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n 
 
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 JO 
 
 .xa.tr 
 
 21 
 
 LOV] 
 
J^l ELEMENTARY 
 
 ARITHMETIC, 
 
 IN DECIMAL CU 
 
 DESIGNED FOB TH 
 
 CANADIAN S 
 
 BY 
 
 JOHN HERBERT SANGSTER. m a, m.d. 
 
 .1IA.THEMATI0AL KASTISR AND LBCTTTaBR IN CKBlf ISTRT AKi> 
 
 KATITBAL PHILOSOPHT IN THK NORMAL SCHOOL FOB 
 
 UFFUR CANADA. 
 
 LOVELL pfeikTINQ AND PUBLISHING COMPANY, 
 
 28 & 23 St. Nicholas Strket. 
 
 1874. 
 
Entered, according to the Act of the Provincial Parliament, in 
 th« year one thousand eight hundred and sixty, by Johw 
 T.OVELL, in the Office of the Registrar of the Province of 
 Canada. 
 
 i 
 
 L 
 
 ^^ 
 
' 
 
 ent, in 
 
 John 
 
 ice of 
 
 ^^* 
 
 ■9 
 
 PREFACE. 
 
 hr presenting the Elementary Arithmetic to his fellow Canadian 
 Tt-Mcliers, the author respectfully solicits their attention to the fol- 
 lowing brief explanation of its arrangement and design. 
 
 First, then, ^Ith regard to the purpose it is designed to serve 
 It may be remarked that the Elementary Arithmetic is eminently 
 a practical treatise on numbers. Every rule in the book is ex- 
 pressed as clearly and as concisoly as possible ; is then illustrated 
 by two or more examples worked out and fully explained ; and is 
 finally enforced by an exercise containing a sufficient number of 
 problems to thoroughly impress it upon the pupil's memory. Thia 
 latter object is still more completoty attained by the miecUaneoua 
 or review problems scattered through the work. It will, I- Dwever, 
 be observed, that, with the exception of Notation, Numeration' 
 the Simple Rules, and Decimal Money, no attempt has been made 
 to give the pupil worded reasons for the processes employed • that 
 except so far as the several rules areexplined by the examples 
 solved, the work of illustrating and explaining is left to the teacher 
 This plan has been adopted chiefly from two considerations. In 
 the first place, young children, those for whom the work is pri- 
 marily intended, learn the reasons of the rules far more easily 
 and expeditiously from familiar and repeated illustrations by the 
 teacher on the blackboard, than they can by studying printed dem- 
 onstrations ; and, in the second place, had these reasons and ex- 
 planations been inserted, they would have increased the size of the 
 book far beyond what was considered desirable. 
 
 It is however believed that in the greater number of instances 
 the rule U so worded, and the solution and accompanying explana- 
 tion of the two or three illustrative examples are so given, as to 
 enable the pupil to master and comprehend the rationale of the 
 process employed. This remark does^iot, of course, apply to the 
 extraction of the equare ana cube roots, but it holds with regard 
 W almoBt every other ruW ib, ^ne book. For a full elucidatiou and 
 
 B«fmm^m&s 
 
4 PREFACE. 
 
 dlBcusBJon of the principJcB involved In nrithrneticfil opemt'ons, 
 tlie att(M)t:on of tlic more advanced etudent is respecifully directed 
 to the uutlior's iN'ational AiMhtiielic. 
 
 With respect to the arrangement, h few words will eutflco In 
 commencing the licnentary the pupil is asHumed to iiHv'o no 
 previous knowledge of arithmetic, and accordingly great care 
 has been expended in wording the definitions, explanations, 
 rules, &c., as concisely as possible, and in making preliminary 
 problems of the very easiest description. The author has also 
 endeavored, at those paits of the sulyect at which the pupil in- 
 variably meets with more or less trouble and difficulty, to prepare 
 him for the considaatlon of the rule and the solution of problems 
 on the slate by a reries of simple mental exercises. It is not for a 
 moment presumed that these mental exercises contain all that is 
 necessary in th. way of preparation : they are rather designed to 
 serve as a sarrMe of the introductory drilling through TThich the 
 class should luiter the rule. The judicious teacher will continue 
 Bomc such ej creiso as a mental training until he is convinced that 
 his pupils can enter into the solution of questions on the elattf 
 without any Buch miserable ariiflces as the attempt to aid their 
 ability to add or subtract by counting on their fingers or on the 
 notches cut in their slate frames. 
 
 The teacher is earnestly recommended to begin, at as early a 
 period as practicable, drilling the pupils on the Mental Arithmetic 
 at the end of the book. He will find it the most efficient of all 
 means for calling forth and cultivating the intellectual faculties ot 
 his scholars, and at the same time the most unfailing and success- 
 ful mode of making them thoroughly comprehend the rinciples 
 of written arithmetic. Although the mental exercises alluded to 
 contain a large number of problems, it is taken for granted the 
 teacher wiU not confine his class to these, but will from time to 
 time supply them with similar questions of his own construction 
 
 The problems throughout the book are all new, and no pains 
 have been spared in reading the proof-sheets to ensure the moat 
 ngid accuracy in every part. 
 TOHOWTO, May, I860. 
 
 J 
 
 i\ 
 
 ^K 
 
 ^' i-i..-"^.-.^ :■ 
 
>pernt!on8, 
 y directed 
 
 iffice. In 
 
 > llHVO DO 
 
 [reut care 
 lanaliona, 
 eliminary 
 ' has also 
 pupil in- 
 o prepare 
 problems 
 not lor a 
 ill that is 
 signed to 
 'hich the 
 continue 
 »ced that 
 the Blatd 
 aid their 
 •r on the 
 
 s early a 
 ithmetio 
 nt of all 
 ulties o/ 
 success- 
 rincipies 
 uded to 
 ited the 
 time to 
 •notion, 
 lo pains 
 ae most 
 
 t 
 
 ■•^ 
 
 CONTEKTTS. 
 
 SECTION I. 
 
 Delirjjtions '^°" 
 
 Numeration .'.'.'.'.'.'.* ^ 
 
 Numeration Table!*..!.. ^^ 
 
 Notation !!!!!!!!!!! -^^ 
 
 Roman Notation ^^ 
 
 Simple Addition 1*^ 
 
 flecapUuladonandExamination'Q^ 27 
 
 Simple Multiplication!!!!!!!!!!! ^^ 
 
 Multiplication Table .* ' ^^ 
 
 To Multiply by a numbe;- noTgreai^r than'l^!!! It 
 
 To Multiply by a Composite Number ^^ 
 
 co"!Ste'.::"!'"* ''''''' *^- 12- 'and-noi '' 
 Proof of Multiplication!!!!! ' '"'^ 
 
 Simple Division *.*' *- '^^ 
 
 Short Division !!!!!! 42 
 
 To Divide by a Composite Number!!!!!!! 1« 
 
 Long Division 45 
 
 Proof of Division...!!!! • ^6 
 
 Recapitulation and ExaminaVion Que^lions'on Didsion 48 
 
•WP 
 
 6 
 
 CONTENTS. 
 
 SECTION II. 
 
 Decimal Currency , 60 
 
 To Jteduco Old Oiumdiaii Currency to Dollars and 
 
 Cents 61 
 
 To Koduce Dollars and Cents to Old Canadian Cur- 
 rency 62 
 
 Addition, Subtraction, Multiplication, and Division of 
 
 Decimal Money 63 
 
 Examination Questions on Decimal Money 65 
 
 Tables of Money, Weights, and Measures 66 
 
 Reduction Descending G2 
 
 Reduction Ascending 63 
 
 Compound Addition 65 
 
 Compound Subtraction 68 
 
 Compound Multiplication TO 
 
 Compound Division 76 
 
 To Divide by an Applicate Is'umber 77 
 
 Miscellaneous Troblema on Sections I., II 78 
 
 t 
 
 SECTION III. 
 
 Greatest Common Measure 81 
 
 Least Common Multiple 83 
 
 SECTION IV. 
 
 Vulgar Fractions — Definitions, &c 85 
 
 Reduction of Vulgar Fractions 88 
 
 To Reduce a Mixed Number to an Improper Fraction... 89 
 
 To Reduce an Improper Fraction to a Mixed Number 89 
 
 To Reduce a Fraction to its Lowest Terms 90 
 
 To Reduce Fractions to a Common Denominator 91 
 
 To Reduce a Compound Fraction to a Simple Fraction 93 
 
 To Reduce a Complex Fraction to a Simple Fraction... K 
 
 'k 
 
 ^^ffl^HSP 
 
 ■iiiii 
 
PAoa 
 60 
 
 61 ^ 
 
 62 
 
 63 
 65 
 66 
 
 62 
 63 
 65 
 68 
 TO 
 76 
 11 
 IS 
 
 81 
 
 82 
 
 • • 
 
 85 
 
 • • 
 
 88 
 
 f • 
 
 89 
 
 2r 
 
 89 
 
 • • 
 
 90 
 
 • • 
 
 91 
 
 n 
 
 98 
 
 t* 
 
 K 
 
 I 
 
 '■k 
 
 CONTENTB 
 
 nan 
 
 m 
 
 97 
 
 99 
 
 1(10 
 
 K I 
 
 K :< 
 
 Kti Reduce a Denominate Fraction from one di nomina- 
 tion to another 
 
 To ;:educe one Denoniinate Nimiber to the iMuctioi *f 
 
 another Denoniinate Nunil)CM 
 
 To rind the Value ofu Denoniinate Fiaetion 
 
 Addition of Fractions 
 
 fciibtraetion of Fractions 
 
 Mtdtiplic.ition of Fractions ' 
 
 Division of Fractions 
 
 To Multipl}' or Divide an Intcgiul Denominate dumber 
 
 by a Fraction 106-7 
 
 Decimals — Definitions, &c ^..., lOg 
 
 Kunjcratiou of Decimals \[[ l(,g 
 
 Notation of Decimals ^/" ](,g 
 
 Addition and Subtuiction of Dccin.nls *.*'... m 
 
 Wultijdifuiion of rx'cin.als .' .....'. ]]o 
 
 Division of Decimals )]«.; 
 
 To Kodiice a Vul^^ir Fuution to a Fccinia! ... no 
 
 Circulating Decimals,— Definitions, &c [ ik; 
 
 To Reduce a Pure Kepetend to a Vulgar Fractio:; .' 117 
 
 To Reduce a Mixed Repetcnd to a Vulgar Fiactioii 117 
 
 Addition, Subtraction, Aiultiplication, and Division of 
 
 Circulating iVci'^ als , jjg 
 
 To Reduce a ' ^nominate Number to the Decima' 
 
 ofanoti .• '^ateNmiibei , "19 
 
 To Find the 'Vjimal of a Denomh ate Kum- 
 
 iH^i" j20 
 
 Miscellaneoua x . 1 JSeotious I.-IV 121 
 
 SECTION V. 
 
 Ratio. 
 
 123 
 
 Simple Proportion 126 
 
 Compound Proportion .......'. 132 
 
 SECTION VI. 
 Practice , ,„., 130 
 
 iA& ^^Eis^aagSMKftBf Bi 
 
^ 
 
 g ' CONTENTS. 
 
 PAO« 
 
 Percentage « i4(^ 
 
 Coramisdion and Brokerage 143 
 
 Inaurance I44 
 
 Buying and Selling Stocks 144 
 
 SECTION VIII. 
 
 Simple Interest , I49 
 
 To Find Interest at 6 per cent 151 
 
 Compound Interest I53 
 
 Discount ,. 155 
 
 Bank Discount , 158 
 
 Simple Partnership 167 
 
 Compound Partnership I59 
 
 SECTION IX. 
 
 Profit and Loss , ^ 181 
 
 Barter .".* iq^ 
 
 Exchange of Currencies 168 
 
 Analysis , , , 1*71 
 
 SECTION X. 
 
 Involution „ „. I'rS 
 
 Extraction of Square Root I77 
 
 Extraction of Cube Root ... 180 
 
 Miscellaneous Problems 182 
 
 Mental Arithmetic 186 
 
 Answers to Exercises ,, 201 
 
 ;i 
 
 A 
 
 f 
 
 :( 
 
^A\ 
 
 W m^jft ' 
 
 PAO« 
 
 14(» 
 
 143 
 
 144 
 
 ?44 
 
 149 
 
 151 
 
 153 
 
 156 
 
 156 
 
 r.>« -167 
 
 169 
 
 ... 161 
 
 .... 16ft 
 
 >... 168 
 
 .... 171 
 
 .... I'!'5 
 .... 177 
 .... 180 
 
 • ••« lo^ 
 
 ...t 186 
 .... 201 
 
 4> 
 
 f 
 
 AEITHMETIO. 
 
 SECTION I. 
 
 OxuPiiyiviV^d, jSUMERATION, SIMPjuiiJ Auu^.l^^J!t, 
 SIMPLE SUBTRACTION, SIMPLE MULTIPLICA- 
 TION, AND SIMPL^ )» VISION. 
 
 1. Arithmetic is the study of numbers. 
 
 2. Numbers are expressions or characters that represent 
 one or more things of tlie same Icind. Thus one^ two^ thrccy 
 seven, eleven, twenty-six, &c., are numbers. 
 
 3. Numbers may be expressed either by words or by 
 characters. 
 
 4. Notation is the art of writing numbers by means of 
 \;haracters. 
 
 5. Numeration is the art of reading numbers thus ex- 
 pressed. 
 
 6. The characters used for the expression of numbers 
 are either figures or letters. 
 
 7. Arabic Notation is the expression of numbers by 
 figures. 
 
 8. Roman Notation is the expression of numbers by 
 letters. 
 
10 
 
 NUMERATION. 
 
 low^*— ^° figures employed iu writing numbers are as fo; 
 
 called one. 
 '' two. 
 three, 
 four, 
 jive, 
 six. 
 seven, 
 eight, 
 nine. 
 
 naught, nothiyig, cipher, or zero. 
 10. All numbers higher than nine arc represented bf 
 writmg two or more of these figures together. 
 
 1, 
 o 
 
 *-» 
 
 8, 
 0, 
 
 a 
 (( 
 i( 
 t( 
 u 
 
 (t 
 
 is written 
 (( 
 
 <( 
 
 tt 
 
 i( 
 
 i( 
 
 it 
 
 t< 
 
 tt 
 
 tt 
 
 tt 
 
 tt 
 
 (( 
 
 Thus, Ten 
 
 Eleven 
 
 Twelve 
 
 Thirteen 
 
 Twenty 
 
 Twenty-one 
 
 Twenty-two 
 
 Thirty 
 
 Thirty-one 
 Forty 
 Fifty 
 
 One Hundred 
 One IJundrcd and Ten 
 One Hundred and Eleven " 
 11. When a number consists of several figures- 
 The first or right-hand figure is called the units' figure, and 
 
 IS said to stand in the ■unita'' place. 
 The second figure from the right hand is called the tens' 
 
 figure, and is said to stand in the tens' place. 
 The tlnrd figure from the right hand is called the hun- 
 dreds' figure, and is said to stand in the hundreds' 
 place. 
 
 The fourth figure is called the thousands' figure, and la 
 said to stand in the thousands' place, &c. 
 
 o- ^v: ^''J« %^^:«« h 3, 3, 4, 6, 0, Y, 8, and 9, are called 
 ^igmjicant Jngiires, because each of them represents or 
 
 10 
 11 
 12 
 13 
 
 20 
 
 21 
 
 22 
 
 30 
 
 31 
 
 40 
 
 60 
 
 100 
 
 110 
 
 111 
 
 4 
 
3 OS fo? 
 
 ero. 
 ited b.f 
 
 N\JMERATION. 
 
 frtnnds for one or more individual 
 called (/i/;iV.s, from a Latin woi'd ii 
 many uneducated peraona are iu 
 the lingers. 
 
 13. Tiic character is itself 
 merely to change the value of 
 making them occupy ditterent pi 
 olone is scven^ followed by one cipli 
 by two ciphers it is '700 or scvev hun 
 it is 7000 or seven thousand^ kc. 
 
 Exercise 1. 
 
 1. Writo nently on your elate nil tlie numbers from 1 to 100. 
 
 2. Write noatly on your ulate all tho numbers fmni 100 to 200. 
 Road the following numbers : 27, 164, 19, 91, 107, 789, 420, 999. 
 I.Vad tlie follow! iij.' numbers ; 10,18,12,61,31, 21,409,717,800. 
 ■\Vrito tho tbllowini; numbers : twoijlyeiuht ; five Innulred 
 
 and sevontecn ; cloven ; sixty-five; two hundred and nine; 
 
 forty ; niiiete(M). 
 Write the following numbers . one hundred and tbirty-(?even ; 
 
 nine hundred nnd six; seventy-ono ; eight hundred and 
 
 seven ; two hmidred and fifty. 
 Rotxd the followini: numbers : 103, 403, 701, 808, 917, 800, 711. 
 
 S. 
 4. 
 6. 
 
 6. 
 
 8. Write the following numbers: seventy- nine ; eiglit liundrcd 
 
 iind forty ; seven hundred and eleven ; four hundred and 
 sixteen • five hundred and five, 
 
 9. Ro!\d the following numbers : 909, 81, 17, 111, 606, 610, 170,919. 
 10. Write the following numbers : llfty-nirie; seventeen; seventy- 
 
 one; nineteen; nine hundred and forty ; sixty-one ; four 
 hundred and twelve. 
 
 e, and 
 
 3 tens' 
 
 I hun' 
 dreds' 
 
 md is 
 
 called 
 its or 
 
 14. To facilitate the reading of hirge numbers, they 
 are divide(i into periods of three figures each, beginning at 
 the ri'rht-hand side. 
 
 't3' 
 
 15. The names of the periods are as follows : — 
 The first or right-hand period is that of Units. 
 The second period is that of Thousands. 
 
 The third " " Millions. 
 
 The fourth " 
 
 u 
 (1 
 
 u 
 
 (( 
 
 The fifth 
 And so on according to the following- 
 
 Trillions. 
 
 i«ataiiB«aw; 
 

 12 
 
 i^i 
 
 V : 
 
 NUMERATION. 
 NUMERATION TABLE. 
 
 
 1st. The names of the period, in tlieir order 
 Example 1 -Read the number 142619 
 
 thus&ti?;:;:.^!;- t:r^v^^ \-v^^ ^-^ 
 
 which the left-hand one IthTn/T'''' '^' ^^^'^^^^' ^^^ 
 that of units T e„ rnl!i- ^^^^^'o^s^ihIs and the other 
 
 «'^<1 th.t thm. a,^";von h?" ,"'f PT'^V^^ separately, we 
 and six hundred „,drvon;'-'^ ""'"'^ ^orty-two thou and 
 whole toffether^Jvon f "^J*^ "/""^^ "">ts, and, readin^.^ the 
 
 hundanl mHl7ev;;;t;!ninr "'^^ ^"''^^'^^^ ^''«"«^"^i^ «i^ 
 
 Hnt''''r:;T:Z^P'\^ *'»« ^"!"bx'r 6V0493278900. 
 
 * I 
 
 ^ 
 
ITOTATIOK. 
 
 18 
 
 
 «*l" 
 
 } 
 
 iSllla^^'^^*"^ P'"^"^' "'" '"^*'' thousands, millions, 
 The 4th period is six hunched and seventy biUions. 
 The 8d period is lour hundred and ninety-three mill- 
 ions. •' 
 
 The 2d period is two hundred and seventy-eiKht thou- 
 
 sand ; and ° 
 
 Tlie Ist period is nine hundred units 
 Zhl T'f "^' f^'"''^ togotlK. , we find that the number is 
 
 ol;^ iunle^^^^^^ " ''"^'"'^ ""^ seventy-eight thousand. 
 
 Example S.—Read the number 67040000000007 
 Here pomthig off into ptM-lods we get 
 , 67,040,000,000,007, 
 
 tf' n2 f" •1'''^';, *^'^ "T^' °*' ^^^^'*»' beginning at the low- 
 
 T ; n o.r '' '^'T''''''^^^ """'^"^''' *>^"'«"^' ^»d trillions. 
 Then reading eaeli period separately, we hive sixty-seven 
 tnl hons m the highest period, fort/'billions in Ue next 
 
 last Fi":,,t^?."\'^*' "r*""^' "^ ^^•^^^^^^' -'^ ^«'--" inthe 
 
 number r^' "'^' "''' '^^'''^^'' ^" ^"^ ^^'^^ ^^'^ Si^^en 
 
 Sixty-seven trillions, forty billions, and seven. 
 
 Exercise 2. 
 Road the following numbers : 
 
 l J???,', ®^^^ ' ^"*^^ ' *°'^ •' ^019 ; 6111 ; 96003 ; 8674567. 
 
 2. 91131140; 967004290; 61300400007023 
 
 3. 1001001001001; 6700000000(59; 81008100810081. 
 
 4. 91234013402 ; 91234207109 ; 100000200003004. 
 6. 67189456713427 ; 9100009134000671001 
 
 6. 71345071913401300041234. 
 
 7. 100001000001000000 ; 203040506070809. 
 
 8. 908007000600006 ; 4003000200001. 
 0. 2046008010 ; llllllllllllii. 
 
 10. 40007 ; 9000000009 ; 870008700087. 
 
 NOTATION. 
 ^^17^ro writ© down numbers, we must attend to the fol- 
 
14 
 
 NOTATION. 
 
 RULE. 
 
 ^ Begin ai *he hf-ha'^d ndc and torite down each period 
 in its proper ord^r, as thounh it were a period of units. 
 
 Place a cipher in each voicani place that occurs in any 
 period; and if any period be wholly vacant, fill it iclth 
 ciphers. 
 
 ^ Example 1. — Write down as one number sixty-seven 
 millions four thousand and eighty-nine. 
 
 Here the left-hand period is 67 millions, the next period 
 to the right is 4 thousand, and the last or ijgiit-hand period 
 IS 89 units. Then writing these together and filling the va^ 
 cant places in the thousands' and units' periods with ciphers 
 we get for the required number 67,004,089. ' 
 
 ^ Example 2. — Write down as one number seventeen bill- 
 ions four hundred and twenty-six thousand and one. 
 
 Here we begin by writing down 17 billions; this we 
 follow by 000 in the period of millions, this by 426 in the 
 period of thousands, and this by 001 in the period of units. 
 Placing these together we get for the required number 
 17,000,426,001. 
 
 Exercise 3. 
 "Write down f1ie following numberB : 
 
 1. Three thousand and twenty-nine ; five thousand and seven- 
 
 toen ; six thousand five hundred; eight thousand and 
 eight ; nine thousand two hundred and seven ; four tliou- 
 sand and ten ; seven thousand and sixty-one : eight thou- 
 sand seven hundred. 
 
 2. Eighty-seven thousand four hundred and eleven ; ninety-four 
 
 thousand and six ; thirty thousand four hundred and fifteen • 
 twenty-four thousand and twenly-four ; seventy thousand 
 SIX hundred ; thirty thousand and one. 
 
 8. Five hundred and sixty-seven thousand ; two hundred and four 
 
 thousand and sixty-three. 
 
 4. Seven hundred and sixty-two thousand seven hundred and 
 nine. 
 
 6. Six hundred and four thousand and ninety. 
 
 a. Seventeen millions and eighty-one ; forty millions two thousand 
 and SIX. 
 
 9. On« hnn«1rr>rl onrl tMf+v mUUm-in -(-.r V.-..»,^--„j j ^ .i ■» 
 
 aud eeven ; twenty millions and eteven. 
 
ROMAN NOTATION. 
 
 15 
 
 6. 
 
 Eight hundred and Beven milliona twentv thousand one hun- 
 TTivrhn"*^ ^'J? ■' ''';?/? ^^^Jired millions and twenty thousand 
 ions mid'ona'' """'''''' ^"" thousand and five ; twenty bin: 
 10. Sixty trillions sixty millions and sixty 
 lo l^r^^'V^t!!'!.*"^^'""^ ^'^'■^'^'" minions and seventy 
 ''• ^iSud'ldrd^t^e'nFy-S""" ^^"^- ^""^'-^^ «^---"d two 
 
 ROMAN NOTATION. 
 
 18. The seven letters used in Roman Notation, with 
 fcjir values, are as follows : 
 
 I. 
 V. 
 X. 
 L. 
 C. 
 
 One. 
 
 Five. 
 
 7'en. 
 
 J^ift^- 
 
 j^ 07ie Hundred. 
 
 -^ Five Hundred. 
 
 ^ One Thousand. 
 
 19.^ All other numbers are expressed by repeatinsr 
 •tfombmmg these letters, as in the following *' "^ » 
 
 or 
 
 TABLE. 
 
 I... 
 
 11... 
 
 Til... 
 
 IV... 
 
 V... 
 
 VI..., 
 
 Vll.. 
 
 VIII 
 
 IX.... 
 
 X 
 
 XL... 
 XII.. 
 XIIL. 
 XIV.. 
 XX... 
 XXX. 
 
 ■VT 
 
 1 
 
 2 
 3 
 4 
 5 
 6 
 1 
 8 
 9 
 
 10 
 
 11 
 
 12 
 
 13 
 
 14 
 
 20 
 
 30 
 
 40 
 
 60 j 
 
 LX .... 
 
 LXX... 
 
 LXXX 
 
 XC 
 
 C 
 
 CC 
 
 CCC... 
 
 CD 
 
 D 
 
 DC. 
 
 DCC... 
 DCCC... 
 
 CM 
 
 M 
 
 MM 
 
 MMM ... 
 
 .. 60 
 .. '70 
 .. 80 
 .. 90 
 . 100 
 . 200 
 . 300 
 . 400 
 . 600 
 . 600 
 . '700 
 . 800 
 . 900 
 .1000 
 .2000 
 .soon 
 
 A;MMD ....8500 
 
 MDCCCLX I860 
 
16 ROMAN NOTATION. 
 
 20. From this table we learn that 
 
 1st As often as a letter is repeated its ralue is re. 
 peated, but no letter can be repeated more than 
 thrice. 
 
 2d. When a letter of a lower value is written before 
 one of a higher, its value is to be subtracted ; 
 but the only letters that may be thus writteu 
 before others are I, X, and C. 
 
 8d. When a letter of a lower value is written after one 
 
 of a higher, their values are to be added. 
 4th. A bar or a dash written over a letter or combina- 
 tion multiplies its value by 1000. Thus X = 10, 
 X = 10000, C = 100, C=: 100000, CCXV=215, 
 CCXV = 215000, &c. 
 5th. The characters for 5, 50, and 600 never stand be- 
 fore others of a higher value, and never suffer 
 repetition. 
 
 6th. A character can never stand before any other than 
 one of the two next higher in value. Thus I 
 can stand before V or X. but before no other 
 letters; X can stand be.v/re L or 0. C can 
 stand before D or M ; and sa on, according to 
 the following scheme : 
 
 :t 
 
 li 
 
 i 
 t 
 
 7 
 
 8 
 
 
 io. 
 
 njMWgni 
 
EXAMINATION QUEST iONS. 
 
 17 
 
 lne is re- 
 more than 
 
 en before 
 ibtracted ; 
 IS writteii 
 
 after one 
 ;d. 
 
 combina- 
 i X = 10, 
 
 S:V=215, 
 
 stand be- 
 ver suffer 
 
 ther than 
 Thus I 
 
 no other 
 C can 
 
 )rJmg to 
 
 DXERCIS^ 4. 
 tbeS?nTomS.ont°r^^^^^^^^ '° ^'"^^'^ Notation, also read 
 
 ^' ^^X^ ViVlxXI V'. ^"^' ^^^^' CDLXX VIII, CCCXXX, 
 
 !?r^T^^^^'^"' ^^^^<^VI, DCCCXLIII, CMX, MI, MCI), 
 a CII,DXI,MDXXXIXJ!kIMMXXX,MMDCCCLVIII CCCI 
 4. CCCXXXIII, X, XC, VM, VCMLXXVII, XXVMMXXVII* 
 XLCDXLIV. ^v.vvii, 
 
 6. MDCCCXCIX, MMCCXXII, MVDV, MXDOIV 
 VMMMDCCCLXXXVIII. 
 
 Exprees the following commoti numbera iu Roman Numerals • 
 
 6. 202, 47, 91, 80, 20, 77, lOl, 10, 111, 606. 
 
 7. 437, 908, 899, 763, 497, 829, 827, 999, 888. 
 
 8. 2233, 3232, 3333, 4321, 12:34, 5078, 8766. 
 
 9. 9999, 25071, 891347, 912342, 16713. 
 
 JO. 191919, 29134, 23476, 912345, 1678942, 3456713. 
 
 UECAPITULATION AND EXAMINATION QUESTIONS. 
 
 1. Ques/ion. Wliat 'b Arithmetic? 
 
 Answer. Arithmetic ie the study of numbers. 
 
 2. Q. What are numbers ? 
 
 A. Numbci-a arc expre-ssions or characters that represent one 
 or more things of the same liind. r^^preseni one 
 
 ^* 9* ^^^^ J8 "*'"«■'!/ oi" the unit of a number ? 
 
 f^^tTh*^'®""u °^ *" number is one of the equal thi.iifs 
 that the number expressen. * liiiiiKs 
 
 4. Q. In the number 19 horses what is the unit? 
 A One horse. 
 
 6. Q. What is the unit In the number 26 shilliDcs? 
 A. One shilling, ^ 
 
 6. Q. 
 
 7. 
 
 A. 
 
 8. Q 
 
 9. 
 
 in. 
 
 What is the unit in 16 days? 19cowa? 107 beans ? 3 far. 
 thmgs? 198 lbs. ? 607? 43 bushels? 293? 769 iSiV? 
 
 °°a?e"t}^ey'?'"°*^^'' ^""^ *^®'^ ""^ '^'■'"'♦^' numbors, and what 
 There are two inethods of expressing numbers, 1st bv 
 words, and 2d, by characters. umueis, xsi, oy 
 
 .- What is Notation? 
 A. Notation is the expression of numbers by characters .• 
 Q. What is Numeration ? * ' 
 
 ^' "^ a?ter8*'°" '^ ^^'^ '"^'"'^'"'^ ""^ numbers expressed by char- 
 
 Q. What different charafttem am naaA ^^- ♦ho t^.- -* 
 
 numbers? ' '"' '^® cApii;as;ou ui 
 
 A. NumbwB are expresMd rtthfr liy jetton or by figure*. 
 2 
 

 18 
 
 11. Q 
 A 
 
 12. Q. 
 J. 
 
 13. Q. 
 
 14. Q. 
 
 16. Q. 
 A, 
 
 16. Q. 
 il. 
 
 17. Q. 
 .1. 
 
 18. Q. 
 .1. 
 
 19. Q. 
 .<!. 
 
 20. Q. 
 
 A. 
 
 21. Q. 
 
 22. Q. 
 i4. 
 
 23. Q. 
 
 EXAMINATION QUESTIONS. v 
 
 Whnt In Ronmu Notation f 
 
 IJorimu Nouulou m tm iiri of oTproBslnjf numbers by Owr* 
 
 titiii It tioi'B of tlu» iilplialiet. 
 VVImi .•••« li.o Htivo I mtiiu>iiil IcttriH ••inployoii In Kun?un 
 
 .No iitlo", niKl \v)»!it iiio tliolr \ 111 « ^ 
 ^ "T l)^ = 0, X — lu, L = 60, C = luo, D = 600. ami M = 
 
 How uiujiy times inny ouoli of thcso lotfcrB, cucoi t V, L 
 Hiul 1», bo roiu«moil ; uiul wliua thus roiiiutiil, wliut do 
 tlu'y moan ? 
 
 No littor (uih t)o ropt'jitotl more tlmii thvK>. timfs; and when 
 a li'lter Is thurt ruptntod, its valim Ia rop«<nttMl. 
 
 Wlui), a letter of a lower vuhjo Iswrlttou Wore one of a 
 
 hinliiT, wimt dot'8 tlu' iHitatlon Imply ? 
 Winn a letter «»f a lower value Id wrhteii before one of n 
 
 lilglior, Us value Is to be subtraoted from that of the 
 
 bluhi«r. 
 
 Wlu-n a letter of n lower value is written after one of a 
 
 lii^lier, what does the notation Imply < 
 When a letter or repetition of letters of a lower value 
 
 eomert attor a letter of a hltcbor value, the uotutlon i«n- 
 
 plUv that their values are to be addeil. 
 
 What ettbct has u bur or a dash over u letter or combination 
 of letters ? 
 
 A bar or a dash written over a letter or combination of let- 
 ters, increases the value a thousand fold. 
 
 What letters are never written before others? 
 
 V, I>. 1), are never written before letters of a hljitber value. 
 
 What letter is never written with a bar over It, and why ? 
 
 1 ; because we have already m\ e.xpreHslou for 1000, viz." M. 
 
 What are the figures used in .Arabic Notation? 
 
 The ttgures employed In Arabic or Common Notation are 
 1,2,3,4,6,6,7,8,0,0. 
 
 What are the tlgures 1, 2, 3, 4, 6, 6, 7, 8, 9 culled, and 
 
 why ? 
 The figures 1, 2, 8, 4, 6, 6, 7, 8, 9 are called significant 
 
 figures, because each of tbem repreachts one or more lii- 
 
 d.vidual things. 
 
 By Avhaf othtr name are they alHO known, and why ? 
 They are also called disrits, from a Latin word meaning " a 
 
 finger," because many persons h;ihllually count on tho 
 
 fingers. 
 
 What Is the character called, and why? 
 
 The character Is called naught, nothing, cipher, or zero, 
 because It has no value In itself, and is merely used to 
 give the digits their proper place. 
 
 What is moant by the place of a digit? 
 
 A diyrit Is said to o-eupy the tlrst, second, third, fourth, 
 fifth, sixth, &e., placey aceordinu: as it is tlio lust digit to 
 tho right hand ot the number, Inst but one, lust but two, 
 laai but taree, last but loiif, laii but. nve, 4m. 
 
 I 
 
SIMPLE ADDITION". 
 
 nhprn by o«f 
 
 M 111 Kwipiin 
 >00, niul M = 
 
 cxovyt V, L, 
 iti'U, wliul clo 
 
 >»; and ■when 
 rbre one of a 
 
 tore ono of A 
 tliut of llie 
 
 lor one of a 
 
 lower vjiliio 
 iiotution iin- 
 
 oombinatlou 
 atiun of let- 
 
 iK^er value. 
 , and why? 
 1000, viz." M. 
 
 rotation are 
 
 called, and 
 
 sifinifieant 
 or more ia- 
 
 vhy ? 
 
 iiicaiiing " a 
 >unt on tbo 
 
 ler^ or zero, 
 fcly used to 
 
 ird, fourth, 
 itBt digit to 
 ist but twu« 
 
 19 
 
 S4. Q 
 A. 
 
 25. Q. 
 A. 
 
 ?6. Q. 
 A. 
 
 17 Q. 
 A. 
 
 28. Q. 
 
 .1. 
 
 20. Q. 
 
 80. Q. 
 
 81. Q. 
 A. 
 
 What tmrnpR are given to the different ordera or dace. 
 »ru'ini,ii.« at tlio riKlit-liand hide? ^ '°*^®' 
 
 UniH.UuH liundriMlH, lhn,iH:uids, tcnn of thousand, hnn 
 
 ''Mm -"VI? r*^'''' *• '"• V'">""'*« "^"^'^ ot,o of tlio order of 
 ti'iir, ten ton« onoof the order of hundreda • t..n him 
 dre.lH, one ot the order of thoimnnds Ac ' 
 
 B»ind, &cr ^' "^''" hundred, OOOO = 9 thou- 
 
 WJiiit is a period of numl)er«? 
 
 ol- ciph^rs.""'"''"^'- '" " "*^^ ^' Bcquenoeof tLreo digit. 
 Why are periods used? 
 ) er.udB are UHod to facilitate the reading of numbera. 
 
 HKht"h^u!!i^o."?ZMr "^" ^^'^^'^ ^^«^""-^ -^^^^ the 
 UnliM, thoneaodB, niiliio.s, bilHonfi, trillions, &o. 
 
 or"?!.?;.': tnroVSiV'"" "''''^ ='"^' P°'"^ '^«' «nto periods 
 ^''dSo'i' ;;f"\'"«"*^''.K at the left-hand Bide, read each 
 
 Jp.X ZSZ ,r;;\^ ,!,:»■ -o"^* -u 
 
 What iB the rule for writin^r numbers? 
 
 SIMPLE ADDITION. 
 
 n.ore numbfrf " '"""^'' "' ^'" '^ ^"^ '^'' '^^ «f two or 
 
 MENTAL EXERCISES. 
 1, Count aloud up to one hundred 
 
 '• "TreS??^"''""^^'' 2'andn3andl»4andn6and 
 ^' ^SoTi? ' •^"'^ 2 "^«*'^"' * ^"'^ 2? 6 and 2? 8 and 2? 10 
 4. TTow niany do 1 and 3 make? 4 and 3? 7 and 3? 10 and 3? 
 6. novv- many do land 4 make? 6and4? 9and4? 13 and4? 
 
 6. Ho^v many do 1 »ad 6 makof 6 aad 6f U aud 61 16 and 6f 
 
 
20 
 
 SIMPLE ADDITION. 
 
 
 1. 
 
 8. 
 
 9. 
 10. 
 11. 
 .12. 
 
 13. 
 
 14. 
 16. 
 
 18. 
 17. 
 18. 
 
 19. 
 
 20. 
 
 21. 
 
 22. 
 
 23. 
 
 24. 
 
 25. 
 
 How many do 1 and 6 make? 7 and 6? 13 nud 6? 19 and 6? dco. 
 
 How many do 1 and 7 make? 8 ami 7V 16 and 7f &c. 
 
 How many do 1 and 8 make? 9 and 8? 17 aiid-8? &,o. 
 
 ♦How many do 1 and 9 make? 10 nud 9 ? 19 and 9? &.O. 
 
 How many are |7 and $8 and .*5 and $9 and |2? 
 
 How many aro 6 api)le8 and 2 apples aud 3 apples and 6 applei 
 and 7 onpk'B and 9 appli's? 
 
 HoAV many aro 6 pegs and 8 pegs and 7 pegs and 4 pegs and 3 
 pegs and 2 pegs and 9 pegs ami 6 pegs and 1 peg ? 
 
 Ifland 9 are how many? 26 and 9? 37and9?42and 9? 71 and 9? 
 
 How many are 16 aud 8? 23 and 8? 35 and 8? 39 and 8? 1> 
 and 8? 
 
 How many aro 6 and T ? 21 and 7 ? 32 and 7 ? 43 and 7 ? 54 and 7 ? 
 
 How many are 6 and 6? 7 and 9? 18 and 4? 23 and 8? 37 aiulSf 
 
 How many are 11 and 9? 13 and 8? 15 and 7? 17 and 6? 18 
 and 5 ? 
 
 How many are 9 and 5? 13 and 7? 27 and 9? 82 and 7? 93 
 
 and 9 ? 
 
 How many are 7 and 2? 9 and 8? 17 and 6? 23 and 9? 32 and 
 
 7? 9 and 9? 
 
 Jane paid 7 cents for apples, 16 cents for cakes, 9 cents for 
 nutp, and 8 cents for candy ; how much did she pay for the- 
 whole? 
 
 William gave 23 cents for a ball, 9 cents for a top, 5 cents for 
 a cord, 8 cents for a bat, and 9 cents for pencils ; what did 
 he pay for the whole ? 
 
 A farmer receives $67 for flour, $9 for potatoes, $7 for butter, 
 $6 for turnips, $9 for straw, .1>8 f)r clieese, and $9 for 
 poultry ; how much does he get for the whole? 
 
 A gentleman bought 27 books in January, 8 in February, 9 in 
 March, 6 in April, 9 in May, 8 in June, 9 in July, 7 in Au- 
 gust, 8 in September, 9 in October, 8 in November, and 7 in 
 December ; how many did he buy in all ? 
 
 Fa'iny worked 7 problems on Monday, 9 on Tuesday, 8 on 
 "Wednesday, 9 on Thursday, on Frid.ay, and 8 on Satur- 
 day ; how many did she work in the entire week ? 
 
 22. The numbers to be added together are called the 
 addends, and the result of the addition is called the ftuni. 
 
 — ■■ M ■■ ,. , . — I II I ■ 1,111.1., I I. II ■■ — !■ ■ II ■■ . I .-.■■^ M . I I 111^ 
 
 * The pupil should be continued at this exerciee until he can 
 count by 2's, 3'8, 4'8, 5's, ifec., with as much facility as he ca'^ by 
 I'.s. For example, beginning at any number, say 17. he phould be 
 ible to count r.apidly bv firos,' thus, 17, 19, 21, 23, 25. Vc. ; or by 
 threps; thus, 17, 20, 23, 26, 29, 32, &c. ; or by sevens; tlms, 17, 24, 
 31, 88, 45, &o. In fact, he cannot add with ease »ud comfort uutll 
 he has been well drilled iu some suoh exercise. 
 
filMPLE ADDITION. 
 
 >and6? dso. 
 Ac. 
 
 &c. 
 
 ? &.O. 
 
 and 5 apple* 
 
 1 pegs and 3 
 ? 
 
 »f 71 and 9? 
 
 9 and 8? !> 
 
 ? 54 and?? 
 8?37aiul3f 
 7 and 6? 18 
 
 2 and 7? 93 
 
 i 9? 32 and 
 
 9 cents for 
 pay for tho> 
 
 , 5 cents for 
 ; what did 
 
 ' for butter, 
 and $9 for 
 
 bruaiy, 9 in 
 ly, 7 in Au- 
 er, and 7 in 
 
 esflay, 8 on 
 
 3 on Satur- 
 ? 
 
 called the 
 
 le fmni. 
 
 ■ -■— — ■_.■. I ■■ .1 m t^ 
 
 ntil he can 
 » he ca" hy 
 e phould be 
 ■Vc. ; or by 
 time, 17, 24, 
 tmfort until 
 
 SI 
 
 RULE FOR SIMPLE ADDITION. 
 25. Write the addends under one another so that ..^ v 
 
 o6tomerf before. ^^^ *^ ^^^ *«»*« «« f^«i 
 
 EXAMPM l._Add together 42T8, 1610, aiid SOOl 
 
 ISIO fi"d thelu™ nV^^h'"*' ""."'"^'' '^»'"""- "e 
 qnni „ i ?. , " "''* <'"■'«' does not exceerl o ' 
 
 8889 
 
23 
 
 SIMPLE ADDITION 
 
 
 
 EX£ 
 
 RCISE 5. 
 
 
 
 (I) 
 
 (2) 
 
 (8) 
 
 (4) 
 
 (6) 
 
 ("') 
 
 128 
 
 I2;a 
 
 1111 
 
 1000 
 
 11000 
 
 0()iio40 
 
 201 
 
 8412 
 
 2222 
 
 218 
 
 1200 
 
 90:]40 
 
 222 
 
 nil 
 
 8100 
 
 4010 
 
 ()U0 
 
 laoo 
 
 888 
 
 8031 
 
 810^^ 
 
 1201 
 
 20104 
 
 2 
 
 7. now many are 713 + 80 + 3 ? 
 
 8. How many aro 12100 + 2210 + 1001 + 421 + 10002? 
 0. How many are 1020 + 304 + 1111 + 3212? 
 
 10. flow many aro 222 + 1111 + 3333 + 1212 + 90000? 
 
 11. How many aro 60004 + 8000 + 741 + 21000? 
 
 12. Aud together twenty-three, four buudred and sixteen, ana 
 
 throe thousand and sixty. 
 
 478 
 693 
 492 
 
 1663 
 
 Example 2.— Find the sum of 478, 693, and 492. 
 
 OPERATION. Here we set down the numbers according tc 
 the rule, and, adding up the first column, we 
 find its sum to be 13, of which we set down the 
 3 under the first column and carry thel to the 
 second. The sum of the second column, with 
 the one carried, is 26, of which we set down the 
 right-hand figure, 6, under the column added, and cany tlie 
 2 to the next. The thi'd column, added, amounts to 16, 
 which we set down in full. 
 
 Example 3.— Add together 7149, 7182, 614, 9187, 
 
 1234, and 79813. ; 
 
 operation. Here the first column amounts to 29, of 
 
 7149 which we set down the right-hand figure 0, 
 
 and carry the 2 to the second column. The 
 
 sum of the second column, with the 2 carried, 
 
 is 17, of which we set down the 7 and carry 
 
 the 1 to the third column. The sum of the 
 
 third column, with the 1 caviled, is 20, of 
 
 which we set down the right-hand figure, 0, 
 
 105079 and carry the 2 to the fomth cohniin. The 
 
 sum of the four'^' column, with the 2 caniod, is "r-, of which 
 
 "we set do\vn 5 right-hand figme and can y tlie ;> to the 
 
 fifth column. Tlie sum of the fifth or lasl column, with the 
 
 a carried, is 10. w4uch we set down in full. 
 
 7132 
 
 614 
 
 9187 
 
 1284 
 
 79818 
 
SIMPLE ADDITIOlf. 
 
 23 
 
 0()ii<»40 
 
 9o:]-to 
 
 130G 
 2 
 
 ixtcen, ana 
 
 492. 
 
 cording tc 
 olumn, we 
 t down the 
 el to the 
 Uimn, with 
 t down the 
 1 cany the 
 ints to 16, 
 
 il4, 9187, 
 
 ) to 29, of 
 
 figure 0, 
 
 imn. The 
 
 2 carried, 
 
 and carry 
 
 iinn of the 
 
 is 20, of 
 
 figuic, 0, 
 
 liTiii. The 
 
 ', of which 
 
 le I) to tlio 
 
 1, with the 
 
 12^45 
 67134 
 91317 
 19134 
 
 (9) 
 80476 
 
 90i)7 
 
 986147, 
 
 91067 
 
 86 
 
 4071 
 
 937 
 
 (13) 
 987654 
 32109 
 8765 
 482 
 10 
 9 
 87 
 654 
 8210 
 98765 
 432109 
 
 EXEHCISB 6. 
 
