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A.M NAT. -'UILOSf.Pl.v IN TUB RAhT LSIUA OH.!.,., ilBH"/?' I.KViv.u, IMPKOVf .i.VU SlMVLII-IKl., Dv THOMAS ATKINSON. M. A,, ^*^*""'>I.iB OF COr.f. OU. COLL., CAJtBHlDSR. .front t^; /gi,u$ ll.«:!j>jn Eoittnn, iuitt) abliitfar.i fcg ({)( tSrottrrj of :j(. D. A J. SADLI NEW ^A.^'JL.V\' S'livKl.T. '(V .' : k K No. •^- N'^ )rKi'-l >.\M r :'rKK.:T ■»* .i^iiiSt !^-/7//y//^ H e- ^ ■\ 4>i W ' eg t AN .3KLEMKNTARY Tin^ATISI^ ' ^ ^v OH 1 .iLG^EBRA. • • a • ••!>> < « a • « f Bytiie L^tV; /, ^ RET. B. BRIDajj^kD, P.R.8., .. ^J«J*y/J/«^Ao;;...-.'.-c««M^ j-.'lz^. IUlosoi,h'jimhEa,tLxdla Coll., Ua'tf:>ra BEVISfeD, IH7H0-/3D, AND filMPLIFIED BY THOMAS ATKI&X, M.A,, LaU ScUlsr of Corp. Ch. foil. 'ccvibridQe. i'nm t!,c £au,t imHn eMtian, «.if, .^bMti.n^ l>i, ii;e I3r.ti;cr3 of r ^ '>..j|« *«5 7/ \- * Kew York: n. & J. SABLIER & CO., 31 BARCLAY ST. Montreal: 275 Notre Dame Street. 1S7C, accomp] the man ^ period professei fjropriat cvjnced in comp These p for tlieni ion. This s *Drised ir ADVERTISEMENT. %' % The excellence of "Bridge's Algebra," as an do- ,icntary treatise, has long been ^ well known uid extensively recognised. In the. Preface to the Second ^Edition the author expressly stAt^;, that "great pains |werc taken to give to it all the perspicuitv and sinipli- |City which the subject would a\im of, and to present It m a form likely to engage th'd- 'attention ofyoun<. f persons just entering on their jiWhematical studies'' ihe design, which he thus proposed to himself, was accomplished with singular fem^^,~-foT not one of the many publications on Algebra, which have durinc. m period of forty years issued from the press, with the professed object of producing a more simple and ap- propriate introduction to the study of the science has evinced such merits as justly ent J . it to be placed in comparison with the performance of Mr I3rido-(. These publications have accordingly failed to secm-e flH' themselves the same measure of public approba- This small compendium embraces all which is com- ^'^ ~ " '""--•' ''■^'i/(^ ana cu-jjcnsive editions, that ^- Iy ADVEimSEMEXT. ether practicallv useful or theoreticn;iy valuable By .ntroducng Eyuation,, and Problems at e e "" . .est stage po.ible, a nove and in.s.r X V fure »n.ch the eCtor is ,,ersuaded eannot fail to ex to tu ounos,tyand stimulate the ardour of the yom?,W l-raist, so as to induee him to pursue his studio t f. t more than usual alacrity, intelligence, and tc ol- nas been g,ven to the work. A great ;ariety of"! tSdt r' ""';=T P™!':^'"^. which are'^not Z amed m former edit^ors, have thus been interspersed the several chapters.;; Besides these addition'm^ t. m.ty of arrangement, or of rcnderin<. ,he subiee. boShatlis';Ir'^''''-''*">""' ""•^■'' -' -""-' bope hat h,s ilc,n,„M-y iVeatise on Algebra' would h"d Its wa.y ,nto <^;P>,Uk Schooh; where it vvas .very well known, tWs branch of eduction was hi but little attended ts-i' and if ,V „ "" ^™H'hen] honprl tl,„t.i,- ,.'. " "°**' confidently Goa»oBD, December, 3347. ^' A. y valuable, at the ear- e feature — ' excite the oiing alge- ■udies with success — ty of new, not con- terspersed ons many ic of uni. ic subject ^ds. Mr. vithout a '«' would e, it \va3 as [then] nfidently ice of its to many ^ntlj at- ing tills n. T. A. CONTENTS t*aAPTKR r DEPrN-mONS. P*®* •^HAP. J I, HnAi, On the Addition, Subtraction, Muitiph^tior; '/uij 'l,-;'. " sion of Algebraic Quantities.... , Addition *' Simple Equations '\\ ^ Ou the Solution of Simple' Equation" ' .1 iVoblems "-' Subtraction .. . .!!.'!'.".'*'.'.* ^*"^'^ On the Solution of Sirn*ple*EquatioM .*.' 11 Multiplication ^^ On the Solution of SimplVEqnVuons; *.'.'. ]! Iroblorns. -° Division. _ _ -9-35 HI. On Algebraic Fractions . ^ "^ ^ \^LtT7^ "^"^ 'M^iipiicati;,;: ;;.;,■ d^: ^ ' sion ot fractions On the Solution of Sim^e Equaiions'. ." ^ Problems f55 On tlie Solution 'oV Siniple 'E^uat.c more unknown Quantities.. 61-Ct} .ons, containing two or Probl ems. 60 '^a-vr. \\. On Involution and Evolution ^^'"^^ '^^.^:^,^ ^-be. anj-sim;^;':^,;^; '' On the Involution of Comnnnn^ * \ I. !„V_'- L' V.' "H '^^ oa the Evoiuto of Aig.srQu^s;::.^!!!!!':?:;; ,'j Tl CONTENTS. PA«t Oil the Investigation of the Rule for the Extraction of the Square Root of Numbers 81 Chap, V Oii Quadratic Equations. . . , 83 On the Solution of pui-e Quadratic Equations. . , , 81 On the Solution of adfected Quadi'atic Equations 85 Problems On the Solution of Problems producing Quadi-atic Equa- tions .*. ...92-93 On the Solution of Quadratic Equations containing two unknown Quantities 98 Ciup VI. O41 Arithmct'c, Geomotric, and Harmonic Progression... 10"2 Problems 105-109 On Geometric Progression 10*» On tlie Summation of an infinite Series of Fractions m Geoniotiic Pi'ogression ; and on the method of finding the value of Cu'cuhiting Decimals 113 Problem , 116 On Harmonic I'rogression 117 Chat. VII. On Permutations and Combinations. II9 AJ'i'KNDix. On the Different Kinds of Numbers 125 On the Four Rules of Arithmetic 126 On the Two Terms of a Fraction 127 On Ratios rdxd Projx)i'tion3 128 On the Squares of Numbers and their Roots. 128 On the Factors and Submultiples of a Number 13G On Odd and Even Numbers 131 On Progi'essions .......131 On Divisible Numbers without a Remainder. iaS Properties and various Explanations. 181 Mi*cellaueous Problems. , 187 4 BRIDGE'S ALGEBRA CHAp-pjilR I. .* * • * DEFSNKTIONS. pressed by melns of Written S o.- Zboh Th^ "'", "" u^d^todono. „u.nbo. o. ^UUies^a^'te I^.t:« semed by the/., lette. ofl^^^^X^^r' 3. Unknown or undctermlhH quantities' aro „«, oil * pressed b, the to. letters ortlii>,%abet "as":, ^X:'' 4. 1 no m?//W^5 of quantities thnt ;« +k^ z . quantities are ti be i^l^S^&'^X^'Z^TfJr' 34- sl: "Th""" f^^ P'-i"S. timbers Tefore'thi/f Is "C rtear„ tfr ^ ^"-"''' » '^ ^^a^l^^Se^^ .Vu^ .« = G«. This ,vn>bo. ist„ed the' tV4«%.'"*'' '" rore-&^+pt»:t''ir taT'i'^^ ^ siiinc th ntr as 5 • and a ^ h a. 1 , -"-""^ »^ + 2 is tiie ., whatever be the "fl^e^of at^'aXr " "^ °' "' '' ""' titios^r^^i^neltoSEl^Xhai:;^^ ^-?^<' --^ or^ or ././.,./«.«,,/ quantities ?_3. llmv i« LiL, f''*"^"''.*^ rei>rcscnt Aown •-6. WhatKs the sign of equality ^-o. W In \TJtror urt"'+?'^ ' ^^LGEBRA. 7. Tlie sign — (read minus) signifies that the quantitv tc which it is prefixed is to be subtracted. Thus 3 — 2 is the same thing as 1 ; a — b means the difference of a and b, or b tal.-en from a ; and a + 6 — .r, signifies that x is to be sub iracted from the sum of a and b. 8. Quantities which have the sign + prefixed to them an* called positive, and those "which have the sign — set before them are termed tiegative quantities. When there is no sign befoi e a quantity -\- is understood : thus a stands fbr + a. '^ 9. The symbol X (read m^oj' is the sigii of multiplication. and signifies that the quan^itje^ between Avhich it 'is placed are to be multiplied togethcji-; - Thus, 6x2 means that (J IS to be multiplied by 24,:aud a X b X c, signifies that a, 6, c, are to be multiplied together. In the place of this symbol a dot or full-point h min used. Thus, a.b.c, means the same as (a X 6 X c. T'h©. product of quantities repre- sented by letters is usually ^fekpressed by placing the letters in close contact, one after another, according to the position in which they stand in the alphabet. Thus, the product of a into b is denoted by ab ; of a,, ^ "and .r, by abx ; and of 3. a, .r, and y, by '6axy. """<> 10. In algebraical computations the word therefore often occurs. To express this \vovd the symbol .*. is generally made use of. Thus the sentence " therefore a -{• b is equal to c + G?," is expressed by " .•! a + i = c -f c/." •EXAMPLES. Ex. 1 . In the algebraical expression, a 4- b — cAet a — Q i = 7, c = 3 ; then a + b - c= 94-7--3 = 10-3 = 13 Ex. 2. In the expression ax ■}- ay ~ xy, let i — b x — % y n= 7 J then, to fnid its value, we have ax-\-ay -xy— 5x2 + 5x7-2x7 =. 10 + 35 — 14 ^45-14 = 31 M :, ii ':):*'''^^' rc:ia f — o. Wiuuirt iiioaiiL by wwrni-e nnd whal oy mgative qunnlitiea i—^ 5X3 (2) ?l±i.= ?i<_?_!^''^' 8a -3/; 8X2 -^Vs IG — "~ 7 ~ ^• Ex. 2. If « = 3, i = 2, c = 1, fn,d the numerical values of (1.) 8r/ 4- c -^W.9. 10 (2.),-+"'-" (3.). 3« + b~bc' oJt -f- ac—bc ir Ans. 1. . 7 Ans.--. 8 2ab—2ac + ic ' of tnnes, the product is called a now.r nf fh. ..,„..tZ. __^ J "- - — • jtutiititV. 11. By wli.it symbol is d......^., division over expressed in any other ;«'">?/• of u quantity? ^ ivision denoted? W] ninnncr?— 12. Wi int is its nnmc !i«t id menut i 7 tli« ALGEBRA. 13. Pnnmrs are usually denoted by placing above the quan. tity to tiie right a small figure, which indicates how often th€ quantity is multiplied into itself. Thus, a - - - - X\iQ first power of a is denoted by a (a'). «Xa- - - the 2d power or s5'?/are of a " a*. «XaX« - the 3d power or CM^^- of a " a\ « X a X a X a the 4th power of a « a\ The small figures \ \ \ &c., set over a, are respectively called the index or exjponent of the corresponding power of a. 14. The roots of quantities. are the quantities from which the powers are by successivf5.t«vltiplication produced. Thus the root of the square niim|?<;r46 is 4, because 4x4= Ig' and the root of the cube num-b^o: 27 is 3, since 3 X 3 X 3 J 15. To express the roods' }^ quantities tne symbol ^/ fa corruption of r, the first le#f.>in the word radix,) witti the proper index, IS employed. Thus, . ^ W or y/a, expresses f he square root of a. /« " '• " •'•".thec^/ierootofa. y« " « "■"■fhe/or«-^/.rootofa. ■«l EXTAMPLES. Ex. 1. If a = 3, 6 = 2 ; then a« =3x3 = 9 a^-^^ 3X3 = 27,4^ = 2X2X2X2 = 16. •'^ « - '^ X , ^^x. 2. If g = 64 ; then -^/a = ^'64 = 8, \/a = V04 = ^"4 X 4 X 4 = 4, Va = V64 = 2. Ex. 3. In the expression ax'' + /;» 6:^^rrc'^^ta=3,i = 5,c=:2 What is the numerical value ? Here ««' + 4' = 3 X 6 x G + 5 x 5 = 108 + 25 = 133 and />.r - a« - c = 5 X 6 - 3 X 3 - 2 = 30 - -- 2 = 1 ii ax'^ 4- h bx -—a^ — c 133_ 19 IK^wJ^rf^^TT*''"'",''*'''^^-^*- Whntaro tl.o roots of qunntities ?- "^lilot^^l^SllT ''''''''' '''''''''''''' ^^P--^^ ^Virat'irtbe ovc the qiian. how often the respectively 5 power of a. s from which luced. Thus, 4X4 = 16, X3X3^ ymbol -/, (a ix,) with the fa. 9, a' = 3 X a = V04 = = 5, c = 2 - 25 =^ 133, D - 2 = 1<) DEFTXITXOXS. £x. 4. If a = 1, i = 3, , ,^ 5^ ^^ ^^ ^^^ ^j^^^ .^^^j^^^^ ^^ (1.) «= + 25^.c:. . Ans. 2. (2.) a' + 3b' _ c>. , „ (3.) «• + 24' + 3c' + 4d'. ^„, 9j (4.)3a'i-25'c + 4o'_4a'A ^„,' ij,; ■^ 3 ^ 3 ^ 3 • -4'«. 51. Ex 5. Let « = 04, 4 = 81. = 1 : M the values of ' ' '■ ^''+V^"- An. n. (2.) v/a+^i+y,. ^,_^_jg (3.) V^ 10 When several quantities are to bo taken as one onan tj^!,, they are enclosed in bracket,, as ( ) M f 1 semed tt +1 '^ ■ ^'' -. \ ^'S"'«- *••" *e qua. tity ri J. = 12. '^^' *"'' •■• ('' + ''-<^)-(<'-0 = 4x3 .1.-fltcnVel;ts2nt';rr\;«t%:,+iV;i-'>,t- -O- (1.) (a+4) . (.+rf) = (3+2) . (3+5)=5i8=4"o (2.) ("+4) <:+rf=(3+2) 3+5=5x3+5=20. (3.) a+6c+,fc3+2x3+5=3+0+5=14 iectuely. Thus, a-i~c is the same as a-(b-c) ofi^fr^^io^tlT-r:,:.?/. "--- ™^.de"on,iuat„r ___ " '•-' ' "' ^ ^''^'■'^ oi viiicuium, cor. llnl^vSr"orn!;rtt r ^'"•"•'^^ ^-^^- ^hnt ,« a t ^ardcda.a:rue;;r„;;'° '^"■"•^•••^^-r "nd denominator i.H a vinculum f Mny tlio f>t'u liucUoii be re- r, ALGEBRA. responding, in fact, in Division to the braokct in MuUlpVca, Hon. Thus, implies that the whole quantity a-f- b-^ i 5 is to be divided by 5. 18. Z^•A•e quantities are such as consist of the same, letter or the same combination of letters ; thus, 5«, and 7«, 4«A ^nd 9a/;, 26.c^ and 66.r^ &c., are called like quantities ; and 7//*- like quantities are such as consist oi different letters, or oi' clif jerent combinations of letters ; thus, 4a, Si, lax, 5bx' &o are unlike quantities. ' ' » 10. Algebraic quantities have also different denominations •according to the number of terms (connected by the sic^n + or — ) of which they consist : thus, _ a, 2b, Sax, &c., quantities consisting of one term, are called mpiple quantities. a4-.r, a (jtuantity consisting of two terms, is called a fjinomial. bx-hy-z, a quantity consisting of three terms, is call-d a trinomial. CHAPTER II. ON THE ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVI- SION, OF ALGEBRAIC QUANTITIES. ADDITION. 20. Addition consists ii. collecting quantities that are like bUo one sum, and connecting by means of their proper signs those that are unlike. From the division of algebraic numv titles into positive and nef/ative, like and unlike, there arise Uiree cases of Addition. Case I. To add like quantities with like sif/ns. 21. In this case, the rule is "To add the coefficienls of the Bevcral quantities together, and to the result annex the com- 13. Wl.at are lay nnrt what aro iinUh qnntititles ?_19. Wliat is a oimnL quant. ty? \\ Unt hnhho,»lal and wl.at u //v;«.>,;»;«rAo. fwh^ Si tudaion ot al-ebra consist ? Iiuo how many casos is ic divided i ADDITION. in MuWiplkcu ntity a-f-i'v— < mme letter., or 7«, \ah and ic3 ; and vn- ers, or ofdif 'X, 5b£\ &e.» enominations Y the sign + M, are called Js called a I, is calkd 8 AND DIVI- ;hat are H^-e proper signs braic quan- there arise icnts of the X the CO lu- II at is fi simpU III what dc«« led^ «ion sp, and the common letter or l-.t'ors •" f . -f • • i from the comrnon principles of AH Im ic if +o / '! ^"^-"I be -153^ ' -^^ ' ^' "^^^^ ^^g^ther, their .mn must Ext ]. S^-f 2a— 5i 4^-j- 8a— 9^+ 4a- 5x+ 7a — Ex. 2, c __Ji+__^/-. 2bc ^^£+23xi^m^ Ex. 5. '7a'-3a'b^2'>b'-~3b' 4a8- aV>+ oii_ /,3 «'-2a''6+3a62__5// 5a»-3a7>4.4a/;2_oi« Ex. 3. 4a'- 3a-4- I 2a'*— a-'-f]7 5«'-- 2a24- 4 3a«- 7a*-|- 3 loa^^— I4a-.f35 Ex. 6., 2^V--3^+ 2 4^V-2^+ ) 3.ry- 5^-1- 10 xy— ar-j_]5 ,u,ttrter4S^^,ob--d tha. s„,.e of .ho always undentood STZ.A r "u""' "«''•" «■• 1 " Ex. 2, wo say, 1 + 1 + ,1^9+7 fo^P- "if >»' '^"''™» "f ^'+5-19; andsoofthtri "' ">"'»""«', 2+1+., Case II. To add like qnanfUics with unlike si<;ns. mfrlvf''^^ ^''^ compound quantity a4.b-c4-d . « • positive or negative, according as the turn oT f f ' '•.' '' torms is greater or less than thS sum of * Will ii.i Ai-2 . 92. Sento tl,c rnio in ,!,„ ,,, „^ I 8 ALGEBRA. Serins above the negative ones is 2a, and k the latter 4/.' Hence this general rule for the addition of like quantities with unlike signs, "Collect the coefficients oHhe positive terms iuti^ one sum, and also of the negative; subtract the Zm^r of these sums from the ffreater ; to this difference, annex the sign of the greater together with the common letter or letters, and the result will be the sum required." If the aggregate of the positive terms be equal to (hat of the negative ones, then this difference is equal to 0; and com. scquently the sum of the quantities will be equal to 0, as in the second column of Ex. 2, following. Ex. 1. 4ar- — 3x--|- 4 ^x^—hx-\- 1 7.i:-^+2x— 4 ~_^4.r+]« 1 Ix'^— 9.r4- y Ex. 2. — 7ai+3^;c— xy — ab-\-2bc-\-4xi/ 3a6— ic+2jry —2ab+4:bc^Sxi/ 5ab — Sbc-^ xy -~2ab +3x// Ex. 3. -fxr'-f 13.r' — 2.r'— 4r'^ 7.r»+ .r' 9.c^- t4.i-^ -133:^-2.y'- — 4.r*— (),r^ E\. 4. 4/''- 2.r+3// — x^-{- 4x— y 7a;'— x+^y ^x^+2\x~ 2y Ex. 5. 5a»— 2ai4- /^« ~a'+ ai--2i' 4a='— 3a-f- i" 2a='+4ai-4/'>» Ex.6. 4.iV+2a7/— 3 ■ xhf— xy—\ Sxy-}-4xi/-^5 ■9.ry-2./-y4-9 Case III. ^ 23. Tiiere now only remains the case where imlike quan- titles are to be added togethei, which must be done by col. lecting them together into one line, and annexing their proper signs ; thus, the sum of 3jr,-2a,+56,-4?/, is Sx-2a-\-5b-. ~-' ' r ■ ■■^'■-■- «ii-^ z.!:tiftiv 4Ljaiini,ius are rnixeu to. 23. State the rule in tlio 2cl nnd 8d casos. 1 f% '•"mm the latter 4^)' quantities with itive terms into 3 lesser of these !X the sign of letters, and the fual to fhat of to 0; and f.'(>;i. Lial to 0, as in Ex. 3. — 4'Z-().r^ Ex. 6. U'y4-2A7/-3 XY— X1J—I •2.;'jy+9 unlike quan- ! done by col. J their proper re mixou to- SIMPLE EQUATIOXS. q gether, as in the foJlowin Ans. x=i 0. f eqiifirions, nncj . \\ hat is 11 ttiui- the suLjoincd cvaranles h, 1 1, ?^ , •'' "'"^ " '"■" ^'« ^<'™ I" '" " -""ci : as jor Histi i^ ALGEBRA. (.V,) Dut the ost of 5 lbs. by the .juestion=30*. M.) TFicrefore, the price of 1 ]b. in shiHings X5=305. r/nilt. '^'''^'''' "•' ^'"^'^'"^ ^^ ^' ^'" ^^*^'" tJ^« F'*^ Thy ...v-ral steps of this soluti. a expressed a].rebr-,If...!iu Hould take the following more compendious fonn-"^ (] .) Let a;=the price of 1 lb. in shillings. (2.) ITien 5.r=the price of 1 lb. in shillingsx 5 (3.) But the cost of 5 lbs. is by the questbn^GO,?. (4.) .-.bxzzzSOs. Jvl"'"^ ''■"^'^'' ^^''^ ^^ '^' P™^ «^ l^b., as was re- It will be seen by steps (2) and (3) of this example that here are two dist.nct expressions for the same tl^S^' and hat m step (4) these expressions are made equal fe' each other. In Iraming equations from problems, this wil in all .ases take place. As a second exaSiple let this p'blem be A house and an orchard are let for£2S a year, but the rent oHhe house is 6 tmies that of the orchard, -"l^'d the rent "f The rent of the house is equal to that of G orchards- we maj^ therefore change the house into G orchards, and w^shlu Tnkn.u the sum of the rents of the orchard, we get 7 times the rent of the orchard=£2S and the 7th part of each side of the equation being t^kvi^ The rent of the orcnard=£4 • ui:i /. the rent of the house=6 times the rent'of the orciiard =6 times £4 INow to i-ve to these operations an alffebraioal shani> i-;et -z^i'ie rent oj the orchard in £ * ' then 8.. . '' « j^ouse « hiMPLE EQL'ATIOXS as was re- But by the condition of the Hent of the orchard question times ri o/^he orchard— £28. or, 7x=£2b. «iHl, dividing each side of the equation by 7, x=£4, the rent of the orchard and ... Gx=Gx£4=£-24, the rent of the house - ..h^.r., .hat 'onr-pa.^^^; tells r ^^ "^^ h> 9. A person unacquainted with alcreb.a mllu ^Z .;.^^rcat_dimcu,„. s„,vi this quosLfi.'-L'"'!!;.?..;,;;^ (2.) The greater part .nust exceed the less by 9 (ii.) But It IS evident that the greater and less n^-ts ,,i i i iogother must be equal to the whole number 35 ^ ^"'' Jii^^f^ s^i^:^ ""''^ "-^ '<=- P-' with the (T.) Hence, twice the less part is equal to 26 2,^'iZX af ?;:!olf' ^'^' """ *^ ^'''^x"-' - «9ual to oJi^i^:^^^, '"» ^™'- P-t -eeds the .... Lut hy adopting the method o/ algebraic wo/^/Zn,, ft r.r (1.) Let the /f-ss part - - . __ (•^.) Then the greater part =a-4-9 (3.) But greater part+1 part =3; "" ■' lum-mt^j^y^^i^ ^T-^.- ' I ll II i i ln i K wii J U ALGEBRA. iliii l!ll (4.) ,.