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Tha following diagrama illuatrata tha mathod: L'axamplaira filmi fut raproduit grica A la g4n«rositi da: D.B. Wtldon Library Univanity of Western Ontario La« imagat suivantas ont At* raproduitas avac la plus grand soin. compta tanv da la condition at da la nattat* d» I'axampiaira film*, at on eonformit* avac laa conditions du eontrat da filmaga. Laa axamplairas originauK dont la couvartu ra an papiar aat imprim*a tont film*t*an commandant par la pramiar plat at 9n tarminant toit par la darnlAra paga qui comporta una amprainta d'impraaaion ou d'illustration. soi? par ia lacond plat, aalon lo cas. Toua las autras exemplairss originauK aont fiim*a w commandant par la pramiAra paga qui comporta una amprainta d'impraaaion ou d'illuatration at an tarminant par la darni*ra paga qui comporta una taiia amprainta. Un daa aymbolaa suivants apparaltra sur ia darnlAra imaga da chaqua microficha, saion la caa: la symbols -♦• signifia "A SUIVRE". la aymboia V aignifia "FIN". L^a cartaa. planchaa. tablaaux. ate. pauvant Atra filmAs A daa taux da rAduction diff Arents. Lorsqua la document ast trop grand pour Atra raproduit an un saul clichA, il ast filmA A partir da I'angia aupAriaur gaucha. da gaucha A droita, at da haut an baa. 9n pranant is nombrs d'imagaa nAcaasaira. Las diagrammas suivants illuatrant la mAthoda. n 22% 1 2 3 1 2 3 4 5 6 /W9^ '^ i ■^ ■% ^^' .^^•^^^^ //5Z^>4^5f^^i^^ I m i ■•!!■.. t.l ri « i me •7 t 'A. _ "• S^ *'»V Co.'. JRXhematiniJ 3etit.. PUBLIC SCHOOL ALGEBRA ON THE INDUCTIVE METHOD, INTENDED AS A.V INTRODUCTORY SERIE8 DEVELOPMENT LESSONS To for. « ,„«. ,. .„, ,_,.„^ ^^^ ^ ^^^^^ ^^^^^_^^ to largtr tcorkt. n C. CLARK80N, A., />/««>/ of Sea/orth Colfegiate InltUnte AND rOHUKKLY Princi^l of iU Provincial Model School, Torant. TEACHERS EDITION. Thk W. J. GAGE COUfPAMv /t>^ . TORONTO. I -40- K,.t..r„| «,-....„||„^ to Art of Pnrll„M,.„, of CnuMln. I„ ,h. offlre „f ,he Mlnl.tfr "• AKriniltHr-. by Thk W. J, (Uok (^oupanv .L«mlt«|,. |„ fh. v..«r«„e tlioumiiHl Hfrlit hundred niid iiliipty thni'. f «l INTRODUCTION. f A» Uhh iNMik id iiitoniluil Ut 1» an ni orul teaching, all doflni- tioiw, and sU oxplauatioiiH of iiiorely inechauical iiiatterM, am omitted The author in convinced that in a first book of algebra, printed explanations aio comparatively UMolesii and are never read hy the pupil The exerci>«e«i are the only parts of much coniie.|u«nce U, the learner, and accordingly this book contains almost nothing but exercises. The pupiU' previous knowledge of arithmetic is a sufficient basis without ptjstulating a aeries of abstract definitions. Upon this basis it is possible to begin, and the pupil may be lead by a pmper «et of questions to find out the facts and the generalisations of algebro for himself. Thb ouidino principles arb to follow tub LINE OK LKAHT REHIHTANCB, TO HEBK PRACTICAL APPLICATIONS FROM TUB COMMENCEMENT, ANI> TO TOHTTONE ALL MATTERS THAT ARB ABSTRUSE tu a second and more advanced coursa Within these lines it is quite practicable to commence algebra much eariier than IS usual a^ .; .«ent, and the pupil's progress in arithmetic is assisted by the li^jht thrown on the rules of interest, discount, alliga- tion, proportion, etc. No teacher who ia the slave of the text-book will be likely to approve of this one, which contains no provision for mere memory work and aims to appeal to reason and intelligence at every ste|w Very few such teacheni, h<.wever, will probably teach algebra, and therefore the author has hope that these exeroises will find favor To be useful is his highest hope j to have aooomplished that end will 00 of itself a rewari ih ••f I 3 .t 4 I II PUBLIC SCHOOL ALGEBRA. FUNDAMENTAL CONCEPTIONa metio and to genoraliM il bv m«„V!j fi? . •«l»"«l in •rith- The obj«t olfl^iT.!^i,^t:^^,:^J^S^-^J->l^^ the connection between th« nnf«»i«« * -Ii. .*'"'*' '"^nnw general notation oTXbrt T>.i . "/ »"f »»n>«tic *««! the more do"™? In buying lSt» Mf I 'j "T""* «««» by the «i2/l« ^.U^-S^PlSvl' I"'":."'™, choosing th.t WT.r.1 9 .PUBLIC SCHOOL ALOBBRA. I i f! »m.Ul ones mate one Wg,Jt'*«™™^y?^ How many of the flock fe too diet.ntTb^'loJn't^''^" r"'"!?- ^i*' '"^ • third thep:^ntth.tth?tS|:rk«^^-*-j C„we„rf« yet. Will it do forX'D«»?f ^,'""'"*^'«''e™-l«enmSwSrf mil«, from ^ .„d y See^^m i»? "^ """ "» ""UBSto'tal (") If three townn lia in • i. . « . rthlJSiS'*"" *=«^' ^"«» ste*t:^d7'^i^9f;_'S3' (1.2) PindthevalBeofa+v+i+frj.. i. f f FUNDAMENTAL CONCiaTIONS. ft sound proems 5fLdu!t?ofi;i'^""i^^? "' *' ^^ '*'' «*«' »>3^ » lead UD to a df^fini^T^^A ■ '., ^® ®^J®<^* o^ Exercise 2 is to aKic ways^ ren^^^ ^^'* ^* multiplication and the itSf a ceH^inst^Z^^^nZ^Tt^^l l' ""'*' that contains in ^u-irrnV:„^;^^^P-^^^^^^^ 13 thi« Inm^sIS^f ^f^'*^ ^^ ^^^y * concise method of findinir the 20 ?n 4x5-0^' unilV*^*'-*^" expression corr^ponding to e ^, in 4 X p - 20, unless we give a and 6 particular values Sxerdfe 2. titioM of the unit doe, T^pie^lv ' "^"°- ^""^ """^ "•^- n9??"4t.T??*'""™'"«'<«' ">>»»•>«- 2!/ mean? 4x? What does cd mean f ' » ><» o*^ -» wh«t la meant ? of wlSlSft w' ^^JTw-^ the number of cubic feet in a pile (7) What is the price of x horses at $51 apiece? (8) What is the price of x horses at |y apiece ? <9) How nwiny pounds are there in x tons ? he troveUed ? ' "'"«""'»y '•' »*« ne»t 7 days ; how far has PUBLIC SCHOOL ALGEBRA Iar^^r«n?^^Xn r ,t^* l!"^^^ --- »--er quires a ex«t between e.,^;^^,^^;^^^^ specimen of the kind of drill r«^?S^?'*- Exercise 3 is only a ^«^rf on , between Z^^^ tint . \*-' *^^ *^«^«' be should g?ve8^U^r ^^"?^^«- « tbe topic, taking extreme and iw^JI *^*^ ™**™ Wessons on this impressionof ExX It^^J V *^ examples to re-infor^ th! rjsi^nd time in con£tn^T^;^Z^,r^^^^^ Ideas are not grasped thofoughlv Ihe nf '-i \^^^ fundamental factory progress in the later sfaX "*"* "^^^^ ^^^^s* SzeroiM3. (3) Is the 6 to 6« riltefr ^ '" ^^^"' '««'or ? (4) What .r. .^f ! " • n»i»«rl«l hctor f <*. ' of the cube Will it do f^r shoAni t'i'c.U^^TCrcX'.ir '"^""'^°™- (20) Find the value of x' + y' +z; when x=5, j,=e s = 7 power," or simply "^n toJSe W^4 '"™'' *" *** ^•"«» t«t''cS!-^r*^orhrSi\"-,T^^^^^^^ (23) In 5» how many equal factors? Whaf tir.^ ,»f * * does an Exponent, like 8^ 5», indicate? ''*'''*^ numheTotthe^XqAiu^f'' *^*P«'»«'»t *» «8ed to show the the difference between a Coefflclent and an Ex,inent f '' (29)If lx2 = 2», • 1x2x2 = 2', 1 x2x2x2 = 2% etc., explain fully what is meant by a', a» a does If a»=l xaxa 1"W^ ther^ Verify his solutions by 8nb«t?*nfi**'"?' •'«• *« tMiiV?: Jar value wij, turfehetJu'S'llTa'S^ntttr' •" •^'«" is added tolch side oTthe ej;'.l?JSrwf r 'r'"^ ^^^^ 45 (3) If 6..3a^2., what is the" lueTf ..r '^'"^'"' (4) When a-6, 6=3? of a: when a-4, 6=6? (5) In the eq::uiion S^-xi^i a . ^« equal quantities, and nex1subtrif« ?' *"***'***'<^ ^^ from the the value of ar/f' °®" ^"'**'^»«t 8« from each side, ^atis (6) In the equation 2ic- 7=^ 4 _, (7) In the equation Sar + fi j. 7 ^ ^' *'• erical quantities to the right In^n.,.!^^: *«Mpose the num- and thus find the value if j.*^^ *^^ ^^^^'-^I e 100 year! Sr nuaiuer. "» —— »«e eqaaiion and find the THE SIGN +. » Sn** . y.H and o, .be^X^f. JJj! "^nd 'x'^^^ c.i^^8t™t".nTl^b^tf S * '».'■»■'« '•y '- P'-fi''. which (17) Multiply every term of the equation «g^^j^:j?Mj«-i-rnt^j-^»'«-- po!^i,,^httJ;'2^' Sl'l^ P-fr'! •'" «!: weighed 9 and half the hSi .- j'^i^^^'j '"" *''* *'"^ »» °«'°1> «« "le t»il t^C Pin? Hh. •^".'^y ^ '"»'='' «' ">e head and tail together. Pmd the weight of each part and of the whole fleh Of the new language the pupil is learning. grammar IsoroiMa. I, 2?3^4f 5^ 6,%T 9^^ ^^ ""^"^ "^^ *•*• **^ ^'"* t'^* sum of 8 "^ niBhlC SCHOOL AJXJBBRA. of i.„ + .%+4e+5rf";fe ''' ""*' '"•' «»d the numericl v.l„e "'-'*^'I^''<.^:r•' ™'"'' "" "■« -™'«^ «.«.i to .he ™™ of J']3'!i'?4ltr"' ™'"" "-^ »•" ■"""'ric, exp^ion for («) With 'he «me val„e^fl a the „„„her e,„al to (7) W,th the same values «ad the number equal to numUaf ^ft"?™'"" °' «»=+»«*+«*«+«* with the same (10) Express x pounds + j, ounces in ounces. (11) Express 6x+3j, dollars in cents. (12) Find the numerical value of the product of % *x £ . " /rite fhie «^,.« ... . . * c d e* ■ttr ., ,, . — -F*"«UUI, or T x-x- X-. '-'?.""' '^'"^""'■' '" • »-»'«' '»™ -ta« some^other s^nTw ^^mlXZ S.tnd'^ht'iX^ ^-^ »- '«' -"' Oe "f X = 18, if X = 20. ' """ nw father's present age, if a, = 16, (t4)Solvetheequ.tio„JxH.ix=x-3. the learner into the difficulf 3 « *5^^ imprudent to plunue negatives quantitfes i^ ^T'^-^^^ "''i?'*'^ doct?toe S Exercises 6 and 7 civethAnwi ^^"^ *" «o'n»;ot departure? ' when his debts L Srpafd? * "'"»' *''" •"« would have oh£U'ho^!i™h"'L"vel'';;^e^/ J*^ ^"o •'"ck^.doiu™,. day received f 18y and paid t?rt T « !^ * *"*^ P*^^ »lly / 2nd Paid|75y; 4th (£y,^*^ed ,^^^^^^ I4lx and received |18a: and paMllUv^ 2?? S"^ ^*'^ ♦^^i'' ^^^ day, paid$41y. ^ " •'iiy, 6th day, received 1253a; and proceed without very ^^^J^^}" *^ff?>'» to.enable him tS of arithmetic comes to his a?d^? -^ . * Previous knowledge prepared to use this new ijlStl^^S: |^^^^ *«d he wiU soon te problems formerly solved bvtlS^^fi!?,!^ «?"®"^ solutions of Exercises 8 to 14 supp^TfurtW^^i?- '^^^^ metic into algebra, ayiJvea^S't-5 *^^^^^^ simple equation and eMvlLVl^ ^'^^"^ knowledge of the ?^e method of exp^i^;^he?£^^«»^^^ ^^e^tions m a problem by means of a«^ ??^ ^t *^® numbers involved exhibited. Eirc^Sli^L^''-'''' ^"^ ^^^ been^Sty Most of the equations in ffite? i?li* *ii**«|*"»« equation. of this kind, and a goodS^fllf^^h"^ *«*°«^e«»i8e directed to this translfTnofprobl^^^ should be lation of equations into problems ti."^?***''"'' *"^ *'ans- the equations without muSh^ ^"^*^ '"''' «*«"y solve (2) Bx«+3;«3+2^,!^,f +3«-^-4a-to -4ar«-3aj»-^» -Saj' + Ha-* _6a.» + 7ar=' (3) 4a& -4x* •5ac- 3a:»-2a;». 66c-76c-6a6-2ac + 86c + 25uc-n6c+6a6. ADDITION. 11 m (4) 14a« + 156* + 16c« - 12c« - 116« -9a* +Sb* -9a* +2lc* + (6) (o + ft + c) + (2a-26-2c)-(3a-36-3c) + (4a + 4ft+4f). (6) Ha + b)+S{a + b)-4(a + b)-3(a + b) + n(a + b). (7) *a-J6-lc + la + Jft+Jc-Ja-Jft-|-Jc. (9) o-26 + 3c + o-J6+|c + 4a + 26 + Jc-Ja. (10) tK+£jj+« shillings + w cents + a dimes + $135/^, taking A 1 = »4, Is. = 20 cents. Give the answer in cents. (11) -2506 + 0+ •996 + 3a6+2c+ -016+ -Ibab + b. (12) Ah.+B yd. + C in. Give the answer in inches. (IS) From 6;c» + 4a:+7 take the sum of 2a'+4a:»+9, and ^jJ X + 4iC — 2. Iztrolie Q. (1) If ac + 45 = 4x, find the value of x. (2) If I had $45 more than I now have in my purse, I should have four times as much as I have at present. How many dol- lars are there in my purse ? (3) If 4a;+|a; = 176, what is the value otxf • ^f^ \^"Y?5^ ®^ ^y "*°^®y *^<^®*1 *o one-seventh of my money just makes 1176. How many dollars have I ? (5) A horse cost three times as much as the bugiry, and the rig was worth $300, including a set of fine harness worth $48. a ma the price of the horse. the contract, but if I pay them $3 per day, I shaU lose $18 per day How many workmen have I, and how much do I receive per day ? (7) The owners of a steamboat make 130 per day eacf., but if the number of partners were decreased by five, the profits would De f 4U each per day. How many owners are there ? SstroiM 10. (1) A man walked 71 miles in three days. He walked 3 miles more the second day than on the first, and 5 miles more the third day than on the second. How far did he travel the first day ? (2) Divide $6,900 ^ ong four sons, so that each shall have $100 more than his n younger brotl^ •, 19 fWLW BCHOOL AI^BBRA. ^) Two fields toirether nMumuM ra 10 ac,^ le«8 than Zlf tieZZ ^fnTTW '»?« "T^' confiBs M^ To« .^11. . "»rger. j P''^ » three time, as much m Browa mi ■. 'J'if'." "»" «»«'' (7) A father i, now tC L J? . i*" t' ''"' *"=•• '*' ^^^ ? ago he w«8 seven tZJ ^u " h " "'l'.'" '"'" «""■ hut 8 yeare father is twioe as old m his son ? °'^ ""'" *»°'' •» ^l"- the IsnolM IL (1) Evaluate ^J+«^ 13 « ar + 2 ^ (2) If x^W, find the numerical value of the expr««ion 7£+5 9a;-l 23 ^-Jo (3) When a = 4 fcasS « o spends to the exp^ssion ' ««<^»m what number corw. a-b ^^01) Express m shortest form ia- J6+ic+Ja-i6-jc+4a+ (5) Add together to get the sumbydiSgTtowk^k„ 'J?i"°^'*»'«' "tUimpted was too smaU bjs. "S t^^^bH^^i; »^i;;^»"» "^ SsBtrdio 12. 'J 1+5," :!^>-f. ('*-«)• 1 t 2 P a tl n SJ Li li >ntain8 f them to pay toost? i; but •ice of worth 1892? years m the orre- a + •by >ted iver THK hlMIM.K ►X^UATIOX. (3) 3aj-2(5a; + 4)=2(4x-»). (*) 6(3-2a;)=.24-4(4jc-6). (6) ib-4(x-2) = b(x-^2). fa\ X X X i 3 XT « ., . 2 "a "4 a"^!* ^•''•--^""ip'y both Miden by 12. ^^) R + 1 + .t - g "" 17. Multiply through by «0. 18 6 4 (8) ??-??+l = a;_ 14 21 8 4 *• 4 5i 2 5| (11) i(«+6)-A(16-3^)-4i. (12) T*»(3aj+3) + A(7ar-4)-^,(7a.+ l)^2. Ixtroist 13. (1) Divide a rope 33 feet long into four narta ho t)i«f ♦!,« j^econd piece is 18 inches longer than the fiTst, ?he JhirS 30 fnciL lon^^r than the second, and the fourth 3* f^et longer thaSTh^ (2) The ages of a man and his wife amount to 80 years and ^ years ago the woman was two-thirds the age of her TJLband What was the sum of their ages at that time? i^usband. rSi J^^V n"»^'^ such that if it be divided into two equal *».^1^ ^ butcher bought a number of sheep for |376, but 7 of them were lost on the way to market. He sold one-foirth of the remainder at cost for $80. How many had he left ? (5) Divide 15,000 into two sums such that beinj? placed at years at o^, the amounts may be equal. J^l^Vl^ "^«* «7'^ «»» acre; I reserved 10 acres a-d «oM ^nir' w^'"^ ''*'" **'''^' ciearing ^40 more than the cosf^f the whole. How many acres wexie thei-e in the farm ? (7) T»r(3x + §) - f(4x - 6§) = \{ox - G), find x, (Multiply by 14.) 14 PVHUC K'UmL AUiKHRA. ^''•""■■Kj^^rciseM J4 fo 9i i ".ore completely the d^trine of mn'rir'^ '/''""'^•^ **» ''^^•Jop ajfficu ty in doi«K this . oL H8 the '^ «r'^°"- '^^'^ '» «» when this Hign occurs it becomes nJi ~ l"*" '* "*** "«»^- But has ru.rio.1 us beyond ihTordZlTTZ"^ *?"^.'"'^ '^*^ •^K^bra and J,at new canes have SnTo wh l-h ''** multiplication, meaning not contradictory of Z ln>T. ?• "^^ *""*•* »»^« «ome agreed upon. Analogy s^^owsfuatiu^V"^?*"'"* *»re«dy SSri^lth^fAr'^Vlh^^^^^^^^^^^ «.-.ns to in^luVrse J^^'T ^.^n^l^ ^A\V^»« '^'^- necosHary whenever a ^^n J^?"^ °', ***'» '^'"♦1 becomes employed in new cases ^hrt^didnoTl^u"'"^ K arithmetic is' 18 no proper proof of thesp «wl • u"' *" arithmetic. There pies what mu/SeZ^ZTsZt^^''''''^.'^'''^'^^^^ -se, .ferring always to thrrt:;aitSX ^^eT;'^^ I SxiroiM 14. (3 F«d the viae of „6 when „. 1234, .od»=MT8. When « = 777, a. . 888 ,The^ „ f ^f J^j'- wh«, a - 1, ;» „ 2 : (7) Multiply x+ V by '^ ^ ^ "*' '**^' ""> ""«'"'"• ;■ «"";W «- by y; by 2^,. ,y 3^ . ^^ .:;,;,. ,""'"'^' •''«='<"««. ««. «nda^ ;; M--*'''^'"-°'''-'* '■'•"" ''**»'''->' «t«5..„iec * -' ^""W a- +i;a6+ft» by «" +Sa6+ft.. ' MULTIPLICAriOir. 16 IztrdM It. a T / if!" ,"**' ^^»^d«^thiH become when*-r + rf^ (-ft)(+c), (-6)(+rf)? "*^ *=*«««' ( + «) ( + ^'X (+a) (+C/), m JJ ^'^'":,w^T**' ^»"»^'»«««*»»'«l»««>Me when fc = e-.|/ (1 1) Multiply a^b-chyd-e-f. I! '■I orn^ative. "+c -a -c+/ a positive 0) Review Exercise 15 and from the examples make a rule for the signs of the prodoet of .Igebraie qoiutles wlthLlkl 8l,ra«, a.^ also for the product of qiuntltlHlth UnllTe S^^^^^ (3) If I walk five yards north, and then turn back an { walk five mile, south, how far am I fr^m the starting potaf? (4) If I gain 15 and lose $5, how much richer am I ? (5) If X people orae in and x people go out. how m.«^ iwrsons ai 6 liiere m the room than before? (6) What is the sum of ( + 5) and (-5)? Of +x-xf xxtrtjs.^ 16 PUBLIC SCHOOL ALGEBRA. ^ W P~ve that (a) (-») = (_,) (^,), „, ,,^^ ^ J^^ ^_^^^ ■ztiolM 17. I2) If ""■?"'"''■''■'"' '"""^ »'"" «'•«'• letU .nione fl^-^-""' '""'^ '"» P-^""* ky m«.Bs of one to the product ot"»' "'•'« "» thepr^uotof 18f.ct„rs,'eacre:ualtoi:;''°'' "'"^ ■" ^"«''8 aultipledby-9vi X - 9a3 » ? * the product of a* . aV ; of aK» . y6 ; of a^ an ' ' ^""^"^^^ A^at^Iitl; "^siate what' r-^^^"'*'.^ ^^'^^^ ^-^ write down the product ^"""^ "^'^^ ^^''^ when we iMrelMls. (1) Find the product of a + 5 multiplied by - 6 wh^f^ittCi"aurt™?"^ •"" '™ """°". viz.,«-»andc-d, ia^A^U^*" " *" ""^ "' ("+»)■ -d of («+»)= ^he„ „ritt«, «^'l% t^«lI;i»l-'']?*B':^i.S!:»!!'i* '^'^ »' ("+ »> -'*out 5- — - ^-^inssxxru lurai Of (d 4-6)5 p ■ » - = d, MULTIPLICATION. 17 (7) Find the expanded values of (x + vV te+«^' /'^^ ,.\* JOD can set them down by copying from memory. (8) Multiply Aa* +3a» +2a^ +a by 4a« - 3a' - 2a^ +« (9) Multiply 263a '"'+ 259 by 1251a » ^ »«+ 407 (10) Multiply - a' ^ ^ by Ilia. (11) Multiply - 555aa: byaa^'**. nq! ^^^' i' *^' ^'^"'' °^ - 2*^ and 2a.- ? Of a X «- « ? Ci<») Simplify by expansion BxeroiMlO. ')» (1) Find the price of 25aj sheep at $5 apiece. Poland sTuXorr '4 2! "Z^l^'^^' "^ ' -*^ »«' Find the price of the ^t sWn, ./^r^?^ ''•'"°"/ ""^ ^I""*? and , centa apiece f« tSl Stens *' "" """** '" "" °"» «"'^ this strr^i" "'„'„sT.t/dauir' '■"?,* "^"^"•"o™ ' -«h of grandson had a, twy J «,d I Iri^J .1^ ">\"'ird generation each « boys and 6 gWs M ouill^L **"'' ^''-''i-dMghter had this happy oldcoinle W i? i • * 1?"'^ Kreat-grand chUdren place in J^,S^'°'"^ *' ">*"• 8°'^ ""Jding which took (5) Said Dick to Harrv ' ' Mv mnna^ „^«i worth Id, and the mS is ^^Zkti^^^'l^^^'^''^?'^^''^^^^'^ Find the Value of the puri! ^"''^ *"' ^^'^ P""^^'" Izoroiie 20. Multiply the following pairs of factors :- (1) 3a2 -4a6+56' and a» -2a6+362. (2) .«« - 3a;» + 2a;+ 1 and £c3 - 2ip - 2. (3) a=-a6-ac + 5=-6c+c-' and a+fe + c (4) 3a?»-2ic-5and2a;-5. 18 PUBLIC SCHOOL ^LOGfiRA. (o) a- b + iic and a + 2b. (7) 3x* -x*-l and 2** - 3a;' + 7 (8) a«+6»+c«+«6 + «e-fccanda-6-c. (y) aj»+4y +32» anda!»-2y»-3«* (10) «'+6=4.9-.« + 36+«ft^^^^_^*^3 (12) Find the fourth power of 2a + 56. (13) ar^ + 5x* + 15a:' + 15a:» + 5a:+ 1 and a*-oa:3 + 10a;'-5a;+l (14) 42a« + 105«36^.23«»6' + 5«.3^,. and 2^^ _5,.^,, (15) ^^ + 5a.« + 15a.3+30a.»+24a.+21 and a?*-5a;3 + iOa;»_5£c+l. Szerolad 21. Find the continued product of the following sets of factors - (1) (aJ+1) (a;+2) (a; + 3). (2) (a; + 2) (a;+3) (a; + 4). (3) (a;+4) (aj+5) (a;+6). (4) (a:+5) (x+7) (x+9). (o) (x+a) ix+b) (x+c). (6) (a-a) (x-b) (x-c). (7) (aj-10) (ar + 1) (aj + 4). (8) (a^+l) (ar + 2) (aj + 3) (a; + 4). (9) («-5) (a; + 6) (a;_7) (a; + 8) (10)(a;»+aa; + «') (x'-ax + a^) {x*-a^x'+a*) 11 (. + 8) (0.^5) (. + 3) (.-3 .-5) (Lm ^• (13) (a;'-ia;+§)(JiP+2). «.,!-. J.-.'-.**"" *'""• "»*>" proc««I.K to tk. .«« MULTIPLICATION FORMULAS. 19 litKf tliUwe^Tf *^^^^^ should begin to learn a with ArithiSetic. AlKdbrafc ^Z.^ ™^ *"^ ^*^' «« compared cases, because he la^^a° Ts ^iZ^r T ^^' «*" P^««'»>'« solutions are true only^Tcerta!^"!*^;. ^^»^« arithmetical . extremely useful formulas are tlC -^ .u'^'h, numbers. Five 22 to 25 inclusive Tnd the Jnn^ ''' ^^^ following exercises, i^ults when the^;trSs ar^ vSS in '^"*'*^. ^ T^'^ ^°^« ^hi do this without actual ^lTfn^!lf ^'^r^l '° ^**"«- He must unchangeable throughonHach ca^''"- ^^^ *^«^^'*^^ '•«•»» >« items may vary Wh^n fif« ^'^.f' "^ ?'***«^ *»ow much the this he ha^s llZdyZ^^'on^^L^T^^^^^^ appreciated and has made a g^aTSe of ;r?rrl fn^^^^^^^^^^ eral.at.on. Part of 25 and the S^Kof '^ at rewtwVx^fcS Istroiit 82. («l'L)"wttho!r "^' +^^+«'- -«« down the exp.„,i„„ „, ia+») without actual multiplication. fo^ Pi„j fr?^'' <'"+''>"' <''+i)'; ('•+2)'- ^^it!:^i:n^\T:rL':t?'^^ ""■ »«».«. of e«=h„f r« ^a Vh "" """1 "' '25U225y> Of 313m= +512„. . - », ^ *""'™ "' 729.K'+512xS .nd of 17a" + 18a"-' («% " ^h^'t i;t1,ar^^iSC"« ^»™ '"e «P«nsion of threC"nLz*r-r»Tt-si*:t"-r *"" ^"*™ ■"-'=•"" 9) Find thesquare of Tofe' -512x>, and of 17a" - 18a" 10) Showthatx(a:+l)(ar+2)(a;+3) + i«te'+<(-4.n" " X a"!^-'>'-^'("-*>' = (»-^) (*-19)+42, show that (13) Prove (a-A)»xrft_/.^»^/^_-v? = 2[(a-6) (a-c)V(&-«)76-c) + (c-«) (c- N.B.— Show eacn didea2(a^ +6 + c^ -ab-bc-ca). I 1 20 PUBLIC SCHOOL ALGEBRA. SzarolM 23. out ^tua. »„UipUea.i„„ .he tZt^ct^fi^ ;%"-» -""■ (2) Set down the product of Mo+»;. (^+5) (^-3).' <*+^>(^-*); («+»)(x-2), (10) Solve the equitU ^' ^^"'*+<"') (217«6-2a»). (4) (A + B - C). ! A= ; B*:% 1 '■ ^""' •''•'''' ^*«'*)"- MULTIPLICATION FORMULAS. 21 (9) Expand 2;2tft+ftt ^''"^*- W""* the product by a+6, andbya»+2a6+ft». answers. ' •°** ^y « +3a»6+3a62+63^ ^j^^^ (5) Set down the expansion of (an-ft). Divide this by «' +3a4+33»' +6a I! iii "•ifi 24 PUBLIC SCHOOL ALGEBRA. SzeroiM 89 (-sj Une factor of IfJ-r' m^i . on the other factor. -46aj» +39a:-9 is 2a:« -5.^^3, find (3) Arrange 12 + a«_7a2_0/,3.o , -a+a+2a'. " o'-^a'+ea'-lB and divide it by answer by inultiplicat^n. ""^ divzdend. Verify your (8) aj« + l divided by a;'+a:»+:r-i-i p- j .i. Give proof. *^ ^* +«+l. Find the remainder. Quotient? +"+0-l^(a3 _2a-i). Ja-^^lr^^'e. • l«+-J6+3c). Quotient? SsereiM 30. thy^t^t'':."-'?:.-^^^* ^e"; -fl-3aj6. and divide verify the quotient. i-ut a = l throughout and thus (3) Prove by multiplication that ^--+.x + i;==a(a + l) (a+2) (a^3) + l when;«= a;=a. BBVIBW IBXISRCI8158. . gg . (5) If theproduct of 4a;'+3a.» ia^_^o'r ^ iBequalto4a:» + lla.«+8l find thl if "^F, ^""^ ^^^^her factor /'«\ Q' 1.. , ^ ' "^" '*!© second factor -^wt (6) Simplify the expression « wctor. exactly divisible b7«?+2a^+2a?r5^ '""^ ^^' ^^ *^ "^^^^^ it Pas^'^'ivef '"^^ of all the work somewhat mor . difficult combinatS t^? T ^T W"*^ i" tions are excluded. In nuZro^« 1^ ' 1 *^* "^^^'^ ^«d ques- fre sareested, and thes? shouM J. • ""^^^^ "*•**•*»» •' testing Pnni?. ^'*12 ^^a'nina^iots /heH^^^ attention* t^n!l'- *iS*"'* ^ enconrwed to kfvlt ^«'^"'^*'® importance, those m Exercise 32 and ff ^i!?,u "*^®"* "«^ problems like metics which may ZnX soW hv^''''"^'^, ^'^^ their arh- tne arithmetleal analTslH o^^A i * P™Wem Is easier than can do questions "pj^itai d^ ^"'^^ ^"^ «^" find that he analysis that wouldTTyo^i hf^""'' ^"v'^^^*' »»<* general He should be led by the pJth of l^asHlS.^ ^"^ /'itSietic of any formal arrangement n*f if v*^*****"*® independently learns the easiest th?nT&st difficn??