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SCJIOOL, TOHONTO, I> WILLIAM 8C0'n\ B.A., Mathkmatical Mabtkh, Nohmal S(ui<.)oi,, Ottawa. Prescribed by the Board of Education in New Brunswick. Prescribed 6,7 the Council of Public Instruction for exclusive ««« in Nova Scotia. Authorized for use in the Schools of Manitoba. Recommended by the Univnroity <>/ Halifax, Nova Scotia. • Authorised by the Council of Public Instruction, Quebec. Authorised by the Education Department, Ot^tario. Aidhorised for use in the Schools of NeivfoUmland. Atdhorised for use in the Schools of Prince Edwanl Island. Authorized for use in the Schools of North-west Territories. Authorised fjr use in the Schools of British Columbia. Elghteentli t:«lition. Price 60 Cents. ^if W. J. GAGE & COMPANY, TORONTO. ■S£Sw (A ^ ^ - — Entered according to Act of Parllamont of OanRda. lu the Office of th« Minister of Agriculture, by Adam MrLLEK dc Co.. In tbe year 1878 =r^^^ PREFACE ^ I The present edition of Hamblin Smith's Arithmetic >. not amply a reprint of the English work. Several article, hava been introduced with a view of making the book more useful and complete. Some of these wiii be found on pages 33-40, 45, 48, 64, 92-95, 115. From Simple Interest to the end, the work has been almost altogether re-written. In the English edition, the treat- ment of the subjects in this part of the book was too simple for our Canadian Schools. In this edition, the important subjects of Discount, Stocks and Shares, F change, &c., have been treated at greater length than in the ordinary text-books on Arithmetic. Some important articles of a practical business nature have been intro. duced ; amongst these are Equation of Accounts and Partnership Settlements. While the scientific character of the work has been preserved, special care has been taken to adapt it to the wants of the business community. Examination Paper., have been added to each chapter These have been carefully selected with a view of securi.-,g variety and avoiding sameness. They are designed to stimulate the student to think for himself, and to assist him in preparing for the different official examinations ^J-'- ly •»« •"«'*°'- A method of h..d,ng the Cube Boot by .ubstitution in a .in,ple and ewily-remembered formula is also given. Toronto, 1890. possible, ibjects of Annuities >r which, ^ method uple and CONTENTS. .ta of "the iwult ia." -M •* MJ. or Hi 11 CO J^i"^^* ^^^^^ "'"'y *• written ttoict. • -3+-1 -If 1^14.1, Hh«re unity Ja JtriiUn /onr iimm • and 80 on. ' .-^' N""»b«r8 lH5tw«,en nino and a hundred are rapi^ tented l.y dvo hgures, tho one on th« !«tt-hau(I aiffnifvinir how many ^ranj^ of ivu unih are contained in the riurn lier represented, and the one on the right4.and signify- ing how ,„any «,ngle units are contained in the number, m addition to the groups of ten • lits. Thus, in th^ oxpression CU, the figure 6 represents ^ix groups of tea units, the hgure 9 represents nine single units. These groups of t«m units are for brevity called Tens and the single units are for brevity called UnHt^ NumharH between ninty-nine and a thousand are represented by three figures. In the expre?sion 74.5, the figure 7 represents sev^n groups of a liundred unit^ the hguro 4 represents four groups of ten units . the ligun; 5 represents five single units. ' In the expression 3175, the iiguro 3 represents three groups of a thoimiad units. In tho expiession 23475, the figure 2 represents two groups of U>n ihomand units. In the expression 123475, the %ure 1 represents on3 group of a hundred chomand In the expression 9123475, the fig-ore 9 represents nine groups of a million units • and so on. ' mm 7. To put the n.AtU,r briefly • wh«ii wm — -s number in tisurm, aiul t^w 4- *L *i ^ •xprewi « the ihini the/ofi-fA th«MadA of ten, of miUioia, I fnUtioni,) Jgare reprcents a number of un<<, ^ flgur« r«pr..««„i. ^ number ^ f^ ' 1 fi«ur.repre.enfnumUrof/.u,!i^rf^ / figuio repreeenu a number of irn, ^Zm. figure repreeent. * number c* ' .,./„ ., ,^'*^' th, « *^7235682719435/•2■03056300672■0I0. 6 KTOTATION, NOTATION. . well JtalKwtK"*"'" " ""■"""^ -P--" ■■• The method to he employed is this • Trillions niiii«„- ' ' ' Billions Million 8 I Thousand Thousand j A.u ^'^ \ 309 and the number expressed in figures is 47309. Again, to Represent bv fic^urpq fr..,n vn- K:;v;-:^ -""- -'^^ '—" - '^- Billions 4 Millions 302 Thousand 018 053 and the number expressed in figures is 4302018053. Examples, (ii.) Express in figures the following numbers •— tein^jde^er'K *"''"'' '^'^"*^^" ' "^^^^^-n; thirteen; six- eiiht^ f^'^S^Z-: i^T::!^:'.^ \ ^f ^ ^ ^^en ; ' twenty- three. / ' lor.y.nme; eighteen; ninety; seventy- twfLPnTredTntfttTix^^ -^ thirty, six hundred and e ahf l! \^""/''^,^ ^"^ ««venty-two hundred and nTnety;^' ^""^ ^""^'^^ ^"^ forty f nine NOTATION. . J (6) Eight liundred and thirf„ .• orty-seven ; four huXd and -"^ ' "*'"«" ^""dred and ur een ; seven hundred and fiftv" ' .^^ t^^^^^^ and tighty.four. ""^ ^"y; three hundred and ei^h|,^th„t"4^1„r,?..itT^'''«i'* ">""'"'■ and hundred and twelve , ".von huSS?„"i"'- ll"^ ""irty ; fi™ , (8) Seven thousand oJtTZ ^ T^^*^'^^^"' thousand eix hundi"d "nd tht.v "'' ""* '''"J--fi''« ; nine eight thousand four hundrld '^" ■'"'t'i ' '"«'" thoiMnS e'ghty-five thousand and forty • "" """-""'^ ""d 'h""; aand^surnte^i°?f ''""^.«-<» and five bUlions six hundred million^ r*",.'^ and three ; thirty . m Seven billions ,• fivTt'ilHl u ''"" *"'' '"«»ty- «« hundred thouikd arfd J^/ *'«'" ^''"''red billions forty^three thousand and stven '°'*^-"''''' ■ eight trillS n»irio«;'S^h'o™S :„"d' thL^e 'I^Tk '^^ "'"-- four "»«e . ...ons «ty.three tUlZ' aM^trhr""" «'^- ait' 8 ROMAN NUMERALS, (it)) Nine triJliuus and nin » • nJn » i. n- hundred: nineteen triij^!. ,. "^^^^ trilhona and nine J^^ i /203/^iV NUMERALS. 14c In the Roman system of Notation. M'hich is bHII employed wero J, V, X^L, c{ A M. '^" '"'"' five hunf]r*»rl .,«^ .V "*''"^®' ^*^"» hfty, a hundred. standing „, the n,jkt li a l.i^C syn, Ul "wt^ Z^Z added to the number renre^entpH Iw ti Z ? , ° "® when standing on the iJZZX '^ .^ISL^ tmt Thus: and VI /epreaentod the number six, IV represented the number /onr, LX represented the number sixty, XL represented the number forty. W?„Vt'7SLt!r."'" "="'""" «■« -'Hod to. num. 1 1. U XI. • 2 JI. 12 XII. 3 III. 13 XIII. 4 IV. U XIV. 5>. 15 :^v. r> VI. IG XVI. 7 VII. 17 XVII. 8 VIII. 18 XVIII »IX. 19 XIX. Y 21 XXI. 80 XXX. 40 XL. 44 XLIV. 50 If. GO LX. 70 LXX. 80 LXXX. ' 90 XC. loo c. " 110 ex. 150 CL. 188 CLXXXVIII. 200 CO. 300 COO. 400 COCO. 600 D. GOO DC. 900 DCCCO. 1000 M. BOM AN NUHKRAIJI. Write ill words: (1) xxvn. (4) LXXIIL ISxainples. (ill) (7) CLXIII. (10) MbccCLXXIi. (2) XLTX. (6) XCII. (8) CXCIX. (3) Lxvm. ((5) QXIJV., (0) DCLXIV. Write in Roman Numerfils: (1) 37. (2) 50. (3) fi2. (4) 87. ^5) 95. (8) 179. (9) 846. (10) 1763. (6) 139. (7) 145. 11. Addition. 16. If we combine two or more groups of units, so ns to make one group, the number of units in this single group 18 called the Sum of the numbers of units in the original groups. To find the sum of 5 and 3, we reason thus: (Art; 5) (Art. 4) Since 3=1+ 1-} 1, 5-f8=5-f-l-hl-j-i =6-1-1+1 =7+1 =8. 4? 16. By practice we become able to express the result of adding a number less than ten. to another number, without breaking up the number which we have to add, into units. Thus we say and so on. 7 and 6 make 12, 15 and 8 make 23 ; Again, if we have three or four numbers, each less than ten, to add together, we perform the process men- tally; thus, to add 4, 7, 9, and 6 together we say 4, 11, «U, 26. 17. We now proceed to explain the process of addition m the case of Iiigher numbers. nuts berg 10 ADDlxioy. We arrange them thus: 2476 397 48C 3007 Placing the figures thaf ^^^^ ;n the same vitical it^n'T "f^ ^" ^^^ --ber »n the same vertical Jh.e 7l °'^. ^^^'^ ^^Present tTnl represent /,«,,rf,,,^, and V" 'T^^'^y ^^^ ^^ose that Adding 7 6 7 o ^ r ** ^* the' L"' 1r"''^' -'I "-r/rthe"' T""'/'"' ''^^ ""^I^ tile line of tens. ' '"^ ^ *ens for addition to Adding 2, 8 Q j » tens, that is t^o hund "ed, »^"'i *''^ ^"^ « twenty-siv 1 thousand for addition to letfof^'"' "^f^ '"' "•« Adding 1 3 and o .u thousands. -''^. and We P.ao:'e'ufcrein:'"oVt"h:.iL^: "•- Examples, (iv.) Add together a) 4 a„0, 3 and 13, 6 and 16, « and 27. 62 36 (3) 40 27 (4) 36 24 ^ number 5senfc tens ^lose that draw a ider this ^ of the vvay ; ^nty-suc B the 6 eds for hirteen s : we on the (a) i8) 237 340 823 459 G 237 4260 (12) (11) 429 347 425 269 • 538 (15) 6842 (1(1) 5079 8526 5037 2409 ADDITIOIV. (6) 209 140 600 (») 5462 723 8004 0217 364 (13) 253 629 189 488 567 976 278 863 384 8750 (17) 8604 4623 4007 7988 5290 6543 3040 5729 7250 (7) (10) (14) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) 64 + 43 + 7 + 85 + 9. 247 + 356 + 28 + 423 + 97 + 12. 425 + 3742 + 4236 + 39 + 847. 7288 + 976 + 45 + 623 + 4000. 8 + 97623 + 3407 + 5260 + 86. 41537 + 9215 + 48 + 6077 + 23 + 2413. 275413 + 3120 + 725 + 5007. 74259 + 346274 + 30000 + 1000001 + 207. 4692 + 72430 + 80000729 + 40 + 600000000 46243 . (29) 748325 35297 54297 825649 632684 246728 20047 815 4207 42376 C17043 645980 3025 (30) 11 562 70 106 24600' 3470 40052 6207 6843 4297 32(i 62 7008 5629 426580 37259 506 670^.92 37987 6493 »n!? J2 (31) "tnimiiHTioN. (^4) 260497 Qsmso 407240 8f;4f)28 264384 7402594 8025837 4398025 C702403 5124917 02 19800 4390143 7409425 (32) (36) 064297 248043 380109 472580 682987 030468 498408 4097498 527 430:'040 27209 152372 4058 7205204 4372943 (38) (36) C25493 76802 5410 87294 4859 862 13 0572043 2809257 430 C98206 45297 3526084 57002 852968 j-4.ev„„ h„„a.ea th..r.T«'„sti ■'i^l:^iz^ .nn|v,e"f ;ii"„;^ irLt ra"ri -^^^ '-o™" thousand and fortv • ««vlr; u j , 'orty-seven ; sixty three hundred t&dTaflft™"'' ""'' "'^ "•"^"^^ th^^L'nZdTntt^-tr «;'"r' ^r-*^-- ♦■'-""d dredand four; eiehteen mill! '^ .^"""'"<' *•"■«« huu- five hundred thou^nd andt" v eth/'"'n'""' ""'* "•""> " lorty , eight milhons and eight. HI. Subtraction the larger number Thu! f composed, sepa -ately from reason thus: ' ^^ ""^'^'^^^^ ^ ^''0'"\ we 3 = 1+1 + 1, II ri '• 4 ntrvnucftiov. 13 62fi493 75802 5410 87204 4859 802 13 C572043 2809257 4;JC C98206 45297 3526084 57002 8529C8 id fifteen ; ind three dred and nd nine ; ^nd nine- sand and hundred ^ ; sixty lousand ; housand ree hun- l three ; fight. p num- nits, of y from 5, we if w« take away one of these units from 5, we hav« 4 left • . wo tHke away the «*Hx,nd unit from 4, we have 8 left • It wo tako awRv thn thi^A ..»:* / « .... •' ^. . .. **"'' » , -J — - -•-«-»..»* uiiib iiuiu « we nave If wo tako away tho third unit from 3, we have 2 left. The Bymho] - , read tniniM, k used to denote the operation of Hubtraction. Thus the operation of guh tractjng 3 from 5, and its connection with the result may be briefly expressed thus : 6-8 = 2. 10. By practice we l)€con.e able to subtract a num- i)er, less than ten, from another number, without break- ing up the smaller number into units; thus we say, 7-4=3, 18-5=13, 49-8-41; and so on. 20. Before we proceed to explain the process of Subtraction m the case of higher numbers, we must notice the principle on which a certain step in the process is founded. If we aro comparing two numbers, with a view to discover the number by which one exceeda the other we may add ten single units to 'the greater, if we also add one group of ten units to the less ; and we may add ten groups of ten units to the greater, if we also add one group of a hundred units to the less ; and so on. Suppose, for. examole, we want to find the number f>y which 56 exceeds ^, we might reason thus : 56= five tens together with six units. 29= two tens together with nine units. To the former add ten sifigle units, and to the latter add ofie group of ten units. Then the resulting numbers will be, in the first case, five tens together with sixteen units in the second case, three tens together with nine units. Hence the excess of the former over the latter will l>e the number, made up of two tens together with seven nniT..?. a«'>« ■will +ke~fi*—-i t~~. ^ ' j » '-■r- 14 AVBTltAmOM. S-PPOM w. We to t,.ke 889 fro.. 926; ■ From 020 Take 580 Hemftinder 037 r^pllTuTu tZTlr,lf •-'''« "■" ««-«« that doing the .a,,,,, wf^ . .L""!, ':""' ^"^'i""' '"'«. and Imnd-cdd. """ ropresent tens and 'i^feen units, and we tot "'"' ," "'"." ""'"• "'■'kin» and .Mown tHe r^^JU:^^ 7^^^^!^^^^ "■e 2 te„», making" ,!"".;„? ^^T/"'" '^'^ '"» '«»* to 9 tens, and set down^e * "iV""' • ?'" "■"'« ^^ »»'■« tho line of tena ^'""' *'""'' » ^ tens, under hundreds. " '' hu..dreds, under the line of Examples, (v ) Find tl,e dirtfe,,,,,, b^^^^^,^ ^j_^ ^^_,^ numbers (i) 13 and 0. (5) owing pairs of 07 23 («) (2) 15 and 7. (4) 3 and 32. 42 39 (3) 23 and 4. («) 87 58 (9) 02 47 10) 313 247 (14) 6239 4127 MtTLTIPLIOATIOV. (11) 704 (12) 630 106 648 (16) 4729 (16) 6268 601 36 1ft (13) 7490 8618 (17) 66472 4001 ^8) 367 249 (11>) 4626 1846 (20) 72649 (21) 20004 43821 17243 (22) 437-^56 (23)629-483 (24)827-706 (26) 3000-958 26 7040-683 27 6269-479 ml 5S~'f.^J^?^ 04296 -53290 l5o 700^0 ''^904 I .^oi ^"""^ i^^'''^- <'*2) 42466 and 102479. /^?k <^24H^ »"*! 14(X)0702. (34) 99909 and 100000. (oo) A million and a thousand. ^^l ^ ^""ui^.^^ millions and a hundred thousand. All %ifi" """""« an^ ft th(.UBand and one. f'^\ wu. """'I^'' '"""^ ^^ ^•^^^ f"-"'" 20 to lea re 18 ? 40 WW """'^' "*"'^ \^ ^"^^'" f^^"» *27 to leave 401 ? a,.H fiff ^^*Sn"'n^«r "'"«t be taken from three thousand and hfteen to leave two thousand four hundred and five? and seven^ "'^"^ ^''^'' ** thousand exceed four hundred (42) The greater of two numbers is 427 and the sum nf (4S)""wTat"nf \"'^' '\ ^^ ^r"^^ of the two nuXrsI (4J) What number must be added to 7428 to make 8047 1 IV. Multiplication. 21. Multiplication i? the process hy which we find the I'^ual ""' ^°"''' '''' "'''''® '^"nibers, which are Thus, if we have to find the sum of three numbers o/'r % 3 ^"^ '^" ^^''^ P'""'"^'' *^'' Multiplication of™' s"'"" '' ""^"^"^ *^'^ Product of the multiplication The number 3 is called the Multiplier The number 7 is called the Multiplicand. rne ioiiowing table must be committed to memory. in n I Three "iw,nn.H!4Tioir. Th# MiUUpUcftUon T»ble. • ? viir tlBM lb 4 2. . 6 3. .19 4. .10 n .20 fl. .24 7. .28 8. .32 0. .30 10. .40 11. .44 12 .48 111 6 «.12 «..J8 4.. 24 5 .SO 6.. 36 „ J 7.. 42 8. 40 ^ 8.. 43 9.. 45 10. 60 11 .65 12 .60 timm lia 7 2. .14 3., SI 4..28 6. .35 6.. 42 7..4» 8. .Be ^..6,3 10.. 70 U..77 12.. 84 Tirelve timer 1 U 12 2 .. 24 12 ..108 I 12 ;;i2o 4 is twelve we alert Tf T/'' T/^^^ ♦'^^^^ ^ times together, th'e TesuTis 12 '' '' ' ""' ' ^"^ ^ ^^ ^^^^ Each of the numbers " - • n i the product 12. '^ ^alW a Faotoh of »iXolrthut '" "■' *''" ™""' "' * *"- 67, w. G7 67 67 67 2aR I j Btnn M llniM « 1» 7 2 1 2..U « 3.. 81 ]^ 4.. 28 JO 5, .35 W 6.. 42 \k 7.. 49 a 8. .B6 4 »..63 10.. 70 « il..77 2 12..84 1 Tirelve timet 1 i» 12 2 .. 24 3 .. 3(i 4 .. 48 5 .. 00 . .. 72 Jl- 7 .. 84ir 8 .. 98- ..108 ..120 I ..132 2 .. U ioation i.-: 1 3 times by a the first, and the whole operation in the example just given is briefly expressed thu.s : - » / 4x67=268. 24, ^ext observe that the multiplier and multipli- cand may change- places, without altering the value of the product. "^HU' ^o^ > = 4 X 3, or 3 times 4 = 4 times a *or 3 times 4 = 4-f 4-f 4, = l+l + l-fl) 4-1+1-rl-M }l. . -f 14.14-14.1 j And 4 tunes 3= 34-3-f-3+3 * =14-1+1^ + 1+14-1 I r. + 1+1+1 /■"• + 1+1-f-lJ Now the results obtamed from I. and 11. must be the same for the horizontal columns of the one are identical wjtn the VArtlOal cnhimna rtf i-h^ .s-.fk^.^ 18 MULTIPLICATIOX. 26. If we multiply a number by 10, the product is obtained by annexing to the number, that is, 10x364=3040. If we multiply a number by 100, the piodu' t ig obtained by annexing 00 to the number, that is, 100x364=30400. So by annexing 000 to a number we multiply it bv 1000, and so on. ^ If we have to multiply a number by 20, we may first multiply it by 2 and then annex to the result, and the final result will be the product required. Again, if we have to multiply a number by 200, we may firsb multiply it by 2 and then annex 00 to'the result. The method of expressing the result of multiplications of this kind in practice is as follows : We multiply 4276 by 700 and 1 4239 by 6000 thus : 4276 700 2993200 14239 6000 85434000 P :S '¥ Examples, (vi.) Find the Product in the following cases of Multipli- cation : . (3) (6) 5 timea 98. 7 times 123. (1) 3 times 15. (2) 4 times 76. (4) 10 times 87. (5) 6 times 114. (7) 11 times 843. (8) 12 times 947. (9) MuHiply 25 by 2, 3, 7, 9. (10) Multiply 127 by 5, 8, 10, 11, 20, 70. (11) Multiply 2467 by 4, 6, 11, 12, 500, 7000. (12) Multiply 37429 by 9, 11, 12, 50000, 80000000. 26. Ex. (1). To multiply 347 by 23. MULTIPUCATlOir. 19 The form of the operation io : 347 23 1041 694 7981 The explanation it this : The multiplier is made up of two parts, 3 and 20 • we there. two rTuS ^ ^^^ ^''' ^^ *^'''» ^^*^"" ^y 20, aAd add t^ • Now 3x347 = 1041. and 20x347=6940. In setting down this second result we omit the zero, because it will have no effect on the addition which bai to be performed. Multiply (i) 23 by 15. (4) 70 by 26. (7) 205 by 43. Examples, (vii) (2) 37 by 29. (5) 125 by 24. (8) 307 by 98. (3) 45 by 36. (6) 327 by 42. (9) 2684 by 35. (10) 57296 by 27. (11) 84293 by k (12) 762^302 by 76. Ex. (2). To multiply 34007 hy 213. ^ 34007 213 102021 34007 68014 7243491 68mTon''^^^ ^"^ """J^^P'^ ^^^^'^ ^y 200. the result is b801400, and tve omit the two zeros at the end, beinjr careful to put the 4 in the place of hundreds ^ UbservG thaf in all ^oc/i,. +u„ /j j. n „ , of each partial product will be in the same-veitical Ihie 90 MULTIPUOATION. With the fipre by which we are multiplying., thus m the example just given, the 4 in the thiVd pfoduct i^ " the same vertical line with the 2 by which we mulli^ed. Ex. (3). To multiply 30047 by 21009. 30047 21009 270423 30047 60094 631257423 Here the first figure on the right of the second pro- duct stands m the pla^e of thousands, becau.eTe C Inen multiplying 30047 by 1000. Ex. (4). To multiply 70407 by 3700. 70407 3700 49284900 211221 260505900 .t h if- Examples, (viii.) Multiply (1) 320 by 532. m (3) 8^9 by 506. h) ^^Ki^^L7Jl^' ..J^^ ^846byyo, (8) 85639 by 598. (10) 30207 by 6060. (12) 8637406 by 40300. (14) 980740 by 3406. (1 6) 870120506 by 700403. (18) 742349 by 947. (20) 428734 by 8067* (22) 8724C0 by 7004B: (24) 423796 by 57243. (26^ 27040.'? Kir KAQnzia 704 by 176. 917 by 406. (7) 7859 by 5006. (9) 79302 by 4007. (11) 642867 by 90807. (13) 970052 by 40072. (15) 9864302 by 300071 ' * (17) 32804070 by 409300 (19) 5:8028 by 6205. (21) 984236 by 5009. (23) 385704 by 36479 (25) 620072 by 400205 v-«/ ^*iJO* Dy obij9'/8. MLLTiPLlCATlON. loo we are 81 ^^^Ex. (5). To find the continued product of 14, 8, and hr^Z Ty %'\^:^''' '''y'' -^ *^- -Itipi, th. ' 14 8 112 70 I that 18, 14x8x70-7840. Examples, (ix.) Find the continued product of (1) 18, 19, and 20. (2) 4.^, T'^ 10 . - (3) 847G, 2300, 70010, and 2003 ' ' ' *°^ ^• 27. When a number is multinlipd ;.« v/.^/^ twice, three times .17^,?? ^^ «^««^/ once, pZ^tc, wh^^T"'"' '? "^'^^ Involution, and the factor in the operation employed as a [power. *''"' '^'''■' ^' "'"^^^^ "^P^^y^^ i^«^«^ of second .power. *''"' '"'' ^' "^"""^ '°^P^«^«^ i««tead of third Thus, 144 is the square of 12, because 12 x 12=144 64 8 the cube of 4, because 4x4x4=64 81 13 the fourth power of 3, because 3x3x3x3 = 81. Examples, (x.) Find the squares of U) 15 , (5) 69. (Q) 114. (13) 789 And the cubes of (14) 11. (15) 13 (18) 68. )iq( fno (22) 356. (^23) 539! (2) 24. (6) 72. (10) 237. (3) 40. (7) 87. (11) 625. (4) 67. (8) 100. (12) 897. B. 26, (17) 47 v^U; 300, (21) 267. (24) 704. (25) 987 l>lVUI01f. V. Division. , 28. Division is the process by which, when a t>rndn^ The given factor is called the Dmsfla. 29. The operation of division is d^fed by the sign + . Thus 12 - 3 signifies that 1 2 is tMliviitn by 3 The same operation is denoted by writing the Div! dend over the Divisor, with a line drLrS^en tfc thus, _ 3 ^dV^'-P^^'' "^.^ '^^^^ *^«^* °°ly of cases in which the J)ividend contains the Divisor L e^act number of afr^9" *?°'' ^°*°^' numbers, the Multiplication Table affords the mean. >f solving questions in Divisioi For inatance, sir e 12=4x3, . , . 12-^4=3,' and 12-r3«4 ; andsmce 96^12 x 8, ' 96^12=8, and 96h-8=12, 31. When we divide one number by another, we find how many times the latter is contained in the 7ormer; and therefore any process by which we can discover how many times one number is conteined in another vdH furnish a rule for division. Such a process is expLined by the examples, which we shall now give. ^''P'^"^^ Ex. (1). Divide 408 ^>^ 17. Since 17 x 20=340, and 17x30=510, it is plain that 17 is contained 408 more than twenty times, and less than thirtij times. ^ If then we take awav 340 frnm A.nA o«^ ^^j !.._ many times 17 is contained in the' numbed Thlt ^^emains" DIVISION. )rocep«i, called is called the 23 /.i tir - ''' -^ ''' ^"^ ''^^ number 'coatains 1 7 Ju.t times?tT ; ''^ tr Quolnt? TV"''''^ ^"^ ^^^ ^-r This process is jepresented more briefly thus : 17)408(20 + 4 An 340 ^ ^' ^ 17 Hence 408 ^ 1 7 = 24 ^^ are enabled to omitleror, " "P'''""'^'^' "d ^'^ 17)408(24 68 68 Ex. (2). Suppose we have to divide 89012 by 10 : Divisor. Dividend. Quotient. 17) 89012 (5236 85 40 34 61 51 102 102 W in th« qnntk-'^^ "i- ' • ^' '^'"'" 5 »« the first fi»„,o - q-ot,e„., .hen mu«,piy 17 by 6, and aubtriot J- -'^ f4 Divisioir. Ex, (3). Divide 020575 by 23. 23) 920575 (40026 92 * V .;l^^|l 057 46 115 116 Bividf Examples, (xi.) (1) 18 by 6. (3) 84 by 7. (5) 182 by 13. (7) 456 by 19. (9) 3996 by 37. (11) 431376 by 817. (13) 19249470 by r;2. (15) 224009433 by 489. (17) 2880376 by 1369. (19) 98955005607 by 4123 (21) 13312053 by 237. (23) 360910856 by 83. (25)- 218860161 by 689. rm) (27) 39916424548 by 1001. 28) (29) 26540638445 by (2) (4) (6) (8) (10) (12) (14) (16) (18) (20) (22) (24) (20) 27 by 9. 132 by 12. 238 by 17. 3708 by 36. 6499 by 493. 076272 by 94f . 86306784 by 358. 40903'-'5214by618 10781526 by 6142 4076361 by 2019. 505350360 by 89. 4600304 by 907. 337403025 by 861 152847420 by 5060. 7649. I DIVISION. ;30} 1 1 firir.ft430800O by 17072. 31) 35()8800H82;{4;U by 74201 ,32) 3691870'22085112 by imiiZ 26 (33) 837741350162461) by 98U89. (34) 5837r»823(i(;!) by C428()7. (35) 2!)5U9ividend is 1171C92, the Divisor 342. Find the DiWsor.^^'^ r>ividend is 149201, the Quotient 23. Find the DiWden^d^^ ^^'''''°' '' *^^^' *^^ ^"°*^^"* ^^^^2. Find the SHOUT DIVISION, 33. When the Divisor is not greater than 12, f-.e proces,- of division may be greatly abridged. Suppose we have to divide 92368 by 8. The operation is set down in the following? form : 8 92368 11540 Quotient. The following is the process : the 2, reading the result as 12- then sinM s f.Tv;f • ? r/ "' l^' f'"" t *" '^"•^'"der, we set down one nnS the 2 andpvetx * to the 3, reading the result as 43 ^ tTen since i •'-' "^= "^"''•^•» »«a preiix iS to theO, reading the 96 m n I ITVIMOir. the 8, reading the result aTifi. A ^*' ^' *"^ P'*"^^ 4 to «m.. in 48, with nrrolTnaer' we'LTf ^ '» ^'^ntained ai^ •"d our operation i. oompletd! " ^ ""^"'" ^^»« «» Next, suppose wo have to divide 11042:304 by 12 The operation is Bot down thus : 12111042304 Q20102 Quotient. The following is the process : whSrerain^![^ ^tn t^^Z ^^« ^^'^-^^-n « "umber in 110, with2toc;rryon the^li^" "''^^''^^^ *"'^'' '*>^«* and there is nothing to carn^ on . ?>,! ^'i^^^'^^^ ''''»<-•« in 24, f in 2, we therefore BeTdoCd unT \^k"o**"* <^ontaiv^d at 2 ; .then 12 is contained n 23 once wit'h' \t ^' *"^ ^'^"y «» 12 |8 conta ned in nOninetirZ\7thoL '*''^^^ ' *^^^» 12 la oontiiined in 24 tuHce exZy '° "^'^^ "" ' lastly, Examples, (xiii.) (^) 7652 by 2. l3) 86502.':2 bv , (5) 7463424 by 6* (2) 725061 by 3. (4; 8749320 by 5. D^Uo'T"; /"' '''^''*«' («) 27540918264. (10) 46528920. ri7) 981754200. (18) 234567000 «et :"°" "' ''^ '""""'"^ "•■™'- ".V 7, „. „„aT2 09) 7971348. (20) 29574468. (21) 073638781^ »MOtUTlOK OF NUMIBRS INTO FAOTORH. 27 VI. On the Resolution of Numbera into Factors. lactors; in the operation of wh-,»h «;« u K'v«a ti.» product U giv^n. Ji tttl^JeTo rfou„r '' thus, the factors of 21 are 3 and 7 the factors of 65 are 5 and li thtie 9 and C are factors of 54 ; *"j .'!"' J*"'"" "' " *»'••« 3 and 3 and the factors of 6 being 2 and 3 the number 54 can be split up into /o«r factors. 2, 3, 3 3 div^Vf^t.retranTu*';- '''"'' "^ ^ -«' tl.us, 2 3, 5, 7. n, 13. 17, 19 are Prime Numbem i^Sz^^^-yZ i^- t-itnT '^ '-- Numbl'' '■ '' '■ '"• ''' '"• ''' !«' 18 «- Composite w;.^r^j:S;CbtrtC:^^ --'-^ '- 4 = 2x2; 6=3x3; 8=2x2x2; 9=3x3 it i. exaetfyTvSe and t.r'"]""'-.''^ ^^'"^ "« l^""^ any small prime'rmCV'XcfH^I: *tr H "'\^ ^^.iP'--^.'" .'hi. >vay tiflThTtuoulnT?: 1^ t:r^l.l' -tiviByr- ctre luo factors required. ' - - - , -...-.xi t,«^ Thus, to tind the factcrH of 2520 : 2 2520 2 1200 2 030 3 316 3 105 6 35 7 7 Hence 2520 -2x2x2x3x3x5x7. Ill i)racticttl aritlnnotic wo seldom require to Hnd all the factora of a componite nuinl)er, but very frequently wo want to know whether a number is exactly divisible by a particular number. The student will find it of use to remember the fol- lowing properties of numbers. A number is exactly divisible by 2 wlen its last figure is or an even digit, as 426 ; 3 whfcu the sum of its digits is divisible by 3, as 679 ; 4 when its last two figuros are divisible by 4, as 2364*; 8 when its last three tij,'ures are divisible by 8, as 26256 5 when its last figure is or 5, as 30 and 135 ; * 9 when the sum of its digits is divisible by 9, as 276265 • 10 when its last figure is ; ' 11 when the dili'eronce between the sum of the digits in the odd places (reckoning from the right) and the sum of the digits in the enen places is either or divitlble by 11. Thus 24794 and 829191 are di- visible by 11. Examples, (xv.) Find whcMier the following numbers be exactly divis- ible by 2, 3, 4, 5, 8, 9, 10, or 11. ^ MII0I.UT10W Of NtfMBIRH IIITO rACrfOlli. 99 (1) (*) (7) (10) NOTK. 117. 1000. 27404. (>480. (2) 288. (5) 23472. («) 324f)6. (U) 019182718. -Wo have inserted theae rAiiinrlr> .. »i.i. bocau.^ in attempting to re.«lvo l" rrmmC info'fS' The Htmlent umy „ow, following the iustructionH giveu 'M Art. 38, work another sot of Kxainplen. 496. 42345, 847H2. at thi« point, Examples, (xvi.) Resolve into jtriin*- factors : (1) (5) (») (13) 07) (21) (25) 18. 30. 54. 91. 108. 288. 729. (2) (0) (10) (14) (18; (22) (20) 24. 39. 67. 99. 112. 432. J)99. (29) (7 (7) (U) (16) (19) (23) (27) 5760. 27. 42. 72. 00. 132. 526. 1296. (4) («) (12) (10) (20) (24) (28) 32. 61. 86. 106. 17(i. (525. 1700. 39. The process of Multiplication may often be niad« shorter wher» the Multiplier is a composite n »X r b v I'^'^olvmg It into two or more fcuitors. '"'"oer, Dy Thus if we have to multiply 2579825 by 56 wo mav resolve 66 into the factors 8 and 7, a„u proceed th^s ; ^ 2679825 8 20638000 7 144470200 The advantage of this method will be more apparent we^L7' T"" '" ^^^Itiplication of sums of money weights, and measures. "iuney, Examples, (xvii.) 2l^}^'1^:^l'^' ^^««^^i"«' the multiplier into factors not 30 I iJfiXAcr rnviMoN. (9) 110725 by .'WO, ill) 36729 by 1320. «407 by m. 2W72 by 64 ,, 90728 by lat. (10) 40207 by loa (W) 704075 by 14400. 40. So ftleo we may oft«n simplify the DroceM at £ rXuT w /'r ""'^T ^^^"^'' Uvr Linre:, oe made u by fttctow each not creator thnn 1 9 i?^l J then divide the Quotient by • a822 by 33. . . 4157028792 by 66. (9) 22030902 by 108. (11) 4726;{4r)00 bv 126. (13) 63706'^ <0O oy 14400. (2) 67910^40 by 15. (4) ll4309r)886 by 27. (6) 2005501072 by 49. (8) 1200130008 by 84. (10) 57607632 by 132. (12) 565».a60 by 720. VII. Inexact Division. DitLr^l^^onn.J^J'^''^ '^'^''" examples, in which the Dividend "^ '" '"^^ '"^^^ «^ ^i™«« '" the Now suppose we have to divide 23 bv 7 ir^^'^3 'ni^ -r^^'t^ ^"""^^ '^'^^ ^« ««^^ divide 23 unit« in.u 3 parcels, each conta ninL' 7 units and whfln JL! have done this, 2 units out of t^.e 2ZLtnt:^''' "' RemVnder. ' "" "' ^^' ' ^^^ ^^^^^^^^ -^ 2 the -ife WIXACT DITIUOir. 81 thuif '"' *' """ **•'" *° ^'"•**« '2469 by 53, w prociHKl M) 72460 (1367 63 194 169 "366 ai8 889 871 IT Hence the, Quotient in l;jfi7, anr, a„d Divide Examples, (xix.} 3492 by 37. BT'^'m by 47. 74\;273C by 71. 87407 by 103. 827626'^ by 723. 48237654 by 4821. 48629C by 41. 67(M>2 by 65. 82740325 by 98. 978402 by 409 0740045(52 by 1009. 08725042{X);i by 0871 luomuu or ouort Division, after hrflaL-inf, *u D.vi8ori„to component factor,, as ii Art 40 , hi' P mamder will bfi f.uitw' k„ * *"' ^"® -Ro- er wiH be foun. by a process iioi»/ to be explained. ^'X (1). Divide 4327b Iv 21. f?' 43276 14426 and 1 unit over, 2m and 5 parcels of 3 units, or 15 units over ; wJieiice the Quotient is 2n«n .^a .u_ -d _ . . iij+l^ or 16. ' " ""' ^'^'^*"^»''-i' r is 32 ki^*.: i ■i'^4.i INEXACT DIVI8I0N. Ex. (2). Divide 572948 by 125. '5 672U48 125/5 114589 and 3 units over, 22917 and 4 parcels of 6 units, or 20 units o'.oi-, 4583 and 2 parcels of 25 units, or 50 units over ; whence tho Quotient is 4583, and the Remainder is 50 + 20 + 3, or 73. Examples, (xx.) Divide, employing Short Division, (1) 4153 by 15. (3) 42813 by 18. (5) 724972 by 26. (7) 2825780 by 33. (9; 35C599 by 48. (11) ^30047914 by 77. (13) 44487 by 105. (15) 1194477 by 210. (2) 587595 by 16. (4) 423672 by 21, (C) 569024971 by 27. (8) 8642396 by 35. (10) 8274913 ^y 64. (12) 419421 by 99. (14) 95379 by 189. 43. In dividing a number by 10, we have mereiy to mark off the />'st figure, the other figures giving 'the Quotient, and the figure marked off the Remainder : thus 2460197 + 10 = 246019 with remainder 7. Again, to divide 42395675 by 20, might pnnjeed thus: 10 42395675 4239567 and 5 unites over, 2119783 and 1 parcel of 10 units over ^ whence the Quotient is 2119783, and Remainder iv/-f-5 or 15. ' V But the operation is written more briefly thus j 2,014239567,5 Again, 2119783 and 15 remainder. in dividing by 100, we mark off the last two figut^B, o "v '•••—.-J TT%,- iiiiiic OH cue lusu vfircs uguri es, m 'Nfe iti^iy PRACTICAL METHODS OP SHOKTBNING LABOB. 33 from Divisor and Divide.id, and find tho Quotient and Kenhiinder by a similar process. Sjuonent ami 44. If any thref, of tlie four numbei-s iUf f,.rr« *u from the Dividend, and you have the Remainder Mnif^'i ^f ^^'''''■' Q»o^ient, and Re.u^inder be given Multiply the Divisor by the Quotient, add the Rema^nde; to the result, and you have the Dividend. "^^"'^^^^^ Snbfrlf^H ^i7''"''- ^^^i^e»^. and Remainder be ^iven 2 't bt th^ n- '^^^ '^' ^^^^dend, divideTe ies..t by the Divisor, and you have the Quotient result by the Quotient, and you tve^^h^^Lfvi^^^^^^^ ''^ Ejtamples. (xxi.) Kni KSdef'' *" "'"'*''"'' ''2'«- «"« Quotient 171. 67.*'^S fhJS'nf ' *•>" **"""-' 1381, the Remainder 24^ MhXSf ^ ^'"''^"■» ''''*'■ *•■« Kemaiader (i) The Quotient is 2010, the Dividsnrf ffinmin »i. t. mamder 317. Find the Divisor! "*"" «66237, the Re- Vra. Methods of Verifying the Oner-nH^na Hnd so«e Practical MTthfdfof sS^ Labor m the Fundamental Rules. ^"^ up^L^ZTown^tLTtdtf ^tU '" ''' '"^^ Thi» is generally sufficient ino her Lthod u"? T"' ahonzontal line across the middle of t"f tl '1^° J?)! '" v^o »epa,^te parte, then find the^surn'orurtVo '#--... 84 PRACTICAL METHODS OP SHORTBWINO LABOR. '%• wf answers, which must agree with the work it i.- to verify If it be a very long sum, it may be divided into thiee parts by two horizontal lines, and the three separate sums found, &c. * 46. Subtraction. The correctness of the result in Subtraction may be tested by adding the remainder or diffe-^ ace to the Subtrahend, when the result ought to be the same as the top line or Minuend. 47. Multiplication. The proof of Multiplication by casting out the nines depends on the following property of numbers : — Any number divided hy nine will leave the same re- mainder as the sum of its digits divided Iry nine. This will be evident from the following example : 6783 ^ 6000 700 80 3 ^ 9 9 ■^T"^9'"^9" = (666 + f) + (77 + i) + (8 + f+f) 6 = 666 4- 77 + 8 + 8 9 ^ 9 ^ 9 + 9 Hence i.' is clearly seen that the remainder, arising from the div:s?on >f 6783 by 9, is the same as that arising from the division of the sum of the digits by 9. This test may be given in the form of the following rule : Divide the sum of the digits in the Multiplicand hy 9, and set down the remainder. Divide the sum of the digits in the MultqMer hy 9, and set down the remainde Mul- tiply the two remainders together, divide the result by 9, and set doim the remainder. If the process be correct^ this remainder toill be the same as the remainder obtained by taking the sum of the digits in the Product and dividing it by 9. ^ For example, if we multiply 76371 by 854 the product is 65220834. PRACTICAL METHODS OP 8HORTENINO L4BOE. OO ree separate Sum of digits in Multiplicand = 24, Sum of digita in Multiplier =17*""^ ^^^^ ^^'"" '''^^¥' ^ First remainder X second remaind^il^ ^'"' ''™""'^^ '' Sum of digits in the Product = 3o!^ ^^'^^ ^'''' ''°^"'^'' ^• This .« n J *"^ ^^-^9 gives remainder 3. as 37 for 73°''" °' '=«""' "' "'« P-d«ct be displaced. the same. ' """ "' '^'8"' "■ «ach case being or Lerte?to„"'"fl:;;/°^ "' "' " '°'' ^' «' -"-er be omitted to the product. If thf res If T'"'"^T' '^ ''^"^ '' »"«. we have a verification o/tl e LTT '?• '''^ '"^'''»'J may also be proved by c,s«„„ „, "Pf"''""'"- ^'^'«»" proof is less direct than in Mu ifnllV" T"'' •"" «>« If ive divided 41 7 bv 20 i," ^""'P'"=''.t»on- For instance n. The most coTvLicnrrm'f ''/^^'''•'•^"''''"de' proof of nines ,s to wrUe this Z." 7'""^ '" "PP'? the = ■«■ .■ema.nder 3, which therefot ptoL t'tork'^ '' '"" » |eontS;mtr„?™be?rd\r^ between any given numJ, 5"!^ '" ^ 'be difference huperior "rder f t^tus 6 if^t -.v"'" • """ «* t^e next K 47 CI- 53. 8468 of 1532 ir *""'"""' complement of (encesrespectivelvof 4 63 I «9 '° ■""J'n'**"^ ">« differ- |.;ext^s«perior units of !Ll"±Tl'V'^' 'O^OO, the e Sd PBACTICAL MBTH0U8 OF SHORTENINQ LABOR. The arithmetical complement of a number may be found by the following ruie : — Begin at the left hand and auhtrad eveni jigure from 9 until the last ; suhtrad that from 10. The arithmetical complement may be used to find the difference between two numbers, thus: if 239 be sub- tracted from 676 the remainder is 337. But if 761, the arithmetical complement of 239, the less number, be added to 576, the greater, tha sum will be 1337, one unit (1000 in this case) of the next superior order greater than the difference of the two numbers. By removing this unit, the number will be left equal to the difference of 239 and 576 ; so that the difference of the two numbers can be found by addition. The arithmetical complement may be written thus i761, with the subtractive unit on the left,j which when added to 576, the sum will be 337, the additive and subtractive units being together equal to zero. This method is employed with great advantage to find the aggregate of several numbers when some of them are additive and sonr subtractive. Tiius, if we have — 3795 - 1532 ~ 2019 + 8759 - 5104, we arrange them as follows : — A. C. of 1532 is " 2019 " ({ 5104 " 3795 18468 17981 8759 14896 the aggregate required. 3899 60. Contractions in Multiplication. — The multipli- cation by any number from 12 to 19 inclusive, may be effected as follows : Multiply by the figure of the Multiplier in the units' place, and to the number to be carried add the figure of the Multiplicand just multiplied. ' r'P figure from 9 PRACTICAL METHODS OF SHORTlf.NINO LABOR. 37 Ex. 1. MuJtiply 2384 by 19. 2384 19 45296 4 X 9=36 ; set down 6 and carry 3 The back figure system, as it is sometime-, called mnv <1 tiinei «7503001» 378 times tho multiplicand "i ^ 9 times 42 times tho I o^.f.ino.4Qfl multiplicand - 9 times f ^'^^'^^'^^2486 405018054 ) 3649280169549 To Squarr any NtiMBKn endino in 5. Square tho 5 mid write down tho ropult ; thon increase tho numhor to tho loft of f) by 1, and multiply this sum by tho nu,mbor to which tho 1 was addod. Sot this product to tho loft of tho 25 and tho number thus formed will be the result required. Ex. Find tho square of 75. 5 squared = 25. Add I to 7 and multiply by 7 and place the 50 to the left cf the 25. 5025 is the result retpiired. 51. Abbrrviations in Division. — Since 4 x 25 is 100, and 8x 125 ia 1000, tho division by 25 will be effected by multiplying tho dividend by 4, and cutting off the last two figures from the product. Tho division by 125 will be elFected by multiplying tho dividend by 8, and cutting )ff tho last throo figures from the product. In each case he figures cut off, when divided respectively by 4 or by 8, ivill bo tho remainder, and those left will be the quotient. Any number can be divided by 9, 99, 999, &c., by successively dividing the given number by 10, 100, 1000, &c., ri spectively, and taking tho sum of tho successive remainders for tho true remainder; except when the sum of the latter exceeds the next higher unit ; in that case both the quotient and remainder must be increased by unity. FUAOTICAL METHOLH Of SHOItTUNINa r.AflOK. I'v Divj.lo 05874 by 99. 100)058,74 0,58 6 3U 006,39 Hnro tho sum of the partial 'remainder is 138 *nr1 omitteS. aiS X tho 1^^^^^ J^'?''^'^' subtrahends are working. ^ ^ ""^ rernaniders retained in the Ex. Divide 1084 1 971 Gl 21 by 6783. 5783 ) 108419716121 ( 18748005 50589 32121 Tu« n . J. . ^^^ fi'^al remainder. 8 times 3 is 24 4fl„, cf ""' "^=5' '.'*P '» «« Allows: kwliich put down) and carrv 6 8 f ,!. ' 7 '"T^ ^ ^ives 2 6783 ) 50581 (8 4317 44) BXAMINATION I>APBRH. \Vj' sny : 8 tiiKos 3 JH 21 ; 4 from 11 gives 7 (put ilown| ami <;uny .'i (in.stoud of 2). Then 8 times 8 uml 3 gives G7 ; 7 from 8 gives* 1 (put down) and curry 0, &c. Examination Papcrn. I. (1) Express in words, 4237400 ; and in figures, six hun- dred and fifty-three thousand eight hundred and twelve. (2) Find the sum of 24753, 86729, 4237, and 804(32. (3) Find the difference between 8(>21)3 and 78464. (4) Multiply 8627 by 493, and 50042 by 307. (5) Divide 8423703 by 9, and 2650582 by 368. II. (1) Write in figures, twenty-five millions two hundred and fifty-seven thousand six hundred and thirty ; and in words, 402050407. (2) From seventeen millions and seventeen take eight thousand and eight. (3) Multiply 0549 by 4037, and 27004 by 3700. (4) Divide 32450780 by 96, first by Long Division and then by Short Division, and show that the results agree. (5) Find the sum of one million and six, fifteen thousaiid and eleven, one hundred thousand and ten, and sixty thou- sand four hundred ; and divide the result by 9. III. (1) Write in words, 10010201401 ; and in figures, one million twenty-three thousand and one. Add together the two numbers, and from the sum subtract their difference. (2) Multiply 740206 by 2080, and 420004 by 3704. (3) Divide 78297426 by 35, employing Short Division. (4) From one hundred and twenty-six millions four hun- dred and six thousand and three take ninety-five millions and four. (u) Divide the product of 723 and 347 by 48. IV. (1) Express in figures the number represented by MDCCCLXXXVIII. (2) Divide 987654321 by 132; using Short Division. nUMINATIUN PAPBRH. «1 (3) P^upe to prime factors 66, 78, and 114. (4) Multiply the sum of 80297 and 40025 by the diflerenoe between 780 and CIH. (5) By how many does one million exceed one hundred unu one i (1) Divide three hundred and Hfty-three billions eitfht millions nine hundred and seventy-two thousand six hun- dred and two by 5400. (2) Multiply 8970589 by 9876. 40^^90^rnd 120"*^ elementary factors (i.e., prime numbers) (4) Express in Roman Notation 24, 47, and 178. , /^^ ^^Jr,!"^'!^ y^^^'^ "^*y ^® *»^en away in 24 carts, eaoh taking 600 br»oks ? VI. (1) Explain the method for the multiplication of two qm7i T' ol^^an ^'^^^i^^^lff o^ several figures, and multiply dOU71 by 20690, explammg the reason for each step of the process. ^ p^<^2)^^ultiply 70894754 by 112756 in three lines of partial (3) By what number must the product of the sum and difference of 8376 and 5684 be increased so that the result may be exactly divisible by 7859 ? (4) A drover bought 527 shppn at $2 per head : twice as maiy calves at thrice as much per head, 19 cows at $29 per ftead, and thrice as many I^orses as cows at four times as much apiece. How much did the whole drove cost him ? *v,^?^ i^"®-^*^^ ^h^ 8"«" o^ *^o numbers is 4331, and one-half their difference is 3363. Find the numbers. VII. (1) Eight head of cattle at $23 each, and 7 horses at $89 wo^rpra^riT ''^'^''"" ^' ^*"'- ™^* ™ *^« ^-d ta^e\? JiLTr::p" tXmetll" '' ''''' '^^ ^°"^ ^'''' '' ^JiL^ man bought an equal number of sheep and cows for V---V.-W-. £.ttcn oneep coat $6.dO, and each cow $21.50. How many of each did he buy ? * ."v/. avw 42 BZAimVATlON PAPBKa. Ha \% HE£ I Find the number ^**® rewttuuler wm 362. (6) The age« of three brothen are 10 1? ....i ik ^ ««d their father will, them h?« propLy V^Hh U^^ •coording to their age.. What doe. eaih get ? ' '^^ vm. (1) There i. a number which, when divided hv a .nj *k quotient diminished by .%^ and the re. Jt multi^lt;^^^^ in' and the product decreased by the ditfereZ Sween th^ ?he ' umC* implement, of 7846 and 347. give. m^'^LI <2) If 5 lbs. of tea are worth 15 Ib^ or cofTee anH a. ik- < coffee are worth 8 lbs. of sugar, how rLiv mu,nH.^# °' are worth 76 lbs. of tea ? ^ ^""^" '^^ ■"K*' (3) Find the number from which if l'?fl'7R k^ ♦-u^ xi. remainder wUl Y^ 45209 less 27C45 ^ *^ (4) A horse is worth 8 time. a. much as a aadHlA «r,^ Vw.*u together .re worth «261. Find thTvalu^ ofihe hi^^ ^'^ doing lost (2 per head. For K^m^ hl^he^d i^^heL^U the remainder to gain «800 on the whole ? '®" IX. JnLt7I'T^T^ be multiplied by 5, 25. 125, &c., by ^3* * I' ?J-^' *^\' *^*P^®" respectively to the numbeT and then dividing it by 2. 4. 8 &a. v^r^L;,. *i! ""'"o®f» this rule. ^ "y ^> % o, «a ij^xplain the reason of (2) Of what number is 99995 both divisor and quotient ? {6) A person bequeathed his property to his 3 sons Tn the youngest he gave «1789 ; to the second "SthneiM much as to the youngest ; and to the eldest 3 times rmudhTto th^second ; find the value of the property ^ (4) In walking a certain distance John talfA« ^>7rqa .* how many steps will James take in w«?k^L half S-^^^^" ' John taking 3 steps for every four Tf James^l ''**"*^ n ^?^ A?®^?^*^'' ^^^^^^ *» HlOam COMMOH FACTOl. X. 43 urouKor* t" which it i.*li.bl« ^* '"'"• "« •how the .Ron •in44-M4^lr,St*2^,62?4T«r3SS™-''''^*+ (4) Divide 7804643457 hy 9999. .mount to 51« ; find the^Sn"' ""™ '"«°""" IX. On the Method of Finding the Hiirheat Common Factor of two or more nS^ and 15. ^ " ^ ^°">°^°» Fac^r of 9, 12, [bnlfl; Zf p. ''*'"" '^°"""°" ^""^ -» "W write Ithe foilo'wi„« IZU'S^t": I'lV'J^'''} ."•"y work Igiven m Art. 38. ' ' *"'v-o "-- -^£-3 or m visibility ^»M»^to«.f&^ I ** HIOHRWT cOMMOHr VAOtOB. * Find tho ii. c. r of (7) 1ft, 27, 106. (8) 32 48 12H U 16,64.260.1024. (lo) 2i; m! ISft. 720. 63. In lar^e nuralKsni, tho factoin cannot often \^ aotonmno,M,y .nnj^ction an.'. U wo luwo to find th^ folL'-ng'rur """*"""' "' ^"^^ "^°""« ^'^ ^^^ />itfVi« M^ frreaiet of the two numhers by the less, awl no rZTr' ^heremaifuler^ r.j>eatin.j the pn.Z\?M ^ J^hus, to find tho H. c. F. of 689 and 1673, wo i,rocee' fro» dividing thH^hTr »XhL*^n.r i^r '■■« Find the L. C. M. of (1) 27 and 54. (3) 633 and 844. (5) 1000 and 2125. (7) 936 and 2025. (9) .*>443 and 4537. Examples, (xxv.) (2) 88 and 108. (4) 195 and 735. (6j 3432 and 3575. (8) 2304 and 4032. nn'^hi Jnd^t^'^; '" '^ %'^ *^^^^ '' ^ove numbers, we 1^. c. i':t til: r:;uit"gimbVri^^^ ftr f f .v'^ [proTe'd'thus'?' '"' '• '• " °' ''' ^^' ^^' -d ^4» - might The L. c. M. of 12 and 20 is 60 of 60 and 36 is 180, of 180 and 54 ia k^'s = tne L. 0. M. of 12, 20, 36, and 54"" is 540. 48 LOWEST COMMON MULTIPLE. ImH m. iamt But in practice it is generally nioio convenient to pr^ ceed by the ^lk>wing Rule : Sf^t doicn the given numbers side by side ; divide by any number^ ccmtmcncing with 2, 3, 5 . . . ivhich will exactly divide tivo at least of the numbers ; set down the quotients and the numbers that are not exactly divisible by the divisor], side by side ; and proceed in this ivay till you get a line of numbers ichlch are prime to one ariother. Then the continued product of all the divisors and the numbers in this line loill be the l. c. m. required. Thus, to find the l. o. m. of 12, 20, 30, 54. 2 12, 20, 30, 54 2 6, 10, 15, 27 a 3, 6,15,27 5 1, 5, 5, 9 1, 1, 1, 9 .*. L. c. M. =2 X 2 X 3 X 5 X 9 = 540. The following is somewhat shorter : Set down the numbers in a line, then strike out any that are contained in any of the others. Divide those not struck out by any number that icill exactly divide one of them ,' under any that it exacfhi divides, place the quotient ; under any which contain some factor mmmon to it, set down the quotient, after striking out this factor ; and bring down all the other numbers. Proceed in this way with the new line ; and so on, until all the numbers left in any line have no common measure, but unity. Then the continued product of the numbers in this line and all the divisors is the l. c. m. of the given numbers. Thus, taking the numbers in Section 57 12 1 12, 20, 30, 54 h 5, 9 ;. L. c. M. =9x5x12 = 540. To find the l. c. m. of 4, 8, 10, 12, 16. 20, 24, 25. 30 Again, ^EXAMINATION PAPERS. 4, ^, ;?, 25, ^ .'. L. c. M. = 26 X 4 X 12 = 1200. 49 Examples, (xxvi.) Find the l. c. m. of (1) 6, 9, 24, 40. (3) 12, 18, 96, 144. (5) 84, 150, 63, 99. (7) 17, 51, 119, 210. (9) 44, 126, 198, 280, 330. (2) 8,12,22,55. (4) 16, 30, 48, 66, 72. X (6) 27, 33, 54; 69 132. < //n J^' 26, 39, 65, 180. (10) 60,338,675,702,975. / Examination Papers. I. (1) Find the least number which, divided hv ia- ik a 17, gives remainder 12 in each case ^ ' ^^' ^""^ v.ilUhey'^st^n'at^ fy^T^^ ^^^^^^ i°«h«« together, how often atisTe;^r2?, anic3oT "'"*'"'' ^ ^^^^^^ '^ "-^es 17765 feef wid7 fh^f ''^^ T-^^'" ^ ^^^^ ^3023 feet long by KV A wo cog-wheels containing 210 a.nA qqn «^^ ively are working toafifh^r a*? u '^^ ^^^^ respect- IT. I /I) Explain how to find (1) the h. c f and (9\ f», . fcr^ of numbers by^Uing^h^L.^tt^^l^^^^^^^^^ . JaTleisS baS "^tf^' "1'^^* ^ "^^* -« *he three Bxact number o?'buret tLt^L^^^ ^^"^' ^^^^•"g -" without a remainder ? ^^'^^ ""^^'"'^ *^« «ame (3) What is the smallest sum of monfiv wifT, ^k- 1, t 3uy sheep at |5 each cows a? ^99/ I ^^*°^ ^ °»n feach? ' ^^ ^* ^22 each, or horses at $75 s^^ *'«m 60 EXAMINATION PAPERS. 4 862 yards, and the third 264 yards ; find the time between their onoe coming all together, and their coming all together again. , (5) Find the least number which divided by 675, 1050, and 4368, will leave the same remainder, 32. III. (1) Explain how you would find all the divisors which a number has. Find those of 1800. (2) The L. c. M. of 2, 3, 4, 5, 6, 8, 9, and another number prime to them is 10440. What is this number ? (8) How do you determine whether a number is prime or composite ? Which of the following numbers are prime and which composite :— 3391, 2699, 14787, and 1477 ? (4) Three men. A, B, and 0, start together from the same place to walk round an island 60 miles in circumference ; they walk in the same direction, A at the rate of 5 miles per hour, B at 4, and C at 3. In what time will all be together for the nrat time after starting, aad how many miles will each have gone ? (5) Find the greatest weight, in grains, that will measure both pounds Avoirdupois and pounds Troy, there beir^ 5760 grains in one pound Troy, and 144 lbs. Avoirdupois ccmtoin as many grains as 175 lbs. Troy. IV. (1) Define Factor, Measure, Multiple, and explain when a number is Prime, and when Composite. In what digits must prime numbers end ? (2) The product of two numbers is 1270374, and half of one of them is 3129. What is the other? (3) The fore- wheel of a carriage was 11 feet in circumfer- ence, and the hind one 13 feet. There being 6280 feet in a mile, how many miles had a carriage gone when the same spots which were on the ground at the time of starting h d been on the ground 360 times at the same instant ? (4) A can dig 36 post-holes in a day ; B can dig 32, and 30 in the same time. What is the smallest number which will furnish exact days' labor, either for each working alone, or for all working together ? (5) How many firkins of butter, each containing 56 lbs. , at oo *•«> Kf%^^.A*J*~ .J .i. U, r.^._ 1 A V.KU u containing 276 lbs. , at 8 cents per pound. AUCTIONS. V. 61 and half of i:SBr^^^^^^'^^' ana »a.ne place, when og^ODot/tUttr tt: ^ttt '" ^^^ Tli^ll" Te"'" lid tthT^iilP'*;^ -"> 5 row. of t«e.. I; 8, 9, 10, and Alee re^^e^ f^r* 'r,', «' ""> di-tanoe. of Hie Mrae .traieht line f nT^ ?;'• .;,*?® ''°«'» "^rt from .here being 62Kt In^'iS "a^d r2f h^ '""" ''^ '" ""■">' there be in the avenue I *^^ '*""' "•"y '««» "ill Ssllt^Tr'T'CTh'eCll*'' '""--^ '"tor. : ^, ^, XI.— On Fractions. 58 Numbers are the measures of quantities. ^''rrdlr/jirsjr^wiir- - -^ quantir/rthe^Zir"*^ "" ''^ "P- »"'« known , 4e NuL4 whTcrexprrsses'h " ^'""'^'"'^ °^ U""' ™'' is contained in the S??! • n "P^ *™«^ *>■'« ^"it the quantity. I^^nt'ty. « called the Mbasube of folLi^n^'ilSrltio^'n^We^:?''™'?'' '"' -« «-a the I'-y the Unit which we y»nn"^'*'»'"""'«°*»'Oney that a nian>»i„comr. Z, !, * ^''"'"'- """^ ''•'en we say \e receive yeaXl Lro"^""'^ " r'' *« "-» ^hat [nity two tlfousiK"s,ld ri^ihe """'."'"V'^ Thousand the meaa«re of his^nclme '^'' ^"° -fbe ^ZeT'iZ:::^:'^' \ "t °* measurement For instance, if we ta^e !^ n' '^'^^ '"''«"'"'^<'- mauBfr.. ^-- _ ' ^^ ^® ^a^e a dollar aa *J^" i"?:"--' i^ „• . • gHH - -"v.«oui« sums of ninnoxr ^.^^ ' .