-. a;+9-4-.r (5.) or2.r-j-9 .... (6.) .'.2x (7.) or2a; . . . (8.) .'.X {less part) (9.) and x-{- 9 {r/reafer part) =35, - =35. . =35-9. =26. 20 =^=13. - =13+9=22. 29. Hjivmg tlnis explained the manner in which the several 5|-eps ni the solution of an arithmetical question may be ex- pressed HI the language of Algebra, we now Droceed to its ex- emplification. PROBLEMS. PROB. 1. A dessert basket contains 30 apples and pears but 4 timps as many pears as apples. How many are there 01 each sort ? "^ Lot a;=the number of apples ; then, as there are 4 times as many pears as apples, 4.r= the number of pears. But by the question the apples and pears to(rether=30 .-. a,-+4^=30. .■idding the terms containing .r, 5.r=30. Dividing each side of this equation by 5, x=G, the number of apples, .'. the number of pears =4.r =4x0 =24. TRon. 2 In a mixture of 10 lbs. of black and green tea, there was 3 times as much black as green. Find the quaiititv Ot each sort. ^ -^ Let .t=the number of lbs. of green tea thcn3x= " « « |,]j^ck^ ' But the black tea -f the green tea = 10 lbs. ^„ . .'.^r 4-3,^ = 10 lbs. Collecting the terms which contain x, „. .^. , 4.r = 101bs. .r=r4 lbs. of green tea, .*. the black tea -3i'=3x4= 12 lbs. s I ,1- t SIMPLE EQUATIONS. 15 Prob. 3. An cqunl mixture of black tea at 5 ihUWucr^ ■, iK «nd ot green at 7 shilllnffs a lb costs 4 .rnhnn! jj^"" '^ "^• lbs. were there of each sort ? ^ ''''' ^^"'^' "'""J' Let _a:=the number of lbs. of each sort • then o.r= the cost of the black in shillings and 7x= « « j^ ,, » ' But cost of biack+the cost of green=4 guineas=:84. .-. 5a:+7.r=:84 12jr=84 .'.a; =7 lbs. of each sort. Prob. 4. The area of the rectangular floor of a school-rpom |s^^^hO square yards, and its breadth yards. ^r^H a; X 9= the area of the floor .'. 9.r=180 ' and .-. ar=:20 yards, the length required. Prob. 5. Divide a rod 15 feet long into two n-irts .<. fl. ,t the one part may be 4 times the length of the other.' ' Let ar = the less part, 1 , 15t> then 4a:=the greater part, '~r' "~ir 1 i>ow d each man's share =2 time's 3x=Qx; )m that of tli« 1 as A and B SIMPLE EQUATIONS. Hence we have, the share of the 4 boys =4. <-''^' «lure of the 3 wo,n.n=37inios 'U o ,^ and the share of the 2 inen -o , ' ^'^^■^ ^iit the sum of nil tL 7 ^"^".=^ times Gj:=12x : sum of all these shares is to amount to £100 • ..4.r-f9.c+12^=£100, ' 25^ =£100': .*. each boy's share — :tU. nn.h , =/]'> wl":fl' t ^^^oman's=z3 times £-1 -^^- ' ^^^d each man's = tiiues £4=i:o4 «hm 'S' altf t^nr^rtTe''? ' f r p™^ ^^-- «- third thrice, and to thV four h f ? ^' ^''"" ''^^''^'^' ^^ the 'i'-st. Whai did he give to each J'"' "'' '' "^"^''^ ''^^ *« ^^'^^ PKon 14 T.- .. ^"'' '^'" '^^'" ^^'-^ "^''^ respectively.' thatrmfddl^re\,:,l;Tel^bleT T^^ ^^^^'^ ^-^^-"'^ tn-p]^'. the least part. ^^ *^^ ^''^'^^ '^^"^ the greatest ^'^^^^^ 5 tinSih: ^oUr^z H ?^ p«'ti l. -ce between the first ^^^t^::;^!'^^ '"'' li;^ ^> ^^ f ^^^"- -ixture contain^ t ame quant^'iV "' ^ ''^^ "^- ^i.^ "ow man,, lbs. arc there of eS S °' "^^ T*^ T, *'' Pkob. 17. A bill nf ^nn -'''«• 8 ibs. sovereigns, and cmwnsTaS .Z, Ijf , '" ^^^''f'S""' ''"If l-iiid tlio numl)cr. ^"'" ""'"•'« of each was nscd. Paon IS T . „ ^■'"''- 400. ^^.ndfb;d^^;^;™^- ^^ the same tin. n-on, Hks 4 miles an hour and the n!h r^ '"^l'"' ''i'^'^' the on. ^ours will they meet?' "^^^'^ ^ '"•^«^- ^i how manv PiiOT! 10 A , ^'**- ^ hours. luv£lV\u P^''''" ^'^"^^^t a horse, chaise nn I f I'" AUO; the price of the hor^o u^no . • ,' ""'* harncs-. '•■n-ncss, and tho'priee of the 'Sse 11^''?, ''^' ^r'' "^' '^"■ h-- and harness; what ^^:^Zl^:^J!:ir'^ ^^ ''<'^'' 17 .1 nswe)\ •I'l'ice of harness = .£'1.3 <( 8 2* chaise = = 80 18 ALGEBRA. • ■} phi! i I iiij! SUBTRACTION. 30. Subtraction is the finding the difference hctwoen twc filgebraic quantities, and the connecting them by proper signs, so as to form one expression: thus, if it were I'equiied to su})tract 5—2 (/. e., 3) from 9, it is evident that the remainder would be greater by 2 than if 5 only were subtracted. For the same reason, if i— c were subtracted from a, the remain- der would bo greater by c, than if b only were subtracted. X~4 — .?/ * +4^ Ex. 6. From 7xr/-\-2x—Sg Subtract S.ri/~ x-^ y Ex. 7. \4:X-\-y—s- x-^y-\-z- 5 11 Ex. 8. 13.i;8-2.c'^4-'7 iimkL i^ ' iM^i2i Lh a^r^n s^i'. ^V!)at is mihtraction ? Stntc the rule for tlio snbtniotion of algcVniic mMi.ti'io!^, and c.\pl:un the pnnoiplc on which it nsts. MVrLE EQUATIONS. 19- cc Letwocn two by proper signs, Bre I'equiied to t the remainder ibtracted. For I a, the rernain- ere subtracted, sr is a— i ; and the remainder • the subtraction ;he quantities to ' another, as in rhe quantity to h4?/ ; therefore cted fi'om each } row under the ve the signs of id then .proceed Ex. 5. 6// — 4y — a -r -f-4o Ex. 8. 13i;8_Oc'J4-7 .MutUi EM. ruction of algcVraic j OF THE SOLUTION OF SIMPLE EQUATIONS, C0XTA1N'IX(, CKLV OMV UNKNOWN QUANTITY-. Rule II. 31. "Any quantity may be transferred from or.e side of ' ;.e equauon to the other oy changing its sign;" and it il toundea upon the axiom, that "if equals b? added to ^r - equal" '^"'^'' '^'' '""'' ^^ remainders Zm be • Ex. 1. Let ;r+8^15; subtraci 8 from each side of the equation, and it becomes :.+8-8 = 15-8j but 8-8^0 0,,f;iV^^'l-^=3»; f<'7 to eachside of the equation,,, mciii 7+7_,i0+, ; but -7+7=0; .-.^=20+7=27. tnenrf.r-2.r=o^_o^.4.9^_5 but 2.r-2j; = • 3ar-'>r-Qv^ + 5. Now 3^-2^=;., and 9+5 = 14; hence ;:! 14. On reviewing the steps of these examples, it appears (1.) That a; 4-8=15 is 'identical with a;=15_8 (2.) « ;r-7=20 « with a:=204-7* (3.) « 3.r-5=2x+9 " with 3.r-2.r=9-f 5. Or, that "the equality of the quantities on each side of thp - equation, is not affected by removing a quantity from one s ^ of the equation to the o-:her and chauffinr^ its s^nP he Idl out ff \ 1- ""'' ^'f' '''^''' ^^ ^» ^q^'-ition, it n)ay . I'e. lUt out of the equation; thus, ii'x+a=c+n then x-cl '^~a; huta~a = 0,,\x=c. ^tu, men x^c-^- h further appears, that the simis of all th. *««.„. ^c> >n may be changed from -j- to -or from " "in T ""."^'i"'*; «, .hen, by the l.ule, .r^c-a+i; change the siirns of ■) lion e sijijns of alt • mm 20 ALGEBRA. Ex.5. Ex. 0. Ex. 7. Ex. 8. Ex. 9. Ex. 10. -a \-C = Z, or T7: Ans. xz=l4. Ans. rrz=29. Ans. x~G. Ans. a;=9i~-2a. Ans. x=2. Ans. ar=10. A71S. ar=5. -.4;t5. a: =3. a ' Ihe terms, then b -x-,-,, -r, in wliich ca^c b t—a-\-b, as before. Ex.4. 2i;+3=a:4-17. 5.K— 4=4ar+25. 7ar-9=e».r— 3. 4x+2a=3a;+9i. 15^+4=34. 8a;+7=6.r4-27. 9,r-3=4.r4-22. Ex. 11. 17x-4jr+0=3i,'+39. Ex. 12. ax—c=b-{-2c. Ex. 13. 5^-(4.r-G) = 12. The sign »— before a bracket being the sign of the wholf quantity enclosed, indicates that the quantity is to be sub. traded; and therefore, according to the Rule, Avhen the brack- ets are removed the sign of each term must be chantrcd rLiis, the signs of 4.r and of C are respectively 4- and — ,''but when the brackets are removed they must be chann-ed to — :tind -f respectively. The equation then becomes 5^-4.r+G = 12. By transposition, 5x—4x= 12— 6; .-. .r=6. Ex. 14. 6.r--(8+.r)=4.r-(x-10). By remo^'ing the brackets, and changing the signs cf the . terms which they enclose, the equation becomes Ox— 8— a:=4=:4,r— .r + 10. Transposing, Gx~z—4x-\-z=l0-^8 ; .-. 2x=18. .. Dividing botli sides of the equation by 2, Ex. 15. 4x—{Sx-\-<)=S. Ex. la Si;-(Cu'-8)=9-(3-T). Ex 17. 4x~(Sx-Cy)~(4r-m = l9 Ex. 18. 5a:--(8-l-3r) = 8^(-ir— 1). Ans. .r=::12. Ans. x=-^2. - /-.,_ If.. Ans. x~ :2. Ans. x = 12 SIMPLE EQUATICXS. 21 ! sijnis f f the ^9. X — 12. IS. x=. -% IS. ! (it }■ 2. IS. X — 12 PR0JJLKM8. ■ Let therefore x=.m^ Jess number; then will a:+15 = the greater: iJut their sum =59; .-. a;+a;-rl5— 59,' or2.r+15=,59. ^ And transposing 15, 2.r=59-15, or2.r=44; .-. ar=:22 the less number -rxo — ^^-i-io~j7 the greater. J3ut together they receive 27 ; '^''^ * .*. .r+a:+5=:27, .,, . or2.c+5=27. lrans]-/,Kwig, 2.r=:27-5, or2.r=:22; ••. .r=ll, the No. James receiver] .6 -r o _ 1 o Kichard received Let a-rrthe number; then 4.r=4 times the number, . 2.c= double the number, .h.refoie, b;y the equality stated in the question _ 4a;=2ar+12. ^ »,y trjpsposition, 4.ir— 2;r=:]2. Oj. TO . .'. arr= 6. •aon. 4. At a, election 420 persons voted, and th le succcs.* i ill *|!1 1' : ! iilli 22 ALGEBRA. ful candidate had a majority of 4(5. each candidate ? How many vut hrcmlstitLTK T ?"' '" ''''''^ "40 miles in 4 dav. He murg?rmnes he sl"Tf '^'?''^''" ^^""^^ ^^at' fourth da.^ less t ai the S" it^' ' '" '^^'■'' ''^"^^ ^^ ^^e travel each day? ' ^^""^ "^^"3^ ^"'^es must b- then^rllZ^'^^ number of miles on the 1st day , ^-9= « « « ?^ and .r- 14=: u „ ^^ ^^ 4th .^T the number of miles whioh hn I'l a 13 lU aajg ^+x~.5+x~9+x~U~2. Collecting the terms, 4x -28- 2 40. 240. u ALGEBRA. Jjy transposition, 4a;=240-f28, 4.r=2G8; .'. ar=07, the number of miles he goes on 1st Oav, ar-5 = C2 " " " " " 2d " g. 9—58 " " « " « 3(] " and a; -14=53 " " « " * 4th '• Prob. 14. It is required to divide the number 99 into five such parts that the first may exceed the second by 3, be less than the third by 10, greater than the fourth by 9. and Itss than the fifth by 10. Ans. The parts are 17, 14, 27, 8, S3 Prob. 15. Two merchants entered into a speculation, by which A gained £54 more than B. Tlie whole gain was £49 less than three times the gain of B. What were the gains ? J ^;is. A'sgain=£157; B's=£103. Prob. 10. In dividing a lot of apples among a certain num. ber of boys, I found that by giving to each I should have too few by 8, but by giving 4 to each boy I should have 1 2 re- maining. How many boys were there 1 Ans. lOi MULTIPLICATION. 32. Multiplication is the finding the product of two ot more algebraic quantities ; and in performing the process, the four following rules must be observed. (I.) When quantities having like signs are multiplied to. gether, the sign of the product will be + ; and if their signs are unlike, the sign of the product Will be — .* * Tlii.s rule for tlie multiplication of the Signs may be thus ex- plained : — I. If -fa is to be multiplied by +ft, it means, that -\-a is to be added to itself as often as there are units in 6, and consequently the product will be -f ab. IT. ]f — a is to be multiplied by -^b, it means, that — a ih to ho added to itself as often as there arc units in b, and therefore the product }b — ab. f5<>. "What is multiplication, and what are the Rules to be observed In multipl cation i I MULTIPLICATIOX. 25 Mt (2.) Tlic coefTidcnts of the facfor^ mu^t Im r,,.,ii- i- . g«hcr, ,o.brm the coefficient o?Zp'o2cl "'"^'^""' '" forming the operation, the kule i, " Tn n ,u- ? J P° ..gns, ./,« the coefficic'its, J^^^ar^: ^I^H^J-'' '^ r- the ay be thus ex- be observed In Case I. ^^'■il'lTJtX^,:::. «'"^" ^™--"- ^ '■o- wi.ich the U - »J. ""' """* "■ '. ""tl ~n»equently tl,o proiluct Or, those four R.le, ,„i„.,,t b„ ,„ e„™pr„,,„„jej ,■„ „„, . „,^, I J?.s"i;r'/i- '„t;w;L'3r;' :^4'ef."- »^" - *- But a — . /j, arfc?<'(/ c ^•»^^ft«- . , = a^ — ^r and a — 6, su btracte d d times = __ «^ |: f,j i. e., -f- a X + c =-- 4- «/• + « X — (/ = — ad — bX~d = i.bd. Ex. 1. i)a Ex. 5. Aabc 3ac ALG EBRA. Ex. 2. Ex.3. — Oax//* -15aV/'c Ex. 6. Ex. 7. dxhf — 4c*f^ —2?/ 2c Ex 4. — 5a*4c -~_2iV 4-lOaV/cl' >:x. 8. — 7c/.r*?/ — 2f/A /J^-^/b ^^^ ^cW^ Case II. 34. When one factor is compound and the other simple^ ''Tlien mc/i^erm of the oompound factor must be multiplied by the simple factor as in the last Case, and the result wil' be the product required." Ex. 1. Multiply Sab—2ac-^d by 4rt Product V2a^b—Sa^c+4a(f Ex. 2. Sx^ — 2x'-!-4 — 14aa? — 42ax~-\- 28a^?^5«^- Ex. 3. Multiply Ix^ — 2.C +4a by — 3a FVoduct Ex. 4. 12a^-2a' + 4a-l Sx -21a.r'-f-(k,';c— 1-.V r Ex, 5. Multiply dah-^3a—x-^] by — x^ Product ZJ^,^^ ^ y^^t- [ Ex. 6. ! 4a;V + 3.r— 2i> — 3.r?/ Case HI. 35. When both factors are compound quantities, each term of the multiplicand must be mu'tiplied by each t erm of the £x 4. — ha*bc >:x. 8. MULTIPLICATION. 27 multiplier; mid then placing; Ike qnantitu- nmhr ea,\ ot\t» Uie sum of all the terms ^viJl be the produo. required. Ex. 1. Multij ]y a -I- h by a -f h 1st, by a qS^- ah 2d, by i /7A-i -// Product aH^oHhO"" Ex.2. a" * -/^« Ex. a fl. 3 i/c D Other simple, : be multiplied the result wil' J A* /s^* -2a;»-h4 +28a.r^-50c/- Ex. 4. Ex. G. 4a;V+3.r— 2^ ities, each term ach ' erm of the Ex. 4. 3.r*+ 2ar 4.r + 7_ i2?+~a? ±?lfM-14T 12.cH29x^-fT4j Ex.5. 3.C'- 2x +5 •-21.g^-f-14.r-35 18.r''~33.r2+44,r"-^^ Ex. C. 14a c — 3a * -f 2 a_c — ah -}- I -Ua'bc +3a%'^2ab •hiiac — Sab + 2 — !tiZ!f!^±l?^+3a-^A«'^(^:f2" Iv -i-2 Ex. 7. jr + 2^.*^-- ar+4 + IT ~4 Ex. 8. Multiply a'^+3a«34-3a/y^+63 ^y a-^b. 28 AliGEBllA. Ex. 9. Multiply 4x'?/-f-3xy— 1 - - Ly 2.c'— a-. Ex. 10. « a^—x^^x-b - - - by 2x'-\-x-\-l. Ans. 2x'—x*-^ 2x''-l0x^—4x^b, Ex. 11. « 3a«-f-2a6-i2 by Sa'^-2ab-i-lj\ Ans. 9a'-4a^lj*i-4aP-bK Ex.12. « ar'+xV+^y'+y' by a;-?/. -.4ns. x*—y*. Ex.13. " ar'-Jar+l by x'^—lx. A /IS. a;'— |-.c^+y.r— ^.r. ON THE SOLUTION OF SIMPLE EQUATIONS CONTAINING ONLY 0NJ5 < UNKNOWN QUANTITY. Ex.1. 3.c-}-4(jr4-2)=3G. The term 4(a;+2) means, l ,.it x-\-2 is to be multiplied by 4, mid the product by Case 2d is 4j;+8; .-. 3x-^4x-\-8=S6. Adding together the terms containing x, and transposing 8, •7x^m-S, 7a;r=28; • • •C' ■^— Htm Ex.2. 8(a;+5)+4(T-f 1)=80. Perfurming the multiplication, 8^+404-4.c+4=:80. Collecting the terms, 12x4- 44 =80. Transposing, 12.f=80~44, 12x=:30; Ex.3. C(.r4-3)+4.ir=58. Ans. xt^4. Ex. 4. 30 (x_3)4-6=:0 (.t-f 2). Ans. x^i. Ex.5. 5(.i;4-4)-3(«-5)=49. Ans. x=:l Ex.6. 4(3+2.r)-2(0-2.i-) = 60. Ans. x-^5 Ex.7. 3(x-2)4-4=-.4(3-.r). ^«*. .r^^o; Ex. a 0{4-a-) -4(G-2,r)-12=0. Ans. a-^O. SIMPLE EQUATIONS. 29 riNO ONLY ONfi PR0DLEM8. suin shall be 35 ? '''''^ ^^ ^ ^""^« the less.. tJie Let ;r=:tho less numLer • \n^ q/- ,,^^^"-^+9=thc greater. ' ^^.t by the problem, 3 times the greater + 5 times the le.. .-. 3.r4-274-5.r=35, . 8.r+27.-=35. Iransposmg, 8.r=35-27=8; .-. ,-r=: 1 , the less number, and .r+9=10, the greater. .'•"d in order to do thif o'"i. ol w''"* ^^ ^^^'^^'^*^ ^^i"^- ^-r: |uho.man;t,i:j::^K,:-^ then^tS"" """'" ^'^?"" ^^ ^^^-^> land 7(.;+5)'^'!^' """"^^'^ "^"^''^^ the "M " |num^.'^i,:'fP^^'^'^'" ^'^ ---,;; b!^LrLl the same .'. bi-=7(^+5), ^ 12.c=:7,c+3o. iransposmg, 12.r-7.r=r35, ^on-ier is in overtaking tL'f^'sf ""n.b.r of hours the .ocu,rl kJlr^t^etLtS^;;^^^ Mow- nnmy passengers were ty/c oTleh dllf "^'"'' ^^^ Let «= the numl)er of nas«nnr,<>.., 6a' = 4>1C* ;-vi* f 1-.- "i ,,i. _» kn.J . p . , " 2d " Mi 3* tiutmitmi SiSkd pl! 1' hi 1 1 (■■ 1 5<' ALGEBRA. But these two'sums amount to £3 12s., or to 72«. .-. 6.r+4(15-rr)-72, 6x-\ (30— 4.r=72. Dy transposition, Gu;-- 4a; =72— GO, 2.r = 12; .'. x = G No. of 1st class passengers ; /. the number of 2d class pas8engers=15— a;=9. Prod. 4. What number is that to which if G be added tvvic€ the sum wi'l be 24? Ans. C. Piion. 5. What two numbers are those whose difference is 0. and if 12 be added to 4 times their sum, the whole will be GO? ' Jns. 3 and 9. Prod. 0. Tea at 6s. per lb. is mixed with tea at 4.9. per lb., and 161bs.jof the mixture is sold for £3 18s. How many lbs. were there of each sort 1 -.-Ins. 7 lbs. and 9 lbs. Prod. 7. The speed of a railway train is 24 miles an hour, and 3 hours after its departure an express train is started tc iHUi 32 miles an hour. In how many hours does the cxpres;-' overtake the train first started 1 Ans. 9 hours. Prod. 8. A mercer having cut 19 yards from each of three equal pieces of silk, and 17 from another of the same length, 0)nnd that the remnants taken together measured' 142 yard^. W hat was the length of each piece ? Let a:=the length of each piece in yards ; .*. X — 19 = the length of each of the 3 remnar.ts, and .r— 17= the length of the other remnant; then 3 {x-l9)-^x-]1=U2, o^3a;-57^•a:-17=:142, 4.r-74 = 142. Transposing, 4.r = 1 42 + 74, 4.c=216; .', .'rr=54. Prod. 9. Divide the number G8 into two such parts, thntl the ditference between the greater and 84 may equal 3 timesj the diubretit* Dotween tno less an.u 40. Let ir=the less part, then G8— ar=thc grsfiter; SIMPLE EQUATIO: ^, ^ . |4oi:lLi't:ri^!,v^!i«^ ?■■ ^ «.o ...at.. 31 n.n b, '-."X^^e^^f^ - e,ua. . each .W . By transposition, ^+3^=120+68-84 4a;=I04; ' and.-.thegroater=42:""''''P"''' to two'upon etl^Tal "CL"' <=^.■•'^^ Netted three shillm«, •tags. How ma„; deais mV^^^f "'"'^ "« ^™» Ave sli ..•:0-:r=the number he lost; iosfwafl,""^^"-^^ ''^'-» *e -no, won and the money .•.2^-3. (20-.r)= 5 2^--60+3ar= 5,' 5x-.Q0= 5, .'. «=13. to i^iff Lt 'tfan^t^^^^^^^^^ P^^y^^l packs of cards so as off twice as many as B ill and R .T ^ happened that A cut '^'"^|;^=the number he'lefl, . 4 ^^=^,*Jo number B left; 3"t the number Bci^n^ """'^^^ ^« «"* <^ff^ ' ^ ,,ft . "^^^r B cut off was equal to 7 times the number .••52-a.=7.(52-2z) 52— ar=364--14ar transposing, |4;r-ar=364~-5o la^=312; ' a:=24 i Aou off48, and B cut off 28 cards. A^-ms m S2 ALGEBRA. pROB. 12. Some persons agreed to give sixpence each to a waterman for carrying them from London to Greenwich; but with this condition, that for every other person taken m by the way, threepence should be abated in their joint fare. Now the waterman took in three more than a fourth part of the number of the first passengers, in consideration c.4* whicli lie took of them but fivepence each. How many persons were there at first 1 Let 4x represent the number of passengers at first ; then 3 more than a fourth part of this number = a: -f 3, and they paid 3 (a? +3) pence. .-. the original passengers paid 6x4a;— 3(a;+3) pence. But the original passengers paid 5x4a: pence ; \ by equalizing these two values, we get 6x4a?-3 (a:4-3)=5x4ar, i 24a; -3a; -9= 20a;. Transposing, 24a; — 3a; — 20a; = 9 ; . • Xzz^tj ; and .'. the No. of passengers were =4x9 =30. Prob. 13. There are two numbers whose difference is 14, and if 9 times the less be subtracted from 6' times the greater, the remainder will be 33. What are the numbers '? Ans. 17 and 31. Prob. 14. Two persons, A and B, lay out equal sums of money in trade; A gains £120, and B loses £80; and now (\.'s mo^iev is treble of B's. What sum had each at first 1 ^ Ans. £180. Prob. 15. A rectangle is 8 feet long, and if it were two feet broader, its area would be 48 feet. Find its breadth. Ans. 4 feet. Prob. 16. William has 4 times as many marbles as ITiomas, but if 12 be given to each, William will then have only twice lis many as Thomas. How many has each % Ans. 24 and 6. Prob. 17. Two rectangular slates are each 8 inches wide, but the length of one is 4 inches greater than that of the othw. Find their lengths, the longer slate being twice the area of the J each to a nivich ; but taken iii by joint fare, th part oi' )n (.f whicli jrsons were first ; =x-\-S, and 3) pence. rence is 14, the greater, rand 31. al sums of ) ; and now at first 1 ns. £180. ere two feet adth. IS. 4 feet. 3 as ITionias, e only twice 24 and 6. inches wide, of the othcj'. e area of the SIMPLE EQUATIONS. 