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Millkb, in the oflftce of the Registrar of the Province of Canada. ; ■f-, .■'^, Y. % 4. PREFACE. ^«»^^^^^^^^v^^^»^^^>^^ xlia- and 'the The object of this Elementary Work is not to. displace any of the valuable Treatises on Algebra generally used in schools, nor does it assume to rank with ^be||^ 'It IS intended simply as an introduce tion to tn# study of this most interesting science, and as a first book so to initiate the pupil that he may in a very short space of time enter upon the most complete and advanced text-books on the subject, undeierred by any apprehensions of great difficulties to be encountered. The scholar who has duly attended to his instrac< tion in Arithmetic will find that there is nothing difficult to comprehend in the princii)le8 of Algebra ; he will see that there is notMng occult to maater, f if^al^g^-^-^^i -'-•'% ■"^p^ ^T'"^^-^ it PBBVAOl. but that his arithmotioal knowledgo may bo appliod and exercised upon a study of progressive interest and satisfaction. The Author has endeavored to make his Treatise as far as it extended demonstrative, and thus to al)breviate the teacher's labor in explanation, as well as to fix the mind of the pupil on the principles upon which algelforaio rules are founded. With so humble an object in view as here indi- cated, it would be out of place to enlarge upon the benefits to be 'derived from the study of Algebra, but those fo\whom the education of youth is en- trusted, conversant of these benefits, will hardly fail to welcome a book having for its object initia- tion and guidance, if it be found, to answer its Montreal, July 1, 1862. --i(*t- --i(*t CONTENTS. t CHATTEBI. TFaoi Definitions, ^ 7 CHAFTEB II. Addition and Subtraction, 15 CHAFTEB ni. Multiplioation an^ Division, 23 CHAPTER IV. Qreatest Oommon Measure and Least Oommon Multiple, i 84 CHAPTER V. Fractions, ........./... 89 CHAPTER VI. Involution and Evolution, 46 CHAPTER Vn. Simple Equation^, 04 CHAPTER Vin. ' Problems producin|^€imple Equations, 66 CHAPTER IX. 4, Quadratic Equations, ,73 Answers to the ExetciseSy* •......•....«.. .^ ,. ... 7t j&^ . ■ ■•■;■.■ H*' *..t •- ..-, • '" • .< t 1 ■ t ■ . . ^ . ] , '... .'■ ---..■ ' ',-' " ' * " ' , . ■ ■ ■ » - < ..* .. ■-. 1 j . . ■ :'-':■ ' 1 1 ^^-^ < r/ " 1 ■ ■ ■ . .- i ' ' 1 ^ 1 J / " *«- . 1 ! ^'V-s 'i^ir f%t ",-?^ ^ / * RUDIMERTAEY ALGEBBA. CHAPTER I. ■ * ■'■ vDEFiNiTioKs. ■■ . ■..;:.;/; 1. We learn by Arithmetic how to calculate with numbers. Algebra teaches us how to perform calcula- tions by means of the letters of the alphabet. 2. Numbers possess a particular and a relative yalue, while letters have no value in particular or in relation to one another. Since then letters possess no particular value, if we can perform calculations with them the results obtained will admit of general application. For if, calculating with letters, we arrive at a certain result, our calculation will apply to any value we may assign to the letters. For example, we shall presently see that if we multiply the difference between a and b by the sum of a and 6, the product is equal to the difference between the square of a and the square of b. Now, as we may use a and ft to r e present any quantiti e s we choo s e, W 9 have, by a simple algebraical operation, arrived at a mi^ma 8 DEflNITIONff. 'j£a' rery important result, for we learn from it that if we take any two numbers whaterer and multiply their difference by their sum the product wfll be the difference between their squares. 8. It is then rery desirable to ascertain how to per- form calculations which give us results admitting of unirersial applicaUon." Before proceeding, however, to make algebraic calculations^ we must become familiar with the meaning of diflBarent signs or symbols which are used for the purpose of abbreviation. 4. = is the «fgw o/e^tto/tfy, and is read "equal to A It indicates that the quantities between which it stands are equal to one another. Thus 4 = twice 2 means that 4 is equal to twice 2. a == T means that the value of a is 7 in some particular problem to which the statement relates. ^ 5.'4.isthe«%no/aWi«bn, and is retfd "plus." It signifies that the quantity before which it stands is to he added. Thus 4 + 3=7 means that 4 added to 3 is equal to 7. a+ 6 means that the quantity represented by a is to be added to that represented by b. If a is equal to 2 and 6 to I then a -I- & =r 3. e. — is the ngnofiubtraclion, and is read "minus.V It signifies that the quantity before which it stands is to be subtracted. Thus 4-3 = 1 means that 3 deducted frc«n 4 is equal to or gives a difference of 1. a^b means o less ^ or with b subtracted. If a is equal to 6 and 6 to 4 then a-6 IS 2. 7. X is the *ign o/fmdtiplication, and is read « into '* It signifies that the quantities between which |t occurs are to be mnttiplied together, thus 7 x 3 = 21, means t ha t 3 tim e s 7 ore 21. MuItipUcation la also indicated by a dot between the qaaaUt|jSB» or (the more usual | Hi^-i to DEFINITIONS. ^jrsT'S ■ 9 way) by writing the qdantities together; thus axb ora.b or (the usual mode of expreMion) ah all mean a multiplied by 6, and if a= 2 and 6.=«, then ax hot o6 = 2x3i=6. 8. o-T- is the sign of divinotif and is read " by " or " divided by." It signifies that the quantity after which it occurs is to be divided by that which follows it. Thus 6-^3 = 2 means that 6 divided by 3 is equal to 2. Division is also indicated by writing the quantities in the form of a fraction. Thus we may express the divi- sion of* by y, thus x-s-y or thus j. Each expression means x divided by y, and if ar=6 and y = 3, then ar.y or J = - = 2. So —^ means o and 6 added to- gether and their sum divided by c ; whfle ^ means a divided by the difference between 6 and c. 0. .*. is an abbreviatipn for there/ore and •.• for because. -, ^^' ( ) f i I I brackets, indicate that the quantities enclosed by them are to l^ dealt with collectively and as forming but one quantity. The same is sometimes indicated by -• written over the quantiUes. Thus 2 x (4-1) or 2x4-1 means that 1 is to be subtracted from 4 and the difference 3moltiplied by 2. The result is 6, but it would have been different if there had been no bracket, for 2 x 4 - 1 = 7. So a: - y - « andar - (y-«) are different in value, for the former means x with both yand z subtracted fr om it, while the latter m eans x with only the difference between y and « subtracted from it. > 10 DEFINITIONS. The fritcUonal line has aliio the game effect « a bracket, since ^^forexample,mea|is the division of the entire quantity a- 6 bjc. _ U^ When* quantity is multiplied by itself any num- ber of times the product is termed a power of the quan- tity and is expressed by writing the index or exponent of the power, or figure denoting the number of tintfs it is repeated, above the quantity. 3 « means 3 x 3 or" the second power or square of 3, and «» means the third power or cube of a, or d X a X a. 12. The roorof a quantitj/ is that quantity which nmltipUedby/tself acertaip^umber of times according to the index of the root Will produce the quantity of ^hich the root is sought. Roots are indicated by the BymbolV called the radix, with a small figure written to, the left (the index), expressing the root to be ex- tracted. Thus ^8 means the cube root of 8. ^x means the square toot of x, for wherey occurs with no small figure written to the left it always indicates the square root. The root of a quantity is also indicated by writing a small fraction wfth the index of the root for denomina-i tor above the quantity ; thus arJ and Vx are equivalent expressions.. ' ■ , . • _J3. Having now become acquainted^i^h algebraic signs we^must investigate the nature of algebraic quan- tities and -we shall then be able to pass on to algebraic calculations. ? ^14. If no sigrnf W prefixed to aquanU^ + is underw stood. All quantities to which + Is prefixed or which gave no sign prefitfid ue t a iUd posit iv e o r additi v e quantities, and all quantities to which — is prefixed are r^^X DEFINITIONS. I f ' It called negaiive or sulitractive quantltieg. In the expres- sion 8 ^ 6, 8 is positive but - 6 is negative, for - 6 means 6 subtracted. In a - ft, a is positive, - 6 is negative. 1^. The coeffieieru is the number prefixed to all alge- bi^lc quantity. In the expression 8xy, 8 is the coeffi-^ clent, and the expression denotes 8 times a^ ; when no^ ^fi5efficient is expressed 1 is understood : thus x meanr- once X. 16. A Quantity not connected with any other by the sign + or - is called a nmple guanUty. Thus ah, - a, X If are all simple quantities. But if coupled with any ' other quantity by the sign + or - the whole expression . is called a comjnmnd quantUyi thus Aft 4.2c is a com- • pound quantity, consisting '^f the simple quantities ab and 2c added together. Thef several simple quantities ' Which make up a compound quantity are called its . term,; thus the expression x + 2^ is a compound-) quantity; x is one of its terms and 2y the other. A quantity which consists of one term only is called a : mmumval; if it consist- of two terms a 6t«omia/ ;" and if of more than two terms a mvltimmial. 