ANSWERS. 397 clviii. (Page 321.) I. 5221. 2. 12232. 3. 2139e. 4. 104300. 5. 1110111001111. 6. atee. 7 6500145. 8. 211021. 9. 6^12. 10. 814. 11. 61415. 12. 123130. 13. 16430335. 14. 27^ I. -41. 4. 12232-20052. Clix. (Page 327.) 2. -162355043. 5. Senary. 3. 25-1. 6. Octonary. I. 1-2187180. 4. 4-740378. 7. 5-3790163. 10. 2-1241803. Clx. (Page 336.) 2. 7-7074922. 5. 2-924059. 8. 40-578098. II. 3-738827. 3. 2-4036784. 6. 3-724833. 9. 62-9905319. 12. 1-61514132 Clxi. (Page 339.) 1. 2-1072100 ; 2-0969100 ; 3-3979400. 2. 1-6989700; 3-6989700; 2-2922560. 3. -7781513 ; 1-4313639 ; 1-7323939 ; 2-7604226. 4. 1-7781513; 2-4771213; -0211893; 5-6354839. 5. 4-8750613; 1-4983106. 6. -3010300; 2-8061800; -2916000. 7. -6989700; r0969100; 3-3910733. 8. -2, 0, 2 : 1, 0, -1. 9. (I) 3. (2) 2. 9 10. a: = 5,i/ = ; ANSWERS. 11. (a) -3010300; 1-397940C; 1-9201233; 1-9979588. (6)103. 12. (a) -6989700; -G020600; 1-7118072; 1-9880618. (6) 8. 13. 3-8821260; 1-4093694; 3-7455326. 14. (i) x=r {2)x = 2. (3)a; = (4)a; = (5)x = (6)a; = 6' ^ / • V J/ Yog a + log 6" los c log a + 2 log h' 4 lo" 6 + log c 2 log c + log /) - 3 log a* log c J log ft + ??i lo^ 6 + 3 log c clxii. (Page 343.) I. 17-6 years. 2. 23-4 years. 3. 7 2725 years nearly. 4. 22-5 years nearly. 6. 12 years nearly. 7. 1 1-724 years, APPENDIX. The following papers are from those set at the llatrinulatioL Examinations of Toronto, Victoria, and McGiil LTniversi- t^es. and at the Examinations for Second Class Provincial Certificates for Ontario. UNIVERSITY OF TORONTO. Junior Matric., 1872 Pass. 1. Multiply ^x'-lxy + y^hjlid' + lxy-y'. Divide a* - 816* by a ± 36 and {x + af - {y - by by X + a — y + b. 2. What qnantity subtracted from x^ + px + q ''ril] make the remainder exactly divisible by a; — a .? Shew that (a + b + cf- {a+ b + c) (a' + b'' + c' -ab -be - ca) - 3abc = 3 (a + 6) (b + c) {c + a). 3. Solve the following equations : (a)^{2x-3) + Hi^x-7) = Ux-^). 4a; — 7 3a; — 5 (^)i.^r=Tn-^r=r2=20- i<^) X — 1 ^x — 2 11-11 4 X — 5 X — 6 2/+I y x + 2 11 (d) « + -2-=l' 3-^-5- =18- 4. In a certain constituency are 1,300 voters, ind two candidates, A and B. A is elected Vjy a tt APPENDIX. certain majority. But the election having been de- clared void, in the second contest {A and B being again the candidates), B is elected by a majority of 10 more than A's majority in the first election ; find the number of votes polled for each in the second election ; having given that, the number of votes polled ior B in the first case : number polled in the second case : • 43 • 44. Junior Matric, 1872. Pass and Honor. 1. Multiply a; + y + 2* - 2y^ zi + 2zi a^ - 2ar4 yi by X + y + zi + 2yi zi — 2zi x^ — 2x^ yi, and divide a' + 86' + 27 c^—18abc by a* + 45» + 9 c'— 2ab — Sac — 66c. 2. Investigivte a rule for finding the H. C. D. of two algebraical expressions. If X + c be the II. C. D. of af + px + q, and x* -f p' X + q', show that {q-q'Y-p {q-q) (p-p) + 9 (p-pY^^^- 3. Shew how to find the square root of a binomial, one oi' whose terms is rational and the other a quad- ratic surd. What is the condition that the result may be more simple than the indicated square root of the given binomial 1 Does the reasoning apply if one of the terms is imaginai y 1 Show that *y/ — 4m' = ^m + ^ -m. 4. Shew how to solve the quadratic aquation aa^ 4 6a; + c = 0, and discuss the results of giving difierent values to the coethcients. If the roots of the above equation be as p to 9 , 6» {p + qY show that — = -• ac APPENDIX. iB 6. Solve the equations xy +y»-10 = 0. (c) a:' + 6ic + 2 £c'+6x+6 a:*+6a + 4 x* + 6a; + 8 a* + 6 a; +10* * • (i) 6 x' - 5 af* - 38 ic* - 5 aj + 6 = 0. 6. Shew how to find the sum of w terms of a geometnp series. What is meant by the sum of an infinite series ? When can such a series be said to hav? • sum % Sum to infinity the series 1 -j- 2r + 3 r* -(- Ac. and find the series of which the sum of n terms i& aF — . a — \ 7. Find Che condition that the equations ax-\-hy — cz — ^. a, a; + 6, 3/ — Ci 2 = 0. e«, a; + 6, y — c, » = 0. may be satisfied by the same values of x, y, z. 8. A number of persons were engaged to do a })Iece of work which would have occupied them m hours if they had commenced at the same time ; instead of doing so, they commenced at equu! intervals, and then continued to work till the whole was finished, tne payments being proportional to the work done by each ; the first comer received r times as much as the last : find the time occupied. APPENDIX. Junior Matric, 1872. Honor. 1. There are three towns, A, B, and C ; the road fi'om B to A forming a right angle with that from B to C. A person travels a certain distance from B towards A, and then crosses by the nearest way to the road leading from C to A, and finds himself three miles from A and seven from C. Arriving at ^, he finds he has gone farther by one-fourth of the distance from B to C than he would have done had he not left the du-ect road.* Requii-ed the distance of B from A and C. 2. If ay -\r h x jx + a^ _bz + cy ^ ^^^^ ^^ c h a ay* a h e 3. Solve the equations x* — yz = a*, y' — zx-b*, «* — a:y = c\ 4. If a, h, and c be positive quantities, shew that a« (6+c) + 6» (c + «) + c« (a + 6) > %abc. 5. Find the values of x and y from the equations o 5?/ + 3 2y + -^ = 1, x ' £c* + 5x + 2/ (y - 1) = 24. 6. A steamer made the trip from St. John to Boston via Yarmouth in 33 hours ; on her return she made two miles an hour le.ss between Boston and Yarmouth, but resumed her former sjieed between the latter place and St. John, thereby making the entire return pas- sage in 11 of the time she would have required had her diminished speed lasted throughout ; had she made her usual time between Boston and Yarmouth, and two miles an hour less between Yarmouth and APPENDIX. T St. John, her return trip would have been made in i-J of the time she would have taken had the whole of her return trip been made at the diminished rate. Find the distance between St. John and Yarmouth and between the latter place and Boston. Junior Matric, Honor. 1 Senior Matric, Pass. ) 1. Solve the following equations : , . i x^- 2a;?/ + 2?/" ^ xj/ (a) .... I 1874. a? + xy + tf = 63. 4a;— ?)xy = 171. Zy-A:xy= 150. 