-.tCu Vj wiuuh ■ '"''"^>'' ^^ suppose this Unit to be 52 FBAOnONB. AlH divided into one hundred equal parts, and we call ea^'.i of these parts one-hundredth of a Dollar; two such parts will be two-hundredths, thrfn will be three-hundredths of a Dollar. Such parts are called Fractions of a Dollar, or other Unit, and we give the following definition : Dep. — A Frac ION is an expression representing one or more of the ei^aal parts of a Unit. The number of equal parts into which the Unit is divided is called the Denominator of the Fraction, and the number expressing how many of these parts are taken to form the Fraction is called the Numerator of the Fraction. These operations are denoted by the following symbols: we represent a fraction by writing the numerator above the denominator, and separating them by * horizontal line. Thus I represents the Fraction of which the Nume- rator is 3 and the Denominator 4. Such Symbols are called Fraction-Symbols, or, for brevity, Fractions. 60. The symbol J is read one-half. The s) abol J is read one-third. The symbol f is read three-fourths. The symbol j is read six-sevenths, and so on. 61. The Numerator and Denominator of a Fraction are called the Terms of the Fraction. A Proper FracM u is one in which the Numerator is less than the Denominator, as f . An Improper Fraction is one whose Numerator is not less than its Denominator, |. In our explanation of the fundamental operations performed with fractions we shall make use, as far as is possible, of proper fractions only. A .* FRACTIONS. irg 62. To show that f = |^. Suppose a Unit to be divided ito 3 equal parts. Then § will represent 2 of these parts (i). .V^l pll:' ''^' °' *^« ' P-^« ^>« -^divided into 4 Thus the Unit has been dividoH int« to i ,-. will repre.e„t 8 of the"e ,„bdWi.bn. «1""'^P»t«- ""d ■iv^orrn'S."" P"'' '" ('> " «"-' *° "' °f the sub. /. 2 parts are equal to 8 subdivisions. We dra^v from this proof two inferences : \\-^l^\'^. numerator and denominator of a frap^mn K^ Tr T, . Thus ?=« and A=^. tei bt theT''"'''°'' ?'' d™°™inator of a fraction be ' Palte^d "" "'" '""^^''' "•« ™'- 0* *h« f^-tion i^ mt 4=tt bv tl- ^^r*?' g'^« ^practical proof fngth ^' ^ "'"^ ^ ''™'8ht line as the unit of iLet the line AC be divided into 6 equd parts. "" ^fext, let each of the parts be sabdivia«d'int?4 equal then AC contains 20 of these subdivisions, AB contains 16 of these 8ubdivi.,v.„. . ••• ^-B is is of ^0 .. ..' "" ' ,„, W 1 t- o. M. of The same rule hold, for three, four, or more fraction. ' wiiLt^rom\r:7'r/"-«- -^^^ «"« Denominators 8, 7, ^ c. M. 56. Quotients 7, 8. New numeral^^ors 21, 32. Equivalent fractions f ^, g| i»enommators 3, 9, 72. • I- c. M. 72. Quotients 24, 8, 1. New numerators 48, 32, 13. Equivalent fractions f f ,' ^|, j.3. Examples, (xxviii.) |«.o!der,^:rr'™""'* '"'="°»' -* *"« lowest com- - ctv?rrthr!^t%T^r;ii*r ? """^ *-'--. »on denominator • thj ^h- f':»ctions with a com! ,^e original Wiot'^ca^^^^'Tr;^^ "' ""' ™'"«« <" [umeratorsof thenewfrZtio^,'"'^^ ^ comparing the For example, to compare th« v.i ifte «qaivalent fraction, are |J, | J, |£.' ■-■- a> 4} 3,na f. 66 ADDITION OK rRACTlOlMU The dwcanding order of value of the numer&tori it 63, M^ 66; • the detcending order of value of the given fraotione ia h h f 67. We may also compare fractions by reducing them to fractions with a common Numerator, and aHsigning the greatest value to that one of the resulting fractions which has the lead denominator. Thus, to compare the values of h II. »nd 8i- The eS2 Y^ he equal to the difference of the onginal frZiinT For example, to find the difference l,etween § and ^ f-Handf = |f, Examples, (xxxl.) Find the difference of the following fractions • (1) |and^. (2) ^andl§. MULTIPLICATION OF FRACTIONS 70. A fraction is multiplied bv a wholp n„^K.., u.. [Uinpi3^iiig the numerator ]>v that numhfir «nH"T""" •"*'' ^e denominator imchunged. ^""^ ^"^'''"^ M MiTLTOUoATioif Of nucTioim. This I muUipliad by 3 b«ooi»M f , for etch of the iymlwU ? and f inipUw that a unit has been divided into 7 «qtml parti», and three tiinea as many of tho«e p . In are taken to form the fraction repre- sented by the latter as are taken to fomi the fraction represented by the former. 71. Toprovnthat of l-A- fof |-|of H- Art. 62. Now, Buppose a unit to be divided into 1ft equal parts, then I of I - 1 of 12 of such parts, -T«^ of 12 of such parU, Art. 62. ^ 8 of such parts ; Art. 62. but ^ « 8 of such parts ; Art. 62. . . I of ^^^. Hence we derive the Rule for what is called MuLTiPLl- OATION f^' FUACTIONB. We extend the meaning of the sign X, and define §xj^ (which, according to our definition in Art 21, would have no meaning) to mean | of ^, and we conclude that I^l~ax5» ^vliich in words gives ua this rule : " Takfi thejmnhict of the numirators to form the nume- rator of the renultinij fraction, and the prodttct of the denominators to form the denominaior." The same rule holds good for the multiplication of three or more fractions. Before effecting the multiplication, common factors should be removed from the nuraerato. and denominator. It will be well for the learner i.o be familiar with the principles laid down in Art. 38. For example, to find the value of ^ of f | of H we | proceed thus : M of M of H = l^^r^ and removing denominator, 2X7X5X7X17 ** 5x6x3x17x7x7 common factors from numerator 5x3 IS andi led MuLTiPLi- umer&tor and i wvmiow or fiu^oifa. H«Hiuce ♦heir tiiaplett form : m (1) «f of |. (7) ^ of }| of H of jfi. (») i?8t of ,^ of mi. (2) |xfx/j. <*) tlMfifx^ «^) Wxffxfs. (8) fHif of ifH of ,Vft. m VISION OF FRACTIONS «"g the numerator unchanged ' *"'* ^**^- Thu. HK:ded by 3 become. ^, parts, and hence each parriu th" 1 '^""^ >"^? ^^ «^i"^* as preat as each part fn f hi l^f ''''"^'" *' ^**^ ^^^^^ , number of parLiftAen in ^^^^^^^^^ ^"'LT''^ **»« «*«« I is one-third of the former ' ''^''' '^' ^'"'^^ ^^'^^'^^o^ 73. ro«AoM;Ma<3 3 ,. " ^0 9X26 9X13X^ = f = Ij. iio »r 11X10X7X13 "• lox? ^ fa* «e )«ay in all oases change the Mived NiimV,-™ i„i.„ Improper Fractions, and proceed as7n til t " Examples. I„ Division weCSprocoed thn: : *""«"'"» For example, In Multiplication it is usually the best course aus - In Addition it is often advantageous to proceed thus : 4§+3H4+| + 3+^ =4+3+f+f =7+ff =74.l3,V fe/f ^!!{:.T^r..!^-? °!^-ore numbers are to be fndmakea^^ti^^L;i:;re:::^iC ^'^ ^^^^"' G8 FBAonoira. m. It In Subtraction we can employ the same method, but a little care is necessary. Suppose we have to take 3f from 4|, Reducing the fractional parts of the numbers to equiva- lent fractions with a comii'on denominator, we have 31fand4Jf We can now take the integral part of the first number from the integral part of the second, and the fractional part of the first from the fractional part of the second, and we have But suppose we have to take 3f from lOf, • Since f=|fandf = Jf ^ is greater than f , and we cannot take away the fractional part of 3§f from the fractional part of lO^f. We escape from the diffi- culty by the device of adding unity to each expression, to 3§| in the form of 1, and to 10^^ in the form of ff. Thus 10M-3M =10^-4||=6t|. TaKe another illustraticfai of a practical nature. From 5^d. take away 3^d. We add four farthings, i.e., f of a penny, to the former sum, and 1 penny to the latter, and reason thus : b^d. - 3|d. = 5f d. - 4|d. .= Ifd. = lid. Examples, (xxxv.) Simplify the following fractions : (1) 4^-3J. (2) 8f-6|. (4) 6fx9f. (5) 14x3^V (7) 2H3i. (8) (9) l^+^+17lf. (10) (11) 14f-6^. (12) The following examples should be carefully noticed : L From 17 take 4^^. 17-4^=16-fl-4A=16-4-4-l-A. ""-12-|-H=12M. (3) 104f-r53^. (6) 9Axl9f 4f-2i FRACTIONa. 63 II. From 317 take ^j. , 317-A = 316+l-^=316+^=316f|. HI. Multiply j^^^% by 397. Since Vm = l-TcW 397x,^ = 397-^,r,=396 + l-Mff =396 + ^^^ = 396,^^^^. ^^ ThuB § of f and | of 2^ of 5f are compound fractions. ^lu'iSiX:'"^^' *° ^^"^P^^ ^-^^-- ^>y *h« P-ess of Thus I of 2i of 5HI xf X y- = ^21^! = |oo^8^^^, 80. A Complex Fraction is one of which the Nume :Zhe7 ^^^^™^"^*°^ ^« ^^-^^ - fraction or a S Thus - I and g| are complex fractions. DivfsioV ''^"'1 '" '^'"P'' ^'"^*^^"« ^y *h« process of Thus 1=1+7=^-4-1= 7=l-7=|+f=|xf=3^ and -=2+|=fx|=Y=3f. Examples, (xxxvi.) Simplify the following fractions : (1) |of5Jof7l. (2) 4^ of 11^ of m (5) ?! . 4| 16§ (3) ^ of 2| of 3^ of 90. (4) i 30J <«) 3T (7) -^:3. («, ^. Q1 rjM _ round br;.;*sideA„rthat .h"e i:::^!:^^:^ 64 FKACTIONS. name of so many units represented hy the numerator No difficulty is ever experienced in finding the H. c. p. or L. CM. of $12 and $16, or of 12 apples and 16 apples. In fractions the name is written imder the number repre- senting the collection of units of that name. Thus to find the H. c. p. of if and M, proceed as in whole numbers; find the n. c. p. of 12 and 16, which is 4, and call it by its name, which in this case is thirty- sixths. Hence the h. c. p. is ^j^. Similarly to find the l. c. m. of i| and if, find the L. c. M. of 12 and 16, which is 48, and call it by its proper name. Hence the L. c. M. is ^f. Hence to find the H. c. P. of fractions we have the following rule : Change them to others having the same name or denomi- nator^ and find the H. c. p. of their numerators. This placed ove¥ the common denominator will he the h. c. p. of the fractions. To find the L. C. M. of fractions : Change them to others having a common denominator ^ and find the l. c. m. of the numerators. This placed over the common denominator ivill be the l. c. m. of the fractions. The following is somewhat shorter : Find the L. O. M. of the numerators, aiid under this i^lace the h. c. p. of the denominators of the fractions. The resulting fraction will he the L. c. M. required. Examples, (xxxvii.) Find the h. c. p. of the following fractions : (1) landf. (2) iiandfoV (3) i 3i, 4i, and 5f . (4) f , f , H, 4^, and 5i. Find the l. c. m. of the following frar*' ions : (5) fandf. (6) 2| and 7i (7) 4i, 5f, andSA. (8) \ of 2| of ^ and ^ of r^ of 2^. H H OW XH» USB OF BaAOKETS. ^5 ON THE USE OF BRACKETS. 82. When an expression is inclosed in a bracket ( \ It IS intended to show that the whole of the expressio J I affected by so.e symbol which precedes orlSZHhe combining liandnbvJ^jr^"' 't'''\ ^® °»*y ^^^ct by by 24. ^ ^ ^* ^^ addition, and multiplying the result will be * * ' ''°" tnerefore the result 2H2i,orV+f,orVx*,orJ|. » d-r[2+3~{4+5-r-(2+A)n » =3-[2 + 3-{4+6-rjn ^ =3-^[2+3^{4+V5|J =3-j-[2h-3-5-^J ='3^[2 + |y = 3-r-^/-=l^ wil7tetrtrk:™e?,!'''' r^^'^t* '^"Sth because it peculiar cC of wf *°,f' fP'"? "'i* neatness a Which ap;!rin°lt"X tKuSr;^ ^-«-». 2- A 4tJ M k! ON THl! USS OF BRAGKITS. the aid of brackets, may be This fraction, by the aid seiited thus, l-f[4 + l^{l-l-r(2-^)}], and then we can simplify it by the gradual removal of the brackets, the final result being ^f. 84. There is another method of simplifying Complex and Continued Fractions, which we may explain by the following examples : Ex. (1). To simplify ^q^- Multiply all the terms of the fraction by 7, and it becomes _31_ or n 14+8 "^ Ty« Ex. (2). To simplify ^. - Multiply the terras by 30, and we get Ex. (3). To simplify |^' Multiply all the terms by 42, and we get 28-18 .. . 35n6 ^"^ *S «' ^' 3 Ex. (4). To simplify 3+ 9+^ 3 195 195 9+^ Ex. (5). To simplify 1 + A + i+i 3+ff" 195 + 28 1 ~223 1 + 1 1 . 1 1+ j_^ 1 1 1 + 1 1 1+f + f 6 8 ON THE irsB OF BRACKETS. Examples, (xxxviii.) Simplify the follow ng fractions : m (1) (4) (7) 6 5 + i (2) (6) 19-^r l.~ 'A. /it -/it (3) rS '•' AT^ 6 + 6 (8) 9 + 1 2 + 2---L- 4-* (10) 3 + i 1 1 + — ^"i^ 86. If two brackets stand side by side taih nn s»nu i^zV'^c ^:^ ^^-\!' '' '^ -plied 'thif^re :z of the other ^^ '"'" *"" ^' m^^/^W by the contents The following cases will illustrate the generally re- ceived usage in Arithmetic respecting these signs : (1) The operations indicated hi ''of J' x and ^ \hn..hi be performed before adding or mhtraikng. ' ^"^^^ Ex. (1). I + f of /r - i H- i + f X ^r = I + ( J of i?r) - a - i) + (f X ^^) = I + A ~ I + A = H. (2) The operations indicated bii x and -^ i»7i/,«/^ Ao I performed in the order in which they ocZ ' ^ ^ • Ex. (2). Ex. (3). Ex. (4). — 16 |xf-f xi -=|x|xfx| 68 MI^ELLANBOUS EXAMPLES IN FRAOTIONS. (3) The operation indicated by **of" should he performed he/ore that indicated by -r ; this is the only case in which custom makes a distinction between x and ** of." Ex. (5). f of 2^-f-l^of I = (t X V) - (V X I) Examples, (xzxiz.) Simplify the following expressions : (1) 3J-(2J + lf). 2 (3) 1 + 5-»- 7 + f 3 4 - (5) (7) fof^ + ^-f (9) (I - A) (2| + 3f). .jj. (2 + ^) ^ (3 + \) (i - i) X (4 - ^) (2) (4A + 2J)^35f. ^*> 7731. (6) ^ + f of -^ - ^. (8) (H - ^) of Th - If. (10) (T^-^)-(7'V + Tk) ^2N (3^-2^)^j^off ^ 2|4-(i + i) 86. We shall conclude iiis Chapter with a set of Miscellaneous Examples on Fractions. Examples, (xl.) (1) Ad«^ together H» T%» ^f> yV* M- . (2) Add ^ of f to ^ of 2i, and multiply the result by (f of |)-(| + |). (3) Subtract f ©f I from 1^ of I, and divide the result by _ 4 I) X (f - t). MWORLLAIfBOUS BXAMPLEJI IW FnACTlOM& 00 (4) Simplify tho fractions Uh 'f^V/ and find their product. tiJluu^lV^-Tr" ''' "'"'"" '' '* '^' ^"' ^'' (7) Multiply the differenco between ^ and m l>v the sum of 4,V and If ; and multiply the result by the difference between lOf and 5§. ^ (8) Simplify (9) Simplify ^ ^ ^^ divided by ^ ^'* -^ '* - ^> (^* ~ 3i> (10) Simplify (11) Simplify ^*y + li/. divided by <^^-^H-A)(2i-|) (4i - A) - (2& - /^ - ^V). (12) Simplify 2j^and(^ofl/^)^|. (13) Simplify 1 J 1 and 4 2 - 4 + 1 - -s IIT 1 1 2 - B (U) Simplify 10| - 1^ , u 7TTl^^"^(^°^2^V)-i|- (15) Simplify 07 ^1% - 'm + h -k ''''^ ^^^ "" ^^^- 70 MfRCILLANIOUN IXAMPLM Iff FRAOTIOir*. (16) Sirapiify X /f of 7 and 3 - 3-i (17) Simplify 4 - ^ X lOf. (18) Simplify (19) Simplify Aof(H3of24|.^-.tH x3|t?^ni}^ ^^ m ^m^m-7Ux 5H 4- 14H ^ ^Tftr- (20) Simplify 19 7 X (21) Simplify Y- X «^/tw - (1^ - n\ 3 -If 2 + ■3— X ^?2? -f- (H - if). (22) Simplify 5 ^-^ «-^ , i-U - 13 4 2 7-^ ^4-* ,(23) Simplify 19 - i-A 2 3 + 3-1 ^-^ ^ $-»^ H-lt J 1^_ "^ 1 2 • 2-i"3-f l|-r6F2S MXAmuAnojn tAnut, n JSsecmination Papert. I. (1) Explain how to reduo« » miiixi numbor to an iniDroMr fraction, and ahow the reaaon for each itep. »«»P«>p«r (2) Bou|ht 1H| yardi of .ilk at |2f a yard, and 27* Ib^ of cho«ae at f^ por lb. ; how much money did I spend f thS liZZ:^ ''"**" '"^ '^^ "•" «' ^2J and 8i contain II. tionln[S«th«r'?''*S^ by.eipresaing one number a. the frac- of iV ^ '" ^'^ ^^^^"^ ^* " **»« ^'»o^i«n (2) How may the relative magnitude of two or more frao i^oXr^^T^tU:-""^" ">« '-'-• A. A. T/lTn Older thJ.h "J'"""""*'" 1000 which mu.t be addVdta order tli»t the sum may be greater than unity. (4) Show tlwt the value of |±|: lie. between | and f At'^if th^'ahtia'^^^alTo itln.JJfrtk*"'' « »' find the value of each. * °' ""* '*^'' ' m. .h„i^.£**'"' N"™*"*""- »nd Denominator, and exolain whv vaS of Ia bU'y"* ''•'''• °' '•"" '• ""'*'' «n, what i. the 2 -^ mnfu»l^?hi''^-i' *• *• r** i- »■«> ""»>''»«* the .urn from n IXAMXNAnOir rAPIM. IV. (1) Before addinj? fractiotu toj\toh A ean or pig8at*i2| h this ium ? at o > i'toe, der, and now 1 ir the mud ; tlier) are 6| hall be 3, G,, multiply tiie im to B, who him. What 7^ fee*- long ;h wK i thiL> (1) Giv fractioni (?) • proper tur. Ail vn. ' Inition of multiplication that will apply to .un diei worth $40000, and leavos l of hit ta wife, f t<, !!• ton, and the rest to hia dautfh- . .. ftt her duttlh leaves j| of her legacy to the wn and tiie re*, .o the daughter; hut the .on add. hiJ fortune to '1?'" "fhfjr: ^'"• 1 1 .^'^ "''"•«• «•- --»: wm ti: whole*? ' ''•^ *" *' ^'**'^'"" ^*" *^«' ««»'» ^'^ o( the caf re?d"an^d wrfta*.^'?"'?*'!? ** ^'^ '""? ' ^* "' ^''« remainder can reaa and write ; j^ of the romainder can read writo Ai.rl c.pher, while the 'eat. 243000, can neither read wr « «1. cipher; what i. tho population? ' ^"'®' °**' (4) Three men, A, B, C, run round a circle in 5 6 and 7Jl .mnute. .^.pectively. If they .tart from the wme iZt at hej«„,e tune and run in the .ame direction! Tow 1^5 Uev run before they are all together again ? and how1,fTen will each have goue round it ? vni mav k1 i ,®^^i« how .ruction, expresoed by lanje numbera (2) Is ^V more nearly equal to J or to 3^ - 2|? + It of 2J i - If, and by now much ? "^^ *h?\J^^ **^® sovereign, who have reigned in England .in«A in'otSLTTofTTh '• *^'? *^\*^^ "^ °»^ nant IthTS lea^h nf ' fV«r *""'*'^'' ^ ^^ ®**^h «^ *^o other., and X of leach of thi-ee others, and there are 6 beside. fim1^^„ Imany .ovoreig^i. have reigned in England .teethe 'onque:r (4) Three horses start from the same point and al fht une time, upon a race course 300 roS in^d^iu . th« fi^? TZT:"'LT ^ *^^ """•^' *^^ Beocmd I The UirdV h -i^-m^Mi^r^^i ^ti^^ - ''A) X 1^1 and X3i a D2CIMAL FRACTIONS. XII.— Decimal Fractions. 07. The mult-^Ies of 10 are 10, 20, 30, 40, 50, and oo on. (Art. 39.) The Powers of 10 are 10, 100, 1000, 10000, and so on, and these are called the first, second, third, fourth Powers of 10. (Art. 27.) 88. A Fraction, which has for its denominator on*^ of the Powers of 10, is called a Decimal Fraction, or for shortness* sake, a Decimal. All other fractions are, by way of dist'Tiction, called Vulgar Fractions. 89. To save the trouble of writing the denominators of decimal fractions, a method of notation is used, by which we can express the value of the denominator in every case. This method will be best explained by the following examples : •3 stands for ^, and is read thus, three-tenths. •25 stands for ^^, ana is read thua, twenty-Jive hundredths. •347 stands tor ^Viy> *^^ ^^ ^^^ thus, three hundred and forty-seven thousandths. The figures which follow the Point • are those which form the Numerator of the fraction in each case. The number of the figures which follow the Point corresponds to the number denoting the particular Power of 10, which forms the Denominator of the traction in each case. Now, as che first power of 10 is 1 followed by one zero, and the second power of 10 is 1 followed by two zeros, and the third power of 10 is 1 followed by three zeros, and so on, we can in every case write the denomi nator by affixing to 1 a number of zeros equal to the number of figures that follow the Point. Thus, -426789 stands for i%W#iJ» six zeros being affixed to the 1, because the number of figures that follow the Point is in this case six. Again, '07 stands for y^^, •006 stands for yi^, •00025 stands for TTy§ W DECIMAL FRACTIONg. 78 0, 50, and oo e number of I l:o zeros which come between the Point and the figures /, T) and 25, not being aet down in the r^umerators of tiie fraction, as having no effect on the value of the nuDerators, seeing that 07 and 7 stand for the same number, and that 005 and 5 stand for the same number. But these zeros affect the value of the denominators. I as tor instance, ' •7 = /^, while -07 = x^, and -007 - j^^. 90. Zeros ajfixe^ to a decimal have no effect on its I value : that is, •7, -70, -700 are all equal : for -7 = ^, •70 = ^^ = ^, 91. The method of representing Decimal Fractions is Imerely an extension of the method by which Integers are represented, as will be seen from the following con- |siderations. ° As the local value of each digit increases tenfold as ^e advance from right to left, so does the local value of 3ach decrease in the same proportion as we advance from jlett to right. If, tiien, we affix a line of digits to the right of the mits place, each one of these having from its position a ^alue, one-tenth par^ of the value which it would have It ifc were one place farther to the left, we shall have on kf Jl!^ 1. .r'^f ^^^ ""'''^^^ P^^^« ^ ««"«« of fFjactions inno denominators are successively lO, 100 w^L' Q* ' 7^'^^ *^® numerators may be any numbers between 9 and zero. Thus 246-4789 - 2 X 100 + 4 X 10 + 6 + A + ^7 + 8 ■ a io ?^' ^ ''"^^^^ "lade up of an integer and a decimal, L .u S^^ ® expressed in a fractional form by writincJ ^ the Numerator all the figures in the number, and m ►re ngures a/re?' the ijotnt. 76 DECIMAL FRACTIONS. Thus, 4-5 = 4§, !or4-5 = 4 + ,%= 18 + t''. H- for 14-075 Again, 14-075 = VoW. Examples, (xli.) Express, by means of fraction-symbols in their lowest terms, (1) -5. (5) (2) -26. (3) -75. (4) 00243. (G) 0000726. (7) 14-8. (8) (9) 50 0004. (10) 100 001. Express in the abbreviated form (11) ^^. (12) ^V (13) (14) T^. (15) ^U^. (16) 25 7 1) /1Q\ .TJr.TOS lOO •375. 104-235. AW- (17) T^tUh' (18) n^hU- (19) 0©- tttdMotjo- 93. We call •5, 3-7, 15-9 decimal expressions of the ^rs< order, •25, 4 39, 143 73 decimal expressions of the second order, "043, 5 '006, 27 "009 decimal expressions of the third order, the number of the order depending on the number of figures that follow the point The number denoting the order we call the Index of the order : thus 1 is the index of the Jirst order, 2 of the second order, and so on. 94. From what is stated in Art. 90 wo learn that a decimal of any order may be made lUto an equivalent decimal of a higher order by affixing one, two, three zeros, according as the index of the higher exce.3ds the index of the lower by 1, 2, 3. Thus '43 may be made into an equivalent decimal of the^/#/i order by affixing three zeros, thus, -43000, and -047 may be made into an equivalent decimal of the seventh order, by affixing four zeros, thub, '0470000. their lowest ADDITION or DECIMAL PRACXIOKS. ADDITION OF DECIMAL FRACTIONS. 77 95. To add -27 to -45 we might proceed thus : .•.•27+-45 = ,?o^^ + ^<^^=.^^==.72. But we obtain the same result if we set down the deci- mals one under another, point under point, add the ligures as if they stood for whole numbers, and place the point in the result under the other points, thus : •27 •45 •72 96. If the decimals to bo added be not of the same order, as for instance '37 and -049, we reason thus : •049 is a decimal of the third order, ^37 is a decimal of the second order, but it can be made into an equivalent decimal of the third order bv athxing a cipher, thus, -370. ^ Then we proceed to add the decimals thus : •370 •049 •419 "^ow suppose we have to add more than two decimal expressions, as -0074, -72, -05, and -123456. . Of « four expressions the last is of the sixth er and we may make the other three into equivalent uecimals I of the sixth order, and set them down thus : •0D7400 •f? 20000 •050000 •123456 •900856 78 ADDITION OF DECIMAL FRACTIONS. m^ When the learner is thoroughly acquainted with the principle on which this process of addition depends, lie may omit the affixed zeros, since they have no effect on the result, and may write the sum just worked out in the following way : ^q^^ •72 •05 •123456 •900856 If the numbers to he added bo made up of integers combined with decimals, we keep the points in a vertical line, and proceed as in addition of integers. Thus to add 4-27, 15-004, '9007, and 23, we proceed thus: or thus, 4-2700 i5-0040 •0007 23 0000 48-1747 4-27 15 004 •9007 23- 43 1747 Examples, (xlii.) (2) -007 and -2394. Find the sum of (1) -275 and -425. (3) -001 and 0002. (4) 13 •279, 3 00046, 742 000372. (5) -000493, 3 24, 15, 42-6, 324-42037. (6) 49-327, -458, 8317-05, 341-875, 32^4962. (7) 700-372, 894 0009, -347, 00082, 5370-006. (8) 560-379, -45687, 350-0036, 7074, 52-257. SUBTRACTION OF DECIMAL FRACTIONS. 97. If we have to find the difference between -47 and •35, where both decimals are of the same order, and -47 is the larger of the two, we proceed thus : From -47 Take -35 Result -12 SUBTEACnON OF DECIMAL FRACTIONS. 79 I-ciforming an operation like that of Subtraction of In- to^'ors, and keeping the points in a vertical line. That this method gives the correct result is evident, for •47- -35 = ^7^-^^ = ^=. 12. 98. If we have to find the difference between -888 aiKl -9, we may make the latter into a decimnl of the thud order, thus, '900, and since this is larger than -888. we proceed thus : ^rom -900 Take '888 Result 012 If we have to find the difference between -998 and 1 we observe that 1 being an integer, must be greater than J08 which IS a Proper Fraction, i.e., i^, and we pro- ^■^'^^^^^"«- From 1000 Take -998 Result 002 Examples. (xliU.) Find the difference between (1) (3) (5) (7) (9) 56-429 and 6-218. 53-316 and 6-0867. 6*047 and 5-9863. •0000086 and -00001. 10 and -0002. (2) 9 -005 and 7 -462. (4) -799 and '8. (6) 850-007 and 2708796. (8) 00537 and 000985. (10) -09999 and -101. MULTIPLICATION OF DECIMALS, Plotted ^thusf'^'''^ *^' ^'""^^'^ "* '^^ "°^ •^^' ^^ ^^g^* ■12x •ll=i\fex^ = ^li_^^g^^.0132, tae result being a decimal of the fourth order. •Oo'oi?' '^ ^^ ^^^® ^"^ ^""^ *^® P''^^'^^*^ ^^ ^'^2 and 4-32X -OOOlP^fg^x^M^^^^g^^ -0005184, the result being a decimal of the seventh order. 80 MULTIPLICATION OF DEClSiMlM. And, generally, tho product of any two decimal ex- pressions is a decimal expression of an order whose index is the sura of the indices of the orders of the two expressions. Hence, we deduce the following rule for Multiplication of Decimals : Multiply as i7i the case of interjevH, and mark off in the product a number of decimal places equal to the sum of the number of decimal jdaces in the two factors. For example, to multiply 2*4327 by 4 '23. 2-4327 4-23 72981 48654 97308 10-290321 -Again, to multiply 43-672 by -00000047. 43 -672 •00000047 305704 174688 2052584 We have now to mark off eleven decimal places from this product, and as the product contains only seven figures, we must prefix four zeros, and put the point on the left of these, thus, '00002052584, and this will be the required product. One more case must be considered. Suppose we have to multiply -235 by -48. •235 •48 1880 940 -11280 DIVISION OF DECIMALS. 81 This decimal of the fifth order is equivalent to a deci- mal of the Jourth order, -1128 (Art. 90), and this is the smiplest form of the result. Examples, (xliv.) (2) 3-62 by 5-23. (4) -562 by -00074. (6) -0009 bj Multiply (1) 7-6 by 4-7. (3) '427 by -235. (5) 300704 by 4-0205. (6) -0009 bv 1000 (7) «23.4075gy 24^0259 (V -SS?i«Vr6235. (9) 1432 C749 by -00004030706. (10) 50704 042 by 004007090061. Find the value of the following : (11) •407x4-03x006. (12) 101 X 1000 X -001- (13) •52x007x4-3x-02. Find the continued product of (14) 07, 4 6, 009, and 52-47. (15) 42 6, -795, 4-03, and 00074 (16) What is the cube A 2-74 ? (17) Raise 3-5 to the fourth power. DIVISION OF DECIMALS. 100. If we have to divide -27 hy 3, we might proceed proltlhusr """^ *° '•'"'"' •''''' '^y '"■ ^^ -"'g'" •00625-25 = „Vo'TO-25 = „^g^= '00025. Hence we derive the following Rule : \DiIi^^-i7 n.^" '""^'^''^ P^'-f"'^^ ^^'^ operation of Idee mil ll^^^^^^^ '''.;"^?^ ^^^^'^^^'^ P^^''' ^ *^'^'' ^re vieumaL jplw^es m the dividend. 62 DIVISION OF DBCIMALI. For example, suppose we have to divide 0086761 hy 243 ) '0086751 ( 367 729 1385 1215 1701 1701 The Quotient is to be a decimal of the eighth order, .'. the rebult is O0000357. 101. Next observe that, if the divisor be a decima/ expression, wp, can in every case change it into an Integer by a process which we shall now explain. If we n^ultiply a decimal expression k^ J^**u ^^I'^K"'- *° ™"''® *^® P«^"<^ «ne P^ace to the right : K^ iXX^ .l^^^S-*' '! *.° ™"''® *^^ P**^"* ^^« Peaces to the right; ri ht • " *** ™°^^ *^® ^°^"^ *^^®® P^^^®" '^ *^' and so on. For instance, 123456 x 10= 1 24*56 and 123 456 x 100 = 12346 '6.' The reason is obvious, for 123-456 X 10=ie^^x 10=1^^ a = 1234 -56, and 123- 456 x 100 = if gj^a x 100 = las ^la = 12345 6. Hence we can transform any Divisor into an Integer by multiplying it by 10, 100, 1000, .... according as the Divisor IS a decimal of the first, second, third order. For example, if the Divisor be -000492, and we* mul- tiply It by 1000000, we transform it into the Integer 492. Now, we may multiply a Divisor by any number, if we multiply the Dividend by the same number. For instance, if the Divisor be 8 and the Dividend 32 we may multiply each by 10, ' so that the Divisor becomes 80, and the Dividend 320 ; DlVmON OF DKCIMAL0. 68 and whether we divide 32 by 8, or 320 by 80, the (Quotient will bo the same iiumljer, that is, 4. 102. We can now lay down a general Rule for Division of Decimals. If the. Divisor he a decimal, change it into an Integer by rnnoving the point a eujkient number of places to the ntjhi, and also re^nove the point in the Dividend the same number of places to the right. Divide as in the case of integers. Tien, if the Dividend he an integer, the Quotient will he an integer, and if the Dividend he a decimal, the Quotient will ha a decimal of the same order. The process will be better understood from the follow- ing examples. Ex. (1). Divfde -626 by -025. •625--026 = :g|fx=8fJ = ^ 25 ) 625 ( 26 50 125 125 Here the Quotient is an Integer, because the Dividend is an Integer ; /. the Quotient required is 25. Ex. (f,). Divide 108-997 by 2-3. 108 997 -■- 2 "3 = l^^-^^t _ 1080-87 _ 1089'97 2'3 23« 23 23 ) 1089-97 ( 4739 92 169 161 89 69 207 207 i4 ^ DlVIftlOM or DIUIMALI. Here the Quotient h a decimal of the second order, bf- Cttuae the Divulcud in u ducimul of the gecond order ; .'. the Quotient required is 47 '30. Ex. (3> Divide -025 by 00025. •025-i-00026-.oSg.?5«8gg3Jf>ii^y». 25 ) 02500 ( 2500 50 125 125 00 Here the Quotient is an Integer^ because the Dividend is an Integer ; , .'. the Quotient required is 2500. Ex, (4). Divide -00169 by 1-3. 13 ) OICO ( 13 13 39 39 Here the Quotient is a decimal of the fourth order, be- cause the Dividend is a decimal of tlio fourth order -, :. the Quotient is 'OOIS. Ex. (5). Divide 625 by -25. 625-j--25=^=^^^ " 25 ) 62500 ( 2500 50 125 125 00 mviiuoN or oivimaia 85 he Dividend Hero tliu Quotioiit in an Integer^ because the Dividend in an Integer ; .'. 6254- -25 « 2500. Those lire casee of cxad division, that in, when, on the process of division being lurricd out, there u no remainder. Divide Examples, (xlv.) 1-290 by 108. (2) •00109 by 13. (4) 15()aa0002 by 302'9. (0) •0a09(J by 000072. (8) •0000015228 by 307. (10) •24204501 by 30 9. (12) •20980305 by 3500. (14) •00131053 by -0005. (10) •830070 by ■()002'M. (IH) 1 0191 by 00079. (20) 241 10047 by -627. (22) 4700400 06583 by 00518003. 17-28 by 0012. 2921 by -23. 1 by •OOOl. •7044 by -0052. 740-44808 by 7-58. 03987 '42 by 000073. 2098(1 14 by 00009. 0173'25 by 00025. •00019517 by 073. 2078 01 by 579. •05220834 by -00864. 103. We next take the following example : Divide 347 by 64. Here 347 -^ 64 and we proceed lius : ■14 7 «*7 00. — 84700 II"' ■" SB II H I II I , I (}4* 64 64 ) 34700 ( 542 320 270 256 140 128 12 We have then the Quotient 642, and Remainder 12. If we wisli to carry on the division further, we may do 180 by placing a decimal point at the end of the Dividend, land affixing as many zeros as we please, observing that iall the figures which will come after those already in the [Quotient will be decimals. ^, .^^v^ .0 * IMAGE EVALUATION TEST TARGET (MT-3) // y. J / -« 1.0 I.I lAAMM |2.5 Ki 1^ 1 2.2 2.0 1.8 1.25 1.4 1.6 ^ 6" — ► v] W DIVISION OF DECIMALS. The operation, completed from the outset, will stand thus: 64 ) 34700 0000 ( 5421876 320 ^ 270 266 140 128 120 64 660 512 480 448 320 320 Divide (1) 7'45by32. (3) 43-26 by 12-5. (5) 1-2 by 625. Examples, (xlvi) (4) (6) 14-327 by 12-8. 7432-976 by -225. •217 by 1260. 104. The student is now to observe that, by employ- ing bhort Division, the example just worked out may be put m^a very concise form. Thus, taking up the work at the point where we have to divide 34700 bv 64. we proceed thus : "^ ' 8 8 34700-0 4337-5000 542-1875 Quotient. So, also, if we have to divide 43672-509 by 36 we proceed thus : "^ ' 4 43672-50900 9 10918-12725 1213 12625' Quotient. 3t, will stand DIVISION or DECIMALS. -^y Again, to divide 0000013932 by 32, we proceed thus: 4 I 0000013932 8 I ooooooailsooo" •00000004353^Quotieni NoTE.--Divioion by 10, 100, 1000... is effected by noving the decimal place in the Dividend one, two three. . .places to the left. ' ' Thus 24 6 -i- 10 =2 46. •47-7-100= 0047. Examples, (xlvii.) (1) 426-478 by 16. (3) 362-47 by 026. (5) 42-007437 by -24. (7) 2-4715 by -000016. (9) -001 by ICO. (2) -07849782 by 72. (4) -00007203 by 4-5. (6) 00463 by 50. (8) 9000 by -00036. (10) 001001001 by 2000. 1 ^' ■^.T"-^^ process of Division may often be shortenpd by multiplying the Dividend and Divisor by a number ipo of 10; thus if we have to divide 24-46927151 bv 12'5, we multiply both by 8. ^ Then ^-^-^^^^11^1 = 105-78417208 12-5 100 = 1 -9575417208. In all cases we may proceed with the division till there recur again and again in the same order in Art*iM^'l!,^fTf "P"' "^ *'' rec^ne^oe of figures Safe "' '^'"'*""* "^ '" " -'•'«'•» i-'"-^ "/ aris^r» f^r^Jif' ?■ ''P?'^ "^^ •>"« '" find the Quotient ** DIVISION OF DECIMALS. 2*47 -f *37=a2;^47_„_»iT_ "87 87 37) 247-0000(0 6756 v 222 250 222 280 269 no 185 250 222 Hence, the Quotient, correct to four places of decima s, is 6-6756. Examples, (xlviii) Find the Quotient to three places of decimals when we divide (1) 42-5 by 0023. (3) 37-9 by 409. (6) -0269 by -281. (2) 197 by -79. (4) 27100 by -00313. (6) 229 by 007. 106. If we continue the division further in the example given in Art. 105, we find the figures 756 coming' again and again in tlie snme order in the Quotient, so that the Quotient is 6-6756756756 . . . without any termination. Let us now take this example. Divide 90 by -0011. Here 90 -j- -0011 = .^g^^ = ^^TI^ - ' 11 ) 900000 81818 Up to this point the Quotient is an Integer : but, if we proceed further with the division, we shall obtain a decimal expression : thus, if we affix two more zeros, preceded by a decimal point, to the dividend, we shall have 11 ) 900000 00 81818-18 DIV18I01C OF DECIMALS. 80 lals when we If we carry on thr division to any extent, we shall have the two iigures 18 coming again and again in the same order. A decimal of this kind is called Periodic, Girculatinff, or Recurring. 107. The extent of the Period is denoted by placing a (lot over the Jirst, and another dot over the lout of the figures in it Thus 18 denotes a decimal of an order such that it can he reiiresented hy no finite index, since it runs on -181818- 18 ... to an inlinito number of figures. So also, 6 TSC stands for 6766750756 .... •047 stands for '047047047 .... •4372 stands for '4372372372 .. .. 26 0479 stands for 26 04797979 .. .. •00026 stands for '000266666 .... 108. A Vulgar Fraction may be converted into a Decimal Fraction ^ y the following process : Reduce the fraction to its lowest terms, and then find the Quotient resulting from the division of the numerator by tho denominator by the rule for division of decimals. Thus, to reduce J to a decimal, we proceed thus: 8 ) 3 000 •375 .-. § = '376. Again, to reduce H to a decinml, we proceed thus : 32) 47 00000 (1-46875 32 160 ♦ 128 220 192 280 256 240 224 160 160 /. M = 1-46876. mm 90 Taruaov or dboxmals. Or, we might work by Short Division, thiw t 4 8 4 7-00 11-76 1-46875 Again, to reduce f to a decimaJ, we proceed thus t 7 I 1-00000000 •14286714.... .-. I = -142857. » 109. To show tha>% token a Vulgar Fraction is reduced to a Decimal, either the operation must terminaie or the figures of the Quotient must recur in the same order. Consider the operation by which such a fraction aa f is reduced to a decimal The oniy remainders that can occur are 0, 1, 2, 3, 4, 5, 6. If the remainder should occur, th^ division terminates : if not, we can only have six different remainders, and when any of these occurs a second time, we must have a recurrence of the former remainders in the same order. When a fraction in its lowest terms is reduced to a decimal and produces a recurring decimal, the extreme limit of the number of places in the period of the recur- ring decimal is one less than the denominator. Thus \ produces a recurring decimal of 6 places. yV produces a recjirring decimal of 18 places. 3^ produces a recu ing decimal of 28 places. 110. When a Vulgar Fraction is in its lowest terms it (ian only be expressed as an Exact Decimal when the denominator is composed of factors, each of which is one of the numbers 2 and 6. Thus f can be expressed as an exact decimal because 8= 2x2x2. ^j can be expressed as an exact decimal because 20= 2x2x5. T25^ can be expressed as an exact decimal because 125 =5x5x5. ^ttta DIVISION OF DRvlUAtM. »i The reason for this is, that no Vulgar Fraction can be expressed as an Exact Decimal unlep^ it can be trans- formed to one which has 10, or some puwer of 10, for its denominator. Now, no number can by multiplication be myde a power of 10 unless it be composed of factors each of which is 2 or 6. b 'fi^^S^ ^" ^ "**^® ^"^^ * ^^^^^ "^ ^^ ^^ multiplying it , « ^?^ ^^ ^ ™*^® '^^^ * Po^^er of 10 by raultiplyinff it by 2x2x2. fjs b 5 5^ ^'^ ^ *^^* ^"'° * ^'^^'^ ^^ ^^ ^^ multiplying it Hence # = — 2__ == ax«xgx5 «-- _ .^^ o axaxa 2x2x3x6x5x5 looiy — «'70. "'f 5x6x5 ~ 6X6x8xaxixa ~ ToOff = '056. iW(F = -225. A 2x2x10 2X2X10X6X6 But such numbers as 7, 12, 30, cannot be made into powers of 10 by multiplication, and hence ^ /j, Ur can- not be reduced to exact decimals. It may also be remarked that, when a Vulgar Fraction in Its lowest terms is reduced to an exact decimal, the order of that decimal is expressed by the greatest number of times that either of the factors 2 or 5 occurs m the denominator. Examples, (xlix.) Convert into decimals the following vulgar fractions • (1)' /o- (2) H. (3) f (4) ^V. 5 j^. (6) ^. (7) ^i. (8) ^g. (9) lifj. (10) w CONTRACTIONS IN MULTIPLICATION AND DIVISION OF DECIMALS. 111. When the number of decimal places is great the figures obtained by the ordinary mode of multiplication Ao%7.?2 unnecessarily numerous. Thus, in multiplying 62'37416 by 2-34169 by the ordinary' method, ^therf 92 COHTRAcmONH OF DKOIMALII. would be ten placfg of docimals in the product, while for all practical purposeH three or four are quite enough. Ex. Multiply 62-37416 by 2-34169 so ai to retain only 4 places of decimtvls. ORDINAUT METHOD. 62 37416 2*34169 CONTBACT^n METHOD. 62-37416 96143-2 66 374 623 24940 187122 1247483 146-0609 136744 24400 7416 G64 48 2 1247483 «= 623741 X 2 + 1 187122 = 62374 X 3 24960- 6237 X 4 + 2 624- 628 X 1 + 1 374- 62 X 6 + 2 56- 6 X 9+ 2 4C7304 146 0609 By comparing the contracted method with the ordi- nary rnetliod, the reason of the preceding operation will be readily understood. Since the product of any order of units by units is of the same order as the figure multiplied, the units' figure of the multiplier is written under the place to be retained. For convenience, the other figures are written in an inverted order. Now (Art. 99) 4, a decimal of the third order, multiplied by 3, a decimal of the first order, will give a decimal of the fourth order ; also, 7, a decimal of the second order, multiplied by 4, a decimal of the second order, will give a decimal of the fourth order, etc., etc. Now, to the product of 2 and 1, 1 must be added : since, if 6 had not been rejected, there would have been 1 to carry ; then the other figures are multiplied in the usual way. Next, multiply 4 by 3 and set down 2 under the 3, and multiply the other figures by 3 in the usual way. Next, multiply 7 by 4, and to the product add 2 : since, if 416 had not been rejected the product would have approximated to 2 thousand, etc. BM to retain C'UNT&ACTIONS Ot OBCIMAUL Hence we have the following rule : Write the Multiplier with the, onhr of its figures ,evermd uniler tliti M uUiplicaHd, an that th^ unit^ figure may be under that figure of the Multiplicand which is the lowest thcimal to be retained in the Product Then viultiply by each finnre of the Multiplier, neglecting all the figures of the Muitiplicand to the riyht of it, except to find what is to he carried, and carrying one more when the rtjected part of any product is 6 or greater than 6. Arrange the parffol jmHlucti so that their right-hand figures may stand in the same vertical column. Their sum will be the product mpiired. From thisjrroduct cut of the desired number of decimal plat^ix 112. V >bi» fciiG divisor consists of several fi'jures, the woiK 'vili oe much shorteTif^'i by cutting off a figure from tl'6 a: visor av each successive step of the division, instj?'*, of .nnexfng a ilgure to t. e dividend. Care must b<3 tft'.iR y: iroreosH ea h. prwlacl by what would have been caaitd if tlv-^ figure or '-gure.j had not been cut off Ex. (1). Divide 3-784T)0 by 2 7 1641 8 correct to three places of decinuils. 2716418)3784169(1393 ' * ' 2716418 1067751 314926 252826 244478 8348 8149 199 By comparing the units of the highest order in the divisor with the units of the same order in the dividend, It IS evident that there must be one figure to the left of the point in the quotient ; hence the answer is 1 -393. Ex. (2). Divide 763-14163 by 21-3642 correct to four places of decimals. ■ ■■ OOMTKACTIONM OV DBCIMAIA 213W2 ) 70;U41«3 ( ;«i7206 640920 122215 100821 15304 14U54 430 427 12 10 6 «3 04 (JOO 284 213042 ) 70314103 ( 367205 * • 640926 122215 100821 15394 14955 439 427 40(500 08210 12 11 1 72390 1 Here the figures of the quotient are 367206, and by comparing the '2 tens of the divisor with the 76 tens of the divi^nd, it h plain there must be 2 places to the left of the point; hence the quotient is 35-7205. From conridering these cases we have the following rule: Compare the left-hand figure of the divisor mth the units of the same order in the dividend, and thus detemiinn the jmsiti to 04726. The decimal -•04726726. From 100000 times the decimal, or 4?26'726. take 100 times the decimal, or 4726! '. \ \ Then 09000 times the decimal « 4722 000 •■• **»« decimal = ^*i^% - ^ |r,. Ex. (3). Find the A'ulgar Fraction equivalent to 3'i4. The decimal - 3 1444 From 100 times the decimal, or 314 44 ... , take 10 times the decimal, or 31 44 ... ^ Then 90 times tho decimal = 283O0. . . . .'. the decimal «= ^^. Examples, (lil) Convert into Vulgr.r Fractions in their lowest terms : (1) -426. (2) -4769. (3) 4-253. (4) 00426. (5) 63 00243. (6) 7-2dll. (7) 2-53d6. I tqfiivateni to alent to -237. KSOUftmiMO DICI1C4IA fV 117. Heno« wa dcduo., tho following rule for rt^odnir i Mixed Uemrnuft D«c;.iiul to a Vuigar Fraction: Fonn in, NHm^at,>r hy taHnrj /rmn he flaures up in }^r,..i anrl /onn the Dmumina*ar by Jung ,hnm fl ""/''T.; T.*" ''"^' ar.;f^r«. l^wJn the^ml poi nt ami th e Hrat period. *««^w»«t Thai -246 •ootti S4fl>3 ftlSav • 90 4TJk--4 UtfUUO » »■ 7*346 - Ii^i*rI2.i . «'»ii »oo "poo" wi/h W '^^'V"«|^'^'^"^r«''^«rr.nngaritnm«.i.al operations ith KocmnK D.cuuils will bo In^at oxplainod by tukinff I. Addition. Find the lum of 3 40, 4-(k7. and Udl Fiwfc n.ake the d^ciiuHla all of the same order, thua : 3 ^Wi), 4 0476, 14(5^. Then, hince the periods consist of 1, 3, 2 fiffurea resoGo t vely, and the l. o. m. of 1, 3. and 2 i^ 8 carf v on Ilf^« decimals si: places further, thui : ^ ^ 8-4999999999 4 0470470470 •1403(136363 7 69341066 II. Subtraction. Hero wo proceed on the same principle as in Addition. Thus to subtract 5 247 from 8 -659 : 8059059 6-247777 2 -^1126 In both operations some care in requisite in observing P-r* KECUBRXNO DMCIMAL&. what figure would be carried on if the columns omitted were taken into account III. In Multiplifation and Division the recurring deci- mals should be converted into vulgar fractions, and v/hen the product or quotient of these fractions has been found, it may be converted into a decimal. Thus, 4-6x3-7 = ^x^-*^-»*-^»** and '05 -r -042 »0 38 000 ttO 38 ^^ 34 9^9" 81 > 000 6xio_a« 38 19' We may then, if it be required, convert -igfA and |f into decimals by the process explained in Art. 108. Examples, (liii.) Find the value of the following expressions : (1) 2-57+ -043 +13-2. (2) 14702 + 3 549+ 2 •20i 15,025 -13-247. (4) -0246 - -00397. 3-7 X 5-49. (6) -0072 X -45. 3'4+409. (8) -074+ -59. (3) (5) (7) 119. When vulgar and decimal fractions arc combined in the same expression, it may usually be simplified in the neatest and easiest way by reducing the vulgar fractions to a decimal form. Thus, if we have to find the sum of 476|, 13f , and 10*375, we should proceed tnus : 476^=476-25 13|= 13-375 10-375 Sum 500000 Examination Papers. I. (1) Show that liny decimal is multiplied by 1000 by removing the decimal point in the multiplicand three places towards the right. mns omitted EXAMINATION PAPBSS. gg In caaea when the diviaion does not terminate, explain how JL?"^ "' the^following statements is more nearly oi = 9 009. W009 — 111 (4) How many times can "0087 be taken from 2 '291 ? What fraction wiJl the remainder be of the former ? (5) Whence does it appear that a vulgar fraction may always be reduced either to a terminated or a^irculating delmmlY Calculate the hmits of the error made in taking M^b an approximate value of 3-1416926 to seven places oVdllimals II, deiima^f.^^*'" ""^^^ ''"^^' ^'^'^^°"* '^'^ ^« expressed as finite Which of the following fractions can be thus expressed ? of^L'^or^rb'arrlre^^^^^^ ^«> '-' *^^ -^"e (3) Whether is 3-714635 more accurately represented by 3-7l5or 3-714, and why? ^ Ja\ .2^** ^A^'Yu ^""f^^.^^" ^« equivalent to the sum of 14-4 ahd 1-44 aivided by their diflFerence ? tei'^Sldt.f'""'^ "'"' «^^^^ "«* ^^^- ^-- ^r by a III. »ifP^^-'^**,^? *^® advantages and disadvantages of working with decimals instead of vulgar fractions ? working (2) If a business produces an annual return of |6,000 and nrofi^^\^*'*"^'^i,*'"^ ^^" "^^^ *"^ ^"other -38 share of the ?artler?^"' "^"'^ "^°"^^ ^^"« *^ *^^^ «h-'« -^ the thiVd fafiiirS^" T^? ^"^"^ « ^^ ^ steamboat sells -7 of his share (4) A »a„ paid $320 for a horee ; for a buggy $36^ more (5) The product of three vulg-.r fractions is | ; two of tnem are expressed by the decimals -as o«^ .iq« u-aotion wiu the third one be expressed ? ' ^""" 1 _.i. . iOO EXAMINATION PAPBB8. IV. (1) How do the Decimals differ from Vulcrar Fra^ti-n* f In^««ll^n'*r^^®^P®' ^"y* ^'^^ y*^^* «^ ^''"th at $-36 per yard In selling he uses a measure which is ^ of a vard too .Wf and charges ^50 per yard. What is hU net g^n 1 ^^''' ^n% ^p"® !®*'®^ contains a mixture of 18 pints of brandy and 7 of water ; another contains 34 pints of brandv and 1? lL"^i\}ll *^' f ^"«*.? °^ '''' fi-t^mixture irr"etrerented' by 423, what number will represent that of the second ? (4) Write in figures four millions and four, and ten billions ninety thousand and seven hundred quadrillionthi Exn^^^^^^^ m words 74000300 000060000007. ^ '*"""*°"'>'^s- ii^xpress w^^found'?£atl«l°*^' Ti '*'^-*^ "°"**^" «4 y^^»' ^»t it was tound that the so-called yara measure with which it was measured was 02083 of a yard too short • what was fh^ correct length of the cloth ? ' ^^ ***® V. (1) When a vulgar fraction is changed to a decimal explain ho^ many figures there will be in the d^imalTf t wnTconsiTf ' '^ \' \^ r^r**"^ ^^^^'"^ «^Pl*in when will consist of a part which does not repeat, and show how many figures there will be in this part > »«« sno'«r now (2) The French matre is 39 371 inches in length. Express the length of 25 metres as a fraction of an Endish mi\7 there being 6280 feet in it and 12 inches in a foot ^ ' (3) If a steamer makes a passage from New York to Liverpool (say 2700 miles) in 230 hours, and a train goes from London to Edinburgh (say 406 miles in 18 hours; low much does the one go faster than the other? Give aiswer m miles and decimal of a mile. »«swer o«l^\w l^*^?* *^® ^""^ ""^ *^® '^i^^**'^ and quotient is 7-6 • and that the divisor is f of the quotient ; aho that the remainder is |f of the divfsor. Fin3 the divided $46.70 less than A, and $34.59 more than C. » « ^ VI. r«iiL!^^** vulgar fractions must be represented by mixed repetends, and what by pure repetends ? (2) Show that no recurring decimal can have more places less on ^ ^ ^''^ ''"'^^ '"^ ^^ denominator SQUAKJ8 ROOT. 101 (3) A man spent $2.50 more than 7^ of hi. «, timo, and $1 U leas than J>M oitL . "'^ ** ^"^ and now has $2 -609 • hoVbth H , k ' ' '* ""'^^^^^^^ fl'^ uuif , now much had Jie^t first ? , ; "^" """ ae*t nrst ? 15 3 53+5 5^-7-5,+ .... I -^ (5) Simplify ^ >< |i_ 3x4 10«"*"JU2^ 239 3 X 4 X 6 lO^'^'ixTkS XIII. Square Root. the square of 12 anrJ 99n ;« ^.u """^^^r. ihus 144 is i. TiT '"^ "^•*"™ '•"" "' 1^* « 12, because the s^Z of 12 V25 i, read ..the.<,„*at' Z of 25!" '° "' *^''^» ^ '"- J'L^eSrptK.:rs,tr '"'^«^^'^°^ >*» ="-- root of 81 i^g a,W f ' «■« know that the square than 100 we know tl/ "''"^ "'*'"" ^•'""^^ «'•«'"" for i..stanorwe k^ow thatT ™'"' ''^ «-PorieL, as, and the sq„;re root of 400 fs 'o ''"Tr°' "^ ^^^ '^ ^^ 10000 is IGO Rjt wl i ?' *.'"' ""* ^I'lare root ol Root of any number as r^h if *°'' '^'"'"'8 *''« Square ^^',v„^ ' ™''" "ow explain. j^Jirst, suppose we have to find the Square Eoot of froI^htX' iwo'Cf "=" ""= *"° ^S«res on the right 12J25. 109 8QUARB ROOT. »: The figurei 12 make what is called the Jird peruxl The figures 25 make what ia called the secorui period. We then take the nearent perfect square not greater than 12, that is 9, and place it under the 12 and put its square root, that is 3, as the first figure of the square root we have to find, thus 12|25 ( 3 9 We subtract 9 from 12, and annex to th« remainder 3 the second period 25, to make a dividend, and we double the first figure of the root, and set down the result as the first terra of a divisor; thus our process up to this point will stand thus : 12|25 ( a. 9 6 I 326 Now we shall have to annex another figure to the 6 and we must therefore reckon the 6 as six tens, or 6o' and then we seek the number of times 60 ia contained in 325, and this being Jlvc times, we set down 6 as the second figure of the root, and annex 5 to the 6, so that our process up to this point will stand th^s : 12125 ( 35 9 65 I 325 We then multiply 65 by 5, and set the product down under the 325 ; and subtracting the product from the 6Z5, we have no remainder, and we conclude that 35 ia the square root of 1225, the full process being : 12|25(35 9 65 325 325 V 35 is the root required. 