 (2) 
 2233 
 
 4567 
 8912 
 8456 
 
 (6) 
 
 inn 
 
 2222 
 
 r>33 
 
 44444 
 
 6555 
 
 (10) 
 123456 
 789123 
 456789 
 123453 
 789123 
 456789 
 987654 
 
 (14) 
 8000700 
 600090 
 1129000 
 47896 
 8104906 
 23427 
 9867 
 999999 
 88888 
 710 
 9184761 
 
 718645 
 191371 
 234716 
 918130 
 
 (11) 
 
 84667 
 
 8000 
 
 69 
 
 470000 
 
 1096^-7 
 
 48001 
 
 290 
 
 (15) 
 8147137 
 913714 
 9100070 
 8000000 
 667755 
 44332 
 8355778 
 986754 
 71347 
 981675 
 19198 
 
 (4) 
 
 91600 
 
 7149 
 
 86004 
 
 19 J 30 
 
 (8) 
 18456 
 
 7 
 
 987 
 
 29 
 
 98613 
 
 (12) 
 728 
 
 674 
 
 1674 
 
 19006 
 
 191( 
 
 98698b 
 
 97979 
 
 (16) 
 
 987654 
 
 137867 
 
 149167 
 
 891S71 
 
 919198 
 
 171296 
 
 147867 
 
 182371 
 
 929292 
 
 292929 
 
 777777 
 
 17. Find the sum of 1247 + 91679 + 27 + 1987 + 1800 + 1798, 
 
 18. Find the sum of 13147 + ft 4. 6ifJ^« 
 i«i37 + 100 + 76649 + 8 + 967^ 
 
 O"? 
 
 1 il t o . nn A . ^. . 
 J.:rj.u T 90% + i«, ,1 + 
 
24 
 
 SIMPLE ADDITION. 
 
 19. How mnny are 6 + 27 + 93 + 47 + 679 + 496 + 9999 ? 
 
 ^' " 98d0?'''^ "''^ 12 + 21 + 679 + 976 + 709 + 9198 + 4617 ♦ 
 
 '^" '^'nL'?!fr^^'7 ^'',"'" *"'"^'«<.^ «"«! Bi'^ty, Bovcn thousand and 
 nn..lec!, fourtiK,U8a.d djrl.t hmulr. .1 .-.ral liftv, nine thou- 
 r: ..!» ■" l^^'-^n'.V^B X, btwn tlioiiKi d nine liundro'l ;in(l 
 ninc't>-iu, f,o.,c thou.sund lour hundred, h.x thouBu.id .liid 
 
 L\m A:'''r.'.^rH''""f''''^ '^"^ cighty-8cvon,and tour thou' 
 Baud tive hundred utid sixty-aoven. 
 
 22. Add togetlier twenty-seven thousand and sixteen, eicht thou, 
 
 eai.d and seven, sixty thousand four hundred and twenty, 
 nve, eighty-tour thousand six hundred and eleven, nineteen 
 UmuHaud and nineteen, tiftv-tive tlmusand seven 'hundred 
 and ninety thousand seven hundred and four. """i«"i 
 
 23. Add together sixty-seven tliousand and nine, forty-nine thou- 
 
 sand s'x hurdrcd and eighty-six, live hundred and tweuty- 
 flye thousand and sixteen, three thousand and eleven, eiglitV- 
 five tliousand seven hundred and twenty-seven, and sixteen 
 tliousand and seven. , 
 
 24. Add together two hundred and seven thousand six liundred 
 
 und nine, eleven njillions ami Hixteen,tlve niilliouH four hun- 
 
 tvtr. h„n?i?^i "'■ w" ''""•^'■«.'' ""\J twenty, s^xty-nx millions 
 t\vo hundred and twenty-nine tliousand and eii»hty-seveii 
 11 ne hundred and eighiy-seven millions six hundred an< 
 seventeen, and tive Ihousa^.d seven hu..dred and thirty-five. 
 
 25. An apple woman 80ld forty-seven apples on Monday, eiglitv- 
 
 nine ..n 1 uesday two hundred and sev, lUeen on Weclnei 
 (lay, one hundred and four on Thuivday, oi e hundred and 
 twenty on Friday, ami two hundred aid eiirhty. seven on 
 Saturday : how many did she sell during tlio week ? 
 
 26. A farmer sent five loads of oats to market. The first load 
 
 contained 63 bushels, the second 68 bus) i els, the third 79 
 bushels, the fourth 57 bushels, and the fiflh 63 bushels 
 How many bushels were there in the five loads? 
 
 27. The imports of the six principal ports of Canada for 1855 
 
 were us tollows : ^•^- * — ■ vi.->i-^.<--„ ^ 
 Quebec, .*456S376 
 
 and Port Sl;inley, . „ ,. 
 
 imports at these six places? 
 
 28. Dunn- 1S48 there wore exported from Canada 224801G bushels 
 
 Vsr,n .r']-.l>V\^f l^''''.''Y'''? '--M^o'-ted 3G45320 bushely ; in 
 5io-'i\? /-^'i ^"^li«lf ; >" 18»1. 4-275896 bushch; ; and in 1852, 
 ;,f." i^f ""'';;'^- ,"T "^='''>' »'"shels of wheat were ex' 
 poi ted trom Canada during the five years endii g 1852? 
 
 ^^•'^bn'hlll^ir ^''"' ^l'';\'."V"\'"'l^'"^'' *^« fi'-^^ fields him 749 
 buslK-lg, the second 1147 bushels, the third 890 bushels, and 
 
 vuu iimrini2iy uuaiwln : how many buehela of turuiug did 
 ho obiain from the four fleldiB? » ««» ui turmps qiq 
 
 - ~'"''»iWynii|| ii pi;iy i !i i)i i i ip|ii| p| |i||^ ^i j ^ h^ 
 
eiBfPLK ADDITION. 
 
 26 
 
 8tt 
 
 hoasand and 
 
 y, niiie tluni- 
 
 en, nineteen 
 
 SI 
 
 J^^t ?' *^®, large boot and shoe factories In Montreal the 
 work turnod cut din-ing a Avock w.8 as follown E ulay 
 1467 paii-H of 8l.o(8,Tuc8Uay, 1509 pair<s Wedncsd^ llfii 
 PH.r. Tlu.rsday, 1447 pairB, l^HcJay, ihli pnirs'a i Safurdav 
 
 ^Jc'Ugr?'' '' ^"''■'°" ''■^''' '''''' ^"""^ ^° *^^ yea»* 1703 l;c 84 years 
 
 <32) 
 
 12 
 S4 
 
 66 
 
 78 
 
 91 
 
 23 
 
 45 
 
 67 
 
 89 
 
 98 
 
 76 
 
 64 
 
 32 
 
 10 
 
 98 
 
 70 
 
 64 
 
 32 
 
 11 
 
 23 
 
 45 
 
 C6 
 
 77 
 
 88 
 
 89 
 
 36 
 
 79 
 
 24 
 
 68 
 
 90 
 
 81 
 
 86 
 
 71 
 
 86 
 
 98. 
 
 (33) 
 
 987 
 
 613 
 
 479 
 
 813 
 
 271 
 
 986 
 
 129 
 
 333 
 
 400 
 
 916 
 
 713 
 
 934 
 
 716 
 
 291 
 
 816 
 
 999 
 
 816 
 
 654 
 
 735 
 
 613 
 
 421 
 
 916 
 
 818 
 
 397 
 
 491 
 
 378 
 
 618 
 
 491 
 
 351 
 
 673 
 
 916 
 
 814 
 
 718 
 
 537 
 
 981 
 
 (34) 
 
 27145 
 91913 
 16719 
 91871 
 49181 
 37162 
 34507 
 891C4 
 66789 
 12345 
 
 67f:no 
 
 12845 
 
 67591 
 
 23456 
 
 7^912 
 
 84567 
 
 89123 
 
 45678 
 
 912S4 
 
 66789 
 
 71G42 
 
 97581 
 
 24G80 
 
 90406 
 
 71480 
 
 61311 
 
 44433 
 
 91671 
 
 32916 
 
 67137 
 
 01346 
 
 13471 
 
 91S99 
 
 12916 
 
 71307 
 
 (36) 
 
 753 
 
 197 
 
 631 
 
 976 
 
 319 
 
 864 
 
 208 
 
 642 
 
 186 
 
 421 
 
 987 
 
 666 
 
 321 
 
 123 
 
 456 
 
 789 
 
 808 
 
 707 
 
 606 
 
 404 
 
 606 
 
 660 
 
 770 
 
 880 
 
 990 
 
 169 
 
 178 
 
 144 
 
 916 
 
 723 
 
 444 
 
 718 
 
 999 
 
 806 
 
 437 " 
 
 (36) 
 
 16 
 
 61 
 
 81 
 
 47 
 
 29 
 
 87 
 
 46 
 
 98 
 
 68 
 
 42 
 
 17 
 
 93 
 
 82 
 
 64 
 
 70 
 
 62 
 
 16 
 
 17 
 
 91 
 
 87 
 
 63 
 
 22 
 
 71 
 
 33 
 
 42 
 
 71 
 
 93 
 
 45 
 
 67 
 
 16 
 
 49 
 
 98 
 
 86 
 
 73 
 
 34 
 
 
*4K 
 
 2d 
 
 ii i 
 
 EXAMINATION" QUESTIONS. 
 
 RECAPITULATION AND EXAMINATION QUESTIONS. 
 
 1. Question. Whnt, in AH/^i♦^r^r,^ 
 
 2. 
 
 Question. What is Addition ? 
 
 3. 
 
 4. 
 
 6. 
 
 6. 
 
 10. 
 
 11. 
 
 12. 
 
 13. 
 
 Q. Wliat are the numbers to be added jalU-d ? 
 ^. 1 lie nuaibers to be added aro called addenda 
 Q. What i8 the ro8uU of the addition called? 
 A. 1 ho result of the addition is called the sm«i. 
 
 A. When numbers aro to be added, w • first write them under 
 ?en«,Tc "'' '^"' ""''' ""''" ''"^"'" '^"i^^'^^"^ "^deJ 
 
 Q. What i^^ the next part of the rule for addition ? 
 
 from the sum? ^*"'' """^'^ '^' ^^^'"^^ ^° ««P^^^*« ^^^^ 
 Q. What is tl)e next thing done ? 
 ^. We next add up tlie units' column, sot down the ri^ht-hand 
 
 the other tiarure or tigures to the tens' eoluma ^ 
 
 Q. What is next done ? 
 
 canied fiom the units, write down the right-hand tiaure 
 of the mm uiiaer the column of tens, and carry the S? 
 figure or figures to the hundreds' column. ^ 
 Q. Why do we set down the riulit-hand figure under the eoT 
 
 The sign of addition .e written thus +, and h culled X. 
 iKi;,', °.**"„=/'l"'5<'> »'"' "hat Ams it mean » 
 
 oneSothir "^'""n *""'' " ^ wditen are equal to 
 
 Q. now may addition be proved ? 
 
 • "^ftStK; SoSarS' "'"'"^ *^^ ^°^""^- ^-^ -S-'n 
 Q. In what other way may addition bo proved ? 
 
 We "lay prove addition by cutting oW the top addend add- 
 ^^!; i'ii;f!!^"^«J.°«^^^he'-'«ndthe?, to their s^umaddl^V the 
 
 withS'fourii."b; thrruii':' ""'' "^''*''*''' ^^^^"^^ ^^'-^^ 
 
 Q. 
 
 
SIMPLE SUBTRACTION. 
 
 SIMPLE SUBTRACTION. 
 
 27. 
 
 26. Subtraction is the process of finding the difference 
 between two numbers. 
 
 27. The sign ■— , called wmw.?, written between two 
 numbers, indicates that the one following the sicru is to be 
 subtrar^ted from the one preceding it. Thus 10 — 9 read 
 16 minus 9, means that 9 is to be subtracted from 16.' 
 
 28. The number to be subtracted is called the siibtra 
 trnid, and the other number the minuend. What is left 
 "iter the subtraction, is called the remainder or dijfevenct. 
 
 1. 
 
 2. 
 
 3. 
 4. 
 
 6. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 iO. 
 11. 
 12. 
 13. 
 
 14. 
 
 16. 
 
 16. 
 
 17. 
 
 MENTAL EXERCISES. 
 
 ■^^°™ ,^^, takea, and how many remain? From 99 take 1, and 
 
 From 98 take 1, and how many re- 
 
 how many remain ? 
 main ? &c. 
 
 From 100 take 2, and how many remain ? From 98 take 2 and 
 maiyZ?" '^'"^'''* ^"^"^ ^^ *^^^ 2, and how mlny re- 
 
 ''L^\Z'.ry'?.e'^"ain /'S "'"^ '^"^^'"' ^^°^ «^ '^^^ 3. and 
 
 ^ w'many rc'nS.'r. ™'"^ ''™^^" ' ^''^^ ^ *^^^ *, and 
 
 How many are 100 -5 ? 95 — 5? 90 — 5? 85— 5? ifeo 
 
 How many are 100 — 6 ? 94 — 6 ? 88 — 6 ? &c. 
 
 How many are 100 — 7 ? 93 — 7 ? fee. 
 
 How many are 100 — 8 ? 92 — 8 ? &c. 
 
 How many are 100 — 9 ? 91 — 9 ? «tc. 
 
 How many are 87-2-3-4-5-9-8- 7-6-4-5? 
 How many are 80-9-6 — 7-1-2-4-5-6-7-4? 
 Howmanyare9 + 3+4 + 9 + 7 + 6-7-4-3-7-9-i_4» 
 
 How^many are 6 + 1 + 6 + 2 + 1 + 3 + 7-2-1-3-7 + 9 + 8 
 How^manyare6 + 7 + 4-9-8 + 2 + 9-8 + 6 + 7-3-2 + 7 
 Howma^nyai-e 19 + 6 + 7-3-4-7 +8-6-2 + 1+9-3- 
 
 ^li^Fl^mieLrMartha?'''^^"" '' ^-- --'y more apples 
 Emma wnrkod 7 qnestionR in arithmetic oaeh davof t' e wo^k 
 . ^t^l^w:!^:rfJ-'•'^.^13 questions on Mondny, \ on 'h.^.^!:,' 
 ,-,..1 - "-rrucy, , ui! liiurB'iay, ii on Ki-i(iav, a; d 9 on S-ft! 
 
28 
 
 BIMPLE SUBTRACTION. 
 
 20. A fnrmor l,nd a flock nf'^^T^ J rcnaining? 
 
 ^ 8, killed 4, aiKrS^pA'lfe'/cHr'^-IIc^;^;^^ '' "i^^ «' ^«-« ^^vny 
 
 21. Lizzie ha8 37 cents Si.n 1 ^' . "" ^' "^"^ ^^^' ^' ep ? ^ 
 
 paper 3 L^s'^r 1 k ro'c^n 'ibr' a' n"" '^^ ^^'V^'^' ' ^^^^^ ^or 
 ma.nderforaelate. How much %^«i^'yv'"^^'' «"^ <J'e re- 
 
 22. A man has 35 cord^ of wood to ! ' -'^''''' ^^" "'^ ^^^^e? 
 
 ^voek8. He .aws ? oing^ the £".1^^ l^''?^" *« «"'«h it in 6 
 8 cords Ihc third, and 9 co.ds the fSh' ^ ^"""^^ "^« ^^^^'^''l. 
 remain for Jam to saw in the fifth week ' °^ ""^"^^ ^°^<^» 
 
 RULE FOR SUBTRACTION. 
 ^^^fo.MPL. l.-ri„d the difference between 167947 and 
 
 OPERATION. 
 
 16'794T Minuend. 
 32717 Subtrahend. 
 
 135230 Difference. 
 
 167947 Proof. 
 
 Here setting down the ffiven mm^ 
 bers aecording°to the rufe we sT 
 
 fi om the one above it, and set dovvn 
 
 *^^Sr;'^^^nthesa'mecolunr 
 ., KvZ^ t« Piove our work we add 
 
 since the sum thus obfL.H •'"'''' *^ *^^ subtrahend ; an^ 
 conolnrl.fh"wr-' i ^^*^'"^^ ^^ equal to theminn.n^ J^J 
 ---Trv_'i^'^ vrOiiv is correct, ^"'"» ""' 
 
 
)aid 7 dollarB fot 
 j.or fruit, and 3 
 
 Florence and 8 
 
 >Jd 9, gave away 
 1 ho k( ep ? 
 
 'ncil,7cent8 for 
 5K, and the re- 
 for the Hlate ? 
 tofjnieh it In 6 
 rds the second, 
 ow many cord» 
 
 SIMPLE SUBTRACTION. 
 
 d9 
 
 '^end so thai 
 
 ract each fg. 
 ^guve of tht 
 ome colmnn. 
 iter than the 
 the latter by 
 nibtrahend. 
 
 ABEND, and 
 and the re- 
 
 ^67947 and 
 
 ?iven nimi- 
 e, we sub. 
 subtrahend 
 d set down 
 column, 
 k, we add 
 bend ; an*! 
 
 Exercise 7. 
 
 <1) 
 
 *914793« 
 18003932 
 
 (5) 
 
 9876543 
 643210 
 
 C2) 
 
 4013598 
 1203008 
 
 CO) 
 
 129147 
 
 20034 
 
 C3) 
 
 619145 
 19143 
 
 (7) 
 914718 
 212206 
 
 6191890 
 
 6190840 
 
 (8) 
 
 898906 
 287103 
 
 10 
 
 From five luillions seven hundred and four thousand nin^ 
 hun,lrcd and eighty, take five millions tl^ree hunted inS 
 four thousand six hundred and forty Jiuncired and 
 
 Example 2.— From 723 take 571. 
 
 Here in the second column, we cannot 
 take 7 trom 2, so we increase the 2 bv 10 
 and thus make it 12. Then we sav 7 from 
 12^ and 5 remain. Next we add 1 co the 
 5 m the subtrahend, and say 6 from 7 and 
 1 remains. 
 
 -From 71006 take 9867. 
 
 Here we say 7 from 6 we cannot, but in- 
 croasmjr the 6 by 10, we say 7 from 16 and 
 9 remain. The 7 (i. e. 6 with 1 added) from 
 J we cannot, but, incrcasinj^ the by 10 xve 
 have 7 Irom 10 and 3 remain. Then 9 from 
 > we cannot, but 9 from 10 (i. e. and 10) and rema h™ 
 '' .^'? LZ\ ---^ ^-\ ^0 from 11 ind / ieS; 
 
 OPE ATION. 
 723 
 
 571 
 152 
 
 Example 
 
 operation. 
 
 71006 
 9867 
 
 61139 
 
 o.- 
 
 (vastly, 1 from 7 and G remain. 
 
 Exercise 8. 
 
 (1) 
 
 670043 
 12571 
 
 810,140 
 914067 
 
 (2) 
 
 816427 
 13518 
 
 (6) 
 
 1910 142 
 
 191008 
 
 (3) 
 
 10134 
 
 5317 
 
 O.) 
 
 816663 
 79S67 
 
 C4) 
 
 291800 
 119137 
 
 B000470 
 916439 
 
30 
 
 li- 
 
 ce) 
 
 80000007 
 9149J36 
 
 SIMPLE SUBTRACTION. 
 
 (10) 
 
 8043007 
 34291(38 
 
 01) 
 
 960007008 
 99S99S6 
 
 (12) 
 
 600400070 
 19140007 
 
 16. F "d he d ffl ^"""'" *°™» "»•' "soil 
 18. «•„„ twenty, Jj^"'Zl. T"" l""""^ '"^ ^"^ 
 
 22. Henry has 276 marhioa . i, 
 
 S»™ ; after pa,i„, fori]lre.e'',^„Xrw-^'„';J'g; /^ 
 
 What wiJl remain ? ^""^ *^"" «"btract 708 from the sum 
 Shall be ]««a fil"./ ,^i*^ ^i^to any four addanria », ^ , . 
 
SIMPLE SU^TRACTIOIQ. 
 
 81 
 
 ^^ '^BubtShc"K]?' ^^^^^ ""*^ *^" remainder 6943, what Ib the 
 ^^' "^'^ mTuucf: d f""^ ^' ^°^^ ""^ '^'^ rer..aincler 7143, what is the 
 63. What Bum will leave 1727 when 917 is taken from it ? 
 
 ^' "voi'">d olj^r^ /r"^^'"' T^^'^ »169 !1 8. The first 
 2fiU Ih. . S7 t ■'■' !'° ''^y^'^ 19^3 U.S., and the thlrq 
 
 "^ goufermarS'? """"'' ^' """^ ^^«« ^^^^ ^^«' ^^" ^ 
 
 (30) 
 17S9437 
 — 1-2371 
 —29867 
 —14371 
 —3198 
 
 C37) 
 918471 
 
 — 6i;n2 
 
 —91314 
 —6713 
 —9147 
 
 (S8) 
 
 167142 
 
 —9347 
 
 —91671 
 
 —9181 
 
 —76 
 
 (39) 
 
 987671 
 
 —81432 
 
 —134-27 
 
 —90000 
 
 —6714 
 
 17^9630 " 
 
 frnm i7o, <? A . •'' ' ^^ ^'"^"^ ^ ^e can't, borrow 10 tlien 17 
 
 (40) (41) .49. 
 
 194362 734713 ^^27 
 
 ~;?JS ~I?1^"^ -67142 
 
 —21713 —61714 QSiqA 
 
 61429 9131? 0% 
 
 -21347 -23916 -6179 
 
 (43) 
 
 814267 
 
 —267 
 gg 
 
 417132 
 —98067 
 
 205588 " 
 
 ^1^:??- !rri.^^ t^^r^i^tr^.^4ra;;d^t^e^^f 
 
 anH X n _ T J o, 3 + -s — — 3 and — 1 = — 4 and — 4 — — s 
 and + ? = - 6 and 5 from 10 = 6 to set down, &c. 
 
 (44) 
 
 (■45) 
 807911.S406914718918291346129131421986 
 
32 
 
 "EXAMINATION QUESTIONS. 
 
 J??mfI?^°^'^^230OO4500O0789O0OO048OO7 
 1119180167452;j7l342912^6678«1126i569 
 
 (47) 
 
 80910020003(}004nno05U(X)OU678009720003 
 914bG71%6714918i;J 47108(J7]49S601427l 
 
 (48) ~ 
 
 9818inoonoOOOO(}Or,0(K)70004 0098167007100 
 lL;i45(i7«9012007]9()130041U910U07180981 
 
 
 8. 
 
 4. 
 
 5. 
 
 6. 
 
 •7. 
 
 Q. 
 A. 
 
 8. 
 
 RECAPITULATION AND EXAMINATION QUESTIONS. 
 1. Question. What is Subtraction ? 
 
 ^'"Te,;vSr?;!i",','„;»b',';^.'"--» «' «■■<""« tho amr.... 
 
 ^' ?■ .S*"" '" 'J"" ""mliiT to be pnbtractcd ciilcd « 
 
 * ^lake'n'ca.toci ?""""• '■'■'""«*"='■ ""= Bub,rabc.„d 1, ,o he 
 ^- ■^'oallcT.Kr™^''* ""' ""'"■•»>">'"» " '"be taken ie 
 
 ^^ mmed?" ''^" °^ subtraction written, and what Is It 
 
 '^mS °^ «^btraction is written thus-, and is call^cd 
 
 Q. What is the first part of the rule for subtraction ? 
 A. Wo are tirat directed to write tlie Ribtrahc nd nnd^r th« 
 injnue;,d so that units come under'^unils^'ter^u^de? 
 
 Q. What is the second part of the rule for subtraction » 
 
 A. After drawing a line below the subti4hS to finkrntp tt 
 
 Q. Wlien any ficaire in the subtrahon'' in avo^it^y *\ „^ ft, a 
 „„^ ,^^ „^. .^^^ ^j(,jj^j suBiittiiisttu-jHgure lo liie ieft. 
 
 -•^tm m mmmmm^ 
 
flfl the pupU in 
 
 t in the subtrii. 
 
 larro tho figure 
 
 •4; 9. .17. .8; 
 
 QUESTIONS, 
 tho difference 
 
 Irahend. 
 iheiid is to be 
 
 to be taken is 
 
 fHon called? 
 .s called the 
 
 id what Is it 
 and is called 
 
 id under the 
 , tens under 
 
 ction ? 
 
 o separate it 
 t-haiid figure 
 n' in the same 
 f tlie piihtra- 
 lext the hun- 
 
 an the figure 
 "roceed ? 
 ! con-espond- 
 
 10 and then 
 
 iett. 
 
 SIMPLE MULTIPLICATION. 
 0. Q. rrowmny subtraction bo proved? 
 
 iiii. uenU. ' ^'' ■="'" fehould be equal to the 
 
 '^-'^^^^X^^Zi^^Z ''" ---'"' i the ,^ 
 
 SIMPLE MULTIPLICATION. 
 
 30. Multiplication is a sliort di-ocpw «f .„i • 
 be,, as „,a,,y ti,„es as tl.ce are u^sTn a^oot"*^ °"' ''*"°- 
 
 plieal-/''" '""'""'• *" "« '""'"plied is called the «„«. 
 
 ««l>fe ' """"^ "^ ""^-i' -« "'"'t'W i« called the 
 call'd^h?;„S.'^'" '''"'""^ '■""» *« -ultiplieation is 
 /aofo^i oflhc^tflli.:: ""' "" "'""■''"'-° "- e'J'ed the 
 
 iu4L t iiuirpL':':? i^r:: '^ » ^^'"-'^ -""- 
 
 actlvdi-vided''irj;ttelil " "ro'" "•"^'^ ^■'-'" "e ex- 
 bers. ' ' ' '' ". '-*. '', 19, ^3, &c., are prime num- 
 
 numbers. ' ' ' "^^^ ^^' 2o, &c., are composite 
 
 ten betwppp f^o^irrA^'^"'''^- "'"" ^^ multiplication. Trrit. 
 
 tiplied to,cetlior... Tlivts 
 
 ^''1'."''^^:"^^^' *^'-'-' ti^ey'a-e to be 
 
 t 
 
 I 
 
 16 X % rc&d le 
 
 mul- 
 
 m«m» tdiat 16 is to 1» multiplied by 7, 
 
 m 
 
 uitij)Hed by 7 
 
 m 
 
u 
 
 SIMPLT?! 
 
 ■ nPLICATION, 
 
 MULTIPLICATrON TABLE. 
 
 Twioo 
 
 1 are 2 
 
 2—4 
 
 3—0 
 
 4—8 
 
 5—10 
 
 6—12 
 
 7—14 
 
 8—16 
 
 9—18 
 
 10 — 20|10 — 30 
 
 11 —22 11 — 38 
 
 «{ ti.m.'.- 4 
 1 are J 1 
 2—02 
 3—93 
 4—12 4 
 
 5 — lo 6 
 
 6 — 18 6 
 
 7 —21 7 
 
 8 — 24 8 
 
 9 — 27 9 
 10 
 11 
 
 12 — 24|l2 —30 12 
 
 — 30j 9 — 
 
 — 40 10 ,- 
 
 — 44 11 — 
 
 — 48 12 — 
 
 45| 9 
 50 10 
 55 11 
 60 12 
 
 10 times 
 1 are 10 
 2— 20 
 3—80 
 4—40 
 5—50 
 0— 60 
 7— 70 
 8—80 
 9—90 
 10—100 
 11—110 
 12—120 
 
 1 1 times 
 1 are 1 1 
 
 2—22 
 
 3—33 
 
 4— 44 
 
 5—55 
 
 6— 06 
 
 7—77 
 
 8—88 
 
 9—99 
 
 10—110 
 
 11 —121 
 
 12—132 
 
 12 times 
 1 are 12 
 2—24 
 
 3—36 
 
 4—48 
 
 5—60 
 
 6—72 
 
 7—84 
 
 8— 96 
 
 9—108 
 
 10 —120 
 
 11—132 
 
 12—144 
 
 MENTAL EXERCISES. 
 
 2 uZ ZT^ '"'' '7^"' ^ ' *^'^^ 3 ? twice 4 ? twice 5 ? twice 6 ? &c 
 
 2. "«^niany are 3 times 2? 3 times 3? 3 times 4?. to. 
 
 3. How many are 4 times 2 ? 4 times 3 ? 4 times 4 ? &c 
 
 4. How many are 6 times 2i 5 times 3? 5 times 4? &c' 
 
 6. How many are 6 times 2? 6 times 3? ft times 4? tc 
 0. How many are 7 times 2? 7 times 3? 7 times 4? &o* 
 
 7. How many are S.times 2? 8 times 8 ? 8 tir.es 4 ? &« * 
 
 9, uow many are dtfmes 2? .? times 3 f 9 tUjae« 4? ^ . 
 
SIMPLE MULTIPLICATION. 
 
 85 
 
 1> 
 
 11 
 
 7 tiiijca 
 
 •e ( 
 
 1 are 7 
 
 - 1: 
 
 2 —H 
 
 - It 
 
 3—21 
 
 . 2-J 
 
 4—28 
 
 -30 
 
 5 — 85 
 
 ■8H 
 
 6 —42 
 
 .42 
 
 1 —49 
 
 ■48 8 — 6ii 
 
 54 9 — 63 
 
 60; 10 — 70 
 
 66 11 — 77 
 
 72 12 — 84 
 
 3 1 12 times 
 
 I 
 
 1 are 12 
 
 5 
 
 2—24 
 
 t 
 
 3—36 
 
 
 4—48 
 
 
 5—60 
 
 
 6-- 72 
 
 
 7—84 
 
 
 8— 96 
 
 
 9—108 
 
 
 10—120 
 
 
 11-132 
 
 12->i44| 
 
 ?twice6?&c. 
 
 c. 
 
 c. 
 
 c. 
 
 :o. 
 
 
 
 9. flow ninny are 10 times 2 ? 10 Hmes 8 ? 10 tfmpg 4 1 &c 
 
 10. How many arc 11 times 2? 11 tiiniM 3? 11 tinu.« 4 ? &o 
 
 11. IIow m:iny wre 12 times 2? 12 timoH 3 ? 12 tliuoH 4 ? &c 
 
 12. How many :,re 3 timoK 7? 7 times 3? How many fs In oi j 
 
 How many 3 b in 21 ? "i""^ < b in .i f 
 
 ''• "^'iillw j;,:;;^?J^^?5r ' '"^^ «» now many D's in 72» 
 14. Kow^m..3^a^6 «n,es^7l 7 times 6» How many 6's in 421 
 
 16. How many aro 8 times 8 ? How many 8's are there In 64 ? 
 
 18. How ma,, y are 12 timos 9 ? 9 times 12 ? How many 12'b in lofi » ' 
 How many 9'8 in 108? ^ ^"^' 
 
 17. How many aro 11 times 11 ? How m.-my IPs in 12^ ? 
 
 18. Howmar.yare 8 times 6? 6 times 8? ^IIow many 8'8 in48» 
 
 How mai,y O's in 48 ? xuaiiy o s in 48 If 
 
 19. How many are 9 times 9 ? How many O's in 81 ? 
 
 20. How many aro 7 timoe 84 8 times 7? How many S's in S6? 
 
 How many 7'8 in 56? i^^uy o s m oo/ 
 
 o^2 xiT2V2'i'ro?-lVi'^3';^r'^24';l')"^ 2x2x4.3, 
 
 22. m.Ht^aryho factors of 18? of 20? of 24? of 32? of 36? of 81? 
 
 23. What are the factors of 72 ? of 84 ? of 56 ? of 39 ? of 108 ? of 121 ? 
 
 24. What^arethefactoreofl5?of35? of 42? of 27? of 88? of 100? 
 
 RULE FOR MULTIPLICATION. 
 
 39. When the multiplier does not exceed 12. 
 
 ^.,?f/T ^^'7 "^""fP^^^^ '^^der the right-hand fqure of the 
 multiplicand, and draw a horizontal line beneath. '' 
 
 Begin at the right-hand side, and midtiply each faure 
 muuilZJ t\P^'^'f P^y^^ot vnd^r that fgure of the 
 
36 
 
 SIMPLE MULTIPLICATION. 
 
 Example.— Multiply 71497 by 12. 
 
 Multiplicand 71497 set (lown 1 T' ^ '^'^ ^^' '^"^ ve 
 Multii,lier 12 are 1 « Tu "'"'''^ ^ ' ^^ times 9 
 
 r.«L V ^V"""'^ «'■« 4« «ncl 11 
 down the 9 and carry the CL.""'' ''' '' "^^'^ ^^ ««^ 
 
 Exercise 9. 
 
 (6) 
 
 018765421 
 6 
 
 671491345 
 10 
 
 ^2) 
 
 01818947 
 3 
 
 («r" 
 
 879165498 
 7 
 
 (10) 
 7801491S91 
 11 
 
 (3) 
 91134719 
 4 
 
 (7~ 
 
 12367986 
 
 8 
 
 (l~ 
 
 4291498671 
 12 
 
 (4) 
 67143917 
 5 
 
 (8~ 
 
 087166498 
 
 9 
 
 (12) 
 78674918 
 8 
 
 13 What is the product of 791876x3? x 2? x4? x 197" 
 
 34. What i« the product Of 818619847 . 7? xsVxoVxll, 
 
 35. vV liat is tlie product of 6179 V '^» ''x»'x9?xn? 
 
 17. Multiply 714719 by 12. ' ^ '' x 8? x6^ 
 
 18. Multiply 1913476 by 9. 
 
 19. How many are 8 times 76598 ? 
 
 21. What is the product of TM x „ . n» x e? x 6, . 12, 
 
 "if 
 
 RULE. 
 
 Tvoducih, the tUrfZofsflh y^^^^ ""'"'''" ■■""■""'I 
 
 IL. 
 
8 ; 12 times 9 
 
 SIMPLE MUr/nPLlCATION. 
 ExAMPLE.—Multiply 671908 by 50. 
 
 37 
 
 OPERATION. 
 071908 
 
 8 
 
 63752tJ4 
 
 7 
 
 37026848 
 
 lactois aio 8 x 7, and, accordiii-r to tlio ink. 
 vo urst inuUiply tho .ivon nu.T.ber by "Lj. 
 iactor, uuU tiicu the le.ult by the othe. LZ, 
 
 Exercise 10. 
 
 Multiply 710867 l)y 48. i 7 \r u- i «, 
 
 ^ • '• Multiply 71G914 hy 144. 
 
 2. Multiply 916704 by 84. 
 8. Multiply 7143G7by 27. 
 
 4. Multiply 1G1714 by 10. 
 
 5. Multii^ly 71(J9Sby 81. 
 
 6. Multiply 81897 by 121. 
 
 8. Multiply 167149 by 54. 
 
 9. Multiply 191878 by 42. 
 
 10. Multiply 891476 by 64. 
 
 11. Multiply 918978 by 108. 
 
 12. Multiply 7654^9 by 132. 
 
 15. What Will 49 l,o,.e. cct nt $147 c-j.; ; ' ' ^^'veUy-six? 
 
 16. Wiiat will or. ( OW9 co-t :.t ^48 earh ? 
 
 17. What will l),S7 iiocjsheads of ,.iu.;,r cost it «8n . t, 1 . « 
 
 18. Suppose a book to ontv^n dior ^ Jingshend ? 
 
 there in the vvholc book ? ''' ''"''' "^''">' ^^'^^^''^ are 
 
 ly If an apple-woman sells 12I apj)lc3 a rinv h..^, 
 
 ^^eli in a ye.r. which, omi'? g^jf ^i^'^^^-.f '^"y ^vill sl-e 
 
 41. When tho multiplier excppflq 1 9 ov.^ • 
 posice number:— exceeas 12, and is not a com- 
 
 RULE. 
 U. Multlphj the muUirdicand by each i!m,r^ «/• /7 
 
38 
 
 SIMPLE MULTIPLlCx\TION. 
 
 .1 i 
 
 thus ohtainedao that the firsit figure falls directly under 
 that f (jure of the viidtlpiier by ivhich it was obtained. 
 
 III. Adit the several partial products together as tliey stand, 
 the sum will be the entire product, sought. 
 
 PROOF OF MULTIPLICATION-. 
 
 42. First MEtnoD. —Mult/phj the multiplicaiid by onb less 
 than the maUii'hu; and to the product thus obtained add the multi- 
 plicand. I he renult should be the same as the product obtained bu 
 inc rule. •* 
 
 Skcond Method.- Cgs/ the 9's out of the multiplicand and set 
 doicn the remainder, also out of the vmltiplier and set dawn the re- 
 maimtrr; mnltiplij tlicse tuo remuinders together, and cast the 9's 
 out of heir product. The rtmainder thus found should he the same 
 asthatohtarned h;/ casting out the 9'sfroni the product of the muUi. 
 phcand by the mulliplier. .y « "= ,nuut 
 
 Thus to prove Example 2, we proceed as follows : 
 
 7 + 4 + 9 + 6 + 3 = 29, and 29 + 9 dvcs a remain- 
 der 2, \vh eh we wriio down to tiie left of a crosd, as 
 111 the maruiii. ' 
 
 N^oxt, 2 I- 9 r: 11, and 11 -f- 9 prives a remainder, 2. 
 which we write t;j the right of ti'e cross 
 
 Next, 2x2 = 4, and 4 + 9 ^ives a remainder, 4. 
 which we write above the cro.i>d. ' 
 
 Lastly 2 + 1 + 7 + 3 + 9 + 2 + 7 = 01, and 31 -4- 9 crivos a re- 
 mainder, 4, wh^ch wo write be;H':itii the ero;^. Theu, since the 
 miniher a'.ove the cross agrees with that bolowit, wo conclude the 
 wori: is correct. 
 
 X 
 
 .1 
 
 I' 1' f 4 
 
 Example 1.— Multiply 74903 by 29. 
 
 OPERATION. 
 
 74963 
 29 
 
 674067 
 149926 
 
 2173927 
 
 Here we first uuiltiply the given multipli- 
 cand by 9, settino- 7, the first figure of the par- 
 tial product, directly under the 9; next we 
 multiply the given multiplicand by 2, and set 
 down the partial product .,o tiiat its fiist fieure. 
 0, falls directly under the 2 by which we are 
 multipIyiYig ; lastly, we add the two partial 
 products together just as they stand. 
 
SIMPLE MULTIPLICATION. 
 Example 2.— Multiply 1U981 by 23004. 
 
 89 
 
 OPERATION. 
 
 714987 
 
 2you4 
 
 2859U48 
 2144901 
 1429974 
 
 1&447560948 
 
 Here we first multiply 714987 by 4, 
 setting the first figure of the paitiai 
 product under Ue 4 ; wc next niUltiply ly 
 3, setting the first figuie of the second 
 partial product under ^the 3, and so on; 
 finally, we add the paitial products to- 
 gether as they stand. 
 
 Note.— Since tie mu!tipi'carcl multiplied 
 by is equal to 0, we pass by the O'e in the 
 multiplier. 
 
 (1) 
 
 7191486 
 28 
 
 Exercise 11. 
 
 (2) 
 314976 
 89 
 
 (3) 
 
 819715 
 698 
 
 C4) 
 
 7819164 
 908 
 
 (6) 
 
 6540910 
 
 8040 
 
 (6) 
 
 7190867 
 8046 
 
 (7) 
 8491791 
 91008 
 
 (8) 
 28700046 
 90870 
 
 71400600 
 000708 
 
 (10) 
 
 123456789 
 98067 
 
 (11) 
 
 91845067 
 90U004 
 
 02) 
 
 987064(1987 
 9060409 
 
 13. What is the product of 71476 x 9187 1 
 
 14. What is the product of 91476x8190? 
 16. What is the product of 8100070x81009? 
 
 16. What is the product of 6858857 x 606007 ? 
 
 17. Multiply six mill'ons three hundred and seven thousand nine 
 
 Huudrt'd and eighteen by twenty thoueaiid seven hundred 
 and ninety, ""«icti 
 
 18. Multiply seventy-eight thousand four hundred and eiehty-six 
 
 by twenty titnes seven thousand and nineteen. 
 
 19. Multiply seven hundred and forty times nine hundred and 
 
 seven Oy thirteen times tAvo hundred and eeveiiteeii. 
 
 20. If an acre of whe^t yield 29 bushels, how much will 149 acres 
 
 produce ? 
 
 21. What will 217 horses cost at $106 each ? 
 
ttl 
 
 J! 
 
 40 
 
 EXAMINATION QUESTIONS. 
 
 S' M ^ ^^'^\ f^l"^'"' ''"'^ ^^^' ^^^* ^1^ ^149 hhds. cost ? 
 
 euta-e populaton onirS^ c^^^cn hou.e. what will be tha 
 
 307 to a volume. 1 Sv manv ^v, JiL ^ ^'^^.^'^^^^ "* reading i, 
 contain ? ' "^ ^'"'^'"^ "^ reading does the libilrj 
 
 ''p'uJZ"^S;?rny'eWl1rrn at'tS"^ ^^f ^^^°«' averages 4) 
 township? ^ cwidren are there attending school in th«J 
 
 C2Y) 
 987671813407198787988699753 
 
 8 
 
 (28) 
 
 817614923569871908147634567 
 
 H 
 
 26 
 
 (29) 
 
 130579864213579843212345678 
 9 
 
 (30) 
 
 811476193457899986888776G54 
 
 12 
 
 5 aro 40 and 4 nmko 44 set dovvn'/o.,^ •"'''" ^ '!."^. ^='^'•>' ^ i « » ^"'^a 
 taught to eimpy touch each^^^^^^^ B^uld bo 
 
 name the digit in the muSr.mH fv? Ji'^ pencil and merely 
 to bo set down, as follows :^ 8 " 6 ' ^J^^^^^^'P'^f. «n^ the figurj 
 
 JtECAPITULATION AND EXAMINATION QUESTION^ 
 
 I Qnnsf ion What is multiplication? 
 
 'L ,^'"t'V,"'^^'''" '?^ ^^'^'-t method of taking one numbe, 
 as many times as there are units in another ^ 
 
 % rrx ^^ '^ '1^^ number to be multiplied ealled ? 
 
 2. 
 
 8. 
 
EXAMINATION QUESTIONS. 
 
 41 
 
 4. Q. "What in the result ot Hie multiplication called ? 
 
 A. The tiumbtn- resulting from the multiplication is called the 
 jji'oduct. 
 
 5. Q. Wlmt avo the far.tovfl of a numliev? 
 
 A. Tlie factors of a nuinlier are those numbers which, multi- 
 plied togetlicr, produce it. Tlius the multiplier and 
 multiylicaiid are the factors of the product. 
 
 6. Q. "What is an integer or integral number? 
 
 A. An integer or integral iiumbcr is a whole number. 
 
 7. Q. Of liow many kiiids arc integers? 
 
 A. Integers are of two kindrj, priwe or cowi!/?os;7e. 
 
 8. Q. What is a prime number ? 
 
 A. A prime number is a number which has. no integral factors 
 except itself and unity. 
 
 9. Q. What are all tL.; prime numbers less than 100? 
 
 10. Q. What is a composite number? 
 
 A. A composite number is the product of two or more integral 
 factorri neither of which is unity. 
 
 11. Q, What are all the composite numbers less than ICO? 
 
 12. Q. IIow is thesiorn of multiplication written? 
 A. The sign of multiplication is written thus, x. 
 
 13. Give the rn'e for multiplication when the multiplier does not 
 
 exceed 12. (See Art. 39.) 
 
 14. Q. In tbi-; and the other rules for mitltiplication, why do we 
 
 heuin mu'tipjying at the r/o-Az-haiid side ? 
 A. Wc Ixgiri at the right-hand side in multiplication for the 
 same reason that we begin at the right-har.d side in ad- 
 dition, i. e. in order to take advantage of the principle 
 of carrying. 
 
 15. Q. What do vou understand by the principle of carrying? 
 
 A. Wiien we have obtained the product of any two digits in 
 multiplication, or the pum of any column in addition, we 
 set down the right-hand figure in that column and carry 
 the other figure or figures to the next product or next 
 colunii', and are thus enabled to do by one process what 
 would otherwise require several. 
 
 16. Give the rule for multiplication when the multiplier can be 
 
 broke" up i:^to two or more factors, neither of them 
 greater than 12. (See Art. 40.) 
 
 17. Give the rule for mulfnlieation when t^o multiplier is not 
 
 compot»ite and i.i greater than 12 ('See Art. 41.) 
 
 18. Q. In this latter rule, why are von duvcteci vo write the rignt- 
 
 liT d figure of each partial product diixvtly u"der that 
 figure of the multiplier by which it was obtained ? 
 
SIMPLE DIVISION. 
 ^- '"^l^^I^^^'^^ZZ^' -I'J-r'ij-O %.ny order 
 
 died/fur the pruduc?, fc ^ "multiplier, give huu-' 
 
 19. Q, TTow do you multiply by 10 100 inon innnn t o 
 
 11 
 
 Hi 
 
 SIMPLE DIVISION. 
 
 43. Division teaches the method nf fit,H!.,„ i 
 
 times one number is containedTn another ^ ^"^ """^ 
 
 44. The number to be divided is called the dividend 
 divu!;. ^^' °""''^' "^ "'"«'' "« divide is called 'the 
 
 conttfneJt tTet^Mtdt^i ^l^^"^ '^^^''^^ 'a 
 
 exacf „u^i« o;resNS:r?r,r^^^rrill'!^--'' » 
 number called a r^mami^r. *^^ ^'^'«^^° a 
 
 tweelf th?°Set: ^:Z^:S tt t'"™' ^^i"^" ''^• 
 si,cm is to he divi.ied bv th4 f„?L • "?^ P^wdmg the 
 -cad ,« .«. „, ^^^nfeLrtJafte i'e S^, '^.V' 
 hv w°J";rJ!;° ."'il''^'.';" "<■,""<' '■■'mbr-r bv nnolhe,- i. „i„ >r-^:...„^ 
 
d by any order 
 e units of the 
 ] c.n ultiplei- 
 iif riiht-I'MJKi 
 ut "ts ; so t) e 
 e tet ^ of tl e 
 we tlirefore 
 ' ie B in t1 e 
 rnii't'iJicai d, 
 ier, give hxiix- 
 
 ly annexing 
 nnd. 
 
 illuBfrate the 
 "12 in Exer- 
 
 SIMPLE DIVISION. 
 
