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 Price Fifteen Cents. 
 
 CAMPBELL'S 
 
 CANADIAN AEUHMETIC, 
 
 4i 
 
 IN DECIMAL CURRENCY; 
 
 ■fcV; ..^ 
 
 OR. 
 
 a 
 
 THE FIRST BOOK OF AllITlDIETrC " 
 
 X 
 
 SUPEHSEDE X 
 
 M0>7REAL & TORON'''0 : 
 
 JAMES 6 A M P B E L J & S O N 
 
 1866. 
 
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 11 
 
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 «im^jhHHHI 
 
CAMPBELL'S 
 
 CANADIAjy ARITHMETIC 
 
 IN DECIMAL CURRENCY, 
 
 OR, 
 
 "THE FIR3T BOOK OF ARITHMETIC" 
 StJPEESEDED. 
 
 •t 
 
 
 TORONTO : 
 CAMPBELL & SON. 
 
 .TORONTO STREET. - 
 
KKKATA. 
 
 ana'S''' hotline, for ' Mi visors tii and 25." r.ad 1« 
 
 Page 106. Kxample 5i. 2iul. for -4 + 2," r«u,l -4 ^ o 
 J'age 146, for Section V., read Section VII * 
 
 No«. in thJ Amvvers "^' ^" '''''''Von<\ with th. 
 
 Enteml accordiug to the Act of Provincial Parliament, in the 
 year one thousand eight hundred and sixty-one. by James 
 P^r !!t'/r "^ '' ^- - "- ^«^- o^ the Regi^trL of th 
 
," load 16 
 <ul "4 -T- 2. 
 
 * the next 
 fie number 
 1 with the 
 
 nt, in the 
 by James 
 trar of the 
 
 PREFACE. 
 
 nrri^ntern h'°f ?/ ^^^ *^L*'^"^'"t"«*X Arithmetic now 
 A»5fr l* *? ^"^ fHlow-trachers un«l tho pubMe, the 
 Anthor has bpoi. giiidod by a somewliat lengthened ex- 
 ?«!?«?" '" *''" '-'''K ^^' ^^^* «'^«''-»-oom, and by hin « nnd 
 
 Sent n^ '"any objects wliich an author, iu this do- 
 
 DumHn«:r ^*^*/'rPV'''^'l''"P'''''t" ""»"*' '" the pre- 
 paration of a text book. Yet, thin, unliappilv, \h all hat 
 
 i« too frequently either proposed or attemptcHl. 
 utmo«?'^«n^r? "'""f '■"^'<' *''« text J)ook as to secure tlie 
 Deration?" V -^ and accuracy in performing the various 
 
 wishes and assist the efforts of the earnest teacher • but 
 Sa te^r'' fa" /ar short of comprehending ali t i 
 accomplisb all that he proposes to effect 
 uffh fi! ' •" f^^^'tion to this, to tamiliuiize the pupil 
 If the principles which shall enable him, at once to 
 deterniine, not merely how to perform -. . ai i oneralVons 
 n obedience to orders, but to determin ... in every cX 
 iM of tl.e^ various operations should bo, m/Jmel 
 
 Scw'o'f anln'^"' ''h •" tbeeflbrtto ^S^The 
 leacDerof an unnecessary <uY upon his time and injre- 
 
 «nd^ve^t fhn ^ J*"^"' .f ^ operations already r)t.«sed ove^ 
 
 .ZunrtL nn'l^^VK^^"^' ''^"^"'''^" '^ ^^^^'^y to gather 
 around the pupil if he docs not frequently retrace his 
 
 RurvlV'^7fiL"''^\ "i'' •*" CMJiignieaeu and comprehensive 
 Tn 2hor st1l7f''^f.^'*'l'"^^ *^.^^*' occupied should lead 
 want! of the fii.^''*^''';.*1 ^"tkipate the purposes and 
 wants of the teacher. It should certainly determine him, 
 
 *wWIM««»».««iii,, 
 
iT. 
 
 PRKrACK. 
 
 extent the «r^p,^„Xlo,,i^^^^^^^ '"«''«' «■> 
 
 pwently distinct and .Hff' „„?'"** '" » ""'■'e'y of «P- 
 
 ?«1ou» dtatanm from o^HS-L"'"'"*,""'"' »'««<> «« 
 different n»me. • or ^„^.k " "' f"^ <=»'>ce«led under 
 
 Bbo"l?d\L"Se1;''author"rr ^"^ *« '^-^^i-. 
 more. It 8™uTd IcaSi^ .J" »"TP' y*' ""•"""■ing 
 ">« pupil, M far a» no^hi. "i'P'j"'-'''"" •« ^nde? 
 P«.d(iSraakinKC«°cu?at^^i*'.'h'^^"'^''i' »''*'«''' «»« 
 to combine SwZ^AT:' 'J*' ',"• "»''»,<'''* '""d Wm 
 
 tealher IhouM be ,o?rmiri".T *" "<•"' '"'"o^-nd 
 that the puDil cainot .?i*'*.K* *™""'"' ''"'' ""e "i, 
 gating, unStaSg anni"a«^fvinrZ"' ."'.'l^""' 
 which the rule, are blied (sX«K Sb^t: «''i"l°? 
 In the preparation of an Arithmetic for ' .h^ **' *"-l 
 
 entt ^r ff^fo? ^"'1 ^ ''^ ttVtprtan" 
 
 enable the Racier totara out Sr ""t" T^'^ "> 
 ticiana. This, however Thv^„ 1 * practical arithme- 
 
 aDDlTA'L*!*'^ Author has attempted to embody and 
 WW leave ius feUow-teachers aud the public to judge 
 
J»? 
 
 CONTENTS. 
 
 SECTION I. 
 
 NoUtion and Numerat Jon, . . . . '*•■• 
 
 Arabic NoUllon, .'."]* » 
 
 Numeration Table,. W 
 
 8lgn.u*edlnArithmeu;;7daiVlin";;;',, " 
 
 MontalExercigesinAddiUon,.. " 
 
 Slate ExerclMB In Addition * >* 
 
 Si'cond Serle. Mental ExercliM^'inAddiiion' " 
 
 Subtraction, Mental Exerciset ^''"'°"' 18 
 
 Mixed Question8 in Addition and SubtVacVion " 
 
 Mixed Mental Exorcises,.. "^•c"on, „ 
 
 Multiplication Table and Extended MuiunV.« W' " :;:. ** 
 
 ««mple Multiplication, Men J^e'c^"'*''""'"^'''' 2» 
 
 When the Multiplier is a Comnn.it* J"'J 26 
 
 Dociraal8,..!^.....'".'.*^P«»'^ Number, with and without 
 
 When tlM Multiplier contains sevlrli '«.' ! 28 
 
 Decimals,.... ....^/"^"^•"'fl*'''**, with and without 
 
 When the,e I. a Fraction in the MaUipVler* *• 
 
 Wvi8lon,MenUlKxerci8« -"'"P"*', „ 
 
 When the Divisor does not exceed ii *» 
 
 When the Divisor is more than 12 withanH -i*/k:V;:: ** 
 
 Division Of D^lmals-RuieTr iiX D<««"»I.,.. 38 
 
 When there is a Fraction in ;i:S;ttr''"'""^^ ^ 
 
 MiscelUneoiw Mental Exercise ' V: ••••• « 
 
 SECTION n. 
 
 Tables of Weight, and Measures 
 
 Wailii«»l<»». *\ .. ' ••••••«•.•••.--, M^ 
 
 Reductton iRcmidiag, 4* 
 
 ' " 47 
 
^' OONTJBKTS. 
 
 Redaction of the Old Currency to the New, &c. . ^ 
 
 r,0ncral Rule Applicable to all Addition ^J 
 
 Mental Exerciso ta<fe|apOjind Addition.'.cl '.i'x " i r? 
 
 ""' 'S'''rr!'. '^.'r .*:' '"« "^•"^^^ importanc;'di;eVted' 'to'bc 
 
 (teneral Rule for all Sobtracilon, .', ?? 
 
 Montol Eiercto la Compoufi.l Sufctraclioo , '.'. 1 
 
 '^"KnrnlSXXle"""'""""'"" »»^™^ -^ '.^w.; ^.^ 
 
 »ulo l«XompJuml Multiplloatta wlwathe MulVlpHor McVcd, i^i A 
 Mc„t»l Exerctac i„ Cv,m|»„„<i Multlpllciton «"»«""'- « 
 
 Htilo for Comijouml Division, ' *'"",' *••••• • " • ' • :' • • *^« 
 
 A Miscelktucons Mental Exerciec,. . , /. ....'....]""" ' ' ' " 'i^"^^'^ 
 
 SBCTIOfRini 
 
 GreatfBt Commou Measure. 
 
 Least Common Multiple, ....;.. ' 
 
 Vulgar FractioD8-tlje variowe kinds^ '..11 ^^ 
 
 "^ ."air;;rHSsa:.'.:'"' ^'^<^^^^ ^ 
 
 Au Iiitrodoctory Kxorcise, obiclv menul. „ri,i>,iM.^ JJ.'.'l-.V' 
 
 >i 
 
60 
 
 V" ■■ 
 
 02 
 65 
 
 COKTBKtS. yij 
 
 ^sJS-c";!'*. :*'"**''" ^^«^«'^ ^ ^ 'X'ok. if thoroughly 
 
 Reduction of Fractions,...,.....' ' """ ^ 
 
 A Mixed Exercise in Fractions .. * ' '^^ 
 
 "Problems in Fractions b v AnalvJiL *^ L " ", " ' ' ' • ^^ 
 
 and conducted witK sSSvin^t^.^^^^^^ 
 
 accuracy and neaVnossTu^^xm^'^ifoS T" «' '''"'"^^ 
 rwisoning, 'U- expression, and corrortnt'ss in 
 
 licamples of the method in which fh-l 'L"\l' " *^ 
 
 before the class, . ... ^^"^ P^^" '« ^ »' - «e orally 
 r>ccimals more fully treated " " '••••• lOl 
 
 To Add, Subtract, Multiplv Ind W^k^ p ^ *' "ct,un.,,. . . . : . io8 
 tends, .V. . ^ ^"^^ **"'' '^•-'««^' Repc- 
 
 To Find the Value of a Vivot, nll;\l"i ' " " *, • • • • ^^^ 
 
 DenomioatioDs"..^. '.*:". ^.''J'"'*' "* Integ..rs of tin, Lower 
 
 ToReduce one Denominate Number L/HU'n "•" ■,", ^^ 
 
 •;3 gJven Denominate Number,^ :^^.. ^'«*^^»'n"l ''f another 
 
 Mixed Kxeroise on Decimals ' ' * *•••'•.. 109 
 
 V" 110 
 
 „ .. , SECfi6}q IV. 
 
 ^ Problem expSned ^^ * determine the sUatemcnt of a , 
 
 pies i^^ecMa Z ^miZnSl'.L''^ «tated~the p„«ci- 
 
 Miscellaneous Exercise " rdo^^TbTp" "1"'"'*'' ^^ 
 
 hen, by Analysis : erab^cinc l'roblon,«^ Proportion, and . 
 
 'edge of VHlirar and iwVl„?iJ^!'*ll'-"*« fcn'nniJg a kuow- 
 
 siating the Problems in ftoMr^,'ioiV'"a"nT"fI'':K''"f **"•"« '» 
 
 be performed orally befo;J tKS *'' ^^^ ^''^^''^ »« 
 
 Compound Proportion-its relation to SimpleProi^rVi^n;::;;:: no 
 
Simple Partnership, ' •• ** 
 
 Compound Partnership ^^ 
 
 *^' 125 
 
 SECTION V. 
 
 Practice-'Rible of Aliquot Parts, &o.. . . tL 
 
 Ml«ceilaneou8 Problems to be done by Practice,:; il ":::::::;:: I^ 
 
 SECTION VI. 
 
 Per-Centage-the terms per cent, and per unit explained iqa 
 
 Commission, Brokerage, and msuranco...... '^ ' !?? 
 
 Profit and Loss-Analysis of Rules, '^^ 
 
 ^"•ifnW^r.". '" •^"^'^^ *»»evaH;;.ope;a;ions-inp;;i; "* 
 
 Barter, ' ' 134 
 
 foterest-HJxplanation of the terms' emptoyei Jff 
 
 To find the Interest for one or more YeLsV "' l^ 
 
 To And the Interest for Months, Days, 4c.... J?! 
 
 B^^t^^^''^''"''^'' '''' ««*^' Time,-andInteVesV,;e 
 
 To ^^^y^mi'^;^;^^ ^^ 
 
 Compound Interest,. ^*** 
 
 "'^/Si^^ei^.r^r^^'^^-^'H^^^ 
 
 Discount-Bank and True DiscounV.'.:;;;:;::;;;;;^,:;;;;;;;;;;^ 
 
 SECTION vn. 
 
 Extraction of the Square Root,.. ' *•' '" 
 
 Brtractkm of the Cube Root,... ^** 
 
 A Shmer Method of Extracting toe'cibew;.' ' ]t 
 
 AppltoatiOB of the Square and Cube Root,.. V J?? 
 
 A Final Comprehensive Miscellaaeoai Exaf cise ^It 
 
 A Final Miscelteneoaa Mental Ex^ciae.. ' ,f? 
 
 Answeni to Exercises ■*•* 
 
 "™' ^^^^^ ^^ 
 
 i kl 
 
 4 
 
 IV 
 
ARITHMETIC. 
 
 JJ 
 
 SECTION I. 
 
 NOTATION AND NUMERATION, SIMPLE ADDI- 
 TION, SUBTRACTION, MULTIPLICATION AND 
 DIVISION. 
 
 NOTATION AND NUMERATION. 
 
 Notation is the expressing of numbers by cer- 
 tain characters. 
 
 Numeration is the reading of these characters. 
 
 There are two systems of Notation in use among 
 lis, viz. : the Roman and the Arabic. 
 
 THE ROMAN NOTATION. 
 
 The Roman Notation is chiefly wed in inscriptions, and 
 to denote chapters and sections of boolcs, <fec. Only seven 
 numeral letters are used, viz. : tho caj^tal letters I, V, X, 
 
 When standing alone, tlie letter I. denotes one, V. de- 
 notes five, X. ten, L. fifty, C. one Inmdred, D. five hundred, 
 H. one thousand. All other numbers are expressed by 
 eombination» of these letters in the following manner. 
 viz. : 
 
 A* 4«j3 wiicu as 
 
 ttuy iviivr in repealed, so many times 
 "" ' * "1. is two: X le 
 
 Mavflne is repeated. Than, L is one 
 ten, XX. twenty, &o. 
 
 #^. 
 
 5 
 
10 
 
 ABITIIMETIC. 
 
 2. When a letter of less value ia placed before a letter 
 
 «eSf; 'h 7^T 'V^^^' r''^ ''' «^° value froi the 
 greater ; but when placed after it, it adds its own value 
 
 eix , X. IS ten, IX. is nine, and XL is eleven, &e. 
 
 ^I\A Ik "^ "^ ^^5 ^^^^ ""^^^ ^'^y le"er increases its 
 va^ue a thousand' Mnes. Tbu., V. is five, T is five thous- 
 and ; X. 18 ten, a. is ten thousand. 
 
 KXEKCISB 1. 
 
 Express the following numbers by letters : Eiffht — 
 ninety-nine,-one hundred and «ixty,— four hundred and 
 
 hu&'lnd"«i^'t""'*r^^-°!/'^'' ^^^ *^ou^td oS? 
 nunared and 8ix,--two thousand and twentv-flve— six 
 
 hundred and ninety-nine,~one bousand nine hundred 
 and tweuty.fiv«,>-two thousand six hundred and eLhtv 
 four thousand nine hundred .ind sixty-five^-t^otCS 
 "red aSStrV^' «-^^-e,-one Ihous'and reeTun' 
 
 THE ARABIC NOTATION. 
 
 haJa^h ^''^}''' ^*l**^'o^ ^'8 *^e method of expressing num- 
 S^e^ie'Ltvlz'r'''''^''"'^'^'^''^ Thechafaoters 
 
 1 2 3 4 6 
 
 Oao. Two. Throe. Pour. Five. 
 
 6 7 8 9 
 
 Six. Seven. Eight. Nino. Naught 
 
 n Ji fir«* "ine are called the simple units, the significant 
 ?l?f ^';k^"*^.^?°*^*''"^^ ^^^ "»°° ^'g'*^- The last Ode (0) is 
 
 5^. V« ""^ ^^ '/'^'t! ^"* *« «'"»P^J ««ed to keep the si*- 
 mficant figures in their proper plaees, by filling up anv 
 place or places for which there are no significant flgfr^^^ 
 
 _ All numbers we^expressed by these ten simple cbarao- 
 ters, thus : The f|ffiir«« are written ns-^at^:, ."--TT "^.^ 
 
 the ngbt and left of a certain poio I called the deoimS 
 
 ] 
 ( 
 
 1 
 i 
 c 
 
 f 
 b 
 
 E 
 
m.' 
 
 '•r;. 
 
 
 
 i 
 
 TUB ABIBIC NOTATION. 
 
 poJQt thua, 124.32S. The first figure to the Ml 
 point IS so mnnv «i,.v._!i !. .;_!?■"„ """ '5« 
 
 11 
 
 point is SO Wn7«„a^it ifs S^f "" ^''^^ "^ «•'» 
 S;?t^?X%«^Ve W r? V'^'w^^^^^^ 
 
 the fourth Is ten ttaLSs-leneela^V?.^^"''*^- 
 and two hundred and flf-j-seVeSTortiS^ 
 
 From what has been paid, we loii-n ihn* tu^ i 
 from the diimS pohf. 'J'Jj Zt "*°f ' "' ^'^''">*« 
 SS fart'hVtolheZl? " ,?:^"> *"? »'«^r "« vZSt 
 
 In the second plaie is o? ten ?^ J " 'f'' ^"^ "K"" 
 m 80 onl '"^ ^"^'^^^-^fls &^i«» tea or the thpuamdthi^ 
 
IT^F? "T*''^1*" 
 
 n 
 
 ARITHMprrC. 
 
 ■XBKCI8E 2. 
 
 foSr t"j;t^r^"r'y-^°«/-««ven hundred andSf^C 
 «idth8,-8even hundred and sixty thouindZ -Lvp^SJ^ 
 
 ^ In reading figures, we group them into periods of thPA*. 
 figircs, (one for units, one for tenp, and one for huS^ 
 
 WHOLE NUMBERS. 
 
 ts 
 
 DECatAl^. 
 
 Increasing to the left "«« Ij»®"Decrea6ingtotherl«ht 
 
 •^^' s;aa -'a- :^-a --!.. «p5!- 
 
 OQ 
 
 CI CO 
 
 : 6 
 
 •2 § i -a I g i 
 
 t) B H S S ^ {3 
 
 4 
 S 
 2 
 5 
 4 
 G 
 5 
 
 I I 
 
 Illli 
 
 n H t=> n ^ 
 
 00 
 
 1 
 
 t3 M 
 
 1 
 
 2 
 5 
 
 5 
 
 
 2 
 
 6 
 3 
 6 
 4 
 4 
 
 6 
 
 2 7 
 
 8 
 
 7 9 
 
 7 
 
 6 5 
 
 6 7 
 
 6 
 4 
 
 6 
 
 4 
 8 6 
 8 4 
 
 6 
 
 7 
 
 5 
 
 7 
 
 8 
 
 7 
 
 2 
 
 5 
 
 4 
 
 in is 
 
 2 : : 
 
 8 3 ; 
 
 6 8 
 
 6 
 
 8 6 
 
 6 4 
 
 8 6 
 
 6 
 
 4 
 2 
 3 
 
 
 i 
 
 7 
 7 3 
 3 7 
 
 8 5 
 
 iiXERCISE 3. 
 
 in if 1!^® P^^l! *** 'e^""^ *o reaa and write this exercise 
 !?JS."^?A'«^'^"L* W« »« *he figures c^ter^"^ 
 w;«iwii, iue seaQuer iiiuHirating on the blaek brimi nu^n^ 
 cially with reference to the decinSs ^' **P®" 
 
 ^^ 
 
 * 
 
 f 
 t 
 t 
 
 ij 
 
■■ 
 
 .. 
 
 It 
 
 ADJ)ITION. 
 
 SIGNS USED IN ARITHMETIC. 
 -f named plus, signifies Addition, as 4+2 equal 6. 
 — named mmuB, signifies Subtraction, as 6-2 eaual ^ 
 X multiplied b,, BigoiflesMultiplicatl'o" t^x^llis 
 -^ divided by, Bignifles Division, as 10-:-2 equal 6 
 
 : :' so is i ''^"tk"' ^'^PO^Won, Is 1 : 2 : : 3 
 
 . to S ^''^^^ .^^"'^^ "•« *b«s read- 
 
 _ ^ Iw to 2 as 3 is to 6. 
 
 l/^arks the Squai-e root, as |/ Tz= 2. 
 #' marks the Cubfj root, as ^'jb ~ 2. 
 
 6. 
 
 2 
 
 3 
 
 2 
 
 2 
 
 4 
 
 4 
 
 3 
 
 5 
 
 6 
 
 4 
 
 6 
 
 6 
 
 4 
 
 5 
 
 7 
 
 and 
 
 1 
 
 1 
 
 2 
 
 3 
 
 1 
 
 2 
 
 3 
 
 1 
 
 2 
 
 3 
 
 1 
 
 2 
 
 4 
 
 3 
 
 1 
 
 ADDITION TABLE. 
 
 are 
 
 3 
 
 7 and 2 are 9 
 
 4 
 
 6 ~ 4 — 9 
 
 4 
 
 6 — 3 — 9 
 
 5 
 
 8—1—9 
 
 6 
 
 8-2-10 
 
 6 
 
 5—6—10 
 
 6 
 
 6—4-10 
 
 6 
 
 7-3—10 
 
 7 • 
 
 9-1-10 
 
 7 
 
 9 — 2 — 11 
 
 7 
 
 6—6—11 
 
 8 
 
 7 -- 4 - 11 
 
 8 
 
 8-3-11 
 
 8 
 
 8 
 
 9 ~ a - 17 
 
 9 
 
 8 
 
 6 
 
 7 
 
 7 
 
 8 
 
 9 
 
 7 
 
 8 
 
 9 
 
 8 
 
 9 
 
 9 
 
 8 
 
 and 3 are 12 , 
 
 4 
 
 6 
 
 5 
 
 6 
 
 6 
 
 4 
 
 7 
 
 6 
 
 5 
 
 7 
 
 6 
 
 7 . 
 
 8 ■ 
 
 12 
 
 12 
 
 12 
 
 13 
 
 13 
 
 13 
 
 14 
 
 14 
 
 14 
 
 16 
 
 16 
 
 16 
 
 16 
 
 ^^^mI^"rB^%'''^^^^ the,;dditionof all the 
 
 berf may KdedXXT^&VStS?^ ' ??^'^'^" 
 accuracy : for thesp rva ♦ h« ^ i ® ntmoat rapidity and 
 
 together^^nanyaddiUo^^^^ 
 
 then every 9 1 4 mu"t ^iv/ 4 f?L ?u**°*^?' 9 and 4 are IS, 
 
 90 ^^A A f^, «:, » must give 3 fw the unit &mrma . !.«»««-' 
 
 tter. Will be fS la adl-^t^™"" >*"■« »^«^ fg"^ 
 
'WMrtM^^ 
 
 ^ ARITUMBTIC. 
 
 ADDITION. 
 
 fwifnf ^i""" '' ^^l ^'*^^^ ^^ ^"^»"g *»^e sura of 
 two or more numbers. ^ 
 
 MENTAL EXERCISER. 
 EXRRCISB 4. 
 
 ^'^. 
 
 
 y 
 
 
 
 
 ■' 
 
wmmmmm 
 
 mm of 
 
 apples, 
 rat? yn^^ 
 
 avs, 3 " 
 egavo 
 
 aula, 
 
 lother , c; ^ 
 
 g ' 
 
 for 4 
 
 > give /, 
 
 kP, to 
 
 her 2 , 
 
 and: ]^r 
 
 lars ; r« 
 
 dol- 
 uaiiy 
 
 and 
 I he 
 
 e(f to 
 
 
 t 
 
 ADDITION. 
 
 i# 
 
 EXERCISES FOR THE SLATE. 
 
 " 
 
 423 
 134 
 
 267 
 
 824 
 
 Ex.iMPLE.— Add together 423, 134, 207. 
 
 Write the numbers under each other, so 
 that units may stand under units, t^-ns 
 iindei tcna, hundreds under hundreds, &e 
 Draw a line under them. Add the figures 
 m the right hand column together, thus, 
 7, 11 14, --14 units make 1 ten and 4 units 
 
 add the 1 to^hnScSumf fhuVl' V'^H2^^^^^^^ 
 the fl^e 2 of the 12, and add S^l/e 1 ?iiei??olur 
 
 aiWoS^^l'"^ * column of figure?, the pupil should not be 
 allowed to repeat each separate figJre, but simply to 
 name each amount once, and only oncf, as he proc^s.^ 
 12 2 8 1 4 % ft 
 
 » 4 6 4 2 4 6 J 
 
 6 
 
 2 
 ] 
 
 3n 
 
 12 
 23 
 
 46 
 
 $ c. 
 41;27 
 24.1^ 
 
 mm 
 
 9 
 
 4 
 2 
 8 
 
 8 
 
 6 
 4 
 2 
 
 8 
 
 3 
 4 
 5 
 
 4 
 3 
 G 
 
 23 
 24 
 35 
 
 57.29 82 
 
 84.173 
 24.429 
 53.836 
 
 2€ 
 42 
 6& 
 
 5 
 4 
 
 7 
 
 21 
 34 
 75 
 
 130 
 
 42. 
 5G 
 85 
 
 11 
 
 3 
 7 
 
 8 
 
 12 
 
 4 
 6 
 9 
 
 -r~-T7m5 j-rr 
 
 42.286 
 23.457 
 97.983 
 
 164.726 
 
 $ C. 
 23.25 
 50.50 
 64.75 
 
 'SO . 
 
 4l /a."7"W-v . . "z — nJ3T5 i-rr j. . 1. — 
 
 mast be put down so that the decimal pofnti 11 ftani 
 directly under each other. 
 
 nals, tne figures 
 
/'bk^''*I'''""»*W* 
 
 •I^^Mr^ 
 
 U 
 
 ARITHMETIO. 
 
 $412.( 
 
 EXKBCI8E 3. 
 
 .03 C. 
 
 346.58 
 427.76 
 
 (6) 
 
 264.64 
 368.003 
 762.74 
 865.693 
 
 1 
 
 1^> 
 16.0C 
 
 246.006 
 78.0423 
 
 604.178 
 40.0764 
 7.2135 
 
 It 
 
 (10) 
 467 
 608 
 
 92 
 400 
 
 78 
 
 (3) 
 $623.29 c. 
 146.03 
 579.42 
 
 547.804 
 653.673 
 865.407 
 276.306 
 
 I 47.12 
 602.35 
 
 68.27 
 720.84 
 
 79.36 
 
 (4) 
 354 
 236 
 
 876 
 
 (13) 
 
 $5120.23 c. 
 7142.45 
 9687.67 
 4312.64 
 
 8687.21 
 
 (14) 
 
 4268 
 2426 
 4276 
 8507 
 2300 
 
 (15) 
 
 $3687.17 c. 
 
 4215.12;^ 
 
 708.17' 
 
 9362.28 
 
 96.19 
 
 (17) 
 5126.246 
 1472.688 
 6826.002 
 9687.426 
 2764.158 
 
 (18) 
 2427 
 
 768 
 9412 
 
 893 
 4026 
 
 475 
 
 (19) 
 
 5036.14 
 
 784.27 
 
 6070.19 
 
 85.19 
 
 7507.17 
 
 687.16 
 
rnrntH 
 
 ii&i^''fc\ 
 
 ADDITION. 
 
 ft 
 
 $i28T4.27c. 
 34126.19 
 C8768.15 
 28642.14 
 65768.29 
 74387.36 
 
 * 96728.48 
 
 (22) 
 24785 
 65843 
 26879 
 43653 
 68754 
 66287 
 05423 
 
 (23) 
 48763.473 
 86270.102 
 4687.643 
 678.170 
 49060.492 
 18709.876 
 70471.218 
 
 (24) 
 
 46536 
 
 54263 
 
 43986 
 
 5079 
 
 81 
 
 641 
 
 98076 
 
 411094.88 351624 278540.974 248653 
 
 m^e ?^^'^ '°*°^ ^^ ^^ "^""^ ^^ *°^ ^^ '"^ ^24 and $62 
 
 79^0.*?^'^'' "^^°^ '^''^ ^^ ^* """^^ ^^ °' ^"^^ ^^ ^* ^°<^ ^8 <5- »na 
 
 $6^ mSe ? """"^ ^"^ ''^^^ ''''^ ^^^ ""^^ ^^^ *'''^ ^^^^ »n* 
 ^^^^^owma,ujao$m and $67 and $240 and $742 
 
 and^$7l26.*''^''^'"''^^^^^^''^ ^^*^ """^ ^^^ *°^ *^^^ 
 
 64fc;+?9Vc%tcf "+' "+'*^ ^-^^^^ ^-^^^ «• + 
 +$3326^"^ ^^260-f-$1404+$8496+$2413+$46+$1786 
 
 $74689^^'^ ^^1264-$27304H- $2687+ $426+1876846+ 
 
 +$^42^6'^^^^^^^^"^^^°*^+^^^^^^*"^-^^^^^^^^+'^*268 
 
 +T,^i' ^^*M® '^® amouat of four hundred and sixtr* 
 
 68. AM togethbr Beven Imndred and ninety-six —Are 
 Jhonaan.! fonr hundred and f«pty,-nine hSotod anS' 
 
 2* 
 
^ 
 
 J^Rnuuht 
 
 «J«ht,— flvo tUtmt^ M four hundred and nlr. i-. ^ 
 iwd Md two tbuu uid and flfU^J^iine v^; ~r ^"'»' 
 •nd nIne,-four hundred and one "'"ety-six thousand 
 
 •37. How much do the followim^^ ann^a «r 
 
 to. »ben «ld6d ■togcVher trW SLT"' S""""* 
 «20.«,.-«468.97. "•'•««.-«H6..i,-$4.g__ 
 
 MENTAL KM'RCISLS. 
 EXERCISE 6. 
 
 •impounds of coffee? "'" "^ ""J «ost of 
 
 l5'cem^:»7'o^:^'£;sr„f«r„v»^'^■ «">". --^ 
 
 ftem WoII«'^So?e th!'?' " ''?'""•«' "-"> ""'M 2 of 
 
 to moS"' """" ''°*'' *« ''»«»"«■ Of « clock strike 
 
 fchadafterm«Sist-'?,~™-'^'/? America, and tint 
 HowoMireshet * '**" '" England 
 
 A»,8beaTS'. n'i l^h.'*"' "'-"''•' 28 bear 
 fcA«s««i*i."3Jvi™*' '! ^ i bear aoDlea. H^w «»«- 
 
 
 
I^^^ll^«_»f 
 
 «UBTBiC?TIOW. 
 
 1^ 
 
 forM'ce? ^-'nV^ " """^ ^^^ ^^ ««°*«' »* «Pool of thread 
 tom^lve/ ''"*'' ^^'^' '""^^' change, ought she 
 
 the other fivi miles an h on J wl^T '"'*'« '^^ ^'O^^'' a°« 
 at the end o? the I't hour " II^w f/f. W ^'"i '^'^ ^ 
 hour, Of 4tb hour, or\7U^7i ^ti Lf;ir%t ',[^f 
 
 anJV?«rcb?7hf / '"^^ ^«'«^«^ «°« hundred 
 
 four pI^^nT ea^h ftVe hid JwLTed ^^S^v'ttf J^' °'°^^- 
 and the tallow seventv-H v nnm,Ho g.^^X-three pounda ; 
 weight of the cow r * ^ "**'' ^ ^"* ^«« *he whole 
 
 iSk 
 
 SUBTRACTION. 
 
 Subtraction is the method of findir- the difler- 
 ente between two numbers. ° ^^ 
 
 "•I ^i|"5 
 
 MENTAL EXERCISES, 
 KXERCISB 7. 
 
 how mrrhTfe Ift r"""' '^ «'^o •^'""' ' of '"em , 
 
'■■^^'? 
 
 T*f », «^w^ ■*.. 
 
 '*i>r-<f * vj**i!ai| SfuMr^ < 
 
 BSS^i|giif«iM. 
 
 ili 
 
 ?0 
 
 A^ITHMEriC. 
 
 5. I gave 15 dollars for a cow, and 6 dol'ara for a sheen • 
 how much more was givea for tie cow thaa for the shee? I 
 What wae given for both ? «« >>ux,anwp i 
 
 q f; M^^f "" haying 9 apples, gave 4 to bis mother and 
 3 to his sister ;-for his generosity his father gave him 
 13 more ; how many had he then? 
 
 7 A m-iQ owing 48 dollars, paid IQ ; what had he to 
 pay • ' -^ 
 
 8. A man on beiiig asked how old he was when he was 
 S^^l'^V^'^'T^'*'^ *^^*^^" present age was 64 years, and 
 ^hLL^^^ been married 37 years"; what wL hi age 
 when he was married ? ^ 
 
 an^'«:^f -t^*^^*"* ^.°"?^^ ^ P^«°® <^^<5lo*b for 115 dollars, 
 aad^fiold It so as to lose 23 dollars ; what did he S 
 
 I 
 
 5 
 
 ^ 
 
 I' 
 
 EXERCISES FOR THE SLATE. 
 
 ExAMPL!5.~From 6237 take 4895. 
 
 6237 
 
 48q'^ «n w ^\^^^^ ^w^l^er under the greater, 
 4895 RO that amts may stend under uniti, tens 
 
 7^ Z^"" ^f^'Jl ^- . ^^^^ ^ l'°e under them. 
 1342 Pegia at the units' place, that is a* the 5. 
 
 imiiprfhA n ^"l^.u"'^ ^ i-emains. Put down th^ 2 
 SfrnL Q ^;k- ^"^ "^^ i^J^^ "'^^ fi^^^^' ^ Jiich is 9. Take 
 fn f?? ^ ' *^'' *'^l''"''i ^^ ^^'^^^ When the lower figure is 
 m this way, greater than the figure above, add to the up^ 
 P^ figure as many ofthesame as U takes to make one of the 
 higher, that^is here as many of the tens as it takes to 
 ??vf r/ ""^ *,¥ ^""^"-eds, which is 10 - 10 and 3 are 13. 
 Take 9 from 13 and 4 remains. Put down 4. Whenever 
 the upper figure has thus been increased, 1 must be added 
 to the next lower before subtracting. Thus, add 1 to 8 
 which makes 9 Take 9 from 2, it cannot be toe ; hen 
 as before, add 10 to the 2. Now take 9 from 12. 'and 3 
 
 T^'s^^rnrr 'Ti*'" '•. Add^lto4,itwilliiake5 
 rxms. The number from which the subtraction is made, 
 
Hmim^i^ 
 
 a sheep : 
 ,e sheep ? 
 
 ml' 
 
 ther and 
 ave him 
 
 d he to 
 
 I he wag 
 irs, and 
 his age 
 
 dollars, 
 he sell 
 
 su]*xa^qx|o^% 
 
 n 
 
 greater, 
 'S, tens 
 * them. 
 
 the 5. 
 a the 2 
 . Take 
 Igure is 
 
 the up' 
 e of the 
 ikes to 
 are 13. 
 lenever 
 J added 
 
 lto8, 
 ij then 
 
 and 3 
 mke $. 
 . The 
 
 or the 
 I made, 
 
 viz., 6237, is called the Minuend. The number which is 
 subtracted, viz. 4891, is culled the Suhtrahend. 
 
 426 
 214 
 
 $ c. 
 647 17 
 423.14 
 
 754 
 621 
 
 $ c. 
 827.86 
 403.27 
 
 S c. 
 968.12 
 412.86 
 
 212 J24.03 
 
 133 
 
 424.59 
 
 556.26 
 
 ■.^"■drilH3.ff^-i;f '.>.;: 
 
 ',»* •■. ■ 
 
 ,• 
 
 
 $ C. 
 
 
 $ 0. 
 
 
 643 498.17 
 
 783 
 
 869.83 
 
 548 
 
 411 132.28 
 
 172 
 
 $ c. 
 
 217.15 
 
 213 
 
 
 
 
 423.234 742 
 
 834.14 
 
 546 
 
 643.425 
 
 279.586 489 
 
 478.27 
 
 29S 
 
 169.843 
 
 143.648 253 355.87 248 473.582 
 
 EXERCISE 8. 
 
 (1) 
 
 74603 
 37684 
 
 (2) 
 
 $91020.24c. 
 
 12647.58 
 
 (5) 
 $42681.520. 
 19697.43 
 
 $741026831.060. 
 278904896.60 
 
 (12) 
 861264981 
 248600989 
 
 (6) 
 42890 
 27601 
 
 '3) 
 $41021.720. 
 768.31 
 
 $81000.10 c. 
 2641.07 
 
 (4) 
 
 40000 
 
 1001 
 
 <8) 
 45301 
 20009 
 
 $6141020.130. 
 1789068.44 
 
 (11) 
 14812.0718 
 741&.8648 
 
 (13) 
 92100.2461 
 19800.7049 
 
 (14) 
 
 $1812010.410. 
 
 898901.22 
 
 15. From seven hundred and nine thousand fnnr h»!n= 
 dred and twenty-seven, take two hundred and fifty-one 
 thousand eight hundred and seventy-two. 
 
>^J^|«*t^:)f:t,,;^,jj-'«;*r-r^-^?^ 
 
 '"' 1 "nmr-^irfma 
 
 ■ " r~-niTiii> i PTi iiiii II 
 
 32 
 
 ARrTHMETIC. 
 
 h 
 
 thoMand seven tenths. "* "•""^'"^ »»« ninety^ 
 
 dr^^ kn?nine?J?,g''!?i,'»««>-fl™ thoasand seven hun- 
 ye«?-1837' '"'"■° '" "" ^''' "28 ; how old was I m the 
 
 MIXED QUESTIONS. 
 
 :' Wl*- i, KXKRCISB 9. 
 
 I 
 
MIXED QDKSTIONS. 
 
 23 
 
 T 
 
 battle ? "* ' ^"^"^ ^*"y returned from the 
 
 y^^s^Ksr--^«^^ 
 
 th J «Wr^ ' ? *^^ ^^^^^^'l t^vo were washed overboard^ 
 eito, the population of it being l,776,556Tth" yw 
 
■># 
 
 u 
 
 ARITIIMETIC. 
 
 MENTAL EXERCISES. 
 EXERCISE 10. • 
 
 ^w.^I'/'l'^?^'' ^M ^' ^' ^^^"' a^d ^as 84 years old 
 when he died ; m what year was he born ? 
 
 2. Johnhas 34 marbles, and Albert 25 ; how many 
 th J othe7? ' ^""^ *'°'' ""^"^ °'^'' ^*' *^' ^"^ *^*° 
 
 3- Having 27 dollars in ray purse, I lost 8 of ihem 
 
 Tt^V'L^r.^^^ "'^'^' but afterwards a man paid me a 
 debt of $4.50 ; how many had I at last ? 
 
 4 Stephen, at a game of marbles, won 4 and lost 6 
 and then had only 8 remaining; how many had he at 
 
 ,u ^uH?®^ ^^ ^^ ^^<^®P i'i one field and 15 in an- 
 
 w'Jit H^-f.*'°i'^^*/^^'^S^ *^ "^^^^ ^^' number 50 : 
 how many did he buy ? ' 
 
 6. A man bought a yoke of oxen for 97 dollars their 
 services amounted to 40 dollars, and their keeping to 13 
 dollars : h^ then sold them for 80 dollars : did he lain or 
 lose, and how much ? ? ^ xtc guiu or 
 
 7. Matthew had 9 nuts, Mary gave him 10 more, and 
 John gave lum enough to make his imm^r 39 ; how 
 maay did John give him ? , vyt 
 
 8. A gentleman gave 853 dollars for a carriage and 
 two horses • the carnage alone was valued at 387 dollars • 
 what was the vtilue of the horses ? How much more were 
 the horses worth than the carriage ? 
 