17. Simple quantities often consist of more than one letter; these letters are called the /acfm which make, op the quantity. We have seen that 06 means a mlti- plied by ft^ a and 6 then are the factors which form the quantity oft; and whenever a compound quantity is cpmposed of two or more quantities multiplied together. . these quantities are similarly called factors, the term factors being employed to represent any quantities, simple or compound, that are multiplied together, v ^ .1^ The valu e of asimple quantity remains the same m whatever order its factors be written ; ab is just the same asja, for both mjan the product of a and fr K ^^-v .^CsS- i"»Tt ..;,j^, ^r '*■'. 12 DiriNITIOlIB; u Ui« to™!r. !? ^ *•"" "• "H'**". «> long tor. Vk T^ ****"■*** ®' dissimUar letters or iL- Uri^Ui. .tad..t with ^g.b«io rig™ a»d the wto, ,/*"?• ' * = » >< 8 or 6; Mrf »«J _ 2 X 2 xSol .\ \ 'i, ^-^ '■hi'^-m7"^0p^ V - i DlfnOTIONg. * jg If these lettett^hare the ..me ralue m before, what totiieTalueofi!!i:! + 8* «a. * ... « on. q^nUtyMti, «cond thT*-. I, .K tS" ^^?'' '»>W "bstitat. (30-3) f«r^ t^.T J u '"-^^'V""^ •»ClOHd W^to tinmn, tto rimpIMction «^ it. no^eri Jv*°. BZIBOIBI 1. In the expression ^+ 2ab - 8^ At which a^*!.. terms are positive and which neeaHre^ W^^^ like and which are nnlike? '"''^^h'^^^^^ ^^2. In the expressions a^ «•«,«• A. ■ ^14.2^, which are compound and which simple qSUL 7 oJ ^t terms ar e the compound quanttties Zi What «e the flOjtors of the simple quantiUes? iwirr * / , ^«-ff* "«/d''^ '/-^i^^ ^?^T-ii«^«i*i''^„*g fU plFlNITIpNS. *'t tt 1 8. If a = 7 and & = 6 what ig o + ft equal to ? 4. If x= 2 and y = 3 what Is y-x equal to? 6, Write down the equivalent of a + ft 4. oft, ^^ere ,a==2andft=:3. 6. o :s a and * -= 3 : what ia a«x» equal to 7 . V. What is the value ofy/ax where o = I8 and x=z21 In the following «zerci8e8 « = 6, ft = 2, c = 8 *» 2 8. Find the value of 2tf -|- ft-- c + xy. 8. Pindtheyalueofa«-f Vy, ' 10. Find the value of 3ft* -|- aft -- ^^y J^ax. 1 1. Find the value of 8ft - ex + 4aft -oc, 12. Find the value of - 2aftcx. 13. Find the value of oftcxy X 2 -f- 4. 14. Find the value of 2 (oft- c 4. iaxy). 16.' What is the value of 8a - 3ft -f- ex 7 18. What is the vsilue of Vex -^-iab-m^yj 17. Find the value of 2(x+y)-Vy. 18. Find the value of 2ftx-c4-ft«-c. 19. Find the value of 2aft + 2a - 2xy + (x -f y). _ a*b* "iae ' ' aOi Fii>a tly mine of . ^ , ■ V . y " : 21. Find the value of— 4- ^"^^ * ft 22. Find the value pf 2a*ft> +^xy-8^Z£L^. •ft ,xy—ft abc c* 34. WluktistliaTalaeof 3?^ ?, , . . --. 1 L^ f^ \ :.^AJ k«... fi\ ■' -:■ ': .•■.■'.vf-tfe^!»V.-:.l«^ r !J*P«i ■-^J -:/*■ , « ^i^^'"^^"* ,**r^*Tir'* i /. where a:=27 '> "" CHAPTER II, ADDITION AND SUBTBACTION. " 1. The qnautitiei »y and 2xy m we hare seen are Uke qtiantities, and we can readily add them together. They mean once tey and twice xy and their sum ktlvee ihnefi aby or 8 xy. 2; The quantities jnat added together are both posi- «Te; If th^y were hotii aegiitiTe^ * «y and - a«y^ they would indicate once ary to be subtraeted front aome quantity and twice xy also to be subtracted, and their Bum would be Zxy to be subtracted, or - Zxy. , 8. Hence when the quantities are like and the signs are like also, algebraic qdanUtieB are added by the fol- "lowiAg ,,■;.;■ V.' ' AM together the coefficiente and set daum the fm^pn- yi^^ ihe^fitfi i»M mimxing thtqmmtUy, (1) Sox 4ax !6ab ax '19^ axaxpLia. (2) 80-%^ (« + y) ' 9a-46y^3(a:+y> 4«-»*!f+ (* + y) «♦' 1l2*.206y-f »(«+^ -*•••— «VI/y -^ f ^JC "I" l^a In the tot e»am^e we add together tb< icTeHl co- effiolenta, 1 (for a^m jl ajp) iu>jj^4 i i^d S^and if •••v'-tf-'Vi^pip* ■ "i' V''» .3f 16 ADDITION AND 8TTBTBA0TI0N. ^• ■■■§ ■6t down their mm 19, annexing tlie qnantitj our; we do not prefix anj siffn, because in the case of a quantity by itself with no sign prefixed + is understood. In the second example we begin by adding together the coefficients of a, the sum, 22, we write down, annex- ing the quantity a. We proceed to add the coefficients of 6y, which we find to be 20 ; we set down 206y, prefixing the sign - , and then dealing with the quantity {x-\-y) which being within braclcets is to be regarded as con- stituting one quantity, #e complete the addition. 4 In setting down compound quantities for addition we so place their terms that lilce quantities come under 0|i€^ another, and we are thus able the more easily to mid their sum. ^ BXIBOIM II. 1. Add together 8&, 46, 66, and 6. 2. Add together - «ary, - 8«y, - 7xy, - Gary, and - 2xy. 3. Add together 4a6-36y, 8a6-26^, ob-by and 2a6 -26y. ■■■ 4. Add together 8«V;-h*"y, 4«V + 2x*y, and «*y"-H«*y. 6. What Js the sum of 3a*& - 2ec2, 2a*h - Sed, and 4a*6 -Serf? 6. What is the ram of xyx -^xy^ yz, Zxyx -f- 2j;y - Syx, 2«yx -f3xy - 2yi; and a:y« -h 4a:y> 4y« 7 7. What is the Bom of 8ax -|- Sdc 4- /, 9a«-|-7ac-H V;2a«-f-4oc+^t 8. Add together 2aa;* .|- 3y + 8, f-2aftcd, 6ax*4-9fty-aftcd^- 16, 4ax«^aftc^ 6c, 2a6c-4a« 4* 86c, and 8a6c - 8axd + y. ' ' 14. iidd together a* + 2a6 + 2a>6* + 6% 3a*-a64. 86«, 4a* - aaft-f a«6«, and 3<»« + 06 - o«6«. ' . .1 *^*' -^^^ together a;*y + 3a64-4ax-<6, 2a;*y-2a6-|- «, 3a?«y - ac + i, and 4a;«y« - 2aa: -f 26. 16. Add together SVxy + 66c -^ 8, 2a:y - 36c - ary *, 2V«y •- «y4. 6c 4. d, and - V«y 4- xV + 26c -ary», 17. Add together 2a6 - V (<>+«) + 2 (a 4- x) 4. xy Zay+ 4(a4-*)-2x«, 2V(a 4- *) - (a 4- a:) 4- *y, and 06 4.3(o4-«)4-x«. 18. Add together x*y 4- Sax 4- 3V«*y-4, 2xy-6ax + V**y, T-2«»y4-V«*y4-»yi 3a:y* + «aa:4.yx«y4-6, »nd x*y -afy* - oo: - a < 7. If we want to take 6 from a we express the recfult by writing the minns sign before 6, thus a -6, because the nign minus indicates subtraction. In subtracting 6 lirom a then we perform the same operation as in adding, only we change the sign of the quantity to be subtracted, and so, if we take a from 2a the result is a, just as it would be if we changed the sign of the a to be subtracted and added it. Again if we want to take -a from a+b tiiere8ultis2a4-6. Fora4-6 = 2a^a4-6,andif-abe taken away or subtracted there remains 2a 4-6. The Bune result is attained by changing the sign of -a and addingit. " ^ _ T jRULl FOR SUBTBAOTIOH. '' Change the sign 9f the quatUity to be eubtractedf or imagine Ulo be ehttiiigedf(md proceed aemaddU^ £14''-- '"^■555" 5 If", t^,']>!k?!»I«''7'" t^f ADDITION AMD BUBTftAOTIOH. 21 S^ . ■' ■ZAMPLB. ,^- /:■/.-. Soar + 66 V - 1a^C'8a*h* 6ax - 26 V 4- ga6c--8a«&»~6 j aaar+86V -12a6(; +6 Here we take the Box and changing the sign to ^6ax we profceed as in .addition by deducting the smaller coefficient 6 from the larger 8 and setting down the difference 2aa:. Passing to the next term in the sum we chknge the sign of 26 V, and then adding the 26»y« io the efr«y« set down 4he result 86«y«, with the plus s^gn prefixed ; then 6abc with the sign changed is - 5a6c, and - 6a6c and - labc are - 12a6c, which we set down ; then -8a«6« with the sign changed becomei ,.|Bo*6* which cancels *8o«6«, for 8a«6« added to -^80*6* is equivalent to 8a«6« -8a«6« |y,then -6 with the sign changed be^comes + 6, and there being no other similar quantity we set down the result with its sign prefizedi and thus complete the answer. ■MS EziBOisa y. 1. From8a6-4-46c+7take6a6-6c-4. 2. Prom6y«-4y4-atAke6y«-4y-a. ' 3. From x» + 2x* - 6j take -«» + 3a?» 4. 4a?. 4. Ph)m 9a;»y« + 7a6 - 2o»y - 6 take - Ix^y* + 6ab-{- 2a«y-6. * 6. Prom - 8« + abc + 2d*~ 4ax take e» - a6c + ad* +«»-^. ■:•■■;■.; ■"■■■■■^■■^•■^■'■-■^: •■•■>/,■. - ^ 6. Prom x*sf -«y» * Bx*y* + 26c t«ke 2«»y + »«y« ^ «Sf"-J-W>c- a:.:- ..:■.;_;'. 1 Take ta6+ 86c-. 9cdE- iodex from 806- 76c- lOoi 4- 8< g + 9a a g. — — —r- •i f 5i-^1-3t>«- •<", "1^. ^ ^MTION AND SUBTBAOTION. ^ 8. Take eab-^2xy -»- 8a:«y + y« from 9y« - 2xy + 2x«« ,^' ''^^^'^ + '^*J' + 4y« + y« from 9ar-6xy + 8y« 10. From i08 + 6a-96+10a:y-f 8i deduct 7*+6a + 86 + a:y. 11. Take Irit+abeyi-Sy-dyz from 8n»*-7c6y + 6y 12. From 10aa?