 43 
 
 how many 
 
 vidend. 
 Jailed the 
 
 dividend, 
 ! divisor. 
 
 itten be-. 
 3ing the 
 
 16-f-4, 
 
 by 4. 
 
 i'-j.v:;;iL-Cl 
 
 en them. 
 
 49. When the divisor does not exceed 12, the rule is 
 called f^hort dlvUion ; but vvheu the divisor is greater than 
 12, it is called long divinion. 
 
 MENTAL EXERCISES. 
 
 1. How many timed ie 2 contaiaod in 8? in 10? in 18? in 11? in 
 
 2a? 
 
 2. How many times is 3 contained in 9? in 15? in 27? in 33? in 
 
 in 
 
 8. Hnw many times is 4 contained in 20? in 28? in 44? in 36? in 
 19/ 
 
 4. How many times is 5 contained in 35 ? in 10 ? in 60 ? in 25 ? in 
 
 28? 
 
 5. How many times is 6 contained in 18 ? in 42? in 64? ia 38? in 
 
 40? 
 
 6. How many times is 7 contained in 35? in 7? in 21? in 63? in 
 
 25? 
 
 7. How many times is 8 contained in 24? in 72? in 98? in 40? in 
 
 57? 
 
 8. How many times ia 9 contained in 81? in 45 ? in 18 ? in 72? in 
 
 60? 
 
 9. How many times is 10 contained in 10 ? in 40? in 100 ? in 120 ? 
 
 in 97 i 
 
 10. How many times is 11 contained in 33 ? in 77 ? in 121 ? in 88 ? 
 
 in 100 ? 
 
 11. How many times is 12 contained in GO ? in 132? inS6? in 96? 
 
 in 117? 
 
 12. How many times is 7 contained in 17 ? in 3 ? in 38 ? in 62 ? in 
 
 29? 
 
 13. How many tmesis 8 contained in 63? in 7 ? in 71? in 90? in 
 
 21? 
 
 14. How many times is 9 contained in 23 ? In 100 ? in 48 ? in 80 ? in 
 
 10? 
 
 15. How many t'.mes is 12 contained in 10 ? in 37? in 140? in 101 ? 
 
 in 92? 
 
 16. Florence has 47 questions in division to work in the week ; 
 
 how maiiy mnet ahe do each day ? 
 
 17. Geori?o has 56 apples and ho wants to make thoin last 7 
 
 weeks ; how many may he eat each week ? how many each 
 day? 
 
 18. Charlie wants to rend a book, containing: 135 page?, through in 
 
 11 days ; how many pagey must lie read each day ? 
 
 19. Emmi has 78 book:=», and wis'^ios to divide them as nearly a3 
 
 ]io-tsible (.'(luaily among 7 shelved? how many must she put 
 oa each shell? ' 
 
 20. A farmer has 107 sheep, and w'shes to divide equally, or as 
 
 nearly so a» jiosaible, among B fields ; hov7 many murit be 
 placed in each ? 
 
44 
 
 SHORT DlVLSrOK 
 RULE FOR SHORT DIVISIOK 
 
 Jormcd a. h/fore. When k elf, -"'^ * "«»4«' M J 
 '- <"»,'/ >>«4 of th diuJrZ,:" "<!' r'^oincd once 
 fiyure and consider that fytf^TrlZtt?: "'"'" *"" 
 
 Example l.-Divide 271406 by 5. 
 
 OPERATION TTo 
 
 the in tl e onh-^ '"' """^ ^« "«* ^^t down 
 
 ^4281i the 7 make T'^^X l^^' ^ ^"en before 
 , , ^ before 1 maijes 2i k • '.^ "^"^ ^ ^ver; 2 
 
 1 before 4 makes U 5's ?n 14 J' V" ^^' ^ ^"^ 1 over- 
 end we have a remainder of '.' w^ ' «^^^' &«• At the 
 divide 1 by 5, we indicate he dvlr'^'" "^""^^ ^«^««"y 
 (^See Art. 48, Note.) "^'"^'^''^^ ^3^ writing it thus i 
 
 EXAMPLE 2.-Divide 704653 by 8. 
 
 OPERATION. 
 
 8)704653 
 880811 
 
 1 and 6 over, and we sot dowi tli' 
 five el^lSf '"*'*"'"'<' read 4 
 Exercise 12. 
 
 <Tl) (2) r-i. 
 
 2)714f3987 S^SOlir.fi, .,n,,L ^^^ 
 
 ^ -^^fJl^oG? 4)91^097 5)130047198 
 
 •'i'™ »-2? »•"« .m'?™, ^ 
 
 i! 
 
(9) 
 
 10)22^rtr! 
 
 ;i>o'3444 
 
 SHORT DIVISION. 
 
 (10) (11) 
 
 11)3121315161 12)914556677 
 
 45 
 
 13. 9146291-4-2 
 
 14. 714632 -i- 3 
 
 15. 12C4G10-H4 
 
 16. 7000000 + 5 
 
 17. 8100406 .+-6 
 
 18. 9001029 -f. 7 
 
 (12) 
 9)111111111111 
 
 19. 100610067 
 
 20. 99999999 ■ 
 
 21. 88888868 • 
 
 22. 123456789- 
 -3. 938276543- 
 24. 2C0OO0O0O 
 
 ■ 8 
 ■10 
 •12 
 
 ■ 9 
 11 
 
 7 
 
 2?' Jr V^™rr" *"-' ^■'"' " '>'^ "">■ of ono hereof 
 the j.k.[,I i« r'r.?,"? " ""'"' P^auces 740 bushels of oata, what I, 
 
 fac."- Jgret'frthan' 11°^^''^ ""-b-- "one of whose 
 
 RULE. 
 
 pivide the given dividend by one factor of tho /»,•„.« 
 
 To obtain the correct remainder mulfitih, ih. i * 
 Example.— Divide 714G9 by 35. 
 
 OPERATION. 
 
 »* 
 
 Here the factors of tlio divisor 
 
 ^^"^ ^ and 7. In dividing by 5 mo 
 
 '7)14293.. 4 = 1st rem. ^^* •'^ I'emainder, 4, and ir dJvid- 
 
 2041. .6 =: 2d rem f^ *^^ 5'^^ quotient by 7 we get 
 
 6 X 5 = 30 + 4 = S f '^"^^'^^.*^^' 6. Then, to get The 
 
 An^. 204Ht f'T remainder, we multiply 6, the 
 
 last remamder, by 5, the first divi 
 
 , , , "•' i!isi remamtler, by 5, the fii 
 
 sor, and ada 4, the first remainder, to the product 
 gives us 84. which we wrim «hn... ,].. .n'lJLf'^' . . 
 
 plained. 
 
 iCu we "wiifco above the divisor 
 
 This 
 
 as belbre ex- 
 
wmmmim^ 
 
 46 
 
 LONd DIVISION. 
 
 1. Divide 714667 by 16. 
 
 2. Divide 100901 by 27. 
 
 3. Divide 9186713 by 81. 
 
 4. Divide 16151712 by 144. 
 6. Divide 1671932 by 42, 
 
 Exercise 13. 
 
 6. Divide 22222222 by 108. 
 
 7. Divide 617149324 by 121. 
 
 8. Divide 8182838485 l)y 100. 
 
 9. Divide 667788991 by 64. 
 10. Divide 78998778998 by 54. 
 
 11. Divide nine hundred nnd seventeen millions forty-oiaht thft • 
 
 Band and six by one hundred and ten. •^ 
 
 12. Divide seventy millions four thousand and nineteen bu slxtv- 
 
 13. How many times is fifty-six contained in seventy-nino times 
 
 four hundred and eleven tiicuaand six hundred and nine? 
 ^*- -^ Jn"n496°fblV^^ weighs 60 lbs. ; how many bushels are there 
 
 15. How many bushels of rye are there in 918674 lbs., one bushel 
 
 of rye weighing 56 lbs. ? 
 
 16. How many bushels of barley are there in 291717 lbs., one 
 
 bushel of barley weighing 48 lbs. ? 
 
 17. If 48 coAvs coat $1774, what is the cost of one cow? 
 
 1^- -^^ifughef?^^* °^ ^®^^® ^^^^^ "^^^^ ^^®'' ^^^^ ^* *^® weight of 1 
 
 19. Divide 71496 x 7 x 17 by 66. 
 
 20. Divide $71498 equally among 45 persons. 
 
 RULE FOR LONG DIVISION. 
 
 ^ 62. Set down the divisor to the left yf the dividend, as 
 in short division, and the quotient to the right, thus : 
 Divisor) Dividend {Quotient. 
 
 Find how many times the divisor is contained in the few- 
 est figures of the dividend that mil contain it once <yr more^ 
 and place the figure thus found in the quotient. 
 
 Multiply the divisor by the figure put in the quotient, 
 wnte the product under the figures divided, and subtract. 
 
 To the right of the remainder thus obtained bring down 
 the next figure of the dividend; divide the number this 
 formed as before, and proceed thus till all the figures of the 
 dividend have been brought down. 
 
 ^ When there is a remainder at the end of the process, 
 write it over the divisor and annex it to the quotient. 
 
 Proof of Division. —Multiply the quotient bi/ the divi- 
 sor, and add in the remainder. The sum should be equal to 
 ikd dividend. 
 
 P 
 
 y.....j 
 
LONQ DIVISION. 
 
 47 
 
 2222 by 108. 
 49324 by 121. 
 ^38485 by 100. 
 SS991 by 64. 
 3778998 by 54. 
 forty-eight thft - 
 
 aeteen by slxty- 
 
 rcnty-nino times 
 ■ed and nine? 
 
 usliels are there 
 lbs., one bushel 
 !91717 IbB., one 
 
 the weight of 1 
 
 Example 1.— Divide 714986 by 613. 
 
 OPERATION. 
 
 Il6re the fewest figures that \\\\\ 
 
 contain Oi;j, the divKsor, an three, 
 
 viz., 7H. 613 in 714 will go 1 time, 
 
 we thcrefoie set 1 in the (luotient ; 
 
 then once 61o is 613, which we set 
 
 down under the 714 and subtract. 
 
 AVe thus get a remainder of 101, to 
 
 which we bring down the 9, and thus 
 
 obtain 1019 as the new number to be 
 
 divided. Next, 613 in 1019 will go 
 
 one time ; we therefore set down tlie 
 
 in the quotient, multiply and subtract as before, and thus 
 
 •btain 406 for remainder, to which we bring down 8, the 
 
 ie.xt figure of the dividend. This gives us 4068 for the 
 
 aext number to be divided; 613 into 4068 will go 6 
 
 times, etc. 
 
 513)7l4986(1166Jf5 
 613 
 
 lOl'J 
 613 
 
 4068 
 3678 
 
 8906 
 3678 
 
 228 
 
 \e dividend^ as 
 , thus : 
 
 >ed in thefew- 
 once or more^ 
 
 L 
 
 !. the quotient^ 
 i subtract, 
 id bring down, 
 number thus 
 figures of the 
 
 ^ the process^ 
 otient. 
 
 it bif the divi' 
 Id be equal to 
 
 Example 2.— Divide 896714 by 8842. 
 
 OPERATION. 
 
 8842)8967l4(101f|]-| 
 8842 
 
 12514 
 
 __88^ 
 
 36"72 
 
 figure, viz., 4, to the 
 
 12514 as the number to be divided, etc. 
 
 Here we say 8842 into 8967 will 
 go once, and we thus get a re- 
 mainder 125, to which we bring 
 down the 1. This gives us 1251 for 
 the next number to be divided ; 
 8842 into 1251 will go times, and 
 we accordingly put into the quo- 
 tient and bring down the next 
 right of the 1251, and thus get 
 
 Exercise 14. 
 
 1. 8916749+227 
 
 2. 8161413-i-lllQ 
 
 3. 1498706-t-2108 
 
 4. S222SoO-i-8i01 
 b. 7142847-*-23 
 
 6. 6171112+17 
 
 7. 8891876+28161 
 
 8. 11223344+3344 
 
 9. 91929394+81007 
 10. 18167123+10123 
 
'•! 
 
 r 
 
 48 
 
 EXAMINATION QUEBTIONH. 
 
 11. Divide 9167402 by 7x I7x 03 
 l'^. 0:v!(le(il49Sn l.yl.ix 15x11 
 13. l^v:dc8lS27UOi)y«i7tin,os"3 
 
 16. "\\ hat iiumbcM' multiiil pHliv fiico,... •„ 
 
 n. 7.3 ti...es 417 i. how it^U™ T^' "'"'^ '''''''''' ' 
 
 IS. 238 time. 1476 i« how many times OH 
 
 2,t:'Vr'''^''"«'^°^"''^''y times 1027? 
 
 1 If n In '' ^^^'^ '' ''^''' """"^ ''"^^'^ 4U7 X 11 1 
 
 Plete'i' journ! y o'l' 9H2 l^n^^.^r'' ^""^' '"''^ '' '^'l"^''^ ^o com. , 
 c"THcS'r>r°ei798? '''''' '^^ «l'illingf«, how much can be pur- 
 ''• ''u^Bani'S?:^^:;;,l^i!^-« four thousand and nine ,, four 
 ''• ''^ifhSiS.ifllliniSr^^^'"""^- -^- ^^ ^Hy thousand 
 
 f- ''StELlli'lilJTUlS.if^^yil^r^ -^ -^^ ^^-enty. 
 26. If 29 ton. of hay cost ,*677, what will I'ton cost ? 
 
 (27) 
 
 11)123450789123456789123456789 
 
 (29) 
 
 (28) 
 ^)91i^68134298764714155986777 
 
 (30) 
 
 7)^0^34005600780091400671478 
 
 12 )77818899906442 2118332700614 
 
 as follows : ii^.12.. ' n!?^^";^?^^::^ i^.^i^.^-ji-^^ 
 
 RECAPITULATION A NdI^minaTION QUESTIONS 
 I- Question. \Vh.,f i. T^:,.:..,.„ ^ oiiursw. 
 
 1. Q«M/2c;,7 Whnt is Division ? 
 
 ^'"Xr^^|;;!;il^:^-;.SS!:,?'^'^^'"^^--°"-onenu 
 
 Q. WhaMfi the dividend? 
 
 A. I he dividend is the number to be divided 
 
 Q. Wl'at is the divisor? ^^i^oa. 
 
 4 Q m ? -""n ' '' '^' ""™^^^ ^y ^^'«h we divide 
 
 T r^^''^^ '« the quot'ei t a 
 
 •^M Iho quotient ie «-h" v^—if -" - 
 
 2. 
 3. 
 
EXAMINATION QUESTIONS. 
 
 49 
 
 (64714155986777 
 
 780091400671478 
 
 I. 
 
 («( Whnt 1b tlic vemninderf 
 
 A. Tiio rcnmlntlfr iB what is left -when the rllvlBor Is r cob- 
 taiiud mi ( xact number of times in .je dividend. 
 
 Q. IFow ifi the nniuiider urii'cn? 
 
 A. Wi! write the riin:iin(ler aboxe n short horizontal line with 
 the divihor bt'iieath it, jiiul ani.e:; tlie exprcBbiou thua 
 lorined to the Integral part of the quotient. 
 
 Q. Can the remainder hv as great a« the divisor? 
 
 J. The remainder cannot be aa great as the divisor. 
 
 How many modes are there of expressing the division of 
 one iiumhcr by anntlier? 
 
 We have three modes of exrreEPing the divie'on of oia 
 number l)y another, viz. : by writing between tVe two 
 numi)er8 the rign of diviMon, -i- or vither of itB paitH • 
 or — . Thus, if we wif^h to cxprt sh the division of 1798 
 
 A. 
 
 by 16, we may do it thus, 17t<«+lC, or thus 1798 : 10, or 
 
 A. 
 
 7«S 
 
 10. 
 12. g. 
 
 13. 
 
 15 
 
 ita 
 
 17 
 
 thub m 
 
 What is the Jistinction between short division and lonir 
 division ? * 
 
 It is short division when the diviror is not greater than 12 
 
 and long division when the divit^or Ir greater than 12. 
 Give the rule for short division. (Sec Art. 60.) 
 
 Give the rule for division when tho divisor can be broken 
 up into two facioiB, neitlier of which is greater than 12. 
 (Bea Art. 51.) 
 
 In this last rule, when there is a remainder after either 
 
 rr °l^''^'""> ''O"^'"' is the correct reniaindcr fonnd ? 
 
 To hnd the true remainder we multiply the ilret divisor 
 
 by the last remaii der and add in the tirst remainder. 
 <5ive the rule for long division. (See Art. 52.) 
 
 ^ In long division, how can you tell how many times the 
 
 divisor is contained in the part of the dividend under 
 
 consideration ? 
 A. By asking how many times the first figure of the divisor 
 
 will go into the first figure, or first two figures, of the 
 
 dividend. 
 
 2- How can you tell when the figure put in the quotient is too 
 
 _ large or too pniall ? 
 A. If it be too large, the product of tho divisor by it will be 
 
 greater than the part of the dividend need : if too small. 
 
 the remainder will be greater than the divisor. 
 Q. How do we prove division f 
 A. To prove division we multiply the divisor and the qnotlent 
 
 together and to the product add the remainder, if there 
 
 be any. The result should be tho dividend. 
 Q. How do we divide by 10, 100, 1000, &c. ? 
 A, We divide bv 10 bv cuttintr ofVthp vioht-iio,.^ Ao-"'-" ^^ *i~- 
 
 dividend :' by 100, by ciTtting off the last "two "<Jit,^t8' to 
 .... ^* "«^.^ J ^y 1000» ^y cuttlig off the last three digits, &^ 
 
ffQ 
 
 OKCIMAL CUIlRlSiS'CV 
 
 SECTION II. 
 
 mom Ah cvnuFmy, tables of mc'a^ey' 
 
 WKIGHTS, AND MEASURES, REDUrrilON ' 
 AND COMPOUND RULEH. * 
 
 DBCflMiL CURRENCY. 
 I. ng dwominationD of Canadian nionoy ^re dollnL. 
 
 The ibilowir. ^ 
 sums oi' mo y 
 
 nnd cents, and 100 cents make 1 dollar 
 explains the n?odo of writing and rcuduy* 
 expressed in the decimal currency : ' 
 
 $T'O0 h read 1 dollai-g. 
 
 t^TL II ^ ^''^^'''^ »"^ 20 cents. 
 t\t^L 1« 'dollars and 8<) cents. 
 
 $417-23 ** 417 dollars and 23 cents 
 423 dollars and 17 centa is written ^423 17 
 94 dollars and 99 cents " $94-90 
 
 6149 dollars and 67 cents " $G149-fi7 
 
 ciphers^''"'''" *'^ converted into cents by annexing twa 
 
 Thus, $69 = 6900 cents. $419 = 47900 cents 
 $17= 1700 cents. $2101 = 21Q100 cents. 
 
 ♦wn ; w J*^ T'^ converted into dol/ars bv cuttin.r oif the 
 two ri??ht-hand figures. These figures arr cut olf bv d^acin^ 
 a sn^all dot between the second mid thi. I fiWn^^^^^^^^ 
 nght-kmd side When thus cutoff, the figure, to re loft 
 of «ie_dot are dollars, those to the right oi'the doV c^nS! 
 
 71934 cents = $719-84 
 4290 cents = $42-90 
 29X671 cents = $2916-? « 
 I 
 
 EXER'^ISB. 
 
 Road $n-12 ; (ITl^-OC ; f^l'iig; .fPC6-6d. 
 Reucl $78-40 ; |<>;(?-8Tf ; fit* ; |79 C9. ' 
 
 V. Bead $11220 : toasdi : lU'ia i A«i7niaL 
 
DECIMAL CURRENCY. 
 
 51 
 
 
 loy fre dollnu; 
 Tlie ibilowii.j,' 
 luuia oi" mo 7. 
 
 3. 
 
 ?. 
 I. 
 
 [49-^7 
 annexing twa 
 
 cents 
 00 cents. 
 
 ultin;-: oJf the 
 oil" by placing 
 u-es from tlio 
 ro.fi to the \ci\ 
 he doU CAQtSb. 
 
 4. 
 
 5. 
 6. 
 7. 
 
 8. 
 9. 
 
 10. 
 
 n. 
 
 12. 
 13. 
 
 15. 
 10. 
 
 jl7. 
 18. 
 
 JV rl 6 down in figures ninoty-throe dollar, forty-novcn ccn.s. 
 
 V\ r. . d.>wu .Ix hundred a..d nine .lollar« »n.| twelve ccnli,. 
 
 U n o , own four l.uudrod and thirty .lollar. and cikU c<nt9. 
 cJntfl ''''" ''■''"" ^"^'"'""^ ""^i ««vc.ity dolluis and «evc. ' -,« 
 
 IIo'.v many cputs are tlioroln three doil.us? 
 
 "o'gl!t"c"n.H7'^' **'■" ''^'"■" '■ «'''^«»'««" ^ollarB and ninety. 
 
 How many conts aro there In $0194 171 
 
 Rcduci' $471-29 to contB. 
 
 Reduco $17-43 to cciitB. 
 
 How many dollars are there In 1714 cents? 
 
 How mimy dollars aro thcro in 0000 cents i 
 
 How many dollars are there lu 6714927 cents f 
 
 Reduce 17147 cents to dollars and cents. 
 
 Rt'duco 0147 cents to dollars ai- ' '-enta. 
 
 Reduce 08705 ceuts to dollars and cent's. 
 
 4. To reduce old Canadian money (pounds shlllinp^ 
 land pence) to the new or decimal currency :■-. ^^'^'^"g«. 
 
 RULE. 
 
 , . fff^i^I/ ^{e pounds bj/ 400, t/ie shillinns bu 20 and tha 
 \fartlungsm the giucn pence and farthinm bu^^ 
 
 U.ifi-it:'r ^"'"'^ '^^^^'^'•' «'^^ ^'^^ -- -^^ *^ ^^^ 
 
 |thoTZiu7tTy I'r'"^^^ ^^ ^^ ^^ multiplying by 5 and dividing 
 Example l.~Reduco £79 4s. llfd. to dollars and cents. 
 
 OPERATION. 
 
 79 X 400 = 31600 = cents in m, 
 J^ f , = 80 = cents in 48. 
 
 bum _ $316-99jV = dollars and cents in £79 43. ll^d 
 Rkason.~£1 = $4 = 400 cents; Is. = 20 cents- and 
 12 fartlnngs = 5 cents, or ono farthii/g = ,^- of a^eni. 
 
 cents. ." "— ' :^.r* 'ofM- .w ^Q%!» aud; 
 
62 
 
 \ 
 
 i/i 
 
 CBCIMAL CUUKENCr 
 
 i 
 
 OPERATION. 
 
 'nx2o' Z%^?,^ =-ntsin£217. 
 38x6-12- ^^^, = c*^"tsmlis. 
 
 bum >. $b/0-35f = dollars mid cents in £217 lis. 9^ 
 
 Exercise 16. 
 Kcduco ^719 m. 4id. to dollars and centa 
 I^cduce X671 12.. 8d. to dollars and cent* 
 
 cduco ^107 0«.10id. to dollars and cens 
 Reduce ^17 17.. 7^d. to dollars and cents 
 
 educo £655 19s. 83.d. to dollars and c^s 
 Keduco £777 lis. 3d. to dollars and cem 
 Reduce £111 n.. lid. to d<.llars and cent 
 
 educe £507 Ss. OJd. to dollars and cents 
 now many dollars and cents are there in £'^7 «« ... , 
 Ho. many dollars and cents are the;: " J^^f ^^. , 
 
 1. 
 2. 
 3. 
 4. 
 
 6. 
 7. 
 8. 
 9. 
 10. 
 
 RULE. 
 
 obtained by 20 arul/nlHir JH ■f^'''''"' of ''"nts thm 
 ef the quotic,Ui"%nce! " ' """^ ''''"* '''* -P™*"' ^i/ 
 iene^^'""'" 1-Red„ca $2M-10 to pouads, shillings, and 
 
 OPERATION. 
 
 Is-iot-^f/^^ f^'^ ^'^ remainder of $3. 
 *^ilO_3iO cents, and 310 -*- 20 - 16 ahni.W „ a 
 
 mainder of 10 cents "" ^^'"'"g' and a re 
 
 10x3r.30-5 = 6d.' 
 Hence 1279-10 = £69 16s. 6d. 
 
 cents = 3 pence. ' "' ^ ®^^"S * a^id fi 
 
PEGIMAL CURRENCr. 
 
 53 
 
 f lis. 9|d. 
 
 shillings, and 
 
 8«?«^ pound>i. 
 and to-them 
 i cents thun 
 Lastly^ mid- 
 product by 
 
 lillings, anc! 
 
 ^^^ExAMPLE 2.~Ileduce $71-29,^, to pounds, shiUings, and 
 
 OPERATION. 
 
 Illfr^ ^^ofoV"^ *" remainder of $3. 
 
 ' '"'"^l^l^ S ^''-'' «™«^« -<^ a remain. 
 
 „"'^-l3=.:28|-5 = 6|d. 
 Iieace$7l-29-i^ = £17 l6s. 5|d. 
 
 Exercise 16. 
 "^ ''1l;«s'';,y,V„r •'°' ^■''"'•^' »- »""-.-° pound,, el.,,- 
 
 andlols"'! '•~^''" '"pother «719-42, $917-87, «429-84, 
 
 "s^^O-^r- 1 ""'•" ""^ '''™ "f 'h" fl-^' or right-hand 
 
 4''9 84 fl ' i ''•' !;"'" °' *" '■"^o"'' "ol"""' with he 
 
 OIMP 9 , '■■ " ^^; ";" "^' ''o™" 'he 8 and carry he 
 
 -^ - rildt" ■^'"" "^ "r !'"'•" ™''™" ''"h the 2 car! 
 
 $2985-89 &c ' '^''"" ""^ " ""<' ^■""7 'he 3, 
 
 s and a ro 
 
 1{ and 5 
 
 Example 2.-From f;9147-80 take $871-94. 
 
 ^I'KIUTION. 
 
 5n-i)-i f, 
 
 nm 
 
 Hero we sav 4 frm-n a o.,,i o . 
 
 cannot ' - "'"^ ^ ^v.-.^-m • y 
 
 SiS2 
 
 75-92 1 li 
 
 tlii-.d column; then 9 from 
 ' "qu^ fi (or 2 from 7) and 5 
 
 •ow 1 fiom the 7 in tl 
 
 18 and 9 rcmaiji; 
 len. jn, &c. 
 
 le 
 
II Ji 
 
 "I .1 
 
 M i.i 
 
 64 
 
 DECIMAL CURREK^CY 
 
 Example 3.— Multiply $67-42 by 24*7. 
 
 OPERATION. 
 
 $6'7'42 Ucre we consider the .$r)'7-42 as be'.n;*- CVlS 
 241 cents, i. e., we pay no attention to the scpli'-ulirig 
 
 point in the multiplicand, and merely p<^int oi! 
 
 47194 the two right-hand figures in the produet, fo? 
 26968 cents. 
 ' 13484 
 
 $16652-74 
 
 Example 4.-~Divide $7149-80 by 19. 
 
 OPJJRATtON. 
 
 19)7149-80(^6-30}^ 
 57 
 
 144 
 
 133 
 
 119 
 114 
 
 Here we divide without regard- 
 ing the separating point, except 
 that, when we bring down the first 
 figure to the right of the point in 
 the dividend, we place a point in 
 the quotient. 
 
 5.8 
 5.7 
 
 10 
 
 Example 6.— Divide $7194-76 by $29-34. 
 
 OPERATION. 
 
 29-34)71 94-76(245/9435^- 
 5868 
 
 1326^ 
 1173-6 
 
 153-16 
 146-70 
 
 6-46 
 
 Hero we divide without re* 
 garding the separating point, i. e., 
 we consider the question as being 
 how often is 2934 cents contained 
 in 710476 cents. We get as a re 
 suit 2452V./V times, or 245 timer 
 with a remainder of $6-46. 
 
 Exercise 17. 
 
 "WTiatis tlie sum of $749-86, $614-91, J91C714. ^mPMo i<n->7A 
 3'U14-29, iijul $29-78 V ' ' , --- ^.) .,-1 .-i 
 
 ^' ^'$9U'9y;Lrd iJmVqT''"* ^^''''' ^"'''''' ^S1^6'-42.$1919' 
 
 18 
 
'i 
 
 X 
 
 examhstation" questions 
 
 as be'.ni' {','li% 
 
 tlic scpji'-atirig 
 
 crcly p-oint oiF 
 
 e product, for 
 
 ithoiit regard- 
 point, except 
 dov/n the first 
 f the point in 
 ce a point in 
 
 tion as being 
 nts contained 
 B get as a re^ 
 ov 245 timer 
 6 -40. 
 
 55 
 
 3. 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 10 
 11. 
 12. 
 13. 
 14. 
 15. 
 
 10. 
 17. 
 
 13. 
 
 19. 
 20. 
 
 21. 
 
 22. 
 
 83. 
 
 Add together $61749, |74-27, $23-32, and £9 8s. 7id 
 From $6714-98 take $982*49. 
 From $4218-23 take $2437-86. 
 
 What is the diflercnce between $914-71 and jC471 16e. lO^d.f 
 Wliat is the ditterence between Jt;29 18s. 9d. and $649-321 
 Multiply $671-21 by 48. 
 Multiply $519-26 by 789. 
 How much ie 529 times $16-83 f 
 Divide $6149-73 by 67. 
 Divide $18793-67 by 149. 
 Divide $1714-86 by $71-42. 
 Divide $9167-42 by $14783. 
 
 ^"/''^'Pro*!^ ^?''^" for $147-80, a carriage for $217-20, bflrness 
 ine f ^ ^°^' ^^^'^' ' ^^^^ ^^^ ^^® ^'^'°^® *^°^* 
 
 "What is the twenty-seventh part of$916-'/l? 
 Divide $^71-90 equally among 13 persons ; what is the share of 
 
 I went out to make purchases, hr.viiifr with me £11 16s. 7*d. 
 1 boiid^t and paid for groceriei^, $17-80 ; dry goods, $21-63 : 
 furniture, $128-76 ; and books, $37-26. Hovv much change 
 did I bring home ? 
 
 What is the cost of 17 tons of hay at $17-45 per ton ? 
 
 From $723-86 take $297-42 ; multiply the result by 63 and divide 
 the product l)y 217. 
 
 In 1858 tl'o ext orts of Canada were as follows: Produce of the 
 nnnc, $314823 ; produce of the flsliciies, $718296 ; produce 
 oi the toreM, $944<727 ; ai unals a u tiieir products, $2462765 : 
 agricultural products, $7904400 ; manufactures, $325376 ; other 
 art 'cles, $1 12538 ; and goods not reported (estimated) $1443044 
 ic'o'o'^ ^^^^ ^^'® ^°^^^ ^^^"® ^^ ^^® Canadian exports fo^ 
 
 '^^IfooSVlo^^V*"*'' ^■''1}'*?.^ ^2'' I^® y^^^ 1858 amounted to 
 $29078527 ; how much did the Canadian imports exceed the 
 exports in 1858? 
 
 If the population of Canada be 2954600, what was the valup '^f 
 the imports for each individual in 1858 ? 
 
 EXAMINATION- QUESTIONS. 
 
 Note.— The answers to these questions are found as indicated 
 
 after each quc';«tiun. 
 
 i. ., .!.,i ,i,t? ijiG tOi_j. Jill nuns ui vj:uiad;an monoyif (Art. 1.^ 
 
 2, What are the Oanailian t<ilvev coiiiB ? (Table on next p:it;e.) 
 
 3. What is the diameter of the Canadian cent? (Table on next 
 
I 
 
 60 
 
 TABLES OP MOITET, 
 
 6. How are dollars converted into cents? (Art 2^ 
 
 7. How are cents converted into dollars ? (Art*. 3 ) 
 
 I ^;in» s;iir ^ - s^ss ^r ^ ^^^^^ 
 
 10. How do we add and subtract, „.ultiply and divide do.Ls and 
 ^^t'^^nXZT^'^^^^^^^^ as so ruany cents, and 
 
 TABLES OF MOj^^EY, WEIGHTS AND 
 
 MEASURES. ^^^^-L' 
 
 CANADIAN DECIMAL MONCY 
 100 cents (c.) make 1 dollar, marked $. 
 
 twcS^^ce-t^'p^Lt a^o VT^^-'' P^^-, anu . 
 
 wiiich is bronze. ^ ^"^^^ 5 ^^^ a one-ccnt pieo^ 
 
 Wellh^lT AvoVtup^^^^^^^ *"^^ ^'^ diameter, .ud loo c. J 
 
 OLD CANADIAN CURRENCT. 
 
 TABLE. 
 
 4 farthings make 1 penny, marked d 
 
 ^2 pence » I shilling, " g 
 
 6 shillings " 1 dollar, " *' 
 
 4 dollars " 1 pound, " I.' 
 
 ENGLISH OR STERLING MONEY. 
 
 TABLE. 
 
 4 farthings (qr.) make 1 penny, marked 
 ^2 pence " ^^hni^ u'^ 
 
 Te« Jn. " "^' Ipoundr " X. 
 
 to 20 shuiinSlSn^^ H ""^^ ^ ^ »^""g« and the Sov^i^ 
 
 d. 
 
 s. 
 
WEIGHTS AND MEASURES. 
 UNITED STATES MONEY. 
 
 TADLE. 
 
 10 mills ^m.) make 1 cent, marked ct. 
 10 cents " 1 dime, " d. 
 
 10 dimes " 1 dollar, " $. 
 
 10 dollars " 1 eagle, " E. 
 
 67 
 
 AVOIUDUPOIS WEIGHT. 
 
 TABLE. 
 
 16 drams make 1 oauce,. marked oz. 
 
 I ft ounces *' 1 pound, 
 
 J5 pounds " 1 quarter, 
 
 4 quarters " 1 hundred vrvVtU 
 
 10 cwt. ** 1 ton, 
 
 »v \p.— This weight isiised in weighing- Vv^vy articles, as meat-. 
 >coi-%v, vegeiab!e>i, grain, etc. 
 
 n 
 
 lb. 
 
 t( 
 
 qr. 
 
 (( 
 
 cwt. 
 
 t( 
 
 t. 
 
 ^jOCOi-% 
 
 TKOY WEIGHT. 
 
 TABLE. 
 
 24 ^ains (grs,) make 1 pennyweight, m^vked dwt. 
 20 pennyweights " 1 ounce, ^' oz. 
 
 12 ounces " 1 pound, " lb. 
 
 Note.— Trw weight is used in weighing the precious metals 
 *na atones ; also m eoientiflc investigations. 
 
 APOTHECARIES' WEIGHT. 
 
 TABLE. 
 
 20 grains (gi^s.) make 1 scruple, marked scr. or 3. 
 
 3 scrupUs " 1 dram; " dr. or 3 . 
 
 8 drams) « 1 ounce, " oz. or ? . 
 
 12 ouncea " i pound, " lb. 
 
 Note.— Apctthecai-Jes and Vhygiclans mix their medicines by 
 ihls weight, bi^ii ihqy ^jv,- uni *eJl ly ^kvoirdupois. 
 
I 
 
 i 
 
 if; 
 
 iit' 
 
 'Mi 
 
 58 
 
 12 1hos(l.) 
 lii inchoji 
 
 8 iniloa 
 
 TABLKS OF MONKY, 
 tONG MKASUUE. 
 
 TAIU.R. 
 
 / 
 
 mnko 1 inch, 
 1 foot, 
 
 1 ro<l, j)«lo, or pcrnli/* 
 1 loii^ue, 
 
 41 
 it 
 U 
 ti 
 tl 
 
 injul<0(! In. 
 
 It 
 t( 
 
 (t 
 it 
 tl 
 
 a. 
 
 I U. or nor, 
 
 <\ir. 
 
 m. 
 
 .«rn?;i'f„h.H!;'~,4''K;,"'' "*"' ""' "' "•" <"«'<'>,f«r,.„..., „f „A 
 
 « ft'Ct 
 
 120 fttthoirja 
 
 tl. 
 ti. 
 it 
 
 1 plU'O. 
 
 1 Ihthom. 
 1 oublo-longth. 
 
 SQUAUE OR LAND MEAS'JKE. 
 
 TAMLK. 
 
 144 sqtmro Inohos luako 1 Kqu; ,> foot, iMnr'«od sn ft 
 «;squaro /bet - l s.nuu . ynrt!, - ^* • 
 
 " 1 S(|\uuo i'od, ' 
 1 rood, 
 1 ttcro, 
 1 square milo, 
 
 "■ ■->|>i «».«%-' #V'\JW 
 
 1*^0^ square ynrds 
 40 square rods 
 4 roods 
 640 acres 
 
 it 
 it 
 it 
 
 it 
 it 
 ti 
 li 
 
 tt 
 
 B(|. yd. 
 pq. rd. 
 r. 
 a. 
 sq. m. 
 
 
 JOO links or 4 rods " 1 chain '« n 
 
 80 chains " i n.ilo.' m !: 
 
 lUOOO square links , " 1 sq. diain ♦' ' ' 
 
 10 square eliains " 1 aoro, 
 
 «( 
 
 aq. 0. 
 
WiCKJHTU AND MMAHJIHICM, 
 
 60 
 
 RfJ. 
 
 ft. 
 
 B(|. 
 
 vi 
 
 pq. 
 
 I'd. 
 
 r. 
 
 
 R. 
 
 
 ^q. 
 
 m. 
 
 M, 
 
 nn, for 
 
 «. I 
 
 tjavfj-n, 
 
 cnnto OH Hoi.ii) mkasuhm. 
 
 TAfU.W. 
 
 1Y2H mi)>lo liu'ljrn (t'uh. In.) imiUtt I (mi1»u» foot, iiuiilitMl <miI). 
 It 
 27 nihlo li'ot, iniikn I (mblo ynnl, inmluMl cub. yd. 
 4i> (Md»lu rpot (>!' iouikI tiiiilMM< i ninko I loii, imiiUrd 
 50 (Mil)io I'vvi of liown tlmlior j* idn, 
 
 12H fuhUi I'ot't. (»r llrcwood mnko I cmiiI, nmH<i'«l o. 
 
 A pllf or ooitl Nvooil 4 r.ol hWili, 4 fool, wl.lo, iumI H fn'i |(»nw, 
 poiilulu* 12H (Mililo loot, Of 1 .ui'tl. Oiitt f.KM Iti Ifiiuth uf micIi n |ill«» 
 In <'i»IItM| u ,'unl fimt ; It |m I'limtl lo |n ndIIiI A-oI, iukI Im «iiii«»Mii|rjii|y 
 ••lUlvuUmt totUuiilultlli jmri .'rmitictl. 
 
 CLOTH MMASimE 
 
 TAill.li). 
 
 2i InrlioM (In.) nudto 1 null, iniiihod nii. 
 
 4 niillM ♦♦ 1 (|nmt(M', " {\r. 
 
 « quurt,(M'« •♦ 1 FltMnlnli cll^ " Fl. cv 
 
 4 qtuirti'i'M •' 1 vuril. »• yd. 
 
 5 nuurtom " 1 ICiigiJHh <'II. •' K. o. 
 
 6 (|unrtt>i'8 " 1 Fmujb t?ll, «• K o. 
 NoTK.— Tho Sootch ell ooiitjiIuM 4 qiiitili-m IJ inoU, 
 
 DKY MKAHUUR. 
 •tahi.k. 
 2 pints (pt.) tnako 1 (jniirf,, 
 4 <n»iirt« " I gidlou, 
 
 2 gallonii " 1 piMjk, 
 
 4 mxiK-N ♦' 1 fjuntbuj, 
 
 S(i bi!;Mii('lrt " 1 chnldion, 
 
 lilQUlJ) MKASlJJtK 
 
 TAHLK. 
 
 4 0\h (gill) rnrdcp 1 pint, 
 
 marked qt. 
 
 •• pk. 
 
 *' bn. 
 ** d). 
 
 2 pintH 
 4 (pmrtfl 
 
 1^1 ^ gHlioTJH 
 
 *^ but ruin 
 2 nniTHlieads 
 2 pipvS 
 
 t4 
 i( 
 t( 
 
 ;( 
 (I 
 (I 
 
 1 (|Ullll., 
 
 1 gullou, 
 1 bttrrol, 
 1 bo^/Hhuftd, 
 1 pipe, 
 I tun. 
 
 trmrUf'd pt. 
 (It. 
 Kfil. 
 bar. 
 hltd. 
 pi. 
 
 tun. 
 
 it 
 it 
 
 tt 
 
60 
 
 TABLES OF MONfiy, 
 TAfiLJ. 
 
 ii ft. 
 
 marked min. 
 " h. 
 " d. 
 " wk. 
 " mo. 
 
 24 hours .. 1 ';<""•. 
 
 ^ <iav9 „ ; "V, 
 
 13 lunar months or /'"'""• "'onth, .. *«; 
 
 ^'^«* month, JHnnnry 
 March, 
 
 May, 
 June, 
 July, 
 
 Auerust 
 
 eeptember, 
 October, 
 
 November, 
 ■December, 
 
 First 
 Second 
 
 Third 
 
 Fourth 
 
 Fifth 
 
 Sixth 
 
 Seventh 
 
 Eighth 
 
 Ninth 
 
 S'^nth 
 
 Elevf-'th 
 
 Twelith 
 
 a 
 u 
 
 it 
 
 (( 
 
 (t 
 
 (( 
 
 ({ 
 
 (( 
 
 (( 
 
 (( 
 
 <l 
 
 has 31 days. 
 " 28 " 
 31 
 30 
 31 
 30 
 31 
 31 
 30 
 31 
 80 
 31 
 
 (( 
 (( 
 i( 
 (( 
 
 (I 
 
 « 
 
 (I 
 « 
 
 « 
 
 (( 
 u 
 i( 
 
 <{ 
 
 (( 
 
 tt 
 
 It 
 
 tt 
 
 ^he number of davs in *i. 
 
 The number of dava ,•« 
 flr"t an/ '• •'"""'"•yon fl„t fl„l, l?:** ""' ""■'' inlerveniJ 
 
 oavinfr sn ^nj,. ^ ,. / '^""le on the fino-p« ..y„. *^^ ^*^ 
 
 - -- vnlj, iuii iutcthe spaceg. ""' '^"^ -^^Oi'i^aJ! *^ <« 
 
^narked mlxK. 
 h. 
 d. 
 wk. 
 mo. 
 
 t{ 
 (i 
 
 t( 
 
 -ar, marked yr, 
 <Jay8 In each ari 
 
 « 
 (i 
 
 K 
 
 t 
 I 
 
 t 
 
 ay be recalle* 
 
 W)irected by 
 intervening 
 'ce betweeo 
 'J iii eeeoncj 
 
 
 WEIGHTS AND MEASLT.ES. -"V 
 
 CIRCULAR MEASURE. 
 
 TABLE. 
 
 ^0 seconds (") make 1 n^inute, marked '. 
 
 JOiimiutes u 1 degree, " «> 
 
 80 degrees *< i g| jj ' ., ' 
 
 12 sigus or 30J degrees " the circumference of 
 
 a circle, marked c. 
 
 *.,Je^W.dfc-/reckou\njflamude'and1on^^^^^^^^^^^ ^^^ measuring 
 
 MISCELLANEOUS TABLE 
 
 12 iwUvidual things make 1 dozen. 
 ]l^^^^n " 1 gross. 
 
 ll^-T-rrv:-" " 1 great gross. 
 20 indiviOual things " 1 score 
 
 24 sheets cf paper. « 1 quire] 
 '^<^ quires " 1 ream 
 
 f^^^T^^ ;; 1 barrel ofpork or beef. 
 
 ,: ,, " 1 barrel of flour. 
 
 ^* • " 1 stone. 
 
 BOOKS. 
 
 A ^hcpt folded mto two leaves iajcalled a folio. 
 
 *' J." «ed into four leaves is called a quarto, or 4to. 
 
 lolded into twelve leaves is called a duodecimo, or 
 
 « ?1^^^ i"*^ ^'^*^^" leaves is called a 16 mo. 
 lolded mto eighteen leaves is called an 18mo. 
 
 HEDUCTIOK 
 
 ^ Reduction « the process of changing a number from 
 mo denominatiuA X) another without alterilg its Xe. 
 
s 
 
 • ( 
 
 63 
 
 REDUCTION. 
 
 8. Reduction Ascending Is the process of reducinc i 
 number from a lower to a higher denomination. 
 
 9. Reduction Descending is the process of reducing' a 
 number from a higher to a lower denomination. 
 
 RULE FOR REDUCTION DESCENDING. 
 
 10. Multiph/ the highest giveii denomination hu that 
 pian/ih/ tokick expresses the n amber of the next lower con> 
 tanned in one ofitn units ami add to the product that number 
 
 Proceed in the smne wa?/ with the remit, and coniinue 
 the process until the required denominaiion is obtained. 
 Example l.—Reduce 427 miles to yards. 
 
 OPEHATION. 
 
 427 = miles 
 8 
 
 3416 = furlongs 
 40 
 
 136640 = perches 
 8* 
 
 683200 
 _68320 
 
 751520 = yards. 
 
 Example 2.— Reduce 6 bushels 
 quarts. 
 
 OPERATION. 
 
 6 bush. 3 pks. 1 gal. 1 qt. 
 