83 hZZf"' ^'''' ""^ ^ '''*^^^' '' '^ ^'"S^^ multiplied by its Rnf :i: ^f ^"^ ^ ^'^'t'*) ''''^ ^^^ ^^^as of the slates. But the larger slate is twice the area of the less .•.8a;X2=8(a;+4), 16x=8x-{-S2; .'. 8ar=32 ; .••ar= 4, the length of the less slate, and 0^+4= 8, « " « greater slate. h 71' ^f'.x^"'"' rectangular boards are equal in area • the breadth of the one is 18 inches, and that of the other 16 mches, and the difference of their lengths 4 inchef F^d tJe length of each and the common area. p ,^ , . , ^^*- 32, 36, and 576. Frob 19. A straight lever (without weight) support^ in equilibrium on a fulcrum 24 lbs. at the end of'the shorter arm and 8 bs. at the end of the longer, but the length ofthTS r .1. %r^^' ""^'^ *^^ ^^^t «^ the shorter. Find thi lengths of the arms. " ^"® Let ^=length in inches of the shorter arm • thenar+6= - .« » j » ' Now the lever will be in equilibrium, when the weiVht at one end multipl ed by the length of the correspond i^Sm ?s Xg InnT"' '' '''" "'' "^^'^^^^^ b^ it--reL .-. 24a; =8 (x-\-e>), 24a; =8^+48, 16a;=48; V f =3 -inches, the length of the shorter arm • anda;+6=9 " " » « longer '^ * Pkob 20. A weight of6 lbs. balances a weight of 24 lbs on H lever (supposed to be without weight), whose lenSh is 20 .r^hes ; if 3 lbs. be added to each weight;, what addittn must be made o each arm of the lever, so%hLt the fulcrum mav Ed? '^'''"^ ^'"'""' ^"^ equilibrium Btm be ;i: This problem resolves itself into two other problems :~. :■..■ --.-.uru uic; i.:m,r,r,.=. ..r rr.. „„ j^ ^-^^ Original posi. »• f°"»-tom.draWyfr„„ tl.at in Mul.ipli If +«X+4=+„S. then +1*=+J, „„d ±±=+^ 4 a X-b^-ab, - - - =1^ =_6. and =11* ^^a — aX~6=+a5, - — a »■ *., /j/lrff eigna • produce-f , and unlike signs — ^. ■f -L iltj ButtaXLT"" '''""' "'"'""" o'lS"""^" 9»"ti,i«,( s,... th. SQ ALGEBRA. In the divisor must be subtracted from its index in the dividend to obtam its index in the quotient. Thus, (1.) +abc divided by +ac - . or ±^ =4-5 4-ac (2.) +Qiibc - . - - . -2a - - or (3.) —lOxi/z . +5y - or 6abc ~2a — lOxyz +5y = —36c. =z—2xz. (4.) ~20a'A3 4a^y or =|?.^' =+5aa:/. Of Division, also, there are three Cases : the same as ia Multiplication, ' Case 1. 37. When dividend and divisor are both simple terms. Ex. 1., Divide 18aa;« by Sax, IQax'- =6a?. Ex.2. Divide 15a'b^ by -5a. + 15a''i« 3aa; Ex.3. Divide —28a^y^ by — 4zy. Ex.5. Divide — 14a^6'c by 7ac -14a«6'c — 5a = -3a6«. -h7ac Ex.4. Divide 25aV by —5a»c. +25aV — 5aV "~ Ex. G. Divide — 20a;y2' by — 4y«. ~20a;'yV — 4y2. Case II. 38. When the dividend is a compound quantity, and the divisor a simple one; then each term of the dividend must be divided senaratelv. and iha -refinlfina- rmonfiViVo «,:n v^ i.i-_ quotient required. 88. State the rule for Case 2d. DIVISION. Ex 1. Divide 4iia»+3a5-f-12a« by Sa 42a«4-3«A+12a» 87 3a - = 14a + i+4a Ex. 2. Divide 90aV-18a^»+4a'^~2ax !jy 2a*. 2<.'a; =45aa;'--9.r-i*i*«-|. Ex 3. Divide 4x*-^2x'-^2x by 2^:. Ex. 4. Divide — 24a«a:V— Sflrary-f-Gzy by ■ —24a^x^y—Saxy -f- Gary — 3a:y ~~" -aty. Ex. 5. Divide 14ai»+7a'6»-21a'i»-f.35a«5 by 7ab Hob "— I 11 \ Cask III. titits' Tn^T• ^'"^'^^"i and divisor are io^A co^j.o.mc? quan. titles In this case, the Rule is, «To arrange both dividend and divisor according to the powers of the sime letter Wn mng with the highest; then find how often the fim tirS he divisor IS contained in the first term of the dividend and place the result in the quotient; multiply each Irm of the «h;; n^ andplacJe the p?oict unlTth'e'cor! it fi-om the^'. 'to tt^ terms in the dividend, and then subtract ^c .^'^.r'fV' K *« the remainder bring down as manv f^rm» V. .uu uiviaenu as wiii make its number of terms equal'to AnVr ^''^''^""^ "°^ '^^^«°'- '^^^ both compound quantities, what u. : 1 1 lAej O } oO ALGEBRA. that of the divisor; and then proceed as before, till all the Ex. 1. Divide a^-Sa'b+Sab'^b^ by a—b. * * In this Example, the dividend is arranged accordincr to the Powersoft the first term of the divisor." HaX do'ne this we proceed by the following steps :— ° ' (1.) a is contained in a\ a' times; put this in the tient quo- (2.) Multiply a-i by a\ and it gives a'^a^'b. JSJ^ Subtract a^^a^b from a«~3a'6, and the remainder is (4.) Bring down the next term +3a5'. («.) JVM7/?>/y and sM^^rocjf as before, and the remaind ab\ er IS (7.) Bring down the last term —b\ ' JS.) a is contained in ab\ +5« times ; put tliis hi the qua iicnt. (9.) Multiplif and subtract as before, and nothing remains PIVISION. 89 •■•f2tfH-« la' "(n^-Sa', Ex.2. }-6aa;» aV+2aa;*+a;» i43l li^ " Wlion there ia a r«mam«fcr, it must be made the numerator of a Fraction whose denominator is the divisor ; "this Fraction must then li6 placed in the quotient (with its proper sign), the same as in common Arithmetic. iO ALGEBRA. Ex.6. ''■''~')::T::i::-^+^^-'f--^+'' -.+1 * + ar'+ar-l Ex.6. '"^'felffs+^^^-^^^'-l^+l 4- x'^-ix * * Ex. 7. Dh ide a*+4a»5+6a=6«+4ad»+i* by a+b Am, a'+3a»6+3a6'«+fi». £;c. 8. Divide «''-5a^^+10aV~10aV+5aa:^-;i^ Am. o*— 2videa^+8a3;r+24aV+32a.:3^.ie^.j^ -4ws. a«+6a«ar+12aar«+S^. DIVISION. d\ Ex. 11. Divide al^—a^ by a — x. Ans. a*+a*a:+aV+aar*•^x*. Ex. 12. Divide Gx*+9j^-20x by Sj^-Sx. Ans, 2aJ'+2x+5- 5a? 3a;'-3jr. Ex. 13. Divide 9x«-46.r»+95a^+150x by x'-4x-5. Ans. 9.J:*— 10ar^+5.»''-30^. Ex. 14. Divide «*-^a^+««+|x-2 by ^x-2. Ans. |z='— ^:r»+l. ,, PROBLEMS PRODUCING SIMPLE EQUATIONS, CONTAINING ONL"? ONB UNKNOWN QUANTITY. Prob. 1. A fish was caught, the tail of which weighed 9 lbs. ; his head weighed as much as his tail and half his body, and his body weighed as much as his head and tail. What did the fish weigh 1 Let 2ar= weight of the body in lbs. ; .-. 9+ar=: weight of tail+| body = weight of head But the body weighs as much as the head and tail j .•.2a:=(9+ar)+9, 2a:=a:+18; .•.a;=18, and .-. 2a;=3r), the weight of body in lbs., 9+a:=27, the weight of head in lbs., and the weight of fish=36-f 27+9=72 lbs. l*ROB. 2. A servant agrssed to serve for £8 a year an^. finds hLseif I160 inSr^^rdti iT:^^^ Ans. £100. Prod 5. A met two beggars,'B and C, and having a certain sum m his pocket, gave B A of it and C i nf Vhl ^ cemm A now hflrl 9ft./ iLa k Ju J , ' ^^ » ^* ^"® remainder : A now had 20(1. left ; what had he at first ? Ans. 5s. £i^^7f\t \^rT ^^ ^V *'^^'^'' *>^^ '^ saddle worth £0^: If the saddle be put on the first horse, his value wiJl b«v come double that of the second ; but if it be put on thelecond: lZuf7 ^;\^^^T^ ^"P^'^ *^^^ «f th« first, wtat is tl^ value of each horse? ^„,. ^g^ and m PRon. 7. A gamester at one sitting lost J of his mm.Av and then won 18.. ; at a second he lost i of the remS' and then won 8.., after which he had 3 guineas left How much money had he at first ? ^ ^^ Let 15-1! = t}lO rillTy»>»ow /->r «U!li: 1_ _ I 1 _ . - having lost j of hismoneyihe had Tof iro"l2r,^m':i'iS-' he then won 18... and therefore had 12x+18 in handTS^ i ALGEBRAIC FRACTIONS. /l.t^ .•.(8a;+12)+3=C3, 8a;+13=:60; /. 8a; =48, ar=: 6; Hence I5x=90s.=£i 10s, CHAPTER III. ON ALQEBRAIO FRACTIONS. o,i?i; ^""^ K"les for the management of Algebraic Fractions Tt^ TT ^' '^'? ^" ^'^^^^^^ arithmetic.^ Th^priSes ?oIIowtg:!!lf '"^" " '^^' "^^"^^^ -^ established aTtS (1.) If the numerator of a fraction be multiplied or th^ ^^ nominator divided by any quantity, the frac fon t Tend^^^^^^ so many times greater in value. rendered (2.) If the numerator of a fraction be divided or the d« hJtlu- ^? t^Q^nV?^?^^o»' and ^denominator of a fraction arc the foundation of tlTScit K. Hoicar'''"'' ''*' ^''""'P'"* ^^^*^'' i 44 ALGEBBA. ON THE REDUCTION OF FRACTIONS. 41. To reduce a mixed Quantity to an improper Fraction, Rtle. "Multiply the integral part by the denominator -f Uie fraction, and to i\iQ product annex the numerator with iti proper sign; under this sum place the former denominator, luia Uie result is the improper fraction required." Ex.1. 2x Reduce 3a+— , to an improper fraction. The integral part X the denominator of the fraction + the »wwrator=3aX5a»+2a:=15a'-f-2ar- „ 15a»4-2a;. , Hence, — ~- is the fraction requu-ed. . Ex.2. Ax Reduce hx— ^^ to an improper fraction. Here 5j:X6a'=30a«^; to this add the numerator with its proper sign, viz., -4x; then ?2f!^±' ig the fraction re- quired. Ex.3. Reduce 5a; —-1. to an improper fraction. Here 5a;X7=35ar. In adding the numerator 2x-3 with Its proper sign, it is to be recollected, that the sign ~ affixed to the fraction —^ means that the whole of that fraction is to be subtracted, and consequently that the signs of each term of the numerator must be changed when it is combined with 35ar; hence the improper fraction required is ooa; — 2.g-{-3 33.P -f- 3 41. How U A mixed quantity reduced to an improper fraction I ALGEBRAIC FRACTIONS. 45 Ex. 4. Reduce 4a6+-- to an improper fraction. 12a'6-f-2c Ans, 8a Ex. 5. Reduce 36'— — to an improper fraction. 156»ar-4a ^7W. Ex. G. Reduce a~a;H -— to an improper fraction. X Ana. 5z !.ion Am n Ex. 7. Reduce 3a;' rrr- to an improper fraction. X 10 .^n^. 30a;'— 4ar4-9 10 42. To rerfwcc an *ot -r Fraction to a mixed Quantity. Rule. « Observe MHu^^^h terms of the numerator are divisi- ble by the denominator without a remainder, the quotient will give the integral part; to this annex (with their proper signs) the remaining terms of the numerator with the denom- inator under them, and the result will be the mixed quantity required." ^ -^ Ex. 1. ^ a'-^ab-^b* i\eauce to a mixed quantity. TT a^-^ab ^^^ — a — """J"* ^^ ^^® integral part, and — is the fractional part; .*. a-f-5-f J. is the mixed quantity required. 48. W hat is tho rule for reducing nn improper fhwtion to a miied (^ua^ m td ALGEBRA. ■ El. 2. Keduce ^ to a mixed quantity. 15a» '6a ^ ^® '^® ^^^^S'^^'^ part, , 2a:-3c . , ana — = — is thejraciional part; 5a 2«-3c . .-. ^H -^ jg the mixed quantity required. , Ex. 3. Reduce — ^_ to a mixed quantity. Ans. 2x— Ex. 4. Reduce 'Jf±^^ ^ a mixed quantity. 5a 2ir* 4a Ans. 3a+l-_?i. 4a ^«s. 10y+3j;-~. ar 43, To reduce Fractions to a common Denominator, Rule. " Multiply each numerator into every denominator tlZ^rt '^' "'^ """aerators, and all JZSfr fof/ether for the common denominator." Ex. 1. p ■, 2.r 5a; 4a neauce — , -, and — , to a common denominator. f^rv-^iv;— TK- f rtence the free 5^X^X5- /5a; f, new numerators ; I tions required are r Hence the free 4^X3X^1^* \ "''' """^^''^^^^^ ; J tions required are 4aXrfX6-12a&) S jp^^ ^5^ j^^^^ 5 X 0X5=156 common denominator; [ isT' \Kh> YbF 48. How aro fractions reduced to a common denominator f ALGEBRAIC FRACTIO^^S. Ex.2. tveauce —g— , and — , to a common denominator. 47 Hore(2.p.f 1)X4= 8.c+4 ) new nume. ^^^^~}2:^___S rators; 5X4=20 common denomi- nator ; Ex.3. 'Hence the fiac- tions required are — ^TTT, and -— — . 20 ' 20 Reduce — j— , ~^ and — , to a common denominator. Here 5a;X Sx2a:=30j:» > .*. the new frac 30ar* • '.1 {a^x) (a+x)x2x=2a'x-2x^ tions are 6^*6^ 1 X(a+:r)x 3 =Sa +Sx ha'x-2.^ , 3a4 3.r (a-i-x) X3 X2x ==Qax +Gx^j QaTTO?^ ^"^ CllJ+e?' Ex. 4. Reduce — , g^,and -^, to a common denominator. "":'' & l5^' ^"^ 157- Ex. 5. Reduce --^, and — g~ to a common denommator. Ar.j. —- — , and 3a: 3a; Ex. 0. 4a;'-|~2a; 3a;' 2a; Reduce — - — , — , and -, to a common denominator. J 4Sabx^-{-24abx 45ia;» , 40aa; C0a6 ' mab ' ^'"^ QQab' Ex.7, o , 7.r'~l , 4i:*— ar+2 Ze aa" An, '±'.f=K .ni?^Zl-:+*^ 4a*x , and 4a'at I : 18 ALOEBKA. 44. To reduce a fraction to its lowest terms. Rule. " Observe what quantity will divide all the terms both of the numerator and denominator without a remainder; Divide them by this quantity, and the fraction is reduced to ^ts lowest terms." Ex. 1. Reduce 14a;»+7 q ar-f21a ;» to its lowest terms. The coefficient of every term of the numerator and denorai- nator of this fraction is divisible by 7, and the letter x also enters into every term ; therefore 7x will divide both numa- rator and denommator without a remainder. Now 14a:»4-7aar+21a;« iX =2a;«+a+ai?, and ---=5a;; 7x * Hence, the fraction in its lowest terms is ^^'"^-JJf. 5x Reduce Ex.2. 20a5c~5a-+10 ac 5a^c to its lowest term. Here the quantity which divides both numerator and de- nominator without a remainder is 5a ; the fraction therefore in its lowest terms is — ^~"+ f ac Ex. 3. Reduce — — -. to its lov/est terms. (r—o* Here a—b will divide both numerator and dcnominatoi, for by Ex. 2 Case III. page 27. a^—b''=(a-{b){a—b)i hence — -— • is the fraction in its lowest terms. tt"f~0 Ex. 4. Reduco -— to its lowest terms. 2jr Ans. — , 44. Show how fractions are reduced to their lowest tenon. \ ALOEBKAIO FKACTIONS. 49 Ex. 5. Reduce — -^ to its lovrest terms. Ans. ~. (it Fx 6. Reduce ^ -^ ^ . to its lowest terms. 2y--3jry Ans. X Ex 7. Reduce r^-^— to its lowest terms; Vtx^ Ans. 3j:«--ar4-2 a^ ON THE ADDITION, SUBTRACTION. MULTIPLICATION, AND DIVISION, OF FRACTIONS. 45. To add Fractions together. Rule. "Reduce the fractions to p. common denominator and then add their numerators together; brin' 12a6» ~ 12«6^-" 6x36x4a=i2a6'J = (dividing by 6) ^^^^f is the sum required. 46. State the rule for addiiiji fraction«i. i lis ■ i; 'I 50 ALGEBRA. Ex.3.Add?^,?^i, 5 ' 2x ' i2x-\-3)x2xx7=2Sx' +42z (3a;^l)x5 X7=105a;~35 4arx5x2ar=r40a;^ 5x2a;x7=70a: 4x and — , together. ^ 28a;»-f42g-fl0 5ar— 85-i»40r» 70a- 68a:''+147ar-35 , 70a? s the Ex. 4. Add --, ~, and ~, together. Ans. ir-x. o. Add 2^' "5' ^^*^ 7"» together. Ans. sum required. 934a; 693* 105a'-f28a'6+3C6« Ex. C. Add ?^i lf+?, and |, together. 70a6 Ans, Ex. 7. Add 5a«+6 169a;+ 77 105 • 4««4-2^ Ans. 36 ' *^"^ "sT"' ^^g^^h®''- SToM-m 156" ' Ex. 8 Add -1^, and — , together. . 4«»— 7a?— 3 Atis, - Etj, i>. Add --— , and — — together 6« Ans. Ex. 10. Add — -.. and — r- . v»o-PtliAr 2a?« tt— a + o ^n*. ^M-26« a»-6» ALGEBRAIC FK ACTIONS. 5J 40. To Subtract Fractional Quan.ities. RiTLE. "Reduce the fractions to a common denominator • and then subtract the numerators from each other, and uudw the difference write the common denominator." 3xXl5=45a; UxX 5=70a r 5 Xi5~75 Ex. 1. Subtract ^ from — . 5 15 •*• 75 — if^ ~ 3 ^^ ^^® difference required. Ex.2. Subtract ?f±3 from ^"^^^ 3 7 (2.r+nx7=14ar+7) 15ar+6--14.r-7 ar-^1 (5a?+2)x3z=15ar+6 \ •*• ^l ~" = "sf ^^ ^^ 3x7=21 ) fraction required. Ex. 3. From i5|r2 subtract ^1, (10u:-9)x7=70a;-63 ) 70x-63-24a:+40 4Gr-^3 (.3^5[x8=24^-40[ .-. -^-±-=:15l^ " ' "^ '" is the fraction required. Ex. 4. 8 X7=56 From r subtract . a—b a+b (a-i) (a+6)=a«~"65 J ^^ j^ ^j^e fraction required. ( Ex. .5. Subtranft If frnm ?? 5 10' "'^•lo- 4«. Give the rule for subtracting fractious. >n vt: il 62 ALGEBRA. ' I Ex. 6. Subtract 5^ from ^i^ fix. 7. Subtract --±1 from If x+l 5* Ex." 8. Subtract ~-=:3 from 1^ Ex. 9. Subtract • — - from ^ a-\-b a~b' Ex.10. Subtract --Hl from - 8 7* ^,«. lE^iZ Ans. Ans. 28 5ar-{-5 4x'+3 l_l£+49 66 • 8 * 47. To Multiply Fractional Quantities. Rule "Multiply their numerators together for a nrw numerator, and tW denominators together for a new d^ L'rmsT' '"' '^'"^' *'^ ^""^^^"S ^-<^^-- to its Wt" Ex. 1. Multiply ^ by 1^. 7 x9 =63 f •*• the fraction requireu is -- ' ^63 Ex.2. Multiply 1^+1 b/' 3 Here C4«-f-l)x6«=24z»-|-6 and 3x7 =21 r 24ar«-f Gar , la; I * * 21 — ~ ^dividing the nu 1 JTierator and denominator bv 3'' arM-2ar * ' ij; — IS the fraction required. (« 47. State the rule for the multiplication of fractio:, M, ALQEbRAIC FRACTIONS. 53 Ex. 3. 00 " a-\-b By Ex. 2. Case III. page 27, (a^-i»)x3a'=(a-t 5) (a-i)x3a«; hence the product is 3a'X (a+^)(a -j)^_ 56x(a+6) (dividing the numeratorand denominator by a+6) ^^'^C^ -^) 3a»-3a»6 ^^ 56 Ex.4. Multiply 54^%y '^« 14 Here (3a:«-5a:) X 7a=21aa;'-35aa; and (2««-3a:) X 14=28a:»-42.r 2a;''-3ar* ^ 21a.r«-35aar , the numerator and denomi- nator by 7x) ^=| is th. J fraction required. 2x Sx Ex 5 Multiply -5. by ~ Ex. 6. Multiply ?^i:f by ^^ 2a:*-4aj* Ex. 7. Multiply ~ by —1'. Ex. 8. Multiply --5fL by ^^^-30 2a: Ans. Ans. Ans, Ans. Ix-^' 3ar-~l ar-2" 4 2' I-' ^ 48. 0/j the Dlvmon of Fractions. tion "^'''* "'^'*^^^^ ^"® divisor, and proceed as in Multiplira 48. Enunciate the rule for division of iVact^ona. »jira )ll>fca wiLMWtoi i*»w» M ai nwn«M 'i i i ;/.;:*'TT^ ■--'—y jwjtwi ■- M ALGEBRA. II Ex. 1. Divide — b/ -. Inter t the divisjr, and it becomes ~ ; hence — x — - JqJ ~ "s ^* ^'^'"^ ^"® numerator and denominator by Qx) s tlie fraction required. Ex.2. Divide ^^^ by ^^^-^ 14ar-3 25' X 6 5 -^ 25 • (143: -3)x5 _ 703?- 15 lOr -4 10^-4 ~ 10x^=^4* Ex. 3. Divide ^^' by lli±. 3a^-5&«_ 5 X (a+&)(«--^) f. 5x(a-f-^^)(a~&) ^. 2a ~ 2a 4a-4-46_ 4x(a + ^) 66 " 66 ' 66 JS Ex. 4. Divide ^ by ^. 2a ^4x(a+6) i _3 06x(g-6) 15a6--15^" . ~" 8a "^ 4a the fraction required. \ ^'^^- 6l- Ex. 5. Divide ~±2 by ?^i. o 5a; Ex. 0. Divide ~ by ^. .4n5. ^n«. 10.g 3 • 4ar--12 5 "' _ _ _ 9r«— 3rr rs* fix. /. jnvide — :; — by — . Ans, n« o SIMPLE EQUATIONS. 6u- ON TIIK SOLUTIOif OF SIMPLE EQUATIONS, CONTAININO ONLf ONE UNKNOWN QUANTITV. Rule III. Inc?!;.^"-!*^"^..^'?" "'^>' ^^ ^^^^^^^ «^ fractions by mullinlv.. SsTnVutlt^ ^^"^'^" '- ^'^ denominators^of thS eac^'Llde Tth«''" ""? ^e cloared of fractions by multiplying^ s^iititf trSiot^^^ '^" ^^^'^-^'^ -/.>/.y thf : This Rule U derived from the axiom rd\ fi,of ,•<• i 2"rjf^^ "^"^^!p"^, then (since tha- multiplication of the fraction | by 3 just takes away the de. nominator and leaves x for the product) we have ar=6x3=18. Ex. 2. Let |4-|=7. Multiplj^ each side of the equation by 2, and we have ^Ag,m,«„/.>/yeach^,HerfJhi, equation by 5.^ „. 7a:=:70'; Var=10. Ex.3. Let^ + -=i3__f Multiply each side by 2, then a:4.- =2G-?? 3 4 Mullinlv oanh oJ/la !»». O j c!._ . ^ __ 6^ 4' Multiply each side by 4, and 12x-\-8x=zm^^(kc. S8 ALGEBRA. By traiispt iition, 12j;-f 8a;+-G.r=::312, 26a:=312; .-. ar= 12. Tliis example might have been solved more simply, Ly mul rL'])l3ing each side of the equation by the least common multi pie of the numbers 2, 3, 4, which is 12. Multiply each side by 12, -7; — |- 2 :156-1?A or, Qx-^4x=zl56—Sx. By transposition, 0a;-f-4a:4-3a?