-l06x + lOcaf+Soyz + ft* take ttJ ex^lhx+ayz. ■ :*gl 13. From2a«x-2ar*-2a?y + yt take a««+J^| Try • 14. Take86« + 2ay + 6a: + dfrom2fly4.96*-6a:-tf. 16. tAke 2^/x - xy -{• lab ^2cd + :f from 2xy^3ab 4^ cd-y. V 8. Snbtraction we have seen is performed by chang- ing the sign of the quantity to be subtracted. If we want to subtract (b-e) from a, since (fe-c) being within brackets is to be regarded as one quantity, we write the result a- (6-c). But if we desire to remove tiie brackets and so break up the quantity (6-c) into the simple quantities composing it, then bearing in mind that (6-c) ig to be subtracted, that is, both b and -c are to be subtracted, ire must change tiie signs of the several terms, and write the result a-d+c. Hence the removal of brackets where preceded by a minus sign necessitates tiM changing tiie signs of att the terms which were in the brackets. ». To show that a- (6- c) la equal to a^b+c, it te only necessary to observe that tiie expression signliles not ttiat A is to be subtraeted from ahnib lessened by g. Now if we Bubtraot b we subtr a ct to e mn e fa by g, •+t»' wnrnPLioATioii AMD Drrauoir. 4 and we fltiiit add e to mule* the resolt eoneoi. Ttioi it becomeB a^b •4-«/ An Mitiimetical ttlustrntioii ihoira «his more plftinly itiU. To snbtnct (4 - 8) ftom 8 we most Bubtract not 4, tat 4 less 2, or 3. The difltoevee is 6, and will be found to be so if the signs are changed #8 directed. Thii8 8--(4-2)3:8-4 + a=:6. .. OHAPTKB III. lIULTIFLICATIOir AKD DIVISION. 1. We hare seen that ab denotes a mnltitj^lted by Kg and therefore if we wanted to multiply a by b we should express the resalt as a6. ^. If We want to mnltiply 2a by 6 we reqidiM to add 2a b times ; the result is 2ab. But if we W4nt to mul- tiply 2a by «-& we in fact require to subtract 2a 6 timeai land the result is -2a6. If again we require to multiply - 2a by ^b we in fiiict want to subtract - 2a b times ; . but we know that the subtraction of -•2a would be ez- pMteed by 20^ so the result of the multipIieatiM is 3ab. fien^ in teultiplyteg algebrale qUantitiMf We haye nM ^iily te regard the quantitiefl thettiselteB but the sigaa Which precede thett^ and it must be carefb£lyMtedthat Uke tign8pp»ive9 pim mnd ui^iketigm minm* 8. tt lanlliplied by « is o squaMi or a to tte tad f&^ntf wUch we hiiTe isea ik etpressed thuto, a* f and M mmm a x « X « X «» which aiay be otherwiia hic fMssed ai tfi x < ev «* X a^ Where ther^inrft we Imyo «he satiie teiteri la bath ntultipNeaiid and qhdtipllMr, it by i ^ laiiB g 84 MTTLTIPLIOATIOMf AND DIVIBIOir. •;^^^ing 88 the index of iti power the inm of Its indices in the multiplier and mnltiplioind. ^Me .^ i„ ««ft,rp^ ^ »it«ft.>,/icanrf are like, the ^t^c* ta«^6ejH«i«r*, 6«/ if they are mlike an^'eign ^''^bf prefixed to the product. ^^ J^^'^y^f^^rithmetiaU coefficient ktten comporing the qumOitiee. M« wm.fc««., add the exponents orindicee of kT^^- MMt exponent in the product. y ««» ««wer ■ ■XAMPLIB. . Multiply 7«6 i^2«:« : -,8*yl^2*: and 3«6 b72(a^ : a> ' m - (3) _ In tte first^ample we multiply 7 by 2 and set down the product 14 J we then find a both in multiplier W iniiltii»licand, and theref5re add together its indices which Me 1 in each case, and giye 2 for tiieindexi^ tte product, ora«, to this we append the letters bz' which remain in ti»e multiplier and multiplicand. No s gn need be pfefized to tiiis, since tiie signs of tiie multi- plier >nd taultipUcand are aimMar, and consequently ttep«duct is positiTe. In multiplying, in Uie second •Mmple, -a*y by 2«the signs »r» unlike, and it becomes neeessaiy to prefix - to tiie product . In multiplying in Fto-gSif'tl' jgtS^j^^ii^.WTtf Si . V\i / HUXTIPIiHUTION AND JMYIBIOK. 26 the last example, we regard the (x -y) ae one qnantitT and iiffix it in the product as we should any other qualify. BziRoisa VI. Multiply 806*4; bj 6a6y. 2. Multiplj 3xy;e by -4««;r. 8. Multiply 8x by -26c? 6. What is the product of 8(a-^x) by 26 ? 6. Multiply-7 (6*-y) by-8ac. . 7. Multiply 3«Ar by Sao:'. 8. Multiply -2«y by -3oy». 9. Multiply 2a%x by a6y. 10. Multiply - 3aW 1^ -2«ty, ^5. As compound quantities consist q£an affweffate of simple quanUties, we mustwhen we ire to3^ a compound quantity by a simple one, multiply ewh tBrm of the multiplicand by the multiplier fld^ multiply a compound quantity by a compound quanti^ we mu&t multiply each term of the muItipliLd by each termof the multipUer. Hence th^ multiplicatiii of compound quanUties is regulated by the foUowing mthejntdtn^lter.aecarding to the rule already gilen, a^ Mthe^ajarmprod^e togeth^^ the product of the entire muli^Hcfftion. V. ^ ^_ ". ■ZAMPLH. : Multiply tag-2y« by a» ; io«,.3yi by a«-<,i / ^n^ a+6byg-6, ^ ^ ^ * ^ ^ m — ti^F*S ' KiitlSils;^. '* J/i'' ^ ifciJ^'W ^^W^[j Ip .y ft6 iim[.t»£ioA«ipii AUD Dmnow. 0) 7a'«-2a*y» 4o«-3y« at- yi :h^ In the first exionple we tnnttiply 1 ax by a\ imd set down the lesnlt; we then mtdtlplj -2y* bf o' and append the lesnlt, luid thns form the pn)dn6t of the mnltiplication of the whole quantity lax - ay* by a*. In the second example we multiply the whole ((Qantily by a* and set down the product ; #e then multiply the whole quantity by -y* and set doirn the product j the addition of the two partial products fives tos the product of the mnltiplioatiQn of the two compound quantities, f tn the last example we find on adding ihe purtJai products that +ab and -oft citneel one anotheri and consequently that (i^*-6^ is the product Of t^multlpU- eation of «+6 by 4-6. A ^^ BXIBGBWB VII. • 1. Mttlt^^ 4- a» 4* » by 8tf. 2. Multiply <^-2fl*+6«bjr a -ft, 8. Multiply 2a:«+ « 1^ « + 2. ' 4. Wbat is the product of a^ - b + x^ by a -I- a: t ^.Hultlply-r6-|-ca:byJ8^-c^ / 6. M^lgply ** - 8«+ 2 br« - 4. > 7. MiUliply *« -2xy + y« by ajs^^. 8. Multiply 4a.- 4ft -^ 4tf by j( + yi a. What li tli> ipitixLti of M^l:^ gity 4^ t by » + «y? .^ v-i &> • V . •. '1' '^*'/'^'» 4f 'T*- ^* »-f pjKWJ r" *■ -6« "I? id set ' and »ftfae o«. tntilj J the ; the sdtot ^/^ iieB* ' , and tiplt- : V. X? i±^ 10. + 2? 11. 12. -«•; 18. 14. 16. 16. 17. XULTIPUOATIOlf AND JHTDROll. H What ii the prodact of »»« - S^A- ?• bj S« -f.^ Multiply 6a* -f 8aaf- 2x" by 3a-«. Multiply a* - 4a»x ^ 9a*x* ^ 4ax* -«♦ by a* - 2a» ■ ■ •' -■'■",■■■■ ■ ■ '■-''- Multiply 2«?+ 6 4-6 by ax^b + e. Multiply 2o» - 26»4- c^ by ^^i _ ja Multiply ar« + «y + V* by x»- y«; Multiply «» + oA + 6« by a'-oft + ft*. * Multiply « + y by * + y and the product by « + ...»•■ ". ■■■,:.-■» 18. Multiply X* + 2«y+V» by «■- axy -I- 1», 6. Since in multiplication like signs prodnee plus aad nnlike minus, it follows that in dirision where the signs of diridend and divisor are similar, the sign of the qno^, tient will be plus, but where they are unlike it will be minus; 7. If we have to divide 6ab by 26 we require to aseOr- taln how often 26 is contained in 8a6. Evidently 4a times, since 4o X 26 == 8o6. We attain the answer then by dividing the coelBoieiit Of the dividend by that of the divisor, and then dividing the letters of the one by the other, by oancelling any letter that is c^tained in both dividend and divisor. If now we havdfc divide 2«r by 6 we are unable to proceed as in the JKe exaaple, for the divisor consists of 6 only, and there is no 6 in the quantity to be cancelled and thus efhet the division. U this case we can cmly indicate the division by writing the quaatities in Pactional form, thus ^ 8. Af divided by a gives o ibr a x atss «■. Hene^^i^ '^aisSOf and theiefbre when the dhrlsot and ttvidead ^»*- ^.^-.SifttJiK^^ ' 28 MULTIPLIOATION AND DIVISION. •ontain dlftrent powers of the Mme letter we mbtrRct the smaller inderfrom the greater, and place the diffe- rence as the index of the letter, either above or below the fractional line, according as the dividend or divisor oon tains the higher power. ^ 9. Hence the division of simple qnantities is performed by the following '■ • ■ .\; ■ . RuLi. . ■ ;.; ' ^ike iigtu of the diviior and dhid^ are like, the quo- tient vriU be poeiHte, but if the tigru are unlike the quotient vrill be negative, and mmt be prefaced by a minus sign. Write the divisor under the dividend, infractunuUform. JHvide the eoeffieietd of the dividend by that of the divisor] or reduce the coefficients of both dmsor and dividend by dividing both by the highesLnumber that vnll go into each without a remainder. \ Cancel any letters that are common to both divis&r and dividend. Where powers of the same lettei'are contained in divisor and dividend, subtract the lesser index from the greater and the difference UnUbethe index for the Utter in the dividend or divisor, whichever has the higher ^ V ■ ; ■■- • ■;■■.■ ^ ■ZAMPU8. Divide Sa*x by 2ax, and - lOofr by Sa*x 2ax =i4» -10a6 \ 6a( In the first example the signs of bo^ divisor and dividend are simUar, and the quotient is therefore posi- tive- We write the d i vide n d an d divi s or 1^^ fractional e write I tibial form, and tiito find that the coefficient of th^ dividend /• -•;?