 Ilere we first multiply by 8, 
 because each mile is equal to 8 fur- 
 longs ; next we multiply the furlongs 
 by 40, to reduce them to porches, 
 because each furlong is equal to 40 
 ::9rches ; lastly we raultij)ly the 
 perches by 5}, to reduce them to 
 yards. 
 
 3 pks. 1 gal 1 qt. to 
 
 27 = pks. in 6 bush. 3 pks. 
 65 = gals, in 6 bush. 3 pks. 1 gal. 
 221 3=qts. in 6Jt)ualk Splca. l gal. I qt. 
 
1 
 
 REDUCTION. 
 
 ' reducing i 
 
 ' reducing a 
 
 NO. 
 
 ion by (hat 
 :t lower cwi' 
 that number 
 n the quail. 
 
 nd continue 
 iaincd. 
 
 iply by 8, 
 lal to 8 fur- 
 the fm longs 
 to porches, 
 jquul to 40 
 ulti{)ly the 
 ZQ them to 
 
 tta 
 
 il 1 qt. to 
 
 1. 
 
 a 
 
 4. 
 
 Here we first multiply the 6 bushels by 4 to reduce to 
 p<jri{s and add in the 3 pecka given ; next we multiply the 
 resulting pedes by 2 to reduce them to gallons, and add in 
 th« 1 gallon given, &c. 
 
 EXKHCISE 18. 
 
 Rodnco 47 cords of wood to cubic feet. 
 
 Reduce 6497 lbs. Avoir, to ounces. 
 
 Reduce £97 16s. ^(i. to fjiribings. 
 
 Reduce 127 a. 2 r. 17 per. 19 yde. 8 ft. 121 in. to inches. 
 
 Reduce G09 toriB 4 cwt. 3 qrs. 17 lbs. 4 oz. 7 drs. to dranMfc 
 
 6. Reduce 4 pipes 1 hlid. 1 bil. 19 z^U. 2 qts. to quarts. 
 
 7. Reduce 17 miles 7 fur. 7 p«r. 2 yds. 2 ft. 4 in. to linos 
 
 8. Reduce 6° 17' 49" to Rcconds. 
 
 9. Reduce 2 cli. 17 bush. 2 pkt?. 1 gal. 1 qt. to pints. 
 
 10. Reduce 9 Frencli ells 1 qr. 3 na. \\ in. to inches, 
 
 11. Reduce 17 weelss 4 day j 9 hours 29 miii. 17 boc. to Beconaa, 
 
 12. Reduce 29 E. 9 dollavp C dimes 2 cents 4 milla to mills. 
 
 13. Reduce 17 lbs. 9 oz. \Q dwt. 11 grs. to grains. 
 
 14. Reduce 37 cub. ydfl. 9 cub. ft. 1111 cub. in. to inches. 
 
 15. Reduce 129 lbs. 4 or. 2 ecr. 11 grs. to grains. 
 
 13. How many square feet nre there in 127 square perches? 
 17. How many inches are there in 127 Eng. clls 1 qr. 2 na. ? 
 48. How many cub. ft. of wood arc there in 17 cords 63 cub. &9 
 
 19. How many quarts are there in 714 gallons ? 
 
 20. How many scruples are there in 71 lbs. 11 oz. 3 drs.? 
 
 21. In 16 cwt. 1 qr. Id lbs. how many ounces are there? 
 
 22. In 11 miles 2 ft. how many inches are there ? 
 
 23. In 123 acres 17 per. how many square yards are there f 
 U. In 27 years 16 days 4 min. how many seconds are theroS 
 26. In 161 days 14 hours how many liours are there f 
 
 26. In £1978 17s. ^<^. how many farthings are there f 
 
 27. How many pints are t-:ere in 17 bush. 1 pic. 1 gal. ) 
 
 28. How many grains are here in 9 lbs. 17 dwt. ? 
 
 29. Rednce 9 sq. m. 1 a. r. yds. to square Inches. 
 <J0. Reduce jClTl lis. IJd. to farthings. 
 
 RULE FOR REDUCTION ASCENDDJTft 
 
 11. Dirirk the given number by that number which it 
 takes of {fie-.^tven denomination to make .om of the next 
 
 
64 
 
 REDUCTION. 
 
 'I I 
 
 
 hiffhcr. Set down the remainder, if (iny, and proceed in 
 the same manner with each mccesnve denomination till you 
 come to the one required. TJu. last quotient, with the several 
 Temaindera annexed, will he the answer required. 
 
 Example- 1. — IIow many pounds Apoth. are there in 
 IGVlUsciuples? 
 
 OPERATION. 
 
 8)16719 scr. 
 8)5573 drs. 
 12)696 oz. 6 drs. 
 58 lbs. oz. 5 drs. 
 
 Uere we divide the scr. 
 by 3 to reduce to drams, 
 because every 3 scruples 
 malce 1 dram. We thus 
 obtain 16719 scr. = 5573. 
 drams. Next we divide 
 the drams bv 8, because 
 every 8 drams are equal to 1 oz., and we thus find the 
 given number of scruples to be equal to 696 oz. 5 drs. 
 scr. Finally, we divide the ounces, by 12, and thus obtam 
 68 lbs. G oz. 5 drs. for the answer. 
 
 Example 2.— Reduce 719864 pints to bushels. 
 
 OPERATION. 
 
 2)719864 = pints. 
 4)369932 qts. pts. 
 2 )89983 gals. qts. pts, 
 4)44991 pks. 1 gal. qts. pts. 
 11247 bush. 8 pks. 1 gal. qts. pts. 
 
 Exercise 19. 
 
 Ana. 
 
 1. Reduce 71989 inches to miles, furlongs, &c. 
 
 2. Reduce 6142 minutes to weeks, days, &c. 
 
 8. Reduce 81427 grains to pounds, ounces, &c., Apoth. weight, 
 4. Reduce 9141762 cubic inches of wood to cord-feet, &c. 
 
 6. In 7177 pints how many chaldrons, bushels, &c. ? 
 G. In 914 cubic feet how many cubic yards ? 
 
 7. In 61479 inches how many French ells, qrs. &c.f 
 
 8. In 89 days how many weeks, &c. ? 
 
 9. Huw many ious.'cwts., &c., are there in 1714tyt54 drams? 
 to. How many acres, roodt, dtc, are there ia 1714961 Inches f 
 
COMPOUND ADDITION. 
 
 60 
 
 nd jproceed in 
 tiatton till you 
 vifh the several 
 'cd. 
 
 . are there in 
 
 divide the scr. 
 iuce to drams, 
 ery 8 scruples 
 m. We thus, 
 9 scr. = 5573 
 !xt we divide 
 by 8, because 
 thus find the 
 6 oz. 5 drs. 
 id thus obtain 
 
 lels. 
 
 U. How many tuns are there In 171439 quarts? 
 
 ^"^ ^ZI^u^l ^''"''.' ?''^ ^^ ^^^'«'> ^'^y-' *°-' *r^ there In 
 
 13. rteUuco 171490894 Jartl,ingH to pounds, Bhillinsr,, raid pence. 
 
 14. K.duco 2987149 miJlri to eagles, dollarn, dimi-s, &r, 
 
 15. Roduco 21111496 inches to roods, square porches, Ac. 
 10. Rodnco 17498 cul>io fout of wood tu cords. 
 
 17. Roduco 919817 ponco to poui.ds, shillings, &c. 
 
 18. lieduco 999 dwt. to pounds, «fcc. 
 
 19. Reduc'j 1771 gaUoiis to bushels. 
 
 20. fiorlnee 91666 Flemish elk to French oils. 
 
 21. How many cwte. qrs. andlb.s. arc there in 1714^ ,. . f 
 ^ How many miles, fur. per. &o., are there in 17110 .eet ? 
 
 23 How many degrees, min. and sec. &c., are there in 1111111 bcc t 
 
 24 Reduce 667789 cubic inches to cubic /ards &o 
 2d Reduce 7891427 grains to pounds, Apoth. ' 
 
 i» Reduce 678846 grains to pounds, Troy 
 
 27 Reduce 298714 dram., to pounds, Avoir. 
 
 re In 61479867 square Indies how many acres, ro-ds &c f 
 
 29 In 91990 yards bow n.,- .y leaguen ? ' ' 
 
 80. In 714986 inches how many fathoms? 
 
 pts. Ans. 
 
 COMPOUND ADDITION. 
 
 12. Compound Addition is the addition of apoiic - 
 cumbers of more than one denomination. ^^ ' 
 
 )th. weight 
 ;, &c. 
 
 drams? 
 L inches ? 
 
 RULE. 
 
 Set doion the addends under one another so that unin 
 of the same order shall be in the same vertical column. 
 
 Begin at the right-hand side and add the first column • 
 divide the sum by the number of that order wlichmakTon. 
 
 ?r:^^.!!^^^'fe^f '• -^ ^o.njhe remainde::t:::^::z 
 
 i ^""TTC' """/^' ^''?' *^^''yif^e quotient to the' next columri 
 ^"■^^iiM?^ through all the collmm to the last 
 
IP 'T f 
 
 4 I 
 
 ; 
 
 , 
 
 66 
 
 COMPOUND ADDITION. 
 
 Example 1.— Add together 9 weeks 4 days 17 hours 
 11 mm. ; 6 wks. 3 days 11 his. 49 min. ; 9 wks. 2 duya 
 hrs. 53 mm. ; and 11 wks. 5 days 21 his. 35 min. 
 
 OPERATION. 
 
 Here the minutes added up amount 
 to 148, wiiich we divide by GO in order 
 to reduce them to hours ; this gives 2 
 hours to carry to th(, next column and 
 28 minutes to set down iu the columa 
 of minutes, and so on. 
 
 wks. ds. hrs. min. 
 9 4 IV 11 
 
 6 
 
 9 
 
 17 
 
 3 
 2 
 5 
 
 11 
 
 
 
 21 
 
 49 
 53 
 85 
 
 43 2 3 28 
 
 Example 2 
 lis. ll|d.; £1 
 £10 lis. ll^d. 
 
 operation. 
 
 £ s. d. 
 
 917 16 4| 
 
 216 11 11^ 
 
 160 14 7 
 
 916 1 9f 
 
 100 9i 
 
 70 17 Ui 
 
 16 16 9| 
 
 2399 6 3 
 
 —What is the sum of £917 16s. 4|d • £216 
 
 60 14s. 7d. ; £916 7s. 9|d. ; £100 Os.' 9M. ; 
 
 and£16 16s. 9id.? * 
 
 Here the farthings added amount to 12, 
 which we divide by 4 to reduce them to 
 pence; this gives us 3 pence to carry and 
 no farthing:, to set down. The pence col- 
 umn, with the 3 carried, amounts to 63 
 which we divide by 12 to reduce to shil- 
 hngs ; this gives us 3 pence to set down 
 under the column added anr" 5 shillings to 
 carry to the shillings' column, and so on. 
 
 (1) 
 
 £ 8. d. 
 179 11 4i 
 
 96 2 0* 
 
 297 8 llf 
 
 9 lOi 
 
 607 19 2* 
 
 9*^ 17 8f 
 
 (4) 
 
 cwt. qre. Iba. oz. 
 
 01 2 22 12 
 
 Exercise 20. 
 
 (2) 
 
 miles fur. per. yds. 
 
 63 7 16 2 
 
 19 6 11 4 
 
 7 36 5 
 
 29 2 6 2J 
 
 11 6 22 4* 
 
 63 7 2 1 
 
 (3) 
 lbs. oz. drs, scr 
 16 11 4 1 
 
 9 
 
 126 
 
 91 
 
 9 
 
 27 
 
 8 
 7 
 8 
 
 4 
 
 5 
 4 
 
 7 
 2 
 6 
 
 
 2 
 1 
 
 
 
 2 
 
 13 
 9 
 
 29 
 
 1 
 
 
 
 t 
 
 X 
 
 
 
 24 
 11 
 
 o 
 
 7 
 
 15 
 
 7 
 
 6 
 
 (5) 
 yds. ft. in. 
 2 11 
 
 27 
 
 16 
 
 98 
 
 7 
 
 9 
 
 1 
 
 2 
 
 1 
 o 
 
 2 
 
 6 
 
 lu 
 8 
 
 bush. 
 9 
 17 
 19 
 37 
 96 
 
 (6) 
 
 pks. gals. 
 
 1 1 
 
 1 
 
 1 f> 
 
 1 i 
 
 1 
 
COMPOUND ADDITIOJS-. 
 
 4 days 17 honrg 
 9 wks. 2 daya 
 
 5 min. 
 
 idded up amount 
 le by GO in order 
 Lus ; this gives 2 
 next column and 
 ni in the columa 
 
 16s. 4fd. ; £216 
 £100 Os. 9id. ; 
 
 id amount to 12, 
 reduce them to 
 nca to carry and 
 The pence col- 
 amounts to 63, 
 ) reduce to shil- 
 nee to set down 
 nr" 5 shiUings ro 
 nn, and so on. 
 
 (3) 
 s. oz. drs. Bcr 
 3 11 4 1 
 5 
 
 8 
 7 
 8 
 
 4 
 
 4 
 7 
 2 
 6 
 
 
 2 
 1 
 
 
 
 2 
 
 (6) 
 I'jsh. pks. gals. 
 
 9 11 
 
 17 1 
 
 19 1 n 
 
 37 
 96 
 
 (7) 
 
 coffls cord ft. cub. ft. 
 
 39 7 H 
 
 ^? 6 8 
 
 37 4. 14 
 
 (10) 
 
 Ibe. OT. dwt. era. 
 
 16 4 2 17 
 
 93 11 17 23 
 
 la 10 16 14 
 
 2? 9 12 
 
 (13) 
 
 ,'k8. dayr. hrs. 
 
 27 
 19 
 11 
 21 
 19 
 
 4 
 6 
 4 
 3 
 5 
 
 23 
 17 
 9 
 12 
 14 
 
 (16) 
 
 Hiig. tolls qris. na. 
 
 43 2 2 
 
 91 3 2 
 
 16 1 3 
 
 (19) 
 
 .^ ^- ^• 
 
 127 19 8 
 
 67 4 11* 
 
 91 16 2* 
 
 127 11 d 
 
 63 12 ici 
 
 a. 
 
 297 
 96 
 11 
 
 27 
 
 (8) 
 
 I'- per, 
 3 16 
 
 
 2 
 8 
 
 9 
 39 
 16 
 
 (11) 
 
 gale. qte. pts. 
 
 12 1 1 
 
 16 1 
 
 10 1 1 
 
 9 10 
 
 67 
 
 (9) 
 
 yfle. qrs. na. in. 
 
 6 2 12 
 2 3 3 1 
 
 6 2 12 
 
 7 13 3^ 
 
 (12) 
 sq. per. sq. yds. sq. ftj 
 II 23 ^6 
 
 98 16 7 
 
 81 so 6 
 
 27 27 2 
 
 (14) 
 
 £ s. d. 
 
 129 6 lU 
 
 17 14 2i 
 
 93 11 7^ 
 
 16 19 2* 
 
 9 2 9f 
 
 (17) 
 
 pks. gals. qts. 
 
 12 1 
 
 6 1 1 
 
 12 1 1 
 
 19 1 1 
 
 (20) 
 
 127 16 20 
 
 19 17 30 
 
 63 27 16 
 
 47 35 9 
 
 63 10 25 
 
 
 
 (15) 
 
 
 qrs. 
 
 1^8. 
 
 oz. 
 
 di"s 
 
 16 
 
 24 
 
 11 
 
 14 
 
 93 
 
 10 
 
 14 
 
 7 
 
 27 
 
 21 
 
 13 
 
 14 
 
 21 
 
 16 
 
 15 
 
 9, 
 
 9 
 
 2 
 
 10 
 
 n 
 
 
 (18) 
 
 07. 
 
 dwt 
 
 127 
 
 14 
 
 93 
 
 5 
 
 91 
 
 17 
 
 127 
 
 12 
 
 grs. 
 
 6 
 
 21 
 
 17 
 
 18 
 
 (22) 
 07'. di-8. !3cr. grs 
 2 17 
 
 a. 
 
 27 
 
 4-6 
 
 323 
 
 86 
 
 (21) 
 
 yre. %vks. dtiys 
 
 27 50 2 
 
 93 16 4 
 
 11 2 6 
 
 23 14 
 
 67 47 6 
 
 (23) 
 
 r. -per. 
 2 36 
 
 7 
 
 ft. 
 
 7 
 
 in. 
 
 3 37 
 2 4 
 
 8 
 
 
 
 
 126 
 107 
 
 13 
 
 27 
 
 6 
 
 23 
 
 "**•*—-—■ 
 
 ■ ■ i>i^ 
 
 ■* — ■ m^ 
 
 
I i 
 
 68 
 
 COMPOUND SUBTRACTION. 
 (24) 
 
 lea. miles 
 
 fur. 
 
 por. 
 
 yfis. 
 
 ft. 
 
 in. 
 
 linos 
 
 14 2 
 
 7 
 
 23 
 
 4 
 
 1 
 
 10 
 
 7 
 
 18 1 
 
 3 
 
 10 
 
 2 
 
 
 
 6 
 
 10 
 
 7 1 
 
 6 
 
 33 
 
 5 
 
 2 
 
 6 
 
 4 
 
 17 2 
 
 6 
 
 17 
 
 3 
 
 2 
 
 7 
 
 11 
 
 16 2 
 
 T 
 
 15 
 
 2 
 
 2 
 
 8 
 
 9 
 
 
 
 
 
 
 
 
 COMPOUND SUBTRACTIO.^, 
 
 13. Com,yOund Subtraction is the subtractioa of appll- 
 cate numbers of more than one denomination. 
 
 ll 
 
 RULE. 
 
 Set the subtrahehJ, under the minuend so that units of§ 
 the same order come in the same vertical column. I 
 
 Begin at the right-hand side and subtract the first terwM 
 of the loiocr line from the corref^ponding term of the uppeA 
 line^ if possible ; but if not, inaease the term of the upper \^ 
 line by the number of units of that denomination which | 
 make one of the next higher ; theyi subtract and set the re-', 
 mainder under the first column and carry one to the given i 
 number of the next denomination contained in the subtra- 
 hend. 
 
 Proceed thus th'ough all the cohimns to the last. 
 
 Example 1.— Subtract 27 miles 7 fur. 6 per. from 93 
 miles 2 fur. 1 per. 
 
 OPEKATION. 
 
 miles fur. per. 
 93 2 7 
 27 7 6 
 
 65 3 1 
 
 Here we say 6 per. from 7 per. and 1 perj 
 
 remains, and wo set down this remainder 
 
 under the column subtracted. Next, 7 fur, 
 
 from 2 fur. we can't; and since 8 fur, 
 
 make 1 mile, we increase the 2 fur. by 8; 
 
 - men t ^ui, iiuui i\) tin: mm a lur. remain. 
 
 Again, adding 1 to the 7, we say 8 from 3 we can't ; bu| 
 
 8 from 13, &c. 
 
>N. 
 
 lines 
 
 7 
 10 
 
 4 
 11 
 
 9 
 
 TIO:^^, 
 
 traction of appli- 
 on. 
 
 so that units of 
 lumn. 
 
 act the first term 
 rm of the upper 
 rm of the upper 
 wiination which 
 't and set tJte re- 
 one to the given 
 d in the subtra- 
 
 ) the last. 
 
 . 6 per. from 93 
 
 7 per. and 1 per. 
 1 this remainder 
 3d. Next, T iur, j 
 nd since 8 fur, 
 the 2 fur. by 8 ; 
 id 3 fur. remain. 
 3 we can't; bu| 
 
 COMPOUND SUBTRACTION. nr^ 
 
 by 
 
 Example 2.-rrom 27 lbs. 4 oz. 1 drs. 2 scr. take 9 lbs. 
 I o OZ, J. o crrs» 
 
 OPERATION. 
 
 IlK oz. drs. scr. grs. Here 16 grs. from 0, we rrn't; 
 
 -^ t> 16 2 and 1 remains; from 7 and 7 
 
 remain ; 6 from 4, we can't, but 6 
 from 16, i. e. 4 + 12 and 10 remains. 
 &c. ' 
 
 Exercise 21. 
 
 (2) 
 
 miles fur. per. 
 
 107 3 
 
 93 7 27 
 
 (3) 
 
 qrs. lbs. oz. 
 
 13 13 6 
 
 11 22 11 
 
 (6) 
 
 £ p. d. 
 
 279 li 
 
 114 17 ll| 
 
 (9) 
 
 hrs, inin. secj. 
 
 274 52 9 
 
 i^8 57 14 
 
 fur. 
 16 
 11 
 
 (12) 
 
 per. yds. 
 
 23 
 23 
 
 4 
 5 
 
 (15) 
 
 yrs. wks. dnv8 
 163 42 6 
 
 93 n 2 
 
 (18) 
 07.. dwt. L'r-i. 
 274 6 ^ 2 
 193 7 
 
 17 
 
 fi?!) 
 
 1T3 
 
 20 
 
 91 27 
 
^0 
 
 COMPOUND MULTIPLICATION. 
 
 I :i 
 
 (22) 
 
 drfl. Bcr. cr». 
 167 7 
 93 1 19 
 
 ^28) 
 
 Flem. e. qrs. na. 
 
 16 
 
 9 2 3 
 
 eq. per. 
 167 
 119 
 
 (24) 
 
 Bq. yds. eq. ft 
 14 3 
 
 27 7 
 
 ■M 
 
 COMPOUND MULTIPLICATION^. 
 
 14. Compound Multiplication is the multiplication of 
 cvpplicate numbers of more than one denomination. 
 
 15. When the multiplicand does not exceed 12: 
 
 RULE. 
 
 f%t down the multiplier under the rirfJit-hand term. 
 
 Multiply every order of u?iits in the multiplicand in 
 mcceHnon, begimmirf with the lowest, by the mtdtiplier, and 
 divide each prodnet, ao fonned, by the number of that de- 
 nonmiation which makes one unit of the next higher : torite 
 down each remainder under units of its own order, and carry 
 the quotient to the next product. 
 
 Example 1. — Multiply 6 hrs. 40 min. lY sec. by 8. 
 
 OPERATION. 
 
 hrs. min. sec. 
 
 6 40 17 
 
 8 
 
 — ■ 
 
 53 22 16 
 
 11 
 
 Here we first multiply 17 sec. by 8 
 which gives us 136 sec. = 16 sec. to set 
 down and 2 min. to carry ; 40 min. x 8 = 
 820 min. and 2 min. carried make 322 min. 
 = 22 min. to set down and 6 hours to 
 carry, &c. 
 
 Example 2.— Multiply 7 lbs. 4 oz. 3 dr. 2 scr. 16 grs. by 
 
 OPERATION. 
 
 lbs. oz. dr. scr. 
 7 4 3 2 
 
 8. 
 
 3 16 
 
 grs. Here 16 grs. x 11 = 176 gra 
 
 16 — 16 grs. to set down and 8 scr. 
 11 to carry; 2 scr. x 11 — 22 scr. 
 iiiu 8 scr. carried, iiiiike 80 ,sci'. — 
 
 scr. to set down and 10 am. to 
 
 carry, &,c. 
 
COMPOUND MULTIPLICATION". 
 
 Exercise 22. 
 
 find the value of— 
 
 1. 5 days 4 hrs. 17 niiii. 4 eec. x 8. 
 
 2. 6 qri^. 17 lbs. 4 oz. x 11. 
 
 3. 22 busli. 1 pk. 1 gal. 1 qt. x 6. 
 
 4. jei79 14ri. llfd. x 12. 
 6. 11 gal. 1 qt. 1 pt. X 11. 
 
 6. 167 lbs. 7 oz. 10 dwt. x 5. 
 
 7. 29 milea 6 fur. 17 per. x 10. 
 
 8. 164 years 11 days 17 hours x 7. 
 D. 46 cub. feet 319 cub. inches x 11. 
 
 10. Ill cords 7 cord ft. 7 cub. feet x 12. 
 
 11. 26 r. 16 per. 4 yds. x 8. 
 
 12. 19 cwt. 1 qr. 23 lbs. x 12. 
 
 13. £127 16s 8id. X 9. 
 
 14. Ill per. 4 yds. 2 ft. 7 in. x 7. 
 
 15. 19 eq. per. 7 yds. 8 ft. x 3. 
 
 16. 179 yds. 2 qrs. 1 iia. 2 in. x 11. 
 
 17. 16 lbs. 11 oz. 2 drs. 1 scr. x 10. 
 
 18. 14 qrs. 14 Ibn. 11 oz. x 4. 
 
 19. 278 miles 6 fur. 11 per. x 2. 
 
 20. 64 weeks 17 hours 38 minutes x XL 
 
 21. 17 pecks 1 gal. 1 qt. 1 pt. x 7. 
 
 22. 109 cwt. 2 qrs. 11 lbs. x 12. 
 
 23. JE169 178. njd. x 9. 
 
 24. 74 a. 2 r. 7 per. 4 yds. x 9. 
 
 11 
 
 scr. IG grs. by 
 
 10. When the multiplier is a composite number, none 
 ^ Its lactors bemg greater than 12 : — 
 
 RULE. 
 
 Murtiphj the given multiplicand by any one factor, tUn 
 mulhpbithe resulting product by a second factor, this second 
 product^ by a third, if there be any, and so on. The last 
 
 proc;ae. is tnc one souynt. 
 
 Example I.—Multiply 1 bush. 1 gal. 1 qt. by 490. 
 
i 
 
 f^' 
 
 n 
 
 COMPOUND MULTIPLICATION. 
 
 OPERATION. 
 
 bush. pks. gals. qts. 
 7 11 
 10 
 
 n\ 2 
 
 
 
 2 
 
 1 
 
 '600 8 
 
 1 
 
 2 
 
 n 
 
 Here the factors of the ro^x^v'^ier 
 490 are 10 x 7 < 7, .md, accordhig 
 to the rule, we multiply the givt;n 
 quantity by any one of them as 10 
 then the product by a second factor! 
 and this last product by the thud. 
 
 8606 2 
 
 72. 
 
 Example 2.~MuItipIy 1 mUes 7 fur. 19 per. 4 yds. by 
 
 OPERATION. 
 
 miles fur. per. yds. Here the factor, of thp multiplier 
 7 19 4 are 9 and 8. We first mnltiply the 
 
 ^ gj^en multiplicand bj 9, and then multi- 
 
 ply the result by 8. 
 17 3 ^ 
 
 63 8 
 
 ^ « J!^^f{-^^?™'e:ht^8ve first multipMri' by 
 
 427 3 20 2 ^*^^ o^Jtained the sfaiwfl i;e8?il 
 
 Exercise 23. 
 Find the value of— - 
 
 1. ^74 198. 4f d, X 16. 
 
 2. 75 lbs. 4 oz. 7 dwt. x 18. 
 
 3. 16 days 4 hours 17 min. x 2L 
 
 4. 37 Flem. ells 2 qre. 1 na. x 3& 
 6. 63 miles 4 fur. 7 per. x 56. 
 
 6. 71 gals. 2 qts. 1 pt. x 77. 
 
 7. 43 hours 19 min. C6 sec. x 84. 
 
 8. 16 a. 3 r. 17 per. x 108. 
 0. 91 o7.. 6 drs. 2 scr. 19 grs. x 121, 
 
 10. X116 lis. llid. X 42. 
 
 11. 115 sq. per. 4 yds. 7 ft. x 144. 
 
 12. 93 cwt. 3 qrs. 17 lbs. x 9a 
 
COMPOUND MULTIPLICATION. 
 
 13. 16 years 110 Jays 11 hours x 60. 
 
 14. 29cub. ydb. 1', cub. ft, 1110 cub. Ux48. 
 
 15. 126buHh. 1 (jt. 1 i)t. x54. 
 
 16. £21 l(5-<. OJd. X 100. 
 
 17. 74 per. 4 yds. i, it. 11 in. x 600. 
 
 18. 93 liours 17 mil,. 67 see. x lioo. 
 
 19. 6 a. 2 r. 7 per. 9 yds. x 560. 
 
 20. £63 143. 9id. X 8100. 
 
 73 
 
 17. When the multiplier is not a eomposite numbw 
 ttkid ss greater tuan 12 : 
 
 RULE. 
 
 Hesohe i/U multiplier into tivo or more composite 
 numbers. ^ 
 
 Find the pr,duct of the \nultiplicand by each of these 
 separately, and add the results together for the required 
 
 inf ,w ''^•"'^^i ^^® .^^"Itiplier u rot proater than 100, we repolve it 
 \nto uns and umfs ; if greater thanlOO and Ickb than 1000 into 
 
 &o'?nf^ f "' ""'? T'y V ^^^«^'^^»- tl'^n 1000 and lees 'than 
 10000, into thousands, hvndreds, tens, and units, &c. 
 
 Thus 89 = 80 + 9; 76 = 70 + 6; ^c. 
 
 147 = 100 + 40 + 7; 747-700 = 40 + 7; &c. 
 6497 = 6000 + 400 + 90 + 7; 9162=9000 + 100+60+2; &o. 
 
 Example 1.— Multiply £'71 16s. 4Jd. by 19. 
 
 OPERATION. 
 
 £ s. d. 
 11 16 4f 
 
 10 
 
 Here the given multiplier, 19 
 10 — 9, and the factors of 10 are 
 10 X 1, &c. 
 
 718 
 
 8 
 
 6027 
 646 
 
 1 
 
 in 
 
 1 
 
 Si 
 
 6f: 
 
 TO times the multiplicand. 
 9 
 
 u 
 
 u 
 
 ti 
 
 u 
 
 «673 16 3i = 79 
 
 ExAMPiB 2.--Multiply 16 cwt 2 qrs. 17 lbs. by 867, 
 
74 
 
 COMPOUND MULTIPLICATION. 
 
 j 
 
 
 OPERATION. 
 
 ,- - cwt. qrs. lbs. 
 
 16 2 nx7= 110 2 19 = 7time»mulK 
 
 cwt. qrs. lbs, 
 17 
 10 
 
 106 2 20x6=1000 20—60 " 
 10 
 
 1667 0x8 = 13336 = 800 ** 
 
 Sum = 14452 3 14 = 867 " 
 Exercise 24. 
 ITind the value of-— 
 
 4 hush. 1 pk. 1 qt. x 718. 
 
 jei6 14s. lljd. X 867. 
 
 9 days 4 hra. 17 nihi. x 263. 
 47 yds. 2 ft. 7 in. x 83. 
 
 C lbs. 4 oz. 7 dwt. x 19T. 
 
 7 a. 4 per. 3 ft. x 985. 
 
 16 yds. 3 qrs. 1 na. x 1149. 
 
 23 oz. 7 drs. 2 scr. 16 grs. x 6472, 
 
 X9 lis. 4id.x 8298. 
 
 73 cwt. 1 qr. 16 lbs. x 67. 
 
 Multiply 7 miles 4 fur. 16 per. 2 yds. 2 ft. 6 In. by 647. 
 
 Multiply 17 Eng. ells 4 qrs. 2 na. 1 in. by 217. 
 
 Multiply 6 cwt. 1 qr. 17 lbs. 4 oz. 7 drs. by 982. 
 
 Multiply 8 a. 2 r. 14 per. 17 yds. 6 ft. 117 in. by 2345. 
 
 Multiply 11 years 217 days 23 hours 47 min. 18 sec. by 667. 
 
 Multiply 2 cords 7 cord ft. 14 cubic ft. by 103. 
 
 Multiply 7 bushels 1 pk. 1 gal. 1 qt. 1 pt! by 3218. 
 
 Multiply 67 lbs. 4 oz. 5 drs. 1 scr. 11 grs. by 975. 
 
 Multiply £174 IAb. Oid. by 780. 
 
 Multiply 23 lbs. 11 oz. 16 dwt. 11 grs. by 859. 
 
 1, 
 
 2. 
 S. 
 4. 
 5. 
 6. 
 7. 
 8. 
 V. 
 10. 
 
 11. 
 
 12. 
 13. 
 14. 
 
 15. 
 18. 
 17. 
 18. 
 ■V9. 
 20. 
 
COMPOUND DIVISIOK-. 
 
 COMPOUND DIVTSTON". 
 
 15 
 
 1 8. Compound Division U tho division of applicatc num- 
 bers of more tliiin one denomination. 
 
 19. Compound Division is divided into two cases: 
 1st. When tho divisor is an abstract ntunbor. 
 2d. When the divisor is an ai^piicate number. 
 
 20. When the divisor is an abstract number and not 
 greater tlian 12:-- 
 
 RULE. 
 
 Set the divUor to the left of the dividend. 
 
 Then, beginning at the left-hand 8ide, divide the first 
 term bj/ it, put the quotient tmder that term, reduce the re- 
 matnder, if an}/, to the next lower denomination, and to the 
 number thus obtained add the given number of that lower 
 denomination. 
 
 Divide the number thus obtained by the divisor, as be- 
 fore ; and so on. 
 
 21. If the divisor is composite:—* 
 
 RULE. 
 
 Divide, as in Ride 1, by each factor in succession. 
 
 22. If the divisor is not composite and is greater Ihau 
 
 12 
 
 RULE. 
 
 »/• thTdividend'' ^""^^ ^' ^''^ ^'''^ ^^^ quotient to the right 
 
 Example 1.— Divide 679 lbs. 4 oz. 1 dwt. by 11. 
 OPERATION. Here we say ll's in G7, 6 and 1 over- 
 
 ly ""a' ^""h^- ^^'' ^" ^^'1 ^"^ 8 "^^'■; 8 'hs. rr 96 
 
 U)6'79 
 
 07,-. and 4 oz. mair" "lOO 
 
 9 and 1 over ; 1 oz. = 20 dwt. and 1 
 
 61 9 2i\ dwt. make 27 dwt.. &c 
 
 n 
 
 oz. ; ll's in 100. -S 
 
I 
 
 i 
 
 I 
 
 10 
 
 COMPOUND DIVISION. 
 
 ExAMPi^F 2.--Dividfe £ir^ 166. 9d. by 84. 
 
 OfEKAi'ION. 
 
 £ 8. a. 
 •7)!/ 9 16 8 • 
 
 12)25 13 9...6rem. 
 
 2 2 9...9 rem. 
 
 7 X 9 = 03 4- 6 = 09, true rem. 
 Then £2 28. 9«5d. ^n«. 
 
 Here the factors tn 
 the divisor are T and 12, 
 and we divide by each, 
 as in Example I. From 
 the two partial remain- 
 ders we obtain the true 
 remainder by the rule iu 
 Art. 51, Sec. II. 
 
 Example 3.— Divide 723 yds. 2 qrs. 1 na. by Ua 
 
 J OPERATION. 
 
 yds. qrs. na. yds. qrg. na. 
 146) 723 2 1 ( 4 3 S,^, 
 684 '^ 
 
 • 139 
 4 
 
 668 
 488 
 
 120 
 
 I 4 
 
 481 
 
 438 
 
 4S 
 
 Exercise 26. 
 Find the value of— 
 
 V fo^ I'' t^' * *• ^- 290 ^q- P^^r. 7 yds. 8 ft. .^ S. 
 
 2. 127 cwt. 2 qr.. 17 lbs. + 11. 7. Ill Ibe. 7 oz. 4 dr. 2 sc, ^ ^ 
 
 5. 172 days 16 h. 29 min. .+ 7. 8. 69 trale. 1 qt . 1 pt + 12 
 
 6. 4179 miles 7 fur. 9 per. + 6. | 10. 796 cwt. 1 qr. 16 lbs. ^ 10. 
 
Compound division. 
 
 11 
 
 n. £196 7«. 8(1. + 24. 
 
 12. 149 fur. 17 per. 4 yds. ■*- 
 
 13. 1479 hri. 47 min. 16 Ht-c. 
 
 14. 1890 Ibri. 7 oz 12 dwt. ■*- 
 
 15. 679 nq. ])er. 7 ft. 107 In. - 
 
 16. 3fir.-.. 19 1brt. lloz 7dra. 
 
 17. 1107 yrrt. 119 days llhra. 
 
 18. 987 oz. 7 drs. 1 ecr. IC g?i 
 
 19. 1679 r, 4 per. 7 ft. 'J6 in. h 
 
 20. 7967 wlcB. 4 days 17 ne?. h 
 
 21. i;uU749 16ri. Hid. -hin. 
 
 
 22. 
 
 35. 
 
 23. 
 
 -v81. 
 
 24. 
 
 108. 
 
 
 ^ 132. 
 
 25. 
 
 *-72. 
 
 20. 
 
 4-144. 
 
 27. 
 
 .-f-07. 
 
 28. 
 
 1-117. 
 
 29. 
 
 »-91G. 
 
 
 30. 
 
 479 0. 7 0. ft. 11 cub. ft. ■*■ 8ft 
 
 7171° 17' W-H147. 
 
 1467 French ells 1 qr. 2 na. 
 
 1 in. -4-267. 
 
 916 mi lea 6 fur. 4 yds. •♦- 67. 
 
 £1911 173. 0|d, + 101. 
 
 9134 1. !. 4 oz, 17 dwt. ■*■ 9631 
 
 7149 bush. 1 qt. 1 pt. ■*■ 41',, 
 
 2716 days 14 hours 17 min. 
 
 9 see. -t- 603. 
 
 4000 cwt. IP loB. 11 oz. -*. 347. 
 
 ^ 23. When t^e Oivisor is an appncate number the quo- 
 nent in sm ab.»,tract number, and means so many times, and 
 wo proceed auuo.ding to the foUowmg 
 
 RULE. 
 
 PcedHce fxith the divisor and the dividend to the lowest 
 '*^«omvnafion mentioned in either, and then proceed as in 
 
 i:oinmo7i aioision. 
 
 Example 2.— Divide '73 oz. 4 dwt. 17 grs. by 9 oz. 1 
 
 -awt. 
 
 OPERATION. 
 
 ^3 OZ. 4 dwt. 17 grs. = 35153 grs. 
 9 oz. 7 dwt. =r 4488 grs. 
 
 4488)35153(7iii^. Ans. 
 31416 
 
 3737 
 
 Here we reduce 
 both the dividend 
 and the divisor to 
 grains, that being tlie 
 lowest denomination 
 contained in either. 
 We then find that the 
 
 iivisor, 4488, is contamed in the dividend 7 times, witb a 
 -•emanider, 3737, and. according to tlic rr,othods ah>eady 
 cxplauied, we set down this remainder above a line with the 
 j.vi.^ur beneath it. We may, however, read tlie answer 7 
 times with a remainder of 3737 grains or 7 oz. 15 dwt. 17 
 grains yet to be divided. 
 
m 
 
 fjQ Miscellaneous problems. 
 
 Example 2.— Divide £703 U^ ry,. by £17 Hs. 9d. 
 
 oPKi; . .* s 
 
 £793 lOs. 5id. = " . " 'iQ farthings. 
 £17 lis. 9d. ~ ' : " 
 
 17028)7r)20G.'(44t.oi4- ^««. 
 08112 
 
 80949 
 08112 
 
 12837 
 
 Exercise 26. 
 Find the value of — 
 
 1. 739 <ihy8 4 liour.- 16 miii. -«- 23 hoara 14 min. 42 boo. 
 
 2. £4907 08. tiii.-f-jeoui?!?. 
 
 3. £1192 17.-. 8vi. -«- jt;9 17.-. 4J 1. 
 
 4. 986 cwt. 2 (irs. 17 lb«. -»- G cwt. 1 qr. 17 Ibe. 9 oz. 
 6. .426 a. 1 r. 23 per. -h 2 a. 8 per. 17 ydn. 4 ft. 
 
 6. 71 fur. 16 per. 3 ydn. 1 ft. -t- 27 per. 3 yds. 2 ft. 7 in. 
 
 7. 1122 cords 3 cord fi. -*■ 12 oorda 11 cub;c ft. 
 
 8. Ill lbs. 4 oz. 7 dwt. ■*■ 9 oz. 7 ilwt. 17 grs. 
 
 9. 14G8 Eng. ellri 2 qre. 1 nii. -«- 73 FIliu. e'.ls 1 na. 1 in. 
 10. 476 bush. 1 gal. 1 pt. ■*■ 3 bush. 1 pk. 1 qt. 
 
 11. Divide 6 lbs. 4 oz. 1 dr. by 1 oz. 7 dre. 1 scr. 7 gra. 
 
 12. Divide £9 4d. 7i 1. by 3^=. Hid. 
 
 13. Divide 7 acres by 17 tq. ydr. 6 ft. 4 in. 
 
 14. Divide 927 miles 4 fur. 7 ]ier. by inilcB 3 irchea. 
 
 15. Divide 11 year^ 47 days 1 hour bv '23 wtekb 2 days 7 hoy**, 
 
 16. Divide 167 bush. 1 i«k. by 9 bush. 1 qt. 
 
 17. Divide 17 tons by 14 cwt. 3 qrs. 
 
 18. Divide 126 yds. 1 qr. 2 na. by 17 French ells 1 qr. 1 in. 
 
 19. Divide 963 tiiik s 420 yd.^. by 7 fur. 03 yds. 
 
 20. Divide £1111 lis. lljd. by £12 13s. 4id. 
 
 Exercise 27. 
 
 Miscellaneous Problems. 
 
 1. Rcdisec 17S9 Flornish cIIh to feet. 
 
 2. Add to^rether 97 lbs. 4 oz. 7 dwt. ; 16 lbs. 1 oz. 16 grs. , 43 Ibg. 7 
 
 dwt. \i grs. -, 19 lbs. 4 oz. 11 dwt. ; aod 11 oz. & grg, 
 
 1 
 
MISOBLLANEOUS PROBLEMS. 
 
 19 
 
 8. 
 
 0. 
 
 t. Express 714, 1111, 2704, 91671, 813471, and 31917169 in Roman 
 
 liumcralu 
 
 4. Divide X179 6». Hid. eqimlly nmon^j 11 pcrHorm. 
 
 6. A HovureiKH woi«:JiB about 123 grains : what iii the weight of 
 i;75fX)ingold? * 
 
 6. Ruua the fullowiiJg nutrbers : 
 
 l()()2fiOOW7006 
 
 000011110011110011 
 
 i()7i4yo7m)4 
 
 71300400200 
 
 7. Sound traveln at tho rate of 1120 fett per second ; !he flash of 
 a cunnim, tired on onu nidi- of a nvcM-, is ol>acrvcd by a person 
 Btandingdinctly opposite on tho other fide 11 Bccoi.de Woro 
 he liears the report, Ilow many miles, lur., &c., is the 
 river in width? ' 
 
 The new Canadian cent Is exactly 1 inch in diameter and 100 
 wegh cxuctlj' 1 lb. Avoirdiipoirt ; what would he tlie weight 
 and wortli of that number of letitu wl.ioii would r- ach com- 
 pletely round the earth, the ci re u inference bein;; 2490.'^ miles ? 
 
 How long would it require to count $794071 in twcnty-cent 
 pieces, at the rate of 108 coins per minute ? 
 
 If a person spends upon an average $2-17 per day, liow much 
 does lie spend duriijg the year? 
 
 What is the weight of 3 dozen silver forks, each weighing 4 oz. 
 1 dwt. 6 grs. ? 
 
 12. Bought 1 lb. of tea for 75 cents, 3 Ihs. of cofTee at 14 cents per 
 
 lb., C lbs. of rice at 5 cents per lb., 27 Ibs.of Hugar at 11 cents 
 per lb., 13 lbs. of raisins at 15 cents jier lb., and a I'arrel of 
 flour for $720 ; how much have I to pay for the whole ? 
 
 13. Write the following expressions in common figures : JSI, LI), 
 
 MMM CCCXXXIII, MMDCLXC, LXXXMXXLIV, 
 
 CDLMDCCIX, MXCMV, and MMVCMDCCIL 
 
 14. How many times can 167 bo subtracted from 271496 ? 
 
 15. When tho multiplier is 714 and the multiplicand 9167, what is 
 
 the product f 
 
 16. What is the ninth part of 67 a. 4 per. 17 yds. ? 
 
 17. Divide £16 lis. among 3 persons, bo that one shall have £i 2b. 
 
 more than each of the others. 
 
 18. Divide |744 among, four persons, so that the first shall haro 
 
 one-sixth of the whole, the second one-fourth of the re- 
 mainder, and the other two, each lialf of what then remains. 
 What 18 the share of each ? 
 
 20 
 11, 
 
 19. 
 
 
 If A has £176 49. 5fd. and B has $694-70 : which has most, and 
 how much? ' 
 
 A recr'^ment of soldiers contains 1147 men ; how much cloth 
 would It require to mako coats for the whole, each coat 
 tfJting 4 ydg. 1 qr. 3 iia. ? ' 
 
r" 
 
 80 
 
 MISC2LLAKE0UB PSOBLSICI. 
 
 ■i'l 
 
 21. 
 
 2". 
 23. 
 24 
 35. 
 
 9k. 
 
 37. 
 
 28. 
 29. 
 
 30. 
 31. 
 82. 
 
 83. 
 
 ft 
 
 
 1 
 
 ' 
 
 i 
 
 ■ 
 
 \ 
 
 
 
 
 ! 1 
 
 
 
 Hi 
 
 i' 
 
 |ii 
 
 
 S6. 
 
 17. 
 SI. 
 
 30. 
 40. 
 
 41. 
 
 What Ib the weight of |7196'40 in cent piecei, Canadlai 
 money ? 
 
 Reduo.e 7 mile-^ 4 fur. 17 per. to fathoms. 
 
 The quotient in 749, the divisor 47 ; wh^t is the dividend? 
 
 What is the dift'erence between XMMCI and 167011 
 
 Tiw» minuend is 71467, the remaiiider 61794, what is the snl) 
 vraht'iid J 
 
 Divide $679 among two persons, so that the firat shall havt 
 ^14G more than the accond. What is the share of each ? 
 
 What irt the product of 714 -t- 16+179 + 42 + 93, multiplied b< 
 914G7-2:i4-946-1127-80040 + 27-67 + 83? 
 
 How many bushels of wheat are there in 71496 lbs. I 
 
 Write down aa one number six trillions seven millions ninety- 
 six thousand four hundred and live. 
 