8QUAAE BOOT. Next, to find the Sqaare Root of 622521, Drawing a lino to mark off tlie two figures on the ngnt, and another line to mark off the next two figures our proo^ for finding the first two figures of the root will be the same as that explained in the first example, and It will stand thus : *^ ' 62|25|21 ( 78 49 148 1336 1184 i 14121 We now annex to the remainder the tldrd period 21. and we double the part of the root already found, 78, and set down the result 156 as a partial divisor, and proceed, as be ore to divide 14121 by 1560, and annex the quotient 9 to the root and to the divisor; and multiplying 1569 by 9 we set the proauct under the 14121 : thus our process in full will be 62 25|21 ( 789 49 148 1326 1184 1569 14121 14121 .*. 789 is the root required. NoTB.--.In practice, instead of dividing 1325 bv 140 thn. ib?^ f ■^' i° ^'^'^^ ^^^2 ^y 1^«- The quotient inus obtamed is, however, sometimes too great, asViU be seen m the next examples. \.^^^''^ ^^® ^'^'^ examples in which the first period |ias only me figure, which must always be the case when iiie proposed square has an odd number of figures m it. i--- ii-iu. -^„ K^quai.6 xfcuui. ox i6b>4/ 52:^0. 104 :■'€ SgUAKB BOOT. Marking oft" the figures by paiw, coinmencin« from the right, we have 1|89|47|52|26( 13765 23 267 2740 27626 89 69 2047 1869 17852 10476 137626 137626 If'T^^- dividing. 89 by 20 the quotient is 4, but if e/e added this to complete the divisor, it would become 24, which, being multiplied by 4, would give 96. a lumber larger than 89. To find the Square Root of 39601. 3|96101 ( 199 23 389 296 261 3501 3501 Note L— The division of 296 by 20 illustrates the •emarks made on the last example. Note II.— The second remainder, 35, is greater than 'he divisor, 29, a result not uncommon in this operation. Eyamples. (Iv.) Find the Square Roots of (1) 196. (3) 1024. (5) 88209. (2) 529. (4) 5625. (6) 119026. 8QUAAB ROOT. 105 10(>029. 193(K)0. 3()3721H>1. er )183J)36. 41241)01. (8) 751089. (^-0) 097225. (12) 22071204. (14) 5250260000. (16) 54(51 210(KK)00. (18) 11»1810713444. (17) 32239084. 124. Tojind the Square Root of a Decimal Fraction, When the given number has an even number of riecimal pkcjs, we proceed to find the Square Root as if the r uriber were an integer, ar.d mark off in the root a numbed of decmml places equal to ha^fthe number in thesq^^r Thui, if the square be a decimal of the nxth order the root will be a decimal of the third order! For example, to find the Square Root of 6-322241^. 5i32|22|49( 2-307 4 43 132 129 46 I 322 ' Since 46 is not contained in 32, we annex an ft f« fi, ^ "*• 4607 32249 I 32249 Examples. (Ivi.) Find the Square Roots of (3) -9025. (6) -000729. (9) 44415-5625. ^,}^.'Jl fii^diiig the Square Root of a Decimal Fraction •'^.^*..., order, thus, -"40, -4000, -^OOOO,, /. 7 '''''''^'''''' *'''"*** **^ 103 iQUAkB HOOT. Thi« is done in onler thnt the denoriiiimtor of tli.' oquivaifltjt fraction may be a perfect wjuarn, which is the case ii^ tho fractiotiB 40 4000 400000 100* ioooo* ioooooo"" but not in tho fractions 4 400 40000 10' lOOO' iooooo "" Also, since for every pair of figures in the square vrc have OTW figure in the root, wo shall have to take a number of figures in the decimal part of the squaro double the number of decimal places we are to have in the root, Suppose, for example, wo have to find the Square Root of '144 %Q four places of decimals. We must have eight decimal places in the square, thus, '1000000, and we mark off t'.ipse and proceed as in the extraction of the root of whole numbers, the root being a decimal of the fourth order, thus : •14|40|00|00 ( -3794.... 9 67 540 469 749 .7100 6741 7584 36900 30336 5564 Note. — The Square Root of a decimal of an odd order is a non-terminating decimal. Examples. (Ivii.) Extract to four places of decimals the Square Roots of (1) 20. (2) 30. (3) -9. (4) 121. (6) -169. (6) -016. (7) -00064. (8) -0^121. an odd order n9\ j-Q-ft." nQiTAiui moor. 107 126. If ..-G have to find the Square Root of a Tuljrar ractmn we can always, by nmltipHcation. make the i.-Mornmator a perfect square, if it J>e not .iTeadjr i n.ultiplymg the numerator by the aanie number. ^ 7« then find the Square Root of the denominator and hnd, exactly or approximately, the square root of the rmmerator. and maJce the results respectively the denon^ nator and numerator of a fractio^ which is the Zt required, exactly or approximately. 3x3 V6 V'9 3 ' We can now extract the square root of 6 to. sav three places of decimals, tJius : * o^ o lo, say, 6i00|00|00( 2-449.... 4 44 484 200 176 4889 ■■-4' 2400 193G 46400 44001 2399 2-449. 3~ - -816.... •66'^6666^' ""S ^^"\''^"''^ ^ *" * ^^^^^> *hus: this decimal "" '^''^'*^^ '^" ^^^^^''^ ^^«* «^ ^^•(^)-N/4I = >Jlf = V529 _ 23 7 V64 -J -^8* Ex. ^4V Tn finr? fV,« O, "D _' 1-28 12 5 108 ■QUAKB ROOT. H«Te WO can reduce the fraction to lower terms ; •32. V«-26 2'6 127. An integer can always be chnnged into a perfect ■tjuare by multiplying by a numl>er equal to or less than the proposed integer. For example, 7 is chantred into a perfect square if multiplied by 7, 18 is changed into a perfect stiuaie if multiplied by 2. Examiiies. Gviii.) Find the Hquare Roots of (1) U' (2) ^. (7) m (8) 3^. (10) 38iJ. (11) 17\l and find tio four places of decimals the Square Roots of (13) f. (14) II (15) 6f. (16) % (17) 7CH. (3) m- (6) 5^. (9) 66|t. (12) llfj. XIV. Cube Root. 128. When a number is multiplied by itself twice, the result is called the Cudb of the number. Thus 27 is the cube of 3, and 216 is the cube of 6. 129. The Cube Root of a given number is that number whose cube is equal to the given number. Thus the Cube Root of 343 is 7, because the cube of 7 is 343. The symbol f, placed before a number, denotes that the cube root of that number is to be taken ; thus ^125 is read "the cube root of 125." 130. A number which has an integer for its cube root is called a Perfect Cube. The numbers, less than 1000, which are perfect cubes should be committed to memory ; they are 1- 8. 27. fi4.. 12R 91 « 5U.<1 Kio hroo . CU»1 IMJOT. 109 ai iKl the (.^ul^ Root, of theee n«r«Ur« *i^ reapeotiviy 1» 3, 3, 4, 6, 6, 7, 8, ft 131. To find tho CuIkj Root of .. ru»^« * i ""'" 1000. w„ prooe«l by . 'I'wCh w°. T' ir*^ explain. ''^ Which we ghall now Ex. To find the Ou»)e Root of 91126. 4 12 6 91fl26 64 4800 I 27125 626 5426 27126 JoXll'K '""" "'■ '"'* *" **" ""»!■"»« atUch the tha''::tr^'r fera'ndTh™: f "' "f."" <" ""> '«>'. 12. to %'"re of the r™t 4^ » th t ' ™ ""^ "I"'" "' '^e fir,t the left of the 27125 "" """"""^ '° "' J"" on u{:^n^r:,fa%''|^nhr^^^^^^^ r'""'"/v^> "•■■"-y this by 5 ; put the resuU A9R j .1. ^ »« 125 ; multiply 4H00 ; this gives LTmufLr"^^' * u^ ^^ ' *^^ i* to the which i. 27f25, under th^ fi 1? ^r *^'- 5^ ^ ' P"* *^« '«B«lt, there is no r^mSei H,« '^^'«**nd«r; subtract, and m v%ti8 45. '^^"'*'"^«»' *J»« Proceaa la complete, ind the Examples, (lix.) Find the Cube Roots of (1) 4096. (2) 32768. 4 493039. (5) 614125. '2 ^^^J- <^^ 389017. 1 ill) Q'5'AOf,\n -^ . . _ -/ -.v-p-. m^ 09319. (3) 74088. (6) 262144 (9) 614125. (12) 250047. 110 01T1I BOOT. Kext, I«t at toke th« om0 in which tho cutw root h»Mi ihroe Ugarm, ftmi Mtraiil Um ouImi root ol 42tftiai0d4 a8|eei|064 348 21 2i6 4 14700 1076^ 16776}" §6661 78876 1087600 9016 6780004 1696616 6766064 We ■epareto the number 428661064 into throe periodB, and take the nearest perfect cqbe not greater than 428. which is 343, an4 we set down ii» cube root, which is 7. We then subtract 34" from 428, and annex to tho remainder 661, the second period. Then we .et down three times 7, which is 21, and three times the ■ |uare of 7, which is 147, and add two zeros to it. Then we dirid*^ 86661 by 14700, which gives the quotient 5, and this we put down midway between 21 and 14V00. Then we multiply 216 by 5. which gives 1076; we add this to 14700 ; we multiply the result, 16775, by 5 ; and subtract tho product, 78876, from 85661 ; and to f ^e re- mainder we annex the thiM period, 064. We then set down three times 75, which is 225, and three times the square of 75, which is 16875. hf* obb^inAd by setting the he root M*ider the second lumbers coupled by the N. B.— This last resul' ' square of 5, the second fir;n divisor, and adding th ^o bracket We then annex two zeros to 16875 and repeat the process explained above to find 4, the third figure of the cube root, which is in this case 754. Next, tfl^ce the case in which the root has four figures and find the Cube Root of 14832537993. CUBS HOOT, « 4 1800 I466f 16/ 72 6 172800 m2b ] 176426 /' 86j 736 7 18007500 61490 14|a»|537i905 6m 6^ 1008637 882125 1264129tt3 18068900 I 126412993 Hbnoe the ^x>t recraired is 2457. g-e 7626, a number too large to be aubtL^/rom 68^1 Examples. (Ix.) Find the Cube Roots of (1) 14706126. (4) 300763000. (7) 99252847. (10) 194104530. (13) 32:^828850. (Ki) 13421V728. (18) (}733730i>7l25. (2) 149721291 (6) 2097152. (8) 1092727. (11) 84027672. (14) 364894912, (17) 122615327232. (3) 2893444a (6) 6736339. (9) 16777216. (12) 180323843. (15) 700227072. 132. To extract the Cube Moot of a Decimal Fraction In order that a Decimal Fraction may be a Perfonf ^^^ZoJ^^' *^^ ^^^' ^^^' 9th'.'^c:dr^'^ inaex ot the order being some multiple of 3. "W^ 4-u__ .^ . . . . - - i ^xwocu iu the foiiowing way : 112 OUBK ROOT. I'Ki Ex. (1). To find the Cube Root of -343. '343 1000 10 f-343=J Ex. (2). To find the Cube Root of -039304. -J ^ 039304 39304^ '10000(^ 34 100 34. Ex. (3). To find the Cube R^ot of -012812904. ^•012812904 =;Jfc L2812904 ^34 1000 = -234. 1000000000 the cube root of an integer or 133. To extract _ decimal expression to a particular place of decimals, we must take three times the number of decimal places in the expression. Thus, to find the cube root of 4-23 accurately to three places of decimals we extract the cube root of 4*230000000, making the given expression a decimal of the ninth order. In working this example we find the cube root of 4230000000, regarded as a whole number^ and mark oflf three decimal places in the result. 134. The Cube B^oot of a Vidgar Fraction may be found by taking the roots of the numerator and denomi- nator, or by reducing the fraction to a decimal of the 3rd, 6th, 9th. . . .order, and proceeding as in Art. 133. Examples. (Ixi.) Find the Cube Roots of (1) -389017. (2) -048228544. (3) 27054036008. (4) \m (5) fl^. (6) 5^^^. (7) 405^V and find to three places of decimals the Cube Roots of (8) 5. (9) 576. (10) -121861281. (11) 15-926972504. (12) |. (13) | .(14) i. (15) 7f. (16) 3^. 135. The fourth root of a number is found by taking the square root of the square root of the number. ThusV4096= V64=a L integer or decimals, we places in the CUBE ROOT. 113 The sixth root of a number W fnur^A v „ x i ■ root of the square root oTthl uu^^let. ' '"^ ''^ ^"'^ Thus V64 = f8 = 2. ExMBples. (Ixil.) Find the Fourth Roots of (1) 531441. (2) 4100625. and the Sixth Roots of (3) 1575 -2961; (4) 4826809. (5) 24794911 9Qfl /^x «o« W ^4/y^yil296. (b) 282429-536481. r, and mark -..c<^ COMMEECIAL ARITHMETIC. XV. On English, Canadian, and United States Currencies. 136. Having explained the principles and processes of Pure Arithmetic, we proceed to show how they are applied to commercial affairs. Measures op Money. 4 farthings are equivalent to 1 penny. 12 pence are equivalent to 1 shilling. 30 shillings are equivalent to 1 pound. The symbol £ placed before or over a number deii.^fces pounds. «• after shillings. " after pence. £i 8. d. Thus £14 5s. 7d, or 14 5 7, stands for fourteen pounds, five shillings, and seven pence. Since 1 farthing is one-fourth of a penny, 2 farthings are one -half of a penny, 3 farthings are three-fourths of a penny. Hence the symbol H. is placed for 1 farthing, 3^ 2 farthings or a halfpenny, |c^ 3 farthings. The symbol q., placed after a number, is sometimes used to denote farthings; thus, Sq «tands foi- three farthings. 137. We call £14 a simple quantity, and £14 5g. 7d. a compound quantity, because the former is expressed with reference to a single unit, whi'e the latter is ex- pressed with reference to three different units. 138. The unit m Canadian and United States cur- rencies is called a Dollar. The tenth part of this unit is for fourteen or a halfpenny, OAHADIAK «,„ „„„„ „^^ ^^^^ ^^ ».ay conceive the unit then ?n L ^- -i'^f •* **'"• '«'« parts, each of these partsTntX; other «'' '1° '^ ""»"»' ^0 on. Hence Can^ian and Un.tecTS:' ^^' ■*?" are based on the DeamnJ «„., I, „ °'*'^' currencies fore, all opemtions „™h1,f^t™f •^'''''''™' »'«'. "-ere- ■neansof the rules in De^CiTr- "*' '*t*™'^ "^^ crcunistance that the, oSi/grrs'lUitr ^'^ TABLE OF CANADIAN AND UNITED STATES COINS. CANADIAN COINS UNITED STATES COINS. Gold. BntUh Weign. worth »„ ^e E^^. . . ,20 British Half-sovereign. m>?te;;.: i; -'i" Three Dollar Piece. Quarter Eagl«, or. . $2^ Silver. „'! Oliver. 50-cent piece, answers to.. S^j}*!' „ 2o-cent piece, answers to. Oualft ^*';, ■iv^-cent piece, answers to .-.Dime f^ oar.^ • Nickel. '•J-cent piece, answers to k x • 5-cent piece. 3-cent piece. oronze. 1 1 cent. ^''^^ Mill, not coined. l°f,"*- I Mill, not coined. E^. (1). $251, 7 cents, 3 mills = $251-073. # 116 REDUimON OF MONET. Ex. (2). 155 '923 « 0(55 + ^J^ + t3u + TT^on) = 8 65 + 02 cents + 3 mills = I 55, 92 cents, 3 mills. The English gold coinage consists of \^ pure met^i and of ^ alloy. The gold and silver coinage of tho United States consists of ^^ pure metal and jV alloy. The silver coin in Canada and England is |^ pure metal and ^ copper. Gold and silver thus alloyed are called standard. The gold or silver before it is coined is called bullion. The term carat is employed to denote the fineness of gold. Pertectly pure gold is said to be 24 carats fine ; a mixture of 18 parts pure gold and six parts of some other metal, is said to be 18 cf rpts fine. This latter is termed jewellers' gold. REDUCTION OF MONEY. 139. The expi ssion 56'. Id. stands for a sum of money which is n^ade up of five shillings and seven pence. Now, since one shilling is equivalent to twelve pence, five shillings are equivalent to sixty pence ; and therefore five shillings and seven pence are equ ivalent to sixty-seven pence. The process by which we change the compound ex- pression 5s. 7cZ. into the equivalent sim2)le expression 67<^. is arranged thus : s. d. 5 7 12 67rf. and we describe the process l: aS : We change the 6 shillings into pence by multiplying by 12, and add to the product the 7 pence. ure metSil and States consists IJ pure metal ard. The gold DEDUCTION O*' MONBY. jj- umoer ot tarthings, wc proceed thus • * «• d. 20 * 87s. 12 4 42182. S :". ""Ts^ 1° ;':-^;:;^. -a „<,a 7.., r^,^, «,.. , Examples, (ixiil.) Reduce to farthings (1) 3|d; 7id. i 9d. ; ll|<^. (3) ^3 12.. ; £5 ; £2 17. 6K; £17 4!. 51.;. Keduce to pence (6) £174 10a.; £432 15,. m.; £1274 17* 9d. thus: 9farthing8=JpCet2W ' ■^«'»'»nder ,s farthing,, set down the quotienf f« \ n-^''^'^^ ^^ P^"*^*'^! shillincT) [pence, thus :yreriSSL"5^ ^^J^^ remainder t -^- (•i). AUc, 75 »hmings=JJpou„da=£3 15.. 118 COMPOUMO ADDITION. -ifi Ex. (4.) To express 4275639 farthings in terms oi£8.d. farthiniffl. 12 20 4275G39 10C8909d and 3 farthings oven 8907,5s. and 9 p«noe over. £4453 and 15 shillings over. /. 4275639 farthings =£4463 16a. 9|d. These methods of expressing a giveii sum of money in another, but equivalent, form are inulud«l in the word Reduction. Examples. (Ixiv.) Reduce to pence and farthings the following numbers of farthings : (1) ' 57. (2) 173. (3) 197. Reduce to shillings, pence, and farthings the following numbers of farthings : (4) 357. (5) 479. (6) 747. Reduce to £> s. d. the following numbers of farthings : (7) 4238. (8) 376289. (9) 642380. 141. The copper coins in use in Great Britaiit are the Farthing, the Halfpenny, and the Penny. The silver coins in use are the Crown (5s.), the Half- crown (2s. 6d), the Florin (2s.), the Shilling, the Six- pence, the Fourpenny piece (or Groat), and the Three- penny piece. The gold coins in use are the Sovereign or Pound, and the Half-sovereign. TLe Guinea (2 Is.) and the Half- guinea (10s. 6o?.) are not in use, but reference la frequently made to them. COMPOUND ADDITION, 142. In adding compound expressions together, we follow the principles which regulate the process cf Addition in the case of pure numbers. terms of Xtf.d >wing numbers COMPOUND ADmnoN. ^g tha?t"^eSu^dtLrdT„dtr^^^^^ "^ ^--^« *»^- - Bhillinga under shiJIinJ^nni?^' '" '^^'^^"*^ ^'^^""•ns, under lrthings.C'eramnl«r ^r* ^"^ ^'^^^»»i"«« . t^M ^^. d|,/., 6s. 4rf., and 17& 9J./., we arrange them thus: «. 4 3 5 17 d. 3 3 4 9| I. A ,,. ^ £1 10 8^ farthings, we place I undpr Tk^ i *^ ^ P®"*^)" an^^ 2 find ': ^: Vint ::'d"r f P--. .-reased by ,, .« cany on 1 for ad^Hion to t^l^^^'r^fc'* '«><' pound and 10 shillinl^^e 1 In'"'"?'''!'''™'*''' ^ I we arrange them thus : ^'^•' ^^^ ^^' s|t^., £ 26 32 245 7 8. 4 12 15 4 d. 9| 2 8^ si £311 18 0^ P f-tt|s*tT\l,^^^ "« fi-d its sum to be \ ' "" " '^^ «^d^«on to the'colun^rofy™^^^' ^^^^ 190 COMPOUND ADDITION. The sun^ of th© colunm of pence, increased by 2, "we find to be 56 pence, asul this being cquivivlent to 3 «)'.illing8, WO place under the colunm of pence, and carry ou 3 for addition to the column of shillings. The sum of the columns of shillings, inc.eased by 3, we find to bo 38 shillings, and this being equivalent to I pound and 18 shillings, wo place 18 under the eolurrins of shillings, and carry on 1 for addition to the colunnis of pounds. The sum of the columns of pounds, increased by 1, we find to bo 311, which we place under those columns, and the sum is complete. Examples. (Ixv.) Perform the operation of addition on the followiitji; sums of njoney. £ a. rf. £ i. (/. £ ir. d. £ «. d. (1)3 6 2 (2) 5 8 3 (3) 6 8 7 (4)7 8 4 6 8 7 9 6 4 6 3 5 8 4 7 9 3 3 4 9 8 9 10 9 6 2 4 10 6 5 2 !) 7 7 4 i: 4 9 2 9 4 4 3 d. 2 6 10 £ «. d. £ g. d. £ s. £ «. d. (5)3 4 U (6) 4 7 5| (7) 7 8 4| (8) 6 2 5 2 6 tl 6 8 91 : 6 9 2 5 3 2 7 6 9 5 2; ■ 5 2 7^ 7 8 4 6 9 0] 8 7 4 ■ 6 3 9" 8 9 r 4 7 9| 5 9 (10) ^l i 7 6 (11) ^ 5 3 3i (9) (12) £ 8. d. £ «. d. £ 8. d. £ 8. the con.pound expression Hrst by one of the factors and then multiply the product h/another of the Cl ^f 1.1 the case of Simple Multiplication. ' ^ Thus, if we have to multiply £12 is TU hv l« ^- ...ultiply flr,t b, 6, and the product ly'i, X,: ' 12 4 7i 61 3 H Product by 6. . ^183 9 4^ Product by 16. proceed thus 17 14 177 9 10 G Product by 10. 6 1064 5 Product by GO. 3 £3192 15 Product by 180. Eramples. (Ixvil.) Find the value of (1) 4 things at 7s. 3d each. (3) 6at7K (5) 8 at 2s. 4d (7) 11 at £2 Is. U, (9) 14 at 17s. 6rf. (11) 16 at 27s. (13) 20 at £5 lb. 4d (15) 22 at £5 lU 4d. (17) 25 at 4s. Qd. (10) 28 at 2s. M. (^'i) 3o at £1 2a. (2) 5atl4d (4) 7 at 9«. Qd. (6) 10 at 2s. 2hd. (8) 12 at £1 4». 3d. (10) 15 at 7«. lOAd (12) 18at.^''s. 63. (14) 21 at 5«. 7Ad (16) 24 at £4. 7s. 2d (18) 27 at 5». lUd. (20) 30 at :P1 19« (22) 35 at £1 2* 6d. 124 OOMPUUMI) MULTIfUCAnOW. 146. Wli«ii the multiplior cannot be split up into factor*, w« inav pro<»e{l m in the following example* : Ex. (1> To multiply XI 7 IJ,*. d^tl. by 79. 17 12 4. 10 176 8^ Product by 10. 7 1233 13 Multiplying 1st line by 9 lb8 14 Adding last two results £1393 lU Product by 70. til Product by 9. 10| Product by 79. Ex. (2). To L.ultiply X3 17*». OJ^/. by 3296. I 3 «. 17 10 38 17 11 Product by 10. 10 388 19 3889 a 2 Product by 100. 10 8 Pn ct by 1000 3 11068 15 Multiplying 6th line by 2 777 18 Multiplying 3rd line by 9 350 1 Multiplying Ist line by 6 23 6 Product by 3000 4 " by 200. 3 " by 90. 9 " by 6. Adding last four results £12820 4 Product by 329(5. 147. The following is a method by wh;ch the process of multiplying a compound quantity by a number greater than 1000 is somewhat shortened. We take as .- n illus tration the example ju.st worked. The process is so simple that no verbal explanation is necessary. •cess IS so a i 8SM 0Okf>UUND DIVUIOK. I ^ d, 1648 ^^^^ ^*»« '•««"lt «f multiplying lh« top lin« by 9. 12 31319 f. 2(101) ftnd U. 23072 \ ,. 329^ j the retuit of multiplying the top line by 17. 20 I 5804 1 126 £ 2932 and 1*. ^^^ **»« "^e-uit of multiplying the top line by 3. £12820 1,. 4rf. (2) (4) (6) (8) 39 at 12j. 6*rf. 7^ at U. 8d. 123 2154 at 5s. 6U. I at £7 1». 3d. (10) 2176 at £2 16... 4^^. Examplea. (Ixviii) Find the value of (1) 29 thinj^s at 4«. 6d. ench. (3) 47 at U OU (6) 89 at Gs. 8d. (7) 145 at £1 Sh. 2d. (9) 3210 at £1 18s. <5M (11) 3084 at £2 (is. i)fd. COMPOUND DIVISION. 148. The process rf dividing a compound quantity y a number IS based upon the principles explained in J e case ot Simple Division, as will be seen from the iullowing examples : Ex. (1). To divide £13 Us. ly. by 9. ^L11_J7 1| ,v , ^^ ^^ ^ Quotient. We reason thus : t'f ~ ftrilv.r^ '^ ^''""^f f} as quotient and £4 remainder ; OTrrliJ?^ ^ k"??' -""^ 17 shillHigg added gives 97 shillings. ^' ^. divided by 9 gives 10.. as quotient and 7s. remainder: H< r/ r^^l®' ^""^ ^ P^""y ^^^^^ gives 85 pence. «0'/. divided by 9 gives M. an nnnfiAnf. ar.A L =:_.i__. 18a~divM Jf k"^«' ^"^ o ' ''^^"^•88 added gives fs'faTthbg;. ^H divided by 9 gives 23. as quotient and no remainder. i\i^ ':■.:■ 126 COMPOUND DIVISION. Ex. (2). To divide £51 15.s. M. by 35. The factors of 35 are \7 £ s. 51 15 d. 5 10 £19 7 Quotient Ex. (3). To divide £53 Us. Sd. by 112. r The factors of 112 are < 4 It £ 53 «. d. 15 8 13 8 11 7 2J 9 71 Quotient. Ex. (4). To,divide £119232 \s. lOy. by 346j. 3405 ) 119232 10395 «. cf. 1 10^ ( £34 15282 13860 1422 20 3465 ) 28441 ( 8s. 27720 721 12 3465 ) 8662 ( 2d. 6930 1732 4 3465 ) 6930 ( 2q. 6930 .-. the Quotient is £34 8s. 2^d. OOMPOUIO) DIVISION. 127 I. Divide ^^*°^^^''- ^^^'^ (1) £1 3s. 7^d by 3. (3) £11 3.. 6d. by 12. (5) £6 2s. lid. by 10. II. Divide (1) £98 11a. 9d. by 54. '" £29 Us. Od. by 108. £39 7a. 6d. by 7. £43 12a. Sd. by 11. £22 Us. 6d. by 12. (5) £3 9s. 4^d. by*'46 III. Divide (1) £167 19s. 2d. by 145. (3) £453 lis. 9^ by 365. (6) £93 Is. 2id. by 291. (6) (2) £13 7a. 9d. by 63. (4) £15 8s. by 132. (6) £43 12s. 8d. by 44. (2) £40 8s. 4^d. by 241. (4) £40669 2s. Id. by 9652. (6) £139 3s. 6d. by 117. 149. One quantity is contained in another of the sarae^xnd as often as the measure of the first is con- tained in the measure of the second, the same unit of measurement being taken in both cases. ir^^'o^P' -^^^ ^^"^y *^°*®« is 1* 1^- contained in lbs. 3c?.? Is. Id. = 13c?. ; and 16a. 3d. = 195£;. Now 13 is contained 15 times in 195 ; .•. 13d is contained 15 times in 195d 4^^7''a ^V'. F"""^ ^^""y *'°'®^ i^ ^^ 3& 2d. contained in £4 35. 2d. = 998d ; and £87 6s. 6d. = 20958d. Now 20958 ~ 998 = 21 ; .-. £4 3a. 2d. is contained in £87 6s. 6d. 21 uimes. Examples. (Ixx,) (1) How many times is £346 16s. contained in £34680 ? ^11 £5 lis. 4(1. £122 9s. 4rf.? ^ £112a. 6d £68 5s.? ;r=( ; £i7i2s. 9|i. . . . . ^1393 8s. io|d? (6) Among how many persons must £641 14s. IIM be divided, so that the share of each may be £2 15s. 6mT (6) Divirie fiT? '"+« «« 1 i i- . . .. c,^..«l.„: 1 %^' """-' -'" ~4-»i iiiiii -jur or sov6xdiL<;iis, halt- ^^overeigns, half-crowns, shillings, and sixpences. 128 PBACTIONAL MULTIPUCATION AND DIVISION. FRACTIONAL MULTIPLICATION AND DIVISION OF MONEY. 160. Ex. (1). Find the value of f of 14«. B>d. i cf Us. M. = ^!^ = 3«. 8d. .-. I of 14a 8d. = 3 X 3s. M. = 11a. It is '-imaterial whether we divide by 4, and then multiply the quotient by 3, or first multiply by 3, and then divide the product by 4 ; thus : I of 14s. 8d = 'JiH'^iii- = !!!:== 11, * 4 4 AJ-«- Ex. (2). Find the value of | of f of £43 is. 6d f of f of £43 4a (ki. = i[Q of £43 4s. Qd. 21 « 10 X £2 U 2c;. = £20 11a. M. Ex. (3). What is the v lue of 2f of 14s. M ? 2f of 14s. 9d.'= y of 177d _ 17 X md. 3009rf. ^^^„ , = 7 — = ~^r- = ^2^d. = £1 158. 9?d. Note.— To find the value of f x 2s. 9.? , we extend the meaning of the sign x (as explained in Art. 71N and replace it by the word of. '' Thus \ X 25. M. = a of 2s. 9d = ??ii^ = i,. 74^. Ex. (4). DivHi; 4.'^. M, by |. 4s. 2d. -T- f = 4s. 2d X f ' = f of 4s. 2d. = 8 X lOd. = ^s. M. Ex. (5). Divide £4 3s. U. by 2|. £4 3s. 9d. -^ 2f = £4 3s. M. ^ § = § of £4 3s. M. = ^^-IH^Ji = £1 11,. 4^^^ Examples. (Ixxi.) Find the value of fj) |of4s. 9d. (2) f of 7s. 2d. m ti A''^\^^^^• ^^^ I of f of £83 16s. 3d. yX X .nk^y-a^ <^^ 5f| of half a crown. ON MBARtTRBs. 129 (11) £mu.8d.~l /i2^ X»or n^ the mixed numberT^to «n • ^^^^^^ »««««sary to turn in Ex. (3), ^u:t:v::i:^^^^^^^ more neatly by multiplyiL first bvth^f!*'^^'?**'^" and then by the ^hoirLm^rln^JT'^^^^ results. "uiuoer, and adding the two Thus, to multiply £427 VZs. 9d. by 6* £ #, d. "^ *' 427 12 9 •^ |^55__^j the result of multiplying the t.p line by 2 285 2138 1 10 3 9 £2423 5 7 Multiply Examples, (borii.) |) i1lVil1.X1^ g) 1^9 18, 3. by 7, (5) £7258 17. 6.. by^ 4'. g> 1^^ ^ «.. by 3|. 6 151. XVI. O^ Measures. Measures op Time. 1 second is written 1 sec. , or Is. 60 seconds make 1 minute, written 1 min., or i™. 60 mmutes make 1 hour, written 1 hr., or Ih. 24 hours make 1 day, written X da., or Id. 7 days make 1 week, written 1 wk. I'n rZh !!!'t!""' ' ^''' '' *^^- *« -««-t of 365 days in rough ^.Iculations a month is taken to consist of 30 days is ^^^r:J^T.;.V^^^^ ^^^-- *- -- -on. 130 on MKA^4LHE8. 7 of them contain 31 days,, 4 contain 30 days, and February has 28 days (and in Leap-year 29). The names oi the 4 months which have 30 days are given in the old verse : Thirty days have September, April, June, and November, To find whether a particular year is a Leap-year, we divide the number of the yea-v by 4 ; if no remainder be left, the year is Leap-year, but to correct an error in our present Calendar, the cetituries which are not exactly divisible by 400, as 1900, 2100... are to be taken as common years, and not as leap years. Examples. (Ixxiii.) lieductton. (1) Reduce 6 hr. 17 min. 25 sec. to seconds ; 17*^- 0™- 438- to seconds. ^ (2) Reduce 3 yr. 143 d. 16 hr. to seconds ; 1 yr. 13 d. hr. 4 min. to minutes. (3) Reduce 48567 min. to days ; 23567 sec. to hours. (4) Reduce 742392 sec. to ditys ; 174296 sec. to weeks. (5) Find the number o' days, reckoning from noon of the one to noon of the other, between the following days in the year 1872 : 1st February and 29th May ; 4th July and 2nd December ; 3rd January and 15 ch October ; 24th February and 23rd June. Also between 25th December, 1872, and 25th May, 1873. Addition. (6) (9) hr. min. sec. da. hr. min. wk. da. hr. 14 21 37 (7) 1 23 15 16 (8) 4 3 16 17 13 32 57 12 38 2 5 17 9 47 43 13 17 43 3 6 9 12 53 54 24 22 7 10 4 13 22 17 50 16 5 58 da. 4 2 19 yr. da. hr. hr. min, sec. hr. min. sec 3 137 15 (10) 14 43 13 (11) 42 14 30 31 4 243 6 32 36 40 65 22 19 42 1 56 7 10 12 53 74 11 42 15 6 135 12 16 38 47 24 18 58 57 7 Sb 9 2 62 8 43 3 29 48 3 taken as hr. min. sec (12) 7 14 26 4 19 38 ON MEA8URBS. Subtraction. f**' hr. mill, (13) 123 IG 4 39 22 17 131 ,^ ^ wk. da. hr. (14) 4 6 18 3 6 20 (18) Multiply 13 hr. 14 min. 43 sec. by 35 ; 17 hrs. 13 min. 39 sec. by 43 (19) Divide 15 wks. 5 dys. 17 hrs. 26 min. by 49: 14 hrs. 56 min. 41 eeo. by 73. 162. Measures op Length. 12 inches make 1 foot, usually written 1 ft.. ^/««*; lyard, lyd 5^yarda 1 polo i J^ *0 poles 1 furlong ' ' "l fur 8 furlongs....! mile.. .. J !^, . „ 3 miles 1 league .'.:;:" 1 lea Hence 1 furlong = 220 yards, and 1 mile = 1760 yards. Cloth Measwes. 2 J inches make 1 nail. 4 nails 1 quarter. Ex. (1). Reduce 3 mi. 5 to inches. mi. fur. po, 3 6 17 8 4 quarters make 1 yard. 5 quarters 1 ell. fur. 17po. 4 yd. 1 ft. 3 ia yd. 4 ft. 1 in. 3 hr. min. Bee 14 30 31 22 19 42 11 42 15 18 68 57 3 29 48 29 fur. 40 1177 po. ^88^ the result of dividing 1177 by 2. 5889 6477^ yd. 3 12 9^-^905 inche' 132 OK MXASURK8. Ex. (2). Reduce 47293 yards to poles. %;293 yd. - (47293 -4- 5^) polet. - (47293 - Y) pole^ - (47293 X ^) polei. Wo may procoed tlius : 47293 yard* 2 11 94586 half.yardt 8598 poles, and 8 half-yards over. .-. 47293 yd. -= 8598 po. 4 yd. Examples. (Ixxiv.) lieductimi. (1) Reduce 3 yd. 2 ft. to inches ; 4 mi. 3 fur. 4 po. to feet. ^J-?) -Reduce 7 mi. 14 po. 3^ yd. to inches ; 27 po. 4^ yd. (3) Reduce 74325 yd. to poles ; 2423694 in. to furlongs. (4) Reduce 723964 ft. to miles ; 82976432 in. to miles. Additum. yd. (5) 4 ft. 2 in. 7 mi. (6) 13 fur. po. 4 20 19 1 9 43 3 9 5 2 10 66 2 13 2G 2 8 4 7 32 36 1 6 16 3 15 17 2 4 19 5 11 Subtraction. yd. (8) 134 ft. 2 in. 7 mi. (9) 235 fur. po. 19 59 1 11 184 5 24 fur. po. yd. (7)2 19 2 4 26 2h 6 11 st 5 23 4 3 n 1 21 i| fur. po. Vd. (10)5 23 H 4 27 4 (11) Multiply 7 yd. 2 ft. 9 in. by 11 ; 16 mi. 5 fur. 7 po. by 56. '^ (12) Multiply 32 po. 3 yd. 1 ft. by 57 ; 36 mi. 3 fur. 6 po. 3^ yd. by 49. '^ ^ (13) Divide 25 yd. 1 ft. 8 in. by 4 ; 17 mi. 3 fur. 7 po. (14) Divide 14 po. 2 yd. 1 ft. 8 in. by 32 ; 11 ml 7 fur. 7 po. by 65. . po. yd. 23 H 27 4 ON MBASUHlg. j«„ 163. Mkasuuks op Surface i44 square inches make 1 square f„ot written 1 .n #* 9 square feet i ^.^e yard 1 S' 1.. m square yards 1 s;^uare pole .' t S* J 40 square poles i rood 7 •^* P^' 4 roods il^^^ Iro. Hence 1 acre = 4840 square yards. ^ ~'" 040 acres «= 1 square mile. Land surveyors make use of a Phnin oo j • length, dividecfinto 100 equal pttl. caH^ ^L^'' '" The square of 22 is 484, and therefore Sn., ("hains make an Acre. "lereiore ly bquare i.We^g-thT'^'"' ^'""" ^'■°'' '' • "1"«« whose side is .„ i„„h 3 'I- r? V y- V 14 ro. 40 587 po. 30^ 17637^ *^^ '^'"^* ^^ *^® ^^^'^^^'^ of 687 by 4. 177831 sq. yd. 9 160054 ^ *he result of multiplyiug | by 9. 1600601 sq. ft. 144 640263 640240 160060 108 the result of muHi"«i«;«~ a i-_ * ^ . 23047771 sq. in. 134 ON MIABUREH. Ex. (2). Reducfc 74237 sq. yards to pole*. 74237 aq. yd. -= (74237 -r- 30i) poles - (74237 + J-f ^) poles - (74237 X xfr) poles. We may proceed thus : 74237 yards 4 121 ] 111 206948 quarter-yards. 2G095 and 3 quarter-yards over. 2454 po. and 1 parcel of 11 quarter-yards over. The remainder is (11 + 3) quarter-yards, or 14 quarter- yards, or 3^ yd. X 74237 sq. yd. = 2454 po. 3^ sq. yd. Examples. (Ixxv.) Reduction. (1) Reduce 5 ac. 3 ro. 17 po. 13 sq. yd. 6 sq. ft. 15 sq. in. to square inches. (2) Reduce 7 ac. 13 po. 6 sq. yd. 3 sq. ft. to square inches. (3) Reduce 250 ac. to square yards, and 73 sq. yd. to square inches. (4) Reduce 6239 sq. in. to square yards, and 16376 sq. yd. to uores. (5) Reduce 34729 sq. yd. to poles, and 562934 sq. in. to square poles. Addition. ac. ro. po. sq. yd. sq. ft. sq. in. ac. ro. po. sq.yd. (6) 47 2 13 (7) 19 7 42 (8) 46 2 16 22 72 1 24 27 5 52 17 3 14 13 89 2 32 32 8 124 7 1 39 14 4 2 23 5 2 72 24 2 15 19 27 3 8 21 6 98 12 17 22 42 2 5 56 3 135 4 1 9 lU Oir MBAflURIR. 135 •0 ro. (») 67 2 29 3 Subtraction. PO- iq. yd. iq. It wi. In. 80 (10) 42 8 124 34 36 8 139 •e. ro. po. (11) IC 2 14 3 24 M. ro. (12) 247 1 243 3 po. «!. yd. tq. ft. iq. In. „. „. -^ 14 (13) 39 7 12 (14) 245 3 19 24 32 8 134 178 3 23 b/sa ^"^"^^^ ^ *°' ^ '''• ^* P°- ^y ^^ ' ^7 *^' 2 ro. 13 po. ^^(16) Divide 7 ao. 2 ro. 18 po. by 21 ; 29 ao. 2 ro. 37 po. 164. Measures of Solidity. 1728 cubic inrliea make 1 cubic foot, written 1 cub. ft. ^7 cubic feet make 1 cubic yard, written 1 cub. yd. A Cube is a solid figure contained by six equal squares. Hence a cubic inch is a six-sided figure, each of whose sides IS a square inch. The lines that form the boundariee of the sides are called the Edges of the Cube. Examples. (Ixxvi.) Reduction. fn^to cubic toohea.'"- *° "="""' '""-' ■ " ""''■ ^^- '''* ''^ JUbu'yZZ''*^^^ ""''■ '"■ *° ""'"'' '*•"; *^^^ <"•"• in. (tl c^alicfncUs."''- '^^ "" ""'"'' ''"'''™ ' ' «'"'• y*"- «« «<"'. Addition. cub. cub. yd. ft. (4) 57 13 32 25 46 19 76 8 oub. in. 572 493 374 587 4 26 1249 RO f A t nt\ 1 cub. cub. cub. yd. ft. in. (5) 43 7 1638 26 22 472 19 16 1384 45 13 427 26 5 1286 0'6 18 27& cub. cub. cub. yd. ft. In. (6) 528 16 432 237 19 583 764 10 1359 446 1275 729 11 346 852 5 1473 180 ON lilAHUKM. '■ SublraCfimi. oub. oub oub. oub. oub. cub, «ob. eub. eub. ytl. ft. in. yd. It In y»J. ft. in. (7) 47 17 643 (8) 247 19 1274 (0) 627 38 23 726 239 18 l'M\S 4i)9 10 266 (10) Multiply 26 oub. yd. 6 oub. ft. 49 cub. In. by 27 ; 472 oub. yd. 17 cub. ft. 238 cub. in. by 63. (11) Divide 78 cub, yd, 13 oub. ft. 262 oub. in. by 12 ; 472 cub. yd. oub. ft. 1416 cub. in. by 69. 155. Measures op CAPActTY. 2 pints inako 1 (juart, written 1 qt. 4 quarts 1 gallon, 1 r/xU, 2 gallons 1 pock, 1 pk. 4 pecks 1 bushel 1 bus. 8 bushels .... 1 quarter 1 qr. 'Examples. (Ixxvii.) \ /{eduction. (1) Reduce 3 pk. 1 gall. 3 pt. to pints, and 214 qrs. 3J bus. to pints. (2) Reduce 4234 pt. to quarters, and 3047 gall, to quarters. Addition. gall. qt. pt. bush. pk. gall. qr. bufih. pk. (3) 4 3 1 (4) 4 3 1 (6) 42 6 3 3 2 1* 6 2 H 27 7 2 12 30 131 6431 14 1^ 4 2 li 4P 6 2 621 310 12 40 Subtraction. , gall. qt. pt, bus. pk. gall. qr. bus. pk. (6) 6 2 (7) 6 3 (8) 36 7 2 431 631 29 73 (9) Multiply 5 qr. 3 bus. 2 pk. by 63, and 15 qr. 2 bus. 1 pk. by 73. (10) Divide 13 gall. 1 pt. by 16, and 348 qr. bus. 1 pk. by 43. 156. Troy Weight. 24 grains make I pennyweight, written 1 dwt. 20 pennyweights make 1 ounce, written 1 oz. 12 ounces make 1 pound, written 1 lb. Chiefly used for weighing gold, silver, and jewels. oub. cub. ft. In » > 10 250 n. by 27; n. by 12; rs. 3i bus. ) quarter^L bufih. 6 7 3 6 4 -1- 2 1 2 bus. 7 7 3 qr. 2 bus. bus. 1 pk. ildwt. 1 1 oz. lib. Olf MRAKVltU. o^ 1.T7 Examples. (IxxvlU.) /itdnriiirfi. (3) Reduce 3 lb. 10 o^ 7 dwt 5 «• . t lu ^ IS gr. to gmins. ^ '*''*• *^ K'-i 7 lb. 4 oa. 17 dwt (4) Reduce 3145 gr. to ounces ; 42672 gr. to lb. Reduce 7^400 gr. to lb. ; 3246 dwt. to lb. Addition. - I II ^ ^ J '0 }? ? 7 6 8 3 ic iS 42 7 16 21 — — — ^" ^^ 12 11 19 23 Subira 'Hon. '" LU »» li^f ™ » ,; X i b/n! ^""'''^'^ ' ''- ' - ' <*-*. by 12 ; 6 IbTT:^!?^ dw^'^l^f b^^^^ - 1« ^-t. 23 gr. by 37 ; 3 lb. 7 o. 10 7 Hy^'lf *^ '' ^'- ' °^- ^« ^-*- by 8 ; 7 lb. 10 o. 17 dwt. „/^^) J^'^ye 9 oz. 17 dwt. 8 gr. bv 37 . IK IK « o j 1^ gr. by 63. , ** ^ ' ' ^" ^°- ^ oz. 9 dwt. lo7. Avoirdupois Wiiorr. 10 drachms. make 1 ounce, .,:,, , J""»<'^« 1 pound written loz. itT""^ l»tone,' .......lib. IPT.^" 1 quarter, J «*' 20 Srndrdweight:;;; ht-<^-«. ; 17 lb. to dr. ; 6 toni to lb. (2) liedttoe tout 7 owt. to «>/.. ; 1ft toni 2 or. to lb. (3) H«duoe 3 owt. 6 lb. 5 oz. to dr ; 3 tont 16 cwt. 7 lb. to lb. (4) Ileduce 4703 ok. to owt. ; 3740 lb. to ton«. (6) Keduc« 7432 oz. to owt. ; 247294 dr. to cwt. Addition. lb. ML dr. (7) i lb. (NU cwt. % lb (0) 3 3 9 10 8 (8) IB 24 10 8 6 4 7 12 11 3 5 7 10 13 16 10 5 29 1 10 14 6 7 8 20 13 16 2 8 15 14 12 6 9 17 7 Subtraction * lb. 01. ' dr. qr. lb. 01. cwt. Y- lb. (0) 16 13 5 (10) 17 13 3 (11) 19 4 14 11 12 14 15 11 17 3 18 tons. cwt. qr. (12) 37 19 2 29 19 3 owt. qr. (13) 10 16 3 lb. 3 25 tont. cwt qr. lb. (14) 74 16 1 13 39 10 3 2B (16) Multiply 17 cwt. 23 lb. 14 oz. by 7 ; 4 owt. 17 lb. by 45. (10) Multiply cwt. 3 qr. 6 lb. by 23 ; 10 oz. 9 dr. by 37. (17) Divide 14 cwt. 2 qr. 8 lb. by 12 ; 32 tons 16 cwt. 1 qr. by 40. (18) Divide 16 cwt. 3 qr. 9 lb. by 05 ; 37 tons 4 owt. 3 qr. 7 lb. by 17. 168. Apothecaries' Weight. 1. Measures of Weight. 437^ grains make 1 ounce, 16 ounces make 1 pound. The grain is the same as the grain Troy. The ounce is the same as the ounce Avoirdupois. This is the table given in the B"itish Pharmacopoeia. The Avoirdupois ounce and pound are taken in prefer- ow MiAiirmit. 130 ftt. qr. lb a 2 24 1 3 6 !9 1 V.) 6 2 U 7 7 MTt. 9 t lb. 4 .7 3 18 owt qr. lb. 16 1 la 16 3 26 owt . 17 lb. ) dr, . by 37. L6 owt. 1 qr. once to tli« ounce and pound Troy of tho old table, »,#. cauw the forni«r are used by wholetale dealers in druus and medicines. In proi,cribin;(. many physicians still employ the scruple (9) of 20 grains and the drachm (3) «^' 60 grains, ^ ' 169. 2, Me ^^ y^- 2 ft- 3 in. by 43f. r\ 9? A% ' P^; \ ^^'^' ^> 25 ac. 2 ro. 15 po. by 29J. (0 27 aq. yd. 7 «q. ft. 36 sq. in. by 2^. ^ * 161. Division of Compound Quantitiec when the divisor contains a fraction. CSee page 128.) Divide Examples. (Ixxxi.) nl 7 m'l^'/f^"' ]l ^^' \7 ?t- ^2^ 7 ^^- ^ °'' 14 dr. by llj. (•^) 7 mi, 2 fur. 12 no. bv 4J*- -'^N 1^ ..^ 1 *i o :_ lZ ^ (0) 26 ac. 2 ro. 12 po. by 4 J" "2' (7) 107 sq. yd. 4 sq. ft. 132 f . (6) 14 ac. 3 ro. 8 po. by 8| sq. in. by ISf I M 140 FRACTIONAL MKASURB8. 162. XVII. Fractional Measures. Ex. (1). How many shillings and pence are there in I of a pound ? f of a pound » ^ of 20 shillings. » H20 shillings. - 128. 6d. Ex. (2). Find the value of ^ of £15 5s. 8d. i- of £15 5». 8d. = 3 times ^ of £15 5s. 8d = 3 times £2 Ss. 8d. = £6 lis. Or thus : £ ». d. 15 6 8 3 45 17 £6 11 Ex. (3). Find the value of 2f of ^ of 5 acres. 2| of ^ of 6 acr' = ^^ of ^g of £ acres. 1 1X3 c K = § of 5 acres. = 1^ ac. = 1 ac. 3 10. 20 po. Examples. (Ixxxii.) « Find the value of the following : * (1) I of £1 ; § of £2 10s. ; f of £5 18s. 5d. (2) f of a mile ; ^^ of an acre ; | of a cwt. (3) 2^ of £54 9s. 8d. ; 3j\ of half-a-guinea ; f of 3f of a mile. (4) ^ of ^ of 1^ of If of 2470 guineas ; f of ^ of 4^ guineas. (5) § of £1 + f of Is + f of 16s. 4d. \ (6) f-^ of £1 + f of 2s. Qd. + f of a guinea. (7) I of 5 ac. 3 ro. + ^ of 7 ac. 2 ro. 20 po. + f of 3 ro. 15 po. (8) T^ of a year -f- ^ of a week + '^ of an lioijr. (9) ^g of a mile + | of a furlong + f of a yard. FIlACndNAL MBARURES. ^^j . vT- I f«JJowing are examples of an ooeraLn vvhich IS the corwerse of that just explained °P^'**^°^ Ex. (1). Express 14*. 7d. as the fraction of £5. 14a. Id. = 175d., and £5 = 12004. Now Id = ^^ of 1200(i. •■• 1754- isi^/rfVof 1200rf.; Hence the fraction required is ^/,«„ or ^, or A. 3 I'^lfi^'"^"""' ^ ^^'' ^ "^' ^^^i^d- ^« t^^e fraction of 6 lb. 5 oz. = 101 oz , and 3 lbs. 12 oz. = 60 02. ; . . tne fraction required is W Ex. (3). Express | of 6.. 9^. as the fraction of is Id ^ ,pa. 9d = 69o?., and 4s. Id. = 55d. .-. 55. 9d is f I of 4«. 7d .'. f of 55. 9d isf of ff of45. Id. ••. the fraction required ia ?-^--l or 4« of f of ^ii' ^fT: ^ '' '^ °' ' ^°- '^ ^^^°- ^"^'^« ^-*-n 5 ac 3 ro. = 23 roods, and 14 ac. 2 ro. = 58 roods ; . . fraction required is (f of ^ of 23) ^ (# of 58) • 3X14X23X6 _2X23 oo JNOTE.— There are several modes of demanding fl^o Tdlrnr ^^"^' ^" *'^ ^^^^^^^^^ exaTpTerC: i»ay be put in the following terms : m wi?"/® 3 shillings to the fraction of 6 shillings n WW f "t°^ ^ "^^"^"^« i« 3 shillings ? ^ 4 iT fi J -iT*'"" u^ 6 shillings is 3 shillings ? Bhillgs?^ '"'"^' ^" *^" ""^*' '^^^^ i« the measure of 3 Examples. (Ixxxiii.) (1) Express lU as the fraction of Qs. 8U 3 i?'^'' f ^^J'- ^^- ^« ^h« fraction of £11 65 5d !r< ^«^uce 95. lO^rf. to the fraction min, iOTI nt St ctiiim YfiS "D J — « , 2"^- ^^ me iraction ot 13s. 2h (5) Reduce 2 days 3 hr.. 5 min. to the fraction of 135. 2^d. of a week. 142 DECIMAL MEABURKB. (0) Reduce 2 roods 20 poles to the fraction of an acre. (7) What fraction is 8 lb. 1 oz. 19 dwt. 9 gr. of 13 lb. 7 oz. 5 dwt. 15 gr. ? (8) What part of 2 qr. 10 lb. 7 oz. 9 dr. is 1 qr. 7 oa. 13 dr.? „ ^ (9) What fraction of 4 lb. 1 oz. 8 dwt. 16 gr. is 1 lb. 1 oz. 9 dwt. 16 gr.? (10) If the unit of meaaureraent be 2^ yd., what is the measure of 2^ feet ? (11) If the unit of measurement be 6 inches, what is the measure of ^|^ of a mile ? (12) What fraction of 2 ac. 37 po. is 3 ac. 2 ro. 1 pc? 164. XVIII. Decimal Measures. Reduction op Deoimals. Ex. (1). How many shillings and pence are there in •375 of a pound? ' •376of £l=:(-376 X 20)s. = 7-6«. and -5 of Is. = ('5 x 12)d. = Qd. :. -375 of £1 - Is. Qd. The operation is performed more briefly thus : ^•375 20 8. 7-500 12 d. 6 000 Ex. (2). Find the value of 3-16875 of £1. ^3-16875 20 s. 3-37500 12 d. 4-50000 4 q, 2-00000 .-. ^3-16876 = £3 3s. 4^ an acre. )f 13 lb. 7 oz. are there in DECIMAL MEASUIIEH. Kx. (3). Find the value of -4250 of 12.y. 8d. •4260 of 12.. 8^. = .4260 of 162d == (-4256 x 152W •4250 152 8612 21280 4250 143 04 0912 :. value required is 64*0912d. Ex. (4). Multiply 27 ac. 3 ro. 14 po. by -235. ac. ro. 27 3 4 111 ro. 40 po. 14 4454 po. •236 22270 13302 8908 40 4 1040 090 20 09 po. ac. 2 ro. 0-09 po. Ex. (5). Find the value of -25 of £1, •26 of ^1 = i|za of ^1 = |3 of ^1 ^ co^, ^ 5^^ Or thus : ^•2555.... 20 s. 6 11li, 12 d, 1 -.3.333., .'. value required is 5s. l^d. 144 DKCIMAL MRAHUKKfl. Examples. (IxxzIt.) iH' m Find the value of (1 (3; (6; (7 (0 (11 (13 (16 (10 (17: •025 of XI. X 009705. •040875 of 1 lb. avoir. •425 of 3». 4(i. '8^ of 53. •36 of 2 qr. 14 lb. 2 1372of 2toiis6cwt. (2) i;i6-275. (4) -9375 of a cwt. (0) 2 003125 of ija (8) 2-40875 of XI 3«. (10) 41^ of 12a. 0(i. (12) 2 125 of a j( guineas. (14) 5 '247 of £^ 2h. 6d. •46 of £3 10<«. + ^75 of 4«. &(i. + 3-245 of 3s. 4d •7 of £1 4- -A of Is. (ki. 2Ab of Is. 8ii. 285714 of £3 3». + -142867 of £3 17».+ U of 10a. Od. 166. The following examples illustrate the operatioi] which is the converse of that already explained. Ex. (1). Express 5«. Od as the decimal of £1. 6a. Od » OOd, and £1 ^ 240d ; .•. 6a. (kl. « 4\% of £1. Now 4% = Ji =- -275 ; .-. 5a. (id. « -275 of £1. Or more briefly thus : 12 20 0-0 d. 5-6 a. •275 £. Where we first express Gd. as the decimal of a shilling, t.e., *5,^ a^id then express 6 -6a. as the decimal of a pound, ie., *276. Ex. (2). Express £7 lbs. lOjrf. as the decimal of £1. 4 12 2 10-5 20 15-875 £7-79375 DECIMAL MEANUKM. 140 Ex. (3). ExprftsR £3 5- q.; „„ ^l. i • » * £5 7n Qd uBcirnai of * •• rf. £ • ,( aS ^ ® 6 7 6 ^0 20 66 12 Now 789 -Wa li$ 107 12 1290 •611.. Ufa ^ tW -If •'' ^'3 S«- 6^. is "611 of £5 7«. (W. (J2d. ^^^' ^""^'^^^ * °^ ^'' ^i''- *"' ^^^^ ^^"^""'^ "^ )^ «^ 5«. 9|d = 277^., and 6«. 2(i. = 296g. ^"^ ff il = 1^1^ = mf = 1'039 Examples. (Ixxxv.) (1) Express 6 cwt. 2 qr. 7 lb. as the decimal of a ton {2) Express 12 grains as the decimal of a lb. troy (3) What decimal of 10 guineas is £1 19«. Ud ? (4) Express ^ of 14a. 4d. as the decimal of £1 (5) Reduce 3-46 of half a guinc. to the decimal of 2.. 6rf. Express f of 2 qr. 14 lb. as the decimal of a cwt t4; ^^'''' ^* '^ ^ °'- ^ ^^*- ^ *h« d«°i«^al of a pound ^(8)^Reduce 3J of IJ of 5 cwt. 2 qr. 21 lb. t ^he decimal (9) What decimal of a pound troy is f of a dwt. ? (10) Reduce 3| guineas to the decimal of £2 15a (11) Reduce 2s. 6d to the decimal of ^ of £1. (12) Express 18s. 4^d as the decimal of £1000 ^.(Jl3)^Reduce £2425 + 3-4125s. + 9-25rf. to the decimal (14) Express -43 of 8.. 3d as the decimal of -Oi of £9 (15) Express 04 of^^i 5s. + -23 of 3s. 9d. as the decimal of "-zi.- ui rtrfS OS. oa. 146 EXAMINATION PAPKRH. EXAMINATION PAPERS. I. Measures of Time. v (1) A sidereal d&y is less than a solar day by 3 minutes 56 seconds ; in how many days will the ditFerence amount to 24 hours ? (2) If SiriuB, orie oi" .»e brightest of the fixed stars, which is probably 592200 times farther from the earth than the sun, were suddenly extinguished,, for how long would it appear to shine to the inhabitants of the earth, supposini; the sun's mean distance from the earth to be 91713000 miles, and that light from the sun reaches the earth in 8 min. 18 sec? (3) The exact length of the year being 365 days 5 hrs. 48 min. 49-7 sec, and computing time as at present, find the error in 12000 years. (4) The Olohe. newspaper of Monday, 18th June, 1877, bears the nuniber 8505. Supposing the paper to have been published every week day without intermission, and num- bered consecutively, give the day of the week, month, and year when No. 1 was published. {^) There was a full moon on June 26, 1868, at 9 hrs. 13 min. a.m. The interval between successive full moons has since been on the average 29 days 12 hrs. 47 min. 30 sec. ; how many full moons happened until December 31, 1873, and -when did the la t take place within that period ? II. Measures of Length. (1) Reduce 9 mi. 7 fur. 39 per. 5 yds. 1 ft. 9 in. to inches, and show that the work is correct by changing it to miles, etc. , (2) The fore-wheel of a carriage, which is 11 ft. in circum- ference, makes 718 revolutions more than the hind one in going 7 miles ; find the circumference of the.hind-wheel. (3) A train, which travels at the uniform rate of 66 ft. a second, leaves Toronto foi- Montreal at 6. 25 a. m. ; when will it reach Montreal, the distance being 333 miles ? At what distance from Montreal will it meet a train which leaves Montreal for Toronto at 8 a.m. , and travels one-third faster than it does 1 (4) From Ephesus to Cunaxa, Xenophon, with the army of Cyrus, marched 16050 stadia of 202 yards 9 inches each in 93 days. Find the average length of a day's march in miles and yards. bxamination papers i4y whael once round 7 ^ ^ ' "^ ""' ^ ""''" ""■" the III. MEikUKKS OF SURKACB. when the lineal Zit"XLZ ''°'""" ' ^""'P'»y >vhi/h i« 7 f«et 6 incL :Lt^S''',"iri''''l:''' °" "'"We "hat i. their amount * ^ °°' * '"'=''«» "'^x J ""d (3) Divide 17 ao. 2 ro. 38 per 19 vd« » « jk • A, B, and C. ijivino tn n ... i '^ .' ' "• *^ '"■ among J of ;hat A anTij lot? '""'"' *«"" " "> A, and to 3 .•.mtii"S^^r a'riTn'r'" ^X'^^y-^^' "nd each bale car'f^, when th:rar:metetteStt !" ^ "k"'-* <>' *<> the square inch whTtT ti,„ j!"''' *' ?" '""'">»' *^'' 16 lbs. on 'vould be the difference of Ihf ^® '.I"*''® f««* ' What stands at 29 inches? P™°""" "''*" *''» l^fometer IV. MEASUBE.S OF CAPACITY eSiSr"'""'^'- iowhtn*re-o?eh:i^.^fh1 43fLi«^er"ano^f^rt'!t^ii' ^*- ' P*l°' ''"""y' »' quantity o? ryfiid'he thufatt' l„| ™"'' ^'^ ''""'"''• ^'"^ (4) I wish to put Hi hu '^' r^l^ A ^t. e . . that shall contain 2 bu 1 oh ^ 4 nf.^V* f"^'" ^"*° ^^« will be required ? ^ * "l*- ^^""^ ' ^^^ «»any bags anf eaVr- ^ainef 9^ t,rjrS:tlS\'f """• average 4i ears of corn If it t' „,« '""\ •*'" '^'1 "" »» fart, whit is the prSuce of t!;:''l!.5 !*?_?' .Til »° """^o » wuoiiei J ~ """ ' ■ "* •** **•' *** Coiittt per U8 PRACTICl. V. Mbaburkm or Weioht. (1) If John buy, by Avoirdupoin wuight, J 2 lb. of opium at 37$ cents per ounce, and sell by Tri>y weight at 40 cents pf$r ounce, should he gain or lose by so doing, and how much i (2) A person purchases goods at the rate of $1.80 per pound, Troy weight, and sells them again by Avoirdupois weight ; at what rate must he sell per ounce so as exactly to reimburse himself ? (3) By multiplying a certain weight by a whole number the result is 8 lbs. 20 srAins Avoirdupois weight, and by multiplying tho same weight by another whole number the result is 8 lbs. 11 o/.. 16 dwt. 10 grs. Find the weight. (4) A row of cent pieces is laid from Toronto to Hamilton. Find their weight, the distance being 30 miles 1 fur. 1 per. 9 in. (5) Find the value of 500 times the difference between an eighty-fourth part of 2^ cwt. and a thirtieth part of 1 cwt. qr. 3 lb. (28 lbs. to the quarter). \ XIK. Practice. 166. Practice is the name given to a method by 'which we find the cost of any number of articles of the same kind when the price of one is given, or the cost of any quantity of goods of mixed denominations, when thf^ cost of a single unit of any denomination is given. I. Simple Practice. When the articles are of the same kind or denomination. Ex. (1). Suppose I have to find the cost of 2478 articles at 3s. 4:d. eadK Knowing that 3.i. 4d is one-sixth part of £1, I reason thus : If the articles had cost £1 each, the total cost would have been £2478 ; 2478 .'. as they cost ^ of £1 each, the cost will be £— 7r-> or £413. The process may be written thus : 38. 4d. is ^ of £1 £2478 = cost of the articles at £1 each. __ _x q_ A J -~-V. wS «*V kJSa "XifSa t?*i«W«** t'MACTICl. 149 Ex. (2). Find tbA coat of ^qa^ ,. , '•ach. ' °' *^*^' articles at £2 12*. 9c/. **• » * "HO*. 1448 10 n *2 . . . . Not. a u ^^^^ ^^ »-<'«»t«t£2l2.. Orf. each tlu!!?'""-^ shorter method would be to taice the p^rt ^^^^^ (^). Find the coat of 425 articles at £2 1%,, ^ Since £Q ta aj (which i, ,,, ot\T): the'ih„«Sr„";r,'"""*''" ^^ -"o !»• s^. J-^. 8rf. i« A of £1 I 1275 = 35 8 4 = !«• 8ol«. at £Z 12 ac. it 12 X 1 AC. 6 2 ro. it 1 ro. is 20 po. is 6 po. is 1 po. is of 1 10. of 2 ro. of 1 ro. of 20 po, of 6 po. 12 16 8 2 d. « 6 3 15 » 375 - 4 876- the rent of 1 aor«. .12 ac. . 2ro. . 1 ro. .20po. . 5 po. 1 po. £41 19 3 '75 «= the rentof 12a. 3r. 26p. vnuTP' ' ''r"*"'''^ ^ ^'"P'">' ^**'^'»'"^»« instead of ^lgar fractions to express the result of the division after the line of pence. ut vision Examples. (Ixxxvii) 6 ac. 3 ro. 4 po. 4| yd. at £10 per rood. 10 «o. 3ro. 26 po. at £2 18.,. 10.^,/. perac?e. C3 c": 3 a'r 174Tb ' rM> '^^ ^''^ '^^- ^^'- ^^ ^^ oj cwt. d qr. 17| lb. at 12 guineas pc r cw^. f« *''• a ^"\^£?- **^ ^^ «"'"«*» P^-r acre. 16 oz. 6 dwt 20 gr. at £3 17*. 6d. per oz. 25 ac. 1 ro. 10 po at £42 2«. 4d per acre. (m{ l^o^K ^.^'•- ^1 ^^' «t ^'22 8... per cwt. (10) 319 cwt. 3 qr. 16 lb. at £2 12.. 6d per cwt. Invoices and Accounts. ^}2^: t"" ^r''^''^ '^ ^ statement in detail, sent by a io 1 e But' Tl "' ''^ *""^ '^« ^«^^« -^ d-liveTed the goods.^ ' ^"""'''^' ^««^"Pti«»' -"d price of V,^Ztru^!:7A % '^T'^^''^ «««t ^>y the Seller to the ■ nd d?l f l^ T *"'"' ^^ '*^"^''' «*>«^i»g *he totals ^ I8i oouroxjm nuonon Bach Mparat« artiol« or amount in an Invoice or an Aoootint ii called an Itim, A DiTAiLRD AOWUNT is a full Btattotnent, M«nt by the Seller to the Buyer at the end of a term of credit, show- ing the dates of delivery, the qimntitii>s, description, pnoM, and sum total of the goock (l(»livRred by the Seller to the Buyer during that term of credit. When an account has been made out it is rendered^ i,e.f «ent in to the Buyer. Spbcimen of an Invoice. John Smith, Esq., Toronto, June 20, 1880. Bought of J. Jones A Co. , 21 Front St. 6 lbs. of Tea at 76 cts. . Bibs, of Loaf Sugar, .at I2h ota 2^ lbs. of Butter. . . .at 30 cts. . s cts. 3 75 1 00 75 5 60 Specimen of an Account. John Smith, Esq., Toronto, July 21, 1889. To J. Jones & Co., 21 Front St. 1889 June 20 To Goods, as per invoice June 23 To *' '•' July 3.. To " " July 12. To " " 6 7 3 2 ^ 19 cts. 50 80 60 27 17 lot or Ml nt by the [lit, show- ISC ription, the Seller rendered, 0, 1889. ront St. s ots. • • 3 75 • ■ 1 00 • t 75 6 60 1, 1889. ront St 5 7 3 2 ^ 19 cts. 50 80 60 27 17 X>KPOir!fD FRAOriCI. Bpmtnm of a iMaihA Account. 153 John Smith, q. t( June 20 " 20 •' 20 2a 23 3 3 12 I i2| 12 Toronto, Juiy 21, 1889. To J. Jonei n9fifiR PROBLKMa igg men work J !