 9. Aman^scrfdacQw for 20 dollars, a calf for 4 dol- 
 lars, and a sheep for 3 dollars ; andJn payment received 
 a waggon worth 17 dollars ;- how much remains dae ' 
 
 •■ I 
 
1 
 
 1 
 
 MULTIPLICATION TABLE. 
 
 MULTIPLICATION TABLR 
 
 Twice 
 tare 
 % .. 
 
 3 .. 
 
 4 .. 
 6 
 6 
 7 
 8 
 9 
 
 10 
 11 
 12 
 
 2 
 
 4 
 
 6 
 
 8 
 
 10 
 
 12 
 
 14 
 
 16 
 
 18 
 
 20 
 
 22 
 
 24 
 
 3 times i 4 times 
 lare 3| 1 are 4 
 
 2 .. 6j 2 .. 8 
 
 3 .. 9 3 .. 12 
 
 4 .. 12 4 ,. 16 
 6 .. 15 5 .. 20 
 
 6 .. 18 6 .. 24 
 
 7 .. 21 7 .. 28 
 
 8 .. 24 8 .. 32 
 
 9 .. 27 9 .. 36 
 
 10 .. 3010 .. 40 
 
 11 .. 3311 .. 44 
 
 12 .. 3612 .. 4812 
 
 5 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 10 
 11 
 
 times 
 are 6 
 .. 10 
 .. 15 
 .. 20 
 .. 26 
 .. 30 
 .. 35 
 .. 40 
 .. 45 
 .. 50 
 .. 65 
 .. SO 
 
 6 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 10 
 11 
 12 
 
 times 
 are 6 
 .. 12 
 .. 18 
 ,. 24 
 .. 30 
 .. 36 
 .. 42 
 .. 48 
 .. 54 
 .. 60 
 .. 66 
 .. 72 
 
 7 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 10 
 
 u 
 
 12 
 
 2$ 
 
 times 
 are 7 
 .. 14 
 .. 21 
 .. 28 
 .. 35 
 .. 42 
 .. 49 
 .. 66 
 ..63 
 .. 70 
 .. 77 
 .. 84 
 
 8 times 
 lare 8 
 
 2 .. 16 
 
 3 .. 24 
 
 4 .. 32 
 
 5 .. 40 
 
 6 .. 48 
 
 9 times 
 
 1 are 9 
 
 2 ,. 18 
 
 7 
 
 8 
 
 9 
 
 10 
 
 56 
 64 
 
 72 
 8i> 
 
 11 .. 88 
 
 12 .. 96 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 27 
 36 
 45 
 54 
 63 
 72 
 81 
 90 
 99 
 108 
 
 10 
 1 
 2 
 3 
 4 
 5 
 6 
 7 
 8 
 9 
 
 10 
 
 11 
 
 12 
 
 times 
 are 10 
 .. 20 
 .. 30 
 .. 40 
 .. 50 
 .. 60 
 .. 70 
 .. 80 
 ..90 
 .. 100 
 
 ..no 
 
 .. 120i 
 
 11 times 
 1 are 11 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 22 
 
 33 
 
 44 
 
 55 
 
 66 
 
 77 
 
 88 
 
 99 
 
 110 
 
 121 
 
 132 
 
 12 times 
 
 1 are 12 
 
 2 . . 24 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 36 
 
 48 
 
 60 
 
 72 
 
 84 
 
 96 
 
 108 
 
 120 
 
 132 
 
 144 
 
 13 times 
 
 2 are 26 
 
 3 .. 39 
 
 EXTENDED MULTIPLICATION TABLE. 
 
 4 
 5 
 6 
 7 
 8 
 9 
 
 52 
 65 
 78 
 91 
 104 
 117 
 
 14 times 
 2 arc 28 
 
 42 
 56 
 70 
 M 
 98 
 112 
 126 
 
 15 timos 
 
 2 arc 30 
 
 3 .. 46 
 
 60 
 
 75 
 
 9Q 
 
 106 
 
 120 
 
 135 
 
 16 times 
 
 2 are 32 
 
 3 .. 48 
 
 4 .. 64 
 
 5 .. 80 
 
 7 '.'. 112 
 3 .. 128 
 9 .. 144 
 
 17 times 
 
 2 are 34 
 
 3 ,. 51 
 
 4 .. 68 
 6 .. 86 
 
 ,. 119 
 
 , . 136 
 
 . 153 
 
 18 times 
 2 are 36 
 3 
 
 54 
 
 72 
 
 90 
 iys 
 126 
 144,. 
 16219 
 
 19 times 
 2 are 38 
 3 
 
 67 
 76 
 96 
 114 
 183 
 152 
 171 
 
^^n'vtv >^"'<:m;:^v^ ■■— ■ 
 
 ■«fj?<,»;.;ijr.,»-r 
 
 •"'•TTtfir- r 
 
 26 
 
 l' 
 
 ■> 
 
 ARITHMETIC. 
 
 SIMPLE MULTIPLICATION. 
 
 Multiplication teaches us to find what a number 
 jnIUmountto,whenit is repeated a number of 
 
 MENTAL EXERCISES. 
 \ EXERCISE 11. . 
 
 1. At 7 cents a-piece, what will 9 pine-appks cost ? 
 
 2. What cost 15 yards of cloth at 8 dollars a yard ? 
 
 costrr^c^i^Tf ?o%«^oi ??tci'jr"^ ^ ''^ 
 
 how many has each, and ho^^ many have they all ? "^ ' 
 6. What cost 794 barrels of flour at 9 dollars a barrel? 
 
 there, and what did he giv/f^ihe who "? ^ ^ """"" 
 
 Case 1.~ Whea the Multiplier do>is not exceed 12. 
 Example.— Multiply ^3 by 7. 
 
 AQ ?*^xu*^® number by which you are to multinltr 
 53 under the number to'be multiplied ; then .av 7 
 Sr%^.."^^^ 21. Put down the I undCT thf 7 
 — Then 7 times 5 make 35, and the 2 of 21 make 37 
 
 fheVt'^^HpSfh- ^^^«3i8calledtheiJ^S^^:. 
 thn pi V m.^ *^® Multiplier; and the 371 js 3d 
 
 oLl^ef SecSM^ 5?^ *^^ multiplier token 
 
 wgeiner are calied^the Factors ; thus 53 and 7 are factors. 
 
 669 427.5 642 7fa?5 396 
 
 ^ 2 2 2 2 
 
 13l« 
 
 855.0 
 
 1284 
 
 1496.30 
 
 792 
 
iMiiliitto^: 
 
 $ c. 
 
 *486.25 
 3 
 
 SIMPLE MULTIPLICATION, 
 
 968 
 3 
 
 $ 8. 
 
 687.32 
 . 4 
 
 983 
 4 
 
 $ c. 
 
 768.69 
 5 
 
 2T 
 
 80G 
 6 
 
 4030 
 
 $ c. 
 
 742.13 
 10 
 
 (1) 
 
 $ c. 
 
 703.66 
 6 
 
 4761.3G 
 
 8G0 
 11 
 
 378 
 
 $ c. 
 597.27 
 
 $ c. 
 
 596.36 
 8 
 
 $ C. 
 
 903.40 
 6 
 
 EXERCISE 12. 
 
 <2) (3) 
 
 974 
 9 
 
 2646 4770.88 8766 
 
 (4) 
 
 $4270.16 c. *67287.43c. $8fi453.26c. $75268.05 c. 
 . * 6 3 
 
 (5) (6) 
 
 9i68 $84076.50 c. 
 7 8 
 
 (7) 
 
 43256 
 9 
 
 (8) 
 $74873.16 c. 
 10 
 
 45687 
 11 
 
 (10) 
 
 *$96844.14c. 
 12 
 
 (11) 
 63875 
 9 
 
 (12) 
 
 $47389.12 c. 
 12 
 
 
f!S1lf'Wff^,w^^rr.^- 
 
 11 
 
 Jf 
 
 *® ARITIIMETIC. 
 
 13. Multiply 875.46 bj .4 
 
 14. 7 
 
 16. • A 
 
 16. " -^ 
 
 17. 
 
 18. 
 
 19. 
 
 20. 
 
 21. 
 
 .. .6 
 .. 3 
 . . 6 
 . .10 
 . 11 
 . .12 
 
 22. Multiply $983.27 c. by 2 
 
 23. 7 
 
 24. ■ 
 25. 
 26. 
 27. 
 28. 
 29. 
 30. 
 
 4 
 
 .8 
 .6 
 5 
 9 
 12 
 11 
 
 13952 
 
 Case II.- When the MuMj>lier is a Composite .^umher* 
 
 Example.— Multiply 436 by 32. 
 
 f.J}^ f "^?g^^^^' ^'^^" 32, 18 formed by two 
 factors, 4 and 8 J therefore iWead of multiplying 
 
 n»w f : ^^tf • Multiply this product by the 
 other factor, 8, and you obtain 13952, the Dro- 
 duct of the 435 multiplied by 3'> ' ^ 
 
 436 
 4 
 
 1744 
 
 8 
 
 31. 
 32. 
 33. 
 34. 
 35. 
 36. 
 
 $ c 
 4264.78 
 7436.87 
 9687.48 
 6748.67 
 6430.67 
 4264.56 
 
 EXKRCISte 13. 
 
 X If? 
 
 37. 
 
 X 18 
 
 38. 
 
 X 24 
 
 39. 
 
 X 27 
 
 40. 
 
 X 36 
 
 41. 
 
 X -^9 
 
 42. 
 
 $ c 
 3687.45 
 2468.76 
 7849.78 
 2040.74 
 4368.76 
 4968.76 
 
 X 54 
 X 56 
 X 72 
 X 108 
 X 132 
 X 144 
 
 the number of decimals in both factors i<; 7 ,.»i ik 
 
 figures in the product, then two cipS mull S'o prSd"'"' '°'^ ' 
 
 .36 
 
 444 
 
 222 
 
 2664 
 
 Here the product has only 4, but it requires 7 ; it should be .000266|. 
 
 EXAATPIJ? :— 
 
2 
 
 7 
 4 
 .8 
 
 .6 
 5 
 9 
 
 12 
 
 11 
 
 SIMPLK MULTIPLICATION. 
 
 29 
 
 Cash III. — When the MuUipliei' contains several figures. 
 
 342() 
 342 
 
 Example.— Multiply 3426 by 342. 
 
 Place the multiplier under the multiplicancl, 
 units under units, &c. Multiply by the unit 
 figure of the multiplier, viz. 2. Then mul- 
 tiply by the next figure of the multiplier, viz. 
 4 ; thus, 4 times 6 malse 24, but take notice 
 that you are to place the 4 of the 24 directly 
 
 under that figure of tlie multiplier by which you 
 
 1171692 are multiplying. Proceed in the same manner 
 with the figure 3 of the mwltiplier. Then add 
 together the products obtained. 
 
 6852 
 13704 
 10278 
 
 Multiply 6487 by 230. 
 230 
 
 194610 
 12974 
 
 1492 10 
 
 Multiply .'§64.87 c. by .203 
 .203 
 
 19461 
 12974 
 
 .«13.168Gr 
 
 EXERCISK 14. 
 
 1. 
 
 2. 
 3. 
 4. 
 6. 
 6. 
 7. 
 8. 
 
 Mult. 
 
 $ c. 
 984.76 
 
 by 
 
 64.2 
 
 7.68 
 
 2.95 
 
 .496 
 
 85.7 
 
 .4368 
 
 789.6 
 
 3654 
 
 9. Mult. 65839 by 
 10. 
 11. 
 12. 
 13. 
 14. 
 15. 
 16, 
 
 958 
 
 627 
 
 369 
 
 426 
 
 704 
 
 8743 
 
 6007 
 
 9864 
 
 17. Multiply sixty-four thousand eight hundred and 
 fifty two, by nine hundred and eighty-seven. 
 
 ♦ Hero tho .16, that la the hundreds, aro cents , the remalniiiir 
 figures, are parts of a cent. 
 
"t ■ I f 
 
 so 
 
 ABITHUKTIO. 
 
 page contain ? ' """"^ '"""s docs tl,e Whole 
 
 ;*■ [' 'H,'"'°" "' '""'' """ ""-y "»""=" «ro .hero r 
 fortune ^Kfc'iJi'^dTv'rH'.S' '''-''" '" '■"' »'" '"at bi, 
 each recei^ea *«^S.'7rJrhrecrre;^^a^fe'to 
 
 anfth'-SVot^n tLT/r''? ""■ P"-" «"« bouses 
 
 co^Lit |'„'»oe„^?e'l7h"dT-7i^'';'L'^^ »"<■ "•<"«"« 
 of the children come to in The vcw? ' ""^ '"PP""-* 
 
 op rpi ' 
 
 and t^ch rs^requ^ .^ |k8 **«r the school, 
 
 required for all the desks T ' "" ""^"^ ""^'^ ^^re 
 
 31. In a school, there wppo p.-v —• j 
 room, and four in the SrTs^ rn i ^/»^°T« in *he boys' 
 
 •*! 
 
 m 
 
'liiffiiiii -'1,. 
 
 SIMPLE MDLTIPUCATIOX. 
 
 n 
 
 ^ ;.ii M^^^VV^o ';'oy». one of them was lazy aod Iny In 
 bed till nine, the othei- was an actWo little fellow whS 
 roEe every morning at six ; how many hours did the aclive 
 boy gam in a year that the other lost ? 
 
 of 'I's/tirjo: aly'r ** ^^°'' ^^^'"^ '"^ " ^'^^ '^^ *^« «^t« 
 
 34. IIow many pins may a boy point in G davH wlm 
 works 8 hours a day, and Joints 16,?U0 pi^ in au\our ? 
 
 Cask 1V.~ When there u a Fraction in the Multiplier, 
 ExAMPLB — Multiply 672 by 27|. 
 
 27| 
 
 mi After multiplying by 27, we then multiply 
 *^ b^i^-l by the l (that is get | cf 672,) and add 
 
 18G48 this to the former product. 
 
 4)672 
 
 168 
 3 
 
 We multiply by J, or by any other 
 
 »o,hr»n hv finaf ''ividir~ '' ' 
 
 fraction, by first divldfing by the chiomina- 
 txyr, or lower figure, (thn 4 in this case.) 
 then multiply the quotient (168) by the 
 "^ a °"™^'**''';,?r "PPer figure ; or, which Is 
 604=1 the same thing, we may multiply hv the 
 
 A .V. .X, , ^""'.^'^'^tor fir^t, and then divine the ,,ro. 
 duct by the denominator. To divide by a fraction do 
 ju8t the opposite. That is multiply by the denominator 
 and dividt ihe product by the numerator. "^''^"''°'*^<^'^' 
 
 Example: 12X1 = 9, butl2-i-| = i6 
 
 20X^== ? 20-f.f=: ? 
 EXERCISE 15. 
 
 1. Mult. 4864.3 by 37f 
 2- 264| 
 
 6. Mult. 48643 by 25 
 
 8. 
 
 9. 
 
 10. 
 
 
-'"■,'» « . - 
 
 ¥ I 
 
 •• ARITHMKTIO. 
 
 MENTAL EXERCISE8. 
 EXERCIHK 16. 
 
 home gpllled 4 timca 2 quarts, lacking 3 quarta • how 
 many quarts had ehc remaining ? ^ ^ ' ^^^ 
 
 2. When Jjeef is 6 conts a pound, and pork l) cents • 
 
 oTh^n '"'"'^ ""'" ^ ^''""'^' '''" P'"''' «'^«* ^i^a" « po»Ddi 
 
 3. Henry iy 4 feet in height and John is 5 : and 5 
 ilTiwnT>.^-^ *^1' ^'^«^'«' considertd as a number, 
 fathSage? ''' ^^ ^ ^^ ^'""'"'^ ""^^^ '' ^« 
 
 1 J' ^ nf .^*^ ^l ^°"^ '^"^ ««^"^e 12 men 8 days, how 
 long will It serve 1 man ? ^ ■ 
 
 6. A man failing in trade is able to pay only C8c. m 
 a dollar ; how much can he pay on a debt of $5 ? How 
 much on a debt of $20 ? "» •^ j now 
 
 6. If an apple cost 2 cents, an orange three times as 
 much lacking 4 cents, and a pine-apple three times as 
 
 rc'osTo^llTtter "^"^^-^ ' ^^'^•^' -^*^»» ^ 
 
 DIVISION. 
 
 Diviaion is the method, first, of finding how 
 often one number is contained in another; and 
 eeoond, of dividing any quantity into a certain 
 number of e4uaJ parts. 
 
 MENTAL EXERCrSES. 
 EXKRCISB 17. 
 1. How manv inf>lnna matr Ka hnA f/%> 10 j.^ -j. a 
 
 •ente a-pieoe ? ' , 
 
^^ 
 
 SHORT DIVISION. 
 
 m 
 
 2. How many pino-upplea at 8 cents each, can be ob- 
 tainod for 40 cents? for fi6 ccnta? . v u u* o»- 
 
 ^tATl ^""'f^^ ;V^'"-''^'l« «f flour for 20 dollarH, and 
 grrrelt of "^^IT^^? '^^"'"^ « ^^"^^ ' ^- ™-J 
 
 m'n c\m'r25 da"^s/' '^^' ''" '' '"'^ ^""'" "« °^-^ »« ^ 
 
 men ridl r'sS a^nl^' ' ""^' '^^ ' ^^"^' ^^" ^^ ^'^^ ^ 
 
 Case I.— When the J}ivisor does not erce^i 12. 
 SHORT DIVISION. 
 ExiMiM.E.— Divide 252 by C. 
 
 ft^9^o ^"', *^® numbers down according to the an- 
 
 6)252 «exed example. Find how often the flgnre^ 
 
 -— which you are to divide, viz : 6 is contained 
 
 42 m the first, or fii-st and second figures ; tW 
 
 ^^°t<> 2, no times, then 6 into 25,4 times and 
 
 1 over Put down the 4 under the 6. Si pposo the 1 
 
 ?^fwiJ'f ' P *^.^^o^'^^ ^'«^^1^ °^«ke it 12. ^r; 6 into 
 
 ,'i*yif'^^"' *^^ 2 under the 2. The number 6 fa 
 
 caUedthe Da.W; 252 the Dividend; amli^S^^otie^. 
 
 2)4628 
 
 2314 
 
 X $ c. 
 
 2)476.58 
 
 2)6824 3)6039 
 
 $238.29 
 
 3412 
 
 3)76389 
 25463 
 
 2013 
 
 $ c. 
 
 4)857.36 
 
 $214.34 
 
 4)8408 
 2102 
 
 6)76590 
 12765 
 
 *K^ ^*!3^ ^^''^ ^® decimals, point of from the riahi of 
 the quotient as many for decii^iols of tie deZ^^Z 
 
 o 
 
•mmmmmlmmmm 
 
 u 
 
 ARITIIMETrc. 
 
 ^^^^^^tV^^f^^t^l^^^^^, 
 
 (1) 
 
 EXERCrSE 18. 
 
 i^) (3) 
 
 c. 
 
 (4) 
 
 *)27U45 5)687^- ^^^g^HT 7)8^20 
 
 (6) 
 $ Q, 
 
 8)764.26 
 
 (9) 
 12)76416872 
 
 (12) 
 3)642687.62* 
 
 (15) 
 6)76002041 
 
 (8) 
 
 (6) (7) 
 
 9)28676 10)642.'68 ii)46267 
 
 (10) 
 8)426876.4^2 
 
 (13) 
 12)46^76^.7*6 
 
 (16) 
 $ c. 
 9)^302^.01 
 
 (11) 
 7(0640268 
 
 (14) 
 8)46876400 
 
 (17) 
 7)41260602 
 
 15. Divide 56472689 bv '* 
 
 19. « d o 
 
 20. '« ,. 2 
 
 21. - .. I 
 
 22. '• .* 6 
 
 23. Divide ii;680.23c. bj 7 
 
 25. - ... J 
 
 26. " .,d 
 
 27. .. , } J 
 
 Caso IL- H'A«H </.e Divisor is a O^mposUe number. 
 (4)6789 ^'^'^'^--^»^^<i« 6789 by 28.' 
 
 17)1697....! JZ\h^l^^^^^,V^^^ 
 
 complete rom.Jnder,r ''"'^'' '^^' 
 
i^n 
 
 c. by 7 
 " 8 
 '• 9 
 "10 
 
 '' 11 
 
 LONG DIVISION. 3Sf 
 
 Rule .—Multiply eachtMiit^'Mc^t mp^^ bifthe pro- 
 duct of aU the divisors previous to its own, and to these sev- 
 eralproduds add ths first remainder— thus, 4X3-|- 1=13, 
 the result, 13, is the true remainder. 
 
 
 
 KXERC'TSK 19. 
 
 
 
 1. 
 
 368745 - 
 
 - 54 
 
 4. 
 
 204076 
 
 H- 108 
 
 '2. 
 
 246876 - 
 
 i- 56 
 
 5. 
 
 436876 
 
 -r- 132 
 
 3. 
 
 784978 - 
 
 - 72- 
 
 6. 
 
 496876 
 
 -f- 144 
 
 Case III.— When the Divisor is tnore than 12. 
 
 LONG DIVISION. 
 
 • '^ji®/»ply difference between Long -dnd ShmiBivision 
 IS, that iQ Short Division part of the operation, (that is, 
 the muUiplying and siibstrading) is done mentaUy, while ip 
 Long Division it is all pnt on the slate. In order to see 
 this, let us compare them, hj doin^ the same queptioB 
 
 bOUl.W^.^ ■ , .;., ^ .^W.. ,;^, ■. ^^ 
 
 ' ExAMiim^Divide 43176 by 8. i. 
 
 ^"«f7oH^y^^'®^- ^^^^^ "^^ say, nientaUy, 8 into 43, IB 
 »)4J17b times (Dutting down the 5) then 5 times 
 
 — -— 8 is 40, 40 from 43 leaves 3. Then we 
 
 o6d7 suppose the 3 that is over to be placed 
 
 . » « . before the next figure 1, and a«ain say, 
 menfo^y 8 mto 31 will go three times, 3 times 8 are 24 
 ftom 31 leaves 7. Then 8 into 77, and so on, doing the 
 multiplymg and substracting in the mind. 
 Same Example by Here we do the very same, only ttu't- 
 sZul^'^i'"'' ting down the prodJt wheix wo mul- 
 in ^ . tiply, and putting down the retnainder 
 W when we substi'act. 
 
 ^7 .. ^® say 8 into 43 goes 5 times, put- 
 
 ^^ tm^ the 5 in fie quotient, then 5 times 
 
 ^ f »s 40, putting it dowur-then 40 
 
 ;: from 43 leaves 3. nuttinir it down. 
 
 ' men bnng the 3, which is over, and' 
 
 finwr, ♦!,« 1 fru ^°^ °®^* ^^^« (1) together, by bringing 
 
 ^0^3 Jfnia'^*'!?. '^PS?*^'^^ *^« «*™^ oppratiinf 8 into 31 , 
 goes 3 times, setting the 3 in the quotient and multiply- 
 
■■WSf n'JJ.-J-iSf • 
 
 86 
 
 ARITHMETIC. 
 
 iSS^et^ff^'15^'' the product 24, then substracting 
 jna siting dow» the remiOnder 7— then brimrinff down 
 
 RITUJ FOR LONG DIVISION. 
 
 i!^J<F*r^^. "^ ^1"' ^^ ""^ ^^^ Dividend pul how many 
 ^^J^.f^visoT IS cordaimd in the fewest figures that will 
 
 ^T/' ^^ ^^""'J^' ^^^ W^^ ^« i^^ right of tha 
 dividend mih a line between them. ^ '' 
 
 and write the 
 
 Multiply the divisor % this quotient figure 
 prodvd under Ouit part of the diwdend taken. 
 
 Si^stract this product from the figures above it. 
 ih^^iS ^*' '^■^'"''^ "^ ^*« dMdmf6m right of 
 
 Divide this as bef^f (facing h cipher in the auotient 
 andjmngtng dmm the next figure if Ju any tiZtheM- 
 fnd wdl not contain theJ&or,)and80or^tillXtL 
 figures are brought down and divfded. 
 
 EsLAMPm^J^iy^^ 6W326 by 427. 
 
 OPERATION. 
 
 427)6i432^(H38i§^ 
 
 **7 
 
 ^ '1673 
 1708 
 
 \ 1281 
 
 Compare this operation with 
 the rule. 
 
 3716 
 3416 
 
 300 
 
 ■■Kf 
 
LONG DIVISION. 
 
 8t 
 
 1. 
 2. 
 3. 
 4. 
 5. 
 6. 
 
 KXERCISK 20, 
 
 Divide 87403 by 857 
 
 n 
 it 
 u 
 
 8427.86 -f- 
 9768.42 -j- 
 
 521 
 403 
 
 684 
 
 78 
 
 946 
 
 /. 
 
 420.1076 
 
 8. 
 
 6416879 
 
 9. 
 
 2864976 
 
 10. 
 
 2876.407 
 
 11. 
 
 64129.30 
 
 12. 
 
 98.00147 
 
 -f- 438 
 
 -^ 648 
 
 ~i- 396 
 
 -f- 4107 
 
 -5- 7481 
 
 -f- 3076 
 
 WHEN THERE ARK DECIMALS. 
 
 hnTT. ^^''^A^'^'l ^^^'»»*!« '^ the divisor or dividend, or 
 ft the dividend must either have the same number of 
 decunals as the divisor, or it must have less, or it must 
 have irwre. ' uiutti 
 
 ihl'F^^^ <?ividend has the same number of decimals as 
 the divisor, no decimals are to be pointed off 
 
 2. If the dividend has less decimals than the divisor 
 add ^rmny ciphers to ike dividend as aremcessa^ti 
 
 mvisor, It will then be like the previous case and nf 
 cou^e no decUmls are to he pointed o/froTthe qut 
 
 in \^^^I • '^ ^"^ H^'^'T}^ ^^ ^^ dividend exceed those 
 m the divisor, point off from the quotient as nmmi asZ 
 
 foTfinci'^if rh?:5^^ l^rTh'^tt 
 
 SL?^ '.u ^ *^® ^^"^^^^^ ^^ three and the dividend 
 airee from the quotient, point off xom If the d iVteor 
 has four and the dividend two, add two to the div dend 
 to make them equal, and point off none. If thedivfsor 
 has two and the dividend five, point off three - 
 
 oration with 
 
 • 13. 
 
 4078948 
 
 
 1 ^^• 
 
 7198641 
 
 
 m 15. 
 
 364.1201 
 
 
 m ^^' 
 
 248070.8 
 
 .0008 
 
 .2864 
 
 1407 
 
 .2600 
 
 17. 
 18. 
 19. 
 20. 
 
 78.64126 
 
 3002602 
 
 402026.4 
 
 9687600 
 
 7410 
 .8000 
 .0069 
 .4300 
 
 21. Divide rIy millinna o/m^.. I J .I -. 
 
 thnnoanfKa iv~T —•''"; =\-^" uuiiuruu ana nmeiy-iour 
 
39 
 
 ARITJIMETIC. 
 
 f- '*''"" '""""'iatl. of $6037.15. 
 
 gallon a minute? *" '^"' "* ^^^ rate of .18 of a 
 
 "any pe«o„s to eichS j""'"'^ '""'^^*' "^ ^"<C-hZ 
 ^e«u^. .,,„,,/,,. 4;jo«M^^ 
 
 r. 
 
 *9u rpi. 
 
 0' 2/ ...lies per dav ? ""' ™"""' it at tJie rate 
 
 'Vjf 
 
 Ex.»MPi.E._Divide 426 by 4^ 
 
 WRRiTIOX. 
 
 3 
 
 t? the BZont^tJllZ' ""'J -J'^Wend 
 « here, will, of eo.nl"'K??,"^»""»te- 
 
 -7 rj-v' J ussriis. 
 
how much 
 
 iN lioH lone- 
 or .JH of a 
 
 t millious, 
 face ; how 
 
 stantfrom 
 
 reaching 
 
 five miles 
 
 3 «arth in 
 a:ht move 
 |i^jbeing 
 
 >0 miles,- 
 the rate 
 
 1. Divide 7534 bv 
 
 2. " u 
 .a ** a 
 
 4. 
 
 «< 
 
 LONO Divrsjos 
 
 KXERCISE 21. 
 
 23f 
 
 46| 
 
 28ft 
 
 59-^ 
 
 99 
 
 5. Divide 7534 by 154A 
 J- " " 265« 
 
 ^- " '* 564|f 
 
 6'OJ'. 
 
 MENTAL FXERCI8KS. 
 EXERCISE 22. 
 
 bought L 4 meW ' '""' """'-^ "'■'"'*^«' ""y ^ 
 
 how L^\%i^ rc ;:o:jT;„s:/p?:«itr«r'^^ 
 
 of sugar were worth 56 cents ? ' ^ ^ ** P"""* 
 
 rotlfZZ'JSft^il^';^ -eh, »„,t be g,.«. 
 
 vidend 
 that is 
 le four 
 are, of 
 B two- 
 Then 
 bring 
 > divi- 
 of the 
 under 
 
40 
 
 ARITHMETIC. 
 
 SECTION 11. 
 
 TABLES, REDUCTION, AND COMPOUND RUIZES. 
 
 I 
 
 TABLES. 
 
 STERLING MOXEY. 
 
 4 farthings = l penny. 
 1^ pence = i shilUiur 
 20 shillings ^IpouilS^- 
 ^ 21 shillings = 1 guinea. 
 
 ^ denotes pounds, .. shillings, and d pence. 
 t one farthing, or one quarter of anythini^ 
 
 I fj^^lfy ennj, or a half of anything. 
 
 J three farthings, or three quarters of anything 
 
 OLD CANADIAN CURRENCY. 
 
 4 farthings = 1 penny, 
 
 lfM?r^ =1 shilling 
 6 shillings = 1 dollar 
 4 dollars = i pound 
 
 marked. 
 d 
 
 $. 
 
 _^ NEW CANADIAN OR DECIMAL MONEY 
 
 ^i!^^T:tZ\^e%^^^^^^^^ cents. The coins, 
 twenty ceit pieces ' ^'^''^^' *^" <^*^»* Pieces, and 
 
 / 
 
 1 dollar nf fJia *.^_ 
 
 «^TT, 
 
t ut.es. 
 
 r thing. 
 
 corns, 
 3, and 
 
 / 
 
 SVIT, 
 
 TABLES. 
 FEDERAL OU UNITED STATES MONET. 
 
 41 
 
 anl'millt"'"^'""^'*"^^ ^^^ ''^'''' ^o»^^. <ii-.s, centH, 
 
 TABIJi:. 
 
 10 mills (m) = 1 cent 
 10 cents = 1 dime 
 
 JX ^^^^^ ~ 1 dollar 
 10 dollars = i eagle 
 
 AVOIRDUPOIS WJKIGHT. 
 
 marked 
 ct. 
 d. 
 
 $. 
 
 marked 
 oz. 
 
 U drams (t?r) = i ounce 
 
 lU ounces — l nnnn^ 
 
 25 pounds. r^Sr ^ 
 
 2000 pounds "^ =iton t 
 
 14 po,mds make one «tone and 8 stone 1 hundredweight 
 
 Of 112 lbs. 
 This weight is used for bread, meat, grocery for .^on^a 
 m general, and for all the metals exc^^gottsCr 
 
 TROY WEIGHT. 
 
 24 grains (gr.) ^ i pennyweight '1^^^ 
 20 pennyweights - 1 Sunce ^ oz 
 
 12 ounces = i pound %\ 
 
 This weight is used for gold, silver, jewels, and liquors. 
 
 APOTDECARIES' WEIGHT. 
 
 20 grains ^ i scmple .or." 
 
 3 scruples = 1 dram drZl 
 
 I drams =- i ounce Z.Z\ 
 
 ^ 12 ounces = l pound »!o/^ 
 
 ' ■■"■■■'■ I ■■ __ Ml., i- -^ 
 
 v- 
 
41 
 
 2^ iaches 
 4 nails 
 
 3 quarters 
 
 4 quarters 
 
 5 quarters 
 C quarters 
 
 ARirmfETio. 
 
 CLOm MBA8UBK. 
 
 =t 1 nail 
 =? 1 quarter 
 = 1 Flemish ell 
 = 1 yard 
 = 1 English ell 
 = 1 French ell 
 
 marked 
 nl 
 qr, 
 
 Fl. eU. 
 
 IS. e. 
 F.e, 
 
 SQUARE OR LAND MEASURE. 
 
 Hi square inches 
 square feet 
 
 30| square yards 
 40 square perches 
 4 roods 
 
 640 acres 
 
 1 square foot 
 
 1 square yard 
 
 1 square perch 
 
 1 rood 
 
 1 acre 
 
 1 square mile 
 
 mariod 
 sq- ft 
 
 aq. yd, 
 
 00. 
 
 sq.nil 
 
 u ?Xwxs^; ir=^isn.^--pj?fe 
 
 i^IfQ MEA3D11B. 
 
 12 linos 
 12 inches 
 3 feet 
 ^ yards 
 
 40 perches 
 8 furlongs 
 3 miles 
 tfO geographical miles. ) 
 or B9^ British or sta- f 
 tute miles j 
 360 degrees 
 
 , , , markod 
 
 = 1 inch in. 
 
 •■ I foot y^ 
 
 } yard yd, 
 
 ■■ 1 perch, polo 
 
 or rod per. 
 
 I furlong fur, 
 
 I mile rni, 
 
 1 league ig. 
 
 1 degi-ee d^. 
 
 the earth's ciroumfereaca* 
 
 iafenl?h^ 'VT^XT^ ^ ^" ^^"^^ *<> t*ree bariey-ooms 
 J^if"^'"' * lochee make one hand. ufl«d in ,»J^^Isz 
 
 ^ 
 
 '' 
 

 plyinflr 
 of li 
 
 ;^ 
 
 Donu 
 
 tables; a 
 
 CUBIC. OK Solid huasvuv. 
 
 48 
 
 1728 cubic Inches 
 
 27 cubic feet 
 
 40 cubic feet of lougli timber, or | 
 
 50 cubic feet ol hewn timber \ ~ 
 
 42 cubic feet __ 
 
 16 cubic feet j-_ 
 
 8 cord feet, or » 
 
 128 cubic feet I ~ ^ ^^''^ ^^ y^'ooCi 
 
 ZlHnliH* Ji ' .^ ""-^ "^'y nnmhor is obtained by 
 tlSecube of 12. ^^ itself^-Uivis, 12X12X12--=:1728^ 
 
 ~ 1 cubic foot 
 - 1 cubic yard 
 
 1 ton or load 
 
 1 ton of shipping 
 1 cord foot 
 
 M11\SUB2 OF I'jlPACITV. 
 
 4 gills 
 2 pints 
 4 quarts 
 2 gallons 
 
 4 pecks 
 
 8 bushels 
 
 5 quarters 
 
 1 pint 
 : 1 quart 
 1 gallon 
 1 peck 
 1 bushel 
 1 quarter 
 1 load 
 
 marked 
 pt 
 
 qt. 
 
 gal 
 
 pk. 
 
 bush. 
 
 %: 
 
 ured. Ihe gill, pmt, quart, gallon, are chiefly used for 
 Mquids. The peck, bushel, quartei% load &c Ir^J^A 
 fordryarticles. The gallon contains 277?274ti^^^^^^^ 
 
 hTllt'^nfTJv ^""""f ^? ^*"^^ ^^P«*l ni^a»'"-« is now, 
 by Act of Parliament, declared ille^l. "^w, 
 
 Ale, wine, and beer were formerly measured bv differ- 
 ent measures In some places a ba^rrel of beer coS 
 32^ in some 34, and in othei-s M gallons. A hoXad ^ 
 
 wtTslLTff' *^ ^^^^- 54^.allons, aS^al ll 
 
 2 hogsheads make 1 pipe or butt. 
 2 pipes or butts make 1 tun. 
 
44 
 
 ARITHMETIC. 
 
 TIMli!: 
 
 : 1 minute 
 1 hour 
 1 day 
 1 week 
 
 1 year 
 
 marked 
 min. 
 
 hr. 
 
 da. 
 wk. 
 
 60 seconds (sec) 
 60 minutes 
 24 hours 
 7 days 
 
 12 months, or 
 62 weeks and 1 day, or 
 365 days 
 
 ^^Every fourth year contains 366 days, and is called hap 
 
 yr: 
 
 lUYwS IN EACH MONTU. 
 
 * Thirty days hath September, 
 April, June and November : 
 All the rest hare thirty-one : 
 February twenty-eight alone, 
 But m Leap- Year twenty-nine. 
 
 DIVISIONS OF THE CIRCLE. 
 
 60 seconds" 
 60 minutes 
 30 degrees 
 12 signs or 360® 
 
 1 minnte 
 
 1 degree 
 
 Isign 
 
 1 circle of the zodiac 
 
 marked 
 »HMi or ' 
 deg. or ® 
 
 S. 
 
 a 
 
 QUANTITIES. 
 
 12 articles 
 
 20 articles 
 144 articles 
 
 24 sheets paper 
 
 20 quires 
 200 pounds 
 196 pounds 
 
 14 pounds 
 
 = 1 dozen 
 
 1 
 1 
 1 
 1 
 1 
 1 
 
 score 
 gross 
 quire 
 ream 
 
 marked 
 doz. 
 
 8C. 
 
 gr. 
 qr. 
 rtn. 
 
 barrel of poj* or hee£ 
 barrel of flour. 
 
 == 1 stone. 
 

 REDUCTION. 
 
 i$ 
 
 yr: 
 
 S. 
 
 a 
 
 
 REDUCTION. 
 
 Reduction is the chan<?ing a liighor denomina- 
 tion to a lower, or a lower to a higher, without 
 altoring the value. 
 
 REDUCTION DESCENDING. 
 
 Case I.—^ higher to a lower, as pounds to pence—weeks 
 to days—qmrts to pints. Hedudim descendmg is always 
 aon^ hf multtpltcation. ^ 
 
 Example.— Reduct £4 Os. C^d. to pence. 
 
 £ s. 
 4 9 
 20 
 
 d. 
 
 89 
 12 
 
 1074 
 4 
 
 4298 
 
 Here we first multiply the £4 by 
 20 to bring them to shillings, be- 
 rmise it takes 20s. to make 1 pound. We 
 add in the 9 shillings io get all the shil- 
 lings together : this will give 89s. Then 
 we multiply the 89s. by 12 to bring 
 them to pence, because it takes 12 pence 
 to make one shilling ; adding in the 6d. 
 this gives 1074d. Then we multiply the 
 pence by 4, to bring them to farthings, 
 because it takes 4 farthings to make 1 
 penny, addmg m the farthings ; we have now 4298 farth- 
 ings. Putting this into the form of a Rule it would run 
 
 Rule FOR Reduction DEscEXDim—jBesrinnmo tcithihe 
 highest deimmmxtion, mvUiply it by as many of the lower, 
 [whatever lower you icish first to hnng it to) as it takes to 
 mtkeone of t/*e satne, adding in whatever nurnber there may 
 he of thai lower ; mdHply this resuU as before, (by as many 
 of the lower as U takes to make one of the same, addina in 
 tf^ low^y, and so continue tUl the required denomination he 
 
 Example.— Keduce 4 cwt. 3qrs. 17 lbs. to oimces. 
 ExiMPLE.— Reduce 16 cwt. 1 qr. 19 lbs. to ounces. 
 
4d 
 
 ARITHMETIC. 
 