 Tlie sum of two numberb is 1746 ; one is 974, what ie the other! 
 
 What is the coet of 23 pair of shoeo at 63. ild. per pair ? 
 
 A gi'Mop. of water weicjhs 10 lbs, and a cubic foot weighs 62* 
 
 lbs. ; how many gallone are therein 748 cub'c feet? 
 Two men, A and B. run a race. A yives B a start of 17 vards, 
 
 but gaiuri on him at the rate of 2 feet in 5 var-ls ; how'much 
 
 will A be in advance of B when B has ru-j one mile? 
 Divide $749-60 among A, B and O, so that A shall have at 
 
 much as B and C toirether, and B and C equal shares 
 
 What is the share of each ? 
 
 2366 cubic feet of wood are to be divided among three charit. 
 able institutions, so that as often as the liri*t receives 2 cubio 
 feet the second shall receive 6 and the third 7 : how many 
 cords does each receive ? 
 
 A farmer owned 247 acres of land and disposed of it as follows : 
 he gave 1 a. 1 r. 17 per. for a school ^ te, Pold I'r a. 23 per. 
 gave 21 a. . r, to his wife, -^nd divided the remaii'der equally 
 among his 3 B(/ns ; how much did each son receive ? 
 
 If 17 seconds elapse between the flash of lightning and the 
 arrival of the report • allowinir sound to travel at the rate 
 of 1120 feet per second, how far off is the thunder-cloud t 
 
 The London Times has a circulation of 12000 per day ; ]f it be 
 sold at 5d. per copy, express in pou'ds, shillings, and P^^nce, 
 and also m dollars and cents, the sum realized by its sale for 
 one entire year (313 days). 
 
 The greater of two numbers is 710 and their difference 207, 
 
 what is the smaller number? 
 The Jewish shekel weighed 219 grains Troy, and was worth 
 
 about 2s. 9^d. Canadian currency ; what was the weight of 
 
 a talent, containing 3vf)0 shekei?*, and what was the value o^ 
 
 500 talents i^n dollars and cents ? 
 
 A wished to exchange 297 yards of cloth at £\ 7h. 4id. vfT 
 yard with B for flour at $317 per barrel; how mvxj 
 
 DArrela at flAUf should hfi Psssi*''* i 
 
ploeei, Coaaftai 
 
 93, multiplied bj 
 
 start of 17 yards, 
 ards : how much 
 
 r difTerence 267, 
 
 GREATEST COMMON MEASURE. 
 
 SECTION III. 
 
 GREATEST COMMON MEASURE AND 
 COMMON MULTIPLE. 
 
 81 
 
 EAST 
 
 GREATEST COMMON MEASURE. 
 
 «u/ uiviae u . that i^, leaving no remainder. 
 
 2. A Coninion Measure of two or morP iinr«K« • 
 number that will e.actl;, divide each ot t""'^"'^' ^^ ""^ 
 
 3. Tiio Greatest Common \f<^i'ion%.a ^p * 
 
 NoT'^ -Tim rrn.f Ir, "^ ^^ Without a remainder. 
 
 *o initral lettZ,S^:^Cu''''' ^'^'"'' '' "«u^"y indicated by 
 4. To find the G. C. M. of two numbers :- 
 
 RULE. 
 
 VMinaer. Ihe last diiisor will be the Q C M Z ".'^^*^" 
 Example l._What is ti.e G. C. M. of 1825 and 2556? 
 
 OPERATION. 
 
 Here we divide the greater 
 number, 2565, by the lessf 1825 
 and tlius obtain a remkinder; 
 
 divisor, and 1826, the former 
 dmsor, becomes the dividend. 
 We hnd that 120 goes into 
 
 ^ . ' , -> a^''^ gives a re- 
 
 mamder.a^SjandsSon. Wlia 
 
 1825)2656(1 
 1825 
 
 730)1825(2 
 1460 
 
 365)730(2 
 730 
 
'M 
 
 1! 
 
 1 
 
 82 
 
 LEAST COMMON MULTIPLE. 
 
 a C.'y^'^' '* ^^^^^^ ""^ remainder, and ia thei 
 
 Example 2.-.What is the G. C. M. of 647 and 2750? 
 
 OPERATION. ' 
 
 64'7)2'750(4 
 2588 
 
 162)647(3 
 
 161)162(1 
 161 
 
 1)161(161 
 161 
 
 Here, in following the 
 rule, we find that the first 
 divisor tliat will go into 
 the then dividend is 1 ; or 
 in other words, the numbe/ 
 have no common measure. 
 
 ExEsciSE 28. 
 Find the G. C. M. of the following numbers : 
 
 1 1024 and 2240. 
 
 2. 1902 and 24409. 
 
 3. lG24ai!fl 14500. 
 
 4. 8^93 and 4609. 
 6. 714 aiid 1176. 
 
 6. 219 a d 11476. 
 
 7. 194706 a d 289913. 
 
 8. 2925 and 29484. 
 
 9. 27525 a! d 1725 
 10. 2254 and 7I(X)1. 
 
 11. 
 12. 
 13. 
 14. 
 
 15. 
 16. 
 17. 
 18, 
 19. 
 20. 
 
 11256 ai d 19899. 
 5161 and 7755. 
 87147 and 178871. 
 1261 and 663. 
 918 and 1347. 
 187 and 255. 
 1914 and 35786. 
 21671 and 22111. 
 82159 and 582. 
 462 and 212. 
 
 LEAST COMMON MULTIPLE. 
 
 It o^n?""" ''T^''^ '' "■ '^^ ^^ ^^ a "^"ItJple of another when 
 of tSes5 "' "" ^"'■■'"'- '^' ^^^^^ ^ certain nuX 
 
 any numocr that .- -cly oontams each of them as «i^isor. 
 ber7iJ]i%^T ^''-°'"'? Multiple of two or raor^ num- 
 
f? and 2750? 
 
 LEAST COMMON- MULTIPLE. 
 
 8. To find the 1. c. m. of two or more numbers :_ 
 
 RFLE. 
 
 2£f7£?'^S .-fSf ;;:?;- ts; 
 
 OPERATION. 
 
 3 
 
 Hr 
 
 1. c. m. = 20 X 4 X 3 = 240. 
 
 ^^her of the m^^Z^Z'Vo^ltlZ''^^^^ ^" ^^' -" 
 ►utJOand 15 (i.e b-causV fu!,' "^ ^'"'''^'^^ ^^^^^on, we strike 
 ^0)- .^-xt we issun.^20 one of Te J^ '^^^ ^ontai. od in 
 ^ divisor. The hiVhost kcfor of i a !, '?^=^'"'^' ^ numbers, 
 'e assumed number 20 is 4 LI ^^^'"^ '' '"'■'" '^ ^''^^to'' of 
 ^y 4, and for aXfi ar rea.o; wo f .r«»'di«gly divide 16 
 e. Th^ ,j....... ,/'f /^'^"O" ^'e divide 24 bv 4 muj .^o hr 
 
 ~" ' --^--<i ^*-<^i^^« are now 4, 6, and 3;of which 
 
 Wm^'- The 
 
84 
 
 LEAST COMMON MULTIPLE 
 
 !> 
 
 |i! 
 
 i! 
 
 8 is exactly contained in 6, and wo therefcre strike it out 
 This leaves 4 and 6, of which we assume 4 as divisor, and, 
 as this contains 2, a factor of 6, we divide the 6 by 2. Then, 
 multiplying the 8, remaining, by the assumed divisors 4 and 
 10, we get 240 for the 1. c. m. 
 
 Example 2.— What is the 1. c. m. of 16, 24, 28, 30, 32 
 36, 40, 44, 45, 48, and 60 ? » » > . 
 
 OPERATION. 
 
 40);^. .^^. .28. .80. .32. .36. .40. .44. .45. .48. .50 
 
 6)7 
 
 7 
 
 3. 
 
 11, 
 
 6, 
 
 Then 1. e. m. 
 
 .. 2 11.. 3.. .. 5 
 
 40 x6x'7x2x 11x3x5=; 554400. 
 
 JTere we strike out at once 16 and 24, since they are 
 contained exactly in 48. Then we assume 40 as divisor, of 
 which 4, one of its factors, reduces 28 to 7, 36 to 9, and 44 
 to 11. Also 8, another factor, reduces 32 to 4 and 48 to 6. 
 Also 10, another factor, reduces 30 to 3 and 50 to 5. Also 
 5, another factor, reduces 45 to 9. Next we s^r?k<^ out 3 
 and 9 in the second line, since they are each contained in 9 
 another number in that line, &c. 
 
 Exercise 29. 
 
 Find the 1. c. m. of — 
 
 1. 6, 9, and 30. 
 
 2. 30 and 55. 
 
 3. 7, 21, 35, 4, and 20. 
 
 4. 2, 9, 16, 35, 56, and 63. 
 
 5. 2, 4, 6, 8, 10, 12, 16, 18, and 20. 
 
 6. 8,9,11, 22, 72,32, and 99. 
 
 7. 6,10,14,18,2?. .a, 11-132, 
 
 9. 1, 2, 3, 4, 5, 6, 7, 8, and 9 
 
 10. 3, 6, 9, 12, 48, 21, 24, p'>d 16, 
 
 11. 8, 21, 63, 40, 160, 240, ai^d SO* 
 
 12. 16, 41, and 38. 
 
 13. 9 and 16. 
 
 14. 112, 200, and 72. 
 
 15. 90, 36, 63, 12, and 7. * 
 
 16. 3, 5, 7,9, and 11. 
 
 8. 5, 10. 15, 20, 25, 30, j, and 40. 
 
 17. 2, 4, 6, S, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, and ^2 
 
 18. 25, 7, 44, 60, 68, 55, 9, 11, 28, 70, and 4. 
 
 19. 720, 396, 252, and 540. 
 
 20. 15, 12, 128, 30, 16, 4, 320^&aa 80. 
 
VULGAR FRACTiOifS. 
 
 85 
 
 6, 24, 28, 30, 32, 
 
 ..45.. 48. .50 
 
 sectio:n^ IV. 
 
 VULGAR AND DECIMAL FRACTIONS. 
 
 .1(19. 
 I 100 
 
 VCTLGAR FRACTIOJfS. 
 
 1. A Fraction is an expression representing one or mors 
 i» %v equal paints into which any quantity may be divided. 
 
 0-e third is written ^ Four ninety-eigliths is 
 
 One fifth is written { written.1 -gi. 
 
 One seventh is written... | Seven hundred eleven- 
 Six sevenths is written . ..f hundredths is writ- 
 
 Kineteen twenty-sevenths ten 
 
 is wriuea ^^ &c. 
 
 2. If a quantity be divided into 3, 5, Y, 11, &c., equal 
 parts, then one of these equal parts is called one third, one 
 Jifth, one seventh, one eleventh, &c., as the case may be. 
 
 3. Every fraction is expressed by two numbers, called 
 terms, written one above the other and separated by a line. 
 
 4. The number written below the line is called the De- 
 nominator, because it shows the denomination, i. e tells 
 mto how mauy equal parts the quantity is supposed'to be 
 divided. 
 
 5. The number above the line is onlJed the Numerator 
 because it numerates or tells how man> of these parts are to 
 be taicen. ^ 
 
 6. Every fraction expresses the division of the numera- 
 tor by the deaominator, and the little horizontal line which 
 eoparates the two terms is derived from and stands for the 
 £ign of divisum. 
 
 Thii.^ f meaiifl either the \ part of 2 or 2 times the a part of L 
 ll means either the ,V part of 13 o. 13 time, the ,», part' on, &c. 
 
 7. Since every fraction expresses the division of the nu- 
 Lucraior oy the deuominator, it fc«o>v« tkajt— . 
 
 ;ll 
 
86 
 
 I ■ 
 
 VULGAR FRACTIONS. 
 
 1 
 
 : I 
 
 ' 
 
 : li I ;!• 
 
 i 
 
 m 
 
 divi^hf "If"'' ""^ ^^'"^ ^'^'*'^" '^ *^e 9^'oii^^t obtained ^t 
 dividing the numerator by tlie denominator, and hence ^ 
 
 Multiplying tlie numerator of a fraction by any number 
 inultiphes the fraction by that number. ^ ' 
 
 Multiplymg the denominator of a fraction by any number 
 divides the fraction by that number. ^ ^ ^""^^^r, 
 
 Hultiplymg both numerator and denominator of a fraction 
 fmct!«,r"' ""'"^''■' ^""^ ^'' '''^'''' '^' value of tSe 
 
 ^'''"^'lelirbl't,'? ^"f'^" ^y ^y ^"°^ber, divides 
 i"t? 11 action by that number. 
 
 Dividing the denominator of a fraction by any number 
 multiplies the fraction by that number. ^ ' 
 
 Dividing both numerator and denominator of a fraction by 
 the same number, does not aflect its value. ^ 
 
 Decfmaf '''''"' "'' "^'''^'^ "^*" *^« ^^^«^^> V^^g^r and 
 
 inato; it ?tuf ^M '^^'" ^' ^ ^^'"'^^^^ ^" ^'^J^t^ the denom- 
 matoi IS 1, loliowed by one or more O's. 
 
 tions.^' '^" '''^'' ^'^'"""' ^'^ ^"^S^^ «r C«"^"ion Frac- 
 ^^NoTE.-The word *« vulgar" Is hero used in the sense of " com- 
 
 11. There are six kinds of Vulgar Fractions— P»-,vr,.>. 
 Improj>er, MM, Simple, Compouul a^^^i^lj^'^^' 
 
 "Tlius ^V, S, ri?, tVt. &c., are proper fractiong. 
 
 r,,..^^' "^"l ^^^P^OP^^': Fraction is a fraction whose denomt 
 nator is not greater than its numerator. 
 
 An Improper Fraction may also be defined to be a frao 
 tlon whose value is equal to or greater than 1. 
 
 Thus I, J^a, I, II, aej.^ 14,^ 1^ |.|^ ^^^ ^^^ hnproper fractious. 
 
 '.^.5.^..„ 
 
VU1.GAR FRACTIONS. 
 
 87 
 
 t4. A Mixed Number is a number made up of a whole 
 lumber and a fraction. 
 
 Thus 16|, 193J, l|f, 999^, 6^, 2f, &c., are mixed numbers. 
 
 15. A Simple Fraction expresses one or more equal 
 parts of unity. 
 
 ThuB I, f , a, 1^, J, j«|, &c., are simple fractions. 
 
 16. A Compound Fraction expresses one or more eqrial 
 j^arts of a fraction, or, in other words, is a fraction o/ a 
 fraction, 
 
 ticuB^^^"*' i ^^ ?. I of 5 of \l of f of If 8, &c., are compound ti ic- 
 
 17. A Complex Fraction has a fraction or a mixed num- 
 ber m its numerator or in its denominator, or both. 
 
 Thus 
 
 4 ICi 3«- i 10 
 
 SJ) 7 , 9j»r, 9, § , dec, are complex fractions. 
 
 18. Any whole number may be made a fraction by 
 placmg 1 beneath it for denominator. 
 
 Thua 5 = f , 17 = -'f , 11 = .y., 217 = aj^, &o. 
 
 EXERCISE. 
 
 I. Read the following fractions : ^, 6|Sf , ^irV^, i^^, j^h, iVr. 
 * Read the following fractions, 7J, llxV, |, 217tV, 603^, 
 
 11379ij«yWV 
 8. ReacT the following fractions : f, i^, Hf , If, SrMir, 11, HII- 
 A. Read all the proper fractions found bt the above. 
 
 5. Read all the improper fractions. 
 
 6. Read all the mixed numbers. 
 
 7. Write down on your alatw any six proper fractions. 
 
 6. Write down on your slate any six improper fractioiw. 
 
« ■ 
 
 • 
 
 i \ lili 
 
 >! m 
 
 % 
 
 88 
 
 REDUCTION OF FRACTIONS. 
 
 ft Write down on your Blate any six mixed number«. 
 
 10. TV rite down on your elato any six decimal fractiong 
 
 11. Write down on your elate any six simple fractionB. 
 .2. Write down on your elate any six compound fractions. 
 
 13. Write down on your slate any six complex fraotioni 
 
 14. Express 7, 9, 4, 23, 17, 34, 109, and 207 as fractions. 
 
 REDUCTIOX OF VULGAR FRACTIONS, 
 a givfn l:^:^^::,^'^,!^'''^''' --b- ^« - ^-tlon having 
 
 RULE. 
 
 of the remlHng c^presHon by (he given denominatoT 
 denomkltor. ^-^"•^^'^ '^3 to a fraction having 20 &, 
 
 123 = -— -, and 
 
 OPERATION. 
 
 123 X 20 _ 2460 
 1 X 20 ~ 20 
 
 Ans. 
 
 deno^m'naL'n '-^^^"^^ ^^ *^ « ^^etion having 29 fo. 
 
 n 
 
 OPERATION. 
 11 X 29 
 
 1 1 X 29 29 ^***' 
 
 ' Exercise 30. 
 
 l K.dn«e 7. ^ 27 and 40 to fV.no,i„,„ „avl„g n for denominator, 
 a Eednce 2, 207, 440, and 8 to fractions haviaf A0» t.. denomia, 
 
 I Ked^<, 22, 47, 69, «,d 100 to fl-acUoa, havla, »3 f„, d,„„„i^ 
 
REDUCTION OF FRACTIONS. 
 
 89 
 
 4. Reduce 217, CS, D27, and 4 to fractions having l3 for denomina- 
 tor. 
 
 6. Iloduco 27, 304, 617, and 93 to fractions having 248 tor denom- 
 inator. 
 
 6. Reduce 209, 407, 789, and 5 to fractions having 611 for denom- 
 iuator. 
 
 20. To reduce a mixed nuiribor to an improper frac- 
 tion :— 
 
 RULE. 
 
 Multiply the whole number by the denominator of the 
 fraction^ to the product add the given numerator and place 
 the sum over the given denominator. 
 
 Example 1. — Keduce 7iJ to an improper fraction. 
 OPERATION. Here we multiply the whole number, 7, by 
 1^ the denominator, 9, and to the product,' GJJ, 
 add the numerator, 4. This gives 67 for the 
 numerator, beneath which we write the given 
 denominator, 9. 
 
 Example 2.— Reduce 167-ff to an improper fraction. 
 
 OPERATION. 
 
 167 X 19 = 3173 and 3173 + 14 = 3187; 
 hence 167^^ = HF- 4ns. 
 
 Exercise 31. 
 Reduce the following mixed numbers to improper fractions : 
 
 9 
 
 V 
 
 1. 16f 
 
 2. 9,2.,-. 
 
 3. 14A-. 
 
 4. 7lf. 
 6. 161f^. 
 
 6. 
 7. 
 8. 
 9. 
 10. 
 
 27-1-3- 
 186^f. 
 
 91H. 
 
 11. 6/3. IG. 111,^,1,^ 
 
 12. 2097A. 11. 214,^7. 
 
 13. 617i|f. 18. 63fg. 
 
 14. 417H. 19. 2345|t^. 
 
 15. 6170,VoV 20. 1919-i^V 
 
 ^1. To reduce an improper fraction to its equivalent 
 mixed number : — 
 
 RULE. 
 
 I)iinde the numerator by the denominator, and ths 
 juotient ivill be the required mixed number. 
 
•M 
 
 00 KEDUOTION OIP FRA0TI0N8. 
 
 Example 1. — Redupp izii * 
 
 neauce J-^i to a mixed number. 
 
 OPERATION. 
 
 ^¥^=1714^9 = 190}. ^„.. 
 Ex.Mn. 2.-Keduco ij;i. to , „,„, ^^^^^^ 
 
 OPERATION. 
 
 HF =14716^ 109 =13s,i,.^„.. 
 
 Re 1 .1 Exercise 32. 
 
 mixed nu"rn'beAV°"°^'°^ improper fraction., to their equivalent 
 
 1. ¥. 
 
 2. ifiai 
 
 8. iiT^ai 
 6. W. 
 
 6. a^. 
 
 9. aip. 
 10. i-^fi.ii 
 
 11. -U,t,9^^. 
 
 12. iia 
 l; 
 
 14 
 
 10- VA 
 
 4 8* 
 1 3 • 
 
 16. Hfl^^. 
 
 17. H*/^. 
 
 18. HV^. 
 
 20. %¥. 
 
 i 
 
 22. To reduce a fraction to its lowest tem>s:_ 
 
 Riri,E. 
 
 ~ftttr""'''"- '""' ''""'"''""or by tkeir neatest 
 EX.MP.. l._Reduee ^i to ita lowest terms. 
 
 p . OPERATION. 
 
 = -iV ^W5. ' ^^ ^t) -- 216 = 11 ; hence ^| 
 
 Example 2.— Reduce ^4«i +« v , 
 
 vi^uuce jT^fi^ to Its lowest terms. 
 
 TI,-. n n ,, OPERATION. 
 
 TheG.CM.of481and26377isl3. 
 
 Then 4 1-^13^37. and 26377 -M3 = 2029 • h 
 
 F(ia7-T = 2g^5. ^WS. -^^--^029; hence 
 
REDUCTION OF FKACT10N8. 
 
 91 
 
 374U' 
 
 4. mi 
 
 6. Vi^„i. 
 6. i'tUh- 
 
 ■1. 
 
 8. 
 9. 
 
 iU.ll 
 T It 9 I • 
 
 3 9 8 b' ?^' 
 
 !• illl 
 (il 19' 
 
 10. W,^. . ^ 1 
 
 11. iViWa-. 
 
 12. mgi. |j| 
 
 Exercise n3. 
 
 Roduco the following fractione to their lowest terms 
 
 1. 
 
 2. 
 
 8. 
 
 NoTR.— A fraction can Romc'tiiTiea be reduced to its lowest terms, 
 ai d the work may alinoBt ahvayn bo materiuliy jensei.ed, by divid- 
 liiir both numerator ami denominator by a/jj/ number which will 
 go into each of Ihem without a rcmaind<r. In order to fticilitate 
 this mode of reduction, it is necest»ary to remember the IbUowinK 
 factrt : " 
 
 1st. Any number that ends in 5 is divisible by 5. 
 
 Any numt)C'r tlnit ends in is divisible by 10, 5, or 2. 
 
 Any number th:it ends in an even number is diviHihle by 2 
 
 When tlie two right-hand lagures are dividible by 4, the whole 
 
 in diviwiljle l»y 4. 
 When the three ridit-liand figures are divisible by 8, the whole 
 number is divisible by 8. 
 6th. When the sum of the ditrite of a number is divisible by 9, the 
 
 sum itself is divisible by 9 or by 3. 
 7th. When the sum of the digits of a number is divisible by 3, the 
 
 number itself is divisible by 3, 
 8th. When the sum of the jritg standing in the even places ia 
 equal to the sum of the digits standing in the oc/t/ places, the 
 number is divisible by 11. ^ i 
 
 Thus the number 7416 i^ divisible by 4, because 16 (the last two 
 digits) is divisible by 4. 
 
 is divisible by 8, because 416 (Its last three 
 
 digits) is divisible by 8. 
 
 is divisible by 9, because the sum of its 
 
 digits (7 + 4 + 1 + 6 = 18) is divisible 
 
 is ttivisible by 3. because the sum of its 
 
 digits (7 + 4 + 1 + 6 = 18) is divisible 
 by 3. 
 So also the number 4567321 is divisible by 11, since the sum of the 
 digits in the odd i)lare' , 1 + 3+0 + 4 = 14 = 2+7 + 5: the 
 Bum of the digits in t* . even places. 
 
 2d. 
 3d. 
 4ih. 
 
 6th. 
 
 23. To reduce two or raore fractions to equivalent frac- 
 tious having a common denominator: — 
 
 RULE. 
 
 Fi7id the IcaM common mullipU of all the denominators. 
 ^ Midtiply both terms of each fraction by the quotient ob- 
 tained by dividing this least common multiple by the denom- 
 inator of the fraction. 
 
^aj 
 
 
 .o^.. \^^^^ 
 
 IMAGE EVALUATION 
 TEST TARGET (MT-3) 
 
 1.0 
 
 I.I 
 
 1.25 
 
 1^ 
 
 IIM 
 
 1.4 
 
 M 
 
 20 
 
 1.6 
 
 
 y 
 
 Photographic 
 
 Sciences 
 
 Corporation 
 
 23 WEST MAIN STREET 
 
 WEBSTER, NY. 14580 
 
 (716) 872-4503 
 
 iV 
 
 (V 
 
 % 
 
 V 
 
 ^^ 
 
 
 \\ 
 
 
 ^NS 
 
 6^ 
 
 o^ 
 
 %<^ 
 
 <> 
 
 '^^ 
 
^ #? ^ A 
 
 W.r 
 
 z. 
 
1 I 
 
 92 REDUCTlON^ OP FRACTIONS. ^ 
 
 OPERATION. 
 
 The .e.t common .uUip.e of the denominators, 4, 1, ». 8. 
 
 Then 604 ^ 4 = 126, the multiplier for both terms of the 
 KHA H u ^^* fraction. 
 
 604 ^ 9 = 56, the multiplier for both terms oi the 
 
 ^d fraction. 
 604 ^ 8 --= 63, the multiplier for both terms o( the ' 
 KrvA c 4th fraction. ® 
 
 604 -^ 2 = 252, the n^ultiplier for both terms of the 
 
 5th traction. 
 504 -^ 3 = 168, the multiplier for both terms of the 
 A«,? *!, c X. oth fraction. 
 
 Ana tne tractions are -i^^^ ^ifi i2.si 126 7k« , „„ 
 **^*^ 6 04, 6*4, Uh nh lUy and fa^. 
 
 tionf L"^^Ve7m"::rd^eU'il;^^^^ '■^^ '» equivalent frac 
 
 OPERATION. 
 
 ^el.c.m. ofY, 5, 9,andlOis630. 
 
 Multiply Doth terms of the 1st fraction by 90 (i e i^a^ 
 
 2d - by 126 i! e.' i'H' 
 1% " by 70 (i. e. ^F) 
 
 -d the fractions become f^, 4 l^,,^^. ^ 
 
 Exercise 34. 
 Reduce to e-euivalent fractice having a con..on denominate 
 
 2 1' r r r t; I ^- - - - ^^^ t. 
 
 ^- 5, f, t, f,and i. 8 I fi s ii 13 ^ ,, 
 
 8. -,i,- fi 5 r, ^^,, 3 °- 8, 9, -10, It, If, and |f 
 
 4 r II -1 ';.'"• ^- ii i^, and If . 
 
 • f4, .„ „ f, /,-, and If. 10. If, .9^^, 1^^ ^„d _,^^ 
 
 ^- H, A, A-, ft, i, and A. 
 ^- i» i» i> i, i, ani -^V 
 
 ^^- To, T.. .„, _ ,,. 
 
 12. 1^ 4 -fi- 1 a. 11 A ,* 
 
 - H, If, and ^fi 
 
REDUCTION OF FRACTIONS. aq 
 
 24. To reduce a compound fraction to a simple one :— 
 
 RULE. 
 
 Multiply all the numerators tor/ether for a new numer- 
 0stor, and all the denominators tocfether for a new denomi- 
 nator. 
 
 ,11 ??!? flv7<S!^"y!"^r'PP^^''"^ *^^^ rule, wo may cast out or cancel 
 all the lactois that are coramou to a uumerator and a denominator 
 of the compound fraction. ^xixiuatui 
 
 Example 1.— Reduce ^,- of f of f of ff of U to a 
 simple fraction. 
 
 nominator 
 
 STATEMENT. 
 
 6x4x3x22x35 
 
 6 4 3 -22 ^35 
 
 — of - of - of -- of — = 
 
 11 ^ 5 27 16 11 X 7x5x27x10 
 
 CANCELLED. 
 
 ^ ;? ^ • 
 
 ;;x''x^x>!7V^-8- ■^''*- 
 
 3 i 
 
 Hei-e 6 and 27 contain a common factor, 3, which is cast 
 out, anii these numbers thus reduced to 2 and 9 Next 
 this 2 reduces 16 to 8, and the 9 is reduced to 3 by the 
 third immerator, which is thus cancelled. Again, 1 1 can- 
 cels 11 (the first denominator) and reduces 22 to 2, and this 
 2 reduces the 8, leforo obtained from the 16, to 4. Next 
 this 4 js cancelled by the 4 in the numerator. A<?aiii, 7 
 cancels tha 7 in the denominator and reduces the Soln the 
 numerator to 5, and this 5 cancels the 5 in the denominator. 
 All the numerators are now reduced to unity, as also all the 
 denomiuaiors but the fourth, which is 3. The resulting 
 
 fraction w therefore --^ - /-ii^L. ; but as 1 is never con- 
 
 ■IXlXlXoXl 
 
 eidered as a multiplier or divisor, we write the result simply 
 
] 
 
 III 
 
 EEDUCTION OP FKACTIOHS. ' 
 
 E.^P.B 2._Eeduce.fr of |of J „f |J to a simple f.c. 
 
 STATEMENT. 
 
 ~ Of 4 of i- of- -. J^x 3 X 66 
 
 ^^ ^ ^20 iixeTsirio 
 
 CANCELLED. 
 
 s 
 
 2 5 
 
 number J?- m^ustTe *S^^^^^ fraction be a mixed 
 
 before applying the rule ° '*' equivalent improper fract^oa 
 
 Exercise 3o. 
 Reduce to equivalent simple fractions. 
 
 1; ioffoff^ofi^of^ 
 2. i of i of A of If of f^. 
 
 '7. 6|of9|ofioffof8^. 
 8. 27iof^ofHofMf. 
 
 8. f of J of H of ,^ „f 33. , ^^ «« 
 
 !• J» 7 "'^^^ of -^ »f ^2. 10. *of iof Jof Jof it ,V 
 5.fof^of2nofi. n..i„fa,of4;„f5of]^ 
 
 6.Aofa|of.s*of^, 1 12. 3, of S, of., of H of 
 
 if Of 
 
 re- 
 
 26. To reduce a comElej faction to a simple one :_ 
 
 RULE. 
 ^^educe hoth num^at^ and denominator to eimplcfiac 
 
KEDUCTION OF FRACTIONS. 
 
 95 
 
 40. 
 Example 1. -—Reduce -^^ to a simple fraction. 
 
 f f " f 1- ~ 2 ">ry 
 
 99 
 
 14 
 
 Y-jV' -^ns. 
 
 Note.— Factors that are common to one of the extremes and 
 one of the means, are to be struck out or cancelled. 
 
 Example 2.--Reduce tt^ to a simple fractioD. 
 81 9 Y 
 
 7-A- Ti ^; X JJ 7x9 
 
 Hf '90 ;; X ^0 
 11 10 
 
 10 =Yo = ^'^''^'''- 
 
 1. 
 
 Exercise 36. 
 Reduce to equivalent simple fractions ; 
 
 1. 
 2| 6| 
 
 V V| 
 
 2. 
 
 5. 
 
 11 
 
 8. 
 
 1 
 
 A 
 
 9 
 
 6. 
 
 19^ 
 
 8. 
 
 1! 
 
 9i 
 
 10. 
 
 11. 
 
 9. 
 
 9 
 
 it 
 
 3i 
 
 2fr 
 4i 
 
 12. 
 
 I 
 
 6| 
 
 9i 
 
 26. To reduce a denominate fraction from one denomi- 
 nation to another : — 
 
r r 
 
 
 
 I 
 
 'i 
 
 1) 
 
 If 
 
 fin 
 
 ill 
 
 06 
 
 KEDUCTION OF FRACTIONS. 
 RULE. 
 
 Tjr ^Ae reduction be from a lower to a higher denomina- 
 tion multiphf the denominator, but if from a higJier to a 
 lower denomination multiply the numerator, as in reduction. 
 0/ whole numbers. 
 
 Example 1.— Reduce f^- of an hour to the fraction of a 
 week. 
 
 -i^f 
 
 h. = 
 
 17x24 
 
 d. = 
 
 OPERATION 
 
 1 
 
 17x24x7 
 
 wk. = 
 
 1 
 
 17x24 
 
 Or^ iefly, 
 
 408- ^^• 
 
 17ir2T^ = 408 ""^ ^ "^^^^^ ^'*^- 
 
 Example 2 -Reduce | of f of f of 35 oz. to tue frac 
 ..on ot a pound Avoir. 
 
 operation. 
 i of f of ^ of 35 oz. = % 02. (by Art. 24.) 
 Then ^ oz. = ^^ = -i. ib. Ans. 
 
 Example 3.— Reduce I of an acre to the fraction of » 
 yard. 
 
 operation. 
 
 X «f «r, „ 7 X 4 X 40 X 301: 
 
 i of an acre = * of a yard 
 
 5 
 
 33880 „ ^ 6776 
 — — - of a yard = , 
 
 Ans. 
 
 Example ^.-Reduce f of ^ of f | of ^ of 25 furlongs 
 to the traction of f of f of § of 7 feet. ^ 
 
 operation. 
 
 i ^J P ?i ^t 2^ ^" ^^ 2^ ^"^^^"SS = ^ of a fur. 
 f ofa-off of 7 feet = i of a foot. 
 
 Then f of a fux. = iZL^l^-^i^ ^ ^ _ 
 I of a foot. Am, 
 
 8 
 
 H^ = fraction of 
 
 
REDUCTION OP FRACTIONS 
 
 m 
 
 he fraction of a 
 
 Exercise 87. 
 
 1. Reduce ^- of a day to the fraction of a week, 
 
 2. Reduce 2V of a cwt. to the fraction of a quarter. 
 
 J. Reduce f of ^ of f of a yard to the fraction of an ell 
 Flemish. 
 
 4. Reduce i of f of || of a mile to the fraction of a 
 perch. 
 
 a. Reduce f of | of 3^ mchea to the fraction of a linear 
 yard. 
 
 «. Reduce f of f of ^ of 6 oz. to the fraction of i of f of 
 
 f of a scruple. 
 T. Reduce -,^ of -,\ of ^ of ^ of a pint to the fraction of 
 
 4 of f of -^ of a bushel. 
 
 B. Reduce f of i\- of 6f shillings to tne fraction of one 
 
 pound. 
 !'. Reduce ^- of 4| hours to the fraction of a week. 
 
 Id'. Reduce f of a lb. to the fraction of | of f of ^ of 
 -5- of a dwt. 
 
 *1. Reduce f of 4| of -^ of f f of an acre to the fraction 
 
 of f of a square yard, 
 lii. Reduce — of — of "I- of J of a farthmg to the fraction 
 
 of a pound. 
 
 27. To reduce one denominate number to the fraction 
 
 m another: — 
 
 RULE. 
 
 Redwe both quantities to the lowest denominaiion con- 
 tained in either. 
 
 ^^^ /'^^ce that quantity which is to be the fraction of 
 the other as numerator^ and thfi remaining guantitu as de- 
 nominator. 
 
if - 
 
 i , 
 
 i 
 
 i 1 ! 
 
 1 1 
 
 1 ' 
 
 1 , 
 
 I 
 
 
 
 mm- f^^ 3 '" 
 
 lii 
 
 98 
 
 REDUCTION OF FRACTIONS. 
 
 Example 1. — Reduce 4 lbs. 2 oz. to the fraction of 9 Vmi 
 7 oz. 11 dwt. 
 
 OPERATION. 
 
 4 lbs. 2 oz. = 1000 dwt. 
 
 9 lbs. 7 oz. 11 dwt. = 2311 dwt. 
 
 Therefore 4 lbs. 2 oz. is 
 
 1000 
 2311 
 
 of 9 lbs. 7 oz. 11 dwt. 
 
 Example 2. — Reduce 168. 4|d. to the fraction of £91 
 9a. lid. 
 
 OPERATION. 
 
 16s. 4|d. = VST farthings. 
 
 £91 9s. lid. = 87836 farthings. 
 
 787 
 Therefore the answer is -,--. 
 
 Exercise 88. 
 
 1. "What fraction is 2 hours 17 minutes of 1 week It 
 
 hours ? 
 
 2. What fraction is 19 lbs. 7 oz. 21 grs. of 11 lbs. 7 oz. 9 
 
 dwt. ? 
 
 3. What fraction is 6 per. 16 yds. 2 ft. 11 in. of 7 roods 
 
 14 perches? 
 
 4. What fraction is 3 qrs. 1 na, 1 in. of 3 Eng. e. 1 qr. 
 
 2 na. V 
 
 6. Reduce 27 weeks 2 days 4 hours 7 min. to the fraction 
 of a year. 
 
 6. Reduce 2 qts. 1 pt. to the fraction of 7 bush. 1 pk. 
 
 7. Reduce 1 lb. 1 oz. to the fraction of 3 cwt. 3 qrs. 
 
 17 lbs. 
 
 8. Reduce £176 ISs. 7^d. to the fraction of £217 19s. 
 
 lid. 
 
 9. What fraction is 17 farthings of 6s. ll|d.? 
 
 10. Reduce 27 square yards to the fraction of an acre. 
 
 11. What fraction is 7 di-s. 1 scr. 17 grs. of 7 lbs. 4 oz. 7 drs. ? 
 
 12. Reduce i of § of f of £7 8s. S^d. to the fraction of S of 
 
 f of ^ of £6 7s. 8id. 
 
 I,iife 
 
ction of 9 Vw. 
 
 »z. 
 
 11 dwt. 
 
 action of £91 
 
 8. 
 
 
 S. 
 
 
 in. of Y roods 
 
 of £217 19s. ■ ^' ^ of f of a bushel. 
 
 REDTTCTION OF FRACTIONS 
 
 99 
 
 " " -To*ffo>?|lf1ltf / - '^ "»•*<>*» ruction 
 "• '^^t: ' ■ "' ^ "' '^0 "' « ™ods ir per. to the fraction 
 
 fmct4 of A of sl of t of li "', " ""'^^'^ '0 th« 
 n "' oj 01 J ot 3 cords 56 cubic ft. 
 
 .flower dtnominttij" " "^""""""'^ fr-^'ion in terms 
 
 UTILE. 
 
 Ex.MP„i._Whatfathe.aIueofHofannIef 
 
 OPERATION. 
 
 """r8,^inf.'«-°)-^-30per.4,d.o 
 .Kx^P..2._Whati.thevaIueofHofacwt.? 
 
 iH , ^ OPERATION. 
 
 .7cwt.^29 = 2qrs.8 1b3.9oz.,4|Jd,u^„.. 
 
 Exercise 39. 
 Find the value of the following factions : 
 1. t Of a week. 
 
 8. a. 
 
 9^ 
 
 toffof|ofahhd 
 K ff of 8|- lbs. Troy. 
 p-f of -a- of?|of 
 
 17 
 
 an acre 
 
 i off, of efof -,A- 
 f^renchell. 
 
 of 
 
REDUCTION OP FRACTIONS. 
 
 09 
 
 " "'^nnVof^!^- '^""-'o the .action 
 "• ""'"1 '■ ■ " ^ "' t "' « 'oods n pe. to the fraotioa 
 
 fmctiL ofYof el of t rf IL"^ 17 oord-foet to the 
 I J "i o? 01 ^ ot 3 cords 56 cubic ft. 
 
 .flower dtnominttio™?.:' " "^""""""'^ *«««»" in terms 
 
 HULE. 
 
 Ex.„P„i._wi„u.the.aIue„fHofa.nIe? 
 
 OPERATION. 
 
 Ex.„P..2._Whati.theval„eofHofacw,.f 
 
 1 H , OPERATION. 
 
 .7cwt.^29 = 2qrs.8 1b3.9„z.MJM:u^«. 
 
 Exercise 39. 
 
 Find the value of the following faetiona: 
 i. i Of a week. 
 
 2- i off of a bushel. 
 
 2, fof^of|ofahhd. 
 
 [4- A- of 81- lbs. Troy. 
 .8^ 
 
 5. f of-.-of?|ofanaere. 
 
 ^« f off ofa£. 
 
 8. ^^ of 3^ of ^^ of an acre. 
 
 9. n of 9^ of X of a mile. 
 
 10. f offofg^ofSScwt. 
 
 11. -.\off ofalb. Apoth. 
 
 |i -M 
 
100 
 
 ADblTIOW OP PRACTIONB. 
 
 ADDITION- OF FRACTIONa 
 
 RULE. 
 
 2^ Seduce the fractions to a comirion denominator^ add 
 the nur.urators together for a new numerator, and beneath 
 their sum write the common denominator. 
 
 Reduce the resulting fraction, if it be an imm-oper fra^^ 
 (ton, to a mixed number. ^ r j 
 
 ^\.?t^'^\~^^ ^"5^ mixed numbers occur among the addends, add 
 grL portS.'^^^^ ^' ^° '^®^'" '^ *^^ '^*' ^•'"^ of the inte. 
 
 Example l.-Add together \, f, f, -i^, and ^. 
 
 OPERATION. 
 
 B7 Art. 23, these fractions, reduced to a common do- 
 nommator, become 
 
 60 ^ 45 
 
 126 "^ 120 "*" I20 "^ 120 "^ 120 
 293 
 
 120 ~ '^'^^' ^'^' 
 
 if 4. ^ - ^^+'^2+45+84+32 
 
 120 .^ 
 
 Example 2.— What is the sum of 6^+19i\+9f+17C 
 
 • ' OPERATION. 
 
 6f+19-,^+9f+l'7f+23if 
 
 = 6+19+9+17+23+f +Tif+f +f +11 
 
 6+19+9+l'7+23 = '74. 
 
 f +l^-+f +f +|f r= Mt+fU + IH + ni+m 
 
 __ 264+224+2oi + 528+6'72 ,„,„ 
 
 = ^6l6 = V^V = 2if J. 
 
 Then U + 2\\\ = 76tfi. Am. 
 
 -«*M*t*i»r4. 
 
 \ 
 
1. 
 
 2. 
 
 8. 
 
 4. 
 
 6. 
 
 6. 
 
 1. 
 
 8. 
 
 9. 
 10. 
 11. 
 12. 
 13. 
 U. 
 15. 
 16. 
 
 SUBTRACTION OF FRA0TIO;>?a, 
 
 EXKRCISB 40. 
 Fl*»<l ihe ^aiue of :— 
 
 ? + 1 J + 2| + 9f 
 
 6i+n§H-196i+29^. 
 8^ + IIH + f + ^ + 16i 
 196^ + 1615 + 20, \ 
 200^ + 763^ + 91 6f 
 
 iO! 
 
 IP 
 
 17. 
 
 18. 
 
 im+19|§+20M + 21f^. 
 6i + 8^ +111 + 9^^.16^^ 
 
 iof f + ioff + f of 6^ 
 
 fofiofi + 9i + 6A + fof^of^^. 
 
 n+9i + 16^ + 20i + ^of§of^of-A- 
 
 ^!.t,^ ^^'^ ^^ ^^ + A ^f A of 242 + i6A. 
 1 1 Hi + 22 J + 3f + I of § of I of ^ 
 en + 89,^^ + 90f^ + I01|g|. 
 
 iof, 
 
 19. 2* + ^ 
 18 
 
 '9'? + 18^ + 1^. 
 
 20. ^of?| + 9^+n^^l6^^y 
 
 + Aoffjoff^. 
 4i 
 
 SUBTRACTlOlSr OF FR^^ CTIOl^S. 
 
 RULE. 
 
 nator: ^'^'^' ^''^ of th, fractions to a common denorm^ 
 
 the common denominator, m^rence write 
 
 I 
 
1 A 
 s it 
 
 102 
 
 SUBTR^ .'i'lON OF FRACTIONS. 
 
 it) J'j 
 
 i-*iP ! 
 
 NoTi.— In tho cfliBe of mixed numberB, it frequently h«iT>penj 
 thai (liv frikclional part of tin Mubtrahend ih grcnler lliaii t.,e fratJ- 
 tioiial partof tiic minuend. Wlicn tliiH occurs, inwtind of iiMluciig 
 botli quantjtiort to ImprDper fractioui* atid tlicn applying tin* vy\W. 
 It iH much l)ct!er lo borrow one from tlio integral jiaft of the mill' 
 uend, and. considering it r.B a fraction liavi; g the couimoa deiiOiuU 
 nator, aaa it to the fraciiondl part o/ the miuuoud. 
 
 £xAMPL£ 1. — From -f-f take ^V 
 
 OPERATION. 
 
 tf - ^ = m- m = ^§?}^ = 5^r. Am. 
 Example 2.— From llf take 9H. 
 
 OPERATION. 
 
 11^ -» 9H = iiff - m = 10 4- m-^ii = low ^ 
 
 Hi = Uf ■ Ans. 
 Example 3.— From f of |f of 1^ take ^^. 
 
 OPERATION. 
 
 ^ Of n Of u -f/o = M - -h\ = m -^"u^ 
 
 Ans. 
 
 */!2XERCISE'41. 
 
 Find the value of. 
 
 -1 4.. 
 
 11 17- 
 
 1. 
 
 2. fof^-lofi 
 
 8. -,\ of 65 - i of 2}. 
 
 4. 169f-23H. 
 
 5. 229 ,V - C7ji. 
 
 9. f of Aof 18,i-of2|f 
 
 10. i+ t + fo--T^,-+l 
 
 11. 16^ + 4i + 3Gi + 20^ - 17-,V 
 
 12. 4iof ^^^ 
 
 6. 1116:^-229Hf. 
 '7. lli^ - lit' 
 
 |of loffi.^ofV. 
 
 I 7 
 
 1 6' 
 
 m . 7 17f 
 
 18. tof^ off of 162-^111 4- l^-hU-h-h 
 14. 96| - 4- of ^ of i of 63 + -i^f + 18^ - 1^10. 
 
 
ntly happenj 
 liiiii t..o friio- 
 (lof loduciiig 
 iiiji lilt' nik'. 
 t of llio mill' 
 won deiiOiuU 
 
 = 10l«\a ^ 
 
 ^"u = HS^ 
 
 4f. 
 
 of 
 
 13 
 n 
 
 
 MULTIPLICATION Oi' J^HACTIONS. 
 10. 41 of 6i of Vf — t of 8f of 7.1. 
 