=156, * 13x=156; ,\x= 12. Ex. 4. Let 2x X 3 "^4" :22. Ex. 5. Let 7x 4" 5x a' 55 -6' Ex.0. Let ^.+5=31-1. <« «> o Ex. 7. Let ^-~+f =44. 5 () 2 Ans. a: =24. Ans. ar=:10. Ans. x=SO. Ans. ar=GO. 60. In the application of the Rules to the solution of simple i.juations in geocral containing only one unknown quantity, il will be proper to observe the following method. (!) To clear thj equation of fractions by Rule III. (2.) To . ol'ect the unknown quantities on one side of the -Cfjuullon, fjid ike known on tl e other, by Uule II. (3.) j'o find the value of the unknown quantity by di- viding TAcU g/iie of the equation by its coefficient, lis in Rule I. 60 F.) I A -' ■: tie thrco Btups Vr which B'mpie equation containing only oivj •.'•-'/> -vn iiaantity rany be BolVoU. SIMPLE EQUATIONS. 57 Ex. 1. l^uvi 'Jie value of a: in the equation — - l=f-|-l?. 7 5 5' Multiply by 7, then 3^-fV=-+-. Multiply by 5, then 15j;4-35=7ar-j-91. 'X)llect the unknown quantities on ) one side, and the known on. the [ 15.r— 7;i:=9I— 35 other; 1 '^' or 8z:i=5G. Divide by the coefficient of x, ar3=— — 7 ' 8 ~ * Ex. 2. Find the value of a: in the equation ^i?- 1=:2— - 5 7* Multiply by 5, then ar+ 3- 5 = 10-—- 7 * Multiply by 7, then 7x4-21 ~25=70-5ar. Lollect the unknown, quantities ) on ont side, and the known > 7ar-f-5ar=70— 21-f-35. '.•n the other ; \ orl2ar=84; ••''■-l2~^- Ex. 3. Find the value of x in the equation x—\ 2:e~2 ~5 -24. Multiply by the least) .^ r , ^ ,^ (vmmon multiple (10), f 40«-5ar+5=10.r4-4.r-4+240. Uy transposition, AQx-^bx—lOx— 4a'=240— 4— 6. or40a;-19.r=231, i. e. 21a:=231 ; 231 ,, IWA As the Jirst step in this Example involves the case " where the sign - stands before a fraction," when the numerator of ./ -M^^mmm^ ^ ALGEBRA. ! that fraction is brought down into the same line with 40ar the signs of both its terms must be changed, for the reasons as. Signed m Ex 3, page 44; and we therefore mai«e it --5.r-f 5 and not 5af— 5. ' Ex. 4. Find the value of a: in the equation 2a?— - -f 1 =5ar~2. Multiply by 2, then 4ar--a;+2= 10a?— 4. By transposition, 4+2=10a:--4a;-f ar, or 6= 7x; 6 6 t or x=-. Ex. 5. What is the value of a; in the equation 3az-{-2bx=Sc+a'i Here Sax-{-2bx={Sa+2b)xx', .•.{Sa+2b)xx=:Sc+a. Divide each side of the equation by Sa+2b, which is iha coefficient of a:; then a;==?^il^ 'Sa-i-2b' Ex. 6. Find the value of a? in the equation Sbx+a=2ax+4f, Bring the unknown quantities to one side of the equati(A. ui/^ the kno%on to the other ; then, 36a;-— 2aa;=4c--a; ' but 36a;— 2aar=(36--2a) Xar ; .'. (36-2a)a;=4c-a. Divide by 36-2a, and «=|^. Ex.7. Find the value of a; in the equation 6a;+.r=2a;+8a Transjose 2a;, then bxA^x 2a?=i3a, or bx— a;=:3a ; but 6a?— a;=(6— l);r; . (6—1^ a;=3a. and x-rz m* w^tmmmmi^m-'r'nk ssasfi Ex. 8 a:4-4.-~ii SIMPLE EQUATIOXS. ^j^ Ans. xz=.Q. " 5^4^3 2^ • ^ 2^3 4~2' -E:x.\12. 3^+i *+5 9~"~F'* — Ex. ^3. -f«5=:,2^_2^ - Ex.|l4. C^-^-.9=.5:r. "Ex.15. 2x~^^+i5^1?i±20 3 6 -^ Ex.lG. ^Z:?+^-2C-^-^ 2 ^S"-"^^ ~F"- 7 -4?w. ar=rl4. -4««, ar=3G. Ans. «=12 -4n*. a?=:18. 2ar--l Ex. 17. 5^~ffll^4.1-3^.fi:2.^ . 3 ^^--«>*i ^ — f-7. Am. z=8. Hoc. 18. 2az-{-b='6cxi-ia, Am. z=^. Ex. 19. Multiply by 15, 45x-.60-2S^4'm^7oj_A^ a 45ar-28ar^4r=72l4+ 60-30 13i;=:92; ' "; t^ ""^B^w^!*". eo ALGEBRA. Ex. 20. 4«-y 7a;-29_8ar+10 18 ; fmdx. 126ar~522 Multiply by 5;r-12, 126:^- 522=65.^-150 126a?-65;r=522-150! 61«=366; .-. ar=6. * Ex. 21. Given ~? L - 1 «~1 a:-|-7"~7"(«-l)' *<> fi»«* * Mult, by 7(x-l), 7-.lifei:l)^i^ »* "f* 7 6=:iiferJ) a;-H7 • Divide by 2, sJltnl} 3ar4.21=7:c-7, 7«— 3ar=2l4-7, 4aJ=28 ; .-. a?=7. Ex.22. Let ?^i4-2ir?-.I«^+J5 . 2J • ^ 14 ^6«+2~' 28 r-^;find« Mult by 83, ,6.+ 10+l^r|f=i«,+ ,5^„] 196;r— 84 -==14 196ar--84=84ar4.28. * ~^-\ V. V. 0, the iJgfflSWft*- SIMPLE EQUATIONS. 63 V. 4 3 -^i 9~-- Ex. 24. ?f±?2_l^-12, ar 36 ~5^34^-4' Ex. 25. ?2^±^ . 5f+20__4.r 80 ' 25 ■^9jr-16~"5'^25* V Ant. 0. ^n*. 8. An9, 4. Ex, 26. ?^S_i2fr5 , f _7« ^+16 9 17a;-32"*"3~12 36~* ^"*- ^• Ex.27. 4(5.-3)-G4(3~.)->3(12.~4)=96. Ans. 0. Ex.28.10(.-f|)^0.(l^j).23. ^,,^ Ex. 29. -?±?f4.5£±?f_,.^ 48 ^««. 3. t:H-3 PROBLEMS. Let a;=:the number required • then ;.+ loathe number, Wth 10 added to it. Now Jths of (ar+10)==?(a:^-10)=l(f±i£)_3£^-30, But, by the question, Jths of (ar+ 10) =66 • ^ • Hence, 5f+22^ee.' Multiply by 5, then 3af+30=330 ; .•.3^=330-30=300; or.r=?^=ioo. the quotient shall be 20? ^ ' ^ ""^^ ''''' ^'^'^^'^ V » Let ar=the number required ; tjien 6^=the number multiplied hv fi. Ox-t-io=ihe product increased by'is/ 6.r-}-18 , •'^ °» ana — = — -ih^it sum divided bv 9 ^ 9 6 : 1! J 62 ALGEBRA. Hence, by the qu^ ion, ^i+l^^g^^ Multip]j^by9,then6:r+18=:l80 1 ROD. 3. A no«!<- io i*k • 1 6 ~" * ^et out of tho C fe tSf ' ^^^j^ -ate,^ and 13 I^et .^.length ofThe1>o:t": j.^^^ ^^^^« P-t ? then |=the part ofit in the earth. 3ar y=^the part ofit in the water, Butparur'ii^nr'^'""'''^^^^^^*^^- whole post ; ' ^ ^ ■ ' " ^al'^ + part out of water = I -^ (t) '• ^■2 s= X. Multiply b.y5, then. + '•-_, 65^5^. Multiply by7, then 7^+ ,5^1:455^35; Prob 4 Afv . 13 -"^^ ^^"gtli of post in ft. had^«- f V ^fte^ paying away kh and ith nf ftad i^5 left ,n my purse. WhJ rnnnlv^v-^^ r "'^ ^«««>^ » Let ^^.>oney in purse a? fim ; ^ '' ^'^'^ ^ then j+-=money paid away. (4+7J = 85, Multiply by 4, then 4z~.x-^^S40' Multiply by 7, then 28.-7.^41=2380. .-. 17.=2380: X ■=27. , and 13 t? ^aterss iinft. ney, \ ? nirg; SIMPLE EQUATIONS. 98 ^rol^ll^fZsl^^^^ « , » ^"''- •*"• ^'*= "'"J 13- mcome. t. l oi it ^^m« ij^s. p,e(|uired his ^/<*. £150. Prob. 12. A gamester at one aittina lost Ifh nf k.-c guineas left. wStr„e^ fa'^R fit,' ""'^'^t: '^^ ' lo the same quantity 7 ' "''•'' "" ''« «q«»l ^ •'^' -^'"- 18, 23, 10, 40. »/«TpeHb^bT^t"^^^*" Jy■."f.^«^ «~...<'- worth -u™ he take ,» fi.m a oheVtV To4 ^:.^^CZS, Ana, 33 at IS*. M. 1 ''1 at 9*. 6ci i# i I !i:i 64 ALGEBKA. <; (( (( 1 uoB. 15. Three persons, A, B, and C, can separately reao ft i.eJd of com m 4,8, and 12 days respectively, lii how luanj days can they conjointly reap the field ? I [.et ;r = No. of days required by them to reap the field; then It 1 represent the work, or the reaping of the field, ■J^=the part reaped by A in 1 day. j_ « ;; « •' g ^» T^= " " " C .'.i+l-\-j'^=z « « « all three But the part reaped by all three in 1 day multiplied by the number of days they took to reap the field, is equal to the whole work, or 1 ; •••(i+i+A)^=l: Clearing of fractions by multiplying by 24, (6+3+3) a;=24, llar=24; i .'. x=2^^ days. Prob. 16. a man and his wife usually drank a cask of *»eer in 10 days, but when the man was absent it lasted the wife 30 days ; how long would the man alone take to drink '^' Ans. 15 days. Prob. 17. A cistern has 3 pipes, two of which will fill it in 3 and 4 hours respectively, and the third will empty it in C hours ; in what time will the cistern be full, if they be all set a-running at once? jins. 2h. 24m. Prob. 18. A person bought oranges at 20c?. per dozen ; if he had bought 6 more for the same money, they would have cost 4c/. a dozen less. How many did he buy ? Let ar=rthe number of oranges ; thena;+6= « " " " at 4d less per dozen. Price of each orange in lstcase=fft=4d'. and « " *' « « 2d " =11 =k ,*, the cost of the oranges =- 5x 3* ■r But we have also inv cifSi ui tne orauaes=* ia;-f-0}. Two independent vrlues have therefore been obtained foi 9 f\\ SIMPLE EQUATIONS. 65 the cost of the oranges ; these values must necessarily be coual to each other ; •' ^ to each other ; 5^ .••3=1(^+6). Multiplying each side of the equation by 3, p ... •'• «=24, the No. of oranges. rROB. 19. A market-woman bought a certain number of apples at two a penny, and as many at three a penny, and sold them at the rate of five for twopence ; after which she found that instead of making her money again as she expected, she lost fourpence by the whole business. How much money had she laid out] ^ Ans.Ss.4d'' Prob. 20. A person rows from Cambridge to Ely a di«^ tance of 20 miles, and back again, in 10 hours, the stream Howmg uniformly in the same direction all the time : and he mds that he can row 2 miles against the stream in the same time that he rows 3 with it. Find the time of his ffoin.5 X Multiplying each term of the equation by 3x, 2+3=3^; .'.x=-=ll and .-. 32- =5, miles per hour down - .t .. ._ . . 2^ ,-. ine :ime m gomg down rhe river=:— =4 ^ours 5 ne of returnin on wquciitly the 0* : 10-4 =6 hours. 0li 4 ' . ■ 'Wl^^ «?E»«^ m ALGEBRA. Prod. 21. A lady bought a hive of bees, and found that the pr,ce came to 2.. more than |-ths and Jth W the price fI^^ Ans. £2. Prob. 22. A hare, 50 leaps U^c<. .. .oyhound takes 4 eaps for the greyhound's 3 ; but W.., of tUe grVho^id's leaps are equal to three of the k. .'.. How mafy iapTwIl ?he greyhound take to catch the hare ? ^ Let X be the No. of ieapn taken by the greyhound ; then - will be the corresponding number taJ, a by the hare. Let 1 represent the space covered by the hare hi 1 leap; then - « 4i «4 u u , , 2 " . " greyhound « .4a: 4x , . . g X 1 or y wil' be the whole space passed over by the hare before she is take. ; and :. x | or | .vill be the space passed over in the corresponding time by the erevhound Now, by tho problem, the difference betwo'n ?l7spaces Srps^;^"'' ^^" '^ the greyhound and harl LTx" .3a: 4x _ • ' 2 - 3 =^^^' Oar— 8a;=:300; .'. ar=300 leaps. ON THE SOLUTION OF SIMPLE EQ. ^ONb, C0NTAI^.VO TWO OR MORE UNKNOWN QUANTITIES. 51. For the solution of equr 'om. coutaining tw , or more u known quantities, as many independent equations arrrl quired as there are unknown quanUties. The two It;.m necessary for the solution of the case when t^vn „nl n «es are concerned, may be ^sI^V ^ ^ ^"^ ax-{-by:szc a'x-\-b'y=:c\ Where a, 6, c, a\ b\ c>, represent known quantities, and z, y, SIMPLE EQUATIONS. 67 thQ unknown quantities whose values are to be found in terms ot these known quantities. There are three different methods by which the value of fMic of the unknown quantities may be deterniin d. J^ ml FIRST METHOD. ^i*^'^ t^e value of one of the unknown quantities in tenns ol the other, and the known quantities by the ruJ. s already given. Find the value of the same unknown quantity from the sofond equation. Put these two values equal to each other ; and we shall then have a simple equation, con^Mning only one unknown quantity, an hich may be solved as before •gx [■ to find X and y. a) Ex. 1. Given x-\-y=S - ... (1) x—yz=4: - - From (1) y=S~x - - " (2) y=.-4 Putt'ng these two values equal to each other, we get ar~4=.8--ar, 2^=12; X=z:6. By ( y=8-a:='8--6=2. Ex.2. Let 4y=.16 (1) 4. , y~M (2) From equation (1), v have a;=16— 4y. (2) « «« 34— y 16 in (( u « « XT-' Hence by the rulo^ ^ =10 -'4y, 34-y=6 -16y, 15y=30j . . y=2. It has already bee shown that ar=:I«-_4y=/'since ?/ -2' and .-. 4y=8) 16-8 8. ^ ^ ' /d '/ 51. For tlio soiution of equations containing titios, how uany independent equations arc niHliod ol'auiution. •- more unkno 1 qaan S^^asawr**- 1 Q8 Ex. 3I Ex.4.! ALGEDRA. \ 5z+3y=38 ) 4x— y=10J" 2^-~3v = — 1 ) 3x-2y=6 f Ans. 1^=4 SECOND METHOD. 1^1 j From either of the equations find liie value of one of (h,. unknown quantities in terms of the other and the known quau. Jties, and for the same unknt wn quantity substitute this value (n the other equatior, and there will arise an equation which Jontams only one unknowi quantity. This equation can be solved by the rules already laid down, I Ex. 1. i/-'X=2 . - ' From (1) y=2+x. (a) This value of ?/ being substituted in (2), eivet 2a!r=6; And by (a) y=2+a;=24-3=5- Ex. 2. x+2 , ^ -3-+8y-31 (1) ~ + 10a:=192 (2) Gearing equation (1) of fractions, a;+2+24y=93, or a;-f-24y=91 Uearing equation (2) effractions, y+54-40a;=768, or i/-\-40x=z'7QZ From (a) ar=91--24y. Substitute this value of a:, according to the rule in equation ^3)1 and ^ y+40(91-.24y)=763, or, y+3640-960y=:763; .-. 959y=:3640~763=2877, and y=3. By referring to equation (a) we have «= 91 -24y=f since '=3; and .-.24^=72) 91 -72=19. !/ K ^ Enunciate the second method of solution. ■mtm^m^x^ SIMPLE EQUATIONS. Ex.8. 4:j;4.3y-_3i Sar+Sy =3!) =22 f 6» Ans, =4 5 THIRD METHOD. Multiply the first equation by the coefficient of x in tho Bccond equation, and tfien multiply the second equation by the coofticient of x m the first equation; subtract the second of these resulting equations from the Jlrst, and there will arise an cquution which contains only y and known quantities, frhm which the value of y can be determined It must bo observed, however, that if the terms, which in the remlitnff equations are the same, have unlike signs, the re- suiting equations must be added, instead of being subtracted, e'uatbns) "" ""^^ ^^ ^Uminaied {i e., expelled from tho Ex. 1. Given 5z-f 4^=55 . . . (l) 3a:+2y=31 - - - (2) To find the values of a: and y Mult. (1) by 3, then 15ar+12y=165 " (2)^7 5, " _15;»+10y=155 .'. by subtraction, we have 2y= 10 .*. y= 5. Now from equation (1) we have 55~4y x=z 5 55-20 Ex.2. _35 ~5 =7. Let the proposed equations be ax-^byz=c - - - (1) a'x-\-b'x-c' (2) Mult. (I) by a', and aa'x-{-a'by=za'e " (2) by a, " aa'x-\-ah'y=ac' ', How are equations solved bj the third method f 16 in /5 i-.f- ■j^-mm": 'SI ' i I ' I I 6 70 ALGEBRA. a'c-^ac' \ t c Mult. (8) by 4, and a'bx+hb'v=bc' B7 subtraction, (ai'Z'b)lJtl_M; JtL ■■ ab'-a'b' m'^'"'-*- Let3.t+4y=29 (n >«!». rn by 3, tl,e„ 9?+l|z^? (2) , *l'-(2)by4,thenC8:.-lg:-fl4 4-rfroi\^i:rtr,r:tr -^ --'t'- •••" together; and then ^ equations must be added 77ar=231 ; rom (]) ve have 4y=29~3i?, =29--<), (smce ar=3 ; and .-. 3^=9) .•.y=:5. ' ilx. 1 'x. 5. ^x. 6. =31) =22 f u^) !-x. 7. q :.v, 8. n k. 9. Let 6.r+3y=:33) 13^~4y=19f 4ar+3y=:31 3ar+2j/ 3ic+2y=40 r 2x+3y=35 f 5a?~4y==iai 4.r+2y=36 f ai?+7y=:79 ) 2y--|a:= 9f ^l?^+3 r • ■ If ^ 'J |y=4. \ y ~7. (a;=ll /; J- y )V -f o 5 /i^ ^!?. ^i>.. li^ SnrPLE EQUATIONS. Ex. 10. x-\-y 3 2x—^y 23 Ex. 11. H^^+y=7 ^^7-^^^;= ^ h'l^W^ti 9 5a;~-13y= 67 ^r''/§./- -<4«s. Ex. 12. 2iziZy=2^+H-l 3 8 ~^=6 5 mn^.Lr^^f /'""'^ unknown quantities are concerned, t'ne most gor-oral form under which equations of this kind can be expressed, is ax-\-by-{-cz^d (I) a'a:+6V+c'«=c/' (2) a"x+b"y\-c"z^d" (3), and the solution of these equations may be conducted as in the follow^ig example : Ex. l.# Let 2a:4-3y4-4s=29 (1) ) , ^ ^ ^ , 3ar+2y+5^=32 (2) ' to fmd the values 4a;4-3y+2^=25 (3) of X, y, r. r. Mi^^tiply /I) by 3, then 0.r+%+ 12^=87 (4) Multiply- X2) by 2, then 0.r+4y 4-1 0-^=04 (5). Subtract (5) from (4) then 5^+ 2^=23 (a). Multiply (2) by 4, then 12ar4-8y +20^=128 Multiply (3) by 3, then 12a;4-9//+ 6^=75 'ijj Subtract - - - . -y4-14z=53 (/3). ^■^ JI. Ilenrjo the given oquations are rednnfirl tn 5//-f 2i'=23 (a) /5 J ^ <> '' 9 L 1ft 7 > ,[**;""" <» '^ 5nip ;»i->i,-/r ^^^ ^ ALGEBRA. 72 • Mult (S^f\ I ' ^^+ 2.=23 By addition . . . 70^—900 ^ «, I'rom equation (/3) . . „_,. ^„ " \^) y= 142—53=56-53=3. 29~3.y~4s III. From equation (1) Ex.2. 29-2: Ex.3. ir+y+2=90 2^+40=3y+20 ^ 2ar+4O=40 4-lO ) ^-h y+ 0= 53 ar+2y+32=105 a:+3y 4-4^=134 -o J*R0BLEM8. greater kdd'ed toldThe t's is7^u7l T> '^^ ' *'"^^« '^- greater be suhtraL* from 6 timl. L f^' ^"^ ^^ ^^^'^^ ^^^ der divided by 8, The q^otien? ^ ^V""!' ^"*^ ^^^ ^e"»^'»- numbers ? -^ ' '"® quotient will be 4. What are the Let ar= the (/reaier number y = the /e«5 number; ' Then 3«+|=:36 ^jj. 9:c+ y=108 ' 6y-.2a;= 32; Or, y 4-90.= 108 (1) W then6y- 2a:= 32; then 56,r=616- 610 ' 56 — From equation (1) j,= 108Jo.=,08-09.,9 ... ru"""' ^- ''^eie is a certain fraction. ,,..1. ,i,„. .-,• , . ., - ". "- r.um.™to,-, ,« ™i™ will be jd ■ and if'r;ubtmro,« 4. -".='WMC(*BrtWMHi«*H', , , SIMPLE EQUATIONS. 73 !^ from the denominator, its value will be |th. What is the traction ? Let a;=its numerator, ) , .,/.,. . x y= denominator; \ *^^" ^^^ ^^^«*^«" '^ - rr+3_l ^ ~3 Add 3 to the numerator, then y 3^+0= y ^ .r 1 Subtract one from denom'., and — ^ — =- y—l 5 J By transposition, v—^x—9 (1) y-5x^l (2). Subtract equation (2) from (1), and we have 2x==8 ; .*. 2r=-=4, the numerator. From equation (1) y=9+3j;=9+12=2l, the denominator. 4 Hence the fraction required is — . 21 Prob. 3. A and B have certain sums of money ; says A to B, Give me £15 of your money, and I shall have 5 times as much as you will have left ; says B to A, Give me £5 of your money, and I shall have exactly as much as you will have left. W hat sum of money had each % Let a;=A'a money ) , ^ • i k_ j what A would have after y=B's money f ^"^^ a;-|- lo- ^ receiving £15 from B. y~ 15= what B would have left. Again, y+ 5= \ ^^^^ ^ would have aftei ° ' ^ ( reccivmg £5 from A. x—- 5=:what A would have left. ITencc, by the question, a;+15=5x (2^— 15)=5y— 75, ) and ?/-}- 5~iK-— 5. j By transposition, 5y— a; =00 (1), ) andy— a;~ — 10 (2). f 8ct down equation (1) 5y— x=90. Multiolv eo". ^2A by 5, 5?/— !>i __ n{\ Subtract (2) from (1) 4ar™H0; n 'I 74 |i ' ALGEBKA. • . ,7* "-": 0(r 1 f A — 'J», A s money. 4 •-"? ^1. s money, ■^rom equation en ^//-ort . . .. 125 J^KOB. 4. Whaf ^ ^ -•>. Rsn.oney. ,,„o ? '^'--"- ' -btract o«., ,h, reUld:?^i;I„';7'' PROB 5 THaj-o • "^"*'- ^» '^''^J 3- to its mmevator itV '''''^'''", ^^^^^^'^n, such th.,t if i 7 , ■"^ggarsdid he relieve ? '" '" '"^ P^^ket; and howf,„V Tl-en 2ixy, o,f = J No of .,;«,. ,,1,,, ,„„,^ "d 2Xy, or 4= ' '"'".^ S'*'™ »' 2»- 6i each Hence, by the q„ostio„,|'=,^^3 (1) "' ^4, each. Sub'. (2) fr„„ (,) „, y_ ^ > ^??.?. 56, and 33. It) Proc. 8. Thoro k ^^''- ^^» ^ <^'git«. Tho...,of thnJS;"."'i'"^^«'S consisting of ?.«i.-rt ^0 added to t '****'««*'™i»»««**««««i,i ^ngmmmt^-^sm,,;: SIMPLE EQUATIOXS. 75 io: A. What is the St Zh^""" '"""■' *' *8its will be inverted. right.ii.d diguTl's: ^JiTxi'A!''^''"" '^«" i^'"^ "- Let xz=le/t.hand digit, y-riffhi./iand digit. ? ,^^-^+2'=the number itself! Hence bvfht !."";' ""^"^'^ ^^'*^ '^'g't^ ^>^''^^e<^- on.i in f '^-^ *"® question, .r+y-5 nr Subtract (2) from (1), tl.,n 2y=G, and ;~|"-~' ^^^^ .rzrS— y = 5— 3—2 • Ajji^ ,•'• t-^c number is (10^+vW23 ' Add 9 to this number, and t betomTs 30 whJ.^ • number with the diffits inverted ''''''"'^^ *^^' ^^'^^ is '.ne ^the, ..eater, the remaind^'^ajT^J^r;^^ the digits will be '■»..« wLTtttLefuSr?""'''"' "'• -t be ta^e to r.™ » 2^ ^'JZ. SSttS ^«5. 33 at 13.V. (id. Prob. 12. A vessel CO imininr ion n ^.^ ^' ^■'''- ^*''- mimitesbytv.ospol^rng^^^^^^^^^ '^ ^0 gallons in a minu c, the other O^lull' ' ^"^ ''''''' ^'^ what time has eac-A'spout run ? ^ "' '" * "^'""^^- -^'"• Ans. 14 gallon spout runs minutes, Ptton. 