*'^*.,^^ * scien^ I?he approaches them. ^ ' difficulties will melt away as he ii«tto^'"fn'^'jSof "''•"' ""' '^^ '^P^^ at m e«ml. ■ztrolMdl. , S' t^ Ifc'fl;^;^ (-^7), find. JW. . .. Share, td^^^lt rm^ ^«,- -« -- /s tW, ^4) ^ 15 TQ T«4^«»» ^V W.s-flvrtJLrS*oT«%"'l" ?■ >-«' o-Jy *wo present age. AR n ™«„ 7 — •'^ ''*^" years as -B was four years ago. Find years ago ■ mil i 26 PUBLIC SCHOOL ALGEBRA. fMnIt by putting a. j_e„i ° "^ -t- F+^. Verify the (*>) A farmer sells 9 horses anrl 7 . t ^ ■ Sztrolat 32. ^Pt TrSi: »X'JZ'T' •""^. '» «"«> V two minute more than the other hZ™/** "Ti"' '» " B'Uons per the first pipe convey y "°* """y B^'lons per minute did remamder y„d half . doJu? mot Hrr^'"" '""^ •'"" *^ how much he spent ' ^* '"'^ "ow »2 left, find If the huliet travelled at 1 430 w «! ^^^ 'eport of the rifle feet, fin4 the length of the laniT ^ "^"^^ •"'^ «°"«d at 1,?S) (4) Iq a battle the general lost 4 of i,- wounded, besides 4,000ta^nri^^ ^i? *""^ ^» »^''"ed and ment of 8,ooo ^en,' but in ret^tlT?' ^* "^^^^^ » reinfo?^ leaving him only 18;0(S) FSrtr/J'''^'.t*^r * «' ^i« troo^ (5) An investor p aces 113 ^.! f^**^ "' ^'« ^'^^ «* fi«t 2 £Lf Af a^rS ttrr s^^ '^ '^^^'' ^ -- 3 mUes in hours In how -anf CStKeTST? ^'^^ ^ ' "^^^ ^'^ " (8) ^AS - 2aj^ 4- lYft^ _ ;^. , ,^ «ey meet ? ' \- ^3)=^ -At'? 4izJC + 0)-f(2ar), findaj. HBVIHW BXBRCWE8. j- Sxtroift 33. (1) Find the numerical value of ♦»,« /«n • ?! + yJ^» a,» ! y! ! J ' '°"°^'"« expression (2) Show that when a; = 5, y„3z»4 (3) Express in shortest form the sum of "^ "^ ' (4) Combine into one sum Jaj«-laj»^..„,. ,? " .vo!.A^'wl.tSaiot-'>+«(-+2)=27(.-3). Verify SsereiM 3i. (1) Divide 8a;* - 2/1-1.3 ,q^2_j « , ^e^f^f E^-4a;f*.. .-^2^ . f ana .:X\: ^'ll:^X^:^ J^4-,^^«' - «2-' 'ron. the product o. .„ ,: . *»• ^' your product W m,f«„ „ i Without „, ^AZ.^t^T^-^^^^^^^^^^ III 88 Pvnua mtooL Ar/jKnRA. (^) In the ©xpression (m + n\» ^f*» ^ \ , »« - n, what is the value of y^ Vhu ,' ^.1^°' *^-''' «"PP««e expression ? ue or y . what m the value of the whole P^^^^^tofV^^^ Principle to find the plication. ' w - n -p + ,^ without actual multi- (8) Get the product of x' - m»»yx«i3 „„^ , out ac.„a. mump„„..,„„. C. yVu? rult^' 1^'^-^, J'^Sf SxarolM36. (2) Divide 3«.v by Jak and O^nl Tl''^ ^'' ^°"' *"«^«''' (3) divide .^^2i3*;4ff,^r 3^^^^^^^^ n"by,r. (4) Divide 0.^+2,187 by^^ 3^ ""^^^ " ^^ «^ + 2jy-3z. ot§^t\?Yh^e::^^^^^^ is an. truth of your result by makiSg ^^y =il l ^ ^*'*'^^' '^^^^ *^« . saS p'*rXtt^2?X-/,^;M^^^^^^^ gives the your answer by putting^r-y^l! ^^'"^ ^ ^'* " ^^'' Verify (7) Divide (x + w)' +3^3. . ,a-1o., , pansion in this way : J^pS^7*"t^? ^^ ^ + y+« without ex- write a. + y form in this qu^JJient^' ^'^^ *^« «^*^' or iess than iab+cd)\ when Bx«roiie 3a. stepi, eVp^t\y tiL'nsTf'ji*^^ "-*^' *h«« 6 steps, then 7 . starting ioint. ^ ""^ *^^ "'^ + "^^ distance fim The a s?4'Ve!^a;ri\Tvl wXd 3^^ rr« ^^ "^«-« of in which I have walked "^ ^^''^ ^^^ direction (south) (3) If +a means l + l4.l4.ij.af« i.^, m U (M^\ /..,^^ + ^+®*°' what does -a mean? -^= ^ T ^; V - *J mean ? - < t t 1- <1 ii I-' 11 ) (I ■1 e: i Cj RKVIKW K.VKUCIHKH. 29 HuS .X^uc^t<;iC;:ir " '""^'^'••-^■- «»'ow that ti. -^-:^-(^U:i^^^i^T .n"' •^•"^''•'"■'-' I times or the exp.ision^to^;i^^4i.-;~^^;^- -^ ^;^^^^^ used to represent nunlr;,";; '.r;;;'.,:";^ J%«^»v,.Vv,/ ..hHract.-s (8) What is the name given to un expression like He + U ^ :n "^ *^+i^+ia.+ ia.+,i,a.^2187? State the ttxiou. u.^,a v ofS^su^^^VaH^^^^^^^^^^ sake of presenti,.K the siEt „ t ff^"" '^'•'"r^^^^ f'^'' ^''^' requirements of a 1^^/^^ v^ ^ . form sm tod In^st to th.. him first to the iSt Ss of'th«''' *'''^''" '''^'''^' ^^'^' '"^'•o^^'x'" what difficult, nnTuroT^ tTtCt^l.^^^^ maturity of thought and mo«f «t?ii • i ' .*^'*^ ''^*'U"'o most gradnally from the pnDlI's kimwilHi. ^r *»*•*'*« proceed HHknowii abstractions'^^? aLbS^^ t«*J«^'l;'T***'r^ /« "'« lorois, to select the easiest 'thljrs flrst "^iL^^^ "'*• *." «**'^ common sense of irood teachlii^ n i vl**'* '* ®"''*'''y the therefore introduceTe simpL ^u^tio^^o^^^^ ^^ experience, we the first things to CleS Th« aii ?^ "nknpwn among unknown qnantltle^ in also V. J*!® V"*'***' eouatlon of two into the practical annlication« n T l^'^l^'^'i']}''^^ «n insight defer to about page 15^ lu l^^'^'^'*^f*'™"l t^'^'^tises excluded as foreign to our nurniL ^^^^ ^^t®'"^^ questions are ca. the pupil can^rtrexCS\:rttr'm;;j^^ Sxereiie 37. contain Lre LTl, " j!'!.?.."-*.."™™' «<»'"ions whid 2-+8.y=lB, with commoivaines" aTi'-o' .f ft^"*^' '""' t.ons are called Slmnlt.neons i;,„',ti,„8.~-- *=^' ''•^^ «)'»- 30 PlTliLlU fSOJKJOL ALUKKUA. (^). Solve the simultaneous equations 03 4- //-c • Oy-.o, -,. m this way :~Muhio'lv tho Uf W 9 f i ' 2^ + 3.v = i(. from the 2nd. Nex mulS.ll th^^^ ^^ '"^/'^? ^^^ P^^^"^^ duct from the 2nd ^ ^ ^ ^^*^ ^^' ^ "^"^^ '^"^^ract the pro- 2 A = S • 2;^^" •• °' * '"^ ^ ^" ^^^ ^^"°--g equations :- (5) 7a;-3;y = 19; 4ic + 7^ = 37. Hint. -Multiply 1st by 4, 2nd by 7. (6) 7ac-9y = 5; 13.r + 42/ = 30. (7) 4a? + y=ll; cc + 4y = 14. (8) 2a? + 3y = 21; l]x + by = 34. id)Sx = 2S-2u;10 + 2x = by. Hint. —Transpose a; and 2/ to one side. ^^^^2'*'3"^' S"*"!"^- HiNT.-ciear of fractions first. (11) 3a?-2y=3(6-«); 3(4a;-3y):^72/. '' Sxeroise 38. lo? !;!"VT! """''^'■' "^^^^'^ difference is 7 and sum 33. (-j l>ivide loO into two parts such thai three times the o-^^fp,. par exceeds seven times the less by 15. N. B ,^7^ 1 1 50 clc (jij 1 can get 7 pounds of tea and 5 pounds of coffee for .4 ^.o' or I can buy 6 pounds of tea and 3 pounds of coffee foi- £"' o" What IS the price per pound of each ? ' ^^-^• N.B. 1x + by = bbQ; 6x + 3y = 4O0 uitofeetner J|^-,dou. ± ind the amount each now has (5) A man bought 100 acres fox- |2.450 navint? «on nr, «,;: for one part and §30 an acre for the remaSr Hnw ^^ acres were there in each part? '^"^ ^^^aindei. How many (6) ^ and B have $9,800. A invests i of his monev and » 1 of his, and then each has the same sum left. Find tScTJtSs' {() A man invested 14,400, partly in bonds navin.^ q*'/ iL and partly in bank stock pay ng "*yTnL?Atf ^U°^-^ "^^""^^^ the same from P.anh • iir./T^^.Zti'll''^/^^^' , ^^f mcome was stock, which were both bougJtlTpr ""' '""^ "''""" ^^ ^' '^" t h h 0( ol REVIEW EXERCISKS. Bzercise 39. ^n (1) Solve the equations 28.« - 23u - oo „„ i no . . W.-B.— Add; divide by 24 and yli.,/ 'i ' , (5) 15ponndsofteLndl7l ""i^^l' "• ^^•^+11^=11. etc. 25 pounds of tea and n .. ^' ^^""^^^^ ^^ coffee cost $7.8fl and the price of each ^r^nlT'' °' ^'^^^ ^^^^ ^^^"^^ ^^^ ta\ A ^-^--^"^tiply 1st equation by 5 o^a u^ o ^, (6) A man invested .$100,000, partly af 5°/?!, \ ' His interest amounted to ^4 640%i:v n ^ ^"^ ^'^'•^^^^' «* 4%. he placed at each rate. ' ^' ''""""^- ^^"'i how mucli (7) Bought a number of estr^ of o <•«,. more at 3 for a ijenny SoW H.! i/ T "" ^^""^'' ^'"^^ «s many ost 4 pence by the tinsacln Hnw ^' ^' '^ ^"^^ ~ I^^"^«' and Szerolse io. many did he buy at each ^fce" ^ "''" =''"«<»• Ho,v ?lone can do it. Ho'w1r/wiH B aVd ';! '""1."^° "■»<' '^o' ^ job? * " ■" ""a «»>«- a* have three tinTes^as mucTas "^ Va/,''f /", T™ ''" ™"ld wonld have twice what B has How^ ■,'] "' ''»'' *'« '^« he had each at first 9 ^ow much does /I win ? What S2 PUBLIC SCHOOL ALGEBRA. (5) Bought two gi-ailes of sugar ; 7 pounds of the first cost the same as 9 pounds of the second, and 9 pounds of the first and 7 pounds of the second cost together 65 cents. Find the price per pound of each grade. (6) A room is twice as long as it is broad, and it takes the same quantity of carpet as another room ,10 feet shorter and 9 feet broader. Find the length of the room. (7) Bought 12 cows and 20 lambs for $335, also 10 cows and 26 lambs for the same sum, paying |2 more apiece for the cows and 75 cents more for each lamb. Find the price of each cow and lamb in the first lot. (8) If Tom gives Harry $10, the latter will have three times as much as Tom has left. If Hany gives Tom $10, Tom will then have twice as much as Harry. How much has each ? (9) How much tea at 72 cents and at 40 cents respectively must be taken to make e^ mixture of 50 pounds worth 60 cents per pound ? (10) I have $12,750 to invest. I can buy ^% bonds at 81 and 5% bonds at 120. How much must I spend on each kind so that the income may be the same for each ? (11) At simple interest a certain sum amounted to $a it m years, and to $6 in n years. Find the sum and the rate. Szeroise 41. X . X (1) If -+- = c, multiply both sides by ab, and thus show a o th&t X - abc-i- {a +b). (2) The sum of two number is 85, and thr6e times their differ- ence is 81. Show that three times the smaller number is 87. (3) Divide $42 among three persons, so that B may have $5 more than A, and C may have as much as A and B together^ (4) How many pounds of tea at 24 cents per pound, and at 42 cents per pound, must be mixed to produce 100 pounds worth 30 cents per pound. (5) Loaned $1,000, part at 4% and the rest at 5% ; interest on the whole was $44. How much was loaned at 4% ? (6) A hare takes 4 leaps in the same time that a hojind takes 3 leaps, but 3 of the hare's take her only the same distance as 2 of the dog's. 'J he hare has made 50 leaps before the dog is loosed ; how many springs will the dog uiakc on a Htiaight course to u d. tl >ws and ^ be cows ich cow REVIEW EXEItCLSES. 33 catch puss? N B — T*f J . i •. i W hai-e and hound, and tu Td Zl t h" T'^' ^^ '^''^l'« '^^^^ lience hare goes ovei 8^0/ wl? i^ i^ ^ *'"' '^"«*'^ ^^ each lean, (7) Solve '*'-^_* + 3 tV — o '.TTi)" • ^- --^"itiply by (.^. _ 7) (.X. + 9). Bzoreise 42. (1) Two men own tOireMiPr 17."» .1, i'lixl the valuo of ,, shaU. B TO ; B g„.es a .,|oo. e.I>.«Ua value to one gold coin """'*'' °' »"™'- ■^o"'" (3) Blown sets outs from X to Z -.t "t ...i fo«y minutes later Jones start, f.-l, V ! i^'"^ l*' ■'»"■■• """1 l>our. When tl,ev me^r.Tone, \' 1 T u "' ^^ '""^« !»■■ ""•Idle point betwee,rx a„Tz f'in?',^ >" " '""<' '"'y™'' '•'« two places. ' -v ana A imd the distance tetwoen the by a t..ight train soin";' U S' ».' irr'^Vt.'^r .'^"'■^ •.) Solve the equations U^~uZxX' .Zf'''' f '?"°°- .ait.K;r„"£f '^ -"'..;t^i;::-rthei. £Ml''dSJ'tre^;- %^^ -J-- ;,-^ as^raii?'o7Si^rte;s*''™rtfe°'''7,»™ all her children. ' '""' """' """'"■ -^SiW more tlnm (t'j If $1,02-4 amounts tr> iiti i.v • interj^t whatis the rate „,°inte;^* ^l- iLV^T "' «°»P°™'' ^^. ^';«uhrsL't'e'?;t'cht;a^defir;;i:f'^'«^--i "«. il 2^li""£» t'inTcrs tr '""" T™ "■"" " "■■"" the hor«« (t~.. ;» "' "_. . /'°'".?.'™ !■'«"' socs 2 mUes mn^e .).... 41x+373,= 17; solve the equations. (13) 1.3a;-I7.v„n; 29.^ •''%=17; solve for ar and »• 34 PUBLIC SCir(X)L ALGEBRA. Note.— Exercises 43 to 50 nre intended to sum up and test what the pupil has now mastered. They are of the iiatnre of a written examination. He is recommended to do tliem over three times before entering on the Second Stage :— The first time to work them out accurately and to master them, the second time to go through them thoroughly at moderate si)eed without any break, and finally to work the whole at hl^h speed to make a record against time. This practice in writing out solutions at high speed is a fine piece of mental gymnastics and soon produces a firm, rapid "touch," and gives the student the indomitable courage and "attack" that secure success when he has to face new problems on the official examination paiiers. J he habit of doing: all lessons in a certain limited time is of the greatest assistance in every branch of study, but particularly in mathematics. It economises time wonderfully and is in fact a useful stimulant to the growing powers of the mind. Try it and see. Szercise 43.- General Review. (1) If « = 1, 6 = 3, c=.5, findthevalueof:!- '' ' a- +62 (2) Shew that i^l _l+a^cr 4a+b'' +b^c'' a"" +2ab + b^ a^+c^ ,- + b^+c ,.2 6»-26c + c-^ \{b + c)-{d-a)\^ + {{c + d)-(b-a)V + {ib+a; + 1 10)Divide„.+2a'63+4.by„,;2„j:,„, (12) PM the remaL; vvh „ J^;'';?^ *'-'.+ ^ £C_rt. ''"^" ^ -/>a3 +(7a;-ris divided by (13) Divide^to fo„ t..s of quotient and «ive U.e .maindev in (14) Solve the equation if 3:c 24. 'y.\ 1/0 •> . ^ N.B. - Multiply tWough by 60? ^• Szereise 46. (1) ?^?-£zi2 3x4-1 ^, 4a; + 3 ge7i1?2*i^^^^^^^^^ r-l^ect ail like ter.s and (7) (x+i\ r.'Tj.sN ^-iKN_- -> - (8) Water expands ino/^~i!'^^.'^^^''^^"^'^'^~^^'^^«^- ■«. ^^^ 100«^-volume of water; ■liOa..etc., etc. 35 36 im:hi,ic school alokbha. ;M Sxoreise 40. (1) Write down tho square of 1 +'2x-x--i,x^ witliout actual multiplication. ( 2) Multii)l.v (a + h + c) (a + b- c) by (<( -h + c){b + c- a) . {•V) Divide 1 - ^x by 1 - j^a? - Ix^ to five terms of quotient. Hint.— :^tultiply both divisor and dividend by 12 before dividinio:. (4) Remove the brackets and simplify the expression kHo-(f>-n)]-^[{b~^a)~i{a-^4{h-ta)}]. (')) Expand and simplify the expression {a-(b-c)\^' + \b-{e-a)Y' + \c-(a-b)]\ Hint. — Remove the inner brackets, then write down tho expansions under one another and add. (fi) Test the accuracy ^f your result in No. 5 by putting (T) If a? = |, and x+i/^x + y + z^O, find the vAluo of the expression (y' - z-) {//^ + z' - y{x - s) }. (H) Divide $5,000 among A, B, C, so that B shall get twice as mucli as C, and A three times as much as B, after paving «-^0 to2>. N.B,— Letcc-C's. '*'''"" (0) If « -6=02 = 3, and a + /> + ic = 2, what is the value of (a-b) \x^-2ax-+a^x- (a + b)b^ \ ? (10) A-(6a?+18)-4i-^tf(ll-3a;) = 5ic - 48 - i2(13 - X) - iig(21 - 2x). N.B.— Multiply through by 36, and then collect terms, next multiply by 13. Szeroise 47. (1) Find the difference between these two expressions ;— (n + 2) (n + 3)(rn-4) and „ 24{«-J(n-l)n,»-§(H-2)nn-f(n-li)]. Hint.— In the 2nd quantity 24 = 2x3x4. Multiplv the first bracket by 2, the next by 3. and the last ))y 4, and reduce. (2) What multiplier will give with r>ir + 4 the product 602c2+53a;+4? (3) Find the coefficient of ac in the product of (.r + O) (.r + IO) (.r--5) (x-^\ REVIEW EXERCISES. 37 It i8 requiml to form a m^xt Z ' ^ • ""^ r^*'*^^" *^ "^ «f^"'e. and 11 of wine. Cw manl '«n^ "'"^x-^ ^^^^°"« ^'^ "^^^^ B respectively ? K B J^^^O^. 1?."/ T'\ ^\'^^^^ ^^om .1 and each, .-. 4^ + 3y=:l, etc. ^ ^^^ ^ *^« numbers from w£ 4jr5^^£?^^ 60 cents, but iiud the prices. ^ '^" ^ *^*^"^'6^* * ^'^y cost 70 cents • Sxeroiso is. (1) If 2a = 3& find the numerical value of 'iZ^ and of 'I b- (-^) BzerdM M. (1) Simplify (2x-l)f23.!j.i^/,^j.,, • 38 PUBLIC SCHOOL ALUKBRA. (3) Show that a: »2 is one solution of the equation. (4) What is the L. C. M. of iix'y ; 2xp' ; bjcyz ; and 4x-,/? (5) 21(x-2) + 28(2a?-6) = i4(a;-G), find sc. (6) What is the G.C.M. of Six*y''z'^ ; GOaj't/^z" ; in2x^y'z'" ? (7) Without niultiplying out the brackets, show that //^iw"V^^^ ^{ ^' ^^i ^f ^«-'^) («-^); (&-«) (6-c); an.l (c - «) ((• - b) as the product, of three factors. (9) TJie populations o( 5 to*vns are a + b ; b + c; c + d ; d + c ■ andc + a. Find the average population of each town. (10) Sold a cow for $42 at i loss ; if I had sold her for $h7 the gam would have been foiir times my present loss. Find fho cost of the cow. (11) Ifa: = <2+6,- y==a-b; find the value of (^*-y')^(aj2+2/^). li Sxeroifli^ 50. (1) Show that {x + 1) (ac + 2) {x + 3)(a; + 4) - 24 - (aj + 4) (a; + 2) (02* + 5£c + 8) = 0. (2) Show that {x+\) (a;+3)(.r + 5) (a;+7) + 15 divided by (as + 2) (05 + 6) (a:=« + 8ic + 10) = 1. (3) Solve the equations, 45a; + 8?/ = 350 ; 212/ - 13« = 132. (4) Fifld the Values of x and y from the equations, Ja; + iy=42; ia: + |t/ = 43. (5) Three pounds of one sort of tea cost as much as four poiuids of a second kind; nine pounds of the first and eight pounds of the second kind cost together $5.40. What is the price per pound of each kind ? ^? ^ ^cTo/*T ^""""^ °^ T*^^ ' *^'Q ^^"^^^ ^o^S^^s 30 pounds and contains 80^ of pure gold; the second contains 907 of gold How many pounds of the second must I fuse with the first bar to cast a third bar containing 87% of pure gold ? (T) A boy swam half a mile down stream in 10 minutes ; in still waterit would have t-aksn him In minntei — " h© take to ewim back again? F TT 1 .. XiW.V iX3iX^ WUl I SECOND STAGE. FACTOIinsTG. Sxeroise 61. X (3) What are the facto, of 2„. -'^.;,;,' ot O^vLi (4) Factoriso3a;3+G«2£c^ z''+nxi/>^ oa^x; and Ip'- _ 7y>^ +34np\ 6 Separate a^^y _ ^.^ 3 ^ o^^ .^^^ ^^^^ ^^^^ (S) Pactor Som'y + oSm'y^^ - 14wy3 (9) Factor 12x»2/* -24a;*2/3 + ,^^^3 a', 10.^2^3 0) Factor Sx^yz^ +6^*y,. _ 15,,. 2,3 ; j^ ' ,. fii/ll^ti;r^^?\^\l:,- ^^tr^ ^'" - ^^^'^^-'-^^ x = h °^ division. Test your answer b^- putting F/'Ci;-;ot-art'*"-'-"-(^-^) (--«) (- + 4)n And Szeroise 62. ro! t' T ^', oi'^ + B) ( A - B) ; what are the factors of «« - «3 2 Factor 121 - Wy^ j 121 _ 36«^ j and 4y^ - 25.- ^ ' aritLtS *l!!J°^"^^ ?».No. 1 to reduce the following and 936=' -64 (4) Factor lOOa^'y^ - 121«^6« ; and (3a- + 5)^ - (ox+sy 40 Pirnr.K^ HonooL Ai.OEnRA. (6) Factor (« -6)2 _,.^. and u-- - («-&)^ (6) Find tljo factors cf (a+b)^ -{^c-d,^ (7) Factor (ax + hf/y-l; and 2ah + o' +h^ .,^ (9) Express a: » • - y « « in fi ve factors. (10) Factor a2 + 126c -462- 9c2. ni MP 5'"*"; "'^^^^ *^^ ^"^^^ ^ *^""« together. (11) Factor aj» + y^ _ ^, _ ^^^ _ g^^ _ ^^^ /i «. -r, "^^ ""'^^''^ *^'^ *°^n^« " together. (12) Factor aj»-y2 +22 _rt2..9.^2.j.o«y. (13) Factor 4(ab + rdy - (a^ +b^ _ c^ -.= ; '^^ ^ ' 256«=^6^'-544a62c + 28962c2. (12) 324m«+432w3n«+144n». (13) 36 laSw _ 684^m+„ ^ 324^2w/ 5» and 68 oyer a,raln several times before proceeding farther. Szeroise 54. (1) A'+A(B + C)+BC=(A + B) (A + C)- a + 7a +10 = (a + 5) (a + 2); find thefactor3of a;=+n *rt\ \-; xactortt +IUJC + 56 .'W-+14a.' + 40 «'-7a + 10. in as / P.WTI.HlNci. w 'o'-^B^+m j:lZ''\Z'' "•;+'»»'+«• (9) a'."-ft.^. {..+4^; .'^:(:' 3;: J•-''^•+9•'- (12) Apply («4.6) (^_M^ „2 J,:^^: ^ ' »*; * ind the common factor of il X' 17a -38; andx'2 + iia;_57o. Szerolse S6. tl.i'irl'e?^^^^^^^ Observe that to the pvodnct Of J "To ^wl?' ^^^ "^''^ ^«' '« e^nal each product being abcdT "'' ''''^- ^" «'"» »C, ^'^""""^^^iffjli^^-'-et the product (0) Factor 0^3::;:^;^ ^^-vethat^.^,,,^,^^^^ whose product shall be I (J.;"^ x 1 '>^^!.'^% V" l"*° ^^^'^^ Parts only be 8aj and 9^. ''^-' °^ '2a:^ These parts can (4)Factor4a.^ + 13^.3 _' + *(-^+3) = (o^. + 3) (.,^^4^^ (5) Factor S.^ ^.34.^ + 01;^"^ ^' ^'^^-^^'^^^C-^^)- (6) Factor 2aj^ + 5*y+Ov^ Hrv. - . ^ (7) Factor Sa^^ull\,,. ^'1" -^ = 4.^+.^; 4.^^, hence split 14a6 into -^20al%ab afif ' ^^^^= " l^Oa^^S f8lF«rfnr.«' _ ^-"««> **«*' and proceed as before. _W J?actor6aj-+oa?^4. Hiv-p AT>r.T._«. „ c^o parts Whose product shalfbe -ToIi^T^'u ' ^^'^'^ ^^ ''^^o as before. "e --4a? , e.e. 8a; -Sx. Proceed 42 i'UUUc NtlluOL ALUliiBJU. SzerolM 60. (I) I actor 12ac»+37a'4.oi Hint. 12x21 -.ix'] xTxO^^og ^c, houce «pl,t 37^- into 28^- + 9^. ' C^J Factor Jo,.2_37^^2i (•»; 1' actor 5r)j;2 + 137^ _07HHr. ^,, ■^-''-« . f"ir.r,ity,) Hint. 27885 1 15 xU /rff 't^/''^'"'^''' ^«^o«/^> (lU) f«'-yS^. + 2l- )j»-i. /..'•+?•';•'•■- 100.-39. '") 2*' + n*+12 W„=H.i,;-' f . /"'',^'+"^-'»■ H'ay be written a^ +Ou/^fy^^lf!^^' ^.«^' ^^^mple a» -6^' two parts whoso product is 1 T (or aTc^^ ''' f ^^^^^ « into l^ts ure evidently M _i and w« ? 7^ *'• '^'^^^ « Lern and deserves careful Lt.tlL ™T f "f"» *han the of inathemati.s, -o factoring ifthe !onl *.f f ''^^^^'^ ^*« *^« S'^-'l cannot be j ae.l exhat.st^vdy in a Iflf^rt"^- , ^^^ ^^^J^ct exe..... but what i. here gi^en^ilXlo.'r/o?^! .^S fACroRixo. 43 Bxtrolie 87. (') Factor 8„4o; nb;"''"*'- (T) Factor a'+b' NB ^!'" " (")' + (*')'• (9) A'-I1»./A %;/A. VJ' '■''■'"«'»+1708,/- .0)matarelt-,fir:/J:5?-,.f'-'''^- V^ -3^ What are tho. factors of a.^ - u^ :\J' CI) Factor' ^.fl 3 «-. , '^ ' "* ^" Szercise 58. Provethis. "^"^"^^^^^'^^^'-AB + B^). I a PUBLIC SCIIOOr. ALGEliUA. (10) Factor SU- - .ga'l,' + m' by Miug lOOa',,-- - tWa'b' i^^) factor 9a* +21 a^Sc-+2ox*. ^^^a u . (12) Factorise 16a;« - I7a;2 + 1. (13) Factor 49rt« - Iba^b- + 1216* (14) Factor as* - lla;2g^2 + y4" (15) Factor £c* -63£cV +81y«. Szeroise 69. power throughout, e a hovh '"ri*"'^ ?»« «»^ the Same (2)Factor2a:2+5a.y_3y2_4 9 n b w (3) -A3 '^-----' -^^^^^^^ 5 t"'' .f^^^- N'^-I^-kout.,a„dgoup^ '' (6) 2a.2 + 2(«2, - 122,2 ^ 6aj^ + i8ys. ^ (7 (a + 36)=^-9(6-e)^ ^^ee ^«;.rme 52. (8) «=-a;y_6y2-4ica + i22,g. (9) 18a;2 - 240^2, + 8//^ + 9^^ _ g^^ :r^^.^ ''I' ^"^ ^^^^ ll^'^*' into IS^'^fe' -4^. .11. .0 f«"P "1 three brackets, factor. ''''*«* N.B. 0Da6=64«5-9«6; group in 8 brackets and factor. 15a= + 10a=^c^ Bzoroise 60. (1) Expand (a-36+c) («+6-3c} 2l'",fl"''^''''''''"--^- ^*^*^^ thi« expression. «S^rc~luriik?brci''ah"H%*'**' ^,»**'»» «»ly terms like it can be expressed a?f ht' *S * ? "/"""^ ^^^^ *« «»«*«»', ,-. c. i? No. 1. iraTthe si^n arf! >'• ""^ T *"^°^^i«l« like those in terms like a\ T c^ and «nt. '' °°/^' n*«=essary to factor the terms like tl etc ' ' T^f/lf ^i^.^^*^^«J ^y trial as to get the - • "IX .^-rrsjcnu/y De tioijc b^' iu§pection PACTOKIXO. fb ^^^^^^^^"^^ ->' in ca.e so.eof .,. contain (a -3b) (a+b) '(a + cUn^^ the fin^j product must Hence group alVthea^an&l^^^^^^^ and aU the c and a terms, and {JtnJJ.^{ ' " ^^^ * *"*^ "^ ^^rms ThisshowsSlwl'3tl,ist^tl^^^ is onefactor: Similarv it i!-?^-. '^'l*"^ +^' »•«• « -Sft + c and +6, and that a+T-3c ^^^^^^ -^' goes with +a can be t'ested by Littplication. '^'^ ^"'*'^' ^'^^ l^^ ^««"lt (y-«); whici is eLul^rol'UtfaTC^/^^^^ ^'' (-y+"^ and y- 2a; -a the other factor TT.! -^T'^ + aiH one factor. (o) ^a'-6ab + 2b^-na + m+oi ^' (6) 6a;2-37aaj+Ga2-5a;-5rt-l (7) Sm2-5my-6y2-m-6w-l ' (8) 5k' -8km+3m' + 7k- 5m + > (1 1) 2a5» - ajy - 3y2 _ 5^3 _ 22^ • (13) 6«»~7ajy+2aja-20y3+64ys^482=. SseroUe 61. (1) A^+B-'+C-'+D^+etc +OAi> . ow. + 2BC + 2BD+etcX2CD;tt^J(l';^^^^^^ , ,, 2AB, 2BC, 2Cd7 etcV, the exurtslo?! ^^S**'^ PrJdncra* in itself and is 'easii; factorX ^The me?ho^ i- ^'^^^J^narl 60 will generallv a»^»>i- u.JTC. . ,, ® method given in ExfimJ^ sqaare root of each 800^6 TflrmT.Ii'"*'' ^'' ^'^^'^^'^ • Take "the 46 PUBLIC SCHOOL ALGEBRA. (2) factor 05* -I- 2« 3 .3^2 ,0,., . t,, ^ (3) Factor 4a* —4/» 3 0^2 . n (4) a»+2a56 + 3a*62 4-4«3»,3 , o 27 4 «,. 4a363 ^2a3^63\t«3j^3 ' '^"^^^«^1>' ^o^' 3«^6* ; -Sibl'tl^ofXnt'^^^^^ ^-B. T.