 I 
 
 Lot the pupil be required to go throuRh these find <hn 
 
 i.Jif'^l/^'*'^ *^® question distinctly a« it is in the 
 havo'hflf? P^'?^,^^' *" *^^' ^1»«« attending :'' Here I 
 
 XSo^^TX'! "' ^^««^«°^inK^ donebymuhi- 
 
 a by as many of the lower as it takes to make one X 
 
 ^o^r^^ ;' ^ , ^* 4--//ien <Ae (/uarfers 6v as many wnrnds 
 as ti takes to make a qmrier, which is 25,~then tiTZZht 
 as many ounces as it takes to make a ^.m^ SS^Tg?^ 
 
 M J r .,^^ n.w EXERCISE 23. 
 
 1. Reduce £264 9s. lod. to pence. 
 
 2. Reduce 3 qrs. 13 lbs. 12 oz. to ounces. 
 
 3. Reduce 241b8 to pennyweights. 
 
 4. Reduce 6 wka. 3 days 14 hours, to hours, 
 
 5. Reduce 76 j^iles 6 fiir. to perches. 
 
 6. Reduce 9 ^. mis. 1 a. r. 9 yds to square ineheg. 
 
 7. Reduce 47 cords of wood to cubic feet 
 
 8. Reduce 4 pipes 1 hhd. 1 brl. 1 9 gals. 2 qts. to quarts. 
 : ^9. Reduce 71 lbs 11 oz. 3 drs. to scruples. 
 
 10. Reduce 123 acres 17 perches to square yard^ 
 
 11. Reduce 9 ha 17 dwts. to grains. 
 
 s.si. iiwawvw i/u» vons ■* Cwu o urn, 
 to drains. 
 
 u 
 
 ^M7 mum. 
 
UBDCCTION. 
 
 41 
 
 UEDUCnON ASCENDING. 
 .n!SLT^' ^y^^}^ « ^^^> as, pence to pomds-days to 
 
 Example— Reduce 67G0 farthings to pounds. 
 
 ^ll^^'rF' • "^u® ^7 ^^'^^ ^'"''^'^ the 67C0 farth- 
 ^^^^^ "'fif« by 4, to bring them to pence, b^ 
 
 rr^ci "^^"^ '.' ^^?. f farthings to make I >enny, 
 12)1440 this givm 1440d. Then we divide the 
 
 20^ Itt ^^}^\^'' H''"» ^^^"^ to shillings, 
 ^^^^^ *^o<^au3o It takes 12 ponce to make one 
 
 TT fhin;ng-thiH gives I2O5. Then we divide 
 
 £6 the 120.V. bj 20. to bring them to pounds. 
 
 because there are 20^. in a noiind-.fhi'« 
 
 gi^os £6. Ilonoe the Rule would run thJ: P'^"^^""^"* 
 
 Roi,B FOR Reddction AiiGKmim.--I>hn'h , „i,,.^ ^ 
 nommUionby as imny of the sam uLJtoZX 
 
 Kilo M ^ ^^ «« ftforc, (6i/ a.<r many 0/ </«, ^ame oi « 
 ^^iomajce o^^ of the higher) and soonUiUU is rXJd 
 up to the denomination required, ^<m.^ 
 
 BXBKOLSS 24. 
 
 Let the pupil be required to go tlirough each of the 
 following questions, first orally, In the sime manner a! 
 euggeatod /or the quesUons in Reduction deeocnZg! 
 
 1. Reduce 1427 ounces to pounds, 
 
 2. Reduoo 42768 farthings to pounds. * 
 
 3. Reduce 2487 grains to ounoos. 
 
 4. Reduce 4786 nails to yards. 
 
 ft. Reduce 4796 pecks to buAhels. 
 
 7. Eeduoo 714986 inches to fothoms. 
 
 t.. 
 
*® ARlTHMETr(\ 
 
 a Reduce 61479867 square miles to acres, roods, Ac. 
 9. Reduce 667789 cubic inches to cubic yards. 
 
 10. Reduce 91666 Flemish ells to French ells. 
 
 11. Reduce 17498 cubic feet to cords. 
 
 - 12. Reduce 2987149 mills to eagles, dollars, dimes, &c. 
 
 KEDUCTIOX OF THE OLD CURRENCY TO THE XEW, AND THE 
 NEW TO THE OLD. 
 
 cente'^dSaiS''f^'/*'"^W ^^"^^ *« dollars and 
 cents, ana dollars and cents to pounds, shilling a.ni\ 
 
 pence, proceiid in the same manne^-that is sTwhether 
 the reduction is ascending ov deseeMiZ^Z aT^^ll 
 multiply accordingly, as the Rule direct^ 
 
 ExAMPLE.-Reduce £24 8s. 6d. to the decimal currency. 
 
 OPERATION. 
 
 £ CtS 
 
 24X400 = 9600 
 s. 
 
 8X20 z= 160 
 d. f. 
 6=24-j-^ == 10 
 
 $97.70 
 
 Here the bringing the 
 pounds and shillings to cents 
 IS Reduction descending. We 
 first multiply the £24 by 400, 
 because there are 400 cents in 
 a pound. Then multiply the 8s. 
 by 20, because there are 20 cts. 
 m a shilling. Then bring the 
 
 these farthings 1. cents. luTc^nte l"*'^^^^^^^^^^ 
 mgs, this therefore is Reduction ascending Wo S 
 fore divide ^y 2f , or, which is the same, by J Ke 
 three results added together give 9770 cente* Thon 
 dividing by 100, to brini them to dollars" wS is S 
 
 llVmfotnt' ^u'^f^^r^ fi/u?esTSjt"tsf we' 
 nave «97.70 cents. The rule, then, would read thus : 
 
 dortelTceSr'"' ''"''°^' P^"^ ^^^ '^^^^ '- 
 
 and 
 
 "Mr. .ijf. y 
 
 Tnh^r.^'^T^ ^"^ i>o«m« (w 400, i/ie shUlirMS dw 20, 
 {changing the pence into farthings) divide ihffaHKings 
 
BKDUCTION. 
 
 m 
 
 ^ J^i. ^f t^^^^« ^^l^^ r^^s together, andpcbU of ifie iioo 
 ngm-hand figures for cents, the figures to the left of the 
 point wiU be dollars. '' •' 
 
 Or some may prefer this method : 
 
 OPERATIOX. 
 
 24X4 = 96 
 8h-5 = 1.60 
 
 6 = 10 
 
 $97.70 
 
 'RvJAS.— Multiply the pounds by 4, 
 and divide the shillings by 5, to get the 
 ^l^rs. Then to get the cents, multiply 
 ail the shillings under 5 by 20, and ccm 
 the pence and farthings so many centSr 
 at sight, 5c. 6c% Sd., 10c.==6d. 15c.= 
 9d. I2^c.~7^d., etc. 
 
 To reduce the dollars and cents back again to pounds, 
 shillmgs, &c., just reveree the operation. Divide th^ 
 Si«i ^ t' 1^5^^* the pounds, and to get the shillings, 
 multiply what is over from the dollars by 5, and divid^ 
 
 ^eLrafsiX''' """' the remaining lenis so many 
 Example.— Reduce $97.70c. to pounds, shillings, &c. 
 
 OPERATION. 
 
 Here we divide the dollars^ 
 by 4, to bring it to pounds: 
 the quotient is 24 and one dollar- 
 over,— this one "dollar multiplied 
 n«/ V:: by 5, to bring it to shillings, ;:ives- 
 £24 8 6 5s. Then divide the 70c. by 20, t^ 
 
 ♦,-^«f • o ^ -.^ ^""^ t*^®"* ^ shillings ; the qu<K 
 
 tient IS 3 and 10 cents over,-thi8 ten cents is 6d. 
 
 KXERCISE 25. 
 
 1. In 264^. 9s. lOd. how many doUai-s and cents t 
 
 2. Reduce 364/. 19s. 9^. to farthings. 
 
 3. In 247i, 12s. 8^d how many dollars and cents ? 
 
 4. How many dollars and ponfaa aro fi.««« j« afro 
 
 5. In 298 crowns, how many dollars and cents? 
 
 $ £, 
 
 97-^4=^24 
 c. 
 
 70-5-20=0 
 
 8. 
 
 5 
 3 6 
 
*® ARITHMETIG. 
 
 6. Reduce 3648 sixpences to dollars and cents. 
 
 7. In 42768 farthings how many dollars and cents ? 
 
 8. How many pounds are there in 67890 shillings? 
 
 9. In 426876 farthings, how many dollars and cents? 
 
 10. How many guineas are there in 36789 shillings? 
 
 11. In 68794 pence, how many crowns ? 
 
 12. How many fourpences are there in 37689 shillings ? 
 
 13. In 2470^ how many crowns? 
 
 14. How many dollars and cents in 39076 half-crowns ? 
 
 15. In 296|5 twopences, how many dollars and cents ? 
 
 16. In 43687 crowns, how many threepences? 
 
 17. How many fivepences are there in 4796 crowns ? 
 
 18. In 76971 halfpence, how many fourpences? 
 
 19. In 798302 pounds, how many five cent pieces ? 
 
 20. How many crowns are there in 7968 guineas? ;' 
 pieces?^" ^^^^^ half guineas, how many seven shillings 
 pelce.^" $276.19, how many pounds, shillings and 
 
 anf pencl? '^''"^'' ^^ "'"*'' ^^"^ "^^""^ P^"^^^^' '^WHings 
 Xufnea^"^^ ""^"^ half-sovereigns are there In 7642 
 
 25. Reduce 3010/. 1 1*. Sd. to farthings. 
 
 26. In 7324 guineas, how many ten cent pieces ? 
 
 27. In 7690 fourpences, how many ten cent pieces ? 
 
 «^®P*'T?°° PP" snouii bo required to prove these ezereUM h* 
 reducing them back again to their given deSomination* **f'®''*» ^^ 
 
COMPOUND ADJDITIOX. 
 
 SI 
 
 COMPOUND RULES. 
 
 COMPOUND ADDITION. 
 
 Tlie pupil will now be able to undtrstand, and 
 do the Compound Rules without almost any further 
 explanation. The trr . i that the Compound and 
 Simple Rules are pre- ii^ly the same. Let us com- 
 pare them in order that we may see this. 
 
 SfMPL?^ ADDITION'. 
 
 fJids. hds. ins. imit$, 
 
 2 4 6 Ci 
 
 7 8 6 4 
 
 2 3 5 6 
 
 4 1 
 
 a 2 
 
 16 8 1 7 
 
 VviUi, 
 
 10)17 
 
 The units 
 rerlnced 
 1-7 ( to tensi 
 
 Tena. 
 
 10)21 ( Tern* 
 to 
 
 
 2-1 (hundreds. 
 
 Hmnindt, 
 
 10)18 ( Hundreds 
 
 j to 
 
 1-8 ( thousands. 
 
 COMPOUND iDDrriox. 
 
 JtL 8. d. 
 
 14 3 9 
 
 2 6 8 
 
 7 15 4 
 
 12 6 3 
 
 .^6 12 
 
 12)24 (The pence 
 
 I reduced 
 
 2-0C( to shillings. 
 
 20)32 (TheshiUfags 
 
 < reduced 
 
 1-12 ( to pounds. 
 
£2 
 
 ARITIIKETIC. 
 
 nUs.fur.per, 
 
 4 6 20 
 
 6 5 13 
 
 7 4 9 
 6 7 12 
 
 TIME, ' 
 
 yrs. wks. days. 
 
 24 6 3 
 
 12 16 5 
 
 41 24 4 
 
 32 13 6 
 
 25 7 14 
 
 Perches, 
 
 40)64 { The perches 
 
 •< reduced 
 
 1-14 ( to furlongs. 
 
 8)23 r^The firlongs 
 - — ■< reduced 
 2-7 ( to miles. 
 
 110 9 4 
 
 The days 
 reduced 
 to weeks 
 
 IFetka. 
 
 52)61(1 ( The weeks 
 
 52 J reduced 
 
 — ( io years. 
 9 
 
 ^aSTq*^^ ?^^' example (see preceding page) is what is 
 
 Si tey "S^'afdt^^^^^^^^^ others CoiSpVy Adllon^ 
 *^ui, mey Me all done by the very same rule, viz : 
 
 Rule for ADWTioN.-Ptoce the numbers to he addedso 
 ^hoS '^ "^ *"^ '^"'"^ "^^y ^'«^ di^l^Sm^ 
 iU^n^^rt^^/'' "^ H 'y^'* ^"^ «o'**'Wn (yr lowest dermnina- 
 
 (1) 
 s. d. 
 
 £ 
 
 12 16 
 16 4 _ 
 64 17 2| 
 43 12 7^ 
 
 4§ 
 6^ 
 
 EXERCISE 26. 
 
 (2) (3) (4) 
 
 P^ydS'fi' dys.hrs.min. cwtqrs,Jbs. 
 
 }? 2 ? 35 16 6 4 ^2 12 
 
 17 4 1 24 18 14 2 3 14 
 
 24 5 52 12 5 6 17 
 
 33 2 2 64 13 3 3 2 24 
 
 mSi^pT^tSuri^t^^slVbeJS^^^^ *^? .^'Vision 
 
 figure, and carryin'g the 0°^^^!^ J^^!"! divWin^l/'fo.* '"*" 
 

 COMPOUND ADDITION. 
 
 53 
 
 /&. 02. dwt grs. yds, qrs. nls. ac's rds.phs. c.yds. c. ft. cin. 
 5 9 8 27 2 3 35 3 27 ]^ 26 667 ^ 
 
 3 2 16 16 39 2 1 48 2 39 24 18 1468 
 
 4 6 17 32 3 3 620 3 15 36 11 246 
 1 8 19 22 47 3 2 17 1 20 39 20 1294 
 
 9. A brewer bought five bags of bops : No. 1 weighed 
 1 cwt 2 qrs. 14ft). ; No. 2 weighed 1 cwt. 2 qrs. 22 lb. ; 
 No. 3 weighed 1 cwt. 1 qr. 23 ft). ; No. 4 weighed 1 cwt. 
 3 qrs 21 lb. : No. 5 weighed 2 cwt. 2 qi-s. 20 fl>. : what 
 was the weight of the whole? ' 
 
 10. A man rode 35 miles, 2 furlongs, 34 perches ; walked 
 :t «\V*'«' ^>rlo°gS' 25 perches, 2 yards ; then rode again 
 42 miles, 7 furlongs, 4 yards : then walked again 15 miles, 4 
 turlongs, 38 perches, 3 y ards ; what was the length of hk 
 journey ? * 
 
 11. Sold to one man 27 qi-s. 6 bushels, 3 peck;; ; to an 
 other 38 qrs. 4 bushels, 2 pecks ; to another 49 qrs. 6 bush- 
 els : and to another 58 qrs. 7 bushels, 3 necks ; ho^v much 
 did I sell m all ? ' 
 
 12. I bought four fields ; in the first there were 6 acres 
 3 roods, 12 perches; in the second 7 acres 2 roods; in 
 the third 9 acres and 13 perches ; in the fourth 5 acres. 2 
 roods, ij6 perches. How much in all ? 
 
 13. The bricklayers were engaged about a house 23 
 weeks, 4 days, and 8 houre ; the cai-penters 14 weeks 6 
 days, and 9 hours ; the painters 12 weeks, 5 days, 7 
 hours, and 34 minutes ; the upholsterer 5 weeks, 10 hours 
 and 42 minutes ; how long were these different workmen 
 engaged about the house ? 
 
 14. A silversmith made three dozen spoons weighmg 5 
 m. 9 oz. 8 dwt. ; a tea-pot, weighing 38>. 2 oz. IC dwt. 16 
 grs. ; two pair silver candlesticks, weighing 4 !b. 6 oz. 17 
 dwt. ; a dozen silver forks. \*eifirhing 1 lb- 8 />z lo /iw* oo 
 giB. ; what was the weight of all the articles? 
 
 16. A tailor bpught four pieces of cloth ; in the first 
 
 r 
 
54 
 
 ARrTHMRTlC. 
 
 "sliort tons?" ' *"*'*"» 'ho entire freight in 
 
 gr- 1 qt ; h^w^uol, did ftey ail how"' ""''"'"' ' "'"'• ^'^ 
 19. Find che sum of 4^; m *>i a,.. i>t * , 
 
 2 ft 6 in. ' ^ "• ^ "*• ' »"^ 8» w- 16 p. . yd. 
 
 c. in., in a fourth 61 cord ^'i S n -' t^ ^^"^« '^ «• "• 127 
 tlic 4 piles? ' ^"^ ^"^^ ''• "*• ' h«^^ mwch Moortiu 
 
 MKNTAL EXERCISK. 
 KXEROWK 27. 
 
 1. A man bought one load of hav fnv -o? q. t 
 other for £0 8s. «. ; how much did Lglvffov both"' '^■ 
 
 buVpiSr^taerM:;;; ' fvMHriW'^'' > « 
 
 
 Row 
 
COMPOUND ADDITION. 
 
 9m 
 
 6. A man bought a cask of raisins for £7 m 4d • l 
 fc. of coffee for Is. 6d. ; 1 cwt. of cocoa for £3 17s •' i 
 keg of molasses for 13.'. 7d. ; 1 box lemons for ll I I ] 
 
 are; for the stcond, 384 dollars- for tL.^rd,^, 
 ofjoth d.d he buy? How much did he gh^tlTo 
 
 in wlStreC^ln'^y,"' X"^ *'""^ ^'««''<'». ""d show 
 
 -Wii^r'anTfiStlSr''''''""" """ '■■^»'« '^-''-- 
 5. Name the denominations in cacli of tJi«^ tahinw ft.««, 
 
 .al.es ^XTu7ome %£"'''•"'' ''""""""""on it 
 aBddMnv*e^;lm^„V^«•^"L<""yy»« '"n '-"ad »P 
 
56 
 
 ARITHMETIC. 
 
 COMPOUND SUBTRACTION. 
 
 From 
 Take 
 
 EXAMPLE 1. 
 SIMPLE. 
 
 thds. hds. tns. nnits. 
 7 8 16 
 4 9 9 3 
 
 Here again, as in Additim, we shall see bv onmn<,v\n^ 
 ^e operation in eao' , that Simple and C^A J Sc^ 
 ^m are both done in precisely the same wky^e hv^e 
 ve^ same rule-so that if a person can do s'm'ilV^ 
 
 4%Z'y:^^^^ --^- -«^ opera- 
 
 ^ Here in the first example, 
 r. e., m the Simple Subtraction, 
 as the lower figure 3 is less 
 than the upper figure 6, we 
 say 3 units from 6 unHs and 
 3 units remain, setting down 
 the 3; and in the. second 
 example, 4 pence from seven 
 pence and three pence re- 
 main, setting down the 3. 
 So far they are exactly 
 alike. "^ 
 
 Then proceeding with the 
 first example, we say 9 tens 
 from 1 ten, I can't. Here, the 
 lower figure is greater than 
 the figure above. Now, in 
 
 y?,^o h^. * .i ®"^^ ^^^^' w® " «<?<^ to the 
 
 tnas. h(^. tns. units, viyper figure as many of the 
 
 ^ ? Q n ' . "?® ^ '*' ^^^''^ ^0 inake one 
 
 ^ i a 9 ofthehighei',"(&ndthi3wedo 
 
 a/iA;e,both in Simple and Cbm- 
 pound)—i. e., in our first ex- 
 ample as many tens as it 
 takes to make a hundred, 
 which is 10, and in the 2d ex- 
 ample as many shillings as it 
 takes to make a pound, which 
 IS 20. Proceeding with the first 
 J A ^^ ^ ''"J'* ^^ ami i is xi * 9 
 
 and 2 remains, setting down the 2 ; then we add 1 
 
 J^om 
 Take 
 
 2 8 2 
 
 EXAMPLE 2. 
 COMPOUND. 
 £. S. 
 
 U 13 
 12 17 
 
 o 
 
 1 16 3 
 
 EXAMPM! 3. 
 SIMPLE. 
 
 I'i'om 
 Take 
 
 JP^om 
 Take 
 
 from 11 
 
 EXAMPLE 4. 
 COMPOUND. 
 
 cv}t' qrs. lbs. 
 16 2 12 
 12 3 24 
 
 

 4^ 
 
 COMPOUND 8UBTBACTI0M. 
 
 51 
 
 to tbe next ttgure in the subtrahend before sabtractinff : 1 
 to 9 18 10 ; 10 from 81 cau't ; then do as before, and bo on^ 
 Proceeding now with the second example we eav, 17 shU- 
 bngs from 13 1 can't, then - add to the upper figurlcu. rmm, 
 qf the same as it takes to make one of the hUjf^r just m S 
 the topJe i.e as many .shillings as it takes to make a 
 pound, which m 20 ; 20 and 13 is 33 ; 17 from 33 and 16 
 
 2:?J*^?k' rfu oi'^^^ly "^^ "^^*^ "0'- proceed further in order 
 to see that the Simple and Compound aio done in preciselT 
 
 S!?*T.TK^y- ^^' ^^"^ P"P" himself do and compare the 
 drd and 4th examples. A plain statement of what we hare 
 
 ft\'?Sr.'^r ?:-"'' " *'^* '^"^"^"^' "«"^^ »*« ^^^ 
 
 Rule for SuBnucnoN.- W^r«e down the numbers to bt 
 subtracted, the less under tlie greater, so thai thiru/s of ihm 
 sarm name may-stand direotly under each other. 
 
 Begin at the right-hand or lowest denominalion, take 
 V^ lower figure fi-om Vie figure ahove and vorUe the re- 
 mainder directly beloio. 
 
 If ike lower Jmure is greater than the figure above, add 
 io the upper figure as many of the same as U takes 
 to make one of the next higher, then subtract the lower 
 
 Jigure and set down the rejnainder. But when the 
 minuend has been thus increcised by adding to U, the next 
 
 figure m the subtrahend musf be increased equaUy by addina 
 one to ttb^ore subtracting. "^ 'i ^ if «<«*»F«r 
 
 (1) 
 £, s. d. 
 73 10 5i 
 48 18 91 
 
 fur. per. yds. 
 7 10 1 
 2 19 4 
 
 KXeRCISB 28. 
 
 cwl. qrs. lbs. 
 17 1 10 
 10 2 27 
 
 yds. qrs. nts. 
 37 3 2 
 18 3 3 
 
 (3) 
 . oz. dtU. 
 18 6 14 
 14 9 15 
 
 («) 
 ac. rd.per. 
 
 1A 
 
 42 1 
 
 16 2 25 
 
 j | l . IB 
 
ii- 
 
 AftltHSfRTICr. 
 
 «* I'o^' ^v** ? ^*^" ^'* London weighs 6 tons 2 cwt 1 
 
 weight? '" *''^' ' *•"" ^"» ""« di* rence g 
 
 <., ?• ^<'" "j? <le«- W statute m. 210 r. 3 yds 1 ft tnkA 
 «3 deg. 25 m. 305 r. 4 yds. 2 ft. ' ' ^^ 
 
 Sued ? ^*- * '»'^- ^ »'"• ; ''»" "'"Ok fe- 
 
 paBtures, the first containing 12 air ?f SSl'L?'^ 
 containing 13 a. 3 r., and th^thiTd b Li r ^39 „ f bTh "^ 
 many aci-es docs the pasture exeeed tie mi^ow'lKf 
 
 I3J bu"^2''p™3 ot'l n.'.\"; ' J"^ 11*- "^^l^** ' "e sold 
 ion «u. i pK. d qt. 1 pt. ; how much remained unsold f 
 
 goTSXanf^n "•'wte^Td'avf V7,TJ «"' »*^' 
 much less^ime Z'i^^V^^'^l t^i^Z ofc" ' "»'' 
 
 on""ch^'^''f°%s''7d' P"^'?^'" ?' "**?''<•' ^2 7s. 9d. ; 
 
 rrnirijSt3rF^^^^ 
 
 ri.i^'stld?ie''mV*«?--'* ^ '^''»- of ^ooa in hi- 
 wiea, sold 16 0. 117 a 65 in. ; how much had he leftf 
 
 .nd^retrai^d April 'ui "uu^^^'lT' ^"«- ^.*' !««■ 
 
 a5'l8M*'°owmIir^fl''.*'*^''S.^*' ^^^' "as paii April 
 «, ioa. , _ow many days was the note aMnterest! 
 
 • •"••X. 
 
IMB^%I 
 
 ^••Tt 
 
 COMPOUND BfBTRACTION. $§ 
 
 MENTAL EXERCISES. 
 KXERCI8E 29. 
 
 1. A bought Of Ji a bushel of wheat for 78. 6(1 He 
 pve him 1 bushel of corn worth 5«. 3d. and pa d the rest 
 m money. How much money did he pay ? ^ ""«»«•*'' 
 
 2. A man sold a box of butter for 178. 4d., and in det 
 received 71b of sugar, worth 9Ad. per ft,., iul the vmt 
 in money. How much money dfd he receive? 
 
 ^Jt' ^ '^^'•chant bought a piece of cloth, containing 19 
 feft ? '^''•' '^^^ ^ ^'^'' ^ ^'- ^^ '' 5 ho^ much had he 
 
 4. A grocer drew out of a hhd, of wine 17 trals 3 ots • 
 how much remained in the hogshead ? ^ ^ ' 
 
 . 6. A smith bought 17 cwt. 3 qrs. of iron, and after hav- 
 ing wrought a fevv days, wishing to k.iow how much of It 
 he had wrought, he weighed what he had left, and found 
 he had 8 cwt. 1 qr. 13 lb. How much had he wrought ? 
 
 u ^' P.^^"I^' of B a bale of cotton for £18 4s and R 
 bought of a 4 barrels of flour for £9 ^. C naid B the 
 rest in mom-.. How much money did he pay ? 
 
 hJ^J^^^ the minuend and the subtrahend are given 
 how do yon find the remainder ? «« »*« given, 
 
 8. When the minuend and remainder are given how 
 do you find the subtrahend ? ^ ' 
 
 how ^r„*li®^''ll:*''^^^"^ ^"^ ^'^^ remainder are given, 
 now do you find the minuend ? » ^"; 
 
 10. When you have the sum of two numbers, and me 
 of them given, how do you find the other ? 
 
 ih^wJf^^ ^^""^ ^""""t ^^^ ^''^^^ «*■ ^^'« numbers, and 
 then- dvfftmwe given, how do you find the less number? 
 
 .7;iL5^®^ ^®" l^^'^ ^^ ^« ®^ ^^ numben, and their 
 ihfftrmct given, how do you find the ^eo^er number ? 
 
 13. Wluitn ih<k at-..^ »^Jl jlj>» --. . > . 
 
60 
 
 ARimVETIC. 
 
 ,, COMPOUND MULTIPLICATION. 
 
 Case t-WTien the miiUiplicr doe^ not cjcceed 12. 
 
 KXAMPLE 1. EXAMPLE 2. 
 
 L™^''®' COMPOUND. 
 
 tnds. Ms. iris, units. £ s, d 
 ^^64 78 6* 41 
 » 9* 
 
 70 7 7 6 70rif~4l 
 
 10)36 (UnUstotens. '" ^ 
 
 ■ Reduction. 
 
 3-6 -j Same as in 
 Addition. 
 I 
 10)67 r T&ns to hds. 
 - — Reduction. 
 6-7 ■ 8ame as in 
 Addition. 
 
 10)77 (Hundreds to 
 • — - thousands. 
 7-7 ■ Reduction. 
 Same as in 
 L Addition. 
 
 4)13 f 
 
 4-2 
 
 12)40 
 
 a-4 
 
 2(0)5(7 
 2-17 
 
 I^hrthings to 
 
 pence. 
 Reduction. 
 Same as in 
 Addition. 
 
 Pence to shil- 
 lings. 
 Reduction. 
 Same as in 
 Addition. 
 
 Shillings to 
 powids. 
 Reduction, 
 Same as in 
 " 7 I Addition. 
 
 -iHm> (the sum li" AddTuon:ttl?p^„;r*i mMp^ 
 
 
COMPOUND MULTIPLIOATIOH. 
 
 61 
 
 tlon) hy as nmny of the savieaJi it t<ikcs to make one of th9 
 mg/ier, -sotting down the remainder and cari-yinir tho 
 quotient to the next product, and so on— inaltiplyin» 
 each figure, and reducfng each product till all are done, 
 jbet the pupil himself do and compare examples 3rd and 
 
 ExAMPi4fl 3KD.--Multiply 98766X12. 
 
 Example 4th.— Multiply 7 mis. 6 fUr. t,: wr. > 12. 
 
 The rule then would bo as follows : 
 
 Rut.K.— When the multiplier <he.9 not exce^. . 12, heainninq 
 <UUie hnoest, multiply each figure by the multiplier, andthm 
 reduce eacJi product, as tms directed to reduce each ammint 
 m AUdition, settiyw down the remainder and carryina the 
 qvatient, as in Addition. ^ ^ 
 
 (1) 
 
 cwt. qrs. lbs. 
 
 4 3 16 
 
 EXERCWE 30. 
 
 (2) 
 
 lbs. oz. dwUi. 
 24 3 12 
 
 8 
 
 (3) 
 yds. qrs. nls. 
 16 3 2 
 7 
 
 
 (4) 
 
 
 
 (5) 
 
 
 
 (6) 
 
 
 acs. 
 
 rds. 
 
 ^r- 
 
 qrs. 
 
 hu. 
 
 pks. 
 
 hrs. 
 
 mt'n. 
 
 ser„ 
 
 3 
 
 2 
 
 27 
 
 7 
 
 6 
 
 2 
 
 9 
 
 25 
 
 30 
 
 
 
 7 
 
 
 
 9 
 
 
 
 6 
 
 7. Multiply 74 a. 2 r. 7 per. 4 yds. X9. 
 
 & Multiply £169 178. llfd. X9- 
 
 9. Multiply 111 corda 7 cu. ft. 7 cu. in. X12. 
 
 10. Multiply 278 mis. 6 fur. 11 per. X7. 
 
 11. Multiply 11 gal. 1 qt. 1 pt XH. 
 
 12. Multiply 109 cwt 2 qrs. il lbs. X^. 
 
 '.'--flJiT- 
 
82 
 
 ARITHMjrriC. 
 
 Cask II.— Wfwn the multiplier exceeds 12. 
 V Example.- -Multiply 4625 X 36. 
 
 wJmav^ln^'fh^^-^^ 7^ ""''^ ^ ^«* 36 times 4625. Now 
 
 a^^fi^e srtiS^r'rti^jr^^^^^^^^^^ 
 
 FIRST WAY. 
 
 4625 
 36 
 
 27750= ^ times. 
 13875 =30 times. 
 
 166500=36 times. 
 
 Simple Multiplication is 
 usually done in the first 
 
 way as in this first example, 
 (that IS, by separating the 
 multiplier into its parts, then 
 finding each part separately, 
 and adding the results toffe- 
 thel^) though it may be done 
 m either way. Here we first 
 
 get 6 time., «.en «■« get 30 ^S,^^^'^ t^^^^«' 
 
 SECOND WAY, 
 
 4625 
 4 
 
 18500= 4 times. 
 166500=36 times. 
 
 , Compound Multiplication 
 18 usually done in the second 
 way, as in this second ex- 
 ample—that is, by resolvinff 
 the multiplier into its fal 
 tors, then multiplying by 
 the first factor, and that pro- 
 duct by the seco^nl, and so 
 
 Example. — MniHniTr ia — a 
 
 (ioxiox4+32:r"''"'- 
 
 
 ■^ 
 
 1 
 
 J 
 1 
 
 c 
 
 oqrs. - Too. 6 oz. x432: 
 
COMPOUND MULTIPLICATION. 
 
 M 
 
 cwt. qrs. lbs. oz 
 14 3 
 
 First we find the high- 
 
 7 /X2=2 times. est number of times— 
 
 1" that is, the 400, for 
 
 T I rrrrr wWch the factors are 
 
 < 8 17 12X3=30 times. (10 X 10 V 4) =. 400 
 
 74 6 9 
 
 8 
 4 
 
 296 
 
 22 
 
 1 
 
 6 1 10 0=400 times. 
 4 1 25 4= 30 times. 
 9 2 14 12= 2 times. 
 
 times. We first multi- 
 ply by these factors to 
 get the 400 times. Then 
 we get the next highest, 
 viz: the 30 times, in 
 the most convenient 
 way we can. The fac- 
 toi-s for the 30 are 
 
 fw^ hiv^^ in^i!: 'll:^t^^2:% « *:&«: 
 
 therefore, we multiply the second line b^S and g^ef 
 30 times at once. We have now, yet, to get it, v^^ the 
 multiplicand (twice). This is got by multiplS thS 
 first line (i.e.,the multiplicand) by 2. Now weliavl 
 
 ^nSrJaJiT^'.u"*^ ***^^' 2 times These severaf 
 r^ultswe add together, and get 432times-the result we 
 
 * The Rule then would bo as follows :— 
 
 Rule for Compound MultiplicatioxV.— ir;i6/i the mw/- 
 ^Zter ex-ceed. 12 resolve the muU^lUr into its fa^sXn 
 ^Plytht multt^cand by om'^faUm, and the re^im 
 
 been used. The last product tciU be the product required. 
 
 th/{JtTfff'-n '^''f \directly resolved into factors, get 
 thefa^s that wxU produce the highest numt^ of times inU 
 and hanng multiplied by these, get tiie products of each ^ ^l 
 
 »«awn«r/)o«»i6fe, byming the product^ already obtained ; then 
 add the several rewHs together. - '^ «*,*««» 
 
 36=6 X € ,- 100=10 X 10 ; 10000=10 X 10 V 
 10X10, Ac. ^ '^ 
 
ARITHMETIC. 
 
 Example.— Suppose the multiplier to be 7862f . 
 
 ^a part 800, =loo ^ which we have already 
 in our second product.) VS— 800 tim«»i 
 
 7852* ~ J '^p«T?^^!^^^^^^^^«^-^« «^"^^^^ 
 
 #i8>Zf _ j first product) X5=:50 times. ^ 
 
 X 2*^2^times^'' multiplicand muluplied 
 
 ^^J^fKh"" ^^"^ multipHcand multiplied 
 . X ?==? times. ^ 
 
 These five results added together = 7852f times, 
 
 EXERCISE 31. 
 
 nnt^ii"?!^'*"'? manner as in the above example write 
 ?S2?;UTe?Jr54^ following numbers, J^^'i^^ 
 
 2. Multiply 17 lbs. 7 02. U dwts. by 478. 
 
 3. Multiply £476 168. 8d by 647, 
 
 4. Multiply 4 mis. 6 ftir. 20 per. by 7426. 
 fi. Multiply 36 yds. 2 qrs. 3 nls. by 8047|. 
 6. Multiply 42 acres 1 rd. 10 per. by 57ef . 
 7„ Multiply 27 qrs. 7 bush. 3 pks. by 807«. 
 & Multiply 32 yrs. 3 wks. 4 days by 236i J. 
 9. Multiply 4 cir. 7 signs 14 deg, by 723f.. 
 
 10. Multiply 7 reams 15 qrs. 9 sh«et« by 36/ . 
 
 11. Multiply $476.17 by 865j^. *' 
 
 12. Multiply $750.25 by 73Cf 
 
 15. If one spoon weigh 3 oz. 5 d^* i* -=^ ^u - - 
 ««ight of 120 spoons ? " ' ^ ^~ ***'^' **"** '^ ***** 
 
 
 f 
 
COMPOUND MULTIPLICATION. 66 
 
 i,.l%^* ''Tf-l'''''''^.'''',W*'''2^°*- 2 fur. 10 r. per hoar, 
 how far will it go in 15 hours ? r vai , 
 
 17. How much cloth will it take to make the clothea for 
 a regiment of soldiers containing 1143 men, if each suH 
 requires 7 yds. 3 qrs. 2 nls. 1 in. ? 
 
 9i}^:^^/*®^o?*^^^P »n going round the world travel 
 da 8 ? ^"''*' ^ ^^' **^^^ ^"^^ ^^^^ ^^^ ^^ ^" ^^^ 
 
 19. In 27 barrels there was on an average in each. 29 
 gallons, 3 quarts, 1 pint ; liow mucli in all ? 
 
 20. I can go to a certain town by the railway in niae 
 hours, ^26 minutes, and 30 seconds , it would toke me, 
 t}l ' fi^f t»nie» as long to go by the stage coach ; hoW 
 long would the coach take ? ' 
 
 21. How much water will be contained in 90 hogs- 
 heads, each containing 62 gal. 1 qt. 1 pt. 1 gi. ? 
 
 22. I bought 375 bales of English goods for £9 lis. 
 did f ^linf *^' ^^^"^ *^^"* ^^^' ^^'^^^ "^^^"^^ ' ^^*^ 
 
 23. How much molasses is contained in 25 hhd. each 
 hhd. having 61 gal. 1 qt. 1 pt. ? 
 
 24. How^mvch wood in 12 piles, each containing 7 
 coixls 5 cu. ft. 12 cu. in. ? ^t^iiuug < 
 
 25. A farmer has 18 lots, and each lot contains 41 a. 
 z r. 11 p. ; how many acres does he own ? . 
 
 MEOTAL KXERCI3ES. 
 EXERCISE 32. 
 
 1. At 7s. 4d. per bush., what cost 18 bushs. of wheat? 
 
 2. What cost 32 lb. of coffee, at Is. 8d. per E>. ? 
 
 3. What h the weight of 5 casks of rasin^s, each cask 
 weighmg 2 cwt. 3 qrs. 25 lb. ? 
 
 3 nis^^of^do^?^ ^ "*'^' ^^^^ ^^ ^^^ ^"^^ ^^ ^ ^^' ^ ?*'• 
 
 i^h^ market-womfttt bought 4 quarts of strawberries 
 for 29 cents, and sold them at 5 cent» a pint: what did 
 she gam ? 4* f > «.. wiw 
 
^j^^^^wm^w ■ 
 
 '^^'^^ml 
 
 66 
 
 ARITHMETIC. '• 
 
 COMPOUND DIVISION, 
 question m division, whether iSf/ior< 7>»i^ c;**l»./ ^ 
 
 CbrrtpOMwd But it is in f^n ynJ^T^^' --^^^^ ^^ 
 
 that all the ^snrJ\^)\ ^^i^^'f -^5'-2>m«ion alone 
 
 SuLh? ^*'''?i^^^^'*' *^ next figure of the diFidStY 
 iJiwt ^hak It would come to if Nos. 4 and 5 were pSS 
 m full,) see annexed examples periorme^ 
 
 .J^.f';'^^^9^'9mvision, the dirisor being more tbaff ■ 
 12, the operations Nos. 2 and 3, ure put dowl in foil Zt 
 bemg simple numbers, where 10 of the lower make one 
 
 San?itld«i"'l*?^^ ^^ are either E 
 
 mentally or aside frotii the question, to serre a tpmn^i-i 
 purpose, and be immediately rubbed off ^nf^^^^ 
 
COMPOUND DIVISION. 
 
 '67. 
 
 f 
 
 The following example, done first by putting the work 
 of each step fully down, (as is done in Compon-ivd Long 
 Division,) and then by performing some of the operations 
 mentally, or contracting two or more into one, (as is 
 done in Compound Short and in Simple Long and Short,) 
 will best illustrate these reraarky and this whole subject 
 of Division. 
 
 Example.— Divide 4893 by 9 
 
 I. • n. 
 
 operation in 
 
 Compound Lono—No. 1. 
 
 9)4893(543fi 
 45 No. 2. 
 
 3 No. 3. 
 . 10 No. 4. 
 
 30 
 
 9 No. 5. 
 
 39 New dividend 
 36 No. 2. 
 
 3 No. 3. 
 
 10 No. 4. 
 
 30 
 3 No. 5. 
 
 33 New dividend. 
 27 No. 2, 
 
 6 No. 3, &c., &c. 
 
 Same operation in 
 
 SiMPf^ Long— No. 1. 
 
 9)4893(54S§ 
 45 No. I. 
 
 39 Nofi. 3, 4, and 5. 
 36 No. 2. 
 
 33 No8. 3, 4, 5. 
 27 No. 2. 
 
 -^ 
 
68 
 
 m. 
 