13. To find thr^^ • ■ ' "^"i"^ f""^ ^ minute., I the sum of tiL . If^^^r ';::?'is \ ^'lU'^ &'' -^^' with Jfh fhn,liff!.....,„„ /"^ M " ®"«^1 ^« 120; the secoml I thoi.unr;^7he\h;;;— --^^ -d ^l«5. 50, 06, 75. i t 76 ALGEBRA. CHAPTER IV. ON INVOLUTION AND EVOLUTION. ON THE INVOLUTION OF NUMBERS AND SIMPLE ALGEBRAIC QUANTITIES. 53. Involution, or " the raising of a quantity to a given power," is performed by the continued multiplication of that quantity into itself till the nu|ttUer of factors amounts to the number of units in the index-%fthat given power. Thus, the square oC a=aXa—ri^ -, the cube of b=bxf>Xb=b^; the fourth poioer of 2 =2x2x2x2-- 16; the Jifth power of 3 =3 X 3 X 3 X 3 X 3p=243 ; &c., &c. ^ 54. Tlie operation is performed in the same manner for simple algebraic quantities, except that in this case it must be observed, that the powers of negative quantities are alter- nately + and — ; the even powers being positive, and the odd powers negative. Thus the square of +2a is 4-2a x +2a or 4-40"; the square of — 2u. is — 2aX — 2a or +4a^; but thp. cube of — 2a=— 2aX— 2aX— 2a=+4a*X — 2a=-8a^ The several powers of - | And the several powers of——, are, a a a* a a a a* cube =^X^X^==^a-' 2c' Sq>i. > = -2cX |4l»lil>— -— A7-A7A7 »Si:c.=&;c. a '6 a V 2c b -+4c*' b Cub. t)^^~''()v^ 2c~ °-" 4th power 2c'" 2c' ___b__ _J_ ~ 2c^ 2c^ =+___, &e.=.&c. b_ "2c *1 8c»' b ON THE INVOLUTION OF COMPOUND ALGEBRAIC QUANTITIES. 55. The powers of compound algebraic quantities are mj^ TiTi Involution performed for simMe al)?8braic quautitioa ?— 55. How are tht powofii of comp mnd (luivntitJ m raised I GKBRAIC to a given tion of that 3unt8 to the Thus, the <6=6'; the potoer of 3 manner for } it must be s are alter- and the odd iX +2a or a^; but tho. s of—--, 2c~ 8c»' b b 2c %c UNTITIES. mtities are iiui iimiiner la Howr are tht INVOLUTION. ^ it^lK: (Anr^^f "?^3" o' '■- I^"'« <•- expound E-v, I. What is the square ofa-f-26? a +23 . a'+2a6 ^+2ai+4i» Square =a^+4a,^q:4^5 Ex. 2. What is the cube of a^—x a^—x - a.'x+ x» Square r=a*~2a«a;-fp;,. a^—x «'~-2a*^+^ -- «*ar+2aV— a^ Cube=a''~-s„-..j.:j:3;;3-^rr7i -Ex. 3. What is the 5th power of a+5? a -i~b «'+ afi + ab +5« «*+ 2oi+i^=:Square g + b a»+ 2a'5+""^ a«+3a^6+ 3^F^b'=:Ci^- g + 5 a*+3««5+ 3a'^^~^3 __+_5!'^+ 3««6''+ 3a//Vi* «^+4aV/Tl?^4a53+^/=.4th Power t±J^il:i2S!±iO«V.«+5a6^+i^=:5th Po«. T8 ALGEBRA. Ex. 4. The 4"^ power of a+Sb is a*+12a«6+54a*A'^+108ai» Ex. 5. Tlie S!7Mare of 3a;'4-2r+5 is 9x*+12a;'»+34x'+20* +25. Ex. 6. The cwJe of Sx-6 is 27a:«-135a:''+ 225a: -125. Ex.7. ThecMieof a;'-2jr+l is z'-Qx'+15z*-20x''+l5x* — Cx+1. Ex. 8. The square of a+i+c is a'+2ab+b^+2ac-{-2bc+c\ ON THE EVOLUTION OF ALGEBRAIC QUANTITIES. 56. Evolution, " or the rule for extracting the root of any quantity," is just the reverse of Involution ; and to perform the operation, we niust inquire what quantity multiplied into itself, till the number of factors amount to the n\imber of units in the index of the given root, will generate the quantity whose root is to be extracted. Thus, (1.) 49=7 x7 ; .-. the sq. roo^ of 49 (or by Def" 15,y'49)=7. {2.)—b^=—bX—bX—b\ .'.cube root ot—b^{^:z:j;z) = —b. . 16a*__2a 2a 2a 2a 4 /16a*_2a ^ ^^8r6'^~3^^36^36^3^;**' V816"*~36' (4.) 32=2X2X2X2X2;.-. ^32=2, (5.) a^=za?Xa?Xd?', ,'.^a^=a\ Hence it may be inferred, that any root of a simple quan- tity can be extracted, by dividing its index, if possible, by the index of the root, 57. If the quantity under the radical sign does not admit of resolution into the number of factors indicated by that sign, or, in other words, if it be not a complete power, then its exact root cannot be extracted, and the quantity itself, with the ladical sign annexed, is called a Surd. Thus -y/37, }/a^, V//, J/47, &;c., &c., are Surd quantities. .«. Wha" {mlMluMon? How is it performed ?— 57. Wlint ie a Surd qiumtity f EVOLUTION. 79 derived from thos" of ZZlT PoTin" "" T' f ■''^ "^^^ Ex. 3 , the square o{a+!ua'+ "^"-^ ?-- root (a). Put a therefore for the fir^f «s_lo a . w / term of the root, square it, and ubtrac J+^"^+*' (-+' that square from the first term of the 2^aT;.» 1"? ^^J" f^^ °'^^^ t^^« terms 2^4T|2«i+i« -2«6+Z> , and t/oM6/e the first term of the l2ab+b' root; set down 2a, and having divided ~^^ he first term of the remainder (2a6) by •==— • It, It gives b, the other term of the root- and smce 2ai4-A«— r9«a.;.\;. -c 1 « ', being subtracted fr&Vo^tL'"';l:ht'^'+'" \*'"'' remains. ™^ oi ought down, nothing 60. Again, the square of a+b4-r ( Av^ f^K v c. x . 2ab-^b'4-2ac4-2bc4-r^- in 7^- (Art. 55, Ex. 8.) is a"-+ deriTed from t ' ' '^''' "^^" *^^ ^^ot may bo the power, by '','+2«*+*'+2ac-f 25c-f c'/a-f b-i-c continuing the « "■ 7.^ \ ' process in the *^" + X"* "^ *' iast Article. goi+i' Thus, baving 2a-^2bi-d2ac+2bc+c found the two }ftac+2bc~ -'■ first terms (a+i) of the root as befor 4'i\t%ry^t f»_- ri ' 'iOC f-HOf »i?SVttSl«t»:; jr* '''''""'^^^*^ '\ 80 ALGEBRA. i;,creased, and it ^f ^X+st + J ^hiXSg subtrac.od iklhU manner the following Examples are solved. Ex. 1. 4** 4:c'4- |>'+T 89 , 6x»+|a:' \ 20.tM-15a;+25 Ex. 2. ^^ . 2a^+2a;')4a;'+2a;* Ex. 3. Tne sqaare rootof4x'+4a:.y+2/'is2a:+y. „. . rn,_.n„.«rnotof25a' + 30a6+9/ris5a+36. ^" " ^"^' " " ■" ' root of 9.*+12.«+22.'+12.+a Ex. 5. Eind the square ^Ins. 3x'+2a;+3. __ *^^^^^^^ ■-■ai. •»--^ ►.-wfci-j EVOLUTION. 81 », it gives f, the term (6) of the the divisor thus iltiply this n*vr being subtrac.od res no remainder, solved. -|x+5. Ex. 6. Extract tlio square root of 4a;*~-16x»4-2 ic'—Wx-^ Ans. 2x^—4x-T 2. Ex. 7. Find the square n jt of SCir^— 3Gi-^+17^«--4jr4l y o Ex. 8. Extract the square root of «*-f 82;'+ 24-f — -f-i?. a;- «* 4 ylns. ar'+4-j — . 2a;*-a:-f2 ■4 4 is 2;c4-y- -9P is 5a +35. Ins. 3x^4-2x+3. . ON THE INVKSTIOATION OP THE RULE FOR TJIE EXTK N OF THE SQUARE BO^ T OF NUMBERS. Before we proceed to the investigation of this Rule, it wili be no'cssary to explain the nature of the common arithmeti- cal notation. 01. It is very well known that the value of the ficrures in the common arithmetical scale increases in a tenfold°propor. tion from the right to the left ; a number, therefore, may be t xpressed by the addition of the units, tens, hundreds, &c , of which It consists. Thus the number 4371 may be eipressed in the followmg manner, viz., 4000+300+70+1, or bv 4^ 1000+3x100+7x10+1 ; hence, if the digits* of a nuU^r hanHhen ^ "' ' '' ^' '' '^°'' ^^S^nning from the left A No. of 2 figures may be expressed by lOa+i « 3 figures ' by lOOa+lOJ+c. 4 figures « by 1000c +1006+ lOc+cf. <^c. &c. &c. 62. Let a number of three figures (viz., lOOa+105+c) be * \^^^^/!!^'J^ of ^ number are meant the figures which composa n, cons,(?ered independentlv of the value winch they possess in thHSh^ metical scale TJius the cTi^it. of the number 537 aK simply the num- Zn'nf' ^1^^' ^\^''^^' **^" ^' ^on^i'lered with respect tJits place n the numeration scale, means 500, and the 3 means 30 ^ HiS;? S Gi ^\?'"™?"."'''*""®*'°''^ s^"'^ of notation. What ig a digit ?-b2 ShoNV the relation between the algebraical hfid ml nerioJ methoi of ox^raot.ng tbo square root, and tliut tfey are Jden icJ /I: f r m ill MICROCOPY RESOLUTION TEST CHART (ANSI and ISO TEST CHART No. 2) I.I 1^ 13.2 1^ Vi m i4.0 •uuu 2.5 ""Z2 zo 1.8 J /APPLIED IIVHGE inc 1653 East Main Street Rochester, New York 14609 USA (716) 482 - 0300 - Phone (716) 288 - 5989 -Fox ./ 'Wsmnmmmixmm'-.' h:\ 82 ALGEERA. squared and its root extracted accordiiig to the Rule in Art 00, and the operation will stand thus ; I^00a'+2000a5 + 100i'+O00ac+o05e+cXl00a^ 20(k+l0Z;) 2000ab + l00b' 2000ab_j-im^ 200a+206+c) 200acH-206c+c« 200«c+20ic4-c2 * * * II. Letff=2] , , 6=3 [ ^''? ^'\^ operation is transformed into the fol. c_lj lowing one; ^^^^^+12000+900+400+00+1^200+30+1 400+30)l2000+900+400 / 12000+ 900 400+G0+l\400+60 + l 7400+00+ 1 * * 5330i/231 II. But It IS evident that this operation would not be affected by collecting the several nJimbcrs which star d m the same luie into one sum, and leaymg out the ciphers which are to be subtracted in the several parts of the operation. Lot this be done • and let two figures be brought down at a time, after the square of the first figure in the root has been sub- tracted; then the operation may be exhibited in the manner annexed- from whwili If ni-»i->/->r.-..„ «^i,„i. iL _ ' ;. trj-j."^«i3 tiiui, uic square loct of 53,361 is 231. 4m urn |l29_ 4011401 1401 ] . ^»— «?<#|«l|!l,'««!i-; '''^»«^<^>^^^^- QUADRATIC EQUATIOXS. 83 square root of 100 is 10- nf lAonn • inn ' r ' ^'"^® ^''^ 1000 &c &.ro . v. f 11 ' ?, ^T^^ '^ 1^^' of 1,000,000 is ^""^t" 100 ;, , t^^^^^^^ T'^<^ root of'a n'un.bor listing of rr%ures '^^' ^^"^''^ ^«°' ^^ ^ '--^er ccn £x. 1. Find the square root of 105,625. . Ans. 325. Ex. 2. Find the square root of 173,050. A71S. 41Q. Ex. 3. Find the square root of 5,934,090. Ans. 243G. CHAPTER Y. ON QUADRATIC EQUATIONS. /. ^o^m- oHhc un I It i ill ■^'.:>:?mf^ mw rjm m.. : 84 ALGEBKA. r ON THE SOLUTION OF PUUE QUADRATIC EQUATIONS necessary by the coefficient of;.'; then extract' ttf ^ root of each side of the equation, and ifwrnTvl the tC Ex. 1. Leta;'+5=54. By transposition, a;^=54— 5=49. -Extract the square root ) of both sides of the f then xz= + W49--^7 equation, j —V^^-—'. Ex. 2. < L'et 3j;'— 4=71. By transposition. 3;c'=71 4.4=^75^ Divideby3,a:''=^=25. o Extract the square root, x=j^^25=+5, Ex. 3. "~ Let ax^—bz=ci then ax^=zc-{-bf and :r*=f±i a V a =244 - - Ans. x= +7, =3a;'+63 Ans. x=z-{-^^ Ans. ar= + 10. Ex. 4. Ex. 5. Oar' 4. 9 4a;'+5 9 =45 Ex. 7. i;j 65. State tha rn\a ^nr. o«I..: . ~~ — ._ .-.. ^..-iTiuj,' juiro qxiauruiio oquatloug. of the QUADRATIC EQUATIONS. flg ON THE SOUTIOK OF ADPHCrED QUADRATIC B0UAT.O.,-S y or / actional. Divide each side of this equation l.y a: + -a; =-. Let --=;9, - =q ; then this equation is KULE I Let x^^px~q. Add |! to each -<3e ) ,^ ^^^^^. ^,^^^ of the equation, then j 4 4"^'^ 4 Extract the square root of each side of the equation, then andar=il:5!+47+7> 2 will arise, on the lejjnd sld/'of tt "Sn' f ^' "^•''•^* which IS a complete square; and by pvI^?;"! Z^' '' '^"•'"'^'^>' binco the square of +« fg ..„? „„j », of P=+4, ™ay be expressed by ± /,?:^ ' ''"" "" ^^"-^^^ '-* reM^irr^fry:;- Pjj'^JlJ^n^ Can U b. 8 86 ./, !i -ALGEBRA. one corresponding to .he sic^n 1 i f, ^^ill have two values ; of the radical quan% ^" "^' '"^ '^^ ^^^^^^ *« the sign-,' Ex. 1. . . , , Let x'+8x=Gr\ ^^ Add ft. y^<'Jl^J^^O)^a^ ,Me of the equation, E.U.-aeUHos,„a;etor„JtKrrl e,u.io. .1 ar-f4=+ V'81 = +a and a; = 9 — 4 — "s". ' or a; =—9-4- -13. Ex. 2. Let a;'-4ar=:45. Add the square iof ) 2 (e. e. 4), then f ^ -4a;+4=45+4=49. Extract the square root, and x-2= + y'49= +7 anda;=7+2=9; ~~ * or, ar=2-7=:-'5 - ■ ^«*- ^= 6 or -18. X — 14e — '1I J . — 01 . . . , ji^g^ ar=17or— q a:'- Gx= 40 . ^ ' or J. ^^ - - - - Ans. a;--=10or — 4. Ex. 0. a;« 5a;=::(J In .his exa„,plo ,ho oo.^ri«, „f , ;, 5, „„ ^,^ _^^^^^^^^ ts hains -; a„d ... adding to each .ide of the eq„a«„„ (2) "" T' "'" e«t I x'-~-5x+(^Y=G4-^-^+^^ 40 \2/ ^ 4 -~4 -^ J. 5 _l7 Ex. 3. Ex. 4. Ex. 5. extracting the sjuar* root, ar-5 - +7 • 2 "~ — o ' 2 ~2 _54.7_^ 2-2 r— O, or —I. ■pwnciisiaWkiS ■■., .«?-i^i^S^«ft^^ QUADRATIC EQtJATIOXS. 3) Extracting the square root, x~l=-h^ . ^ — 2' 1-4-5 • • ^=^±2=3 or-S. «r-^-/ar_7» Ans. xz:zQ or ^IS. Ex. 9. a;24-3^=28 . . j . Ex. 11. a;2j- aj—on , Ex.12. Let7^«-.20^z=32; find or. Dividing by 7, x^-^^x=^-^. ^:pJ;; \ .^-^.+ (^y.L!52.^^4 100,324 I J 7 V7/ 7^41) 40^49 -4ir* Hence, :.-:L?= + ,/??4_ + 18, anda:=— ±l?=4or-i|. Ex. 13. • 6x«+4ar=273. Dividing hy 5, a;«+-^=?Z2. 5 5 .idfadd' ^ I (%v± and .'+ t+l-?I? . 4 __ 1369 ^ W 25 5 ^25- 5 ^25-~25^ "• Extracting the square root, «+?- +?Z ' ^5~— 5* Ex. 14. 3a''+2^— ini A ii:. S8 ALGEBRA. Ex. Ex. 15. 10. -5z= -2x= =117 . . . =280 . . . Am. Ans. ■• Oor :10or - 0* Ex. 17 4a;'- -7x= :49? , . . Ans. X=z ■12,or 3 -101, A qt.adratic equation seldom aonpnr^ in o ^ . . as those of the preceding exaTples^ ft I t)f 7"" '^ ''""^'^ foundnecessaiTtoemnrorinT 'i V ^^^^^^^^'^ generallv following reductiol.^ ^ '^'' '°^"^^^" ^^ ^ ^^^'^^''^tic the (1.) Clear the equation of fractions. (2.) Transpose the terms involving ^« nnrl ^ f^ .u , .v ha«d and th„ numbers to the ri«hth^„/.4"V.he%;t:^ ^^,V3.) B:v,do all the tornas of the equation by the ..V.C (4.) Complete the square. Ex. 1. 4a;« 1 3 ll=r- Multiply by 3, and 4a;'-33=.r. By transposition, 4.r— a; =33 Divide by 4, and x'-.\x~^ 4 4' Complete the ) 1 1 33 1 ^oa , square, \ ^"~7^+;rr=^+i-=2l24.1_ ^ 4 04 4^64 64^64- 64 Extracting the sq. root, x—-=: +?? ■.8—8" 529 I4-23 Ey. 2. 9 4 9+- 4rJ-J. a; •=5ar-f-5. •«*«^:«i»«SR%g|^:-?^, Ex. 3. Ex. 4. Ex. 5. Ex. G. Ex. 7. Ex. 8. Ex.9. Ex. 10. Ex. 11. Ex. 12. both —4 of QUADRATIC EQUATIONS. gft 5 25 5^2o~25' 5 —5' andar=-+-=2or-l 6 ^--*+lI Ans. x=:zl2or-6. 2^ 1 7 3 ~*"^'^3* • - • - - Ans. x=3 or J. 3 2~" ^ns. x~G or —1. 2 a;+l'^a:"~ -^w*« a?=2 or — ^. a:»-34=|^. Ans. x =6 or -^l 5+- =5] Ans. ar=25 or 1. , 24 ^ '^'^^^ZiT^^''''^- - - - ^«^. a:=5or-2. a;+l a? ~"6" ' * ■ ^"*' ^=2 or -3. x-\-2 — ^"-^•- - - Ans. ar=10or— i Given a;'4-8a:= -31; find ar. ^+8a:4-lG=lG-31 = ~15, x-\-4= + V—l6; / — 15, & ar= —4 — V — 15^ lich r-re impassible or imaginary values of x, - 8* 90 ALGEBRA. Ans. ar=1+^V — 1^ Ans. x=l5 or 1. Ex 13. a;'- 2:,= - 2. . . Ex. 14. £'-lQ£=z-lo. . - Ex. 15. Let 132«+2t=G0. Divide by 13, 3;'' + ?:?=^. ^ ' ^13 13 square of ^ j - +T3^1oi)=l3+nu>=I(i9+loi)=10 Extract the ) 781 109 1 .4- -V/T81 square root r^"^i3~ — 73" =,+ • • aC ■ I- ±27.94-1 20.94 27.94 • 13 13 ^ Is"^^-^^ ^^ -2.226. Ex. 16. ar'-6.r+19=rl3. - Ans. a:=4.732 or 1.268. Ex.17. 5.c=-f4.r=25. . . Ans. ;r=1.871. _ Any equation, in which the unlcnown quantity is found onlv ni two terms, with the index of the higher power double that of the lower, may be solved as a quadratic by the pre cedmg rules. J i Ex. 18. Let a;'— 2^^ =48. Complete the square, .r*— 2.c'4- 1 =49 • Extract the square root, .r^— 1 = + 7- .-. x^=8 or —6 ; and .-. X =2 or yZTT Fi. 19. 2x~7^x=m. ^ 7 , 99 ^-2V-=2-' •7 . . /7\« 99 49_841 2 "^lO^Te* 2» ^U/ '>- i«~^^ 4 4 2 •. by squaring both sides, a- =81 or ~. :-^mLmm QUADRATIC EQUATIONS. 9). ; 781^ >r -2,226. or 1.2G8. 'oiind only er double y the pre Ex. 20. Ex. 21. ^+4,r'^=12. - Aus. z=±^2 or ±y/Z:?, • Ans. ar=3or^i:TJr Rule IJ. Multiply each side of ) . ^^^ ^^'±^^-^=<^, ' tlie equation by 4a, \ *"^" 'i«'-i'l:4a4.r=4ac. Add b- to each side, ) . , , we have f 4a V+^ 4adz-i b^=4ac-i-l,\ Extract the square root as before, 2ax±6 = ± ^4^;^^ and X =i V^^^TZ"'^:^ rom which we infor thni- wp , . U multiplied by /„",•;« 1 "^ T^ ''^^ °^ "'O "q^t^n . side there be Idl{ rJl^^T' "^f: ""<> *° '»'* ' quantity on the left.h„„d sWe of 1 ""■^""" "^ "' "'« square of 2a.+4. Ex net tt f ''™"°;' "'" ^^ ""-' tho equation, a"Sd there Irifes! PI ''''"' "'^ *^''«''' ^'''^ °<" the value of'. „ay be determLd "^'^ ''"'""'"• '■™"' "'''»'> ' i..'tL:7a!;*heSre°?he'p1"'="* '".^^ '■°™ -•+P-^=?; Pl>ing each sTde ofThe equatt Z\ ''' ^^'^^''^ ^T^-nuk of the coefficient of " " ' ^^ *' ""•* '"'''"'g ">e square spondingto theSL + Id'lV^l "^ ™'"'^; <""> corre- radical luantityT ' ""^ ""'"• '» *« ^'g» -. of the • Ex. I. Multiply each side of the ) equation by (4„) ,3 ; j 36.-'+ 60^=504. «ISt£S;I° ,"!!:'*,'?.„«•■> -M.bo found In .!,„»«, ^„„. . I ^H2 ALGEBRA. -^^cM (/>«) 25 to each side ) of the equation, we have f ^Cx'-f-GOjr + o-.^g^j^ ^25=520 .-. 6jr= +23-5=18 or --28• 28 (J or a-=-r-= _42 Ex.2. 14 3 Leta;'— 15jr=— 54. Multiply by 4, then 4^^-C0^-_oig ■ Add (i«) 225 ) ,,, to each side [ ^"^ 4x»-60x+225=225~216=9. Extract the square root, 2ar— 15=+ ^9—4-3 . 18 12 •'•2^=lS±3=18orl2. and;r=_or~==9orG. \ ON inE SOLUTION OF PROBLEMS PRODUCING QUADRATIC EQUATIONS. values of the unknown l;„t ™'r ""^ "f ""^ ' lonniVa,! m, . . "™' quMtity, Will answcr the conditions . 3r.et:^l:.\7rnr: :?;t -h,t i^i>r - Problem 1. : ,.-Ltsi;lVb: cV '" '"*° '™ ^-•' p''^'. "■»' 'h^-- Let x—one part, then 56— a? = the o//<^r part, and X (56-a:)= ;;rocf«c; of the t%Yo parts. Hence, by the question, x (56-.r)=640, or 56x— a:':=640. gUADRATIC EQUATION-: H f 25 = 520. equation, whick i:23; 8 or —28; -n- 1=9. [13; 8 or 12, ADRATIC ve quadratfc ■ one of the e conditions Mys be verv L itself. s, that theii 3. 9S^ ,j , . % transposition, a;»-5Cj:=: -040 ii>- completing the square, ) „ . (UuleI.) j-^-^^-^-f784=784-.040z=M4 anda:=^8 + 12=40or 10. Ill th,^ case It appears that the two values of the unknown I If".?"- ~' ^"^ "^ '"■" ""mbcrs whoso dilTerenoo is 7 ai.fl nun^oei. VV hat are the numbers? Let a;=the less number, then «+7=the r/reaier number, and - +30=3 half their product plus 30. Hence, by the question, ^±Z)+3o=^. (,j,,,, ^f ^^^^^^ or +30=ar». Multiply by 2 . z'-{.7x--].Q0 = 2x\ nf u?!' transposition . x"--7x=Q0, Multiply by 4, and add ) , ,, ^„ 49 (Rule 11.) f 4^-28^;+ 49=240+49=289, /. 2a;-7=y'289=17 2^=17+7=24, or ar=12 less numL3r; hence a:+7=12+7=19 preaier number. Prob. 3. To divide the number 30 into two such parts. Let a'=the /ess part, then 30--ar=the greater part, and 30-ar-a; or 30-2a:=their c/^/e,-e«ce. 1 Tf^nna ^\\T +Vi . V > /r>/\ AV'^v— a;j=«x(oU— 2.ir), or30x-a;°=240-lCj:. I 'i m I i ALGEBRA. Bj transposition, a;«-w46^-- _24Q^ 'bmplete the square.) , ,^ (UvLEl) [^-46^+529=529-240=280; .-. ar-23= + ^289= + 17, and ar=23±17=40 or 6=/m part , 30-ar=30- 6=24 =<7rea/*r part, (n thi8 case, the solution of the equation ijives 40 and .<0, we take 6 for the less part, which gives 24 for the ZlaZ i^da/red ' ''' '"^ """^'^'*^' ^ ^"^ «' ~ ^h^ tSS: sofen'at&'T n''"^^^ '^^'l^^' ^^'^ ^^^^ ^h'^h he .oi V i'^ 85., per piece, and ga ned by the barZ faster than B, and arrives at his journey's end 8 hours aiid or minutes be/ore him. At what rate did o Jh nToL"";!,!^ per iiuur '^ ~ * '" •''"■=-'■ .>vmm.r^i«m: 240=289; !89=i:l7, G=less part , realer part. fives 40 and '' be a pari of for the (/reader the conditions 15s., which he )y the bargain le number of Be cosif the whole for ; QUADRATIC EQUATIONS. 95 5_65025 '""1024* 255 32' 15. a place at 1 hour fasier )urs and 2C Then 'fUT"" r. ^"'"^ "'„*"-'' B < -vels. And — =number of hours for whi^h B travels. 150 x-\~S~~ " ** " A « ney\" e^d't^L'er "' "'"'" ^'' '""^^ '''"''' '' ^'^ J^^ i50 150 150 25 150 By reduction, x^+Sx=54. Complete the square, ;i:«-f 3^-1- -=544.