^ and v= ^zeroise 32. (3)«=» + 126c + 463+6ac + 9c^;4a6. (4) 9f + 16y3+4.y2_o4;gaj+16s^>lo^^ (6)aj*+4a33+6«2^4ic+l. (7) «♦ -Oc^f^+fto 4.07,3,7 ,^4 J.,, j, (8)m*+8m' + 18m2+8m + l (II) a«+Ga^ + 15a* +20a3 + 15a»+Jl 1 (14 ia-4-a*6'+i&3^.^.,, 3^- (16) 25a3^ + 9w* _ .onon«.2_L'fA •> ^ . Bxeroiso 63. I'A(JTOKl\o. ""y of these th4?^tors w IlT -f v'''"' ''^ 'Uvfaiw.tSthe^. •■«;- Observe, hat i^L^:Se„%,'::,^^„^-/J^«.•d,4t .-) 3*'+nx+6; sXnl^^ff* *-/^»*J'-«%- N.B. aels^'S.+B^f +«f+3• •^ 1- - , -X + 5 are evidently of no i,se. or OH o, i' *^" show that thi« «r;ii « U T'^ having a f I 28-21. Give A and B fivl J!l , ®''*''*^y ^^^vide 28 + oi fJ'e proposition is true ^ ^"^^^^ ^^^"«« each and provothat ^ (2) Suppose A = »»-y. -R _ ^ (3) In Question 2, show t W ' '' *^^ ^- ^- °^ ^ and B ' (^oad A plus or wm^.^ B)^ "^ '" ^^^^ °^«« measures A±B "r :J t t±??:'H" =■ -" ""' " • Q, show that X must be an exaof T- '^ "?= ^ ^^^^^r of P and >y putting P = 7, Q = 5 1 l^^f t^^^««" of A and B. Test tS« ietters five other VaTueslafh^^' B = 2G,. and by givlng'thte . ('; If A = 2a;»«ift^j_,. ^ ractor this suKi. Determin; noT. iT '^^ ~ ^*- Find A + B and B, if they have any. """^ *^® common factor of A and 47 i« VVUhhi SCniOOL ALQEBHA. ft i (8^ If A-3a- + 14a.+ 8, B^ia^ + Ida + 12', lind the value of 3A - ^B and factor this sum. Can a be a factor of A and B ? What is the common factor, if they have anv ? Test them for a +4. (9) If A = a;'-.v3, B = x' -2xy + y' -, find A - Ba; and factor this difference. Can xy be a factor of A and B? Can 2x + y? What is the C. F.. of A and B if they have one ? Is a; - y a factor ? flO) If A=a2-3a-18, B^a'^ + ll.r + 24 ; what is B-A? What are its factors ? Which one is a C. F. of A and B ? (11) In Question 10, find B-A as before, also find 3B+4A. What is the common binomial factor of these two quantities ? Since 7a cannot measure A and B, what must be their common factor? (12) If A = 3fc='+21fc-132, B = Gk^y-^Mk^y-13Sky-eey, can the y in B be a factor of A ? Can the 6 in B and the 3 in A have any common factor ? Strike out the factor 3 in A and Qy in B, and set aside their common factor, 3. Call the quotients C and D and find their C. -F. . a- + 11 . What must be the C. F. of A and B ? Zzeroise 65. (1) When A = 3x'--2x-l, B^Gx^-x~\, show that B-A -= x{Sx + 1), and that B - 2A = 3£c + 1 , and hence infer the highest common factor of A and B. (2) When A = 5c3-35c + 2, B = aj=« + 2a; - 3, show that B - A is x-1, andthat2B-3Ais bx{x+l) {x-1) and hence find the H. C. F. of A and B. (3) If A = Sx^-2x^-2x+l, B=^x'-x*-3x^+ix-1, f5nd their H. C. F. N.B. A = 3ar;-(a;-l) + (£c2_2«4-l); B = x^x^ -l)-(3x^ -4x + l): i,e. A = 3ac2(a;_i) + (a;-l)2; B -a;»(a;2 _ 1) - (3£c - 1) (se - 1). (4) If A==6a«6 + rt3ft2_rtft4 J B=-4rt=«-6a»6-4a62+3fe3, find the H. C. F. of A and B. N.B.— Strike out from A the simple factor ab which is not contained at all in B ; call the quotient A I . Then A , + 3B = (22^2 -Sab- ib^)a, of which a cannot be part of the common factor ; .-. eitfier lla+46 or 2a -&, or both or neither will be the H. C. F. It is plainly useless to try the first ; .-. try 2a - b. (5) nx*-9ax^-a^x^-a* and 13a;* -10a£c=» -2a*a;* -a*. ivr-R-R* .,«,.'. . ... ... A — /mZ fOnn t ^\ to try the first two factors ; .-, try the third 1 nrOHRST VOMMON FACTOR. 49 Szeroise 66. factor wm a,«, di^Me thrs^roftfiitL""'"?'-'' " ""■""on of the two given quantities Thi. »^ ."5"°" "' ""y nrnltiple, or tffo iionibers. whivi, process or flndinar thA ri n i» A and B will divide E, At fi'' "i*™'^ oo-nmon factor ^f ' '^ 2.» - 1 I 6aj2 _ ^ _ J I ^, '.*{.^='+« - 3* - 1 mo., convenientbf .s™.r ,'he";^rt™; "Tj^ ""'^ ^ Ranged -1 3a?+l -3.^-1 A modification of thi^ ^i coefficients alone te^g t^,""'''r'«'>ed coefficients (,> the of proceeding with larfe eSssii^i';i;''f ">« """^t rapid 'meih^ Tn nsing the above nlan 4?^.'J?1'? '^' ™ not easilv fac.!«T -.. «. etc., anvsimpieteo^whLrirXwfrd^ •""?■■ »t- .s not n.e.. divifio^'fCpH^^o-^A and 60 Public school ALutunA. oml In Tlew h to find the H.C.F. of A and B, and if any one of the remainders contains some factor manifei:t]y not contained in A and B tiiis factor is of no present xi«e and may be set aside. Also, a simple factor may be introduced at any stage witliout cliangmg the final result. For instance, if the preceding example contained 4x^ instead of 3x'% we might multiply Gx--x-l by the simple factor 2 so as to make it exactly divisible by 4x^ and thus save a line or two of work. The student may apply this method to find the H. C. F. in the following examples. (1) x^ - 7x + 10 and 4a;'' - 25a;= + 20a; + 25. (2) Sy- + Uy-lb and 8^' +301/+ Vhj - 30. (3) 2a*+a'-20a--7a + 24: and 2a* +3a=' -13a= -7a+ 15. (4) 2w° - 8m* + 12m=» - Sm^ + 2w and 3m» -6m' + 3m. (5) 3a8 + 15a»6-3a362-15a=63 and 10a''-'60a*b-10a-b'+S0ab\ (6) 3a;=' + (4a - 2h)x - 2ad + a^ and a2='+(2a-6)x2-(2rt& a^)x ■a^b. (7) 300a3+265a=+50a + 24and60a2 f J3a + 4. (8) X* +7x^ +7x^ -lox and x^ -2x'' -Vdx + UO. (9) ib'-c)x^ + 2{ab-ac)x-i-a-b-a^c and [ab -ac + b'- bc)x + {a^c-{- ab^ -a^b -ahcV (10) 20a«+a=-land25a*+5a3-a-l. (11) «'+3a*-8a2-9a-3anda«-2a*-r>rt3+4a=+13a + 6. (12) 2a;*+9a;3+14a; + 3and3a;*+15a33+5x= + 10a; + 2. After applying the method of division to these exercises the student is recommended 10 work them over again, applying the principles of factoring as far as possible, and at any stage Thus in question 1 and 2 factor the first expression, in 4 and 7 the second expression, in other cases factor the sum or the differ- ence as the particular case seems to require. One of the most useful exercises that can be devised is to search out various ways of solving the same problem. Szoroise 67. (1) What is the H. C. F. of ax and bxP What is their pro- duct ? What is their L. C. M. ? ^ -, ^^^. J!':^14?1*^® product of ax and 6ic by their H. C. F. ; how uoes this dilTer Iroui their h. CM.? I^EAST COMMOX MULTIPLE. (3) Divide ax by tJieir H r t? i ^^ / what is the result ? G^ve slmrok™"^'^^^^^' '^^ '^^^^''^nt by ^- ^- ^; ^ ^^'^^ *^*^er way of finding their their a C. f!1> ^* ^' '^- °^ «*^' ^c*/, ce^e, and t/e/. What is (o) The H C F f * find the proauct Of .ttVeTpSj^. '"»'- ^> ^- ^^- '^^ a*x, ^ 0)^FM the L. ,, J,, „, ^^ ^^^^ ^^^^ ^^^^^ ^^ ^^^ (8) Find the L. C M nf .,a. i. (») Fi„a the L. CM „/:?^'j:f' r '5- °- ^-^ ('"' ^in^ ^he L. C, M. It Zl^ ^ l^^'^r tl . (1) The n.C.-F. x-L C' \r t . (a^-b^)', what is the quotieri' wl T^'^"^"*^'^''°« ^'^^ («=-&=) a' +06+62 ? quotient when tbeu- product is divided by (-^) Find the L. C. M. of a;^ +5,, . 4 . ■> „ , (3) ^Vhat is the lowest nit f.f' ^'+2a:-8; ^= + 7a: + lo following ^^sio::^^:^i^J^^^^r. each of the and Gjc2- 7^ + 2 = ? -'»'+"; . (5) L. C. M of 6a)= - 13aj+ 6 ; 10^2 - „ (7) Each of the foUo J +''"«~3, and Oa:' >9a;=+ 5^.-0^9 of expressionVl^i^^^^^^^^^ nun^^r ' expres^ons, the ; 3u:' + la:^«Ga:=-.i2a;_5 6? PUBLIC SCHOOL ALGEBRA. FRACTIONS. aze«oiie 09. (1) In the fractions ? and ^ chan^^e tlio denominators to 20 and yz respectively without altering the vahies of the fractions. ^vi:l Jj; *^'^ same fractions find what the denominators become when the numerators are changed to 21 and tvx respectively (3) Reduce ^.^^ to lowest terms, and do the s* abcde same with tlu; fraction /cdfcfe" 21 . (4) Convert ^ into a mixed number, and express in that form the fraction ^'ti^+J''. a-b (5) Express 27 + — in the.form of a fraction, and put .]«- 11^ into the frat Honal form. ^ (6) c^-Gx + S i ^.a_23a;+28 ' •^^*^*°^' ^^^^ ^^^'»»'* of thi.s fraction and then Y( uce it to its lowest terms. Xztroiie 70. Eeduce the. following fractions to tl»©ii- lowest terms. I^jj a'+3a' -4« ^0) 2a^+^a^-l a* -i 2a- +2u ' + 2t/ -t 1 Hint. Factor the N. Hint. .3«»=9«2 FRACTIONS. CI) "' ■"'-^+1 Hint. In^., 2c^=>3c--ri. Hint. D-N -/^ i\ / « „ w-m4-— 1- Hint. -4w'=_5w'+,. f%' factor. 63 factor. no c SxerciM 71. 2 ,^ ,- the .san.e denominator and find their (2) Reduce ^ ^ f^ *i »<» -, J- to ,he same „enom,„a,„r and fl„d thei ir sum. ('») Re %„,.•, > 1 . '^4 f,' 7 ^ *o their least common — „.ato.a...«„a.He3u.„r.e,:.et„„„,,„„3 W What J.S the sum of 5?4.3« . 26„ ('>) What iK the difference oil -I 2 li Of I-l? («) P.n.l the s„m and aU„ the difference of ? and £. (7) Make -^ nnd 3^^ - "^ fh, « , ."""^ ^' + 1 ""^''^ "'" "^'"^ denominator and thus find their sum. ana in;UI,IO HC'IIOOL AUtt.«KA. 11 fi (8) Find the sum of -•^-1-^ + '. ft h <• (9) Add together ^ + ^J' + "''. a b V (10) Add -L-f/-^ ; ud also ^ +.1.. x+1 x-3 x+a x-a (11)^ + 1 + A; 'L-J+^lT^ + '^Z^ ctb ca be ab be ca (12) — fL +_j!_ . ^ ^+^y+y ^_^x'~xy+j/' 2a - 2b 2b - 2a ' x^-^ii^y 2 + :iJl^^:^y. fizeroise 72. (1) What is the product of i and ? ? Of ^ i\m\ *' V 5 7 b (l (2) Multiply I 3 4 5 ^^^^ a 6 c d (3) Multiply IJ by 2J ; also a + l hy 2a + h b c (A\ ««» Cy2 rf^2 ,^,^2 , ^ . , orni. 5c^ a' +2/' •^ -y a^ X (6) -'>^'' -6a; + 8 ^ ag' - 5ag + 6 ^x- 1 x'-ix + 3 x^-2x + 8 x^-4x~+l' CT) ^' •*• -^ ~J^ V. ^ag' - 1 2a; - 40 ^ g' - b- x'-+bx + 6 3x^ -ISx + 16^ a^-b^' ' ' (x'-6)»-«»''ic='-(rt--6)=- Ezeroiso 73. a V . c «^AAv« — M I b d PRACTfOXH. <^>°'-^«(|x^)B.(f.«); 56 I and ^ (•^) Divide^ by?; ana ^bv 3.- ' :-by?.f. c c d -'y 7y (^) Reduce i''^*J.'/'^Ww=/i2 cy Sacorcifo 74. — Beduce the following expressions to then. ' , (I) ^l + --2^.-3. j^;'"*''7«-P^estform:^ (I) 2rJ + «-2 a._ X-i l+JC + x ■J/ (2) (^_ft5^6>w^a " '"• -/^' ^/l'^ iN.J3.-Se6pagen^^ No. 5 andFv. • - ^•^ + 7^2 + a^_« ^' ^«^l Exercise ot;. ^,^a^^9.^.,,;:^""--^' -^Exercise (T) -,_ J 4.64 - -'rt « « 0? + --.— + ■1 N.B — Add 1 a 'r + 2n ^'+-">th; ?nd+4f] m PUBLIC SCHOOL At.fJliUUA. i I «»-. (8) .-. "7^,. + c-a a~b Sxfrolao 76. (I) Fiiul the value of fizl + ^"2 . a;+8 , ^^(5i) Kxpi^ss * HhillinK +./ ,^nce +,, fartl. .ga as a fraction of (3) Find the numerical valiiu of the fraction u'+ft^'+c^ra,,/^.' ^^'''«" -^f-'-'^ + c, y- c + a, z = a + 6. Hint. Factor N and D.^; obHorve that x+v 4-. --(« + /. + .. and that ..- + ,/.+,= _^.y_,^,j^^f -(« + '' + 'J, (4) Find the A-aluo of the following expression when y 1 • N.B. ;y=-l, y» = .oi, y^=.m, etc., ,. iv» = -000002. ot,-. li^^ ''-'' by l-« to 5 terms, and write down the (fi) If.T+ =land« + l = i, prove that y + 1 = 1. J X 2 Sxeroite 76. le&t your answer by puttinf? « = i. ^ (2) ^« + -e _3tf 2a: 16a;2 Te„i .._ ,*• t. .:n,^%vpr n^ jHirfini,' a -- .r ._ I. k ^ *'ttACTIONK. (3) 1 _i__ ^.+n ^"'*-2'»"^^«rify your result. ^»*«-^-~^l, and verify your ...suit. Puta-' ." , VV-*-Jir> ^ and verify your answer. 67 f^ + c-a c + a-b^r+fjrr,' ^uta-6.e.Und verify your answer. (7) -3- J + 2a ^ 1 '' +3a + 2 a'+4^?3+„T^P55-^-g. Test your result. SzerolM 77. (I) ^i±^'+^:i5= 26» «+« a-aj°*^3rZ^a' ""^ substitut.on. ^ + b iTft ' ""^ *' *^"d verify the vahie. «--6^ c(« + 6)+,^i^y ^"'«-2;6«e»l;andverify HiXT. Divide each N. by its D-. , the vern^U,^,,^^ taking each nar^^ ^"^V"""^"^ ' "^'^ -« -"^•+12==2a;^-9x-+7 •^^'i^«^«tely, and got ( < ; Verify t ho results in 5 ««.! <- u , ♦ 58 PUllLIC SCHOOL ALGEBKA. Po^Sfout^^^^^^^^^^ Hints Lnve from making partial additions or «nKf^^ *^^* frequently arises number of fractions, instead nfJ.f*'"^*'^"^ ^" dealing with a oi^eration at one ste. W off, nT^V"^ *° P^^'^^^ "^ the whde K ^^ "*' a series of fractloifg .Tif /''® *T™» «<' an eqna- high dimensions or of compl?catocl f«r„f ** n^***** " ^- ^' ». of mathematical studies is to fdvecnn^'' ^"^ ^^'^^^ benefit of ment of the student in choos nVth*'""°"^ ^^^^^^'^^ to ti.o jndg- Phshmg the end in view. TlL^oweHfe "^l^^^ «^ ---" Zntll^ «<>nstantly illustrated in ^hese !fT''*.t''*"* »"-«^« J*i?"«« «f a problem being always fh/""^'"."; ^1® "^'"Pl^'st perfect knowledge and the moJf Tifi^ ¥ '*®®"^* «^ the most thmkmg necessary to produce thAi'^V^^^ ^^^^P ""^ ^^^ ^^s^act expense of mechanicafworr ZiZt'T^^' ^or at the leS find that Algebra gives aim oKt, f?«.f*ndent proceeds he will tm he has mastered the ^^implSs&ZT'^ ^'' ^"««<'«« Tj f jj n® ""^^ produce. oftl^subSftt^^T^fScU^^^^ (^KXERAL RkvIEW fractional equations of ^,ro nn tl ''^l"»tio"s. In a few caSl dnced in order to Cp^^^^^^h tTp7v7fr'^ -^ ^''^^'^ ^^"^^ntrt theoriguial invention of thrSpnfV"^^^" ^"^^ *« exercise examples will be found within th«- -.^"^"^ ^" ^«ses the capable of solution and fh i « ^™'*« ^^ ^""8 power nml sr^rsed with the r^C of ^he oM ^^l'/ ""T ^"^ S in^e"^ interest and encouragehhxi wkh tbl "^'^ °"^^ ^^^^'^ to whet h s intellectual power. ^^ *^^ consciousness of growth in -Sjc-l Bxereiao ^8. I2'IWI=';\S'"-to;rt,4-3«-+^f Will ,i™ the product a'^-li^'-'^: ir.""' "' ''^'- + 2'-^ - 2^.V' ; and the factor „, (o) One factor of «3 +863 ,„3 >. , . apply some kynrforwr/f^'-^^"'' *^® other factor and t-v J ufir answar. a re in of the tioi (4 M \'7 (6) c»t:xj£itAJ. itjiviisw. i>'J iTlvn Tt— _ ... iK •■Simplify and test. x^ . „, Hint. Expand bracket nn^ .''' Bxerclae^9. (1) Divide G0a;3_17..2 4^.-., remaznder reduced to Wltti'^^^ -^+^-2 and give the • (2) Divide the product of 0^2 . ^ . ^ "^Tt^ . (.") A certain .ort,a.e w n ''"' ^^""^- ^^^^^^^^^^^ ^n -^months to C iTdZ / ^"^^"»* ^" «^ "months to $u « 7 of sample interest per™^ '^^^ °^ ^^^ Mortgage a nlTt!;^' 4Te \, .^•^' ^* '« = «^ortgage, . = rate per cent n. then in 1st case w + », _r. a^ _ ^'' ^^"*- P^r annum ; ^ ««, 2nd by y, and subtract. (4) Reduce ^zi_££l ^ ^ti _ ^- 1 s^naplest form, ic-l a;^i ^•B. Multiply the N. and the D fi. x. (5) Divide $100 amon^ , . "^'^°"' ^^^ "' - 1- -h a^pr/-^^^^^^^^ boy and a ^i^ :^^^^-^ /if) /"I ^ n Sxepoise 80. ■ and fiO ^vuua sorrooL algkhua. i^-^')' + «j^£c-jr)2 (2) '^L^fLzDl.y- N.B. Tost your answer by putting x = y = z^i ^'^'^"3,^^^;;Jt^;)^ + (--)' subtract cc. >, (r>) ?+'»_«. 5 A ~ t.' /J- Szerciae 81. (1) Add together a4./j_o,. . q ■™amderby3«-66+5c '* "°™ ""o snm; and multiply the b 'OF.nd the factors of 9.,= -8a^_4,J •'^■ (4) Find the L. C. M, of 3x' - te» _ L,' ,,. „ JX-- il ''"'' " "r"r "•" - »^«1 *o these three together :- ^-^^^^^1-^^^ ^^^r^ ^^^,, ^^,, (fi) Reduce to lower terms the fraction v^-t 15a;^ + 13a;.|-2 ^ (7) .1 2 Hint. Add Ist and 9„^ . w! .u.. _•*• then add the 4th '"' ""^ ««°':rac,. ord ; / OENSRAL UEVIEW. (U SxercUe 82. » (I) Solve the equation |(27 - x) + i{Sx-4)- J (o,r - 2) = ?. (V ?^' ^ (2) Find the three factors of 25G6* - Shi*, (t' b *t. *.) (u^, - ^aj l/i ^^ f a*j > (3) One factor of ia"^ -^a'b-\Qah^ +'6Qb'' is ^<-2/;, find tlie other two factors. j (4) If rt2+2a& + 9 is divisible by «+/>, what must be the numerical value oih'i K - A (.')) If a+6 + c=:0, show that a- +fo= +c^ = -2(/»/:=: 369. Hint. Subtract ; multiply difference by 7. 62 l-UliHO SCHOOL AI.UKBKA. u 'A K-^^' 7^ Bxerciae 84. (1) Find the H. C. r. otx^-^r^ l^^^i- ^•^^ ^-:6"^6r^"*"^-' find the sum. Test your answer by putting a=:0, 6=1, c=0. ^6) If .^-^^+|, ^^«_-| ,^, ^^^ ^^^;^ ^^ ^3^^^ Test your answer by putting a = o 6.^1''' ^' (7) *(5a;-9) + ,ii,(7a;_i)==.wij ' ' V } ,6^? '^^ ^"^ ^^ and the length of the street ir"feet ""^ *'^'' '"^ '^^^ ^''^^ (7) Beduce !f±f?£lzi!^ f„ in,.. . * « - ct'^^^ao^ lowest terms. A. GENEKAL KEVIEW. (8) Simplify (-.^l-^'zly)^ '^^±11 Put 05=1, y = 0, and test your answer. C'J e numeri- ir result. if they li by $5 he vow Sxeroise 86. (1) A(5a;-13)— i^f(a; + 3)+4 + i(5-£c)=0, find a?. (2) Reduce g9(«^* -4 a^^+4x-'» -1 ) 65(aj*+4aj2 -4a;-l)* (3) K3aj-2.y)+^(2£c-3','^=2i; and Sly-SOte + lH =0 ; find a; and y. ^ (4) Resolve the expression (1+a?) (l+v)-a;v(l -cc) (l-v) into two factors. (o) A pound of salt is added to 31 pounds of pure water ; how much pure water must be added to the solution so that 32 pounds of the new mixture may contain 2 ounces of salt ? (6) A capitalist receives an income of $4,640 from $100 000 the greater part being invested at 5% and the rest at 4°/ ; what IS the amount at each rate ? . ^° ' (7) Divide a number a into three parts so that the second sHail be n times and the third m times the first. Sxeroise 87. (1) Prove that if c be a common measure of a and 6, it will also measure ma ± rib. (2) Find the H. C- F. of Qx^ - \Qx^ + 10a;-4 and 9a33 +3as2 *- 3x- 6 by applying the principle of Q. 1. (3) Examine 81a;* -hlOSa?' -24a. , 1, and divide it ir-o two equal factors by comparing it with the formula of Exf>r -se n page 20. oo _ , (4) Divide (_L + _^ ) by ^ _i_ - J^ \ (5) Twentv-four nersona aiihor»r'i*lw3/l oo„,«ii,> i t _ boat, but four of them failed to pay up and consequently each of tho others had to increase his subscription bv «^'>. What did the boat cost? fil A '•VBUC SCHOOL AU>EU„„. «- 2*. Pa«e 20, and th.^ ';itat^;:^'^^^' ^^'^ ^- <''^) Simj 'lify _J _7__^ _ 4 - 20a; r:r- =>V %-2 3a^+4y=31, find ;x. and y. iM ^..73 o ^ 6 7 3-^ 21' (<^) -i, A and C can diir « f ,, • ' ^"''^ P«^* ^a^d. 'IS much again a^ » „ i^„* *^®"^^ ^n 30 hourer hour and arrives jubt in time. Find his rate at first. N. B. 205^ +,jc- 36 =r 0, when x- = 1st rate, i.e. (J5 - 4) (Sa- + 9) = ; hence either ic - 4 = 0, or 2^- + 9 = 0. If .r- 4=0, a? = 4; if 2aj + 9 = 0, £c= -4J, which would have no meaning here. Szerolse 00. (1) If a certain class-room were 7 feet shorter and f^ feet wider It would be an exact square and of the same area as at present 1' -nd the length and breadth of the class-room. . < ®, ^ *"? ^ ^^"^ P"" --^ square rods of flax in .') davs ; B and ( J2.5 ; and A and C 230 in the same time. How many square rods can each do in one day ? . (--J) A man invested $330 in two parts ; on one part he gained l.i/, on the other he lost 8%. Ho received from both $345; liud the amount of each investment. (4) A Mississippi steamer takes IGO minutes less time to go from .1 to B than from B to .4. In still water she could make 14 miles per hour, but the river runs at the rate of IJ miles per Jiour. Find the distance from .4 to J5. •./i'lln^^®^ "* *^'^'^ ^ dissolved partnership, A got $2,070, 7? M,.»20 ; the capital of the firm was $3,400, and J's monev w«s put lu for 12 months and i3's for IG months. Find each man's sliare of the profits. ordinary ordinary north at fn how 47.) B would ' on the wuiiid ice and Szerolse 01. ( I ) Prove that a" = 1 ; and that a'» x a" = a"'+»' (2)-«-?^=l,.^ + y=«;fi„d«.andy. a b bub ^ (3) A speculator paid §3,000 for some railroad shares • ho reserved 10 shares, and made U a share on the remainder b>- scaling them for $2/700. How many shares did he purchase ? (4) If a lake steamer could go 4 mile an hour faster she could make a trip of 420 miles in 2 hours less time than she now makes it. What is the siearncr's usual rnte? Uy y "^i^UC SCHOOL ALGEBRA. the hor.es Was ^'oo'^ """^'^^^ '^ Kain of 16^ ^'u' ' «P««t the Steerclio 92. (2) Find the acre . ' ^*' (3) J and B perf„„ ' °\'^ + '"». *»+aw, 6'„_^.„ 1 then in II case ^ .. ~ ^"'^' ^ = ^'« (4) Factor a;2 _ 9^ o-- ' 2x v 9( (1) Prove that »»WU. 83. »■-'. '^ '*''» "^ '"« -hc,e. FLd\l^C^„S"'y«t3%,and (4) -^..^._.-5 .._. "-'"'-h invest. '^^i^h, : (r «-C .^TTj- N.B. Add A««i, '^'"-*>-f^--n.r-7,,,„„.' =5^' tie sepa- OKNEUAI- UKVIKW. on the cost; >nt ; spent the 5 my profit on tie? '> + c-a). ^ert starts iniies more at C- 9 days nCtoD. stances tra- V> B goes "fcif^had nave com- tch require etc. ts square etc. . fil %> and invest- (5) If I sell my horse for $200 I shall lose monev, but if I gef J2o0 for him my gain will bo three-fourths of the former loss. Find the true value of the horse, (0) Between 2 and 3 o'clock, I mistook the hour hand for the mmute hand and therefore thought it was 55 minutes earlier than the true time. Find the correct time. (7) A mass of lead and tin weighs 180 pounds in the air, but if It IS suspended in water it weighs 21 pounds less. On exjieri- ment it is found that 87 pounds of tin weigh 82 pounds when immersed in water, and that 23 pounds of lead weigh only 21 pounds in water. Find the number of pounds of tin and of lead in the given mass. Sxeroiie 94. (1) If the sign ^ placed over a quantity means '*the square root," so that ^4=2, ^9 = 3, ^16=4, etc.; simp' 'fy the follow- ing expressions, ^81 - ^25 ; Va^-V6^ s!{a- -r2ah-{-ir-). (2) If ^ denotes the cube root of a quantity, so th,. 8=2, ^^.'^S ^^^ = 4, etc.; simplify the expressions ^'a' /ft' 2"'. V8l-^12o; ^(a' + 3a26+3a62+63). ^(27£c^+27x- +9« + l). N.B. In the last example compared with the preceding observe that 6=1, and a = 3a;. (3) If (a+6)2 =a2 +2a6 + 7^^ what is J(a^ +2a6 + 62N? Of £c2+4.>o + 4? ' (4) Reduce J( "" +h^ -\-c^ +2ah-2bc-2ca) to simplest form. N.B. Revidw Exercise 24 and Exercise 61, and do this question by inspection alone, (5) What is the value of y' 81 - ^ 16 ? Of ^3* j + p 256 ? (6) If 9 . 9 . 9-729, express 729 with the Radical Sign ( J) so that its value may = 9 ; 1331 so that its value may - 11. (7) Find the value of 5 ^(62 + 3ic) - 1 V(95| - occ), when x = 6^. ^8) Evaluate ^ - ^/Lt5, when x = l,yA sepa- (1) Find the value of [ VCa^ +h^) + c-\ [sl{a' ^-h')-c\ whena = 4, 6 = 5, c=&. ^* • ■ \ A v^^' >»- 3 ['h G8 -'' ^'>«» /> = '> p.. "•J".no lOOt of '' -it.r ^.2 ^,. • ^rove this (•^) ('H-6xe)3 3 •^•^'''^(^' + ^>). lean, this I,v h iV =«' + 63 + r3,<, ,., "'" '>J heart. ^^ca(c + a) + i,,;,/ '■ + -^'Hrr+ h) + nhc(h + r) + ''') In question o , w "^^^« + '') (He) rc + «^ i)iovo '. + /, 4 " 2^substitute c + rf fo *L -^^ '■") Show ,1, ,, ,, ;, 3= ■' '»»">ad of „ ."'^^o;; . «")]. (II) Kactoi«='+fe^ -ca+3a6<-; «' -6' +c' +J j6c • (1-2) In «^ + h^ 4- c^ - '^ahc substitute c + d iov c and arran>je' the result III a form similar to the original exprossioi Sxeroise 97. (1) Prove that {a^b + c)' =a= -^-b'' +c' +2ab-\-2bc-^2ca. x2) From que.stion (1) find without actual multiplication the sfiuaresof -a^h + c; a h + r; a+b-c; :mda~b~c. (3) Ifx'+hi+c^ is a perfect square, find the value of b in terms of x and c. (4) The difference betueeu the squai. s of any two odd num- bers IS always exactly divisible by 8. N. B. Let x and v be anv two numbers, then 2x and i>y/ are even numbers, and '2x+ 1 and i\y + 1 are odd numbers. Then if .r and // are even, cc - w Is even and divisible by i> ; if x and // are odd numbers .r - w is even • if one IS odd and the other even r + y/ is odd and :c + y+l is even. T.'i'l.T^f ^3"*^'*' °f every odd mimbor with 1 subtracted is divisjljle by 8. ((i) The area of a rectanj,nilar lield is (V ucres J»(;o yards, and lis length IS three times its breadth. Find the distance round the held. (7) Find the diaj,'onal of a common field 143 yd. by 1 U] yd. (8) Two boys start from a treet corner, one west, the other north, the one at .'J miles per hour, the other at 4 nii!.'si>er hour • iu.d til- lengtli of the straight line h weeis '.em ui the end ol two hours and tluee-quarters. (SeeEu«li< ,17.) Exercise 98. (I) A earns twice as i.uich and B three times us much per dav us C earns. .1 worlds 7 days, B 4 days, and (' 10 days, and their total earnings amoui to $72. Find tlie daily v. aires of each. "^ *' iJfd ^^T ""[ '*f " ^ ^'1 ''■'"■''' ^'*^^' '^"^^ ^''<3 <-uirlage is worth ?