1 1 1 a? xy y^ 1 1 1 o — . + — r-„+ -.= 133. X* u^y^ y" Am i find one solution of the equations {d) .. (&) y* — x*^ 68. a^ + sf x = y. 2. Find a number whose cube exceeds six times the next greater number by three. 3. Explain the meaning of the terms Highest com- mon measure and Lowest common multiple as applied to algebraical quantities, and prove the rule for finding the Highest common measure of two quantities. 4. Reduce to their lowest terms the following fractions : \a) . (6) i Sar* + i^^^^x — 10^ ' x' + lOx' + 35a;- + 50a; + 24 x' + IBa;^ + irj,/2 + 342.^- + 360 ▼1 APPENDIX. 5. Find the sum of n terms of the series — |, \, — \, (fee, and the ccth term of the series a; + 1 2 2> — x 6. Find the relations between the roots and co efficients of the equation ax* +;yx + ^ = 0. Solve the equation a;* + 6a;'+10x« + 3a;=110. 7. A cask contains 15 gallons of a mixture of wine and water, which is poured into a second cask con- taininif wine and water in the proportion of two of the former to one of the latter, and in the resulting mixture the wine and water are found to be equal. Had the quantity in the second cask originally been only one- half of what it was, the resulting mixture would have been in the proportion of seven of wine to eight of water. Find the quantity in the second cask. 8. What rate per cent, per annum, payable half- yearly, is equivalent to ten per cent, per annum, pay- able yearly. 9. A is engaged to do a piece of work and is tn receive $3 for every day he works, but is to forfeit one dollar for the first day he is al)sent, two for the second, three for the third, and so on. Sixteen davs elnps • l)efore he finishes the work and he receives §26. Find the number of days he is absent. Cliange the enunciation of this problem so as to apply to the negative solution. Junior .Vatric, 1876. Pass. 1. Explain the use of negative and fractional in- dices in Algebra. Multiply, -A by i/'«'' and the product bv V" APPENDIX. vii Simplify , writing the factors all in one line. 2. Multiply together a* + ax ■¥:!?, a + x, a*-ctx + 3^, a-x, and divide the product by a^ - ar*. 3. Divide 1 by 1 - 2x + x^ to six terms, and give the remainder. Also divide 27a;''-6x^ + ^ by Sa;^ + 2x + J. a, ^ n a»-« ■> + • 4. Multiply a +6 by a + 6 . 6. Solve the equations : (!)• 3a; + 4 7a; - 3 5 2 a;-16 4 • (2). ( X (y + z) = ■ly{z + x) = {z{x + y)-- .24, .45, = 49. ^ior Matric , 1876. Hnvn 1. An oarsman finds that during the firet half of the time of rowing over any course he rows at the rate of five miles an hour, and during the second half, at the rate of four and a half miles. His course is up and down a stream which flows at the rate ol three milus an ho\ir, and he finds that by going down the streani first, and up afterwards, it takes him one hour lojiu:er to go over the course than by going first up and then down. Find the length of the course. 2. Shew that if a^ 6^ tr" be in ^.P., then wilJ h - <■, « + rt, a + 6 be in II.P. Also, if a, 6, c be in A. P., then will he 7 ca ah a + , b + , cr - -^ c c + a a-^ It be in U.P. APPENDIX. 3. If s = ffl + 6 + c, then y/{as + be) (bs + ac) (cs + ah) = (s-a) (s — b) (s - c) 4. If a, + fta + +a„^ —, then (6- - «,)*+ + (s - a„)^ = a,2 + a/+ +a''«. 5. If the fraction - — - — - , when reduced to a re- 2n + 1 petend, contaijis 2n figures, shew how to infer the last n digits after obtaining the fii-st n. Find the value of -Jy by dividing to 8 digits, 6. Solve the equations X — y + z-S, xv + xz = '2 + 1/z, Junior Matric, 1876. Honor. 1. Shew that the method of finding the square loot of a number is analagous to that of finding the square root of an algebraic quantity. Fencing of given length is placed in the form of a rectangle, so as to inchnle the greatest possible area, which is found to be 10 acies. The shape of the field is then altered, but still remains a rectangle, and it is found that with 162 yards more fencing, the same area as before may be enclosed. Find the sidea of the latter rectangle. 2. Prove the rule for finding the Lowest C^ommon Multiple of two compound algebraic quantities. Find the L.O.M. of a» - 6=» + c' + 3a6c and d-(b-^c) -^{c + a)^^ (« + b)+abc. 3. If a, p be the roots of the equation 3^+px + g = 0, shew that the equ;ition may be thrown into the form (x — a) (x- f3) - 0. APPENDIX. Lx 3 + v/2 is a root of the equation aj* — 5a;' + 2a^ + a; f 7 = : find the other roots. 4. (1) Shew how to extract the square root of a binomial, one of whose terms is rational, and the other a quadratic surd. (2) Find a factor which will rationalize x^ — y^. 5. a, b are the first two terms of an H. P., what is the nth term ? JI a,l, che in. H. P., shew that h^ci - cf = 2c-{b - a)^+ 2a"(c - by. 6. A and B are to race from M to 'N and back. A moves at the rate of 10 miles an hour, and gets a start of 20 minutes. On A's returning from N, he meets B moving towards it, and one mile from it ; but A is oveiiaken by B when one mile from J\I. Find the distance from ]\I to N. 7. Solve the equations (1). ar' + 8-2a^+lla;+U. X 51 (2). V JC 12 a:y Second Class Certificates, 1873. 1. Multiply +_+iby f+--l. b a •' b a „ ^, , a'- 3ab + 26- a^ - lab + 1 26' 2. Shew that —. — „, 0.-26 a- ob ;an be reduced to the form 36, APPENDIX. 3, Reduce to its lowest terms the fraction, . b^ 1 "^ "*■ 12 + 9 ^ 1 4. (a) Prove that a^ - ?/" is divisible hj x-y with- ut remainder, when m is any positive integer. ih) Is there a remainder when a;""- 100 i>i ^ided by a; - 1 ] If so, write it down. 0. Given ax + by = 1, ,xy 1 and - + , = -^' a ah Find the difference between x and y. 6, Given 3 - n^^zM _ j'^ t! - ^x-\) 3(x+i) "• Find X in terms of wi, ^. X 2 ^ 7a; + 16 /. Given - =-5. Find the value of ;= n*' y 3 73^ + 24 -<. Given = 1, x-y X -¥y , 6 10 and 3. Find x and v. X ~y x-\-y ^ 9. There is a number of two digits. )^y inverting :.Iie digits we obtain a number which is less by 8 than hree times the original number ; but if we increase bach of the digits of the original number by unity, and invert the digits thus augmented, a number is obtained which exceeds the original number by 29. Find the number. 