«,: liull^'ir^S 7;^;;""™ " '^' '"-' ' ^ I^^Smoe 27 ...en can d„ the work i„ (14xiO) hour,, or 140 1 "'an can do the work in (27 X 140) hr ■ ■ ^'' ■"«» "an do the work in ^ililio i ' o, . . Now 316 hours have tobe dist.butei\,„ali; over 45 dav. • . . the number of ^^ they work each daj^^^ or ^ ' cosfof f2^b s" ^ ^^'' '^ ''' '"'' ^•'•^^' ^h^t ^i" fee the Since % c' tea cost $5.60, a. of tea coats «l^, or 80 cents. . . 1^ lb. of tea -nat 12 x 80 cts., or |9.60. Ex. (9). If 9 horses can plough 46 aereq in o «^«* • t.me, how ,„a„y acre, can l^^i.orL pV^Ti'n" rhe^'s^ni: , , ^ 6»»^u wiiit) piougn 4o ac 1 horse can in the given time plough ^ ac ' •• ^^ ^^™«« «a'^ »n the given time plough"* 12X46 ac. Fx Cin\ fir orMJac. of land in five duvs hTr'" ?'""^" " "^'""■" l''''""'^ plough it inlh^i^d'aysr "^ '""" ""' *"« '■^1"'-^ '» In 6 days the land can be pIo«ghed by 16 ho«.es ; Tn \ A^' I'u t"** **" ''^ P'""Kl>«l by (5 X 15) horses • In 3 days the land can be ploughed by' »-f-', or 25 horses. t^ j,iven, m the detnand a niacrnitiKlft of ««^ i • j" ffivpn an/^ +K, • "'"oUiiuae 01 one kind t^ °wh:r'kind^ hTtTrCnT^Tt"! fr^'^f °^ •solution contains the nm..„it",dl; J tt ''' ^'"^ ?^ **•« :™«ed that ai tke .^^/t'^Ll'';!??^^'^'-. *^. -"ng oj „Mcn the magnitude is reguired in th^Z^ ■'". 150 PROBLKMH. Thus in Ex. (10) the order of the supposition is changed, and the magnitude, J 5 horses, put at the end of the line, because wo have to find how many horses will be required in tlve demand. Examples. (Iszxiz.) (1) If a man walk 62 miles in i days, in how many days will he walk 93 miles ? (2) If 12 men reap Beld in 4 days, in what time will 32 men reap it i (3) If 350 acres of land cost $61250, what will 273 acres cost? (4) How many men can perform in 12 days a piece of work which 15 men can perform in 20 days ? (5) The rent of 17 acres ia $297, what is the rent of 86 acres ? (6) If a man walk 116 miles in 8 days, how far will he walk in 14 days ? (7) A farmer sells a flock of 270 sheep at $240 a score, what does he get for them ? (8) A servant 8 wages being $108 per arnium, how much ought she to receive for 7 weeks ? (9) A clerk's salary is £191 128.Jdd. per annum ; what ought he to receive for 60 days' service ? (10) A ship performs a voyage in 63 days, sailing at the rate of 6 knots an hour ; how long would it take her if she sailed at the rate of 7 knots an hour ? (11) A bankrupt's effects are worth $860, and his debts are $4300 ; what does he pay in the dollar ? Note II. — To one of the magnitudes in a supposition there is a corresponding magnitude of the same kind in the demand, and these magnitudes must be expressed in units of the same denomination. Ex. A man walks 1 m. 1 fur. 7 po, in 20 minutes ; how long will he take to walk 41 m. 2 fur. 12 po.? Here 1 m. 1 fur. 7 po. =367 poles, and 41 m. 2 fur. 12 po. = 13212 poles. Then hfc walks 3G7 poles in 20 minutes ; he walks 1 pole in ^^^- mm. , 20 36 7 he walks 13212 poles in 13212X20 367 min. , or 720 min. PROBLEM8 INVOLVING FRACTIONS. 157 Examples. {lxx:xix.)-~Continued, (14) If 3 owt. 3 qr. cost »27, what will be the cost of 2 owt ? Jt of 9?wt!f ' "'■ ^ "■• °°" ^' "'• «i''- ""at i. the 170. Problems involving Fractions. Ex. IfJ of an estate be worth $1500, what is the value of i of the estate ? Since f of the eatate is worth 81500, ^ of the estate is worth $Mp j .*. the estate is worth $^^i^ or $3500. Hence ^ of the estate is worth ^lii^oo q, $2800. Examples, (xc.) (1) If f of an estate be worth $7520, what is the value of If or tne estate ? ^lom^ ^rf"? !r' ^, °^ */?^P ^'^'^ ««"« t «f l^is share for !tPl260 ; what is the value of the ship ? ^iWlld^' ^^ *^* ^"^^^ ^^^' ^'^*' ^^^ "^"'^^ °*" ^ ^"y ^*^^ (6) A man walks 18 m. 2 fur. 26 po. 3f yd. in 5A hours How long does he take to walk a mile and a half ? (6) A gentleman possessing ^^ of an estate sold f of — of his share for $603. 12| ; what would i of ^ of the estat! sell for at the same rate ? «qTn^l'^%''*"'*^l?^o^ol^ ^'^*- °^ goods for 60 miles cost mone ? ^^ ''^^' ^® ^'^^'^^ ^^"^ *^® ^*™® (8) What is the value of ^-^ of ^V of a vessel, if a person who owns f^ of it sell i of ^ of his Ihare for $1400 ? ?^ T^^f" *^® ^^^^'^ o^ gol'^®^- In 8 days 1 aero can be ploughed by |~J| horses. In 8 days 64 acres can be ploughed by -g^-^j^" ^<*'^*'®*- .•. the number of horses required is 6. Ex. (2). If 35 bushels of oats last 7 horses for 20 days, how many days will 96 bushels last 18 horses 1 35 bushels last 7 horses for 20 days. 1 bushel lasts 7 horses for |f days. 1 bushel lasts 1 horse for 7X20 35 days. 96 bushels last one horse for iMl^iao days. 3 5 96 bushels last 18 horses for 9«2LT>^ days. 18X35 /. the number of days is 21^. Examples, (xci.) (1) If 40 acres of grass be mowed by 8 men in 7 days, how many acres will be mowed by 24 men in 28 days ? (2) If $60 will p&y 8 men for 5 days' work, how much will pay 32 men for 24 days' work ? (3) If a regiment of 939 soldiers consume 351 quarters of wheat in 168 days, how many soldiers will consumb 1404 quarters in 56 days ? COMPLEX PROBLEMS. 159 (4) U two horaet eat 8 bushels of oats in 16 davs how many horses will eat 3000 quarters in 24 days ? ^ ' iRf?^ ? * t*"'®' ''^^'''^ ^^2 ^^'^ *he carriage of 3 cwt for in 16 dlys.™®" ^" ^^^ '"^ * ***y*' ""^^ '""^ ^^^ 18 »»«" «»"> aJS} 5^'^ fjny buahels of wheat will serve 72 people 8 days, when 4 bushels serve 6 people 24 days ? ^ (9) If a man travel 150 miles in 6 days when the days are * ^^PL"^^ *M ® carriage of goods weighing 5 cwt. 2 or 12 lb for 160 miles come to $16.70, what will be theclari or r^'2*lb'"';r^^'"-^?^' "^ *^^ «*"^«' '^'^ weightgfcwt cwt. ? ' "^""^ *^''**"'^' *^®'^ ^^^"« 112 Ibs.*^in the ai) If $120 pay 16 labourers for 6 days, how manv labourers at the same rate will $270 pay for 8 days ? ^ dats\\V«l%rhnw ^ ^'"T™' ^ ^'^"^^ ^""^^ ^»y' ^«r 10 aays, cost ^1.20, how many burners may be lighted 4 hours every evening for 16 days kt a cost of $21.60. (13) If a travelling party of three spend $190 in 4 weeks how long will $475 last a travelling party of five a? the s^m^ ,(\*> ^ !* «««* ^120 to keep two horses for five months what will It cost to keep three horses for eleven months ?' (15) If it cost £29 7s, 6d. to keep 6 horses for 6 weeks how long may 3 horses be kept for £20 lis. 3d. 1 ^^^^ ^K^,^^^ ^*" '®*P » field of 12^ acres in 3i davs working 16 hours a day, in what time can^ men reap^a fiTd of 15 acres, working 12 hours a day ? ^ (17) If 858 men in 6 months consume 234 quarters of 7^^1i ^o *"*"? ^o*'*^™ r" ^^ ^«^"i^«d for the consump- tion of 979 men for 3 months and a half ? ^ (18) The wages of 5 men for 6 weeks being $315, how manv weeks will 4 men work for $231 ? k * » "ow many JVLl!^^LT7.1^ ^'^'V- L uoes _., or 17 bONE IN A CBRTAm TIMR. 101 •■ together thoy do ^+,^. or HI daily; heydothewo.Wi„|||day,or2yd;ys. anda „'^3|fr'.lt5r-^%°/r^^ - 4 hours; ^ can A do it aloneV ^ ^ ^'^ ^^ *^°""- ^^^ ^>»-^ ti^ne ^ and i^ can do ^ in an hour ; ■ *7J»^.of ^'s strength, assisted by ^ and ^ t + iV »i ai^ hour; «u oy xj and 6^ can do • two men JT ^r^ ^ '"" ^° ^''^ ^" *" h«»r 5 • . two men of A'b strengt h can do i- + 4. 7 • or 4& _7 , ./ / + iV - /ff in an hour, or ^f - ^, or Jjf, or ^ in an hour ; •"• ^ can do ^ in an hour ; _ .-. A can do the work in 6 hours. part mfJbr" : / iLTtm'b! ^^^''" - « '--. the J " 1 noui will be represented by } what timeVill they flir t whe^r^,rP^''*'™'y- ^^ time 1 0"!" u When they all run at the aamo ••''•"y^'HIf^'O'i^Unl minute; ■ th. cn'iu**^ *" riff in A of a minute ; .. they hll the ve„el ia %» minute., or 9^ minute,. wiU tfe hath^l. fuH ffi ;U\t-; neL^e^ One pipe 611, ^ of the bath in a minute. ■ t a Xt':! ™""'"«' A-=^ - Tfe of the bath i, filled •■• the bath i, filled in 120 minute,. Examples. (xoU.) hou«. In what'ti^=.ll;; d"„ t t^Zi :oT^±^r 4 lea PBOBLBMS tMLktlVQ TO CLOCKS. (3) A can do a piece of work in 36 days ; B can do it in 40 dayi ; C can do it in 46 days. In what time will they do it, all working togethor 1 . „ , j ^ . (3) A and B can reap a field of wheat m 3 dayi ; A and C in 3j[ daye ; B and in 4 daya. In what time could they reap it, all working together ? (4) If three pipes fill a vessel in 6, 8, and 12 minutes respectively, in what time will the vessel be filled when all three are open at once ? , « xv n y (5) A does {fs of a piece of work in 14 days. He then calls in B, and they liuiah the work in 2 days. How long would B take to do the whole work by himself ? (0) A does a piece of work in 3 hours, which is twice the time B and together take to do it ; ^ and could together do it in 1^ hours. How long would B alone take to do it ? (7) A can do a piece of work in 27 days, and B in 16 days ; A works at it alone for 12 days, B then works alone 6 days, and then finishes the work in 4 days. In what time could C have done the work by himself ? , «^ . x (8) A cistern i* filled by two pipes in 18 and 20 minutes respectively, anA emptied by a tap in 40 minutes ; what part of It will be tilled in 10 minutes when all are opened at the same instant ? 173. Problems relating to Clocks. The minute-hand moves 12 times as fast as the hour- hand, and therefore in 12 minutes the minute-hand gaips 11 niinnte-divisions on the hour-hand. Ex. (1). Find the time betweei- 3 and 4 o'clock when the hands of a watch are together. At 3 o'clock there are 16 minute- divisions between the hands ; we have therefore to find how long it will take the minute-hand to gain 15 minute-divisions on the hour-hand. The minute-han^ gains 11 minute-divisions in 12 minutes ; 1 minute-division in ^f minutes ; 16 minute-divisions in ^ \Y" min. ; .-. the time required is — ~ min. , or 16^ min. past 3. Ex. (2). At what time between 2 and 3 are the hands of a clock at right angles to each other 1 When the hands are at right angles there is a space of 15 minute-divisions between them. PROBLBMS KBLATIKO TO OUXIKH, 163 Hence »i„ce at 2 o'clock thero are 10 minote-ditrinion* between the hands, we have to find how long it willtako The minute-hand gain. 11 minute-divUion. in 12 minute. ; 1 minute-division in |« minute. ; 25 minute-diviMona in ^~*^ min • .-. the time required i. t-JOJ ^j,,^ ^, ^^ J^] ^^ '^ of f*rlolt\/\?i T^'^ *'?'' ^'*^'^" ^ ^"^^ ^ »«^ t^^e handH ot a clock at right angles to each other? minute-hand haa over-taken the hour-hand • s«r.nnHlv after the minute-hand has passed thrWhand ^' betwel; tlThrn/ '''^r^ *'^^''. '^^^ ^^ minute^ivisiona oeiween the hands, we have to find : First, how long it will take the minute-hand to gain JO - 15, or 15 minute-divisions on the hour-hand 30 + Ifi" n/'^^"^- ^°^^it' ^^^ t-ke the minute4iand to gain JO -f 16, or 45 minute-divisions on the hour-hand. The process in each case will be similar to that in the 'Sn. p'tT ^'* '""^ '^' '""''' ''' ^'^ "''"• ^"^ f ), "^f • (f ^' T'"'^ ^^? ^^^'^ between 7 and 8 o'clock when the hands of a watch are opposite to each other. When the hands are opposite there is a space of 30 mmutes between them, and at 7 o'clock there is a space of 35 minutes between the hands. ^ Hence in this case we have to find how long it will take oxamDleriTflT''^^ ^'''^^f ^ *^"* '"^ *^« P^^^^^ing examples, and the result is Sy^r min. past 7. Examples, (xciii.) At what time are the hands of a watch tngatha. i.«f,„^^ the hrurs of "^ • -~^^ii 164 KXAMINATIOH I'APMIS. (1) 4»&dS. (2) Oand7. (3) "and 10? At what time are the hands of a watoh at right angles to each other between (4) 4 rnd 6. (5) 7 and 8. (6) 11 and 12 ? M < it. < Me are the hands of a watch opposite to each <« ' w I . »an (7) land 2. (8) 4 and 5. (») 8and0? m ' Exdminatum Papers. T. (1) If for a given sum I can have 1200 lbs. carried 36 inileM, how many pounds can I have carried 24 miles for the same sum ? (2) If I of a ship be worth l$13C63, what is the value of f of the ship ? (3) A silver tankard weighs 1 lb. 10 oz. ; what is its value when a dozen spocms, weighitig 3| oz. each, are worth $64 ? (4) A man spends $01.60 every 35 days, and saves $400 a year. What is his annual income ? (5) When the income-tax is 6(/. in the £, a man pays £15 7«. 6d. What is his income ? II. (1) A man's income is reduced from $2720 to $2640. 6«i when he has paid his income tax. What is his tax on the dollar ? (2) If 10 horses and 132 sheep can be kept 8 days for $202, what sum will keep 15 horses and 148 sheep for the same time, supposing 5 horses to eat as much as 84 sheep ? (3) A man receives 75 cents in the dollar of what was due to him and thereby loses $602. 10. What was due to him ? (4) If 15 men can perform a piece of work in 22 «iays, how many men will finish another piece of work 4 times as large in a fifth part of the time ? (5) If 72 men dig a trench in 63 days, in how many days will 42 men dig another trench three times as great ? III. (1) The wages of A and B together h 7^ days amount to the same sura as the wages of A alone for 12f days. For iiuw iimuy uayb wUi lue aUiVi ya,y buo wngos kh £j oiv/uo i XXAMINATION. fAVntUi. 165 man pays (2) If 1(X> mon can i)erfonn a piece of work in 30 dayi now many men can perfor.n another pieoe of work thrice ai Urge in one-fourth of the lime / (3) If 6 men or 7 women can do » pi«o0 of work in 37 davi how Ions will a piece of work twice a« great occupy 7 men and 6 women ? •» f / • (4) Two peraoni A and B, finish a work in 20 days, which B by hmiwlf could do in 50 davs. In what time could A hnish It by hmwolf ? How much more of the work ia done by A than Ji i (6) If a oiafcern when full of water can bo emptied in 15 nunutea by a pipe, and when empty can be filled by another m 20 muiutet; If the cistern be full, in what time can it be emptied by both pipes being opened at the same time ? IV. (1) A and 5 can do a piece of work alone in 15 and 18 days respectively ; they work together at it for 3 days, when B leaves, but A continues, and after throe days is joined by and they finish it together in 4 days. In what time would do the work by himself ? (2) If a man can do treble, and a woman double, the work ? ^^,*" *"® ®*"*® **"*°' **"^ **^"8 w»"'d men, 16 women, and 18 boys take to do double the work which 7 men 12 women, and 9 boys complete in 250 days ? ' ' (3) A and B walk to meet each other from two places 100 miles distan*. A walks 6 miles an hour and B 4 miles an hour. At what point on the road do they meet, and at what two times are they fifty miles apart from each other ? ^^r^^^j^ 7^^"^^ ^^^^^^ ^' ^^ minutes too fast at noon on Monday loses 3 min. 10 sec. daily. What will be the time indicated by the watch at a quart- r past ten on the morninff of the following Saturday ? * (6) A watch set accurately at 12 o'clock indicates 10 min to 6 at 5 oclock. What is the exact t.me when the watch indicates 5 o clock ? If it indicated 10 minutes past 5 at 6 oclock, what would be the exact time when the hands indicated 5 o clock ? V. (1) A labourer agreed to work for 60 days on this con- dition : that every day he worked he should receive $2, and At fk??---?-^^-™ ^^^^'^^ ^^^"^^ P^y ^>i^^- ^"''^is board. -£= ;..c e-puaiiuii VI tiio time uv loutjivtju $9z. How manv days did he work ? ' 166 II1IIPI.R iNTiuirr. (5) A phset of work can b© done in a day of 11| hoars by 2 m«ti, nr ft wom«n, or 12 hoyi ; in what time ouuid it M dune by 1 man, 2 women, and 3 boya together ? (3) A oittem haa two suppIyinK pif>ea, A and J3, and a t»p. C. When the oittern ia empty, A and ti are turned on, ahd it ia filled in 4 houri * then B in shut and tur «od en, and the ctatem ia quite emptied in 40 hours ; when, L stly, A ia •hut and B turned on, and in GO hours afterwaraa the oiste**n is attain filled. In what time oould the cistern be filled by each of the pipes, .4 and It, singly t (4) A clock is set at 12 o'olook on Saturday night, and at noon on Tueaday it is 3 minutes too fast. Supposing its rate regular, what will be the true time when the dock strikea four on Thursday afternoon / (6) A contractor engages what he considers a sufHoient number of men to execute a piece of work in 84 days ; but he aaoertaiua thai throe of hia men do, respectively, 4, |, -vnd I less than an average day's work, and two other » J and ^ more, and in order to complete the work in the 14 weeks, he procures the 'help of 17 additional men for t' « 84th day. How much less or more than an average day's vtork on the part of these 17 men is rti<}uired 7 ' XXI. Simple Interest. 174. Interest is that which is paid by one who borrows money for the use of the money. The money lent is called the Puincipal. The Borrower agrees to pay at what is called a certain Rate of interest, which is usually reckoned by the ,\im paid for the use of $100 for 1 year. Thus, if I borro'V $500 for 1 year, and agree to pay $25 for the use of the money, I am said to borrow at the Bate of 5 per cent,, per a?mum ; that is, I agree to pay $5 for the use of every $100 in the loan at the end of tlij year. The sura mado up of the Principal and Interest dxlded together is called the Amount at the end of the time for which the nr/oney is borrowed. 170. The solution of questions relating to Intei-est depends on precisely the same principles as those ex- plained in the last Section, and it is only because oi tfif" MMPUI IJITMRWr. in? neeesiity of explaining UjchniomI t«rni« thai th«re it any • on u, wpsmfci, thii or the gucoeeding Seotioni froiu ^^ - ' we rmmm al)out the question j Wh T pav 4a ^ I pay fur the hire of 4 hoi^ for 5 month., if the hire of 3 hortet for a month ? »o <1< '• <4K)u about the quettion : 16 for the une o f|100 for a year ? ^^ ' •^ ^ -»» " i p»y Ex. (1). To find tJie Simple fnteregt on |2676 for 3 years, at 8 per cent., we reason thus : Interest on ^100 for I j'ear is $8 ; on II for 1 year is $j go ; on $2675 for 1 year i»$a«7axt . 100 * on $2675 for 3 years is S«2«il£21 wtAlfi • Too •«•« ( .'. the interest is $642. Hence we derive the following rule : Multiply the Principal by the Bate per cent, and the The piocess stands thus : $ 2675 8 21400 3 $642.00 .*. the interest is $642| ^ Ex. (2). Find the interest on $3200 for 2 veaj^ ^d I iiioutiis at / 1 per cent. 1: 168 8IMPLB INTEREST. Since 7 months is iV of a year, the time is 2^^ years. Tnterest on $100 for 1 year is $7. 50, t( (i $1 for 1 year is r^^ it $3200 for 1 year is ^11£PJLL±? . $3200 for 2,\ years is 2^^^ X $32ooj ; m how many years will it amount to i$1406.25 ? (8) The sum of $500 is borrowed at the beginning of the year at a certain rate per cent. , and after 9 months $400 more 18 borrowed at double the previous rate. At the enu of the ^f ""t, .? '2*®f^^*^ """^ ^''^^ ^'^^^^ ^» ^35 ; what is the rate at which the first sum was borrowed ? K ^Va ^n \T^ ^'''^^ ^^y^ ^^^^ *^6 interest on £243 6s. 8d. be £4 Os. lOd. at 6} per cent. ? (10) If £556 17s. 6d. be loaned for 125 days and then amount to £565 18s. 9d., what was the rate ? (11) The interest on $8000 for one day is $2 ; find the rate per cent, per annum. (12) Bought 5000 bushels of wheat at $1.25 a bushel payable m 6 months ; I immediately realized for it at $1.20 cash, and put the money out at interest at 10 per cent. At the appointed time I paid for the wheat ; did I gain or losr by the transaction, and. how much ? (13) The interest on a sum of money at the end of 61 veart 18 thre ,ths of the sum itself ; what rate per cen. ^s charged ? ir » (14) A suni of money at simple interest has in 4* year?, an^^unted to $735, the rate of interest being 5 per cen* , 3r ar . m; what was the sum at first, an 1 in how man, years more will ifc amount to $1140 ? ^ (15) The interest on $1805, loaned on May 13tb. at 51 Per cent, per annum, is $37 905 ; on what day was the raonev returned ? " j 172 PARTIAL PAYMENTS. PARTIAL PAYMENTS. 177. A partial payment is the payment of a part of the amount due on a note or lK)ud. When partial pay- ments are made they are endorsed on the note or bond. To compute the interest on such a note proceed according to the following rule : Compute the inte'^est on the principal to the time of tha first ]>aymenty and if this payment exceed the interest then due add the interest to the principal, and from the sum take the payment ; the remainder will form a new principal, ivith which jyroceed as before. Bat if the payment he less than the interest, compide the interest on the prijicipal to the time rvhen the sum of the payments shall first equal or exceed the interest due ; add the interest to the principal, and from the sum subtract the sum of thti ^myments, and treat the remainder as a new principal. This rule proceeds on the ground that in all cases the payment should be applied first to the interest due, then to the principal, and that the principal remains unchanged until the sum paid exceeds the accrued interest. Ex. (1). $4000. Toronto, June I, 1872. Two years after date I promise to pay William Smith, or order, four thousand dollars, for value received, with interest at 7 per cent. Richard Paywrll. On this note were the following endorsements : Sept. 15, 1872, four hundred and fifty dollars. Dec. 15, 1872, fifty dollars. Mar. 1, 1873, five hundred dollars. Jan. 1, 1874, one thousand dollars. What remained uue June 4, 1874. % / ' 17S 00 89 89 PARTIAL PAYMENTH. Principal on interest June 1, 1872 o^/ Interest to Sept. 16, 1872 ........'..■'.'. ' Amount "IT: Less Ist payment.. ........[ i Remainder for a new principal . aonqn oo I Interest from Sept. ll to Dec. lo,' 1872; i,l ^''^^ ^' I «W. 44, winch exceeds ihe payment f Interest from Sept. 15, 1872, to March 1? 1873. ' 117 20 Amount •• ^^ j^ ^" Leas the sum of the 2nd and 3rd payments 650 00 Remainder for a new principal.. . ijqioq no Interest from March 1, 1873, to Jan. 1,1874 .'.V.: l 180 4? Amount "^qooT^ Less payment Jan. 1, 1874.;;;:;:;;.;;; ;; 1006 00 T . I^emainder for a new principal. . . !»2qsi fift Interest from Jan. 1 to June 4, 1874. : : : : ' : : : : ; * 70 94 Balance due June 4, 1874 ^455^ Examples, (xcvi.) (1) $1500. Hamilton, Jan. 1, 1877. ordPr' Fn' ""^^t' 'Y^ '^? P'°""^^ *« W S- White, or received ''^ '^""'''' ''''^' '''''''''■ ^^^^^ George Brown & Co. The following payments were made on this note • ISnlm' ^^^^' ^^^^' '^""^ ^^^ 1^77, 400; Sept. 1, What was due Jan. 1, 1878, interest at 6 per cent. ? (2) $3500. I^ELLEviLLE, March 15, 1876. to pay Win Smith, or order, three thousand five hundred uollars, with interest. uuuieu James Jones & Co. 174 COMPOUND INTSBI8T. tlndorstd ns followa : June 1, 1876, $800; Sept. 1, 1876, $100; Jan. 1, 1877, $1560; Mttfch 1, 1877, $300, x What was due May 16, 1877, interest at 6 per cent. (3) $1200. Toronto, Oct. 15, 1859. One year from date we promise to pay James Smith, or order, twelve hundred dollars, for value received, with interest. Wilder & Son. Endorsed as follows : Oct. 15, $1860, $1000 ; April 15, 1861, $200. How much remained due Oct. 15, 1861, interest at 6 per cent, t XXII. Compound Interest 178. Compound Interest' is that which is paid, not only for the use of the original sum lent, but also for use of the interest as it becomes due. The interest on $500 for 1 year at 4 per cent, is $20, If then $500 bo lent at Compound Interest for 2 years at 4 per cent., the interest for the first year is $20. Now, as the borrower has to pay for the use of this $20, the interest for the second year must be calculated on $520. Hence interest for second year >=$^~^ =$20.80. To put the matter in a more simple way, we have supposed the bori'ower to retain the interest due at the end of the first year, but the reasoning will be the same if we suppose the lender to receive the interest at the end of the first year, and to put it out immediately at the same rate of interest. 179. We may calculate Compound Interest by the following rule : Find the interest for the first year : add it to the original principal'' call the result the Second Principal: find the OOMI>OUlfD INTBRKSI'. 176 intered on this /or the aet.md year: add it to the mcond mncrpal : call the result i,. Third PHri^pal /Jt mtered on this for the third year^ and so on Ex. (1). Find the Compound Interest on $7500 for 3 years at 4 per cent. $7500 Ib the principal for ihojirat year. 4 $300.00 The interest for the^r«< year is $300. Add this to the original principal, $7500. Then $7800 is the principal for the secwid. year. $312.00 The interest for the sec(md year is $312. Add this to the second year's principci, $7800. Then |8^ 12 is the principal for the third year. 4 $324.48 The interest for the third year is $324. 4a /. Compound Interest required is $300 + $3l2 + $S24.48=$936.48. If the Amount at Compound Interest be required add $936 48'''^^ P^ncipal, $7f:00, to the Compound Inte'rest, Then amount required = $8436. 48. Ex. (2). What is the Compound Interest of $250 for 2 years, at 7 per cent. ? $250 Principal for Isfc year. $250 X 0.07 = 17.50 Interest for the 1st year. fto£j^, Kr. ^ r.^ ^^^- ^ Principal for 2nd year. $267.50 X 0.07 = 18-725 Interest for 2nd year. w ^ T> • - , ^286-225 Amt. at Comp. Int. for 2 yrs. First Principal 260.00 ^ $36,226 Com. Int. for 2 years. 170 COMPOUNt) INTKkfS»T. I Examples. (xctU.) Find the Compound Interest on * (1) '$375 for 3 years at 5 per cent. h) $564 fur 4 years at 7 per cent. (3) $1154.37 for 4 years at 5 per cent. (4) $740 for 6 years at 7 per cent. Note I. — When the Conjpound Interest is required for 3^ years, it is usual to Hnd the Compound Interest for the whole of the fourth year, and take half the result as the Compound InCorest for the half year. This really implies that the interest is paid half-yearly, but the approximation does not differ much from the exact truth. 180. The process for finding the amount of a sum at Compound Interest may be presented in a very brief and neat form as follows : — If the rate of interest be 4 per oent., Amount of $100 at the end of 1 year is $104, of $1 at the end of 1 year is }g^ of $1. Hence it follows that : Amount of any mm at 4 per cent, in 1 year = \%ji of thc*t sum. Again, Amount for second year = \^^ of amount for the first year ; .'. Amount of any aum at 4 per cent, in 2 years = Ig^ of jg^ of that sum. Suppose, then, we have to find the amount oi $540 in 3 years at 4 per cent., Compound Interest. The amount is }g^ of \%^ of \^^ of $540 = $540 X (1-04)8 = $607-420. From the above example it will 140 noticed that the amount -of $1 for a year at 4 per cent, is raised to the power indicated by the number of years for which Compound Interest is to be^calculated. Hence we have the following rule :— - . * To find the sum to which any principal will amount if put out to Com,2)owid Interest at a given rate in a given COMPOUND INTBRBRT. m number of years, Jind the amount of $1 for a ymr at the given rate, raim that mm to the power which is denoted by the givmi number of yearM, and multiply the result by th§ number of dollars in the given principal. Ex. (1). Find the amount of $850 in three years at 6 per cent., Compound Interest. The Amount = $850 x (I'CXJ)" - $850 X 1 191016 ^ - $1012aGa The Compound Interest «= |!1012-36 ~ |850 - $102 36. No IK II.— When the number of years is large, the student is recommended to employ the contracted method of multiplication, explained in Art. 111. ? Interest may he payable either yearly, half-yearly, or quarterly, or at some other stated period. In finding the Compound Interest on $2000 in two years, when the interest is payable half-yearly, at 5 per cent., we reason thus : 5 per cent, for \ year = 2h per cent, half yearly, 2 yeari. = 4 half years. Hence we have to find the Compound Interest on $2000, for four timen of paiimenf, at l\ per cent. The Amount = $2000 x (1 025)* = $2000 X 1 10.38127 = $2207 '626. The Interest = $2207 025 - $2000 = $207 025. Ex. (2). What principal will iiniount to $1012-363 in 3 years at 6 per cent, ? Principal x ,10G)« - $1012-363 .-. Prineipal = ,'H^» Examples, (xoviii.) (1) Wh^t is the Compound Interest on $1000 for 2 years at 6 per cent., payable half-yearly ? ' (2) What is the amo' t of $200 for 3 years, at 6 per cent., payable half-yearly. ^ ' 178 fwmawt WORTH AND Mtooinrr. (3) Find ih« Oonip«und Int«rMt on f675.7& for 3| jMli, mi 6 per oenl. per aunuiu. (4) A money dealer biirrowed 91000 fur d yeere at 6 pe^ oent. intero*t ; and loaned the taitie in luch a manner aa to emnpimnti the iuteriMt every utunths. What profit did he make in 8 yeara by thia proceeding 7 (ft) Find the dineronce in Compound Interest on £{M)00 for 2 ywira at 4 per oent., aoeording aa it ia reckoned yearly or halfyeurly. (6) What is the difference between the Compound Interest on $4(K)00 for 4 years, and on fdOOOO for 2 years, the rate in both *<:aMi>s btiing 5 per oent. ? (7) A and 3 lend each $248 for 3 years at ^ per cent, one at BimpU), the othfer at Compound Interest ; And the difference of the amount of interest which they respectively receive. (8) What sum at four per oent. , Oompound Interest, will amount in 2| years to |ll«9807728. (0) What Riim will amount to $27783 in 3 yean At 6 per cent., Compuiind Interest. XXIII. Present Worth and Discount. 181. Suppose A owes li $105, to he paid at the end of a year. If 4 be disposed to pay ott" the debt at once the sum which he ought to pay should be such that, if put out at interest by B^ it will amount at the end of a year to $105. Suppose, further, that B can put out bis money at 5 per cent, interest; then, if he put out $100 at intereatj thia is the sum whicli will amouut at the end of a year to $105. Hence $100 ia the sum, which A ougbt to pay at once, and thia ia called tlio Present Worth of the debt, and is evidently such a sum as would, if put out to interest for the given time and rate, amount to tbo debt. The diflFer- encrt between the Debt and the Present Worth, which is in the case under consideration $5, is called the Discount. Discount is therefore the abatement made when a sum of money is jiid before it is due and is equal to the interest on the Present Worth of the debt. Ex. (1). Thus, to find the Present Worth of $1781.40, due 4 years hence, reckoning interest at 6 per cent iiao hM fo- ill pftMiit Worth « •I hiM for iU Pr«««iit VVortl- • m 9100 ... $\n : ..•1781.40 h«« for iet Prei«i»t Wirth f V!£JL*aiiJlS -11484.50 .'. IVwent Worth M

^ Pnotograpnic Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 4? ,v ^^\ '^^\ ;^ t80 PRE8RNT WORTH AND DISCOUNT. .-. $50 ia the interest on $50JLi», or $460 ; .'. $460 is the sum required. Again, the interest on $46 for 2 years is $6 ; .'. the interest en $45 for 1 year is $| ; .'. the interest on $1 for 1 year is $^^ ; . . the interest on $100 for 1 year is $——/ = ^^5 *» .*. the Rate is 6f per cent. Note I. — From the above it will be seen that the discount on any sum is the Present Worth of the interest of that sura for the same time and rate : thus $45 is the Present Worth of $50 for two years at a certain rate per cent. Ex. (5). If $20 be allowed off a bill of $420 due in 6 months, how much shall be allowed off the same bill due in 12 months? $20 is iJhe discount off $420 for 6 months ; $20 is the interest on $^00 for 6 months ; $40 is the interest on $400 for 12 months ; $40 is the discount off $440 for 12 months ; ;. $^^ is the discount off $1 for 12 months ; . $i2o_>ilo is the discount off $420 £or 12 months. Now $120J<^ = $38 A ; 440 ^^ .'. the Discount required is $38^. Note II. — The student will observe that the Discount is not proportional to either the time or the rate. Ex. (6). If $15 be the Interest on $115 for a given time, what should be the Discount off $115 for the same time ? $15 is the interest on $115 ; .-. $15 is the discount off $130 ; •"• ItVtj is the discount off $1 ; • i»ii6 X 15 is the discount off $115. 130 Now $ "f X IS = $13^ 130 Tff ' the Discount required is $13^^. PRIUFMT WORTH AND DISCOUNT. 181 Ex. (7). If $10 be allowed off a bill of /|110, due 8 months hence, what should be the bill from which the san J sum is allowed as 4 months' discount ? 110 is the discount off $110 for 8 months ; •'■ f iJl ^® *^® interesf on $100 for 8 months ; .-. JlO 18 th« interest on $200 for 4 months ; .-. $10 18 the discount off $210 for 4 months ; .'. the sum required is $210. Ex. (8). Find the I Vesent Worth of $842.70 for two years, at 6 per cent., Compound Interest. • Im £""»P<*"«d Interest on $100 for 2 years at 6 per cent IS $U. OO. .•. $112.36 has for its Present Worth $100 ; $1 has for its Present Worth $-i 100 -i2'86 ' .'. $842.70 has for its Present Worth jfe ^^^-^oxioo . ^ 112' 36 > = $750. .'. Present Worth required = $750. Examples, (xcix.) Find the Present AVorth of (1) $5520, due 4 years hence, at 5 per cent. (2) $84.70, due 2 J years hence, at 9 per cent. (3) $615, due 1 year 4 months hence, at 7 per cent. (4) $1120, due 10 months hence, at 5 per cent. (5) £618 2s. 6d., due 3| years hence, at 4 per cent. Find the Discount on (6) 1^^036, due in 9 months, at 8 per cent. (7) $1884.30, due in 31 years, at 10 per cent. (8) $637.50, due in 5h years, at 5 per cent. (9) £1165 163. 3d, due in 2^ years, at 6 per cent. (10) £252 19s. 3d., due in 9 months, at 4^ per cent. (11) Find the Present Worth of $6945.75, due 3 vears Hence, reckonmg compound interest at 5 per cent. (12) Find the Discount on $245.25, due 1^ years hence, at Ot per cent compound interest, payable quarterly. ofi^Poati*!?^^"'"^" *^°^P*« f 19-3125 in payment of a debt ot ;tP^U.,^J, d^ie m 12 months, m consideration of being paid at once. What rate of discount does he allow ? 182 PRK«KNT WORTH AND DIJH^OUNT. (14) Find the Present Worth of a bill for SI 127. 10, drawn Jan. 1 at 4 months, and discounted Feb. 20 nt 10 per cent. I>er annum. (15) 'rtie Discount on ^275 for j? certain time is f 25 ; #hat is the Discount on the same sum (1) for twice that time, and (2) for half the time ? (16) A tradesman marks his goods, with two prices, one for cash and the other for credit of (> months ; what reh ion should the two prices bear to each other, alh ing interest at 7i per cent.? If the credit price of an article be $33.20, wnat is the cash price ? (17) If $98 be accepted in present payment of $128, due some time hence, what should be a proper discount off a bill of $128 which has only half the time to run t (18) A certain sum ought to have $20.80 allowed as 8 months' interest on it ; but a bill jfor the same sum due in 8 months »t the same rate should have $20 only allowed ofTas discount in consideration of present payment. What is the sum and the rate per cent. ? 182. The Discount of which we liave been treating; is called Mathematical Discount, or True| Discount, to distinguish it from Practical Discount," ol^ which there are two kinds : (1) The deduction made by a trader, when an account is paid to him before the time when he proposes to demand payment. It is then calculated as interest on the account. Thus, if a trader gives notice on his bill that he will allow 10 per cent, discount for immediate payment, and if the amount of the bill be $25.60, he deducts $2.55, and the customer pays him $22.95. (2) The deduction made by a lender of money from the sum which he proposes to lend. Thus, if a borrower binds himself by a bill to pay $100 a year hence, and a discounter advances money on .the security of this bill, at the rate of 5 per cent., he gives to the holder of the bill $95 and takes the bill. True Discount is the Interest on the Present Worth of a debt. Practical Discount is the Interest on the debt itself. Hence Practical Discount is greater than True Discount. PK««ST WOHTll AND DMXIVm. JSS o" Oct." ti^^ : '"i-f »• -'"Id H« ,K„ni„all, due "lw..y8 rPc^koned s< that , I m H"'"'"'- "">»'l>8 are l«> AtJU». 186 (4) What muit be the rate of intereit in order tha* ih- ducounfc on #10292 Dav&hlB At ih »«,i « i vLo , ** *°* be 1872 ? ' payaoie at Ui and of 1 year 73 dayt may (6) A iradesnian who is ready to allow R ««.. «««* annrm Compound Interest foT^ldy nfone^ iarkedt TI ^'^f: /?*• ^^-y^^rn. U he charge «1 26 in his Ml ihit ought the ready «»oney price to have been ? ' ' ^* m. ' iJllt^d?,'\tT' tr"^^^^^ ^^'"^ ^« immediately for «7fiAf» j^'^ months afterwards he sold the land for #7500 on a credit of 12 months, with interest Mnn«S being i,t 6 per oeut., what is the 8peiukt<^^'s S kt th« Tn J of the 12 months' credit, at which Le he Jetu'rns fhe $mm {4) A merchant bought 43 cwt, 3 or of Rn^ar if an ox :r^;'"' ^5 immediately sold at'#7 peTfw'tTon^a cre^dU Silted ?nth«t:^^i:*^ J^!f P»^«^«^'^«r« note for the amoun .tXnfmlke''^"^ '^^ ' ^^^ °"^^- ^^'^^ ^^^^^ ^^^^ ^^e 5 n?r ninf *^* ^'^■*"*^ ^°''*^ "^ ^1^ *J"« n years henc afe «iven sum i.'I^mrf""!/ ""^^"^ that the Mcount of the £ven sum IS equal to the interest of the Present Worth for *he same time and at the same rate of interest ? able'^ hllHearrJ'"^-^ k'* ^}^^ ^* ^ ?«' *^«»*- "^t^^est, pay- SyrLns m^onS; ""'"a^- ^""^ """"'^^ ^'' "'*«*•«»* in equal C^revT^^Lo'nth ?^ "^ "^^^"^« ' ^^- "^-1^ -ght h2 to minth! aH/nei c«nf '"*"'"'* °" ^^^ ^^a. 4rf. for three S- m? fnr it ? ^"i^o ^^"^ *""""*' '"^ ^^^"'^l to the discount « A83 for 16 mos. at 3 per cent, per annum. / IV. ^''(1) How much may be gained by hiring monev at K y *^ r^r^^ohh- ^'2?'.*^"? "* « "^"»^^«' alirngVe prefei? rnidis^^lTntT *^ '^ "°''^"^' ^^ ^^^-'"« « Her iiUe^est^of f 'fnn?°f ^^*''"^!J *^o^ "^^P^^ «"^\ay bd paid together, without injustice to A or B. When great exactness is demanded, interest must be added to the sums paid after they are due, and discount subtracted h*om the sums paid before they are due. But in practice the following rule is sulficiently accurate : MulHph/ each deU hy the numher of days [or months] after ivhirli it is dun : add the results tofjether : divide this sum 1)1/ the sum of the dehfs : the quotient will he the number of days \or months] in the equated time. Take i,he following examples : Ex. (1). If $300 be due from ^ to i5 at the end of 5 months, and $700 at the end of 9 months, when may both sums be paid in a single payment without unfairness to A or to ^ ? i..r , r ., • i. J i- 300 X54-700X0 N?:imber of months ni equated tnne == --300+700 _ 7 800 "■ 1600 _ 12. ~ 10 = 7^ .-. the whole amount of the debt should be paid at the end of 7f months. The principle on which this solution depends is, that tne interest of the mouey, the payment of which is delayed beyond the time at which it is due, is equal to the interest of that which is to be paid before it becomes due. KgDATIOH or VAYUUtn. Jgy a .'".l!it",u.^th„"l^'''r ^^^j' ■">•" 2« "."..th, after II iM (me, una tho mtorest on t for that timi: i. h.« but 8r00 in paid H mnntlm before it is due ami th. fiOO X 6 = 3000 COO X 7 4200 «00 X 10 - 8000 il^O 1900 ) 15200 8 .'. "Jhe equa eJ time is Q months. :ru^'u^!''Z^^\^Tl^ '' ^"^ -^ '^"«'^ approximation. H 1« 1; ^ -^ ^^^''^ ^'^ equitable when the various ^::;;fef;ro eirf^f^KL t till' ^^ It is also to T <> observed that the error involved in f hi« .rea;:^,rted til "' ''"""' "'' "^ ^«^-- ^" - <— =t„ 1 77- ' """.Payments made before they are due Examples, (ci.) What is the equated time of ^^^a)J250 due 4 months hence, and 8350 d:ie 10 months Find the equated time of anf«a*(^"Sut"6'4St„or' ^""'^ ".ontha hence. tlild'o? ?mo„fh*S'^t'S " "r 'ff-^-J^tely, 8600 at X inontn, IHOO at the end of 7 months, and the 188 EQUATION 07 AOOOUlin. rernnimler fti the end of a year. At what time might the whole debt fairly be paid in one lum / (4) A grocer ought to receive from a ouatomer $50 at the ©nd of 2 njositha, $30 at the ejid «f 4 months, and $20 at the end of 6i months. What would be the proper time for receiving thu wholo lum together i <6) A debt ia to Imj piiid as follows : One sixth now, and one-sixth eve^-y three months tiiitil the whole is paid. When might the whole debt be paid at once 7 (6) If 9450 be due in 10 months, and $260 be due in 13^ months ; rind the sum which, if paid now, would b« equivji- lent to the whole dubt at the equated time, interest at 4 per cent. i<{7) There is due to a merchant $800, one-sixth of which Ik to bo paid in 2 months, ono-third in 3 months, and the remainder in 6 mouths ; but the debtor agi ob to pay one- half down. How long may ho retain the other halt so that neither party may sustain loss ? ^ (8) A sold goods to B at sundry times, and on different terms of credit, as follows: Sept. SO, 1868, $80.75, on 4 months' credit; Nov. 3, 18G8, $150, on 6 months' credit; Jan. 1, 1869, ^WK).80, on 6 months' credit; March 10, 1809, $40.60, on 6 moi.ths' credit; April 26, 1809, $60.30, on 4 months' credit. How much will balance the account June 2, 1809 1 (9) A owes B on the Ist of March the following sums : £140 due on 20th of April, .il20 due on 14th of May, £380 due on 15th of June. On what day may B pay theee debts together ? (10) M buys goods of N^ and has 6 months' credit* from the date of invoice. The goods are delivered on 6 diflferent days, to the following amount : iJlOl 14«. lOd. on Aug. 8, ;ei44 2«. lOd. on Sept. 5, £303 18s. lOrf. on Sept. 18, £757 0«. Sd. on Nov. 13, £123 ILs. 6d. on Nov. 28, £123 11«. 6(i. on Dec. 5. On the 13th January, iV, who desires to receive all the debts in one payment, reckocs that this paym«3nt should be made in 100 days. Show that this is approximately correct. EQUATION OF ACCOUNTS. 185. Equation of Accounts (also called " Averaging of Accounts" and "Compound Equation of Payments") is the process of finding at what time tlie balance of an account can be paid without gain or loss to either party. ■QVATIOW OF AtXfOimil. 1^ ^« "I II, niKi 18 what om owm thi» othur Kx. Dr A in Aooount with B. Cr 1877. •/ttn. 1 Feb. 4 Mar. 10 To Mdto it It 1600.00 000. (K) HOO.OO 1H77. jFab. 10 I Mar. 4 ByOaah Jan. 1, 500 X 0- o Feb. 4,(i()0x34=>2040() Mar. 10, 800x08-64400 Jiao.oo 600.00 iWO) 74800(39A 6700 17800 17100 700 |F6b. 10, 1000x00=. Mar. 4, CKK) x 22 * 13200 1600) 13200(8i 12800 400 Feb. 9 to Fob. 18 bete td'r' °Th " ""l", ^'» been a losa of interest to the c «Ht de an 1 7 "^ ''"? ■ng gain to tlie debit side Now 1„ tl,! Tl ™""'P'>n'l- bo one of oauitv wo (i,w^ i i ' • '" ^""leDient should .lays. * *"° "'''"""^'' """ «1900 would in 9 If «1900 gain a certain interest in 9 days and «J00 w,Il ^..n the same interest in l^ss, „r 57 d,y^ '..e'f;^t^/::!;rJ K;:%Tr"j;:x^^ t'^ints another day. ^ upwanis, it then • 190 EQUATIOK or A«}0001IVM. Hence w« have the following ruU : Fird find the ^uai^l iimf for Mieh itidf. of th^ aemmni mtimraiely. Theti inultiply thf (nuMtnt due. fm thai tido which falhtim vvmt hf/ the %uwbet of daif^ Itehiwm thf dateg of the equated titnfn, ivid dividf* ihejmnluH hy the tHilaucr nf the acroHfd. The iiiioticnt will hf the numlm- ofdaf/H to Im counftfd ton^kHufrornthn hkimt date whrn the mnaller $ide of the aretmnt fall* dun riRiT ; and BACKWAHD tohen the larger side fallt due firht. Examples. (oU.) (1) Average the following account : l)r. J. Hughes in account with S. Adams. Or. 1875. July 4 Aug. 20 Aug. 20 Bdpt. 25 Deo. 6 To Hiilatice •• J.Idae. t4 815.58 178.26 .387.20 418.70 1876. Aug. 10 Sept. 1 gnpt 26 Nov. 20 Dflc. 1 By Oaiu. Mdiie. Oach. (t (« laio {AS 07»').<)0 512.26 1«1.76 100.00 (2) When is the biilanc.3 of the following account due? Db. a. B. Conron. Ob. 1877. Sep. 12 Oct. 15 Nov 18 Dec. 1 To Mdse at 30 days $927 30 Oct. 10 By Oaah it (I (I it (I 30 00 30 342.76 212.13 175.50 Nov 20 Nov 30 I $600. 00 .'iOO.OO 250.00 (3) When did the balance of the following accounts become due, the merchandise items being on G months 1 J. Green iu account with Adam Miller & Co. Or. Dr. 1876. March 1 " 20 AprU U " 30 June 15 July 18 Aug. 30 Sept. 25 To Mdse. $720.75 815. .30 587.80 300.00 625.25 560.00 684.00 «( <( 365.30 1876. April 1 May 30 July 20 Sept. 25 «' 30 Oct. 30 Nov. 20 By Cash Mdse. Cash (t Mdse. (( (< $700.00 569.89 500.00 100. CO 750.20 329. 9<) 600.00 vVkRAOM AMD PBHUIitTMH. I»t rJ^7 ^'*"»«" »°«» Percentagea. J^S;:^"'^ '"^ """»""'•"• ^'■i.h ,„.j. occur, Examples, (c. .n^'^sl."" ""* •""""> "' 'f'i. 3«i, ^78? O.'lor'MJ, 2Bi, PERCENT AUJS8. "emte h,„,B«lf fo, the trouble of coUectk;^ * ' ""°"" m both cases. K^^^ups wouia ^e u»o same Ex. (j). How iiiuch is 3 per cent, on $1479 ? Since ^100 yields $3, «1 yields «^g„ 11479 yields fi^^«, or ^.37. 192 COMMISSION AND BKOKEHAOE. ■ t ■ 1' I "I i Ex (2). The number of boys in a school increases in a certain period from 125 to 180; what is the increase per cent.? On 125 the increase is 65. On 1 the increase is ^^i^^. On 100 the increase is > ooMl, or MIP or 44 ; .'. the increase is 44 per cent. Examples, (civ.) (1) Find 5 per cent, of $2400 ; 8 per cent, of 3475 horses. (2) How much per cent, is 26 parts out of 75 ; 178 out of 8900 ; J out of *? (3) The population of London proper decreased 33*11 per cent, between 1861 and 1871. In 1861 it was 113,387 ; find what it was in 187 1^ (4) How much per cent, is 9d. in the pound ; 12^ cents in the dollar ; $3 in every $20 ? (5) Find the number of which 21 is 7 per cent. ; 750 is 3^ per cent. ; 215 is •005 per cent. COMMISSION AND BROKERAGE. 188. Commission is the charge made by an agent for buying or selling goods, and i%^OTerally a percentage on the money engaged in the transaction. Brokerage is the charge made by a broker for buying or selling stocks, bills of exchange, &c. In computing Commission, care must be taken to calculate it on the money actually employed in the business. Ex. (1). My agent has purchased wheat, on my account, to the amount of $18768. What is his com- mission at 1| per cent.? The Commission on $100 is $1.75 ; IS Jl-75 100 $18768 is $lH «8xi-75 ^ ^ 100 = $328.44 Commission required. Hence the following rule may be used : Multiply the given sum by the rate per cent, and divide the product by 100., and the result ts the Commission or Brokerage. OOMMISHION AND BROKRRA(JK. I93 baiance in si,r How ;li*,£ll"v'. -"^ '"^«"' «"> Since the Commiuaion on $100 is 81 GO out of $101.50 he can in veat $100; ' $1 •« <( « 100 . 101.60 ' $1827 $1927X100 lot. 83 - $1800, Bum required. On $101.50 the ComHiisaion is $1.50 ; ** $1 ** " ■ o l.go 101 .50 = $27. •• the Commission required is $27. Examples, (cv.) Find the Commission on (3) «2364 at i per cent. (4) «375 at -6 per c'ent se^ iroi;r.ro^2"i:r^^^^^^ He . to in silk, after deducting hi« r.^^ ■ ' • ^"^®f* *^® P'^^ceeds af/i"^.Sr.!!°??° ,'" ? •'-'^» to invest with instruction baIano;7^G„1eS;;SSr'^lPrwirt'?."^«»'"*'^« vested, and how much will be ihe brokerl^" ^ *""" °'"» '»- 194 INSURANCE. INSURANCE. 189. Insijkanoe is security guaranteed by one pArty, on being paid a certain sum, to another against any loss. The PREMIU".: is tht» sum paid for Insurance. It is always a certain per cent, of the sura insured. The Policy is the written contract of Insurance. Note. — As the Premium is always so much per cent, of the sum insured, it is found by the same rule as Gommisaion. Ex. What sum should be insured at 4 per cent., on goods worth $2940, that the owner may receive, in case of loss, the value of both goods and premium ? Since the premium on $100 at 4 per cent, is $4, $96 worth of goodo would be covered by $100 ; .-.$1 " '♦ " $VV>: .'.$2940 '* *' *♦ ft204oxioo \ 96 = $3062.50, sura required. Examples, (cvl.) (1) What will be the premium of insurance on the furniture of a house valued at $2500 at ^ per oent. ? (2) What is the premium for insuring a cargo, valued at $21350, at 3^ per cent. ? (3) For what sura should goods worth £4384 Os. 3d. be insured at 2^ per cent, that the owner may recover, in case of loss, the value of both goods and premium ? (4) A person at the age of 40 insures his life in each of two offices for $5500, the premiums being at the rate of 3J and 3|^ per cent, respectively. Find bis annual payment. (5) What sum must be paid to insure a cargo worth $26400, the premium being 1| per cent., policy duty \ per cent., and brokerage | per cent.? (6) A trader gets 500 barrels of flour insured for 75 per cent, of its cost at 2^ per cent., paying $80.85 premium. At what price per barrel did he purchase the flour ? (7) A company took a risk at 2| per cent. , and re-insured f of it in another company at 3 per cent. The premium received exceeded that paid by $10. What was the amount of the risk ? /Q\ k - 1- ! i _C 1 - .. r J -i fkJI i A_ cover f of its value. The premium was f.71.26. What were the apples worth ? TAXE8, DUTIES OR 0USTOM8. TAXES. 195 190. A TAX is a sum of money assessed on a person in proportion to th« value of his property, amount of income, etc., for public purposes. ^ ^ ^' In order to levy a tax persons, called assessors, are hrst employed to ascertain or ppraise the value of all the property taxed. When this has been done the sum to be levied is apportioned amongst the property-owners according to the value of the property of each. lliX. A certain town has property valued at $1,560,000 and levies a tax of $23,400 ; what should B pay whose property is valued at $7500 ? f J' "»« Since $1560000 pays $23400 ; ••^lpay8$Tlim^; .'