>:ilU. V\ hfwi tiie better horse is in the^ fnn-.ao.r. +k^ ,.:,. : _..ii three tini , as much as the second horse together with 107 o£ the vahio of tlu iirst horse. -Find the valtio ol e^ch horse. - I 70 v.»- JU»~ (..I '•^W^IO SCHOOL ALT^KBR^. (4) U. t "'""•"'' '''J« before the t**; After paying out I a„a ' nt ^«^*: «-v:„uehhaar;%.,,? "^ ™°"«^'' ^ >- the '"o f«st, and Jo, c7«irrf- """ B'-own's w»,.> ■'"'* ""is ca„ t-- hour did oacHf ^tl .'• '"ir '«"> "» X V? il""'""«' . (2) Two sums invested ,^o "■*"' ' ^J"'"* ""« times and meet in 7 k "^^^^er start from f «r« * le djd go, the" wo,,ld i "".V"' ^"ster, MdT,/f' S"' """> -B sion ^a,t i"^3*^° "-"^-i^^I value of « thaf ■„ , ^^) A Jwtxn'^r'""'"^''"-^-^"*'"'^ wo station,, X and Y p*^ *»""'''» « miles an 1, . . - ^"■'"^^"'-«''«l«lo time, etc. ill f pev how aiid .^« wuh tbon •rjved nt home '•"lo before the Jiad only 16 Jn a straight OKSKRAh MKVIEW. 71 (6) A liquor dealer buy8 |.roof Hpiritn and mixes u terfaiu quantity of water with it. He sells the mixtuie at J shillhms a gallon niore than ho paid for the spirits und thus Kuins y.ljv on his outlay. But if ho had us.d twice the tnuintity of water that he did use, he would have gained 'M^X on h is ino'nov. Find the proportion of water in the mixture he sold. N.B. T^t * = No. gallons of spirits ; y sliillings = cost per gal. ; and ac»No. gallons of water added to form the mixture ; cost = sy divide tlie equations. selling price = _.9i/; supposed selling price = ii-.v,/ 80 HO ■' forth at the 'r Watches, -ach other, '8 drives ii Btween tlie w this can '^minutes vhafc rates les. '■oduce an •changed, line time, than B. ^e speed t- Pind ' 6xpres- ^tween ion, z, » down Com- h etc. I Zzeroise 100. (1) Find the H. C. F. of l+x + x'-x" and 1 -a?* -«» +x". (2) If a + 6=1, provethat (a»-62)2^^3^j^3_^^ (3) Solve the equation ( or + 2) ' + (./? + by^{x + iy. (4) divide x' +Sy^ -O'Jz' + isxyz by x + 2y-^z. and tost your answer by substituting oc = 5, y = 4, z^^li throughout. (o) Find the square root of 36a-'-120aic-12a*ic+100a2+20rtVf.a«. (6) Find two equal factors of the expression {x'+y'){oc'+z') + 2xix'+yz)(y^z) + 4x\t/z. (7) Show that ix-yy + (y-zy + (z-xy is equal to Hx - .y) (y -z){z- x) (x' +y^+z^- xy - yK - a*). (8) Show that {a■>rhY-\.{a^cY^{a^■dY^■{h^cy^{h^•dY^{c^dY = 3(a + 6 + c + d)(a2+6»+c2+d2). (9) Show that 8(a+6 + c)=' - {a-vhY - (6 + c)3 _ (c + «) ' = 3(2a + 6 + c) {a^-'^h-ifc) (a+6 + 2c). (10) Show that (3a - 6 - c)» + (36 - c - a) ' + (3c - « - 6i =• --3(8a-6-c) (36-c-a) (3c-o-6) = l6(a'+6='+c'-3rt6c). C>>^*^«5 urn J*-*- & THIRD STAGE. 'J'yPE SOLUTIONS. Note.— ^g fu- , ever, intended m f T^ ^^^e. The foU • ^^® te^icher's wn^f '^^ hole class should TVPK .SOLUTIONS. 78 frequently do this work witli the crayon instead of the pen. The following exercises should all be reproduced by the pupil inde- pendently of the book a considerable number of times. If thp\- are written on the blackboard, each pupil may in turn road over his work aloud and demonstrate the solution, the rcmaininu members being required to cro.ss-examine and '-quiz" him at the conclusion. In this connection the teacher may occiusionally write down false or inaccurate solutions and require pupils to detect the fallacy or the mistake and to correct the work. To secure thorough, solid progi-ess, review work of some kind should be allowed about one-half of all the time that can be allotted to algebra on the time-table, and the rapid reproduction of former work is the essential part of review. an aid to exhibit the Mns much ^er's work are, how- the cieai- 'trate cer- lapiditj- examina- achieve- ■t^ei', and exercises the best hig-hest r many >i«tions eed and imounr e more e year, 'k t\\o ^'aclier ' niost ^'n on n dis- >f the ^rfect iCcii- ould (1) Express [a-{h-c-{d-e) -/} - g] in simplest form. Solution. -!--++- + _ a-b + c+d-e+f-y. N.B. In the first line write down the signs onlv, changing those that require change. In writing out this line follow out the changes caused by each minus sign separately. The line is to be written piece by piece, not continuously, thus the signs in this question are to be written as here numbered, 1, 2, 3, 4, (}, 7 r>. (2) Simplify the expression 2x^ + {ix^ - {Ix^ -2x)}~ [»ac2 -ix-{i>x^+ {Gx- - Tx^) - 1 lx\}. Solution. + + _ + _+^. ^.__|__ + 2x^ +ix'' -Ix' +2x-Sx' +ix + bx' +(yx^ -4x + l -Ux Result = 2.^2 -9a; + 7. N.B. The line written over 4£c - 7 is called the vinculum and has the same meaning as a bracket. (3) Multiply a?" -bx''+ 5x^ - 3 by x* + bx^ + 3. Solution. l+O-S + O-h 5 +0- 3 . 1 + -H5 -1-0 1+0-5+0-1- 5 +0- 3 -t-5 + 0-25 +0+25 +0-15 + 3 +0-15 -t-O-t-15 + 9-9 a;'»+O + + O-i7£c''+0 + 7ic*+0+O +0-9 A m II !i W Multiply ^.^5 +10 1 .7n:.'.°r»??-i2o -5 - 5 - 23-7^" + ?f0+210 «n?i+f ">« "onti^ed product „f a,^. Solution. .... ^^"^ «-7, a?_9^ Ji*- (^i I!! iil!: ^•^- This arrant, + ^OOi^T^rgo^o- o{ the missing ; to save the -^Y«* shows other powers se having zero Ex. 20. TYPE SOLUTIONS. 75 -ICo t£4 +21 %«re that rst term is iat««,£c7^ 7, a;-9, Ihetln (6) Explain by an example Horner's Synthetic Division. Demonstration. As division is simply the un-multiidyinff of a product of which one factor is given we require to reverse the jprocess of multiplication in each particular, i.e. to subtract where we formerly added and to divide where we multiplied. Let us take the factors a^ +2ab + b^ and a + b and first form the product, and then divide the prodvict back by a+b. The quotient must be a^ +2ab + b^. The operation will stand thus :— Example (a) 1+2 + 1 1 + 1 1 -1 1+2 + 1 1 + 2 + 1 = partial product added. 14.34.3 + 1= the product. - 1 - 2 - 1 = partial product subtracted. 1 + 2 + 1 = quotient. In order to change the signs of the partial product as required for subtraction we change the sign of every term of tlie divisor except the first term. The quotient is found by dividing the sum of each column by the first term. A second cxamiile will make this more apparent. Let us take a^ +2ab + b- and -2 ft +06 and proceed as before. The operation will stand thus :— Example (b) 2a + 36 2a -36 a^+ 2ab+b' 2a='+4a26 + 2a6' + 3a-6 + 6a6^ +36' = partial product added. Ha'^^la^b + Sab^ +36^ = product. - 3«^6 - 6^6^^ - 36' = partial prod'ct siibtradcd. a^ + 2a6 +6^ = quotient. The last two lines are found thus :— Dividing 2a into 2rt' we get a*, the first term of the quotient. Multiply this a' into 36 with its sign changed we get - 3a ^6 and place it under + 7a ^ 6 : the sum is 4a'^6, which we divide by 2a and get 2a6, the second term of the quotient. Multiply this by - 36 and we have - 0a6% and the sum is 2a6^ in the third column. Dividing this by 2a we get 6^ the third term of the quotient. Again multiplying by -36, adding and we find the sum==0, which ends the operation and shows that there is no remainder. Using detached coeffi- cients only the division would stand thus : — 2 - o 2+7+8+3 0—0 1 + 2 + 1 =^a'+2a6+6= If I ) 76 """'■«-' «=»00'- '^u^m^,. (6) Divide IW .0 = ,^— J|'^/-^'';r/^-V---.-.,.' / a " Solution. i-k ** "^^^^-^ by .^;3^9 » + 3 I +0-0+0 ^—| -5::-ryp-_ + 10-30 + 6 qi'otient ^vill Z of /^ '';.'''''«'"', for ii".'''»''«>- is drawn after ""^ torms. "' "<■•" d.me„sio.s „„d J,?, "f • that i;^ (8) Find th '°" """fain ^ J ^n+;o'r "'"'''""-'--'01 •• expression .^-'^ tnt^ . ~ ^ + ^ +~2~ Vr — ,- - -'«'"-' the e^:--:"-ore the remainder.. ,e„,„,^ f'^T^Cr^fc^e-u**,.^ '^il+l ^een quotient °f from the the divisor, to illustrate -3. TVPio soLirno.vs. u awn after lat is the ■e contain »en js' :^ s. -.:>=o. - 101. 1-0 -lOl ■lor ii'-'ri- tlO) What vahie of x will make ^= -2a: + 3 an exact divisor of the expression Ix* - 'Sx' + 1 ? Solution. By division we find tlia remainder = _ 34a; _ 1 1 . To make the division exact we must have thi^ = 0, whence x= -^^. (11) Find the value of x, when lOoj ' » + lOx" + 10.i;^ - 1000 is exactly divisible by x' {x* + 1) - (x - 1). Solution. Dividing by x'+x'-x+l, we get the quotient lOaj-^ and a remainder of lOjj* - 1000, and this must=0 as in examples 8 and 9 ; i.e. 10a:* = 1000; a;*=100; a3-=10; ic- ^10 =3-1623777. (12) Find the value olx^ -ix + S, when x = 3. Solution. Since x' =x-.x = x\n, x' =3^= =lix.x^^.c Take in - 4x, and x^ -ix = bx = W. Add +3, anda;^-4a!4-;5 = 18. The work may be arranged as in Horner's division, and is thon an exemplification of that method. Instead of bringing down - ix and +3, carry np 9x and 15 thus : - - x'^+O -4x +3 ox-+dx + 1.) + 1« i 3x + b (13) Divide x^ +}/''+ z^ ~ 3xy?: by x+t/ + z. Solution. -(.V + 2) 0..X'- - .r(:]f/z) x^ - x{y + '/^'jriy- -yz + 'J) Here tlie dividend is arranged in descending powers of x and all the coeflicients are put in brackets. (14) Reduce to simplest form {a-k-h + c-^rdy-\-{a-b-c + dy-+{a~b + c~d)' +(a + h-c-dy Sorvnox. The briickets are alike except the signs : and for every u t jo.^i is a &, a c and a d, i.e. the expressions arc syn.. metrMfal. hach bracket is a square, and therefore the cxmin- sion T,:!! certain only #wo kinds of terms, viz.squarf^mc, a'. and i Hxuvts -ike 2ab. Following a^' througli we gel !-/-, and w« therefore know that Acr ^Ab^ +Ac' +AiP is par, of the aaiiwer. t^ext following 2ah through the four brackets, wo ^m u, eaun imir carjxiung, and we therefore know , that 2«c, 2ad etc., wjU ftloo ouucsi. lUwJi, 'J(a^' +6^ ^^^ -vd'yi^ the rosulL ' 78 Uy-i e /-BUO SCHOO. ,,«^,^^ / " ■■- each b^a'S "fe «« chan^t a'^SoT,"'-'" **"««'' 'or it second'. '''V^W heoom!! hi"'"' ■"■'"''h is the " ""^ '" ">« Result „4(«T;^;^?'-2«6-0 an ax there « . " ^^""netry i. „„„ ^'''*+*4'+os)». , we«et: «4^7i?™=::?^/o"f t ^041; eth rf^^^^ f'- Which we eee .ha^ptfevi^r^^*-^ + -'-.VCfe) + .o„(4^) n«^ *!- 2a2^^2 j/l\, *^e result is J^oilowir.o- n-i +1, . f"Ose hke 'An-h f '*^'^c sorts nf + i?(a^+6';^a.^"'^"gl^out, we ^et A"""^ ^'^ '^oseliljT' + -} i« ofte parfc of til .! + « +«^-«^==o:!r ,^«*^- — i-Buio. Uni/rw^,' ^ ' "'^«Ce C*y+y2 + ;^a?). TyPE KOUJTIONS. 79 (b + c-ay, trical, lor' it > c, and c into + « and is the ' same as the same as the ' t«e same as » ft'om which =«iating one ace We get, 'ymnietry. '^' ihero is o^.i^hout. diet and • 9, -««=%) ■ea). ceding. *^^owa erms, I. _ 'i'iiiCe V We haveoa=/> + 3a-ft-3«=6 + 3a7> = 0«='6. Hence part of the answer he remaining part must be of the same tyi)e as ahc. To find this part put a = 6 = c = l, which makes the ^iven expression = oO, and the part found = 42 ; and from this we see that - 1 ^abc '^ o/%'*??3"""f P^^\°^ *^'^ ^^«^'^' ^"'i the whole expressio'n is (19) Simplify (a + b + cy -(a+b-cy -{a + c-by -(b + c-ay, :„ Tcd + S?j^{c + d + a)+6cda + 30=^(^ + 6 ^b)+Sdab + Sd\a + b + c) + 6abc N.B. If we write a, b, c, c? m a circle and draw all possible straight lines from a to 6, a to c, a to d,b to c, b to d, c to d, and so on if there are more than four letters, then the straight lines will indicate the double products in the square, the triangles will show the pro- , , ,., ducts like a6c in the cube, the quadrila- terals, the products like abed, etc. Thus with three letters there IS only one product like abc for only one triangle can be drawn : but with four letters there are four triangles, with five letters there are ten triangles, etc. Let the student form the cube of «+04-c-|-a+e by symmetry. (21) What is the value of a^+b^ -\.c^ -3abc when c + d m substituted for c? See Exercise 96, question 12. Solution. The given expression is symmetrical for a, b, c, and contains all their cubes and all their products three and 80 ^vuuo ficooL AunmiA. t-hree togethei-. Tho ,« • i ^•B. By norjcin.. tl.n «,.. . ^ '^ + « J« «"bsfif„tp,] fn,- ,. (-'-'; Ealuco to sfmjilost f„ ■^'h -;+: + - + -;: + (23) Factor «3^,3^^._3^^^^ ' . • ^^- the expS'S^^ ^^)- I^ooking to the sy„.„.etry we can arrange ) + 63'~1'( ! + «(«= -6c), l+c' Now «= _; • '*"'" ' ' "^^^^" ~^^) ^ (a + b+c) („' J.;,! r.r " ™^ of terms caii,-^i„ „.!? 3' . tt'ii the samo o. 20 we can Let th^ tp(I for r. "<.>' io covivct ' monsioMs hixie (limeii- ike. Everv try. ) ' three pjo- gn in each 'ame way, c into - c et eg Ic^. arrange TVJ>K Soi.irtlOXH. 81 ,v:'j./.3 V^^^'''/:^- ^T'"''^ ""^ ""^ '' synunet.-ical expression " V '^'"«0'»Pieto, an(lw^> a)• seeKxon-,sef)i; No 1 ; anU Huhrnw, the sani;, term a thctn l' Ihis j^vos usia + hy+c-^-tiah(n+h + r), of whirl. .,4-/-hV is evidently one factor The quotient is (,. + z])^ - („ + /,), + . ' _t- Jr wh.eh reduces to «= +6--" +.^ _«,,_,;_,„/ the otter fic'tor. ' _ Solution (3). Looking at a' +h^ +c^ -i^ahc and com panngjt with Exercised, No. 4, we see that it is s^'mmetrioa h' incomplete, and we add and subtract tlie missing term (a+6 + c)^-3(«+6 + ,o (aft + />c. + r«), of which a+b + c is one or «2 x 7.^''? ' "".^ ^r + '> + — 3(r/^ + hr + rvO, T^f ?L , A t'''l~" . '"' ~ '*'' '^ ^'^^ °<^^'e^" f fi^^o'- a« before. Son «^"^y>ns the symmetrical form of the given expres- (54) Prove that >i 1 .orfect square and find its square root. + ' '' ^""^ + ^ ^- ^«) '>^ Solution. I^t a'- +b^- +r^- ^.r ; ami ab + bc + ca = v and ■:f f J^*^ ^ „^^? .y ?. ^'^^ «.iv«» q"-"tity. We get in "his ' way ti acting 205^ .v- , Restore the values of x and jy and we get Ihe square root of this perfect square is therefore a'^+b^+c^-P^ahc. ^-ich the omplete id sub- le first 3 have row. (25) Express S(a + b + cy - (a+b)' ■- (b + cy - (c + aV m factors. / v i / ot X, y, z, and this = a(rt + 2ft + c) («+&+0c) (Oa^.6^.e). (26^ Simnlifv tliA ovriVoaoirtTi (3a-6-c)=' + :.36-c-«)-^+(3c_a_6)3 -a(3a-6-c)(3& c- a) (3c -a -6). Ex. 100, No. 10. 82 m 1 1 I'tfULIC SC'iltHjf ,, .^ e to a ,. . rst! ana r-ef™^ "^ «"4t/;-'i;*^?rc? „ which ispIaMytt*Tz,tff -<;'*-«='^-ac3-a6n Hence the whole exprfaaion +«'-«ft-»e-ca)'. "* ^ SoumoN. Letx^«- ' ^ aoc. valuet?h?°^* - *"+*+") =3»- '/="/' MrA''"'^ »-»" these, result. (B) Exercise 96, No i vfK ^ + «) (2 + a?) + 3a6c is f ].^ 7 o .3 *_. iium (A) TyPK SOLUTloJfS. 88 a-b) cal. Let + 6 + r)». -2£C), of 1(1 factor and the expand \ b to c, = 0, and ab^) Sabc)\ these, r these Udto ch is is the !umed z = a, ivhole (A) ,• ml^jJ"^'- ^^"^ '-el'ition^l^t'tween r aw\ ^ which is necessary to make x" - o xactly divisible by {x - m) » . SoLUTiox. l^ivlde x" -etc. by (a:-m)» and put the remainder = 0. 1+0+0 +0 2m + 4w3+(;,H' — mi' '^m^ ' l+27/?+:(»i It is therefore necessary tiiat ( . sei>iirately. -07 + 4/' + H7»^ -3m* -^m^' ('"»*-07) + (4/'-4?/t*). of tlie last two cohunns^O, of from the last of the th«f«,. " i ,^^*=^^7^ m'- I 'H=^"=7% from the iirst of the two; and 4;-.^ |,m' ; /• = m'*; r*=m'-" two. Hence the relation must be /•♦ =q'', ' (30) Fin(i the H. C. F. and the L. C. M. of apx"^ + {aq ■rbp)x + 6 + c)6e-&(6 + c)c, which=-0, and this IS the remainder. As the expression is symmetrical, fe and c must also be factors if a is one, therefore abc will divide the expression. Now each term of th« Avnroca.-^r. ,-f ,v,.,u.„j:-j -,..l would contain <^res letters and no more, like aab, 'ctiZaia^^c^l it IS therefore evident that thoro cannot be any other lUerd, n IT* IMAGE EVALUATION TEST TARGET (MT-3) /y .^f-^ ^ ^ « 1.0 1.1 2.5 ltt|28 itt liirii tt Itt Sf |i& 12.0 11:25 i 1.4 i ■ 1.6 '> ^ ^ / f %'/ // HiolDgraphic Sciences Corporation ^ '^ \ \\ ^. 23 WIST MAIN STIHT WnSTER,N.Y. I4SM (71«) •73-4503 <0 O^ ,^ .^ { 84 PUBLIC SCHOOL ALGBBR.v. I number like ;{, IXJ^'']''T:^'^' j-^^^or :"« the,-eL^ {^Z rherefore .),o wl.ole e.M,,«»i„„ .!_ , .,V" " » " » = >• X B T r" -• to assume\hem"SicM'^'at eS K^r 1?"^^^''''*^ *-"•*" ^^ '"'o »ot Will vanish altogether and we shall 7«i ^'?"'**^ '" ^''"^ ^"-^ N *or example hero it would nr/J* i^'} *° <^etermine its v„h,o wemighttake./=i7"o ^ .' *" **^^ '"' °^'''' «rr»0; hm not make the expreUion'li"" Vi„?"f ^/^^^ Raines that'wi Wish to prove that ^"^ illustrate further, siipposo we oiily /Arec dimensio^ If tir^*^-' ^^'^»«« the expressionla^ number, hence "^^ *^'^'^ ^« «ny ofher factor it is some not cause (a+b) ( A^r( 'Ta W o T '' ^"^^' f«»' tl.ele vaLs do ffiit'U' + ^'^^^' + "A« =0 Hid marr '^•"^ ^'"'-^ ^«ue Taking these values wo have '^ «l'snpi««r altogether. '-^ or A = i, as required. : (^2) Show that SoLtrnox p,„ „ „ , , ^'"""on "nd division. «"t. .heretore the™ ^^li^ri "^nX'^r?" ™""""''/''«'-thH>ugh sjon vfif. fn Vw. / 1 ._. "® «"othpr hfpm/. fo^*,^- ^i- _ ,y"ii'« . -"•:-^vuna. i ii is factor ,nnst oW ■:::"■' T^ *^""«"- Jnu8i oljoy ^ymmetrj', must TYPE SOLUTIONS. 85 Imvori. />, (• similarly involvod nu.l imist tl.ovofoio 1^. .,+/, + ,■ for t us IS tho only expression of om- .lin.onsioii thai is svnune- Jrienl for a, h, nm\ c. • nmv.o (a+h + r)* (f. + ry -vw. ^ SotHio-^h + r). Put n^h=^v ^ I ami .P - 2* - :>^ - 2' + P + 1* + i ' I \ j . 1 XT ,.. »>«i-i'5-H'.-ir. + :} = ;)N. :u;; ' hence N = 12, and 12«W«+/> + c.) is the mlucert expression rs may be verified by expanding the terms and adding them up. (.•J3) Find the factors o{ {x-yy + {y -zY + {z- j-)\ Exercise 100, No. 7. / v / Solution. Test iorx-y by putting .r - »/ = • « ^ a; = « Write y instead of « throughout and get Q + iu-zY' iiz-vV'' ±"1m.v T^ 'w^'' '^^' -^ - ^ ^^""^'^ "° rem.:inder: Hence by symmetry (ap-y) (y-z) ^-o,) is a factor of the expression, and eiccounts for L ALOhJaHA. vl + c-by-{h+c-aV. (d) (a + b + f)-' - („■'' +/>•' +r3). (e) (rt + O + c)'' -(«'•+,'/• +<•'), (F) (a+b + cy-(rt+b-cy'-(a + (■{(! + b) (a' +h^-,r.^^ ^ (H) «='(6 + c-,0='+ftV + «-6)'+c^(« + (»-r)» + (^t'+b'+c'-bc-va-ab)ib + c-a)(c + a-b)(a+b~c) (L) (<^Y^ + c + cir-ia'^+b^+c^+d'^)-(a+b+d) + (c + 4-(6 + rf)'* + (a + c) ^)'+(6 + c)' Besitlts. (a) - {a - fc) (h 1 c) (c - a). (H) -(«-/>) (6-c) (c-rt) (a-hb + c). (C) -(a-/>) (6-c) (e-flr) («» +6^ 4-c' +«6 + ^>r + m). (D) 3(« + />)(/> + (.) (c + a). ^ (E) b{a+b) (/, + ,) (efa) (a' +6» +0=* +«&+/x- + r«). F)24.,/>. (o) 2«M«+^ + c). („)2r,6c- ^>e+ca). (K)24«6«/. (L) G0fi6ct;(a+6 + c4-d). HINT8. In A and b N= - 1. In c, N= - 1 and P= - 1 InE, N = 5, P = 5. InH, N = 0, P = 2. (34) Find the H. C. F. of 2.x* +9^' + 14a; + 3 and 3j3 +15a;2+5a.2^i0ar^ 2. Exercise (J6, No. 12. Solution. Apply the principle of Exercise 64 No 4 i« this way :-Take multiples of the two exijn^sionH M.u^h ihi? when added or snbtracted the first teriis will l„«Ji ^**I take nmlllples such that thi iSt & wi Tnce ' ?h?* use the two reniainders in the same way; and ?he ne^t two remainders m the same way, until the o^iation s compl S Thus in the given example call the first exnrps^lnn a j .^' second B Tako op qa^ n .V . expression A and the feci^onu u. lake 2B - dA, call the i^raa nder C Tatr, qp o a call the remainder D. Hence everv measmp nf 1 I i>~"^', measure C and D. Take 3D- ^^^^.^^1°!^.-^"^ ^J'P^ 2C + 5D, call the remainder F, and obsex^e' thit F^nS ^ ^^^Z TYPE SOMTIONS. .S7 b + cy 4, in that next then two leted. 1 the -2A, will XttKu e the Argument: Every measure of A and B will ,n(.Hsi.re (; nnd I'i every .measure of C and D will nieasnro E and F l.nce every measure of A and B will measure E and F. But E a IF ranTp^?e'thTll!rFTf ra?dT'"^^^^^^ '"'^''^^•^°' "^'^'^^ 6+27 + 6 + 30 + 0+ 42+ 9 10+ 20+ 4 3+ 10- 22- 5 = C Operation: Using detached coefficients the work mav exhibited in the following form :— ' 1 9 a ^i 5 6 7 8 9 10 11 12 18 14 13 16 be 4 + 18+ 0+ 28+ if 9 + 45+ 15 + ;;0+ 6 5 + 27+ 15"+" 2 10+ oo-iio-*S" 15+ 81+ 45+ 6 D 31 ^1 + 155 + 31 1 + 5+ 1=E 6+ 20- 44-10 25 + 136+ 75+10 31|3 1 + 155+ 31 1+ 5+~T But E = F, .-. a?' +5£c+ 1 =H. C. F. of A and B. f'^P^anation : In line 1 a zero is put in where x' is raissimr In line 8 the quantity written in full would be 'n'ssnife. ^^* +27x' + lox^ ^2x=:x{ox^ + 27x^ +lox + '>) but as a- is plainly no f-ctor of A and B it is droiTwd and in line 10 we take 3D = 3(5«j3 +27a.^ +etc.) instead of ' T I- i. .,- 3(5a;«+27a;='+etc.) Jart o'f tie H ^^'^'i^^^^K?^' +»•« + 1)» and 31 is evidently „o wedroD it Ihll/q iT^i'-*" r ^'^ searching, and therefore r«/i ? A ^ ^•^' ^*' ^°' 1^ *^e necmary, because if A thatElVr Z7''''' '^'"irV' T"^^^ ^ ^'^^'^^y to show struck out for tb« «a"°"'' ^" ^'"^^^i' ''^^ ""^e'-i^^al factor 31 is sirucK out for the same reason as in line 11. nm^lnl'? •* ^! *^^ P""?'P^« of Exercise 64, No. 4, has been com- prehended, actual experience shows that the a oungest Duoils can apply it intelligently by the preceding method tX most diffi" 88 PUBLIC SCHOOL ALOEHRA. .m!^ JJ. 1 '""** ^''^'"PJe can be understood bv junior in.i.iJs wbo bare mastered Exercise <;4, page 47. (.*)5) Reduce to lowest terms — -IzHI^l + Sj 21.c^~r377a;«+8' Matriculation Examination, Toronto Unir^sity Solution. A B TTW Hx'^-377.T-'+21 21a' -377j;«4.8 ^}i>.rJ-'mx^-^^^1x'^+^9 Divide by 29. x'- \nx* 18^3 +^1 « c. 13a' -377^^^* +377a' - 13 Divide by 13. 29x*+ 29*=' 1 - D. Divide by 2a;'. Divide by 2. 16a*- 42a3\,. 2 ^ 8a*- 21a-' f 1=F. J|aJ- Ma--21a + 9 Divide by Jl. •ta* - 7 a-^ - 7a + 3r G. ( G-3H)-r2 (3L-K) -i-S (3K-L)^.s^. 7a;- 21.;;s-^-2urr7 Divide by 7. a; - i\x'+ 3x- 1 = H. '>a^- i(ja'+ 2a Divide by 2a 3a-^-8a-'.fl-.K a;' - 8a+3^ L. a'- 3a+l=M. a-- 3.r+l = N. = H. C. F. of A and B. 1 + 3 -1 01 "^ i\ + ^ -^^'+ +0+21 24 + /2 + 192 +504+189 +63 - 8 - 24 - 64-168 j-63-21 '1 ^ f* - '>4 - li>H - 63 - 21 8ar' +24a* +64a-« + 168a^T(^+ op 'r^^^^^^^j^j^ ^'r -'I + V) 4. _;nv + 0+0 1+6 +"8 "T +3 -1 _ _ ~ ' ^" — *\HJ — M-] 2\x^ + 63a* + 168aMT64a»T24^+ 8 ^.?, "^ " -•'••-l-u+UI+U <•.. + 189 +504 + 192 + 72 ' + ">4 - -^ 1 • 63 - 1G 8 - 64 I - 24 - S rod. denominator TV I'M hOUniOXM. ami (W) If .r^a is ,1,0 H. r. F. of the rp.antiti.s .r' + n,+u i-emnimlers are each ==0. But an in tho pv...... ''"*"-^ ' ^L'\ „,, I 77 ♦! . , -"Ml, n» 111 ine e\auj|»l«'s on i.aires i( i'M In. tlio ronmni.lors are found l.y jMittinK j' + ri=:0, /.