10. A student takes a certain number of minutes to walk from his residence to the Normal School. Were the distance ^th of a mile greater, he would need to incKMs. his pace (number of miles j^er hour) APPENDIX. by ^ of a mile in the hour, in order to reach the school in the same time. Find how much he would have to diminish his pace in order still to reach the school in exactly the same time, if the distance were ■^^ of a mile less than it is. Second Class Certificates, 1875. 1. Find the continued product of the expressions, a + b + c, c + a-6, b + c-a, a + b-c. a' + a^b a{a-b) 2ab 2. Simplify ^,^~ y - ^^^ - -^-^,- 3. Find the Lowest Common Multiple of 3a^ - 2a; - I and ^a?-'2.x'-^x+\. 4. Find the value of x from the equation, ax — a* — '6bx 6bx — 5a^ bx + 4 i - — ab^ -bx+ — K — — -f a 2a 4 5. Solve the simultaneous equations, 05 V -jrt. c d - +— = x y -n. 6. In the immediately preceding question, if a ['U})il should say that, when nb — md, and be ^- ad, the values of x and y obtained in the ordinary method, have the form f, and that he does not know Iio'a- io interpret such a result, what would you reply ? 7. Two travellers set out on a journey, one with %^ 00, the other with $48 ; they meet with robbers, who take from the first twire as much as they take from the second ; and wliat remains with the first is 5 times that which remains witii the second. How i;aMir'> mor^ey did each traveller lose '? APPENDIX. 8. A and B labor together on a piece of work for two days ; and tben B finishes the work by himself in 8 days ; but A, with half of the assistance that B coukl render, would have finished the work in 6 days. In what time | could each of them do the whole work alone % 9. P and Q are travelling along the same road in the same direction. At noon P, who goes at the rate of j?i miles an hour, is at a point A ; while Q, who goes at the rate of n miles in the hour, is at a point B, two miles in advance of A. When are they to- gether \ Has the answer a meaning when m — n is nega- tive 1 Has it a meaning when.m = ?i? If so, state what inter^jretation it must receive in these cases. 10. P is a number of two digits, x being the left hand digit and y the right. By inverting the digits, the number Q is obtained. Prove that 11 (a; + y) (P— Q) = 9(a;— 2/)(P + Q). Second Class Certificates, 1876. 1. Divide (1 + m) o^ — {m -¥n) xy {x — y) — {n — 1) y' by ar^— iy + 2/«. Shew that {a + a^U + hY—{a — aiM + 1)^ is ex- actly divisible by 2ai6i. 2. Resolve into factors x* + layii (z* — t^) — y*, 11^ {h — c) + b^{G — a) 4- c-{a — h), »ind 25a;* + bx^ — X — 1, 3. If x^ -hpci? -k- q.r + r is exactly di\'isible by a:' + mx + n. then nq — io^ = 7in. 4. Prov»^ that if m be a common measure of p and APPENDIX. xiii q, it will also measure the difierence of any multiples of p and q. Find the G. C. M. oi x*—^x^ ^ {(i—\)x^ + yx— q and x* — qoi? + {p — l)a^ + qx — p and of 1 + x^ + x-¥^ and 2x + 1x^ + 3a,-* + 3a;^' 5. Prove the rule for multiplication of fractions. a* — (y — zY 2/* — (2 — cc)^ «^ — {x — ?/)' Simplify (2/ + zf—zc' (z + a;)2— y* (x + yf a? and -5— Ts, — -^ — r2 + 6. Wliat is the distinction between an identity and au equation ? If a; — a = y + b, prove x — b = y + o. Solve the equations (2 -^x) (7/1 — 3) = — 4 — '2ii'x, 16a;— 13 40.x— 43 32a^-30 20:^24^ 7. What are simultaneous equations ? Explain why there m\xst be given as many independent equations as there aie unkno'vn quantities involved. If there is a gieatei- number of squations than unknown o'^.^.n- tities, what is the inlerence '( Eliminate x anci v irom xhe eor.atioiib ax ^ by = c, ax + bv = c , ax + U'y = <~. 8. Solve the equations — ( 1 ) y/n + x-^ 'Wn — X = m (2) 3a; + w + s=i;j 3'y + 2 + a;= lo 'iz -^ x + y - 17 9. A j)erson has two kintls of foreign money ; it takes a pieces of the first kind to make one £, and b pieces of the second kind : he is ofieied one £. for c pieces, Low many pieces of each kirul must he take 1 rfr APPENDIX. 10. A person starts to walk to a railway station four and arhalf miles olF, intending to arrive at a certain time ; but afiur walking a mile and a-liaif he is detained twenty minutes, in consequence of which he is obliged to walk a mile and a-half an hour faster in order to reach the station at the appointed time. Find at what pace he stai-ted. 11. {a) If y = ^ then will ^j^, = ^,. (6) Find by Homer's method of division the value of a*+ 290a;'+ 279ar'—2892j:*—586a>— 312 when a; = —289. («) Shew without actual multiplication that (a + 6 + cf — {a + 6 -^ c) (a* — ah + 6'— 6c + c*— ac) McGILL UNIVEE.SITY. First Year Exhibitions, 1873. 1. The clifFerence between the first and second oi four numbers in geometrical progression is 12, and the difference between the 3rd and 4th is 300 ; find them. 2. Find two numbers whose difference is 8, and t])e harmonica! mean between them 1|^. 3. Prove the general formula for finding the sum of an arithmetical series. 4. The diflerences between the hypotenuse and the two sides of a right-angled triangle are 3 and 6 fe.s]»ectively ; find the sides. 6. Solve the equations ce^ + 2/^ = 25 , x + y=l; X a; + 1 13 a; + 1 X ~ 6 ' x + y + 2; = 0, x + 2/ = s-t; x-7)=y \ z 03+4 3x + 8 +11= . 3a3 + 5 2a3 + 3 6. A cistern can be filled by two pipes in 24' and D '' respectively, and emptied by a third in 20' ; in what time would it be tilled, if all three were running together. 7. Shew that aj^ ^}? - ^ (a + b + c) (a + b-c) ^^ 2ah ~ 2ab tvi APPENDIX. 8. Prove the rule for finding the gi'eatest common measure of two quantities. First Year Exhibitions. 1874. 1. The sum of 15 terms of an arithmetic series is 600, and the common ditierence is 5 ; find the first term. 2. Find the last term and the sum to 7 terms of the series 1-4+16-&C. 3. Find the arithmetical, geometric, and harmonic means between 3| and 1^. 4. The difierence between the hypotenuse and each of the two sides of a right-angled triangle is 3 and G respectively ; find the sides. 5. The sum of the two digits of a certain number is six times their difierence, and the number itself exceeds six times their sum by 3 ; find it. 6. Solve the equations : — X- y = l; «'-?/'= 19 3.C - 7 4a;- 10 .^, X + x + b -^^' x-\{u-1) = b\ 4?/-i- (a;+10) = 3. 