« >trftct diffe. mIow lyJBor irmed / /■ moTf \dbf eath 'and mnr r and '\dtnd and KMli- iend IfVLtlPUOATro^ 2i ii exactly dirislble bj that of the diTieor, and girei 4 for the quotient. We And * both In dividend and diri- ■or, and therefore it becomes cancelled, a is contained in both dividend and divisor, and subtracting the index of a in the divisor iVom that in the dividend gives a ai the quotient We thus obtahi 4a at the result of the division of 8a^ by 2ax. In the second example the sigus are dissimUar, an4 the quotient requires to have a minus sign prefixed. We reduce the coefficients bjr dividing by 3, and aa there are no letters in the divisor that are contained In^ the dividend, we can only express the quotient as- ^ V ■ -;^' ■ , .* iy BXIBOISI VIII. 1. Pivide4a^by2a6. ' 2. Divide 8xy by 3«. 8. Divide - 16a%c by 2a6^ / 4. Divide 9«*y by - 3te. 6. Divide 2a% by flftc, - 6. Divide 6a6« by -Sary. ^ 7. Divide 16a»6% b^ 26^- 8. Divide 3flAcy by oary. ^ 10. Since compound quantities consist of an aggre- gate of simple quantities, we must, in order to divide a componnd. quantity, divide each of !♦« terms by tiie divisor. Hence where the dividend is a compound quantity and the divisor a simple one, we proceed by thefbllowing 1 - "Rmja, . ■ muU each tern of the dividend by the diviior aeconUng to ^ preceding rtOe, and prefix to eoM term in the aw- tient Ui praperiign. * t. "3 ,•4 1 \ \ HVliV WiM ATioN AHD wrvum. DiTid* 8tt% - 7y by - ia. Vy -ao. =-4*r + 35. Tlaolng the qoAiitiUefl in flraetionftl fbn^V "If^ ^i^^^ that 8 |^IVIiXINi« ' Sk figmtt «i6» m mat^, frttk ttrvu m «r# me$uary/or tlu ntxt dUntum, and contmm iht optratiom athng a$ practkabU. Vth»r*Umyr*mawdtrplac€iiintht/brm9fafhieiim fa tkeqwHuni wUhth» divmr/ar iUdm^mmaton IXAMPLV i: Dirlde ex* - 96 b7 - 6 + 8«. 8* - 6)6«* - &6(2a:« 4- 4«i 4- «x 4. 16 ex* - 12«» :b 12x» - 24a;« a4Ar* - 96 24r* - 48^ 48a; -96 48a; -96 tot in the dirigof nsill* In »»> . -, nnt thh 1. .T product, btlnghqr-doTWrae .at tem of th. dlTl*^ ih« to win go into ia«. 4.. ti.^. w. p.t+^' to «^9«ot.e««k multiply th. «Ti«» h, it J,p,^a Divide -8ax + 4aj« + 4«« + 2« by-2a:+2a. ^ 2a- 2ap)4a» -8ar4. 4x« + 2fr<2a^aa»4^s ^ ^, -^'■ 4a«-4fl* ^3a-2ar' 4a «-4ga; -4aiP4r4a;« IS' 62 HULTIPUOATION AHD DlVWIOir/ • We here arrange the term, of \he dUteor «id dlrl. dend according to the Indloe. of a ; harlng ^^^^j^'f^'f^ M m the la.t example we find a remainder 26, which I. »0t dirlBlttle; we therefore write this r^ma^^der^^^ the quotient In the form of^a fraction, with the dlTl- ■or for denominator, thni 2a- 2« ■XAMPM a. Dltldeo»-«»bya-a:. «-*)••-«•(•■ + «* + *' ax' \'* 1. DlTlde a*-2ab + b* by «-*. 2. Divide a« + 2«t + *" ^^« + *- .^' ^ 3. DWlde a1a» - 12a»a: + 2$y*'-^>': 8 Divide 2a6 + 8ac.2a6«-16a6c + 3yby a-2a6. 9/ Divide 2ar+«» -2a!^-*«»-1f"J>y*-yv 10. divide - 8a« + 8a»6 + !««**" »»y ^«* + ^«- 11. Divide x*-y* by *-y. 12. Dividea^>3«?ft + 3ay-ft»bya-ft.— ;_ 14. Divide 8a*-8y «'-*'• ' i> ' f I '*■ "'*" MtTLTIPLiOATION AND DITIfllOlf. 83 •* la. HATing now MoerUined how to add, •nbtraet ■lulUpIj, and diTide algebraio quanUUef, we may note •ome points before proceeding. Ai in the exprenioa Sab, 8, one of the faotorg, is oaUed the coefficient, lo a another of the factors, is sometimes ddled a UtmU eo^ ciMt. Bob means 8 times oft or a times 86 or b Umes 9a. Any factor of a qaantity then maj be regarded at « coefficient. If we had to add Sax and 36*, or subtract 80* from aftor, according to the rule, the qnanUtiet being nnlilce, we shoulc^ express the results as 8a«4^ ibx and 2bx^8ax. But if we regard a and 6 as littral coefficients, the result is expressed as {8a + 2b)x and 1(26 -8a)x. ^18. We are obliged to place do-f 26 within brackets because they are both regarded as coefficients of the quantity x, and the result of^the addiUon Is 8a+ 26 times X. So if we remored the bracket from the expres- sion (8a 4- 26)x it would be necessary to mulUply each term in the bracket by x. 14. We must be Tery oarefbl in remoring braoketi to remember that anything aflhctinc^ the quantity within brackets affects the whole quantity, and theivfore * affiBctf e^ch term when the bracket hi i;emoTed. If the whofe quanUty is to be multiplied then eaah term mast be multiplied on removing the bracket, and if the whole quantity is to be subtracted the signs of each term must be changed cp its removal. 16. The following results; the truth of which inay b^ ascertained by actual multiplicati8n, should be here noted •§ general formulas of considerable practical raln^ (g+ft)x(o4-6) = a«.f2a6-h6«. .■ ' ■ ■■' (a-6)x(a-,6)^a^-2a6+6«. (a-.6)x(a+6)»-6>, "- -, 9 ^^ "— ^ --■ OHAPTEB IV. 0RBATE8T COMMON MEASURE AND LEAST COMMON MULTIPLE. , 1. A truasure of a quantity is any qaantitj #hioh will diylde it and leare no remainder ; in other irordd olie of its factors. A contnum tneawre of two or more quantities is anj qnantitj Ifrhieli will diride aU of them without a remainder; in' other words a faetor common to all of them. The greatest c&mmon meoMure of two or more quantities is the greatest quantity whieh will diyide all of them without a remainder ; or, in other words, the product of the highest common factors. 2. Thus a is a measure of a6^ for abis compcsed of the factors a, and h ; so it is a iQcasure of a*h which is composed of the factors a, a, and h\ a; is a common measure of d;'y« and «'y, for they are respectiyely com- posed of the factors x, x^ y, «, andor, «, y^ ahd it is appa- rent that X is a factor which is common to or contained in both quantities. The greatest common measure of £'y«and «'y is 9^9^ for the highest factors that are comAOtt to both are j;' and y, and their product x^y is the hic^st factor which is conttuiied in both the quan^ . titles.' ^ ' ■ ' '>■•■ 8. We AseertAin' 4. But if the quantities of which the G. 0. M is to b^ ascertained are compound, we must proceed as in « arithmetic, by dividing one by the other, treating the remainder after the first division as a new divisor, and the former divisor as the new dividend, and thus con- tmumgtill there is no remainder. The last divisor nsed will be the G. CM. If we have to ascertain the G.O.M.of more than two compound quantities, we . firstascertam the G. 0. M. of any twoof them, and thin ascertain the G. C. M, of another quantity, and ^he G. 0. M. already found, and so on. 5. Where any of the quantities contains a fiictor common to all its terms, we may simplify the quantity by eradicatmg or striking out the factor. While, how- ever, we may eradicate factors common to the different terms in one quantity, and factors common to those in another, we must when we strike out the sam, factor " from all the quantities be careful to note that it will form a part of the G. 0. M., and that the last divisor ^ed must be multiplied by it to obtain the correct > Whenever we havi a remainder brought into use as a divisor m the course oi ascertaining the G M cImuZ '""^ ^"*^ «<«»«9n foctor thai itsjermi . 7. Whenever in tiie course of ascertaining^e G ' fxaltw h' tt?' V '^'^ '" ^^ ^^"^ of the dividend is no; exactly divisible by the first term of the divisor, we ¥^ ^ ■T^'J'^. ' ;e-*^ " -v"i,r'i' "^' 'T:¥T"i^"i^"'*S^''S'T^""^T**'^'^ ^""1^'W"*CT ^'K^'^^^rw^-'-J^-'frf^ 36 OaBATUST COMMON MBASUBB AND mayinultiplj the dividend by each a namber as will mi^e it so divisible. ^ . ■XAMPLI. What is the G. . M. of 9a*b - 266 and 9a9 + 3a - 20 f 9a«6 - 256)9a» 4- 3o^ 20(1 9a«-25 9a« - 25. ~ 3a4-6)9a«-25(3a-6 9aH-15a ' -15a-25 -16a-25 . We find that 9aH -^256 contains a factor (6) common to all its terms *, this we eradicate, and simplify the quantity to 9a'— 25. If, however, the other quantity were 9a<&-)- Soft- 206, it would b)e necessary to note when eradicating the 6 from the divisor and dividend, that 6 would form a part of the G. 0^ M., and that thp G. 0. H. found by division would require to be multi- plied by 6 to obtain the true G. CM. We now divide one quantity by the other, and obtain as a remainder 30+ 6, which we make a divisor, placing th% preceding divisor as the new dividend. We find that 3a -4- 6 divides it exactly, and that it is conse- qaently the G. O. M. of the two quantities. If instead of seeking the G. G. M. of 9a'6 - 266 and 9a'H- 3a - 20 we had to ascertain the G. . M. of 9a' - 26, 9a* 4- 3a-^ 20, and 6a6 -\- lOfr, we should, having ascer- ^ined thd G. 0. Bf. of the two first quantities to be 3a + 6, take the two quantities 3a+ 6 ' and 6a6*4- 106, and proceed to ascertain their G. O.M. This we should find to be 3a + 6, which would consequently be the [.lof the three quantities. ^ , LEAST COMMON MUIiTIPLl. 37 8. A midtiple of a quantity is Unj quantity that con- tains it as divisor, or as one of its factors. A cmman multiple of two or more quantities is any quantity that contains all of them as divisors, in other words that has all the quantities in it as factors. The least common midtiple of two or more quantities is the lowest quan- tity that contains all of them as factors. 9. The least common multiple of two quantities is ascertained by finding their G. 0. M., dividing one of them by it, and multiplying the quotient by the remain- ing quantity. Or we may strike out the factors that are common to any two of the quantities of which we desire to ascertain the L . 