 Same operation lv 
 
 COMPOLTSP SUOHT,; 
 
 9)4893 
 
 54^1 
 
 ARITHMETIC. 
 
 IV. 
 SaMK OPEitATlOX r\ 
 
 Si,vrr.K Shokt. 
 
 9)i89;5 
 
 3 No. 3. 
 10 No. 4. 
 
 30 
 
 9 No. 5. 
 
 543g 
 
 3 No. 3. 
 
 10 No. 4. 
 
 30 
 3 No. 5. 
 
 39 new div'd. 33 newdiv.d 
 
 ThiH example, as we .ee hero, can be done ip any way 
 tvomta.. most expanclea to the mo«t contracted, but if 
 
 fh^rl """' ':^r °^'°"^ ^'-^ ^"°" ^ ^^^ »"»<!' «»d it could 
 
 f the dividend v. re of different denominations, so that 
 
 he remainder couid not be reduced by multiplying con- 
 
 stantly by lO, then we could not reduce and add by 
 
 hnngmg dovrn, or calling the remainder so many tens • 
 
 we would consequently have to do it in the fiiit way! 
 
 Rule for Compound Division. -ProcmZ as in Simple 
 Division but as the reirmiM^^ cannot he reduced and added 
 to the ^mt loioe^' by bringing down the ne^ figure, m^vply 
 ^by as ^mnyof the nexi lower as U takes to make one^ 
 the same oMmj in the givm nun^ei^ of the next low! 
 Divide the mmber thus obtained by the divisor, a^ befo^-e 
 ana so on. j •> 
 
COMPOUND DIVISION. 
 
 % 
 
 i -EXAMPLE. 
 
 CoMrouND Short. 
 ciot. qrs. lbs. oz. 
 9)17 3 U 3 
 
 1 3 
 
 23 12J 
 
 8 cwt. Ist rem. 
 4 
 
 32 red. to qrs. 
 3 given qrs. added. 
 
 35 new dividend. 
 
 8 qi-8. 2(1 rem. 
 25 
 
 200 red. to lbs. 
 14 given lbs. added. 
 
 214 new dividend. 
 
 7 lbs. 3d rem. 
 16 
 
 112 rtd. to ozs. 
 3 given ozs. added. 
 
 115 new dividend. 
 
 EXKKCI8K 33. 
 
 1. Divide 19 cwt. 3 qrs. 8 fes. hy 3. 
 
 2. Divide 18 lbs. 6 oz. U dwt«. by 17. 
 
 3. Divide 16 per. 2 yds. 1 J ft. by 9. 
 
 4. Divide 64 yds. 2 qra. 3 nls. b/42. 
 |. Divide 36 acr. 3 rd. 27 per., by 31. 
 e. Diyide 9 qrs. 7 bush. 3 pks. by 9. 
 
 2.— EXAMPI.E. 
 
 Compound Long. 
 £ s. d. 
 19)25 18 6(1 7 3J y: 
 
 6 Ist rem. £ 
 20 
 
 138 rod. to 8. and s. ad'd. 
 133 
 
 5 2nd rem. s. 
 12 
 
 eered.tod.andd^ad'd. 
 
 9 3rd rem. d. 
 4 
 
 36 red. to f. 
 19 
 
 7. 127 cwt. 2 qrs. 17 lbs 
 
 11. 
 
70 
 
 ARrTHMETIC. 
 
 S, £96749 16s. lljd -^ 117. 
 9. 69 gals. 1 qt. 1 pt. -^ 12. 
 
 10. 679 sq. per. 7 ft. 107 in. --- 132. 
 
 11. 290 sq. per. 7 yds. 8 ft. -f- 3. 
 
 12. 1467 French ells 1 qr. 2 nls. 1 in. 
 
 13. 172 days 16 h. 29 m. -j- 7. 
 
 14. 916 miles 6 fm*. 4 yds. -^ 67. 
 
 267. 
 
 questim. ^oT^tlce ff*^T' ''f '?'''« "«'w« of the 
 
 (pn-ceo/on«)$5. 2na P w7«?U/S'"*^'T?'i' ""• 0- 
 and 3ra%. t inunZofmn^^!f^:{^^''<^> ««»-- 
 
 P. aJdN t' 2'±\"/ ^"^ ^'"''«) "e rsjuired, then, 
 thus Tf 1 h™^ ii"* «"'•"'' »"'' «>e auction reads 
 
 N ^T**mn«f A?* -^^"^^ '^^'*"®> ^® rcywired, then P. W and 
 
 ?fir;5*,^rih.^^Z«^' ^^<i. the queXrnorre^?; 
 
 do f deceive •''"'Tn^L'fJ^ * "^^'^l' '^^^ ^«*»'y barrels 
 receive. To do ttiis, we^nc^ how oftenihe price 
 
operaiio7i, 
 of every 
 the other 
 Division 
 're of the 
 St $5, 20 
 St, P. O. 
 $100,— 
 
 kinds of 
 
 kI, then, 
 
 n reads 
 
 barrels 
 
 '<^ one, 
 
 -MULTI- 
 
 mmber, 
 J multi- 
 
 W. and 
 is t/i.U8 ; 
 nee per 
 of the 
 IS, i. e. 
 r into a 
 erefore 
 e mtme 
 
 iP. O. 
 reads : 
 mrrels 
 e price 
 
 COMPOUND DIVISION. 
 
 n 
 
 and ttie dmsor and divideivl must be of the same n^e' 
 This 18 Division, ou finding how o^ten onk ramo r? r?v 
 
 TA^'ED IJf another OF THE SAME NAME. ^ ^^*'" 
 
 Here, then, are two cases in Division quite distinct Jn 
 their very nature and^everv question in Diviln must be 
 liKG the one or the other of these examples. Observe in 
 the one P. O. is required, and P W and N T ^!! ®' 
 to the other N. ? fa r^mviaTd pi'o'^andr^JZ 
 
 EXERCISE 34. (MENTAL.) 
 
 nu^?\L^^^r^'' ^"^ ""^ ^""'^ ^^»* ni^t be given in the 
 
 teg ixXlt '*• ""' *'' "'^'"^ ^^*^ ^^^^ -f your fol- 
 
 must'b'e^SiW ''ftT ^-^ '^"'^* ,^^"«^* or sold, what 
 before ^ ^"'"^^ **° example, and do with it as 
 
 anL^i""*^ examples like the last in which the mice o^ cm, 
 
 «w f***^*^"* J^'^^^^^^'Wngs are here anv twA nf 
 which may be given to the find the ttOrd. ^ ^^ 
 
 an^tSir^' this fr-res rise to three disUnd kinds of^^' 
 S?n^ o^ 1' *^* e»7e an ex^pie t f . ach^ atataoB of what 
 kind each examplo is, and Aowi: .Wbedtone! 
 
 *• What ia tb<* m£«m« /v» r~«^jt .» .< ... * .^s. w . 
 of tte.?«o««*, inoacb UM.'^'diiwoif^,*!! ' 
 
72 
 
 ARITHMETTP 
 
 fln^S « • " ""-^TP^f.."^ ^*»*^^ ^f» ' price Of 3 m givon to 
 find the price of 7. First stat- aow you would Dmpood 
 in doing it stop by step, then .lo it ac^ordi^g"y ^ ^''*^^ 
 
 4K^" ^,^* ^^"^'^ y^" subHtituto fur the word mV^» - 
 toe different kind« of question« you havehin examin. 
 
 10. Give examples in Avl.ich the price or weight, Ac.* first 
 
 %tl o'r weiihf J; 'n^;' V*^^W^' '' «^^^" *« «"i the 
 Srlntf?- i^.'*^'f* ^'- I^^»"st, State how vou would 
 proceed in doing eaoh, ilen do each accordingly 
 
 11. Give an oxanjp.e in which the price of Hi nmi th^ 
 price of the whole are ^ven to find tj^^nmube? 'of things 
 Proceed m explaining, and doing it as before. ^ 
 
 ih}\H^^^ <^xample8 of questions in which the ice:jht of 
 the whole and the .weight of 3, and of41 an ofo^ «r{ 
 ^ven torn the numbef of th&s. Go thVough ^ou^fan 
 of doing each, first, orally, an^thcn do each as befor^e 
 
 ln^hr«:;;^«*'Ji5n*'"^ w''^*'!.^*'^'"8ana Iho miscellaneous exerciser 
 mvde same manner before doing them. 1 irst KtatiniTiJhoi ^t • ' 
 
 JTv J'cth7tet?n 'if"' *^^°' "f yo" i»i!!^5 ^ PJK iSVoin7,? 
 giving each step in its proper order, then work ItnccordJ-.gly ' 
 
 EXERCISE 35. (STATE.) 
 
 1. If 36 bags of cotton weighed owt. . are ih« 
 how much did one weigh ? ^ ^ ^^*-' 
 
 2. A gentleman sent a silver tankarri ^- a silvei-^mlfh 
 ?? dw?''f ^^ *" '^'''' '' ^°*^ ^i''^««^' ^^^- ?o weight. ;S: 
 weighli-^fftlrT? '^'^"^ ^'^ ""' -ake,-the4k. rd 
 
 A J: F^^"^ P^'^l'"' ^ ^^'"^ ^» **>o»^t 130 miles ; T . of^ n 
 does a coacb wheel turn round betwe >n the t'v h . ■, 
 the circumference of the wl eel being ] 4 feet ? 
 
 rwA?r'i?r^^'''^J^^^^ ^^ ««g^^- «*^<^h containing 13 
 b^dcnT' "**^ ^^ ^"* ^" ^""^^ ^ '^^ «^ -"24 tons 
 
 1 
 
 i 
 
 A 
 
 
 
 P 
 
 ai 
 e< 
 
COUFOVSD DIVI8I0N. 
 
 73 
 
 6. If in ao dajcs a man travels 746 ml. 6 fur., travel- 
 
 Cd. l^^'Z.^'yS^Il ^, P- ^-^' -t ^^7 12«. 
 
 which Tm?vi'm/.''T' ;\-rr^'' 17 perches ofland, for 
 wnicn I pay £1764 ; what did I pay for it per parch ? 
 
 9. The area of a tract of bu^h country " 1823.3 Acrnu 
 ing eacn 7b a. ? r. 25p., of how many lots will it consist? 
 
 10. Bought 65 yardB of cloth, for which I paid £72 lin 
 ^a, : what did it cost per yard ? *' ^ ^4fl. 
 
 11, A rich man divided lfi8 bii. 1 nk f^ nt ^xC n^tn^ 
 among 36 p.. ^ men : how much did each receive/^ "" 
 
 ^ilhY? f"" ^^''''l -^ 2^** '""^''^ in a day, how long 
 m n ifes V "^ ^' ^^^""' *^" ^^^^^"^^^ bein^ about 12? 
 
 of wintco^r''' ^' *"" ^'^"^^ ^^'^" ^ ^'^^*- ''^■''- 3 q*«- 
 
 fo/$94!50^^^^'^^ ^^'**"°' how much wine may be bought 
 
 16. At $2,215 per gale., what cost a^ qts. ? 
 
 17. At $1.80 for 3J qts. of wine, what is that per gal ? 
 
 fe/$i.to ^^'^^^ ^^''' ^^**' ^^^ "^"^^ '^'°*' *"^^ '^^^ ^^°e^* 
 
 KXBRCrSB 36. (on PRECl':mNO RULES.) 
 
 ^^ f«*^^^<^ earnestly recommended that, when going 
 ZZ^\ ^^ s^^««qH«nt exercises iiie first thm^^ 
 2^'l' S ^: "^J"^!^^ *? «^^e dlBtin^tly. in g.oeral 
 ««„"^' ~xi — ,"',*^.,"" *^^ CiuoD, ueior> attempting to do 
 
 each problei , as it comes before the ciagfl. 2ld. How 
 
74 
 
 ARITHMKTIC. 
 
 it is propwwd to do it, ffivinir eo h a^«« « *. 
 briefly in its proper order. ^ ^^^ "^'^"^ »nd 
 
 If properly conducted, this will orovn «^ k^ 
 
 Spealaryj, in Grammar, Composition, mdt>nic "^"^^ 
 
 in a great measure determined and certein "^' ^ 
 
 ^rTt^tTtKa te^^^^^^^^^^^ ^ -con. 
 
 pupila in tlie variounTperS! "^ ^''"'"'^ ^^^ *^*^ 
 
 EXKRCrSE 36. 
 
 592f«^j;':^^?4"'°"''"'''« "'•""""« '" Konmn Notation : 
 
 4. Reduce $569425 to old Canadian currency. ' 
 
 by ae tranJcUon ? *®*'''* ' "''»' ^'-J '"' B^in 
 
 6. Multiply 749 lbs. 10 0,. (avoirdupois) by 725. 
 '• huppoec a man had 98 lbs. 2 oz. I* dwt r »,. „» <i 
 
 are $35^6^ ?nl**" ^^^l^^'* ^^^^'^^ J »« the fli^t tliere 
 rmcj/ia then a'eaclir" ' "'*''*' """"^ ^nmkm old ewr- 
 
 »^ 
 
 < 
 c 
 
 a 
 
 s 
 1 
 
 li 
 e: 
 
 fl( 
 e£ 
 
 ta 
 
 wi 
 
COMPOUND DrVTPION. 
 
 in 
 
 *i.i?* ^ P*?* ^*^^ contain 9000 bar1«^v corns rhiI r, ^f 
 
 97 m'ni^'!i*« ^"""^u ''^^ hundred and twenty-nine billion 
 dred and nfnT .k"^^'"^ ^^^ «^^^" thousand PourS 
 
 14. Reduce 95679.367 Indies to acres, roods, &c 
 
 r^p^t}^ ?"?' ^.^y. ^^ ^''- «»««*•• at 7d. a pound : 36 drums 
 abol^':i.^twlth.^r"^' '' ^^^*^« ofrasi'n^,a;6r7T 
 currency? *^^ ^"*^""^ ^*' ^^"^ ^^»1' ^^ <i«cimal 
 
 soldTcwri nrTin' ^^- Vl^?' *^»d *^* anotheSe he 
 15 cte p^r lb ? ' "^^^^ '' ^''^ remainder worth at 
 
 18. The circumference of the earth contain.^ one bll 
 express tne same m miles. •#*v«vp , 
 
 fl Jl* ^^ ^.fields there are 113 a. 3 r. 25 p. of land • If th«» 
 
 4*CrV"" ' "*"'v yarus or eaipet, 25 inches wide will if 
 take to cover a room 19 ft. 17 inche, long, by 18^9 ta 
 
 /-... 
 
 / 
 
76 
 
 ARITHMETIC. 
 
 ie^rrc.Hfrr^^^^J^Y''*^i^ nu.„ consumed in a 
 
 IS lb. 6 oz. m the same time » How min» J,!ii !. ^ ' 
 It in one day at the same rate ? ^ *"' ™"™"'® 
 
 P«cto^q^™toof°tK''"''7,'''''''?l'''"« 233 bushels 3 
 
 Lgs, i.onfch"'L;;^ttarb%n^^^^^^^^^ P»' '^ - »•« 
 lofmiL'i2^J^^Xtt„^""r^•' ^r " ^-^^^ <" 
 
 02. spin ? ' "^*" ®^ * s*^^^*»r size will 5 
 
 the whale be 8o&%^i^1'g'ai^$t7^' ^'' '"''• '"^^ 
 
 anf ofVJtlf ^!r^^^^^^^^^^ ir "' ^'^-^-^ iB 770 a, 
 
 oontain respectively V^ What i8 Z t n J ^'Tu ^^ ^^^ 
 per perch ? '^^^r ^^^^ ^^ ***® *»"<! worth at $460 
 
 30. If 8 horses consume two-thiivia nf ceo u o i - 
 ot "/a^r '" '*' <''^^' wharw5?,"otr^ S^..^^' 
 
OOMPOl7N§'l)lYt8ION. 
 
 11 
 
 Mi?!:/^ I ^^^^ consume In |of a day, 1 pk. 1 qt. 1 pt. 2 
 days ? ' ^^^^ ^^"^^ ^ *^°^^^ cnnsumo in 3C6 
 
 , * ■ 
 
 anf *R^ft^So"r''f";S^* '^''"*'' ^^ ^^^* ^^"^' 25 feet 'wide, 
 amount of his bill m decimal currency ? 
 
 34. In one pile of wood arc 37 cords 119 cu. ft. 1462 
 Tin V '" ?"^'¥^'' 9. cords 101 cu. ft ; in a third, 48 cords 
 
 much wood m the four piles, and its value at $8.84 a 
 cord " 
 
 35. Bought 50 casks of molasses, each containino- 58 
 gais. 3 qte., at 60 cts, per gal. ; afterwards 215 ga^. 2 
 qts. leaked out, and the remainder was sold at 38. Id. ner 
 gal. ; what was the result of the operation ? 
 
 36. Bought a piece of land 68 rods long, and 25i rods 
 wide, at £b 4s. Od. per acre j what did it amount to? 
 
 37. A man lends bis neighbour £135 6s. 8d., and takes 
 I^^^ u P^y^^cnt 4 cows at $22 a piece, also a horae worth 
 *&0 ; how much remained due ? 
 
 .38. How muuy suits of clothes can bo made from 3 
 pieces of cloth cnch containing 39 yds. 2 qra. 3 nls. every 
 4J suits requiring 13 yds. 1 qr. 2 nls. ^ 
 
 39. What weight of water may be contained in a canal 
 whose depth is 8 ft., width 25 ft. and length 25 miles 
 when a cubic foot of water weighs 1,000 oz. 
 
 ^t\'i^ ^^y many persons may £60 15s. 6d bo dis- 
 tnbutod, giving £4 I3s. 6d. to each ? 
 
 41^Reduco m 8)8. 3 oz. 3 dr. 1 scrapie 14 grs. to 
 
 42. Divide £1694 16r. 9d. by £9 198, ll^d. 
 
 43. Multiply 6 weeks 4 days 3 hrs. 17 min. by 429. 
 
 Am. \^r\\w *^*'«^«ikMr A-S-^^^^^ -J :-..:-.^j* , _. _ 
 
 ^r^^r].*- '*«"'' •i"^"'^^^*'?*** a eoacn wheel turn in a 
 
 S**1«^^I P °^'^*"' *^^ circumference ol the wheel be- 
 tag 16 n. 6 in. 
 
7e 
 
 ARITHHRTIC. 
 
 «. Reduce 64622.716 cub. in. of firewood to cords 
 
 allowmg 2 quire, ofVaste papTiu "SSf ?■«««' 
 ow*t Jt? 1" "' P"'-""''*^ «"«' ^27 108., what will U 
 pe?ton" " "''*• "'Po'-'^he, cost £19 5s., what i, that 
 
 heL^uZ *19 Ss*!?" ''"'■ l'"*-*""^' ^I-*' qo^ntily .nay 
 4 qte. cos??''"''"" "' ^''^'" ™^' «-»2. '"«" will 1 pt 
 bushel" ^ ''^- * '""• "^ ^'""" <'«'" ^2 cte., what is that per 
 
 bo«ghfL*7f c^l' "'"''""• ""^ '"""' '^''«»* """y be 
 
 Let the pupil .,et down on the slate 
 * ««'«'»° »f 20 or 30 numbers, for 
 adition, each number containing 5 
 or 6 flgures, and let him read un ttoir 
 increasing sum to the (»p7f t£e S 
 b eorre';^;'^™ • beglSr^^S^? f™' to^see'K 
 
 - ^-^;« ;iuc;..xo« uuuer m reBpec^'e holding ;i^ ^ 
 
 56959) 
 481304 
 910567 
 814312 
 916727 
 
COMPOUND DIVISION. 
 
 79 
 
 Division, (EquaZ Parta) Nos. 7, 23 &c 
 Division, (Sow Cfkn) Nos. 20, 23, \^' 
 Reduction, (Asceruling) Nos. II, 18 Ac ' 
 
 EXBRCISB 37. (MENTAL.) 
 
 theSt? ' """ "' ''""^'' ^y 8' '"•'I l»i what U 
 to/ndf/nte'? '"'' """"y ^"""8^' How many d„l- 
 pockoTtyhii^ i^ll',"' ll C^*^ Of "ops, each 
 equVtSlrTa^LfSis-'/^fwr, '^ ' «'• -'->«. - 
 
 c4fl7„r„7i ^s ^.^^o^ToSt"' ^ '''■ °^''-*- 
 
 sheep dW he tfen have? ^'""^'" '"' """"" i "ow many 
 wJghi'4''^3'ci:t.t,rH''lb"i/' "*' »^'-''' «-- PiK 
 .^.^Ti^yT "M^"^r:l.?iX"Tol^e^^^^^^^ What is .hat per 
 
TC;"*" *'il'?"l^o^-J'l !l"l' 
 
 
 ARrTUMETIC. 
 
 nr^t f ^'^ "" fl?^"" P^ ^^*'*^' ^^^^^ contained 43 yds. 1 
 qr., ft tailor cut 3 suits, containing 6 yda. 2 qrs 2 uls each 
 How much cloth was there left ? ^ ^ ■ 
 
 14. At 17 cents per lb., how many p&nds of "chocolate 
 may be bought for $5021 ? How mai"y Ibrfor m ? 
 
 ft. wiw™T- ^^^^^t* ^<>rse for 80 dollui-s, paid 2 dollars 
 his wo^-k "I .'5'''^'^°^' ^^*^ "««^^^^^^ * *^«"^rs a week for 
 70%^?! '~f ^^'"^ expiration of 10 weeks he sold him for 
 70 dollars ; how much d|d he gain by the operation? 
 
 ^J^' ^ ^"^.^ '^ ?^ ^^^'^ ^^^ore a hound, and the hound 
 
 gams 5 rods on the fox every 10 minutes : in how maSv 
 
 , minutes will the fox be caught ? In how many hoTiS?^ 
 
 anf ^ofrSX'?] ^i^if ^^ 1 "^"^i?^^*^ ?"^°^ ^^'' 84 dollars, 
 ma tliey sell it lor, and what was each man's share of the 
 
 ar.1^1 A %rmer sold 18 bushels of wheat at $2 a bushel 
 received ' "''' "'^°^ 'carrels of flour did he 
 
 *irl.l n^.^^'^^tx^'fht a cow for £13J and sold her for 
 
 hnoln'i 1 '^^"^ ^^^ y^\* ?** Saiiied in oats at 20 cents a 
 bufihel ; how many bushels did he buy? 
 
 n«l^^ t f"^^" ««ld «ome wood for £2 8s., and received in 
 
 at lOd^nir ^ T^T^l ^'>' ^^- P^^ ^^^- « ^- of raisins 
 mnnot ^ w ^ ^i^' ''^ '^'"^ ^* ^^^- '^^- ^^d the rcst in 
 money. How much money did he receive ? 
 
 bt^hels^cost^'"'^''^" ""^ "^^^"^ '''''* ^^-^^^ "^^^^ ^'^^ 28 
 
 C08t^? ^^ ^^ ""^ ''' ^'"''^^^ "^""'^ ^ ^^ ^ ^' '"^^^^ ^^" ® ^^^«^1« 
 
 23. If 16 men can finish a piece of work in 12 davH in 
 what time can 20 fmish it ? *^' ^ 
 
 mmspJdV^'''^ "^"""'^ I o^ a $, how much will's per- 
 
 ' 
 
GREATEST COMMON MEASUHE. 
 
 81 
 
 SECTION III. 
 
 IP I 
 
 GREATEST COMMON MEASURE; LEAST 
 
 COMMON MULTIPLE; VULGAR 
 
 FRACTIONS; AND DECIMAL 
 
 FRACTIONS. 
 
 GREATEST COMMON MEASURE, 
 
 A »«a,«r« or fador of a number is ,my number 
 which divides it without a remainder Thus 2 TTl 
 and 12, are.,, measure. 0^2. A^ mfi 'that wiU 
 
 is III e,":.: " ""'"■* -umbers without a remaind 
 IS called a comn«„ „u>asure or factor, and the weaW 
 number that will do this is called th ,re2t eltl 
 me^^r. rep.wented by the letter. G. CM. Th^Hfea 
 C. M. of 4 and 6 ; and 15 is the G. it H m »d 1 
 
 To And the G. C. M. of two numbt r* ■ _ 
 Ruv^.^mviJe ike .jreater by IM tc , and the last dwh«r 
 
 5 
 
82 
 
 ARITHMETIC. 
 
 Example— What is the G. C. M. of 132 and 312 ? 
 
 132)312(2 
 264 
 
 48)132(2 
 96 
 
 36)48 1 
 36 
 
 Here we divide the greater, 
 312 by the lesH, 132, and obtain 
 the remainder 48. by which 
 we now divide 132, the last 
 divisor, and get the remainder, 
 o^. __. The last divi- 
 
 36 ; and so on 
 
 sor 12, is therefore the G. C. 5 
 
 '■!. 
 
 i / 
 
 12)36(3 
 36 
 
 ' Exercise 38. 
 
 Find the G. C. M, of the following numbers : 
 
 1. 
 
 224 and 336. 
 
 6. 
 
 348 and 1024. 
 
 2. 
 
 175 and 2040. 
 
 7. 
 
 1225 and 625. 
 
 8. 
 
 2121 and 1313. 
 
 8. 
 
 429 and 715. 
 
 4. 
 
 377 and 1131. 
 
 9. 
 
 2431 and 770. 
 
 5. 
 
 900 and 3474. 
 
 10. 
 
 1379 and 2401. 
 
 LEAST COMMON MULTIPLE. 
 
 A multiple of a number, is any number that will con- 
 tain it without a remainder. Thus, 12 is a multiple ot 
 1, 2, 3, 4, 6, and 12. ^ 
 
 A common multiple of two or more numbers is, any 
 niimbar that will contain each of them without a ivinain- 
 der ; and the least number that will do this is called the 
 Imst common muUiple represented by tlie letters I c. m. 
 Thus, 16 is a c. m. of 1, 2. 4, and 8, but 8 is the /= e. m. of 
 these numbers. 
 
J2 and 312 ? 
 
 ie the greater, 
 1.32, and obtain 
 48. by which 
 
 132, the last 
 the remainder, 
 
 The last divi- 
 de the G. CM. 
 
 ers: 
 
 i 1024. 
 i 625. 
 1 7ir>. 
 i 770. 
 i 2401. 
 
 LEAST COMMON MULTIPLE. 83 
 
 To find the I. c. m. of two or more numbers :— 
 
 RiTLE.— Wnte the given numbers in a line, mid strike mi 
 any that arecoMained in any of the others withmt a re- 
 nmtnder. Set aside any one of the numbers not struck out, 
 as a factor to he retained, and divide each of the renwininq 
 numbers qf the line hy the greatest number that will exactlu 
 divide it and the number set aside, bringing down the mo- 
 turds and any numbers tJmt cannot be thus divided for the 
 next i'ine» 
 
 Proceed with this line as withthe first line, and so on with 
 each succeeding liw until no number loill exactly divide anu 
 two numbers m the line. ^ 
 
 Then, muMvply all the factors set aside, andaUthe numbers 
 left m the last line together, and the product toill be the 
 t. c. m. 
 
 Example.— What is the I c. m.. of 4, 6, 8, 10 12 16 
 20,24, 25, and 30? ' ' ' ' ' ^**' 
 
 OPERATION. 
 
 16)4 "^"k •' ig| " 1^ •• 16 " 20 • 24 • 25 • 30 
 ^^ ^"^B " 25 •• Ifi 
 
 E. 
 
 that will con- 
 a multiple of 
 
 ibers is, any 
 out a r^jmain- 
 } is called the 
 etters I c. m. 
 
 the /. /». m. nf 
 
 16X15X5-^1200, /. c. m. 
 
 Here 4, 6, 8, 10, and 12, are struck out at once as each 
 Wa f^"" '^ contamed in some other of the given nurabera. 
 VVe then set aside 16 as a factor to be retained, and then 
 
 . S^l^fr-*"^ t' Sn'^*'^^ ^ '^ *^^ g''^»*««t nwniber that' 
 divides IG and 20 without a remainder, for the same 
 
 IZ!" r ^'""'^^ ^^y^ ^ ^"^^ ^^ ^y ^' ^nd ^ there is no 
 factor common to 16 and 25, we bring down the 25 
 
 ^r^ TLf" '? ^^^.^V "°^ ^' 3, 25, and 15 ; we strike 
 ^'^^^ '^^''^'.^i^^^^^J^. We then 
 product 1200, ibercm.'"'^ -e-rs leii, wmcii give the 
 
! • I 
 
 84 
 
 ARITHMETIC. 
 
 Find the /. c. m. of ; 
 
 3. 15, 20. 
 
 2- «, 4, in, 
 
 y- i2. 15, 16'. 
 f », 15, 18, 20. 
 »>• ^ 12, 15, 20. 
 
 J. «, 12, 1H,18. 24. 
 
 7. 2, 4,8,1(5.10,48. 
 
 I' i^i'^' ^7,35,40,45. 
 1 • o' ?.' ^^' ^^' 18, 20, 21. 
 !«• «, 9 10, 12, 25/32, 
 75, 80. 
 
 VULGAR FRACTIONS 
 
 orlpSr'l'Srs/;: " ■"'"«"' ^-^P'--^!""' «fth«r proper 
 
 ^4 Comjjottnd Fraction U i f.-u^*: 
 several fractions contU bV^ won? / ^^^Tof? 
 
 A (hmplex Fraction is one whinh h„c 
 any of the other Mnd« of CLtt ?tr„re,lro?fo"; 
 
 its deaominator, or for both. ^6,t~,. 5* li^l; 
 
 7 . 9 *^ 
 
 3 
 3 
 
 iu a iracuoii. As, I4t|, I3|, ^c, ' '" "'' "'^^'^ "Uiuper 
 
VULGAR FRACTIONS. 
 
 85 
 
 1HJ8. 24. 
 
 ^ 16.10,48. 
 
 15, 27, 35, 40, 45. 
 
 Oi 15, 18, 20, 21. 
 
 !»' 12, 25, 32. 
 80. 
 
 ne or more O's 
 
 Common Frac- 
 proper, Simple, 
 
 lumei-ator less 
 
 its numerator 
 
 either proper 
 )arts of unity. 
 
 ' traction, or 
 As, I oil 
 
 i or more of 
 erator, orfor 
 
 A, fe.,'*.* 
 
 ^^^■' ttui. 
 
 If 15 apples were divided equally among 4 bovs how 
 many would each boy receive ? * ^ ' 
 
 If it were 12 instead of 15 apples, each bov would of 
 course get exactly 3 ; and if it were'l6 apples^ 3^07 
 woudreceive exactly 4 ; but as there are 15 apples to be 
 
 each and 4 is more, hence, each boy must get more than 
 3 and less than 4. After each of the 4 boyf ha??ece^eS 
 3 apples there are still 3 remaining (not yet divided.^- 
 LrTsamiT^'^f ?."'"■'' ^^'^ be^dWided into feqial 
 
 Th«M-« .^vf ^'^^?\^PP^^'^ ^« ^«« already received^ 
 
 thel nar?nrf * '^','.f f^?? ^^^ ^^^^ l'>^ 3 apples and 
 tne i part of 3 or (which is the same thing) 3 and the # 
 part of one. This f is called a Vulgar or Com7non ^ao- 
 twn The upper figure 3 is called the NmiKRAXoR and 
 the lower figure 4 is called the Denominator. 
 
 We learn from the examination of this question— 
 
 I. How fractions originate. 
 
 II. What a fraction is. 
 
 th^/^^f^'l}^.^'} ^f^'^^'ion representing one or more of 
 the equal parts into which any quantity may be divided '^ 
 
 in. What the Denmiimtor represents. 
 
 «,W f^jf '"^«J<>^j («cco»'d% to the manner in which it is 
 
 ^^^^,)repr^enis1Mdivis(yr-ihemmeofthefracti(m-. 
 the number of equal parts into which the qiiardUyisSed 
 -the nuf^er it takes to make an integer o,- tide tSz 
 the size of the pieces. ^' 
 
 IV". What the Numerator represents. 
 parts taken^"'''^ ' tprese/W5 tut uiviaend, also, the nninl^ of 
 
'ytm^^mmm^f,. 
 
 n 
 
 86 
 
 ARITHMBTIfr. 
 
 BX8JRCI8K 40. 
 
 priDoiple, and question under . '-5 t? '"««''-at.> C4ch 
 01.M on it before pv<^LXr'o\C^''''' '""''"* ">* 
 
 3- Wliatdoe. the Denonjinat..,- represent' 
 pi^ea"^""- """• *« denominator represent, the^fo of.h, 
 «• What doe, the Nnmemtor i-eprcent 7 
 
 ^^iwece*. This beln^^i;;^^; ^''''"'" *^' '^'""^^' ^ 
 
 upon^^Sl J^l^^^o^^ ^vhat effect it has had 
 inator by 4. "^ ^'^^^''^^ ^ *« multiply the deuoub 
 
 upoMhTfX^f Slie'Kl^^^'/^^^ '' ^«« had 
 .'or by 3. '^•^^^^'^'n t\j to d.vide the denomina- 
 
 tor by 4. *'*^"®'* f to multiply the numera- 
 
 4« it ha. 
 
 merator by 2. °^ traction f^ to divide the nu- 
 
 ^S'by'^Xl^^^^^ ^W«,.^ or 
 
 crease or d^creasJ^y%Zi,J'r "^^^ ^^' ^«I«e be i^! 
 
 ,J^;.9?nipare|with JL an^3L5--- ^ 
 
 euec. „.» had upon t^i^^^ih:'^^:^!^^^ 
 
 
TUT^OAR PRACTrONS. 
 
 m 
 
 Iho mimora, ,r and denominator 6y the sam. number, and 
 what effict ipon t e value of the fraction W to divid** 
 the numerator and denominator hy the same number 
 
 H. Remember, tha , the denom,mio rtpresoxl. th^ 
 nwnber of pieces it takes /o make a whole me, md 
 
 Zi^glhe case "'*'^*'' '^^"*^* ''* '**.^"^««^^- ^ 
 
 .«2^i*'' t« r«?«c/nf7 l/rtpro/>e^. actum to a whde or 
 m\xed numbm'. Wnu- .«t a rule for doing it. 
 
 12. How manv fiths in 7, 12, 5, whole numbern? How 
 many 12fhs in 1,'), '?0, 46? 
 
 T/iAj is r«<?«ci»fy 7 10^0/6 mn^r to a fraction hmnM a 
 cftven denmninator. ^Vrite out a ru^e for doing it. 
 
 >Jk* """T-,?," " ^*^' *" ??' '" ^^ 1' »« 17§ ? TTow many 
 
 TAw is reducing a miaeed w.-nber tuan improper fraction. 
 Write out a rule for doing i 
 
 III. Remember that dti'Wiw./ the mtmeratorand demmi- 
 vaiue. This being the case 
 
 .Jt ^^^^ ®*<'^ ofjhe following ft ctions the least nnm- 
 erator and denominator it can have, without altering 
 the value, viz. : I ^, j§, j|, j^, ^^. 4. ^'^'^"»« 
 
 oiJf ^.ir^^'f'*-^ ''.•^''*'''^" '^ BfWoM'M^ ^m«,9. Write 
 out a luie for doing it. 
 
 .JZ^S'^'^T^S'' *?.** ^^^Itiplying the nunwrator and «fe- 
 *w voviee. This being the case— 
 
 af^fh^^rrSZ ^<>^'/c<:?r«ling to this principle, each four 
 r.J^^I^^J''^}''^ f'*»ll«?8 may all be made t^) have the 
 I1r^a"A^"""?1.'**t ^**^""'*' aitering their value, via.: 
 
 %,^: ^' ^"^ ^' ^' ^*' *^' *"^ ^' ^' ^*' ^«' *"^ H' t&. 
 

 IMAGE EVALUATION 
 TEST TARGET (MT-S) 
 
 II 1.1 
 11.25 
 
 UIU 125 
 2.0 
 
 ■u 
 u 
 
 U 11.6 
 
 
 flhotographic 
 _Sciences 
 Corporation 
 
 23 WEST MAIN STRCET 
 
 WEBSTER, N.Y. 145S0 
 
 (716) •72-4503 
 
 5V 
 
 k 
 
 # 
 
 <^ 
 
 
 
 
^mA^^^^^mi^^^i*^^mmm'-^mi<i^!mm 
 
 r-^\ 
 
 AltlTIIMETIC. 
 
 pot a rule for doinir it «n,i „i ^ "^ subtracted). Writ. 
 «« the black-board.) (^™«''e'«plaiii to Uie cto 
 
 nominator or d'rlddlfi' ""'' '*«< ''"<«W«i«/ «. 2-' 
 « "-y ». M by 1. "^ =^ ' »8 ''y » -(« isfgetf „?g)' 
 
 wne rule. ' -"" "»« wree foi- division in 
 
 Ijv0f2, of-i|of C. 
 
 "-'" '^^ '"^'^''^ *»«"»«« cannot be (uWeS %^'^; 
 
^rVJ.QAH FRACTIONS. * gg 
 
 Z.'f'^'"^' "- or.fror. tke o,k.r. This being th« 
 18. A,M the following- fractions, i + t,., . , , 
 ly. Subtract | — f, ^ — A, 5} — jl a 
 
 rnltrti^Zk!^ '^"^^^^^^^MctU^'l^ out a 
 
 20. Reduce 5 ofa lb to the fraction of a cwt 
 . f «*/ year to months, weeks, &c. 
 H of an hour to the fraction of a week. 
 A ofa£ to shillings, pence, &c. 
 What part of 8 4i is f of a shilling? 
 What part of a month is | of a day ? 
 
 the van'ous operation??n iSlL- P^nciplea involved in 
 mind of the Kr^^^^^^ <f exercise the 
 
 teacher. If thiTL'. Wn L„w *^,t '"'^^^^'^^^ens of the 
 nothing fnrth^^^is LceSa^^^^^^^ ^f ^«"ed, 
 
 carefully worded an«l S'^h Sf ^^ ^*'"®^<^ ^^e rules 
 practice. The pupil sho,^^^^^ exercises for 
 
 ready written^^s XecStiT^^ f "^^^^ ^^ has al- 
 notice partial larlyirwha^^^^^^^ ^^'^ ^^^*^°' ^"^ 
 
 what respects they di^r ^ ^ *^^^ "«'"^'^' »»<! J» 
 
 RULES WITH EXERCISES. 
 
 nulU^'"^ a» improper f™otioato a whole or mixed 
 
 5* 
 
 Example. ^ = 2*. 
 
90 
 
 ARITHMKTIO. 
 KXKRCWK ^1. 
 
 no»»««no/o,-. ' °^^ ^^^^ ^« »'c*K« place th^ de. 
 
 J BXKRCrSK 42. 
 
 Reduce to improper fractions ; 
 donomSlrr-'''^''' ''»»"'«^«o a fraction having.gi,» 
 
 ExAMPLK. 12 to 6tha, 12 ^_- T^. 
 
 Red ^^^^^^ ^^^ ^*^ (^^ («> 
 
 To reduce a fraction tol^^^est term*-^ 
 
 ^^y divide Hum^T^ /^ «f^ ^f^^r (hat Zi 
 
VULGAR FRACTIONS. 
 
 91 
 
 d Qumbers : 
 
 ' fraction— 
 ^ce th4 de- 
 
 ^8^ given 
 ffor. 
 
 to 9Hw, 
 
 hat toio 
 cr than 
 
 OBrtKRVF. -^.Bt. Any number Oiat ends In 5 is divisible 
 by 6. 
 
 2nd. Any number that onds in is divisible by 10. 5 
 01* 2. ^ > > 
 
 3rd. Any number that ends in an oven number is di- 
 visible by 2. 
 
 4tb. Wh«nthetwo rigbt-han.lliguree are diviflible by 
 4, the whole is divisible by 4. ^ 
 
 6th. When the three right-hand figures are divisible by 
 8, the whole number is divi.sible by 8. 
 