^. _ 225 , . ox-t ^__o4i-- ----(Rule I.): Hence -^l^Clj_fii_ ^^" or 2 V 4 2 ' 4 - ■ 4 225 15 f , 15--3 and x^ — — =G miles an hour f)r B, 2 a:-|-3=9 « ^ flight'?h;L^e^'oo^tTfh^.t? f.f '^ "P^^ * *^^^5 at 'one Iths of ther two bees ttn 5 *^^ ^^"i^^vay ; at another on the tree ?* " remamed. Honv many alightc^d Let 24;»=the No. of bees then xi- ~-+2=2x\ or9ar+16.r»+18=18.c» .-. 18u:'-.16a;«— 9x=18, /r. or 2.^''— 9a;=18 (UuLE II.) Multiply by 8 16i:^--72.r=144. ^^^./S'lb Jl!^^- .'^•- M^ Strachey's tran«lation of t,.. pudson. be foumi^t.;acr:mrwmw£%/;-'7'" 7^7"?" r"'' "P*'" ««'°' 8ta..d. h, that tnu.slution. p 02 '^''"'^ °^ '^^"^•°"' «» i< 96 i I <■? II I ! i ALGEBRA. Add 81; then lG^»-72ar+81=225, or4x~9=15;' .•.4^=15+9 =24, * „ ^ 24 and ir= — —a. 4 ' ••• 2^'=72, No. of bees. tnlTrolIt''^^^^^^^ ^6^ "'" '' "^^^ ^^^. -^^ P-*^ '^-^ P o WL ^iw«. 27and6. JlZLY^I '"" """"'"" «^^ ""'- -h"^ -m is 29, p Q rn. ^''*- 25 and 4. Of tS AS f '^rwtTo rn\^x^.V"^ *'" ^-^ p .- _, ^««. 13 and 8. .?xa„ „/«/« ^.,„,,,. Vhat fro th:"'u™beri ?' "^'"" '° "'" P 11 m. ^«s. 17 and 11. p ,„ . ^ ^««. 15 and 8. B gives £5 aV„e to each '^I ^'IP^hT;" H " ''' "'"^ persons were relieved by A and B rj^ecttve^-j " """-^ ^«». 120 by A, 80 by B. We cost him TshmLsTss H^' '""^ '^^^ """^ there? sumings Jess. How many sheep weru for^S'7.- HavinStsTs o?"f^ " ^r'" "'"»''^' "^ ^''eep 8 shillings a.ird%'„fit\:^-:''.rC- ."''.''•f ""■"'""'^ «' many sheep did he buy? igjiin. How Am. 38. 4 ' bees. such parts thai Ans. 27 and 6. hose sura is 20, 4aw. 25 and 4. s 5, and jth part ers? 1/is. 13 and 8. is 6 ; and if 47 be equal to the i? !«. 17andll. sum is 30; and uare of the ksj- ns. 21 and 9. product is 120. >m the greater, 1 also be 120. ''«. 15 and 8. mong a certain >re than B, and - How many A, 80 by B. ibcr cf sheep sheep u-ould Y sheep were Ans. 40. iber of sheep remainder: at argttin. How Am. 38. QUADRATIC EQUATIONS. 97 rhe'^di^tlcLfso^^ to a place at an hour faster than T) . d i vesttl' ' '''' '' "" ^"^" hours before him. A'" ,'\^^fllZ%A^ ^u J^"^"^^ « end 10 Let ar=6ji^ part, t^enl6-rc=t't^' 'other part. Hence x (16-a:) or 16x-^^?rt70. Transpose, and ar'—l'diii— 70. Complete the square, ' *.'.'.' a:«~ 16^+64= ~r0.4^iij4== _6^ .-. a;--8=: + -v/irO, or a:=8+>/IlG» Let ar= one factor;f .'* then -=the other factor. Hence x'^+^—h x" * or ar'^-f a'^ar*. fro;|tV;Sp;SttTt?e^*J;;^^^^^^^ -'-^^ can „n. may be divided is when these two lZ\l iT 'f ""''J ^'^'^^ ""'"l^^*' jluct therefore/winch couIdTrir?romn,rr''?""^' I''" ^'•^»*^''t P'"» •nto two part., is when each of Is 8 'Tn/^- *''' ""•"?'"'• '' (livide the number 16 into two si c , Trt- ♦! * *i "''''• '" •"^q"'^"^ "to 70 ' the solution of the querontr/r*):' ''''' P^^^^^ should be Plitd tL« two numbers which bdn. n,.,,,:. I ! ; 9 1 ■ II II 98 ALGEBRA. By KULE 17. 2 ^ ^.... -^^5. 6 and 3. ON T,« acuTIO^ OP QnADRAT,0>SQCAT.O»S COKTAi™ ,we UNKNOWN QUANTITIES, ver/ well kiown these cli.es, the two following are * V Case t,' equation;" i ^h ch case X P„^- ''t'!™'"''''' '^ ^ "''"^'^ then submitute Lit Z v„i ")■ ""*.' .'™P''' '^1»^'»». "d -Mvcd fey th^e :?S; rI;;? ^ ■J^^^^^'-- -W^h may be *)lwtion of these equatiotcrntnf^t/'^^^ ^''^ ^^"^'•al of higher dimensio2rtharqt!;,Vralfcs '*'^ ^^ '"'^'^"^ °^ '^'^''^'» equatio,.. to one quadr^i^^f 'ti;^ IZ^ZmT'^'"'^ ^' '"^"^"'^ ^'«' *^ ) i "^^■^^^'^^^mmk.:'^^0mm.^ o two such fee ^ns. 6 and 3. CONTAINING TWC n quantities, in in a quadratic I by means of ) following are the values of 3, is a simple a value of one equation, and ther equation ; ^hich may be equations cm his: t the general na of eqnationa Jclucing the two QUADRATIC EQUATIONS. T * . « ^X. 1. Let a:+2y=7 ) and x^-\.Zxy-y =23 [ ^^ ^"^ ^^^^ values of ^ and y. From 1st equation, .=7~2y, ... a:«=49-28y+4r ; Substitute these values for x and x^ i„ the 2d emmtion then 49~28z,+V+21y-.63,«-y=2S, ^ ' R I? ir °r 3/-+7y=49~23=2G. By Rule II. 3G/+a%+49=312+49=361 :"./0y+7=19 6y==19-7=12, ory-2 ar=7-2y=7--4=:3. 99 J * • 1 • Let|±-^= 9 Ki, 2. to find the values of x and y. and 3a'y=210 From 1st equation, 2a;+y=27 • .-. 2^=27— y. apd xJitry ' ' 2 Hence, 3^y=3x^^Xy=210, or3x(27-y)xy=420 8ly-3y«=420 27y- y»=140; R T> ,T ory'~27y=_l40. By Rule H., 4y«--108y+729=729~500=109 ; .•.2y-27=13,ory.?Z|:!3_.^3,>^ andx^^Lz^^^^ ill! ! If! j 1 1 III I' ! i ! l\\ I !' nil I 100 ALGEBRA. .to'nuXl '° "" """" °' "-^ '"^•'-•^ ^!g!'- What I. Hence, a:=3y ) , and lOjr+y— 12= a;'' | ">^ "^^ question ; .•• by sub.) „. . .' . stitution r^^+y-12=:9/,ffi,i.Mo^=,30y,and ;r»=0/) 3 31 -12 J: .-. 2,'--y= ByRuLE I.,y'-?-v4-— -2^' ^^ 061-432 529 9^ 324~3^^^=-3oT-=324- , Hence, y-?^=??. or v--^'*~q « '•'is 18'?^ ^-18=^' a^=3y=9^; and consequently the number is 9^.. E<4. Let2ar-3y= 1) *'*•' 2a;*4-ary-5/=20 j ^^-^"^ ^^e values of x and y. •°' ^ws. ar=5, y=3. 35; and if four times theelJ^rhl'^--/J"^""'r *"' l^» tl.o less plu, one, the quotifr^n vf w \ ^^''^ "'mo^ ber. w'hat are 'the nSo™ /'" "' '''™'i'l *?,'^" """" -d«5. 13 and 4. ^wj?. 78. Case II 71 When x\ y\ or a?y, is found in every term nf fJ,. . equations, they assume the form of ^ ^^^ ^'^'^ n.ay ho effected :-as in the following E^ampiesr" ^"^"""" ^'^^^^fS^**''^' digit. What ia ^rt. 61, lOx+y liber. question ; rj, and x^—^xf) -432529 24 -324' =3, es of X and y, ar=5, y=3. if the less be ainder will be by three times the less num- B, 13 and 4. i^hose digits is digits will be Ans. 78. w of the two fk QUADRATIC EQUATIOXS. Ex. 1. Let2a''-f3ary+y''~20 5^+4^'=41 ; AMumex=.y, then2.y+3y/+y^==oo or ,/»- ?? 101 and 5i;V+4/=41, or v'- -i] 2i;-'+3y-f i 41 20 41 Hence 2y*+3?;+l ~~^^p^^ which reduced is, Qv'-^^\v^~.\i. 41v__^I3 " 6 ~ "6"* 5z''''+4' .•.i>' % KuLE I., z/'-iJf .lggl_13C0 6 144 144 > .•.t.-i^=:±2!.or.-41±37 13 . 12 lo>or._-__^^_,,.. Let v=5, then v«=-^L__ ^1 _369 v=2;y=^X3=l. , Ex. 2. theg.tlH:77Ta'„TXse'dT^ ^"^ -"^^'P^-d by i« equal to 12 ? ' ^""'^ ^'^^^^«^« multiplied b^' the less Let ar=greater number. ' 2/= less Then (^•+y)X:r=;^r»4:a-v=.77 Assume x-=.vy' Then j;y+^y.^77 ) or /-^ j andt,y«-y^=12'[ if" . • . ^. _ ' Which contain two unknown nnantiMV- tern*. "^ "'"'"* •^-■jiutaicai j-ror. I or 42 11 7 24 -=-3°' 4- ns of the question , 48 48 ■==3-=16, <4=7. the square of the iy be 56 ; and the ly be 40. Arts. 9 and 5. , that 3 times the of the less is 110; fie less is 4. What Ans. 6 and 1. IIARMONJCAL )r decrease by the Tie quantity, then 'Ml Progression. quadratic eqiiationa mmrnm^'.^^l^i^ AHITIIMETXCAL PHOGKEobiO.,. J nils the numbers \ o. n a r n the addition of 1 ^o ,\nh f' ' ''.' ^' '^^'- (" ^icli increa,^' U ^-- V^2 from :^ach%u Lt ;i J^^"^^ '"''^r V che .;./.,..; H'o series itself bo exprelerf h """"'? ''i''"""^- ""m r ,v "eco.-di„g «s rf i. po.mlltZ^ZT'"' " " *-™"«/one; «"'e:, 1. e., the coefficient of j/^r' ;"""=>'"■'« it is ttr«, ««.Jy than the number Vhieh ITJ '!,™ 'f """^'^ '« '" '" '^•^ »«■«• Henee, if the numhel /t'^'"" "y'""' '"■»' be denoted bv (,A tV ' ,u """iibei- of tenns in the serie, w"i be aH'L^'^i't:%iJzt 'r r ^^^ ^^^ ^- then ^ "''' " "le /ith term be represcnte.t by I? , T.. ^=«+(«-iK i-x. 1. Fjiid the 50th term of H,n • , TT Luxixi or the series 1 3 ^^ -y ^ Ex. 2. Find the 12th term offk '" '^^^"^ or the series 50 47 aa a,^ Here a= 50 1 • / rt\ , /,« ' ' ^' ^^- Ex.3.Fi„dthe25th~.e™„f;;,„,3_,_3_^j_^_^ '^"- " ^'^'" ' IM,,f " '>»f4n-l!;S'4| ""'"'"""•'' '"»- ior intennedftTte™!)- . ^^ere the number of terms i^ fi ,.0 1 , !!!:!!::::^^;:^fs|~m^ «■ -™%. ;h„ c^te„ns to b. V fi 104 ALGEBRA. ! It m m I II I ')i; I : I J I ! I i 0= 1 / But a-f (n — 1) d~l l=^4S[ .-.1+7^=43; . n= 8 y .-. c/=0. And the means required are 7, 13, 19, 25, 31, 37. Ex. 6. Find 7 arithmetic means between 3 and 59. Ans. 10, 17, 24, 31, 38, 45, 52. Ex. 7. Find 8 arithmetic means between 4 and 67. Ex. 8. Insert 9 arithmetic means between 9 and 1 09, 74. Let a be the Jirst term of a series of quantities in arithmetic progression, d the common difference, n the num- ber of terms, I the last term, and S the sum of the series : Tiien ^=a+(a+(/) + (a4-2(/)+ +/ and,, writing this series in a reverse order, Sz=l+{l-~d)-{-{l-2d)-\- ... -fa. These two equations being added together, there results 2 >Sr=(«+Z) + (a+/)4-(a+/)+ --- +(a+0 =(a+Z)Xw, since there are n terms j .n .-. fc(a+/)^ - (1). Hence it appears that the sum of the series is equal to th« sum of the first and last terms multiplied by half the number of terms : And since l—a-^{r: — 1) d; .'.S=.i2a-^{n-l)dl^ (2). From this equation, any three of the four quantities a, d^ n, s, being given, the fourth can be found. Ex. 1. Find the sum of the series 1, 3, 5, 7, 9, 11, &c. continued to 120 terms. Ilore a= n .., g^ | 2a4.(n_l) ,/ j- x| d=z 2 «=120J =|2X1+(120-1)2[X-^. :=:(2+119x2)x60=240x60=14400 f=43; . J5, 31, 37. u 3 and 59. 7, 24, 31, 38, 45, 52. sn 4 and 67. en 9 and 109. 3ries of quantities in iifference, n the num,' be sum of the series: . - - 4-a. er, there results ... - +(a+0 ^ terms; (1). e series is equal to th« cd by half the number ■'-- (2). e four quantities cr, d^ d. 1,3,5,7,9, 11, &c. 44 -1)2[X-^. 0=240X60=14400 ARITHMEIICAL PP.OGRESSIOX. I05 ^c^^tolo SLV'^ «"- ^^ ^^^ -ies 15, 11, 7, 3,^1, ^5 .= -4l^^^ ^■'^"-')<'l^- =|2xI5+(20-l)x-4[x« flf=- 4 «= 20J =|2X^5+(20-l)x~4[x?? = (30-.19x4)xl0 ^ = (30-76) X 10 = -40X10= -460 14, &/ ^"' ^^^«"-«^25 terms ofthe series 2, 5, S H 34,V ^''' ''' -- o^^« -- of the series^- ^30' li^%f"^^^-""^«^^^^--o^theserie^r^t|! Herea=|. ''■ ^= ^^+{n—l) dl!!, ^=i i =|2XH(150-l)xj[i^ n=150 =/?4.149\ 151 ^ ^ ^ I3+T"/ '^=-3-X75=377.5. 2it &c. ^"' *'^ ^""^ «^ 3^ terms of the series 1, 1^, 2. ^««. 280. p PROBLEMS. term? ""J ^^^^^ ^0. What is the /rii 2& Here A8'-„i 240 ) . c f« , ^ » •^^" .. >S'=|2a+(n-l)rfl^ '^"■" ^( ^240= J2a+(20-l)> ./ J ?( -= 20 J =(2«-19x4)10 ^^ l24=2a-76; •■• 2«= 124+76=200. Hence the series is 100, 96, 92, - Hit }.t ."}>er of - terms P • « -a.^.. 20. | Prob. 7. The sum of an arithmetic series is 54, the ^rst ifer»n 14. and common difference —2. What is the number m tei ' « i Ans. 9 or 6. jeries I's 1455, thajirsl What is the common 30 1455 5=1455; ic/=87; cf=3. JC. series is 567, the Jirat the number of terms. = S =507 567. ►=570, 1= +24 ; (=21 or -27. series is 950, the com' f 25. What is the first Ans. 2. series is 105, theirs/ What is the common A71S. 3. ;ic seiii'- ' ■ 4*0, Jirst ''hat '*•■■ •■; , J>er of } series is 54, the first What is the number ni Ans. 9 or 6. AHITILMETICAL I'ROGKESSIOX. j^? tfnrd, and so on. Jn how man t ^ ^-^H *''^'*^» 20 the journey's end? ""^"^ ^*.^=* ^'^ he arrive at his Uere is giver. «= 30 i SZ ~'iQly''^'''^'^^'^-^^^r of terms. .-. |2x30+(n»l)x-2|^^=198, (31— «)n=i98^ w'-31«+ ©'=? ■198= 109 n- 4 - 4 31 ,13 ••"=2-±2-=22or9. pos1tL"?lir of rS it"'^ r '"^^ ^-- 'he two the traveller's arrival at his ifJrn/' '''^ '^i^^-^^^ ;^^r^o^. of that if the proposed serieiHs^r^'' \^ "^"^^ ^^^-^-' terms, the 16th term will htL;; ' ' *^?'' ^^ ^""^ed to 22 terms will be «^^«^eW L wh nh "'^' !"^ '^^ ^^"^^'"'"g ^^ traveller on the 16th 'day anfh' ^ ^^.c^^OM during the six days foIIowin^"'^ ''u/^'' 'P^''^^' hini again, at the end ofX 9oTa^' ^""^ ^'^'^ ^^^^ brin« ^vhieh he was at the end of the l^h '^' '? '^^ ^^"^^ P«i»t at place whence he set out '^' '''"•' ^^^ "^"^^ from the pthering up 200 sSiS pdntl"^ f!?'" ^^'^ ^^^^' h> -.^ 2 feet from each other js^tl?^'!?. k"'' ^' '""''''^^^ ^l^Oly to a basket stand nffaf-f^iw ^^^™g^^^«h stone [he first stone, and C^X^Zt^^T^ ^"^ ^"^'^^ ^^^^ basket stands? ^^"^^ '^^^^ the spot where fh^ 3 jj ft: t /* % 108 ALGEBRA. i-ii M III '//Sb."- "'^"'')' "«-»''(#•-- 2 feet, and «„„J„ Here a= 60 ) « f n ». «=200) =(120+398)X100. =518x100=51800 feet, ffenee the distance required=103"600= I'q .""'i""' .'"'• What i, ««-a^:;'eSsf' ' '"' '"^f 5?ro-- the .a4, aifd so o^' "wfft' ml'^ H"? '*™>-«4" ohaHt,at.he,endofihe^\™"^^t..''lcrir6Vi''' before B; B follows him Tl !^ I '"""''^ ""'• 20 mhiutos « the second, 7 the Mi™ and so? ' f 1"""^^ ""'>"' hour B overtake A» ' '° ""• '" '"'»' many hours «-i|l ^M. in 8 hours. arithr«e' proIL'slo^' LT"^ """"'f "^ 1™«ities u, whose «»» is ela to ll?!- T'.' ''^^'•^«« ^^ 2. and if 13 be added to heZ„!;r''"'°'';T""'"-' """-"'ver; by the numiJofV^, S Lto« "; ""?,? ?'^ ^-^ ''<' divided «™>. What aro^Smberrr """ ''^ "l"*' '» *«>« Let the/r«< term=x, } and iVb. 0/ tenns^y; ] ^nen the wcowrf term will be a'+2. '^ ''' ^^^''"^^'^'^ ^^^+(^^^X|, substitute .for .,2 for ft, and.for., and it becomes 2l+(^riy3^|(^,y..__ ' " li>r the swm of the scries. By the problem, a^y+y^^y^^^^ ^^ ^^^^^^ unci — : — -:. '-!-., GEOMETRIC PROGRESSIOX. et, and numitef 109 ftirbngi, feet. G40 ?ave 1 shilling ird, and so on. w. £110 95. giving away a hree farthings disposed of in ^ lis. 6fd of 6 miles an 1 20 minufes hQ^first hour, 17 hours \vill n 8 hours. [uantities in « is 2, and ; moreover, 1 be divided 1 to thojirsi 2. •r a, 2 for b, Hence, ^^?=;r, or ;t«-8x=-15; .*. «'-8^+lC = 16-15=l, anda:--4= + l;...;r='5or,1, Fmm which it appears that there are ^^^J'sets of numt,. ...•^"'^'':- ^"^^ ^''''''' '^ ^ ^^^^a^n number of quantities in arithmetic progression, whose >.^ term is 2 and wW Lm Am. 2, 5, 8, 11, 14. ON OEOMETHIO PROORESSION. ra. If a series of quantities increase or decrease hv .!,„ contnmal ,«,,//,>&„&„ or division by tl,e same qum.Utv then continual ^^.-Wiif ' ^AfanlVet^^^^^^^ Y '•>; m,dl,pl,cal,on by J), are in Geometrical Progression 76. In general, if a represents the first term of sncl, . series and r the common mulliple or ratio then mw h! series itself be represented by a or or' n^' „Ta. ""y '''« will evidently be aL-„«..«„/o"r' ZZ^i^; 'ser es „e;,„:S eri;s"u,e1l;':f '••"^ ''^™^^'--^'™''°'' '" "» °'^°g" g ""'" " "' geoirietncul progression found f ~5 tiie common "' -mmiikm ei-ii),, no ALGEBRA. iJere the common ratio -?-o "~1 — ^• ^ 2, rind the common ratio of the series ^ ^ ^ . 3' <>■ 27' ^^ I" this example the common ratio = ^^2_2 9'3-3' •^x. 3. imd the commo,, ratio in the series 5 1 ^ ^ ^ TO. Lot S be the sum of the aeries a, „' a^ J „ Mu]t,p„ .he eWtion b, ., a„a itbeeot; " ' = ^^ S"i'-ttho.,,.e,„..„,„„.:,:-j-=^ •••'~a=.^^^,orO.-.l)^^„,.. -a; and therefore, S=^^^~:Zf^ _« the convenienee of ea,cu,ation, .he-efore, it is better '" *" '"'"' '^ "-"-pose the equation into S - «-'"■" t '"i'ltiplj-ing the numerator and dnn„ • . ~ ^^^' ^ — - hv -1 denominator of the fraction r—1 y ^' ™- "■ ' bo the last term of „ series of this Icind „ W"tion.lheref„re,if„„y,h_^/f7'^, '-l' ^""^ *^'» l, ke given, the fourth maVbefoun *"/"'"' '»"'""■«- ^' «. ^ 78. Whnt i« fi,« „ ^ 7 ! ~~ ■ — — ___ ^ «'^«'"«trieal K^gr^^i^t" '"^ "'« «»'» a^" « terms of u series of „„n,bo« cal progression 4 8 2 3^ 2 3 5' 25' *^^- A 3 ns. r. o ^, ^c, then = S. ar" =rS. we have, ^rs are Jess it is better he fraction kind, then >om this ?s ;S; a, r, >nnimb«n GEOMETRIC PKOGRESSIOX. HI Ex.1. Find the sum of the series 1, 3, 9, 27, &c. to 12 terms. Here a = 1 r= 3 «=12J .^_^"^2^_ 1X3*^-1 r-1 3-1 _8P-1 "" 2 • 531441-1 531440 ■ 2 =~"2 — ='*^5'^20. Ex. 2. ^- Find the sum often terms of the series l4-?4.^-. ® ^ 3^9"^27' ^ a..- 1 2 r_= - 3 f?=rlO 3 Now L2)'"=2L»^ ^4 ■■-sr- 3'« "59049 ' 1024^58025 59049 ~59049' 3x58025 174075 and S=z 59049 59049 * 81^^^- ^"'^ '^'' ^""^ "^ ^ '''"^^ ^f ^h- series, 1, 3, 9, 07 Ex. 4. Imd the sum of 1, 2, 4, 8, 16, &c. to 14 terms. , , , -^ns. 1C3S3. Ex. 6. Find the sum of 1, -, - -1 &« m ft f 3 9' 'n' • '^ ^ terms. '3' 9' 27' . 3280 Ana. . Ex. 6. Find three geometric moans between 2 and^f Herea= 2) And ar-^'^/ /=32 I • 0..4 2r* -32, :JG. And the means required are 4 8 10, rii 112 ALGEBRA. iiii ! :ii:|iiF 11 ^. 7. .Find two geometric meana between 4 and 256. P « -P- J , * ^^^' ^^ ^^^ 64. J^x. a Find three geometric means between | and 9. p^ -p. , "^^^- i» 1, 3. T «? tSf,"^^t"« "lean between a and /. Ihen a, x, /, are three terms in geometric progression. and -= _ a X or x^=zal .'.xz=y^ Ex. 10. What is the geometric mean between 16 and 64 1 Ex. 11. Jr^ert four geometric means between -^ and'sT Atis. 1, 3, 9, 27. PROBLEMS. squares to 21. ^ *^ ^ » ^"^ ^^^^ «wm o/M«V I'-fc p ^, :^y, be the numbers. Then by the problem, X X* '~,+x'+xy=.21 . . (2) Trom equation (1), x{l+l^.y\ ^^ '• Vsquanng, -'g+?+3+2y+y]:=49 .% by subtra«tir.ii, a:'^^^^2+2y\ ~I^ or 14z=28 *. X=: 2. 4 and 256. 'S. 16 and 64. >n I and 9. Am. ^, 1, 3. ndl. ' progression, en 16 and 64? Ans. 32 1 } and 81. 1, 3, 9, 27. 5 progression, 3 sum of their problem, GEOMETRIC PROGRESSrOJf. Tliis value of a; being inserted in (1), us 5±3 ^ 1 Hence, the numbers a: e 1, 2, 4; or 4 2 1. .•.y= Ans. 2, 4, 8 ; or 8, 4, 2. nhosT";^- islr^nll'/^'' """J^f ' '" S^^"^^^'« profession Ans. 3, 6, 12. ^>,o . °°' ^;P°If ''''^ **'''^^ numbers in geometric procrression 's 100. What are the numbers? Ans. 2, 10, 50. Prod. 5. There are three numbers in geometric proffressio,. atrr„'^er 'r^ *" ^^ o' *^>-™^ 'j«°i«S -4/js. 1, 5, 25. I OS THE SUMMATION OF AN INFINITE SERIES OF FRACTIONS ,H ThTvaZ oTr """"'' ^•"' "^ ™--™0O0F Z,N„ THE VALUE OF CIRCULATINa DECIMALS. «nJ?' ^"^ ^^"''''^^ expression for the sum of a ffcometi-ln sct^es^hose common ratio (r) is a fraction, is flTl^) '^'=T=7"' ^"PPOse now n to be indefinitely great, then '•• ('• being a proper fraction) will be ii defin tely ma//,» »j * When r is a proper fraction, it is evident that r" dccrea^n as » increase: let r=- for instance, then r^=l- ..--J= ^-__I . 114 ;l I * 'i: ■ I fl ALGELRA. tt^CraTor'::^^^^^^^^^^^ ^-^^ -Peot to a in v^hen the number of terms is infmite is-^ ^ ' 1-/ Ex. 1. Find thesum of the series l+I + U^ a,, , • ^ . Here a=:l ) _ , 2 Ex. 2. Findtho value.ofl-L-ij _L,. _, . , Ex. 3. Pind the value of H-i. -,-1 . ± . c , . ^ . Ex. 4. Find the valueof l+l+i. . 27 ^ . ^ Ex. 5. Find the value of 1+1+ A . ... . • t""'' ^' bers composing whiIhL<,''"^f^"^ ^'"■"^«^^' ^^e num. _____^_»J^^^^ progressions, whose 80. W»iD» to *u . \ ~ ' ■ ~ — ■ ■ uun.bcr ortenua^; alter" '"''"' '""' of ^geometric series, when th. ■ha4i respect to a in [ing the valua '>S' approaches, infinitum. ' infinitum. Ans,^, infinitum, dna. 4, '■nitum, Ins. ^. tJous me. the 11 um- »s, whose 'i when thi * GEOMETRIC PROGEESSIOX. llo 1 1 1 common ratios are — ~ * » •,. 10' 100' 1000' ' *c^<^rding to the num. ber of factors contained in the repeating decimal. ^- 1 rind the value of the circulating decimal .33333 &c llils d^ecmal is represented bj the geometric series ' TO'^'IOO'^IOOO"^*^''-' ^^^'« >•«' '^^»* is A and com»ao« 1 ^" 10 Hence a= — . 10' -2 **~10' .-. aS'= 3 10 1-r 1— L ^^-^ 10 9~3 Ex.2. Find the value of .32323232, (fee. at/ m> //«;;,. 32 Here a= . 