0. M. ; multiply the quantities so simplified together and the product by the factors struck out. <» 10. In seeking the L.d. M. of quantities we are en- deavouring to find the lowest quantities that contain them all as measures ; obviously then all we have to do is to resolve each, quantity into its factors, and the L, 0. M.Will be composed of all the factors peculiar to each quantity and of the factors common to any two or more of them. ■ ■ .. IZAMPLB. , - Find the L. 0. M, of a«6«c, Babe, lind 2d. Here striking out the fectors abc, common to ihe first and second quantity, and 2, common to the second and third, the quantities are reduced to aft, 4, and rf j their product, 4flW, multiplied by the factdrs struck out. 2a6c, gives BaHt'cd as the L. 0..M. — 11- yor a s ctot a ining the G; 0. M. of t w o quantities we may use the following » iSmi&iS£M< ■ . ; _ * .;." ; ^*.^'i '♦^jf^ f'-j— -^^ ^4-*-^vjj^^. fi ^f^^^j^p^jj^i^i^^ ■; . ■.'■.■-. 3f 0R1AT||9T CplI^OM UEASUBK AND • RnLB: ^ : If tt# ^ntitie$be simple find by inspection the greatest c?*+ 2«y+y». ■•■^ . 'yfx''m^^W ■^^•'^"Wt^fSKS LSAST OmmW HULTIPLB. at 8. FlidtMa.0.JI.Qfp«(a'-aj«)wi4#»«4.^ a»-« Md IQ. Whatig tlieL.G.M. of 6a^(S 3,a>6, $ab* and 6a^ 7 11. II II la, « II 13. 11 II 14. II If 15. II . II 3a:-|-3y? . 2(o4-6)and3(o«-6«)T 6(a;«y + a;y»), 9(«34-a:«y), 4(y» Hr«y*)7 8«y, 16a:«y«, 46xy, and flicKcy* T a«y and a(xy - y * )T CHAPTER V. , X. By the arithmetical expression I wemean one half or one divided by 2, and in like manner we hare seen that the expression r means the diyisibn of ^ by ft, imd is an algebraic fraction. \ 2. Tlie qijantity above the line is called the ntfmcfa/or, that below the derummaiory aiid both together op^t|)- tuUthibUmuofthefracHon. " ■ . ^\, 8. By multiplying the numerator or dividing th»^ denominator pf a fraction, we in effect ninltiply tha fraction j by diyidiug the numerator ormiUtiplylng'l^he .^^enoikunatoi; we in efect divide the fraction. . « 4. Qii^oa if we B^ultiplytha ^1^n^|:a1;Q^ »nd. 4ei^i|)4« ^p»t » r^ » frmtion ly t^ yaglft ;^^t| |y w^Jw vO i^ multiply ««4 »^ ^ 89^ tMi»MiW4# % e»jB%» bj — 1 «- ■7 ki 40 ^BAdnONS. / the Mme quantity, it foUows that multiplying botii numerator and denominator of a faction by t^e same quantity leares its yalue unaltered, and simil&ly that diriding both numerator and denominator by the same quantity leaves the value of the fraction unaltered. 6. To reduce a fraction to its lowest terms :<-• ,, . ■-■% . . - Rum. JHHdethetermofthe/racHonbytheirq.C.M. ■ ■ XXAliPLlB. ■ ^ _ _ a*b*x ^ 2a?+4o6+2W. ' ■ terms. ■ ■ '; ■ The 0. 0.^. of a*&*« and ofty is a6 ; dividing the ' terms of the fraction by this we obtain — aa the lowest terms of the fraction. The value of the fraction is unaltered, for we have divided both numerator and denominator by the same quantity. In the second example the fraiction may^e expressed 2(a»4-2«ft + ft*) " 3x(a-M ) — , but since (o + 6)>=o» + 2cift + 6>^ Hre fiirther simplify it to 2i^^^^ii\then steiWng out a-4-&, which is a fiiictor common io both tvrms Of the firaction, and which is the G. 0. M. of the terms, ve reduce the fraction to its lowest terms ■ ^^"*" ^ ' 3a? ■ ;■ 6. It is therefore apparent that we need not ascertain the G. 0. if. of the terms df a fraction if we are able to split up te reiiolve them into their several factors, for by cancelling or dividing the terms of the fraction by tiie common flteton we reduce it to its lowest tertas.. ■~f> FBAOTIOifB. 41 tk. ....■■• ^ 7. A tailed quantitj, that is • quantity oonsisting of a #^qle quantity and a fraction, toay be reduced to fractional foiwfey multiplying the whole quantity by the denominmot; of the fraptioii and connecting the product with the fraction, placing underneath the de- nominator; ■' ■ 1XA1IPI.I. ■ ■ Rednce 26 - -^^ to fi»ctional form. 26 multiplied by 6 gives 26», and annexing the frac- tion we obtain ^^^^ or ?*!=^'±l%,?*!r^^ the equivalent fraction. 8. Where the denominator of a fraction will divide the numerator, or divide it leaving a remainder, we can reduce the fraction to a whole quanUty or n mixed quantity (as the cAse may fee.) Thus ^-i^-fei?* can at Once be reduced to the whole quantity a?4-y and 2o?64-86a: ' a «*» be reduced to the mixed quantity 2a6-j- 86* * ~. The student wai perceive that this is only apply. ing the rules of division In cases TThere the numerator of the fraction is divisible ^ the denominator, or divi- sible leaving a remainder. ■ - ' ' . ' ' '■ 0. To reduce fractions to a common denotainator;^ - ' V \ RULl. — - yirf^/y eac h ntmeraior by the dmmmator of the oiher fraetiom, and aU the denominatort together for a eamman denominator. ^ . 49 FEAOVIONB. UXAMfhM. ■ , a« 26 '3 Beduce --» -- and - to a common dtnominatolr. sc . aiy z . Maltiplying the first numerator a^ by the denominatort of the other Araetions we obtain a'y« for the numerator of the first fraction, and similarly 2bxx for the numera- tor of the second, and 3axy for that <5f the third ; then multiplying all the denominators together we obUin axyz for the common denominator, and the fractions* . o'yjg 2ltxz 3axy become , — , • . axyz* axyz* axyz • 10. The common denominator of any fractions is not necessarily their least common denominator ; this, is obtained by finding the L. 0. M. of the seyeral denomi- nators, and the fractions may be reduced to their least common denominator by multiplying the numerator of^ each by the quotient obtained by the division of the least common denominator by its denominator. X 2y Thus to reduce^ and ^ to their least conunoa de- nominator, we find the L. 0. M. of 2a& and 35, whichiis 6ab ; 2a6 will go into 6a6 3 times, and we mnltlply the numerator a; by 3 ; 36 will go into ^oft 2a times, and w« multiply the numerator 2y hf 2a; and tltius obtain 3x '■ 4ay ,^ r-^ and g-^ for th0 fractions rednccid to their leiuifi eoffiT mon denominator. li. To add or subtract fractipns we pbserre the fol- lowing - , . • ; ■; ; ;"• ' - 'Rpu. . •; "• '- ; '. ; Mm thefieactiom to a common cknan^iiKtfor, 4(f^ V nAtract the fwnuratort (a« the com may he) fofi «f (i^ mmtero^or, WMfor uhith jitore Me eommmdmomiiiator. >.■. '. mcnowi, « f**»rj*- ^d together^ .n«;^.i.d .,btr,ct ^^ ft,„ x — fi* y . /iv + « - 2 a: 4- 3 2x« y «^ + 2« ' irV2» ' ««+2«4.a:»-2« a:"-* x*^4 a;*-.4 «* - 4 1 . 1 x + y . ■ V \ * + y ar-y a?+y-a:+jt 2y i?«-ii« *-y a; + y x*-y* x«-.y« ^.^-y brinX*^ h'*'^*" '""^ "^' «"°»Pl« th.t whea 1 fenng the fractions together under the common denomi. nfttor It « necessary to change the signs of both termi -'m"^* ^"'^■*■*• ^""^ *^^ «"*"« quantity is to be subtracted and may be regarded the same as if it was withm brackets. ;, ' ^- To mqUiply fractions :^ ■'• Bulb. ^v^iply the numerators together Ma newnumeri^ar, and the denomtnatars together for ft new denamimUar. \ %r IX. , '^ 3a^ Multiply — by -^. ^ 3a»^7a6x3a« 21a«6* X X xy yxay x*y i4- l.f«J?° «»i»Jo» of ftaoton. i, iw«>rm.d by tawt- i»« th. tormi of Xb» dirtaor .«d tlje, »rt*iplytag. . V ^^~'~*"»'* * , ;v-^.' ■^=-'' i , 44 { FRA0TI0N8. Divide ab c X by ;- • 1 ^ « 06 y — . a«.— X ~ =» vex ex we find one of the factors in the nnmera- 14. In mnltiplying fraotions we may cancel any fac- tor that is common tOy either of the numerators and either of the denominators. For example if we required to multiply ^37 by — r— by placing the fractions in order for multiplication^ thus 2a X (a»"6*) (0-6) X 6 tor, a* *- b*f may be resolved into the fiiotors (a J 8* — y x — y "®® -^ and -^ to their least com. denom. 9, .Add together f,f, and ^-. lo: Add together^ and ^. 11. Add together and — . ■ ct a 12. Add together ^.»^,„d'»Z». 13. Subtract -^ from ~. , ■ - - --^ .' ■ ■ ■ » ■ ■ .. 14. Subtract -?^ from '^■'■® 15. Subtract 6 a a ^om^. ,16. Subtract?^ + ^ fro« !f^. l7.Multii»Iy^by|. T V ' .a i 3ax 18. Multiply -~ by ^^^. I T^fi^^ ■^frr^')!^-'"''^* ** ;j" ,""5* ' • f V\ *-"TS 't.''^^ ^ 4* INVOLUTIOH AND INVOLUTION. . ««-a« 86 19. MulUply —2— by o+x' 8a 76 8c ao. Multiply together -— , — , and -r- a-x' ox' 8x 4x 11. DiTide -r- by -r-. 3 6 H. DiTide 8y»-y a byj. 33. Divide 8a. 86 2a ^ 4a-46 ^^ 36 • » >/ OHAPTKR VI. INVOLtmONAND EVOLtJnON. 1. Inyolatfon is the process of raising quantities to any required power, and is performed by multiplying the quantity into itself as many times (less one) as there are uniti ifi the index of the required powei^ . 