. $7500 pays $Lfi »o^2.3 4oo I 5«0000 = $112.50, tax required. Examples (cvii.) ftinmnn" ? ®l^°°l ^f*\''" containing property valuod at l8Srand «2ir V ^v." ^r^i *" P^y ^^' *«^^^«r's salary o mam etc ^ pfn/ J '^ ^^^ \''" exj^^nA^^i in purchasing maps, etc. Find ^'s tax, who owns property, real and personal, worth $5400. ^ ^ ^' ^ MIa^I^''^^^'' *^^"^ ^®^0^ worth of prcperty pays a tax of $144.50. Find the rate on the dollar. P^^^ P^^^ » t»^ JP^^tl^ property of Toronto be valued at $75000000, Sl40(f fiL f).P?! if T ^^T^ ^^^^h «f P'-^Perty, pays i$l 400, find the total tax levied in Toronto. J'* f^j- (4) In a certain section a school-house is to be built at an expense of $8400, to be defrayed by a tax upon nropertv valued at $700000. What is the rate of taxation to cover both the cost of the school-house and the collector's com! mission at 4 per cent. ? DUTIES on CUSTOMS 191. Duties or Customs are sums of money required by Orovernment to be paid ( nearly all imported goods Ihe law requires that all goods entering Canada shall De landed at certain places where CrTSTOM HousE= a— - established. These pla^jes are called Ports of Entry^"^ 106 HTORAOK. Duties are of t.wo kinds, ad valttrem and ^peeijfr. An Ad Valorem duty is a certain percentage on the cost of 'the goods in the country from which they are imported. A SpKCiFid duty is a sum computed on the ton, yard, gallon, etc., without regard to the value of the goods. NoTB. — Ah ad valorem du'ioH are perceutagea, they are computed iu the same mauner as ComniiHHion, etc. Ex. Find the Specitio Duty on 760 lbs. of Sulphuric Acid at J cent per lb. Duty on 1 lb. is A cent. " 700 lbs. is'-^i'." cents - ^3.80, duty retjuired. Examples, (cviii.) (1) What is tha Duty on 7035 Ibn. of tea, valued at $3600, at 6 cents per lb. 1 (2) Find the Ad Valorem Duty on an invoice of books which cost $1700 at 5 per cent. (3) Find the Specific Duty on 750 gallons of wine worth $2150 at 60 cents per gaHon. (4) Find the Duty on 8400 lbs. of sugar worth 7i cents per lb., the specific duty being ^ cent per lb. and the ad valorem duty 25 per cent. (5) Paid $1602.50 duty on an invoice of cotton at the rate of 17^ per cent. What was the value of the cotton ? STORAGE, 192. Storaor is a charge made by a person who stores movable property or goods for another. It is usually reckoned by the month of 30 days at a certain price per bushel, cask, box, bale, etc. The owners of the goods pay for putting the goods in store, stowing away, and the expenses of delivery. When goods are received and delivered at the pleasure of the consignor, the dues for storage are usually deter- mined by an average. Ex. What is the cost of storage, at Ic. per bushel per month, of wheat received and delivered as per - S ' '■^k.- KXAMIXilTION PAFKRS. m Accou.U cUmd October Hint, 1877, Account ok 8torao« or W.. rat, reokivkd and deuvbkeu Koii AcuoDNT OF John Jones, Toronto. Dati. July, 1877. « (I August (( 2 11 IG 21 10 15 ,20 September 5 110 16 Bal. on hand Oct. 2 oohed. 200 350* 'm Dell, verod. 160 200 100 300 '456" 60 1260 200 1J50 100 Bni. ance. 200 60 400 100 500 60 200 800 100 Day« Products. P 6 •5 20 5 5 6 5 17 1800 260 2000 20(X) 2500 250 000 1000 J 500 1700 30)13000 433J 1260 I 1250 433j^ X 1 cent - H.Z^, The Btoroge of 200 bu. for 9 days + 50 bu. for 6 days +400 bu. for 5 days + 100 bu. for 2o'day8 + 500 bu. for^6 dZ dtv« 4- 'Jnn K ^ ,<^*y« + 200 bu. for 6 days + 300 bu. for 5 fcV * ^"; ^'^ ^^ ^^y* '^ ^^"^ «*»"« a« th'^ storage of 13000 bu. for 1 day, or of 433^ bu. for a month of 30 days The storage of 433^ bu. at 1 cent per bushel is $4.33^. Exammation Papers. I. (1) If a grocer's pound weight is 10 drams too light, find his gam per cent, from this source alone. . » > "" 8228 wh^flflJ'^'u^^fu* deduction of 6 per cent., becomes ^228 what should It have become if a deduction of 6i- per cent, had been made ? * ^ minLfii"*! *'\%lf "^ ""^ *^^ ^^°*^« imported when an ad valorem duty of 17A per cent, produces $637. (4) The population of a city has increased by 5975 persons between 1860 and 1870 ; this increase is 12^ per cent.^of the population of 1870. What wrh the population in imo ? (5) In 1850 the populatioi. of a town was $7600 ; in 1870 ^98 ■XAMINATIOK PAPBRfl. it was found to be OlOtJ. If the increase per cent, during the finit decade v/as the same as during the last, whut wm this per cent. ? ' n. (1) A, after paying an income tax of 1^ per cent, on all his salary over $400, has $1739,00 left. Find his salary. (2) A town has levied a tax of $7340, which sum includes the amount voted for building a bridge and the collector's fees at 3 per cent. What was expended on the bridge i (3) The average of ten results was 17*5; that of the first three was 16*25, and of the next fo'ir 16 5 ; the eighth was 3 less than the ninth, and 4 less than the tenth. What was the tenth ? (4) The gross receipts of a railway company in a certain year are apportioned thus : 40 per cent, to pay the working expenses, 54 per cent, to give the shareholders a dividend at i.he rate of 3h per cent, on their shares ; and the remainder, $42525, is reserved. What was the paid-up capital of the company * (5) A can do 5 per cent, of a piece of work in 3 days of 10 hours each ; B can do 7| per cent, of it in 6 days of 8 hours each. If both men work together and the whole work be worth $85, what does each get ? III. (1) A cargo is valued at $7005.45 ; the premium of insu- rance is at the rate of b^ per cent., policy duty at \ per cent., and commission at /g^ per cent.; what sum must be insured to cover the cargo and the expenses of insurance ? (2) Received and delivered, on account of James Smithy sundry bales of cotton, as follows: Received Jan. 1, 1877, 2310 bales; Jan. 16, 120 bales ; Feb. 1, 300 bales. Delivered Feb. 22,^ 1000 bales ; March 1, 600 bales ; April 3, 400 bales ; April 10, 312 bales. Required the number of bales re- maining in store May 1, and the cost of storage up to that date, at the rate of 5 cents a bale per month. (3) If the increase in the number of male and female criminals is 2| per cent. , while the decrease in the number of males alone is 7^ per cent.,, and the increase in the number of females is lOf per cent., compare the antecedent numbers of male and female prisoners. (4) A person takes a railway return-ticket for a month, payingf 25 per centr more for it than he would have done foi* a single ticket. At the end of a month he obtains an ex- RXAMINATION PAPERS. 19P tdnaion of time for a week by paying 6 per cent, on the monthly ticket. The wrhole sum paid is $10.60; find the price of the single ticket. (o> The paper duty was l^d. per lb. , and the weight of a certain book U lbs. The paper manufacturer realized 10 per cent, on his sale, and the publisher 20 per cent, on his outlay. What reduction might be made in the price of the book on tl:e abolition of the paper duty, allowing to each tradesman the same rate of profit as befo-e ? IV. (1) A merchant bought 37 yds. 2 qrs. of cloth at $4.87* per yard, and 40 yds. 2h qrs. of silk at 93f cents per yard For what sukn must the whole be sold to make a profit of 33^ per cent. ? (2) A commission merchant is to sell 12000 lbs. of cotton and invest the proceeds in sugar, retaining !» per cent, on the sale and the same on the purchase. Cotton selling at 7 cents, and sugar at 5 cents per pound, what quantity of sugar can the merchant buy ? (3) In an examination of 750 candidates, '22 on the whole do well, -34 barely pass, and the rest fail. How many do iveill, barely pass, and fail, respectively ? (4) Sold grain on commission at 6 per cent. ; invested net proceeds m groceries at 2 per cent, commission. My whole commission was $70. What was the value of the grain and groceries ? (6) A commission merchant receives 125 bbls. of flour from A, 150 bbls. from B, 225 bbls. from 0; he finds on inspection that A' a is 10 per cent, better than B'b, and G'a 5jAp per cent, better than ^'s ; he sells the whole lot at $7 per barrel, and charges 4 per cent, commission. How much does he remic to each ? * (1) A broker charges me 1| per cent, commission for purchasing some uncuri^nt bank bills at 25 per cent, discount: of these bills three of $10 each and one of $50 became worthless ; I dispose of the remainder at par, and thus make $520. What was the amount of bills purchased ? (2) A ^hole8ale merchant sent a quantity of goods into the country to be sold by auction, on a commijision of 4* per cent What amount of goods must be sold that his agent V J .*"'""' -■—■-•" ..ivii iii», avails is> tiii; cilUUuuXi OI vjpiyi'jL after retaining a commission of 2 per cent. ? 900 PROFIT AND LOUR. (3) A factor receiver f30O5C, and is directed to purchaae cotton at f 280 per balo ; he ia to receive 4 per cent, com- minion. ' How mHuy bHle« doei he buy ? v (4) Bold goods to a certain amount on a commission of 6 per cent, and having ruutitted the net prueeeUs to the owner, received for prompt payment ^ per cent, which amounted to 91C.16. What was the amount of commission ? (5) A man obtained an insurance for life at the age of 37, and died when 51 years old. The policy required annual payments during life at $2 8074 per $100, and secured to the jbeirs $1700.0^ more than the amount of all the premiums paid. What was the face value of the policy 1 XXVI. Profit and Loss. 193. If I sell *or $lOr} that *or which I gave $100, 1 gain $5 on an outlay of $100. If I sell^for $95 that for wh'ch I gave $100, T lose $5 on an outlay of $100. The following Examples will show the method of 8ol"ing questions relating to Profit and Loss, the prin- ciples laid down in Section xx being followed. Ex. (1). I sell for $6 that for which I gave $5. What is my gain per cent.? On an op.tlay of $5 my gain is $1 ; On an outlay of $1 my gain is $| ; On an outlay of $100 my gain is $1M, or $20 ; 5 /. I gain 20 per cent. Ex. (2). I bought some goods for $17. How must I sell them in order to gain 17 \^ per cent.? That for which I gave $100 I must sell for $1171| ; That for which I gave $1 I must sell for $iijj~-^ ; That for which I gave $17 I must sell for $^J2< . 20 I buught for ^li lao f or 90. FKOWT ANI» Lots. 201 Ex. (4). If by selling coffee at 1,. 7rf. per lb I kiM •» per cent., what must I aell it at to gain 5 per cent ? That which I lell at 9W. I bought for lOOd.; that which I sell at hi I bought for ^od. ; * that which I lell at lOd. I bought for l±2Llood., or 20d. Having thus found the cost price, we proced thu. : To gain 5 percent., that for which I gave iOOd. I must sell for 106d ; tliat for which I gave Id. I must Bell for {^d. ; that for which I gave 20d. I must sell for apilio?^ ^^ i , o . 0., 100 '' r thug : In the first case, that which costs lOOd. sells for 95rf. In the second case, that which costs lOOd sells for lObd.; that which sells for \)5d. must bring 105rf. ; Id. must bring ^d.; 19d, must bring llJUMd, or Is. 9c?., as before. ** Ex. (6) A quantity of tea is sold for 834 cents per pound; the gam is 10 per cent., and the total gain ia $48. What IS the quantity of the tea ? That which sells for $110 costs $100 ; " $1 " $ljg; " 83Jct8. " | 83^ X 100 . 110 * .-. the cost price per lb. = |J of $0.83 J ; .-. the gain on 1 lb. = ^ of $0.83 J ; but the gain per lb. x No. of lbs. sold = total '^ain, or ^ of $0-83^ X No. of lbs. sold = $48 ; ' No. of lbs. sold = $- (( (( 48 tV of $0-83^ .'. 633f lbs. is the qr.antity sold. 202 HlOrtT AND CMM. 194. When tea, ipirit^i, wine, aiu! «uch commotlitim ur« mixed it inuHt Ikj obgerved that ({UAiitit^ of ingrtidiontt -^ '.|URntity of mixture, cult uf ingredients <^ cost of mixture. Thus, if a nnxturo is ni»do of 1 gallon of ale at 8 ota. a gallon, 3 at 15 ctg., 4 at 20 ct«., and 12 at 7 cts. quantity of ingredionts-(l+3 + 4+ 12) gallH., or 20 galls, cost of nigreiients- (8 + 45 + 80 r84) cts., or ^2.17. If T want to k* »w what gain per cent. I shall make by selling this mixture at 26 cts. a gallon, I reason thus: 20 gall, at 20 ots. will sell for $5.20 ; .'. that for which I gave 82.17, 1 sell for $5.20 ; .-. $2. 17 gains ($6.20 - $2. 17) = $3.03 ; :. $1 gains $2;^ ; .'. $^00 gains $i"»-XAii>5, or $131). 63. y •! 7 .*. I gain 139-63 per cent. 195. In solving questions on Profit and Loss the student must be very careful to notice whether the gain is calculated on the selling price or cost price. Thus, it is sometimes said that a retailer's profit is 25 per cent , meaning that he gave 75 cents for an article which he sells for $1. His profit in this case is 33j^ per cent, on his outlay. Care must therefore be taken to express distinctly what is meant. The profit on a single trans- action or set of transactions by no means represents a net profit, as it is not charged with a variety of expenses which belong to the business in general rather than to the set of transactions in question. Ex. If 100 articles of a given kind can be made in a week out of $40 worth of raw materials, cost of labor, etc., being $10, fixed charges for rent, etc., being $250 a year, find (1) the cost price of each article, (2) the invoice price in order that a profit of SO per cent, on the cost price may be realized, the following allowances being necessary, viz,: 10 per cent, commission to agents on money received for sales, and 12 per cent, for bad debts, and (3) the amount o£ profit in a year. PROnr AND LONH. ^03 (1) The fixed chargei rauit be referred to the eiime unit of time aa the rest of the eetimate, viz. : 1 week «< 9^ « $^. Coit of 100 articlei - $50 4- •VV» |54'8077; .'. coat of 1 article =« |0 548077. (2) The profit on capital may be regarded aa part of the cost of nroduotion. It would be §o, in fact, if the money were borrowed at .'JO per cent, irrterent. l\0 per cent, added to fO-548077 gIvMM Hiaox 84B0TT *^ . 100 Again, the coinrniB»»< >n is paid on the money actually received ; to provide for it wo must take the ^ of | H80X'g4§0T7 oj. e iOXl» O X'»4a077 100 • ^ 9X100 ~' Next : 12 per cent, on bad debts means that 12 do not pay for 88 who do. To provide for it wo take ^ of the selling price. The invoice price will therefore be $ t pox 10X1 SOX • BJ, 8 077 q-. g • g()0 (3) To find the profit we must take j% of the cost price and multiply by lOO x 52. Annual profit =- |t3oxiooxgax'g4go7.7 « ^5. Examples, (cix.) (1) If I buy an article for J^;}.20 and sell it for |4, what is my gain per cent. ? (2) If I sell goods for $2240 and gain 12 per cent., what was the cost price i (3) If 375 yards of si'k be sold for $1960, and 20 per cent. profit be made, what did it coat per yaiu ? (4) If, by soiling wine at 17«. &d. a gallon, I losa 5 per cent. , at what price must I sell it to gain 16 per cent. ? (5) If, by selling goods for $544, I lose 16 per cent , how much per cent, should I have lost or gained if I had sold them for $672 ? (6) The manufacturer will supply a certain article at Ud. If a tradesman charges 2d., what profit per cent, will he make ? (7) A tradesman's prices are 20 per cent, above cost price. If he allows a customer 10 pei cent, on his bill, what profit does he make ? ^^(8) A tradesman's pricet are 25 per cent, above cost price, ir n« allow a cutttouter 12 per cent, on his bill, what profit does he uiake ? 904 «TOfTt« AWn ABAftM. (9) A inAii huf (roods at £23 ft*, bd. and a«II« th«rd »i £98 9«. i|(i. How much doet h« lote p«r o«nt.? (10) A kuAii buy* guodi at i'lS (k :W. aiid mIU th^ia again at Ull ]5i. 9|d H»>w iiiuoh do«m ho Iimm por cent, t (11) A man buyi gjMidii at thu rate of |9C iwr owt. , and Mill 2 ton», U owt. 3 r.r 12 lb». for ftMXKX How much Iiaa h« gaitiud or loat }>«r otmt on his <»utUy ? (12) If 8 iwr cent, b Kainod by aollinst a pioco of groan 1 for $4126.00, what would b« gauitKl p«r cent, by stiQiutf il for $4202? ^ / "• (*3) If 3 p«r cent. mor« Ikj gained by aoUiiig a horsa for •333 than by selling htm for $324, what must his original price have be«n ? (14) A grocer mixes 12 lb, of i^a at '2». ^d. per lb. with 4 Iba. at 3a. 2|d. At what price must hu noil tho mixture so as to gain ,'W^ per cent, upon his outlay / (16) Uik: many pouiuls of tobacco at $1.05 a jxjund must a tobaoconiat mix with 4 lb. at $1.30, ihat he may soil the mixti«re at $L66§ per pound, and gain 354 per cent, upon hi*» outlay ? (18) A spirit merchant buys 80 gallons of whiskey at $3.00 per gJlon, and IBO gallons more at $3.00 per gallon, and mix( « them. At what price must he sell the mixture to gain 8| per cei)t. upon his outlay ? (17) I mix 80 gallons of gin at $3.10 per gallon with 90 gallons at $3.41»5, a"d sell tho mixture so as to gain 10 per oent. At what price per ;^ullon do I sell it ? (18) A grocer buys two sorts of tea at 55 c^nts and 01 Ij cents per lb. respectively. He mixes them su ris to have 3 lb. of the dearer for every 1 lb. of tb.^ jheapi^r tort, and sells the mixture at 80 conts per lb. What does he gaiU per cent.? * *^ XXVIi. ..Gc jks fciud Shares. 190. The Gov ....lienu of a country, the authorities of a city, etc., often find it necessary to borrow money to cai^iy^on public works, etc. A loan is then contracted and the borrower pled^s the credit of the country, city, etc., to pay a fixed rate of interest on the sum borrow«*d until the debt is paid off The term stock is applied to any such government lrkn.n If \Aaf\ Ac^nni-aa i-t..^ ;r...^:i...l _£ .;i...i \ ^Z^ 1. —i _ *• »* jOittv "S^Ov^ ^>OJDpai3 ] ikif»li»^ Railway Ooinpai., Mnless he cun buy at 75. Henc.i if // wiHhod i.^' seil ).^omini.,n 6 per ceni. stock he would have to sell it at a (iMCoiint. Again, i'i money could only be loaned at 5 per cent Ji wouU bo .*Iile to seb ^100 of HUch stock for raoi-e than f 100 money, in this chhh he would sell at a l*re,uiu7n. Among the other causeH which determine the vrdue of stock, we u'ay mention its desirability of a safe invest- ment, commercial and political changes at home and abroad, etc. 197. Stook 16 at Par when it sells for its nominal value, as when $100 stock sells for $100 money. It is at a Premium when it sells for more than its no7ni7ial value. Thus, when $100 stock sells for $109 money it is at a Premucm of 9 per cent. It is at a Discount when it sel.'s at less chan its nominal value. Thus, vhen $100 stock sells for $85 niuney, it is at a dmount of 15 per cent. The purchase and sale of stocks are usually effected by means of a stock-broker, who is paid a -erta'n per- centage on all 8tooic that passes th.ough his hands. Thus, if stock is at 92^ and the broker chs-s/es i oer 206 STO0K8 AND SHARES, cent., the buyer will have to pay $93 ($92^ + 1) for $100 stock, and the seller would receive $92 ($92| - |) for it. 198. Stock is often named from the interest which is paid to the owners of the btock. Thus, the Dominion Government stock, paying interest at the rate of 6 per cent, is spoken of as the Dominion 6 per cents., or Dominion 6's. Consols are a part of the National Debt of Great Britain, so called from the Consolidation of the stock of various annuities into a joint 3 per cent, stock. The National Debt of Great Britain, which now amounts to abont 773 millions, has been incurred by loans made to the Stato by individuals. Interest is paid upon the main part of this debt at the rate of 3 per cent. The names of the persons who have a claim on the nation ft>r such interest, are registered in books ke^jii by the Bank of England on behalf of the Government. Such persons are called Fundholders ; the debt itself is often called The Funds ; and the interest, which is pay- able half-yearly, is called Dividends. Suppose ^ to be a Fundholder in that particular part of t;he National Debt called The Three Per Cent. Consols, and suppose the amount of the debt, which he is ack- nowledged by the Register to hold, be £5000, he is then said to hold .£5000 stock. A cannot demand the payment of 5000 sovereigns, or any smaller sum, from the Government, as a redemption of the debt, but the Government undertakes to pay him (or any one to whom he may assign his claim) 75 sovereigns, every half-year, that being the amount of interest on £5000 for half a year at 3 per cent. Now suppose A to be desirous of sellinj^- his claim to B. The value of the claim does not vary much from time to time in the '^ase before us, for England is known to be willing and is acknowledged to be able to pay the interest on her debt, and the security of the claim makes the Fuiidholder satisfied with a low rate of interest, STOCKS AND 8HARB8. 207 punctually paid and easily obtained. The value of ^eiOO Stock in Consols is at the present time (July 12 1877) 923, tiiat is, A can obtain ^92^ for each £100 Stock ^i'fJ'i o 1?^^^' ^"'^ ^^' ""^ *^® payment of 50 x £92^, or i/4bl8 15.V., can have the £5000 Stock, now held by A transferred to him, * ^'s name is then removed from the Register, and B'a name is inserted in it, and the process is called a Trans- fer. A IS said to sell out of the Funds, and B is said to invest in them. 199. United States securities are of two kinds- Notes and Bonds. United States 6's, 5-20 are bonds bearing interest at b per cent., and payable in 20 years, but may be paid in 5 years if the Government choose. When it is necessary to distingdish different issues of bonds bearing the same rate of interest, the year at which they become due is mentioned ; thus U. S. 6's, 5-20 of '84 • U S 6'fl 5-20 of '85. o*> u. <5. OS, Notes are of two kinds : First, those payable on demand, without interest known as United States Legal-tender Not^s, or "Green Backs." Second, Treasury Notes payable at a specified time with interest. Of this kind are notes bearing interest ai 7t% per cent., and known as 7-30's. These have all been redeemed. 200. Currency is a term used in commercial Ian- guage, First, to denote the aggregate of Specie, Bills of Exchange, Bank Bills, Treasury Notes, and other sub- stitutes for money employed in buying, selling, and carrying on exchange of commodities between various countries. *^eea7icv, to denote whatever circulating medium is used in any country as a substitute for the Government 208 STOCKS AND SHARB8. standard. It Pometimea happens that the paper cur- rency of a country becomes depreciated in value. Thus, when we read in Stock quotations of buyi»^g at 94} and selling at 95L it is meant that a broker would give $94| gold for 8100 t paper currency, and that he would sell 1100 of paper currency for $95j gold. Also, when we reaci that gold is 105^, it is meant that the paper currency is taken as the standard for the time being, and |105i of such currency would be given for |100 gold. 201. In Canada the liability on all Bank Stocks is limited to double the amount of the subscribed capital. On all other stocks the liability of shareholders is strictly limited to the amount of the subscribed capital. When all the Capital of a company has been paid up, it is often changed from Shares to Stock, because in the case of Stock, transactions can be carried on with reference to any portions of it, wL-:eas in the case of Shares, fractional parts of those Shares cannot be transferred. Three points must now be clearly marked : (1) Wo shall know the amount of money received by A for any given amount of stock, if we know the price of the stock at the time of sale. (2) We shall know how much stock can be bought by B for any given amount of money, if we know the price of the stock at the time of sale. (3) We shall know the amoimt of incoiine received by A (and subsequently by B) on any given amount of stock, if we know the rate of interest payable on the stock ; the income depenfiing in no way on the price of the stock. These three cases we now proceed to illustrate : Ex. (1). What is the value of $2,500 stock in the Dominion 5's at 98 J ? (I (( STOCKS AND SHARES. 200 The value of $100 ■took is $98, 25 ; II stock is $«-«^2_» ; ^ 100 $2600 stock is $ 2goo x 98. gg . 100 ' = $2456.25. Ex. (2). How much stock can be purchased at 921 for 1740 ? 2 For $02.50 I can purchase $100 stock ; for $1 " $-,r" ; ^^^'^^^^ " • «^^ii^, or $800 stock. • ^Li^l What annual income is derived from invest- ing f 3920 m the 6 per cents, at 98 1 Here, the owner of $100 stock has an income of $6. and to purchase this stock he must pay $98 ; v , u .'. $98 gives an income of $6 ; ••• ^3920 « $3«ii2ii, or $240. Ex. (4). What sum must be invested in the Dominion ^foAAo^^ ®° **'*^ ^ ^^y ^^^® ^-^ ^-^nual income of Since $6 is got from investing $95, (( (( (( .-. $1200 '« $120^15^ or $19000. Ex. (5). Wiat annual income is derived from $3550 stock in the U.S. 5's, 10-40 ? Income on $100 stock is $5 ; $3550 «' ^^~~, or $177. 50. This is merely a case of finding%he Interest, where the stock is the Principal. 1 -A ""■ (h} ,^°"g^* ®*««^ in the Bank of Commerce at lliv. The last dividend was at 8 per cent.; what per cent, did I make on the investment ? $120 gives an income of $8 ; 120 ' - o > :. the per -uit. required is 6f . (( <( 210 STOCKS AND HHAKR8. must Ex. (7). When stock is at 84, how much sto be sold to raise $462 ? Since $84 is got from selling $100 stock ; •*• ^1 " I'sV stock ; ••• UL«o stock ; or $550 stock. Ex. (8). What is the price of Ontario Bank stock when $6000 stock produces $r)880 ? Since $6000 stock is worth $5880, ••• «i " mu ; .-. $100 <* $ LOOJ< i, 880 or ^9g » (1000 ' ' .'. the stock was selling at 98. Ex. (9). By investing in the Dominioxi 6's I nmke 6i per cent. What was the selling price of this etock ? " Sinie $6.50 is got from investing $100, ''■II " «<^>^*^> .*. the selling price was 92^*^. Ex. (1 ). Which is the more advantageous stock to invest in 6 per cents, at 95, or 6 per cen:s. ab 87t|, and how much per cent, is it better Income for $95 in the 6 per cents, is $6 ; .-. Income for $1 in the 6 per cer.ts. is $^'^. Income for $1 in the 5 per cents, ia $^-|p, or $jJ^. We have now to compare the fractions ^% and ^VV Reduced to a common denominator those become XV .-. Income for $1 in the 6 per cents, is (X%\ - JOHr) of $1 better than in the 5 per cents. ^ ^^^^^ ,oW J"f^?^ ^"^^ ^^^ ^" *^® ^ Pe^ cents, is 100 x (^,'^A- ^\\) of $1 better than in the 5 per cents. ; Now 100 X cms - ''' ^ = -91. . . .per cent, required. Ex. (11). A person transfers £5000 stock from a 3 per cent, stock at 72, and invef?ts the orof^eftdp. in n. 4. -.ev cent, stock at 90, Find the difference in his income ^"* STOCKS AND 8UARE8. 211 mm' ^^ "*""' ^^^ '^"^^ ** '^^' ^"'^ «^** XX72X60) or Now his^rse income on the £5000 stock was i'«ooo^» 100 ' .... oriflSO. And his .ieamd income on the £4000 stock is £ipnoyi 100 ' . u • ,.• . or £160; . . he increases his income by £10. Ex. (12). A person invests .£1075 10«. in Consols a^t t?^''^":^ ^'^i' '^"^ ««"« "»* ^*»«" they are at y^t. What IS his gain, brokerage at i per cent, on each transaction? £{f^-^)!' *"""'^^ '^^'''^ "'""^^ *W + ^) is sold for .-. on £89f the gain is £3f ; .-. on £1 the gain is £^^ or £y>j»y ; .-. on £1075 10s. the gain is £1076-5 x ^j, or £43 10a. Ex. (13). A person invested in Bank stock at 894 and sold out at 103^, and cleared $397.60. How ranch did he invest, brokerage being J per cent, on each transaction ? Here what cost $90 is sold for $103^ ; ••. he gained $13.25 by investing he gained |1 by investing $~^^ .-. he gained $397.50 by investing ^^lli^oxoo^ op ^2700. 1 3 .25 Ex. (14). A person havinor to pay $3606^% two years hence, invested a certain sum in the Toronto 6 per cent city bonds, to accumulate jr^ierest until the debt be paid and also an equal sum next year; supposing the invest- ments to be made when the stock was at 99, and the hrst year's interest also invested in stock, and the price to remain the same, what must be the sum [invested on eac-ii occasion that, there may be just sufficient to pay the debt at the proper time ? 212 8TOOKH ANH HHAKK8. Every $99 invested will give ?!»] intereat ; .-. every |1 invested will give H,;'„ interest ; .. I sum invested will give $ surn x ^ intf/rest. interZt ^ """* "^ "^ invested will give $ sum x s% x /^ Hence at the end of the second year there were on hand the two sums invested. Two years' interest on the first investment - 2 x sum x {,% ; One year's interest on the second investment = sum x g"^ ; And the interest on the first year's interest » sum x A ^eoeXr""* ^^"^ '""' "" ^^ "^ """" X A X VV to meet ••• (2 + if + jsUt) sum - $3606^ ; ,3606^ sum = $ V^W = $1050. Examples, (ex.) Find the value of (1) $7645 stock in the 6 per cents, at 95. (2) $9800 stock in the 6 per cents, at 80, (3) $7650 stock in the 7 per cents, at 118i (4) ;fi3850 stock in the 3 per cents, at 95 / (5) -?572 10s. stock in the 3 per cents, at 91J. How much stock will (6) $8400 buy in the 4 per cents, at 75 ? (7) $3757.50 buy in the 8 per cents, at 125| ? (8) $994.50 buy in the 7 per cents, at 117 ? (9) £2199 buy in the 3 per cents, at 91f ? (10) £5527 lOa. buy in the 3 per cenli. at 92| ? What income is got from investing (11) $934.25 in the 6 per cents, at 101 ? (12) $4147 in 4 per cent, stock at 72^ ? (13) $6720 in 5^ per cent, stock at 96 ? (14) $3725 in 3 per cent, stock at 74i ? (15) £8475 10s. in 3 per cent, stock at 92| ? STOUlCa AND MHAEKS. 218 "What amount of stock miiRt be sold (10) In the 8 per cents, at 125 to produce $760? (17) In the Dominion S'b at 92|^ to produce $6291 (18) In the 6 per conta. at 101 to produce $069.50 ? (19) In the 7^ per cents, at 128 to produce $4096 ? ^ What per cent ie made by inveating in the (20) 8 per cents, at 120 ? (21) 5 per cents, at 9B ? (22) 6 per cents, at 104 1 (23) 3^ per cents, at 75? When Greenbacks are at (24) 90, what is the price of gold ? (25) 92^, what is the price of gold ? (26) 84, what is the price of gold ? When gold is at a premium of (27) 10 per cent. , what are " Greenbacks " quoted at ? (28) 25 per e«nt., what are " Greenbacks" quoted at ? worth\^^ P®^ cent., what is $5700 of American currency \ Wliat sum must be invested in the $64?f? ^ ^^^ '^^^^' *** ^^^ ^"^ ** ^"^ produce an income of a.oSn\^ P^"" ''®"**- ** ^ ^^ *« *« produce an income of . 35o7oO f $271)0 ? ^^ ^^"^ °^"*** ** ^^ *" *" ^° produce an income of , What is the selling price of stock when (33) $550 stock in the 6 per cents, produce $558.25 ? (34) $7840 stock in the 4 per cents, produce $6664 ? (35) £840 stock in the 3 per cents, produce £773 17«. u (36) What must I pay for U.S. iO-40'8 (Interest at R V) that my investment may yield 6 per cent. ? (37) Which is the better investment, the buying of 9 per cent, stocks at 25 per cent, advance, or 6 per cent, stocks at /?!f ^^^^' "^^^°""^' ^^^ how "»«ch per cent, better ? vSS) The difference between the incomes derived from mvestmg a certain sum in 6 per cent, stock at 126, and in 9 ntft 214 HTCM'KH ANH MHAKRH. per cent, stock at 210, ii £22 10a3«1750i,, caeh and owe. «9350i dur ntthe pr"eada JiT" K «"?«"»*". »1600, are paid out 7 he proceeas or tlie business, and on Jan 1 ift'7« i,; i i • valued at |397P'^ 1-e hal isftrfo ;. "' V ^^'J' **^« »*"ck is Whftf la fvl V'l ^. ^^r*^^ "* c^h and owes $7550 SL?!J^ whole proht or the year's transacti<:.« «f^; bega;i;'1h|yei;i '^"'' """'^'^ °" ^'^^^ °^P^**i ^ith which" he *if ^ f sit IHVimON IJtTO PROPOMTinNAf. fARIt. (1) I rM«i?«d AH 8 {Mr o«nl. diviUand on niilw»v mUtok^ Mid tnvMted the money in the iMme stock mt 80, My ttook havinf^ inorMMd to #13750, what waa the amount of my dividend i (2) Hovr many ahari'^^ of $')0 eiiuh niUHt be bought at 25 per cent, discount, l>rokura«u I'f por cent., and »old at U\ per cent, discount, brukurugu 1] per cent., to gain l|121.G0)( / (3) What «un muRt bo investod in United Htatea 10-40'h hearing inttuuiitat 5 por cent., [)iiyahlu in gtild piirchaittul at par, to prcKlucu a atMni-unnual income of $4AH) U.8. currency, when gold i» (puitud at 175 por cent. V (4) The chiirtur of a new riulroiid oomfMuiy limits the stock to li!tion is #Hr»(K)00 If the company cail in the Huul instalment of its stock, and assess the stock- holders for the remaining outlay, what will be the rate per cent? I ^(5) A person invests 810380 in the 3 per cents, at 01 ; he sells out !ii2ivi,Je m\7 Hinous three txartn«ra *i .h*r«i ar. to he in proportio,. „f J, nn^^p""^*' "''^^ that 18 th« proportion of 15, la and iC *°' "' Now 15 + 10 -f- 6 «. 31 ; ^•tiiourit of one «httro out of 31 .h»rw - fM; „ ,27 Than one of the i>.rtnur. receive, 16 x $27 „r #405;' the seoond reoe /e« 10 x «27, or $270 ; the third receive, X f27, or |l«2.' Ex. (3). A rate of fllioio :„ ^ i _ . • , , ^. »>.ip», and the ,r„t.ny±:i^^X '^'t^^iif':^^:;: t'.i.«. vv,.f;':;"r;;jT,;::.;:l' "'"' *"•''•"" "■ "■" $37250 $43350 it • 3 47(10X43 13 n» eOflQ $5tAlo^*aia nrfii^on $*Ma « X 4 ajj - jji MOA loaaoo ' ^^^^*^- T.' /*v .^. .. 100300 ' ' t^K (4). Divido .$1000 H,Mo„i/ A, Ji and P «.. ,i.. a Kop.eaeuting O'a part by 1, . Ji'a part will bo IL ana .I's part will bo l| + A of lA « 2 • .... .„ , ^ n- ij + 1 _ 4^ timci G's share. 4^ times 6"s shara = $1000 ; oi u SICOO * . s share = -_ =, $230769 ; Jiy share = | of C'b = $307 -692 A s sha-e = 2 times O'a -- .•$461-538. J/ake thfi ^raf. na..*- ..^ it... - ^,„,, ^^ ^„^. ^^.., , ^.^^^^ ^^. .^^ question 230 DIViaiON INTO TROPORTIOKAL FAKTt. the second paH will be f of tlie first, and the third will bo 9 of the first. Sun^ of the parts - 1 + g + g « Jg times the first. Hence |8 times the Ist « 2."T. ^ The Ist » 2li7 ^ J§ = 120. The 2nd = ^ of Ist - ^ of 120 - 72. The 3rd = j| of Ist = J of IL>0 == 45. Ex. (6). Divide $3400 among A, /i, and (7, so that A may Imve $800 more than § of JTs s)iare, and B $600 less than } of CTs share. Representing Cs share by 1, then JB's share = tt of Cs share - $000 ; A'& share »= H of ii's share 4- $800 -= I (I of CTn - $(i00) + $800 « i of C"s + $400. Sum of all the shares = C's + | (^^^h - $G0O + i Cs 4- $400 " S O's - $200 ; .*. I Cs - $200 - $0400. , J Cs « $3400 + ?2C3 ' - $3000. Cs = SICOO. Fs = I of $1000 - $000 = $600. ^'s - * of $1600 + $400 - $1200. Examples, (cxi.) (1) Divide $60 into two parts nroportional to 11 and 9. (2^ Divide $2500 into parts : .^ortional to 2, 3, 7, 8. (3) Divide $8470 into parts proportional to ^, ^, |, and ^. (4) Gunpowder is made of ealtpetre, sulphur, and charcoal, in parts proportional to 76, 10, and 15 ; how many pounds of each are contained in 12 cwt. of gunpowder ? (5) The sides of a triangle are as 3, 4, 5, and the sum of the lengths of the sides is 480 yards ; find the sides. (6) Divide $640 among A, 5, and G, so that A may have three times as much as B, and G as much as A and B to- gether. (7) Divide 100 apples among three boys, so that the first may receive 7 as often as the second receives *^, and the third may receive 6 as often as the second receives 4. y (8) A bankrupt owes £272 10s. to A, £354 5s. to B, and £490 10.?-. to Ci hia asBeta are £418 IQ.t. 4^d. Wh^f. r??]! each of the creditors receive ? > .--scsanri DIVISION INTO PROPOHTIONAL PARTS. 821 (9) A force of police 1921 atrruK is to be diatributed amona 4 t„wn« m proportion to the nuinbor of inhSntaiirri^ tiveiy. Determine the number of men sent to each .J}^^ ^i!1*^® ^^^ '"^^^ "" «*1"»^ n"n'l>er of half -sovereign, crown, half-crowna, ahiUinga. aixpencea, and fourpenc:;.^"'- (11) A piece of land of 200 acres is to be divided «mnn« four person, in pr.,portion to their rentl from surround-^S property ; supposing these rents to be £T,(^ So i^ and m how many acres must be allotted to each ? ' thi'^e!^:;' ;ifl rUveTbv /' B^ ''' '^ ^''^^ ^^^ ^^^ pennvDiece and fW*K ^ ^' "^ "^^^ '^^'^'V® » ^«ur- recei^S h?r i. H ^^''''^ *'''*' *' '"'*"?^ shillings in the sum '!o\ '^^ ^. "^ **'" '*^^""'^^* ^" ^^^^ ""^ received by^ so^thl J''h^ ^^^'^^ *"^^"« ^ "»«"♦ 7 women, and 14 boy. eacn Doy f of each woman s sliare. (15) A man left his property to be divided ainnn» hi. <« vZe of Ih': ;ro7or?^ ,"'^ ''""^' " »!**»• What' w„ th! «4Jf I "'ri"** f^ "■"""« '^> -B. ond C, .o that A mav Mt S Wh"? ^ "'S' t'""' ■""» f' «800'more tU H I'.' snare. What are the shares of each '( •" i oi jj » Ires'of V=M '"" *"*" ^^ "* ^'» '^'-''' Whatlro toe 6rsJ*23Hj^!^Tfc°' three fractions isJIi; and 22 limes the ursc, ^d times the aeoond. and 24 flmt^l^ho. *ki-,a • i products. Find the fraction """* 8'™ ^'^ „3^^ P'yide the simple interest on JH71 for IS vears at B How' ;„;;nrbo5;^7the« rn"tht'LS;o1iY *" ""^ "' *"• "^* 222 PAUTNfiHHMIP. PAIiTNmiJSHIP. 203. When persons unite to carry on any particular branch of business the connection so foiincd ia called a Partnkuship. The method of working questions fn Partnership is the same as that explained in the prooed- ing article. Ex. (1). A, Bf and O entered into partnership to carry on a mercantile business for two years. A puts in $9000, B $6000, and O $3000. They gained $4500. What is each one's share of the gain 1 The whole capital investod is $18000. Then $18000 gains $4500 ; .-. $1 gains $^«(f(fo, or $i. $9000 gains $oooo ^ $2260. $6000 gains $1000 « $1500. $3000 gams $aoi!il = $760. 4 Hence ^'s share of the gain is $2250 ; JS's, $1500 : and C's, $750. Ex. (2). Aj B, and entered into partnership for trading. A put in $600 for 4 months, B $400 for 5 ujonths, and O $200 for 6 months. They gained $980. What was each man's share of the gain ? 1000 for 4 months = $2400 for I month. $400 "5 ♦' = $2000 " " $200 "6 " = $1200 ** " The whole capital is equivalent to $5600 for 1 month. Then $5600 gains $980 ; .-. $1 gains $^^ = $^. $2400 gains $2ioo^ = $420. $2000 gains $2000x7 « 1350. 40 $1200 gains $1^00x7 ^ ^210. 40 .-. A'b share is $420, iJ's $350, and C's $210. Examples, (cxii.) (1) Two men jointly purchased a house for $2592, the nrst coutnoutiug $804 towarUa the puruaase and the second I PARTNEIWHIP. 223 81728. Thev afterwards rented the house for ill ^2 % annually. What .hare of the rent ought each to have ? * affr!ein1'tf'nav'^«S>^;A"f'^ JT*'^ a naature for 3 month., agreeing to pay f22.50 for the use of the same. A nut in 6 horses. B put n 18 cows, and C IK) sheej, Considering 3 shin :,*7^"7'*r"* *" ^ °°^«' *»"* «*«»» c«^ «« «q"»i t? 3 sheep, wliat part of me rent ought each to pay ? in^fifd' ^;».*"'^ .^. ?*®''®^ ,^"*" partnership for speoulatinft nUh^dT'«rf^T*/.*^fH^r8 ^25780;of whiT/fur* nished hB contributed | of the remainder, and the bal- anoe Heir clear profit was 20 per cent, of the orLina investment. How should it be divided ? * /iJr ^,****^» * business with a capital of $2400 on the 19th Sof'«T4'" ''^^h '^'\f J"^y -dniita aVrtne?B with a capital ot «I800. The profite amount to $943 by the 31st of December. What is each person's share ? (5) D and ^ enter into partnership ; D puts in i40 for tit 18 $410. What is the share of each ? (7) Three graziers hire a pasture for their common use inorh^^'^o'^'y ^'^T ^^i^- ^"« ?"<-« i» 10 o^en for Smiths' mo„^h«' y""""'' for 4 months, and the third 14 ox^nfor 2 months. How much of the rent should each pay ? ^ (8) Two men complete in a fortnight a piece of work for which they are paid $29.55. One of them Lrks altlrnatl wCn«.?%T^' and does nothing on the remaining day What part of the sum should each receive ? ^ «/nn .i if"? ^ ]^®fi!i> *''''^® »" partnership. A puts in 11^ f «'1' *"*ll^.^** **^« «»^ ^^ 2 months. J5 puts n $300 at first, and $000 at the end of three months The divided ? " '"^ ^^ *^ y^"' '' ^^^^' How shoSld this be (10) Johnston &nd Wilson formed a co-partnershio in LTntTn ri ^ y^- Johnston at first contrib^uteSTsoS) o dn ^l^i' ?^ ^.^ *^*^.^"^ «^ 12 months put in $1500 more Wilson at first put m $3500, but at the end of 15 months from the beginning withdrew $1000. At the end of the first $22o0. Their joint profits were SJ124/^ Rn^r «.,„k^ *u.-" if bo apportioned ? " "' '" ""* '^ ■'*iik 224 PARTNEUHUIP HRTTLEMBNTB. (11) A and B rent a field for 088.20 A puti in 10 howea tor 1# months, 30 oxen fur 2 months, and 100 sheep for 31 months ; B, 40 horses for 2\ months, 50 oxen for \\ montlis, »nd 115 sheep for 3 months. If the food consumed in the same time by a horse, an ox, and a sheep be as the numbers 3, 2, 1, what proportion of the rent must each pay? (12) A person in his wiU directed that * his property should be given to A, i to i^, ^ to O, and \ to Z>. Shew that this disposition cannot be fulfilled. If his property amount to 31886.50, dis^HJse of it so that their shares may have to one another the relation he intended. (13) A^ B, and C had each a cask of rum containinc respectively 36, 64, and 78 gallons. They blended their rum, and then refilled their casks from the mixture. How much of the rums of A and B are contained in C"a cask ? (14) ^ rents a house for $187.20 ; at the end of 4 months he takes in .B as a co-tenant, and they admit in like manner for the last 2^ months. What portion of the rent must each of them pay ? Partnership settlements, 204. When a partnership is dissolved, either by mutual consent or by limitation of contrnet, the adjust- ment of the proceeds between the members is called a Partnership Settlement. If the Resources are found to exceed the Liabilities, the diflference is termed Net Capital; if the Liabilities exceed the Resources, the difierence is Net Insolvenoy. The investment of the partners is the Net Capital at commencement. If the net capital at closing exceeds the net capital at com- mencement, the difference is the Net Gain; if the opposite. Net Loss. This net gain, or net loss, is then shared between the partners in accordance with the original agreement between them. This division is fre- quently not made in exact proportion to the amount invested; sometimes the skill of one partner is considered equal to the capital of another; sometimes a stated salary is allowed each partner according to his ability or reputation ; and sometimes, where unequal amounts are invested, interest is allowed each partner on iiis invest- ment; but whatever allowance is made such allowance «M««oj^ Jn/a ^//yf^^^^J «j5 -. 1_^jr^l TSli iA-J!L — ^ VC- «>Sv«7w«^ Us a uiiiouuy anu yo to reduoo ilia yuin. PARTNRRMHIP MRTTLKMENTS. 226 Mercha„d,,e, ^m^atZ^^^-J^'Z^'K^^"^ 93240 ■ on account, *37B Th«»^„r d ?,' ^T^- ''■ Brown owe. »2600, and dreTout durin., h ."'""'"d ?' <»mmeAoing to share equallv in aain. q«^ 1 ""■*"®f"» v^w- They agreed .nd wjjaa^ter^rfrih aT^l^„r '' """ «*'" ' OWNKRBHIP. Dr. «6flO il withdrew. 280 JJ <• Ob. 12500 2500 1840 Total investment $6000 vi^' ''withdrawn 840 *>rms not Investment 4100 ^^Rbsourom akd Lubimtim. **240 .^r. 2676 •1250 860 870 fl il^o SteSel''"'^'"^- 4160 Credit exoeu of Ownership. 1270 Not Gain. 636 ^'s share of net ,fain. S'oSlr*^® ^'8 present net capital ■= $2500 - «560 4. SkMR i &r' ^' ^"«^"* -t'capitall $2m^-Z^ol'ls6 Examples, (cxlii.) not gam, and net capital of each aHlo-t *f " ''' m%f Mdle! im2 'TXe? 838^ k^"!^^ ""^S ^"'•> able, 83486; Bills Receivable ««9fi ,J?"'"'8''«e> Receiv- Payable, $2450; onaccmmt. «i<>!^^®^ •^''^y "."^ "" Billg a d'ebt lohlOob wa» rZed Iv ,h ^ '""*''?'' ^'^°^' ""» business. He drew Z «M^ '^ ""Z •*''";! '"<* P»*" "luring capital invited, 842a ' stl^!^ "^""^ r'/"''' °» $1860, and is allor^^ i„r '"7 <* *r"'"> and drew out "'ug oi each. Ihey have cash, 82263, and Real 220 ▲LLIOATION. Estate worth $5000. They owe on Mortgages, $3846 ; on N tea, $44C2 ; on Personal Accounts, $675. A invested $6000,, and drew out $2860. B invested $4000, drew out $6560, and is allowed for extra services $260. A shares f and B ^ ot the gains and losses. What is the net loss? ^/hat is the financial standing of each ? XXIX. Alligation. 206. Alligation is the process by which we find the mean or average price of a compound when we mix or unite two or more articles of different values. Ex. (1). A merchant has brown sugar v/orth 8 cents per pound, New Orleans worth 9 cents, and refined sugar worth 14 cents. How many pounds of each kind must he use in order to form a mixture worth 12 cents per pound ? By selling the mixture at 12 cents per lb., we see that 8 cents (brown) gains 4 cents on 1 lb. ; .*. 1 cent is gained on I lb. 9 cents (New Orleans) gains 3 cents on 1 lb. ; ,'. 1 cent is gained on ^ lb. 14 cents (refined) loses 2 cents on 1 lb. ; .*. 1 cent is lost on 1^ lb. Now with every cent gain he must combine a cent loss, hence he must have lbs. at 8 cts. 6 lbs. *' 14 cts. lbs. ." 9 cts. 6 lbs. ** 14 cts. lb. at 8 cts. lb. " 14 cts. lb. *♦ 9 eta. lb. " 14 cts. He must, therefore, have 3 lbs. brown sugar, 4 lbs. New Orleans, and 12 lbs. refined. We may show that these quantities \7ill make the mixture required, as follows : 3 lbs. at 8 cts. per lb. = 24 cts. 4 lbs. " 9 cts. " =36 cts. 12 lbs. " 14 cts. " =168 cts 19 lbs. = whole mixture. 228 cts. = value of mixture. Hence, if 19 lbs. be worth 228 cents, 1 lb. is worth 2^8 = 12 cts. Or Wfi mav r«nann fVins Tlio 1 M fV»^ 1 IK brown exactly balances the 1 ct. lost on the ^ lb. of the AIXIOATION. 227 I mix or J*h!l.!fi' ^^^"""o ?l^ "»;«1*»^° i 5b- of the brown and h lb. ol the refined, or 2 lbs. of the one and 4 lbs. of the other: ^miJarly for every 2 lbs. of New Orleans, there must be t^\u u ^^^' ^^V^"- ^^ '■«^"®^ ^e^-e required to bal- anoe the brown, and 3 lb*, of the refined to balance the New Orleans, th ^e must be 7 lbs. of the refined in the compound. 1 herefore the respective quantities are 2 lbs. brown. 2 lbs New Orleans, and 7 lbs. refined. ' From the above we see that in examples of this kind a variety of answers may frequently be obtained, and all ot them may ba correct. To ascertain their correctness we resort to the method of proof given in this example. 206. From the above analysis we derive aa easy practical method of solving such questiona Ex. (2). Ho^ much sugar at 10, 13, 15, 17, and 18 cents per pound must be taken to make a mixture worth lb cents per lb.? We proceed as follows : Differences. 6 3 1 16 1 2 10 13 16 17 18 1 1 1 Write down the prices in a vertical column, and place the differences between these prices and the mean in a second verti- cal column to the left. Now V ^ « fl /fu ® 1 ® 10, 1 @ 13, and 1 ® 15, 7 J' S' ^ . low-^st that can be taken); " J ~~., V ' ' ^ **^^^ would represent a loss of 10 as compared with the mean; and this loss must be balanced by taking the necessary multiples of the differences 1 and 2 which represent gain as compared with the mean. u ^o 'i «,®H®" *^** i^'s loss of 10 can be made up in four wavs • and 1% 18.^ ® ^^' * ^ ^^' 3 @ 18, 6 @ 17, h 1^8 ® 17,' 6 3 1 1 2 Other combinations may be made, as e.g.. 10 13 16 17 18 Here 1 @ 10, 1 @ 13, and 2 @ 15, give loss of 11, which can be made up by multiples of the differences 1 and V q ^ - o (°PP«s^*« i7 and 18) in /w ways, 1, 3, 5, 7, 9 as indicatfii ' * 6,4,3,2,1 «28 ALUOAnOM. [H i Also, .10 3 13 1 15 1 2 6 3 1 17 18 Again, 10 13 15 1 2 17 18 Also, 1 2 1 1, 3, 6, 7, 9, 11 0,6,4,3,2, 1 2 1 1 Where 1 ® 10, 2 ® J.3, and 1 m 15 give 13 lou, which rimy be made up in mx different ways. 2, 4, 6, 8, 10, 12, 14 7,6,6,4, 3, 2, I WhtM 2 @ 10, 1 ® 13, and 1 @ 15 give loM of 16, which may be made up in ieven waya. 6 10 3 13 1 15 1 17' 2 18 1 1 3 Where 1 ® 10, 1 @ 13, and 3 @ 16 give loss of 12, which may be made up iu^jje waya ; and thus an indefi- i»ate number of combination! may be 2, 4, 6, 8, 10 formed. 6, 4, », 2, 1 It should be observed that if the differences opposite the prices lea8 than the mean are together greater than the sura of the other differences (as in the example), we assign numbers (the lowest possible) to the prices less thaji the mean fibst and vice versa ; eg. of the latter case : — ' How much coffee at 25, 24, 23, 22, 21, 19, 18, and 17 cents per pound must be taken to make a mixture worth 20 cents per pound ? Diff's. 's. 20 3 17 2 18 1 19 • • 1 • • 21 2 22 3 23 4 24 5 25 4, 3, 2, 1, 1, 2 1, 2, 3, 5, 4, 2, &c. 1, 2, 3, 2, 4. 3 1 1 1 1 1 Here the sum of the dif- ferences in excess of the mean is greater than that of the differences below the mean ; we therefore wssign Jirift numbers to the prices which are greater tL\n the mean, viz., 1 @ 21, 1 @ 22, 1 @ 23, 1 @ 24, and 1 @ 25 ; this gives a grain of 15, which may be balanced as above by 1 (a; 19, 1 (§> 18, and 4 ® x7 ; or by 2 @ 19, 2 ® 18. and 3® 17, Ac, &C. ^ ^ , Ex. (3). A grocer has 12 lbs. of brown sugar, worth «uLs per pound, which he wishes to mix with claritied iU ALUtiATION. 229 ■ugar worm 16 cftits p«r pound, so that the muture may b© worth 14 cenU per pountl. How many pouudi of claritied sugaf nju«t he take I ' Proceeding m in the piovioua examples, without ^ofe^ enco to the quantity of tlio brown sugar, wo find that there must bo 1 ib, brown sugar to 2 11»h. chirified 8Uj,'ar. But as 12 lbs. of brown 8u<,'ar are required, we nmat multiply each of thoao quantities by 12 in order that the gam and loss may bo equal. We shall therefore have 12 X 2 - 24 lbs. of claL.iod sugar. Ex. (4). A grocer wishes to mix 20 lbs. of sugar worth 9 cents per pound, and 10 lbs. worth 12 cents per pound, with clarified sugar worth 15 cents, so that the compound may sell for 13 cents. How much of the clarified must he take? 20 lbs. at 9 cents « $1.80 10 lbs. at 12 cents « $1.20 30 $3.00 Then, if 30 lbs. is worth $3, 1 lb. " ^^ « 10 cents. The value of 1 lb. of the mixture is, therefore, worth 10 cents. The question may then be read as follows : How many pounds of clarified sugar, worth 15 cents per pound, must be mixed with 30 lbs. of another kind of sugar, worth 10 cents per pound, so that the mixture may be sold for 13 conU. per pound? The questiou in this form has already been fully ex- plained. Ex. (5). A merchant has West India sugar worth 8 cents per pound, and New Orleans sugar worth 13 cents. He wishes to combine these so as to make a barrel con- taining 175 lbs., which he may sell at 11 cents per pound. How many pounds of each kind must he take ? Solving the question without reference to the 175 lbs., we find that 2 lbs. of West India suj^'ur und 3 lbs. of New Orleans sugar will form a mixture worth 11 cents per pound. Adding these quantities we find that they foim a uiiAtUio of 5 ibs. But the required mixture is I 930 ■XCRAHOI. to contain 175 lbs., or 35 timo« 5. Wo hUaU therefore have . 85 X 2 Ibi. « 70 IIm. Went India sugftr. 35 K 3 lbs. mt 105 IIm. Nuw OrleaiiH RUgar. Examples, (cziy.) (1) What quaniitieii of c<»ffoo, worth 23 and Sft oenta reipectivoly pt»r pounrl, muni bo mixed tog<»ther «o that the compound may be sold for 30 cent* a pound / (2) What (juantity of oats at 35 cents per buiih«l, rye at 60 ceuta per buiihtil, and Imrlt'y at 80 cents, must bo takei > form a ntixture worth 55 cunts pur bushel ? (3) How much tea, wort*- respectively 55 cents and 75 cents per pound, mu«t be mixed with 30 lbs. worth 90 cents per pound, in order that ♦he compound may be sold for 70 cents per pound i (4) How much water will it require to dilute 00 gallons of alcchol, worth $1.50 per gallon, so that the mixture may be worth only $1.20 per gallon? (6) How many gallons of kerosene oil, worth 60 cents per gallon, must be mixed with 12 gallons of coal oil, worth 30 cents, and 8 gallons of Aurt)ra oil, worth 5(5 cents, so that the compound may bo sold for 50 cents per gallon ? (0) A farmer has 10 bushels of corn, worth 48 cents per bushel, and 12 bushuls of oats at 34 cents per busht'I, which he wishes to mix with rye at CO cunts and barluy at 80 centi, in order to sell the compound at 50 cents per bushel. How many bushels of rye and barley W'U be required '/ (7) A confectioner mixes three difterent qualities of candy worth respectively 14 cents, 18 cents, and 30 cents per pound, so as to make a box of 84 lbs. ; how many pounds of each sort must he take so as to sell the compound at an average prico of 24 cents per pound ? (8) A farmer has three different qualities of wool, worth respectively 33 cents, 37 cents, and 4o cents per pound. He wishes to make up a package amounting to 120 lbs., which he can afford to sell at 3') cents per pound. How many pounds of each kind must he take ? XXX. Exchange. 