232:c+l 8^5 .^ 7. A man could reap a field by himself in 20 hours, but with his son's help for 6 hours, he could do it in 16 hours ; how long would the son be in reaping tho lield by himself? 8. Find the value in its simplest form of x + y 2a; x' y - x^ ^ y ~ x + y 3^y-y^ tvii APPENDIX. 9. Find the greatest common measure of Sar* -f 3a:^ - 15a; + 9 and 3a;^ + 3a;' - 21a;* — 9^. First Year Exhibitions, 1876. 1. Solve the equations 12a \a + a; + /rt — X = =-,— ^, \ \ O V ff + a; ■X y X y X y - ^ - = 1 _ -; _ f _= 1 +_. a b cab c 2. Reduce to its siuipiest form the expression ; — 7 V54 + 3 VFe + ^' 2 - 5 4/128. 3. Find the greatest common measure of 2x'-h^^ — Sx + 5 and 7x' — 1 2x+5. 4. Simplifjfc 5. A nuoaoer consists uf two digits, of which the lett is twice the right, and tlie sum of the digits is one-seventh of the number itself. Find the n;imber. 6. Solve the following : — X y X z 1/ z _ + 1^ + 1, - + - =2, - + - =3; a b 'I r. be 1 1 X y 7. Find the sum o\' n terms of the series 1, 3; 5, 7, &c. (a.) Shew that the reinprocals of the first four terms, and also of any consecutive four terms, are ir» harmonical proportion. tjNIVERSITY OP VICTORIA COLLEGE. Matriculation, 1873. 1. What is the " dimetision " of a term ? WLtn k ail expression said to be " homogeneous " ? 2. Remove the bx-ackets from, and simplify the following expression : — {•la — Zc + id) - \M ~ {m ^ Za)\ + |5a — {— 4 __j)|_ ^3a — {4a — 5rf — 4)|. 3. P'.ove the " E.ule ot Signs" in ISIultiplication- 4. INLultiply a — hjx + . ax 5. Di vide ax^ + bx'' -*- cx-\- dhy x — ♦•. 6. L^ivide 1 by 1 + a;. 7. ^'''nd the Greatest C mmon Measure of 6a* — aV — \'Ik and ya'' - \'2a^a^ — 6a-.r - Sx*. tt, i'rom 3a — 2c — - ~ _ subtract; 2a — x — x^ — 1 a — X (1+1^42^ 9. Civ en ■: ' ,> to find a; and v. I'l l>ivide tne iiui.i )er a into four such pai-ts that t!if M^coud shall SAceed the first by m, the thiid shall exi-i.(- 1 tilt; >^oooad oy n, ana the fourth shall exceed the third hv p. ) I. .V .sum o^ moi^e^ pat out at siiople intersd APPENDIX. xU amouxits in m months to a dollars, and in n monthi to h dollars. Required the sum and rate per cent. 12. Given a-' + a6 - 5x•^ to find the values of x. 13. Divide the number 49 into two sucn parts that the quutient ot the greater divided by the less niay be to the quotient of the less divided by the greater, as I to |. 14. L>ivide the number 100 into two such parts that thcii- product ma^ be equal to the dilierence of their squares. j" ar* 4 a;?/ — 56, ] 15. Given I >tofind\aluesofa;aud2/. l«y+2/-60j 16. A farmer bought a numbei of sheep for $80, and if he had bought four more for the same money, he would have paid %\ less for eu,ch. How many did he buy 1 Matriculation, 1874. 1. Find the Gieatest Common Pleasure of 26^ — iO'ib'' + 9>a% and 9^4^ — 2>a¥ + Zd^U" — 9rt '/?. rti.! de- moiistrate the rale. a* + 7^ n o* — (UB 2. Add tc'-ether a — x + , 3a — , =^ a + x a + x „ 3a* — 2af , . a- + x 2x — , and — 4a — ;. a — X a — XT 3. Divide + by — =-' 1+a; 1 — X 1 — X i+x- and reduce. 4. Given I (x — a) — lo {2x — 3b) — ^ (a — x) = 10a 4 116 to find x. 5. A sum of mone/ was divided ainoii!:^ tin ee per- sons, A, B, aiid C, at> follows : tJie slian of 1 ©.,.rewded 4 of the sha.-es of h and C >\v §1J0: th^ APPENDIX. share of B, f of the shares of A and V)y 81-0 ; and the share of C, | of the shares of A and B by ^120. What was each person's share? 6. Given | ^3 ^ ]f^_'^^_4^ . 12 | ^^ ^^-^ * ^'^'^ 2/- 7. Shew that a quadratic equation of one unknown quantity cannot hav^e more than two loots. 8. Given ~ .— — ; to find the vahie of x. 4 + V a; >/x 9. The e is a stank of hay whose len-fth is to its breadth ab 5 to 4, and whose height is to its Vueadth as 7 to 8. It is wotth as mai y cents per mibic foot as it is feet in breflcith; and the whole is worth at that rate 224 times as many cents as bhere art square teet on the bottom. Fjnd the cdu^ensions of the stack. y/ocy + b \ 10- Given ^ /" V to find x and y. J^ = y/xy — 4 I x->ry / 11. In attempting to arrange a number of countei-s in the form of a square it was found there wer«t se\ en over, and when the side of the square was incie:ised oy one, there was a deficiency of 8 to complete the square. Find the number of countere. 12. Reduce to its simplest form g' — (6 — cf ^ b^ — {c — ay _^ c» — {(i-^b)* (a + c)« — 6» (a + by — c» (6 + c)' — a"' 13. A and B ciin do a piece of work in 12 days; in hew many days could each do it alone, if it would lake A 1 days longer than B 1 y w I ^ ^^ 1 4. Given ) x— y = A ) x, y, z, \ X* + 1/ -^z' + w' = 62^ APPENDIX. larf 15. Find the last term, and the sum of 50 terms, of the series 2, 4, 6, 8, ifec. ( M ' IG. Writo down the expansion of {x — - > 17. How many u..ut»ir)nc swains may be rung on tea different bells, supposing all the combinations to produce diffe-ent notA" ' ANSWERS- Junior Matric l87'J. I'af^s. {x + ay + {x + a) {y-b) + {y-bY. 2. a^^ap + q 3. (a), 1^; {b), U; (c), 4^; (J), ^ ^. 4. 640, 660. Junior Maine. , 1872. jPcws a/w? Honor, 1. I s-i + (.rJ - yi) I ' I 2i _ (xJ — .vi) I ' = I «j _ (a^ , .,*)» I '; ,, + 2b + 3c. 2. We have c* — ;?ci- 5- = aiid c* — ^'c + g-' = 0, fron? vhicli to elim-Date c. 4. If /3 be one root, - - _ fi ^1 + -Y '' = i8'-^ aad, eliminating j, — = ^ — 2Z . ac pg 6. (a), 4, — 7, ^(— 3± v/277) ; (6), 3, 2, ; — 3,— 3 — _-, -L;- -i, - ~ (c-),-3 ^_ /2. ((f), Divide tL rough bv x^ and put y for jc+ -, and :.y^ — 2 for a:^+ — , then y = _ or — - and a; = 3. i, — 4 or — 3. 3 2 ' * ANSWERS. xxUi 6. ; >a 4-a -J-a -f- ..-. V «• rrr- Junior Mdtric, 1872. Honor. L 8 ami 6 miles. 2. Each of the first set ol' fractions may be shewn equal to X y labc « or 2ahc h or 2ahc b' + c'-a* c' + a^'-b'' s , which are therefore equaL a' + b'-c^ ^ 3. Multiplying the equations successively by y, z, a and 2, X, y, we obtain c^x + a^y + h^z = 0, b'x + c'l/ + a'z - ; thence —. — j^, = . . ^ . , = a —o'c —car and X — 4. «* + 6^>2a6,.-.c(a» + 6^)>2rt6c, &c. 5. 3,0;- 2,- 5; -3, 6; -8, 1. 6. 00 and 2-tO mla Junior Matric, Uonor. \ .„_ Senior Matric, Pass. 1. (a), From first x— 'ly or .//, and then solutions are 3, 3; -3,— -3^y2r, v/21; --V2l,— V2f: (6).>e(41drs/7"69^H-37±/7Gy). (c). i.i; -i.