2. The involution of simple quantities is generally performed by multiplying the index of the quantity by that of the required power, and prefixing the result of the involution of the coefficient (if any) obtained by actual multiplication. For; since a* raised to the 3rd power = a' x a« X o*,= a«, the result is evidently more simply obtained by multiplying the index of A (3) by the index of the power (3), thus a* to the 3rd power = a**' =:o«. Thus we see that in the ease of simple qu a ntities by th e proc e ss above m e ntion e d w o obUun the same result as if we multiplied the quanti^ into itself as raftny times (less one) as. there are units in the index of the required power. rirfc. *mff^ww Ttrt6tttvm ita tvotirttow. 4> S. Bat in the o«f e of oompbniid qattntttl^s we mmt proceed bj ftotuAl multiplication. 4. Since «:• X «• = X* it ia erldent thAt we may Iq lome meaaare abbreriate the process of Inrolring com- pound quantities to high powers. For since the square of a quanUty multiplied by itself gires the 4th power, we may obtain the 4th power by first squaring the quantity and then multiplying the square by itself. Similarly since «» x a:'=x« we may obtain the 6tb power by mnltiplying the cube by itself^ Ac. «. In the case of a fraction we must involve the numerator and also the denominator to the required power, and the results will be the terms of the fracUon raised to the required power. * e, i^ the caae of simple quantities we must note that where they are negative, the powers whose index is odd will b« nefative, while those whose index is evea will be poaitive^ > •■■■:;, '■%•'■ iXAMPLiSi ■ #lmt is the square of 2o»x and the cnbe of ax* f fr by aotnal mnltipliisation. ■ ' W (2o«a?)«-:2a«x x ^a*a? = 4tf*±'* ^mxt IB the i*^ p^er of a - 2^1 •\ Here we multiply o- 2x by itself, and thus obtain the s ^nye, a^ -4 ii» 4- to% and multiplying this agiUn by itMlf weobtein the 4* power required, tf*-8tt»«+24' •^T^Wft'^-*;^ ^t^"^?? "^^r/ 4t iNyOLUTION AltD lYOLUTION. '■•2ax-^4x* a*~4ax-^4x* ■/ a*-^ 4ax -\^ 4x* : ♦■■ -4a»a;-|-16a«i«-16flar» / Aa*x* - 16oj;» -f- iftc* / a* - 8a»* -f 24o«x« - 82o»» + 16*« 7. We may therefore, for the inyolution of algebraic qaantities, proceed by the following ^Intheeau of timpU quantities inw>lve the coefficient to the required power and append the quantity vfUh the mdieee of iteeeveralUtteremultipUed by the index of the required power. If the quantity be negative and the index of the power oddf the product or power muet have a negative eign prefixed. In the cau of compound quantities muU^lythe quantity Mo itself as many times {leu one) as there ate units in the required power abbreviating^ however^ ifpossibU, the num- ber of actual muUytlicationSf as shewn in section 4. In the cau of fractions involve the numerator and also the denonunattfrforthe terms of the fraction raised to the re- quiredpower, 8. Broln^ if the extraction of the roota of qnanti- tiei. Since V)« = »«, it follows that the eqaare root otx* if xK Aid l)ence to obtain the required root of a ■imple qoantity^ we mnft iSrft extract the root df the wm^s^ ^i4l'«J 'Sipi^Un^f^^^^ myOLUTION AND ITOLUTION. . 4$ mimerioal coeffioient (if any) and then diTide the index of the qoantitj by the index of the root. But if it Bhould happen that we cannot do this, the index of the quantity not being exactly diyisible by that of the root^ or if the quantity hare no index greater 4han unity (as oar) then the root required cannot be extracted, and the quantity must be written down with its radical sign preBxed. This expression is called a surd. ' Thus the 6"» root of o» can only be expressed thus, i^a',, and this is called a surd. So the cube of 2je*.s: ^2 X ^x^* But the Cube root of «» is x, for the index of the qnintity 3, divided by the index of the root to be extractedi 3, is 1. Therefore the cube root of 2x* is x;^2, and this is similarly called a surd. 9. We know that + multiplied by + gives +, and that - multiplied by - gives + also. + is produced, therefore,.both by the intermultiplication of positive and of negative .quantities. It follows, therefore, that the square root, or any root whose index, is even, of a positive (^antity, may be either positive or negative; and this is expressed by writing the result thus V^' •S? i 3f • %i » Hence, the even roots of a positive quantity may be positive or negative, and are expressed by ± No negative quantity can have an even root. The odd root of a quantity will have the same sign as the quantity itselfl this last position is evident, for if the quantity is positive, every power of it viM be positive also; but itf it be negative, while the seoond power would \)e posi- tive, the ttird power (and siiniliirly eyery other odd power) would b6 immediately ptqduccid by tfiultiplying ^f <.*!. 'to mw^r^^m^sme «««?»B i-^tfle^ii.Mfi.ltftiin ^^m-'i' "t^> )v ;> ii . aryoLuvioii ▲(«> lyounnoir* *l^tlir« 1^ a a«gwttT« qiiaii«it7,iieoeMmrflj predaefng a B«fatiTie qimnti^4^ the f esnllt. 10. HeBM to^ctraot tlie roots of 8im|Ae qnaiktitl«9 iv» hftfv the following . •_ . RUM*' Iffihe root to be extraettd be eveiifthe reeuH may be porithte or negativeybut if odd prefix the tign of the qum- HtyUulf. Extraet the required root of the coefficient^ and append ^ekttere contponng the quantify, dividing their indicea by Iftdf o/%e rodtfor the indices to place in the root. Bztract tie sqaare root of »o**« ; and the ciA>e root «lf-«a»»». Ihi 1ft» ^rst example ire find the square root of 9 tolie 8, and the square root of o*** is obtained by dividing «ie index of each letter by.2, the index of the root re- quired, and we thns obtain 3a*a;*, which shonld stricHy be expressed ±Za*x\ since 3a*x* may be positire or negatiTe. In the second example we have to extract an odd root, and it Vin therefore hare the same sign as the quantity itself or-. The root of the quantity is ex- tracted in a similar manner to the preceding example. 11. If we multiply a +:b hj a + 6 we obtain d* + 2a* -|-6%andif a--6 be multipliedby a-d tiie result is ifi'^2ah'^¥, Tfhatis the square 6f a quantity of two terms ooQSisis of the square df ea'cli and twice the^ prodfiot, added or subtricte4 (as the case mi^ Im). Vrom this fsrmula we find the tta» for the fxtiMio^ ^ Hie square roqt of eoapouod qaiUtti^es, nfvounfow wfrotArm^. iSttract Me tquare. root qfffisftrd ierm in fhe quantitff undf^e ihat root in the quotient; equare the term placed in Me rootj deduct it and bring down the remainders muS^ t^iyiheterm in the quotient by 2^ and find hotooftenitvbOl goin$othefir$t term of the remainder j and place the reeutt in the quotient toUh ite proper eign ; also couple the quantity placed in the root to the divisor ^ and muUiply the whole divisor by the term last placed in the rod. >^l(f after this is done there is stiU a remainder ^ proceed aM , mvli^lying all the terms already^in the root for M# wt of the new dknsor^ Extract the square. root of 4x* - 4xy 4- y*. 4** - 4a:y + y« ( 2x - y ■%' 4«-y) -4«y-fy« ' -■■■,-. - ■ • We extract the square root of the &8t term and find it is 2x ; place it in the qaotient, sqaare it, and snhtract it f^om the quantity, bringing down the remainder -4a?y + y* ; we now multiply the terms of the quotient by 2/ and obtain 4x for tiie first part of our diyisor; 4g will go into -4xy-y timey ; we place -y in the quo- lient, and also complete our dirisor with it ; now mnl- Uplying the diyisor by the term last placed in the root, wte ebtdn -. 4iry + y^, which deducted leaves no remaia- ^W. Thei»fore 2x > y is the sqaare root of 4v* - 4aey 4: li^ttttw hid deducted .4xy 4 yt there had stiU <>eei| te»ffii roittfUniny, we f^^^^l^ jiaye maltigflied a*"|^^ y'^v't^'T"'*^*''*" ... i nrWHTTIOH AlfD IVOLUTIQir. I hy^ for the ^firat part of a new divisor, and then pro- Weded m before, and if we fbund that theexa^ gqnare root of th6 quantity could not be extracted, we should expreas the refult as a surd. Thus a" -i 36 has no exact «qaare root, and its squar e r6ot w ould be expressed in the fonn of a «urd, thus, yo»-,3A. „ , ■XAMPLB 2. "V. Extract the square rooti of a'-ofr 4. --. a' • * A 2 a" 2a N: 12, If we had to extiract the 4*>» root we could extract the ^quai^B root, and then again extract the square root Qf the root found for ar« =;«r X a;*. 18. By cubing a+6 and a'-b, and invostigating the compdsition of the product we kre enabled to find, for the extraction of the cube root of eompound quabtitiea ^thft^llo#in^Vy ; - ■ V^ ' Take the cube root of the first term mi plaee U in the quotient; cube the' first term, and deduct it /r 1 aAj ^ nnroLUTioN amd svoLtrTioN* 118 '■■*•- ■ * ■ Mk inihik ^Hmt to the term ihereinf mvU^tfying the niiOy the term last placed in the root and,annexing the tehole to the divieor, * , •. /.