 Cth. When the sum of the dijyits of a number is divisi- 
 ble by 3, the number itself is divisible by 3. 
 
 EXERCISK 44. 
 
 CD (2) (3) (4) (5) (6) 
 Reduce JfgJ, ^^^, ^^^, ^|.ag, ^g^.^^. ^^^^ ^„ 
 
 their lowest terms. 
 
 To reduce a compoand fraction to a simple fraction— 
 RuLE.~Trnte the whoie numbers in the form of fractions, 
 and r^uce the mixed mn>hers t» improper fractions, and tlie 
 complex fractions to simple ones, then rnvHiptyaU the numer- 
 als toaeHusrfor a numeratoi% and all the iUmtnimtors to- 
 gether for a denmninator. 
 
 Complex fractions are reduced to simple fractions hy 
 dividing the numerator reduced to a simple fraction bu tJu 
 deiwmimUor reduced to a simple fraction. (See rule for 
 Division.) 
 
 KotK.— Before muUipIyinfr the nmnerators and denominators to- 
 gether, the work may be greatly shOTtenod by cancelling all the 
 factors that are common to any numerator and denominator. 
 
 PXAMPLE. g0f40f G3^0fi| = |XtXf|X^=- 
 
 S 
 
 1^ 
 
 4oa 
 40g 
 
92 
 
 (I) 
 » 0^1? of y of I, 
 
 (4) 
 
 ARITIINKTIC. 
 
 tXEKCKSE 44. 
 (2) 
 
 6fof-|of2off. 
 
 n 
 
 I, 
 
 to^|ofl^, 
 
 (5) 
 
 (3) 
 
 (6) 
 ioffof Uofi- 
 
 . m 
 
 To add and subtract fractions. 
 
 f.&r'''^"'"' "o « oommon denominator .nd .da 
 
 iX6X8=144-| 
 *X6X8=Ii2 
 Operation. J i •^ * X 8=160 , 
 
 ^«X4X8=m^ "'*if»''^» = % 
 7 X 4 X «=1«8 '™"'" when added. 
 
 -8X4X« = i92 
 Or^ftefollowingrule, which U generally a shorter me- 
 
 Jo reduce fractions to their least common denomU- 
 

 th^n,for 
 btract the 
 
 the min- 
 inator. 
 
 * toushole 
 
 'j/" each 
 ind add 
 
 VULGAR FRACTIOKS. 
 
 tlplyihimymnaor of each fraction by the quotient obtained 
 oyaw\a%ng its denominator into this common denominator. 
 
 Example.— (With same fractions as above.) 
 
 Operation, i — = 
 
 r 3 
 
 
 18] 
 24 
 
 4 
 
 
 5 
 
 
 20 
 
 G 
 
 
 24 
 
 7 
 
 . 8 
 
 = 
 
 21 
 24. 
 
 result wlien added. 
 
 Notk.— Boforo finding the 1. c. m. by the rulo, always sec first if it 
 cannot be found by <iMpec««i-that Is, try whether the greatest of 
 the given numbers Is not it ; If not, try twice the greatest, then 
 three times, &c. Observe, In the last Example, is 8 it? No. Is 16 » 
 No. Is 24? Yos. U is the least number that will contain each of the 
 deuominato without a remainder. vi»t,.tui luc 
 
 Example.— Subtract f from |. 
 
 Operation. ■ 
 
 9 
 
 From 7 — 
 
 12 
 
 Take 1 — 
 6 
 
 4 
 12 
 
 A> result. 
 
 added! 
 
 er me- 
 
 u>mia- 
 
 enomi- 
 n mvi-. 
 
 EXERCISE 45. 
 
 Reduco to a common denominator, tlien add and sub- 
 tract the following fractions. 
 
 (& + ii + |)~(i^ + H.) 
 iM + ^jf + ^)-m + l) 
 ( J + f + ^)-(ih + AO 
 (il + tf + «)-J|- 
 (H + A + if)-A- 
 
 2. 
 3. 
 4. 
 5. 
 6. 
 
 "'^Kf^ESP, 
 
H 
 
 ARITHMETIC. 
 
 To add mixed nunibere. 
 To Bubtractmlxed nnmbers. 
 
 21" SJt Prom 124 =r 1211 
 
 23£^2Jj4 R^s«#^^ 
 
 Result 48i|^48f2H = 50H. 
 
 21 
 
 KXERCISE 46. 
 
 (5^ + 17|)~(6J 4_ Ijl). 
 (2^IH-16f+10J)-l3|. 
 
 ??^t«"^^«A-fi0a^Ki33^V 
 JSlX'^'Sa^^^^^^^ 0^ -y kind b. 
 
 1. 
 
 2. 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 mmeratat and deruminat^I *^* 
 
 wc oi,mmm to any 
 
 anJ 
 
 
 
■f.^,... 
 
 VULGAR IRACTfOKg. 
 
 95 
 
 / obtained 
 Hon.9. ih^n 
 
 ACTION. 
 
 3 
 
 Ind bjr 
 
 idioms, 
 to iany 
 
 
 r. 
 
 - ^ 
 % 
 
 Or, 9 xs^gr XT-f ^r"=3 = ^% 
 
 3 
 Or, 2^ X 2 X I = If X ^ V i - 
 
 _, * ., ■ ■ ■ r' "$** :•..-' 
 
 ExERciSK. |X« = what? 1X6--= what? 86 X i^ 
 
 = what? ^5|p X if -what? \ x4^X7 = 
 what? ^ * ^^ 
 
 .6 
 
 •4*. -. 
 
 fr JSo^^''^^^ if''*°*!£"^' ®'' » fraction/of any kind by 
 fractions, or by a fraction, of any kind. ^ 
 
 WW, and t^fto^o ani mlced nwrnhwa to imprcper MctUms 
 
 Divisor, andjproo«rf as in muUiplicatim. 
 
 ExAMPLK.1 -+.i = J.vi-®v^ 8 
 12 4 12 ^ 3 - Ti -^ "3 == 15 
 
 rk- 5 ^ 6 1 6 
 
 Or,n -f- (2 + I) = V + «-*-») = V kft = 
 
 If II' 
 
I! 
 
 96 
 
 ARITHMETIC. 
 
 Exercise. ^-^ 5 = what? «-,. c = whatT 36^. 
 - what? v^ ^ ^5 ^ ^^^^^ Z| ^ / jx 
 
 = what? * ^ 94/ 
 
 .-■> 
 
 EXERCISE 47. 
 
 Find the value of : 
 ^•AX(2JX3J). 
 2- A -*- (2J H- 3J.) 
 
 3-3X7JXiJX3A. 
 4. 3 -- (7J -f- s^). 
 
 *• (llf if- 6|) X (9§ + 7i^). 
 V. 5ilrOf|J of i y S V ir 
 
 9.9JoflJ^C|ofg. 
 
 4ofV4):''^^--^''-^^^^-^^^-^-^+^^ 
 
 12. ( 2L 6 
 
 13. fof^of(i + f)XAof^. 
 
 -^ + !i±17-MJ\ 
 A 2J + 8i / 
 
 14. 
 
 "tJ"^ \-J -^ i of 3^ of 9|j. 
 
 r r»4? ft _!• «- - _ ' 
 
 15. I of f of £1 las. 8^d, 
 
 le- i off of 61 of 5^ of i^^ of 6 lbs. 4 oz. Avoir 
 
 l'7vHo/^of^of6Jacres= 
 
 '5i 
 
 ?*# 
 
36 
 
 -f 2i) 
 
 VULGAR FBACIIONS. 
 
 18. f Of ^ of J J of 3 aerea 1 rood 27 per. 
 
 19. 4 acres 6 per. 5 yds. -h 5A V a v *9 
 20.11cwt.2qr8.17IbB.x4ofJA,t. 
 
 97 
 
 In 
 
 »mt — ,_ 
 
 REDUCTION OF FUACTIONS. 
 
 it f 
 
 sake of convenience ruTerwS beTr!n'vS''"^V^' ?^« 
 of denominate fractions is done b^yZci?^^^^ 
 rules^as reduction of denominate ffi^'Sbe™^?^ 
 fractions, just as in whole numbers red, ^HnnT ' ,.^" 
 Is done by midtiplymg the /rSramfin i "*-'^^**'^'^*'*^ 
 % is done by dividL //i /Sl'> «^^^^ '^'''^' 
 
 com-se supposed n^ to knorhow' to 1./ ^^*'™^'* *« ^^ 
 ation in Multiplication or WWsion ^ *^'"' ^"^ ^^P^^' 
 
 » 
 
 To reduce a denominate fractlnn fi.««» i 
 
 ation to anotlier. "action fiom one denomina- 
 
 ^Example l.-Red«co f^ of an hour to the fmction of a 
 
 OPEllATION. ■ 
 
 10X24 - 10X4X7 280- 
 
 ^^^AMP,^ 2._Reduce I of an acre to the fraction of a 
 
 OPBRATION. 
 
 <■) !, 
 
 I Of an acre = I><iXi?.X30^ 
 
 of a yard. 
 
 33880 
 
 ?*f- 
 
 Of a yard = 6776 yards. Am, 
 
m 
 
 ARrrHMRTlo. 
 
 / ft.. 
 
 FlemSh * ^ "'• ""^ •» "■•• "-"c'lon of ,„ ei] 
 
 '•"1SS.""«"^«"'''™""'o.bcfr.c.io„of. 
 A Reduce I of J ,f sj Inchw to «.„ fr«t,o„ .f . „„, 
 4.ft«<l„c«J„f 6 0,. to the f™„Uoa «f , of,of}.f 
 
 a scruple. 
 7. Bed..* Aoa of^ or. p,„, ^ ,,„ ^^^^^^ 
 
 yT of a peck. 
 
 ..Z?C""' """ "«""■""'»'* """"•«• to ihe frwiioo Of 
 lbf^z':'fodwt''''*'""' '?•'**■ * "^ '» 'he fmcUon of^S 
 
 OraRAITON. 
 
 8 lbs. 5 oz. ^ 820 dwt. 
 
 S lbs. 6 oz. 10 dwt ~ 850 dwt. 
 
 Therefore 3 lbs. 5 0. fa ^ of 3 Ibe. 6 0. 10 dwt 
 
 , _^ FTKRCrSE 50. 
 
 1. what fraction la in h/.„«» t _j_^. . . 
 
 hours? '■ "' •' "*'""«es or i ^eek 7 
 
▼ULOAR KRACTIONg. 
 
 •ek. 
 larter. 
 >n of An ell 
 
 'tion or a 
 of ti mild. 
 
 onoff Of 
 
 ^etion of 
 
 ^(f the 
 ienomin- 
 
 ion of 3 
 
 dirt 
 
 n 
 
 a.WM^fhictionl.9Ibs.7o.2g„.ufll»>.«o. . 
 4.WJ«t^fraeUo„,«2qr.laa.in„.or2Kng.e..,. 
 
 KXKKCfSH 51. 
 
 A)llowr|ng frtictloiw 
 
 Find the raluo of Hie 
 
 1. I of a week. 
 
 2. I of I of a peck. 
 
 3- fo'f of Jofa njilo. 
 
 *. Aof8|bhd. 
 
 »• f of ^ of-| ofa 4. 
 
 6.Hf^of6gof^ofacwt. 
 ^'' i off of an acre. 
 
 8. Add together f cwt., 8| Iba.. 3A 
 
 9. 3J Eng. ells -f 4J yds. -f- ^ nb. 
 
 10. i wk. -h 1^ day -I- 1 hour. 
 
 11. I of 58. 6d. + I of J of 178. 2Jd! -f' 
 
 12. Prom 8j^ cwt. take 2 qrs. 3f lbs. 
 
 13. Prom 7 weeks take 9^ days. 
 t*. Prow 7| oz. take S^ dwta. ^ ■ 
 
 J?- I of 218. -f I of 6s. -f. I of 78. 6d. - I of 2d. 
 
 oz. 
 
 $0f£o 6*j. 7^d. 
 
 ireek 7 
 
ilOO 
 
 ARITHMKTIC. 
 
 PROBLEMS IN FRACTIONS 
 
 BY ANALYSrs. 
 
 ^r^Vb^XeS t (^^7^^^. -r- should be 
 teacher, before doing them & '^ f^ ^?"°« of tiie 
 be dispensed ^ith Lam pietxf ff 3"^^^'^' ^^" «<^* 
 be given to mental training ' '^ "^^^^ prominence 
 
 tb/;;'es^f w"^^^^^ tt Steo^? i«-' *« -l,se 
 whole class might be calle?rfnV\?'* , '^^ <^o 5 or, the 
 and a certain time allowed Ldn^n ^ f ^ ^^^^'^ ** once, 
 ou ; after which, the dass Lw^ "^ questions given 
 feeing the board, each pi piisSf uf ^ their sea(« 
 
 m turn to go through theSSn ^? k^^ *^ *^« board 
 his work, the members oTfirnll'^'' ?.^^' ^"^«*^on and 
 teacher observing the manner f h^l ^'^^^"^^S^ and the 
 as well as the mechanic^ operatbn inTT' *^« l^^^O' 
 
 The class should go over th A ^ ^^'^• 
 even a third time, before hayi^XVT^''''\ * «^«ond and 
 and analysing without peiShifh" ' ''"^ ^.^^ explaining 
 day working out the auif^l^ '*?®''**^o°» 5 anothe? 
 
 wUhoutany%laSrrftunn1-f^^^ "" P^««^We' 
 est number correctly in a ^,vSn ^. ^^"^ ^^<^« *»>e great^ 
 next greatest, second, and S on. ' ™**'^'"» ^^*' *h« 
 
 Example I.— if ii /^f „ i. t / 
 dollar, what wilY ^*„f A^l^f "*"«" «»«' M of » 
 
 »ii«. Now.iVavoTheS 
 
 of 
 
VdLOAR FRACTIONS. 
 
 101 
 
 •n general 
 operation 
 
 i case. 
 
 explained 
 
 hbuld be 
 »g of the 
 i will not 
 )minence 
 
 ' analyse 
 or, the 
 at once, 
 OS given 
 ■ii' seata 
 »e board 
 iion and 
 md the 
 e logic, 
 
 • 
 
 nd and 
 
 mother 
 
 >ajible, 
 
 greiatr 
 
 St, th« 
 
 §ofa 
 
 much 
 ice of 
 
 1 bushel to find the price of || of a bushel. I will first 
 get ^e pnce of ^ of a bushe! by dividing the pr ce" f 
 one bushel by 48, i. e., $IU ->~ 48 — SJLAJL Nr.«. t k-^« 
 the price of 4 to find Se^price'ofl|^-do''?hTs?^^^^ 
 
 ^mh the price of || of a bushd, Slthi^g requijP 
 Or, more briefly, 
 
 11. Analysis BY the Pupil.-'' To do this, I will first 
 get the price of one bushel, by dividing the price of U 
 of a bushel by JJ. Now, I have the pnce of one bushlF 
 to find the pnce of |f of a bushel ; to this, I will 
 multiply the price ot* one bushel by |^, and that wUl 
 give the price of || of a bushel.'' 
 
 OPERATrON. 
 
 % 17 
 
 IIS) 13 B4 
 
 115 ^ 11 ^ ^J^ 
 
 3 5{)^ 
 
 12 
 
 --$: 
 
 221 
 
 396 
 
 ExAMPi^,2.-A post, standing in a stream, has i of its 
 length in the earth, f m the water, and 5 feet above the 
 water ; what is the length of the post? 
 
 Statement by the PupU.-^' Here I have the part of the 
 post whicn is in the earth, and the part which is in the 
 water and the number of feet above the water, given to 
 find the length of the post." ^ ' 
 
 • v^w?^-7"" J ^^H ^^'^^ ^^^ what part of the post ia 
 m both the land and the water by adding the part in 
 each together, t. c., i and f = J|. 4^hen thi 5 feet above 
 the water mupt be the remaining A of the post. Then 
 as 6 feet is^, 2A feet must be g and 2J X 15. t. e° 
 37J feet must be the entire length. 
 
 Operation.- J + 1 = i|, and Jf - 1| = ^, i^21^ * 
 
 •^ = 37^ feet, the entire length. ^ 
 
 i^T^f'^'^'' 5.- A man can do a certain piece of work 
 in S^ays, and a boy can do it in 5 davs fn what tim* 
 can both together do the work ? ^ ^® 
 
m 
 
 ARITHMETIC. 
 
 E»4Mn.K 4 — Alln«,i„ • ™y» = If days. 
 
 ♦ 1- ff » man travels i9x •> .' '^ - W =- 12 days, 
 miles dooi he ^;t'&>'<» '" 12^ hour,, how many 
 
 c« btUI|l,?ft2^'^«Jj;^;;^how many aero, of tend 
 be »'<«'«h*fo?MJ dS«T ' '""'.'"'"'y '•"'« -Jf coal can 
 
 hought rrw'^dolg^^*"*™ "*' '«''• "ow much eau b« 
 1- ute? <'""'•"'" "i fe«'^ W many ™d. a.« «,«, 
 
 '-'«'Sriro%,?'b*"2^or*" "'*""' " ^<"' '^<' I Of I Of 
 &Whatnamb.,.I,U..,,,f^U,^,^^j^ ^ 
 
 .4f ' « »'• 0""" per y«d, What oo,.. J J. ,;^ ^ 
 »«r Ll5***?? K«^*»* Ifl barrel, of aeia- .* oi ^." 
 
*h and the 
 = A in on* 
 nolo work, 
 
 = H days. 
 
 1 a certain 
 
 a day, in 
 
 travelling 
 
 onid take 
 : 10 hoar* 
 ravel 11 J 
 ' wbenGhe 
 12 days. 
 
 ow many 
 1 of land 
 *oal cun 
 t»o equal 
 k can be 
 re them 
 
 L 
 
 mrdaf 
 
 "a, and 
 letliej* 
 
 ▼ULOAR FHACTIOKS. 
 
 103 
 
 |dol> 
 
 wonld be the gain o?XT " *°"'"* ■"■ "o™- "*»' 
 mnoh ought a™ men to plough in 7| dajw ! ' 
 
 SSSinB^^' *"" ^ «"'» »■• '<»« "y these *8everal 
 
 tune could C alone do it? ^ " ^^*^ 
 
 17. What number must be taken from 2fif5i» #h«»* *k^ 
 remainder may be 116f ? ^''^if' "»at the 
 
 18. Iff off of an acre of land sells for Jl!i« laa «-.«♦« 
 what^^uld . ,0, oo».«i„i„g T J% ?' ,*3'^.'?,.iSrS 
 
 h-i?: ^neht » piece of silk containlnK 96J vardu «nd 
 iJWu.e^iSJ^ioITc.rtS't^''''"'''''^*"^ «- 
 
 d.^LnW.it»^,Vo't?» ? ^'-'•••^ ^X ""^ 
 
ARITHMBTIC. 
 
 lS^¥;;>»^?f # Of the Whole mci-'i^u^ 
 
 *08»lBlU8t this man sustain ? ^^' ^ ^^* Part of the 
 
 o.»— ■•-o.H5,...i,^2 
 9ft fp, , ^^^ 
 
 wW Wth^^f/"^'**'^ ^f a room are 2oi « u " 
 
 28. Express 4 bus 1 ni- i , 
 «nd reduce 5 owtXiifj,^/"'- ^ '''«• «» « fr. of a <,, . 
 
 * 29 Tf J. ^ * 
 
 will COS. |,»25* sl^ •"' --* ^6 10s. TJd., What share 
 
 DECIMALS. 
 
 ;;_i^._,:^Ss«2rr.-, 
 
 ^ "^-'^in^'^Sl?'''?^^^'^ 
 
DECIMALS. 
 
 105 
 
 ih}L~^^T'^4i'^' ^'Aa<, in J•m(iMls^ o,nd witing decimals, 
 
 jl. The number (which i» read ami written, just as in 
 ' ledmaf pSr -^*°P*^*^'^ independently of the 
 2. Tie mme, (which is dejxjrmined by the number 
 
 Of figures in the decimal, on the principle that one 
 igure has two {l e. a unit and one Cypher) 
 
 decimal figi„^ ^^„ ,^^ ^,. p. „ unii ana one Cypher) 
 for a denominator, tico h^ three, three has four ; 
 
 fn'fh! 'T.°/^ ^^.^% ""^"^^ <>^ the.figm-e farthest 
 to the right from the decimal point.) 
 
 EXAMFI.V3S.— 15, Js^^^^5, name, is hwwtredths ,- read/^een 
 
 •179,No. is 179, name, is tkoumndths: read 
 
 one hundred and seventy-nine thousandths, &c 
 
 OBSERVE,-Whole numbers and decimals may be read 
 
 e ttier separately or togetJher, giving the whole numbeiS 
 
 tiiename of the lowest decimal : thus, 6'2is^ad6 and 
 
 2 tenths, or 62 tenths,-,764-15 is' read' 764 ^d 15 hun^ 
 
 todredths'^'^^''^^""^ ^^'^^^^ four hundred and fifteen 
 
 EXERCISE 52. 
 
 1. -7864, 78-6237, and 8064-2987432. 
 
 ^' IfeS^^ "^"^^ milUontha-seventy-nine million bil- 
 
 'tS^aSX"' "^ *>»Honths-two"billion ten 
 
 EXERCISERS. 
 
 ^.--Remember, that, the greater the multiplicand and the 
 mulhplverj 1M m^rth^produd ; and the iZTbTmmT 
 ^^jand muhplier, the less the' product This bdSgS 
 
 V 1. Compare the products of 2 V 4,>2x 4, afld -2 V-4 
 and 8how,^accordhig to tffi principle, why the^ 
 
\ 
 
 ^ ARtTHMETIC. 
 
 2. Do the same with 3 X 6, -03 X 6, and -OS X 06 
 
 '«trt^s,rttt-i;;:rerah"'t^ 
 
 torn pages 27 and 28.) P""cipie, (see note bot- 
 
 Find the products of 
 
 4. 002 X 003 X 64-23 X '0007 X 7603. 
 
 5. -5 X -006 X 2-7 X 053 and -03 X '4 X '0067. 
 
 dUHsar and the dsth^di^Idl^L^)^' *^^ ^"^^ '*« 
 being th^ case-l ^^^'^'^w^^ the less the quotient. This 
 
 EXERCISE 54. 
 
 ,?^^*^f*^«^"<>*^«°*8 0f8-j-2,.8--^ 8 • -2 .» 
 
 2. Do the same with 4 ^2, -4 4.2 i^., .„j . „ 
 
 S. Repeat the rule for di^Montf dectoat and "^ ■^• 
 
 its c«rrectne,._by this prl„d;.e,tr'ptge''3rr^'' 
 
 4. Pmd the quotient of -006 ^ 27, 6 ^ -27 -mni „, 
 
 5. 13 -.7-864, 7.862^_786.2,and 002o!l or" 
 
 Example. I = 2 ana iX3 _^ g _ ^ 
 
 .. .,.^..,.^..^;, . BXEHOOB 65. 
 
 : *• ^Z*'"' "^ P''"*'?'* »PP"«« *o pro™ the foHow- 
 
 o^'^"Cl':^rr^r/''lS!''?.<^*<^'»<^. Anne, 
 ampoiniojr&sindivisiin" "^'"""^^ -y «« aw«w«wK»^, 
 
 \ 
 
 I 
 
 ■"■^rseaKisffiasrsd 
 
DECIMALS. ^ 
 
 ExAMPLB—Reduce, f to its decfmai 
 . 300 
 
 Reduce to their equivalent deoimaU : 
 2- h h h i§. 
 
 «• 27H, 96if, 112||. 
 
 r&ion would never temS^ the^quotient, and the di- 
 ExAMPij5,i = .333,&c. ^ = 57U28,&c. 
 
 deSl?ieTal&: Jt^tei;:i,*td'^^^ ?^^"^P^^' *^« 
 by writing a dot ove/th J^l .w * "* «*Pres8ed 
 ^en mofe than one fiLrere^^^ *?if' i'P."'^*^' «»"»' '^' 
 
 expressed by wri W a dot n^^ 3? ^^^^ «°d it is 
 of the perd oTcl^e ^as thH^^Sf S"' *'*\*»«* %«^«« 
 i« called) thus, 'hTmf ^^fi^'^^^tl^at i;5eats 
 
 ^uTotSr^ S^t'of a'Kf r *^Mf--alpomtis the 
 called a p^.re ^petend if nnw^"''^' **'** '^P^^ 5 *his Is 
 oirmiiati4 decir^TmovJ^SlJ'''^ ''^P^**^' and a pure 
 iires do not coSnce ?o^l,S?•f^?® '•^P^**«- ^^ t^eV 
 
 decimal point?Th?decimalSrlr"'^.'^^*i^'^ «««' ^« 
 mixed circulating decS ^^'^^^ '"^edrepetend, or, 
 
 lattg'^i-nrap''5?Ca'a':^^^ -i^^iaapure circa- 
 ^^ circulaliigl'cL:!^^^^ "^^^^' '* ""^'"'^ » 
 
ijii 
 
 ^^W ARITHJfSTIO. 
 
 ^ rs as there are fibres in the decimal f^ a kerumZ 
 
 ExAiiPui!.-8 = §, 1*56 = ^f^. M 
 
 To reduce a mixed repetend to a vulgar fraction. - 
 ,«A^r^'T^'^^' ^fi^^'^ <^ ^ r^t repeat fr,m the 
 
 torwrue as many 9'. as there are repeating Jigures,f6Uowed 
 ^<^rnany0^s as there are figures that do n^repJit. 
 Reduce -346 and •12?^4 to vulgar inactions. nJoB n; 
 |:xAMFLB.->:345 = ||#, .123^ ^ i^^i^ , . 
 
 ■ ■■■/» ■■" """ — T ■ '■ 
 
 ,. , BXERCISE 56. 
 
 Reduce to equivalent vulgar fractions. 
 
 1. -43, -163. 2. 037. 138. vf* 
 
 3. -425, .)6. 4. i678; 716151. 
 
 To add, subtract, multiply, or divide pure or mi*ed 
 repetends, or circulating decimals, 
 
 RuLE-Mrst redwe them to th^r equivaleni vulgar frac- 
 Hms, thm proceed as viUh any other vidgarf radios. 
 
 BXBBCISK57. 
 
 i: 2723 +26 + 7-72 -f.. 297 ^isMi 475: . 
 
 2. •3 — -05, '04—00769238. 
 
 8.37r33X-36,7-74X4-75. 
 
 4 -614-5- 2-766, 6-71 -*- 036*. • 
 
 To find the value of a givfen deeimalm intoirhra a0:«" 
 iow«r denominations— " 
 
yr,andaa 
 ( denomin- 
 
 lon. 
 
 ■ from Vie 
 
 ienomina- 
 
 hfdUowed 
 
 ,;-J- 
 
 DECIMALt*^ 
 
 109 
 
 r milled 
 
 nrfrac- 
 is. 
 
 B air^hi 
 
 Rule.— Mtftipiy #Ae given decimal by as many of the 
 lower defumimUion as U takes to make one of the same, 
 pointing offfr&m the product as in midtipKcation ; then pro- 
 ceed as before with the decimal part, and so on till the lowest 
 denomination is reached. 
 
 Example.— Find the value of 23-43 of a da^. 
 Days 23-43 and decimal of days. 
 
 172 
 86 - 
 
 Hours 10-32 and decimal of hours. 
 60 
 
 Minutes 19-20 and decimal of minutes. 
 60 
 
 Seconds 12-00 
 Hence, 23-43 days = 23 d. 10 h. 19 m. 12 8. 
 
 BXBROISH 58. 
 
 Find the value of 
 
 1. -626 cwt. 
 
 2. -726 gals. 
 
 3. -3375 tons. 
 
 4. '0*5 acres. 
 
 5. 1-86 of 3s. 4d. 
 
 6. 13126 of $6. 
 
 7. -326 of IJ ton. 
 
 8. -USoflfur. 36p. 
 
 2 yds. 5 in. 
 
 To reduce a denominate number to the decimal of 
 another given denominate number, 
 
 Rxjix.-rr--fieginnin^ with the iQwest denomtna<i<m, reduce 
 Uiodie decimal of the next higher, writing U after the higher, 
 
 €mnex U to tike next higher, and ^o on, tucending to the 
 required denomination. 
 
^^ ABITHMKTIC. 
 
 20)6 . 75 g. 
 
 4. 3375 dec. of a ifi. 
 
 I 
 
 Operation. 
 2)1 pt. 
 
 4)2.5 qts. 
 2)1.625 gal. 
 4)3.8125 pks. 
 
 •953125 dec. of a bush. 
 
 EXERCISE 59. 
 
 1. Reduce 9 oz. 2 dr. to the decimal of a lb. 
 
 2. 
 
 3. 
 
 4. 
 6. 
 6. 
 7. 
 8. 
 
 « 
 
 « 
 
 « 
 
 « 
 
 « 
 
 3 for 33 yds. to the decimal of « mile. 
 
 1 owt 3 qrs. 7 lbs. to tie decimal of 2} tons 
 
 3*hL , *','•'"■""" »" ■!"• 12 l""- 
 
 3} inches to the decim al of J mile 
 
 22 guineas to the decimal of ^625 
 
 9. Paid 36c foMf ' I"' '? "■" ""<'• »' """o- 
 yard ? '" -^^ y"^ <"• ■='»«• ; whM was that a 
 
 "*■ ^hl?|o?t!S;??'^* """■ <"«»«-•; how much was 
 
 "• li^Cletlf'' "■"» J- "->". "ow fa., can y„„ 
 
 It Syt ofX^ """T' "* ^- r p'^-^^' 
 
 did ifcoL la? ' ""^ '"'^ «' «' !>« ?™in ; w% 
 ^^^ ^?J^:*«?¥ JcrecfiaW: a, .s doiw . *... . ^, 
 
gal.2qrti. 
 mal of a 
 
 RATIO. 
 
 Ill 
 
 ?. 
 
 SECTION IV. 
 
 RATIO, PROPORTION, PARTNERSHIP. 
 
 f a bush. 
 
 'J tons. 
 I day. 
 
 that a 
 5h ^as 
 n ypu 
 
 
 RATIO. 
 The Ratio of one number to another is the relation ex- 
 pressed by the quotient obtained by dividing the former 
 by the latter. 
 
 Thus, the Ralto of 6 to 12 = i, but the ratio of 12 to 
 = 2. 
 
 The former term of the ratio is called the mUececlent, 
 the latter is called the consequent. The ratio between 
 two numbers is usually expi'essed thus. 6 : 12, which 
 signifies the same as 6 -f- 12. or ^. 
 
 When the consequent is equal to the antecedent, the 
 ratio is called a ratio of eqitmty, as 6 : 6 ; when the con- 
 sequent is greater than the antecedent, it is called a ratio 
 of greater inequality or a nsi)ig ratio, as 6 : ] 2 : when the 
 conseqUi^t is less than the antecedent, the ratio is called 
 a ratitf of' less inequality, or a falling ratio, as 12 : fi. 
 
 Ratios are compounded by multiplying all the antece- 
 dents together for a new antecedent, and all the conse- 
 quents together for a new consequent. 
 
 The Ratios 2 : 4, 6 : 8, 3 : 2, when compounded = 36 : 64, 
 which ib called a compound ratio. 
 
 The vakie of a ratio is found by dividing the antece- 
 dent by the consequent. 
 
 EXERCISE 60. 
 
 1, What is the value of each of the ratios, 7 : U, 2 : 15, 
 
m 
 
 ^^ ARlTHMBfUBT. 
 
 PROPORTION. 
 
 tr^'^'i^^^^^^^^ or union Of 
 
 thus, G : 18 f; 8 , 24; reud 6 ii h? /;*^^''*LH"« their equality, 
 that 6 diviSod by 8 j^^eVZ «i„^ ^ ''i*' ^4, meaning 
 ed by 24, or, ^^J^J ^^ ^*^^ quoUont as 8 dind- 
 
 4/?t^utU^^^^ .P-rit'^ there are ..^ 
 
 by which the anTw^ris'obtVer"'"' ^^* the j^eration 
 
 what ^rcTn' n' r;; Z^^ ' ^-- i-^ 28 days, in 
 
 BimX:o^;o~;tn^^^^^^^^ --^ question in 
 
 fourth. Tvo of the three aralfke'v^^^^ ^'^ ^^ « 
 
 the 7 men and the 17 men fnrmfn * ^"* t^"^*"npJe,) 
 ratio ; the other is ^gi^Z te\2i of ?h'*^ '''^'^'^' 
 ratio, the mate, as it were of ih^f .?® mcomplet 
 term. Thfe odd, or unlated tem tt" *?' ^' ."f T"^ 
 tje same name as the tS reoXJ .*®''"' "^^^^ ^ o^ 
 the term of hn incom»J^r««o^ ?'!l **^ answer, being 
 the third ten. IZ^eJftff,'^'^^'^^^^^''^^ ^l^V^ bf 
 ^noke that thr t,. : v"'^ tK?7ir i"'^^' «^««»y* 
 an^tccr. Ther n-^V.^*^ *•"«/'*« «am« name ew ^c 
 correcUy, c^i.:,; S^ij^^f*"^^ t*»^ other two term^ 
 
 /e*^3~t^a7S4raavi "^* ^'■•^*^'*' ^ ** « rising roHo;^ 
 
PROPORTIOX. 
 
 113 
 
 tnd which 
 
 . 16 lbs. 
 diiya : 13 
 
 union of 
 equality, 
 tneaning 
 » diFid- 
 
 ment of 
 •n; and 
 eratio4 
 
 ays, in 
 
 tipn in 
 find a 
 mple,) 
 >erfect 
 nplet 
 juired 
 h ia of 
 being 
 kysbe 
 Iwaya 
 as the 
 terms 
 
 ifffrM 
 
 Sircars."" '^■^-' "« '■--""" """""- 
 
 KJi'P"^ll'*^h'^'^^'^'"^ the qnostion distinctly in the 
 hearV^ir o? the class, a« fbiJow8 :— '■ If 7 men can build a 
 h'^use m ^d (lavs, in what time can 17 men build it?" 
 J. cecds,-^" Hore I have tlie number of days.'' (or what- 
 Zn,TI ^* ''"''^f'''^ J" the question,) - in which 7 men 
 In which 17 men can do the same work. As days are 
 know that it will take less days for 17 men to do a piece 
 
 The question stands when stated, 
 
 Man. Mok. Days. 
 
 • 17 : 7 : : 28 : Am. Road, 17 men \^ to 7 men aa 28 
 days is to the answer. ; ^in .m:'U'ix<^..' ' 
 
 (julstiSns!''"''' ""'""'''' '*''^"''' ''"^' •''"^ ^^^ *^^ following 
 
 KXEKCISE 61. 
 
 1. If I buy 7 lbs. of sugar for 76 cents, how manv 
 pounds can I buy for $(>. !»>.,» 
 
 2. If a family of 10 persons usoB bnshel? of wheat In a 
 
 r»*S'i„'°cSyr"^'^ *'" '^-'- "«"■>' '"'™ *^ 
 
 3. If a person, whose rent is $145, pays $12-63 t&xt^ 
 how much should a person pay whose rent is $378 ^^ 
 
 4. If I give $6 tor the use of $100 for 12 montha 
 what must I give for the use of $357.82 the sameW. 
 
 i^ovlj the two terms of tho complete rXuSarep^SriSU 
 Tbe term of the incomplete ratio is essoatSSyTflEt SL?S: 
 
 i- i-w «.u qaoofciou, OJ5U1U, aii'udug'u iqo three fi^rmsaroAU LmM^'^l 
 
 6» 
 
mmmw 
 
 114 
 
 ARITHMETIC. 
 
 ill! 
 
 ounces and the otherJaf' m- i. ">»«*'«»5 o^ was 
 In order to understand why we mnsf mni««w *i, 
 
 6-f°l^.,f''.l'5'"/»',»?« dividend. For eiamp5e 
 
 «iLn^g^ihl*'S;Sll|„lJotSSfSoSr^ 
 the means and the extremes ai^nnwc^L^? *^ 
 
 16. Hence • * " X !«' *•>« extremes aUo being 12 X 
 (be unknown term (wWoh ^ tK ^-""^ ""^ "*''«•■, or 
 
 ratio. FoV;.lneeweu;r.Se\=rS?ex"i^^^^^^^^^^^ 
 
 ?feiii: 
 
(^ of 6 feet, 
 « 153 feet? 
 
 ! attended 
 ' is stated, 
 
 r the first 
 1 ; if not, 
 ere could 
 one was 
 the other 
 ations, as 
 d to the 
 Jr of con- 
 
 tiply the 
 
 product 
 
 3er that, 
 
 r middle 
 
 z product 
 
 ; for in- 
 
 = 9X18 
 
 the two 
 
 division 
 
 ns is the 
 
 idend of 
 
 iie pro- 
 
 xample, 
 
 12X8, 
 
 ^ = M. 
 ictuaUy 
 
 dopted, 
 'gl2X 
 
 n, and 
 ther, or 
 roblem 
 ., " The 
 
 '€m€S^\ 
 
 mplete 
 iz., the 
 
 PROPORTION. 
 
 115 
 
 first term, and since, " the product of the means is equal to 
 the product of the extremes," hence, when we get the 
 product of the means we have a product and one of the 
 factors, viz., the first term, given to find the other, viz., 
 the fourth term, of course, we have only to divide the 
 product by the known factor or first term to find the 
 unknoum factor or fourth term. Hence, to do any ques- 
 tion after it is stated and the first and second terms re- 
 duced to the same name, multiply the second and third terms 
 together, and divide the product by the first term,— the quo- 
 tient will be the fourth term, that is," the required term of 
 the incomplete ratio, or the answer. 
 
 Example.— If I buy 7 lbs. of sugar for 75 cents, how 
 many can I buy for $6. 
 
 OPERATION. Here, after reducing the second 
 
 $6 : : 7 lbs. term to the same name as the first, 
 
 100 600 viz., to cents ; as the product of 
 lbs. the means is equal to the product 
 
 600 75)4200(56 o/<Ae extremes, we multiply 600 
 
 75 c. 
 
 450 
 450 
 
 and 7, (the m,eans) together to get 
 the product of the extremes. Then 
 we have the product of the ex- 
 tremes, viz., 4200, and one of the 
 ^ ^ ^ ,^ , extremes, viz., 75, (the first term) 
 ?iAA'\- ^^ *°® ^*^®'*' ^^ course we divide the product 
 4200 by the known factor, 75, and get the other, the un- 
 known factor, or fourth term, 56 lbs. the Ans. 
 
 What has been said with reference to the statement 
 and workmg out of questions in proportion, may now b» 
 expressed in the form of a rule. 
 
 RuTJE FOR Proportion.— Of the three give.i numbers, 
 make that one the third term which is of the same name as 
 iJie one required in the answer. 
 
 7hen consider from the nature of the question whether the 
 answer vnll be greater or less than this term. If greater 
 arrange the two remaining terms as a rising ratio': if less 
 arrange them as a falling ratio. > j f 
 
 y-ien reduce the fit si aitu second terms to the same name, 
 ana, either leave the third term as it is, or reduce it to its 
 lowest denomination, as may he thought most convenient. 
 
ue 
 
 ARITHMETie. 
 
 owm, In the same name as thTlhJi^ *""''"" '°®'« «"« 
 
 EXEKCISE (i2. 
 