<.. ^= -^/,- hence ttj =a' - m + «==0-«(// _ /.) + ., Silitracring these »iuations wc . .-t a{p ^- /•) - {,, _ h) = r^7) Sl.ovv that .r + a is a common factor of the n.,antifieK cxampirr;:::;^.':^:;^'::^.^':^^'^'-^ '^^ '" ^^'^ ^--^''^^ Subtract and « ' -a*(;>lT) =0^ Substitute this value of « in K,, : „ 1 wo m't (p-iy-q(p-]) + ]=0. (.iH) Find the sum of the fractions Jll'l + '''' + -<''*'' nu„.e™Sr"na^Lt-iV':v--',-,u!;'t'V'" 7';/";'"' Tl.«r n • ,<^ + *)+/, which is.r-/r. The C. D. 18 also «» - *», ,.^„^ ^^^^ ^ , ^^^^^^.^ ^, ^^^^ ^^^^^ ^ ^ (•50) Find the sum of the fractions - -JL 4., 2 , (.r - ;{) (.,; - 4) (./,• - 2) (4 - .r) "^ (2 - ./^H^ -Tn' / . i > ^tu^^^^ ^ *''*' ^''^ symmetry, so t hat (.i- - 2 , (^. „ n (jc-4ns the C. D. ;. makes the 2nd fractionU«.^;i J ttic .fu remains iposiiive. Hence numerator = (.r-2)-f Iheroforo whole Kum of the sum iJ) + (./•- 4) -.0. \ 90 PUBLIC SCHOOL ALGEBRA. (40) Sim])lify the expression x+y —Second Clans, 1887. Solution. Tlie C. D. =(.r» -y*) (y' -ga;)^*' -a-wV lience the numerator of the sum will be ^ ^^ ' {y+z){x^-yz) + (^ + .V)(«»-a'.y) = 0. Hence the whole sum =0, as in the pi-eceding example (41) Simplify the expression X 2y X (* + .y) (^ + 2y) (JC + y) (^ + sy) ^ i^^T2y) {x + Sy) " ^^y — Third Class, 1887. Solution. The C. D. is {x-\-y) (x + 'iv) (x4.Zv\ Hence the numerator of the sum j^ ^^ ^'^ ^ '^'Ma^ + ^y). ^(* + 3y) + 2y (.r + 2.y) + a:(a, + y) -{x^y){x + 2y) ••• whole expression «(* + 2^)J^±2y)____ 1 (« + y) (« + 2~y) (a; + 3y)"^:i:3P' Test. Put x = 0, y=l, and § - J = J (42) Solve the equation ^~^ + ^- ^ ■ c-aj 3a; a»-6c 6»_ca c« -afr'afe+ft^T^' Solution. Transpose the right hand side oart hv n«rf and add each fraction on the left to%ne of the partes Thus'fo; a-x X the first fraction on the left -^ a^-bc ab + bc + ca The numerator of the sum of these two is, a(ab+be + ca)-x(ab+bc + ea)-x(a^ -be) .I.eo.h:rrwV:re%1??-^tf-^->:-(''.t»+^)«By»yn.L^.th, <•(" " ")-.-r( *' ' Mc.' ^^ SiTn/rr'i*'" *»«« «'>4-6c + ca in its denominator. Strike thiH nut of tho denommator, and clear of fraction., and we get Collecting tenns and bracketing, wo have by transposition {ah^bc + ca) + ft(«»-ftc)(c»-rt6) + c{a^-bc){h'-ca) n(b'^ - rn) (c' - ab) b{a'-b<) (e*-ab) c{a^ -be) {b^ -ca) I.e. a;-(a6 + 6c + ca)-!-(a + 6 + c). (43) Solve the equation -=12a6{a;»+(a+6)a}. ^ >» t ' i-« SoLirriON. For a+6 write w throughout, and expand thus -«*-4a;»a - 6aj'a» - 4ira» - {-w* + a* +b*) Divide by 10a6 and x'+wxJx' iw^ ^^ x^w~a + b. (44) Solve the equation 3a6c_6a; 2a + 6 = 3ca:-*!^. a (a+by Solution. Transpose ^ and factor, thus ab ( a+b{ 3c + (a+b) ab n + b a f I aL (a+ft)»J/ -x|3c+?...^_\ I a (a +6)'/ i '(rt+6)»/ = ae. I- Oi I'lrUI.IU KCIIIMIL AMJKIINA. 'U-b ; (4r>) Solve iIh' »'.|uuiion .S<»umoN. \Ait j-am- u'-hm.n- • ""'I on Hnhsti,„ti„«theMovnlnes'tlnouKhou;, wo A"'"' Cleun„K of the fractions and tranHiK)sinjr, we ^ot rlence one of t ho t wo hinrknfu ^n^^o¥ n • xi_ • ki..io.r.i '' '^ ,'""^* "= "' s'nf^e their i)ro«0, andar=:*(./+6). and from ea.^h we uet n ,n"„ /! n"*" ^^?' T ^'^ '^"""^ ' otmcu iitN uhicli mlucGH the original Equation to an Identity. ,. .-^^'l 1^. '*^.?''" ^'^'^ '""O'" Hamilton to Toronto bv boat at tbp ra e of i;J m, les ,K.r hour, remains an hour andTlmlf iXroVto nnd returns by rail at the rate of 2(1 miles i^r hotir He iaTonl &ri;^,jT;;]:^,S. "■° "'-'» '™" "»""- ?» SoMTTiON. Let X = the distance expresseortion, viz. fi and i) Distance = 3 X l.T = 39 = 1 § X 2G. (47) A number consists of two diirit«« • if tv,cc^ A' •* i, reversed, ,l,o ,.„mbev ,h„s former's ZtLuitZttZ^ by nv™ ,1,0 greater digit; also, tour tim^ J, o digit exc^I three tinios the other by unity. Find the digits. Soi.UTinv T^t- in-.. . «-_xT_ -^ reversed. '::^^ = 2a7l.^7:J^j^^ '''''=''" '''''^ ^^'«'^« Also 4y = 3.r + 1 ; \vhence x = 9,' y = 7. No. = 97. TYPH tMil.UTlUNH. 93 (48) A compouud of tin and l«)ad weigliH 10 IK tiiiien an iiiueli fts «n equal Uilk of water, while tin weiKhs 7 44 times, and load 1 1 "35 times as much as onual ludks of water. Find the numlx>r of pounds of each metal In the coin[K>uud. SoLimON. Let x be the weight of the tin, and y the weight of the lead in a given cjuantity of the conii)ound. . . *+ 7 -44 « weight of water equal in bulk to the tin in compound and y+lLHs')-*" '♦ ♦' «« lead " Also X 4- y » weight of the comixinnd ; •. ( j; + y) -i- 1 0-4 3 «a weight of water e*_1 ^ ^IW V 1135x13 14755 Therefore the proportions of tin and lead are as 2970 : 14755 in any given quantity. (49) If 76 men and 69 boys can do as much work in 299 days fts 40 men and 33 boys can do in 557 days, how many men will do as much work in a day as 15 boys can do? Solution. Let x stand for a day's work by a man. Let J/ «« '• " " boy. Then 299(7«J7 + 59y) = 557 (40 j? + 33j/), I.e. 22,724a:+ 17,(;41y«22,280a; + 18,381j/; or 444ac = 740y ; or Gar = lOy ; Therefore 9a; «=16y. Ans. 9* men. (50) A man divides $1,300 into two sums and lends them at different rates '^f interest. He finds the incomes from them to be equal. If ; . -ad loaned the first at the rate of the second ho would have received |36, and the second at the rate o! the first he would have obtained 149. Find the rates of interest.— Junior Matriculation^ Toronto University, 1890. Solution. Several interpretations of the meaning of this problem are possible, according to the answers returned to the following questions: Simple or compound interest? For the same time or for different times? Is $36 the income from the whole investment or from the first sum alone? Is $49 the income from the whole, etc.? Taking one set of answers : Let P, and Pj be the sums, so that P, +Ps =1300; and let r, and r, be the rates at simple interrat for equal tiroes on $1. Then wehavePtri^P,r, ; and P,r3=.36; P,r,=49, from which .r,_36r, . „». r, G_i', P, r, 49ra whence — r, 7 Pa JN I HmLIi- svUiMH, ALUBMHA. "■■"' i».r., «y^ o,.',l,.'«;„;,d',^;r' "* ""* '""« I* '% on th. »i..w'ie ,^„,i.™:'r 7';''"" »' "•'» ■"-"■oj- >■'» ^« .0 » 7% on tJie per annum " the th.'w 'K the first five years' '^comes a the funds eceivetl h» 1 he origj. end ///_ ) i^tt=suni per cents sold out r. d at the I only fa; t'ed from tion > Mand n starts ley meet 'rom C. from C TVPB SOLUTIONS. H ilea. MiiltO ter in '1 ... i* wiii tea up with a pint moaRuro, by aid of \vhicli they discover that Alfrod'M pad holds half a KftHon more thuu Edward'H aiir of quarts in Herbert's pail, therefore x + 'i and .r + 4 aro the numljer of quarts in Uie |iails of Edward and Alfred ; and let //- nimiber in the well From the dat% given we have therefore the equations aj + 2 a?+4 x Whence 2y (a; + 2) (a;+4) = 8 ; and 2y ac(a;+2) .40. And from these two we get ?ili«6 or ic = l X Also y = 4(a; + 2) (« + 4 ) - 60 quarts, or 15 gallons. (64) Each of three cubical vessels, A, B, C, whose capacities are as 1 : 8 : 27, respectively, is partially fllkd with water the quantities of water in them being as 1:2 : 8, respectively. So much water is now poured from A into B, and and so much from B mto C, as to make the depth of the water the same in each vessel. After this, 128 f cubic feet of water is poured from C into B, and then so much from B into Ay as to leave the depth of water in A twice as great as the depth of water in B. The quantity of water in A is now less by 100 cubic feet than it was originally. How much water did each of the vessels originally contain?— CawiiwMl«/e, Englmid. Solution. Let 6a3 = total number of cubic feet of water in the three vessels, viz. x cubic feet in A, 2x in B and 3x in C. Then since the volumes of the cubes are as 1:8: 27, their sides ^ *^ \u^^ ^' *"^ *^®'^ square bases as 1:4:9. Hence when aU are filled to the same height, the number of cubic feet in each will be as 1 : 4 : 9 : and each vessel will contain 6a; 14' t.f.. Bx B.L?5;r, 27x 96 PUBLIC SCHOOL ALOEBHA. be 6. Tl„ 7^ "'-'''•»"""■«'"■» of these must ""■1 f.om ,Uta :...wo, o,_,„„„ *, "^+-7- -'-'»*=6a,; - t-e o,.,-.i„., ,ua„t.iera;r:.'e':Sr.rve:S. '"**' '^'"•'='' (00) A fiaudnlent merchaTif „ howev.,., tl,e sc„le.p,„,^rX. .T'™ "" '»''"■« «'^° If ^n per cen, „„ CXu.l%Z:^Z i^^S ''«'-'^« ^ 'wh-e^S""' '^«*'>' '» i»™ds of artMe . """■""^•-'"'"^Me...... TitseUsfo, '""^ ' ' 100/ \ 100 /> and from (2) tell J- * \ '^"'+lW=«'- Multiply these two «„d divide through by „,„„ ,„a /'l^^y_..^^„ expanding and tvansposiug '"^^ loT ^ Henoe.":S:»!?t»-f?-.'0)(.+nO). TYPE SOLUTIONS. 07 Seethe depth *or as 1 :■> 0); and l.y ' these must feet, which ce both ill 11% more ti-ue. If, rJied when he would legitimate 'icie when of article . cle. ost price. cx\ wor • • 0) ) • • (2) ig -w, 'o and ±1. )0 ) ;«0. (56) Simplify )(l+bc) ' thus, (fi-a)i -J- - ^•«-^' \ 'll+ca (l+a6)(H.6c)/ = (c-a) I (L+«^) (^ +^0-a+&^)(l4-Cfl) \ \(l+ca)(l + a6)(l+ftc}' J ^(c-a)i -1±*H±^±^!£- 1 - ca - 6- - ab^'c \ '\(l-\-ab)( " ^f'-rr—-- f . («-6)(6-c)(c-a) (l + aft)(l+6c) (l+crt)' Jr* PVBLW SCHOOL ALOebra. (o9) It f^±y^p+z e+x SO.PTXOX. I-t the tWe,uaI fractions.. . « + y = m(3a - 6) -v »"wons •»,, and we get y+z =»w(36_c( i ^u And (1)^(2) gi^^ ■ ■'■^-^ c+a-6 tT+ft+a' prove that «o~I.teaeh.aJ,ri;''^^''^^ y==mic+a-b) «^»m(a+6c), whence a.+y.._^, , Also, a,y«^(fty^ +y+«=«»(a+6+c) . . . P'^^micz+az-bz) And aa;=m(&s+c2-aa) . y--(ay+.y-e^),whence.y4^,^,, (l)-^(2} gives -^yj:«_ _ a+6+c (1) (2) ^ "sxrc ' University. I and we get King a into cz +c»)..(2) TVta SOLUTIONS. 26 + c 2c+a 2a+6' ** 99 2I(a + 6 + c)(a; + 2y+3z)«(x+y+8)(41a+386+47c). ^Matrict^ation, Toronto University, 1876, and Second Cfass. Solution. 2y-a=»n(26+c) 2z-x = m{2c + a) 2a;-y = «,(2a + 6), /. {C + y + z = w(a + 6 + c)3 .. (1) Also, 26y-13z = w(26+c)13 34«-17£c = wi(2c4-a)17 24x-12y = wi(2a + 6)12, . 7(x+2|^ + 3z) = m(41a + 386 + 47c) . . . (2, Multiply these results (1) and (2) and striking out m we have the relation required. ■cz). (1) (2) 100 ^^'^UC SCHOOL AtOi.1,1,.4. ^XAUmATlO^ PAPERS. HI »0. 1 'Wee, it wonH ^? """*'' ""^ been LTri""?" *■■« <•*« in 7 ^p. a. (1) Simplify 7/1 4A ,- the form Tj!",''V*' ""da;' +ca:+rf 1,. "aysaloTift «ri,« ^. . * P*«ce of wort i« , ■•■> ;' .I'v" ;;:*/" ?,J°i«? him, an7S^. t„'» ^yj -V ^ work, „ I EXAMINATION PAP£R8» 101 ^a ). ^nfo 4 find >»-mer shall lasea are in other in 7 in circum- ^indthe orks « e work ingl.r ? Xre. 3. (1) Multiply 1-x* +x* -«• by 1 +x». (2) Find the value of ^a^h' + fa^b' +a^ J ft» when a = 5, 6 = 25. (3) Find the H. C. F. of x* + 7*=' + dx^ - 32x - 32 andar'=+9.T + 20. (4) Find the L. C. M. of x^-y\ x^ +xy- '2y\ and 05*+ 3xy +2y'. (6) What is the condition that x^-^px + q may be exactly divisible by a;- r# (6) 16 - 4a; -h 8ae = 5a; + 14, find A". (7) The breadth of an oblong space is four yards less than its length ; the area of the space is 252 yards. Find the length of its sides. (8) A square field contains I acre 2 roods 27 perches 23^ square yards. An oblong field contains 6,400 square yards, und has one side as much longer than the side of the square field as the other side is shorter than it. Find the length and breadth of the oblong. Wo.*. (1) Express algebraically : —The fourth power of the sum of two numbers, a and 6, t<^ther with twice the product of thcv squares, is equal to the sum of their fourth powers together witli four times the product of their product and the square of thoir sum. Verify your statement when « = 2, fc = 3. (2) Subtract {x + y) (3a - 26) from (x + y) (3a + 26). (3) Divide a;' + y ' + 1 - 2y + 2.T - 2xy by a- - y + 1 . (4) Simplify (a;+2+ Jl\ / (-^-.a;). ^ sc-2' ^a;'~4 ^ (5) ~17(a;-lzf) = 12(da;-^l?). .'R\ Zx X-\ — ^ _o a;+2 |f.B. (.r-10)(7a;+ll)=0; .-. a;«10-0, or else 7a;+ll = 0. ^VBUO SCHOOL ALOBBRA. 109 .-wi. AWIBBRA. 0) Factorise in simplest forms ar« i. , (2) Show that (a - ft) a o ™ * - ^^ ^' " a^ - 12. (3) ?JL'2+ir5 , . * -"• N.B. (2ar-59)"'(Oa.+25)=o. o^ tq n (4) A purse contained nni i. ' " ' " ^ ^'^ °= "' or 2a: + 25 » *nd the value is £o,o/''7^ 'H^P^nny and . , (7) A train 200 ylrd, T„7 ' '' ""» '«'+8«-105. »o. e. (1) i(2aj + 7) -iron ^'k , v« i.i-.._ a + 1^ if"^ ^®8s 6 + 1' *h«n-L-butgreatei.than«-I N.B. Let«=:t>.. . « I. «j.i I «. 6 + 1 Aj.1' ®'<*' m Simplify 213/1 -./iij. KjiA ' — \ 'v — ucTr 3^-<^!. BXAMIMATION PAPERS. 103 ' of any two ^ three times 2a:+25«0. ^^y pieces, >f the coins 7rf. find -105. 240 yards ack, in 12 > other it he second s which less c^l (5) Find the value of "^^^ " ^0^y±^ - ??y when ac— 5, y = 3. (6) Solve the equation 23 _x + n Bx_Q ac + 4 3 U ■ (7) <4 and B have two guineas between them ; and if A gives to B one shilling for every penny J5 has, A will then have ten shillings less than B now has. How much money has each ? N.B. One guinea = 2U.; B gives nothing to A. No. 7. (1) Divide ia» + }a'a;-2a;» by ha+x, (2) Find the H. C. F. of x^y + 2x^y^ +2xy' +y* and bx^ + 10x*y + bx'y*. (3) Simplify JL - __i - _* t1_ . ^ ^ x-l 2(a;+l) 2(x' + l) (4) H3a:-l)+i(6-ac)-TS-(2aj-4)-2-^(aj + 2). (5) i(aj+3y)+T»,(3ac+y) = ll| a; + y = 8(y-ac) (6) Solve the equation ^^~ ? - 1 + ??zJ ^ a;-2 6 l-x find X and 2^. 0. (7) A grocer bought 224 pounds of sugar at the rate of 25«. for 112 pounds. The sugar having been damaged, he sold part of it at 2d. per pound and the rest at 2Jd. He lost 12 J % on the whole transaction. How much did he sell at each price? (1) Simplify a^-b' iro, 8. -J{a-36-(36-a)}. a*+ab+b* (2) Determine the numerical values of c and d so that in the product of x^+x + 1 and x^+cx^+dx+e the . coefficients of X* and x^ may vanish. (3) Show that the L. C. M. of two quantities equals their pro- duct divided by their H. C. F. -Hv* III Jill! a i', J04 ^ ' (•*) Find the H r p i , (o) Simplify ?£ziy 4ar-5y 1 («} Solve J*' +^^-1 « fr'Z\Io!iduc2ol "'T ""^^^ ^«« paid off b. .. (1) Show that the L P \r * 'I.e.r product divided ty*ii,e"r H. '^%^<^^--'^l exp,^,„,„ ,., (2) Prove that 5? x*'^'*^; , nnn,,^* ^ W'^^^^he letters denote any numbers whatever. ^ N.B. Let?==a. ^-, ft ' 5 ^' • «=&a?, etc. (3) Solve the equation !£+7 _ ^^-S » w "-%-«, ,a._:L,,l;^7• 1^0. 10. n (1) Express 2la>_ 40a. _ 91. i.q , . ' " .--^-iintheformoffactowr"^ ^ \ EXAMINATION I'APEKM. 106 I and \ «© gei ounces of an alloy of silver rfhd gold containing 91 -7 i)er cent, of pure k'oUI, so that the resulting mixture may contain 84 i^er cent, of gold? (7) A can do half as much work as J5, and B can do lialf as much as C; how long would each separately retiuire to do a piece of work that they can together complete in 24 days ? »ny aniontit >^ them ek. and I'ato of No. 11. (1) Simplify a-[26+{3c-3a-(a + 6)} + {2a-(6 + c)}L using only three lines in the solution. (2) Factor ac» - y» + «=» _ ^a _ Ojiz + 2ay. (3) Find the H. C. F. of 5a!» (12a;' + 4a;» + 17£c - 3) and 10a2(24ac' - 52a;» + 14a; - 1). (4) Simplify *_?_. -. -1 Iz^Oa; *^ l-2a; H-2a; 4a;' -1 (5) Find the values of a;, y and z from the equations 10a;+15y-24z = 41; 15a;- 12y + 16z=10; andl8a;-14y-7z= -13. (6) Show that three globes or spheres whose diameters are S inches, 4 inches, and 5 inches, respectively, contain together the same volume as a globe 6 inches in diameter. N.B. Cubes, spheres and other regular solids have their volumes proportional to the cubes of their like dimensions. (7) A train runs 1 hour and then stops 15 minutes to repair a break in the engine. Afterwards it runs at ^ its former rate, and arrives 24 mirutes late. If the break had haDDsnfid !J niilf^ farther on, the tv 'n would have been only 2r minutes lateT Find thd nsoal spe. f the train in iles per hour. iOH ^ / / Wo. la. y into five factors. (■») From ^ ,„btrnct «^ (^) Multiply together ^ " «' l-y^ ■ (7) Two ^ '' ^"^ «*; ; find cc. son he wonM i " **" the bairiraffft K«S il, ^^® '^e weight a-b c-d' prove also that J^l+fft* 3c* +4rf« »0. 13. • (2) Multiply ^a + o«^. 3.^ f^-'^J^- (3) From — i±£_ . ^ l~x (4) Divide 3a:» +iabco^ - 6^3^,^ "*" . ,,, ^ r6) ^ ■^^'i the value of a?. v'^J A can correct 7n ^ . correct 160 pa^s S 01 ?*^^ ^<>r the p^ss in 11 r -g425page'sTL«/?*'^'"^' ^^^ ^o^will^het fc.l^- / ihe weight 'O one per- 8 can \ C) BXAMISTATIOK PAPBKN. 107 (2^±«^+l.x+l. 2ac+l 3a; Find jt'. Find jr. (8) a?»-12a: = Y8S- (9) Find the value of 2 Vrf -~& +3 ^3d + 2c-l+ 4 Va+6 + 2?Td when a»0, 6-2, c-4, and ^i = 6. (10) Find the product of « -fe by a +fo. (1 1) Find the difference between x - 3// + 4* and cc + 2y - fi«. (12) Divide a:Vjy«-««+ 2a:' j/«-2«^"l by «» f }/»-«'- 1. (13) Find the value of x in the equation 4a; + 9 »8a; - 3. »0. 14. Provincial Model School, Toronto. (1) Write in words the meaning of the expression (oftc 4- xifx) • . (2) Add together 5w + 3n + p, 3{m + n + p) and bp + oii + tii. and obtain the numerical result when »= il =a i!L «• 1 . 10 100 (3) Add together (a+6)x + (a + c)y, (6+c)y, .and (c - a)x + (6 - a)y. (4) Prove that c - (a - 6) = c - a +&. (5) Simplify 16- {2-a;- -j(3-^)}. (6) Prove that a* xa^ ^a'', and multiply X* +2x^y + ix^y^ +iixy^ + \Gy* by x-2y. (7) Prove that a" -J-a' =«*, and divide {x^-9x^y + 2'dxy'' - 15y=») {x-7y) by a;' -8*1/ + 72/-. (8) Resolve the following expressions into factors : — 81ac*-l,(4a; + 32/)2-(3j5 + 4.y)', 12x»-14a' + 2. -. (9) Solve the following equations : — (i) I3a;-21(a;-3) = 10-21(3-a:). (ii) (2 + as) (a - 3) = - 4 - 2aa;. (10^ A and B Dlay together for $5 : it A win= he will have thrice as much as B, but if he lose he will have only twice ns much. What has, each 9t first? 1:0 /r 108 (11) ii^uce to its Jowo«t terras (13) Prove that '^ x '^ « *»« 6 rf hd' and solve J^*^ -'^-c-a . ^ ^ ^ ^y-'X + y \ M Wo. la. -P./-?A C/aw, &,„/;, ^.^^^ ^^^ (1) Bepresent the sum of (2) Simplify the expression 2^L»/*+a+ ^/» ex^rLs^or ^''^ " """ *- """tog L. C. M. of two nj,^,-, ^*) ^^ 6 ■*■"."*-' show that ... '^ ^^+*-'^)' + ^(*+c-a)» + (e, t- ,^+,,, -^ inte^J^ S^ ''"'"''" '""'"^" iB true when m and I (6) Factor a^ - 3a«»c« + 2d^ ■'1 %-^- *-'«*^» ^asiitjht hand digit beine'tho^i^aterr'''' "^ "' ""^^*°® I /- \ 8(o*-a»6), ic equt; t.Vi 8- Vi ftlsrobrnic j^ts be uuu toe I f 109 ir«..«r..'„ 0/ Torouio^Pa., Mair. aiaii^, 1881. (f5) Simplify (i) JiJi^* (ii) a^'^-S + x => ..3 .r'+l (") Resolve into factors M.u;t*o?t^hr;i^^^^^^^^^ *^« w co.... (8) Solve the equations (i) aac+b'^bx-^a. 1 («) -r; + -^ 1 1 2 3 (9) There are two vessels A AnA n ^ i UTo. 17. 110 PITBLIC SCHOOL ALGEBRA. / (3) Prove the rale for flndino. ti,. nv, „ y Find the G.C.M of L*I il°.-.f /.'?» ■""»«««• poind ,uamit° "'" '" "*'•'"«"« '"' »«««" root of . eom- (5) Solve the following equations (ii) £+o_a!+6 . (iii)?+«„2+c J a aj a; *ro. 18. , ^cOm University, Montreal- School F. ■ • (1) Multiply 1^2x-a.^ i 3 r. ^--"»»"a«on. the result if 1 - 2aj = 3. ~ ^"^ ^ '*^^^' ^"^ find the value of (3) Simplify |;c(^+l)|^^2_^^^^^^^^^ 2^^^^ 1 fA\ -D 3 ' 2a?+l 2* (4) Reduce the following fraction« f^ .». • , «'«^+«' r«.* ^ '!"' *" *^^"' ^o^e«t terms : (5) Find t. ■ ^''^ ^""^^^'^ lT2S + 2^ri:^- (o) Find the square root of ' «'*+2a;?..jK+i andof ^^'-^^J+l (6) Solve the equations (iii) ix-y-I^i: S.,a.?r2^„ J BXAJIINATION PAPERS. Ill entities, a com- ueof 15&» i I T XTo. 10. Cotlegi of Oitawa, OnL— Matriculation Examination. (1) Clear away the parentheses, and reduce the following expression : rt+6-(2a-36)^4(5a + 76)-(-13a + 26)+3{a-6(6-a)|. (2) Give the three formulas for the expansion of (a+by, [a-b)^, and (a + 6) (a-b), and give an example of each formula. (a) Divide5a;-3-4aj«+a:*+aj' by -3+x- -^x. (4) Find the G.C.D. and the L.aM. of thu tluee following expressions (205-4) (3ac-6); (a; -3) (4a; -8); (2a; -6) (5a; -10). (5) Simplify 2w n n m (6) Solve the equations 2x + 4y-3a = 22; 4a;-2y+52 = 18; 5a; + 7y-z = 63. (7) Extract the square root of 15a*6* +a« -6a*6- 20a'6' +6« i.l5«^6« -6a6». (8) Convert fi into such an expression, not a decimal, as shall not necessitate two ejctractions in finding the cube root of |. (9) Solve the following equation, Ja;' - |a; + 20J = 42§. ( 10) The hypotenuse of a right-angled triangle is 20 feet and t^ area of the triangle is 96 square feet. Find the length of the legs. f ITo. 20. University of Oxford, England— Local Exaniination, Junior Candidates. (l) Find the value of ^^-^"^(^JZ^ j. hlj[±b 4c(a+6) ~ ' \~ c ' when a « 1, 6 »« 0, c »: - i^. 112 I " PVHuv SCHOOL ALasamA. (-^) Multiply a-*- aa^n ^^3^ .. , (a) Simplify (i)(2^_+^_^ . , ^ Hi) J o*' + 2 , a? + 3 a?+i ■^?^^+(^r^.- (4) Find the G r M * , (5) Solve the e,ua.i„„, ^ """*'• ""^ ^^C'*-*'). (u)^±? = o« + l c, y+4 "'2^7'^(«+i) = ll(y+o). H No. 21. (-') Multiply together a-' ^^ , a < express the result f„ atop,: f-;^,+«' »' + 7»-i8, ,,._,, ,^ (3) Find tife G. C. M. of Sa;' + W + 10.^ - and the L Sl'L^/^^r^+S *" ' W Simplify {i)(j^y_ x+y 1 , , , , (5) Solve the equations (i) iz5+ ^ ~ 2 _ ar «. _ i an^'ihi\tr.±,!!?.»-4ti:!f;atr3r:i!!::.':.^^ . - txicn louuu to be half full- "fiT,^'"^'*^"^°"**^' »" luii , find Its capacity. BXAMINATION PAPERS. 113 ? + 3y' ^« + 8; also ttion. -1, and Ko. 22. University of Oxford, England— Local Examination, Senior Candidatea. (1) Prove that (a + b) ia + x)(b + x)-a(b + xy'-b{a+xy-.= {a-byj:, and divide a^ -//' by a^-2afc^ + 2a^6- &-. (2) Besolve into component factors (i)63a5»y-28a;y'; (ii) a'*-a*6-a6*+6-. Find the remainder when a« + 6" is divided by a -ft. (3) Find the G. C. M. of X* - 6a;» + 13a?' - 12ic + 4 and x* - 4a;' + 8x* - 16a? + 16, and the L. C. M. of x' - y', x' +y', a?' - scy», and (4) Simplify the fractions (i) "^ - ±t} - J_ • x + 1 ac + 2 u; + 4 (") 1 1 1-1 1 1 1- 2ac a; aj* ac ac' + l (5) Solve the equations (i) *{(2a:-32)-(ac+l(;)S =v,-j(Je-20)-(2a-ll)},• (ii)(a[:+5)(y+7)=(a:•+l)(y-9)+112, 2a;+5=3y-4; ae'-S ac*4-5 1). K pacity. Ko. 23. Vnivervitf/ of Oxford, Eng.— First Examination of Women. (1) Find the value of ^-Z^*!'