207. The term Kj-chanfff is here used for giving or _ • • •_ j_i i> ^ _ _■ __ _ 1 ".^ Fovoivitiu iti uiio iuuncV OI Oiio CuuiiDi v' dt ouui ouuoti iu MXOMANfll. 93] va1u« to a ium of inonwy of another country. For example, if tui English men Imnt pays to a French mer- chant 100 sovoreigt.M and receives in return 2500 fraaos, it is a case of Exchange. Tn countries which carry on considerable trade with oach ()thfr, the dchta reciprocally due from the one to the otiier are generally nearly e<|ual. In Knglai»d there IS always a large tmmber of pernons indebted to others in Ame-ica, and likewise a largo number in America owing money in England. Now if coin, or specie, as it IS called, were sent from England to pay the debts in America, and from America to England, the specie would have to bo tranHtnitted twi«;e, and would neces- sarily involve risk, loss of interest, and expense of trans- portation. To avoid this risk, cfec. Bills of Exohan(jb are used to liquidate debts reciprocally due between two place* without any actual transmiasion of money. 208. A Bill of Exchange is a written order, ad- dressed to u person in a distant place, directing him to pay a certain sum of money, at a specified time, to another, or to his order. The person who signs the bill IS called the Drawer, or Maker. The person to whom it is addressed is the Drawee, and after the Drawee agrees to pay it, and writes "accepted" with his signa- ture and the date across the face of it, he becomes the Acceptor. The person to whom the money is to be paid is the Payee; if he transfers payment to another he endorses it, i.e., he writes his name across the back of it and becomes responsible for its payment in case the Drawee fails tp ni \ke payment. 209. The Par of Exchange between two countries m at less thon their par value. Now the real rate of exchange, depend- ing on the balance of trade, is called the CouRSR ov Ex- OfiAKOE; and it is at a premium or tiwroftnt, according as it is al)0vc or IksIow the par of exchange. Of course no one would give a premium greater than the cost of transmitting specie, But if the balance of trade is against England as regards America, but in favor of England an against France, the English merchant may find it advantageous to remit to France, and then for France tb remit to America, and this mode is adopted when the course of exchange by this circuitous route is less than the direct course of exchange. The finding the course of exchange between two places, by comparing the courses of exchange between them and one or more intervening places, is called Ahbitration of Exchangb. The arbitration is Simple when only one place inter- venes, and Compound when more than one. Bills of Exchange are usually drawn in sets, three bills constituting a set. These are distinguished from one another by being called the Jiratf second, and third of exchange. These are forwarded by different routes so as to guard against delay or their being lost. The first that arrives is paid, and the other two become void. 210. By Act of Parliament the valua of the pound sterling was fixed at $4|. This was much below its intrinsic value, which is now fixed at |4.86§. The rates of exchange wJiich aro quoted in commercial papers are still ^calculated at a certain per cent, on the old par of exchange. Exchange is at par betweer Great Britain and Canada when it is at a premium of 9i per cent., for 4iAl i ] 1 ftl J. 1 Aj rtno ***-^. wv«w^\A 2 f «.! V.'OiAi/, \;oumo %}z IXCVAJfOl. 9V9M or iiiurr oa tMLAna bili of ucmahoi. iiooa Stamp. Toronto, July 12, 1880. A /* i**?. ^*y«' "'g'i^ P»y to the ordiT H Adam Millor & Co., t)n« ThouMnd D'.lUrt. v*lu» r«llar = ^2 Framtce. — 10 centimes «^ 1 decime ; lo' docime » 1 franc =>^ .jog Orbbcb.— 100 lepta=l drachma ; 1 drachme (iilV'^r) « -166 tlOLLAND.— 100 cents = 1 florin or guilder : 1 finrin (niiver;« „ .^ 234 EXCHANQB. Hamburg.— 12 pfenning = 1 schilling ; 16 schilling =1 marc ; 3 marcs = 1 rix-dollar = "84 Mexico. — 8 rial8==l dollar^ , 1*00 PoETUOAL. --400ree8 = l cruzado ; 1000ree8 = l milree or crown = 1 '12 Prussia. — 12 pfennings = 1 grosch (silver) ; 80 groa- chen = 1 thaler or dollar = "69 Russia. — 100 copecks = 1 ruble (silver) = 78 Sweden. — 48 8killing8 = l rix-dollar sp ;ie= 1"06 Spain. — 34 maravedi8 = l real of old } ate*= '10 8 reals = 1 piastre ; 4 piastres = i pistole of ex- change ; 20 reals vellon = l Spanish dollar = ... 100 TuEKEY. — 3 aspers = 1 para ; 40 paras ^= 1 piastre (vari- able), about 'Om Venice. — 100 centi8imi = l lira^ .... '180 .VALUE OF FOREIGN COINS. Guinea ^5.10 Sovereign of Gtdat Britain 4.86S Crown of Enyriand 1.216 Half-crown of England 808 Shilling of England ■ . .24 Franc of France - .. .18 Flve-fianc pi ce of France 93 LivreTour' , of France . .18j Forty-frail ece of "France 7.66 Grown of l-i.-nce 1.06 Louis-d'Or of France 4.56 Florin of the Netherlands .40 Guilder of the Netherlands 40 Florin of South Germany 40 Thaler or Rix-Dollar of Prussia and North Germany 69 Hix-Dollar of Bremen 78f Florin of Prussia 22f Mare- Ban CO of Hamburg 35 Florin of Austria 48J Florin of Saxony, Bohemia, and Trieste 48 Florin of Nuremburg and Frank- fort 40 Rix-Dollar of Denmark 1.00 Spocie-Dollar of Denmai-k 1.05 Dollar of Sweden and Norway. . 1.06 Milree of Portugal 1.12 Ex. (1). A broker in Toronto sold a bill of exchange on London, the face of which was for ^750 8s. ; what did he receive for the bill, exchange being quoted at llOj? *The old plate real is not a coin, hrt is the denomination in ^»4lich exchanges are usually made. Milree of Madeira $1.00 Milree of Azores 83J Real-Vellon of Spain 05 Real-Plate of Spain 10 Pistole of Spain 3.97 Rial of Spain 12 Pistareen 18 Cross Pistareen 16 Ruble (silver) of Russia .75 Imperial of Russia 7.83 Doubloon of Mexico 16.60 Half -Joe of Portugal "8.53 Lira of Tuscany and Lombardy. . . 16 Lira of Sardinia 186 Ounce of Sicily 2.40 Ducat of Naples 80 Crown of Tuscany 1.05 P'lorence Livre 15 Genoa Livre 18| Geneva Livre 21 Leghorn Dollar 90 Swiss Livre 27 Scudo of Malta 40 Turkish Piastre 05 Pagoda of Indie 1 84 Rupee of India 44^ Tael of China L48 EXUHANOK. 236 $1,00 .83^ .05 .10 3.97 .12 .18 .16 .76 7.83 16.60 "8.63 rdy. .16 .186 2.40 .80 1.05 .15 .181 .21 .90 .27 .40 .05 1.84 . .44* '.'.'.. 1.48 KCl] lange Since £1«$4^ x llOj^, i.e., $4f increased by 10^ per cent, .. £750 -4 =$750-4 x 4^ x 1-10| -13676.96; .'. he got $3676.96 for the bill. Ex. (2)1 What ia the value in English money of 4528*7 franca, when the course of exchange between Paris and London is at 25 3 francs per pound sterling i Since 25-3 francs = £1, 1 franc = £-^: .-. 4528-7 francs = £11? 8_^ or £179 75.8 Ex. (3). A merchant pays a debt of 4379 milrees in Portugal with £971 lis. 9fc?. What ^s the course of exchange in pence per milree ? £971 lis. 9|d = 932727 farthings ; Then since 4379 milrees = 932727 farthings, 1 milree = o||||I farthings, or 213 farthings; .-. the course of exchange is 53| pence per milree. Ex. (4). If 11-65 Dutch florins are given for 24-42 francs, 352 florins for 407 marks of Hamburg, and 58i marks for 32 silver rubles of St. Petersburg; how many francs should be given for 932 silver rubles ? Here 1 silver ruble =.S8-25 marks. 1 mark = | J| florins, 1 florin = I*" J I francs ; ••• 1 «il^«r ruble = ^ X l^ X f i^^ francs, or3-3 francs; .-. 932 ailver rubles = 932 x 3*3 francs, or 3075 '6 francs. Ex. (5). A New York merchant remits 27940 florins to Amsterdam by way of London and Pans, at a time when the exchange of Kow York on London is $4-885 for £1, of London on Paris is 25-4 francs for £1, and of Paris on Amsterdam is 212 francs for 100 florins; J per cent, brokerage being paid in London and in Paris, hnw many dollars will purchase the oill of exchange ? 236 BXCHAMOE. Since 100 florins = 212 francs, .-. 1 florin = f JJ francs. Bdt to buy a bill of 100 fr. requires a bill of I00| fr. ; .*. to buy a bill of 1 fr. requires a bill of |gj fr. Again, 25 40 fr. = £1, • 1 fr = £-J— - • but to buy a bill of £100 requires £100^ ; £1 " £m- Again, £1 = $4 '885 ; ••■ 1 flo"n = ^?U X I8i X 2 5^ ^ ^oh X 4-885 ; .•. 27940 florins = i^'ilMoy 2x2 xsoix aoi x ^ -aas 100 X 800 X 25-40 X 800 = $11420 317, sum required. Ex. (6). A merchant of Toronto wishes to transmit 2400 jnarcs banco to Hamljurg. He finds exchange between Toronto and Hamburg to be 35 cents for 1 marc. The exchange between Toronto and London is $4.83 for £1 ; that between London and Paris is 26 francs for £1 ; and that of Paris on Hamburg is 47 francs for 25 marcs. By what way should the Toronto merchant remit ? By direct exchange 1 marc = $0.35 ; .-. 2400 marcs =• $2400 x 0-35 By circuitous exchange 25 marcs = 47 francs ; .•. 1 marc = |l francs ; but 26 francs = £1 ; .-. 1 franc = £^, and £1 = $4.83; .. 1 marc = $4.83 x ^ x ^l; .'. 2400 marcs = $ 2^00 x 4.83 x 47 26 X 26 ;p; =$838.19. By direct exchange the merchant paya $840 for his bUl of exchange, and only $838. 19 by the circuitous mode ; .*. the circuitous mode is better by $1.81. KXCUAMOS. 237 Examples, (cxv.) (1) When $7300 are paid in Toronto for a bill of exchAnc- on Liverpool for £1600, how .a- sterling exchange q^o^f ^ (2) What will be the cost of a bill on Paris for 236874 francs, exchange being 5-3 francs to the dollar ? ^ (3) If ^1 be worth 12 florins, and also be worth 26 fran«« copecks in Hussian money, what is the value of the naooleon ^?^"^''""P"'^^^ (N.B. -20 francs ==l„4;ieonr (5) The French franc is divided into 100 centimes and th« Frankfort florm into CO kreutzers. When the pound sterling IS worth 25 oO francs in Paris, and IJ fl 54 kr at BV«„rfi".^ f, >^i i" ^^^^ exchange on Paris was quoted in New Yort ^f 512i francs to the dollar and gold was at I35r If a New York merchant owed 12669 francs in Havre! how nfuch the face of his bill m pounds, shillings, and pence ? • l'^'"L*i*® P*' ^^ exchange between the U.S. gold eaffle :^% fb'f ^oCT^^e ^"' *'^ '--'''^' ^' '^^^ Amsterdam is 12 161 florins for £1, aid between Amaterrm and Paris 209^ francs for 100 florins. . Amsterdam (10) If a merchant buys a bill in London, drawn in Paris at the rate of 25 5 francs per pound sterling, and if ?hS JS 8 sold m Amsterdam at the rate of 30 francs for iVflorins and the money received be invested in a bill on Hambum a the rate of IS florins for 20 marcs banco, wha? S?he rati of exchange between London and Hamburg, or what La pound sterling m London worth in Hamburg ? (11) If the exchange of London on Hamburg is 14 marcs banco per pound sterling; that of Hamburg on Amste^S Parfs T^t''''' ^r i^ f''""'' *^*^ of AmsleTdam on 238 KX A MI NATION PAPERS. (12) The exchaiigo at Paris upon London is at the rate of 25 francs 70 centimes for £1 sterling, and the exchange at Vienpa upon Paris is at the rate of 40^ Austrian florins for 20 francs. Find how many Austrian florins should be paid at Vienna for a £50 note. (13) What is the arbitrated rate of exchange between London and Lisbon, when bills on Paris, bought in London at 25 06 francs per 4 are sold in Lisbon at 625 rees per 3 francs ? (14) Given that 1 ounce Troy equals 31-1 grammes ; that 10 grammes of French standard gold are worth 31 francs ; and that the worth of a given weight of English standard gold is to that of the same weight of French standard gold as 3151 to 3100, find what number of Troy ounces of English standard gold the franc is equal to, and what is the fixed number of francs equivalent to £1 ?— the English mint price of standard gold being 77s. lO^d. per ounce. ;< Kxaminatinn Papers. \ I. (1) If three fluids, whose volumes are as 3, 7, and 12, and their specific gravities -96, 1*15, and 136, be mixed together, what will be the specific gravity of the compound ? (2) If f of ^'s money equals | of ^'s, and | of ii's equals f of Cs, and the interest of all their money at 8 per cent, for 4 years 6 months is $6291, how much money has each ? ■^ (3) A Toronto merchant wishes to pay a debt of £1200 in London. How many dollars must he pay to procure re- mittances through France and Hamburg if we allow that 21 francs = $4, 19 marcs banco at Hamburg = 35 francs at Paris, and £7 at London = 96 marcs banco at Hamburg? ^ (4) A merchant in Cincinnati wishes to remit $14331.60 to New York. Exchange on New York^is | per cent, premium, but in St. Louis ^ per cent, premium, from St. Louis to New Orleans | per cent, discount, and from New Orleans to New York 1 per cent, discount. What will be the value in New York by each method, and how much better is the circular ? (5) A merchant in Toronto purchased a draft on New York for $2660, drawn at 60 days, paying $2670.89. What was the course of exchange ? II. /t \ A ^; 11 IV Ti'+U Pi IK interior quality, and gains 16 % by selling the mixture at 87 EXAMINATION PAPER& 239 cents per pound. Allowing that a pound of the one coat 12 cents more than a pound of the other, what was the cost of each kind per pound ? u ^^L4™^ ^ ^^^ *" partnership in a concern in which A has $20000 engaged, and B $30000. The gr.)88 receipts for a year are 112800 ; of this one-eighth part is expended in salaries of clerks, and $120 in insurance. By an arrange- ment between the partners A is to receive 8 % upon his . Dital, and /i 4 % upon his, and then the remainder of the Di jhts IS to be divided in proportion to the capital ^^mploved. Find the net receipts of A and B. «- r j . (3) Bills on Amsterdam, bought in London at 12 florins 15 cents per £1 sterling, are sold in Pi^ris at 67* jflorins for IJO francs. What is the course of exchange bi'tween London and Fans ? (4) On the Ist Jan. A brought into a business $1400. and Z'l^' April $2000 more ; on the Ist June he took' out f 1000, and 3 months after this he brought in $2400 B brought into the business $2000 ; 4 months after thi^s iie took out $600, and on the Ist Nov. brought m $2600 Their clear profit for the year is $4032. How much ought 'each to receive f (5) A cask contains 12 gals, of wine and 18 gals, of water : another cask contains 9 gals, of wine and 3 gals, of water how many gallons must be drawn from each cask 6o as to produce by their mixture 7 gals, of wine and 7 gals, of Ill ^^\^ merchant has sugar at 8, 10, 12, and 20 cents a p()und ; with these he wishes to fill a cask that holds 200 lbs. How much of each kind i mt he take so that the mixture may be worth 15 cents a p d ? ^? Af^o,^^^^' ^""^^^ ""^ Montreal yielded $1190-234 when sold at 1^ / discount, and interest off at 6 per cent. What was the face of the draft ? n ioU^'^o^^'" !P^ '"t ^ "'^^^^^^ ^ ^1^^ »" 5 ^lonths, and b ^400 ?"^°^ ^^ *^® ^^^^® ®*°*''^' ^'^ P*^* °^ ** (4) From a cask of wine one-fourth is drawn off, and the cask 18 filled up with water ; one-fourth of the mixture is then drawn off, and the cask again filled up with water : after cni5 nas Deeu done tour times altogether, what fraction of the original quantity of wine will be left in the cask ? EXAMINATION I'APERS. (5) A penon in London owes another in St Petersburg 920 roubles, which must be remitted through Paris. H» pays the requisite sum to his broker at a time wh^n th'- exchange between London and Paris is 23 15 francs for £1, and between Paris and St. Petersburg 1 2 francs for 1 rouble. The remittance is delayed until the rates are 25 '36 francs for £1 and 1 15 francs for 1 rouble. What does the broker gain or lose '■ the delay ? IV. (1) If, when the course of exchange between Endand and Spain is SSkd. per dollar of 20 reals, a merchant in Liverpool draws a bill of £354 16s. 3d on Madrid, how many dollars and reals will pay the draft ? ^ (2) I wish to pay a bill in Naples of 7500 lira ; the direct exchange is $0.22 ■>> 1 lira; the exchange on London is $4.95 ; of London on Paris is £1 = 26 francs ; of Paris on Naples is 1| francs « 1 lira. What is the difference between the direct and circuitous exchinge ? (3) A merchant in New York gave $1000 for a bill on London of £200. What was the rate of exchange ? '^^ (4) A mei-chant in New York wishes to pay £3000 in London. Exchange on London is at par ; on Paris 6 francs 25 centimes per $1, and on Amsterdam 40 cents to a guilder. The exchange between Franco and England at the same time is 25 francs to £1, and that of Amsterdam on England 12J gnilders to £1. Which is the most advantageous, the direct exchange, or through Paris, or through Amsterdam? (5) How many pounds of sugar at 8, 13, and 14 cents per pound may be mixed with 3 pounds at 9| cents, 2 pounds at 8^ cents, and 4 lbs. at 14 centH a pound, so as to gain 16 per cent, by selling the mixture at 14^ cents per pound ? V. (1) Three districts are to provide according to their popu- lation a contingent of 182 men. The population of the districts is i; .'*>6, 735, and 43G1 respectively. Find as exactly as possible the number of men to be provided by each district. (2) A person mixes 4 gallons of gin at 15s. per gallon with 4 gallons of water and a gallon of base spirit worth 10s. What is his gain per cent, on his outlay by selling the mix- ture at 2§s. per bottle of 6 to the gallon ? (3) The stocks of three partners, A^ B, and O, are $3500, and $1200 respectively. If jB's stock is in trade 2 months longer than A% what time was each stock in trade ? EATIO ASD P&OPO&nov. 241 (4) A merohftnk every year gains 60 % on hu capital, of which he spend. £i200 per annum in house and otherVi- pensea. At the enu of 4 years ho finds himself in possession of 4 times as much as what he had at commencing buwneM. What was his original capital ? ^ «*"«». (5) There are two mixtures of wine and water, the quan- Swo T^'"® n which are respectively 34 and '40 ol the whole, n a gallon of the first Is mixed with two gallons of the second, what decimal part will the wine be in the com- sTrefced ?'" "'"''' ^'^ '''''' ^^ '^' ^"^ ^«*"'« ^ XXXI. Ratio and Proportion. 211. If ^ and 5 b© quantities of the same kind, the relative greatness of A with respect to J3 is called the Katio of ^ to i5. 212. The ratio of one quantity to another quantity is represented in Arithmetic by the fraction which ex- presses the measures of the first when the second is taken as the unit of measurement. T^"8» i{ 6 shillings be the unit, the measure of 3 shillines bV tCfracUo^r "^ ^ '™"^' ^"^ ^ '^'"^^' " represented The words "the ratio of 3 shillings to 6 shiUines" are abbreviated thus : 3 shillings : 6 shillings. 213. Ratios may be compared with each other by comparing the fractions by which they are represented. Thus 2 pence : 5 pence is represented by f , a;ad 3 pence : 7 pence is represented by ^, Now I = ^f , and f = ^, /. f is greater than f , and .-. 3 pence : 7 pence is greater than 2 pence : 5 pence. When we thus compare the ratios existing between two pairs of quantities, it is not necessary that all four quantities should be of the same kind i it is onlv ■ary that each pair should be of the same kind. " 'J ' ■* 842 RATIO AND PROPORTIOW. For exfttiiple, wfi can compare th© ratio of 4 fihiliiti^ii to 7 shillings with the ratio of 7 days to 12 daya, and finding that ^ is less than ^, we may say that the ratio of 4 shillings to 7 shillings is less than the ratio of 7 days to 12 days. 214. When the ratio symbol (:) is placed between two 7iumberiif we may substitute for it the fraction symbol. Thus, if we have to compare the ratios 2 : 3 and 5 : 7, we efi'ect it by comparing the fractions j} and 1^. 216. Ratios are compounden by multiplying together the fractions by which they are represented, and ex- pressing the resulting fraction as a ratio. Thus the ratio compounded of 2 : 3 and 5 : 7 is 10 : 21. 2 an(| 3 are called the Terms of the ratio 2:3. 2 is called the Antbobdent and 3 the Oonsrquknt of the ratio. 216. Ratios are either direct or inverse. A direct ratio is the quotient of the antecedent divided by the consequent. An inverse ratio, or reciprocal ratio, is the quotient of the consequent divided by the antecedent. Examples, (czvi.) (1) Compare the ratios 2 : 5 and 4 : 9. (2) Compare the ratios 17 : 39 and 19 : 41. (3) Compare the ratios 4 : 7, 8 : 16, and 13 : 24. (4) Compound the ratios 5 : 7, 13 : 16, 21 : 91, and 46 : 52. (6) Compound the ratios 3^ : 4, 3^ : 7v l\ : 3^, 2^ : If. (6) If the ratio be 25 and the consequent $1.25, what is the antecedent ? (7) How much does the ratio 36x4x3:12x16x2 exceed that of 60-r (3 X 6) : 20 X 2-J-8 ? (8) What is the reciprocal ratio of ^ : ,\ ; of 2J : 7 *§ ? (9) A owns a farm of 180 acres. There are 36 sq. miles in the township in which it is situated. What is the relation of the latter to the former ? PMOPOKTION. 343 (10) The ratio 03 : 6JJ re.ults from compounding four ratio, together; three of the.e are 7 : 8, 12 • 15 and i 1 Kxpresa the fourth ratio in its Bin.pleat form ' * ' *' termi^of?rati.!f '' ^'^^ '^^'^^^^ '^'' ""'"^ 'A"'^"^*^^ *" ^th PROPORTION. 217. Pboportion consists in the equality of two ratios. The Arithmetical test of Proportion is therefore that the twoJracUons refrrmtnting tho ratios must Im equal. Thus the ratio 6 : 12 is equal to the ratio 4 : 8. be- cause the fraction {\ = the fraction ^ The four numbers 6, 12, 4, 8, written in the order in which they stand in the ratios, are said to be m pro- portion, or xyroportionals, and this relation is thus ex- p resseci — 6 : 12 = 4 : 8. The two terms 6 and 8 are called the Extremkh. 12 and 4 '« Means. The sign of equality is usually expressed thus, : then tlie ratios read 6 is to 1 2 as 4 is to 8. and 218. When four numbers are in proportion, the product of tlie extremes » the product of the means. For example, if G : 12 : ; 4 : 8, 6 X 8 = 12 x'4. For, since -^^ = |, by hypothesis, - JLiUl —. ^ X 12 "12x8 8 X la* Now the denominators of these fractions are equal and therefore the nu.nerators must also be equal, that is^ 6 X 8 = 4 X 12. From this it is evident that if three out of the four numbers that; form a. rk».rtr\/^..f;^v» the fourth. 'I J are given, we can iiad 244 rmopoRTioif. Ex. (1). Find a fourth proportional to 3, 3 : 16 •■ 7 : number required ; 15.7. . .*, 8 X number required ■■ 16 x 7 ; v .'. number required ^ U^ ■> 36. Ex. (2). What numl)er hm the same riitio to 9 ♦iiat 3 haa to 5 1 .< 3 : 6 "■ number required : ; .'. 6 X number required ■■3x0; .'. number required — V ■" ^f • ' 219. Three numbers are said to be in Continurd Propoktion when the ratio of the first to the second is equal to the ratio of the second to the third. Thus 3, 6, 12, are continued proportion, for i •» fy. The second number is called a Mean Proportiokal between the first and the third. Ex. Find a mean proportional between 6 and 24. 6 : required number => required number t 24 ; .*. required number x required number ==6x24; .', square of required number = 144 ; .'. required number is 12. 220. When two quantities are connected in such a way that when one is increased 2, 3, times, the other is also increased 2, 3, tim^, they are in dired proportioi.. For example, if 1 lb. of sugar cost 9 cents, 2 lbs. will cost 2 x cents, 3 lbs. *• 3x9 cents ; hence 7 lbs. " 7x9 cents, and 25 lbs. *♦ 25 x 9 cents ; .'. 7 lbs. : 25 lbs. : : 7 x 9 cents : 25 x 9 cents. That is, the coat of sugar is directly proportional to its weight 221. When two quantities are connected in such a way, that when one is increased 2, 3, times, the other is dimimnhed 2, .S, , , , , , , times-, they are invdraelv proportional; thus, if one man can mow a field in 12 ■iMrLB rftopoRTioir. 94D days, 2 man ran mow it in half the time, or in V davi 3 in«n m a third of tha time, or in "jf day^ Ac ^ Henc« 4 men can mow it in •!« dayi j and 12 " •« ffdayi, .-. 4 men : 12 men :: || dayi : > • dayi ; that ii the ntimft^r »/ m^n roquirofl to do a certain work i. mvrmly proportional to the number of dayt, or™ Jrw B'T^mples. (cxvii.) priportlT"^' *' ^' '' ""^ ^^ '^ '**** '^^y ™»y b* in (2) Find the teoond term when 18, 2-^, and 1-6 are th« other three terms of a proportional ? "' »"« ^ » "» the (3) Find a mean proportional to 038 and -00162 A^toa ^ - ^ °' ^' «»** ^- H of B, and the ratio of (6) Find a fourth proportional to 5, 7, and 16. (6) Find a fourth proportional to |, f and f . (7) Find a fourth proportional to -3, -16, and -09. (8) Find a mean proportional to 14 and 56. (9) Find a mean proportional to ^ and |^ share f^Wel' 8^2. "'"** •* "^ ' '''^'^ " ^ : 3, and 0'. SIMPLE PROPORTION, OR RULE OF THREE. i\!^^f^. /^^? *^J'^ ^J"^^ ""^ ^ proportion are given, to hnd the fouHh, It ,s a Simple Proportion. In a simple proportion we have two ratios given; one of thess has both term«, the other is incomplete, having only one f hi /Ki J'^^'/.u ^® ^'''^'' *^^°"' "^""^^ ^« *>^ <^«e kind, and the third and the answer of another kind. Ex. (1;. If D horses eat 20 bushels of oats in a riven time? '"^"^ ^^'^' """* ^ ^""''^^ ^^' '"^ *^« «^™« nrn^^^f- ^^t ,'^"'?^»' ^^ ^ushels consumed is directly proportional to the number of horses. — v/ ou. .*. bu. required , ou. reuuirea 8X20 B2. 846 HIMPLI FIU}P«>IITI do a pi««<'<» of III o duvH. Ill Ex. (2). If men can do a pi««iij proportioned to the uurt»l>er of iiieii. Hence 9 : fl :: 6 dayi : dayii .wiuired ; .'. dftyii required -^ > " ' - SJ. • Ex. (3). If :i cwt. I und. L hat diatanco will A win at the aame rate of runniag ? ' ^f) A watdll wa« (V/j R,in. gi„^ ^t noon; i* » t2 min m 20h hours ; find tho true time when ifs lianda a.« t<»ether for the fourth time uftor noon. ^9 'M."'^" *'*' ? '^"'"®" ''*' ^ ^y *^an perform a pieot of work m 27 J day. m what timu can (a) 5 men and women lierform it / and (6) 6 men and 8 hoys perform it ? (0) If 14iJ shares of a property are worth ««110. 16, what aw» »;} shares worth ? (7) A floor can be covered by 32^ yards of carpet 7 duar- torj wide; how many yards of Brussels carpet 20 in. width will cover the same room ? (8) Two clocks, of which one gains 4m. ISs. and th« other OSes dm. 16s. in 24 hours, Aore both within 2Am. of the true time, th. former fast aui the latter slow, at noon on Monday ; they now differ from one another by half an hour. *ind the day of the wei^c and the hour of the day. (9) If 6336 stones 3| ift. long complete a certain quantity of^wall, how many similar stones of 2'A feet long wUl raise a (10) A beseiged town, containing 22400 inhabitants, has provisions to last 3 weeks. How many must be sent kway that they may be able to hold out 7 weeks / GOMPOUiYD PROPORTION, 223. Where five, seven, 7nne, &c., terms of a propor- tion are given, to find a sixth, eighth, tenth, &c., teiin, it IS called Compound Proportion or the Double Rule OP Three. In Compound Proportion there are three or more raciua given, all being complete but one. 248 COMPOUND PROPORTION. A Compound Propcrtioi is produced by multiplying together the corresponding terms of two or more simple proportions. Thus, 12 : 6 :: 4:2 9:3:: 6:2 5:4:: 10 : 8 multiplied together produce the proportion 540 : 72 : : 240 : 32. Ex. If 6 men in 8 days, working 10 hours a day, can reap 24 acres of wheat, how many acres could 10 men reap in 15 days of 1 2 hours each ? 6 : 10 : : 24t : acres required, 8 : 15 10 : 12 480 : 1800 :: 24 : acres required ; .. acres required — iiooJLHi = 90. ^ 480 24, thfe term of the imperfect ratio, is put in the 3rd place ; the other ratios are then considered separately and treated as in Sirrple Proportion. After all the ratios have been stated, all the first terms are multi- plied together for a. new fir&t terra and similarly with the second terms. The answer is then got as in Simple Proportion. Note I. — Before compounding the complete ratios it is convenient to cancel all the factors common to the 1st terms, and to the 2nd or 3rd terms. When any of the 1st and 2nd terms are not of the same denomination, they must be reduced to a common denomination before proceeding with the solution. Note II. — Before stating the question it is convenient to write down the terms of the supposition under one another and opposite these to place the corresponding terms of the deihand with an x opposite the term of the same name as the answer required. Thus, in the above example, 6 men 10, 8 days 15, 10 hours 12, 24 acres a^ COMPOUND PROPORTION. :i*i9 Examples, (cxix.) (1) If 18 men in 12 days build a wall 40 feet long, 3 feet thick, and 16 feet high, how many men muBt be employed to build a wall 120 yards long, 8 feet thick, and 10 feet high, in 60 davs ? (2) An engineer engages to complete a tunnel 3| miles long m 2 years 10 months ; for a year and a half he employs 1200 men, an . then finds he has completed only three- eighths of his work. How many additional men must he employ to complete it in the required time ? (3) Two sets of men perform the same amount of work. Each man in the first set is stronger than each man in the s'^cond in the ratio of 7 to 6 ; the first set workd 6 days a •k f >r 10 weeks, and the second set 5 days a week for 7 .ks. If there are 9 men in the first set, how many are there in the second ? (4) If 20 laen can excavate 185 cubic yards of earth in 9 hours, how many men could do half the work in a fifth of the time. * X (5) At the siege of Sebastopol it was found that p. certain length of trench could be dug by the soldiers and navvies in 4 days, but that when only half the navvies were present it required 7 days to dig tho same length of trench. Compare the amount of work done by tho navvies with that do^e by the soldiers. (6) Two elephants which are 10 in length, 9 in breadth, 36 in girth, and 7 in height, consume one drona of grain ; how much will be the rations of 10 ether elephants, which «re a quarter more 'n length and oth. dimensions. (7) How many revolutions will be nade by a wheel which revolves at the rate of 360 revolutions in 7 minutes, while another wheel, which revolves at the rate of 470 in 8 min., makes 658 revolutions ? (8) A piece of work is to be done in 36 c^ays ; 15 men work at it 15 hours a day, but after 24 days only | of it is done ; if three mc^e men are put on, how many hours a day must all work to finish it in the given time ? (9) If 248 men, in 5^ days of 12 hours 3ach, dig a ditch of 7 degrees of hardness, 232^ yds. long, 3^ yds. wide, and 2^ yds. deep ; in hovf many days of 9 hours' each, will 24 men dig a ditch of 4 degrees of hardnoss, 387^ vds. long, 5i yds. wide, and ^ yds. deep ? " »' * -^ (iO) If 5 compositors in 16 days of 11 hours each, can compose 25 sheets of 24 pages in a sheet, 44 lines in a page, ^50 THE MBTBIO SY8TBM. and 40 letters in a line, in how many days of 10 hours each can 9 compositors compose a volume (to be printed in the same kind of type), consisting of 36 sheets, 16 pages to a sheet, 50 lines to a page, and 45 letters to a line ? ^ XXyjI. The Metric System. 224. The Metric System of Weights and Measures is now i use in many countries of Europe. The following is an account of the system as it is established in France, where it originated at the end of the last century. The basis of all measurement is the Metre, a measure of length equal to the ten-millionth part of the distance f roni the North Pole to the Equator. The length of the Metre in English measure is 39-37 inches, nearly. , Units of Metric Measures. 1. Length. — The Metre. 2. Surface. — The Are =100 square metres. 3. Solidity. — The Stere = 1 cubic metre. 4. Capacity. — The Litre = the cube of the tenth part of a metre. > 5. Weight. — The Gramme, which is the weight of a quantity of distilled water which fills the cube of the hundredth part of a metre. The tables of Weights and Measures under the Metric System are constructed upon one uniform 'principle. Prefixes derived from Greek and Latin are at.a,ched to each of the units. Cheek Prefixes. Deca stands for 10 times'^ Hecto stands tor 100 times I ,-, , .. Kilo stands for 1000 times ( *"® ^^^ Myria stands for 10000 times J Latin Prefixes. Deci stands for the 10th part ^ Centi stands for the 100th part \ of the unit, Miia stands for the iOUUth part J THK MBTRIO SYSTEM. 2BX Thus, A decametre = 10 metres. A hectolitre = 100 litres. A kilogramme = 1000 grammes. A myriametre = 10000 metres. Also, A decilitre =» 1 litre. A centimetre = 01 metre. A milligramme = 001 gramme. Note. —In English measures the following are rouch an proximations of some of the French measures : ^ ^" The kilogramme is about 2J lb. Avoird. The litre is about If pints. The kilometre is about 5 furlongs. The hectare is about 2^ acres. MEASURES OF LENGTH, 10 decimetres (dcm. } i metre'(m. ). 100 centimetres (cm.) << 1000 millimetres (mm.) a 1000 metres ''^^i kilometre. 1 inch = 2 539954 centimetres. 1 foot = 3 047945 decimetres. 1 yard = 0-914383 metres. 1 mile =i 1-609316 kilometres. Note.— A rough rule for converting French metres info English yards a to add 10 per cent^ to them '?hus'^ metres are nearly equal to 44 yards. MEASURES OF SURFACE. 100 square decimetres (sq. dcm.) = 1 square metre or 10000 " centimetres (sq. cm.) = '"''*?*"* ^'^' '"•> 1000000 '♦ millimetres (sq. mm.) = «« 100 square metres. i o*.« 10000 " : f®-. 1 lieci«i:c. 1 square inch = 6 4513669 sq. cm. 1 '' foot = %-2899683 sq. dcm. i yard = -83609715 sq. m. 1 " acre = 40467101 hectare. 252 THE METRIC «T8TBM. - MEASURES OF CAPACITY. 1000 cubic decimetres (cb. dcm. ) . . 1 cubic metre or stera. 1000000 cubic centimetres (cb. cm, ) ** 1000000000 cubic millimetres (cb. mm.) 1 cubic decimetre 1 litre. 1 " inch = 16-380176 cb. cm. J 1 •• foot = 28-316312 dcm. 1 gallon = 4-54345797 litres. ^ MEASURES OF WEIGHT. 1 cubic centimetre of distilled water at 4°C. at the sea's level in the latitude of Paris is 1 gram (grm. ). 1000 cubic centimetres of distilled water weighed under the same conditions 1 kilogram (kilo. ). 1000 grams (grms. ) 1 kilogram. , 10000 decigrams . . ** 100000 centigrams . . " 1000000 milligrams . . ** 1 grain = 06479895 gram. 1 Troy oz. = 31 103496 grams. 1 lb. Avd. x= 0-45359265 kilo. 1 cwt. = 50-80237689 kilos. Examples, (cxx.) (1) What is the fundamental unit in this system ? Whence and why was it chosen ? (2) Name the units of weight and capacity, and show how larger and smaller measures are attained. /* (3) Give the English equivalents of a kilometre and kilo- gram. (4) How many millimetres are contained in 5 metres ? (6) How many decimetres are equivalent to 106726 milli- metres ? (6) ^lequired the number of milligrams in 15 cb. cm. of water measured at 4°C. ? (7) How many millimetres and centimetres are respectively contained in 0*437 of a decimetre ? . (8) How many square centimetres are there in lo'o square metres? MSAHDRBMINT OF AREA. 253 (9) How many square decimetret are contained in. 108042 square centimetres { (10) Define the fj;ram and litre. How many grams art contained in 1 725 kilograms ? (11) How many milligrams are there in a decigram ? How many decigrams m a kilogram ? (12) How many centigrams are contained in 2 567 kilo- grams? (13) Required, the number of milligrams contained in 5 cubic centimetres of water measured at 4°0. (14) In an English inch are contained 25 3995 millimetres. How many kilometres are there in a mile ? (15) A gallon is equal to 4 543 litres. How many cubic centimetres are contained in one pint ? (16) Three pipes furnish respectively 30 litres, 45 litres, and 80 litres an hour. What quantity of water do they sup- ply together in 24 hours. XXXIII.— Measurement of Area. 225. The unit of measurement, by which we measure Area or Surface, is derived from the unit of Length. Thus, if we take an inch as the unit of length, and con- struct a- square whose side is an inch, this Square Inch may be taken as the Unit of Area, and the measure of any given area will be the number of times it contains this unit, in accordance with the remarks in Art. 58. Let ABDC be a rectangle, and let the side AB h& 4 inches in length and the side AC 3 inches in length, A '. f'S-J> ■ B * C D 254 MEA8T7BEMBNT O. ' AttRA. i i Then, if the Unit of Length be an inch, the mtiimte of AB is 4, and the measure of AC is 3, Divide AB, AC into four and three equal parts res{)ec- tively, and draw linen through the points of division paralbl to AC, AB respectively. Then the rectangle ABDC is divided into a number of equal aquares^ each of which is a square inch. If one of these squares be taken as the Unit of Area, the measure of the area of ABDC will be the number of these squares Now, this number is the same as that obtained by multiplying the measure of AB by the measure of AC : that is, measure of ABDO = 3x4 = 12 ; .'. the area of ABDC is 12 square inches. Hence, to find the area of a rectangle we multiply the measure ol the length by the measure of the breadth, and the product will be the measure of the area. Ex. (1). A rectangular garden is 48 feet long and 25 feet broad, what is its area ? Taking a foot as the unit of length, and therefore a square foot as the unit of area, measure of the area=:48 x 25—1200 ; .'.the area is 1200 square feet. Ex. (2). A rectangular board is 2 ft. 7 in. long and 1 ft. 4 in. broad, what is the area of its surface ? ' Taking 1 inch as the unit of length, and therefore 1 square inch as the unit of area, measure of the area =31 x 16=496 ; .'. the area is 496 square inches. Or, we might take 1 foot as the unit of length, and then measure of area=2^ X l^ = f|^ = V = 31^ ; .'. the area is 3f square feet. Ex. (3). The length of the side of a square croquet- ground is 49 yards, what is its area ? Taking 1 y.^ra as a unit of length, area = (±2 x 49) sq. yds. = 2401 sq. yds. MIASUWBMBNT OF AREA. 266 Nom-Observe the difference between the expres- sions 49 yards square and 49 square ijarrh. The foJmer referB . a squr.re wf.se side is 49 yards, and whose area « 2401 square yards; the latter to a surface whose area li 4^ square yards. Ex (4). A rectangular bowling-green is 56 yards lonjr comer ^ "^' ^"^ **^' ^^''*"^« from'corner t^ Hence the square of the measure of the side opposite the right angle is equal to the sum of the squares o/ the measures of the sides containing the right angla Thus, in our present example, square of measure of distance from corner to comer = (56 X 56) + (42 X 42) = 4900; .'. distance is 70 yards. Examples, (cxxi.) dimenLnsV''^ ""^ '^'' rectangles having the following (1) 7 ft. by 5 ft. (2) 13^ ft,, by 10 ft. 3) 22i ft. by l^ ft. (4) 5 ft. 4 in. by 2 ft. 3 in ^ !! \ ^ '''' ^y ^ y^' 2 ^*- (6) 5 yd- 1 ft. by 4 yd. 2 ft. (7) 12 yd. 2 ft. by 5 yd. 1 ft. ^ y -^ (8) 6 yd. 2 ft. 3 in. by 2 yd. 1 ft. 5 in. (9) 7 yd. 2 ft. by 5 yd. 2 ft. 6 in. Find the area of the squares whose sides have the lollowmg lengths : (10) 5^ yd. (11) 37^ yd. (12) 17| ft (16) 7 yd. 1 ft. 5 in. (17) 16 yd. 2 ft. 3 in. Find^the breadth of the following rectangles, having given the area and length : a 6 > "o-viug /I Q\ 1 hfn (19) Area 71 sq. ft. 100 , leugtu 11 It. sq. in., length 9 ft. Sin. 266 MEAMTREMENT OF AREA. (20) Area 854 sq. ft. 84 mi in., length 97 ft. 8 in. (21) Area 1 acre, length 440 yd (22) Area 5 acres, length 275 yd. (23) Area 5 ac. 1 ro. 30 po., length 267 yd. 2 ft What are the sides of the squares whose areas are (24) 81 sq. fi (26) 266 sq. ft. (26) 1178 sq. yd. 7 sq. ft (37) 33 ac. 4305 sq. yd. (28) A rectangular field is 226 yards in length and 120 yards in breadth ; what will be the length of a straight path from corner to corner ? (29) A rectangular field is 300 yards long and 200 yards broad. Find the distance from corner to corner. (30) A rectangular plantation, whose width is 88 yards, contains 2^ acres. Find the distance from corner to corner. (31) What is the length of the diagonal of a square whose side is 5 inches ? (32) The) area of a square is 390626 square feet. What is the length of the diagonal ? ^ CARPETING ROOMS. 226. If we know the area of the floor of a room, we know how many square inches of carpet will be required to cover it. Carpets are sold in strips, and when the width of a strip is known, we shall know how much length of carpet will »'-' required to cover a given surface. For instance, if the surface be 162 square feet, and the carpet selected be 27 Inches wide, we reason. thus : 162 sq. ft. = 162 x 144 sq. inches ; .-. length of carpet required = "^^^^^^^ in. =864 in. = 24 yds. Then we find the cost of 24 yards at $1.20 per yard to be $28.80. Examples, (cxzii.) How many yards of carpet, 27 inches wide, will be required for rooms whose dimensions are : a) 15 ft. bv 13 ft. (2) 25 ft. by 12 ft 6 in. (3) 22 ft. 4 in. by 20 ft. 3 in. (4) 27 ft. by 14^ ft (6) 35 ft 4 in. by 27 ft. 3 in. MBASITRKMIHT OF ARRA. 907 Find the expense of carpeting rooms whoM dimen sions are : M6) 18 ft. by 14 ft., with carpet 30 inohei wide, at $1 a yard. (7) 22 ft. by 15^ ft., with carpet 27 inohet wide, at «1.80 a yard. */nl^^ ^^\? *"• ^y ^^ ^^' ^ *"•' ^^^^ o*"^?®* » y*'* '^itJe, at ^108 a yard. (9) 34 ft. 8 in. by 13 ft. 3 in., with carpet | yard wide, at 0*. ijid. a yard. , PAPERING THE WALLS OF A BOOM. 227. To find the quantity of paper required to covek' one wall of a room, we find the area of the surface of the wall by taking the product of the measures of the length and breadth of that wall, the latter being the same as the height of the room. Hence, we shall find the area of the four walls of the room if we take the measure oj the compass of the room and multiply it by the measure of the height By the compass of a room we mean the length of a string stretched tight on the floor, and going all round the room. Deductions for doors, windows, and fire-place must be made in practice. Suppose, then, we have to find how much paper is required for the walls of a room whose length is 22 ft. 3 in., breadth 17 ft. 4 in., and height 9 ft. 6 in. We first find the compass of the room, thus : ft. in. 22 3^ H 17 22 17 } 1 i) dimensions of the four sides. 79 2 compass of the room. To get the area of paper required, we multiply the measure of the compass of the room by the measure of the height, thus : area = {^ x 79J) sq. ft. - IIJ^J^ gq. ft. = 7C2^ sq. a m d 868 MBAHURiiriirF or A NoTi. — Papers, like carpets, ari we know tho width of a Htrip wo shall know' how many feet ill length will Iw requircjd to cover a given 8urfa«!«. Thui, in the room under ooniideration, if the paper be 20 iQoheti wide, length of paper required - (752^2 -^f§) ft. - "HI fi - 451^ ft. Examples, (cxxiii.) How many scjuare feot of paper will l>e required for fooi m "^hose diuienBiona are : (1) Length, 19 ft. ; breadth, 10 ft. ; height, ft.? (2) Length, 24^ ft.; breadth, 18 j ft.; height, 10 ft.? (3) Length, 25 ft. 7 in.; breadth, 19 ft. 4 in.; heiaht. Oft. 9 in.? * ' breadth, 18 ft. 7 in. ; height. (-) Length, 23 ft. 5 in. 9 ft. 6 in.? Find the expense of papering rooms whose diraen- Bions are : (5) Length, 18 ft. ; breadth, 14 ft. ; height, 8 ft. ; with paper lb inches wide, at 20 cents a yard. (6) Length, 20 ft. 6 in. ; breadth, 17 ft. 4 in. ; height, 9 ft. ; with paper 20 inches wide, at 10 cents a yard. (7) Length, 30 ft. 8 in. ; breadth, 26 ft. 6 in. ; height, 10 ft. 6 in. ; with paper 2 ft. wide, at Sd. a yard. (8) Length, 26 ft.; breadth, 21 ft.; height, 10 ft.; with paper 20 in. wide at 9d a yard, allowing for a fireplace which IS 6 ft. 3 in. by 4 ft., a door which is 7 ft. by U ft., and two windows, each 6 ft. by 3^ ft. a Miscellaneous Examples, (cxxiv.) (1) Find the cost of varnishing the floor of a room 14 ft. 4 in. broad, and 16 ft. 6 in. long, ai. 20 cents per square yd. (2) What will it cost to pave an area 146 ft. 9 in. long and 88 ft. 9 in. broad, at 36 cents per square yard ? (3) The area of a square garden is 4 roods 1 pole 29 sq. yd. 6| sq. ft. Find the length of its side ? (4) Find the length of the side of a square whose area is 1 ro. 26 po. 28 sq. yi. 4^ sq. ft. (5) Find the expense « i turfing a plot of ground which is 40 yd. long and 100 ft. wide with turfs each a yard in length and i It. in breadth, the turts, when laid, costing 63. 9d. Der hundred. MBA8i?RKMlNT or AKBjI. Sfif (6) A iouaw room, wh<>M floor iiiiMuitir^ 12 .q yd 1 «, ^JTfhmg It. ceiJmg and walU at 6 ctnu pc^p^wj y^ (7) It c.mi» m to cover the floor of a r..«m 8^ yd. lona bv (8) If the co.t of papering a rtK>m 8| yd. long and «f yd ft mde at 4d per yawl, be £2 m. Sd' wide, with paper 2 .« „.„« find the height of the room. (9) A rectangular field, whoao length is 997 yd 1 ft «nn the'Lld.""' ""'' '"'- ''■ ' ^- '^ *^d tL'brilk'trof of ^whlMrsSTJir" *" '^'" ^" '^ "^"^'^ ««^^ «-»» "d« A /^^^ 1'h« length of a r*Jl® J-'^u ''^ .P*P«"ng a 'oom 21 ft. long, 15 ft. ^^A^ The length of one side of a rectangular field is 679 yards, and the area of the field is 50 ac. 2 ro 32 do ViH the length of the other side and of the diagonal ^ (18) A rectangular field, 300 yards long and 150 broad in separated into 4 equal parts by 2 band« of trees 20 feet^dl lTt\u ''' ''^''- ?r ^^'^' will each part b^, and what will be the area covered by the trees ? , (19) A room, whose height is 11 feet, and len^j, fw-- iu w^'WtZ '*^ ^^'^'f ^^P^*" ^ feetwide lorTto foli walls. How many yards of gilt .moulding will be required ? aeo MIAIIUIIBMBMT Of HOUDITY. (20) Whftt will be the ooet of p«intinff the walle and oeiitng of m room whiMH) height, length, and breadth- are 12 ft 6 In., 27 ft 4 in., and 20 ft. reepeottvely, at 36 ctnU per ■f|uare yard ? (21) Find the expenie of oar( iting a room 16 ft 9 in. long and 13 ft 4 in. broad, with oa-iKit 27 iaohea wide, at 95 oenta per yard. (22) Find the cost of carpeting a room 10 yd. 2 ft. long and 7 yd. 1 ft broad, with carpijt | yd. wide, at $1.08 a yard. (23) If the cost of carpeting a room 1 1 yd. long and 8 yd. ide, with oa of the carpet wide, with carpet at 3/». a yard bo om 11 yd. £11) !&.., find the width (24) How many flag-stunes, each 6-76 ft. long and 415 ft wioie, are requisite fur paving a oloiater which iticloaea a reotaingular court 4577 yd. lung and 41*93 yd. wide, the oloiiter being 12 '45 ft. wide ? yXXIV. Measurement of Solidity. 228. Tho Unit of Measurement, by which we measure the Volume of a Solid body, or the Capacity of a vessel, is derived from the Unit of Length. Thus, if we take an inch as the unit of length, anu construct a cube, each of whose edges is an inch in length, this Cubic Inch may l>e taken as the Unit of Volume, and the measure of any gi/en volume will be thi. number of times it contains this unit. . 229. Let ABDC be a rectangle, and leu the fide AB be 4 inches in length, end the side A' J 3 inches in length. A B £ ...,., < I MBMUftKMBirT 09 mf.mvtY. mi I Then ABOD wiU corjUiu 12 g<|uare inehei (Art. 225). Nov/, ■uppose we conifcrocfc a number of blooki of wood, perfect cuIhjs, whose volume is a cubio inch, and place one o4 these on each side of the squares in ABDC, and then place another of the blocks on the top of each of the iirat set, and so on till wo have piled up 6 layers. Then wo shall have constructed a rectangular solid, •vhose length is 4 inches, breadth 3 inclms, and depth or thickness 6 inches. Now the number of cuolo inches in this solid we estimate in tho following ,vay . for eiioh of ; he squares in ABDC we shall have a pile of 5 cubic inches ; there- fore the number of cubic inches in the solid will he 6x12, or 60 Hence we obtain the following Rule : To find the euhio content of a rectanrnUar mUd^ find *he continuexi prmiuct of the measures of the length, breadth, and thickness, and the result is the msaaure of the cubic content. Ex. (1). Find the cubic content of a rectangular piece of tmiDer whose length is 47 ft., breadth 4 ft., and thickness 2 ft. Taking a foot as the unit of length, and therefore a cubic foot IS the unit of cubic content, measure of cubic content is 47 x 4 x 3 - 564 ; .•. the cubic content is 604 cubic feet. Ex. (2). What is the cubic content of a room v-h(»se length is 22 ft. 6 ia, breadth 18 ft. 3 in. and heiclt 10 ft.? * Cubic content (22i X 18i X 10) cub. ft. ii!l^ cub. ft. = 4100i cub ft. Ex. (3). A rectangular sheet of water, of uniform depth, is 430 yards long, 270 yards broad, and contains 7314300 cubic feet of water. What is its depth ? Reducing the length and breadth to feet, area of surface = (430 x 3 x 270 x 3) sq. ft. ; T Q 1 A 1 e\ft uepta - ^^g^,,o^3 It. = / tc. 202 MBACURKMBNT <>F F-OL'DITY. Examples, (cxxv.) Find the cubic content of the rectangular solids whose dimenBi(. .iS are : (1) 8 ft., 7 ft., 6 ft. (2) 10^ ft., 8i ft., 6^ ft. (3) 5 ft. 6 in., 4 ft. 3 in., 3 ft. 7 in. (4) 11 ft. 8 in., 9 ft. 10 in., 7 ft. 5 in. (5) 6 yd. 2 ft. 4 in., 3 yd. 1 ft. 7 in., 4 ft. 11 in. (6) How many bricks will be required to build a wall 75 ft. lonj<, 6 ft. high, and 18 in. thick, each brick being 9 in. lon^, 4^ in. wide, and 3 in. deep "i (7) A lake, whose area is 45 acres, is covered rvith ice 3 inches thick. Find the weight of the ice in tons, if a cubic foot oi ice weigh 920 oz. avoir. (8) If 500 men excavate a basin 800 yd. long, 500 yd. wide, and 40 yd. deep, in 4 months, how many men will be required to excavate a basin 1000 yd. long, 400 yd. wide, and 60 yd. deep, in 5 months ? (9) A Square block ot stone, 2 ft. in thickness, is in cubic content 6 cub ft. 24 'n. What is the length of its edge ? (10) What weight of water will a reotangular cistern con- tain, the length being 4 ft., the breadth 2 ft. in., and the d',pth 3 ft. 3 in., when a cubic foot of water weighs 1000 oz. ? ^(11) A block of stone is 4 ft. long, 2^ ft. broad, and IJ ft. thick ; it weighs 27 cwt. Find the weight of 100 cubic inches of the stone. (12) A cubic foot of water weighs 1000 oz. Find the length of the side of a cubic vessel whose contents (water) weigh 4 torn. 12 cwt. 3 qr. 10 lbs. 7 oz. (112 lbs. = 1 cwt.) (13) If 120 men can make an embankment | of a mile long, 30 yards wide, and 7 yards high, in 42 days, how many men would it take to make an embankment 1000 yards long 36 yards wide, and 22 feet high, in 30 days ? S^^o^- ^®*^**"g"l»^ cistern, 9 ft. long, 5 ft. 4 in. wide, and 2 ft. 3 m. deep, is filled with liquid which weighs 2520 pounds. How deep must a rectangular cistern be w'.ich will hold 3850 pounds of the same liquid, its length being 8 ft, and Its width 6 ft. 8 in J ^ (15) Find the cost of making a road 110 yards in length and 18 feet wide ; the soil being first excavated to the depth 1 -J o P^'rf"* *,*^®®* ^^ ^^' P^^ ^^^^^ y^^<^5 rubble being then leid 8 inches deep, at Is. per cubic yard, and gravv^l placed on the top, 9 inches thick, at 2s. 6d. per cubic yard. EXAailNATlON PAPERS. 