 103 
 
 MULTIPLICATIOISr OF FRACTIOl^S. 
 
 RULE. 
 
 31. Reduce all mixed number 8 to improper fractions 
 and co.nplex and compound fractions to simple ones. Can- 
 cel the factors that are common to a numerator and a denom- 
 inator of the resulting fractions , 
 
 Multiply all the reduced numerators together for a new 
 numerator^ and all i}u reduced denominators together f<yr a 
 new denominator. 
 
 Reduce the result^ ifi - <:,jary^ to a mixed number. 
 
 Example 1.-— Multiply f by \^. 
 
 Hpre we cancel the first denominator and reduce the 
 geconu numerator to 8. 
 
 Example 2.— Multiply together ff, f , 3^, and f f. 
 
 STATEMENT. CANCELLED. 
 
 frxtxfxf|=— x^ X- x^- = - = l. Ana. 
 
 63. 
 
 Example 8.— Multiply together ^, tV, ^h Hy Hy and 
 
 STATEMENT. .» 
 
 fx-A-x^/^x^/xfx^a 
 
 CANCELLED. 
 
 2 4 ;r 
 
 ^vl-v^ 48 3 ^3_2x3x4x4d 
 
 -=1152.-4n*'. 
 
m 
 
 
 I 
 
 liil 
 
 m&M 
 
 104 MULTIPLICATION OP FRACTIONS. 
 
 STATEMENT. 
 
 _ 1 205 
 
 66S5 
 
 El^-t^ir*! 42. 
 Find the value of— 
 
 2. ^xA. 
 8. Ax2ixfx8|. 
 4. 6^x4Ax'77x4|. 
 6. 3x7ixHx3-iV 
 6. 9txT^fx2xTa5rxff. 
 
 1 8|x9ixlOixl 
 
 8. f of^of(i4-f)x^ofTV. 
 
 9. 2'7fx98A. 
 
 10. 16^x8^x^x1?-. 
 
 11. (ll|-f-6i)x(9|-'7|). 
 io 4f 6A 
 
 fp ^xiofSf of9f. 
 
 18. 6f of 8^x9ioY t|. 
 
 CANCELLED. 
 
 8 8 
 
 ^ S 0^ 
 
 ^^ n l^^ __ 206x8x8x8 
 
 iV9 
 
:^1VIS10N OF FRACTIONS, 
 
 14. ^ of I of f X f X f^ X 693. 
 
 15. (4f - 2^) X 2i of 4^ of (7i - 6^). 
 
 ^6.6^of|oflxf xllf. 
 
 105 
 
 X Multiply f of I, of ^ by ^i X f X If. 
 
 18. What is the product of f x ,\ x ^ x -Sj x 6^ x — x — ? 
 
 19. Find the value of (6f + 4} + 9J) x (6^ + 3f) x 
 
 20. What is the product of (8-,^,- — 2^ + SA ^ 13.\ ^ 
 
 (6A - 2^ + 2 4- i) X (i of hi A)? 
 
 DIVISIOISr OF FRACTIOIsrS. 
 RULE. 
 
 32. Reduce eornpound and cmiplex fractions to simple 
 cnes, and whole and mixed numbers to improper fractions. 
 
 Invert the terms of the divisor^ and proceed as in mul- 
 Hplicattou. 
 
 Example 1.— Divide ^- by ^. 
 
 OPERATION. 
 
 Example 2.— Divide | of f ,- by t\- of 8f . . 
 
 OPERATION. 
 
 offr + Aofa/=:fi-f.M = H X H = Ta,. Ans. 
 Example 3.~Divide 8^ by 3-ft-. 
 
 OPEBATION. 
 
!l: i 
 1 , 
 
 I • 
 
 
 106 
 
 DIVISION OP FRACTIONS. 
 
 E..HP.. 4,-DivideAofAof J . 3|byA of»Jx4|. 
 
 STATEMENT. 
 A X iV X V:f^ X ^^^ ^ ,±, X f II X V. 
 
 TERMS OF DIVISOR INVERTED 
 
 = Ax,irxWxi^!^xi^x|JJxA. 
 
 CANCELLED. 
 
 "■;^ '';;'' 
 
 3 
 
 \3- -^^f^l^ni. 
 
 Find the value of— 
 
 3- '71 -^i off of 61 
 
 4. f of|-^^of^. 
 
 6. S|of8i-^6|of 5|. 
 
 6. ^of3|of9i-f-?i 
 
 9| 
 
 Exercise 43. 
 
 9. ^^-«i 
 
 10. 8fof4|of6i^3fof^, 
 
 11. 4i--61of /,-of4f 
 
 6f 
 
 V. fof8fof6|^4|of2f. 
 13. (fof|of8|)-i-(fof4|--^S). 
 
 12. 9iofy^6|of-|. 
 
 14. 8fof6iof4xfofiof,i,x|-^6fof«i. 
 
 15. 9iof8f of 6f--^lof-5l 
 
 fracto:- """^'^^^^ *" ^^'^^^^^^ denominate number by a 
 
 RULE. 
 
 a A . f . • t 
 
 A«*:;'2^3?.S';:^^'t^if!L=-'- -z^* 
 
 resw/^ 6y <Ae denominator. 
 
DIVISION OF FRACTIONS. 
 
 107 
 
 ExAJ LE.~How much is yV of ^ of f of 4| of 4 days 
 21 nours ' minutes? 
 
 OPERATION. 
 
 t^f of ^ ->f f of 4^ of 4 d. 21 h.1m. = I,- of 4 d. 21 h. 7m. 
 * 1 21 h. 7 m. X 3 
 "" ' ^"il = 1 d. 7 h. 66 m. 271^^ sec. Ans. 
 
 34 To divide an integral denominate number by a frac 
 don :- 
 
 RULE. 
 
 31 'tipli/ the denomiftafe nnmher by the denominator of 
 tJie/r vtion and divide the result by the numerator. 
 
 EXAMPLE.— Divide 7 lbs. 4 oz. 7 dwt. by \ of 4f of 6^ 
 
 OPERATION. 
 
 7 lb t OZ. 7 dwt. -f- i of 4a of 6i of -^ = 7 lbs. 4 oz,, 7 
 dwt. ^ -^3 = . ~ = 4 lbs. 7 oz. 19 dwt 
 
 30 
 
 2f g'^. ^n«. 
 
 Exercise 44. 
 Find the value of— 
 
 1. I off of £1 16s. 8^d. 
 
 2. 31- of 81^ of -33 of II of 4 bush. 1 pk. 1 pt. 
 
 3. \ of f of 6| of ef of H of 6 lbs. 4 oz. Avoir. 
 
 4. Hoff of 38jof6|acres. 
 
 6- |of |of 11 cwt. 1 qr. 11 lbs. 
 
 6. f of ^ of j^ of 3 acres 1 rood 27 per. 
 
 7. 6|ofHof-ja, of£6 lis. 43d. 
 
 8. 6} of 111 of 21 of i^ of 7 miles 4 fur. 17 per. 
 
 ». I cf -f,- of 3 lbs. 5 oz. Avoir, -f 6^ of || of 6 lbs. 1 ^ oz. 
 iO. 23 of } of 6 lbs. 11 oz. 4 drs. 1 scr. 16 grs. 
 
108 
 
 BECIMALS. 
 
 ■il 
 
 I' If 
 
 . 'f 
 
 ' , 
 
 11. 4} of 6^ of i^- of A of 4 yds. 3 qrs. 2 na. 
 
 12. U of ^ of 6^ of 2 qrs. 17 lbs. 4 oz. 
 
 13. T^«of6iofMof21bush. Spks. 
 
 14.-A-of3iof,^of+iof|of7wk.4d.5h. 
 
 15. 21 lbs. 11 oz. 1 dwt. 4- f of"^ of m 
 
 16. 4 acres 6 per. 5 yds. ^ 5^ x f x ^f. 
 11. £169 4s. Hid. -f. 3^ of Gi of j^a 
 
 18. 11 cwt. 2 qrs. i1 lbs. x 6^ of 4f of xiy 
 
 BECIMALS. 
 
 Thus TR, rW, T^^ ^fii« » ^ ^ . 
 
 T6) Toff) Tooo> Tooooi TiTOooo) «c., are decimal fractions 
 37. The or(/ers to the riffht of tho ^om«,oi • x 
 
 to the^oflowlng' *' '''^ ""^ ^''^^^^ ^^ P^^^^^^ according 
 
 RULE. 
 
 Point off into npvin^o ^-e *x , v» 
 
DECIMALS. 
 
 109 
 
 ^der of the extreme right-hand figure. TIten find by sim- 
 
 Example l.—Read -0000006174. 
 
 •000 000 r^'f ';^ ""^.-^K ^^T" expression, it becomes 
 
 Ttha of /! '/I' JTvr^'"^. ^^ ^'^^" *^^^^ ^^'^ denomination 
 s that of tenths ofbilhonths, and the expression represents 
 
 jrx^^ous^nd one hundred and seventy-four tentKf biu! 
 
 Example 2.— Read 41V'000000071461Y4. 
 
 .1,0^^ wr'''f'''F ?^'. ^^ set 41'7-000,000,071,461,74, i. e. 
 the nght-hand digit is that of hundredlL of tr liorUlu 
 and the decimal part is 7146174 hundredths {f trillionths: 
 Hence the whole expression is read—Four hundred and 
 seventeen, and seven millions one hundred and forty- 
 TillioS^ ''''^ '^^ ^^ seventy-four hundredths of 
 
 Exercise 45. 
 Read the following expressions : 
 
 1. -27; -043; '007; '6914; -008196; -00071423. 
 
 2. 6-7; 93-42; 147-1394; 2170000009. 
 
 3. 71-00089; 167-193; 91-0008674 
 
 4. 5674378-000914786; 71300400-000000600407. 
 
 folbwinj^ """^^ any decimal, we proceed according to the 
 
 RULE. 
 
 voitt'Z*^Z ^'f'^^'^y P}^<^^' to the right of the decimal 
 yotntme given den&rnination comes. 
 
 fnen if the given digits do not occupy all of these 
 
 saS^""'' 1— Write down seventy-eight tenths of thou- 
 
 }, 
 
110 
 
 DECIMALS. 
 
 ».ian^f^l/'"Pl*^"*^' ^^ thousandths occurr the foi fh 
 
 bef S then wr^^^^ P^^* -^ ^^- ^ -cl the num- 
 
 •00Y8. 
 
 nnd that trilliont ha oppimv tha. lo^u i """"""*»» <kc., ,va 
 
 tains s«a: d:g,ts, and consequently we must insek behv,?pn 
 the decimal point and the 7 the difference ^^11 if ^3 
 6, 1. 6. 8 ciphers, when the expression becomes^! 
 
 •00000000704091. 
 
 EXERCI. 46. 
 "Write down the following numbers • 
 
 2. Soven hundred thousand and ,txtoon hundredth, of billionth, 
 
 3. Fi>e nj,n,o„, tweuty.niue thousand and eleven tenth, oftrm: 
 
 '■ ""fer^" "">-""""' '°" '""'^'■•ed •».<! three tentha of mill 
 6. Seven hundred and nine thousandths 
 
 ». E,,re,, decimally 749 units + 2 tenths * 49 tenths of „„,. 
 10. EipMM u a dwiau! 7897 hsadwdth, + 7M numouUifc 
 
ADDITION OF DECIMALS. 
 
 111 
 
 41. To multiply a decimal by 10, 100, 1000, &c., we 
 remove the decimal point 1, 2, 3, &c. places to the ri'rht 
 and to divide by 10, 100, 1000, &c. we remove it similarly 
 to the left. ' 
 
 Thus, -006 X 10 = -06; -006 x 100= -6; -006 x 10000 = 
 60, &c. 
 •867 -i- 10 = -0867; -867 -h 100000 = '00000867, &c. 
 
 NoTB.— Affixing ciphers to a decimal, 1. e, writing them to tho 
 I'thi of it, dees not alter its value, thus, -79, -790, and -79000 are all 
 /qual, each representing /j^ ; but prefixing ciphers to a decimal, 
 1. e. writing them between the decimal point and the left-hand 
 figure of the decimal, divides the decimal by 10 for each cipher 
 thud prefixed. 
 
 sandths ; four 
 
 tenths of mill. 
 
 ADDITION AND SUBTRACTION OF 
 
 DECIMALS. 
 
 RULE. 
 
 42. Write the numbers to be added or subtracted, so that 
 the decimal points shall be directly under one another, and 
 proceed as in addition and subtraction of whole numbers 
 oemg careful to place the decimal point in the answer in the 
 same vertical line with the others. 
 
 Example 1.— Add together 78-647, '0078, 9'816, 4-278 
 m-4278, and -0091. ' i*^'^'! 
 
 OPERATION. 
 
 78-647 
 •0078 
 9-816 
 4-278 
 967-4278 
 •0091 
 
 Here we begin at the right-hand side, as in 
 simple addition, and proceed as follows : 1 and 
 8 make 9 and 8 make 17, set down 7 and carry 
 1 ; 1 and 9 make 10 and 7 make 17, &c. 
 
 1060-1857 
 

 I;:- 
 
 112 MULTIPLIOATrON OF DEOIMALa 
 
 Example 2.~Prom Yl-0047 take 9\^'^8l6'r, 
 
 OPERATION. 
 
 ^o'oadLfi'z f ^^""^ ''^'^y '^ ^^^"™ ^ ^e can't, but 1 
 ±0(H)8m from 10 and 3 remain; 7 (i e. 6 and 1 
 
 620038833 carried) from we can't, bV 1 from 10 
 
 A« 1/% and 3 remain; 2 from we oA^'t imt 2 
 
 n,V "'^ ' "^""^''' " *»■" ^ *« «'^^- ^•.S t,m 
 
 Exercise 47. 
 
 Find the value of— 
 
 1. 18-716+ 967 + 34-71 + -271 + 698-7149 + 23-0067 
 ^91T,^^^?^■^^^^'■^^^ ^^'^^23^146 + «78-906+12-9867^ 
 8. 216-714763 + 29 + 9867 + 91-0986+ 7-81645+ -09868 
 
 4, 26-1111 + 11.22222+34-646+1719186 + 11 127 + 816-7142. 
 6. ;9167 + 9-9 + 8-98+7-614+-0986 + 17 + 19n + 963-714 
 6 9-^84 + 9111-77 + 967-769+463 + 7-0009 + 8-61 + 911-1257 
 
 7. 167-914-6-8147 
 
 8. 9161 0098- 7149-16716 
 e. 710916714-27-1471. 
 
 10. 1111-116-22-22222. 
 
 11. 279-00906-117-916. 
 
 12. 627-4-91-7469. 
 
 MULTIPLICATION OF DECIMALS. 
 
 RULE. 
 
 *.hnf^' ^'f'P^y ^^j 9iven decimals as though they wers 
 whole numbers, and mark off in the product asmanl 
 
 fztg&:: " ''"' ""'' ''''''' ""^^^>^- -^ -"S' 
 
 } Example 1.— Multiply -743 by -067. 
 
DIVISION- OF DECIMALS. 
 
 OPERATION, 
 
 •743 
 •01)7 
 
 6201 
 4458 
 
 113 
 
 Here, when we multiplv 743 and fi*? *r...^*\. 
 wc get as the product 4^1 hx^^JJ. ^uf h^^ 
 
 Example 2.--Multiply {^69 by 30-8. 
 
 opebation. 
 
 Exercise 48. 
 t'ind the value of— 
 
 3.' 2- 16x2-06 8. 3-42 X -061 X -OOW. ' 
 
 • ^' lib X 2 06. 9 4i.]4g, g.. 
 
 4. •O07X-O0678. , .0. 80-08 x 6 6 x 20 02 
 ». 67-91X8-.00004. 1,. 1-0,2 x -00719 
 
 «• "-111 . 9-711-6. I 12. .2 , .y ^ .,, ^ 
 
 DIVISION OF DECIMi\iS. 
 
 RULE. 
 
 
1H 
 
 DIVISION OF DEOIMALa 
 
 Example 1.— Divide 716'193 by 614. 
 
 OPEitATION. 
 
 ei4)716-193(M6643, &c. 
 614 
 
 1021 
 614 
 
 4079 
 8684 
 
 3963 
 8684 
 
 2690 
 2456 
 
 2340 
 1842 
 
 Here we divide as in whole 
 numbers, but, when we have 
 brought down the unit's figure, 
 which in this example is in 
 the first step of the division, 
 ve place the decimal point ui 
 the quotient. After bringing; 
 down the last figure in thej 
 decimal, we may continue tho 
 division by bringing do^v 
 ciphers. (See Art. 39.) 
 
 498, kc 
 Example 2.— Divide 7*43 bv -0079. 
 
 OPERATION. 
 
 • 7 43 _t. -0079 = 74300 -5- 79 = 940'5063, &c. 
 
 Here, since the divisor contains decimals, we remove 
 tlie decimal point to the right of it, and also as many places, 
 .. e. four places, to the right in ihe dividend ; this gives us 
 74300 -^ 79. 
 
 45. The* following will illustrate the mode of thus pre- 
 paring numbers for division when the divisor containB 
 decimals : 
 
 67-9 -*- -9 — 679 -V- by 9. 
 
 27-09 -T- -0047 = 270900 -f- 47. 
 
 27-14678 -5- 2-47 = 2714-678 -:- 247. 
 
 114-00672 -i- 6-0437 = 1140067*2 -4- 60437, • 
 
 278 -^ -0147 = 2780000 ^ 147. 
 
 a-«H769 4- 27-I4M =s 2«U7-80 f- 271404* 
 
REDUCTION OF DEGIMALS 
 
 1. 781 + 1071 
 2 91-142 + 7-8. 
 a 31 123 -f. 0146. 
 4. 91234 -I- 000718 
 6. 0467+01471. 
 a 918 -•. 814-71. 
 
 907 104 + 12-0461 
 91-671 -t- -000016. 
 8-8 -♦- 0641. 
 7147-12 + 1127. 
 •817 -I- -9147. 
 213 + 91-614. 
 
 115 
 
 REDUCTIOI^ OF DECIMALS. 
 
 46. To reduce a vulgar fraction to a decimal :^ 
 
 RULE. 
 ^ivide the numerator by the denominator. 
 
 E^MPLEL-Reduceltoadecimal. 
 
 OPERATION. 
 
 8)7 
 
 •8^6 Ans 
 Example 2.-.Reduce if to a decimal'. 
 
 OPERATION. 
 
 14 -r 31 = .461612903. Ann. 
 1^^ Exercise 50. 
 
 *• Tj, s^, and A. /» gi^^ j o,- 
 
 .1 m 2 , ?„ '• "5 W^ and W. 
 
 4 in ii /f, ^* 8 2 and If}. 
 
CIRCULATINQ DECIMAi^S. 
 
 285714, and in the last part the fipnve 4, constantlj 
 recur. In this case the decimal is called s. rept^ter or ci 
 culator. 
 
 48. Decimals which do not terminate, i. *. which con. 
 gist of the same digit or set of digits constantly repeated, 
 are called Repeating or Circulating Decimals. 
 
 49. The digits or set of digits which repeats, is called 
 a repctcndy period^ or circle. 
 
 NoTB.— The terms period and circle are used only when the 
 repetend contains tWo or more digits. 
 
 60. A Single Repetend is one in which only a single 
 digit repeats. 
 
 Thus, -3333, &o. ; '7777, &o. ; -83888, fvre single repetends. 
 
 * 51. A Single Repetend is expressed by writing the digit 
 that repeats with a dot over it. 
 
 Thus, -333, &c., iB written 3 ; -777, &o., la written -7. 
 
 52. A Circulating Decimal or Compound Repetend iSj 
 one in which more than one digit repeats. 
 
 ThuH, -34734:347, &o. ; -202020, &c. -, -123412341234, &c., areolr. 
 culating decimals or coraijound repetends. ' 
 
 53. A Circulating Decimal is .expressed by wiping 
 the recurring peri ad once with a dot over its first and \m\ 
 
 digits, . . j 
 
 Thus, -347341, &c., is written -347; -2020, &c., -20; -12341234, 
 
 &c., "1234. 
 
 54. A Pure Repetend or Circulating Decimal is one in 
 which the repetend commences hnmediaieli/ sSter thQ dec-* 
 imal point. 
 
 55. A Mixed Repetend or Circulating Decimal is ono 
 which contains one or more ciphers or sigmficant figurc-j 
 between the repetend and the decimal pouit. 
 
 Thu. 3, -7, -i are pure repetendB. 
 
 •78917, 0378, -002 are mixed repetends. 
 
 •72, -043, -81376 are pure circulating decimals. 
 
 •1378, -673i205, -071786 are mixed circulating decimfJi • 
 
1234, &c., are oir. 
 
 D., -iO; -12341234, 
 
 OIV ' OLATINQ DECTM A La j j ^ 
 
 fracfiou J- '''*^''''^ "^ ^"'''' '*'^''''^"'* *^ ^'^ equivalent vulgar 
 
 RULE. 
 
 rnV* the period ihdf for numerator, and for denomi^ 
 nator as 7nany 0'6 as there are digits in the period. 
 
 Thus, -^=5; -42 = ^1 = ^^; •no = §^J. 
 •2107 = Hg5; ilSl = ^lai = ^i^.j9^. 
 fracfion .^''^^^"'^•^ * ^^^^^ repetend to its equivalent vulgar 
 
 RULE. 
 
 ASr?<ft^mj^ Me finite part of the rmxed repetend from the 
 *oiole, anl write the remainder as numerator; then for 
 Jmomtnator write a^ many 9'« as there are digits in the 
 (ircle, followed by as many O's as Ihere are digits the 
 ^n.ve part. ^ 
 
 wr?n ''f''i;~T^'^ ^'J'*® P^^^ °^ ^ ™'-^^^^ repetend is the part be- 
 
 OPERATION. 
 
 41'7— 4= 413 = 1st numerator. 
 ^t.??'^o "" K^^ = '222 = 2d numerator. 
 7161423 - 710 -= 7160707 = 3d numerator. 
 
 Also 1st decimal contains 2 digits in the circle and 1 in 
 the finite p-^-t, therefore the denominator of the ist 
 fraction is 9dO. 
 
 Similr.rly the denominator of the 2d fraction is 9900 
 and of the 3d fraction 99990U0. * 
 
 rWefore ^-4^ = -tig . -123^ = i|j^ ^ .4^11^. . .^^q{^^ 
 
 — S9 9~aU0O« 
 
r- 
 
 !iiliPP>«HHH 
 
 il8 
 
 ClilOULATING DECIMALS. 
 
 Exercise 51. 
 
 ExprcBS tho following pure and mixed ropotonda as vtLl||;u» 
 fraction:^ : 
 
 1. -1 ; -78 ; and •4. 
 
 2. -21 \ -347 ; -2178. 
 8. 19867473; -27. 
 4. -126; -214. 
 
 6. -2134 , -216 ; •2li4. 
 
 6. -12346 1078. 
 
 
 7. -6714; 12710. 
 
 8. •9186-, -142. 
 0. -12347; -1278. 
 
 10. -16714 ; -9 ; -86. 
 
 11. 27-43 ; 17-8i6. 
 
 12. 467 12345 ; 1616161.. 
 
 I 
 
 \ti 
 
 t 
 
 58. In order to add, subtract, multiply, or divid jp-j"* 
 or mixed repetends : — 
 
 RULE. 
 
 Reduce them to their equivalent vulgar J'^actions^ and 
 then addj subtract^ multiply ^ or divide then fractions, 
 
 • • • 
 
 Example 1.— From 9 70 take 4-918. • 
 
 0PEP^4.TI0N. 
 
 9-76 = 9^ = ni and 4-9id = 4|^a = 4^^. 
 
 Then m - mh = 9i^a - nu = m^ = ni aw 
 
 Example 2.— Divide -927 by -012346. 
 
 
 OPERATION. • 
 
 O 1 » ,^^^ 107 „^ Kl „„^ .r\i i-^n 4 f!' 
 
 990 
 
 1 1 U 
 
 601 
 
 auu 
 
 n -I n J I 
 
 99 0000 — 000 Off' 
 
 Then U -i- Alh = H X m^^ -= HH^ = 75^-. An$. 
 
Find the ralue of— 
 f © + -66. 
 i. 9-i2 + -725. 
 3. 614 - 2-714. 
 '. 7-9186 - 2-347. 
 6. 7-6 + 1-23 + 7-191. 
 
 REDUCTION OF DECIMALa 
 Exercise 62. 
 
 119 
 
 0. -7 X -12 X -67. 
 
 7. -67 X -914. 
 
 8. 6-7i X 6-713. 
 
 9. -614 -»- 2-760, 
 10. 1-647 + 3 621. 
 
 RULE. 
 
 of ItlT '■"''''''"''' '""^ 1 "'• 1 P'- to the decimal 
 
 <:pkration. 
 
 610875 quarts by 4 to reduce then, to the decimal of 
 ■ a gallon, to which we prefix a 0, as no gallons 
 arepiven; and eo on. 
 
 fraction of"the"other'^L'7tf «'^* ™''uce one quantity to a 
 » ito deoin^';;'S li" '^'° ""^'"'' "»« '^'•J'i^g frLica 
 

 ''i I 
 
 iA 1 
 
 120 
 
 REDUCTION OF DECIMALS. 
 
 Example 2. — Reduce 4b. ll|d. to the decimal of £2 U 
 43. ll|d. 239 farthings 239. 
 
 Here, by Art. 25, 
 
 Then 239 -*- 2443 = -09763. -4n«. 
 
 £2 lis. ~ 2448 farthings 2448* 
 
 Exercise 63. 
 
 Reduce : 
 
 1. 2 days 7 hra. to the decimal of a week. 
 
 2. 7 oz. 4 dwt. 9 grB. to the decimal of a pound. 
 8. 16 IbB. 7 oz. 3 dra. to the decimal of a ewt. 
 4. 116 days 14 hours to the decimal of a year. 
 
 6. 1 rood 17 yards to the decimal of an acre. 
 
 0. 3 qrB. 1 na. 1 in. to the decimal of a French ell. 
 
 7. 168. Hid. to the decimal of a pound. 
 
 8. jE9 148. 8id. to the decimal of de77 Os. 9d. 
 
 9. 2 days 17 rain, to the decimal of 7 weeks 4 days. 
 
 10. 3 fur. 17 per. to the decimal of 2 miles 4 yds. 1 ft. 
 
 11. 17 lbs. 4 oz, to the decimal of 19 lbs, 7 oz. 5 drs. 1 eor. 
 
 12. 2 roods 27 yds. to the decimal of 29 per. 29 yds. 
 
 61. To find the value of a given decunal of a denom/ 
 nate number : — 
 
 RULE. 
 
 Multiply the given decimal hy the number of units o* 
 the next lower denotnination that make one of the given de- 
 nomination. 
 
 Point off as many decimal places as there were in th% 
 multiplicand^ * and the integral portion^ i " any^ will 6* 
 units of that lower denomination. Ilie decimal part 
 may be reduced to a still lower denomination ; and so on. 
 
of £2 11 
 
 239. 
 2448' 
 
 3r. 
 
 a denod^ 
 
 f units cf 
 le given ae^ 
 
 oere in the 
 y, vnll bh 
 iinal pari 
 and so on^ 
 
 MISCELLANEOUS PROBLEMS. 
 Example.— Find the value of -27625 of a lb. Troy. 
 
 OPERATION. 
 
 •27625 = decimal of a lb. 
 12 
 
 121 
 
 8-31600 = oz. and decimal of an oz. 
 20 
 
 6-30000 = dwt. and decimal of a dwt. 
 24 
 
 7-20000 = grs. and decimal of a gr. " 
 Then 3 oz. 6 dwt. 7-2 grs. Ans. 
 
 Here we first multiply by 12, because 12 oz. make 1 lb 
 and we thus get 3-;j1500 oz. Next we multiply the deci- 
 mal -31500 by 20 to reduce it to dwt., &c 
 
 Exercise 54. 
 Find the value of— 
 
 1. -146785 of jei. 
 
 2. -71403 of a woelc. 
 
 3. 2-147 of a pound Apoth. 
 
 4. -C143 of a mile. 
 
 5. -916147 of an acre. 
 
 6. 214Gl7of aFreneh ell. 
 
 7. 9-2645 of an hour. 
 
 8. 4-7177 of a hhd. 
 
 9. 3-33625 of a rood. 
 
 10. 9-914 of £1. 
 
 11. 6-714 of £3 4*. 7id. 
 
 12. 9-1467 of a year,, 
 
 13. -12345 of $2-78, 
 
 14. -66265 of 27 sq. yds. 2 ft. 
 
 15. 7-467-25 of 7 cwt. 2 qrs. 17 Iba. 
 
 16. 6-4715 of jE? 78. 7id. 
 
 Exercise 55. 
 
 Miscellaneous Problems, 
 
 I. Reduce £297 48. S^d. to dollars and cents, and divide the result 
 by -0005. 
 
 J. Find the least common multiple of 9, 11, 18, 15, 21, 22, 42, 36, and 
 
 % Add together $78-90, J427-43, $209-17, |80-43, and $17-90, and from 
 tho sum subtract :£183 15.h 114^ 
 
 4. Reduce h f , W, h and jL to equivalent fractions having a com- 
 mon denominator 
 
 III 
 
f^ 
 
 122 
 
 MISCELLANEOUS PROBLEMS. 
 
 6. How much Is 726 timeg £2 4b. lOid. f 
 
 ^' " enc;"orT^'V'?''7'ir"'^ !" ^•'■'■'■^"^^^ ^''^"'^^ ^'^^'^"^ « circnmfe 
 
 n .« u „ .1- ; ^ '"• 'r;?^^^':J" »^o'"fe' f»'*^"^ Toronto to Stoii. s 
 Creek, a distiince of 44 milts? ^ 
 
 '■ ^*-fo.8?7;'^^''° ^"'""^ ^^f\''''^ ^' ^o that A shall have just 
 $<49 8o more than each of the others. ^ 
 
 8. In 1858 the value of the horses imported into the differont n - 
 
 of ^'ani.da was as follows : Clifton |6880, Conlicool< $9775 
 Mo,-n8burgh $6750 Prescott $58877, 8tan.tead$9105, To.o tc 
 t^lS ^\^^^?' *^^^-^' o«i^r P^rts $58712 ; what wa.' the total 
 value of the horses imported into Canada in 1858 ? 
 
 9. In the samo year the vrlue of the horned cattle Imported fntc 
 
 Canada ^a-; ag foUows : Coaticoolc $2702, Bur dee $3537 
 J^?'<iV;r«««^^ n°' ^'""^"fSUO, Suult Stct Marie $3lS WinJ: 
 sor $10688, other ports $25561 ; how much did ll^e value of 
 the horses imported into Canada in the year 1858 exceed 
 ' that of the horned cattle imported in the same year ? 
 
 10. What is the difference between | of 3i of 6f of |28-28 and -7 of 
 
 2-4 of 3-7 of II of £6 lie. 6id. ? 
 
 11. Express 704, 1111, 9876, 23471, and 9142371 in Roman numerals 
 
 12. Write down as one number seven hundred billion'^ four thou- 
 
 sand and twenty, and six millions two hundred thousand 
 and nineteen tenths of trillionths. mousaua 
 
 13. Find the value of ^^^^^^^ ^ ^^^ "^ 23 x 115>r93 x 144 
 
 25x729x184x27x12x13 
 
 14. Find the value of 2f + J of f of /^ of 37i + f - f + 8* - 4J. 
 
 16. Reduce 2 days 4 hours to the decimal of 3 weeks 3 d-^vt. 
 
 16. What is the Greatest Common Measure of 17810 and t53294t 
 
 17. ^ed^ice^-7, -93, -00045, and -27146 to their equivalent vulgar 
 
 18. How many square Inches are there In 2 a. 1 r. 17 per. ( yds. f 
 
 19. What is the value of -7149625 of a mile ? 
 
 20. Find the value of -7 of a per. + -626 of a yd. + -713 of a ft + 
 
 •91 of an inch. 
 
 21. Which the greatest and which the least of ,-\, .V. 5t ' 
 
 22. Express 3i ells Flem. as a fraction of a yard. 
 
 23. A farmer at a fair sold 229 sheop at $3-73 each, and hr^^^ch* ^^^ 
 coA\« at £11 lis. 7d. eacli ; how mucJi mout 
 
 borne? 
 
 '-i 
 
 <jairy 
 
RATIO. 
 
 c^rcnmfe 
 to Ston< S 
 
 have just 
 
 •ok $9775, 
 5, Toroi tc 
 b the total 
 
 orted fntc 
 iee $3537, 
 56, Wind. 
 ' value of 
 58 exceed 
 r? 
 
 and -7 of 
 
 123 
 
 ^' 
 
 26. 
 
 87. 
 
 from cSf^otir'n" ^'""e yonr thon- wore exports, 
 
 What is the c. m. of 6, 10, 16. 20, 24, 28, 32, 36, 40, and 44 ? 
 How many t'.mes 123 is 746 times 193 ? 
 
 ^""T'f.TV" ^^* ""^ ^^^ ^•^'^«"'- 83, what i« the divide. 
 
 
 wiiat is 
 29. Divide 346 a. 1 r. 17 p;r7b; 2 a. 3 r. 27 per. 9 yds. 
 
 fil :tT ^^"'^. ^ ^^- ^ '''■ '^ '^' ^'•=^«t'«" of 11 ^^"«h. 3 pks. 
 "• ^i^^XtTgSb;''^^^^ ^-"^-^^ '^"^ ^217 4s. 7|d.fand dl- 
 
 8JL jj-ind the value of <^ >< ^7 x 21 x IPi x 2fi4xj625 
 
 a5x81x56x48x617x40 
 
 lumeralfl. 
 
 our thou- 
 thousaud 
 
 SECTION" y. 
 
 A«^7i0, SIMPLE PROPORTION, COMPOUND 
 PROPORTION. 
 
 M 
 
 -4j. 
 
 vc. 
 (3294? 
 
 *^ vulgaj 
 I yds. f 
 
 ^ a ft. + 
 
 'fc <jaiiy 
 
 RATIO. . 
 
 *. The Ratio of one number to another is thp mw.'/,*,* 
 ai-^ng from the division of the former b/the litter!^ 
 Thus, the ratio of 16 to 4 is 16 -5- 4 = 4 
 
 the ratio of 27 to 8 is 27 -i 8 = Sf &c. 
 
 ^ The Ratio of one number to another is mmmnnlTr 
 expressed by writing a colon between them ^^^^^^^^^^ 
 
 Thus, the ratio of 16 to 4 is expressed by 4 
 the ratio of 27 to 8 is expressed by 27 .• 8.* 
 
 ,^f * ^u^ ^'^*i^ ^^,""^ number to another mav also be ev. 
 W^«*o0 by y^ntiug them in the form of a fraction 
 
 Ihus, the ratio :)f Iti to 4 may be expre.'^sed by JLfi 
 .ht, ruac 9x 2y to 8 may be expressed by ^." 
 
 
r' W 
 
 if?; 
 
 fl 
 
 II 
 
 
 124 
 
 RATIO. 
 
 4. The two numbers that constitute the ratio are calM 
 the terms of thb ratio; und the hist term is called the ante, 
 cedent and the other the consequent of the ratio. 
 
 5. If the antecedent is equal to the consequent, the 
 ratio IS called a ro^aco/^yj/a/^. 
 
 If the antecedent is greater than the consequent, the 
 ratio IS called a ratio of greater inequality. 
 ^ If the antecedent is less than the consequent, the ratic 
 ^aaWGd a ratio of less inequality, 
 
 6. A Simple Ratio is the ratio of any one number to ani 
 other number. ' 
 
 ^ 7 A Compound Ratio is a ratio produced by compound* 
 mg or multip ymg together the corresponding terms of twc^ 
 or more simple ratios. 
 
 8. The value of a ratio is found by dividing the ante, 
 cedent by the consequent. 
 
 9. Ratios are compared together by comparing theii 
 values together. .' r & ^m 
 
 10. Ratios are compounded together by multiplyina 
 together all the antecedents for a new antecedent and A 
 the consequents for a new consequent. 
 
 ..♦?^''.^-T^®!f '?^,"^"^"^^J''"^ ^^'6 antocedentfl together for a nexi 
 n^itecedent and the c«r,eequent8 together for a m-w co'ppouen? 
 
 Tequent? ""^' ^"'''^' '^^' '^ """^°^°" '"^ ^" antecedent ° a a coa 
 Example l.—What is the ratio of 27 to 8 ? 
 
 OPERATION. 
 
 27 -5- 3 = 9. Ans. 
 Example 2.— What is the ratio of 83 to 6 ? 
 
 OPERATION. 
 
 Ans. 
 Example 3.— What is the value of the ratio o- ^4 to V , 
 
 OPERATION. 
 
 94^7 = 13-428. Ans. 
 
 83 -=- 6 = 13^.. 
 
RATIO. 
 
 125 
 
 Example 4.~What is the value of the ratio of 17 to 23? 
 
 OPERATION. . 
 
 17^-23 = 0-739. Ans. 
 
 Example 5. — Point out which is the greatest and which 
 the least of the following ratios— 7 : 4, 9| : 6, 27 : 16. 
 
 OPERATION. 
 
 7: 4= 7-H ^*:^l-75. 
 9^: 5 = 9-5-^- 5 = 1-0. 
 27: 16= 27 -^ 16 = 1-68. 
 Therefore 9| : 5 is the greatest and 27 : 16 the least. 
 
 Example 6. — Find the ratio compounded of 6 : 7, 9 : 4, 
 11 : 13, 12 : 55, and 6 : 27. 
 
 STATEMENT. CANCELLED. 
 
 6 9 11 12 5 6 g n 12 3 
 ^ 4 "^ 13 "^ 55 "" 27 - 7 .4 "" 13 "" ^^ "" 12/- 
 
 -r — — = /j- = 6 : 91. 4ns, 
 7x13 "* ^ 
 
 Exercise 66. 
 
 1. What is the ratio of 7 • 4? 19 : 3 ? 26 : 2? 144 : 6? 29 : 2? 16 . 3? 
 
 2. "What is the ratio of 27:41? 71:19? 26.7? 28:7? 129:2? 
 
 47.18? 
 
 Find the value of— 
 
 3. 7:2; 9:14; 63:7; 29:3; 19:27; 34:6. 
 
 4. 91 : 7 ; 16 : 3 ; 28 : 5 ; 111 : 7 ; 222 : 11 ; 167 : 29. 
 
 Compare toiretlier tho following rAtios, and point out which is 
 the least and which the greatest : 
 
 6. 9:17, 16:33, and 47 : 79. 
 
 6. 11 . 3, 17 : 5, 38 11, and 164 : 55. 
 
 7. 49 : 5, 176 : 16-4, 267'4 25-9, and 8 : '89. 
 
 Compound '.ogether the following ratios : 
 
 8. 7 : 4, 11 : 23, 11* : 9, and 9 : 14. 
 
 9. 6 : 11, 12 : 17, Si : 4*, 27 121, and 6i fl. 
 10. 15 : 4, 16 : 7, 9 : 20, 70 27, and : 6. 
 
 U 8:7, G 5,4.5, 2 1, and -21 ^2. 
 12. 2 3, 4 : 5, 6 : 7, 8 : 9, and 16 ■23. 
 
 I 
 
 if- 
 
i 
 
 126 
 
 SIMPLE PROPORTION. 
 
 SIMPLE PROPORTION. 
 
 11. Simple Proportion enables us to find a fourth iiuiri» 
 ber which shall have the same ratio to the third of thrct* 
 given numbers that the second of these numbeis has to the 
 first ; hence proportion consists in an equality of ratios. 
 
 12. In every simple proportion three terms are given to 
 find the fourth, and this fourth term must be of the same 
 name or denomination as the third. 
 
 13. Proportion is expressed by writing the sign :: be, 
 tween the two equal ratios that compose the proper ':ion. 
 
 ThuP, 11)0 proportion exietinsf brtwoen 7, 21, 19, and 67 is ex- 
 pi^essed thus-? : 21 : : 19 : 67, and is read 7 is to 21 us 19 is to 67. 
 
 14. The two outer terms of a proportion are called thf 
 extremes, and the two intermediate terms the means. 
 
 Thus, in the above example 7 and 67 are the extromes, and ;» 
 and 19 ibe means. ' * 
 
 15. In every pioportion the product of the extremes )'« 
 equal to the product of the means. 
 
 us, in the following examples we have 
 9:54:: 2il2, and 9 x 12 = 54 x 2. 
 7: 21:: 19: 67, and 7x67 = 21x19. 
 16: 3.:I2:2^, and 16 X 2^= 3x12. 
 
 16. Since 1st term x 4:th term — 2d term x Zd term^ \{ 
 
 follows that the 4th term = ^^ ^'''''' ^ ^^ ^^^•^^. 
 
 1st term 
 
 That is, the 4th term of every proportion is found bj 
 multiplying together the 2d and 3d terms and dividing theb 
 product by the first term. 
 
 Example 1. — What is the fourth proportional to 7 11 
 and 23? ' • 
 
 OPERATION. 
 
 
 i ^'i7i3t aSuve %iii ivrm or ^m* =^ 
 
 11x28 
 
 7 36f 
 
SIMPLE PROPORTION. 
 
 i2ir 
 
 Example 2.—Find a fourth proportional to 24, 106, 
 
 OPERATION. 
 
 86 6 
 
 14 ; 106 : : 40 : Ans. Hence Ans = ^-5i^^ =, W xfP 
 
 24 '/i 
 
 r.86x6 = n6. ^ 
 
 FxAMPLE 8.— Find a fourth proportional to 80. V6 
 
 OPERATION. 
 
 88 83 
 
 80 : 76 : : ^^ • Ar^. Hence Ans. = ^2il? _ ??iL??_ 
 
 6 
 
 Exercise 67. 
 
 260|. 
 
 Fine* tile fou.-ih proportional 
 >• 7, 21, AiiJ 40. 
 
 Jl, 7, r,rcl46. 
 A 11, 3. and 17. 
 4. 9, 47, and 29. 
 f>. 0, 23, and 42. 
 
 6. Ill, 21, and 184. 
 
 7. 9, 10, and 11. 
 
 8. 13, 14, and 66. 
 0. 17J8, 109, nn.i 72. 
 
 10. 253, 16, and 11. 
 
 11. 9, 891, aid 100. 
 
 12. 9 da>8, 21 days, and |6160. 
 
 to the following numbers : 
 
 13. 11 IbB., 147 lbs., and i;i6 4«. 
 
 Hid. 
 
 14. 3 owt., 20 cwt., and $66-87. 
 16. 9 nii IcH, 17 milts, and 16 days. 
 
 16. 21 acrt•^', 47 acres, and 11 
 
 wks. 
 
 17. 17 bushels, 29 bushels, and 
 
 £6 7f^. 4d. 
 
 18. 211a(rcB,lf.cre,ai d|749fi-40. 
 
 19. 62 miles, 3 miloe, and $42140. 
 
 20. 7 months, 23 mouthe, and 
 
 156-70. 
 
 17. In the prec 'i.^ exercise the first three terms of 
 ihe proportion are g, en in their proper order, but very 
 jCenerally in proportion the pupil is required to mal<e 
 
 ft^k^'^'^"'^"^^ """'^^''* -^^ '*''^^^ ^ ^^^^'^ according to tne 
 
 i-ii 
 
 
lis 
 
 128 
 
 BIMPLK PROPORTION. 
 
 RULE. 
 
 I. Rfidi'ce the two nwnhn't which are of difcrent nam^i 
 from the ininwer to the loiieat dcnotninailon contained in 
 either of thcyn. 
 
 L\. Set the number which is of the same kind a.5 the 
 answer in the third plnce^ and^ when the answer is to be 
 greater than this third term, write the greater of the other 
 two numbers in the second place ; but when the answer i\ 
 to be less than the third term, write the smaller of the other 
 two numbers in the second place. 
 
 III. Multiply the second and third terms together^ and 
 divide their product by the first term. 
 
 Proof. — Multiply the answer by the first term, and the 
 product should be the same as that obtained by muUipLying 
 together the second and third terms. 
 
 Example 1. — If 11 men can mow a field in 23 days, in* 
 what time will 6 men mow it V 
 
 OPERATION. 
 
 5: 11:: 1^\Ans. 
 11 
 
 6)253 
 
 Here, as the answer must be days, 
 we place the 23 days as the 3d term. 
 Then, since it is evident that 5 men 
 „ will require more days to mov/ the 
 
 50Jdays. ggijj ^Yitm 11 men, that is, since the 
 
 answer must be greater than the third term, we place 11, 
 the greater of the two remaining numbers, as the second 
 term. 
 
 Example 2. — If 16 bushels of oats cost $6*70, what will 
 96 bushels cost ? 
 
 OPERATION. 
 
 6 
 
 /^ : ^^ : : |6-70 : Ans. 
 6 
 
 Here, since the answer must be 
 money, we put the IG-^O in the 
 
 third place; then, because the an- 
 
 $40"20 Ans. g^yer must evidently be greater than 
 the third teim, ^6 bushels costing more than 16 bushels, v/e 
 put 96, the 'greater of the two remaining numbers, in the 
 second place. Lastly, before applying the rule we cancel 
 by dividing the first and second terms by 16. * 
 
fniPLtt PROPORTION. 
 
 120 
 
 JJxAMPiK V.» vlf 9 acres of grass will pasture 21 cowi 
 aow many *iuw8 jill 6 acres pasture ? 
 
 OPERA'ilON. 
 
 2 « ol ^ ^^''® ^® P"* 21, the number of 
 
 ;^ : p * : ^/ : Ans. oows, in the third term, since tlie an- 
 
 '7x2;:= 14. Ans. ^^^^ '^ to be cowa; next, since the an- 
 
 sv/er will be less than the third terra, 6 
 Rcres pasturing fewer cows than 9 acres, we place 6, the 
 •mailer of the two remaining terms, in the second place. 
 