-,■' -f, ■ ■ ^, ■■ . , • ■ EInd the cube root of a' - 3a* x -^-.Bax* -'St. \ - Ba*mBaX'^x^) -3o««4-3aa:«-x» .. '^3a*z+3ax*^9^ We firat" extract the cnbe root of a^\ place the resnlt' in the quotient, cnbe it, subtract and bring down the remainder - 3a*x + 3ax* -x^ Theifi^re place Srtimei .thesqnare of a-orda* as the first term of tiie . diiisor, and-^cL it wiU go into j3a*Xf ^» times ; we place ^-x in the qnoUent; we then, coiD^ij|t^ th^ diTisbr5bijr. Annexiiig 3 times a to theJermlM^lacedii| the root, -af,<;^nd mnltipV the sno^ 8^ power.;- 8. WhAt if the 10*» power of 2c«x? .rff"»" f ;■''•■ ' 4^ Whfti ii 111* iqi^m of a-« - te? «. Oab«a-6. ^^. Baiiie 3a*x* to the 4«» power. ^1ni»tlgthe8**po^rof^ v ^ ^, 8. Raii^ a+.2ft to the 4«» power. ' ; 9, What itf the square ^oqt of o**- 2ax +*x* t 10. Kztract thecabe root of e4fl«a:'y«. U. Bztract the 8qii«re root of a* -.4a»«4» ««»««- 12* What if the cube root of •« - 9a.*x+ Ua«»« ^! 13. Find the square root of 4a« + 46«..8a6« 14. Raise 2a&i; to the empower. 15. B»tr««t th. ,^^ ,^^t of ^ ^^"-^ ^ J^ I \ ifil^LB EQUATIONS. ' Jl^#ii^ ieenMittftt the sifrn =r deootttr equally; , Wnere Hds^ sign^ oediirs hetWeen two qnaittRI^ fhn I i^eieezpNffion i8^teni^a»«9iMMbft. If « -= « nntf i| i w equAtion. It does no^of coarse iMMv that sib ywai,$ equal to a, but th*t fii^ the partioular investln. I on we are making^ eftheri5Ntt1|cts we know or from dednc^ons we |uiTema4#^ by aliprtik^ «,-.«. Theitwo ■idfi of Mi equatioa sap areJeAhy ih^ ^n ^i^r pon- «igt of limple or e«ii|ieiai^yuHitities^ smMs^^t^'L^ '^ ±.^ e J:AA..AifcL^.vt, ititt v«friHi,T ^»i^!fct.ii jmiii iqiTianiiiik iirlter l«it«rt of tfa» alphidbtt; while th» iMi tetters ttf ; the alphabet «re vied to mptimni^M^moum futmHiim, that if qnMtities tlMfi^liM of which wo hftTO to dieooTor •iiher nnmerieolly or in tonat of ,tho knqim ^ontHieii Vhns a^xhpBiW hayo an eqjaadon whonin • it aa l^iAnuMM fuofiiN^; the valae of x in the equation if ieadilj fopad, for if 2a; == 8, x most equal 4. We have fUMeriained the Talne of tbennluiown qoaatitf, and bf , i§o doing have (aa it it termed) to^it/M or M<««i the V oqnation. 9o we might reqaire, to find the valne of • where lcssm one Bide Of an oqiiation anj quantity we pleaae, provided that we inaint^ the eqnalitjr by adding to or anbtracting ilrom the other Bide of the equation thetame quantity. The rtason of this is erident, since the two sides of an eqna- tion being eqnal to one another, an^ addition to eaeh ilde <^ the same quantity cannot aflbot the equality • mibsisting, i. We may multiply br divide one side of a^ equation 1^ wiy quantity^ provided we maintain the equaiitv bv ^ ttoltiplring or dividing the other side by the mmo 5to*ntl^ rf «««» . > Any teriii may W trtospbsed irom on^^ri^^^ equation to the other if the sign be changed. For if « + 7 =;: 8^ an4 in order to solve the equation ^o wish to transiWBe the r from the first side of the equation to the othe^, we in fact subtract t from the first side, and therefor^ must be careAU to subtiNict it from the other rtdcrtpo, :thus * == 8- 1 Weliave in reality transposed 7 from one side of the equatio^ to the other, ch^nginir ' Aits'signs^ • ■ ■ . -^•"; ■■-■ 4- '^^ *;- :/ d.^he signs of the several quantities in an equation W be changed, provided the signs of all the quantities on Iboth sides are changed. . id. eHrnp l o eqn a t io nK involving one unkno w n quan- tl^ ar^ lolviBd by the following .^, , ^i, , JTy .■■v > is ■V s^A. — - BtTU. n Vihin an anyflradioru muUiptjftht eqmUm 6y ikt dfnominafor or leatt common dmbmifuUorf tq a$ totradi' etdtthefraetumi, ^ , ^ mfenoitn quantity may bt on tht 10 hand or ftr$t nde of iAe cftialim ami ■• ■■";'>^/, '■ '■.■ ^-^ = 24' a 4" * — r •!■ 9 ■ J.3x=: IB-, '■-.vaRaa-g ''•■-■...,:^" - V^ ■.-•'■./'■ '^'^ ■ ^^■'■ V«^«-^=a4 ^d the Value of «. TranfpoBing ^/x-S^H^b Squaring, to get rid «f tike radiciU efga «-.8'$s 49 In this mmple^ aAep traoflpoiriiig^ we find it neeei- fazy toget rid of the radical liga ; to dd tfaia wtfttral-' tiply one aide by vfi»^i*nd the other bjp its equivalent Yi in other wordvw»i9urt both lidei of the equation. Foriince wem^jMitipfy Ibth lidei of an equation ■IT': 4 *• HIMBUI WVJAtSUXmL ibj the Mune qaAntity, obTiooilj w* maj tqiiart moIi' ■ide of the eqaation if deiirahto, fllnee tn fo doing w« nroltipl7 eaoh lide bj equiralent quuttiUM* —^ ■ ■, "'.. - — ^ BmioimXIV. - jk_. .„ 1. 2x + 8 = '+9; inddft. ifgj am 8.^+ j=;U}fliid». 6. ax-4c^a6-c; findx. * t. 8 (a-») = - -I- 8 ; find X. 8. Sox - 26 + 46x z= 2x 4- 6c ; find X. t. r4>- =s^* fittdx. • W. 2Jc+-g + ^ = 2-xj findx^ / X4-24 ■ «» 12. V^r+ 2 = 8 ; find X. ^ 18. 8 4- 2af =« - V*^** i •■* * W. JC = «"-*"+^J findx.^ ■ 8-x^' .XI* 15. X* 4 — |- + T ~ ''"^^ ^* ""^> *"* *• ■ ;, *-f^e 16-8X 25 . -i % *> 12 6 H. 4*-yi+«— afVi+«^; tod*. - " 1 «x + 2y = 22 ^B j Holtiplying the fint equation bj 3, 12x + 9y = 93 ^B 1 Ifnltiplying tbe eecond « by 4, 12x-|-8y = 88 1 1 ' ^ ^abtracting y=:6 f 'i- And since y=:fi f 7 ■ ■ , '*^' , 4x+16=s81 ' K* . .•i4ar = 16 t ♦, • ■ ■ ' ■ • .•.«=-4..^ ■ \- " - ' * * BnBOinXY. * A 'i l.*-y=:l * > «+y=9; finds and y. : ' ' >. / ' ^ 2. 2«-F8y3s81 ■ /-.r '■' "- ' ' •* ' r i Hat — RV Ifi • ffiui -r anil u •* • *»* •^■~ *• > •■■• • a«o y • ^^l^„ ., . -, . "J"--- , 1 \ f ^. ' ^4Vft. ^MttiMidH^^v^>» SS^^^Ik =■^1 • > f^ S^SW* '''*' mmnm wmoMmmBJ 3. |4-j=U 9 — - (X «y) = 4 ^ find « ami y S^dx + dysslO • , • bXiJ^ey^ 8 ; fiad « and y. iiix *. f^= <{; find « and y. «+ 1 Y- - 2y 3? a ff-^ %■ idij |-^ 4-9= 3 J find « and y, jte 4- 7 = if - 3 J fifid^ « and y 9» «+y i=a • te + cy =;:i(e j find 9 and y; 10. x-fy = 2a " a^-^=6; find X and y. 11. 8a;*-%=sl8 ^ ^. . to+TyssSl; filldxandy ia, fi(p.^-8y + ais= -i±l ^'-T^^jf -I- I ; ftidarandf. -It^ -Mmu n • ( M rnnvha wvoAxiwm. «t 17. If ir* hftTe to find the raiiiet of tbree tmknown qaantitiM, wei miui^ u we haTe jeea, htkye three inde- pendeiit eqnatioiii. '. IB. We jolre these equations by taking two of the ^nations «nd thence obtaining an equation inyolWni^ ::oD^ two of the unknown: quantities j we then take anoiher two of/the equations,- and thenoe obtain an equation iuTolTing the same two quantities ; thus wa^ bbtiin two equations inVolTing two unknown quantities. \ As we alfeady know ^ow to solve these we are able to Mce|tain the ralues of two of the unknown quantities. Bj substitutibn in one of the equations of the values jdready found; we obtain the value Of the third uikknown nIEi.. It>ls not necessary to give a sn^ific rule for the •tolution of these equations. We ji^procet^ to show by examples how readily we may reduce these equations to those involving two unknown quantities only. ^ ,^<- ■r. ' :■■ ' ■ZAMPU. : , R«-2y-fi =8 ; fi' '■■■-•- «4-3f+« =9; find Of, y and if. ing the first equation by 2, lOx *^ 4y + 2«= Ift . By the li^jond eqnationi .3a;-f^2y-2a=s 4 Syadditib% 18x-2y=:20 • By the second equation, ^8x+ 2y-2x=: 4 Multiplying the thlM^uatian by 2, '$»:^ 2y + 2« s 18 : ,/ . Byi^ddition^ (kt4-4ys22 We have thoi eUminated * and obtanred two eqoi- I" " * '■ aniPLl 1QUATI0N8. 13x- 2y = 2Q . giving i^^ we find « = 2 and y = 3, and sabatitut- ing these va^es of x and y in the first equation we obtain 10-6 + « = 8 .•.« = 8-4=4 .•.x=2, y=3 and «=4 BZKBOISI XVI. :/ U2x + 3y+z=:Vt «+ y4-«=, 9 4* - y-.«= 1: find X, y, and 2a: + y 3 +«=ia 2Ltf^,,^ J^4.2«=:13 2 ^+4j{=s:9 J find », y, and «. 8. . -■'3 ±1 3 V, ==2 = a;-|-3 8«- (x4-y)==14 i find «, y, and «. 4/*^* V?l^ Vi=29 * ■; .;*^1Jy+^3«=:'82 .■ ' ';: 2 + I + ^ = 10 ; find «, y, and ar* M -*;•> flWtt OHAPTBB VIII. " ■' '■ ■."■■■ ■* . TBOBLEMS FBOIttrcni<» safFLU BQUATIONS, \ .^y ■■■■■'- . .]- .■ 1. W« eMi MOir pneticAU/ ft|pty our' knowledge df •qiMtioM to the folatioa of Mithmetieftl problem!, OerUiniketf beinf gireft in the qneetioB we hare to find ■ovie foftntitf or qunntitiee unknown, from their ielttion to other qnuititiet aa ibown ^ the problem, S. There it no general mle for the aolntion of these, problem. The ■todent mait read eftrelbll|r eter the torma of the eqnntion, and then pnttiag «, or c and y,: or «, y, and «, to lepreaant the onknown quantities, he most express In algebraie language the relation sobsist- ing between the known a94 unknown qaantities in the problem. He then hat an equation inTolrihg one or more unknown quantities, wfiieh he akaa^y knows how toBolre. ^^^-^ ■TAin>I.Bg> • •' ■...■.■> ■/-•■.■ •■::•■, ■■(»^' ■•^ -■"-/ ■ .■ ■ The ram of two numbers is 20, and one it tiii»-third» of the other. What ate tiie numbers? ;\ Let jTsone of the numbers.- ..»■.■ ■■ ■ -■ mien by Ui^ question -^ = the etiw. - .^ , to ■ ^ .e» ■■■■ > A *'.- «aiPLB IQITATIQ ^ ' \ 5«s=eo . • «r r *.- -■ ■/ \ Therefore It fti^ 8 AM the namlMiii I S spend every year nine-tenthi of my iooome all bai i|40 *, what I save is jatt $20 less than 0|ie-foac]th of my inopme* . Ho^ mncfa do I r^oeire cer annimi 7 ' n ' I»«t IT =sriki^ income. ><: Tto? by the qnestioi^j^-40st: what I spen^* . ^^t by the tnestlon j- - 20 a what I sav** 4 to ^-10 -»-^ lfnlUplylngby20| 6lr<4iti00sr 20«^18x + g()^ . te-20«H-18ar3^0 + 400 ftb£3l200 . |te-'- ;^, # "l=«1^» |rhi •:. ■'*■■«- --ax tim-^ifae sis lliidAtfirii iv f. , ''» . - .'f « • I' ;i ^ iridualB ; if there Wfri^hree nior*.