100' r= 1 100' 1— r 32 32 1 1 100-1 ~99* 100 Ex. a. Find th« value of .713333, &c. ad infinitum. The series ^of fractions representing the value of this decimal are — + (geomet-io series) --1-+-J_ . &. 100 ^ lOQO^lOOOO ^ ^^ 100"^ • Here ,.__ .-. S= 3 1000 3 1 J 1 1000-100~900~306' 10 Iknce the value of the decimal=/~+^)Zi4. .1.^214 \tvy X'^OO /iOO 3oo"^a0(j 107 150' 116 ALGEBRA. ^^' 4. Find the vaJue of .8134^4^4 a 2re a=r-?f 1 -^^^4^434, &c. ad injinittm. Here a=Jl_ 1 10000 \ 1 34 .^=--l_^_iO00O^ 34 '"'=100 I ^~''' 1— i- ^^^o^o-ioo'^y^ ^ 100 And value of the decimal =ii . e 81 , ^ ^x.5..Findthevalueof.77777,,V.n. 100 100^9900-i)900- Ex.6. Ex.7 EjT.a Ex. 9. « « 9 .232323 &c. ac/ m//»7«m. M Ans. 23 99' M .83333, &c. arf ^V/^^Vw;;^. Ans. --. •'^l4l4I4,&c.a«?e;y?;,,V..m. . 707 Ans. u •956806. &c.<,rf ,■„>,•,„„, Ans. 990* 287 Let S=: .813434 .-. 10000 *^=8134.3434 a^dj!00^__8i.3434 .' .* * .* .-. 9900 >S=8053 " . <._8053 ' * 9900' ^^ ^^^"^-e. -"V "-ves over ^ a mile the ^mw" '"-■^"^^Vg cause, it *m;,i«r second, | the M,Vd; fnjiniium. \ f 34 -lOQ-yyoo 8053 A 23 99 im. ns. Q' itum, 707 990' ns. im. ns. 287 300* found 1 -ause, it HAKMOXIO PROGRESSION. 117 and so on Show that, according to this law of motion, the body, though it move on to all eternity, will never pass over a space greater than 2 miles. ON HARMONIC PROGRESSION. 81. A series of quantities, whose reciprocals are in arith. luetic progression, are said to be in Harmonic Pronression. Ihus the numbers 2, 3, 6, are in harmonic progression, since heir reciprocals ^, ^, J, are in arithmetic progression I -4 being the common difference). ^ ' Ex. 1. Find a harmonic mean between 1 and — - 3* Let X be the mean required : ^^" ^' T' "2"' ^^^ ^" arithmetic progression, Andi-l=i_i X 2 X X ^2 5 .*. af= 2 4 5* Ex. 2. Find a third number to be in harmonic pro^re«8loi mth 6 and 4. i o ^ Let X be the number required : Tnen — , ~, —, are in arithmetic progression. All 1 And - — --: 4 6 1 * X 1 1_ X 4 -i-i. 2 6 1— i 1 111 1'-' r.l I: if ar=3 118 V i ALGEBRA. rhe re.,rocak of 9 and 3 arc I and ^ ., • k T and last term of an arithmer ' ^' ''' '^'•^^'■^' amr^Uc means a^tf r^inSeTt' V^^^ ^^^^ 3 according to Art. 73— "^^erted. We have therefore. 3 ^=5j And a+(n— Ijc/:--/^ 3 9 .*. d=z 3^ '9 9 18' 9 Hence i 1 ^ l^' '^6' 9 '18'-^ the arithmetic ..«., to be inserted between ^ and ^, and therefore tkeir reciprocals 6 ^ ^^ are .three harm^^^ -- -.uired. ' ^' ^' • '• ^"' ^ ^^^--- — between 12 and C. Ex. 5. The numbers 4 «n,i p . ^»»- §• progression; find a third te™ ^^ '"° '"™= »'''' ha™onic E^-O.Eindt.o harmonic n.e.„s Ween S4 .:!":;" E..7.InseruhreeharmoniemeansheJ::r5:;fdf S2. Let a. i c t/ ^ ^ T. 2 ' ''' 4 • *«-».p4re:sio;:;hli'i%"r7°'"^™"«"-" « ' b' c ' (/' 7' ^C" are in arith and 3. are thejlrst een M-hlch 3 } therefore. inserted ' 2' 5^' ns. 8. trmoiiic '. 12. 56. 163. is. 15 I* ties in arith, HARMONIC PROGRESSION. 119 a^ From (1) -i-i-i- 1 b a c b ..(I) • ■ - (2) ^ c e d (3) &c. = &c. a--6 h—c ab c a—h b—c a c a a—b c b—c' o converting this equation into a proportion, a:c::a — b:b~c Similarly from (2) b:d::b^c:c-d (3) cie-.-.c^did—s and so on for any number of quantities. . These proportions are frequently assumed a/fh*. /.w •.• to quantities in Aamo^^zc progression nnf,!. 'Y'''^!!''' expressed in words-— if anv//,;.^T!l.. "^^>' ^^ *^"« gression be taken M. f^, / quantities in Aa,v«on/c pro- a«=jy; .'.a—b» c*=b^', .'.c=bi (1). r2\. By multiplying (1) by (2) ac=b> ! + ■J, • ; I ■I ii i-i;!i -s?.«^^M«e.t*-»n*f^»^t«.v;'J^ I I I $ =1 V Vl 120 ALOEBKA. l^ut by geometric progression ac=A» and .-. 2=il+il y X z or i-i=i_l y X Z y' , CHAPTER VII. ON PERMUTATIONS AND COMBINATIONS. 83. By Permutations are meanf t]io «., i, «-lHch any quantities, n i c Te V ""^^^ ^^ ^'''«'^^^' spect to theirorder, when takon'/ ' T ""^'^""^ ^'^h m f , «?., are the difre;ent pe^ml^e^^^^^^ ^«' ^^' < ^«. ^ ,<•, c/, when taken two aTdtZfLf^^ ^T ^"^"tities a, 7^, ci., of the ^Am quantifies Vr/.' "*'', "^•^' *^^' *-« Mre. together, &c., &j"^"^'^*'^ «' *' <^. when taken ^Are, „ J '^'^m:^z:z:t^ th., hy («-!) be substituted for ;. in tho l^^Vo^-T''?''''''^'- *^'^'' '^ of permutations of n-^l Z!t ft ^'^f ^^' ^^^" ^he number will be (.-1) (,^,) ^ jJj^^X '±Vr.f!!' ^- together -> c, a, ^ O.C., taken t^o and two\og^^ a^e'^^J^;:!!^; fsiWi, ^r.^Agmm^^,*^^me'—- ■. PERMUTATIONS. 121 f changet » with m her, three •*, cf/, '0'c, Acer, ?Aree and then, by ations in B(n--l) 0^, ^, &o. ■he form e. ; i. e. d two iS )gGt her, For if lumbei' 3get her /ions of quantities a, 6, c, d e^, &c, taken //.r.. fl;.rf three together ir which a may stand first for the same reason there a?e (n-U il;i ^' ""'"*'' ; ''' '^^ permutations of his kind viU therefore amount to . (/i— 1) (w— 2). r ^fetHer. ^'''^ '^'' """'*'' '>/;>^m./a^/o„, of n things taken By Arts. 85 and 8(5— llie No. taken /m;o together — ;i («- 1) --" " " '/'^^^ " =n (n-1) (,,«o) Sm.dar]y " four « =n (n-1) (n>2) (.-3) supposed 'tX'ttLZ'^^t is ff ^;h^ P""f ^^ ^'^^^•^' '^ taZs of. things rcf !('^t tXi; t ^rgetw r-- n (,.-1) (,,^2) (/.-r+2) will be by putting .-I ^ :):: ^X^^;:-^^^^^^-' {n-l){n-2) («~r+l) Now,^if a be placed before each of these permutations, t^'^ore {n-l){n-2) («-r+l) ft1s"ciraftLl'e>"-ir''"ry *^^^^^"^' ^" ^-^'^h - ^^-^d^/'--^^. ic IS Clear tnat there will m like manner be the same nnm)u.n here arl IS *' ^'h? *!•' ^'»"d mr.o&./y /„, ,. »,„, „, .vel_y_sta„d first,, that is, „ times^ri)'' [t^^g] "^.'•f';'; 'or«(„_l)(„_2) („_,.+ ,) nJ! • *"\''<*™ P>-oved, that if the law by whioh the c^ ^TI^JH^l ".'"""-.of permutations of ^, thln's t^kt!; - "-„„_.:ci 13 iouna, oe true, it is also true for tho nov* uperjor number, or when „ tiling, ,,« tak „ r,i::, ! but the law of .he expression ha? been found to l,f,lj fi',' 5 XI 1:1 p»»«(f^.^«,»««^^jf^,,,,.V.^^^^^ 122 ALGEBRA. m m taken four togetfer k U t.^ T '°«f ''^'•' '"'^ '^ "■"« «to and si on fo?any number no? „ "/""T ''^™-^^'^ '"S'^'her, taken together ^ ' «'''''"*'■ "">" »> »Weh may be mathematical sconces ® ^''"" "''l>°'-t»»'» in the titicI'al^;=%Ln (Itn'-^'tr "^"f "" *^ •)"""- mutation, which might be' fo;m;dV,.^^,'^,*,^""™''''• "'' P" the word •.«,.„... aVflxS/xTxiLlar '°'"''°''"" thefit^t\^trbi::tutSrrTr"'''-°^'''"-. mutations by the number Tf 1? .' "^^hole number of per. arisen if .^/j;j:ZsXScTi;,st::d"Si"°^^ of the same letter. Thus if the samrilft 1 Z,''^ repetition then we must divide by 2x 1 ?/,•!"■ m" °°™'' ""'«' .nust divide by 3X2X I ; if p t ,ncs bt f of ''"" "T' ^ any other letter which mnO „„i ^ ,• • •■?'! »"'' ^o for the^eneral e.xpre2"n forV , IC'o^f ?" Tt^ ^'^™» thmgs. of Which there are , l^'T^lX IZmZ' "' o' another, &e., &e is ^ii!iriijfcl2H!ii:3) • . 2.1 the permutations which may be formPfl Z.r ,i \ .. posing the word "m.m..."iince .TccmVl; ' ^'""'' T^^" 8.7.6.5.4.3.2.1 '''*'^' ^ ^"■'''') '^^'^ 1.2.3. >a„ges=lx2x3x4x5xcx7x8=40320. ^^^^mm. w together taken three smonstratcd * true when i-'e together, ich may be nonstmiive mce in the the quan- ' them will >er of per composing I* of times, ^erofper- ould liave repetition cur twice^ 'hrice, we nd so for . Hence ions of n her, q of . Thus Bi's com- oice) are ts which be rung 0320, COMBINATIOXS. ^23 Jl\I:t " ""S^ °f ''"'■^'-' «<"»"-■ how ™a,.y u.jnaU The number of signals, when the/aj, „e taken- Singly, are Two together =5.4 . . Three . . . =5.4.3 - . Four . . . =5.4.3.2 - Five . . . =5.4.3.2.1 = 5 = 20 = 60 = 120 = 120 .'. the total number of signals = 325 ietfeS,tuti ariiiT""""'"' ■=» ''''''7:^:^^'' ^I'f ^Ai ^^^ ^"f"^ permutations can be formed out of th.. Xn af S ""' ^''"^^'^ '-^-'™'^. ^. thetSf j!^ Aris. 2o20 and 1680. ON COMBINATIONS. w|^!h ?j„ttr:7onrofr;t,rtr/f t"' are the ««6»„&„ which ean hef^tl/lfJl.'/' quantities a, i, c, d, tal'«t.iiu, 20. If the numerator be equal to the denominator the frac- tion IS equal to h Jf the numerator is less than the denoin- n.ator, the fraction ,s less than 1. If the numerator is gi cater than the denominator, the fraction is greater than 1. 21 Of two fractions having the same denominator, the greatest is that which has the greatest numerator. Of two • fractions having the same numerator, the greatest is that which has the smallest denominator. 22. To render a fraction greater, the numerator is multi pi.e. without touching the denominator, or the denominator is divided without touching the numerator. lo ronder a fraction smaller, the numerator I^ divided mth. out torching the denominator, or the denominator is m„lt;. pnca witijuul louciiing the numerator. 23. Tlie two terms of a fraction being multiplied or dlri d.vl by a like number, its value remains the same. irt I f S f lii it 1123 APPEN DIX. 24. Any whole numoer may always be put indiferently vundei- the form of a fraction : ^he only thhig is to give it the unit for denominator. 25. To take any part or fraction of a number, is to mnlti- ply it by that fraction. Thus to take the | of 12=12x| = ^^=8. Then if the fraction to take has the unit for. numerator, \vt have only to divide by the denominator. Thus to take the h- h 01" i, &c., of a number, is to divide that number bv 2 3, or 4, ' *^ ' 20. To incrcfise a number by any fraction of itself, is to multiply it by a new fraction whose denominator equals the sum of the two given terms, and whose denominator remains the same, Ttus to increase^GO by ^^j=60Xi2 = * j§"=:85. ON RATIOS AND PROPORTIONS. 27. The ratio of two numbers is the quotient of the first number divided by the second. Thus the ratio of 15 to 5 is 8. The connection of two equal ratios is called the rjeometrlai, . proporUoa. Thus 15 : 5 : : G : 2 is one proportion, as the ratio ot 15 to 5=3, and the ratio of 6 to 2=also 3. The first term of a ratio is named antecedent; and the second consequent. Ihe first and the fourth term of a proportion are called ihQ extremes ; the second and third are called \kii. mediums. The mediums liiay always exchange places without dis- turbing the proportion. , 28. The product of the extremes alwavs equals that of the mediums. 29. We determine the fourth term of a proportion whnn ' unknown, by diuding the product of the mediums by th fir:^•t ; \erm. m 1 ON TflE SQUARES OF NUMBERS AND THEIR ROOTS. The sqtiare number or second power of a number is that number multiplied once by itself. Thu ,sqi,bserve that the successive difference befwpnn f l.o *iways exceeds itself by two un ^^ "lT.n= f '^''*''?' 30. A number which is i ot a nerfprf cmm^^ • ^^ ^ <^oot, add to 25 the doubll of 5lV^~i i* ^? ^'''■' ^ "' *^^ 25 + 11=36, with tlu^l^lfe'^"'^ ^'"^^ '"''^ ^^^^^ off^trlt-^I^ZrSot'r ^ --^"^-'-t i. Le the number be 53, of which the v/=7+thc rfm.ind.. -15 cut'offfh'' '^' ^'.°'i' '"'"^ '''' do^.bIe of the "ot 7 + 64-15=49, of which theXT ^~"^ = ^^'-^«" ^^''^Ihave given number, ^'ampl . ' '''^' '^' '^^'^^ ^^'^"^ ^he Sujiposc tiio number is 86, of which tho /-o_Lfi >iov 5. Ju order to have tl^ roo -ft f,! '^~ i"^ uf ''''^'","- rf,ot9_i_]7 nr]>^ ihl ^'^.^oot=:8, to tho doublc of tie ^2. Then l£lSf_nI ^^f ''^•"/J^r 5, you will have 17+5=. 34. Knowing the differonco of tw -xtractei M iu:nb.3rs and that /^._Suppose -^10, 20, 30, 40, 50, of which the sum = 150. Then that sum divided by 5, the number of the terms — 5 =30. Hence, as we see, the middle term =30. If the number of the terms be even, the two mea^is taken € r J^|^|-f^^jrams '■- tio a* Iherft v h the ratio ' term? be ortion 3x rogros'ilc n Tiultipliod h the half IS 5-f-thu IS. ' the first ut one, or 1 througli, . Remark Al PENDIX 133 even. \{ eh equals rk ical pit), ium divi. it if tl:al r \ or f the suiw le terina \s takes i tS"' '^'"^ ''^ ^"" ^'^'^^^ ^^ J-'^ the number of the .umlTof lfo?lT5 d'-.'^A'^'^' 2^' 3^' «f -h>ch the terms=:io5-35 lln^Af ^ ^^' '^^ ^'^^^ ^he number ol first'mdtifiS w'h^^'ati'riSrr P^^^""^^'"' ^^"'^^^ -^^ equal to th^ num/er of the termsll " ^^'' '^ "^ ^^^^^ The,,, oWve tlrl^riS— fel^Tx1.^=i^'- term is takon from the product 3lH ^ ■ ""f " ' ""^ '''^' by the ratio, lessened by a uS '''""""'^" '' ^'"'^"'' rJ'>t:t';fsrz:-o!o «/'« = ^, ■■ '«2, of which ,,,0 =4b. From 480 „".;« the fiTt ™ o" '^™ '"^^^ Now tl,is remainder 484 : (thera.ro t-xtZT^ll^!; "^^ "r the terms. " ^^— j — -;4^, sum p.y'^^r'i/V^o" tf:if -' P""f'-P'- o„ progression „p applicable t:^„wrr;pfoS„?iri,o"', '" """'" "■■™ change the words/,.,!^ Crdlf, ilL^r"""^^ '^' ox WV,SIBt„ NUMBERS w,T„OUT A REMAINDER titule o'^ t™nrS,:^S:ferthr '''r '" " """ bcr exactly divisible bv another Th '^"'^^'' ^"^ """' n-de without any remainder ''"'' '"" "^'^-^ '-^'^'^^^ «'' oonsKiered as simple unit ™!w it ^ll^^^rt"^ '-^"■^^' rigl^atV^diSelfr ^^'^^^^^^ ^^^^ ^^^ ^^ %-s on th. n; ?" nil ""'"^'''' germinated by a 5 or an n i>> 0, all even numbers a lH 12 idy divisible by 3. i I iU APPEXpiX. > ■ill I' Jili .if' •lit , I 000 Vc^' '^^^' ^^^^' <^Cm all numbers ending with 0, 00. ^J%}^\fSTJ''' ''''''"" """^bers whereof the sum of :hi 4rh n h ft.. !. «'''f ' ^'^•' ^' ^^"^^ ^'^ the sum of the 5>.] ik 19 II ^^-'OrwhosedilFerenceis 11 or a multiple of IJ.' Ti o- 1 ''^^'' numbers already divisible bv 3 and 4 liy ^o, all <;i;en or «/jei;g;i numbers whereof the two last hgures on the right are divisible by 25. PROPERTIES AND VARIOUS EXPLANATIONS. 4-fhi' SI ^^"^ TT^^l 'iT^'"" ^^^ S^-eatest equal (the sum -f the d flerence) (fiyided by 2, and the smallest equal (tlS ^wm-the dfence) divided also by 2; whence it follow that the sum mcreased by the difTerenee equals .'e./crthe greater number, and that, diminished by the difference il equals twice the smaller. ^ -aicicace, ii ,.t^'Jf ''*^?^ two unequal numbers, wiihoui altering thetr sum the greater ,s dmanished by half the difference, mid the lesser increased by the other half. 50. The difference between two numbers cannot be superior LpiTse" ''' '''-''' """^^-' ^- ^^ -^ ^^-» or 67. The least difference possible between two whole num- bers, even, or between two whole numbers, od I, is 2 58. Knowing how many times A is greater 'than B, wo at o^ice perceive how many times B is smaller than A. pre e7wno the same numerator to the given fraction, and forming « new denomniator from the sum of the two terms. Examples- A being > B by ^, B is B by I, B is < A by /j^. stantly determme how many times B is greater than A pre- serving the same numerator to th. given fraction, and flVrmTno a new denomniator of the iriven ,hmnu>u.., ^„ a, . _.u. , ",\"8 numeratcn- is subtracted. Axamples : ' •— u uio figures on hose ligurc:! vith 0, 00. sim» of :1m of ilie JM, tipleof li. and 4. e two last APPEXDIX. (the sum equal (the it follows twice the fcrence, it nng tketr e, and the e superior equal or lole II urn- B, wo at reserving ng & new npks : 3, we in- • A, pre- formino A benig < 13 by |-, 13 is > A bv t' i&c of the givol fractiot iw«> '°™"^"* "'« '»•" '«■•- Aboing|ofB, (i=i„,^A.' Abcingjof iJ,U=|of A co,fmi„f;h:ii.iit:„rw'„ ir '•■"' r"^ "^ •- ""-"'-«. number to the o.^.^^Z^'Ch.Z^''. P"''""'"" "^ -- «"« its den„n,i„ator. S seco, d .w'™'''";, "'^ ""= ''■•«'' proportion of the lesser m,nbert„ .h *'" ""^'^^^ ">'■• it will expres.s the nroporZn of , f S'™"'"""' """^ "»"^''i i'-^am^k.-T.^vrtT^'^Zl 13 T'"f 'V"? 'r"^- 12 and the difference 2- the ™!„ „ . ' °^*l'"''' thesu,n = the other. Then 6 in 'a ,Ltbn rC'''^ ""T'^"' « "'"'^ second fraction such as it i, we «1> S, 1='' "'"' '^'"«"'g «" t'.at the lesser nn,„b:r=V:fThV",,t-'' *• "»' - to' sav the s„„,=95 ™f Ihe d Le,T-'°" ' " ^'"'^ ^=- "^ -^^ «,„. o.:.__ . . ^ aincrei.ce 2. -ie ,„e contains, there ore 3 tunes J the other. Then t differ bv 9 fh*. ^.-fli; "> by or a multiple of 9 W itditrerby^^fotb;?"^^^^^^ isV' ,f the two figures is 2 3 W r^ \^^'' ^^^<^'fferencebetM.> o fta Tk« II , "'"erence of the figures H nnA i t tW. The smaller the difference -^ hr.u. . ^~^- No. 1. Sum a 3x5=15 Sum 9. ,4x4=l(> ^X / = 14 3X6=18 4X5=20 « ' fu I'if 186 APPENDIX. 1st. The one of the two numbers which, laultiplied one bv tho other, g.vc the smallest product, is alwkys tht un^C^hethe thi; sum be ocW or even; ' ''"*'"'*'^ 'o , f.'f, ^^'° S'-eator product is equal to half the sum; od. if the sum he odd, No. 2, the two numbers M'hich give the g, eater product differ between themselves by the unit. 04. There are some quantities which, according to the lutuie ot the question, can only be fractionary. Thus fov n.stance, when, m speaking of workmen, birds, eqqs &c ' wg mentioix the /.«//; thirds^ Quarters, &c., it is n^cefsarit kp- po ed that those quantities are exactly divisible by 2, 3,4, &c. flf! nf /^ k'!^ Tl ^''"'''^'>'' ^^^ ""»^^«r whereby i quan- tity of his kind becomes divisible, if increased by one of it. parts add the two terms of the fraction which expresses that part, the sum will be the answer. \:ollT' ^''' f '""Ple, if 1 increase by \ the contents of a bas- kct of eggs, I conclude that those contents, at? first exactly di- visible by 7, is now divisible by^+5=A 12 If, on the contrary, the quantit/of one'of its parts be di- « .imshed, you will determine the number by which it becomes divisible, taking the numerator from the denominator. The relTianider will be the answer. I '^^^".^'/^'•/^^a'^Pje, if I diminish by f the birds of an aviarv I conclude hat their number, at first exactlv divisible by 7 Ts- now (divisible by 7— 5 = A. 2. " ":/*,^^ In fv/.^l 'f ?K ^P'^'f «" ^'hich often occurs in my solutions. In Oder that the reader may perfectly understand it, J am ' flbout te give here an explanatory example. Let us suppose that the question is to divide |'r4-4^ i„fn tvjo parts, one of which = the f of the other. Make the su.u o( the two terms of the given fraction, you will have S-f 5 ^ N which indicates that the lesser part ought to have the 2 -»i the number (a:+4) and the greater the |! * Opera* n ; (a:+4) f =:(2f+i^ the lesser number. m^ t>+4}| = 8 the greater number. 1th thcvfol APPENDIX. 1 PT one by tha I'fc, whether 1 numbers sum; which givo he unit. ng to (he Thus, fov S, &C., AVG iarily sup- i, 3, 4, &;c. )y a quan- one of its esses that I of a bas- ixactly di- rts be di- ; becomes tor. The Ml aviary, e by 7 , IS • solutions, it, 1 am f 4) into the sum s 3-1-5^ the I «)i aer. MISCELLANEOUS PJiOB] EMS. lien"- o?S!ft"erS"?„rr "?' *« ^"'" <"■''- 'i-. •nay be 10. ''"« """ 'V 7 and tho other by 3 07. Divide *lfrft 1 . , -^'(s. 28 and IS. t-nally to theif ag^s :'b W t"? TT ^' ^' ^' P^P- ofA,whoisbutL]fofC'rWh^ ;[^?'^^^^^ ^^^" ^'^'^t J A *^ *^*^ ^^^''e of each ?-' What' is the capital V^*' '"''"' '^ ^^^ ^'"«"»t of $8208. 