263 EXAMIKATJOlsr PAPERa tiiil fn -rmrr!;^'' such that if it be added twenty-three times to Ji 001, the sum will be 40200. and retailed it at $'3. 35 per yard. What was his prSfiJ ?^ ' (3) Find the H. C. F. of 372, 837, 248 ; and arrange the three fractions I H. U in orde; of magnit' -de ^ Fii'i ht Ch oft:c\'''' p^^^^ ^" * «^-^^ ^' ^ «"- miS ^iTJ"^^ ^^^^ ^"'*l"T ^"*° *" ^'^"^^ »"™^«r of sover- eigns, half-sovereigns, half-crowns, and farthings. r«i?L^'''''^fu ^* ^^7' ^'^^ ^y *^^' '^"d "^14 by 7000 ; add the results together, and turn the decimal into a vulgar fraction (7) Simplify the expression 7 57 x '36 - 2*345 IlidU'his'tfmilt"''" "' ''' ^''*^ '' "'^""'^ ^-*- (9) If telegraph posts are placed 66 yards apart, and a tram passes one in every three seconds, how many miles an hour IS the train running ? j *^b »n oali^^- ^{u P^^r*^ *P'^"*^« ^" four months as much as he tnat He earns $420 every six months ? ^' rr s 'io^lPJ^'"'! many steps does a man, whose length of pace is 32 inches, take in 4| miles ? ^ Jl!^ ?^'''a^ ^^'^^f ^^*^^^" - "^«"' «o that one may re- ceive a third as much again as the other. th^'riu^'ll'^d^™;,* - A by I + i - A, and express (14) Find the value of m - n) -^ yf ofj 2| - (I + 1) and express the result as a decimal. (16) Simplify the expression 1*3 x (2-4 f 7-6) -f 2-364 - l-69t. fJ}^' ^.t^"''® H/y- V^ P^- ^1 y^- *o inches, and find what fraction the rosult is of 3 acres. (17) A is to receive $1.25 a day every day he works. .'•ft ^1 .41 264 EXAMINATION PAPSIti. and to forfeit ^SO every day he is idle. At the end of 7B days his wages amount to $09.15. How many days was he idle? (18) If 24 men can do a piece of work in 12 days of 10 hours each, how many men can do three times as much in 10 days of 8 hours each ? (19) If -3 of an estate is worth $7600, what is the value tt '48 of the estate ? (20) A, By and C start on a tour, «ach with $200 in his pocket, and agree to divide their expenses equally. When they return A has $37.60, B $50.82, and G $10.71. What ought A and B to pay C to settle their accounts ? r (21) Find the value of H of 9x5 i4x3 -l5* in the lowest terras and redVice to its lowest terms Ml'v (22) Express as vulgar fractions 24 0025 and 0008125 ; and divide 11214 by 6-34 and 1121*4 by -534. (23) What fraction is 7 cwt. 4 lb. of 3 tons 1 qr. (long ton)? How often must one go round a square field of 10 acres to run 1 mile ? (24) A gunboat's crew, consisting of a lieutenant, a gunner, and 15 seamen, captured a prize worth ^6399 7«. ; the lieutenant's share is 10 times and the gunner's share 3 times as much as that of each seaman. What is the value of each person's share ? (25) Extract the square root of 107-9616, and of ^V (26) A clock which loses 4 minutes in 12 hours is 10 minutes fast at midnight on Sunday. What o'clock will it indicate at 6 o'clock on Wednesday evening ? (27) The distance between two wickets was marked out for 22 yards, but the yard measure was ^-^ of an inch too short. What was the actual distance ? (28) What is the difference between simple interest, i -i- pound interest, and discount ? Find the difference bet\. aen the simple interest and the true discount on $1900 for If years at 8 per cent. (29) What is the present worth of a bill of $170, due in 4 months, reckoning money at }i per annum ? * '■ ^^-; W EXAMINATION PAP£R8. 265 " (30) Find the interest on $880 for 1} years at i^ percent, and the discount on $929.60 tor 2^ years, at 2^ per cent. (31) Simplify Oi°'n) 3 0(3!"' )■ 14 (32) Find the vulgar fraction equivalent to 101015 •55 (33) Which is the better investment, the 3^ per cents, at 91 or the 4 per cents, at 103 ? How much must a man invest in the former that iie may have a yearly income of $4851, after paying an incomt tax of 2 cents in the dollar ? (34) Two ships get under weigh at the sam time for the same port, distant 1200 miles. The faster vessel averacjes 10 knots an hour, and arrives at the port a day and a halt before the other. What will the latter vessel average an hour ? (35) Divide $87.50 between two men, so that one may receive half as much again as the other. (36) A man has |3430 stock in the 3^ per c nk. at 83|; when the stock rises 2 per cent, he transfers his capital to the 4 per cents, at 98. Find the alteration in his income ? (37) The weight of the water contained in - rectangular cistern 8 ft. long, 7 ft. wide, b 93f cwt. If a cubic foot of water weigh 1000 oz., find the depth of wt ter in the ciptem. (38) If $S is the discount off |333 for 2 months, what was the rate per jcent. ? What should be the discount off |333 for 1 year ? (39) The height of a tower on a river's ' ^ak is 55 feet, the length of a line from the top to the oppusi*^ • bank is 78 feet. What is the breadth of the river ? (40) How many yards of matting, '^ , eet broad, will oovw a floor tfcat is 27 3 feet long and 20 16 feet broad ? (41) Simplify the fraction n of li - I of J^ 2 "^lii 2 17" 206 EXAMINATION PAPERS. (42) If J of U of an estate .be worth $300, what will be 2A the value of .- of the estate ? (43) Of an electric cable \}f rests on the bottom of the sea, l\ hangs in the water, and 234§ yards are amvAoyeX on land ; what is the length of the cable 1 (44) Extract the cube root of 16777216. (45) At what price must an article, which cost 15s., be sold so as to gain 10 per cent, ? (46) The number of disposable seamen at Portsmouth is 800, at Plymouth 756, and at Sheerness 404. A ship is com- missioned, whose complement is 490 seamen. How many must be drafted from each place so as to take an equal pro- portion ? (47) (a) Find the difference between the simple and com- pound interest of $416. 66| for 2 years at 8 per cent. (6) Find the rate of interest when the discount on $211.60 due at the end of 1^ years is $27.60. (48) What sum will amount to $3213 in ten ye-i's at 8 per cent, simple interest ? (49) The length of a rectangular field which contains 4 ao. 3 ro. 14 po. 26^ sq. yd. is 260 yd. 1 ft. 4 in. ; what is its breadth ? (50) A room is 14 ft. 3 in. high, 20 ft. wide, 24 ft. long ; what will it cost to paper it with paper 2^ ft. wide, whose price is ll^d. per yard ; allowing 8 ft. by 5 ft. 3 in. for each of four doors, 10 ft. by 6 ft. 8 in. for each of tv'o windows, and 6 ft. 6 in. by 5 ft. for a fireplace ? (51) Simplify the fraction ^~:^* of ^-i 34-i 2-H 3+i 2+i 3 - i " 2 - ^ (52) Find the value of . •003 of £1 5s. + -069* of £5 - '8 of 2s. Sd. (53) If I of the cargo of a ship be worth $16000, what will be the value oi ^ oi ^ of the remainder ? (5^'' A can mow 5 acres of grass in 3 days, B 7 acres in 9 da,y«, C 11 acres 'n 12 days ; in how many days can they jointly mow 121 acres ? EXAMINATION PAPERS. 267 .s ^2rr\ tlTf ^; ^J"«^.\i« \«*- 4» «• fast on Monday at noon, aid U lose in'rly r'"^'^' ^'^ '''' tollowingSuniay : what (56) Tlie rent of a farm is $720, and th^ taxes are 14| together r^" '" ""''''' ^°^ '^'' '^^"""'' °^^«^^ ^"^ taxef (57) Three persons divide the cost of an entertainment amongst the- m such a manner that the first pays i of the wliole and the second g of what the first pays, and tie third tirbillV''""'' ''' ^'"^l^i^i^'-^^^: what is^tlie amount of (58) If an income ot «51200 pays $i8 for income tax, how much must be paid on an income of $750 when the tax is lialt as mucli agam ? (59) J invests $552 in the 8i per cents, when they are at 92 ; B mvests $079 m the 8 per cents, when they are at 97. t md the difference of their incomes. (60) What is the cost of the carpet for a room, the dimen- sions of which are 21 feet long, 15| feet wide, a^ 4U cents per square yard ? ^ (61) Simplify: + 3 m •281 ^06 \ 4i + 5i (62) A regiment marching 3^ miles an' hour makes 110 ^teps a minute : what is the length of tlie step ? (63) How long would a cohiran of men, extending 3420 feet m length, take to march through a street a mile long at tlie rate ot 58 paces in a minute, each pace being 2^ feet ? (64) A street being 850 feet long, and the width of the ^'''''^f'TL''.'' ^^''^' ^'^^ ^"^^"^ ^ ^^^^ 3 i^-find the cost of pav- lug It at 37^ cents a square foot ? ^ (65) Two pipes together fiU a cistern in 1 hour- one of tlier tl\\f? '* "''' ^^ ^''''''- ^'''^ ^^"^- ^^" '^ *^^« ^^^ (66) How many hours a day must 42 boys work, to do in t^^ days what 27 men can do in 28 daysof lOhours long ; the nork of a boy being half that of a many ? f S At what rate will the simple interest on $125 amount to ^13.1^^ m J ^ years? (68) What principal will give $616 simple interest in 6* years at 6| per cent. ? r « *« uf 268 RXAMINATION PAPBR8. (69) A log of timber is 18 ft. lone, 1 ft. 4 in. wide, and 15 in. thick. If a piece containing 2^ solid feet be cut off the end .of it, what length will be left? (70) If 8 guineas be expended in purchasing BrusseU carpet | yd. wide, at 3h. Qd. a yard, for a room 20 ft. long and 16 ft. 9 in. broad, how much of the floor will remain uncovered ? (71) Simplify; n + n ^ + H 1 + ;06 •6' (72) 2 + i Find the value of •02 of £1 + -03 of 7«. Gil + 014 of 2s. 9rf. (73) Extract the square root of 30712 •6026 of jM/V, and of •000000133226. (74) A bankrupt owes $7860, and pays 37^ cents in the dollar. How much did his creditors jointly lose ? (76)| If 14 men can mow 35 acres of grass in 6 days of 10 hours each, in how many days of 12 hours each can 3 men mow 24 acres ? (76) If 9 men or 16 women could do a piece of work in 144 days, in what time would 7 men and 9 women do it, working together ? (77) Divide $2849 among A^ B, and (7, in the proportion of -7, -28, and 056. (78) The mathematical discount on a sum of money for 2 years is $360 ; the interest on the same sum for the same time is $400. Find the sum and the rate per cent. (79) Find the gain or loss per cent, in buying oranges at $2.50 per hundred and selling them at 8 for 12 cents. (80) What will bo the cost of papering a room 21 ft. long by 15 ft. broad and 11 ft. high, which has two windows, each 9 ft. high and 3 ft. wide, a door 7 ft. high and 3 ft. 6 in. wide, and a fire-place 4 ft. high by 4 ft. 6 in. wide, with paper 2 ft. 3 in. wide at 9s. a piece ; the price of putting it on being 60?. per piece, and each piece containing 12 yards ? (81) Simplify (1) (2) 2| -H + 9tV ^ - 2i + 13/t (3-71 - 1908) X 7 03 2-2 - ^, BXAUINATION PAPERIJ. 269 (82) A man ^ wnn f\ of a mine, and sells l^si of his share. What fraction tn the mine has he left ? (83) A and B can do a piece of work in 8 days. R and C can do It m 12 days, and A, B, and C can do it in « days. In how many da/s caa ^ and C do it ? (84) A clock which gains 7J minutes in 24 hours is 12 minutes fast at midnight on Sunday. What o'clock will it indicate at 4 o clock on Wednesday afternoon ? (85) Gunpowder being composed of 33 parts of nitre, 7 of charcoal, and 5 of sulphur, find how many pounds of each will be required to make 30 lbs. of powder. xxJ?^l ^l^^^ ^^ *^® difference between Interest and Disoop.nt ? Which of the two is greater ? Find the difference between the interest and discount on f 1639 for 4| raos. at 6^^ per cent. (87) Find the difference between the true and bank dis- counts on a note of $10400 due in 6 months (days of grace included), at 8 % per annaui. ;C3) f of ^'s stock was destroyed by fire, ^ of the re- mainder was injured by watoi and smoke; he sold the uninjured goods at cost price, and the injured goods at a third of cost price. He realized $1155. What did he lose by the fire ? (89) Having given that the weight of a cubic foot of water is 1000 oz., and that the imperial gallon contains 277 274 cubic inches, find the weight of a pint of water. (90) A room is 22 ft." 6 in. long, 20 ft. 3 in. wide, and 10 ft. 9 in. high. Find the cost of carpeting the room at $1. 20 a square yard, and of papering the walls at 20 cents a square yard. (91) Simplify •004 4- 0005 2-423 + 3-576 + 2 0001911 (92) The quotient in a division question equals six times the divisor, and the divisor equals six times the remainder ; the three amount together to 516. Find the dividend. ^93} Add together '60625 of ^1 + 142857 of Us. lOhd., and If of ^T- of £3 5s. Id., and express the result as the decimal of 27 shillings. (04) A clock gains 3^ minutes a day. How must the 270 BXAMINATION PAFBRH. hands bo placed at noon bo ar to point to true time at 7 h- 30 m. P.M.? (05) A person investR $750 at simple interest, and at the end of 3 years and 8 months he ^nds that be possesses $950.25 ; at what rate per cent, per annum vras his profit ? (1)0) A person's half-yearly income is derived from tho proceeds of $4550 at a certain rate per cent. , and $5420 at 1 Eer cent, more than the former. His whole income is $453. determine the rates. (\)7) What will be the cost of enclosing a rectangular garden, 90 yd. long and 30 yd. 2 ft. 3 in. broad with a wall 8 ft. 4 in. high, at the rate of $1.20 per superiioial square yard ? (98) A person invests £10000 in 3 per cents, at 75, and when they rise to 78 he sells out and invests the produce in bank shares at £208 each, which pay a dividend of £8 per share. Show that his income is not altered. (99) What must be the least number of soldiers in a regi- ment H;o admit of its being drawn up 2, 3, 4, 5, or 6 deep, and also of its being formed into a solid square ? (100) If $40 is a proper discount off $300 for 8 months, what should be the 12 months' interest on $360 ? (101) Multiply 5V375 by 729819 with three lines of multi- plication, and divide 123456 by 03, using short division. (102) A French metro = 1 0936 of a yard, and a centi- metre is the hundredth part of a metre. Find a centimetre in decimals of an inch to 4 places. (103) A and B can do a piece of work in 4 days, B and C in 5| days, and A and C in 4| ds^ys. In what time can each do the work separately ? (104) M starts from and travels towards D at a rate of 6 miles per hour ; two hours afterwards JV^ starts from C, and going 10 miles per hour reaches D 4 hours before M. Find th(^ distance from to D. (106) Find the simpk interest on $2733^ at 4 per cent, for 3 years and 9 months ; and determine what sum will amount to $926. 10 in 3 years at 5 per cent, compound interest. (106) Find the difference between the discount on $1622.50 for 14 months at 7 per cent, per annum and the interest on $1760 for 15 months at 6 per cent, per annum. (107) A women buys a certain number of apples i^ 3 a EXAMINATION I'AFRRII. 27i penny, aiid th« H.imo number at 2 a penny ; the then mixea r hem and sella them at 5 for twoi>enci. rfoV much d >^ i^ Ljam or lose per cent.? • u ^o> Bno il^^wi Pf *■«""» J'y.^^PownK of gmxls for fl82, loses 9 per i.mt What ought they to have been sold at to realkTa |)ro6t of 7 per cent. ? *^«*m«» » (100) Find the cost of papeiing a room 14 ft. 5 in. long, hi t. 7 in. broad, and 12 ft. 3 in. high, with paper at !« oonts P«r square yard. In the room a?e 4 wi„cf,wa 4 ft. ht by Vft ■ '"• ^^ ^ ^'' ^ ^"^ ^^ * ^^^^P*'**'^ ^ ^^ 1 ^^1?^^'?'*.®?^''''",'*! ^•'"^n^'on* of a box without a lid are, length 4 feet, breadth 3 feet, depth 2 feet, and the thickneai of the sides and bottom is the same, namely 1 inch. If the makin/flf"K° ^^'1 ''^*I\^ "^*'*'"*^ '" ^*' ^""^ ^^8 cost of making the box » ^^ of the cost of the material, what will tne box cost i «fii o«f *f h^ :'^*^'" ^''"'"^ together at the same in- Btant, and they toll at intervals of 1, 2, 3, 4. 5, 6, 7, 8 seconds ospectively. Aftey what time will they be again tolling at the fame instant ? .^ » ♦*» »v (112) Simplify H- jot U H " (i V 7 + iol 6 18 °S^) 4 7 (113)^, A and O are partners ; A receives two-fifths of tlie profits, B and dividing the remainder enuallv • ^'s income 18 mcreased by $220 when the rate of 'profit rises from 8 to 10 per cent. Find the capital of B and C. (114) A railway train, having left a terminus at noon, is overtaken at 6 p.m. by another train which leit the same erminus at 1 p.m. If the former train had been 10 mile. farther on the road when the latter started, it would not have been overtaken till 8 p.m. Find the rates of the trains. /ii^l^/1''^''" '"""^^^^ ^^^^ i» Turkish 6 per cent, stock at »u ; find the rate of interest he gets for his money. When his stock has risen to 104 he sells out, and buys £20 railway Shares at i'18, which piiy dividend at the rate of U per cent i^ind the alteration in his income. ^il^^ ^^ ^ ^^^ ^^^ ^ ^^y ^^^ ^^*P 13 acres in 2 days, and 7 men and 6 boys c^n reap 33 acres in 4 days, how long will It take 2 men and 2 boys to reap 10 acres ? I 979 KUMINATION FAPimS. Iff (117) The cott price of a book !• $4.75, expenie of thft •ale 6 %, profit 24 % ; what ia the retail price ? (118) Show that the aiinplo intereat on $626 for 8 monthi at 7 % is eiixxaX to that ojs $1CK)3.76 at 8 % for 4 month*. (119) One olook ^aIub 4 niinutei in 12 hours, and another lotea 4 miuutea in 24 hours. They are set right at noon on Monday. Determine the time indicated by each clock when the one appears to have gained U\^ minutes on the other. (120) A rectangular court is 50 yards long and 80 yards broad. It has oaths joining the middle points of the oppo- site sides of feet in breadth, and also oaths of the »«*.me breadth running all round it. The remainder is covered with ?;nuMi. If the cost of tho pavement be 12^ cents per square cot, and of the grass 70 conts per square yard, tind the whole cost of laying out the court. (121i) How many times does the 29th day of the month occur in 400 consecutive years i (122) A creditor, agreeing to receive 1^281.25 for a debt, finds that he has been paid at the rate of 62]^ cents in the dollar ; how much was the debt ? (123) A J B, and rent a meadow for $43. -4 puts in 10 horses for 1 month, B 12 oxen for 2 months, and V 20 sheep for 3 months. How should the expense be divided if the quantities eaten by a horse, an ox, and a sheep during the same time be in the ratio of 4, 3, and 1 ? (124) If the price of 9760 bricks, of which the length, breadth, and thickness are 20 inches, 10 inches, and 12^ inches respectively, be $213.50, what will be the price of 100 bricks which are one-fifth smaller in every dimension ? (125) How many years* purchase should I give for an estate so as to get 3| per cent, interest for my money ? (126) How often between 11 and 12 are the hands of a clock an integral number of minute spaces apart ? (127) A and B walk a race of 25 miles ; A gives B 45 minutes' start ; A walks uniformly a mile in 11 mmutes, and catches B at the 20th milestone ; find B'a rate, and by how much he lost in time and space. (128) A debt is due at the end of 4| months; | is paid immediately, and I at the end of 3 months ; when ought the romainder to be paid ? MXAMINATION i'APRRM. f7» (120) A m»n, by teWma out of a 3 p«r o«nt. itook at 99, pins IC per o«nt. on hb rnvostinent. At what price did he buy, and what wai hi» inoomo, auppoting that he realized S]ftd4ft? (laO) A tank is 8 ft. long, 5 ft. 4 in. wide, 4 ft. 6 in. deep. Find the number of gallonii it containi, having given that 1 oub. ft. of water woigha 100() oz., and that a pint of water woighe a pound and n ({uarter, (131) Simplify /3 2\ /13 1\ 2,3, "^^13-9>'"V3 +0/-^3^^8"^*^- 71 of ihh + ^A (132) In a dormitory j^jf of the boys are in the upper school, f of the reniaiiidor in the niiddlo, and the rest, 8 in number, in the lower. Find the number in the dormitory. (133) The circumference of the fore-wheel of a carriage is 8 feet, and that of the hind-wheel is 10 feet. In what dis- tance will the fore- wheel make 100 revolutions more than the hind-wheel ? (134) A and B receive $1.37^ for digging a garden. They work at it toj^ether for 4^ hours ; B then left, and A finished the work in 3^ hours. How should the pay be divided ? (135) What are the two exact times when the har.ds of a watch are equally distant from fig. III.? (136) In how many years will $320 double itself at 7^ per cent, simple interest? (137) A person invests the present value of £2368, due two years hence at 4 per cent. , in gas shares, which pay at the rate of 9 per cent. ; he gives £144 for each share of £100. What is his annual income, and what rate per cenc. does ho make of his money invested in the gas shares ? (138) At Ibilliards A can give B 5 points in a game of 60, and C 10 points in 60. Huv/ many points can B give G in a game of 90 ? (139) How much money must one invest in 3 per cent. Consols, when they are at 10 per cent, below par, in order to have an income of £2000 a year ? (140) A level reach in a canal, 14 miles 6 furlongs long and 48 feet broad, is kept up by a lock 80 feet long, 12 feet broad, and having a fall of 8 ft. 6 in. How many barges ^<.s^. IMAGE EVALUATION TEST TARGET (MT-3) 1.0 III I.I ^ ^ m ■^ 1^ 112.2 lU lit u 1^ 12.0 1.8 ' 1.25 1.4 1,6 ■* 6" ► p ^. /a ^ 0> V. ^ ^ >> /^ Photographic Sciences Corporation « 1. •"^ \ 4s^ % v ^^^ >J^ ^^^ 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 \ \ 'tl" £74 EXAMINATION PAPEB8. might pftss tlirouffh tlio lock before the water in the uppei canal was lowcrea one inch ? (141) Find the value of 31 ?-i of $5.67. 71 of J -^ l + l (142) i4 can do a piece of work in 6 days, which j> can destroy in 4. A has worked for 10 days, daring the last 6 ol whicli U has been destroying ; Jiow many days must A now work alone, in orde^* to complete his task ? (148) Two cisterns of equal dimensions are filled with water, and the taps for both are opened at the same time. If the water in one will run out in 5 hours, and that in the other in 4 hours, find when one cistern will have twice as much water in it as the other has. (144) If u men, 4 women, 5 boys, or 6 girls, can perform a piece of work in 60 days, now long will it take 1 man, 2 women, 3 boys, and 4 girls, ali working together? (145) Two trains start at the same time from London ami Ediuburgli, and proceed towards each other at the rates of 80 and 60 miles per hour respectively. When they meet, it is found that one train has run 100 miles farther than the other. Find tlie distance between London and Edinburgli. (146) Two persons buy respectively with the same sums into the 8 and 8^ per c^nts., and get the same amount of interest. The 3 per cants, are at 75 : at what price are tlie 8^ per cents. ? (147) Divide $1986.60 among A, B, and C, m the propor. tion of 2*3, 1-15, and '52,4 respectively. (148) If for a sovereign one can buy 11 gulden 12 kreut- zers or 26-5 francs, and for one 20-franc piece gulden 2J kreutzei-s, how much per cent, is gained by buying French gold with English gold before buying German money ? (149) Express 69^ miles in metres, 32 metres being taken to be equivalent to 35 yards. (150) Find the cost of painting the walls ot a square room 14 ft. high and 18 ft. long, with two doors 8 ft. by 4, and three windows 10 ft. by 5, the amount saved by each window bemg .£2 I65. 3d. What additional height would increase the cost by nine shillings. n 2i + 3^ (151) Simplify -^ + g|- --^ + i + I ol A. EXAMINATION PAPBBS. 275 (152) Two lines are 41 06328 inches and 0438 of an inch bHuTTSh ?"" "'^"^i'"*^^ ""' long aa?^ett?e "^t (153) ^ and ^ start to run a race ; their speeds are a« 1-7 to 18. ^ runs 2^ miles in 10 min. 48 sec. ; B finishes the course in 34 minutes : determine the length of the bourse. (154) A boat's crew row over a course of a mile and a quarter against a stream which flowB at the rate of 2 ailes an hour, m 10 minutes. The usual rate of the stream iahaK in^ usuilTtlte f; tL*i V;-^^ ^'^^^ ''' ^- ^-^'^^ What was his original income ? ^ "^ *^- (166) A man invested $14350 in the U. S. O's at 107i the brokerage being i°/ ; what will be his clear income after an income tax of 6% is deducted ? (167) A soldier has 5 hours' leave of absence : how far may he ride on a coach which travels 10 miles an hour To L 1 hour" '^'^'P ^" '^"^"' "^"^^^"8 ^' '^' rate o75 milS (158) Two trains start at the same time, the ono from London to Norwich, the other from Norwich to London S ^hey arrive m Norwich and London respectively 1 hour and 4 hours after they pass ea^h other, show that one travels twice as fast as the other. travels (159) When £170 will purchase 4233 francs, what is the course of exchange between London and Paris ? And f 50? .-old pieces of 20 francs contain as m^ch pure goW as 400 nZI^'' '' *^^ ^'*' °^ ^^"^^°8" ^^*^«^" ^^^on and (160) A hollow cubical box, mad^of material which is 1-3 inches in thickness, has an interior capacity of 50-653 cubic feet : determine the length of the outside edge of the box increase 1+i (161) Simplify (o^ of -S.zJ]&_\ ^ H ^ "^ V ^ 121-7/2/ ■ 2 (162) Gold of the value of £423267 arrive, from Australia • What 18 its weight m lbs. avoirdupois, the price being £3 ISs.' 276 BXAHINATION PAPERS. (163) A can do one-half of a piece of work in 1 hour, J3 can do three-fuurth» of the remainder in 1 hour, and G can finish it in 20 minutes ; how long would Ay By and (J togc^ther take to do it? (104) If 1 pay «760 now for a debt of ^771. 09f not yot payable, and money be considered worth 7^ per cent, per annum, when will the debt be due ? (165) Two equal wine-glasses are filled with mixtures of spirit and water in the ratios of 1 of spirit to 3 of water and 1 of spirit to 4 of water ; when the contents are mixed in a tumbler, find the strength of the mixture. (166) At what per cent, in advance of cost must a merchant mark his goods so that after throwing off 20 per cent, of the marked price he may make a profit of 25 per cent. ? (167) A man receiving a legacy of $34510 invested one- half in Dominion 6 per cents, at 101, and the other half in U. S. 5 per cents, at 84^, paying brokerage at ^ % ; what annual income did he secure from his legacy ? (16^) A piece of work must be finished in 36 days, and 15 men are set to do it, working 9 hours a day ; but after 24 days it is found that only three-fifths of the work is done. If 3 additional men be then put on, how many hours a day will they all have tu labor in oider to finish the work in time ? (169) Of two stalactites hanging from the flat roof of a cavern, one is 102 inches longer than the other, and the shorter one increases in length at the rate of 3 014 inches in a century. Find the rate of increase of the other, in order that they may be of the same length at the end of 125 years. (170) Two men, A and B, start from Cambridge, at 4 and 5 o'clock a. m. respectively, to walk to London, a distance of 50 miles; B passes A at the twentieth milestone, and reaches London at 5 p. m. When will A arrive there ? (171) Find the square root of 10747*4689, and the cube root of 189119224 (172) A person can read a book containing 220 pages, each of which contains 28 lines, and each line on an average 12 words, in 5^ hours ; how long will it take him to read a book containing 400 pages, each of which contains 36 lines, and each line on an average 14 words ? (173) The whole time occupied by a train 120 yards long, 1 ■XAMINATION PAPlRa. 277 nd the cube travelling at the rate of 20 miles an hour, in croasing a bridflt 18 18 seconds. Find the length of the bridge. (174) If 20 men, 40 women, and 60 children receive 84200 among them for seven weeks' work, and 2 men receive as much as 3 women or 5 children, what sum does a woman receive per week ? wuu«wi iJiriJ"? olocKs begin to strike 12 together; one strikes m 36 seconds, the other m 25. What fraction of a minuto is there between their seventh strokes ? l,«n7?L^ speculator bought 43 shares in a mine at 35^, and bcight with the proceeds 6 per cent, railway stock at 28 premium. Fmd hisannual income from the latter investment. oi^WpHnl.^ °^'''^' "^''^^ ^ ^""^^^^^^ «" Tuesday morning. ?hl ^X f^y morning one wants 10 minutes to 11 when hXh.T.J*"^"'' ^\ i^S'" ™"°»' ™"«*^ *he faster be put back that they may strike 9 together on Wednesday evening 1 (178) How much ore must one raise that on losing U in roas .ng and ,^ of the residue in smeltii., there mTy rVsult 50b tons of pure metal? ^ t^ouiu ari^P?nVr/'Jl**'T il''?'^ ten millions, and the births are 1 in 20 and the deaths 1 m 30 annually, what will the population become in 5 years ? (180) There are two rectangular fields equal in area- the sides of one are 946 yards and 1344 y.:r?s in length,' and ha onger side of the cecond is 1134 yards. What i^ the eTch field? "^'^ '"^ *^"^ "^""y *^'«^ *^« *^«^« i" hJr^ l^ , I ""^^n ""^ ""^ ^ ^''^*'®^ ^^® '^^16 of its ^hole num- beckm« n^tr P^^^v,"^^^^?^' t.^^^ ^^^" *^^«^ *he masters were^hprfhlf *^if ""^^^t ^^^ ™*°y *>^y« ^^^ masters were there before the new boys came ? (182) Divide $350 among 4 persons, so that B may have hree times as much as J, half as much again as ^ and J together, and D as much as A, B, and G together ? wPf^^ By selling a house for $3700 I lost 7A per cent. What must I have sold \i for to have gained 12^ per cent.? ^^?^^ ?\"«*i.*^^® diflference between the interest and iia- eount on $1265 for 73 days at 6 %. n»P^^ *^ merchant sells tea to a tradesman at a profit of 60 per cent., but the tradesman, becoming a bankrupt, pays 278 EXAMINATION PAPER8. 37^ oenta in the dollar. How much does the merchant ^ain or lose by the sale ? (18o) What sum must a man invest in the 6 pe'r cent. County bonds at lOij^ in order to have a clear income of $1424.40, after paying an income tax of 1^ % on all over $400 ? (187) A baker's outky for flour is 70 per cent, of his gross receipts, and other trbdc expenHos 20 per cent. The price of flour falls 50 per cent. , and other trade expenses are thereby reduced 26 per cent. What reduction should he make in the price of a 16c. loaf, allowing him still to realize the same amount of profit ? (188) What is the average annual pro£it of a business when a partner, entitled to f of the profits, feceives as his share for 2 years and 4 months the sum of $7890.60? (189) If '\ tradesman adds to the cost price of his goods a profit of 12i per cent., wlict is the cost price of an article which he seUs for $3. 82^ ? (190) .A rectangular piece of ground 72 yards by 45 yards is to be laid out in 4 plots of grass, each 27 feet by 13^ feet, and a pond in the centre 6 yards square, to contain 262 cubic yards of water. Find the expense of gravelling the remainder at 2f cts. per square yard, and the depth of the pond. (191) Find the value of 6| of f of 2| i^(} + h) l-^of U+iof I' vtj } (132) If 12 men or 18 boys can do | of a piece ^' work in 6^ hours, in what time will 11 men and 9 boys dc .he rest ? (193) Find the principal sum on which the simple interest in 2 J years at 6^ % per annum is $1068.75. (194) The compound interest on a certilin sum at 4 per cent, for 2 years exceeds the simple interest for the same time at the same rate by $6. What is the sum ? (195) Two ships are built. Twice as many ship-carpenters are employed about the first as about the second ; the first is built in 9 months, the secord in 8 months ; the wages of each man of the first set are 25 cents per hour, and they work 12 hours a day ; the wages of each of the second set are 18 cents per hour, and they work 10^ hours a day. The cost of the first in carpenters' waget was $300.00 ; what was that of the second t ohfflntTZrbrt&o'rK^ divided ..„,„ hi. 8,e ha. been p.id, ^^ ^l^'J^CJ^'^^uTt'X'T' ""'^ each Drother's. The leanov A,uJ oe twice as great aa one per cent., and oii a tofcherWr * '*'**^'" »*»"« ^»»8 each will rooe/ve brother i three per oeni, find what $360.50 a/ainst ^ fnr w^i t I .^^^' *"^ ^ ^a» a bill of discount [TIM^^^^^^ months' annum, what hae B to ppy /? ^ " ^ P®^ °®"*' Per much does higai^ or1o!.e;"er rnt?'' "' '^'' ^' ''^' «- coul't^y Lto'in>norf YoSL^' ^^^^^ '" f'^^^^^ -^^^^ - itself on its own pr "dLeT^F?"*""*^"' ^"^ ««» maintain quantity of wh^tw: Jn ?&„?;;*^'^^^^ ^^ ^-^> ^^^ ^he thi's^Llrd7eT:rtTnl2t^^^^^ j^i^.^.i'^ ^ ^f. with he rows, and the rate at wh^^the ^tre^nVows^"'^^ "' ""^^'^ (201) Show that 1 + + 2+ 3 + V. +. 4 + 4 + %^;i 8 8 take to do it ? ' *"^ ^ "^ 4 days. How long would ^ and G losnU'pe'fce^^^^^ ^- ^''n''^ ^ P«^ -"* - ^204^ A P I' ^ ^""^^ ^y «^^^^"g ^* for $57 ? l^W4; A Fr^ch metre contains 3q-S?l ^»„r i, ■ i! "f interertS 5 D^r ™„f *■■ ' ^ """'*t*' o™^"' '^e rate '■f an articlblfc ^hanS?;. «'''«'''«'»!* price to be? v-Kiu, wnat ought its ready-money price 280 EXAMINATION PAPRBH. (206) Compound interest reckoned quarterly at 2% is equal to what interest reckoned yearly ? (207) A persou having «9790 in the Toronto city 6 per cent, bjnda sells out at 98|, and invests the prcK^eeds in Bank of Montreal stock at 177^, which pays a dividend of 12 per cent, per annum. Find the change in his income, brokerage in each transaction beino; ^ %. (208) I buy wheat at i9«. a quarter, and some of a superior quality at 6«. per bushel ; in what proportion must I mix rtiem so as to gain 26 per cent, by selling the mixture at 67». Qd. per quarter ? (209) The weight of a cubic foot of water being 1000 oz. , find the weight of a rectangular block of gold 8 inches in length, 2 inches in thioknew, and 3 inches in breadth, the weight of a mass of gold being 19 26 times the weight of an equal bulk of water. (210) The contents of a cistern is the sum of two cubes whose edges are 10 inches and 2 inches, and the area of its base is tne diflference between two squares whose sides are 1^ and If feet. Find its depth. (211) Find the value of -857142867 of £10 14a. Id. accu- rately ; and show that the error committed by neglecting all decimals of an order higher than the fifth is less than j^^ of 'A penny. (212) The sum of $327 is borrowed at the beginning of a year at interest, and after 9 months have passed $400 more is borrowed at a rato of interest double that which the former sum bears. At the end of the year the interest on both loans is $26. 36. What is the rate of interest in each case ? (213) A dealer purchases a liquid at 4.'*. per gallon, and dilutes it with so much water that, when he sells the com- pound at 3s. a gallon, he gains 20 per cent, on his outlay. How much water is there in every gallon of the compound sold? (214) The discount on $666.50 for 9 months is $16.60; iind the rate of interest. (216) A merchant lost a cargo at sea which he had insured; the broker offered him a sum of money for his loss, which the merchant refused as being 10 per cent, below the estimated value of his loss ; the broker then offered $379.75 ^ more than he offered at first, and the whole amount of the aXAMlIfATIOX PAPSBa» Ml - - ti-»i -- «,.Sw , r.e nndWy suia it for lo per cent, leu thAn hit asking price, and gained $5. 76. How much did the hori. cost, and what was the aaking price ? ^H n^l ^! ^? maRons, working 10 hours a day, oan build a iT tikA''i'«^ ""'^ T y*^^ '^«"« »" « ^'^y-' ^ow ?ong will It take 7 masons, working 9 hours a d^, ti buUd a wall 9 feet high and 140 yards long ? (218) A bankrupt's assets are $2700, out of which he pays 75 cents m the dollar on half his debts, and 60 cente on the other lialf. What is the amount of his debts ? fh?«^^ ^n* »^^P f^^tai/ang 150 hhd. >f wine pays for toll at the Suez Canal the value of 2 hhcl. ..aiitini.*^f 30 ; and an of 2Vd'"ari"^ wi'^^- ^^ ' tl,o^memte, ihe value oy hhd. and m bes.de. ; - . v in Jm. valu»» o^ tha i/ine per fl Jf ^^^oA Pi°*"''e-ga"ery oenb'^t. ot *,'ireo Icrge roomn • the two are 20 yd. long, 20 yd :.rc.v;. dui 5"^^ >* ^h ^iuupos- mgthe walls to be covered with 'pici^.c, exJptthT^^t L«'fV" l^ • ^'fK*"^.^ ^^ ''^'^«' ^"d «f ^^^«*^ e^'^h room has two, what will bo the number of pictureSv h« averaae =»ize being 8 feet by 3 feet ? ' average (221) Simplify 8 5-1 83 9-7 - 6-4 71 3 1 X -lA 2 15 (222) Find the square rootb af 15376-248001 and ^^'^^ /ooox S^"^^ « J. * ^.i?T'*l' ^^*®'' ^""""^ * ^***^«' ^«"nd that he had only two-thirds of h.s army left fit for action ; one-ninth of the army had been wounded, and the remainder, 2000 men the bat't'le'^"*"''^* '"^''^ ^'^ *^^ ^'""^ °''"'^'* "^^^^^^ (224) A contractor sends in a tender of |20,000 for a cer- tain work; a second sends in a tender of 1^19,000, but stipulates to be paid $2000 every three months; find the diflerence between tenders, supposing the work in both cases to be fanished m two years, and money to be worth 7* oer cent, simple interest. n < j per 282 BXAMnrATIOM PAPBB& m (226) What turn uf raoaay miut be left in order ihftt, after a legacy duty of 10 % hat been paid, the remainder being invested in the Dominion 6 per cents, at 91 1 may give a yearly income of $450, brokerage at | %. (226) If two boys and one man can do a piece of work in 4 hours, and two men and <.ne boy oan do tho same in 3 hour*, find in what times a man, a b<>y, and a man and a bey to- gether, respectively, could do the same. (227) Show that the interest on $15840 for 3 months at 8 per cent, is eqoal to the discount on $3696 for 16 months at 7^ per cent. (228) A piece of work has to be finished in 36 days, and 16 men are set to do it, working 1) hours a day ; but after 24 days it is found that only three-fifths of the work is done ; if 3 additional men be then put on, how many hours a day will all have to work so as to finiah the task in time ? (220) The interest on a certain sum at simple interest is |>c(60, and tho discount $340 for the same time and rate. What is the sum ? (230) The breadth of a room is twi<» its height and half its length, and the contents are 409C cubic feet. Find the dimensions of the room. (231) If 1 lb. of tea be worth 50 oranges, and 70 oranges be worth 84 lemons, what is the value of a pound of tea when a lemon is worth a penny? (232) > t a certain battle two-thirds of the defeated army ran away with their arms, five-sevenths of the remainder left their arms on the field, and of the rest seven-eighths were missing, the remaining 500 being either killed or wounded. Find the whole number of the army. (233) If gold be at a premium of 20 per cent. , and a person buy goods marked $135, and offers gold to the amount of $135, what change ought he to receive in notes, 5 per cent, bsing abated for ready payment. (234) Show that the difference between the interest and the discount on the same sum for the same time is the interest of the discount. (236) I bought 20 lbs. of opium by Avoirdupois weight at 55 cents per ounce, and sold by Troy weight at 60 cents per ounce. Did I gain or lose, and how much ? (236) By investing a certain sum of money in the 6 per cents, at 91^ a man obtains an income of $320 ; what would 1 'J BXAMINATIOlf FAPMU. um he obtain by iiivotting an xiual ^um in the 6 per cenU. «t 80? (237) A tradeeniAn nmkea » deduction of 10 per cent fot lor L'"'{?*^j **? *,^i" **' ^^ ^"® '» ^2 monthe, reoeiving V-" •u. r Hid tiie aiii«r«m« h«lwe«n thin (lutn aad the preient .Torth of the debt, reckoning interett at 10 p«r cent. (238) Af inveets one-third of hia pioperty in bank •took, one-sixth in Oon»f>l«, and the remumder in railway aharee. When he aells out he makea a profit of 6 per cent. , 3 per cent., and 2 per eent. respectively on the inveatmenta, and realizea ilOKK). JRequired the amount of hia property originally. r r / (239) Mr. A. tent $3681 to his agent with inatructiona to deduct hia com. at 2| % and invest the balance in flour at $7.00 per bbl. If the coat of freightage and insurance amounts to $119, at what must the flour be sold per bbl. so 8« to make a profit of 20 % ? (240) How many bricks, 9 in. long, ^^ broad, and 4 thick, will be required for a wall 60 ft. long, *h ft. high, and 4 ft thick, allowing 6|^ per cent, of the space for mortar ? (241) What is the value of •25 of ^ of I ti of 8 guineas? (242) A work can be accomplished by A and B in 4 days, by A and G in Q days, by B and O in 8 days. Find in what time it would be accomplished by all working together. (243) A man hired a laborer to do a certain amount of work, on the agreement that for every day ho worked he should have $1.60, but that for every day he absented him- self he should lose 60 cents. He worked twice as many days as he absented hiraselt, and received on the whole $72. Find how long he was doing the work. (244) A legacy of $146000 is left to three sons in the proportion of ^, |, and J respectively. How much will each receive ? (245) If $10 is a proper discount oflF $210 for 3 months, what should be a proper discount off the same sum for 1 year? (246) The price of gold in this country is £3 17«. l%d. an ounce. ^ Find the lea«it number of ounces which can be coined into au exact number of sovereigns, and the number of sovereigns so coined. m Kl A Ml NATION rAYIMa. (247) A tnerchmit in Ttirt>itto inttruot«d hit ^,tn% in Mon- treal to 11)11 a ountignnient uf Uuur at 97- N) p«r twrrwl and invMt the proo«edt in Montreal bank stock tit 174|, which payt half-yearly dividends of 7 %■ If thti> merchant'* Ant dividend la $445.50, and commiMioni of 1 % and ^ % He allowed on the traniaotioiui rMpeotively, how nmoy oarreb of flour were told ? (248) Btate the connection between Tmy and Avoirdupoit weight!. A ring weight 1 dwt. 4 gr., and ia wortu £1 2a. If 1050 of inch rings be packed in n box weighing 3^ lb., what would it cott to convey them 144 miles at the rate of (k per lung ton per mile, insurance being demanded at the rat* of I per cent.? (240) How long will it be before $250C, put out to com- pound interest at 10 per cent, per annum, will obtaiii |1727.58|f as interest? (250) The breadth of a room is two-thirds of its length and throe-halves of its height, and the contents are 5832 oubio feet. Find the dimenjioas o£ the room. \ (251) MulUply 32856 by 121711, using 3 lines of multipli- cation only, (262) Simplify 2-8 of 2-27 4-4 - 2-83 , 6 8 of 3 ^ of . 1136 1-6 + 2-62& 2-25 (253) An agent received $21.70 for collecting a debt of $2480. VVhal rate was his commissicn ? (264) A man sells out of the U.S. 6'b 6-20 of '86 at 92^ and realizes $25760. If he invests the proceed«i in Erie R. R stock at 45, which pays a yearly dividend ( i 3^ %, what alteration in his income has ensued, brokerage on ^ach of the two transactions being ^ % ? (255) A farmer bought a horse for a bill of $292, due in 1 month, and sold him lor a bill of $348, due iu 4 months. What did ht) gain per cent., money being worth 4| % ? (256) A man and a boy are to work on alternate days at a piece of work which would have occupied the boy alone 13 days. If the boy take the tirso day the work will be finished half a day later than if the man commences. Jind how long they would take to do it working together. (257) Two men invest $300 and $100 in a machine ; it works 6 months for each of them. Determine what one must pay the other if they would have made 30 per cent, on the money by letting the machine. IXAMIlfAnOIl FAP7IU& im -k *? i"£* ^"•}»"^'y' »»■*««» ot at tho «nd of two ymn, wh«n tli^ debt will b« duo. /* cmi nlwe out the ir mey Which h« wm ree«iv« at 5 p«r o«iit. mt«rwt, and by th.»t in«ini K^in #25 en the trwifootioa. At what r»te iP £e db- counfc calculated ? (260) If 30 men, working 8 houn a day for l« daya. caq d.K a trench 72 yard, bug, 18 wide^Mcf 12 deep, in bow many day, will 3 J men, working 12 Iioiira a day. dig a trench «4 yarda long, 27 wide, ^nd 18 deep / (260) A man diic«.unt» a bill «f £180, dmwn at 4 mcntU at UO per oenfc. per annum, and iniia^n on xiving in part pav- rnent 5 Wi of wine, which he clmrgea at 4 guineaa • dozen, and a picture, which he chargea at Jt:»9. How mu'*** mdy money doea he pay? If the ooat to the man of wine ann the i..cturo be only one-fourth of the gum he Charged for them, what i% thfe reid intereat the man haa bt onarged 7 (261) One-tenth of a rod i« coloured red, one-twentie< orange, one-thirtiath yellow, one-fortieth green, one-fiftie. plue, one-8ixtiyth mdiRo. and the remainder, which ia 30 mchea long, violet What in the length of the rod ? i. 2S?^ The discount on a certain sum, due 9 months hence. ioin^ and the interest on the same sum for the same ti.«e » »^. 70. * ind the sum and the rate of interest (263) T-.VO persouB, walking a the rate of 3 and 4 miles per hour respectively, set off from the same place in onp.mite te ir'i "" Tu^\ around a park, and meet in 10 minutes. Pmd the length of the walk round the park. (204) In a hundred yards race A can give B four and hve yards stait. If B were to race (7, giving him one yard in a hundred, which would win ? j' « »" (265) A man buys an article and sells it again so as to sola It tor ^1 less, he would have gained 10 per cent. Find the cost price. (266) If the diffarenoe between the simple and compound interest on a sum of money fi« AM viksJit. Viiklm v/AA UA40 EXAMINATION PAPERS. 289 imfa'ik'?**''"' '"'"■8 """-one.! at «« rate of 6 per ««at. (Bnxi Ji : '"""""''"'> I'e'^omelialf and half? of the bread pa^tJe other two Vli?' "^'^^ '^^« ^"'^ ^^^'^''e they to divide the money? « l^alf-pence ; how ought (806) If the discount on a bill duo ft rr,««fv, u gej cent. pe. a„„„„ ,. ..^.t's'. ta^ i? tS^^Lturort^* second year he gaied 87"°/ wCl t ,"" "T*^^' «"> capital; tliethirdyeaiheloi'4^ '' 1'* also added to his »200 worse than w'i.eu h^ began ^iskel^'FilT?;? '""^^'f with which he began, "usmess. FmA the capital on the other. 'S'hat Sd Z WscosThii"?^^^ T "''^'- or lose on the whole ? "^ ^ ^^es he gain (309) The difference between the interest nnri fi, ^• on a certain bom of monev for r »^i .? * . "^® discount $2: what is the sm^ ? ^ ^ "'^°*^''' ^<^ ^ Per cent., is (810) A cistern without a ton is 97 ft- i ,« - and 6 ft. 6 in. deep- wha?w?n ,-f J}' ^''"^' ^^ ft. wide, out. at 4i cents a sq^uarlyard ? '°'' '" ^"'^^ ^* ^"^^^^ ^^d (8H) SimpUfy (a) 8-i 2~ d'vided by 1 + 6 + i (6) ?i^^a^.^i|+Ai (318) A Bum of monoir or«/x„^*., :^ ^^ . ^. interest to $787i fi^'i^r^"" ""'''' ''^ "'"' •''^"°' **'' '^% simple $900. ' '° *^**'^- -f" ^10^ ^any years will it amount to 290 IXAMINATIOH PAPBB8. (Sl4) I spent 25 % more than my income in a certain year; for each of the next four years I saved 6f % of it, ».nd then 3 fotmd that I had lived within it and had $50 besides. What was my income ? (315) A school rate of 6 mills per dollar and a general purpose rate of 8 mills in the dollar produce a tax of $101 .40. Find the assessed value of the property. (316) A grocer has 225 lbs. of tea, of which he sells 45 lb. at 72 cents per lb., und only gains 8 per cent, at this price. He now raises the p.Hice so as to gain 10 per cent, on the whole outlay. What iu the price when raised i (317) If I owe $2000, to be paid in 4 months' time, and I pay $500 now, what extension of time ought to be allowed me for the payment of the remainder, reckoning money to be worth 8 per cent, per annum simple interest ? (318) A and B run a mile race ; at first A runs 11 yards to B'a 10, but after A has run a half a mile he tires and ruus 9 yards in the time in which he at first ran 11, B running at his original rate. Which wins, and by how much ? (319) A woman buys a certain number of eggs at 21 a shilling, and the s- no number at 19 a shilling ; she mixes them together and sells them at 20 a shjlling. How much does she gain or lose per cent, by the transaction ? (320) A room whose height is 11 feet, and length twice its breadth, takes 143 yards of paper 2 feet wide for its four walls. How many yards of gilt moulding will be required ? (321) Simplify n + n ^ Ij X (1^ X 6i) + i + jj^, 6} X 3J - ?* X and find their sum. (322) Simplify (-006 of £2 Is. 8d. + 3-454 of £3 6i(.)x 5t«-. (323) Two boys, A and B, come into school punctually by their own watches, which are quite right at 9 o'clock on Monday morning, ^'s watch gains two minutes, and B'a watch loses a minute and a half every day. Find how much later B will be than A at Friday afternoon school, 2 p.m. (324) Two gangs of 6 and 9 men are set to reap two fields of 35 and 46 acres respectively. The first gang works 7 hours in the day, and the latter 8 hours. If the first cane compiete their work in 12 days, in how many days will the second gang complete theirs ? SXAMINATION PAPBRS. 291 at ^ef cento '^'n' t'Jf.'f'^"" '"?** ^® ^«"*» P^*" »»>•' -"^ .ome wh«n Til . ^,0 ^** proportion must he mix them tha* equal. iiviPg m different towns, they are rated at li oAntS ""fJr? u '" *^^ ^^"" respectively^ Whatt J^l ILT, were LS mnr« h?* 'u ^^^ f."^"*" "^ ^^^ ^ > ^^ h»« ^«et8 •ifS.^ ^! ^Pj'"'^?^ '^^'■^ <»" ^ <^one in 50 days by 36 men working at it together, and if, after working at it for 12 dTva Java in whXh"^'"' *"-^^*'' '^^^ ^^^'^^ ^^^ *h« nuJnber^oJ /Qom remaining men could finish the work. R^W ^^^^f^n?"®? ^^^ two-thirds of the amount that mdTn aT/^/*'!'^'!'^ *"^ *« «^"^« «»*"fr, Robert gave owe Oharies'f ' "^ *^'^ ^^^^ ^^*'^^^- ^^^^ ^^^ ^^^^ 4 ft^ 2 J^ s'J?*^' ^'"f-^**:' and height of a wooden box are (331) Simplify 3f X ItV + 4tV - 3A 3^ H-7^^2S^^ + y 4? X (3f X 5f) - 171, and find their sum. Q P^^^ .^ ™*" ^,*^^* * ^®r*a"» distance, and rides back in 3 hours 46 mm.; he could ride both ways in 2i hours How long would it take him co walk both wa^s ? * ^ T P'^li^.'^^I^r*" ^? **^ * "^'^^ain place in a certain time and I find that, If I walk at the rate of 4 miles pS hour I 'shall shall be 10 minutes too soon. How far have I to go ? v334) A,B, 0, and D. enter into partnershin • J an<1 R Ire of eachi *^'^- ^'"'^ «"" '"^^^ W''"' « *!