-i; ii;-i>-i ('0, 4,18. 2. 3. , , , 33x-«+61a;+ 10 ,,, a;^ + 3x + 2 ^ '' a: + 2 ^ T^ + llaj + 30 ixiv ANSWERS. _ X (3— x) 6. a; - 2 and a; + 5 are factors, and roots are, 2, — 5, H-3±^/35): 7. 7^ gals. 8. 4.88 percent. 9. I days. He receives $3 every day the work continues ; he returns nothing the first day he is idle, $1 the second, and so on, and the number of days he works is IG. Junior Matric, 1876. Pas$. 1. a' ; a""'' 6""' c~V 2. a« - cc» ; a'+^, 3. 1 + 2a; + 3ar^ 4- 4a^ + 5a;* + Gar* + ; rem. 7a:^- Gx'. 90.-^ — 6a;+l. 4. a^+^ab)^' '' -^ (ab) + b'^'. 5. (1), 2. (2), 2, 5, 7; or -2,-5,-7. JiMiior Matric., 1876. Honor. 1. 35 mis. 2. (2), These quantities are in H. P. if ,&c., are in A. P., i.e., if a, 6, c ah + ac + be are in .4. P. 5. It may be shewn that the remainder at the nth decimal place is 2n ; hence if the nth digit be increased by unity, and the whole subtracted from 1, the remainder is the remaining part nf the period. 6. a = 4,a; = 2or-3^=3or-2;a--l,« = 2*^ro; y--2-.7l0. ANSWERS. xxf Junior Matric, 1876. Honor, 1. 121 and 400 yards. 2. (a — h + c) {ah + 6c -f ca) (a* + 6* + c* + rtb + be — ca) 3. Iri4,tionaI roots go in pairs/. 3 — i/iT is a rooi , and other roots a"e ^ (-—1 zbyZIg). S 13 2 la. 4. X- + le'y^ + ic-_y^ + .ry -f- xiy^ + j/^. •"»■ i— 7 ^T-7 A\- ^- ^ "'^^• 0+ (?^ — 1) (a — b) 1. (1), Plainly x + 2 divides both sides, and roots are— 2,24- /f. {-2), x= 3, >/ ^ i cv I ; x = — 3, y- — 4 or — i. Second Class Certificates, 1873. ^- Kb^a)-^=b^^^-a^- 2. {a-b)-{a-\b) = 2,b. 5. ('/. — t*) (a; — y) = ; .*. if a be not = 6, x -y — ; it' a — b,x — y may have any value. 4 3-1 4to, , , . , , , 6. , , o- '« • n, provided x be not = - 23 ; 1 i//i- 1 ) "* ^ ' ' then fraction becomes § and is indetei minate. o 1 1 x-y ' a; + / ^ ' ' -^ 9. 13 10. ^ of a mile per hour. «xvi ANSWERS. Second Class Certificates, IS 7f 1. 2{a'h' + hh- + c^a*) - (a^ + b' + c'). 2- -\ a f o 3. (3x + \){\x^- 2^- 3x+l). 4. 2a(26-^ - 5) 4a - o6 5. he — ad ^6c- -a^ 7?ic — na 6. X and y are indeterminate : there Ls but one equation. 7. 8SS, .$44. 8. 1 4 Jays, 1 1 g Jays. 2 9- ^i^ hrs. m — n negative means that thev m — n ° 2 were together hi-s. before noon, m — n, n — n they are neve»^' together. 10. Each side equals 99(x-^ — y-). Second Class Certificates, 187C. 1. {\+m)x-{l-n)ij. 2. (x + y)'(.r-y); {a-h) {b-c) {c-a); (S.yr-l) (5ar' + x-r 1). 3. Let the other factor be x-¥ a; multijjly and eijuato co-etiicients ; eliminating a, nq — 'n- - na; other condition in pn — mn — r. 4. x— 1; 1 ^ -t». g { x + y-z) {x-y + z) {>/ + z- x) . _1__ [x + y + zf a - b 6. -§; 1. 7. a'ih'c - be) + b''[^ac' — dr) + c'{a'h — ah) = 0. 8. (1,) Cube, and 3(« + x)i (>i-x)i (n») = m* - ^jj, q a (c — 6) 6 (a — c) a — 6 a — i 10. 3 miles an hour. AA.i'^l^'ER^* xxvii 11. (a), See §359. {b), 2,000. (c), Snnstitute suc- cessively — b, —c, —a for a, b, c, in tho left hand side, and it appears that a + b, b + c, c + a are factors, and /. expression is of form N{a + b) (6 + c) {c + a); putting a-b-c- 1 , we get iV'= 3. First Year Exhibit/ions, 1873. 1.3,15,75,375. 2. 9 and 1, or V^ and- H- 4.9,12. 5. (a), 4, -3; -3,4. (6),2,-3. (c), 4,-5, 6. {d),-\ 6.40'. l.J^l±^Ll^ = . 2ab Firxf. Year Exhibitions, 1874. 1. 5. 2. {—if; 3277. 3. 2^; 1\; 2^^. 4. 9, 12. 5. 75. 6. (a),3,2;— 2,— 3. (6), 7 or— If (c),5,3. {d),U. 7. 30 hours. 8. JL. 9. 3(a; + 3). x + y First Year Exhibitions, 1876. 11111 1 1 4 3a 6 c a b c b ' b ' \ 1 1 ' 1 1 ,1 a« ^^ 7 «=" 6"* c* 2. — 12 '^/2. 3. X — 1. 4. m. 5. 21, 42, 63, or 81. 6. o, b, 2c; 1, 1 ixviii ANSWERS. Matriculation, 1873. 2. \\a — 3c — 5(Z + m. 4. — ax. 5. ax' + (ar + 6) aj -|- (a/ ' + br + c) -f ar' + 6a^ Jfcr + d X — r C. l—x¥ x'— x^ ^ .. .. 7. 3ft' + 4a--*. 8. (^t — .r) {x'—2)^ g 144, 216. x-^ — 1 10. \ {a — 2>i)i — "In — ^j), lire. ^^ vib~na 1200 (a — 6) ?u — ?i mh — na 12. drJv'^- 13. 28, 21. U. 50 {VI —1), 50 (3-/5). 15. x= zfc 10, ^= =F10; x = ±W2, y ^ ^ 3»/2 16. 16. Matriculation, 1874. 1 1 o 4a* + a'x — 2rt;c' + ar* • . 1. a— 6. 2. ;^ 3w 1. cc* — a^ 4. _ 5a — 36. 5. 600, 480, 360. 6. 2, 4 ; 4, 2. 8. 4 or 9f 9.20,16,14ft. 10. 40, 10; 10, 40. 11.56. 12. 1. ' 13. 30 and 20 days. 14. 6, 2, 41, U,or-2, — 6,— 1^, — 4|. 15. 100, 2550. 16. x^ — 7a^ + 21af'— 35.r + 35a:-' — Sla:"* + 7«-» — x-\ 17. 1023. cE. 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They are arranged in progressive lessons in such a manner as to be available with almost any text book of E iglis"!! Grammar, and take the learner by easy stages from the simplest English work to the most difficult constructions in the language. Price, 30 Cents. Outlines of Englisn Grammar. These elementary ideas are reduced to regular form oy means of careful dcilnitions and plain rules, illustrated by abundant and varied examples for practice. The learner is made acquainted, in moderate measure, with the most important of the older forms of English, with the way in which words are constructed, and with the elements of which modern English is made up. Analysis is treated so far as to give the power of dealing with sen- tences of plain construction and moderate dificulty In the English Grammar the same subjects are presented with much greater fulness, aiiu carried to a more advanced and ditlicult stage. The work contains ample materials for the requirements of Competitive Examinalions reai^hmg at least the standard of the Matriculation Examination of the University of London The Shorter English Grammar. is intended for learners who have but a limited amount of time at their dis- posal for English studies ; but the experience of schools in which it has been the only English Grammar used, has shown that, when well mastered, this work also is sullicient for the London JIatriculation Examination. (lE. J. (Sage S: €o'e. |lctD Cbucatioual gMorks. THE BEST ELEMENTARY AND GRAMMAR COMPOSITION. Revised Ed. of Miller's Language Lessons. Now in iiidcsti-uctible iron biiidini^-. Sixth edition; 2C0th thousand, with Exaniiaation Papers for admission to High Schools. Adapted as an Intro- ductory Text Book to Mason's Grammar. (a TIIOROIGII EXAMIKA7I0S GIVEN). St. Thomas, Nov. 30th, 1878. To the Trosident and Members of tlie County of Elj^in Teacher's .Associa- tion : — In adcordance with a motion pas.sed at tlie last regular mectin<j of the Association, appointing the undersigned a Committee to consider the respective merits of different English Grammars,with a view to suggest the most suitable one for Public Schools, wc beg leave to report, that, after ful" ly comjiaiing the various editions that have been recommended, we believe that " Miller's Swinton's Language Lessons " is l)est .adapted to the wants of junior pupils, and we would urge its authorization on the Goverimient, and its introduction into our Public Schools. Signed, A. F. Bitler, Co. Inspector, .1. McLean, Town Inspector. J. Millar, M. A., Head M.aster St. Thomas High School. A. Steele, M. A., " Orangeville High SchooL N. Campbell, " Co. of Elgin Model School. It was moved and seconded that the report be received and adopted.