69. A capitalist placed the * of V . , "^'''' ^^^^^' ;...o,-e arte,, .hid. \Z2^,^ mo'"fr'' "'" '"" '" *''» hi-.st? "-" ''-'"• Wow many were in ii '2- Says A to B, Give ,„c *10(. i , "'"'• *^^- «l.at thou hast. Ill m^U'ad «lTV,.f 'i-nlf ™/'""'''« '3. Find two „u,„he.s whose dr *''"''""'^ ®™"- -y be to one a^^o^J' :^-J:t:--<'^;;, r ''"""''• 7r, A J r» ^^"'^'^ are the numbers? Ans ^ m^A a '' would ha™ .;s'm™h kJ A Imd'clrr^",^--^'^"^ (ais lii IS each ] --■^^.mnutcr^t ■is I -'J 133 APPENDIX. 76. What is that number whose seventh part multiplied bv Its eighth and the product divided by 3 would give 298| for It. What are two numbers whose product is 750 and whose quotient is 3^ ? Aus. 50 and 15. 78. A person being questioned about his age, replied • My mother was twenty years of age at my birth, and the nun.boV of her years multiplied by mine exceed by 2500 years hgr ige and mine united. What is his age ? Jus. 42. .eJ^'i-^ iffl^T"^ ^,^"S^^ ^^"^^ furniture and sold it shortly after for |1 44; by which he gained as much per cent, as k CA>st. Kequired the first cost. Ans. $80. 80. Determine two numbers whose sum shall be 41 and the sum of whose squares shall be 901. Ans. 15 and 26. 81. The difference between two numbers is 8, and the sum ot their squares 544. What are the numbers ? 8x5. Ihe product of two numbers is 255. and the sum of their squares 514. What are the numbers? Ans. 15 and 17. 83. Divide the number 16 into two such parts, that if to their product the sum of their squares be added, the result will be 208. Ans. 4 and 12. ^■^V^^* l^o^i'^ "'""^^'' '^^'''^ ^^^^^ to its square root the sum shall be 1332? W 1296. 85. What is the number that exceeds its square root by *' Ans.5Ql. ?^L ^i"i *^° numbers, such that their sum, their product, and the difference of their squares, may be equal ? Ans .^tVi> l±-/5 2 ' 2~* 87. Find two numbers, whose difference multiplied by the difterencc of their squares gives for product 160, and 4ose sum multiplied by the sum of their squares gives 580 for pro fit .V, Ans.7md2. 88. VN hat is the ratio of a progression by difference of 22 terriM, the first of which is 1 and the last 15? Ans ^ 89. J here is « nmviKni. ^e t.^.-^ £. _-- « .i _. .^ vide It by the sum of its figures, then inverting the numbci rjultiplied bjj ive 298 1 for Ans. 224. is 750 and •0 and 15. •eplied : M/ tho number years hsr Ans, 42. Id it shortly cent, as it dns. $80. be 41, and 3 and 26. .nd the sun) 1 and 20. " the sum (»f 15 and 17. , that if to , the result t and 12. ire root the IS. 1296. •e root by ns. 56^. sir product, i + v^s 2~* ied by the and whose ^0 for pro r and 3. snce of 22 Ans. |. ii yuu di. i numbci APPEXDIX. 139 lot, I answeirSMhe »Tf ?.?' t«f "-''^ P"' " » ■-"' of «n egg exceeded the f'of the wS h^T '"''''' ^•■»- ">« J all the eggs employed "''" ''>' "■« ^^ ^^""'y- How many ehil- 96. Two numbers are sunh fKof ♦ • 7 , , ^'^"'''' ^^• more than the greater and if ft *"^^' ^t" ^^'^ '« 3 units i\ and the U^s^'tsT the fori h ^''"''' ^ augmented its What are they ? *' ™^' ^^^"'"^'^ <^ouble the latter 97. A brother said to his sister • f m^'"' ^^ ""^ ^^• founds to make £30- eive niHl' T"^^ ''^''^ * «^ vour t^e sister, | of yours'to^ havellt will v""'' """^- ^^^^'''^^ How many pounds had each% ' /^" gT." '"« ^^em ? yS. By means of „ i " . ''• "^^ ""<^ -^^0. Michael L spend $2 a dTan^^^^^^^^ ^"'"^"'^'^'^ ^'^ '— ^ legacy. Whit was it? ^ ^ '^^ "^ ^^«'-'>' ^he |^ of tho 99. A fruiterer )mn„Kf ^''*- *^00. J n 14U APPENi IX jvould have been but 10 shillings less than half the first cost Itoquu-ed the first cost and selling price. Ans. 100 and 120 shillings. 101. Determine two numbers whose difference shall b« equal to one ol them, and whose product shall be 18 more than triple their sum. Ans. 12 and 6. 102. Of three numbers the mean is ^ greater than the less, jnd the turmc-r is i less .han the greater ; now if each was re- rtuced ^, tiieir sum would be reduced 19 units. What are the three numbers ? Ans. 24, 18, and 15. 103. Three times a number, less 20, is as much above fr^V wr^'-"T^^'"^f ^^' ^°"'^^ part, plus 2, is undei Its half. What is the number ? Ans. 24. 104. Louisa buys 2^ lbs. of sugar, at 6d. per lb., andgives in payment a piece of silver such that the square of the Siece returned exceeds trie triple of the expense by a sum equal to the return. Required the value of the piece given by Louisa. Ans. "is. 8d. 105. By what number should 3 be multiplied in order that the f J of the product may be equal to the sum of the two ^^^^^'''^ ' Ans. By 12. 106. When the granddaughter was born, the grandfather was 3^ times the granddaughter's present age, and 10 years af^er, the latter was but i of the grandfathers aforesaid^a«e. Kequired their respective ages. Ans. 90 and 20?^ 107. Determine three numbers, of which the greater, equal U) the sum of the two others, is also equal to the I of their product, and of which the less is but the I of the other two ^^''^''' aI. 24, IS, and G. 1 ^^?"..^" yncle claims the /^ of an inheritance, the nephev* I and the niece the remainder. The product of the parts of the two latter IS 13 millions less than the square of the m- clespart. What was the inheritance? ^ns. $12 000. .,\^^:.y''^'^^'^<'J'»^^^^^^^'^,yvhose sum is equal to4 time?* their difference, and the greater of which, plus the difference ?Kceeds the less by 48 units. What are the numbers ? Ans. ()0 and 3t>. My wafch is very methodical in its time. Were ( to. iiU. ie first costii shillings, ce shall be >e 18 more 12 and 6. lan the less, iach was re- What are '<, and 15. luch above 2, is undei Ans. 24. »., andgives •f the piece m equal to by Ix)ui8a. f. Is. 8d. order that >f the two . By 12. grandfather 1 10 years •esaid age. and 20. ater, equal J of their other two , and G. le nephew e pai-ts of of the lUi- 12,000. to 4 time?* lirteronce, TS? md 3tj. Wan- f tf> APPENDIX. 141 1 1 1 -p^,,, , ^^^' i minute fast or slow od'r\ZZrCti7;.' Z:':*"' T ^ <'^*-^ brother- Tea,. J of the sister's ^eatiesHL/nT -T" «•■« brother will be ofeaoh? ^^O's less than the sister. What is the age 112 HnH I ^ u, ■ ®'"'^'"' '8; lirother, U < -M Hve tripled ^.r^^bral' tt |. J^'J' ^^ ^■^rz\r'tt^-^ttfX-:i.ei 114 I k„ . ^^"•*'- -A^t 72 and at 40 -^- .'ow^^;israri::;rz'^« "-^ '?— net, just as you please nf ,ZT [I *"«'■■""'• the prod- my sister's ai;d Zt^:/'^^ .^;::'JX "^"^ '''--«■"' 115. Subtract /oT.';tot3''^ '--■ 2 years each. ««e,you shall hlem^Lthit,''-''^''-T'J''''<"y^^tor', years will have lessen^lt " Wh^^lhTf ,f„f ^ h f '"^'^ onVo^r teoiro^tf s but »^— '^^ • .u. is but half ,„, ,J, ^Z: I^l^tstelt'' mayi4"„^airjTtl;:rat' '"1 1 ^''™- o^'" I-' lt« i, may exceed'the lessftv I' " ®?'"'' *"""'f-ed US Tk J ^«». 20 and 4. .«m, and thr,ttie"„t Ts " Wh^ ^"^'."'^ '^^^^ «-- '''eir 4 i"«"r IS rf. vv hat are the numbers ? 119. Square I plus 1 of thn . ^'''' ^^ ^''"^ "^ qimrter of a n«n?„f, 1. ^.^*^^.3r««»•« tti.'it I am .hort of. r* ? "" " "' '^"" •> "" ^"' P'-oduce my age. What is 120 A gamester lo8» at the first Am. 16. game the srjujire of ^ o| 142 APPENDIX. the dollars he ha4 about him; but at the second round he quintuples his remainder, and withdraws neither gainer nor loser. How many dollars had he ? Ans. 80. 121. I am going to add 5 new shelves to my library, each of which will hold 20 vol'umes more than the ten already ex- isting and so I shall have 1000 volumes in all. How manv ^'«^^^"«^^ ^...600. ^ 122 Multiply half the father's age by half the son's age and you will have the square of the son's age; this square is equal to double the sum of their ages. How old is each ? Ans. 40 and 10. 123. The breadth of my room is but the f of its length — As broad as it is lo-jg, it would contain 144 square feet more. Kequired its dimensions. ^ns. 24 by 18 124. The difference between the | of my age, less 5, and Its g, plus 3, IS the square root of my age. What is it? Ans. 3(5 years. 125. The product of two numbers is 220. If from tho greater you subtract the difference, their product will lessen yy units, i^md the two numbers by one unknown term. Ans. 20 and 11. 126. What is the number of your house? Tlie sum of its digits, considered as units, is equal to ^ of the number. Find *^ Ans. 54. 127. The square of the difference of two numbers is equal to Its sum, and I of the former is equal to | of the latter. What are the two numbers ? Jns. 10 and 6. 128. An officer gave the following indicatjon of the number of his regiment : One of its factors is to the other • • 1 • 5 and their sum is to their product :: 6 : 25. What was thr luniiber? ^ ,o- Ans. 12a. 129. My age is composed of two figures, and read back- wards It makes me I older. What is it? Ans. 45. 130. The product of two numbers is ^ more than their sum, and is equal to triple their difference. What are they ? Ans. 2 and 6. 131. When the brother's age was the snu.irf^ nf fh^ =-!=f.-.T-.»= /he was | of the brother's present age, and 8 years hencrthe round he gainer nor Ans. 80. irary, each il ready ex- low many ns. 600. n's age and 5 square is i each ? and 10. ? length. — feet more. t by 18. ess 5, and is it? > years. ' from tho nil lessen term, and 11. 3um of it8 >er. Find ns. 54. 's is equal he latter. ' and 6. e number ?r : : 1 : 5, It was thr *5. 125. jad back ns. 45. h.in their ^re they ? and 6. .^ „:_i. »_ hence thr APPENDIX. 143 « their ages will be augmented it. I. What is the age and at the sam? perioj th^eTe of ^ ''''" ^' ^"' '^^ t ^^^''■^ i of the three agL united'^ f hi te the'^^S^ '""^''^^^'^ 134. Divide a baskpf .f ^'''* ^^' ^^' «"<^ 1^2. the part of the' o^t m ;2^thaT^f .t"^ ^'■^^^'•^' ^ ^^^^^ aiid that of the second to the parf f h ' '''^"'^ = = * ^ ^ 5 greatest ^ubmultS ? i d2'"''h *'"*! divided hy t. pounds d'd he gain ?' ^^'I'shed £12. How manj number. What is it f °"^" "' ""^"itude, i, f „f the 137 J gQ]^ , ^ -4wa-, JiH. in anoiher. Triple tto r'mri^,dl"*^1 ''" ""' house, and 25 tlie primitive ccftents. What h^' it" ^"" ^'""' '•''P'-"i''oe ">y'?^t;&,;itS;re?ttsv''^--^^ r^i^eS etir ^■"' — "z:i mZ..i,- I.IQ rr I K„^ -11 ^'**' 1^ and 8. Jave bee^lettr^* CltohaTlf • '1 "'^ -"'" a«d I pay for it? "uuuie wiiat it cost me. What 140. One of the fapfnr« „p , • , ^''*- ^^'- ■■"" " - "■■ ■"-- -att !;ttt IS 144 APPENDIX. pres, ut time, plu^ its root, is half greater than less its root What o'clock was it ? 4^^ g 142. The sum of the four terms of an arithmetical pro. gression is 44 ; tliat of the two first, 18. What are the four ^^""'^- ^1/^^.8,10,12,14. 143 There is 4 difference between two numbers, and theii sum is less tiian their product. Required the two numbers. Ans. 2 and 6. 144. The difference of two numbers equals 4 of the jjreater and represents the square of the less. What are the num' ^^^^- ^ln«. 30and5. 145 There are two unequal numbers; the less is equal to I ot their sum, and their sum equals ^j of their pr )duct. Ans. 6 and 4. ini^^' I^u '"??^"^ ^'^^ numbers, plus their difference, makes 100, and their difference joined to their quotient equals 45 What are the numbers? Atis. 50 and 10. * 147 I received this morning a basket of peaches: I laid i by for myself, and the remainder, a prime number, 1 divided amongst my children in equal parts. How many children ^^^^^^ ;in.. 11. 148. The I of the numerator equals the f of the denomi- nator, and the sum of the two terms is 11 more than the pro. duct of ^ of the denominator by ^ of the numerator. What IS the fraction? j„„ s 149. 1 bought a horse yesterday and sold him at a profit equal to the f , less £11, of my outlay, bv which 1 gained 20 per cent. Required the first cost and selling price. Ans. i>20 and £24. 150. The four terms of a proportion make, together, 100. Ihe first i« equal to the third and the ratio is 4 What are the terms? Ans. 40 : 10::40: lo! 151. The dividend is equa' 10 the square of the divisor and their sum added to their qu nent, is equal to 840. What is the dividend ? what is the divisor ? Ans. 784 ar. d 28. 152. One square is quadruple another, and their sum added to the sum of the t.wn rm^ta \a f^QA \,\7l.„*. .1 — - ... .,._.._., TT iK-tL arc tne twc ^"^^^^«- ^-Im'. 400 and 100. APPENDIX. 3S its root Ans. 5. etical pro- re the four , 12, 14. , and theii umbers. 2 and 6. be greater, the nura > and 5. 5 equal to )duct. > and 4. ice, makes squals 45, and 10. !s: I laid , 1 divided children ns. 11. denomi- 1 the pro- '. What A.ns. |. it a profit gained 20 di;24. her, 100. •Vhat are 0:10. isor, and What is r.d 28. m added the two i 100. 145 153. A milkmaid sells hens' eggs and ducks' oa^. tu • mean price is 16 cents the dozen Now a a ^f '. .^^''"" are worth 10 dozen of the former pin • 1T" ^^ '^' ^^"*^^ dozen. "'^''' Required the price of each .r. „ru '^'**- J 2 and 20 cents. io4 What cost these sl^ pounds of sucrar ? rr .k »^orth 3 cents per lb more mv nnM ^ ,1 . ^^^^ ^^""o more. Whatliditcorre'rT/. '^ would have been I l^r m, . ^ •• ^ns. iDctS. they « ^ *" "^ '"" proportional. What are two numbers? -^ '^^ What are the 1K7 A , ^w*. 6 and 6. . .xth\\etSe'rXbr4:taVt roof "r^^/T square? 'i "mes its root. Required the • 158. An annuity placed at iqi r.. . ^"''' *^^' duces monthly a lenteauJfn-J ^ ''^"*' P^*" ^""'""' P'"^ annuity ? ^ ^^"^^ ^^ ^^" ^^^^''^ '^ot. What is the i^jf. Ihe product of two numbers is 120 A,l^ i . l and their product shall be 150. wLt a '^he dumber; ?'"^ 160. Two numbers are equal If o k. fj'l' ^ ^"^ ^• product will increase 51 What are Th. ! ^^^ '? ''^"^' '^^'^ «» »'i. w nat are the two numbers ? Iflo Tk« « 1- - '^'**' 4 and 12 ho less b, 2. m Jr th'/thtettr ""^ iim : their difference ■ • 7 ■ whole numbers with o*ne unk,?own tenn'^ '^ ^A^Ts^nfT" oflt.^^a^or'nilrgTbl'ittvl'S,"'"'!^^/^'-''- and my expense was tie samf at h iftt;!' tT'' mea,, pr,ce was 35 shillings. Required the 'prtje rf^a' ^JS:? ^'w- 30 and 40 shillings. >„l!!i'lr!'r..';™'=.«' '"M ? Kv, my daughter's a„,. will to-dav ? ""'^ "' ""'*J' '■'^ «as o years ago. 186. The .sum of two numbers is 4 What is her age Ans. 10 years, times their difference; I i B! 148 APPENDIX. and the d.fferenofe ,i, of their product. What .re the two num Jn )^Ia , " °®*'^'" ^^'"& questioned about the number of men wh e sumtTl Zrr' '■ '^t"^""'"^^^ ^- but 3 factors wnose sum 18 31. What was their number? Ans 25 188 Waiter, your bill of fare amounts to so much, does it hat^hl. K^" IT'' r'V T^^^- ^^^"' here is another sum that lacks but i dollar of the J of the first, and let's say no IrinnV*' ^' '"' "^^^-^iheless, hard. sir. to lose the double, plus 1, of the sq .arew.of my bill. What was the nmountt * '" ^w«. 144 dollars. ih«t ft! T\ °^ '^'^ pandpapa and grandson are such, that their nuotient is equal to | of their product, and the sum of eaeh^ q^o^^ent.aftd. product is 320. What is the ^ge ^^ • ■ - ■ Ans. 96 and 3 years.'' lyo. 1 mixed two pipes'-Af Vine ; one cost 180 shilJinffs and he other 140 shillings. The first contained 20 bo ties rr^ e than the second and cost 5d. l^ess per bottle. What is the vZl of a bottle of the mixture ? = ;,. ^,,. ^^^-^'^^ 191. Ihe quotient exceeds the divisor by half phis 1 and the sum of the divisor and quotient exceeds doubeCsoi^ire root of the dividend plus 1. .R-equired the divid!'. id visor and quotient. Ans. 400, 10, and 25. u!hfr ^7," ^*"^et«J"nni»g tog\^ttter, filled a basi,, in 3 hours If the first had run but 2 hours, it would have taken the second 6 hours to do the remainder. What time would it take each tunning alone? ^... 4 and 12 hours 193 T^e father's age has two factors, of which one repre- sents his daughter's age, and the other is 18 less. Sqimie ?h^ iXtZVf^^ ''^'' the daughter's age, and tSis Lu U> the sum of both ages, and the general result will be 100 Required the age of each. a ns. 03 and 2 1 -, lit .'T^e product of two numbers exceeds their sum ov U and the.r difference is 2. What are the two numbers ? ^ ^ Ans. and 4, _195. A number of three figures is a multiple of 11. and rim ".-iiio is quaarupiu the huiiuieds. What is the number? Au,. 154. > he two num 10 and e. Tiber of men but 3 factors, Ans. 25. luch, does it and receipt mother sum let's say no the double, le amount * i dollars. n are such, nd the sum t is the affe 3 years. liliings and ottles more is the value . 2s. 5d. plus 1, and the square id, divisor, and 25. in 3 hours, the second t tali lugi- ■tiv.„u„,her. which Tas:t;r''4"aL''rdrY80T'';H now he reduced J. What was the prim tive nt 1, 'r f w'llil B their present number' ^ I" w j cV"" /!«». I.'jO mid 84. n.nl ik""!"'"'"'"'' oot bylwrenLr "'"^ " 9nn A u ^ .. .'o ^^*«. illo and £25. diJfs i.V""" ri?^'^'''.^^"''"^ ''^ «"«h, that the sun. <.f its digits ,s lb; and by m verting the number, then addin./tt! m." wtt ;:r:itrt'" ^•'" •^'^ '^''*-^ v'^-- of?L,tetroroVits^^^^ their sum is to their product • • 6 • 05 vVKa. ' ' u' ' ^ ber? F^wuuci .. o . .iD. What was the num- 202 Two sisters have unequal .ums for their purchases ;tn«rfa:^^"™^- -- ^^^- - f ?^^*^wi^^ ^08. What IS ihe number, whose scmaie reduced t„ it. ■luarter, excneds by { three times the | ',f the nuXV? tinferaslrari"iU:.!;r .\:°T f ^--<'. ^^i:: is^ squares of inarble of a^^riainXiTirsi^i^St'Z uWt/ n.e mason answered, that if the length was but double Te l.S* 150 APPENDIX. CJD breadth, it would have taken 800 less. What do vou dudo fi-oni this answer ? ' i uo ^ou Ans. That it would have taken 4000. .Jf' ^ T^'^^"^ g^^« 2^ "^' his profit to the poor. At thr >ear s end his alms amounted to $390. I demand what vvas the amount of his sales, if half was at 10, I at 15 and k remainder at 18 per cent, profit ? ^Ans. $6(^000 fhnT' J^^ '""" ""f the'foJr terms of a proposition is 63- the first ,s 4 more than thVk:c.ond ; the quotient of the th rd by the second ,s>8i; and .the product of the means I 136 Required the four terms. ' .;: . . Ans. 8 : 4 • • 34 17 ^ A*jf . * : o : ; 4 • 1 'i . TTDC IWD. do you cjD ikeii 4000. oor. At the nd what was 15, and the . 160,000. ber was rep representing at the sum t what num. No. 210. ition is 63 ; of the third sans i,« 136. : 34 : 17. ^ion i'i 576 ; term k 10, What an> :4- 1','. I ,i> 5' t ■ < \ .?« x '•s < % f- \ h il »V™. apiTroved school books D. k J, SAP! fE'' c<' ;l qi p„ I « f OJii odiiion. ^rui. u # tun in Puor Sc/^oU cnuls■^lA^ uw^muiiis' fikst •in.-" PRAfTJi VI .. -"^- Lie * ■' u ',' ''"«■ Dioct-se of Boston. ",'!' , , ThVam..'^i^!i>rib1e^X?h """^ ^''"«*-"^^ -''^ «« cut-. „." %• Any of thi^boTe Book* will be ^ui hTIaii u * CO.