 Example 4.— If 6 lbs. 4 oz. 1 drs. Avoir, cost $169-40. 
 ^iiat will 1 lb. 11 oz. Avoir, cost? 
 
 OPERATION. 
 
 6 lbs. 4 OZ. 7 drs. = 1607 dra. 
 lib. 11 oz. = 432 drs. 
 1607: 432;: $169-40 :^re«. 
 432 
 
 33880 
 60820 
 67760 
 
 
 1607)73180-80($45-588. 
 6428 
 
 Ant, 
 
 8900 
 8036 
 
 
 865-8 
 803-5 
 
 
 62-30 
 48-21 
 
 » 
 
 ^ 14-090 
 12-856 
 
 
 1-234 
 
 1? -. -_.. 
 
 US. oa. pay lor 17 days' work, ibr 
 now many days will £9 lis. 4d. pay ? 
 
I 
 
 lao 
 
 SIMPLE rUOPORTION. 
 
 m 
 
 
 41 
 
 !? ■ 
 
 OPERATION. 
 
 £2 Oa. 8d. = fifiOd. 
 £i) 11 8. 4d. = 22960. 
 10 41 
 
 41 
 
 17 
 
 68 
 
 10)09 7_ 
 
 69/(y da3^ ^w* 
 
 Example 6. — If 3?- barrels of apples jiay lor '.*-,»> Y^Trg?) 
 els of wheat, how many bushels of wheat can be puvcnased 
 for 17^ barrels of apples? 
 
 OPKRATION. 
 
 ^:ln 
 
 Hence the ^?/s. 
 
 
 •'i I 
 
 ■ I I 
 
 ail N/ iu'i _L. aa 
 
 : Ans. 
 A ns. 
 
 Here, after making the st.itement, we reduce the te^na^ 
 to their equivalent improper fractions, ir^vert the first t^3rm 
 or divisor and connect it to the othflr twn J,7 th« '"•«^ "^ 
 multiplication. 
 
 Example 7. — If 4^ days' work coot 21J f^bHUngs, wliu. 
 will 17-^- days' work cost 5* 
 
 OPERATION. 
 
 4|: 17f,:: 21f :^7i5. 
 or 2j2. : laa : : i^ii . ^„s. 
 
 54 
 193 W 193 X 64 
 
 Hence ^WS. = ^r-r- ■>« — r 
 
 H 
 
 ^ 
 
 11 X 11 
 
 ^ 86-iV>s. 
 
 Exercise 58. 
 
 1. If 28 men h-' dig 27 acres in a week, hew Uiany acr"J» «<"-u 42 
 men dt^ 9the same epac^wf Uiue? 
 
Blltl'f.E r^:OI'ORTION. 
 
 131 
 
 What win 05 lbs. of f«uj?ar coRt if |1;10 pay for 13 lbs. ? 
 
 ITiiw ni!iny moi) would perform In 125 dnys a piece of work 
 wlih'l) 100 men can pcpforin In 14;') days? 
 
 If a ] c-rsoii ran Hiiish ajournoy in 100 d;iy.->, travolliiip 10 hours 
 per (I ly, liow iiiHiiy (l;iyn Avould lie tako - • ~- 
 elltd only 6 lioiira per day ? 
 
 to d J it If lie tiav- 
 
 ucr*t« ^'"'U 43. 
 
 5. 
 6. 
 
 7. 
 
 8. 
 
 9. 
 
 10. 
 11. 
 12. 
 
 13. 
 14. 
 15. 
 
 16. 
 
 17. 
 
 18. 
 
 19. 
 
 20. 
 
 21. 
 
 <?2. 
 23. 
 
 24. 
 
 "'. 
 
 2G. 
 
 If 10| yards of cloth cost |12-90, what will 4^'*^ yards cost? 
 
 A biuikriiprs efVect.M .nmoiintcd to ,<i7149, Avliicli paid his credit 
 ors 80 ccntB in the $ ; to what sum did Ids debts amount? 
 
 What will be the cost of 10714 feet of clear timber at '' per 
 thousaid? 
 
 What will bo the tax on $7149-70 at the rate of 1} cents on 
 
 thci$? 
 
 If -J of a person's Income is $7194-60, what is his income ? 
 
 What is the cont of 127 acres of land if 871 acres cost $8071-40? 
 
 If 702 yds. of muslin cost £48 2s. 4i 1., what will 540 yds. coat? 
 
 If a ship has Avatcr sufficient to last a crew of 35 men for d 
 
 months, iiow long will it last a en w of 20 men ? 
 Wliat is the tax on £1749 16s. 8Jd. at 3^. 4(1. in tlic £ ? 
 If 24 a. 1 r. 17 per. cost $76o'80, what will 7 a. 1 per. yds. cost? 
 
 If tiic railway fare for 100 miles is $3-75, what ouL'ht it to be 
 for63mil(.«8? 
 
 now miicli must l)e civen for 276 bushels of wheat at the rate 
 of $7-90 for 7 bnjhels? 
 
 A ba-.ikruprs debts amo.mt to $71911-40 and liis effects <o 
 $.^3009-80 ; how much can he pay in the $? 
 
 What will ho the cost of (Iraii'iiig 247 a. 1 r. 27 per. when 17 a. 
 1 r. 36 per. co.st £111 17s. 8d ? 
 
 If 16 barrels of flour can be bought for $9780, v.'hat must be 
 paid for 27 barrels? 
 
 Tlure are two numbers in the proportion of 7 to 8 and the 
 
 larger number is 291 ; what is the smalkT? 
 What will be the cost of 71 cwt. 1 qr. 17 lbs., if $-21-60 pay for 
 
 3 cwt. 2 qr. 26 lbs. ? (Allowing 112 lbs. to tl;e cwt.) 
 
 If 3f yds. of lir.en cost $2-21, what will 7f yds. cost ? 
 
 A b(6jeged town containing 15000 i!,habt;iiit8 has provisi'T.a 
 sutHcient to hiBt 5 weekn ; how long will the prov^sior.a last 
 if 7000 of the inhabitants be sent awav ? 
 
 If a stick 7 feet liiirh cast a shadow 5 feel in lor-trth, Avhnt is the 
 heisilit of a spire that cat^ts a shadow 120 fi ot in le; gth ? 
 
 How f;u- c.ui a man travel in 27 dav.s at the r.ate of 149 miles in 
 
 4 d:'.vs. ? 
 
 If a person steps over 4 vc-h. m 5 paces, to how many yarda 
 will 729 of his paces U equal i ^ / ^ 
 
 111 
 
COMPOUND PROPORTION. 
 
 There arc two numbers in the proportion of 6 to 11 and tno 
 
 Bmaller is 29, what is the larger ? 
 
 28. At 29 cents per lb., v.'hat Avill bo the cost of 174 \hc. of raislnsl 
 
 29. ITow much land at $4-75 i)er ncre must be given in exchange for 
 
 243 acres at $3-60 per acre ? 
 
 30. If 4^ lbs. of nutmegs cost jBJ?, what will 27§ lbs., cost ? 
 
 31. If G^ acres of land cost $Q1^\, for how much land will |23f, 
 
 pay ? 
 
 32. If 4-32 lbs. of coffee cost $1-17, what will 9-78 Ibe. cost? 
 
 83. What will 9^- lbs. of spice cost when $17* pays for 19-87 lbs. ? 
 
 M. If 11 cows make 29 pounds of butter per week, how much may 
 be expected from 27 cows ? 
 
 35. If 7 men put up 200 perches of fencing in 2 days, how long 
 would they take to put up 900 perches ? 
 
 86. If SlOO stock is worth $95/,-, how much can be purchased fo> 
 } $100 ? 
 
 37. Wliat will 16 lbs. 4 oz. 2 dwt. cost if 11 oz. 11 dwt. 11 grs. cosi 
 
 $47-90? 
 
 38. If the rent of 73 a. 14 per. be X17 48. 9d., what will be the rent 
 
 of 33 a. 1 r. 23 pir. ? 
 
 39. If h of f- of J of 171 lbs. cost | of ^\ of $38-50, what wih lsj of 
 
 I of I of Si lis. cost ? 
 
 40. Two rumbeis are to one another as 11 to 29, and the greater 
 
 of the two is 107, what Is the smaller ? 
 
 COMPOUND PROPORTION. 
 
 !. =5f 
 
 18. Compound Proportion is an equality between a 
 compound ratio and a simple ratio. 
 
 19. Compound Proportion is also called the Double 
 Rule of Three, because all questions in compound proper- 
 tion, when worked by simple proportion, i "iquire two or 
 more statements. 
 
 20. In questions in Compound Proportion, ^,ive or more 
 terms are given to find another term; that i,\ three or 
 more ratios are given, one of which is imperfect M- wants a 
 term, 
 
 fil. Questions i^ Coi^ound Proportioa are scVed by 
 ^he follow >" — 
 
 I ! 
 
to 11 and tn» 
 
 COMPOCJSTD PEOPOKTION. 
 BtJLa 
 
 133 
 
 inMon as thet/HrduTm. """"■■ '" "" '™« *»<»»- 
 
 mucf ^17 m»'e?ra ^3 d^-l^^ ""> ^"^ «'«-^3. how 
 
 OPERATION. 
 
 tO 
 1 
 
 17 
 3 
 
 :: $78-43 
 17 
 
 Ans. 
 
 54901 
 _7843 
 
 1333-31 
 3 
 
 Here we set $78-43 in the 3d 
 term, because the answer is to 
 be money. X^xt we take the 
 other terms in pairs, first for in- 
 stance, the numbers referrino- to 
 men and of these we place°the 
 
 V0)3999-93($5Y-1418?r^ /^^ . ^^^^"^ Place, be- 
 350 ^* ^418 cause leaving the time oui of 
 
 ■4^9 tr merarihr' ''^^'^.'"° '-'"'y 
 
 uie men and the wages, it is evi- 
 dent that the answer must be 
 greater than the third term. 
 Again, for a similar reisoii, leav- 
 mg the men out of consideration 
 weplace 3, the least of the re' 
 niainmg terms, in the second 
 place. Lastly, we multiply to- 
 gether the $78-43, the 17, ..nd 
 tne 6, and divide thoir r>iod'=-^ 
 by 70, which is tlie product of 
 the two lirst-terms. 
 
 ■f 
 
STATEMENT. 
 
 405 : 522 
 
 8 : 27 
 
 COMPOUND PROPORTION. 
 
 Example 2.~If 20 men can build 405 yds. of wall in 2" 
 days, how many men will it require to build 522 yds. ii> ■ 
 days ? 
 
 OIERATION. 
 
 :: 20 : Ans, 
 
 CANCELLED. 
 ff 29 
 
 ^ 3 
 
 Then 8 x 29 = 87. Ans. 
 
 Here, since the answer is to be men, we place 20 men 
 in the third term. Next, we take the two numbers referring 
 to length of wall, and, leaving the time out of consideration, 
 arrange these as in simple proportion. Afterwards, we take 
 tlie two numbers referring to time, and, leaving length of 
 wall out of consideration, also arrange these as in simple 
 proportion. 
 
 Exercise 59. 
 
 1\ men can (\\s, 7 acres in 12 days, how many acres can 17 men 
 dig in 22 days? 
 
 2. I-:a family of 11 people 8]iei!d $490 in 7 months, how much will 
 a family of 7 persons spend in 16 months? 
 
 3 If 1"'0 i-eamfl of paper make 5000 copit s of a boolc of 15 sheets, 
 how m-ich paper will be required for 4000 copies of a book or 
 11 B^eete ? 
 
 4. If 21 men can mow 93 acres in 5 days, how long will 7 men re- 
 quire lo mow IG a. 3 r. '20 ] er. ? 
 
 6 If 50 men c.-\!i di<,' an excavation in 7 days, working 11 hours per 
 day, how many day^^ will 24 men require when they work 
 o.iy 8 liou'-s prr da*;,? 
 
 6. If $750 {Tfvii! $204 in 23 months, how much will $467 gain in 7 
 months? 
 
 7 If a wall 79 feet Icnc;- * foet hiurh, and 2 fret thick be built by 17 
 men in 11 davs wl.at lentr.b of wall 5 feet high ».,d 3 ieei 
 tliick can be buih by M men in 83 days? 
 
 8 If 3 me'i can cradle 34 ncvs of wbe;it in 5 <l;>yp, hinv many men 
 will i't req-ure to cradle ^'^ a. 32 per. in 6 days " 
 
 8 If a ditch 36't<-et loi-s:, 8 f<jet d«iep. nnd C fr-et Vif'o V.t ■''■lu ^'y '''-' 
 men in 4 davs, in whnt timt wiT. 4S 'n-^i di*< a i>"*.' kH tel 
 long, 6 feet deep, and 3 feet wi'^el 
 
COMPOUND PROPORTION. 
 
 136 
 
 -LED. 
 
 29 
 
 n 
 
 1 
 
 3 
 
 
 i8 
 
 7. Ans. 
 
 \ $467 gain in 7 
 
 hinv many men 
 
 10. If .*' men can saw 90 cords of wood in 6 days ^.nen the dava 
 
 are 9 liours long, how many coi-.l^ can 8 me . eaw in 30 days 
 when Ihe dayrt arc 12 uoura long ? ^ 
 
 11. If 5 ompoHitora in 10 dayh, each 11 J-u.e lontr, cniT composi' 25 
 
 Hh(-et8 o! 24 pagrs ui each sncft, 44 lin, s i., a pnyv, a,;,) 40 
 letters i,, a lin ■, m how many days, eacj. 10 hours luiiu, can 9 
 compos.tors compose 36 sheets of 16 pages to a sheet, 50 lines 
 to a page, and 46 letters to a line ? i ^ co 
 
 12. If 243 men in 5* days of 11 honrs each, dig a trench of 7 degrees 
 
 of hardncHs, 232| yards long, 3j yards wide, ai.d 24 fards 
 deep 111 how many days, of 9 hours eacli, will 24 nun diij a 
 trench of 4 degrees of hardness, aavt yards long, 53 yard* 
 wide, auaSi yards deep? t.i « ^". «• 
 
 13. If .60 men can dig a trencli 500 feot long, 36 wide, and 40 deep, 
 
 in 24 days ot 8 hour.s eacli, how many men wiU be required 
 to d g a ticich 55U feet long, 68 wide, and 90 deep, in 44 days 
 ol 9 liours each ? •' 
 
 14. If 9 Ib^^re OZ.4 dwt. of silver make 5 dozen forkp each worth 
 
 r;,V/-f?n'n'"^-.i"""^,3;'"'^'''.''''''^' '^'^^■^^^ 7^- S*'-'m *ii ^e 'I'ade 
 out ot 11 lbs. 11 oz. 17 grs. ? 
 
 15. If 279 liushels of potatoes feed 4 cows for 60 dnya, how many 
 
 bushels will be required to feed 27 cows for 200 days ? 
 
 16. If 7-3 acres of land arc trenched by 23 men in 27-9 days, work- 
 
 ing 11-4 liours per day, how many acres of land may bo 
 trenched by 4S men in 165 days when they woik 9-riioura 
 
 pel* uftV ? 
 
 17 If -he wages of 11 men for 11 days bo $111-11, what will be the 
 wiges ot 10 men for 16 day.s ? 
 
 18. li\ Clock of marble 8 feet long 4 feet wide, and 2 feet thick, 
 
 weife I 85o0 lbs., Avhat will be tlie weight of another block o/ 
 marbit> 6 teet long, 6 feet wide, and 4 feet thic' \ 
 
 19. If a rectaixgular vat 8 feet square and 2\ feet deep hold 10000 
 
 ;in ^^'"i"'"' l^ow "lany pounds of water will a rectangular 
 vat 10 feet ^ong, 8 feet wide, and 2 feot deep, contain ? 
 
 ^^" -^^ Jf.»?;"^ T"' T^^^'f, 2.^ ^^^- ""' ^^""«^ li yds. Wide, how many 
 Py'S widT? required to make 17| yds! of flannel 
 
 21. If 2U43» yds. of eloih If yds. wide make coats for a regiment of 
 
 Bo.diers containing 847 men, how much cloth 24 yds. wide 
 will be required to make coats for another regiment which 
 contains 981 men ? * 
 
 22. If 8 men. can cradle U7 acres in 4 days of 7| honrs each, ho\r 
 
 many acres will 14 men cradle in 3i days of 9^^ hours each I 
 
 23 If 1450 gain *24 in 12 montns. what principal will gain $97 in 4 
 niontns t ^ » .» 
 
 ^ ^^iSiT'*"''^- """,' ^4 J"?hel9 Of oats in 9 days, h-w many bushels 
 ot oats will last 29 norses 27 aay* ? 
 
 ■ I 
 
m 
 
 li't 
 
 h H - 
 
 ''.SC PKACTICK 
 
 SECTION VI. 
 
 PllACTICE. 
 
 1. Practice is a short method of findin<* tht^ vnlim « 
 
 2. An Ahquot Part is an exact or even part. 
 
 & momh; t^X^^t^l^l^^^^^-> 2 months. U mo^h^ 
 . TABLE OF ALIQUOT PARTS. 
 
 Parts of ill. 
 
 Parts of a 
 month. 
 
 Parts of £1. 
 
 — 1.16 days=: -J 
 
 = i 
 
 50 cts. 
 
 m = i'lO 
 25 
 
 20 
 
 H 
 H 
 
 5 
 4 
 J 
 
 =iV 
 
 • 16 
 
 n 
 
 5 
 3 
 2 
 1 
 
 =- i 
 
 — .1. 
 
 — 1 
 
 —-1. 
 
 — 1 5 
 
 lOs. 
 6s. 8d. = 
 
 6s. =r 
 
 4s. = 
 8s. 4d. = 
 2s. 6d. =r 
 
 2s. ^ 
 
 
 1 
 
 1 
 
 c. 
 
 J 
 
 ,^0 
 
 Parts 
 Is. 
 
 6d. 
 
 4d. 
 
 3d. 
 
 2d. 
 
 Ud. 
 
 Id. 
 
 of 
 
 i 
 i 
 
 »Parts of a 
 
 cwt.* of 113 
 
 lbs. 
 
 
 Note.— The 
 aliquot parts 
 of a year arc 
 the same as 
 those of a 
 shilling.— See 
 4th coiumii. 
 
 Is. 8d. =-1, 
 Is. 4d. = 1; 
 Is. 3d. rr-jl. 
 Is. =,\ 
 
 56 Ibs.= ^ 
 28 1bs.=r I 
 
 16 1bs.=r I 
 
 14 lbs. = i 
 
 8 1bs.r^-,i- 
 7 lbs.:^-,Jj 
 
 Parts of a qr 
 of 28 lbs 
 
 14 1b8.r^ I 
 1 lbs.rx I 
 
 3|lbs.~ ^ 
 
 If lbs.rt-,ig 
 
 •Although we allow but 100 lbs. to the cwt in Tnnnrl^ u ;. 
 Often nec.v.sary to make calculations with the oVdcwVofU'/L 
 
 ^j^i-^'itffi-^^^i^^i^^.i^^^ the latter is sJui ",? c'c;:;:n;;?,VJ'i5 
 
 The aliquot pa.^^SThen;w%^4?'of^ Union,, fco. 
 
 aliquot parts of |l. ^ ^^ ^^^- ^^*^ ^^^ ^^e af tho 
 
one af tho 
 
 PRAOTICK. 
 
 137 
 
 ^^BxAMPLE l.-What i3 the cost of 47^cows at |33-40 
 
 OPEIIATION. 
 
 $33-40 X 47 = $1569-80. Atis. 
 
 ll-S^O^pTlb' '-''''''' '' ^^^ -^- «^ 16^8 lbs. of tea at 
 
 fiOcts.U ,$1,^8= value of 16V8@$1. 
 
 I I 839= " u ^ .g^ 
 
 $2517 = 
 
 (( 
 
 @ $1-60 
 ^^EXAHP.. 3._Fi„d the price of 21C4 articles at $1 87* 
 
 OPERATION. 
 60 CtS. 1 1 1 12164 = nricP of 91 aA „ *. , _ 
 
 25 i 1082 ^P"'?,°^ 2164 articles® 
 
 ^2i U 541 = u u „ @ 
 
 I l___270-60= i« i» u ® 
 
 $4057-5"0= " «t « ^:: 
 
 @$1-87| " 
 
 each. 
 
 •50 " 
 ■25 " 
 I2i " 
 
 each '""'" *-^"«' *« ™>«e of 078 sheep at $3-79 
 
 OPERATION. 
 
 iOctllf *"'! = val«e of 978 rf,cep @ SI 
 
 each. 
 
 12984 r= 
 
 489 =: 
 
 195-60 = 
 
 48 '90 = 
 
 89'12 = 
 
 13706-62 = 
 
 (t 
 
 (t 
 
 <( 
 
 (( 
 (I 
 
 (( 
 
 @ |8 
 
 @ 
 
 @ 
 
 @ $3-79 
 
 •60 
 •20 
 •05 
 •04 
 
 u 
 
 (I 
 
 NoTK^Tn 71 ^^ '"^ '** 
 
 value of\.no a? c?e"fB'S'ven1o fl"'*.*?.*^^ preceding, .vhere tho 
 «HJcl.s of the «ame donomf". f„n "thi^f °5 *» ^«'-^'^^" number of 
 of procoedliiff le to mn /?„]♦! "' t'^® shortest and eimi.lest mn^L 
 oun>b.r of ar^.icle« Thi^L^^^ ^'^'^ °^ °"« ^^ticle by^ the ^veS 
 
 
 Ex 
 
 1678. 
 
 A1.PLE 8.-Jns. ';=$1.87i x 21S4. 
 
 'LK4.-'.4w«. =:|3'V9 
 
 14 • 
 
 if 
 
 X 97SL 
 
138 
 
 PRACTICE. 
 
 Example «.-— Find the value of 1679 lbs. 14 oz. 12 dra. 
 Avoir, at $109-40 per lb. 
 
 8 07. 
 
 2oz. 
 
 I.ll-H. 
 
 tf ird. , i 
 
 OPKRATION. 
 
 nC>Q-40 X 1079 = $2S44l.'2-60 - valiio of 107P 11)P 
 
 " 84-7() ]' lf)S-lC4- " 
 
 f^':l%[ 1284578^0]!= » 
 
 6-29» 
 2-()4G 
 
 $liJ0-164 
 
 14 07.. 12 dra 
 
 iC79 Ibe. 14 oz. 12 dm, 
 
 Example 6.— Allowing' 112 lbs. to the cwt, find thi» 
 value of 229 cwt. 3 qrs. 17 lbs. of tallow at $6*20 per cwt. 
 
 r 'I 
 
 pi 
 Pi-', 
 
 2qr. 
 
 Iqr. 
 
 141 ). 
 
 2 11). 
 
 lib. 
 
 OPERATION. 
 i?'0-20 X 229 = $1419-80 = vnluo of 229 cwt. 
 
 li-10 ' = val ue of 2 qr. ^'""^ = " " ^qr . 17jb, 
 
 1-55 = " ofl qr. |1425'a9= " 229 cwt. 3 qr. 171b 
 
 •775 = " 
 1107 = " 
 •0563 = «' 
 
 of 14 lb. 
 of 21b. 
 of 1 lb. 
 
 ^5^591 
 
 Example 7. — What is the value of 29 lbs. 7 dwt. 10 gra 
 of gold at £3 17s. ll^d. per oz. ? 
 
 i« 
 
 operation. 
 29 Ibp. = S48 ounces. 
 
 6 dwt. 
 
 2i dwt. 
 6 grs. 
 Igr. 
 
 J I £3 17e. Hid. x 348 = jtlSSe 28. 3d. = value of 29 Iba 
 
 i. 
 
 1 
 
 19s. 5|f 
 
 9s. 8U 
 
 = value of 5 dwt. 
 
 -- " 2 dwt. 12 grs. 
 
 6 grs. 
 
 1 gr. 
 
 it 
 
 £1 10s. 41U 
 Then £1356 2s. 8d. 
 
 1 10s. 4||Sd. = 
 
 = " 7 dwt. 19 grs. 
 = value of 2„ ibs. 
 
 (( 
 
 7 dwt. 19 grs. 
 
 £1357. 12s. 7|Iutl- = " 29 lbs. 7 dwt. 19 grs. 
 
 Example 8. — What is the price o 7149 tons of hay al 
 £2 13s. 9d. per ton J 
 
oz. 12 dra 
 
 3 of 29 Iba 
 
 lOs. 
 
 J! £7149 
 2 
 
 PRACTIOBJ. 
 
 OPERATION. 
 : valuo of 7149 tons @jGl 
 
 ' £14298 = 
 
 2h. 6.1. f a574109. = 
 
 Irt. 3il. i 8y;i12rt. fi'l. = 
 
 I I 44tV1fJs. yd. = 
 
 u 
 u 
 
 II 
 
 (< 
 (( 
 II 
 
 II 
 
 $/; 10-1. 
 
 (</^ 2ri. 6d. 
 
 189 
 
 per toa 
 
 u 
 
 II 
 (I 
 (t 
 
 K 
 
 £19212188. 9.1. = .„,„,. 
 
 .E-KAMPLE 9.-Fiiidtlie price onU\jfmi^s i^' \^d at 
 |)^/"4d pur acre. 
 
 OPKRATION. 
 
 tf'tl ^ ^^^^ = $196097-()7 = value of 1 149 acres. 
 
 10-28f = " 
 
 (I 
 
 a nf 
 
 oi an a. 
 
 7. 217} at i»914-70. 
 
 8. 618; at $42-71}. 
 
 9. 907 { J at $16-93. 
 
 10. ?04i at £2 78. 8id. 
 
 11. 604f nt £93 13h. 7d. 
 
 12. 904i5at£16 4s. 9J(i 
 
 %VM\()i"i^z= " " 7149^ acTca," 
 
 ^. , , Exercise GO. 
 
 Find tho value of— 
 
 1. 229 at $2-75. 
 
 2. 743 at $;3-81. 
 8. 7114at$97-86J. 
 4. 213 at £2 16^. 4a. 
 6. S21 at £9 1b. IJil. 
 6- 7147 at £12 123. 2}(L 
 
 13. 017 lbs. 4 (jz. Avoir, at $01-43 per lb. 
 
 14. 2171 a. 2 r. 17 per. at $970 per acre. 
 
 15. 114 Imsli. 1 pk. 1 gal. 1 qt. at 37} cents per bushel 
 16 209 lbs. 7 dwt. 16 irrs. at $1-71 per oz. 
 
 17. 614 yds. 2 qrs. 1 iia. at $2-73 per yd. 
 
 18. 16 a. 1 r. 4 per. 7 yd.^. at £2 178. 6d. per acre. 
 
 19. 29 wks. 4 days 11 h. at $7-40 per week. 
 
 20. 167 miles 7 fur. 8 per. at £9 33. 6d. per mile. 
 
 21. 217 lbs. 4 oz. 6 drs. 2 scr. at £9 68. 7d. per oz. 
 
 22. 9167 sheep at £1 3-i. 61. each. 
 
 23. 21791 burihels of wheat at $l-40 per bupbel. 
 
 24. 1673i\ sq yds.of painting at 2.-i. S^d. per sq. yd. 
 26. 437 a. 9 per. 7 yds. of laud at $2140 r)er acre. 
 
 26. 97 cub. yd^. 4 ft. at $0-73 per cub. yd. 
 
 27. 614v\ cwt. of iron at $1-23 pi r cwt. 
 
 28. 23 lbs. 4 oz. 7 dwt. 11 grs. at Hid. per dwt. 
 
 29. 216 - -- - 
 
 t i 
 
 
140 
 
 PKRCSNTAGB. 
 
 SECTION VII. 
 
 PERCENTAGE, COMMISSION, BROKERAGE 
 INSURANCE, STOCKS. 
 
 s;i 
 
 
 hki' 
 
 :•:!' 
 
 h 
 
 1 
 
 PERCENTAGE. 
 
 1. Percentage or Per Cent, means a certain allowantm 
 or rate per 100. Per Cent, is a contraction of the Latin 
 per centum, and means, " by the hundred." 
 
 Thfis, if a person purchase 100 barrels of flour and some of 
 them become wortlilesH through being damaged, he is said to hav« 
 lost 1 per cent., 2 per cent., 3 per cent., 7 per cent., 15 per cent., or 
 29 per cent., dtc, of nis flour, accord! ug as his loss is 1, 2, 3, 7, 15, 
 or 29 barrels. 
 
 :a. "When the rate pei cent, is given, the rate per unit is 
 found by dividing by 100, or what amounts to the same 
 thing, removing the decimal point ♦wo places to the left ia 
 the number that expresses the rate per cent. 
 
 Example 1. — What rate per unit is equivalent to 6 pe» 
 cent. ? 
 
 Ans. 6 -r- 100 = -05. 
 
 Example 2. — ^What rate per unit is equivaic:^t to 8| per 
 cent. ? 
 
 Ans. 8f -4- 100 = S*16 -^ 100 = -OSYS. 
 
 Example 3.- -What rate per unit is equivalent to 23 per 
 cent. ? 
 
 Ans. 23 -f- 100 = '28. 
 
 Example 4. — ^What rate per cent, is equivalent to '248 
 per unit ? 
 
 Ans. -243 x 100 = 24,^-. 
 
 Example 5.— -What rate per cent, is equivalent t* '010 
 per unit/ 
 
 Ans. '016 X 100 = V-5 = '7K 
 
PERCENTAGE. 
 
 141 
 
 ».,r c^ntT'^ 6— Wivftc rate per unit is en; ivalent to 12-6J 
 Arts. 12-G3 -i- 100 = -1263. 
 
 Exercise 
 
 W*iat me per unit Is equivalent 
 1 e per cent. ? 4i per cent. ? i 6. 
 ? S-7 per cent. ? 29^ per cent. ? ( 7' 
 ^. *J-2 per cent. ? 8} per cent. ? 
 C ni per cent.? 147 per cent.? 
 J- OJ per cent, f 63g*j per cent. ? 
 
 8. 
 
 9. 
 
 10. 
 
 What rate per cent. Is equivalent 
 
 IX -07 p*er unit? -61 per unit? 
 <2. 1-47 per unit ? 056 per unit ? 
 )3. -8725 per unit ? 2*2 per unit ? 
 
 14. ill per unit? 1-107 per unit ? 
 
 15. 1-06 per unit? 007 per unit ? 
 
 16. 675 per unit ? 035 per unit ? 
 
 17. 
 
 18. 
 
 19. 
 20. 
 
 61, 
 
 to— 
 
 8 per cent. ? | per cent. ? 
 . 3 percent.? 2| per cent.? 
 
 i percent. ?9f percent.? 
 
 16-2 per cent. ? -98 per cent.f 
 
 147-2 per cent.? 2612 per 
 cent, t 
 
 to— 
 
 •014 per unit? 016 per unit? 
 
 •0095 per unit? 1-217 nor 
 
 unit? 
 
 •00125 per unit? -135 per 
 
 unit? * 
 
 •0005 per unit? -2775 ty^r 
 unit ? ^ 
 
 /! 
 
 3. To fi; 'T the percentage on any given number:— 
 
 RULE. 
 
 kvi^aU^^^ ^^' ^''''' '''''^^''' ^y ^^' ^^'^ J^^'' ^«*'^ ^^?rt,,e^ 
 Example I.— Whax is 16 per cent, of $6'74 ? 
 
 OPERATION. 
 
 16prT'cent. = -16 per unit. 
 %^U X -16 = $107-84. Ans. 
 ^Example 2.^What is 7 per cent, of 8473 acres of 
 
 ftPWD ATTr»»T 
 — . . ,.j.i,, 
 
 ^"^ V^sof il'^'" *""** - O'^^, '• 17 n«r. 18 yds, 1 
 
 3? f t! i 
 
I 
 
 
 liii 
 
 142 
 
 PEU CENT AGE. 
 
 Example ?. — What is 11 per cent, of 947 bv ihwls o} 
 apples ? 
 
 OPERATION. 
 
 fl47 X -11 = 104-17 bush, Ans, 
 £xAMPi.£ 'k — How much is 23 per cent, of $61-t /"b**^ 
 
 OPERATION. 
 
 23 per cent. = -23 per unit. 
 10147-80 X -23 = $1413-994 An{ 
 
 EXEIICISE 62. ■ • 
 
 1. How much Is 27 per cent, of $00?0-80 
 
 2. Wliat is 871 per cent, of ijl'234? 
 
 3. What is 6i per cent, of $89-40? 
 
 How much IB ITi percent, of $2998-4() 
 
 What is 8i per cent, of 204 a, 2 r. 14 per. 
 
 How much is '7 per cent, of 29 bush. 2 pks. • 
 
 What is -72 per cent, of 429 Ibf. 11 oz. 6 dwtf 
 
 Wliat is 15 per cent, of 227 weeks 4 days 11 hourf 
 
 What is 6 per cent, of £93 14,s. Tid. ? 
 
 What is 29 per cent, of $2947-40? ■ 
 
 From 10 per cent, of $294 take 29 per cent, of $1^917. 
 
 12, Add together 7 per cent, of $94 80, 11 per cent, of $112*^ -utC VJk 
 per cent, of $l'29G-42. 
 
 A person purchases ahouse for $7429 upon tlic foPowinQ: acree- 
 ment :— He is to p;>y 15 per cont. of the pm hase money 
 down, 17 per cent, in 6 months, 29 per cent, in 15 montlis, ft 
 per ceiit, in 20 months, and the balance at the expiration of 
 two years ; what are his several payments, upon the suppo- 
 Bition that he pays r.o interest? 
 
 A firmer works 227 acres of land, which he crops as follows :— 
 20 per cent, in wheat, 18 per cent, in e^rass, 17 per cent, in 
 peas, 19 per cent, in oatt*, ;' ^ " per cent, in root crops, the 
 rest beini? fallow ; wh.at nui of acro-s does he sow to each 
 
 crop and hpwmuch is in fan-jw ? 
 
 A res:iment went into the field 1147 stronar, and after the battle 
 it was found that 23 per cent, had been l<:illed or wounded, 
 and 7 per cent, taken prisoners -what was the number killed 
 or wounded, and what the Qumber taken prlsonerd? 
 
 4. 
 
 5. 
 
 6. 
 
 7. 
 
 8 
 
 9, 
 10. 
 11, 
 
 13 
 
 14 
 
 16. 
 
COMMISSION AND BROKERAGE. 
 
 COMMISSION AKD BROKER.\aE. 
 
 143 
 
 6. To compute comm:^ion or bvokeragc :— 
 
 iec£l^^^ '^^''''^ ^"^^'^^ ^y '^^ r.tc^crunite.presseA 
 ?er ceitt ?'^ 1—WhP.t is Cie commission on $749-40 at 28 
 
 OPERATION. 
 
 18 per cent. = -18 per unit. 
 
 $749-40 X -38 =:$184-b92. Arts. 
 
 rr cemT"" ^-^^^'^^ ^' ^^»« commiesion on $198-87 at 22f 
 
 OPEFUTION. 
 
 22| per cent. = -2275 per unit. 
 $198-37 x-2275 = $45-129175. Ans. 
 
 ^I^RM^i'^ S'—A broker purchases stock to the amount 
 ^^ W67 30; what is his brokerage at the rate oi^^, 
 
 OPERATION. 
 
 2^ per cent. -: -02125 per unit 
 $9867-30 X •OJ125 = $209-680.l25. Am. 
 
 • I ^""l \' T "^"^'"'««^"» «' ^76-80 at ^ per cent ? 
 A wnat la the commissioD ox |)i6-80 at H per e«jsi f 
 
 
 s>;4ii8 
 
I|m»«, 
 
 n 
 
 
 144 
 
 INBCRANCS. 
 
 8. What is tlio brokerage on $10800 at U per cent.! 
 
 4. What iB the hrok.riyo on ♦SSTT-OO at SJ per cent.! 
 , Wliat Is the brokcrnpe on $078-90 at 5 per cvnt. t 
 
 6. What iB tho conimiHfllon on ^(ilWoO at Ff per cent, f 
 7 What Is the conimiH^iun on $47'80 at 26 rer cv.il. ? 
 
 5. To what does tho brokerage on $7654i)2 umoiait at 4i per 
 
 cent. ? 
 g. To wliat docs the comrAlsBlon on $23456 amount at 28 per 
 cent. 1 
 
 el •, what 
 vll2230 ; 
 
 la What is the commiBsion on $655-65 at 18f per cent, f 
 11 An acent BelU Cl7 huHhelrt of wheat at $1-70 per b' 
 
 * l8 hlrt commiB8ion at V2\ per cent. 1 
 12. A oonnnir^sion merchant kcII.^ uoods to the amount 
 
 what is hifl commisriion at S^i pi-r cent, f 
 13 A broker purchanefl ^toek to the amount of $8765-40-, what i^ 
 ^* his brokerage at i',J rer cent.? u * . v,, 
 
 14. An atrent purchnPCH silk« to the amount of $7800 ; what 1« hii 
 
 commission at 7 per cent.? 
 16 An agent collects debts to the amount of $90780 ; wliat is hi* 
 
 commisBion at 15 per cent. ? 
 13 A commiBsion merchant Pellfl 7400 barrels of Oour ft |7-«H 
 
 per barrel ; to what does his commlsBion amount at 8i pel 
 
 17. An ajrent sells a farm for $7450 ; what is his commission at 2\ 
 per cent.? «.^-^^^ i * i. 
 
 18 A broker negotiates a mortcfnge for the sum of $1140 ; what ia 
 his brokerage at 3f per cent, i 
 
 INSURANCE. 
 
 7 Insurance is a written agreement by which an indi- 
 vidual or an incorporated company becomes bound, .mcon. 
 fiideration of a certain sum paid in advance, to exempt tlie 
 owners of certain kinds of property, as houses, household 
 furniture, merchandise, ships, &c., from loss by fire, ship, 
 wreck, or other calamity. 
 
 8. The Wr'Men Instrument, or contract between the 
 parties, is called a Policy of Insurance. 
 
 g rrun s,,,r. nnhl for the insurance is called the Premium, 
 and is iJsmary a'certai^ p«r O^t. o?^ the sum lor wmett loe 
 property is iiiioured. 
 
 8. 
 lO. 
 
INSURANCE. 
 
 Uirio. '^ ^^^ '"^* y^'^^, or o.iKT epeciiied 
 
 Jou^eri;rn!n;;n?.?^^r";4"rs "f;,';"';"!"^;^o"«e«. stores, ^0. 
 tencneut. Vesad. are iu^urSir ^1,:;'^;^^ -l^^I.^!!^" ^^ 
 
 fortu^voyagoorthiycur. 
 
 flroodg, 
 lum 
 tho 
 
 11. The premium to Ik: paid ol a nolinv of .• 
 ^puted l;v tho rnlln.v,M„ ^ ^ ** P^"^y 01 » 
 
 cor^puted L7 the followin*^ 
 
 o 
 
 insurpjice is 
 
 ; between the 
 
 OPLRATIOX. 
 
 li per cent. = -0175 oer unit i « la . . , 
 
 for insurance on each $'. * '' ^* ''"*^ '' *^« «^^rg« 
 Then 17480 x '0175 = |i 30-90. Ans, 
 
 Exercise G4. 
 
 Compute tho insuranoo on— 
 
 1. $789-46 at 2^ per cent. 
 
 2. f8l67-50 at 2^ per cent. 
 
 3. $8900 at 3^ per cent. 
 
 4. $8740 at f per cent. 
 6. $1888 at I j,er cent. 
 
 6. $11247-60 at U per cent. 
 
 10 
 

 H 
 
 
 I 
 
 '■ a; 
 
 no 
 
 STOGES. " 
 
 ft li. 
 
 BUYING AND SEJuLIEG SlOCivS. 
 
 12. Stock is a term used to denote the Capital of mon- 
 eyed institutions, as Banks, llailroad Companies, Gas Com- 
 panies, Insurance Companies, Manufactories, &c. 
 
 13. Stock is usually divided into portions of $100 ot 
 £100 each, called shares^ and the ditt'eront individuals own- 
 Ixig these are called shareholders or stockholders. 
 
 14. The nominal or par value of a share is its originu} 
 cost or valuation. 
 
 15. The market or real value of a share is the sum for 
 which it can be sold. 
 
 16. The rise and fall in the value of stock is reckon*. I 
 at a certain per cent, on its nominal or par value. 
 
 17. When stocks sell for their original cost or valuatioi^^ 
 they are said to be at par ; when they sell for more tha^ 
 their original valdation, they are said to be at a premium oi 
 adva7U'e, or above par ; when they do not bring their orig. 
 iual cost or valuation, they are said to be at a discount^ or 
 below par. 
 
 Note.— Par is a Latin word, and means equal or a state of 
 equality. Stock is at par when a hundred-dollar share fills ^or 
 |l00 ; it is above par when it brings more than |100, and below par 
 when it will not bring as much as $100. 
 
 MENTAL EXERCISE. 
 
 1. Wlien Piock io selling at a premium of 17 per cent., what Is |X 
 
 Slock worth in money ? 
 
 $100 stoik —^117 money, therefore |1 stock = $1-17 money. Anth 
 
 2. "Wnon Ptoctr la selling at a discount of 9 per cent., what is tha 
 
 worth of $1 stock? 
 
 |100 stock — $91 money, therefore |1 stock = $.y'»l money. Ana. 
 
 & When stock ,i8 -i per oent. above par, wha'; is the value of $J 
 
 stock? 
 
 |100 stock =|104-i>a »OQiQ^tk««forB tl atotok == tlOi& mouey- 
 
 k- "... — -■^. 
 
STOCKS. j^y 
 
 of iVS^f ^^ " i^'-^"^^"- o^^4* l'--cent.. What ia the wortl. 
 ''■ ''1% Sr ""' '^ ^^*°°-^ '^^ 2 per cent., what Is the worth 
 
 Of slo^kll^''^ ""^^^ '""^ '""'^ ^^ I^^^'^ ^«r a gi 
 
 given amount 
 
 RULE. 
 
 purchL^l^^^ ^^^^* ^"^^^^^ of «tock a given sum wUl 
 
 RULE. 
 
 Divide the given sum by the worth o/$l stocJc. 
 i^XAMPLE l.—What is the woith of ft'7-iQ.cn * i , 
 It IS selling at a premium of srper ce„t^ ?^ ^ '*^'^ ^^"^ 
 
 OPERATION. 
 
 $100 Stock ^IIOS-^S money, therefore <ftl «fn.i • 
 
 $1-0876 ^.y, "ieieiore $i stock le worth 
 
 Then$l-0875x 749-80 =$816-4075. ^n. 
 
 OPERATION. 
 
 We |100 .<5trtnlr — *no.KA * - - 
 
 , worth lO-^^S."^" ^■■"' """^•'^' ^"^ ^^^^^ ^' stock is 
 u'i«u $1200^ 10-925 = |1297'28:n ^„j. 
 
", r \ mm M it i iMw! 
 
 148 
 
 INTEREST 
 
 Exercise 65*- 
 
 1. How much fitock cnn bo pnrchnsod for $793 when It Is selling 
 
 nt a proinluin of 17i per cent. I 
 
 2. Wliat is the value of $9476 stock at 9J per cent, discount t 
 
 8. If I own 9 shares of stock In the Metropolitan Water Worka^ 
 the |)ar value of each share being $126, an<l sell out when tha 
 stock is at a premium of Sf per cent., what do 1 receive tot 
 my 9 ishares ? 
 
 4 When Upper Canada Bank stock is soiling at a premium of 3| 
 per ot'nt., what must I pay lor 17 sharen, the par value being 
 |lll-216 per share ? 
 
 5. When the stock of the Ontario and Huron Railway is 22 pef 
 
 cent, below par, how much should I pay for $6470 «tock ? 
 
 6. Wlien the stock of the Hamilton Gas Works is selling at a 
 \ premium of GJ per cent, I wish to invest $2000 in it ; wLal 
 
 amount of stock do I receive? 
 
 7. When stock is 27 p(T cent, above par, what amount can be- 
 
 purchased for $7000? 
 
 8. Wlien stock is 8 per cent, below par, what is the value o*' 
 
 $6140 stock? 
 
 0. When Grand Tnmk Railway stock Is selling at 1} per cent 
 premium, what must I pay for 27 shares, the par value beinjf 
 f 25 per share ? 
 
 10. When Montreal Rank stock is selling at a premium of 13| pel 
 cent., how much should I get for $11120 9 
 
 SECTION" VIIL 
 
 SIMPLE INTEREST, COMPOUND INTEREST, 
 DISCOUNT, PARTNERSHIP. 
 
 1. Interest is the sura allowed *br the use of money 
 and is usually reckoned at a certain rate per cent, pet 
 annum. 
 
 2. The sum lent is called the Principal. 
 
 3. The sura paid for the uao of each hundred dollars w 
 called the Kate Per Ceofii 
 
when it Is selling 
 
 t. dtacountf 
 
 SIMPLE INTEREST. 
 
 149 
 
 th« priiSpaf al^hi i^tr ""''''''' '' ^'^"'° *^^^^^- 
 
 $20000 ietho principal. 
 7-00 is the I'ato por cent. 
 
 14-00 18 the interest. 
 21400 is the amount = principal + Interest. 
 tf. Interest is either Simple or Compound. 
 
 ,t amount can b» 
 
 t is the valub of 
 
 mium of 18i per 
 
 ndred dollars w 
 
 SIMPLE IISTEREST, 
 
 i-ggeimT'" '"'"■'"" '' """P"'"'' '^"""■'"'■S to the follow 
 
 nULE. 
 