 
 1. If 6 
 can 
 
 beVollght^fc?'!?''''"'-''' ""-^^ -any yard, can 
 
 '• '^?rf'?toTSrp^?y\r *-'•" " ■""-.^"•«' 
 
 *• wS'l'S?;\^ ?o';f.' "^'- ^ '''■■ ' "- »f.«°"'- i -bat 
 *• 'i^S^.WS^:^^'' ^ -^ -"><• *-= tons 
 '• mlt^S "' P^P- <""' S33.15 , what w„,„a 990 
 #T. If 96 men reap 40 acrp<? nf n-c-r, {« 
 
 many men would i^TesLtr^ 'i ^^J?' ' ^^^ 
 :, yds/in the same time? '■ ^ '•''^'' ^^ P^^^'- 20 
 
 8. There are two numbers in the rotio of -> fn ^ .i,^ , 
 IS 85 5 what is the smaller ? "" *" *^^ ^^'^^^'' 
 
 ^' ^V^Zn'"f^ ^^""^ '^^'^ ^^ S months by 
 
 -anymenmustbeem^^SlSlVr'^^- ^^T- 
 In'lO iours^^'b™ ^'^^^^ "• ^^' ^^^^^ uill empty it 
 ^11 emi;iy it^nVSJS^f ^' *^^ ^«- -P-^*^ 
 
 ^^' i^Z^Z ""l- Jf ^ "'^^^ ^''^^ provisions for 9 months 
 
 „9 ~ ■■ ■ •■" ^a=i II uvi vvnea it is bui 
 
 wake an, acre , 
 
 3^ rods Jong? 
 
 out 
 
PROPORTION. 
 
 117 
 
 letheTf and 
 mUlbe the 
 "educed to, 
 
 lanyraen 
 
 r^ards can 
 
 M, at the 
 
 u- ; what 
 
 95 tons 
 
 »uld 990 
 
 k; how 
 per. 20 
 
 e larger 
 
 nths by 
 
 tl, it is 
 
 How 
 
 tnpty it 
 ipacity 
 
 lonths, 
 le pro- 
 son be 
 
 eadth, 
 is but 
 
 I 
 
 13» If a man perform a journey in 15 days, when the 
 days are 12 hours long ; in how many will he do it 
 when the days are but 10 hours long ? 
 
 14. If a field will feed 6 cows 91 days ; how long mil it 
 feed 21 cows? ' 
 
 If I can walk from Kingston to Toronto, a distance 
 of 180 miles, in 46 hrs. 18 min. ; in what time can I 
 walk from Toronto to Montreal, the distance beinct 
 333 miles? 
 
 15. 
 
 A farmer sold I of his land to A, ^ of it to B, and 
 the remainder, which was 100 acres, to C ; how much 
 land did the farmer own ? 
 
 17. 
 
 18 
 
 A cistern whose capacity is 3000 gals., is supplied 
 with water by a pipe which pours into it 7 gal. per 
 minute. By leakage the cistern will lose, during the 
 time of filling, at the rate of 2^ gals, per minute : in 
 what time will the cistern be filled. 
 
 A post standing in a stream, has J of its length in 
 the earth, § in the water, and 5 feet above the water ; 
 what is the length of the post ? 
 
 19. A person failing in business owes $5000, and, is able 
 to pay only $2000 ; how much can he pay per dol- 
 lar to his creditors, and how much should that cr^ 
 ditor receive to whom he owes $1000 ? ^^ 
 
 20. How riiany yards of linen | yd. wide, will be equiva- 
 lent to 30 yds. of another kmd which is | yd. wide ? 
 
 21. A traveller having gone 375,5 miles on his journey, 
 finds that | of it remains to be travelled ; what was 
 the length of his journey ? 
 
 22. A and B depart from the same place, and journey 
 in the same direction ; A starts 3 days before B, and 
 goes 30 miles per day ; B follows at the rate of 33 J 
 miles per day; in how many days win the latter 
 overtake the former ? 
 
 '■^y*^ — ^"^^ *""* ^^ quegtiOiib of fciii« Exercise may bo done without 
 a Knowledge of v^ulgar and Decimal Fractions. The remaining ques- 
 tioiaare designed for exorcises in Fractions as well as Proportion. 
 
ARITHMETIC. 
 
 118 
 
 together to do the work ? *'™* ®"«f^* '^oth 
 
 24. A can dig a ditch in 5 dava R i„ « ^ 
 
 days ; in what time coulKh^T.i ?*^^' ^°<^ C! in 8 
 ditch ? ^^"'^ *^e three together dig the 
 
 26. Two masons together hniu o ^ n • , 
 
 them could hafe builUhe wj? hl^^V?^' ""« o' 
 
 «-,r^J . 2 "*'*"''> what IS the Talu« 
 
 ^ fw-j "'51 of 31bs.t 
 
 27;if the rent of 20 ao 3 rd. 2 p.. be ,.o ; what „W 
 
 hethejentof Sof|of3ac.3r.? 
 
 28. If 9-35 ]|,g of rice co.f »n no . u 
 
 be bought for $9 73T * "^ ' '""' """y PO'Uid« can 
 
 '"■ va^To?S.o"li^r"^f ''"* «''«-3*; -»«* « the 
 '"■ Of ir.SZr"**'-^^' what.the„,u, 
 
 ^'' "onC^aTeir**? f '"^'""'' -^--ure which 
 
 OiS. What is the vaTna /^f q „ r, „ 
 
 •oooeiooi yds co^t $'0 Ao'r- ' "'«• »' 'loth, if 
 
 ''• * o?'tS'e%e^2St*.S: "iVr '».'"' o-^ «o». 
 «)niainder,whichwas«5ono f^?:-""^'"*''' »"« the 
 the value if his este^f ' ° *"' '"^'"' ' ''''»* ^aa 
 
 '*• "ilsU';^^""- "«' »'«-^«"; -*« cost 7.0. 
 
 ''*'■ "/ 1'"? '"»' *3 ^^ **<!• ; What cost i of Jtofi* 
 ci.^^c^).""'"*"^* "«»' (&pr»ssed in d'ecimlj 
 
COMPOUKD PROPORTION. 
 
 119 
 
 days, and 
 ight both 
 
 nd in 8 
 >r dig the 
 
 » ; one of 
 15 days : 
 »e it ? 
 
 he value 
 
 hat will 
 
 nds can 
 ; in the 
 
 5 ralue 
 
 which 
 easure 
 
 ioth, if 
 
 yson, 
 d the 
 it was 
 
 7ac. 
 
 Of-rf 
 
 iimal 
 
 36. If the freight on a ton of merchandise is £1 3s. 2Ad. ; 
 how many tons can be paid for with * of * of I of 
 
 37. If it takes 40 yds. of carpet to cover A of the floor 
 01 a room when the carpet is 27 inches wide ; how 
 many yds. will cover the same, the width bainlr t of 
 f of 2 ft. 9 in. ? 6 ¥ " 
 
 38. What cost 90 tons of hay, at the rate of 4 of a cwt. for 
 
 $9,325? 
 
 39. A can build a well in 5 days, B in c days. A began 
 ? « ^^YS^* ^^^ 3 days and then left. B commenced 
 to finish it ; in what time can he do so ? (expressed in 
 the decimal of a day and of a year.) 
 
 ■ — »»» 
 
 COMPOUND PROPORTION. 
 
 When the relation of the required quantity to the given 
 quantity of the same name depends upon two or more 
 conditions, we have a Compound Proportion ; that is, an 
 equality between a Compound and a Simple Ratio. 
 
 A question in Compound Proportion is stated just as a 
 question in Simple Proportion ; i. e., each simple ratio 
 found m the question is made a rising or falling ratio, 
 without any reference to its connection with the other 
 simple ratios with which it is to be compounded when 
 all are arranged. 
 
 Example.— If 264 men, in 5 days of 12 hours each, can 
 dig a trench 240 yards long, 8 wide, and 2 deep : in how 
 mMiy days, of 9 hours long, will 24 men dig a trench, 
 420 yards long, 5 wide, and 3 deep ? 
 
 In this example, the rumber of days required depends 
 upon five conditions, viz., the number of men, the length 
 of the days, the length, the width, and the depth of the 
 trench. As days are required, of course the 5 days go 
 m the third term ; then, 1st, will 24 men require more 
 or less days than 264? More, of course.— hfin'^A the mon 
 form a rising ratio. Then, 2nd, will 9 hours long require 
 more or l©ss days than 12 hours long ? More,— hence, the 
 
120 
 
 ABITHMETIO.: 
 
 stands when itated, ' """^ ^^P**^' ^^e question 
 
 Men, 
 
 HourS; 
 
 Length 
 Width, 
 Depth, 
 
 Befor^ arranging each ratio, 
 
 the terms must be reduced 
 to the same name if not so 
 
 5 days ^^^^^^7' and before com- 
 
 ^^'- pounding the ratios, all fa^- 
 
 tors common to the first and 
 
 second terms should be can- 
 
 ^«»«1 5, then, having com- 
 
 6 : 386 ';TT~ PO"P^ed the ratios, we pro- 
 R ceed just as in simple pro- 
 
 portion. ^ ^ 
 
 J 6)1925 
 
 3205 
 
 6 
 
 days. Ans. 
 
 poS^t "^^ ^^^^^ *^« ^"owing rule for Compound Bro- 
 
 nomina/wi, a^-ranae theZ^^f '^ ^f""^ to the same de- 
 halving canceml^Jucha,\n '-If^^ P^m^Uon ; then, 
 ^i^;d term, a^ui aU StZiZlf'Tff^i'^'^^^^ ^^ 
 by the product of the%,Ttmm ' "^"^ '^''^'^' ^^^ ^^^^^^^ 
 
 ■■!!h,r; 
 
 :: EXERCISE 6b. 
 
 4. I£ 14borses eat 56 bushels of na*« i« i. -.- , __ 
 -»n^ migiii be kept on 120 bushels forl^da^ ?""*"' 
 
)thor three, 
 e question 
 
 each ratio, 
 •e reduced 
 if not so 
 fore com- 
 33, all fac- 
 ie first and 
 Id be can- 
 ^i^g com- 
 ', we pro- 
 mple pro- 
 
 *> . . ■ ■ • 
 
 'ltd term 
 h having 
 swne de- 
 «/ then, 
 'ether the 
 M result 
 
 *2 days, 
 
 r ma^ny 
 32njen 
 
 a, UOVV 
 J^S? 
 
 COMPOUKD''¥fe6p(>RTION. 121 
 
 5. If 3000 copies of a book of 11 sheets require 66 
 l^^^^^J P*?^^' how much paper will be required 
 tor 5000 copies of a book of 12^ sheets ? ' 
 
 6. If 12 horses plough 11 acres in 5 days, how many 
 
 will plough 33 acres in 18 days ? ^ 
 
 7. if a person earn 16 guineas in 108 days, how many 
 sovereigns would he earn at the same rate in 270 
 days, 20 guineas being equal to 21 sovereigns ? 
 
 8. If a garrison of 1800 men eat' 100 barrels of flour in 
 
 35 days, how many men Would eat 200 barrels in 
 45 days? 
 
 ^' ?Q^S^^ ^^^' besieged in a town, with provisions for 
 28 days, at the rate 0118 ounces a day for each man 
 be reinforced by 600 men ; how many ounces a day 
 must each man have that the provisions may last 
 them for 42 days ? ^ 
 
 10. If 7 men can mow 84 acres in I^'days of 81 hours 
 each ; how many can be mowed by 20 men in 11 
 days of 7| hours each? 
 
 ? § ™^P ^^^ ^^? * ^^^^^^ 1^^ f*- ^o"g' 3 ft. broad, and 
 4i n 6 m. deep m 9 days ; how many will be required 
 to d g a trench 80 ft. long, 5 ft. broad, and 2 ft. deep, 
 in 5J days? ^' 
 
 If 7 masons can erect a certain piece of wall in 20f 
 days of 9§. hours each ; hQw long would it take 3 
 masons to do 2f of the same work, reckomnff 12 
 hours t( the day? 
 
 13. If 6 iron bars, 4 ft. long, 3 in. broad, and 2 in. thick, 
 weigh 288 lbs. ; how much will 15 weigh, each 6A 
 ft. long, 4 in. broad, and 3 in. thick ? 
 
 How many pounds of thread will it require to make 
 60 yards, of 3 quarters wide, if 7 pounds make 14 
 yards, 6 quarters wide ? 
 
 How much hay will 32 horses eat in 120 days, if 96 
 horses eat 3| tons in 7^ weeks ? 
 
 -"' fr TT'"" ""* F**/ *^^' piiinijnjj a ourraee 2i ieet long 
 and 13f feet wide, what length of surface that is 10* 
 feet wide, can be painted for $31.72 ? 
 
 11. 
 
 12 
 
 14 
 
 15. 
 
m^ 
 
 •^WTHiieric. 
 
 of sugar, weigKrM;*'*""*^ "" «"«« «>« PHc^ 
 io%#f 36 oxen eat the irraHs nf o 
 
 M oxen eat theaaSUr „"S •' h„'' "•"""' «»" 
 J!""i? «»' «>e grass of 20 L7eft in ""^ ,"^y •»«>> 
 '»« the grass to grow nDttb™Ty ?" ^* '"'^- ™PP<«- 
 
 wide, and li ft llf'^"^' «»<"> « ft- long as f»' 
 
 e^V24ft^ C2?T^lSe",';„7??ft"1r ^'" '^' 
 men dig m 81 days, by workW « k ^^^J^' ^an 240 
 20. If the 8d. loaf wei4 5 ?i n, u^ ^ ^f'^ ^^^y^ 
 perquarter : wKii t '^*k^'' ^^^^* ^^ »* «13.50 
 
 deep ; what length oflre^^cMlsr^-' *"*^ ^J 3^d«- 
 4| days of 9S hours lonr7».r; ^^ P^^^eers dig in 
 Ae a'nd 3| /ds deep ? «^' *^' *''"^^ ^^^"^ ^ fds 
 
 '"• ?3ri2^1f f ^2,^^^^^^^^^ miles Of road in 12 
 ^ven month. havrnL^Sf:,^"^^^^^^^^^ i«« ^««- 
 the road have been comTff^ ^u^ ^"* ^ "liles of 
 men must be employed to^^^^^^^^ ^^^. «»^«?^ more 
 prescribed ? *^^ ^ ^^^®^ *^® work in the time 
 
 2aMf 3 men can cradlo Q7.no i « 
 
 hou,. each , how m»; men ^^U o,;^,? .^r »' ?» 
 
 , . "« days, of 10-4 houra each ? ^ ""' """^ » 
 
 ing^ror.\7arh''or^^«"-'"«S.'lays.work. 
 
 ^^ 4}*days, by Sg^Th"j..'j:-:- i° -"mow 
 
 • ,« £"?'?»» l^^el 3*1 miles in 71d,„ ,„-...,_ 
 
''ould f ths 
 i the price 
 
 eeks, and 
 'any oxen 
 'j suppos- 
 
 Ach 
 
 SIMPLE PARTNERSHIP. 
 
 123 
 
 da 
 
 ng, 36 
 
 7' 
 ft. 
 
 that are 
 
 » can 240 
 
 ay? 
 
 It $13.50 
 ^at when 
 
 can dig 
 
 H yds. 
 
 8 dig in 
 
 Hyde, 
 
 d in 12 
 men. 
 niles of 
 r more 
 le time 
 
 , of 7§ 
 ime in 
 
 '* ti. -I't 
 ^ays, 
 3 rein- 
 ; how ; 
 rthat 
 
 work- 
 mow 
 
 Iling 
 
 rate ol 8 tor 6J cents ; how much does he gain per 
 cent, by sellmg them at the rate of 3 for 2^ cents ? 
 
 ^^' Hn^^^^^^t ^^"^®° ^a° ^^ t of a piece of work in 
 
 A T'^^y l^ **®^^ "**°y days will 1 man and 1 woman 
 do f of it ? 
 
 , ., ,«IMPLE PARTNERSHIP. 
 
 pilJP^^ PAETNEiwun' is the method of distributing the 
 Profits or Losses of aFirm, or Company, equitably among 
 the partners, when the stocks or sums contributed by the 
 sevei-al partners continue in trade for the same time; 
 
 The sum of the stocks or shares of all the nartnprs is 
 called the ^ocfc or Capital of the business. ^ 
 
 partnTby Ihl IZ^^ ^^"^^°^ '' '^«*"^^^^^ *^ ^'^^ 
 
 Ruui.-~2he whde capital is to each partner's share of the 
 
 loss "'' ^"*'* '''* ^""'^ ^ '^ ^'^ partner's gain or 
 
 By the rule,— 
 
 whole capital ; A's sbarc : : whole gain : A's gain, 
 or, in this example, $1000 ; $400 : : $540 : A's gain. 
 
 $400X540 
 
 or, 
 
 1000 
 
 ==^216 = A's gain. 
 
 ;tf fs°45o/fnr ?^' '^T' «?«t^t^te his sha^e of the capi- 
 tal (^350) for A's m the above proportion. 
 
 $1000 : $350 : : $540 : $189, B's share of gain 
 and $1000 : $250 : ; $540 : $135,, C's share oigain. 
 
 NoTR-H-In this exaraplo, $ia0O gains $)40. or what in ih« <.«»«« 
 ^'"^K^?^. Pf th«. capital gfins 54 cen\8'HL^toot?naT<5 
 
 Span's SS'f r^ 1« thiB a.Sf,i£Lrjw,de^X ;tt 
 capital and multiply by each partner's share of it. 
 
124 
 
 *><\ 
 
 ARITHMETIC. 
 EXEKCTSE 64. 
 
 ^' B sfso^fvit" '"*v r^'i^'^'^l^'P ; A pays $420 and 
 
 2. A, B and C build a vessel at a cost of $50 000 of 
 which A pays $15,000, B $25,000, and C the re- 
 eTchpl^T""^^ ^"^' ^1>«00.; W muc^of ^hifdJes 
 
 -^tfwh^t"* ""T" ^ ^^^'^ $70;6 $90,*and D $120 : 
 X to each? '"""""* *" ^^' ' ^"^ "^"«»^ "^««* he 
 
 4. A, B, and C make up a capital of $20,OCO • B and r; 
 each contribute twice as much as Afbut Afe to re 
 ceive one-third of the profits for extra sefvi^es. at 
 
 Lla^h't^^teC? ''-' '^^^ ^-^^ ^'^^^T^^^^ 
 
 I2 2oo h; ^ "'^*'" ?? ^ P^y« 1 dollar "heriose 
 $2,400 by the operation ; what is the loss of eLh? 
 
 ^J"'^. partners. A, B, C, and D, shipped 640 sheen 
 of which A owned 120 ; B 80 ; C 200 ; and D the r?' 
 mamder. In a storm 160 of them were thmwt 
 overboard ; how many sheep did D 0^ an&w 
 much was each partner's loss ? ' ^^ 
 
 ^' '^oiP'iP' ^^^J^' hired a pasture for $120 ; A nut in 
 
 8. A, B, and C, purchase a farm for $3500 ; of which A 
 
 "el280'r2n/ ? ^A'^'? ' ^"'^ ^ $50^ Thet 
 JeTsUTearecdve'^ '^'"^ ' ^^" --^ of th^ 
 
 ^' i^oJR^iff' T^^^^""^ ^2 sailors, took a prize of 
 $2240; of which the captain takes 14 shares -thi 
 mate 6 shares ; and the remainder is equX divided 
 among the sailors ; how much did each receiveV 
 
 ''• 1' n^rl,^'.t"!^ 2 ^ ^?^\2l $30,00J) Of which 
 restrthe gin ii Si^T^^l^i^l?^ ^^^ 
 
COMPOUND PARTNERSHIP. 
 
 425 
 
 3 $420 and 
 i^hat is each 
 
 $50,000, of 
 I C the re- 
 •f this does 
 
 dD $120; 
 Qh must he 
 
 ; B and 
 A is to re- 
 r vices ; at 
 )00 ; what 
 
 sonlvibiite 
 ire, uud C 
 they lose 
 of each? 
 
 40 sheep, 
 D the re- 
 
 e thrown 
 and how 
 
 A put in 
 ow much 
 
 which A 
 0. Thej 
 ih of this 
 
 prize of 
 res; the 
 ' divided 
 ive? 
 
 >f which 
 jays the 
 m much 
 
 of this must each receive, a clear salary of $2,047.50 
 per annum being allowed to C for liis services as 
 acting partner ? 
 
 COMPOUND PARTNERSnn*. 
 
 When the stock of the several partners of a company 
 is invested for different periods of time, we are able to 
 distribute the gains or the losses equitably among them 
 by the rule of Compound Partnership. 'f ■ 
 
 Questions in Compound Partnership aw' solved by the 
 following 
 
 Rule. — Multiply the stock of mch partner by the time it 
 is in trade, then as ihe sufn of the products is to each pro- 
 duct, so in the whole gain or loss to the gain or loss of each, 
 partner. 
 
 Example 1. — Three partners, A, B, and C, put money 
 into trade as follows : — A put in $100 for 4 months ; 
 B $300 for 2 months, and C $500 for 3 months. They 
 gained $250 ; how much is each man's share of the gain ? 
 
 A'.^$lO0X4= $400 
 B's $300 X 2 = $«00 
 C*8 $500 X 3 = $1600 
 
 $2500 = sum of the products. 
 $400 X 250.) 
 
 2500 
 
 Then, $2500 : $400 : : $250 : A'a share, or 
 
 ' $40 QO, A's share. 
 
 $600 V 250 
 $2500 : $600 : : $250 : B's share, or ^- = 
 
 $60.00, B'b share. 
 
 $1500X250 
 
 $2500 : $1500 : : $250 :C'8 share, or Z — __<^ =: 
 
 *&ou 
 
 $150.00, C's share. 
 
 It is evident that $100 for 4 months i» equal to 4 times 
 $100 for 1 month, and $300 for 2 months equal to 2 times 
 
126 
 
 If 
 
 ARITHMITIC. 
 
 
 1. A and B enter Into 
 
 EXERCISE 65 
 
 A putff 
 
 for 
 
 - months, ana a »6'in fi^,. ii" 
 with . capital of $2600 ... S'"' ^J""" P«rtnerehip 
 
 ^•To7tra■:^tCr:s3r^'^p^^^^ 
 
 a bill of $10.20. How much I'., u^ ^*^« *<> Pa/ 
 «• A, with a Capital of S^oT i f^ '"'*"' '^"^^ 
 
 and. meetini witS sSs S M? '^""- ^' i«^«^ 
 with a capita! of $60o"on tb; G^^^n?u^ as « partner 
 four months after theradmff A''^^*''^**^«"owing 
 brought $800 stock • ft^Sf 1 ^ ^ * partner, wfio 
 the glin to be $700 • ho^m"; 2^1*^?.^^"' *^«^ ^^d 
 the partners ? "^'"^ ' ^'^^ n^ust it be divided a^ong 
 
 '• rnths'a^dlLL^^^^^^^^^^ '• A put. in $100 6 
 
 4 months, and the^n takes o^t"^^^^^^ ^ P"*« ^° ^200 
 the year thev find thatfhp^h ^80 5. at the close of 
 is the profit of each ? ^ ^^""^ «^^^"^^ ^^^^ ; what 
 
 what ttiffl df^ il "''''.*^-« «»d received «40 fa?" .' 
 
 8. 
 
orefore, we 
 Ino for one 
 ihip. 
 
 1 $840 for 
 ej gained 
 
 iBeiipital 
 
 rtn«r8hJp, 
 
 year the 
 
 puts in 8 
 and C 30 
 pay? 
 
 days; B 
 14 days ; 
 to pay 
 lill? ' 
 
 1, 1846, 
 partner, 
 llowing ; 
 ler, wno 
 hey find 
 • among 
 
 $100 6 
 in $200 
 slose of 
 ; what 
 
 $85 for 
 » for 4 
 
 Al con- 
 gain ; 
 
 oraAv% • 
 
 PRAOTIOB. 
 
 127 
 
 SECTION V. 
 
 PRACTICE. 
 
 Practice is a short method of finding the cost of any 
 quantity when the price of a unit is given. 
 
 An Aliquot part of a quantity is such a part as, when 
 taken a certain number of times will exactly make that 
 quantity. 
 
 TABLE OF ALIQUOT PARTS. 
 
 Parts of a 
 Dollar. 
 
 60 ct8. = $J 
 83J cts. = $^ 
 25 cts. = $^ 
 20 cts. = $^ 
 16f cts. = $^ 
 12 J cts. = $1 
 6\ cts. = $^ 
 6 cts. = $|^ 
 
 Parts of a 
 Poand. 
 
 lOs ==£J 
 6s8d==:£i 
 6s =£J 
 4s =£J 
 3s4d = £i 
 28 6d = £| 
 28 =£^ 
 ls8d = £^ 
 
 Parts of a 
 Shilling. 
 
 6d t=J 
 4d =i 
 3d ==i 
 
 2d =:i 
 
 ljd==^ sh 
 
 Id =^8h 
 
 id=^8h 
 
 sh. 
 sh. 
 8h. 
 sh. 
 
 WQigbt. 
 
 2qr.=:^ cwt. 
 
 lqr.=-|^ cwt. 
 161b.* = 1^ cwt. 
 14 lb. = I cwt. 
 
 8 lb. =^ cwt. 
 
 7 lb. = j^ cwt. 
 14 lb. = J qr. 
 
 71b.==J qr. 
 
 * That is taking 28 lbs. for a quarter. 
 In solving questions, the pupil may, (in order to find 
 out the aliquot part that he can most advantageously use), 
 divide £1, $1, 1 acre, &c., (as the case may be), by 2, 3, 
 4, Ac., and select from the quotients, that are free from 
 fractions, the numbers that will suit the case. 
 
 Example l.—What is the price of 78 cwt. of sugar at 
 £Z per cwt. ? 
 
 78 cwt. @ £3 per cwt. 
 
 £*S ;^ price of 78 cwt. @ £i per cwt. 
 
 ^ ' 
 
 £234 = price of 78 cwt. (^ £3 per cwt. 
 
^^smm* 
 
 Wf 
 
 ;l i 
 
 128 
 
 ARITHMETIC. 
 
 Here the price of 78 cwt. at£I is evidently £78 and at 
 
 68 8d.==-U£ 
 
 £60^priee_of60ft. @£i per ft. 
 
 £20 = price of 60 ft. @ Cs sTJer lb. 
 
 each part separately, will U tttrice r^^l'^^^ ^^ "«'' ^^ 
 ExAMPi^ 3.-What cost 796 cows at £9 12s. 6d. each ? 
 £ 796 = price of 796 cows at Xl eacb. 
 
 iQe. - >^ of £1. 
 2s 6(1. = 1^ of 10s. 
 
 £7164 
 398 
 9910 
 
 
 
 (( 
 
 £9 
 10 
 2 
 
 each. 
 each. 
 6 each. 
 
 £7661 10 « " « u £972~6> ■ 
 
 Of tchtTex^;;^^^^^^^^^ (f well as the price 
 
 which ct, w?p^;S as'fon^^f denominations ; in 
 
 10| cwt. 2qrs. 21 lbs. at £2 4s. 4d. per cwt! 
 
 4b. » 3^ of £2. 
 4(>. =5^ of 4s. 
 ^^S*". "=>^ofcwt. 
 14 b. «.i^of2qr. 
 7 lb.=: J^oflAlh. 
 
 £396 
 
 39 12 
 
 3 6 
 
 1 2 
 
 6 
 
 £440 8 5^ 
 
 - value of 198 cwt. at £2 p. cwt. 
 — '< ti u 4g * n 
 
 — " (« i« 4j" „ 
 
 2 - value of 2qrs. at £2 4s. 4d. p.c. 
 5^ — ■ " 14 lbs. " " " ■ 
 " 7 Toa. 
 
 A IV 
 
 <( (( 
 
 (( 
 it 
 
 <( 
 (I 
 
 Price required. _ 
 
PRACTICR. 
 
 129 
 
 Or, it might have been solved more 
 
 = value of 1 cwt. 
 
 2qr8. -= >^ cwt. 
 14 lbs. = }i of 2 qrs. 
 7 lbs. «^ of 14 lbs. 
 
 £ 2 
 
 simply, thus 
 
 198 
 
 £4a8 18 =« value 
 
 12 2 -« <( 
 
 5 6>^ -. «« 
 
 2 9}-;i =. t( 
 
 of 198 cwt. 
 2 qrs. 
 14 lbs. 
 7 lbs. 
 
 £440 8 
 
 5% =- Price required. 
 
 r I II i 
 
 EXERCISE 64. 
 
 i' ^^* ^®^^' ^^^ ^^^- <^^ ^^ at 6s per lb. ? 
 
 Q Sw ^®^! ?5 ^^*- **^ ^'^Sar a* ^2.90 per cwt. ? 
 3. What cost 120 sheep at £1 10s. each ? 
 
 A ^^l '? *^^ ^"*^ ^° *,^^*- of opium at 8s. 6d. per lb. 
 '• m*6^%*eVardr' ''' ''''' ^^ ^^ ^^otE at Si 
 
 ""• "^^A'Xtll^'J^lr'' ' ^- 12 1^- Of currants 
 8. What cost :)lb8.4oz.6dwt.ofgoMat£312B an n n» 
 
 ^ii^*?* **"' "'° *°- ' '<~*' -^^ "*' »'^^^^^^^ 
 
 10. What cost 9 tons 13 cwt. of steel at £5 198 fid n t/^n? 
 1. What cost 1 yd. 3. qr. 2 nl. of linen a? $1 3:5 p^'yd ? 
 1 q- ^1 '^?i ^^^t lbs of sugar at 13 cente per lb! J 
 13. What IS the pnce of fencing 90 miles 3 fiir. 20 ohs of 
 , . ^fl^^^y^ *^* ^^^^ 9s. 4d. per mile ? ^ "^- ^" P^«- of 
 
 piTacre*?^ ^"'^ ""^ ^" ^'' ^ ^- ^^ P' ** ^^ l^^' 6^' 
 
 ^^* Jerlon?^' ^"'^ "^^^^^ ^"' ^® ^^*- ' ^»' ^' ^« 138. 
 16. The Sun's apparent motion in the ecliptic is 59' 8"-3 
 17 If fay-how much is it in 10 days 7 hs 20 min ? 
 
 1ft' WKaJ r tu^ P^°^ ^SS^f 7^ ^* 13s. lOd. per yd ?' 
 18. What is the pnce of 79|lbs. at £1 2s. lid ner ib ? 
 
 -hoV much in ioj daysl ^'"' '^ '''* "' ^^* P"' ^''^ 
 20. What cost 96 o. yds. 18 c. ft 56 c in. of stone at 12* 
 cents per c. yard? *** 
 

 laii 
 
 ARirUMBTIO. 
 
 I- 
 
 li: I 
 
 SECTION VI. 
 
 ^^AnS^^x?.^^^' COMMISSION, BROKER. 
 AGE, INSURANCE, PROFIT AND LOSS 
 INTEREST, DISCOUNT, AND BARTER ' 
 
 PER-CENTAGE. 
 
 dr Jd " wht« 7i!^'^t ^ "^^ ^^\'^"'' °^^^^ "' by *he hun- 
 dred, when, therefore, we speak of so much " per cent " 
 
 we mean so much in the hundred. If 3 persons W of 
 eyery hundred died in Toronto during the Zr the 
 mortality would be expressed as 3 per cent. ^ ' 
 
 The rate per cent, may be conveniently expressed in 
 decimals to show the rate per «n«,- for eLmpS, % per 
 cent. 18 6 per hundred, or -06 per unit, (the rate per uiUt 
 being of course the 100th part Sf the rate^per hund^i^)^^) 
 2 per cent, is -02 per unit. | ^^ per cent. Is .0376 per unit. 
 
 8 
 
 2>^ 
 
 (I 
 
 is -OS 
 is '026 
 is -0425 
 is 0025 
 is -005 
 
 (( 
 
 (C 
 
 X 
 
 f 
 
 ii 
 
 II 
 l( 
 << 
 
 (< 
 
 is -06125 
 is 0462* 
 is -016 
 is *0075 
 is 004 
 
 
 Example 1 —I bought a house for $800 and paid 5 per 
 cent, down, how much did I pay ? " ^ pw 
 
 ^^.\ ?*.? ? ^V^ ^^°*- ^^^ » ^a' is $5 on every $100 
 that is the hmh part of $5 for every dollar = $Ti?7= '06 
 
 Jnr/n" ^!^. * ^^"^^ b« 800 times as S aa for 
 ?nTfiS*'k*^^* ^ '^^ X 800 = $40.00. Hence the rule 
 for finding the per-centage on any quantity is 
 
 ^^Xt^J,iC^ 2'''VViy'^'' ^'^"^^"yY^"'^ ^^^ P^ Mw»f' expressed 
 deoimatty, and the product toUl be theper^miage required. 
 
LOSS, 
 
 the hun' 
 r ceaV* 
 18 out of 
 ^, the 
 
 )ssed in 
 ', 5 per 
 )er unit 
 red), so 
 
 er unit. 
 «{ 
 
 « 
 
 iSper 
 
 = •05. 
 
 as for 
 
 le rule 
 
 >re88ed 
 Hired. 
 
 PEROBNTAOK. 
 
 Example.— What is 6 per cent, of $618.25. 
 $61826 
 
 •06 rate per unit. 
 
 37'0950 per-centage, or answer. 
 
 131 
 
 EXERCISE 65. 
 
 1. What is 35 per cent, of $14720 ? 
 
 2. What is ^ per cent, of £240 lOa. ? 
 
 3. What is J per cent, of $94.48 ? 
 
 4. What is 9^ per cent, of £10 lis. 2d. ? 
 
 5. What is 3 per cent, of 10 tons 5 cwt. 2 qrs. ? 
 
 6. What is 99 J per cent, of 60 acres 3 roods ? 
 
 7. What is 1^ per cent, of $1 ? 
 
 8. In a school there are 340 boys and girls, 40 per cent, 
 of the whole are girls ; how many boys are there ? 
 
 9. A merchant bought 1000 brls. of flour aud sold 10 
 per cent, of them, afterwards 10 per cent, of what 
 remained, and again 20 per cent, of the remainder : 
 how many barrels had he remaining ? 
 
 10. A gentleman's income is $5000 per year ; he spends 
 
 6 per cent, of it as personal expenses, 30 per cent, 
 for household expenses, 38 percent, for rent, &c.,Soc., 
 and invests the balance ; how much does he invest ? 
 
 11. The deaths in England are about If per cent, per an- 
 num of the population: if the population is 18,000,000, 
 what is the number of deaths annually ? 
 
 12. The copper mines of Lake Superior produce ore coii- 
 taining 70 per cent, of pure copper : how much cop- 
 per is there in 1260 tons of ore ? 
 
 - »»v> 
 
 -f-T^ 
 
 COMMISSION, BROKERAGE, AND INSURANCE. 
 
 Commission is the sum, at a certain rate per ceni, 
 which is charged by one person for his services in buying 
 or selling, &c., for another. - . v* , 
 
 ,irivr.-iasc 13 vr mu samo uaturo, oeing a per-eeni«ge 
 paid to a broker for negotiating notes; bills, buying and 
 selling stock, &c., &c. 
 
il 
 
 'f. 
 
 132 
 
 ARITHMETIC. 
 
 per-centaire hrlnnihl ^ ^ P**? * certain premiun or 
 
 *c. to W wLtevS am?^T?' /° ^?^ «^ »««« V fire' 
 surince. ^ ^^atever amount of loss is covered by in- 
 
 i>.-5"S-5SS^ *** 0"^ -•» «* "- >■«'« per unU «-' 
 pcf^rf ';~^''*' ^ "■« ""'""■teion on $642ft_«t5J 
 
 6i per cent = -055 per unit 
 $6420 
 •055 
 
 32100 
 32100 
 
 $363100 
 
 miSoTon 1^5^21 1? f pVrtI ^' ^ '^'^^^^'^ -- 
 
 i per cent. = -00125 per unit. 
 $62530.20 X -00125 = $78-16275 
 
 $« P«id M BMmi^™ ?^' H' ■" 5*» evidenay lest the 
 that in case of toS boft Si^^i^^'/S "«'"«" »"".«> 
 
 /'EuLB.—;8iiWm<rf <jy.fv«/4 A,As« AiAA ^i_ .. 
 
 «^«? ««u. 0/tte ;;„^ W iSSli' (r^i'S^ISt 
 
 i 
 
^ (an In- 
 emiun or 
 B by fire, 
 id byin- 
 
 , then, it 
 -centage 
 
 "ance, or 
 'en sum, 
 
 unit ex- 
 2fiLat5^ 
 
 J com- 
 
 1 loss, 
 value 
 licy," 
 
 good 
 nt., in 
 Btthe 
 
 how 
 m, so 
 rand 
 
 inihr 
 
 COMMISSION, BROKERAOB AND INSURANCE. 133 
 
 i.J^^ ?r"5^^ ^""^^ °*"** ^® insured at 6 per cent, 
 on $120, so that m case of loss, not only the value of the 
 property, but also the 2>rmmm may be repaid ? 
 
 OPERATION. 
 
 100 --6 = 94. 
 
 Then 94 : 120 : : 100 :' Jns. 
 $127.65|f. Ans. 
 
 «r>Tw^'^®™''^ .^."^ $127.65|f , at 6 per cent, is $7MU 
 
 wJi f '1 *'*'^, ""^ '^^' *^« «^«' i« paid the premium i 
 well as the value Of his property. "i'"m as 
 
 EXERCISE 66. 
 
 1. What is the commission on £90 12s. 3d. at 8 per cent? 
 
 2. What w the premium of insurance on $1000 at If per 
 
 3. What is the jrokerage on $99,999.80 at fg per cent. ? 
 
 4. What is the commission on £942 16s. 3d. at 4^ per ct. ? 
 6. What is the commission on £946 18s. lOd. at ^J per ct.? 
 6. What is the premium of insurance on £1486 13s. 9d. 
 
 at 2| per cent. ? , .* 
 
 ''• Y^jt must be insured at 5f per cent, on £1938 128. 6d.. 
 so that m case of loss, both the value of goods and 
 the premmm may be repaid? 
 
 8. What is the commission on $90 at 17^ per cent? 
 
 9. A broker sells stock to the value of $87,634.75. what 
 IS Ins commission at | per cent. ? 
 
 10. Add to $250 the commission on iteelf at 8* per cent., 
 and find the insurance of the sum at 4 per cent ? 
 
 11. What is the brokerage on $7450 at 2^ per cent. ? 
 
 12. A fector sells 75 bales of cotton at $450 per bale, and 
 18 to receive 2 per cent, commission ; how much 
 money must he pay to his principal ? 
 
 13. An agent bought goods amounting to $2^6 : what 
 IS his commission at 2| per cent ? > «•- 
 
 ^** «fl?*^-^"'S'-?^H,* ^'^^^^'i percent to invest 
 $8450 m United States stocks : how much was the 
 brokerage ? - 
 
184 
 
 ARITHMETIC. 
 
 16. An attorney collected a debt of $3476.50, and charged 
 7t^ per cent, commission ; how much did he receive? 
 
 ^^* ii/S!2°'h\'''' °^*^^<^hant sold goods to the amount of 
 $4536 at 2J per cent. ; what was his commission ? 
 
 17. What is the commission for selling dry goods at 4 
 per cent, to the amount of $746, and groceries at 3 
 per cent, to the amount of $542 ? K^^^eries at d 
 
 18. A commission merchant sells $436 worth of drv 
 goods at 3 per cent., and $458 worth of paper at 2* 
 per cent ; what is his commission ? ^ 
 
 { PROFIT AND LOSS. 
 
 nfl,miw5ft!.l*i''°'i° per-centage as the following, are 
 tisually placed under the head of Profit and Loss. 
 
 o^i??;^^ ^-""^ ^ ^"y c^^*^ a* 60 cents per yard, and 
 sell it at 75 cents ; what do I gain per cent. ? ' 
 
 ANALYSis.--If I gain 16 cents on 60 cents, (which I do 
 
 2S^^t.^:Z^!^1 "' ??. «™*'' »"« » celling, gain 
 ^ per cent ; what a the selling price ? 
 
 lio^i^l^'^~^ ".^'^ *" '*"'»' ""^ gained 26 per cent of 
 S.CX'TheLr'" '• "«■"'*• '"• + l«'-7«een?: 
 
 giru^'S.^d'ri'^efcSe :£'^ "' •°'^ "«' -»'• - 
 -{.^f^rf^f.'** ix^"?^'*^* on <A« jSwn cast, at the 
 
 
i charged 
 ! receive ? 
 
 imount of 
 Ission ? 
 