>« (336) I have a certain sum of money wherewith to buy u 292 KX AM I NATION PAPKBa. certain number of nuU, and I find that if T buy at the rate of 40 for 10 cents I shall spend 5 cents too much ; if at the rate of 60 for 10 cents, 10 cenU too little. How much money had I ? ^ (337) If A has $38940 to invest, and can buy Toronto city 6 % bonds at 98^, or Montreal Corporation Consolidated 7 % stock at 117^, how much will the one transaction be better than the other, brokerage being ^ % ? (338) What must be the face of the note fo- 3 mos., made ou 18th Aug., so that discounted at 7^ % on the day of making at the bank, the proceeds may be"814315 ? (339) If, in a meadow of 20 acres, the grass grows at a uniform rate, and 133 oxen consume the whole of the grass on it in 13 days, or that 28 oxen 5 acres of it in 16 days, how many oxen can eat up 4 acres of it in 14 days ? ^(340) In a constituency, in which each elector may vote for two candidates, half of the constituency vote for ^, but divide their votes among B, C\ A A', in the proportions of 4, 3, 2, 1 ; of the remainder, hatf vote for B, and divide their vot^s among (7, If, E, in the proportions of 3, 1, 1 ; two-thirds of the remainder vote for D and E^ and 540 do not vote at all. Find the order on the poll, and the whole number of electors. (341) Simplify 1^ of 2f + 6| 4- 2| + f 5^ + ^ 2-2 H of ^ 24 + -53 (342) Simplify H^ •64 ) of 1^ of f -^ lOj "^ "* 13§ of 5^ (343) When tho New York gold market is at 104|, what would I get for $2304 50 currency ? (344) A person invests $9450 in 5^ per cent, stock, so as to receive an income of $787.50. What was the price of the stock ? (345) Two pipes, A and B, would fill a cistern in 25 min- utes and 30 minutes respectively ; both are opened together, but at the end of 8f minutes the second is turned oflf. In how many minutes will the cistern be filled ? (346) A man for 5 years spends £40 a year more than his income. If he, at the evd of that time, reduce his expendi- ture 10 per cent., in 4 years he will have paid oflf his debts and saved £120. Find his income. (347) The sum of £177 is to be divided amons 15 men. 20 ■ZA Ml NATION PAPBBII. 998 women, and 30 children, in auoh a manner that a man and a child may roceive together as much as two women, and all the women may together receive £60; what will thtev each respectively recei\"^: • (348) If 8000 rnecies be equal kj b miies, and if a cubic i?il)?!"-i"^ ^*^®^ weighs six tons, and a cubic metre of water 1000 kilogramme«, find the ratio of a kilogramme to a pound avoirdupois. (Long ton.) (349) A. mixture of soda and potash, dissolved in 2640 grams of water, took up 980 grains of aqueous sulphuric acid, and the weight of the compound solution was 4285 grains. Find how much potash and how much soda the mixture contained, assuming that aqueous sulphuric acid tmites with soda in the proportion of 49 grains to 32, and with potash in the proportion of 49 to 48. (350) A room is 21 ft. long, 15 ft. 6 in. wide, 10 ft. high • It contains 3 windows, the recesses of which reach to the ceiling, and are 4 ft. 6 in. wide ; there are in it 4 doors, each 6 ft. 6 m. high and 3 ft. 3 in. wide ; the fire-place is 6 ft. wide and 4 ft. high ; a skirting 1 ft. 8 in. deep runs round the walls. Find the expense of papering the room at 6 cents a square foot. ANSWERS. Ex. (i.), p. 5. • (1) Seven; thirteen; forty-five; fifty-nine; three hundred and twenty-six ; four thousand five hundred and seventy- eight. (2) Ninety ; one hundred and ten ; two hundred and seven ; four thousand three hundred : four thousand and thirty-six ; four thousand three hundred and six. (3) Seven hundred and eighty ; six hundred and nine ; five thousand three hundred and sixty ; two thousand and twenty ; one thousand one hundred and one. (4) Thihy-six thousand fout hundred and ninety-seven ; forty-nine thousand five hundred and thirty-two ; six hundred and fifty-four thousand three hundred and twenty- one ; seven hundred and forty-th je thousand two hundred and sixty-nine. (5) Forty-five thousand ; thirty-two thousand six hundred ; seventy-five thousand two hundred and thirty; five hundred thousand. (6) Eight millions five hundred and seventy-two thousand nine hundred and fourteen ; three millions four hundred and sixty-nine thousand two hundred and eighteen ; four millions six hundred and twenty-nine thousand eight hundred and seventeen. (7) Nine millions ; twenty-nine millions ; seven hundred and fifteen millions. (8) Nine hundred and te.. millions three hundred and seven thousand two hundred and forty ; three hundred and «even millions four thousand two hundred and five ; three hundred and eighty millions five hundred and three thou- sand and forty. (9) Two hundred and forty-three billions seven hundred and fifty-nine millions two hundred and sixty-eight thou- sand three hundred and forty-two ; three hundred and seven billions four hundred and five millions six thousand two hundred and seventy. ANIWIBM. 295 Ex. (ii), p. 6. (1) 9; 12; 17; 19; 13; 16; 11. (2) 23 ; 27 ; 36 ; 38 ; 44 ; 40 ; 26 ; 34. (3) 67 ; 75 ; 62 ; 83 ; 74 ; 92 ; 68 ; 95. (4) 76 ; 22 ; 50 ; 15 ; 28 ; 61 ; 49 ; 18; 90; 73. (5) 107 ; 130 ; 246 ; 372 ; (K)8 ; 740 ; 990. (6) 836 ; 747 ; 410 ; 913 ; 750 ; 384. (7) 818 ; 808 ; 20(> ; 430 ; 512 ; 787. (8) 7845 ; 9()37 ; 120(K) ; 8400 ; 6003 : 86040. (9) 5470; 3650; 8780; 1247; 4808. (10). 6004 ; 7022 ; 3«)0 ; 9047 ; 2017 ; 19402. (11) 70007 ; 60060 : 14014 ; 70017 ; 12303 ; 16005. (12) 366728 ; 64084.? ; 900000 ; 800040. (13) 7000000 ; 4576f,65 ; 75806940. (14) 315000000; 5040000; 8{)00700; 18000020, 700000002. (15) 31683«. (22) Clil(m(UU7. (24) 24250:J6442a (m 13675566747. 17) 1342u706861000. 10) 361)0386740. (21) 4930038124. (23) 1407000681. 25) 248156014760. (27) 249493696792. i Ex. (U.), p. 21. (1) 6840. (2) 1900680. (3) 1121111043844000, Ex. (x.), p. 21. (?) 576. (5) 4761. (8) 10000. (11) 390626. 14) 1331. 17) 103823. (20) 1000000. (23) 156590819. (1) 225. (4) 3249. (7) 7569. (10) 50169. (13) C22621. (16) 15026. (19) 804357. (22) 46U€016. (26) 961504803. (3) 1600. (6) 5184. (9) 12906. fl2) 804609. (16) 2107. (18) 314432. (21) 10074593. (24) 348913664. a) a (5) 14. (9) 108.- , (13) 66283. (17) 2104. (21) 66169. (25) 317649. (29) 3469806. (32) 6642300741. (36) 300071. (1) 8426. (10) (14) (18) (22) (26) Ex. (xi.), p. 24. (3) 12. (7) 24. (11) 528. (15) 458097. (19) 24000729. (23) 4348432. (27) 30876648. (30) 68274025. (33) 8462074231. (36) 29970. Ex. (xii.), p. 25. (2) 6487. 3. 14. 13. 241248. 17663. 5678094. 391626. (4) 11. (8) 103. 12) 1032. 16) 7689628. 20) 2019. (24) 6072. (28) 30207. (31) 472304974. (34) 90807. (3) 64008924. Ex. (xiii.), p. 26. (1) 8826. (2) 241987. (4) 1749804. (6) 1243904. (7)70267440. (8)306547, ,.,__.« ^}nl 4^4513674545. (11) 0047544611. (12) 3007490200467. (13) 2131902, 1421308, 1065081. (14) 310218774, 206812516. 155J.09387. (15) 13770469132. 0180.SOfiOft« f,P.85920Kfl« (16) 9036784, 68i6115, 5169880. (3) 21C2658. (6) 500603. (9) 659372. ANAWMJI. 0*1 (18) <*^ (20) m 196380840, 1227H.'i7ft, 1090S3800. 46in340r), 2tKJ20H6<), 'itKXJaOOO. H;IH7«M, 724<«IH, MH2VX 4224024, 2fUiAA8a, 2404639. m2'Mnm, «l2a»8«9*-?, 561366651. 3.9. 2, 3, 5, 10. 2, 4, a 2, 3, 4, 5, 8, 0. Ex. (xv.), p. 28. (2) 2,3,4,8,9. 5) 2, 3, 4, 8, 9. (8) 6. (11) 2,11. (8) 3, 5, J, 11. (6) 3, 5, 9. (9) 2, 3, 4. <4) (7) (10) (13) (16) (19) (21) (23) (29) 2, 3, a 2, 2, 2, 2, 2, 2, 3, 7. 3, 19. ( 7, 13. (14) 3, 6, 7. (17) 2, 2, 3, 11. 2, 2, 2, 2, 2, 3, 3. 3, 6, 6, 7. Of O, Of O, O, Oi 2, 2, 2, 2, 3, 3, 3, 3. (28) 2, 2, 2, 2, 2, 2, 2, 3, 3, 6. Ex. (xvi.), f. 29. (2) 2,2,2.3. 2, 2, 3, 3. 3, 17. 2, 2, 3, 3. 3, 3, 11. 2, 2, 3, 3, 3. (20^ (26) 3 9 12 15 18) 2, 2, 2, 2, 11. 2, 2, 2, 2, 8, 3, 8. 6, 5, 6, 5. 3, 3, 3, 37. 2, 2, 2, 2, 2, 5, 11. 8, 3, 3. 3, 13. «, 8, Of u. 6, 17. 2, 2, 5, 6. ?, 2, 2, 2, 7. (1) 4868. (4) :«)5892. (7) 3104199. (10) 4342356. (1) (7) (10) (13) 2472. 42370218. 74232667 436876. 37296. Ex. (xvii.), p. 29. (2) 9306. (6) 420077. (8) 1107(5096. (U) 48482280. Ex. (xviii.), p. 30. (2) 452736. (5) 8642934. (8) 14237262. (11) 3781076. (3) 147474, (6) 1594432. (9) 32(UJl()()0. (12) l'>138680000. (3) (6) (9) (12, 41708032. '''28. « w *i) ( CI. Ex. (xix.), p. 31. 94, rem. 14. (2) 11860, rem. 36. 18573, rem. 17. (4) 878, rem. 22. 106631, rem. 35. (6) 844380, rem. 85. 849, rem. 20. (8) 2392, rem. 134. 11447, rem. 72. (10) 965316, rem. 718. 10005, rem. 3669. (12) lou0082i, rem. 1812. AKirWCRA. (1) 376, ram. m 2:178, rem IB. rem. 9. (o; 'zdWm, rem. 22. (D Httm, rem 23. '«) 7420, rem. 7. II) 2»t»7636, rem. 19. [V3) 423, rem. 71' (16) 6687, rem. 207. 8i. (XX. K p. 32. 4) 20174. rem. la '6) 2107491^, ram. 25. {6) 24ti926, rem. 21. (10) 121^295, rem. 33. (12) 4230, ram. 67. (14) 504, rem. 133. Ex. (xxi.), p. 33. (1) 1. (2) 472nan. (3) 034. (4) 3012. (1) Four inilliona f"jr hundred and ninety six ; (Ui3fe'2 on Paptrrtt. (Page 40.) (I.) o hundred nnd thirty-icveti thouMud (2) imuHi. (4) 4253111 ; 15362u.,*. (3, 7829. (5) 93597" ; 7429. (II.) (1) 26267030 ; four hundred and two millione fifty chou- 8and four hundred and seven. (2) l(im>2009. (4) 338001, rem. 63. (3) 2r»438313 ; 99014800. (6) 1175427; 130003. (III.) (1) Ten billions ten millions two hundred and ouo thou- sand four hundred and one ; 1023001 ; 10011224402 • 2040002. ' (3) 2237009, rem. 11. (5) 6226, rem. 33. (2) 1546478344 ; 1577913816. (4) 31406999. (IV.) SW ^^' (2) 7482229, rem. 93. 3 2. 2, 2, 7 ; 2, 3, 13 ; 2, 3, 19. (4) 12000590. (5) 999899. (V.) (1) 66299476, rem. 5^46. (2) 38652792964 (3) 2,2,2,6; 2, 3, b, 5 ; 2,3,3,7 (■») ixiv J xivi« ; dxxviii. [p) r200(J. mfll 300 ANflWBlM. (VI.) (1) 610161890. (3) 7283. (6) 7684 and 978. (VII.) (1) $269. (3) 252. (5) $13300, $11900, $10500. (1^ ?000. (3) 31239. (5) $30. (2) 9999000025. (4) 11796 steps. (Vlll.) (IX.) (X.) (2) 8070344882024. (4) $14541. (2) 72 days. (4) 296237. (2) 450 lbs. (4) $232. (3) $375(59. (5) $21000 ; $5400. (2) 170()80900742874'^'52. (3) 2796219. (4) 786643. (5) Rem. 12 ; Divisor 72 ; Quot. 432. (1) 2. (6) 7. (1) 48. (6) 36. (1)4. (2) 6. (7) 3. Kx. (xxii.), p. 44. (3) 20. (8) 16. (4) 18. (9) 16. (5) 48. (10) 3. Ex. (xxiii.), p. 4b. (2) 32. (3) 3. (4) 3. (5) 3453. (7) 936. (8) 355. (9) 23. (10) 2345. Ex. (xxiv.), p. 46. (2) 2. (3) 73. (4) 29. (5) 41. (6) 37. Ex. (xxv.), p. 47. 0) 54. (5) 17000. (9) 31759. (1) 360. (5) 36036. [if) '^t'i'Mf. (2) 2376. (6) 85800. (3) 2532. (7) 23400. Ex. (xxvi.), p. 49. (2) 1320. (6) 27324. (3) 288. (7) 3570. (4) 9555. (8) 16128. (4) 5040. ^8) 2340. S! ANgWSBS. Bxamination Papers. (Page 49). (I.) 801 8827. (2^ 85 times. (8) 44496 raiLi. (6) 84, 36 and 182. 7. (n.) (2) Bags of 1, 2, cr 8 bu. ecch ; bins of 800, 150, or 200 (5) 982882. bu. (8) ei660. (4) 60min. (in.) AJ^Lh^'3 ^' ^» ^' ^' ®' ^0' 1*^' 15' 18. 20, 24, 26, 80 3« Ko%%'?oJ?8S?: "^' ^^' ^^°- >«»-^' ^^- «So: (2) 29. (8) 8891 and 2699 aj:e prime ; 14787 and 1477 are com- posite. ""* (4) 60 hours ; A, 300 mi. ; B, 240 mi. ; 0, 18C mi. (6) 40 gi-ains. (IV.) ;S fA , ^^^ ^* ^- (4) 70660. (5) 24 firkins. (V). (2) 60. (8) 8 and 6. (4) 44 times ; 9284 trees. (6) 8866000. Ex. (xxvii.), p. 64. (1) A- (2) A- (8) h (4) I (5) |. (6> i' (7) H. (8) M. (9) m. (10) i|. Ex. (xxviii ), p. 66. (1) ih U (2) fl, it, if (8) |3,, ^^^ j^o. '4) Uh m^ aVir. iy¥o. (6) Hh Hh Uh Hi Ex. (xxix.), p. 56. The fractions are arranged in descending order. -»- a '"' * ' A^ rS^ 44. .9- 17 (2) w A. H. ^fxF- (5) A. ^. ii. (6) ^, ^; J,;. If»- (6)M- (1)3^. (1) ^r- (6) f/i/k. (l)f. (6)M- (1) ^• (6) 42^. (1) h\%- (5) 43^. (9) 38^. .(1) 27i (5) H. (6) 3i (1) I^jV (5) hV (9) 2|f. ANHWER8. Ex. (xxx.), p. 67. (2)^i. (3) M- (4) (7) i?i^f. (8) mi (») Ex. (xxxi.), p. 57. (2) T^. (3) Tk- (4) (7) iWV- (8) ih' (») Ex. (xxxii.), p. 59. (2) m- (3) f. (4) (7) nih (8) ^W (9) Ex. (xxxiii.), p. 60. (2) i (3) /«. (4) (7) n. (8) f. (9) Ex. (xxxiv.), p. 61, tffo* 1^- 27* (2) ^' (6) 3m. 0277 (3) ^ ' 29 (7) 31^. Ex. (xxxv.), p. 62. (2) If. (3) nm- (6) 178f. (10) 2^. (7) 5/^. (11) 8ji. Ex. (xxxvi.), p. 63. (2) 744. (6)4. (3) 718|. (7) 2|. Ex. (xxxvii.), p. 64. (2) Tk. (6) 142^. (3) ^• (7) 8350i Ex. (xxxviii.), p. 67. (2) ^Vff. (3) hi (6) 11- . (7) f§. (5) Hf. (B) T^ff. (5) A- (6) I (4) 173010 . ' 1000 ■ (8) 928ff . (4) 65^ (8) 12A'-^5- (12) m- (4)M- (8) 26. (4) T^FTF- (8) 66. (4) m- (8) w^. > VSWBRS. 303 (1) fi (5) b^. (1) m- (4) hi W, 3. (7) M> h (10) 3. (13) I /t. (16) 6, 18. (19) ^• Ex. (xxxix.), p. 68. <2) iV (3) 1^. (6) i (7) fit. (10) 1}|. (11) 7^^. Ex. (xl. ), p. 68. (2) h%, B- . (6)4i. (8) 3. (11) lom. (14) Uh If (17) 66. (20) 1. (23) 2. (4)f (8) 20igg. (12) |. (3) i, 6f. (6) IIH, 20. (9) 7i (12) H, li (15) 1, |. (18) 14^^. (21) i. Examination Papers. (Page 71.) CO (2) $49i. (3) 5^%\. (4) $13860. (6) m ana f^. (11.) <2) H, If, 1^5, ^. (3) j%%. (5) Ship, $24000 ; cargo, $36000. (2) $18§. (HI.) (4) jilh' (5) A, 20 ; B, 48 ; C, 84. (IV.) V, ^^^ ^^\c.. ^^^ ^*tV (4) Horse, $120; carriage, $105; harness, $25. (5) A, $4334 ; B, $1474 ; C, $3080. (V.) )aI ^'a ^S f !^/^ ' ^^^ ^^«^P ^ 390 calves ; 806 pigs. C*/ vJ-oO. (5) 18 ft. (VI.) (2) A, A, Z^. (3) 1. (4) $40. '(VII.) (2) $1333^, ^. (3) 1000000. (4) 30 min.; A, 6 times; B, 5 times ; C. 4 times. (6) A, /^; B,^^^; C, J^^ ; D, f f . (5) A ft. 304 ▲NSWUtl. v2) i ; A-. SeOO rods ; 3000 rode. (Vm.) (3) 36. (5) 252. (4) 30 rain. ; 4500 rodi ; (&) tmnf' /Q\ 125 001 W^ TFo'o"- (13) -4579. (17) -025679. Ex. (xli.), p. 76. (2) h (3) |. (6) iTJ^arnj. (7) V- (10) i;-S|JA (11) -9. (14) -003. (15) 172-95. (18) 3-25793. (19) -0019. (1) -7. (4) 758-279832. (7) 6964-72672. Ex. (xlii.), p. 78. (2) -2464. (5) 385-260863. (8) 970-17047. (4)f. (8) Im*I. ^ ' 200 (12) -37. (16) -0000059. (3) 0012. (6) 8741-2062. (1) 51-211. (5) -0607. (8) 004385. Ex. (xUii.), p. 79. (2) 1-543. (3) 48-2293. (4) (6) 579-1274. (7) -0000014. (9) 9 9998. (10) 00101. -001. (1) 35-25. (4) -00041588. (7) 14977-92625425. •057746898828045. (9 (11) (14) -00984126. -15205806. Ex. (xliv.), p. 81. (2) 18-9326. (3) -100345. (5) 12-08980432. (6) -9. (8) -0 00465131. (10) 203-1756G2750726562. (12) 1-01. (17) 150-0625. (15) -1009981674. (13) 00031304 (16) 20-570824. (1) 12. (2) (4) 12700. (5) (7) 430. (8) (10) 98-476. (11) (13) -0000771039. (14) (16) 2469300000 (17) (19) 1290. (20) (22) 76-371. (23) Ex. (xlv.), p. 85. 14400. 43 078. 147. -0065839. 299846000. 3596. 3-59. 905741000. (3) -0013. (6) 10000. (9) -0000002004. (12) 876540000. (15) -20162. (18) -00000029. (21) 457-61, (1) 23-28125. (4) 33035-448... Ex. (xlvi.), p. 87. (2) 1-119296875. (5) 00192. (S) (6) 3-4608. •0001736. (1) 26-654875. (4) AJWSWBBa. Ex. (xlvii.). p. 87. .nnAni«i7 ^^> 0010002475. h\ 1^4 it' ^^> 175 0:U)99876. (lol 'SI W ^'^ '^^^^- Ex. (; Ha ;, p. 88^ n 18478-260. / . .04^ (4) 8658146-964 g) .095 SOS (5) 14498-8. (6) -0000926. (9) 00001. (1) -35. (5) -ooi. (9) 01236. (1) -09484. (4) 235 104. (7) -0374. (10) -928. a) I (1) m- ^TTTRf- (5) 53^ 9 (1) 15-8430. (4) -02067249. (7) m- Ex. (xlix.), p. 91. (2) -44. (3) -857142. (6) 02439. (7) 623809. (10) 2-345. Ex. (1.), p. 94.' (2) 002521. (5) 26-38702. (8; 426 104. Ex. (li.), p. 95. (2) A. (3) ,f,. (^^^Wt- (7)„f„. Ex. (Hi.), p. 96. (2) mi (3) 4ff. (6) nm- (7) 2^1^. Ex. (liii.), p. 98. (2) 20-51662025. (5) 20|^«.. 37 Tigr- is) -092. (6) 32714-285. (8) -«JL (4) -6i. (8) -216. (3) 165-6995. (6) 1-611. (9) 170-3367. (4) T^iVr. (8) TTirT. (4) zUh- (3) 1-7780052. (6) ^hh- Examination Pajjera. (Page 98.) (I.) (3) 1st. (4) 26S ti.^es ; J. (5) -0000006 and 0000009. (II.) ^\) ^. t1^, hm. Ti^^ff. (2) |;i4.90. (3) 3-715 (8) -7142. 806 AVSWXBS, *2) I816J. (8) 18: $8000. (4) $2821. (6) W/V*. (IV.) (2) $21.60. (8) 425, i) 4000004*00000010000090007 ; Seventy-four millions, three hundred and six, and sixty millions and seven trilliontlis. (6) 82^ yd. (V.) i^) Til fHir. (8) 10-7608 miles. (4) 1C||. (6) ^,$192.28^; B,|146.58i; O, 0110.94^. (VI.) (8) $34i. (4) 8-141692. Ex. (Iv.), p. 106. (2) 28. (6) -00097061, (1) 14. (6) J^7. (9) 440, (18) 28466. (17) 6678. (1) 4-1. (6) -26. (9) 210-76. (1) 4-4721. (6) -4110. (Q) 4-0806. (6) m- (9)8^. (13) 7905. (17) 8-7849. (^) 16. (9) 85. (8) 82. (7) 827. (6) 846. (10) 836. (11) 6031. (14) 72600. (16) 2081. (18) 48796?. Ex. (hn..), p. 106. (2) 16-79. (8) -96. (6) -027. (7) 181-81 (10) 187-66. Ex. (Ivii.), p. 107. (2) 6-4772. (8) -9486. (6) -1264. (7) -0262. (10) -9999. (11) -6026. Ex. (Iviii.), p. 108. (2) A. (8) « (6) 2^ (7) 21. (10) 6^ (11) 4i (14) -6464. (16) 2-6298. Eic. (lix.), p. 110, (2) 82. (10) 99] (8) 42. /r7\ OA \'/ (11) 89- (4) 76. (8) 867. (12) 4698. (16) 78900C. (4) -61. (8) 1-001. (4) -8478. (8) -0847. (12) 6-4883. (4)1^. (8) Hg. (12) 8f (16) 8-0822. (4) 79. /0\ mo (12) 68. (1) (5) <9) (13) (1) 245. (6) 128. (9) 256. (13) 686. (17) 4968. (1) 73. (6) f. (9) 8-320. (13) -908. (1) 27. (5) 64 ANnWMM. Ex. (Ix.), p. 111. (2) 531. (6) 179. (10) 679. (14) 708. aS) 8765. (3) 307. (7) 463. (11) 438. (16) 888. Ex. (Ixi.), p. 112. (2) -364. (3) 30 02. (6) If ■ (10) -405. (14) -693. (7) 7f (11) 2-516. (15) 1-966. Ex. (Ixii.), p. 113. (2) 45. (3) 6-3. (6) 8-1. 307 (4) 670. (8) 103. (12) 607. (16) 512. (4) H. (8) 1-709. (12) -822. (16) 1-473. (4) 13. Ex. (Ixiii.), p. 117. (6) 960 ; 1228 ; 4253 ; 'l4087. U Ii88o! 'iK 305^3. (1) Uid. (4) 7s. 5^d. (7) £4 8s. 3^d. Ex. (Ixi v.), p. lis. (2) 43id (5) 9h. up. (8). £391 19s. ^d. (1) £21 15s. Id. (4) £31 12». 9d. (7) £32 9s. SM. (10) £181 18s. Qd. (13) £200 17s. lUd. (15) £3602 17s. 65. Ex. (Ixv.), p. 120. (2) £31 8s. Od. (5) £23 13s. 8kd. (8) £32 6s. 5rf. (11) £240 19s. Id. (14) £220 6s. 9|c?. (3) 49Jd. (6) 15s. G^d. (9) £564 19s. 7d. (3) £32 15s. 2d. (6) £33 18s. 44d. (9) £169 5s. Id. (12) £168 lis. (1) (3) (6) (7) (») Ex. (Ixvi.), p. 122. ?fi«''?w ^2> -^SSOs. 10(i 7a fL^^'^' (4) £238 17s. ma. it ^i^- (C) £1 16s. 7id ^|«- (8) £1519 12« Ql/* *;jt>iu» 17s. 6|rf. (10) £1219 19s. iS^. Wl 908 ikMMWBiia. ji if (10) (13) (16) (li*) (22) £1 9«. .£3 (b. 6d. £22 iia. Sd. £5 18«. Ud £111 Ga. Sd. £104 12.1 £3 14« 8rf. £39 7«. dd. Ex. (Ixvii.), p. 12a (2) 5a. lOd. (6) IBh. Sd. (8) £14 11«. (11) £21 12«. (14) £6 18«. l^d. (17) £6 12.-*. Qd. (20) £48. 3s. 9rf. £1 2«. Id £12 5« £15 16a. £122 9.1. 4d £8 0». lO^d. £36 6». (1) (3) (6) (7) (9) (11) Ex. (Ixviii), p. 125. £6 10«. 6d £3 12». b^d. £29 13.?. 4/i. £107 19j*. 2d £(5189 5a. 7*d £8616 38. dd. (2) £24 9a. l*d (4) £5 18a. 4d. (6) £34 la. 7id (8) £15212 12a. Qd. (10) £6022 Oa. 7id I Ex. (Ixix.), p. 127. I. (1) 7a. lO^d (2) £5 12a. 6d (4) £3 19a. 4d (5) 12a. S^d n. (1) £1 10.S. e^d (2) 4a. 3d (4) 2a. 4(i. (5) ia. 6^d. in. (1) £1 3a. 2d (2) 3a. 4|d (4) £4 4a. 3id. (6) 6a. 4|d. (3) 18a. 7id. (6) £1 17a. 7^d (3) 58. 6d. (6) 19a. lOd (3) £1 4a. 10|d (6) £1 3a. O^^d 1) 100. 5^ 231. Ex. (Ixx.), p. 127. (2) 22. (3) 42. (6) 10. (4) 79. (1) 3a. 6|d (4) la (7) 10.?. 6d (10) £48 la. 4f (13) £8 3a. S^d. Ex. (Ixxi. ), p. 128. (2) 4a. 5|d (5) la. 9|d (8) Us 8d. (1^) £77 5a. (1^) £8 12a. Id (3) 6.S. 6|rf. (6) £24 16a. 8d (9) £13 (is. 6'/. (1*?) £1 15a. 0|rf. Ex. (Ixxii.), p. 129 (1) £1412 11a. Sd. (2) (3) £28299 la. lOd. (4) £3226 Oa. dd. £31282 8a. M. £27^77 ioa. od. AJiaWMM. il) •i) (3) (4) (5) («) (8) (10) (11) (12) (14) (lo; (17) (18) (19) Ex. (ixxiii.), p. 180. 22646 sec.; 61243 gee. 107020800 wo. ; 544324 min. 8 di4. 14 hr. 13 min. 12 iec. ; wk. 2 da. hr. 24 min. 66 .eo. 18; 15i; 286; 120; 161. 70 hr. 34 min. 30 Bee. (7) 26 wrk. 2 da. 2 hr. Jo 77 hr. 3 min. 41 leo. 260 da. 23 hr. 1 min. 13 sec. 136 da. 1 hr. 42 min. 22 yr. 293 da. 1 hr. 2 hr. 54 min. 48 see. (13) (16) 83 da. 17 hr. 47 min. 298 da. 21 hr. (1) (3) (4) (6) (7) («) (11) (12) (13) (14) (1) (3) (4) ?) !«) ;7) ,8) (9) (10) (in (13) t) (t 6 da. 22 hr. 1 yr. 331 da. 21 hr. 6 da 9 hr. 36 min. 46 gee. ^ da. 6 hr. 14 mm. ; 12 min. 17 sec. Ex. (Ixxiv.), p. 132. 132 ih ; 23166 ft. (o\ aara^q • ^^^. 50 mi 9 t,K. QR* ,^°' '" y^- 8 in. uv/ mi. ^ tur. 36 no. {i'^\ qk «-» o _ j Ills' ' '^-'^/'\r- ' ^-- si po. ' • '"'• 18o8 po 3 yd. ; 1783 mi. 3 fur. 5 po. 1 yd. yd. 1 ft. 2 in. ; 6 fur. 6/^ po. ^ -^ yd. 1 ft. 5^ in. ; 1 fur. 29J J po. Ex. (Ixxv.), p. 134. 163 sq. yd. 7 sq.ft. 91 gq. in. 27 ac. 2 ro. 36 po. * ^ -»' * 5 sq. yd. 8 sq. ft. 129 sq. in. 1 ac. 2 ro. 16 po. (12) 6 sq. yd. 7 8q. ft. 22 eq. in. ac. 3 ro. 3i> po. (14, (15) (li>) 1 ro. IS po. i 1 ro. 2 3 ao. 1 ro. 30 po. "i ac. o ~o. y < po. po. 81U AMtWlBS. Bx. (luvi.). p. IM. ^ cub. ft. ; 1176188 cub. in. ; 604558 cub. m. 48 cub. ft. 21 cub. in. ; 9 cub. yd. 11 cub. ft. 87'icub. in. 244944 cnb in.; 1491)04 cub. in. 270 cub. yd. 20 cub. ft lUn cub. in. 195 cub. yd. 8 oub. ft. 298 cub. in. 8558 cub. yd. 10 cub. ft 284 cub. in. 8 cub. yd. 20 cub. ft. 1646 oub. in. , 8 oub. yd. 1684 cub. in. (9) 27 oub. yd. 7 cub. ft. 1472 oub. in. (10^ 707 cub. yd. 1828 cub. in. ; 25049 cub. yd. 17 cub. ft. 618 oub. In. (11) 6 cub. yd. 14 cub. ft. 1029 oub. ux.i 3 cub. yd. 24 oub. in. Ex. (1«^). P' 188. (1) 59 pts. ; 109792 pts. (2) 8 qr. 2 ba. 1 gall. 2 pt. ; 47 qr. 4 bus. 8 pk. 1 gall. (8) 4b gal). 1 pt. (4) 20 bus. 1 pk. 1 gall. (5) 197 qr. 8 bus. C*) 2 qt 1 pt. (7J 3 pk. 1 gall. (8) 6 qr. 7 bus. 8 pk. (9) 842 qr. 4 bus. 2 pk. ; 1115 qr. 4 bus. 1 pk. (lo) 8 qt. 1 pt. qr. 8 pk. Ex. (lxxviii.).p 187. (1) 12960 gr. (2) 1680 dwt.> 8420 dwt. ' 6186 dwt 22258 gr.; 42668 gr.. 6 oz. 11 dwt. 1 gr ; 7 lb 4q2. 18 dwt. 12 lb. 6 oz. 19 dwt. 18 gr. ; 18 lb. 6 oz, 6 dwt. 7) 80 oz. 4 dwt. 9 gr. 9) 3 oz. 4 dwt. 21 gr. (11) 9 oz. 12 dwt. 28 gz. 74 lb. 7 oz. 87 lb. 7 oz. 12 dwt. 18 gr 7 lb. 9 oz. 18 dwt. 89 lb. 6 oz. 8 dwt. ; 141 lb. 7 oz. 19 dwt 401 oz. 7 dwt. 11 gr ; 148 lb. 9 oz. 5 dwt. 21 gr. 2 lb. 12 dwt. ; 6 oz. 6 dwt. llf gr. 6 dwt, 8 gr. : 2 oz. 19 dwt. 20 gr. Ex (Ixxix.). p lae (1) 17C00 oz. : 4852 dr. ; lOt 0. (2) 208200 oz.; 80050 lbs. (8) 78416 dr. ; 7507 ibs. (4) 2 cwt. 8 qrs. 22 lbs. 11 oz. : 1 ton 17 owt. 1 qr. 24 lbs. 811 (fi) 4 cwt 2 qw. 14 ibs. 8 02. : (7) 45 qr. 19 lbs. Ifl oi. (9i 2 lb. 1 oz, 9 tlr. (}J 1 fwt. 1 qr. 11 Ibt. (18) 8 lbs. 14 dr«. ^ 02. J © owt 2 qm. Ifi Ib^ Jfi oi. fi8 lb. 12 oz. 1 dr. 88 cwt. 2 qr. 14 Iba. 2 (ir. 22 lb. 8 oa. 7t. 19 cwt. a q. 84t. 18 cwt. 1 qr. 18 lbs. 120 cwt. 67 lb«. 2 ()z. ; 187 cwt. 66 Iba. 156 owt. 1 qr. 16 lbs. ; 890 oz. 18 dr. 1 qr. 14 It 02. ; 2U 8 cwt. 8 qr. H^ Ibn. Ex. (Ixxx.), p. 189. (1) 18 cwt. 1 qr. 2| lb. (2) (3) 80 mi. 1 fur. 22 po. (4) (6) 166 ao. 8 ro. 82 po. (6) (7) 7b sq. yd. 7 sq. ft. G sq. in. 18 lb. 14 02. 12 dr. 679 yd. 1 ft. 6 in. 757 ao. 2 ro. 12 po. Ex. (lxxxi.)s p. 189. (1) 2 owt. 4 lb. (8) 1 mi. 6 fur. 8 po. (7 6 ao. 8 ro. 4 po. sq.yd. 7 sq.ft. 87 sq, K I (2) 10 02. 6 dr. (4) 8 yd. 6 in. (6) 1 ao. 8 ro. 8 po. in. (1) (2) (6) (8) (10) Ex. (Ixxxii.), p 140. 18«. id. ; £1 iU. U. ; £2 10«. M. 6 fur. 16 po. ; ^0 po. ; 8 qr. 8* lb. 4'^^ f^l \l'' ^¥^3 ^^^^d.S mi. 2 fur. ifi'^o^' n. . (7) 9 ao. 2 ro. 18J po. A i* o ^- ^^.T' ^) 2 fiir. 87 yd. l/in. 4 nwt. 2 qrs. 11 lbs. 10^ oz. Ex. (IxMdii.), p. 141 (2) m- (8) H^ (6) f. (7) I (10) ^- (11) HfK Ex. (Ixxxiv.), p. 144. (2) £16 68. 6rf. (4) 3qr. 181b. J'>;)a- (6) jei6. Os. 6rf. (9) A. (4) «f. (12) t«. (1) 12s. 6 ^^^^^- . jfj Jg^^^^ 1«- Ex. (d.), p. 187. (2) 4tV mo. (6) 7i mo. (1» n mo. (4) 8imo. mIvS^^?; A11 .^ ..(8) $864.01, The eqiated toTifl May_6, l&SU. All the buls are eauivalent f/> ft8«9 ^ iuio wiii draw uiterest at «y. till June 2. (9) 28 kiy ' (8) 6 mo. (6) $666Hff . I' *■ 816 AMBWXBS. (1) Aug. 6, 1875. Es. (di), p. 19a 1) 21-25. 4) 26-9625. (3) Not. 25, 1877 Ex. (dii.), p. 191. (2) 788-571428. (6) 10-164876. Ex. (civ.), p. 192. (2) Not. 20, 1877. (1) «120; 278 horses. (8) 75844. (6)800; 28487i; 4800000. (1) $106-40. 4) ftl.87i. (7) $7488. (10) $20000; $60. ^3-125. (4) $416-25. [7) $10000. Ex. (ov.), p. 193. (2) $700. (5) 4.J%. (8) $d800. Ex. (ovi.), p. 194. (2) 747-25. (6) $478. (8) $4800. Ex. (cvii.), p. 196. (1) $66.70. (2) $0-017. (4) li cents in the dollar. (8) 66087-6. (2) 88i ; 2 ; 40. (4)8|; J2i; 16. (8) $5-91. (6) $77. (9) $38400. (8)£448816e (6) $9.80. (8) $1312600. (1) $468.10. (4) $199.60. (1) 4-065. (4) 41825. (1) $1760. (4) $10986000. (1)«8- f9\ A a $8400. Ex. (cviii.), p. 7M. (2) $88. (8) $460. (6) $9600. ^ ^ Sxamination Papers, (Page 197.) (I.) <2) J226. (8) $3640. (o) 10. (H.) (2) $7119.80. (8) 21 75. (6) ^,$40; B,$45. (in.) y (2) 418 bales; $328.68 % -»# ^r*^. yvi «i 0(t*. #. • ^^';v^ ANHWEB8. (IV.) (1) 8805.78^ (2) 16222V11 lbs. (4) Gram, 1^1020 ; groceries, $950. (5) A gets $842.80; £,$918.87; C, $1698.88. 817 (8) 166: 266; 880. (i) $2686H. ^4) $256. (V.) (2) $2140. (5j $3000. Ex. (cix.), p. 208. (8) 100 bales. (1) 26. (2) $2000. (8^ $4 86 ^A\ f ^ ^*' ^*^- Pe'^gall. (5) 3|^ gain. (6)' 884. ' f^ll ?„P®' °®''*- (^) 10 per cent. (9) 6. K «Q^n ^^^^ 18-9...per cent. (12 10. (16) $3.46. (17) $8.60. (18) 88i per cent. :i) $7262 75. (4 i'3542. (7) $3000. (10) £6000. (18) $885. (16) $600. (19) $8200. (22) 5H. (26) 108i (28) 80„ (81) $67500. (84) 85. (37) 6 per cents; (40) £24960. (48) Increased (45) Gain $125. CS) 6000. (51) $44092. (54) $4^: 4 A Ex. (ex.), p. 212. (2) $7840. (5) ^52b IS-. 9d. (8) $850. (11) $55.60. (14) $150. (17) $680. (20) 6f. (28) 4|. (26) 119^. (29) $5000. (82) $41540. (36) 92,^. i%.(38) £4726. (41) 90. .66. (46) 6f years. (49) 6 per cents. (62) 89f. (65 $3200000. (8) $9066.25. (6) $11200. (9) £2400. (12) $228.80. (15) £276. (18) $960. (21) 5,V. (24) IIU. (27) 90rt. (80) $9600. (38) lOH. (36) 88i. (89) H. (42) Nothing. (44) $10692 ; $21884 (47) Loss $46.22. (ro) $80800. (53) $96. (56) 6040iHf . (1) $2668. (8) $1.86H per lb. (6) iioss of |lO-i63. Mxaminazton Papers. (Page 216). (I.) (2) Loss 8f per eemt. fA\ ni 1 !■ \-/ -^x r VA V/^XII 818 AMtWKBfl. (1) $2,88i. (4) 90. (tL) (2) in; 3i. (6) 223t. (m.) (1) 4000 IbE • $1.08^. (2) 80. $2.68| ; 40% and 89f f %. (4) $10. (1) 12-99 grtin. $8160; $6528. (1) $1000, (4) 48i. (IV.) (2) Lost$71|* (4) $510.59. (V.) (2) 40. (8) 9;',| cts. per oz. (8) A; $2.67f; (5) 102-723... (8) $6000; (5) $2580. (8) $9142f (5) Loses $60; gains $180. Ex. (cxi.), p. 220. (J) $88 ; $27. (2) $260 • $376 ; $876 ; $1000. M $3300 ; $2200 ; $1650 ; $1320. '^ *9 (4) 9 owt. of saltpetre; 1^ cwt. of sulphur; 14 cwt. of cnia,rcoaI. (6) 120 yd. ; 160 yd. ; 200 yd. (6) SJ240 to A ; $80 to S; $820 to (7. W) ^.j . 82 ; 40. 8 A, ^102 3«. 9d. ; 5, ^182 16a. lO^cJ.; 0, ^183 18«. 9d (9) 118 ; 889 ; 678 ; 791. (10) 80. (11) 67i| ; 401^ ; 91|f. lOff. (12) A, 9s.. B,128.; (7, 24«, (18) Men, $5; women, $3; boys, $2.40. (14) Men, $182.70; women, $182. 70 ; children, $162.26. ^}Sl f^2?\.^ « ^ <^^) ^,6700; 1?,$2600; C, $1800. (17) ^,$1050; B, $1200; 0, $1250; D, $1600. (18) iVvV; 1% ; 2^. (19) $175.60; $21^40; $262.72; $117.00; $149.76. (20) 1200 boys. Ex. (cxii.), p. 222. (1) First, $44.26 ; Second, $88.60. (2) A, $4.60; B, $6.76 ; C, $11.26. (8) A, $2062.40; B, $2320.20; 0, $778.40. (4) 4, $656t^A; B, $236^%-. (5) D, $20; B, $60. (8) 4, $87.60: J5, $120: a $202.60 ' "^ ANSWEB8. iH9 (8) $15.30; $14.26. (V) $30 ; $48 ; $28. (9) J,$245; jB,$226. (U) ^, $118.30; JS, $55.90; O, $ia Ex.(cxiii,), p. 225. m v^* P"'H^^^ » ^'«' ^2312 ; J5's, $2172. (2 Net loss, $3105 ; ^'s, $2830 ; B'b »mt (3) Netloss,$3500; ^'s, $1010; ^s^etLsolvency, $2730 Ex. (cxiv.), p. 230. l\ on u'- °^ ^™*' 7 lbs. of second. •^ tn ik"' ""?*?.' ^^ ^"- "^^ ' 20 bu. barley. 2 g'ali 1e' ostnJ '' ''''' f^^'^M't ^^> '' «*»• -*- 3^6^lSr'\^v^•-^«-^^^^^^^^^ 36 lb. at 33 cts. ; 36 lb. at 37 cts ; 48 lbs. at 46 cts. Ex. (cx\,), p. 237. f^t^ffril^' (3) 2 fr. 13 cent. ill I?' ^^h' W ^3345.44 (W) £1 = 13| marcs banco. J? !?S.^- (13) 53^d. per mil- (14) -0102045 02.; 25 -17 francs. ^i (3) (5) (7) (8) (1) 109^. (4) 1760 copeks. (7) £576 12«. 6d. 25 fr. 45c. for ^1. (11) $4.86; £1. ree (nearly). Examination Papers, (I.) (2) A $6075 ; B, $5400 ; C, $6000. )i} PJ5^^*' ^14224.91; cir., $14476 72- (5) 2 '341 % discount. ' *P **'<>• ^^, (H.) (3) £?-T5%5??f?* '"''*'• , ^,^4912 -5, $6168. (5) lOalidi "^ ^*> ^, »24 . £, $1708. (1) 1-2372. (3) $5774.43. gain, $251.81. /1\ QOJ. It-. '3 (HI.) /2S «i9io~ ?QN\i2k.^i'*^ 12 cents and lUU lbs. of 20 cents. (2) $1212. (3) $1257|. (4) ^. (5) £2 3s, 2^ (neTrty). I li 320 IMSWBM. (IV.) (1) $2211^. (2) 148.68. (s\ An f U640. (5) 1 lb. at 8; 8i lb. at 18 ; 8 lb. at U. (V.) (1) fi9, 17, and 106. (8) 8, 10 and i months. (6) -42 ; 28|V per cent. Ex. (oxvi), p. 242. i is greater. if is greatest : f is least. 112: 406. (6) ♦31.26. i : H' (9) 128 • 1 (2) llSf per cent. (4) 9176|^. (2) H is greater. (4) 46 : 864. (10) 9 : IS. Ex. (cxvii.) p 246. J) 4 : p : : 12 : 9. 4) A :G : ; 26 : 89. 7) -048. (2) 12|. (6) 21. (8) 28 (10) A #662; B $460; C $346; D$280. 51 (1) i81286. (4) 8 h. 26 min. P.M. (7) 78f . (9) Ex. (oxviiii.) p 247. (2) 10 h, 40 m. 86 A sec (8) -0076 (6) if. (9) A. (8) A mi. 7722 stones. (6) lOd. ; 12f d, ^ (6) $47.18. (8) 8 P.M. Thursday. (101 128C0 Ex. (cxix., p, 249. (1)64 men. (2) 1060 men (8)18. !«( ?2 ??®^* ^^^ Navvies did 6 times as much as soldiers. (6) 12^ dronas. (7] 676. ^8' 16» (9) 166, (10 12 days. ^ ' ^' (4} (6) Ex. (C3xx)., p. 262. 6000 mm. (6) 16 milligrams. (8) 165000 sq. cm. (6) 1067.25 dcm. (7) 48-7 nmi, ; 4.87 cm. /I AX i„or - (^) 1086-42 sq. dcm. Jt2? oI?t^^^ ^^^) ^^^ milligrams ; 10000 decigrami. )l! ?^J«^?M^''"''- <^^) ^^^ milHgrams (14) 1-60981 lalomfitrefi. (iR\ fi«'7.fi'7K «« ^^ (16) 8720 litres. (1) 36 sq. ft. (4) 12 sq. ft. (7) 608 sq. ft. (10) 30| «q. yd. (13) 878i aq.^t. m nfnq.ft. (17) 2232 sq. ft 19) 7 ft. 5 in. (22} 88 yd. (26) 16 ft. (28) 255 yd. (31) 6 V2 in. Ex. (cxxi.), p. 2fifi. 881 (2) 135 sq. ft. (6) 452if sq. ft. (8) 150X sq. ft. (11) 14061 «q. yd. (14) 91 sq 81 sq. in, (20) 8 ft. 9 in. (23) 90 yd. (26) 103 ft. (29) 360 5 yd. (32) 625 V2 ft C3) 300f sq. ft. (6) 224 sq. ft. (9) 402^ sq. ft (12) 315^ sq. ft (21) 11 yd. (24) 9 ft: (27) 405 yd. (30) 163-25 yd. nearly. (1) 28|. (4) 58. (7) JJ90.93^, (1) a30. (5) $25.60. (8) £6 6& 9fffd. Ex. (oxxii.)p. 256. (2) 46^. (5) Uiff (8) mM- Ex. (cxxiii.), p. 258. 67. $33.60. ^li 9a. 8cL (2) 855. (6) $13.62. (3) 875§. (4) 79a (7) £6 133. 2^. (1) $4.93i», (4) 135 ft (7) $1.20. (10) 22|ac. (13) $12. (16) $13.50. (18) 10511^ aq. (20) $69. 2d ^ (23) 2 ft Ex. (cxxi v.), p. 268. (2) $520. 96i. (6) £13 10s. (8) 12 ft (11) 17|ft (14) 5952 stones. .x , ^i^)^ 429 yds.; 715 yds. yd.; 29o5| sq. yd. (iq) (3) (6) (9) (12) (16) (21) $29.55f! (24) 300. Ex. (cxxv.), p. 262. 210 ft $5.95. 62 yd. 1 ft 1 ft. 9| in. $9.00. 26. (22) $112.64. UllUuoutit [llmll^-A,, (3)83Hcub.ft (7) 16335 tons (R\ Jm ^'^ °"^- ^*- ^^^ ^^^0. riO^ 90511 (13) 160.' n VJ-a; ^ cwt. (14) ^ ft (12) 5^ ft. (15) £38 19a. 2d, 822 AS^' ExamviMtian Papers. (8)81; H;H;h (6) -02; 2000; -000002 (1) 118. (2) 175. (4) 80 inches. (6 j of each. 2000-020002; --^J—^ol^*- (7) -482. (8) 7899 mi. 1 fur. 26 po. 8 ft. 6 in. (9) 45 miles. (10) 9210. (12) »6670, 17660. (14) }; -76. (16) 14. (17) 12 days. (18) 108. (20) A,nAd; B, $16.81. (22) ^m\ tMts\ -21; 2100. (24) ^142 12a, Qd. ; £42 Us. 9d (Z6) 12-96; If. (27) 21 yd. 2 ft. 2* in. (29) $166-66|. (81) 8i. (88) 4 per cents.; $128700. (86) $86.00 and $62.50. (87) 3 ft. (89) 65-8 ft. (40) 75^ yds. (42) $4200. (48) 2601 J f (46) 16«. M. (47) 1^2f; 10%. (48) $1785. (60) £5 15s. Oid.(61) ^jt (68) $8738i. (56) $828.68. (69) $1.60. (62) 88-6 in. (66) 8 hr. (68) $1680. <71) lU!- (74) $4906-25. (64) 86 days (67) $9.87i. (60) $15f. (68) 1 hr. (66) 8 hr. (69) 16i ft. (72) 7 976d. (11) 9406 stops. (18) -iUn; -0189. (16) 17095260 in. ; fH- (19) $12000. (21) 1 ; H. (28) rWr : twice. ; £14 6/r. Sd. (26) 6 h. 48 rain. (28) $82'66f . (30) $49-60 and $49.60. (82) fiSf J. (84) 7^% knots. (86) 85 cents less. (80) 6jY/o ; «i7i|. (41) 1. (44) 256. (46) 200, 189, 101. (49) 270 ft. (62) 55. Id. (55) 26 sec. loss. (58) $16^. (61) 2|. (64) $8846.87i. (67) 7 %. (70) 11 sq. tt. (78) 89^8^1^ ; .000866. (76) 107tV7 days. (75) 16 day. (77) A gets $1926 ; B $770 ; C $164. ' (Ys] >86()0 ; 6| %. (79) Loss of 40 %. (80) £4 Is, 6Ud. (81) § 6.88408. (82)^. (83) 8 days. (84) 4 hr. 82 min. (85) 221b. of nitre, 4|lb. of charcoal, 8^ lb. of sulphur. (86) 95tV cents. (87) $16. (88) $12706. (90) $60.75 ; $^0.42^. (02) 81116. (94) 1^ min. to 12. tCio \ Ao/ a. eoy '/o (9») 900. '/O' (89) 1-2636 lb. (91) 1. (98) £1 ; -740. (96) 7i %. \iW) 50Y.0U, ANHWHJUi. 323 (101) (103) (104) (10«) (109) (111) (114) (116) (119) (120) (122) (124) (127) (129) (131) (134) (136) (136) (138) (141) (144) (146) 42238274«525. (102) 3037 in. Am Oij^ d». ; H in OH d«. ; in 14 A da. ! S^-H ^^^^^ Lo««»4%. (108) f214. f'^-'^^U- (110) U I9,i. Umin (112) l (113) $8266. 30 m,. ; 2p mi. per hr. (115) 7§ ; £60 lou. On Tuenday km. when one clock marks 9 hr. 11 min •,«*oir3*' ^^^®'' ® ^^- ^* «»»»• 30 Bee. fi238-70. (121) 44J)7time.. |"\ (126)30. f^V^'^'^' .wgp^e.ia,^^^^^^^^^^^ ^5 ^"S- (130) 12t'^)ffal 4.V- (132) 46. (133) 4000 ft.* A gets 88 cents ; B, 49| cents. 13U min. and 16,*^ min. past 3. 13.^ years (149) (151) (153) (157) (160) (163) (165) (167) (170) (173) (176) (179) 181) (182) (183) (186) (I8r) (192) (194) (196) 10 (139) £60000. 81.76. (142) 3idttys. 28* days. 87i. $572; C, $259.50. 111835f metres. 2. (152) 937 7i miles. (164) 8^ Id'i miles. (159) 3 ft. 11 jV in. (161) 48 min. (137) i;136 9a. 2d. • M %. (140) 38. * (143) 3J| hrs. (145) 400 miles. (11 A gets $1155; B, (148) 6f (150) £44 13«. 3d. ; l ft. •02268 of an inch. _ (155) £600. (156) $760 249fr.=£l; 2516fr.«,£l 1. (162) 7442^. (164) 4^ moniihs. 9 of spirit to 31 of water. (166) 56i^ °/ f^- ^\t^\ W (169) 2.ll8t. in a century, t^/-''' 9;l\^ 103-67; 574. (172) 16 hr/ 56 yd. (174) $6. (175) -,V min S^ilH- ^^V^ 13|fmin/ "(178) 1520 tons. lTmilV3T5lys. ^'''^ ^ ^ ^0 yd. ; 262,.^, ac. A gets $17.50 ; B, $52.50 ; C, $105 ; D, $175. $4500. $24360. $3.40. 1tP7 hr. $3700 (184) 18 cents. (185) 40 % of loss. (187) 6 cents. (188) $11835.75. (190) $81.12; 7 yds. (191) 2. (193) $7500. (196) $8400, Each child gets $1920.60 ; each brother, $960.30. 394 ▲IfRWIRJI. (Ii>7) (109) (200) (203) (206) (207) (209) (211) (214) (216) (218) (221) (223) (225 (228' (230' (231) (233) (236) (239) (241) (244) (246) (248) (260) (251 (254) (256) (259) (261) (263) (266) (269) (271) (275) (278) (281) (283) (284) (286) <288) (290) (1981 He (jaint 14,flr %• 3 pinta 1216. 50000000 quarter*. ' 3 milea an hour ; 1^ miles an hour. (202) 4|. 37 A. (204) 1600-306 metrei. $2.00. (2(M)) 8-243%. f 5940 of increase. (208) 2 : 7. 33 lb. 7 oz. Avoir. (210) 2 ft. 3 in. £9 3*. Qd. (212) G % and 10 %. (213) 4%. (215) 92450. $92 ; $115. (217) 30 days. $4000. (219) $116. (220) 474. •06. (222) 124 001 and f. 9000 men. (224) Second greater by $50. $*.U2r>. (226) 7i hr. ; 18 hr. • 51 hr. 10 hours. (229) $6120. Length, 32 ft. ; breadth, 16 ft. ; height, 8 ft. 6«. (232) 42000. $33.75. (235) Lost $1. $306. (237) $i|. (2b8) i;6000. $0.50. (240) 48000. 3«. (242) 3, P^ days. (243) 90 days. $56000, $48000, $42000. (245) $35. 160; o23. (247) 1500 bbls. £1 11«. ^d. (249) 5i years. Length, 27 ft.; breadth, 18 ft.; height, 12 ft. 3998036016. (252) 9. (253) 1%. Increase $160. (265) 17f %. 4jday8. (257) $30. (268) 4 i per cent. 24 days. (260) £110; 150 per cent. 400 in. (262) $563^ ; 6 per cent, l^mi. (264) B. (265) $200. $1200. (267) Loses 5%. (268) $4.81, nearly. $40800. (270) 26 cents. Between 0001 and 0002. (273) A, $3200; B, $4800; C\ $6000 ; L», $7000. (274) $6.50. If cents. (276) $6.60. (277) 96| cents. $12800. (279) $4.30. (280) ^^ix^ in. 760. (282) 478, 369, and 684. S 46 mi. and 30 mi. per hour. 50 per cent. (286) 82^ cents. 16| cwt of nitre ; 1^ cwt. of sulphur ; 2^jf cwt. of charcoal. (287) 4^ njiles per hour. Capital, $1000000 ; receipts, $100000. (289) 3^ %. 1:23. (291) 36|f ; lOAV (292) 1^. (293) £66 13«. 4d. lOA'V- (294) J -8523809 hr. AmrmRmji. SS^ 6 hr. iO^g min. 70-41; 146. •274.12^; $456.26. SA and h. $50000. (308) •5100. (310) $12 31. (296) (298) (1^00) (302) (304) (300) 6!f per cent $3000. 73Jc -^^t. B, by 16 yd«. 20 ynrda. £62 5». 9 days, $1440. 70. i; i; 1. $ 32d*yi. 8 per cant. 28i ; 427i »»>• 4 per o«iit $1023.73. $80; $133|; Lom $131. (311) iimi If. (313) 16 y»an. (3i5N $7800. (317) limonthi. (319) * per cent loss. (321) X ; 6Sf ; 7. (323) 14 min. 43| •eo. (326) 1:2. (327) $100000 ; $4000. (329) 2«. 6d. (330) 88. 2U , - (332) 5 hr. (333) 6 mile*. ^234 ; $266,40 ; $.300 . $3^.5 60. 3i hours. (330) 70 cents. First is $50. (338) $14600. (339) 25 oxen. 2910 ; i>, 2052 ; C, 1944 ; A', 1728 ; in (341) (344) (347) (348) (349) (350) A, 3240 ; B, all, 6480. Ijm. (342) 4^. (343) $2200. SJ- , (345) 18 min. (346) £1100. Man gets £4 4«. ; woman gets £3 ; child geti £1 16«. 27961 ; 12500. 376 grains of potash ; 390 grains of soda. $20.96. i "^^"^^^^^ APPENDIX I. Let P INTEHEsr, ANIVUITIMH, ^c 1. To find the amount of a ffivett sum, in any t/itten time, at Simple Iiittrest. If P bo the principal in doUara, n the longth of time in years, r the mtereat of $1 for 1 year; then the interest of |P for 1 year will be Pr, and for n years will be Pru ; where- fore, if I be the interest, then I- Pm. If M be the amount, we have , M P + Pm - P (1 + m). 2. To find the amount of a given mm, in arty f/iven time, at Compound Interent. the principal in (ioMara; ♦• = the interest (jf $1 for one year ; ** n ■■ the number of years ; " R = the amount of $1 for 1 year = 1 + » • then PR will be the aisiount of $P for 1 year, and thiH becomes the Principal for the 2nd year : .. PR R = PR' will be the amount of $P for 2 years, and this becomes the Principal for 3rd year : .. PR^'R - PR» will be the amount of $P for 3 years, etc. ; hence M = PR'» = P (1 + rY wili be the amount of $V for n years. Interest = PR" - P = P(R« - 1). 3. To show that the fonutda M - PUT is true when n is fractional. If n is fractional we can always tind a whole number suoh tnat /Mi id a whole number = q., suppose. Divide the AFPBNDR. 827 ye*r into a «qual intmrvftli, and bt in be the ainotint ol tl in one of th«M intorvalfi, tlion the atnount of |I in a in- tervalu in m'* , and in ec ual U) H; alHo tlio iiimmnt of 11 in n years, that iH na ii.torvalH, iH equal to !»•»*, and tlierefure equal to R"; hence the amount of 1 1' — PU", tlierefim thu foniiuln is true for fractional vuIuoh of n. ThuM, if r' in the nominal yearly rate of interest of |1 pay- able q iitnm a year, meaning that ■- in tho interest payable at the end of each ^th part of a year, then the amount of tl in a year i* • (A + -) , and the t' (6) In what time will a sum of money treble itself, at 5 per cent. , Compound Interest ? log. 3 = -4771212, log. 1-05 = 0211893. (6) A sum of money, $P, is left among A^ B, 0, in such a manner that at the end of a, h, c years, when they respec- tively feome of age, they are to possess equal sums. Find the share of each at compound interest, (7) Two men invest sums of $4410 and $4400 respectively, at the same rate of interest, the former at simple, the latter at compound interest ; at the end of two years their proper- ties amount to equal sums. Find the rate of interest. (8) In a certain county the births in a year amount to an mth of the whole population, and the deaths to un nth. In how many years will the population be doubled ? (9) A person spends in the first year m times thr interest of his property ; in the second year, 2m times that of the remainder , in the third year, 3m times that at the end of the second, and so on ; and at the end of 2p years he has nothing left. Show that in the pth year he spends as much as he has left at the end of that year. (10) If interest be payable at every instant, in how many years would $1 amount to $6, reckoning interest at 5 per cent. ? (11) A person starts with a certain capital, which produces T "m 4 per cent, per annum compound interest. He spends every year a sum equal to twice the original interest on his capital. Find iji how many years he will be ruined, having given log. 2 = '3010300, log. 13 = 11139434. (12) The population of a county is 35743. There is nc emigration or immigrallou. The annual deaths afu 27 : n yearii. t, time. ly rate of be PR", n t years, le end of taelf in n f, at 5 per in such a )y respec- Find the pectively, the latter ir proper- 3St. )unt to an I nth. In le interest bat of the ;he end of irs he has Is as much how many k at 5 per 1 produces He spends rest on his ed, having here is no - ^_ e\^f :„ APPENDIX. 881 the 1000, and the births 62 in 1000. What will be the in- crease of the population in five year? ? (18^ If the population of a countr3' be P, and every year the number of deaths a -^tb and the number of births -iVth, of the wliole population at the beginning of the year ; find in what time the population will be doubled. .og 181 = «^-25768, lOg 3 = '4771213. log 2 = -30103. (14) On a sum of money bonovsred, interest is paid at the rate of 5 per cent. After a time $600 of the loan is paid oflF, xnd the interest on the remainder reduced to 4 percent., and ihe yearly interest is now lessened one-third "What was the sum borrowed? (15) If a debt « at compound interest is discharged in w a m (1 + rf (1— wr)=l. )rears by annual payments of ^, show that DISCOUNT. 5. To find the Present Worth and Discount on any sum, for a given time, (1) Compound Interest. (2) at Simple Interest. Ths principal difference between Amount and Present Worth is that the former is reckoned forwards from a given date while the latter is ckoned backwards from the same date. Hence it is ev t tliat if V represents the Present Worth, then, V = P(l -{- r)-** P = Ti'TZxn foi^ Compound Interest; expanding and neglecting r* and higher powers we have F ? Tnr ^^^ Simple Interest. f miimsisimiim 882 APPENDIX. If D be the Discount, then . D=P-V p p_7- r-» CompouDd Iksterest. P Pnr . , , ' = ,-, . Simple Interest. 6. If we expand P (I + r)-", an^ neglect r' and higher powers, we get P (1 — nr) which may be called the common prcwent worth. The true present worth is J—. : by division I + nr* •' = P (1 -nr + n^r* — n'r^ + - it is a negative quantity. That i«, the common present worth of a bill for $100 due 20 years hence at 5 per cent, is notliing, and for any period beyond 20 years the holder of the bill would require to pay a certain sum to get quit of it, which is absurd. The true present worth of 1100 due in 20 yeaxs, as given oy lue lormuia ■. . -, is vou. APPINDIX. sss 7. The interest is greater than the discount. Sinoe D - P - . 1 +nr 1 1 + nr 15* Pnr - ^ +1 J f p , 1 1 _., ^ . • D I ' . . I > D. 8. Since ^ l + nr* and D- '•"^ 1 + nr I 1 + nr K7e see that j^^^^>^_, ^^^^ If we expand, neglecting r« and higher powers, we have Px(l - n,r) + P,(l - n,r) = (P^ + P^) (i _ nr) 'T, PiHi +p^„^==(p^ +P„)n; P + P which is the rule given in Art. 184. ^ ' 12. We have seen (Art. 6) that the expansion of (l + ry 1 . V / , neglecting c-^ ^nd higher powers, gives ^mmm present worth instead of true present worth. Ihe above process is, therefore, incorrect. It may easily be seen that we have taken the interest instead of the discount of sum paid before it is due, and thus, since interest is greater than discount (Art. 7), a small advan- tage has been given to the payer. 13. If we write equation (1) in the form. Pi _ Px + P. (1 + r)«i (1 + r)»a (1 + r)* and expand, neglecting r^ and higher powers, we have l + n^r I + n^r 1 + nr ' which is the form of the equation for Simple Interest piSiiwii. 338 APPINDIX. n Solving for n we get P, + Pa +~r(P,ni + Pan/) ^ which Li the oorroot value of the equated time. If r be a very small (juantity, as in practice it usually is, and Pj, Pjj, not very large, wo hhall have ** P, +P, » •*" ^^""- M ANNUITIES. 14. The term Annuity is understood to signify any interest of money, rent, or pension, payable from time to tiiAe, at particular periods ; and these payments may take place yariy, half-yearly^ quarterly^ ^c. 15. To find the Amount of an annuity to he paid for a given number ofyears^ at Coinpound Interest. Let A bq the annuity, n the number of years, R the amount on one dollar in one year, M the required amount. We have Amount due at the end of 1 year = A 2 3 &c. = A + AR « A + AR + AR' = &c. = A + AR + AR2 + . . . . + AR*-» R"-l = A R - 1 R"-l y ice M = A rvi — 1 16. For Simple Interest, expanding and neglecting r^ and higher powers, we get AmXDUL 339 t usually is, signify any from time ments may paid for a jars, R the e required neglecting -H - n(n - 1^ \ 17. Tojirui tha Prettmit Valm rf an annuitif, to be paid for a fjtven numher of yean, at Compound Interest. I. The amount of the annuity at the end of the *: ^t year is A, while the present value is AR~' ; similar, the amount at the end of the nth year is AR"-', and the present value is AR"*. Hence, in order to obtain the present value from the amount, we must first multiply the formula for the amount by R, and then change the sign of the index of R. Multiplying by R we get R-1 R~f^' - R Ohanging sign of index we have A A A R-^ - 1 R-^ - 1 1 - R - - (1 - R-«). II. We may obtain the same result by proceeding on the principle that if the present value P be put out to compound interest for n years, it ought to amount to the same as the annuity for that time. Hence PR** = A P = A P -1^ 1 - R^ R-1 A/ _ = - - i i - R- 1 V 340 APriNDII. TTI. W« will now prooMr) on the principle th*t t.h« prMent value P is the Riim of the pretent valuM of the ratpeotive annual paymt^nts. PrtMnt value of A due 1 year henoe «• AH"^, 2 •• - AK-*» Sto, ■■ ACf U tt M II P - AR-» + AR-« + AR-«. . . . AR— R- - i - AR-* R=rzi - R-»)- 18. For Simple Inierest, expanding and neglecting 1* and higher potcerg, we get A (1 -f r)" - I A 1 + nr + -(^)»^ +....- 1 r 1 + ♦If as A • ; '■ " 1 + nr ^ nA 2 -Kn - l) r 2' 1 + Mr 19. To find the present value of a perpetual annuity. I. Reckoning Compound Interest. P a» AR~' + AR~^ + . ... ad ir\finitv/m. AR-^ "" 1 - R-i A. "R - 1 A Apn£iir4x e thftt tli« //. ^iiKkomny iiimpif IntmnL llf^M of tb« p_fiA a + (n-l)r . 8 l-¥nr l-^, .nA J + a-i)!- <-\ 2 i + r " ft-; Now wh«n n » », the limit of » S + (1 - 5)»- 4. ri Sil -0)r + r - t Hence, the limit of P, when n QcA 2 • ♦n^ i'?!'"' '^7' *^*^ *" *'^'**'« *'*"» «^ n»on«y w required ,v«, -SjI? T**%*" «;'«"»-o an equal Rnnual paymeT.t for corrir mi^hn ^ f^' ^V»"^>«''-^««. therefore, tkt the only inf !1* •^'*,'*^ computing annuitiee ii on the oompouud interMt principle. '^ SK). To find the Present Value of an annuity, to com- mence at the end of p years, and then to continue q years. The present values of the first; second, &c.. g*h payments, evTd^ntly r "'^ ^ ^ '^ ^'^ P ^^l ^^^ rj^olliy^ "will whence the present value P-AR- . il -R-i/ ir - l/ + R~' + ...+R-<'^-^> RlHq If the annuity is payable /or ever after p years have ex- pired, by BumraiLg the above aeries ad in,:nitum, we have P rrs A Rp(R-l)' These formula enable us to compute the values of Revtr- miu, or Ammities in Reversion ■ and the latter determines he value of the tee Simple of the freehold estate, which is 10 tall m at the expiration of v vears. <■ ■., 842 APPBNDIX R| F Ex. (1). A sum of ^a is borrcwflH for a periofl of m years, to bo repaid by equal annual instalments, the first payment to be made after one year. Find ,the amount of the annual instalment. Let A be the annual instalment. Then the amount of this annual payment in w years Again, if the sum a be allowed to accumulate for m years at compound interest, its amount = a R*". N Now these two amounts ought to be equaL EEence we have — Ir'"- 1 i =aR»«; * A ^ -. R" r R»»-l ^a 1__ Ex, (2). The present value of an annuity of $1, to continue q years, is $10 ; and the present "/nluf* of an annuity of $1, to continue '2q years, is $16. Find the rate of interest. Here, and 10 " - (1 - R-«), Art. 17, ?6--(l-R-«*); or 16 • 1 - R-2? •10- 1 ■ - R-« = (1 - R-«) (1 + R-«) 1 - R-*? = l + R-«; ^ 16 ~ 10 1 - -R^; 1 - R-^ = 2 APPEKDIX. 848 or m years Snbtitutdng in the first equation, we get 2 5 7" = 10; 26* or 100 r = 4. The rate is, therefore, 4 per cent. Ex. 8. A mortgage of |6,000, interest at 6 per cent per annum, has 7 years anl IC months to runf find its be half-yearly, willbe300(l4-. ' s'^lar^; thelmS:^, .>f the second payment of interest at the en.l otthelvT^f^ ..11 be 800 r05V- and so on The amount of IheLt ' lyment will be $300. ^*^'' He^nce, the whole amount of the mortgage and Interest 6000 + 303 (1 05)1* ^. 300(1-06)12 + ,, ^ ^q^ -= 6000 + 300 {{104)i'+a.05)i2+ * ^ jl ' = 5000.300 |;j^|!z|| = 8462-06. Mterest at 10 per cem. per annm>i for 7 years and 10 mT •J ought to amount to tiie same as the mortgage for that .-. I (l-05y"^ = 8462-06 ! .-. P = 3940.13. The Present Value of the Mortgage is, therefore, $3940.18. The /alue of (1 •0C)i6» may be fomid (1) By means of a table of logarithms. 1 n?^ ^^ 1*^.""^ ^'^^ *^ *^® 16*^ power, dividing this bv LdadLf^T^*^-^t''^ P^"^'-' ^-^-^tof theliffefenee power ST.(^. ''''' ^'''^''' ^^ ^'* approximately the 151 ■^ *--, -^ ';■ . sa APFEHDIX. 1^ (8) By the Binomial Theorem, as follow? • JExBKcisa IIL uieis tf^/r aZXt .to^rr' -/ » ~-'*'^ pound interest at 6 per cent? P'-^hased, aUowing com- a.muV'p?rd:dl°hTiTm thm^'y."* «'««« p« should dMcend to his onW fj f ' ,«""■ ''''' decease, it daughter for the nevf V" ^ " 'P"' ^° y«"'. *» his Jnly tie/ for ever afte?wfrfr WhTf** *" Vbo-^volent ir,stit„! bequest at the tfmeorhUdeoIIJe* J" •'*'* ™'"« "* ««h est at 6 per cent.™ decease, allowing compound inter- years old.' He ris^colkL^dTh-lT™'.'''' ""'" ^^ "■» ^0 verted the some nto „rilw • wh f"'* *°""*"5'' ""^ ™- means, accumulated? ■ ' """ *"'°"°' ""' •'y 'hose in12 ytrSn' «S t'r." ^If "' * P» ««>"■. *» !«> paid annuJpa^LtT 'nstalments. What wiU be the ing a fineff-lr What 1^1^ ^di •""'l''' T^"^"* °» P^^" this fine ? ""* additional rent equivalent to "^^ ''.the p JSSl rdv^^^^r//^™''* Thi. method wix wUxi a'; the next trial " '^~' ^^ ^**''* "^ ^**^^ 8®^ X 2 f \ ANSWERS. «: ^ I r I (C) ^'sahar© is (7) 5 per cent. aO) n = 20^^Ili. log. e (12) 6708-471. Exercise I. (6) 22-6 years, nearly. PR6+C R«+6 + Ra+c^-p6+/ (1) $706.66^. (4) $900. (8) . J !^?ii (11) 17-67 years. (13) 125 years, nearly. (14) $36oo. Exercise II. (2) $13585. (6) 3ff. (8) -^ a + r (5) 1002-^ per cent. (3) $19.60. (7) aRp-^ - bRp-*^. (9) 11 -463 percent. (lo) ij years. Exercise III. ^'^ ''*S2m'lo"'''"*' ^'''^•^^' ^* ^^«^P«"»d interest, *>^173.10. (2) $802.42 (3) $16666.66f. (4) ^2199.95. (5) $7360.08; $6404.74; $2901.83. (6) ,3000.56. (7) rfrR« R«-l' 150-76 1 - (l-04)-3o (8) ^^^ r=^ (R--R-) (10) ^= 2 (1 - J- 1 , If 1 6 1 R'*/ (11) -ii » _ 1 1 . . Q I (1 + 9\mn / . ^ w interest of iS^l for 1 year. \ (12) $53.63. (13) $2422.85. \ », nearly. ) - log. mn' •a rears. (14) $3600. (3) $19.60. mt. ^) IJ years. nd interest, $3000.66. 11. year.