«iMrft wo«M gei|#J4Ml«r>Mfl thatf he rec^ bii^ if there were^fwo feiiihi^h i^oaid receive ftdoUar moi^; Mbir iufif0notm^ the)re, ftud what 4o«« e^ch no^irel ." .,i\^, .' .I4ei«:=-the namber df |M^ii8| '■■^u/^;-:;.^^^^^ \'£y '^ yfinA%:^ what each reo«^Vpi^ ^ ■ -i ' -^v' ^ > Then the lum to K» difi^^^^ls^y. , From ^e Uniia^ «x«-|- By ~ ^ cj xy ■■'•3 ' "' - r«+.3y^ From the seoond d^::.■ -v^ „-^- :V ^.% 8f ==6«' . ''..^ SubStittttWtlils Value of y ie second f li^-ptss^Vi ^ -■ r '-i-'fft" ■<■" -^-.^ . - ;^ •:-:'„ '■--.4: - ■. :; 0*=: 18- ■-.■;• '■ ■ *e ?^^^^^%^^ tie pumber iiU. > y ; ■fumM v^niere ii a iertain fraction ; if 1 be added io tba niun^toi; itj^gpttt^ii but^tf^^ be added to the deno- IJ'i; What is the ftractioni . V |i#ij-sth^ft«oti^ii| **^>^^ -ife.^': *■ '^\;- :^- ^\. ^% ^. /-^ From the flMt 2« + 2asy Knb the Meond drisy4.8 Mid«att£ ■;:■:■:.•.:;■■■". a -~t"- ■■[.■■ 8y-6 = 2y4.(j 1 ■ y=i2'-"^ and the fraction is therefore '^. BoBaiwi XVII. 1. Wnd a number fuch that i of it ihaU exceed i of 2. What number is that vhich being dk ided br 8, fud 6 added to the quotient, and the iuiff then multi- I^ied b/ 4 giTCB 60 7 > \!* ^ ^"*f^* ^®®^ •* ^-^^ P«' cord; if the amount I iWd out had enabled me to ^rcbase 10 cordi more, itj^uM hare cost me only $3.00 per Cord. How many '»**^ did I purchase 7 ^ paid an aeoonnt amounting to $114.ci6 in BnglMt ■preretens (at $S.00 each)^ American balMollan, aad Canadian twenty cent pieces, ushig an equal number of each Coin ; what was thft number 7 . 5. TheAsum of two numbers is 23; o^thitd the greater added to the less is equal to 19. t4at are the numbers 7- X ,.-. ■,.,.;•.■• ■■[.^'.^.v ._. - ^,Tr Tfhjoie WM i$ ''^ •ad difhrenoe ^7 y gmifiM moafnom. yji T. A mill WM Mfeplioyietf ftr Sd lliyi ; ^1h lUf Im mnrlnd lie Ncflhred a dotltt, «wli d^y >• WU liMl ftv itofMttd SO eentf ; he reeeired %t the eUd of iHit Ittttn 14 doltftn. fiow naay dftjs did %• woA t 8. H&d ft fraeHoii nitSi tbftt t nrtrtrmeted from th« mimerfttor baket U i, bat If SO ti added 19 ttae. Aeoo- mtsfttor it iMcomef |. i^i A penoa hu two hones ud » itelgli worth IftN^. If the fint hor^ if hamefied to the ilelgh they tt* worth three timei m nmeh ai the ee«o&d hoiM ; bat if the eeeond horse he pat to the slefgh th^ are Wt>rth exaotiy the Ti^loe of the Arst honOk IVIutt li ^M^ horse wortlit |K 10. A nomber consists of two iKglls whose sihi H f : add 68 to the namber and the digltl Ij^feiMWi ^ tetted. What is the nomber f > 11; A and B hare e|eh a certain som; A aiAetd ft for 16 dollars, so that what he Woald t^ hfttf night eqaal 5 times what B had. B in repi/ailssd A ibr s' ^narsi BO that the lorn each had might- be eitdil. .What snm ^oes each possesst ^ ^ li. AflPfn po^hased twp biitt^ib#iyf i i^ofatiigl;.^ piUd fbr ona of the top twice ^ as for the other, and fbr the hoiise donlble what he for the bailding lots, while the entire proper^ ccsthhli ). ir^t ^as ti^ price JfeMtt Idt and of the ►T ■:■:;"->>■;: ^'■^^■: ..:;--^^'^^: ^/- V' v.-^-'^ '■■'."- ■: jie nombtt dT totes poiled At a fccent ele^tloii \ the soccessflil candi^te had a minority df gl4 }"how nuuij' totes were recorded tor each c^d^ datet. V . _ ':^___ „.w\ ^ L: ■■■.. . . ib dlifuiatging^ lom^ aooooBte X j^ild gWlV tMh |,i^|,aiii|ofteytiM^. llritlMkiiil riiyiiliidiit^^l^ . '^ . . hoaie' Tf' Il"^ W BIMPJM IQUATIOMtf. *'% :# / V IB. There are two numben; twice the grtAt«r If 8 J0M than four timei th|J^|^iit if to three Umei the gw»ter you add twI^W fei^ ailtdWde the lum bj «1, the quotient wm be 4. What are the Humben T 16. There are three numbers ; the filit added to twice the lum of the other two amounts to 41; the Wcond tnd twice the sum of the other two equals 45 ; the third •ad twice the sum of the other two gires 48. What ■'. :tte.. the numbers 7 . j^ . "■• ■* ■ ■■■ f^ .'' iPl ^*> P^<»« ^ »^hUeB'sisXlOW. What rati ofinier^t does the slockin which they hare iuTested jJeld, add wikt rata of interest are they paying i^jr the ■wmv that the/1ia%bOnowed^ •«i^ 'W'*^"™"**' consists of three dig& If «M be iHid mhB nuAber its digits bf)come inrerted. The middle digit Is equal to the sum .of the other two dlTided b;^|jpd If the i^piber bof rTded by the sum oCits 4lfltt4he quotieiitlSill bjt^'^I'in^i ^ber ^9. The sum of twd njamnig «,* theifdifisrtnce *. IHiatawthvt: '^..W ' - p^ ao. I find that I «|fetefc book printed ^^neh a i>ag^#«nd l|pe AodoUtoi which I can spare a»r the e^Epense. I fttd, howerer, that the book Is so •ztopflTe^titwUlooBtme 800 dollars more than I hai%, and I am compelled to reduce it. This I do by cancelling 2P0 pages. I then find thitt I hare 60 doUars moM la n.. »* + ex + 4 = 44i flid*. Htre trftniflBcrisf Um known qnanUtj | x« + ex = U-isB40 Adding the iquare of half the ooefBoient of « j x* -f* 6x + 9 =: 49 Bztrftoting the root x + 3 = ^7 1 x = ±»-8 = 4or-l0 (3) x'<-iMB=s(f ; findx. Oompletlng the iquare Bztracting the root «*-|w + -4=«+T .*-r=±\/f+T (») •••*"=I±V« + T x + V**+V=llj fiadx. Tranipoeing X in order t6 ■qo^ re Mxd tluu get lid of the radix Squaring Transpoiittg 4x-|.iaaiai-aa«+x« - xt-lf aex := 120 «*Ua6x=s-iao x> -: aex + 169 = 160 - 120 = ^ y^i8=it' ■;' r xf=^i7s20or 6 7. When tiro noknown q^antltiea are inrolred iii ft Sinadratio, the lolntion maj be made according to the i»rm of the equation l^ dimirent .modes, and nie moit — >c ticable mod e of lolntiofl " ' ' Completing i Bxtractini^ th *",,''• •ifc*^* Btion U the IbiBi. WtgMX^M i* lin^ eqwittinf , tity, and thng lolTe the eqiiMloii, orw* maj plOMtl by • iMkUtt method, if pnoUoidUtr ^^^sM. ■ s-'-j'.: 76 QUAORATIO 1QUATION0. J. *:. *' J 0) . **■ „ . / 'y- ^' find* and y. • Here •Inco/ayzz: 6, by adding and subtracting 2«/ =s la from the first equation, we obtain «« + 2aT^ + y«=:26 .-.x+yz^jiS " «i-a*y + yt=-4 .v*-y = il Hence2x=^eandx = ±3 i •ndaysriiandyssia' ' **■ ' <^ . ' ■ l ■« - '' " '' ' < 1. ««+.-te = 21;*finda:. / 3. ««-r8« = 9; finder. „ 8. 2*«-4*+l8-34j find< ■a * . * " *• « + jrii = 6 ; find a:. ■ ■ ^ ■' '■<».■ 5. »»-a«;=aj find*. I C. o«*-ft«=5c; find X, 7. ** + y« = a6 > . . • «y=i2. fihdxalndy. .. 8. ««iyt = 72 i * + y = iaj firidaraidy. ,^'7?^ ^ *^** nnmberiironi the pquare of which if i yon deduct 6 times itself tl^e remainder is 40 ? ' .^i^iK^**"^ *!! °"°»^" ■'^^'^ *^* *!»•« diifew^^^ and their product 240. ; , • ■ I I. ti* ^*^'' numbers are ^o^e the product of which !• 34, and the sum of their squares 148? ™^ Ito^L^t** • T^' "'^'* **** if 7* take 12 from / • A .s- . .1. „ .^■f' ^ ><'?''.■ ■i^y. , Tim nut JJ+^ffWT-i^ '5»f •-Sv ANSWERS to THE EXMCISIS. r « V > 'fc. !..-«. iij \ i. i ; «/ li J 6. 861 1 1 8, 1^1 0. aV; 10. 28; 11. ay la. -82O; as. saO; H. 8«{ w, 80} 16. aaj IT: lai i8.^i x». ioj, aa. 0} ai* *i ia. ITC; as. 18 J i4. 16. / ; ^^ ; BXi U.i-^1, 186 J a: - aaa^f 0. 1606 - 8%|H|. 8*V+ itd^; 6. 90^6 '^ dial*, 6,jr«y»4-l(ll«y'^-il6yj»j 7. 14a« + 14flC;f (^; 8. i^^-t^ 10^4- ae^j. J^ ai0-86 ^ ly- T«^ ^ . Bi^ilf^. aaier^a. 14,3(f'-^y f9, pm'-i^ 6. 8rf6-8*yj 6. 18«y+4ry+ l/ t, 0*?ip.V^i^^^^ aVi; 9. 4a6 + 3Ky4r«y4.8«. * * V^ 4 Bx, IY.^1. Ua4-9fr ; 2. 10x^>«; 8. 16<^.»|- asy-f 16a^ya_asyd + a6c| 4. llo»+ 46*-6a64. ia*c+ a^y"! 6» 64- 16*y +B«V + 10«y« +a»«y 4- «»« } 6. 18a6 - j»ali 4- aia»«+9oc»+96c ; 1. 21ax-\- iay«4- M + 4 I 8. 8*+ 6a:y-6y + 3«; 9, YaS^13a6 + aa6c+6crf+a 24#»-i«a , *«*» \. «*-y / »V "ity 9. w^-f^. ,^ *f±^. ♦- ^'^-^^ lOx ^lt». aofr 18. 8ai 18. \ i ao. 55^JZ«>5 "- *> * ' aa. ^ , »«• 8a' 1 '^ fc XHI.— 4af-4flfr+^; a. M + at CTt-flrf 7. iraslO, y«l; 8. «=£a-6-10, y = 2a-a6>10; Pl^/:nt.^l. tfasi, yasf, ^s4; 8. «=bB, y = l, Mm$i 8. «s4 yi^l^ iittSV i. »m$,fm9,»s: 18. Ix. Jrrn^l. 8«'; a. S?; 8i 98; 4. 18; 5. 18 mnA 8;^. 9ftnd5; 7. 16; 8. {; 9; flOOandfSO; 10. 18^ 11. $86 and $26; 18. $800, $1600 and $4880; 13. 768 flftd 586 } 14. ^Smy 18. 60 and 88 ; 16. 7, 9, and 11 ; ' „;■ * '8; 8. 9 and 8 1 9. 10; 10. 30 and 13; 11. 13 and 3; U, 12, ^ ' THB BNIh -tH. JOBll hO^Mhh, PBIMTin. s> 1 ■ ■' ■* '" ■ \ \ * I . « '* :,,>imMMii y«i',''«w|i =8; 8. LOc- 86 «e-6« > id 26 -10; 11* • m /'.v f »sas 19* Itf and i 10. 1$; 18. 769 •Bd 11 : ad 1 tor,- 2 J r. 4 And I and a ; >-..'.'^ ■■> ..^;' ■^-5 ^ ■ ■ ». > >• ■^■::.-., V ■."\- ■■•:t.-" y ^': x^;"^:.-.. i; ;/, f^ *■■■ y*-;-" " '* "■"-■/ ..-. ■'' '■. ■*•*'■"■ ■*.' ' .' "*'^.'--''' .■■V\. ♦ «" '. ,-'.v>>-.-.^ ■. -■ f. .s •" '• -^■■'- ■.„■■.,"'>.- "'rv*-:'"" -'I' ■-■■■.■■"■ • <>■'■■ .■-•.-: -.5;.,; .■.., ., ■ • • - „ • . .,;fV;:>f^^ \'-: "■ ■ ,.''."' ■ -■ r - ■\ - . . ■ ■ ■ . . ' , T' ■ \ . ^- ■ ' <.•' •■;;■" ''. ■'■ 1 / ■ - ' ■ - ■■ ■ . . : : . y^' ■ ' ' '! ' ■ '''■ ' '■ ■ ; . ^-.^m ■/-/:, . \ • ■■ -"".-■./'■; ■ . ■ ■ '. ■ - • . ■ ' i- . ■ ' ",; ■-.'^'; -■.■"■ ■ ■\ ■ "■ '■ '■ ' ..■ ■ . ■ ■■ ■> '■V ■ - . ■ / • .9.. ;.'_■■■ ... "•-.-"■ \ J - . . I ■ 1 . ' . '■" ' ■• , r ' . ' ' ' ' ■'■ "■ , ■ « .• '- ■ ' ■' :.■ ;'•%■"■.' ,, ' ■' ■ . ■ ^ ■ . ' ', . ...^.■' !^.f,. ::