^ when rt = l, 6= -i, c==0. (2) Takea + 26+3c-4d-5efrom3a-46 + c-rf + «'. (3) Multiply a' - a'ftH- 06=* - ?;^ by « +6, and divide a* +a"6- +?;♦ by a- +a6+&*. (4) Find the H. C. F. of a* +5a' -6 and a« +5a' +4, and the L. C. M. of 12(a» -6»), 16(a» +6'), 20a6(a' -6'). (5) Simplify a»+4a-f3 I f , 114 PUBUC SCHOOL ALGEBRA. I- (6) Ettraet the square root of a' - 2a' +Sa' „ii (7)Solv,(i)7(..i)_6(,_2)_3(,_3|''' *' ^ fK»-3) (a;-13) = («_4)(a,_9), he^P .l^i"ia^ mS^htr Wjr-^-'^ '^""''«»• ""^ """» he has lost twemy-fonr shiuini S,Tfi' T""""^ "" ^^ ^-^s B b.s. With whit s«, did S te^„t ^' ' ""^ °' "'»« Ko. Si. (1) Evaluate (a.-y)»+(y_^)2^.(,_, (2) From the sum of i(2x-3u4.A9\ o«^ i />• « subtract M&%+6zr '"^ *(*'»+%-"') (3) Mul«p„,.+ , + „(,_^, ,^ a=y-a(.+y)+„.. W Express in factors (i) 7x" - 77x - 182 • «M. .. . (")20a!<-60a;'j,+46a!V. 7 f". 1\^- ""• "• "'-■ + "-+80. 9X. + 63..-9X.18 (7) Find the L. C. M. of 15x=(a>-2«a=+x>), 21a'(a'^2ax+^.^, 35a:c(a'.^n (8) Simplify \ + — -1 a x+2 x-2 4 + 1 > and find the value of 2-a 2+a 4- ac' W22C2I a;: +i- certain num- nd thrice the nber of each as four times ;s, and finds I till A finds »ird of what IV'otnen. f-liz) -9a: -18. BXAMIXATION PAPERS. (9) Solve tiie equations . (i)«(a;-l)+3(*-9)-(aj-.13) = ll; 116 x-a x-b x-a-b = 0; (10) The sum of three consecutive whole numbers exceeds the greatest of them by 19 ; what are the numbers? Wo. 25. Umver$Uy of Oxford, Eng. ^Firat Examination of Women. (1) Find the value of x(y + z)+y[x-(y + z)]-z[y-x(z-x)l whena;=3, y=2, « = 1. ^^ (2) Subtract 2x* +ix^-x + i from ^x* + a;= - |a; - 1. (3) Multiply l+2x + 3y+4x^- 6xy +dy*hyl-2x- 3y. (4) Divide x* -^x^ +£c» +ix - 2 by a; - 1. (5) Besolve into factors (i) 4a;* - 3Gx^y^ - iii)i2x-3yy-(x-2yy. (6) Find the G. C. M. of a;» -3a;+2 and x^ +4x» - 5. (7) Find the L. 0. M. of 12(1-05?), 15(l-aj)», and 20(ar+jc»). (8) Simplify (i) 2^ x^xl^x^-^Z£^. c 8a=« cd 4b^-4bd' (ii)('-J_ + _i£_U(^_i 4a; X ^l+4aj l-4a;'" Vl-ia; 1+4^/* (9) Solvetheequations a)-+- --^-* + ^-9q. ^'3 9 27 -Xf-"-''' (ii) x+b x+c~ ' (ui)?^ = ,|, 2y±if = 2i. a: (10) A person-pealks a certain distance at the rate of 3 J miles an hour, and finds that if he had walkfid 4 milpo -» i.' ™ u! woma have gone the same distance in less time by one hourl what 18 the distance? ^"wux, 1 ^^,~. U8 PUBLIC SCHOOL ALOBBRA. ,' ! «ro-ae. CTniver.iU, of Cambridge, England. (1) Prove that (aj + 4)'-(',r4.l\3„q/^^ IN, (2) Simplify (i) (a + O + v) (l + i + l).(tt±(^±^)Ja±V) , ab-a~b + l ' (4) Prove that, ^^ - and n be positive intege.., (..,« ."n): (o) Solve the equations (i) ^(5^ + j) + 2^-3 ^ ^ . 8 . « + X' 6 + a: a6 ' ITo. 27. , ^ ' ^n^.-Aeconel Premou* Examination. (1) Simplify 6 (a -26) ("6-9,,^ /« 'v^ /.^ fromthe8«mofc]«_,)=\^^d(;^2o6^tait\^f"^^-^^^' ^"'J /ON rk«« '^ "^ ^*^ *"® square of 2(a - h\ (4) Eesolve into the simplest possible factors • (1) 6a3-+oa;y-6y2 . (ii) aj'-13;»2y + 42£^y2. riii) {a-\-2h+ncY^i(a+h~cy- (iv)8la;^-G25y». '^ ' ^' /^'^^^stt^' ~--' '-- Of t.o al^eh^io^ Find the highest common f= ^*' "• ^^«' - 7a; + 10 »»;{ 2V-.«^ - 30J+1. )+27. a)ia+b) 8 15' a +6. mtnation. 12a6, and o{2(a-h). c) + 5c' Igebraical BXAMlNATro.V PAPKRS. (6) Reduce to simple fractions in their lowest terms (i) ag*-7ay4-igy' ^ ^' -oxy+4y^ . x^ +nxy + Gy- ' x" +xy-2y^ * 117 (Ui) a- ja6 a +6 a» + as-a 1-1 1 1 a»-6» a 6 (7) Solve the equations ,,x jc+l 2a;-3 3aj-2 -^ (i) ___ -— j^_- _^-10; 8 (ii) _+_sa-14, _+?L = 24: ^ ' 6 5 ' 9 2 ' (Hi) 6a;'-17x+14=0; (iv) aj» + y' = 5a» 4- 562 + 8a6, ary = 2a» + 26' + 5a6. (8) Find the value of ic"»xa:«, when m and « are positive integers. Simplify a^+« x aP+^ -r-aS-P- IJTo. 28. Intermediate Examination, Ontario, December, 1878. (1) Multiply 4a» -laj+Vir by 2a:+i. Prove that (i« - y) ' - (ac - iy) ' is exactly divisible by x + y. (2) Express in words the meaning of the formulse (a+a) (a5+6) = a:2 +(a+6)ac+rt6. Retaining the order of the terms, how will the right- handed member of this expression be affected by changing, in the left-hand member (i) the sign of h only, (ii) the sign of a only, (iii) the signs of both a and bf (3) SimpHfy(a+6)* + (a-6)*-2('a'-63)»: and show that (a+6+c) ip+c-a) {a-\-c-h) (a+6-c)»4a'6' when a'+6' = c*. i 118 (4) Prove that ? + £ = b d be PUBLIC SCHOOL ALQBB&A. ad Simplify (<»!:+*! + iW_?&' \^ia{a+b) -^IVrTn f''**'",T«'*°nto to Niagara, 35 miles, in the steamer . City of Toronto" and returned in the "Rothesay," makinc wenT?n t^i^^'R^H^^^'^r,^ ,'' ™^""1« ' on anoth^eV oTcS m le an ho t /^^it^^^^ (f >ose speed o.i this occasion was 1 , m.le an hour less than usual), from Toronto to Lewiston 4^^ ;rh\iet"sTe:ifcr^«'- ^^^ ''^ ^--' -- ^^ ^- (6) Solve 3 2 1 ^ a X 2 1 2 --- = -. » X y a midl^nulw""'"""™ ''""°'*" "'"^ P"^-"* '» ^» «»" the <'^"xi'-^:4+|;=liP«-veth.t 6' a;'+y»* THIRD CLASS TEACHERS, ONTARIO. m.?^J «r"-^®'® follows the complete series of naners spf f«^ Third Class Teachers in Ontario from 1 BTs! when Sm was woies and Hints seem to be required they will be (annA wiVi. tlie answers in the Teacher's Edition. ^"'^ KTo. 20. Jtt/y, 1878. (2) Prove that a'r^'-±26»^3^^^^ft3^2a3A3 .3 . • « SIAMINATION PAPERS. 119 (8) Multiply together Jp+ ^q+ Jr, v'p- V^- ^r, 'JP- s/q+ Vr, and l/p+ jq- ^r. (4) a;' +y' is divided by x' - y' and the result is divided by the quotient ol x + yhyx^y, what will be the result if Ja;= - }t/? (5) Pind two factors each of a5*+a»+l, x* +1 and three factors of sc" + l. (6) Solve the equations :— ^2 4 ^^T"^ "• ^ (b) «-3_l-2a;^2-a; 2a; + l ^ ^ 6 + x 3 =0. (7) A and B could do a certain piece of work in 15 days • B and C could do it in 20 days ; in what time could A and C do it supposing A c-.n do three times as much as U in a given time ? XTO. 30. Julyy 1879. (1) Find the value of 3ac» + 54a;« + 50aj» - 19a;» - 35a; - 18 when a = - 17. (2) Demonstrate the identities :— (i) (5m»+4mn+n«)»-(3w»+4mn+n»)» =4wi'(2m + n)». (ii) {a+b+c) iab+bc+ca)-abc^(a+b) (b+c) (c + a). (iii) (a-6) (c-d)+(6-c) (a-rf)+(c-a) (6 + d)=0. (3) Divide (w'+an*) (x^ +ay^)-a(nx-myy by mx + any. ^xS*^ Prove that if from the square of the sum of two numbers there be taken four times their product, the remainder is a square. (5) Solve (i) (a; - 1) (a; - 2) - (a; - 3) (a; - 4) = 3. (ii) -A_ +_i_ =___?__ x-1 x-2 x*-3ar + 2' (iii) ix-a)ib-c) + (x-b) ic--a)+(x-c) (a-b) = x-a-b-c. (6) What value of x will mpke x^+2ax+b* the square of x + c? What is the result when a^b<=c? (7) A man is thrice as old as his son, five years ago he was tour times as old ; how old is he ? ! />■! 120 PUBLIC SCHOOL ALOBBHA. . (I nil No. 91. Jvfy, 1880, .S-^ Zo^'}^^^'^ «-5 inches and A. 'Tfeetlllnches, flndthe 'bc-ea. (2) If .. .4 find^eorr^, %y_%^^,^J^, the value of ^'''^ ^^""JlfA^Z^-^^^ «-^ + ^. prove that (4) Divide «»+6'+c'-3a6cbya + 6 + c. (5) Find the factors of (i) 15a;'-19a?y-10y2? (ii) (■^-2n + l)"-(2„_l).„(a--2f.)' (9) Having 75 minutes at my disDo««l h^^^ t ca^ge at .§ .iles ^r hou. h"av1;r^o*•la^rbi'^c'^^^^^^ f.f^^S^ c mlnlt^sT findte ratTt^ IST '" ' ^^°"^S« ^^ "^*"™ «" rate at which the stream flows '"'" ^" '^''^^ ^**«' ^^^ *^« VTo. 92. Intermediate Examinatiou, Ontario, July, mt (1) Factor .^^ ^y, . ^„^ .^a ^^3 ^.,3 .3 Utilize your results to show that 68, find the value of h-bc-ea. r>cV equal I go in a •^i miles return in and the 3a6c)'. KXAMINATION I'APBKS. 121 (2) If a' - Jc-ft' -ca, and a bo not o«,.ml to />. tlicn (.•J) Show ho.v to find the L.C.M. of two uk^brulc expressions. h.ve a L r M'n?^Ii''7' that a:' +«u:^ +/. and ^-^ +r.r + ./ umy ua\e a ij. L. M. of the form x* +px^ +qx- +rx + s. + j£±i:»//' (4) SiraiJify j£+J^)!!_ ^. _( y-t-g)Jg' , (y - 2) (« - a;) (« - X) (a; - y) ■*■ (.,; -"7)17/Z7V (•») Extract the square root of (ii) a:*+x» + ya;' + Ja? + Y. 0») Find the value of a; in ■Lxplam result. '^ (7) Find an expression for k in terms of a, b, c, that will n.akr Ik2 »3 „9 t n .« fc-c , vanish. (S) If for every «3 of income A ha., B has $2 ; for every $\> Hnd the proportion of his income that A saves. (9) Solve the equations : (i)^:+J+£c(ae-l) = (ar-l)». 1 ^ 1 _ 1 iT-y^ x-4a' (ii) o 1 ac - a X - 2rt (iii) ?^t?e^!±?^i^i»l:::^4^i^«^ X'+X'+I (iv) x^+xy + y = 2b x-l ac'-l a; + icy+ fy = 25\ y'=3i/- »0. 33. Intermediate Examination , 1882. ill Forrn An AvnrAfioinn o«vwvx_:.-i •.■%. . . ■>^ 12S PUBLIC SCHOOL ALGBBBA. (2) Factor ax* - (a +6) (x - p)xy - by». e. .. for^^^^^^^^^^^^ the .uaneiUes . , i{ad-bcy - («. 4-d« -6» -c»)» will vanish. (3) Find the lowest common measure, not being a fraction, of the quantities ?!±2£±? and ^I±l5±i2 (4) Reduce to lowest terms the following fractions - (5)(i)Ify+z + u=-a,, + t.+a.=6,n+^+y.e.c.+y+««rf, ■1.,: 1^1 1 1 then — ■fl + — r + -i-+-i-,: 1+1 1 + * i + ? n.rf (ii) If aa;=»6 + c, fry-c+a, c««a+6, then 1. 1 +JL+ 1 1+ac iTy 1+i (6) Solve the equation ajc» +fta;+c=.0 What value of x will satisfy the equation '0. a + a 05+6 a+^ o (7) Solve the equations (ii) Ji--!^=_?_- 1 a5-4 aj-3 a;-5 a^Tl' (iii) /«*+«V+y*«2l. (8) Solve the equations a;+y + a= 6^ 3a;+2y-«= 4 [ 3a;-2y+5a=4^ aj-4i/+2 = i I 4«-% + 62:=:5J ..jiwgief--:. a, b, 1, of d, I BXAMINAl'lON PAPfilU). 123 (10) The hour, minute, and second hands of a watch are on concentric axes, the same divisions on the dial arwerir for Jl vT"^^ ami seconds. Find when first betweTn : a„d 4 the' m^t rndtoirhln^^; ^^"^ '^''^ ^'^ ^^ ^ ^/-n Ho. 34. ^ InUrmediate and Third Cla»$, July, 1883. (1) Divide (i) (a - b)c ' + (6 - c)a » + (c - a)b' by (a ~6) (6-c)(c-a); (2) What must be the values of a, 6, and c, that X +ax^ ^bx+c may have »- 1, a:-2, and aj-3 all as factors? » ««« * o (3) Find the H. C. F. of (i) 3x« - 4a;' + 1 and 4aj* - 6a;» - £c' + x+ 1 • (ii) 8x'-y'+27z' + ISxyz and ix' + 12xz+dz' ^y^ (4) Simplify ^ • (i)(l£!-l)(-iaL_i)+/8a^'_lW 4a.»+2a;y ,x (ii) ae'+(a+6)ac» + (a6-H)ic-|.5 te» + (a6+l)ic»+(a+ft)a;+i- (5) Find the value of x that will make ^c+^ + arf-fftc independent of c and d. a;-3c + 2rf (6) (i)Iia+5+c=0, thenl+i+l„/l.lxl\' (u) If a.=a«+6»+c« and y=«6+ftc+ca, ,..., „ ^^^'^ ac» +2y3 -Sary* -(a^ +&« +C' -3a^K:)^ (m) If 2a = y+a, 26=z+a;, 2c = ar+j,, express m terms of a?, y, and a. V .-- i l'>4 PUBLIC SCIffK)!. ALGEBltA. (T) Find a value of « whicli will make the quantities ^-^L±^(«+i-')and(« + ^)^«+rf) , a + b + c ~ a + c + d ®^"*^ ^° "^^^ another, (8) Solve the equations (i) sfx + 3+ ^'^^^^. 3 — " (ii) 2 = 0; (iii) (a?4-«+6)(c + rf)==(^ + e+rf)(a+M where c+dis not-equal to a+b. feetf ^nei?her'2!.n ' a right-angled triangle exceeds the other by 3 another. Fn/nrwh^t KMn^J^ ff/ '^ ? '^ ^"P^*'^^ ^^ 1.0 inserted that thrcfste^n mav fin 1^ -^^ v""^'' P'P« '""«* did water not flow from ft atT ^« *^'^<^« l^^e time it ^ould Ko.36. Intermediate and Third Clas$, July, 1884. (1) Divide (a« -6*) («; - .y*) - 4a6xy(6»a.3 - a»y») (2) Simplify (i) n+m m+n vn n n-m m-n m n (ii) ^-^Lz^L^£H£z:«) . ft-c^^'-a^a-ft (3) Besolve into linear factors (.i')*(.ab+cdy-(a'+b'-c'-d')'. <*) Show that («'« + <,.y + ^) (»-c) + (6'a;+jj,+») (e-„> + (c-a=+c3,+.) (a-6)-(i-^) (c-^fftii)^."* (5) If !/+-2a,^.+x=2», x+j,=o„, fl„' t^e TAiVd Claaa^ July, 1885. (l)SimpUfya»+6a+e»_(«_ft^^)^ -(6-c+a)(6+c-a)-(c-a+6)(c+a-6) (2) Divide a* +6* +c* - 26»c» ~ 2a»c» - 2fl2fc> bya»+62_c»+2a6. (3) Multiply a!«-3 - x^-Q+x^ _ i by «;» + 1. (4) Fiftd the factors of a» - 6» + c» - d» + 2ac - 26rf. (5) Findthefactorsof (a+6)»>(6^c^j+^g^^^, X x+c ag + 2 c r (6) Simplify . fiC 4" 3c (P) -Uetermine as given 126 PUBLIC SCHOOL ALQmUA, (9) Solve the simultaneous equations X y ' x+2y=xy. ing 18 each moS^for the oxerlft'^*"^^ f°' ""^ ^"»^ «"«»' W- What was the priS p^r o^ anS Ihf. ?i?^ S^'^ '^' *^« «^eep. first lot ? ^ "" "^"^ ^*»** *^« price per sheep of the »0. 37. ^« a? C a; a -r + T— (2) Simplify 1 ^ 4(x-i) 4(05+1) (~aj-l)» (aj+ry (3) Simplify /£+y^g-y\ /ag'4-y» a'-y'v (4) Prove that £±?(l-l\_*+^/^-.l\ c-«/l iv IS tne difference of two squares. (5) Resolve into linear factors ^6M. f ^*'+^^+''*^(*'+*'«+«*+^)(c« + «6+6c+ca). (6) Besolve into three factors (a;+y)» (x'^z')-(x+zy (a?«+y«) (T) Show that «.-^ on., ««ev..« ^ . ,h.e wiU „.to equal to the cube of (» + 2c), and find that value. (8) Solve the equation ^?zi - ^"^ ^ a?-5 aj - 6 «-2 05-3 jc-6~ST7* (9) Solve the simultaneous equations H£:iy.2y-« o«_^ 2„_^ 1 2 ~4 8~"^^- 6y 6 than half of the originarnumbe'r." """'^^^ ""*^" « IP^ter EXAMINATION PAPERS. 127 ,* he pay. leep. fthe a). is it iro. 98. Third Class, 1887. (1) Show that IS divisxbloby 4a?'+4y». ' (2) Find the product of UJL ^^-i^andl+yl±^lz£! i-JL ^ -^ a; y+« (3) Find the a CM. of 2x^-bx*y+6xy'-2y' ^nd2x'+Bx'y-6xy' +2y^ (6) Simplify X 2y («+y)(«+2y) ^^Ty)l^ (7) Solve the equations ;— (i)p(a-g')»g(a;_|>); X 1 (a'+2y)(a:+3y)"(^^+3y) (ii) i(5a; - 6) + §(3 - 2x) = 1(5 - 3 J ). (8) Solve the equations :— (i) aj+y-6, ox+fry^fr* ; rafeVinii!rho^"Lmr^'"\*" ''"'r^ ^y ^"^t^^the and retum^rraU aUhe r^^? ?« ^T *^\* ^*^' ''^ '^^^'^^o altogether %S hoi^ • C^ ,k ^T^ *">"'• ^« *« «<>«« To^Snto ^ ' ^"*^ *^® ^*«<^*"<« from Hamilton to (10) A number consists of two di« the number, the diit/lXb^Tv^i^td ttf^^,^.^^ '° Wo. iO. (1) (i) Define the terms quantitv nnif n»w.>i^ quantity. How is quantity measS ? ' "'^' "'«*"^^ ence)M»r^c^.'Z^rd^S^-n -i;^^^^^^^^ (2) Factor (c - JB) (£c» + aft) + (a + aj) («» - 6c) + (ft - a;\ fa.2 , .^v What values of x will make this expression i 0? ^ ^• ^^ ''' tion; wirthilfn'^*^"^" quantities, under what condi- uons wiu the expression be negative ? (3) If two expressions have a common factor r.^.™ j-i, x .i. sum or the difference of anv mu'tinW n# *l ' P^®^®. **»»* the have that common facto? """^^'^^^ ^^ *^««« expressions will Find the highest common factor of the expressions :~ (4) Add together the following ;— a+x («-^6)(a-c) 6-aj-(Ti7]r(ftr^)+^r^-(^ a )(c-ft)- **® PUBLIC SCHOOL AUIKBBA. (6) Find all the factors of «* +4y* • (6) Reduce to its simplest form where 2« = a+6+c. - ^ '' (7) Solve the equations :— a-4 £c-5 x-7 (0 a;-5 a;-6 a;-8 03-8 . "a;-9' tiini l%ry' ^*'"' "' * ^^" *^^ «"°^ o^ *!>« 'oUowing frac (x-ay _ (x-by (x-c)' {x-b)ix- ey (^T^n^T^)' (^-oHx-Zb) ^ day exceeds that done hv /♦ i« « i v -^ together in a whole work WW ♦• ,^* ''*?'' ^^ one-tv-entieth of the hims^lfT ^' ''"'^ ^°"^^ «^°^ ^«i^« to do the work by oi5i^^^ '^ "^^®^® number, greater than 800 and less thRT, mn • one. FM the nuX. " three-fourths of the original n..y he C^i« J'eiro^rrti^" Tiit ^^e^™"*" ITO. 41. Primary JExamination, 1890. (1) Find the value (in the simplest form) of w»(c-n»)-j.n'(m-c»)+c'(w-w)a + whenn-»»»=o. ;j * i »nnc(»wnc~l)+7, BXAlflNATION PAPERS. 181 (2) (i) Find the remainder when 9a > » + 4a« - 27/,6 a. i is divided by a" + 2a* + 1. *'« +i (ii) Divide, by Horner's method, ,«, „ ^ ■;-^''y'-W«'y' + ta'y+ia*by|y«4.3a.y..ia». W llx + aiaa, common factor of a* +px + l and a: +Pa:«+ga;+i,showthat(p-l)»_^(p^l)^.j„0. (4) Simplify (i) ^^^±y±l}J'^l±£+^)/U + z,z + x^x + yX (ii) ,-i-^±3!Klz52')_ ■ _«?(! -y') + .vf l~xM (l-a:y)»-(a:4-y)» (T-x^) (l-yi)-4ary (5) Resolve in factors : (i) 7a;-42y-2a;» + 9a;,y+18y». (ii) a«-3a»+3a*-a»-8. (^ii) (ax+byy+(ay-bxy+c^x^+c'p\ (6) Solve the equations : (i) ^±j^c=,( ft-c)' . aa;-6+c (6+c)» ' (ii) ^ + _^^~2a5 ~ 7a3-2 H Hx + bl 8 J (iii) (a:-2rt)» + (a;-26)3 = 2(a;-a-6)«. onShkVoT\hrsZuS''tr.f "^"^ *^"^^ *»^« o^her by diife.nceof^l^\r"^^^^^^^^^^^^^^^ ™ ^^If th^ and $1,000 toTheirnn/.^^^^^^^ one-half of the remainder his sh^^to be JoToo ' mif ' ""V^ '^^ '^^^ ^^° '^"»d property? ' " ^''** ^*« *^e *otal value of the divid^ by the sum of .L^-X^^ZuJ^l^, ^^^ eqiliViLlnf f :r the fi^^.e' Vll '' '' ""^ *^^ ^-<^« o^ "^ ^^-^ R ll?i ;? *^i? *««®*5«' o^"^ P doUars, ^ and ^ « ^nlia.„ „«., ^ -«- - - uuiiars. now much money does each own ?""'' ^"^^ 132 PUBLIC SGliDOL ALGBBRA (12) A train starts on a journey of 240 miles : after aoinst 103 miles It reduces its speed by one-fifth, and in ^n^utXll hours late at, its destination. Find the ordinary^i^ of th! Vto. 48. Primary Examination^ 1891. (1) (i) Show^that^6a;»+13a^+6y»+12ar+18y is divisible by (ii) If the product of a and h equal aj' 2x'y -Snc'^v^ +ixy' +2y*f andifa = a.2^0y% 'findft (2) Factor (i) x' +5ax + bx-i-10ab -26'. in)x''-2x''+x*'a*x* + 2xa*-a*. (3) Simplify (0 (ii) (iiij (a+by-4ab a^-b' " x-y y-z z~x «'-(^-y)» x^-(y-z)^-^^rzjj^z^ x^ ,yS i T + 7 y' {x-y) {x-z) (y-x) (y-zj'^(z-x)(z-y) l..ffi ^^ ^'■"^® ^""^ ^"^^°S *^e L. C. M. of two or more alee W expressions Apply your rule to the finding of thTL^C M (6) Solve the following equations :— (i) ^J?+1 ^3fta ~2a + c . ajfl 6(« + l)~a* (ii) fc±^^t^ ^ («+ c)_(a+a} . x+n+q x+c+a ' (iii) (a+a;)(6+as)-a(6+c)«=?!5+jc». b piS wuJ'hTe?,^^'^^"^ ^-*^ ^f"°"« '« ^^^ i'^ 20 minutes by 3 pipes winch let in water at uniform rates, the first nine admite &et^r;^^^"S:^*'''^ r' the second 5 Us^hjfth: t^7t^Zl? ' "^ ™"'^ ^*'"" ^^^^ *^^o«8»» e*«i^ pipe If: 1 03 U he ^y LAMINATION PAPERS. 188 (7) A and B have the same income. A contracts a debt eadh VS:l!''^^^^'^\^J'.T^^r'')^^^ ^^« ^"*^'»«; ^ iives on four- fifths of his. At the end of 10 years B lends A enough to pay his debts and has 1160 to spare. Find the income of each (8) A and B play for a stake of $12. If A win he wiU have thrice as much money as B. If he lose he will have twice as much. Wnat amount of money does each possess at first ? fl/S ^/""''l Y'^l '^ ^"^.'^^ ***'' •^ ^"^^ ^"« »» of them at a pro- ^" ,/A/* ^"^' P"*^ "*"«* *»e sell the remainder that he may gain 10% on the whole? "® «.ii?o^ / *'*'^^! ^'T C' to /) at the rate of 6 miles per hour : B starts from C two hours after A, and travelling 10 miles per cZd "" *'''"'' ^^^"^ ^- ^^'^ the^^distance frSL Ho. id. Primary Escaviination, 1892. (1) Multiply l+»(l-2aj)+aj»(l-2«)»+a;»(l_2a:)»+x«a-2a;)« + by 1 -x+2x\ carrying the product to the term contaShig iv (2) The dividend is y ' . y* + 2y » - 3y _ o evSly^vSiZt'^f.t' *' ^""^'^'^ ial^^^^ca) to make it (4) Put 4a«6« - (a» +6» -c^)» into four factors. (5) Put into four factors {x + 2)(x+6) (a+4+ V6) (a;+4- V6)-15 (6) Find the H.C.F. of 2a5« +«' -3x'» -ai + l and a* -2a;» +a;» +2a:-2 (7) Simplify (l+a6)(l+ac)^(l+(>c) (l+feg) (l + cg) (l+c6) (8) Find a? when (x-a)* (l+ax) = (x^n\* n «o»\ . j prove that the value yiu getUisfies the ^ualn.^ '^""^ ' """^ r.J:^ ^T ^^?^^^ eggs a dozen when a rise of 20°/ in their pnoe makes a difference of 50 eiro.« m *h^ „„^i^\,ZYA.^JrJ^^^ I ll 184 rVBUO SCHOOL ALOBBSA. ' 3io. i4. Primartf Eseamination, 1898. (1) (i) I>ivide4a«4.4a(n-l)d+(„.i)a^a by 2^^(^_ (ii) Divide l-a;»-y»-8a.y by l-aj-y. (8) Solve the equations : /j) 2a;+3 3a;H.4 i -^ (ii) (aj+7)» + (6-aj)(ar+6)-86aj. 25^/\i\^),r^-* '* the price of bread per loaf, if an increase of '^iZ^Tr oTetul^rt^'oV""^^' ^' ^-- "^^^"^ (5) (i) Factor oj* - 64 j x* +a;'y» +y«. (ii) Show that a: +y is a factor of i (*-»»)« +Py} ' + { waj + (1 - p)y} 3. (iii) Factor 16a»+4a6-4ac-12ft» + 17*c-6c«. (6) Simplify (i) (101)* - (9 9)« '^(101)» + (99)»' (ii) (p-a) {p-h) (P-c) (a-b) (a^c) (b-c) (b--a)'^(^Zd)J^)' i ANSWERS, HINTS AND SKELETON SOLUTIONS. SxtroUt 1,-Piff» 1. (9) 8; 30; 722; 16«5 ; 12221. (10) 47; 58; 131; 108. (11) 176. (12) 20. axireiie a,-Pij:a 3. ?2T^'^' ^^^> io^ + By+7z) miles. 9 oS'n ,u (12) «6orf divisions. riOW4 ^7^ (13) «;^y people. (10) (4.X + 7.V) cents. (20) 2642. (26) 36 ; 33 ; 33. (1) 05=12. (2) «=-.3. (3) ac=4. (4) a;=4. (5) a? =4. (6) ic=3. (7) a? =8. (8) a;- 7. (9) a; = 838. (10) a; =38 years Istroifo 3,-Piffe i. (27) 4864. (30) a' = l; 216; 36; 6; 1; 5. Szeroiae ij-Pftfa e. (11) a? = 38. (12) x'= 16 years. (13) a; = 20 yards. (14) a; =5 minutes. (15) a; = 6. (16) 12; a; = 24. (17) a; = 372. (18) aj = 372. •t.U'^. i I i ( (20) 9 +36 + 27 = 72 lb. l.lfl PUBLIC HUHUOL AIXiBBKA. (H) 70. (4) 1(5739. (o) 102. (ft) 91. (")-(•; «-6 + <'; a-ft-c-«/; ri-/> + c + tf.- n-b + c-d. Xsereife 7,-Fft9« 9. (1 ) « + -'>• (7) a- - y .• ; «(y - jc), rs in No. «. (2) o^+\)n\ ^H) $(,/-//). (4 ) ( 1 1 - ;•)) mlle«. (9) .$(.'J.)2u. - 120j/). (:») $(«)000 - 2000). (t)) X'- y ; ; or .»/- J-, tccording an // is »o88 than, or equal to, or greater than / . Zxeroiie 8,— F^are 10. (1) 19,«4-/;. (2) ^x'+Lv'-nx\ (3) Aah - Abr + -J^ar. (4) 21f«N I2b*+:''*e\ (o) 4a + *)b + iU. (6) 15(rt+ft). (') I) — a - 12 12 7 («) .^- 5 12^^ 5 —2. 12 /ft\ OJ.,1 . !42y. " M ANHWBIliJ. II I NTH ANI, 8KKI.KTON 8<)l (10) (I00.f + 400y + m-r-f «•+ I0a + 13500A) HI) iah + 'Jh + dv. (12) (12A+.-)r,B + C) inches. (13) -3x». UTI0N8, cents. t87 ■awTOlM 8,-PMrt 131. (4) 1448. (5) flH}i. (1) a?=15. (8) cc«448. (1) 20 miles. (2) tloOO; $1650; $1800; $1950. (3) 40 and 10 acres. (4) $120. ■««oiii lX,-Piet 12. (6) 5() mon; $150. (7) 20. ■»WUi 10,-P»fff 11. (5)a-=l. (6) $2800; $1400. (7) 48 ami 2' yoars. (1) Vahm^aO. (2) Value -23 J. (•■> .N'Muiber-189i5;«yV. /i\ 13 17, 11 12 «o ;io (5) a^+h-+c^. (6) 240. U) a; = 5. (2) aj«6. (3) «-=!. ■««Oiie 12, Pigt 12. (4) X-6J. (5) x = 4J. (6) ie-7. (7) x^HO. ^8) .«r-a5. (9) 05^4. (10) ic-8. (11) ar=»8. (12) x^7. ■xiToise 13, Patre 18. (3) 63. ; 16H acres. (4) aOsheei.. '^~ '* a»rcise 14,-Pa8r« 14. (8) 7,006,652. TTT rr.r- _ Trrrrrrrrn \~i ^-•" ^•', •> I ij ^^•J (6) Art + fc6 ; t/kn- mft ; ka + Aft + ma + mb. n p " 1.18 M/BLIC SCH(K»h ALC4KBltA. (7) XZ + xk + t/^ + yk. (8) 6a?y,- 10,^y. ig^jy^. 43^2^. joa;'. (9) 2Ut'+S9ab+10b-; STa' +42a6 + 166' (10) Oa'. (11) $(ax + b.ic)', $(i)nx + bbx). (^^) a*+ba''b + 8a'^b^+bab^+b*. Bzeroise 15, -Page is. (2) «c + m/-/;r-/W. (6) ar-m/-fec + W. (9) +30 miles. , lO) -, 'JO miles. (11) ad-ae-af-bd + fHi + hf-ed + ce + cf. Sxeroise ie,-Pftge 16. (3) Oyards. , (fi) +5-5 = 0; +a;-a;=0. Exorcise 1^,-Pafire 16. (4) 12xy: -I2xy; 12xy ; 72a'; -7i>^«; 12a'^x'K (5) «!»•+//; aK«y»| rtW+w- _ Bxeroise 18,-Pa»e 16. (1) -o«-25. (2) ac-ad-bc + bd. (.".) rt^+3a26 + 3aft2+fc3. (4) a=+2rt6 + 6%- «'+.-]« 2ft + 3^62+ 63 (5) a= + 2rt + l. (6) a** +5a«6 -f- lOa^^ft" + lOa^ft^ + 5^6* +6«. (7) x' +2xy + y^ ; a;^ +3ic=y + 3»v=' +y''; x*+ix''y + 6x-y^- +4xy^ +y* ; xl + bx*y+10x'y' + \bx^y'^ +bxy* +y^ . (8) 16a«-9a«-4a«-4«Va^ '^ (9)329,013a'«'*«+324,009a'^='« + 112,301a>='» + 110.593 (10) -777a"«. ' ' (11) - 555a5RS+a-. (12) -4; 1. J.6»8 T6643«ao ANHWBRH, HINTS AN'I> SKELETON SOLUTIONS. i;{9 +W. =0. ,2 1 Bxepoiso 10,-Page 17. " (l)«l25a.. (2)(2«/. + 2crf4-125)o.nts. (3) (a-'^y + as^g) cents. (4) acw + i,,,;^. + 5«c + 56.r + ^,/^ + Had + byz + 6ftv (5) Purse = |![D-rfl-fA-)l. awpoiie 20,-Pajre 17. (1) 3a« - 10«36 + 22a2/>2 _ 22«6a + 15^4. (2) a?' - 5^= + 7.r •• + 2x^ - (\.v - o. (3) rt»+6'+c^-3«6e. (o) «^+a6 + 3ac-262+66c. (6) -10rt«+21a36-21a2/>2 + !