— Carried unanimously. i^" TO AVOID MISTAKES, ASK FOR REVISED EDITION MILLER'S SWINTON'S. PROOFS OF THE SlTEiUORITT C)F MILLERS EDITION' Miller's Swinton's is authorized by the Education Department for use in the Schools of Ontario. Only Edition adopted by the Protestant Board of Education of Montreal' and used in many of the principal Schools of the Province of Queijcc. Only Edition used in the Schools of Newfoundland. Only Edition adopted by the Su,ii of Education for the Schools of Manitoba- Miller's Revised Swinton's is used in nine-tenths of the principal Schools of Ontario. Only Edition prepared as an introductory Book to Mason's Granmiar both haying the same Definitions. J. ^agc ^ €00. S-^W ©bucational cMorks. I EXAMINATION SERIES. Canadian History. Bv James L. Hughes, Inspector of Public Schools, Toronto. Price, 25 Cents. HISTORY TAUGHT BY TOPICAL METHOD. A PRIMER IN CANADIAN HISTORY, FOR SCHOOLS AND STUDENTS PREPARING FOR EXAMINATIONS. 1. The historj- is di\idcd into periods in accordance with the great na- tional changes that have taken place. 2. The history of each period is given topically initead,of in chronolog- ical order. 3. Examination questions are given at the end of each chapter. 4. Examination papers, selected from the official examinations of the different provinces, are given in the Appendix. 5. Student's review outlines, to enable a student to thoroughly test his own progress, are inserted at the end of each chapter. 6. Special attention is paid to the educational, social and commercial progress of the country. 7. Constitutional growth is treated in a brief but comprehensive exer- cise. SS" By the aid of this work students can prepare and review for exam- inations in Canadian History more quickly than by the use of any other work. Epoch Primer of English History. By Rev M. Creighton, M. A., Late Fellow and Tutor of Merton College, Oxford. Authorized by the Education Department for use in Public Schools, and foi admission to the High Schools of Ontario. Its adaptability to Public School use over all other School Histories will be shown by the fact that — In a brief compass of one hundred and eighty pages it covers all the work required for pupils preparing for entrance to High Schools. The price is less than one-half that of the other authorized histories. In using the other Histories, pupils are compelled to read nearly thi-ce times as much in order to secure the same results. Creighton's Epoch Primer has been adopted by the Toronto School Board, and many of the principal Public Sclxjul* in Ontario. J. QSagc ■& €o's Jlcto CSbunttional (iilorks. Authorized for use in the Schools of Ontario. The Epoch Primer of English History. By Rev. M. Creigiito.n, il. A., Late Fellow and Tutor of Mcrton College, O-Mord. Sixth Edition, - - - P rice, 30 Cents, Most thorough. Aberdeen Joi-rsal. This volume, taken with the eight small volumes cojitaining the ac- counts of the different epochs, presents what may be regarded as the most thorough course of elementary English History ever published. What was needed. Toro.sto Daily Gloeb. It is just such a manual as i^s needed by public school pupils who are going up for a IJiigh School ju.tc. Used in separate schoo 3. M. Stafford, PRiPiST. We are using this History m our Convent and .Separate Schools in Lind- say. Very concise. Hamilton TrMEs. A very concise little book that should be used in the Schools. In its pages will be found incidents of English History from A. D. 43 to 1870, in" t^rcstitig alike to young and old. A favorite. London Advertiser. The book will prove a favorite with teachers preparing pupils for the entrance examinations to the High Schools. Very attractive. Rritisii Wiiio, Kingston. This little book, of one hundred and fort}' pages, presents history in a very attractive shape. Wisely arranged. Canada Pre.sbvterian. The epochs chosen for the division of English History are well marked —no mere artificial milestones, arbitrarily erected by the author, but reaj natural landmarks, consisting of great and important events or remarkable changes. Interesting. YARMorTn Tribune, Nova Scotu. With a perfect freedom from all looseness of style the interest is sn 'tH sustained throughout the narrati\c that those who commence t^. will tiiul it difficult to leave off with its perusal incomplete. •'' Comprehensive. Litkrart World. The special value of this historical outline is that it gives the reader a comprehonsjve view of the course of memorable events and epochs, M. 3. (gage ^ Co'e. ^cto Cbitrational cHorks. ; THE BEST ELEMENTARY TEXT-BOOK OF THE YEAR. ' Gage's Practical Speller. , A MANUAL OF SPELLING AND DICTATION. 