 >ods at 4 
 eries at 3 
 
 i of dry 
 per at 2\ 
 
 ingj are 
 
 s. 
 
 trd, and 
 ich I do 
 
 iug and 
 
 and the 
 m to tht 
 
 ?, gain 
 
 2ent of 
 > centi 
 
 nt. are 
 
 ai tht 
 8j sub- 
 
 BARTBK. 1^5 
 
 Example 3.~If I sell at 75 cents, and gain 25 per cent. : 
 what was the coat price ? 
 
 Analysts.— •0126 cents of selling price is equal to -01 
 cent of buying price ; hence, 
 
 When the selling price and the gain or loss per ceiU. 
 are given to find the 6uj/ingf price : — '*> 
 
 Rdle.— Diuide the selling price hy 1 plus the gain per 
 unit, or, {if loss he given), by 1 minus the loss per unit, ex- 
 pressed decirmtty, and the quottent wiU be the buying price. 
 
 Example.— '75 -i- 1-25 = -60. 
 
 EXBRCKB 71. 
 
 1. Bought iron for $980 ; for how much must it be sold 
 to gain 12 per cent. ? 
 
 2. Bought 50 tons of steel, at $45 per ton ; how must it 
 
 be sold per ton to gain 3 per cent. ? 
 
 3. By selling su^ar at $8.75 per cwt., I gain 22 per 
 cent. ; what did it cost me ? 
 
 4. By selling flour at $5.50 per barrel, I lose 8J per 
 cent. ; what did it cost me ? 
 
 5. If I sell 140 bushels of wheat for $350, and thereby 
 gain 24 per cent. 5 for how much should I have sold 
 it a bushel, to lose 20 per cent. ? 
 
 6. If I sell 30 yards of broad-cloth for $132, and there- 
 by gain 10 per cent. ; how ought I to sell it a yard 
 to lose 25 per cent. ? 
 
 7. Sold 15 boxes of damaged raisins for $34.50, which 
 was at a loss of 8 per cent. ; how should I have sold 
 them a box to have gained 33^ per cent. ? 
 
 8. A merchant sold two boxes of goods for $60 a-piece 5 
 on one he gained 20 per cent, and on the other he 
 lost 20 per cent. ; did be gain or lose by the opera- 
 tion,' and how much t 
 
 »»« 
 
 BARTER. ' 1 
 
 The method bf' which two nai-tie^ can e^nbanffA i*ac.{is 
 
 at certwn prices' without loss to either, is caliedlBarter. 
 Thus, if I had 20 lbs. of tea wortti $9, and wished to 
 
 exchange it for sugar, at the rate of $18 per cwt. ; I 
 
136 
 
 ABITHUKTIC. 
 
 R^"'7'^f ■" '" ^'"' *''« '•""""ing fa the '' 
 
 ft« otflo-, and the qmlkJ^aiJ^L **" ^ .'*< ?«o»«% o/- 
 a«d the quMk„t Smqmnt^.'^' " *«*««*!"*« P^«, 
 
 «.fc?elveI?2?ySSfofS ".'.J' '?•» P""" ">• 
 •16 y 20 — *<i nn - • ? ''^ "*"•* ?<"• yard ? 
 
 lbB.,Vaa?u7ot;4P,''°* »f o-". ■>»<■' $3.00 ..'laLls 
 
 EXERCISK 72. 
 
 per lb., for wheat ft ^'inin? ^/ I*'* ** ^ cents 
 bushels did hTieceivef ^ ^ ''"'^^^ ' ^^^ ^m 
 
 '• arf/re'coivc^^^^^^^^ ff ^^ c-ts a volume, 
 
 per ib., 40 lbs. of suiar at n / .^^*^' ^^^^ cents 
 
 balanc^ in tea at 4 Ste ner lb I"^ ^'^' ^^- ^"^ t^« 
 he receive ? ^^^ ^^- ' ^ew many lbs. did 
 
 '• w^te^Jl^^? -^ a farm of 80 acres, 
 how many acres d^tT^ctive"?' ^^^'"* *^ «» *^^« \ 
 
 - returi at ^Cii^H^J^^^"* "^"^* ''^ ^-^^ive in 
 ^. ±low njanjr l^orses worth $65 each nnn t « 
 
 a farm of 200 acres 2 roods, ^th $1? jKcre'l ''' 
 
 INTEREST, 
 per annum. ^ reckoned at so much per cent. 
 
 ifi© Sum paid for everr hnna^^ dsSh. 
 
 tf«* wtuti ra(e 
 
 i>«* <?e>»^. 
 
 
SIMPLE INTEBES8T. 
 
 18t 
 
 It' 
 
 Thus, if I borrow $600 dollars for a year, and airree 
 
 ?n & *^' ^ ^"^^ "^v* ^°^y ^^^' »^»* aJ«o i^36 for inter^ 
 In this example we have,. «*•»?* «f%. 
 
 , f 600 as the Principal. 
 
 J6 " Rate per cent. 
 
 966 « Interest. 
 ^ $636 '' Amount (Principal + Interest). " 
 
 ♦K^^i"^^-®"' .^* i^® *^°^ ^^ » ?'*a*<?fl period, the interesi 
 ^Ztlt '' !^^'^ ^ ^^\ principal; and thi^ amoSit^ 
 cotoesthenew pnncipal, and so on, ^t the exniraSon^ 
 
 SLresT ' ^'^^^^^ ^^'^^^^g P«id^ this is calledXmpoSnd 
 
 ,i-.'-f 
 
 SIMPLE INTEREST. 
 
 number" o/yL^'''"^ '^ ' ^^^'^ ^""^ ^^^ ^^^ ^'^' o^ "V 
 
 ^um^—Mvltiply the principai by the rate per unU tad 
 the product by the time. ^*^y^ unu, <ma 
 
 Example. —Required the interest of $240 for 3 years at 
 o per cent, per annum ? ' 
 
 $240 = principal. 
 •05 = rate per unit. 
 
 $ 1 2.00 =: interest for 1 year. 
 3 
 
 ;$3^.00 = interest ^l^ a yearg. . 
 When the iime is not an evenfitlmber of years, but i^ 
 eludes months and weeks, &c., it will geneklly be foimd 
 most^conyement to get the interest for the rnoVitl.. S™ 
 
 *»Ai., uy rracfcice. 
 
 -^•^ 
 
 For example, if the question is, what is the interest 
 
 of 
 
^^ ARITHMETIC. 
 
 Of $400 for 2 years, 8 monthi, md 
 per annum ? 
 
 6 months = J year. 
 
 2 months ==^6 months. 
 2 weeks = J of 2 months. 
 
 $400 
 •04 
 
 > weeks, 
 : principal. 
 
 4pero€it 
 
 $16.00 = interest for 1 year. 
 2 
 
 $32.00 = 
 8.00 = 
 
 I $43.33| 
 
 tt 
 
 H 
 
 « 
 
 2 years, 
 'tmos. 
 2mo8. 
 2 weeks 
 
 $300 
 •06 
 
 $18.00 = Mterest for l year. 
 
 2 
 
 $36.00 = interest for 2 years. 
 
 Then to find the interest for 90 days, 
 365 : 90 : : $18.00 
 18.00 
 
 mm $40.43 = Ans. 
 
 1400 
 1095 
 
 305 
 
 !► 
 
8IMPLK INTEREST. 
 
 id9 
 
 BXBRCISB 67. 
 
 I' 3?*' }^ ^^® interest of $908.20 for 1 year at 3 pr. ct.t 
 ' a' 2?** 1* *^® interest of $624 for 2 jears at 8 prf cent. ? 
 
 3. What i8 *he amount of $400 for 4 years at S per ct. ? 
 
 4. What is liie interest of $643.70 for 6 years at 8 per 
 cent. 7 
 
 6. Wliat is the interest of $2124.84 for 9 years at iA ner 
 cent? ^ *^ 
 
 1 What is the interest of $5347.62 for 6 years at 6 per 
 cent. *^ 
 
 7. What is the interest of $3217.68 for 11 years at 6 per 
 
 8. What is the interest of $8922.60 for 7 years at ^ per 
 cent. T 
 
 9. What is the interest of $4159.71 for 13 years at 6 ner 
 cent? ^-"^^ *^ 
 
 10. What is the interest of $9754.32 for 8 years at 7 per 
 cent. ? "if 
 
 11. What is the interest of $9112.27 for 12 years at 8 per 
 cent. • 
 
 12. What is the interest of $3V65.38 for 10 years at 7i 
 per cent ? •' a 
 
 13. What is the interest of $80 for 12 years 3 months and 
 
 20 day^s at 10 per cent ? 
 
 14. What IS the interest of $1000 from March 1 to Jan. 9 
 at 9| per cent ? ' 
 
 16. What is the interest of £584 18s. 8d. for 1 year and 9 
 -, months at 3| per cent. ? 
 
 16. What is the amount of $326 for 2 years 5 months aj. 
 4| per. cent. ? 
 
 17. What is the interest of $162 from Aug. 24 to Jan. 1 
 at 6 per cent ? ' 
 
 18. "What is the interest of $1 for 80yeara 6 months at 15 
 per cent. ? 
 
 19. What is the amount of $2800 for 89 days at 6 per ct ? 
 
 There are five things in Interest— the Prvn(^cd, the Hixte 
 pw cent., the Time, the Interest, and the Ammnt — any ihf-ee 
 
 ttf 'which b**'*^** rrirran tJ"-^ i»*lw».. i-sr" Tsva— *w- " _ ' •-• 
 
 most questions, the principal, rate per cent., and Hme are 
 given to flad the interest, as in the last exercise ; but we 
 
140 
 
 ARITHMRTIO. 
 
 I ' 
 
 .SSi,rUmr ''^"^'° ^ «"^ ^« P^'-^'P'^^ rate por 
 iateL"?ar?gl?^^^^^^ ^^^" ***^ '^"^ P^' «««t' ^Imo, and 
 
 $1 at 3} per cent, for 4 years r-, -14, and 
 127.40 
 .J = ^^10, required principal. 
 
 •rJgiven,'^^ ""*"' ""^^^ *^' P"'°"^P*^' ^"^^^ and rate 
 
 > ^l^r^" ''^** ^'^^^ """^ ^240 give $72 interest at 
 The interest on $240 at 5 per cent, for 1 year = $12 and 
 
 $72 given interest ' 
 
 = 6 years. 
 
 $12 interest on principal for 1 year. 
 ^ To find the rate, the principal, time, and interest being 
 
 ^RDLE.—I>iuf(ie the given interest hy the interest «n /A* 
 mtru una be the required rate per cent. ^'^^vo 
 
 ^Slf?'^""^' y^^^i '^*® P^^ cent per annum will 
 »248 in 7 years give $86.80 interest ? 
 
 «*^- $lT36?° ^^^ ^'''*'" principal at 1 per cent for 7 
 
 $86.80 given interest 
 TTT — = 5 per cent required rati- 
 rnM iDt* ,Aston given principal U 1 dm- 
 cent for 7 years. *^ 
 
 if 
 
COMPOUND INTRKE8T. 
 EXKUCI8K 68. 
 
 141 
 
 1. What sum will give $25 98 of interest in 7 months at 
 6 per cout. per annum ? 
 
 2. In what time will $893.56 give $44.68 interest at 6 
 per cent. ? 
 
 3. At what rate will $856.84 in 4 years 9 weeks and 12 
 days give $204.93 interest ? 
 
 4. A gentleman lent $2000 and received for interest 
 $676 ; how long had it been unpaid, computing in- 
 terest at 7 per cent, per annum ? 
 
 5. I lent money for 6 years at 5 per cent., and received 
 
 as amount due at end of that time $1040 : how much 
 did I lend at first ? 
 
 6. Required the rate per cent, at which $100 will a- 
 mount to $200 in 10 years ? 
 
 COMPOUND r ^iEt,T. 
 
 Money is lent at Compound Interest when interest is 
 charged not only on the original principal, but also on 
 the interest that remains unpaid. 
 
 For EXAMPI.E.— If I ? <mtow $100 at 6 percent, per 
 annum, at the end of a year I owe $105 — the $100 bor- 
 rowed and $5 for interest ; if I do not pay the $5, at the 
 end of the second . ear, I owe $105, and interest on it for 
 a year, or $5.25, in all $110.26 -interest being charged 
 not only on the principal, but also on the $5 of unpaid 
 interest. 
 
 To compute the Compound Interest of a given sum 
 for any number of payments, 
 
 RvLR.—Mnd the interest of the given sum for the first 
 period, and add U to the principal. Take this sum as ths 
 principal for the next period and find the interest thereon, 
 and add to U the principal used for that period ; proceed in 
 this manner viUh each period of the given time, a>id the last 
 resiui tviicoe the anwunt of the principal for the given timet 
 from which, of course, if the principal be subtracted, the 
 rmiainder is the interest for the proposed time. 
 
I^^j^-m*mit^m'mt>miffmm- - 
 
 142 
 
 ARITHMETIC. 
 
 $960 = principal for first year. 
 
 ^^ == interest " « « of « «« ^ 
 iTnftfi • . at 5 pr. ct. 
 
 An 4ft "" .P^n^'^Pal for second year. 
 O0-4Q = interest " " • " 
 
 Add 48 
 
 « 
 
 « 
 
 ^^^sI'q? ^ P?'^^.'*^ ^^^ ^^d year. 
 ^^♦■^^ = interest " u * i. 
 
 a 
 
 Add 
 i Add 
 
 Subtract^S^ ^°^"^* "^ "^ "^^^^^ y^^^> 
 $161.32 Ans. 
 
 more convenient :— ' ^"* "** found 
 
 Mnd the amount of $1 far one ncrimi nt yi.^ • 
 per cent, and mvit4>ly itZ itsdf^r^ei^J^ f^ ^'T' »•«'« 
 
 l.oi"ri!^' « at a^p. cenMs .U5 x 1... X 1.05 X 
 
 o ^x X ^765.600 amount of $600 
 Subtract 600 - 
 
 $165,600 Ans. 
 
 1.0« X l.«« = tUOM. 
 
coirpoinn> interest. 
 
 143 
 
 $600 
 
 A^, to find amov.nt at the end of the third period, the 
 principal being now $1.05 X $106, we have 
 
 $1 : $1.05 : : (1.06 X 1.06) : amount of third period, 
 or, 1.05 X 1.06 X 1.05 ^ 
 
 ■ ^ /^ =$1.05X1.06X1.06= « 
 
 By proeeeding in the same manner, we find that the 
 amount of $1 for any number of periods, is equal to the 
 amount of the first period multiplied by itself one time 
 leas than the number of periods. 
 
 EXBRdSB 69» 
 
 Find the amounts of the following sums at the civen 
 rates per cent b "'t* 
 
 1. $500 for 3 years at 7 per cent per annum ? 
 
 2. $840 for 3 years at 6 per cent per annum ? 
 
 3. $880 for 6 years at 2 per cent per annum ? 
 
 4. $3056 for 2 years at 6 per cent half yearly ? 
 
 5. $3 for 10 years at 4 J per cent per annum ? 
 
 6. What is the interest of $2615.20 for 6 years at U ner 
 cent per annum ? '' *^ 
 
 7. What is the interest of $900 for 2* years at 6 net 
 cent half yearly? *^ 
 
 8. What is the interest of $100 for 3 years at 6 per cent 
 quarterly? ^ 
 
 9. What is the interest of $818 for 7i years at 4Der 
 cent half yearly? a j »i8 »i4per 
 
 10, What is the amount of $2329.50 for 5 yeani at 5^1 nev 
 cent per annum ? ^ ^^ 
 
 li. What is the compound faiterest on $200 for 8 years at 
 6 per cent., allowing interest to be due annually ? 
 
 12. What is the compound interest on $1000, for 2 y^ars 
 at 8 per cent, allowing interest to be due half 
 yearly ? 
 
 1 3; What would $500 amount to in 6 years, at 6 per cent 
 Interest, if the interest be comp&mded annually ? 
 
144 
 
 ABITHMETIO. 
 
 DISCOUNT. 
 
 Wed that the L;T;^J«?J^«,*»f «^-onbein^ 
 
 sum which at so muchDer c6n?^^,^iS**'" ^^"""^ '' "*« 
 
 named m the notT iX tofwS'fsTZ' befolT 
 note becomes due • thus if -nte,^^/^- lo lun Defore the 
 fn ««^ ""^«^ , , ' ''"'*s, ir iiitereat is at 5 ner cpnf «inft 
 in one year ^ould amount tn <!Hin« c^ Zl /w t^ent., *iuo 
 
 nnip, fnr "Klft;; "i aiuouni 10 *105, SO that the Value of ft 
 
 fhrft^'^df,isS'„fr„^\T$?*-''- « »-% f'o'J! ^ia 
 
 charge as discount 5 Ur cent ^«i^"'^®/-'J*y.'''>">^ 
 of $3, the true discount «10o-$:..2S, instead 
 
 sent worth, and $127.05 ~ Si 21 -««;;:^ 5-^^' *?^ E®" 
 
 araount of $1 for the time be nJjTol*. hfi'S?"S^« J^*»® 
 drawn fm* «i o^^ ^f^ * "^'^"s^'-Oo ,* had the bill b«e& 
 arawn toi $1.0o. of course its present worth woHld h«^«i 
 but It IS drawn for $127.05. Therefore * ' 
 
 $'-05 : $127.05 ::$i present worth, or 
 $127. 05 X 1 $127.05 
 
 — = $121, present worth. 
 
 1.05 
 
 1.05 
 
 .■TKa l?^y,T^ at 1 
 
 i¥ tr.S^CTL"" '*^'^»'»p« on any aura i» ju8t the interAMt nn 
 it for the time meationedJ,efore it is die | but on iTbilS 
 
DISCOUNT. 
 
 145 
 
 3 days o/§rrace are allowed, which bankers always add 
 m computing discount. A bill dated let January, and 
 drawn at 3 months, is due on 4th April, not on the first. 
 
 .,l^^^^"^.-~^^** ^^ *^® ^a°fc discount on a note for 
 55120, due m 6 months at 5 per cent ? 
 
 Interest of $120 for 6 mo. at 6 p. ot. per annum is $3.00 
 Add mterest of " 3 days " « •' .949 
 
 Bank discount is 
 
 $3,049 
 
 exekciSe 70. 
 
 1. What is the bank dip ount on a bill for $2196 for 32 
 days at 6 per c - per annum ? 
 
 2; What is the tr * discount on a note for $370 for 3 
 months at 7 p«i cent, per annum? 
 
 3. What is the true and the bank discount on $99.20 for 
 
 1 year at 5^ per cent. ? 
 
 4. What is the tme present worth of $139.81, due 10 
 years hence, at 5J per cent. ? 
 
 6. In the last example, reckoning bank manner of dis- 
 count, what is the present worth? How much 
 more than the true discount does the banker charge ? 
 
 6. I discounted at the bank the following bills, at the 
 rate of 8 per cent, per annum : one note for $1000 
 due m 6 months, two notes for $500 each, due in 70 
 days, and one for $320, due in 2 years ; bow much 
 did I receive in all ? 
 
 7. What is the discount on a note for $911.40, drawn at 
 6 months, at 8 per cent. ? 
 
 8. What is the discount on a note, dated March 26th 
 and due June 23rd, for $897.20, at 3f per cent. ? ' 
 
 9. A note of $1651.60 is due in 11 months, but the 
 person to whom it is payable sells it with the dis- 
 count off at 6 per cent. ; bow much shall he receive ? 
 
 10. C owes D $3456, to be paid October 27th, 1842 0. 
 
 D consents 5 how much ought I> to receive, interest 
 at 6 per cent. ? 
 
is*%««#); •<'{ 
 
 146 
 
 iv ; r, 
 
 ARITHMETIC. 
 
 SECTION V. 
 
 INVOLUTION AND EVOLUTION. 
 
 When we mnltiply any number ly UsOf any number 
 of times, Ae several produete are caUed po«^., Tfte 
 
 duct fa 16, ap<m»r of /<«.., if „„,ap„ed ^j '^^ f » 
 4= 64), the product 64 b another potoerof four 
 
 po^,fa"rdtl,'^ "'""'■^"''^«- ""<"- "■« 
 to the e»mple taken above, 4 fa the w( of 16 and 64 
 
 • JJT"*^' "^ '*"*^ •*■*'' »««>^. «>M, *c., accord ' 
 mg to the number of time, the „«< fa taker.; air 
 
 th^Vo'f th'"'!!'' P*^ ""*" *■' '«-«• -<• *>- 
 
 kuuu power tne owdc. Powers are often indicated hxr 
 ^ting after ite number, and a little higher thTn^ber 
 ^t 4? T^r"'" Of times it is tflcen as aXt" 
 thus, 4«== second power of 4 ; 7« =sixth power on" 
 this small figure is called the i«W or e^^Ju ' 
 
 -ae process of finding a power of a number is called 
 Invduiion, and the i,roc««, nf ««^:-^\.— - 
 
 number is called M,ohUu^i "^'^" "^^ '""*' "'" * 
 
 
EXTRACTION OP THE SQUARE ROOT. 
 
 147 
 
 ON. 
 
 ' number 
 rs of the 
 the pro- 
 *X4X 
 
 ices the 
 
 and 64. 
 
 accord- 
 factor ; 
 i power 
 
 and the 
 itedby 
 lumber 
 factory 
 r of 7 ; 
 
 called 
 '* of a 
 
 INVOLUTION. 
 
 To involve a number to any required power, 
 
 RcLJB.— Tafce the number as a factor, as often as is inr 
 dtcafcd oy ihe index of the power.-, the contimed product 
 of these factors is the required power. 
 
 Example 1.— What is the 4th power of 6. 
 
 64 = 5X5X5X5 = 625 ^n». 
 
 Example 2.— What is the 3rd power of 3J. 
 
 3i = J and i X J X i = ''r -= 42-876 ^rw. 
 
 ^ To involve mixed numbers, we reduce them first to 
 improper fractions. 
 
 exercise 74. 
 
 Involve, 
 
 1. 8 to the fourth power. 
 
 2. 23 to the fifth power. 
 
 3. 225 to the tenth power. 
 
 4. 9^ to the fourth power. 
 6. I to the third power. 
 
 Involve, 
 
 6. 1*25 to the second power. 
 
 7. -901 to the third power. 
 
 8. 11 to the sixth power. 
 
 2A 
 
 9. a^ to the second power. 
 
 EVOLUTION. 
 
 EXTRACTION OP THE SQUARE ROOT. 
 
 To extract the sq tare root of a number, is to find a 
 number which if multiplied mce by itself would produce 
 the number. 
 
 The theory cf the ruhs for the extraction (f the Square 
 and the Cube Boot, can be best explained by the teacher, toUh. 
 blocks made for the purpose, and this thou^^ be thoraughbf 
 done in the outset. 
 
 ' ^ .»»._. 
 
 jjxfjCTZ uic pt^*i ptoce&is to iav vuocfvisps, the quKslion, 
 ^*uihy?^' shxnud be put and answered in reference to ta*ih 
 step in the rvk. 
 
148 
 
 ABITHMBTIC. 
 
 BULB. 
 
 (1.)— Poinf off the given number into veriods of /»m 
 fiumoers, and from h^ to right in dedmals.) 
 
 *>Jtl ^^^f }^ highest square contained in the teft hand 
 penod, and place Us root as a quotient in division!^ ^^ 
 
 aJdi SflZ^^'^I ^'^'^''^ number frrnn the left handpeHod 
 (4.) Double the root already fomid for a tried divisor. 
 
 SS'^ "^' """^ *""""'' '"^ peri«J/«. anew 
 
 ExAMPj,E.-What is the square root of 2025 ? 
 
 '4X4= fe''^'^*^"''"*- ,W« first point off from the 
 
 " -T^T- «^^* ^^\ P^"«^8 of two 
 
 425 figures each, and find the root 
 
 4^5 must contain two digits* the 
 
 ^hMt square in 20 fa 16, tte «jt°rf of/ll 1'" ' .t 
 wuVAf * '>''?« down the next period ; wi then dmi 
 
 L°r5'SI,^S'"'«!l"'^*^«' '"»' dw"ioTan5^m^th?W.' 
 Hig 6, find the work completed. *«uiMpij- 
 
 2 
 85 
 
KXTRACTION OF THB CUBE ROOT. 
 
 149 
 
 's of two 
 I in whole 
 
 t^i hand 
 
 idperiod, 
 ^eriodfor 
 
 nsor. 
 
 ed in the 
 mre thus 
 divisor. 
 
 Jti'>^ the 
 d to the 
 r a new 
 
 ew trial 
 ss until 
 
 imerator 
 lare root 
 bion and 
 
 >m the 
 ' t\ro 
 le root 
 s; the 
 
 ; the 
 e 4 in 
 luare^ 
 I dou- 
 9 eon< 
 
 root, 
 
 BXGRCTSB 73. 
 
 What is the eqaare root of the ibilowing numbers :— 
 
 1. 1316, 222784. 
 
 2. 4922894, 5499026. 
 
 3. 67306616,5. 
 
 4. -5, |. 
 
 6. ^, Hi^ 
 
 7. 20J. If. 
 
 8. 207|f , 338-683604. 
 
 10. 23784^, 74786^^?^. 
 
 11. 90374376, 23473783. 
 
 12. 847376^, 783703||J. 
 
 ■*• » -■ 
 
 EXTRACTION OF THni CUBE ROOT. 
 
 To extract the cube root of a number, is to find a root 
 which when multiplied twice by itself, produces the 
 number ; thus, the cube root of 27 is 3, for 3 X 3 X 3 
 
 The teacher shmdd explain the rule for the extractim of 
 the cube root, by using the blocks, b^ore the pupil is re- ' 
 quired to commit or use it : and the pupil shmld be able to 
 answer the question, " «j%," in rtference to each step in 
 the rule, before he proceeds to the exercise. 
 
 RULE. 
 
 (1.) Point off the given number into periods of three 
 Mures each, pointing to the left in whole numbers, and to 
 the right m decimals. 
 
 (2.) Find the highest cube contained iniheUft hand pertod 
 and pkice its root a$ a quotient in division. 
 
 (3.) Sta>ira^> thi cube from the left hand period, and to 
 the rerminder, if any, bring down the next period for a 
 new dttmend. 
 
 JH ^^Ph/^ fQ^re of the pari of the root found ba 
 ZOO, for a tnat divisor. ^ 
 
 ib.j Mnd how often this trial divisor is contained in the 
 dmdend, and put the figure thus obitUmd in the rooi» 
 
m 
 
 ARITHICBTIO. 
 
 pp>ar> of !■• J last i^ni^^T ^ '"' ""^ ■*" «*« 
 
 par/ 0/ /Ac roof now 
 ' and proceed as before 
 
 um aU the periods are brought down 
 
 In fractions, when both the numei-afn,. «n^ a 
 tor are complete cubes extra^^^hl V*^^ denomina- 
 separately I la all other oaT«? ii^® cube root of each 
 decimal, L'd pi^ceed'S inL'tmbers. ''^^"^" *^ «* 
 
 BxABPLK. -What is the cube root of 1963125 f ' 
 .V •^•■^ 1963115(125 
 
 963 =;l8t dividend. 
 
 JJtMaldiv. 1»X300= 300 
 IetincreaselX2X30=: 60 
 2nd « 2X 2= 4 
 
 Complete divisor = 354 733 
 2nd trial div. 12* V " in a "oaa TZIZ — 
 
 Complete divisor =46026 226126 
 
 principle. ° loroswmg, and u toiindod on the same 
 
 Having pointed off Uie nnmbei- i*-*- -wi-L-j. -j- _ - 
 Uf« eaci., « in the former meU.oipl|,i^r»^:SJ^'^: 
 
EXTRACTION OF THE CUBE HOOT. 
 
 131 
 
 tervals before it two ciphers as the commencement of two 
 columns ; having found the first figure of the root, place 
 it in the first column, and having added it to what stands 
 above it, multiply the sum by the same flgi.re, writing 
 the product in the second column ; add in lilte manner 
 in the second column, multiply the sum by the same 
 figure, and place the product in the third column, and 
 subtract it from what stands above it. 
 
 IF Perform a process exactly similar in the first and 
 second columns, add the figure found in the root to the 
 first column, then add one cipher to the first, and two 
 ciphers to the second column, and bring down a period 
 of three figures to the third column ; divide the third 
 column by the second as a trial divisor to find the next 
 figure m the root ; place this figure in the first column, 
 and proceed as before. 
 
 Example.— (Same number as before.) 
 
 Ist Column. 2nd Column. 3rd Column. 
 
 
 
 1 
 
 
 1 
 
 l*963i26 
 1 
 
 1 
 1 
 
 1 
 2 
 
 953 
 
 2 
 1 
 
 300 
 64 
 
 
 30 
 2 
 
 364 
 68 
 
 728 
 
 32 
 2 
 
 43200 
 1826 
 
 225126 
 
 34 
 
 2 
 
 45026 
 
 225126 
 
 360 
 
 3S5 
 
'■ 'aw i l W W y jfifl lff yWf ff Mli ltH 
 
 U2 
 
 ARrrHMBTIC. 
 
 KXERCIBU 7ft. 
 
 (Extract the cube root of the following numbers ;— 
 
 4li 373248, 592704. 
 
 2. 12326391, 15625. 
 % 29859^4, 704969. 
 
 4. iff, 21024576. 
 
 6. 84-604519, 000348. 
 
 6. 2, <>97002999. 
 
 7. -389017, ^^. 
 
 APPLICATIOxN OF THE SQUARE AND CUBE ROOT. 
 
 1. If an acre of land be laid out in a square form, what 
 will be the length of each side In rods ? 
 
 2. What will be the length of the side of a square, in 
 rods,;jttiat shall contain 100 acres? 
 
 3. A cellar is 25 feet long, 20 feet wide, and 8J feet deep ; 
 what will be the dunensions of another cellar of 
 equal capacity in the form of a cube ? 
 
 4. What will be the length of one side of a cubical 
 granary that shall contain 2500 bushels of grain ? 
 
 5. What is the length, in rods, of one side of a square 
 that shall contain 12 acres ? 
 
 ^' ^ conjipany of speculators bought a tract of land 
 for $6724, each agreeing to pay as many dollars as 
 there were partners ; how many partners were there ? 
 
 7. What must be the length, depth, and breadth of a 
 box, when these dimensirns are all equal and the 
 box contains 4913 cubic feet? 
 
 8. The solidity of a cubical block is 21952 cubic yards ; 
 what is the length of each side ? What is the area 
 of the surface ? 
 
 9. A general has an army of 7225 men ; how many 
 must be put in each line in order to place them in & 
 square form ? 
 
 10. Two persons start from the same point ; one travels 
 due east 50 miles, the other due south 84 miles ; how 
 for are they apart ? 
 
 How many Buiaii cubes of 2 inches on a side can be 
 sawed out of a cube 2 feet on a side, if nething is. 
 lost in sawing? * 
 
 Ha 
 
 7 
 
 a 
 
 9 
 10 
 11 
 12. 
 13. 
 
 14. 
 15. 
 
MISCKLLANEOUS EXEECISB. 
 
 153 
 
 »r8 : — 
 >0348. 
 
 E ROOT, 
 inn, what 
 
 q[aare, in 
 
 »et deep ; 
 cellar of 
 
 I cubical 
 ;rain ? 
 
 a square 
 
 of land 
 lollars as 
 re there ? 
 
 tdth of a 
 
 i and the 
 
 Ic yards ; 
 the area 
 
 iw many 
 hem in a 
 
 e travels 
 les ; hoMT 
 
 3 can be 
 [»thing ia 
 
 FINAL COMPREHENSIVE ML^CELLANEOUS 
 EXERCISE 
 1. Write down as one number, five thousand billion 
 four thousand and three, and ninety-two thousandth"'. 
 itt *^«°*y J^'^'isand three hundred and twenty- 
 five milhonths to one hundred thousand and tweutv- 
 five ton-thousandths. "wtmjr 
 
 3. Multiply 9873J by 786-18231. 
 
 *• ^m/n««!T9^" '"i^l'"^ "'"'*^^"' ^°^^ inquiring the 
 time past 12 o clock, received for an answer, tz of 
 
 aftJn'iJL^''™ -T ^ ^"'^"^g^*? wl^at o'clock in the 
 aiternoon was :t ? > 
 
 5. From one half of a piece of cloth containing 82 vds 
 2 qrs., a tailor cut six 8uit« of clothes : how much 
 did each suit contain ? 
 
 6. Multiply 23-564 by 2-65, and divide the. result by 
 2-123-f (-23— '-0062.) 
 
 7. What is the exact decimal value of jfg. 
 
 8. If £100 of bank stock is worth £98|, what is £362 8s. 
 
 7Jd. of stock worth? 
 
 9. If you pay £37 10s. per ton for iron, at what rate 
 
 must you sell it to gain tho price of 1 ton on 15 tons ? 
 
 10. What will be the rent of 35 acres, 2 roods, 10 r ot 
 land, if 46 acres, 3 roods, 14 r. are worth £50 ? 
 
 11. If a landlord deducts § on a shilling to his tenant, 
 
 what will be the deduction on £76 3s. 4^d. ? 
 
 12. If J and ^ of a pasture cost £4 10s., what will the 
 whole pasture cost ? 
 
 13. Bought 840 apples, at the rale of 10 for a penny, and 
 
 2^0 more, at 8 for a penny ; if I sell them at 3G for 
 4d., shall I gain or lose by the operation, and how 
 much per cent. ? 
 
 14. A former having sold J and | of hiw sheep, had 95 
 
 left ; how many had he at first? 
 io. A man having $15750, spent ^ for a house, A the re- 
 mainder for a barn, and 4 of the balance for a car- 
 nage ; how much had he left? 
 8 
 
194 
 
 ARITHMETIC. 
 
 16. Wl L is tho diftWrence beiwocn ig of 27.>. and io{ 
 
 315? 
 
 17. Whfit number is that, ^ of which exceeds J by 387 ? 
 
 18. What number is that, f and | of which make 255 ? 
 
 19. What number must be added to 13754 to make 
 :,'. 811935? 
 
 20. What must bo taken from 11371 that 793i^ «»av be 
 left? 
 
 21. What must bo added to 217i that the sum may be 
 
 17 J times 19^? 
 
 22. What number multiplied ty 45J. will i)roduco 288J? 
 
 23. What number divided by 37 j, will give 193J for the 
 . ! quotiett? 
 
 24. Bought J of a ship, and sold f of it ; h«w much was 
 left ? ■...>) ' \i 
 
 25. A grocer used a false weight of 13i oz.'for a pound ; 
 what was the amount of his fraud in woighuig 500 
 lbs. ? 
 
 26. If ^ lbs. of lard coat Jfs. ; how much will f § ton 
 cost? 
 
 27. What number is that, J of which exceeds I by 428 ? 
 
 28. What number is that, f and | of which is 510 ? 
 
 29. A father gave his eldest son twice as much as the 
 second, the second three times as much as the third, 
 who had $1573 ; how much did he give to all? 
 
 30. A man having 4 children, gave twice as much to the 
 • . 4th as to the 3rd 5 twice as much t(3 the 2ud as to the 
 
 4th ; and to the let twice as much as to the 2nd, 
 which was $7860 ; what did he give to all ? 
 
 31. A man gave ^ of his estate to his eldest daughter ; 
 ^ the remainder to the 2nd ; and J of the remainder 
 to the 3rd, who received $3560 ; what was his 
 estate ? 
 
 32. A and B travelling, A has 5 loaves of bread, and B has 
 S. They are overtaken by C, who says. ** Let me par- 
 take with you, and 1 will pay for what 1 eat." They 
 all eat an equal quantity, and C pay? $8. How shall 
 
 1^ 
 
MIHCELLANEOUS EXERCISH. 
 
 155 
 
 the monoy bo divided equitably between A and B ? 
 
 33. B and C together can build a boat in IH days : with 
 the aaslBtance of A they can do it in 11 daye • in 
 what time would A do it himself ? ' 
 
 34. If A can do a piece of work alone in 10 days, and 
 A and B together in 7 days, in what time can B do 
 it alone ? 
 
 35. A, B, and C can complete a aisce of rork together 
 in 12 days ; C can do it alor..> : , U tt .yg, and A in 
 34 days ; m what time could h i it >^ himself? 
 
 36. A can do a piece of work in '^ wi . .£8, B can do thrice 
 A8 much in 8 weeks, and C 5 times as much in 12 
 weeks ; in what time can they finish it jointly ? 
 
 37. Bought 120 oranges at 2 a penny, and 120 more at 
 
 ^ ^.^^^74 ^^^ ^^^^ *^^'" *" together at five for 2d. : 
 what did I gain or lose by the bargain ? 
 
 38. A man left kis two sons $iqoo ; their ages were 14 
 and 18 years respectively; if their shares were 
 put to interest at 6 per cent, per aimum, they would 
 be equal when each would bo 21 years of age ; what 
 was the share of each '/ © , m*» ^ 
 
 S9. What is to pay for the rent of a house, at $372 203 a 
 jrear, for 5 years in arrears, at 6 per cent, (simple 
 interest 7} 
 
 40. The bead of a fish is 4 feet long, the tail as long m 
 
 ^ . the head and ^ the length of the body, the body as 
 
 long as the head and tail ; what i, the length of the 
 fish ? 
 
 41. The sum of two numbers is 2fi6f , the product of the 
 greater multiplied by 3, equals the product of the 
 less multiplied by 5 ; what are the numbers ? 
 
 42. A military officer placed his men in a square : beinjr 
 reinforced by three times his number, he placed the 
 whole again m a square ; again being reinforced by 
 three times his last number, be placed the whole a 
 
 :.^iI. iT^i°_*_'!2"*^f' which had 40 men on each 
 =:i->i^ , iiv— man J men naa ne ai uivt? 
 
 43. Suppose that a man stands 80 feet from a steeple, 
 that a line to him from the top of the steeple is 100 
 
I»i'4l**^-- 
 
 156 
 
 ARITHMETIC. 
 
 ^. 
 
 feet long and that the spire is three times as hicrh as 
 
 from the top of the spire to the man ? '^ 
 
 44. Two ships sail from the same port ; one sails direct! v 
 east at the rate of 10 miles, the other directly sou J 
 at the rate cf 7J miles an hour : how far are thev 
 apart at the end of 3 days ? ■ ^ 
 
 46. If eggs be bought at the rate of 5 for 2 cents : how 
 must they be sold to gain 40 per cent.? 
 
 46. Three gftlemen contribute towards the building of 
 a church, $657, the church is distant from the fir! t 2 
 miles from the second 2| miles, and from the third 
 3^ miles ; they agree that their shares shall be re- 
 ciprocally proportional to their distances from the 
 Church ; how much must each contribute ? 
 
 *^* ^«»."'^^*?*^Af^?? y^^'* increase his capital by a 
 fifth part of itself, except an expenditure of $400 
 
 9U00 J what was his ongmal capital ? 
 
 48. A's note of $7851.04 was dated Sept. 5th, 1837 on 
 which were endorsed the following payments viz • 
 Nov 13th, 1839, $416.98 ; May loi, 1840, $152 :* 
 what was due March Ist, 1841, the interest beinjj 6 
 per cent. ? » ' 
 
 49. A person dying, worth $5460, left a wife and 2 chil- 
 dren, a son and a daughter, absent in a foreign 
 country. He directed that if his son returned, the 
 mother should have one-third of he estate, and the 
 son the remainder ; but if the daughter returned 
 she should have one-third, and the mother the re- 
 mainder. Now It so happened that they both re- 
 turned ; how mupt the estate be dividedf to fulfil 
 the tiather's intentions ? 
 
 50. If 12 apples be worth as much as 17 pears, and 2 
 pears cost l^d. ; what is the value of 99 apples ? 
 
 51. Place the nine natu J numbers, I, 2, 3, 4, 5 6 7 8 9 
 in such a manner that the sum of the odd d'igite Wall 
 ao equal to the sum of th even ones. 
 