(;/>«. (7) 6ic« - 1 l£c« + 22aj« - Ax"" - 7. (9) ac*+2.a;V-8y«-18v^'?=-92'« (10) a«-/>-^+27 + 9a6. (11) «a?;+27y^+2='-18«y.-. (12) \Qa* + 160«'6 + (;00«^62 ^. joq^^^^ ^^35^, (13) x^ ~ \hx<^+oQx' +56a?« - ISa-' + 1. (14) 84a«-437a«ft2+56 (15) a^'' + 131ar;« + l20.«^-81.r + 21. (4) 6.r«-l9a?2+25. (8) a'-ft3_^3_3^,^^ 93. axepcise ai,-Paffe 19. (3 . +Ioa.-+74.V120. (4) .3^02,,^j,3 . (0) ^'+x^a^I>,.c)^.xiah + bc + ca)+abc. i^) ^'-^'{a^'b^.c)+x{ah+1>c^ca)-ahc. (7) ae'-5aj2_4Gar-40. (8) «* + 10a?=' + .35x'= + 50.r + 24 . + .^^-85u)^-8(Jx + i680. (10) ajs+a:*a«+a«. (11) X* - 98a:* + 2401a;' - 14400. 140 PUBLIC SCHOOL ATX^BBRA. (12) l-]a'+h + hK 4 3 9 '36 (14) a^h* -2a^bc + a'c^ -b'c^ +of,icd-h^dK (15) x\ (13)ia.'4-^a.» + ^a.+ |. SzeroiM 22,-Paffe lo. i\) tr^+2ax + x^; a-+2ay + y''; .r^+2xy + y^; a*+2a + l; a-+42 . (9) 9a«4-24rt'/>+ 12^262. 47089a'fc2 -217a«/>-2a»«. ■H i^.'-^!-v ■*' ^yy ' " ANSWERS, HINTS A:N SOLUTIONS. 141 (13) x = 6U. . (14) ;«»6. (16) (a + 4)(« + 5). (15) (a+6)(a+c). Sxeroise 2i,-Paffe 20. n'+b^-+4c'-+2ab + 4ae + 4bf; a- +^b^-+9c^+4ab + 6ac+12bc • ^a'+db'+c'^12ab + 4ac + Gbi! (3)144a= + ,696J + 196c^+312«& + 336«c + 3646c-; 441aj-+484y2+O60l+9O4^y^2142a; + oo44„. 240le* + 1296y« + 625. « +3528^3,. + 24 ^'^« +'18002,3,. (4) aj=+y2^3»+2a;y-2«2-2y2; ^'+iy'+z'+4xy-2ccz-4yz; ^'+y' + 9a'+2xy-6ax-6ay. (5) 4a=+96=+c3 + 12a6-4ao-66c- 144«^ + im^ + 19r,c= - 312a/> + 336«c - 3646c • 441a:» + 484y » + 2601 + 924a?.y - 2142ar - 2244y. i^)x^+y'+z'-2xy-2xz + 2yz; a'+b^-+4c--2a',-4ac + 4bc; ^' +y' +9z' -2xy-(ixz + 6yz. (7) »'^'--^b^"y''+c^'z^-+2abxy + 2acxz + 2bcyz • It > I tl + _ a _ u (8) «=6^+6=c='+f^i='+2«6^e + 2a^6c + 2a6,-=' .* »'b'+f>^c^+c^a^-2ab'c.-2a^te\2^r ' (9) 4a^6^+96V + 16c=a^ + 12a6V-16a^&c-04«,>,.. . 9^y+16y^.^+252='^.^_24«.y^.-30^..>;40aJV.»- ^'b^+ib^c^+9c^a^-4ab^c-Lbc+12lcV ' (10) «*&»+6;e»+c*a^+2a»63^ + 2«36c'+2a&V: a-6=c=-(a2+ft.^.^u^2rt6 + 2ac + 26c) ft ft + 16« 1 1 7 24a 100 f^ "„4L- .-m-ig 142 PUBLIC SCHOOL ALGEBRA. (11) rt38^.^3e^^34_^^3-_2^30^O^3r.. , 441a«»+484a««+529«*«-924««'='-966a*« + i019a«r.. (12) (i'' + 1^8lb' + WSl-82a*'b-S.Qa*'+WS\b' 308025a' •'•+256"'+25 + 5550a^^^6<*+5550a'''''^ + rt^+a-«+l + 2-2rt*-2a-2. 506"; Bxepoise 26,-Pajre 21, (1) 05* + .V^* + 2^ + m;= + 2ixy + xz + octv + yz + yw + zt^;). (2) f +4ft^+9c= + 16rf=+4«& + 6ae + 8arf + 126c + 166d + 24crf. ' 4a.- +9.y^ +16^=' +25^= + 12.t2,+ Ite + 20xt; + 241,2 + 30 vt; + 40gj;.\ I 9a= + 166=^ +25c« +3GcZ=' + 24afr + 30ac + 36acZ + 406c + 486c? + 60mZ. (3) a:^' + 2/2 + 22 ^ ««2 ; 2£cy + 2*2 + 2xw - 2yz - 2yw + ^wz • a'+b'+c'+d'+2ab-2ac + 2ad-2bc + 2bd-2cd' X +y' +z' +w' -^^y-2xz + 2xw + 2yz~2yw-2zu'. (^) »' +b' +c'^ +d' -2ab-2ac-2ad + 2lM' + 2fjd + ^cd ' u^+ W- + 4c2 + 4rf2 _ 4«6 _ 4^, _ 4^^ ^ g^. _^ g^ ^ g'^^ 4a2 +962 + i6e2 +25x2 - 12a6- 16ac- 20«a; + 246c + 30&r + 406'*. (5) a-x^ +b^y' + c-^z' ^d'-io^ ■¥^abxy + 2acxz + 2adxw + ' 2bcyz + 2bdyw + 2cdzn' ; a'b' +62c2 +c2rf2 +,^2^. +2a62c + 2a26t« + 26c2d + 2acd^+4abcd; .c2 +«= +y2 +62 -2a*' + 2ajy + 26a;-2a.v-2«6+262/. L. -«' 13 2 1 1 ,« T -t* T . 16 >y 'i j #ii mi < i.^a. :,. ANSWERS, jriNTS AM, SKEf.ETOX SOLUTIONS. 143 4*9 1 :,.^^,.,„=^.,„. !„,,„, ,,,^,_. 1 .«,,2 1 4 6 1 .,. . 1 (8) '?!+^+^'' + 2^^.2r 26. fo2 y.2 ~ ' X" » c b M a cd ad = c' a- 7 y^TT' (9) ar'+-L. 125 (11) la,*+jLy^ (10) £c«-2a?'2/»+j/«. (12) l-x'\ 16 8r (13) 18arr» +21a.' +8x« +aj« +-fi3.r^ +96a;* +43arH-6. (15) l+ir■''+a^♦+cc•+ic'^ (14) «« - 57£c* + 2«6£c2 - 1 . (J) 72. Sxercise 26,— Page 22. (2) 4a2+3a-l9. (3) _4i. (8) 35 eggs. (9) 125; $17; $28. (lO) Ab\ ■ . ^ . (4)|a*-la»6» + 2^ft3_2,^. (7) ;« = 2. BzeroUe 27,-Faffe 23. (1) «'• (2) a'. (4) 36; 2a. (6) I24acy;'4y horses; 6a? horses. 1 ' T -r , J IS- ; "', (6) 36; -2a; -36; 2a. 144 PUBLIC (UlUfHU. ALnmtlA. (8) To divicle /e-fc« quantities, subtract the exponent of the > divisor from the exponent of the dividend • •'»=«''=:1. (9) arethe quotients, (4) a*^+4a'b + Ga'b'+4tib'+f,*. a' +na'b + 3ab' +h\ rt' +2aft + 6% and a + b are he three c notients. (5) «°+6a»6 + l5a«62^20a»63 + 15a2^>«+6a6°4.6«j a' +3a''6 + 3rt/>2 +fo3^ and n^-f /> are the quotients! a)a'-b\ (9)9rt'-6a + l. (8) l+5a;+15ac2+45a;'4-etc. 4 2 BzeroUe 20,-Paffe 287" (1) 3x-7y, (4) 3a=«+a*-2rt-5. {7) 2a + 7. (9) rt''+2rt*+rt»+4«2_2a + l. (2) 8JL-3. (5) 4a;2-3. ' (3)^«-a-6. (6) 3aj-4. (8) -2. (10) 3« + 26 + e. (1) 5689-5605i. /C\ -1. Vv/ Mt/TCOt axeroUo 30,-Pa8ro 24. (2) a?»-3. (5) a;8+2a.+3. (7) 24a;y«. (8) 4. AN8VVEUS, HINTS AND SKELETON SOLUTIONS. »««OitO 31,-Pig8 28. 145 W^Oeais. (o)a. = l. (S) a+b + c. (7) Number hours =.!*+ ^ 7 25* («) Cow.«,5. N.B. The words ..„tehe«a™Kic.»,„„a..*.,7o. (1) 15 gallons. (4) 30,000 men. (6) 1,^30 men. («) a;=.2J. axerolio 32,-Pa»o 26. (2)«21. (3) 4, 290 feet. (5) $10,000 at 4%; $3,000 at 5%. (7) 56 liours. (9) ac-120. (1) 4. BxerciBo 33,-Page 2^. /ox ^^4 . - ' ^2) Each = 480. (0) o4. (6) a^-Sa^ 4.23«-26. (7) ^ = 7. ^ *^ * axoroiae 34,-Pa»e 27. (1) 4a;2-3aa' + a%- 8^4 = 2. S *f :'':;-:+««*-"''P-«*'+7»>_l,. 1x4x8 = 32. (3) 2bx'+24cx'^ -4x\ (4) l-a;»ys. (6)a33-3aj.y + 2x-y;y = 0; ic='+2a;. (7)(m-n)«-(p_^)2^^^3_2^^^^, (1) Equal. (7) a;+y+22. (8) ^ = 4 Szer4i8e 36,— Pa^ sq. (2) 9a3; 3a«« ; n*. (4) • - .i.ya o^c-' +»aj*- 27033 +81aj3_ 243^ (9) They are equal. + 729. i 146 PUBLIC SCH(M>L ALGEBRA. {y -« a«roiid 3fl,-Pftjft 28. (I) 4-18Hteps. (2) --IS steps. ,3) - ^/ = _ 1 _ 1 _ 1 _ etj. (•1) -y-3-;3_3, or -12; -( + a)-( + 3)-( + 3)-( + y), or -12. I >'m Like signs give + in tlie i)rocluct ; Unlike signs give - in the product. = +a + a + a+etc. = +ab. (7) 1, 2, 3, etc have always one p,-ecise vah.e, but a, b, c. etc may have any v,Uu. we choose to ^ive them; 'thej; values are variable, but the values of arithmetical fi'jures are invariable. (9) The tmh of an a.io.n is perceived immediately by intuition, NOT v^diately by some chain of reasoning which is necessary to make it understood. Szercise 37,-Pag« 29. (2) 03 = 2; y = 4. (3) arj.7. y^5 (4^ aj=l; y = 2. (5) a; = 4; y = 3. (6) a; = 2; y = l. SxeroUo 38,-Pajfo 30. (1) 13 ; 20. (2) 43 J ; 106J. (3) 50 and 40 cents per lb; (4) A, 1250 : JJ. «390 (7) £c = 2 ; y=3. l8) aj = 3 ; y=5. (9) a; = 5 ' y=4. (10) a; = 6; y=12. (11)£C = 4; y=3. (5) 55 acres ; 45 acres. (6) $4800 ; $5000. (7) «2000 ; $2400. 1^^ AN8WEKH, HINTM AND HKELl-rrUN BOLUTION8. U7 their letical lition, ich is IzerolM 30,-Faffe 31. (1) x = 9 ; y= 10. (5) Tea, 32 cents ; coffee, 18 cents. (2) ac-xT; y-5. («) ^G4,000; $36,000. (3) 05 = 9; y- -7. (7) 120 eggs. (4) ac=:3; i/= -2. Szeroise 40,-Faffe 31. (1) 240; 160 oranges. (2) 9; 7i days. (3) $19; $41. (5) 4 J cents; 3 J centn. (7) $25; $1.75. (9) 18|at40; 31 J at 72c. i)er lb. (10) $6750 at 3% ; $6000 at 5%. (11) Sum = (an -6m) -J- (n-?w) dollars; Rate = 1 00(6 - a) -^ {an - hrn) \yQv cent (4) $24; $12. (6) Length = 22 ft. 6 in. (H) $22; $26. Sac9roiie 41,-Faffe 32. (3) A, $10; B, $15; C, $25. (5) $600. (4) 66§ lb. and 33J lb. (6) 300 leaps. (7) x' = 3. (1) $40. (4) 56 miles. (7) 21. (10) $5; $4. Szerclse 42,— Fa^e 33. (2) 5 silver coins. (5) 23 ; 19. (8) $500; $1000; $4000. (11) 7 and 31 miles. (12) 03= -5; .v = 6. (13) aj^lO; y=7. (3) 29 miles. (6) 8 ; 5. (9) 6i%. (1)3. (2) 120. Szeroise 43,-Faffe 34. . (3) ar«+y«+a«. (5) 4d-4e. (4) 0. 148 VimUC HCHCX)L ALOKBIU. (fi) j:; = ;{i7. (1) ^•-4. (2) 03 = 4. (3) aj = 2. (4) x = 8. axerolir 48,-Pftfe 35. (i>) x'-18. (<0 a; = 20; .shares 240 ; IhO; 200 (^) Vf%. Bxorcii© 4e,-P»je 36. (1) l+4aj + 2a'--5a;3-*^+x'"*+ljf«. (2) 2(a^6=»+62,..+,3,^,)_^^^,_^^,^^^^ ^3)l-la, + _La.2 + ^_^. _73_ b 3() 216 lOBf-"^ +^*^- (4)a-6. C^)^(fr+b'+c-^)-'){ah + hc + va). (6) P + P + 12^9_g^y (8) $553.33J; $110(J.66§; $3320. (10) a; =10. 0)1 ^ 8 t ANSWKHS. HINT!"* AN'll HKPU.RTflM SOUfTIONS. 149 00. 3320. t (1) 2n+H. (2) V2jc + \. (3) 140. (2) a? = 35. (3) 35 miles. (4) ${\Ohd^ac\ Sstroise 47,-Paflr« 36. (4) 24 miles. (5) 2 Kallons from .1 ,• 14 from B. \<*^ Tea r>Oc. : wnitcar .l.^c. |ier ll». SzeroiM 48, Faff« 37. .7 (fi) .X -- ' : // = z. 4 ;j ('7) (rt^i + /«/ + fs) 4- ( >f + /> 4- ,'). (H) OH lb. of copper. Sxeroiie 40,~ Pases 37, 38. (1) 8a;* - I ^ ^ 2 (2) -2af>c. (3) 16 = 16. (4) (iOx-y^z. (5) .'r = 2. (6) 125c\v*2"- (7) 2(/>f/ + 7/- + /•/>). (K) («_/,)(/,_,.) (<'-r/). (9) ;(r,+/» + C + -s-(r?2+62). ExeroUe 60,— Paffo 38. (3) x = (i; y^lO. (6) 70 1b. (4) a? = 144; // = 216. (7) 30 mmutes. (5) 36 cents ; 27 cents. Szerolse 51,— Page 30. (l)x(x + a)', rt(2a-3): a'(a-b): :ni(m-2n): »2f»4.2oV (2) x^x^ - 5) ; 12a?(l +4a?.v) ; 27(1 - Gx) ■ Mx^x - 3). 150 PUBLIC HOHOOL ALOKBRA. (4) 3a.(.x.»+o,,»^.^,). 7,>»(l_;,^.49p«). (9) 12a;V^^''//'-2ar'+3a«.y-l). (10) .V».y«(z + Oa;23 _5a.»y^2'^(.^,^^^ (11) a-+5ar»+3. * ' ,,,^. fi2) a; -5. axtrolM 62,-Pige 30. (1) (P + 7)(p-«7). (2)(ll+4.y)(ll_4y); (ll+6«)(n_6«). (2^ + 5)^2,/-^ (3) 70x8 = 5r.O; 200; 25000; 500.158:70000; 872000 ' (4) (10a?.y + lla6) (lO^y-Uaft); (8a; + 8) (-2a; + 2), or 16(a;+ 1) (i _«.). (5) ia-b + c)(a-h-r); {x + a-h) {x-n^h), {^) {a + h',c-d){n + b-c + d). (7) («a.+6y+l) iax + hy-l). (a+b + c) (a + b-c) (8) («^+ft^)(«+6)(a-6); («^+.^)(a»+ft«)(«+ft) (,.,). (9)(a.-+y«)(x^+.V*)(..^.y3)(^^y^(^ (10) (rt + 26-3c)(a-2ft + 3c): (11) («-y + «H-d)(a;-y_z-rf). (12) ix-z + j/-a)(x-z-y^n). (^^)(^+b + c-d)(a+b + c+d)(c+d + a-b)(c + d^a+b). (1) («+y)'; (2a+ft)»; (3a+26)». (2) (x + Gyy; (ia'x + bx^yy. (3) (.i; + 9)2; (9a»+4)». (4) (2a;»-20y^)^. (6a;V+l)^ (5) ('6a + 7M2- /*»j_«i\j 1 I 2v'). 0. 6). AN8WE1W, III NTH AND «KBLKTON M>LUTION8. l5l (7) (x-yy; (2«;-/y)»; (i]a-hy. (H) (2.r«~3j/')'; (5m'»i-.'U.';«. (9) (%»< -7t/')*; (20wt-.'J0«»)». (10) (7m-^ + lO«»)-^; (0rt'^-5jc»i/)*. (11) (13arv4.t4.y«)»; (IbVi- 17^')'. (13) ( 9v .-18,,n)«. SzoroiM 64,-Pajfo iO. (1) (x + 2)(a; + l). (2) (« + 7)(« + 8); (x' + 4)(x + 10); (a-6)(a-2). (3)(a + 3)(fl + 8); (.r + 10)(a:r+Il); (w + ;j) (m + 16). (4) (a;- 10) (or- 19); (,> + 10) (cc- 19) ; (ic+10) (cc + 19). (0) (a:-10)(aj+19); (,yz-lhth) {f/z- 17 ah). (0) (a;.y _ 1) (.ry - 2H) : (« V/« - 5) (« =6» _ 6). (7) {cd - V^ah) (cd - 1 7 ah) ; («« - 1 Iz) (ac - 132). (8; (aj - 2) (a; + 7) ; (;r - 9) (ar - 225) ; (2a - 3a;)'. (9) {ax + hy){nx''by); 12a' v». (10) aK ' ' (11) i^+m (aJ + 2) ; (a;-.v) (aj-a). (12) 6399 and 1439611. (13) (a +6) (a +6 + 1). (14) a; -19. axoroise 66,-Pftgre 41. (4) (4a; + 1) (a^ + 3). (7) (2a + 56) (4a - 3b). (5) (ix + 3y) (2oc + ly). (8) (2a;- 1) (3a;+4). (6) (2a;+y)(a;+2y). (1) (3a?+7) (4a;+3). (2) rSa;-?^ r4a._3^ (3) (7a; +165) (8a; -169). ■ '' Xh^ ITIILIC SCHOOr. AmBBRA. (4) (4a; + l) (3ac-2);. (4.r-l) (3a;-l); (4«+l) (3£c- 1). (o) (n«- »)(;» + 1); (3a?-2)(« + 2); (3a; + 4) (2£p- 1); (4.x + 1) (.r + 3). (6) (2// + 3z)(3i,-z); (lla-6) (« -2ft) ; (2aj + 7y) (Sac-jy). (7) (4 jr^ + 1 ) (3a; - 1) ; (2ac + 3y) (2£c + y) ; (3ft + .r) (2ft - 3a;). (8) (7.x>-3)(a' + 18). (9) (x - iy) (ox - 9y) ; (4c - 5) (4c - 7) ; (.r + 3) {x - 13). (10) (a; - 7) (5a; - 3) ; (a; - 1) (3a' + 2) ; (a; - 3) (3.r + 2) ; (a; + 6)(.3.i;-4). (11) (x + 4) (2a; + 3); (4i/ + 72) (4y-3) ; 2(2a; - r>j/) (a; - y). (12) (r).v-22)(r».y-32); {7x' +oy) (Ix' -Sy). (13) ( 17a; + 1 1 //) (43a; - 29j/) ; (59^ + 23ft) {la - 136). Bzeroiso 57,— Fage 13. (2) (a; + y) (a;' - xy + .y' ) ; {m -f n) (m^ - mn + n^); (w + z) {w^ ~WZ + Z'). (3) (2a+ft) (4rt«-2flft+ft-). (4) (2a + 3ft) (4a ' - Baft + 9ft' ). (5) (ox + ly) (25' -63i>.v + 49.v2). (6) (a'+ft') (a«-a'ft +ft*). (7) (a + ft) (a^-ab^-b^Jia'-a^b^+h'^). (8) {x*+y*)(x^-x*y*+y^); (x' +y^) (x'' -x'y' +y'); (nx" + \2y'^) (121a;'"- 132.x'\y« + 144 ,y'-). (10) {x y) (x"^ +xy + y^); {m - n) (m • + m n-\-n"); (w - z) (w' + w& + z^). (11) (2u -ft) (4a' +2aft+ft=) ; (2a -3ft) (4a' +6aft + 9ft=). (12) (5a; -7y) r25a;' +35a;.y+49y') ; (6m + 8n) (36m ' + 48m« + 64n' ) ; (9^ - 7*) {Sip- + 6o/>* + 49*-). - 3a;). 3). ■y)' H.v"); ANSWERS, HINTS AND SKELETON HOI.UTIONS. 153 (13) (a+b) (a-b) (a= +ab + b') {a' -ab + h'); (a-b)(a''+ab+b-) (a^ +a^b^ +b''); (Ila;'*-12.y«)(loi.^.o^l3,,^,ye^j^^^,,^ -/"^^ '' (14) (13as^-14y«)(169ic'Vl82«^/« + 196y''. (13j;^ + 14y«)(169a3'«-l82a;^y« + 194'«). SzeroUe68, Faffe 43. (3) (0.* -x^-y^+y*) (x'-+xy + y^) (x^- -xy + y^). (i) ('X:-+nx + d){x^--3x + 9). (5) (4a- +2ax + x'-) {4a' - 2nx + x'). (6) (9«=+(5«4-4)(9a^ -(]a + 4^; {x^- +ix+m) (x^ -4x + 16) (7) («='+2a + 4)(a'-2« + 4); (m*+5,«^+25)(m«-5m«+25). (i>'+^>+l)(/)^-^>+lj(;,4_^2^1^^ -" (8) i2x' + bxy - 3y-) {2x- - hxy - '6y^). (9) (8*= +4./-y + 9.y') (8^» -4a;.v + 9.v='). (lOy (9a= + 10«6 + 462) (9^2 _ 10a6+46»). (11) (3a= +3aic + 5ic=) (Sa^ -3a« + 5a;') (12)4.^±3.-1 (13)7«^±13«6^11.3. (14)a.«±3a.3,-y». (15) a;«±9^.y + 9y». SxereiM 69,-Pa9o n. (l)(-^-26H-a.^+3a). (2) (2. - y) (. ^ 3.y - 2.). i^)ia^x)ia-x-b). (4)(.-y)(.x + y-,). . (o)(a-6)(6-c); 2(x- + .y)(aj + y + «). (6) 2(ac+3y)(a;-2.y + 3aj. (7) (a + 3c) (a-3c + 6ft). («, (. . 3.y)<. + 2.v - 4.). (9)(3a?-2.y)(6£c-<:y + 32). (10) (3^2 -9A2 j.f^^2^ /^o^2 . r.1.2 =^ ( 1 1) (9a + 86 - 20c) (8a -b-\). r- 1. 154 PUBLIC SCHOOL ALGEBRA. SxtpciM eb,- patfo 44. (4) (2a-n6 + l)(.^oft_3^. (5) (2a -6-3) (« -26- 7). (b) (6aj-rt + l) («-(]«-!). (7) (3m + 2y+l^(2a?-3y-l). (8) (5^--3m + 2)a--yn + l). ' • (9) (2A: + .y-32)(3A;-y + 6a). (10) (6cc-y-A:)(ar-2,4.6fc). (1 1) (a? + y + s) (2£c - 3y - 2«), (12) (2/>-3r/ + 2c/) (3p-2y + 3a). (13) (2ac - 5y + 63) (3a; + 4y - 8a). Bxorciso 91,-Pafiro 48. (2)(x'+a;4-l)». (4)(a'+«36+«6'+ft3),. (3) (2a' -a -1)3. (5) (7a;« +4a;y + s,3)2. Bxeroise e2,-Paffe 46. (l)(a + 66)%- (aj-4y)^. (2aa;y-6)^ J![JrC'^^' (^> («-^26 + 3c)». (4) (3a;-2y-4«)2. (6)(x»+2a^ l)2=(£c + i)4^ (8) (m=+4w + ])2. (10) («3_3aj2+2£c-5)2. (11) (l+3a + 3a'+a')2=(i+a)«. (12) (1 - 3a; + 3£c»-a;=')2=(i _«.)». (5) (9a-36 + c)=. (7) (a2+63_c2+^)2 (9) (2m-3/n-5y)=. m (la + a^b+lb)\ (15)(m='+^m+l)'. /<a)=. (17) (3a2-«6 + 56)' r' t. ANSWERS, HINTS AJa> SKELETON SOLUTIONS. 155 (1) x-7. <:2) rt + 4. Oi) x-y. (4) a +3. (o)i>-7. axepdae 83,-Page 40. (6) n + 3. (7) &»-^w=. (8) a^-2ab + h-. (9) a + 6. (10) ic-i-o. (11) m{x-\). (12) 3(fc + n,. (13) £C + 3. (10) a + 3 (1) 3a:+l. (2) x-\. axercise Wj-Pagre 47. (11) a + 3. aaMPelMe6,-Pa»t48. • (3)aT-l. (4) 2a -6. (12) 3(fc+li,. (5) x-a. SseroiM OSy—Pago 60. (l)aj-5. (2) 8y» + 14y-15. (3) a»+2a -3. (4) w(m»-2wn-l). (5)a*-a6^ (6) cc + a. (7) 5a + 4. (8) ic-5. (9) b-c. (10) 5a2_i. (11) (a + l)^ (12) a;'+5a;+l. Szeroise e7,-Piff«B 60, 61. (1) x; abx^i abx. (2) It is the same. (3) The L. C. M. (4) abcde; 1. (6) abx\ (6) cdy^ J dy. (7) abcdefx^. (8) a6c; a6c. (10) a;'(a»-6»)(a='-a6+63). 156 PUBLIC SCHOOL .ALGEBRA. BzeroUe 68,-Pasre U- (1) a^-ab^-a^b + b\ (2) x* +(ix^ +3x^ -26x-2i. (3) X* - ox^ + 4. (4) 603=' - 2bx^ + 2'dx - 6. (b) 120x*-lMx^-l'2i)x'+lix + 2i (6) 18a;*-45«'+a7*=-19ic + <). (7) 1-a*. («*) (iO{x-+'6x + 2). (9) a;=»-7£C + 6. (10) Sic^* -14ac + 6. ( 1 1) (£c - 1) (cc'- + 2a; + 2) (x-= + 3ac + 3). (12) 3a;«-lljc*-8a;«+42ac* + l9a7=-47x-30. ,^, 15. 3C2 ^ ^ 20' yz (2) —• a^ ^ ^ 28' wy (3)1?1; 1 ^ '^ 489 / Zzerois* 69,— Page 62. 46' (4) i,v; (i^'6b+ — -: a-o ,.. 362 3ac-116 (o) — ; . (H) x-2 Ax-f (») a^+4a 7a' -llrt-o Sxaroise 70,— Faffei 62, 63. a^ + 1 (^) a'-2« + l a'-3a + r 5a' -x^' («) a;' +3a; + 5 ac'^+5ic + 3' ,g. 8(ar«+jc"+a:«+a;3)-21(ac'+ac + l) -j:l — i \ O / __ 3 (X'" -f iC" + it; " j "- o^o;" ^ u;' 4 U; r 1) 1 ANSWERS, HINTS AND SKELFrrON SOL UTIONS. 157 1 (1) (2) 13 12* cH-h ah ' (7) ? '+a?4-agy-y (3) 2H fird + b^d + lH' ftrd (8) (f>) (C^-l ab + bc + ca n^b'+b^c^+c.2a- (4) (5) ^ 57a + 8ft ~36 • 1 b abc m -. 2.x + 4 05"+*,^ -21 a (6) 6 a& ad±b c ~~bd~' (11) co; + 6V + az a6<,' 0. (12) 1; i?y(^ + y') 2 a;*+j!:'y2+y«* (1) 1?; e£ ^35' M' (2) ^ ; ?. ^^> ^* ' 6c ^^ ■'"'**"'■ ^"''^ + 2«'^^)' (4) K. (7) - 4(fi-fe) 3(a'-rt64.62)* («) «+a-ft' ■ateroiia *7^~Hf 84. rn 28. «^ie - 46 . Sac' (2) ^+^^'+^* (6) 4d.>ra -i-in (.^..,.4) (^^6) (^^^^^ (7) 24a'-,^,;r(xV. 4«2)(^2_^2)^ (1) |. Bxeroise 76,-Paflre sa. (4) '0953563. (3) a; -" 8 ,r + 8" <-) 2a -6 rt + 6' (») 0. ^-^ 9^0^^^* + ^^ + 9)- (5) l+3aj + 3u;'+etc.,. 3a.^'». (3)2. Bxepciae 7e,-Pa8re 66. ^J^"''- (2) 8aj-f.(3a + 2aj). (3)(a. + 3)^(a.*-l). (4) 2«:.^(a»+o.^). (5) 2(a + 6 + c). ^ (6)2a6c(a + 6 + c)^(«_6 + c)(a + 6_c)(-.a+6 + c) (7)2-(a + 3). ^ Szoroloe 77,-Pftjro 67. ^1) ^(a' )^x. (2) aj=.li. (3)aj=l. (4) -^ : :l±t£). 2a -c <5) .• - ■(6) ic = 4 2a -6 ANSWEKS, a,NT« AND SKBLETOX .SOLUTIONS. 159 Sztrolae 78,-PftM 68: (5) a + 26 + c. ... , ^, (1) a; + 2' axtedM 7Q,-Page 5Q. (2) (2«-3)(2a;-l). (5) Man, MO ; woman, «H,. boy, «« J girl, ,2. axepoiae 80,-Paffe 60. («) aj = sq. rt. of 15, or ^15. (8) 358 yards. (5) X = c-2462 + i5c2. (2) «J*-y^ -2* +42/^2 + 42/2^-62,2^2 * (3) (3j2-72/)(3a? + %). = ^^'-'^^^'+mx' + 120x^.216x + m (S\ (Rr>-0\ /O. "J V-*^ T 1 , (7) (8a;y2_4aj2y_4^3^^^^4_ Ji *> ■""Tf~ 160 PUBLIC MCHOOL ALUBBKA. axorolM 82,-Pnft 01. (1) 03=13. (4) 5^3 (2) (46 + yrt) (4^-a«) (l(;/>2 +9a»). (6) ar = 21. . (a) (« + 2/>)(4«^-9/,). (7)^94, IzeroUe 83,-Paffe ei. ( I ) X7/-r (x- + y) days. (4) a>= Ij ; y = 1 1. (5) Number = np-^ {m - n). (6)x- = 9; y=-3. 4 4 (3) x-+bx + c. Szeroiid 84,-Pftjro 62. (1) a;- + 2ic-3. (2) if = 2711. (3) x=J)G; y = 72. (4) a -48. ^5) <<'-t-^>"+t-'-tffr-&c-( a "{a-b)(b-c)ia-<') • (6) (a*+Ga26*+6<)^4a6(«2+t2). (7)ic«13. SzercUe 86,-Paffe 62. (1) (ac + 5) («,--2); (x+1) (x-1) {x'' +x + l) (x' ^x+1) ; (3x + 7)(3ar-2). (2) x'==ll; y/^7. (3) HO trees ; lengtli = 59 x 20 = 1180 feet. l-3«c i^)^=h y-l (7) i+4x (4) 50. (5) 3. (8) 3£C4-(2a; + y). 1 ( (1) iC=ll. Bz«roUe 86,-Fas» 63. (•*) (l+iKy)(l+a;+2/-a?y). (3) 02 = 13; y = 9. (a;-4)». (4) l-r(l-a; + x»). (i'l) $240. (fi) 04 square yards. (7) (3rt'-2fi/> + 5A)2. (•>; ic=:5; ye=4. (fi) 1J>5 hours. i^)^'^^y'^-z^ + 2xy + xz-^yz. (7)aj-l. (1) 4 miles an hour. (2) 4 hours. azeroUe 80,-Page 84. (3) 15 miles ; 2 miles an hour. (4) 4 miles an hour. 2»roiii 00,-Paffe M, (1) 24ifeetbyloweet. ,4^ ,.- ., (9\ A lea n o^, ^ -^ ^^' miles. 2) A, 38i ; 5, 36J ; C, 38 sq. rods. (5) A $070 • 5 .ooft (3) 1180; $150. ^. *-7n B, $320. (2) x = a] y = 0. (3) 60 shares. ■"•wiw 01,~Pajfe 68. (4) 10 miles per hour. (j>) $1250.« (6) Aa^h\ ■«wela« 92,~Page 88. (1) 150 miles. *'> '^ *o ^« »iays ; B in 45 days. r)(x-15)(ar+17); (3x^-4y)(7a; + 5v). (2) ft+w. (oj3x2-a;+7. ' 1 i f62 t»r?BUc m;iUHii. amucbka. rt + 6 + o-O; transpose, factor, Hubstitute. (9.) a+h-r = 0, factor . ..i .ue. A V8. 0. (3) «12r, ; $375. ^*^ "^ " ^- (") ''^ PO"n) 1 ; 11. (2) r/*-/>*; 4; a+h; 3x+l. (6) ^729==9: fm[ = u (3) nr+/>; a? + 2. » (7) ^^ (4) a+6-c. (8^ 0^ BzerciBO 96,— Faffe 67. (1) ^- m 1?. (4) Show ^" =. b^' , take t e fcth j,ool of both sides ; 2 -ft (5) ^a- 1.-1,. (6) Sq. root otft.a« + i08a"o^+48a63 + 166* = 9a»+6a6 + 4ft2 (7) 660 = 5^x13x2. Ans. 2R. 1a If 96,--Pa?ee8 Note. -No. 2 is der. od !rom No. 1 by substituting ft + c for 6 and arranging; No. 3 comes from No. 2 bv ual\t^XfnlnnfZA re-arranging ; No. 4 comes from No. 3 by addh^;' 3^^?? Teach ciency of Sabc ; No. 5 comes from No. 4 bv factoring (^j.h2.\ («& + ftc + ca)-aftc. The whole exercise is intendJ^^^ practice m substitution and tranof«.»«o*:.„ :_" . ^^. Si^^ glimpse jt symmetry. " '"""" """ '^" '^^^^ " nindsof tin. 1331 = 11. :&. ab + 4b\ ANKWBftH, HrSTH AM. .sKK,,^rrf.N HOl.i flON8. 1(>8 (11) a + 6.0) (.= +,. ^,...„,^,,^^.^ ^ ^r^S^!^;:t?!:;:^ ^-- -»>-in.tin; -.,or. ..for *"« " ana c in tho formula of No H axorclse ©"Tj-Paffo ep. (3) /> = 2c. (-<.,. ;/ = 36_,_„ . ^.3c-«-/,,. hence the expression given = .r-^ +.y' + 23 _g fron. .nestion !.. kx.nise <).;. But ^ + y + .:«+ft + , ,,. Hddn.g the values of x. .y and z. Also ^. - y = 4(« - Jetc hencex.^+2,'+s3_y^^^^ " *'« Hetc, = («+/>+c) r[,„ - /„^ +(,•, _ , ,:• +,^. _^^,. = 16(a+/> + c)Jf(«-/>)-+etc.J = 16(a^'+/>»+r< -,'j„,^., bv No. 9. Exercise 9<;. EXAMINATION PAPERS. No. 1,-P*are 100. (1) For - JlFc read - fb^c. Ans. 110 (2) 2(a+6 + r). (4) x''-x*y^-Ax^y^+iy\ m R\ Rfi *«„* J 1/M n ' (5) 768. •3 icci aiiu iu 23 leet. ■■'1 ip the given es. et -rt + 2/i + r; '. '", and we (l+b + e hj' (t -h), etc., le 9(1. A.V8WKliH, HINTS AXU HKELKTO.V SOLUTIONS. 16& ITo, 2,^H99 100. (1) B6-4a. (3) (/> -f;> / («-,.) ,ueai.8 (/>-./; 4- (,i-r). (2) a^t 15) u;»8,- or ^.-1 5 («) ^mm./(m.p)dHys; i^ in .,«/,,_ ,,^,„ ,,,,,^ Ifo. 3,-P»firo 101. ^^)^--^ (2)755. (5) r-+/>r + 7«0. (7) IH yards by 14 yards. (8) Length = 13 1. 2.3 yards; breadth = 48 -77 yard«. (••l) X' + i. *^o. '4,-Pure 101. (1) («+'»^ + ^>^^^6==a^+.V4{«/>(«+6)M. (4) |^(.. + 2,. ..(2) 4b(x+,,), (3) a-y+l. (5) x=l?t 4(59 ( / (7) 45 miles and 30 miles i)er hour ^'■/U- + yM^'-.y); (a;-l)%- (.r-7)(« + 16). 16fi (1) a?=19. (2) a; = 17, or 14^ (4) Ua-ob + 16c. (1) a'+aa^-2£c^ (3) *il^^i) (4) a; = 6. PUBLIC SCHOOL ALGEBRA. No, e,-Pajre 102. (5) -H 88 ffJ) x = 7, or - 38 J. (7) ^ has 40.V. ,• B has 2.v. No. 7,-Paare 103. (5) « = 7; y = 9. (t>) a; =4, or -1. (7) 70 lb. at 2d. ; 154 lb. at 2Jrf. (1) 2b. 1^0. 8,-Paare X03. (4)H.C.F..2«-3.,. (^)--^';^^ = 0. I^. C. M. =ax(3a + 2x) Oia-2x) (2a -Sx)' (o) .y(9.c + 50.y)-=-(4.:p + 5.v) r5a; + 6.v)20 (<') a? = 6i or3. (7) A paid $50 ; B $30 ; f $16. (3) « = 8. (4) aj = 9; y=:iy. (5) 2a;^4-6ic + 4. Mfo. 0,-Pajre 104. (6) « = 4|, or 2. (7) 9 at 40*. V 18 at 29*. 62(a2 -ft^^. (14) j:, = 1, or lug. (15)a; = lf; 2/ = 2i; z-=-12. 168 PUBLIC SCHOOL ALGEBRA. Ho. 16,-Paw lot. (1) l-i-xyz. (2) 2ac^(c=-f/). (6) (a"» - c»« ) (rtw - 2c« ) (8) 45. MTo. ie,-Pago 10©, (o)b'c'-a^b-cr; -36= -36. (6) (i) a^c^h; (il) -l^{x' + i), (") (a + 6)(«-6); (a + 6)(/> + c); (a + b) (h + c) (c + a) ia+b){a~b) (a + b + c). CI. C. M. = rt+^. L. C. M. = (>< + &) l^-i-c) (ffa) (a-b) {a + b + c). (8) (i) a; ^ 1 , by hisiiection. (ii) Multiply throujjrh by x + 2, and get x-^ -2£c-7, whence x=^l± ^'S. (iii) a; = // = 0; orx-=-A, «=- — li -^ 16* (9) .4 contains Z water aud I wine ; £ ^^ water and 1 wine. Let a; and t/- quantities drawn from .4 and i? respectively ; from which x — 2;y=^l4. Ho. 17,— Paffoa 100, 110. (H) For -4023 2:3^ ^.^^^ -ix^z^ ; G. C. M. =3x^-2z\ (5) (i) x^li: y/=.-j; s = 2. (ii) The e 4. 1 ) r.7. _ n 2 (4) x + 2; 72a263(«2~fe2)(a3_63). (5) a;«13; x = b, y^ -2. 170 PUBLIC SCHOOL ALGEBRA. (1) 24 Wo. ai,-Pajro 112. (2) (x-iy {x-2) (x' +X + 1) {ic-6) (x+d). (3) G.C.M.=a; + l,- L. C. M. = 12a;^y'(x + .v) (cc-y)'. xy* a;"* . 4 3 o (6) 186 gallons. No. 22,-Paffe 113. (2) 7a;y(3a;-22/)(3a; + 2y); (a-6)'' (a+6) («^ +6==); 26". (3) G. C.M.