1 Price, 30 Cents. ' Sixty copies ordered. Molxt Forest .Advocate, i After careful inspect on we unhesitatingly pronounce it the best spell- ' ing book ever in use in our public schools. The Practical Speller secures ■ an easy access to its contents by the very systematic arrangements of the wDrds in topical classes ; a permanent impression on the memory bj' the frequent review of difficult words ; and a saving of time and eflort by the selection of only such words as are difficult and of connnon occurrence Mr. Reid, H. S. Master heartily reconunends the work, and ordered some sixty copies. It is a book that should be on every business man's table as well as in the school room. Is a necessity. Presb. Witness, Halifa.x. We have already had repeated occasion to speak highly of the Educa- tional Series of which this book is one. The "Speller" is a necessity ; and we have seen no book which we can recommend more heartily than the one before us. Good print. Bowmasville Observer. The " Practical Speller " is a credit to the publishers in its general get 1 up, classification of subjects, and clearness of treatment. The child wh« uses this hook will not have damaged eyesight through bad print. I o I What it is. Strathrov Age. It is a series of graded lessons, -lontaining the words in general use, with abbreviations, etc. ; words of similar pronunciation and different spell- ing a collection of the most difficult words in the language, and a number of literary selections which may be used for dictation lessons, and commit" ted to memorj' by the pupiis B\«ery teacher should introduce it. Canadian Statesman. It is an improvement on the old spelling book. Every teacher should | rntroduce it into his classes | The best yet seen. Colchester Scn, Nova Scotia. Itis away ahead of any"speller"that we have heretofore seen. Our public I schools want a good spelling book. The publication Ijcfore us is the best j 1 je ha^ e \ et seen. J. ©age & QTo.s' Jlciu (giincattoniH SBorkg. Gage's Practical Speller. A new Manual of Siielling and Dictation. PriO«, 30 Cents Pl:il.MI.MS.NT Fi^ATuan The book is divided into five parts as follows : PART 1 Coiitaitu the words in common use in laily life tog-ether with ahbrevia- tions, foinn, etc. if a boy has to leave school early, he should at least know how to spell the uurds of couuuoii occurrence in connection with his business. PART fl. Gives words liable to be spelled incorrectly beiauM the same sounds are spelled in various ways in them. PART III. Contains words pronounced alike but spelled differently with diticre'it meanings. PART IT. Contains • larj;e collection of the most difKcult words in common use, and is intended to supply material for a general review, and for spelling matches and tests. PART V. Contains literary selections which are Intended to be memorized and re' cited as well as used for dictation lesson* and lessons in mtiraU. DICTATION LESSONS. All the lessons are gultable for dictation lessons on the slate or in dicta- tion book. RRVIBWa. These will be found throughout the book. An excellent compendium. Alex.McRae.Prin. Jr.ad'ii.Dujby.N.S. I rcg-ard it as a necessity and an excellent com] cmiium of the subject of which it treats. Its natural and judicious anaii-ement wi.^1 accords with its title. I'uinls instructed in its principles, umier the care of dilijrent teachers, cannot fail to become correct spellers. It ^reat \alue will, doubt- less, secure for it a wide circulation. I liave seen no liook on the subject which I can more cordially recommend than "Tliu Practical Speller." Supply a want long felt. J'lm J"hj,^t,-n, l.P.S., Belleville. Tne hints for teaching spelling are exiellent. ! bave shown it to a num- ber of experienced teachers, and they all think it is the best and most prac- tical work on spellini^ and dictation ever presented to tlif public. It will supply a want long felt by teachers. Admirably adapted. Colin rr. liosene, l.P.S., Wol/vHU, A', s. The arr».njren\ent an. I :,'rading of the different classes of woixls I reirard as excellent. Mucli benefit must arise from cnnnnittiii!,' to memory the "Literary Selections." The work is admirably adapted to our public schools, and 1 shall recommend it as the best I ha\ e seen. cE. J. (Sage ^ QLo's. <^eU) (gbucational eMorks THE BEST ELEMENTARY TEXT-BOOK OF THE YEAR. : Gage's Practical Speller. I A MANUAL OF SPELLING AND DICTATION. ' Price, 30 Cents. ! : Sixty copies ordered. Mount Forest Advocate. ■ I After careful inspect on we unhesitatingly pronounce it the best spell- j I ing book ever in use in our public schools. The Practical Speller secures : I an easy access to its contents by the very systematic arrangements of the ! words in topical classes ; a permanent impression on the memory by the ( frequetit review of difficult words ; and a sa\i!ig of time and effort by the 1 selection of only such words as are difficult and of common occurrence | I Mr. Reid, H. S, Master heartily recommends the work, and ordered some | I sixty copies. It is a book that should be on every business man's table as 1 i well as in the school room. Is a necessity. Presb. Witxes.s, Halifax. We ha\e already had repeated occasion to speak highly of the Educa- tional Series of which this book is one. The "Speller" is a necessity ; and 1 we have seen no book which we can recommend more heartily than the one ' liefore us. ! o • ( Good print. Bow.maxville Ob.server. ] The " Practical Speller" is a credit to the publishers in its general get ! ".p, classification of subjects, and clearness of treatment. The child whe uses this book will not have damaged eyesight through bad print. i What it is. Strathrot Age. It is a series of graded lessons, containing the words in general use, i ■vith abbreviations, etc. ; words of similar proiumciation and different spell- ing a collection of the most difficult words in the language, and a number of literary selections which may be used for dictation lessons, and commit" ted to memory by the pupils. | Every teacher should introduce it. Caxadian- Statesman. } It is an improvement on the old spelling book. Every teacher should 1 introduce it into his classes The best yet seen. Colchester St7<, Nova Scotia. It is away ahead of any"speller"that we have heretofore seen. Our public schools want a good spelling book. The publicaticTt \efore us is the best we have vet seen. 8E. S- ®«9^ * ^'^- <^^to (Ebucational SRorke. WORKS FOR TEACHERS AND STUDENTS, BY JAS. L. HUGHES. Examination Primer in Canadian History. On the Topical Method. By Jas. L. HruiiK.s, iti^iwctor of Schools, To. rooto. A Primer for Students preparing for Examination. Price, 25c Mistakes in Teaching. By Jaj. Lauoiilin IIooiies. Second edition. Price, 50c. AIK>PTKS B7 8TATR UNIVRRaiTT Of IOWA, AS A.S BI.EMKNTART WORg FOR 081 OF TKACH8R8. This work discuascf) in a terse manner over one hundred of the mistakes commonly made by untrained or inexiierienced Teachci-s. It is Jesi;;fied to warn young Teachers of the errors they are liable to make, and to help the oideT raembiii-!