 52. What part of TA is a third part of 2d. ? 
 
MISCELLANEOUS EXERCISE. 
 
 157 
 
 ??o T'^ S® ?'°® "^'S'^' 1' 2, 3, 4, 5. 6, 7, 8, 9, be 
 
 placed in the form of a square, so that the sum of 
 
 each row of figures, reckoned upwards, downwards. 
 
 horizontally, diagonally, shall just equal 15. 
 
 44; I owe my friend a shilling ; he has nothing but coin« 
 
 ■ worth |7s. each, and I have nothing but guineafc. 
 
 - How must an exchange take place between us, so 
 
 that 1 may pay my debt, but no more ? 
 
 65. How must a board that is 16 inches long, and 9 
 inches broad, be cut, so that when the two parts re 
 joined together, they may fonn a square? 
 
 66. A has by him IJ cwi of tea, the prime cost of which 
 
 was £96. Now, granting interest to be at 5 per cent.. 
 
 tV^.T''^^ *^ ^?? ^^'^ ^® "»^^«* sell it per lb. to B 
 so that by taking his. note, payable at 3 months, he 
 may clear 20 guineas by the bargain. 
 57. The hour and minute hand of a clock are exactly to- 
 gether at 12 o'clock ; when are they next together ? 
 
 68. There is an island 73 miles in circumference, and 
 3 persons all start together to travel the same war 
 ^out It ; A goes 5 miles a day, B 8, and C 10 : when 
 will they all come together again ? 
 
 69. Sold goods for 60 guineas, and by so doing loit 17 
 per cent, whereas I ought, in dealing, to have clear- 
 ed 10 per cent. ; how much were they sold under 
 their just value? 
 
 60. A hare starts 40 yards before a greyhound, and is not 
 perceived by him till she has been rlinning ^ 
 
 « seconds ; away she goes at tlie rate of 10 miles an 
 Hour, and the dog, on view, makes after her at the 
 rate of 1 8 ; how long will the course hold, and what 
 
 B^ZgoTlLl^T ''"' ^^^"^°^ ^^'^ *^^^-^ 
 
 61. A reservoir for water has two cocks to supply it : br 
 the first alone it may be filled m 40 minutes, and by 
 the second in 50 minutes j it has likewise a dii 
 r~Tv'~Si. "'";■"' .'V -«^^" •« lusj, wacu rail, De etuo- 
 tied in 25 minutes. Now, if these three cocks are aU 
 left open, when the water cornea in, in w^t time 
 
168 
 
 ARITHMETIC. 
 
 7i»i^nf%f^^? ^^ Y*'"^; suppb.i!ttg the influx and 
 efflux of the water to be always alike ? 
 
 m the following proportion ; to A he left h to B i 
 toCi,toD^, desiring the whole might £ distrl- 
 butedfaccordiDgly ; what is the proper share of each ? 
 
 ^Z'^^nl ,%f ^^.P^a°ks will floor a barn 60J feet 
 long, and 33* wide; when the planks are 15 feet 
 long, and 15 inches wide ? 
 
 64. The amount of a sum of money which has been put 
 
 Hm J . '"*'T* '!.^^^°' **"^ *^« P"«ciP^l is just 7 
 times as much as the intei e«t ; what is the principal ? 
 
 65. A tradesman increased his estate annually i Bart 
 and at the end of 3J years,* found that his net estate 
 
 ^oTSl^g/^ ''''' '''-''-'^ -^^* ^^TZ 
 
 ^^^\?a'T/lu' '^^""^'^^ ^ ^^ ^' y^^^Iy i°<^ome plus 
 Smef ^-^"^aming -J plus £15 'what wai his 
 
 ^^' Inl'^'"'' containing 60 gallons of water has 3 un- 
 3pnf«fvl-'il!''^^'4^^ ^*5 the greatest cock 
 ri *K?^S*^ 'o '^ ^ ^''"'* 5 the second in 2 hours : and 
 the third in 3 : in what time will it be empty if they 
 all run together ? F'-j' " tuey 
 
 • 68. Jnm orchard of fruit trees, J of them bear apples, 
 i pears, ^ plums, and 50 of them cherries ; how many 
 trees are there in all ? ' ^ 
 
 69. Suppose the sea allow oe for sailors to be 5 lbs. of 
 beef, and 3 lbs. of biscuit a day, for a mess of 4 
 people ; and that the price of the first is 2id oer lb 
 
 h«t ' wS ??."°^ ^^^- 1 ^«^' ^f ^^^ ship'Icompany 
 
 U ^L^^^^ ^^^"^^^^ ^b^ ^^* «°«* the government 
 12 gumeas per day, what must they ply for their 
 bread per week ? / i j *"i wieir 
 
 ^®* t/M^''°®i'.^''l^.v^''^"^ ** ^^«- a thousand, by which 
 he cleared | of the money ; but he afterw^r/, rS 
 
 '^iT'i? i?' ?"{: * ^'^O'lsancl ; what did ha clear per 
 cent, by the latter price ? '^ 
 
MENTAL BXEBCISK. 
 
 159 
 
 PINAL MISCELLANEOUS MENTAL EXERCISE. 
 L How many are 12 and 9 ? 11 and 17 ? 17 and 19 ? 
 
 2. How many are 28 and 43 ? 46 and 37 ? 126 and 514 ? 
 
 3. How many are 7 + 9 -f 15 -f 27 -f 6 -f 17. 
 
 4. How many are 17 — 6 ? 42 — 16 ? 523 ~ 64 ? 
 
 5. How many are 127 — 69 ? 615 — 173 ? 2986 — 175 ? 
 
 6. How many are 1276 — 14— 123 -f- 24 — 18 -f 154 ? 
 
 7. How many are 6 X 2 ~ 5 + 8? 24 X 17 — 15 -f 8 
 
 X36? 
 
 8. How many arc 176 X 1? ^ 23 -}- 8? 54 X 26 + 19 
 
 — 23 -f- 9 ? 
 
 9. How many are 28 X 15 — 11 -f 18 -i- 14 ? 163 X 17 
 
 — 19 4- 126-^ 15? 
 
 10. How many are 12 X t? 12-5- |? i of 20? y of 10? 
 
 11. 4 times 8 are how many times 5, 6, and 7 ? 
 
 12. 2| of 12 are how many times f of 18 ? 
 
 13. What is f of that^ number of which 63 js | ? 
 
 14. 15 is I of how many times 27 ? 
 
 15. f of 126 is f of how many times | of 617 ? 
 
 16. 17 boolts at $5.2? ? 126 horses at $57.96? 
 
 17. ^ of 624 ? Ji of 826 ? f of 483 ? 
 
 18. How much will | of a barrel of apples cost. If ^ cos 
 75 cents ? 
 
 19. How much will 6 cords of wood cost, if f of 4 cords 
 
 cost $12. 
 
 20. 1231-59 I? f X f X iF f X f - § off? 
 
 2L 18 is 4 times what number? 36 is J of f of wha^ 
 number? 
 
 22. If 15 cords of wood cost $65, what will 11 J cords 
 
 cost? 
 
 23. If $126 pay 17 men, how many will pay 150 men ? 
 
 24. If 17 men can build a barn in 17 days, in how many 
 
 days can 27 men build it?* 
 
 26. A can do a piece of work in 8 days, B in 12 days ; 
 in what time can both working together do it ? 
 
160 
 
 ABITHMETK5. 
 
 I 
 
 ^^* iSd LI $l27^'i*f{' '"? f^^^^'^^ i of the balance, 
 ana nad $127 left ; M-hat was the debt ? 
 
 How much is 11 per cent, off $125 ? 
 
 37. 
 
 28. 
 29. 
 
 654 lbs 
 
 cent« 
 
 30. 
 
 31. 
 32. 
 33. 
 
 « 
 
 84. 
 
 lB6. 
 36. 
 
 37. 
 
 38. 
 
 39. 
 
 40. 
 .41. 
 42. 
 43. 
 
 cheese 
 
 wh'.S'did I 'JuZl ""^ '""' "' '' P" «»*• P'»«' i 
 What is the interest on $763, at 9| per cent ' 
 Interest on $826 for 3 years, at 4J per cent ' 
 
 wh"af ^Xt^'p'^ct^f "■"' "^^ " "' »*•=« •• 
 
 ^nt p'er'SLrt"'" "' *''' *"• * """"^'' «" * P*^ 
 87^ yds. of cloth, at 67f cent, per iyd'f ' 
 
 ^5:"S^^s.^:f^--'^*-p---™ 
 
 The int«jrest on $27 for 3 yeai-s was $5.15 ; what was 
 the rate per cent. ? > wuaiwas 
 
 prLirpalT* '* * ^'' ''^'- ^^^ ^-' ' ^^^^^ ^as a,* 
 wnfboTo^^ tluit, lo^hichif ^ bearded, the sum 
 
 \yhat number is that which being increased bv its I 
 Its I and 18 more, will be doubled ^ ^ ^' 
 
 to'iac\ oler^ Mo'4' ''''' *"° P"^^' *^«* ^^^" ^« 
 
 mh ott-Tlt f *'*' *"^ P^^*^' *^^* «^^» ^^ *o 
 A fishing rod, the length of which is 24 feet, is in two 
 
 Sfn!J ' * i 1 ^®"?®^ P*^* ^<l"*ls the length of th 
 shorter ; how long is each part ? « "» «» 
 
 -~ S "* ^M. o UU.U1UCX ( 
 
 Reduce 9 years and 8 months to the fraction of a 
 
 I 
 
MENTAL EXEBCISE. 
 
 161 
 
 46. Reduce 2 years 4 months and 15 days to the fraction 
 of a year. 
 
 47. Reduce 5 years 9 months and 18 days to the fraction 
 
 of a year. 
 
 48. At 2 per cent., what part of the principal equals the 
 interest ? What part at 8, 10, 7, 6, 6, 1 8, 25 ? 
 
 49. A man sold a quantity of goods, and thereby gained 
 
 75 per cent, on the cost ; what part of the cost equals 
 the gam? 
 
 SO- If a^ of the principal equals the interest, w^'it is ^ 
 rate per cent ? 
 
 51. A book was sold for f of | of what it cost : what 
 was the loss per cent. ? 
 
 52. How many pounds of coffee, at 8 cente a pound, 
 must be given for 9| pounds of sugar, at 10 cente a 
 pound. 
 
 53. What is the sum of 15, 5, 3, 2, 8, 3, 7, 2, 9, 6, 7, 8, 6, 
 
 ^' Po**i^*^^ ^'^^. ^^ ^^' *' ^0' 6' 3, 7. 2, 5, 9, 11, 20, 4, 
 6, 8, 3, 7, 4, 3 times 5, 3, 8 less 5, 7, 4, 9, 12, 7, 3, 6, 
 
 55. What is the sum of 11, 5, 8, 3, 12, 4, 5, 20 less 7, 5, 
 • 8, 9, 3, 5, 13 less 5, 7, 16 less 8|, 4J, 7, 9L and 11 
 
 times 4. i * «7 
 
 56. Benjamin Franklin died in 1790, and was 84 years of 
 age at his death ; in what year was he born ? 
 
 Note— Thp earth turns on its axis from west to east, that is 380« 
 once every 24 hours ; hence it revolves IS** iu i.r^e hour, that Sl» 
 in 4 minutes, that is 1' of distance in 4 soconds of time. As th • 
 earth turns from west to east, the f .rther the place is east the earlfc- 
 It gets the sun, and hence, the later it is in the day. " 
 
 From this we may easily tell the difference in time between any 
 
 two places, if we only know the diffor(?nce in longitude. To find the 
 
 difference m lougitude, if the places are both east or both iront 
 
 longitude, subtract. If one is east and the other, west, addthwn 
 
 ... ... . -r.. ii- i -„-i iavci as, ivruuvy or lireuencEiuJi, and 
 
 how much ? 
 
 58, When ifis 9 o'clock, a.m., at Toronto, what timeui 
 It at Frederickton ? 
 8* 
 
i1 
 
 'f 
 
 II 
 
 
 112 
 
 ARITHJIKTIC. 
 
 w. When it is 6 o'clock 
 
 is It mid-night, and at what p .ces is ^^l JfZ 
 
 laces 
 
 «1. 7 hours iese than 6 o'clock, i> 
 
 X?«J^e?'""'^''-'''-«»'clocr2''o' 
 
 18 What o'clock ? 8 
 
 62, K; 
 
 «3. jiowmsmy U.,, of 
 
 -^«eftoi«th«, 3f|J^o84ths|^to56ths 
 
 f«"«^^in^,^,.v,|,^,and§. 
 
 how much must 
 
 IP«yfor8pou).%..tthe»ameS? ' 
 «-fof30i8|,>fwh,„Humb.r? 
 
 ♦ i. ine distance from A to B whirh ic .<a m ... 
 J ihe distance from C to D • wf«f wu^'j.'" ^ of 
 from C to I) ? """^ ^ ^0 D , what ls the distance 
 
 o'clock faVr^'^l^roX'r' ',"' '^™''™' "'"^ 
 
 «he lastquiSon '^*'' P'*"'* ■mentioned .ii 
 
. ANSWERS. 
 
 ANSWERS 
 
 UB 
 
 EXERCISE 1. 
 
 Vlli, XI, XV, XIX, XXIX, XXXV, XCIX CLX. 
 CCCXL, DLXIX, MCVI, MMXXV Sx 
 
 MDccccxxv, MMDCLxxx, ulmS^kr 
 
 MMDCCCCLXI, MCCCXL. 
 
 
 EXERCISE 2 
 
 74, 140, 999, 750, -4, -27 .014, -760, 76-027, 675-864. 
 
 EXERCISK 3. 
 
 For instance, 1st, Pour and hit i^nth^ ^^ . ._ , 
 tenths : 4-6 or 4J» or M 9nT c- * xu ^' ®^ forty.«ix 
 two hi n^l^fi,.^ ^^" xl"^' Sixty-three and seventy- 
 seventrtwo h^fJ^r.fK'"^ thousand three hundred aSd 
 
 1. 22 cents. 
 
 2. 21 dollars. 
 
 3. 18 marbles. 
 
 1. $1186-37. 
 
 2. 1248. 
 
 3. $1348.74. 
 
 4. 1465. 
 
 6. 2260-979. 
 
 6. 2072. 
 
 7. 2343-190. 
 
 8. 2856. 
 
 5. 976'5it>2. 
 
 10. 1636. 
 
 11. $1617.94. 
 
 12. 1056. 
 
 EXERCISE 4. 
 
 4. 35 sheep. 
 
 5. 13 cents. 
 
 6. 95 cents. 
 
 EXERCISE 5. 
 
 13. $34950.10. 
 
 14. 21867. 
 
 15. $18068.93*. 
 
 16. 10913. 
 
 17. 30155-740. 
 
 18. 18001. 
 
 19. 2017012. 
 
 20. 14.H72- 
 
 25. $105. 
 
 26. $2.93. 
 
 27. $408. 
 
 28. $1476. 
 
 7. 22 dollara. 
 
 8. 15 dollani. 
 
 29. $16388. 
 
 30. 4258. 
 
 31. $24731. 
 82. $1658286. 
 
 33. $7861214. 
 
 34. $536146. 
 
 36. 75676. 
 
 HR Q11A10 
 
 37. $660.11. 
 
 38. 2246 appl«[ 
 
 39. 72 apples. 
 
lU 
 
 AVSWRRa. 
 
 1. 162 cente. 
 
 2. 62 cents. 
 
 3. 281 dollars. 
 
 4. 156 times. 
 
 fcXERCrSE 6. 
 
 5. 66 years. 
 
 6. 109 trees. 
 
 7. $15.24. 
 S. 7 cents. 
 
 9. 8, 16, 32, 40, 
 64, 72. 
 10. 537 pounds. 
 
 1. 11. 
 
 2. 6 dollars. 
 
 3. 3, 11 in both. 
 
 1. 36919. 
 
 2. $7837i66. 
 
 3. $40263.41. 
 
 4. 38999. 
 
 5. $22984.09* 
 C 15289. 
 
 7 $78359.03. 
 
 8. $5292. 
 
 9, $462121934.46. 
 
 EXERCISE 7. 
 
 4. 6 years. 
 
 5. $9, $21 for both 
 6. 15 apples. 
 
 EXERCISE 8. 
 
 10. $4351951.69. 
 
 11. 7392-2070. 
 
 12. 612663992. 
 
 13. 72299-5412. 
 
 14. $913109.19. 
 16. 457555. 
 
 16. 1205999-32. 
 
 17. -00036. 
 
 18. 57955. 
 
 19. $8072. 
 
 7. 29 dollars. 
 
 8. 27 years. 
 
 9. 92 dollars. 
 
 20. 171. 
 
 21. 344. 
 
 22. 172. 
 
 23. 178. 
 
 24. 106. 
 
 25. 135. 
 
 26. 799. 
 
 27. 1386517. 
 
 28. 11 years. 
 
 1. 83. "^ 
 
 2. 2720. 
 
 3. 1557. 
 
 1. 1706. 
 
 2. 69 both, 9 more. 
 
 3. $17.50. 
 
 4. 10. 
 
 EXERCISE 9. 
 
 4. 162. 
 
 5. 415. 
 
 6. 221. 
 
 E3CERCISB 10. 
 
 5. 16. 
 
 6. $10 gaia. 
 
 7. 20. 
 
 7. 1244556. 
 
 8. $287. 
 
 8. 
 
 $466 
 
 
 $79. 
 
 9. 
 
 $10. 
 
 1. 63. 
 
 3. 16,20,24. 
 
 EXERCISE 11. 
 
 4. ^John 10, Jos. 
 
 46, ail 64. 
 
 5. 7146 dollars. 
 
 6. 304 v^ds '■■ 
 2m doHii*s.n r 
 
 1 1 
 
32, 40, 
 
 i, 
 
 >unds. 
 
 IT8. 
 
 9. 
 
 iT8. 
 
 17. 
 krs. 
 
 J. 
 
 Im 
 
 in 
 
 1. $17104.64. 
 
 2. $134574.86. 
 
 3. $432266.30. 
 
 4. $225804.16. 
 
 5. 66276. 
 
 6. $672612. 
 
 7. 389304. 
 
 8. $748731-60. 
 
 9. 502657. 
 
 10. $1162129.68. 
 
 31. $68236-48. 
 
 32. $133863.66. 
 
 33. $232499.62. 
 
 34. $182214.09. 
 
 1. $63221-592. 
 
 2. $7464-4808. 
 
 3. $2905-0420. 
 
 4. $488-44096. 
 
 5. $84393-932. 
 
 6. 430-143168. 
 
 7. $777566-496. 
 
 8. $3598313.04. 
 
 9. 63073762. 
 
 10. 41281053. 
 
 11. 24294591. 
 
 12. $28047414. 
 
 1. 1832219f. 
 
 2. lf«7M«5* 
 
 3. 2087596^ 
 
 4. 3179483f(|-. 
 
 ANSWERS. 
 BXBRCI8E 12. 
 
 11. 574675. 
 
 12. $568669.44. 
 
 13. 350-184. 
 
 14. 6128-22. 
 
 15. 7879-14. 
 
 16. 525-276. 
 
 17. 2626.38. 
 
 18. 4377-30. 
 
 19. 8754-60. 
 
 20. 9630-06. 
 
 165 
 
 EXERCISE 13. 
 
 35. 
 36. 
 37. 
 38. 
 
 $231504.12. 
 $208963.44. 
 $199122.30. 
 $138250.56. 
 
 EXERCISE 14. 
 
 13. 46350656. 
 
 14. 575630377. 
 
 15. 395494873. 
 
 16. 649435896. 
 
 17. 64008924. 
 
 18. 3704412744. 
 
 19. 403576660. 
 
 20. 175320. j 
 
 21. $2941.64. 
 
 22. 2592. 
 
 23. 2303. 
 
 V KBRCISB 15. 
 
 J. 3669180|1 
 7. 70632%*^' 
 
 21. 
 22. 
 23. 
 24. 
 25. 
 26. 
 27. 
 28. 
 29. 
 30. 
 
 105-0552. 
 
 $1966.54. 
 
 $6882.89. 
 
 $3933.08. 
 
 $786-616. 
 
 $589-962. 
 
 $4916.35. 
 
 $8849.43. 
 
 $11799.24. 
 
 $10815.97. 
 
 39. $565184.16. 
 
 40. $220399.92 
 
 41. $570676.32. 
 
 42. $715,501.44. 
 
 24. 3168. 
 
 25. $3243.75. 
 96. 4480. 
 
 27. $18.25. 
 
 28. 2144. 
 
 29. 81056. 
 
 30. 783. 
 
 31. 80 pages. 
 
 32. 1095. 
 
 33. 56940. 
 
 34. 768000. 
 
 8. 3458517^ .• 
 S. 197oi4o#. 
 10. 837d64||. 
 
166 
 
 1. 30 quarts. 
 
 2. 86 cents. 
 
 3. 60 jeara, 
 
 KXilRtlSB 16. 
 
 4. 96 days. 
 
 5. $3.40 and $l:j.60. 
 
 6. 21 cents. 
 
 ]. 6. 
 
 ^•f/or40c>,.7for56ctfl 
 o. 117 dimes. 
 
 BXERCIfciK 17. 
 
 4. 6 bbls. 
 
 5. 6 days. 
 
 6. 10 miles. 
 
 ' 
 
 1. 6911J. 
 
 2. 137'52#. 
 
 3. 13281^. ^ 
 
 4. $116.17i. 
 
 5. $96.63f. 
 
 6. 3186«.' 
 
 7. $6iM.A. 
 
 8. 4206JV. 
 
 9. 6368906. 
 
 EXERciaa 18. 
 
 10. $53359.658. 
 
 11. 137718U. 
 
 12. $71409.735. 
 
 13. $39064.065l 
 
 14. 6859560. 
 
 16. 1266701165 
 I'^ $4780.661. 
 
 17. ."^89437 If. 
 
 18. 28236344^. 
 
 19. 18824229*. 
 
 20. 14118-172f. 
 112945374. 
 94121145: 
 $97.17f 
 $85,02|. 
 $75.68i. 
 
 26. $68.02X. 
 
 27. $61.83^1. 
 
 21 
 22 
 23. 
 24. 
 
 25. 
 
 1. 6828a|. 
 
 2. 4408f|, 
 
 BXEECISH 19. 
 
 3. lo: ^. 
 
 4. 188i,^V 
 
 6. 309^. 
 6 3450^. 
 
 2. 167|||. 
 
 3. 216y.f. 
 
 6. 108-0511 
 
 6. 10-32i5l. 
 
 7. '951 i^a. 
 
 8. 99^23 83. 
 
 9. 723*319. 
 
 10. •700.^3 740 
 
 KXERCWH 20, 
 
 11. 8-571I13, 
 
 12. -03 1 8611.' 
 
 13. 5098685000. 
 14.26134 191984 
 16. •2687i«£9 
 
 16. S54118-46i. 
 
 17. OlOeiaiie 
 13. 37632621. 
 
 10 Kaon A nan KK 
 
 I 
 
 20. 2?- j)302i4 
 
 ?1, 1^ 500001 Ao. 
 ' $''10.8211. 
 
 k 3o. 
 ^5. 266|. 
 
 26. 2066666|f. 
 
 27. 192268315. 
 
 t\r\ 
 
1. 8i7n. 
 
 2. leoflf. 
 
 3. 261^V 
 
 AFBWERS. 
 RXBR0r8H 21. 
 
 4. 127Ui. 
 6. 56i4|f 
 
 6- 28iSJ|. 
 
 7. 13g§| 
 
 167 
 
 !9|. 
 72|^. 
 57^. 
 
 14, , 
 
 h 
 
 1. 40 oranges. 
 
 2. 4 dimes. 
 
 1. 63(78. 
 
 2. 1420. 
 
 3. 5760. 
 
 4. 1094. 
 
 I 
 
 EXERCISB 22. 
 
 3. 21 cents. 
 
 4. 24 oranges. 
 
 EXERCISE 23. 
 6. 24560. 
 
 6. 36136690704. 
 
 7. 6016. 
 a 2472. 
 
 5. 24 oranges. 
 
 6. 4 men. 
 
 9. 20721. 
 
 10. 5ya834i. 
 
 11. 52248. 
 
 12. 291454023. 
 
 1. sr ^^8. 3 oz. 
 
 2. £4v Is. 
 
 3. 6 oz. 1) iwt. 15 gr. 
 
 4. 299 T 2 nls. 
 
 5. 1199 bush. 
 
 6. 51 days, 20 57 min, 
 
 7. 7930 fathoms, .- ft. 2 in. 
 
 EXKRCWE 24. 
 
 a 3934714880 ac. 
 157388459520 r. 
 6295538380800 per 
 190440036019200 yds 
 1713960324172800ft." 
 246810286680883200 in 
 9. 14 yds. 8 ft. 781 in. 
 
 10. 183332. 
 
 11. 1093 cds. 10 ft 
 
 12. 298 ea. $7. 1dm. 4 c 9 m 
 
 1. $1057.96A. 
 
 2. 350390. 
 
 3. $990.54^. 
 
 4. $1159.20. 
 
 5. $298.00. 
 «. $364.80. 
 7. $178.20. 
 
 a £33y4 lOs. 
 9. $177a66. 
 
 10. 1761 g. 188. 
 
 11. 1 =46 cr. 2s. lOd. 
 
 12. U3067 fourpences, 
 18. ^880 cr. 
 
 14. $1963a 
 
 HXBBciSE 25. 
 
 15. $989.50 
 
 16. 
 
 17. 
 
 18. 
 
 873740. 
 57552. 
 9621 l^d. 
 
 19. 63864160. 
 
 20. 
 21. 
 
 22. 
 23. 
 
 ?4- 
 25. 
 26. 
 27. 
 
 33465 cr. 3s. 
 118801 Is. 6d 
 £69 llfd. 
 €182 lOf 8fd. 
 
 anAQ 9c 
 
 2890160. 
 30760a 
 15126 f| eta. 
 
.:»»« 
 
 168 
 
 AKSWBRri. 
 EXKnoWE 26. 
 
 1. jei37 10b. nd. 
 
 2. 92 per. 4 yds. 2 ft. 
 
 3. 177 dys. 11 hrs. 28 min. 
 
 4. 17 cwt. 2 qrs. 7 lbs. 
 
 6. 16 lbs. 4 07 1 dwt. 14 gr. 
 
 6. 148 yds. U jra. 1 nl. 
 
 7. 722 ac. 3 r. 21 per. 
 
 8. 120 yds. 23 ft. 119 in. 
 
 9. 9 cwt. 2 qrs. 
 
 10. 118 m. 6 f. 18 per. 3J yds. 
 
 11. 175 qrs. 1 bus. 
 
 12. 28 a. 21 per. 
 
 13. 66 w. 2d. 11 h. 16 mm. 
 
 14. 15 lbs. 4 oz. 1 dwt. 14 gr. 
 
 15. 148 yds. qrs. 2 nls. 
 
 16. £96 lOs. lOd. 
 
 17. 16 tons, 107 lbs. 
 
 18. 6 hhd. 63 g. 3 qta. 
 
 19. 178 m. 36 per. 3 yd. 10 in. 
 
 20. 166cd8.104c.ft.54c.in. 
 
 1. £13 Us. 4d. 
 
 2. 78 bu. 1 gal. 
 
 3. 18 cwt. 3 lbs. 
 
 KXERCISB 27. 
 
 4. 8.'J ydti. 3 qi-s. 1 nl. 
 
 5. £13 178. 8d. 
 
 6. 228 yds. gave $1482. 
 
 EXERCISE 28. 
 
 1. £24 lis. 7fd. 
 
 2. 6 cwt. 2 qrs. 11 lbs. 
 8. 3 lbs. 8 oz. 19 dwt. 
 4. 4 ft. 30 p. 2J yds. 
 6, 19 yds. 
 
 6. 25 a. 2 1. 25 p. 
 
 7. 192 t. 19 cwt. 3 qr. 6 lbs. 
 
 8. 3 lbs. 11 oz. 11 dwt. 6 gr. 
 
 9. 96deg. 6Um, 224 r. 3*1 
 
 yd. 2 ft. 
 
 10. 19 yds. 
 
 11. 11 a.3r. 18 per. 
 
 12. 436 bu. 2 pk. 2 qt. 1 nt. 
 
 13. 2 w. 19 days. 
 
 14. $96.27J. 
 
 15. 48 cds. 106 ft. 58 in. 
 
 16. 3 yrs. 233 days. 
 
 17. 6 lbs. 4 dr. 1 scr. 16 gr. 
 
 18. 552 days. 
 
 1. 2s. 3d. 
 
 2. Us. 9Jd. 
 
 3. 15 yds. 2 qrs. 
 
 4. 45 gal. 1 qt. 
 
 6. 9 cwt. 1 qr. 15 lb. 
 6. £9 Is. 
 
 If. Subtract sub. from 
 I Subtract rem. from 
 
 EXERCISE 29. 
 
 9. Add sub. to rem. 
 
 10. Subtract one from sum. 
 
 11. Subtract diff. from the 
 greater. 
 
 12. Add dilF. toless. 
 .13. Half the sum -J the d\ft. 
 
 min. j = greater, ar half the 
 min. I sum— the diff —the less. 
 
 1. 
 2. 
 
AN8WKR8. 
 
 I6d 
 
 KXERCISK 30. 
 
 1. 19 cwt. 2 qrs. 8 lbs. 
 
 2. IH lbs. 4 oz. 16 (Jwt. 
 
 3. nSydfl. 2nl8. 
 
 4. 25 ac. 2 r. 29 per. 
 
 5. 70 qrs. 2 bu. 2 pks. 
 
 6. 56hr8. 33 mill. 
 
 I 7. 670 a. 3 r. 24 por. 5| yd. 
 a £1529l8.9fd. ^ 
 9. 1332cd8. 84o.tt. 84c.in. 
 
 10. 1951 ml. 3 fiir. 37 per. 
 
 11. 124gal. Ipt. 
 1 12. 667 cwt. 2 qrs. 16 lbs. 
 
 KXERCISK 31. 
 
 2. 8432 lbs. 8 oz. 12 dwt. 
 
 3. £308478 168. 4d. 
 
 4. 35737 m. 5 fur. 
 
 6. 295251 yd. 3 qr. U nl. 
 
 6. 24404) ac. 2 r. 171 per. 
 
 7. 22594 qr. 2J pks' 
 
 8. 7597 yrs. 31 w. 3Jlday8. 
 
 9. 3346 cir. 4 signs. 
 
 10. 283 r. 6 qrs. } gheets. 
 
 11. $412164.81VW. 
 
 12. $552612.7li 
 
 13. 33 oz. 
 
 14. 211 cwt. 3qr. lib. 4oz' 
 
 15. 393 oz. 15 dwt. 
 
 16. 319 m. 1 fiir. 30 per. 
 
 17. 9032yd8. 3qr8. 2nls. 
 
 18. 77612 m. 1 fur. 24 per. 
 
 19. 806 gal. 2 qt. 1 pt. 
 
 20. 47 hrs. 7 m. 30 sec. .. 
 
 21. 6991 gal. 
 
 22. $1637.90. 
 
 23. 1634 g.-l qt. 1 pt. 
 
 24. 84 cdB. 60 ft. 1 44 in. 
 
 25. 748 ac. 38 per. 
 
 EXERCISE 32. 
 
 1. £6 128. 
 
 2. £2 136. 4d. 
 
 3. 14 cwt. 3 qrs. 13 lbs. 
 
 4. $10-53. 
 6. 11 cents. 
 
 1. 6 cwt. 2 qrs. 12 lbs. 
 
 2. 1 lb. 1 oz. 2 dwt. 
 
 3. 1 per. 4 yds. llj in. 
 
 4. 1 yd. 2 qrs. Of^ nls. 
 6. 1 ac. SOX per. 
 
 6. 1 qr. 3# pkfl, 
 
 7. 11 cwt. 2 qrs. 111b. 11 JLoz. 
 8. £826 18s. 6^d. 
 
 BXEBCISH 33. 
 
 9. 6 f^^l 3 qts. f pt. 
 
 10. 6 per. 4 yd. 3 a 27 in. 
 
 11. 96per. 22yd.7ft. 24in. 
 
 12. 6 F. ells. 2 qrs. 3 nls. 
 
 19 o/ 
 
 hifio. 
 
 *f 
 
 a X.. 
 
 14. 13 m. 6 far. 18 
 
 I lyds, 
 
 ^f eec. 
 per.2f^ 
 
170 
 
 ANSWERS. 
 
 BXERCISH 35. 
 
 155% lbs. 
 
 1. 1 cwt. 1 qr 
 
 2. 21 and 8 dwt. 
 
 3. 57200 times. 
 
 4. 472 m^ hhd. 
 
 5. 3 lb. 14 dwt. 13i gr. 
 
 6. 24 m. 7 fiir. 4 per. 
 
 7. 47 yds. 
 
 a 2s. 5^^d. 
 9. 237Me lots. 
 
 10. £1 28. 4^d. 
 
 11. 4bu.2pks. Igal. U|qt 
 
 12. 53 dys. 13 hrs. 42% Mn. 
 
 13. $94.50. 
 
 14. $302.40. 
 
 15. 1 hhd. 15 gal. 3 qts. 
 
 16. $1.80. 
 
 17. $2-215. 
 
 18. 3| qts. 
 
 EXERCISE 3(J. 
 
 1. 1003-87. 
 
 2. DxbLL, MMMMDCCIX, LXlx, DXXXIV. 
 
 3. Fifty-nine whole numbers, and ten million nine 
 hundred and sixty-seven thousand, nine hundred and 
 ten one hundred milliontbs ; one hundred and thir- 
 teen billion, three hundred and sixty-seven thousand 
 one hundred and eighty-nine whole numbers, and 
 one hundred and ^bur thousand two hundred and 
 fifty seven millionths ; eighty-nine billion, seven 
 hundred and fifty-four million six hundred aad 
 twenty-one thousand nine hundred and thirty-six • 
 ten billion, one thousand whole numbers, and one 
 thousand ten millionths. 
 
 . gr. 
 gland 
 
 4. £1423 11a. 3d. 
 
 5. $26. 
 
 6. 643478 lbs. 2 oz 
 
 7. 1 oz, 12 dwt. 14 
 
 8. 55 ys. longest in 
 
 9. £89 Is. 3d. in 1st. 
 £68 17s. ll#d. in 2nd. 
 £20168.7.M*!iu3rd&4th. 
 
 10. $74.80. 
 
 11. 29 d. 12 h, 44 m. 3 sec. 
 
 12. 60 yds. 
 
 18. 629,697,7O7,498-O0OO0O617. 
 
 14. 11 per. 2 yds. 1.3995 ft. 
 
 15. 
 
 16. 
 
 17. 
 
 18. 
 
 19. 
 
 2« 
 
 21. 
 
 (1.) 67950000-00. 
 
 (2.) 6795-000000. 
 
 (3.) 6-795000000. 
 
 (4.) 6795-000000. 
 
 (5.) 6795000-000. 
 
 (6.) 679500000-0. 
 
 $58-03. 
 
 $6160. 
 
 25000 miles. 
 
 5 ac. 3 r. SCjl, per. 
 
 hi 4- vnH. 
 
 each woman $22.00. 
 " man $66.00. 
 
 
answl;rs. 
 
 . 42f min. 
 
 3qts. 
 
 lion nine 
 idred and 
 and thir- 
 thousand 
 bers, and 
 dred and 
 n, seven 
 ireci aiad 
 Jirtj-'Six ; 
 and one 
 
 3er. 
 
 2.00. 
 6.00. 
 
 EXERCISE 36 — continued. 
 
 22. each man 8 lb. ]3|| oz., 34. 
 cost £12 9s. 10i|d. 35. 
 
 23. 196 jm men. 36. 
 1377^^^- men. 37. 
 
 24. 2 bu. 7 qts. 38. 
 
 25. Factors 3, 7, 5, 6 ; re- 39. 
 mainders, 0, 0, 1, 4. 4 V 40. 
 5 + 1X7X3 = 441.' 41. 
 
 26. 100 lbs. 42. 
 
 27. 2^ fur. 43. 
 
 28. $3194. 44. 
 
 29. (1.) 13 ac. 2 r. 17 per. 45. 
 
 23 yds. 4| ft. 46. 
 
 (2.) 9 ac. 2 r. 3 per. 29 47. 
 
 yds. 21 ft. worth 48. 
 
 $42570.06. 49. 
 
 30. 6| qts. 50. 
 
 31. 312 If days. 51 
 
 32. 1G9 bu. 1 pk. 6 qts. 1 pt. 52. 
 2f gills. 5,3. 
 
 33. $44 729. 54. 
 
 156 cds. 103 c. f 
 Gain $345.91. 
 
 £67 9s. 
 £63 6s. 83." 
 
 m 
 
 $602.13. 
 
 3i%d. 
 
 40;^ suits. 
 
 29464t.5cwt.2qr.241b. 
 
 13 persons. 
 1032694 grains. 
 
 2827 w. 3 d. 16 h. 19 m. 
 
 16640 times. 
 
 36 §1 oz. 
 
 27. 
 
 •246956 cords. 
 
 11666-66, &c., sheets. 
 £19 5s. 
 £27 lOs. 
 
 14 cwt. 
 72 cents. 
 $1.92. 
 
 1 pk. 4 qts. 
 
 EXERCISE 37. 
 
 1. 16. 
 
 2. 7448. or $148.80. 
 
 3. 3 cwt. 1 qr. 6 lbs. 
 
 4. 108 yds. 
 
 6. 120 dinners. 
 
 6. 4 coats. 
 
 7. 65 sheep. 
 
 8. 58 cwt. 1 qr. 20 lbs. 
 
 9. £4 Is., £10 9p. 3d. 
 
 10. 21| barrels. 
 
 11. 32 centa. 
 
 12. 42uf burruls. 
 
 13. 23 yds. 1 qr. 2 nl. 
 
 14. 29535/y and 141X. 
 16. $10. 
 
 16. 160 min. or 2 h. 40 min. 
 
 17. $70. share of each, $2. 
 
 18. 7| barrels. 
 [19. llf bush. 
 
 20. 168. 
 
 21. $35. 
 
 22. $4-444. 
 
 23. 9| days. 
 
 24. 11 of a dollar. 
 
{/ 
 
 // y 
 
 »l f 
 
 172 
 
 1. U2. 
 
 2. None. 
 
 3. 101. 
 
 4. srr. 
 
 1. 60. 
 
 2. 16. 
 
 3. 240. 
 
 4. 180. 
 
 K^BOISH 38. 
 
 *5. 18. 
 6. 346. 
 7. 25. ■ '■•'./' 
 
 a. 143. 
 
 9. 11. 
 
 m 7. 
 
 .;- '.J;.-;^..; V. 
 
 KXBRCISE 39. 
 
 5. 12a 
 
 6. 144. 
 
 7. 240. 
 
 -v** 
 
 W-« 
 
 1 8. 7560. 
 9. 12B0. 
 10. 7200. 
 
 li 
 
 1. B2^} 
 
 2. 8f. 
 
 1. ^\ 
 
 2. ^ir. 
 
 B3CKR€ISH 41. 
 
 3. 9. 
 
 4. 1^5. . 
 
 KXEROLSB 42. 
 8. ^g^. 
 
 BXERCI8B 43. 
 
 5- 9|fti 
 6' 3^^, 
 
 6. 1! 
 6. «i 
 
 
 
 3. ^r, w, ¥^. w, w. 
 
 4., »f4, iif«, i^a, i||a 
 
 6. •V«,o||8,i^6 i5ft*e 
 
 ^ 
 
 '■A'. 
 
 (1) lff|. 
 
 (4) MM- 
 
 HXBRCISl 44. 
 
 4. *Mt. 
 
 
 •*• VW A. 
 
 ¥•