=a:2-4a; + 4; L. C. M. =a22/(a;2 -t/2) (3j4 ^^2^,2 ^2/«). (4) 5£c + 8 X* -x^ -x"^ -2x-\ {x-l)(x + 2) (X + 4) ' x'~^x^~r^^:^ (5) a: = 24; a; = 3, y = 5; x = <2, ora: = l. 2 No. 23,-Paffe 113. ^^^ ^- (2) 2a-66-2c + 3rf + 6e. (3) a* -fe*; rt2-a6 + «,2. (4) G.C.M.=a + l; L. C. M. =60rt6(a2 „t2) (^4 ^^252^^4)^ (6) a^-a + i. 2 (5) (a+3)-^(a-2). (7)ar=7; x = 18; £c = i; ^ = 7, ty = 6. (8) 30, 60, 90 pieces. (10) A, 39 sliiliings ; B, 21 shillings. /■ON /»f» _ „ 1 n^ \i// ua aaa si years. ANSWERS, HINTS AND SKELETON SOLUTIONS. 171 (1) 1. »o. a4,-Paffe 114. • (3) x\y-a) + 2n^x^+x{y'-2a^y-.a^)+ay{y + a)\ (4) a;'+42/= + 25s=-2as// + 10.v3 + 5£«. KB. This is a'+h'-c'+^ahc-a + h-c. See Exercise 96, No. 11. (5) 7(a?-13) (;r + 2); 5.7;=f2.r-32/)». (6) a;-6. (7)105«^a.^(«=-«,3)3. (fi)a64- (a -/>)'; 0. (9)x = 22;«. = ^+|!;^ = 7;2/ = a (10) N.B. Let K-1, 02, 05+ 1 be the numbers. Ans. 9, 10, 11. . (1)1. ITo. 25,~Page 115. (2) hc'-^^ + ^x-\, 2 ;{ 4 (3) 1 - 8oj= - 21y^ - 18xy. See Exercise 96. (4)a.3_2^,. + 4 - (5) ix\x + 3y) (x-Sy); (x-y) (3x-by). (J)C,Ox(l-x')(l+x), («*; ^^^^■M;.x = (2he-ca-h')^(a-t^', x = o, y=4 (10) 2^' miles. (6) sc-l. . (8) bed; 1. Jffo. 26,~Pa«e 116, (2) 1 ; (a + 1 ) ,7; + 1). (3) a. n. M. = .^r = ^ ^ . CO x.^^, 05.0. or -ia'+b')^fa^b); x^l, y^l a 172 PUBLIC SCHOOL ALOBBRA. No. 27,-Paffe lie. (1) llab-Ua^ ; a='+6^ (2) 2a' +3ah + 2h\ (3) 9. (4) {^x-2y)(2x + Sy)', x{x-ly) {x-Qy); -(«-5c) (3a + 464-c); {3x + by) (Sx-by) (dx' +2by'). (5) x'-l. (6)(.r-32/)-^(.x + 3.v); (2x-a)~(x + a); (a' -.h^^)^a'b. (7) a. = 24. In (ii) for | read ?; .x = 36, .v = 40; a. = 2, or ! • a; = 2a + &; y = a + 2b. (8) a2i>+<59. , ■b^c. IZ^t^T ^"' ''^"" ^" *'^ *"^ ^-- expressions we see that they become identical and are therefore eoual for any and every value of .; thus when . = 2 we say that the value ,s indeterminate and may be any number. (7) 45 years. No. 31,-Paje 120, (1) 3141-6 square inches. *'^ '° I'. •■ tz. ■";;■■ "' ° ■*'"'■ «*"• s- ^ ". No. (3) x-y^a-b; y-z^i.e; z~x = c-a; ■■■ iK^-2')'+(s'-^)'+etc.] = l[(„_6).+etc.] See Exercise 90, No. 9, page 68, and No. 26, page 82. (4) See No. 23, page 80. a' +6' +c= -^_6e-c„ (6) ac = lj| a;=:2w2. (7) a;=-4c. See pages 77 and 83. (9) 3 miles. (10) Let a? = rate per minute in still water, ^~ " of stream ; ••• a; + y = down rate per minute, a;-y = up *< << ••• a; = -(sum of rates), y - 1 difference. i^h ^b v ANS\^'KKS, liLSis AN1> SKELETON SOLUTIONS. 176 xpressions B equal for V that the e 77, No. 82. x + ^y) ; h by is, I No. 32, -Pago 120. (1) («) (oo + y) '(x'-xy + y^-). (ft) See Ex. 96, Nos. 8, 9, 10. (i) Let x + z=^m, y-z^n; .-. a^ + .y=m + n, etc. /. expression = m^ + /r^ - (m + n) (m - n) = = (m + n)wn. Divide through by m + n using (a), and (w^ -mn + n')- {m -;,)== ,nn, an identity, (ii) ^tn'-bc = x;b^'-va = y;c^-ab..z; and expression = x'+y'+z'-3xyz; and Ex. 9<>, No. 10 ^-(^ + y + z)[{x-y){y-z)+(y-z)(z-x) + (2-«)(as-y)]. ^ow x+y + z=.a' +b' +c' -ab-bc-ca ; x-y = ia^b){a + b + c); y-z = (6-c) (a + 6 + c), etc. ••• («'-2/)(2/-«) = (a-6)(6-c)(a + 6 + c)Setc. .-. expression=-(a2+6^+c='-a6-etc.), multiplied by -( " •' " " ), multiplied by (« + & + «)' = (a' +6' +c'-3a6c)«. (2) Prom the given relation (a-b) (a+b + c) = 0, whp-3e a+b + c=^0, since a-b is nojf =0; ••• (a + b + c) (ab + hc + ca)=0, which is the required expres- sion factored. (3) (6) See Exercise 67. page 51, Nos. 5 and 6. Since the L C. M. =prodrct -G. C. M., and the L. C. M. has only 4 dimensions, while the product has 6 dimensions, the W. O. M. must have 2 dimensions. .-. Let x^+mx + k=> G. C. M. Divide each of the given quantities by this, and put the remainders = 0, part by part. Hence -fc = m(a-m), and b ^ Ida - m) ; also - m = c - fc, and rf = - mk. Eliminate mandfc, andA: = l(6_d); m==l(b-d-ac) o, a ' Substitute these values in - fc=7ii(a _ rti) and get -d). (4) Sum= a'Cas^-y -^ete (^^^¥ly^)l^) '"- ' (^+3/2+«B). PUBU<; nCAUHtl. AUiKBkA. (5) (i) Factoring up the expression we pet 4T»6»^^'' + ^-^)' i^'b + cy (6 + c-a)», of which the square root is ^,.+6.,)(,.,^,)(,^^_^j^ (ii) Square root -aj» + la. + |, by inspection. (fi) The given expression vanishes when a. + a»0,r.«,4.6-o or x + c = 0. It is an laentit., and not an n^ation Any value of a; will satisfy it. t«»"on. Anj (7) ('ill (^) + (c + a) (c - a) ^ , ^ N.B ^'-c) + (c-a) + (a_6)=.o. Hence the expression would =0, if |±f,,,,e^^,,,«^5^^^^^,^ Ofif6 + c = fc-a, etc.; «.f. if /,==a^.5^^ (8) Let 3a? = ^'s income ; 12.v = ^'s expenditure. .. 3x- 12y^A^s saving. Therefore 2a. = B's income, y = B s expenditure ;2x-y==B^s saving. •■• ^(^^ - ^2y) =H2x- y), A saves half his income. (9)(i)t«=|. (ii)a.=5a. (iii)a.= -3. (iv) Add the equations, andcc + y = 7, or -8. From first equation - 8a; + y =- 26 aj = 3, or -^; y = 4, or -1?. d 3 JTo. 33,-Pjhp» 121. (1) See No. 20 and 21, page 79. ANSWERS, HINTS AND »!K*;i,BTOy SOI.imONS. 177 hich the 0. on. Any sion = 1, i, (2) (i) iax In,) {x^^xy + y^). N.B. Observe that a ami b are involved only tu a single power. See Exercise 59, page 44. {n)lax + ay + hx-cy-hz c2) (a* +6*+c» +a6 + ^c + ac). N.B. The exi ir i i^ evidently (i) with x + y for «, a + 6 for X, y^ h, and ft + c for y. Therefore sub- stitute these values in the first result. (iii) The factors of the expression are a+6 + c + d/ a-6-o + t/; -a+^ + c-t/; and -a + 6-c+rf=;0. The expression will vanish if any one factor =0. The four relations r.ay be reduced to these three : « + 6 + c + t? = 0; a + d = 6 + c; a + c = 6 + rf. (3) Ans. as + 3. N.B. Without the limitation it would have been (« + 3) ^ (ar + 4) (a: + 5). (4) (i) (6a«+a;-'+a52+a; + l)-^(a;*_i); (n) 1. 3 N. B. In (ii) the denominator = 3(a -b) {b- c) (c-a) See pages 83, 84, 85. The foUowing solution is shorter :- Puta-6 = aj; b-c = y; c-a^z; and the denominator = «=» +y3 +^^ Add 'dxyz-^xyz and {x + y + z) {x'^y^+z^-xy - etc. )+ 3a;ys is the result. But as + y + z = 0, .-. Qxyz is the value of the expression. X + 3tC. (5) (i) Ans. 1. N.B. a+x b+y But x + y+z + u=a+x=b+y=etc. (ii)a^=(6+c)-^a; .: l+x^ia+b+c)^a. By symmetry l + y = (« + 6+c)-f-6; etc. =etc. Sum = (a + 6 + e}-j-(o + 6 + c) = l. (6) (1) Transpose e ; multiply through by 4a ; add b^ to each side ; extract the square root. Ans. x = (-b± ^b'^Aac) -•- 2a. 18^, .0 ^ ** y*^ V', IMAGE EVALUATION TEST TARGET (MT-3) // J .0 ^ v^ 1.0 £itt m " m 12.2 I.I u Itt 12.0 u IL25 H 1.4 1I& 1.6 ^Sciences Corporation 23 WIST MAIN STMIT WIMTII,N.Y. 145M (71«)t7a-4S03 ^J^^% ^ J^. n would by ' 1> we need id im; reases is as large nity.j ito. therefore that the 'or«» there mknown dviSnite to have ; in this i AN8WE118, HINTS AND 8K1SLCTON SOLUTIONS. 179 (10) At 3 o'clock the hour hand is at III. and the minute hand and the second are both at XII. Let x - space moved by hour hand before second hand gets midway between the other two hands. Then 12£c» space travelled by minute hand; 720a; = space travelled by second hand. Thus the second hand must be 708a; ahead of the minute hand = distance behind hour hand. Therefore 707a; = distance of second hand from III., and hence 720ar + 707a; -distance from XII. to IIL-if^ «,;„.,*«„ ^„ *i.- ,i._i . _ 15 15 minutes on the dial; minute divisions passed by the hour hand. .*. 1427 the second hand has moved ^^_ x 720 - 7 ^}}- seconds. Ans. 1427 1427 XTo. ai,-Pft«t 133. (1) (i) -(a+b+c). See page 85, No. 33, B. (ii) (a;» + j/2)^a;V. (2) If a;' +aa;» + etc. =(x-l) (a; -2) (a; -3), the expression must equal part by part, le. a- -6 = c, 6-11. (3)(aj-l)»; 2a;-y+3«. (4) 2 ; (a + 6) -!- (6a; + 1 ). See Exercitw 59, page 44. (5) Numerator = (rt+ 6) (c+d); hence c+d must cancel out; I. «. a; - 3<; + 2d must contain c+d. . Put a; - 3c + 2d = n(c+ atitl ^. and zj^ n.-e the times mentioned. I-ty.heighrn.om bottom; .. the equation Wo, 96, Page i2i. (1) ComparinB ti»e dividend with rh« -i» • ^^^hat the quotient ^nnsTlL ""'*' *^^"^ ^^ ^-»"' ^« verified '^^ytuh-nM-Z^^^^ '^*' i"Hl«ction is easily ro, ,.., , ** ^''"' J-emoving the brackets. fractions, and the numerator is ' ^*''*'*' '^""^^ "« can easily bo provetl by p„ttim, 1 "f ^ ''^ '" " '"' method shown on page 84. CZ-^" \ ' "''" *''*^- ^^« Then l:?^=-44 J-45^»^ol^:-ff = ^: ^^-2^»^'. etc. See Exercise 55, TageJ, ^ ^^^^ ->^*— .45«,3, \'*) V* nen ci = f, or *• = A m- h . ^ he wWe «p™.io.. = 0. N.,, ., i^ ^,„. „_ (-) (-+.; (y +,) „+.,=,^, 3^ ^^ ^^_ ^_^; *^«« *■ («) Let fir ''«''-*• «, y-^ «-y'"^'*^='''- Seep»ge98. Multiply the first by 2, add and 3. »...«/„... .._ vSTT-s/, CSC i ANHWRRH, IIINTH ANI> HKELETON SOLUTIONS. l^d. y terin^ we WJ is easily 'ket8. 9 last tlu-ee » after the l-u-6). J 84. I> ' r I 1 + ' + ,, whence (2x-9)(a!--10x + 24) = (2jc-9)(./''-10ac + L>t); je-4J. X -10+-=a!-lO + ?l, or!--, oc a; ac .r whit5h can only be true when jc = x . See soiution of No. 2(a'+6'+c'-a^>-«»c-crt). N.B. Pnt a =/> = «• -^ 1 and test. (2) (a-ft)»-c'. N.B. Pntrt = 0, 6 = 1, c = 2, and test. (3) ac»' -Jc»«-« + x« -1. (4) (o+6+c+-d-l, and test. 4 (9\ 3?= -1: 1/= *. (10) Oxen, $100; sliMp, $7. N.B. Verily the result. ■"X/ 189 '•UBLIC SCHOOL ALGEBRA. (0 (-^-?)'. X y See page 80, No.' 23, Nb^'*''"^*^^) («+*). etc. (8) See Paper 86, No. 7! (9) x~3i; »_49. , M . 00) 84. '-"'-*^''' -l-'-lul.. eti:'"*-*"'- •»-'• (l)<»* + ft' is divisible by a+ft («) '-C-3,, r^:::- ^^--i-er, p.,«,. " *- - 8J- N.B. M,jup^ jj^^g^ ^^ ^ ^^ '" "^ °" *'*"«<» .• No. 46, p,g, 92. I »'y» putsch 2. etc. 5-0. », etc.; this somes an ANHWICIIH. „,XTK yso 8KELfrrON HOLUTIONg. (10) 97. See Solution ; No. 47, page 9-^ Wo. 3«,-Vag« 188. ^'^ t^'1°4i V.'T '*• .7'" '»"»-»« « not ,0 short : :!'!;:''"'?.-'^ + «"'(*-«)'-Mft-c)'t(6-c)« 183 (2)a6c-.270; 1728a;y«-f.m«6. niactor. l'uta; = y = 2»2, andN=l. An8. a?y«(x+y + a). (4) 2(''-« + '-)';^(P'-7'). N.B. A .^-" + "' +«■) i 0. ^.B. Each N. ...d D. i, ide„.i„.,. (6)(6»-x)(3y + 2x-7); (o'-« ow„. „, <«•+!/■)(«' +6'Ve.). * -*'''+»«'-2«+4); (6) (i)(ft'+c')(c-»)+2a6<.-. NB Addl,„« u .. .ubtraot . ,™„ each ,ide, .^ut 1 h si i^tlt" • "I'V'de one result by the other " »'»• «P»«-»tely ,• •-each inanite since, vanrhes"'^ '^''^ (p t2; ,21. N.B. Ut 3., 4. be the bills; 7. = deb. etc (8)112000. N.B.Lee6..propeny.etc. ' (»)84. (10) 18W minutes past « o'clock N b t . minute spaces traversed by hour hand ^r"""**^*- «' traversed by minute hand, • 30- is^ljoii """^""^ (11) ^haslp + i«_l (12) 27i miles per hour. zf; B, ^'a9 + ~r; C, •-» + 2" 1 1 2 186 PUBUO 80HOOL ALOUBA. (1) (8a;+2y+6)(2ac-f3y); (x» -2y«) (x« -2acy-y»). (2) (x + 5a-fe)(x + 2fc); (x« -a«) («• -2x + l). (3) (rt-fc)+(rt+6); 0, Heo pages 89 and 90 ; ar + y + s. N.B. The N. of themim=^ - x''(j/ -«)-//'(«- jc)-«'(x-y) - (a? + y + «) (a? - .y), etc. See page 85, B. (4) (x-2a)»(a; + 2a)». N.B. 2nd = U + 2(32m - 21n) •*• 20m(m - n) to make 10% profit. (10) Let X - fi*s time; X 4*6- i4*8 time; 10x-6(x4-6); 90 miles. STo. 18,-raft 138. (1) 1 - x+2x' - 1 - x(l - 2x). For 1 - 2x write y throughout and (1 ■\-xy-\-x*y* +x'y' +x*y* +etc.) (1 -xy)- 1. Ans. (2) Dividend - remainder - y » + 2y ' +0 ~ 3y ~ 3y^ - 1 ; divide this by y*+0+y*+l, using Horner's method ; y + 2y* + l. Ans. (8) See page 84. Ans. Add -a!ffc; or add 6'<*. AN8WKKK, „,.vrH AND HKELim>M BOLimONIl. 187 > ♦■y+jc. !-ar)-«»(x-y) I, B. 8rd- (as -2«)». N.B. For n-f o are from Prize Problems contributed to the Canaila School Journal. The object of the lKX)k Is to place In the bauds of teachers and pupils a careful selection of Problems suitable to the fourth class in our public schools, thus enabling them to iireimro thoroughly for entrance examinations to high schools and collegiate institutes. The selection is particularly strong in interest and percentage, loth edition. I'rice 20 cents. Decidedly a help. Will be decidedly a help to teach- ers who are preparing pi^iils for the entrance examinations to high schools. The Problems cover a wide Held, and are, most of them, eminently practical.— J . A. Wis- MKR, M.A., Head Master, Park- dale. ■ittercstbiS and i tractleiil. I find the qaestlors in the various departments of Arithmetic pre- sented in great variety, and at once interesting and practical.— J. A. Macphebson, Head Master, Bee- ton P.8. Excellent A»r home work. Your " Prize Problems in Arith- metic " contains a great number of excellent Problems for home work or examinations.— J. 8. Rowat, P.M.8.t Caledonia. Imtend latrodaclng It. I Intend introducing it in my fourth-book class at the beginning of the term. It will make an excel- lent review book.— M. P. McMab- TBR, Principal P. 8., Thorold. More tham pleased. I am more than pleased with it— Thos. HAMiiOND, Head Master, P.fif., Aylmer. Comprehensive and very prac- tical. The problems are definitely word- •d. comprehensive in range, and ni» nSactisaL ' C- Hniiao. Principal P.8., Palmer$U>n. Will nse It. I will endeavor to have my class of next session provide themselves with the work.- A. N. Thornton, Head Master, P.8., Wallaeeburg. Rrst of the kind. I consider it the best of the kind tlint I have seen.-J. C. Stbwart, Principal Pembroke P. 8. An indlspensiblc help. I have thoroughly examined "Prize Problems In Arithmetic," by itobertson and Ballard, and be- lieve teachers will iind it an indis- pensible help. — J as. Duncan, Prin. Essex Model and Windsor Central Schools. Time nnd labor saving. The questions are practical and will Induce a wider range of thought. Any teacher using this work will find it both time and labor-saving In reviewing for ex- aminations. — M. H. Thompson, Principal Aurora P. 8. Shall Introdnee the book. The problems involving Interest and percentage are very numeroni, and Include an almost endless vari- ety of apnlication. I shall introduce the book to our entrance claw,— W. E. GrovES, Principal P.8., Wingham. Vseftal to teachers and pnpUs. I feel certain that the work will not only be Interesting, batalso very meftaUo both teachers and pupils.— i T. O. Stkhi^k, F.M.S., Mar fie. W. J. Gagb ft Co. '8 Publications. Gage-s New Topical Englls;i and Canad ian Hlg tory Notes. This little Primer Is prepared to cover the Public School History Coarse In English and Canadian History, and is printed so as to fum- ish a number of blank leaves to allow student, to make «ldltionri notes. Price 26 cents. 1. The Notes an, arranged Topically under such headings aa bes. liHilcate the True Growth of the nation. ». The Progress of the People, the Struggle for Freedom, the EstabUah- ment of Representative Govenunent, and the Development of Ed«. cation. Uterature and Religion, are given more p«)minence than «. The Colonial Extension of the British Empire is briefly mrtUned 4. The whole History la Classified, so that the Relationships of the G.^t Upward Movement can be understood. 6. The arrangement of the Note^ makes It Eaay. Definite and Thonrngh Reviewing, perfectly simple without a teacher. 6. The Notes supply an Admirable Preparation for the study of larger htatories, and the l,est mean, for Clearly Remembering what his been learned frtan them. 7. Ample space haa been left for Additional Notes, to be written by the stu- dent. toStlmulate the Further Study of the importantsubject with which ■y the aae of this l««l« Book i. Hme la Saved to teachers and pupils. «. Success at Examinations made more certain. 3. Interest is Awakened In the study of Hlatoiy, «. Aslmple,deflnIteMetliodofSt«4yto»Htatoryto,«v«aied. W. J. QrAQV & CO.'S PUBUCATIOMS. Gagre's EnyllBh and Canadian Higtorjr Note Book. ** I kave MldoM Had one plucked In ktotory after uMlnc IkiM iMMik.*' — ROBEBT W. Briuut, Principal Public School, Drayton. •iHHt wkat I kave been look- ing for. The History is somettiinR: excel- lent— lust wtiat I liave l)een loolcingr for. I intend to liave all my En- trance pupils aret them.— Sam. J. Latta, Zurich. Botk convenient and kelpfni. I thinic it well suited " )r the pur- iMMe for which it in Inttuded. The iHisy teacher, and in fact any teacher, who \ia» the subject in hand, will And the notes Imth con- venient and helpful in prefMirinf? his lesson and in assigning; home work.— A. McKkk, Head Master, Uxbridge P. S. Ureal aitsistance to Ike teaeker 'Vhe History >vill be of great as- sistance to ttie teacher in prei>ar- iuK ills notes. The topics are well chosen and the book is a good out- line of the sutaiect.— H. R. LoNou, Prin. Model School, Clinton. rieaited wltk tke arrange* mrnt. I am pai-ticularly pleased with the arrangement of tiic topics in the English and Canadian History Note Book. It will greatly aid iciU'hers in saving time, and all candidates preitaring for examina- tion—from promotion to senior leaving. It is an eminently prac- tical work. — B. C. H. BECKER, Prin. Brighton P.S. MVIH HHvc time. It will save the teacher as well ns the pupils a great amount of valuable time. With present text book on History, pupils require an encyciopeadla and an unabiidged dictionary tu arrive at the meau- ing of the language.— J. G. Cau- i»U"filr:Uf«, Pi-iii. Vayvga P.S. Probable It will be adopted In tke Nortk«weHt. I believe it is well adapted to the use for which it in intendertant. notes along side of your general on&s.— W. A. Stewart, Prin. P.S., Lancaster. Have never met any work to equal It. I am highly pleased with your History Primer. I have never met any work to etiual it for the in- tended purpose. So far as I am liersonaily concerned I find it a wonderful assistance in presenting historical facts to my pupils in a satisfactory and telling manner.— J. A. Hill, Ph. U., Headmaster W. J. «A(;b & CO.'H PUULIOATIONS. OAGB*S English & Canadian History |4ote Book. From P««F. WIIUAM rLAKK, M. A.. ,x .»., y*-iMfii rrfrr lo It." '^ * " * ** »*'^'^""' »»• ■ A loii|c«fip|t wnnl. Your Ensriish nnd Canadian His- tory Notes Hll a lonff.fV.lt want There seems to be no text Iwok at pres.-nt s.iitable for Jun or Kit" and tlio notes will sau; the tSlr imieh tnnu in arranffinir class work, 1 hey are Jnst the thlnj? for ora teachinsr and review, and may l" A vvvy Krpnt »M. nlan History will meet the needs of inany twjehers wlw are pre^^l for time. Entrance puirfls will «,«{ It a \-ery great aid.-L. F. Hvii- noiN Prin. Nianam Fall/ P. S Admlmhiy iKlMpicd. It is admirably adapted for En- trance classes, and ufll nmke the w'ork definite and thomnir -S Y Pm-iB"!"' ''''"• ''"""'- ''^'^^^ Jnst tbe Ihlng. .. I *'!! J>J«asefl when I see vour ! vS"*?"4 ,f"J«* ^'anadlan His^t^r^ Note-book" in the hands of mv !'''"*• ..^J"« of them are usinff Ihe.n with their reirular text-K? ' ^»a .in,i If,,, j: just the thln»f.-G «r«'Mt help to HtnileiitH «r his- «ory. 3 '^"Pl'sli History Note Book and although I fimimmyYivX: Jidieed aKainst Note Booksof any &«»?""'* ^^y "'"* y«»" is the NotP,thL'?'f* carefully selected , wotes that I have seen come from l.'^SiSrf '^'^''V'n^®'' the care of a iw* h!fi.*^*'^^'; they must be a fw- help to students of history.- TH08. rfAMMOND, H.M. Aylmer «iriiMpiBg pointm unppiicd. Canadian and English fllstory a?c .•^"Vrl- This little book is ad l"'™hly adapted to the needs of fhe teacher. Sinn- History can iHJtauffht to y""»K; pni.ils only- by provJdrnJ them ^rith pra-splnfrjwiiltHThcVeby they may rot,. in the Iwld they »et during a l,sj.on. and your%^k nAti, Hiagewny P.S. mi nn Important iil«hr In review. .J.^IV^'^^ to find the Notes on Hlg- i;^VMv^e;^;y^;^esi&rl^ Thb W. J. Gagb Go's Publications. THE PUBLIC SCHOOL ALGEBRA OK THK INUUCTIVn METHOD nv 0. CliARKSON, II.A., Prln. Coll. Instltnte. Seaforth, Ont. Intended a» an introduc-toty Aciies of development lessons to form a ffuide to oral tenchintr and a thorouRh introdnction to larger works. AH delinlttons, all cxpTaii.itious of merely mechanical matters, and all simple examples worknl out ns models are omitted. These matters belonff to the viva voce teaehingr and in a first book of algebra it is com- paratively useless to print long explanations, for they are never read by junior puplK The exercigea are the only parts of much consequence to the learner, and accordingly thib book consists almost wholly of exer- cises. The pupils' previous knowledge of arithmetic is a sufiicient basis and a long li i of abstract definitions is entirely unnecessary. By a properly graded set of questions the pupil is led to discover tlie facts and make liis own generalizations. He Is led to evolve algebra out of arithmetic by careruUy constructed and finely graded exercises, induc- tive questions, comparisons, etc. The guiding principles of the book are these : 1. Follow the Ubo of least reslstaaee. 2. Seek prv:-:*,Ie«I applicatioiiB tnm the boflMilBf. 8. CoBHcet arlthaetie aad algebni as eloaely as pouiUe. 4. latrodaee siaiple testa of mecutwej whorever poaslMe. 6. Avoid all difllealt exaaiples. 6. Grade the steps very eareftilly. 7. Sappljr abaadaaee of review work nad repeat the saaie idea •ader varioas fbrau. order of latrodaelag Poatpoae all dineal- 8. Pay ao att«Btloa to the tradltloaal the topics. Select the easier llrat. ties to a later stage. 9. Sappljr a treasarv of praetleal exaaiplea eoatalnlaf a rich variety of qaestioas. The plan of the book is entirely original. The development of fhe sulject is the simplest yet discovered and the progress of the pupil is proportionally raind. The first fifty pages contain as much as Is usually ^ven in ISO of the common text-books. This book is solid matter. Mo space lost on definitions and superfluous explanations. Short Clear Hints and Suggestions to all the harder examples show the pupil how to begin and what to aim at. The examples are so arranged that the pupil has to work out his own education, but he ia not left without sufficient help to prevent him fh)m making unnecessary sacrifice of time over hard problems. There is no other oook that can rival the Public School Algebra as an introductory text-book. It hag been prMtared with a view to meet precisdy the contemplated atand^ard jor ETinitAKCK EIxaiunations nf Oum A,v|f. JLFTKH IftQ^ Xt wiil liB foimd yif>bt !IQ tO ths i**Mi> i»ly^w*^nfa i q£ the ibis Gbntury. " " ~^ It is Condenaed, Original, Helpftil, and will win its way wherever it is tried. ...y- W tf ■ ^ s ■•■^«^1^ « • "i >•■.,»■'