^ of the profession to discard whatever methods or habits may be preventing their higher success. The niistAkes are arranged under the following head* : 1. Mistakes in Management. 2. Mistakes in Discipliiie. 1. Mistakes in Methods. 4. Mistakes in Manner. How to Secure and Retain Attention. By Ja8. LAuanLiN Huoiibs. Price. 25 Cente. Comprising Kinds of Attention. CharActcristlcs of Positive .\ttention! CSiaracteristics of The Teacher, flow to Control a Class. Developing Men t«l Activity. Cultivation of the Senses. rFrom The School an'd U.vtvkrsitt Maqazlsb, Loyrios, Eno.) "ReiMete «-ith valuable hints and practiail suggestions which are evident- ly the result of wide experience in the scholastic profession " Manual of Drill and Calisthenics for use in Schools. By J. L. HoonBS, Public .School Inspector, Toronto, Graduate of Militar\ School, H. M. 2dth Regiment. Price. 40 Cents. The work contains : Ttie Squad Drill prescribed for Public Schools In On tario, with full and explicit directions for teaching it. Free Gymnastio Ex- ercises, carefully selected from the best Gennan and .American sj" -.-uns, and arranged in proper classes. Gennan Caliethenic Exercises, as *»ught by the late Colonel Goodwin in Toronto .N'omial .'^ohl>ol, and in Enicland. Several of the l>est Kinderirarten Games, atid a few choice Exercise Songs. The instructions throughout the book are divested, as far as possible, of Jnneoessarj- technicalities. "A most valuable book for every teacher, i«rtlcii!arly In oonntry placer It emliraccs all that a school teacher should teach his pupils on this subject. Any teacher can use the e&ry drill lessons, and by doing so he will be con- ferring a benefit on his country."— C. Kadcliffr PRAU.tALT, Ucy "if Mrct Life Guards, Orill Instractor .\ona»l a-xl Model School*. Toronto KIRKLAND k SCOTT'S EXAMINATION PAPERS. Suitalsle for Intermediate Examinations. BEPRINTBD FROM GAGE'S SCHOOL EXAMINER AND STUDENTS ASSISTANT FOR 1881. COMPILED BT Thohab Eibeland, M.A., Science Master, Normal School, Mid Wm. Scott, B. a., Head Master, Provincial Model School. PRICE, - SO CKNTS. Thit vohnne contains papers on ArUhmetie, Euclid, Geography, Algebra, Book-keevitui, History, Statics and Hydrostaiies, English liiUrtHure, French (July, 18S0), Chemistry, English Orammar. FROM THE PREFACE. In reapoDM to the desire of a larxt: numt>er of Teachers, we reprint the KxuninatioD Papers suitable foi the Annual Intermediate Ex fcmination, which have appeared in the number*, for 1881, ot Gage's "School Kxaiiiiner and Student's Assistant." The steadily tncreastng circulation of this monthly magazine, and the numerous letters received testifying to the great value of the questions in the vmHoos cubjeots reouired for the Examinations, plainly indicate that sach k periodical U a mo«t oaeful aid to both MAcher and student. The almost exhsastire nature of the questions on each subject bringv the student Into clot^ acquaintance with every needful point; and the drill experienced in thinlving and working out the answers is of Incalculable practical benefit to those who wish to exo«l at written examiDatloua. When we state that the editors of this department ot tiM SeJwol Examiner are Messrs. T. Sirkland, M.A., and W. Scott, B.A., we con- sider tiiat It is a suihcient guarantee for the excellenos and appro- priateness of the work, 04 these gentlemen have eam«d a wide reputa- tion as specialists in science and hterature. lo consequence ot numerous applications (or the PrwKsh Paper given at the Intermediate Elxamination, 1880, w« reprodoo* it in this book. Hints and Answers to the Above, 50 Cents. W. J. OAGE <& CO.. t. (OTBK, Vm. J. 6iigc & doQ. Ilclu €butational eMorks. The Canada Schooi Journal HAS RKXEIVED AN HONORABLE MENTIOM AT PARIS EXHIBITION, 1878 Adopted by nearly every County in Canada. Reconniiended by the Ministe of Education, Ontario. Recoiiiniended by the Council of Public Instruction, Quebec. Recomniendfd by Chief Supt of Education, New Brunswick. Rcccniniendcd by Chief Supt. of Education, Nova Scotia. Reconinicnded liy Chief Supt. of Education, British Columbia. Recommended by Chief Supt. of Education, Manitoba. IT IS EDITED BY A Committee of some of the Leading Educationists in Ontario, assisted by able Provircial Editors in the Provinces of Quebec, Nova Scotia; New Brunswick, Prnce Edward Island, Manitoba, and British Columbia, thus having each s«H;tion of the UomiiiioD fully represented. CONTAINS TWENTY-FOUR PAGES OP READING MATTER. Live Editorials; Contributions on important Educationa! topics; Selec- tions- Readings for the School Room ; and Notes and News from eacb Pro- vince. I PiiACTicAi, Department will always contain useful hints on methods of teaching different subjects. MAniBMATicAL Departmbnt gives solutions to difBcult problems also on Examination Papers Official Department contains sueh regulations as may be Issued trom time to time Sulscription. 81 00 per annum, strictly in advance. Read TUB F:i!,i,"u ■ i i'tter frov John Oreenleaf Whittibb, thb Fa- ucis AMI.RH-'AN I'ul.l I b"^ve also rt'cei\til a .No of the •' Canada School Journal,' which seems to me the brightest and most readable of Educational ilagannes I am very truly thy friend, John Greenleaf Whittier. A Club of I.Ojo Subscribers from Nova Scotia. (Copy) Edccation Office, Halifax, N S . Nov. 17, 1878. Messrs. Adam Miller & Co., Toronto, Ont Dear Sirs,— In order to meei, the wishes of our teachers in various parts j of the Province, and to secure for them the adxantajje ot vcur excellent periodical, i hereby subscribe in their behalf for one thousaii'' (1,000) ccpits > at club rates mentioned in .\our recent esteemed fa\or subscription!-- -vil, begin with January issue, and lists will be forwarded to your office in a li» days. Yours truly, David Allison, Chief Supt. of Education. Address. W. J. GAGE & CO., Toronto, Canada. i