 P'-S&nnn/''':^ r'''"rf huthc rate pe,- unit ex- 
 
 J^^lii''':tXt7S^^ly^^^^^^^ 
 
 
 «nf frr:;^c.Ir7^'^' '^ *" '""'^=^' »„ $759-80 at 7p« 
 
 OPERATFOX. 
 
 iea^^''ar!u "^ 2.~What is the interest on $777-40 for 
 rears at (.^ per cent, per annunj y «" m »u lor 
 
 OPERATION . 
 
 tW-40x -0625x7 =1340-1 126. Am. 
 
•WW 
 
 ill: 
 
 3= 
 
 150 
 
 SIMPLE INTEREST. 
 
 
 
 Example 3.- What is the interest of $66'7'7 for 8 years. 8 
 months 20 daj^ at 5| per cent. ? 
 
 OPERATION. 
 
 $667'7 = principal. 
 •055 = rate per unit. 
 
 33385 
 33385 
 
 6 mo. 
 
 2 mo. 
 
 15 days 
 i6 days 
 
 .';3G7-235 
 _8 
 
 2937-880 = 
 
 183'6]75 = 
 
 61-20583 = 
 
 15-30145=: 
 
 5-10048 = 
 
 = interest for 1 year. 
 
 $3203-10526: 
 
 it 
 
 (I 
 (( 
 (( 
 
 8 years, 
 6 months. 
 2 months. 
 15 days. 
 5 days. 
 
 8 yrs. 8 mo. 20 day* 
 
 Exercise 66. 
 
 Find the interest of— 
 
 1. $974 for 1 year at 11 per cent. 
 
 2. $1078-90 for 7 years at 9 per cent. 
 8. $142-70 for 16 years at 8 per cent. 
 
 4. $80-80 for 22 yeard at 7 per cent. 
 
 5. $67-49 for 6 years at 2i per cent. 
 
 6. $208-60 for 11 year.-, ai 3| per cent. 
 
 7. $800 for 6 years 5 raontlis 18 days at 8 per cent. 
 
 8. $7400 for 9 years 11 months 24 days at 6^- jier cent, 
 
 9. $9680-80 for 14 yearb4 months at 3 per cent. 
 
 10. $476-76 for 10 years 8 months at 6| per cent. 
 
 11. $8900 f(^r 6 years 7 months 28 days at 11^ per cent. 
 2. $8160 for 9 years 15 days at 7i per cent. 
 
 .3. $41290 for 6 years at 4| per cent. 
 
 a. $127-40 for 3 years 3 moiths 3 daj » nt 12^ per cent 
 
 15. $S0'63 for 4'7S years at 2 97 per rent. 
 
 16. $106-70 for 11-113 years at 13-47 per cent. 
 
 9. Since the legal rate of interest in Canada is 6 pwr 
 cent, when not otherwise Specified by direct agreem«iat, 
 
SIMPLE IXTjEPtEST. 
 
 151 
 
 it ia Important to have some simple rule by which interest 
 at 6 per cent, can be computed. ^ mierest 
 
 U ^%lluCtl^" ^'''''' "^^' foranynumber of months 
 
 RULE. 
 
 -zenh!''^' ^^' '''''^*'^ o/^o;^i^. by 2 and call ths quotieni 
 
 6 pe^-^'ctnt.T ^•~"^^^* '^ *^" ^*^^««t «f $1 for 8 months at 
 8-5-2 = 4 cents. Ans. 
 
 Example 2.— What is the interest of $1 for 7 years 3 
 months at 6 per cent. V ^ "^ 
 
 ^ ^'''■'*n^'?r'¥ = ^^ °'^^*^'' ^^'i 87 -f- 2 = 43^ cent. :. 
 
 Example 3.— What is the interest of $1 for 11 years 7 
 months at 6 per cent. ? <»> "^ n years / 
 
 11 years 7 months = 139 months, and 139 ^ 2 - fiQi 
 cents = $0-695. Ans. » «^" ^^^ . ^ - 69^ 
 
 at eVer lent^ :i'^' ^'''''' ^^ ^^ ^"' ^^^ ^^^^^^ ^^ days 
 
 RULE. 
 
 «>. ,f ^?"^'i^^ '^^^*«»' o/^«y« *y 6 awc?ca// ^A^ result mills 
 or tenths of a cent 
 
 per cenTr^ ^'""^^^^ '" *^' '°'''''* ^^ ^^ ^^^ ^^ ^^^^ ^^ 6 
 18 -f- 6 = 3 mills = $0-003. Ans-. 
 
 T » ■■ I W -r II-!-.. 
 
 6 peT"i^nt.T ''•~'' "^' '" ^"^ ^°*^'^^«^ o*'^l *or 26 days at 
 26 s- 6 = 4i miUs = $0-0043. Ans, 
 
 If 
 

 ill! 
 
 
 ii 
 
 152 
 
 SIMPLB INTEREST. 
 
 Example 3. — What is the interest of $i for 1 yeara ^ 
 months 27 days at 6 per cent. ? 
 
 •? years 4 months = 88 months,^nd 88 -i- 2 = 4^ cents =« 
 $0-44 = interest for '7 years 4 months. 
 
 27 -^ 6 = 4i mills = $0-0045 = interest for 27 days. 
 Then $0*4446 = interest for 7 years 4 months 27 daya- 
 
 Exercise 67. 
 What is the interest of $1 at 6 per cent, per annum for : 
 
 1. 8 moB. ? 7 mos. ? 11 mo8. ? 
 
 2. 2 years 9 months? 
 
 3. 16 years 4 months 1 
 
 4. 6 years 11 months ? 
 ?k 11 years 1 month ? 
 
 6. 10 years 10 months ? ' 
 
 7. 4 years 5 months ? 
 
 8. 6 years 3 months 12 day»f 
 
 9. 3 years 3 m )nth8 3 days* 
 
 10. 4 years 7 mouths 10 daya? 
 
 11. 1 year 9 montha 25 days? 
 
 12. 2 years 7 mouths 17 days? 
 
 12. To find the interest of any sum of money for imj 
 time at 6 per cent, per annum : — 
 
 KULE. 
 
 Find by the iaat two rules the interest of $1 for th6 
 given time and multiply it by the given principal. 
 
 Example 1.— What is the interest of $67 for 2 years 3 
 months 12 days at 6 per cent. ? 
 
 OPERATION. 
 
 for 2 years 3 months 12 days = $0*137. 
 67 = $9-179. Ans. 
 
 Interest of $1 
 Then $0-137 x 
 
 Example 2.— What is the interest of $714-71 for 3 yeaw 
 7 months 11 days at 6 per cent. ? 
 
 operation. 
 Interest of $1 for 3 years 7 months 11 days = $0-216^. 
 Then $714-71 x 0-216^ = $154-97295|. Aus. 
 
 NoTB.— When the number of days is not exactly divipible by 
 6, the intfrcst for the dayn h:id beltei' bo written a-i mills and u 
 fraction of a mill, :md then the interest of |1 for 'lO given timt^ 
 thus expressed, used for multiplier, us in the last eaf;ampio. 
 
COMPOUND INTEREST. 
 
 Exercise 68 
 jWtid the Interest at 6 per cent, per annum of; 
 
 1. *1904 for 7 years 9 months. 
 
 2. 1274-80 foi i years 11 months. 
 
 8. 1671-90 for 2 years 2 months 12 days. 
 4. $213-27 for 3 years 3 mouths 3 days. 
 6. 149-73 for 4 years 4 months 4 days. 
 
 6. $619-80 for 6 yoars 6 months 5 days. 
 
 7. $27-60 for 6 years 6 months 6 days. 
 
 8 $47-32 for 7 yeari^ 7 months 7 days. 
 
 9 $222-22 for 8 years 8 months 8 days. 
 10 $345-67 for 9 years d months 9 days. 
 11- $789-23 for 10 years 10 months 10 dayi. 
 12. $809 for 11 years 11 months 11 days. 
 13 $207-40 for 3 years 24 days. 
 
 i4. $98-20 for 1 year 28 days. « 
 
 15. $76-42 for 2 years 7 months 15 days. 
 
 16. $9146-70 for 2 years 6 months 20 days. 
 
 158 
 
 COMPOUI>/D INTEREST. 
 
 13. Money is lent at Compound Interest when the In- 
 terest, as It falls due from time to time, is added to the 
 pnncipal ; the sum thus obtained constituting a new prin- 
 cipal for the ensuing year, half-year, quarter, &c., aa the 
 casv,' may be. 
 
 14. To compute the Compoimd Interest on any sum of 
 money for a given number of payments : 
 
 RULE. 
 
 ^ Find the interest on the given principal for one period, 
 ». € ONE YEAR, HALF YEAR, or QUARTER, as the cttse may be, 
 ana add it to the pi'incipal. 
 
 Then fnd the intei est on this cmount for the nexi- 
 l^l\l^ ^'^^ ^^^ ''^ ^^ ^^^ Jor^•«c^y5rt/ used for that period, .i^ 
 
 0€j GTS. 
 
 Proceed in this manner with each successive year or 
 period of the proposed ttma. 
 
 Ml 
 
 W 
 
riki 
 
 154 
 
 COMPOUND INTEREST. 
 
 I- fb 
 
 _mM 
 
 Then the last remit mil he the amount of the given 
 principal^ at the given rate for the given time. Subtract 
 the given principal from this^ and the remainder mil be the 
 Compound Interest required. 
 
 Example. — What is the compound interest of $700 fo* 
 2 years at 4 per cent, half-yearly y 
 
 OPERATION. 
 
 Here, since the interest is half-yearly there are four pay. 
 
 menta. 
 Interest of $700 at 4 per cent. = $28. 
 Then $700 + 28 = $728 = principal for 2d half year. 
 Interest of $728 at 4 per cent. = $29-12. 
 I'hen $728 + $29-12 = $757-12 = principal for 8d half 
 
 year. 
 Interest of $757-12 at 4 per cent. = C30-2848. 
 Then $757-12 + $30-2848 = $787-4048 z= principal for 4th 
 
 hal. year. 
 Interest of $787-4048 at 4 per cent. = $?1 -496198. 
 Then $787*4048 + $31-496192 = $818*90 == amount at end 
 
 of 4th half vear. 
 
 From $818-90, the amount, 
 Take $700- 00, the principal. 
 
 The remainder, $118-90, is the compou'id Iniereatr 
 
 Exercise 69. 
 "What is the compound interest of . 
 
 1. $1000 for 3 years at 7 per cent, per annum 1 
 
 2, 1800 for 4 years at 6 per cent, per annum ? 
 8. $900 for 6 years at 6 per cent, per annum ? 
 4. $600 for 2 years at 4 per cent, half yearly ? 
 6. $250 for 2 years at 3i per cent, half yearly I 
 
 6. $880 for 1\ years at 2 per cent, quarterly ? 
 
 What are the amount and compound interest o^ 
 
 7. $500 for 3 years at Tf per cent, per annum ? 
 
 8. $400 for 2 years at 4^ per cent, half yearly ? 
 
 9. $"U-90 for 2 years at 2^ per cent, quarterly ? 
 10. fTQ-i^ 60 for If years at ^ per cent, half yearly f 
 
DISCOUNT. 
 
 DISCOUNT. 
 
 155 
 
 nnv^^'.^r^^""* '3 an allowance made for advancinr. ilm 
 
 17. The 6-we discount on a note or otlior securitv i«^ fhr* 
 jntcMvst cu .ts present worth at the f^iven rate pe cai and 
 for the given tin.e ; but the dank discount (i. e.tled count 
 OH con.putcd by bankers) is the nuercst oJ the sum named 
 in the note, &c, at the given rate and for the given time 
 
 securfty.^''''""^"*® ^^"^ ^^-w^ discount on a note or other 
 
 RULE. 
 
 amount of %\ for the cjiven time, and at the qiven rate 
 The qnohent vnll be the present worth. The dlscomitiH 
 sum ^ *"*^*'«'^^*''^^ ^f''<^ present wmth from the given 
 
 Example l.-What is the present worth of a note of 
 , auo 6 months hence, at V per cent, discount ? 
 
 OPERATION. 
 
 J per cent, per annum = 1-^ per cent, for 3 months 
 1| per cent. ^lO-OlTS^^ interest on $1 for 3 months at 
 7 per cent, per annum. 
 
 ^^'"=1^0175/^' ^1 ^' fc^iven rate and f,.r given time 
 Then $409 ^ 10175 = |401-i)65. Ana. 
 
 i^l 
 
 '''« 
 
 -H 
 
 kJ 
 
 il 
 
156 
 
 BANK DISCOUNT. 
 
 I^i 
 
 «jii 
 
 ■ijih 
 
 Example 2. — What is the discount on a note fnr 
 $794-6o, due 27 days hunce, discounting at 8 per cent.? 
 
 OPERATION. 
 
 Amount of $1 at 8 per cent, per annum for 27 days =» 
 
 $1 -00591 7. 
 Then |794-()8 -=-$1 •005917 = $789*955 = present worth. 
 And $794-63 -r- $789-955 = $4-675 = discount. 
 
 Ex']:rcisk 70. 
 
 Whnt is the dlecount on : 
 
 1. A note of $740 drawn for 3 months, discount nt 7 per cent. ? 
 
 2. A note of $90 drawn for 2 months, diecount at 9 per cent, I 
 
 3. A note of $250 drawn for 6 months, diBcount at Q per cent, f 
 
 4. A note of $714-20 drawn for 11 months, discounting at 11 pe» 
 
 cent. ? 
 
 6. A note of $911-40 drawn for 5 months, discounting at 8 per 
 
 cent. ? 
 fi. A note of $671-43 drawn for 4 months, discounting at 7 per 
 cent,? 
 
 7. A hill of $94760 drawn at 2 years, discounting at 4 per cpnt.? 
 
 8. A bill of $88893 drawn at 1 year 4 months, discountinfi: at 7 
 
 per cent.? 
 
 9. A bill of $7146'90 drawn at 47 dayp, discounting at 10 per^^nt.f 
 
 10. A bill of $710 drawn at -.^ months, discounting at 7 per cent.? 
 
 11. A bill of $1100 drawn at 1| months, di^countirg at 7 per cfnt.? 
 
 12. A bill of $G71483 drawn at 2i months, discounting at 6 per 
 
 cent. ? 
 
 BANK DISCOUNT. 
 
 19. As already remarked, the hank discount on any 
 sum is the same as the interest on that sum, and hence to 
 compute bank discount : — 
 
 RULE. 
 
 Add 8 daijs to the time which the note has io mn before 
 it becomes dite^ and calculate the interest for this time at (he 
 
 given rate per cent. 
 
 ■jjrvTw Tlie 3 d'lys added are the dnt/s of^rnce-.ovty'Q 3 dnvi' 
 
 'Tirhich, bymereai'tik' usago, SH^Allowedio ^lai se, after a bill ft' 
 flue, l-iefdrc it is payable, Binkers always add these 3 d^s to'*^ 
 lime for which they compute diBcount. 
 
 I 
 
 2. 
 6. 
 1. 
 
1 a note for 
 per cent. ? 
 
 or 27 days =» 
 
 sent worth. 
 
 it. 
 
 7 per cent, f 
 per cent, i 
 i3 per cent, f 
 mtiiig at 11 pel 
 
 nting at 8 per 
 
 nting Rt 7 per 
 
 1 4 per opnt. ? 
 scountiDfi: at 7 
 
 at 10 per^^ut.f 
 7 per cent. ? 
 at 7 por cf nt. f 
 nting at 6 per 
 
 iount^n any 
 and hence to 
 
 SIMPLE PARTNERSHIP. 
 
 167 
 
 IS840 ""drrQ V""T^"* ^'v^^^ ^'^"^ diHcounton a bill of 
 f 840, due 69 days hence, discounting at 7 per cent. ? 
 
 OPERATION. 
 
 Interest of $840 at 7 per cent, for 1 year = $58-80. 
 
 69 -+- 3 = 72 days, and 72 days = i of a year (300 davsV- 
 
 Hence bank discount = i of $58-8') = $11-16. Ans. 
 
 $47? X'^Vn'T^^^"* '' ^^' ^""^ ^•^^^""t «n a note of 
 f47l, due 3 montlis hence, discounting at 7 per cent. V 
 
 OPERATION. 
 
 taterest of |471 for 1 year at 7 per cent. = $32-97 
 rime for which discount is charged = 3 months 3 days. 
 B moa.| i $32-97 Interest or bank discount for 1 year. 
 
 ^ days 
 
 ^h 
 
 82425 
 •2747 
 
 
 i( 
 
 (i 
 
 3 months. 
 3 days. 
 
 3 mos. 3 days. 
 
 $8-5172 
 
 Exercise 71. 
 Find the bank discount on : 
 I A note of 1700, due 42 days hence, discounting at 7 per cent 
 
 0* V u!l! °1'"^^^' ''"^ ^^ ^^^^ ^'^"°^' discounting at 8 per cent. ' 
 B. A b.l of $790, due 4 montlis hence, diecouutlng at 5 per cent. 
 ^ "^°;f^ of $614-30, due 2 months hence, discounting at 7 per 
 
 ■ ' ^ cint!"^ ^^^"^^' ^"^ ^ "^^^^^^ ^«"^«' discounting at 9 per 
 «. A note of $94-80, due 20 days hence, discounting at 10 per cent. 
 
 io run before 
 lis time at (he 
 
 . nr tVir> 3 dnv;i 
 ift<va biil ft' 
 e 3 da;ysto'*^ 
 
 SIMPLE PARTNERSHIP. 
 
 ^s P^Vf?"'^^-® w'S'^'^^riP' "^^"^^ ^^'^ ^^"gle Fellowship 
 ^.^^j f,'^'P^l?^^2* ^"^^> ^"'^^^^^^ "« to distribute the 
 To,>f«. "V — 7.' ' ";'" "' vmxjpiuij uquiiauiy anjoni( its 
 

 
 
 ill 
 
 I 
 
 •I' 
 
 158 
 
 SIMPLK FAHi- iiRSHIP. 
 
 21. The whole money employed in tue busincmis called 
 the capital or stock. 
 
 22. The profit or loss belonging to each member is 
 calculjucd according to tiic following 
 
 RULE. 
 
 As the whole stock is to each mail's share of the stocky 
 so is the whole gain or loss to each maiCs share of the gain 
 or loss. 
 
 Example. — A and B enter into partnership as grocers, 
 with a capital of $14000, of which A contributca $850(> and 
 B the remainder. They gain $4*740 ; what portion of thid 
 must each receive ? 
 
 OPERATION. 
 
 Whole stock : A's stock :: Whole profit : A's profit. 
 
 That is, $14000 : $8500 :: $4740 : A's profit, which is equal 
 
 $8500 X 4740 
 *°— 14000— = *28m57. 
 
 Again, whole stock : B's stock : . whole profit : B's profit. 
 Or $14000 : $5500 :: $4740: B's profit, which is equal 
 ^ $5500 X 4740 ^ 
 
 '' -- Tiooo— = ^'''^ '^^' 
 
 Note. —After A'b profit ha«3 been found by the rule, B's may 
 be determined by subtracting A'b share from the whole profit. 
 
 Exercise 72. 
 
 1. A, B, and C enter into buslnees with a capital of $7000, of whi'h 
 
 A contributes $2700, B $4200, and <J the balance, and tncy 
 gain $1700; how must this be divided among them? 
 
 2. B and C together own a Btenmboat worth $29000, of which B 
 
 contributed $17400 and C the balance After paying all 
 expenses for running her durinj? tlie season, they find that 
 they are losers to the amount of $904-70 ; what portion of this 
 loss must each sustain ? 
 
 8. Three persons rent a papture for the summer ; the first puts in 
 21 cows, the second 17 cows, and the third 47 cov.b. The 
 rent is $307 ; what portion of this must each pay ? 
 
 A. Three persons are to nViare $7493 in the followinor manner, viz. ; 
 Hs often as A gets $4 B gets |7 and gets $9 ; wha^ is th» 
 •hareof eaohl 
 
 M ' 
 
 II 
 
 ,t 
 
OOMPOUxXD PARTNERSHIP 
 
 o. A iToniiomaii bonneathod ilTr-no ♦„ »,, ,, 
 
 Of „, 2,, „,„ 3, . w!;...;'wt^,r.Er,';';?:„"i;T '" '"•-•■°«'« 
 
 K:uu Wiuo ; what i, the .;i,';n,'„"f S',""'' " """ ""'■ l" 4 
 9. Thre« persons liiivo iralned wtiWMi «..^ 
 
 C $7-49 ; what Im the sliaie of each? ^'" ^''^''^ •^ ^^ ""d 
 a A yoBBd worth $14900 ie cntlroly lost i nf if k i . 
 
 ' " »,t'a^n' •"'" '"'- »"•"' «-"»>■ "tail be ,0 one another .. 
 oroditor receive ( ^^ ♦'**' i l""* much should each 
 
 COMPOUND PARTNERSPIIP. 
 
 profits or losses oF ^^r,vh„2T' T?^'"^ ""' *^ '""''« 'he 
 cording toThefoilowitfrP""'''' ^"'"^^'^P -" «'"-" -- 
 
 to 
 
 r: 
 
 RULE. 
 
 &«/ar product \Ti\,lf^r', "" F""^""*' u to each par- 
 
a 
 
 ii 
 
 lUf! 
 
 160 
 
 COMPOUND PARTNERSnil 
 
 OPKUATION. 
 
 I 790 X 8 r= $()820 . 
 1145 X 7= 8016 
 966 X 10= 9000 
 
 Sum = il»996 
 
 
 $23005 : $0320:: $880 : A's share, whHi i, 
 
 = $231 7816. 
 
 8v46 X i.8f 
 $23995 : $8015:: $880 : B'a share, whHi ia - -/yycis^ 
 
 $293-9445. 
 
 ORfiO X 83< 
 
 ^ $23995 : $9000:: $880 : C's share, which ia — 2399^" 
 
 = $354-2738. 
 
 ^oTB.— When two phtires hjive 'o^pn found, the third nrmv be 
 found by fubtractinir their euni from the whole pro'it or losa. Sc 
 when there arc four prtituers, and three sliares have lec, foirid b> 
 tlie rule, the fourth nu«y be obtained by Hubtractiug t\o sum ^ 
 iheae three from the whole prollt or loss, &c. 
 
 Exercise 73 
 
 1 Two persona contract to make a road for $4000. A furnjahea 4C 
 
 laborers for 37 days and B 36 laborers for 62 days ; what part 
 of the $4600 should each receive f 
 
 2 *Thre© persons enter into businoes as hardware merchants. A 
 
 put-T in $2000 for 7 moutlis, B $1800 for li months, and C 
 $1600 for 12 months ; if they gain $2400, what is the share of 
 each i 
 
 % Two men undertake tp drain a field for the sum of $2400. A 
 furnishes 10 men for 34 days and B 15 men for 36 days, nnd 
 they have to pay $400 out of the $2400 for material; how 
 much of the remaining $2000 should each receive ? 
 
 4. Three porpone rent a pasture for $120. A puts in 27 cow? "or 
 4 months, B 20 cowf* for 6 montlia, and (■ 24 rows for li 
 months ; what portion of the re.it should each pay ? 
 
 F^nr persons begin bus'ness with n capital of $10000, of whic) 
 $2400 belonj? to A, $3000 to B, $2000 to C, and tic bala c^ ' 
 D A ullowr*' his coiiiribiifio:; {u fciiinin i 11 mrriui'- . ''■ 4 
 months, 9 month*, and D 12 montlis, and they gaiu*260Ui 
 litTW Bbouhi thiB l>e aivMed omoug tbomt 
 
2J.Vl'3 
 
 8v.l6 X i>8( 
 
 \& 
 
 23995 
 
 tht' third mav be 
 iro'it or loBa. Sc 
 ve ">cc"i foirid b> 
 !ting t\o Bum >» 
 
 A fumjaliea 40 
 days ; \vl)at part 
 
 e merchants. A 
 i months, and C 
 it is the BDure of 
 
 mm of $2400. A 
 I for 36 diivB, nnd 
 r material ; bow 
 ccivc ? 
 
 lis in 27 pow? w^r 
 ;' 24 rows for l\ 
 cli pay ? 
 
 $10000, of -sxhirf 
 ni tie Viala c ' 
 
 ••1 I. ! ' T 
 
 1 1 IWi'Ilt I- . ' t 
 
 tlieygumi^25U0i 
 
 PROFIT AND LOBB. 
 
 161 
 
 ** \?jJ!^r ^'' 'V T>«^"er8h«P, have mado $950 ; what aro their 
 
 iso f om V'^t'' "^"'" ^i r"rch:,?,.B $4000 wh o^^^^^^^^^^ 
 
 SECTIOlSr IX. 
 
 l^EOFIT AIS-D LOSS, BARTER, EXCHANGE OF 
 CURRENCIES, AND ANALYSia 
 
 PROFIT AND LOSS. 
 
 1. Profit and Loss enables us to ascertain how much 
 ^e gain or lose on any mercantile transaction, and aJ^o hoW 
 much we must increase or d/minish tlie price of our cooda 
 m order to make a certaui gain or lo^ per cent ^ 
 
 CASE i: 
 
 «P ^' F^ 'f^^ !^ ^^^^^ S^^^ ^' ^^^^ on a certain quantitv 
 of goods when the prime cost and selling price are g?ven:- 
 
 RULE. 
 
 ^,.^ ^„j,, ^.. jy^^. omnti,, !b., yard ol hn 
 
 ^^ wnole gam or los8^ - ..- v. . -.^Hf. • - 
 
 11 
 
 1; 
 
 it^ 
 
 > 
 
JIC[2 PROFIT AND LOSS. 
 
 Example 1.— What do I gain if I buy 704 barrels ot 
 flour a'- $i-25 and sell it again at $4'93 per barrel V 
 
 OPIRATION. 
 
 From ^4-93 the selling price, 
 Take |4-25 the buying price. 
 
 The remainder $0-08 is the gain per barrel. 
 Then $0-68 x 104 - |4'78-72, the whole gain. 
 
 Example 2.— If I buy 1G40 bushels of oats at 37^ cent* 
 per bushel and sell them at 8D^ cents, what do 1 lose on thei 
 triuisaction ? 
 
 ii Si 
 
 k 
 
 OPERATION. 
 
 From 37^ cents, the bviying price, 
 Take 35^ cents, the selling price. 
 
 The remainder 2^ cents = the loss per bushel. 
 Thp-u 2A cents x 1640 = $36-90. Ans, 
 
 Exercise 74. 
 
 1 If I buy 209 yards of flannel at 6 ' cents per yard, RAd gell It 
 again at 70 cents, what do I ga . on the transaction ? 
 
 2. If I purohrtse 8900 bushels of wheat at ;>l-29 and sell it again at 
 $1-42, what do I gain on the transaction ? . ^ . 
 
 a. Suppose I sell 780 cedar posts atl2i cents each, which I bought 
 at 16 cents each, what do I lose on the transaction ? 
 
 d Bought 1142 thousand bricks at $492 per thousand and sold 
 them at $5-47, what is my entire gain ? 
 
 6. Boucrht 17 cwt. 2 qrs. 11 lbs. of butter at 18 cents per lb. and 
 
 sold it at $23 per cwt. ; what is my entire gain ? 
 
 §. Bought 1143 lbs. of maple sugar at 11 cents per lb. and gold i» 
 at $13*50 per cwt., what is ray entire gain ? 
 
 7. If I purchase 63 ♦ona of hay at $17'49 per to« Jxnd hsvp txj «eD 
 
 it et 112*94, what is my entire loflts ? 
 
 d. If I purchase 47 sheep at |3'37^ eftct an«i ft«li\ Uk«2£i w^ $& 11 
 what is my entire gai^ 
 
sell it again at 
 
 mnd and Bold 
 
 ts per lb. and 
 
 PROFIT AND LOSa 
 
 CASE ir. 
 
 163 
 
 gi^^u:— * ^^"lii-jS i"c to&fc price being 
 
 RULE. 
 
 must I cl.argo per bushel" ^ P®"' ''™'- ! "'•»' 
 
 OPERATION. 
 
 I want to gain $14 on $100, or 14 cents on «i 
 
 Hence my selling pricem«,stbo$l-U f,.''6"o!l$,.824. An. 
 
 for!iXrii;«,»;;t:r^'^^t;rSnf^ 
 
 p OPERATION. 
 
 'ITfid^v^r/"""^ '» set $11G, therefore for every 
 
 ^1 paia away I require to receive ^1 -Ifi ^ 
 
 Hence I must sell for |M6 x 7437-80 l*j8627-848. An,. 
 
 «^4?oTeif atrS'Sf Vnr™'^'' "'."''"'• '■"^ «*"'" »"<» 
 whole ? "'^ ® P""" <"""■ ; ''ha' do I get for the 
 
 OPERATION. 
 
 Therefore I get for the wliole $094 x 7190 = $8768-eO. 
 
 ExERciste 75. 
 to gain 10 jx*r cent. I ^^"^ * iatjautt I eelil la oider 
 
 m 
 
104 
 
 PROFIT AND LOSa 
 
 2. Bought a quantity of Irntlicr for |890 ; for what muet I sell It In 
 order to yaiii 17 ptu- ct-nt. I 
 
 8. Boiitrlit 030 biisholH of wlioat n\ ^1-23 per buslicl. and ai^roo to 
 si'll It at a luBB of 8 per cent.; ^vLat do I receivo for the 
 whole I 
 
 4. Boudit 050 saw lopfs at 44 ccntp Oiich : for what must I eell the 
 
 lot iM order to gain 38 per cent. ? 
 
 5. BoiiKlit 411 barrels of flour at $5-22 per barrel ; for what must 1 
 
 Bcll the wholo in order to gain 12^ per ecnt. ? 
 
 C. Bou 1,-1 it 512 dozen brooms at |2 80 per dozen, and aproe to boU 
 at a loHH of 15 per cent.; what do I receive for the whole ? 
 
 ?. Bon.cht (54980 dozen c^crs at 7 cents per dozen, and bcH bo m 
 to Kain 24 per ceal.; what do 1 receive for the whole lot of 
 
 J 8. Bongbt 908 tons of coal at $5-22 per ton, and agreo to eell at a 
 loss of H per cent. ; what do I got for the whole f 
 
 h-m^ 
 
 
 CASE III. 
 
 4. To find the rate per cent, of profit w loss when tho 
 buying price and the eelliug price are given : — 
 
 RULE. 
 
 Find the difference between the hiying price and the sell- 
 ing price ; this will be the whole pain or loss. 
 
 77icn say, as the buying price is to 100 so is ths whole 
 gain or loss to the gain or loss per cent. 
 
 Example 1.— If I buy a house for $2700 and sell it for 
 $3060, what is my gain per cent. ? 
 
 OPERATION. 
 
 From $3050, the selling price, 
 
 Take $2 700, the buying price. 
 Tlie difference, $360, is the whole gain. 
 Then $2700 : $100 :: $360 ; the gain on $100, i. e., the gam 
 per cent. 
 
 „ 100 X 360 
 
 Hence gam per cent. = — j^^— = 12Jf . Am, 
 
nust I sell it in 
 
 PROFIT AND LOSS. ,«^ 
 
 loo 
 
 sold^it roA75lr!:"''f ' " ^"T^'^y «^ '^^^^^ ^or $^790 ana 
 omu It lor ^7oO ; Viiat was my Io.sh per cent, v 
 
 oi'K!:ation. 
 $"790 - $750 = $40 = wl.olo Ios.<. 
 
 Then .$790 : ,$100 : .- 40 : ^*^*^ "^ ^^ - k i nn. n. * ^ 
 
 • /-/t^Q ~ oyV per cent. ^ns. 
 
 7SS when tho 
 
 d sell it for 
 
 Exercise VO. 
 '■ "".".I'lV'S-on,'^' P'^'' '" ""O "•"" " "' ♦26'25; What wa. „y 
 
 ' ''"SUi-c;;;;;, ""■«'"» """ -'"" f<"-«o»oi wh.tw., my 
 
 CASK IV. 
 
 _ 6. To find Ihf. cfM prim when the sellhi" nriee and thA 
 (am or lo« per cent, me given :— " ^ 
 
 KULE. 
 
 ^0 $100 .vo IS the selling price to the cost price. ^ ^ * 
 
 I SOW fJ^si!-'"'^^^'* '.T ^''^ ^ ^^y ^^' ^ ^^rriage which 
 1 sold tor ^31 <, guinnig U per cenLP ° 
 
 OPERATION. 
 
 10O+ $11 =&^M 
 
 TJ- - — — 
 
 Then $111 ; ^loo : : $317: l^L^l^ ^ |286 
 
 111 
 
 586. Ana» 
 
' *is a L.j^::z 
 
 f i 
 
 1. f 
 
 r«f» 
 
 Ijl ) 
 
 166 
 
 BARTER. 
 
 Example 2.--SokI a quantity of butter for |>14T 
 losing tl.ercby 7 per cent, oii Mic uansactiou; what did it 
 
 tost iiin V ' 
 
 cost mc 5* 
 
 Then §9a : $100 
 
 OPERATION. 
 
 .$100- $7 = $03. 
 
 $2147 : ii*5Lx_ 2m 
 93 
 
 Exercise 77. 
 
 = $2308-602. Ana. 
 
 1. "NVIint did I pny per bushel for wheat which I sold for $1-70 ct 
 
 a g:i:n of 18 y.cv ceiit. ? 
 
 Wt'iit did I p!i3 
 
 $324 
 
 guiiui t' 29 percent.? 
 
 or !i quantity of rhingles which I sold for 
 
 1 Sold 356 l)tislu'Irt (if cl 
 
 <n 
 
 cent. ; wliat did it c-o.il rac per hv ^i i 
 
 VI- ed for $11^0, losing thereby 11 pe> 
 
 4. What did I p.iv lor 1. inter 
 iiig it at 8/j, ec t'i]Hr 11). y 
 
 U! :)ii which 1 lost 14 per cent, sell- 
 
 6. Sold a uriet-niill for mi)0 and gained 43 per cent 
 
 action ; wliat did 1 i v for it ? 
 
 on the trans' 
 
 6. An aueiit sells 209 I 
 
 Now this was 11 per ceit. above tl 
 
 an-els of Hour for mo at $6-72 per barrel 
 
 to pay my agent 20 per co' t. lur commits 
 flour j^tand me per barre! ? 
 
 e cost price, Imt I liave 
 
 Oil 1 whu*. does tho 
 
 7. Sold a horse for $145 ai 
 
 v.'hat dill tho horse cost 
 
 d giiined 9 per cent, on the transaction 
 
 me 
 
 8. What did T pay f( 
 
 $12 per 1000, gaining SI per con 
 
 )i;fw().iieh draining tiles which I sold fo» 
 
 BARTER. 
 
 6, Barter enables two parties to make an exchange of 
 goods at prices agreed upon so that neither shall suffei 
 loss. 
 
 7. Questions in barter are solved by the following 
 
 RULE. 
 
 Find the value of the covimodity whose price and guan' 
 tity are given. 
 
BARTER. 
 
 107 
 
 ^ .^iff" li^i ^? '^' ^''■'^ ^^ ^^'' "^^'"^ commodity and the 
 ^ottentmtl he he quantity; or divide by the guaniiiy and 
 the quotient zcill he the price. ^ ^ 
 
 Example l.—IIow ranch tea at $0-85 per lb inu«t a 
 famer^receive for 211 bushels of to^nips a' 23 cei^spe? 
 
 OrEriATION. 
 
 211 bushels of turnips at 23 cents = ^48-53 
 
 Ihen $48-53 - $0-86 = 67-094 lbs. = 57 lbs. 1^ oz. 
 
 Example 2.--A has 307 yards of linen at 63 cents per 
 jard and b|uteis it w,th B for 20 cwt. of sugar; what does 
 h get per lb. loi- his sugar ? 
 
 OPERATION. 
 
 307 yards at 03 cents? rr .$193-41. 
 
 20 cwt. of sugar— 2000 lbs. 
 
 Then $193-41 - 20'J0 .= $oo%7 = 9-^0^ cents. Ans, 
 
 Exercise 78. 
 
 *• ^ B fo?VA'i,r/Sf ' ^' V V^^ 1^7 ^^•' ^hioh he barters with 
 li loi o4i lbs. of tea ; what does the tea stand B per lb ? 
 
 ^' '^''fn mvS77^f«^^^^^^^ o^g^atlOceiitspcTdozen nndtako8 
 in px> nu nt 47 H'S. ol raisii.s at 18 conls per lb , 9 Ibn of 'o-.f 
 su!.'ar at 14 cc-ut. per lb., 23 Ibe. of rice at^6 o n s , er'l ai d 
 
 ^' ^ KvShh'} ^^ '^^'- "^ ^^1 "* 27 cents per lb., and bnrlered them 
 AMth ft dry-goods merchant for drugget at 43 cent«i per vavd 
 hoY/niuch drugget did ho receive? per >aid , 
 
 '■ ^ t^™thenrfoV.rV''''''\^'^ ^'"^ «^'d'>ar- 
 
 shtop? '*'^' how much does he give for each 
 
 ^ ^ r fi!^?'*^',"^ ^}^Y '''"r*^' ^1-73 .l-er yanl ard b.,rtere It with 
 ^:^^{;^rr:'^}:^ -^ ^^^-^^ '« -oney ; whit £ " 
 
 A farmer has 409 lbs. of chec 
 
 SB whicli ho bartcTM with a neigh- 
 
 ^S?^^ti^S-J2f--^-— -^- 
 
 ho 
 
jG8 
 
 CURRENCIES. 
 
 llii 
 
 
 M& 
 
 11 
 
 i ^ 
 
 SI 
 
 ill 
 
 hl< 
 
 11 
 
 7. A farmer cai-rles to a £fr1«t-min420 buBhels of wheat, worth fl'Sd 
 
 per Inisbel, find rccoivoM in paynu-iit |207'50 aiidllOgS* Iba. 
 of flour ; how much does the miller charge per cwt. for t^^ 
 
 8. B hus 423 lb». of Biigar which is worth 11 centa per lb. and he 
 
 imitcM It with C for ^oldon Byrup worth 23 cants per quart; 
 bow much eyrup does he receive ! i i" » 
 
 EXCHANGE OF CURRENCIES. 
 
 8. Tablo of Currencies in Canada and the United States. 
 
 In Canada, Nova Scotia, New Bruns- 
 wick, &c $1 = 6s. or£i 
 
 ^n New Yoik, North Carolina, Ohio, 
 
 and Michigan ^x = 8s. or ^i 
 
 In New England, Virginia, KentJickj, 
 Tennessee, Indiana, Illinois, Mis- 
 sissippi, Missouri.. $i ^ ^g. ^^ £.%. 
 
 In Pennsjlvania, New Jersey, Dela- 
 
 ware, and Maryland $i ^ ^g. ed. or £|. 
 
 In Georgia and South Carolina $1 = -is. 8d. or £^^, 
 
 ^*''^"-,~X^®?^ pounds, Bhillinps, and pence .ire not coins but 
 are merely the denominations employed iti keeping accouX The 
 remaining States use the Federal money excluiiv^ly. 
 
 9. To reduce dollars and cents to old Canadian Curren- 
 cy, or to any State Currency : — 
 
 RULE. 
 
 ^fumph, the ffiven sum by that fraction of £\ which ex- 
 presses the value o/$l ; the product will he pounds anddeci- 
 "mals of a pound. ^ 
 
 ihim'f'^ '"^''^^ ^^*^ rf««w«^5 to shillings, pence, and far- 
 Example 1.— Reduce f;20'7-43 to old Canadian Currency. 
 
 OPKRATION. 
 
 And £51*857() = £ol 17s. l|d. Ans, 
 
CURRENCIES. ifjg 
 
 Example 2.— Reduce $294*80 to Kentucky currency. 
 
 OPERATION. 
 
 $1 = £ ,a,. Then $294-80 x '?^j = £88-44. 
 = £88 83. yf d. Ans. 
 
 10. To reduce old Canadian Currency or any State 
 Cu.Tency to dollars and cents : — 
 
 RULE. 
 
 Express the given sum decimally and divide it by the 
 value of $1 expressed as a fraction of a pound. ' 27ie 
 quotient will be dollars and cents^ d'c. 
 
 Example 1.— Reduce £227 8b. 4fd. old Canadian Cur- 
 rency to dollars and cents. 
 
 operation. 
 
 £227 83. 4fd.=£227-419'79. 
 
 Then 227-41979 •- i = 227-41979 x 4 = $909-679. Ans, 
 
 Example 2.— Reduce £411 6s. 7^ Michigan Currency 
 to dollars and cents. 
 
 operation. 
 £411 6a. 7|d. = £411-33125. 
 Then 411-33125 -J- f = 411-33126 xf=$1028-328. Ans, 
 
 11. To reduce dollars and cents to sterling money:— 
 
 RULE. 
 
 Divide the r/iven sum by the value of £1 sterling 
 ($4-867). 27ie result ivill be pounds sterling and decimals 
 of a pound 
 
 Then reduce the decimal to shillings < nd pence. 
 
 Example 1. — Reduce $1479-83 to s'.erling money. 
 
 operation. 
 $1479-33 -f- 4-867 = £304-06888 = £304 Is. 0-»^d. 
 
 „!l 
 
 \ ■'- 
 
 |i 
 
 W^ 
 

 I 
 
 
 170 CUBIIEN0IE8. 
 
 12. To reduce sterling money to dollars and cents :— 
 
 RULE. 
 
 Fo'prcss the piven sum decimally and multiply it by //*# 
 Ugal value of £1 dcrliny ($4-807). *^ 
 
 Example 2.-Ile(iuce £29 4s. Yd. sterling to doUarf 
 and cents. ** 
 
 OPERATION. 
 
 £29 4!3. 7d. -£29'2291G('>. 
 Theu£ii9-i^29166 x 4 80? =:= $142-2588-5. 
 
 Exercise 79. 
 
 Ana. 
 
 4. 
 5. 
 
 c. 
 
 7. 
 8. 
 
 9, 
 10. 
 11. 
 12. 
 13. 
 14. 
 15. 
 18. 
 37. 
 28. 
 19. 
 
 m. 
 
 Reduce $714 93 to oM Ciinadian currency. 
 
 Reduce $014-90 to North Carolina currency. 
 
 Reduce $611-20 to New Englaij*: ourrouoy. ' 
 
 Rcduco ij43.92 to Ohio currency. 
 
 Roduco £203 178. 4d. old Ciiuadian currency to dollars and 
 
 ^^.i!!'! ^294 lie. Hid. New Jersey currency to dollars and 
 
 ■ ^' cents. "^^^^ '"■ ^^- ^°"^'' C^»-«"»» currency to dollars and 
 Reduce £89 lis. lOid. New Brunswick currency to doUara and 
 
 Rtduce $994-70 to sterling -joney. 
 
 Reduce |896 93 to sterlin;; money. 
 
 Reduce $1020-11 to sterling money. 
 
 Reduce $89-74 to sterling money. 
 
 Reduce £29 Ws. Hid. sterling to dollars and cents. 
 
 Reduce £-294 16s. 2id. sterling to dollars and cents. 
 
 Poduco £411 16s. 7d. sterling to dollars and cents. 
 
 Reduce £843 9s. Od. sterling to dollars and cents. 
 
 Roduco £-294 11.^. lOd. Delaware currency to dollars and cent 
 
 Reduce $2947-80 to sterlinf i^onp.". 
 
 Reiluce $129110 to New York currency. 
 
 Reuucc £470 19.^. 8}d. sterling to dollars and cents. 
 
cents :— 
 
 hj it by th9 
 to dollurr 
 
 An8. 
 
 dollars and 
 dollars and 
 dollars and 
 dollars and 
 
 and ceni 
 
 ANALYSIS. 
 
 ANALYSIS. 
 
 171 
 
 13. Anurysia ni nnthmetie is M.c process of solvin- 
 pj:o)kMus nulopcidontly of set ruIcH, l,y trucin. tl.o n^huions 
 o tl.(. given innuberH and rrasonin-yVo../ ll,r <,ivcn uun.bcr 
 to unity ami from, unity to the required numOcr. 
 
 n.m^wT.lr.^''*A'y ?^ llie prercHlIn^' oiicratioiis are worked bv tho 
 rrobk.ui" '""'^^ "^ "^^^''^''"-^ ^*'" '"^'^^'"d to a vuricly of oihc!; 
 
 ^ni^/oV? ! """^^ "^ ^"^^'^^^ ^^ peaches cost $28, bow much 
 will 12 bushels cost y 
 
 OPERATION. 
 
 \W^lfi'\' T,^^^^' 1 b"«^^^'l^i» cost I of $28, that is, $4. 
 
 Jvow It ] bushel cost $4, 12 bushels will cost 12 times I4. 
 
 that \H, $48. Alls. * ♦ 
 
 ExAMVLE 2.— ,\ of 88 are hew many times 6 ? 
 
 OPERATION. 
 
 h Of 88 is 8, and therefore f ,- of 88 is 8 x Y = 56, and 6 
 IS contamed in 5G 11^ tunes. Ans. 
 
 ExAMi'LE 8.--A person bourrjit a horse and naid ^To 
 cash, anc' this was .\ of the price%f it: what dU it^co't y 
 
 OPERATION. 
 
 If :.'72 13 /v of the price, $72 -f- 6 == ^12, must bo X. of 
 the pr,eo, and therefore the price is ^12 x 11 .- 
 ^1 ■:>/!. Ans. 
 
 U.,>.rr,E 4.~rf 28 men can do a pioce of woik iu 42 
 '* /*, m uan- many days can 21 men do it ? 
 
 'fi 
 
s 
 
 '1 
 
 T 
 
 1 
 
 i 
 
 ) 
 
 
 * 
 
 1 
 
 •' 
 
 I 
 
 ' 
 
 li(l 
 
 1„ 
 
 i; 
 
 172 
 
 ANALYSIS. 
 
 ^ ERATION. 
 
 If 28 men require '-2 dn s, 1 mau will require 28 tiuaj 
 
 days. 
 
 Then, if 1 mau requir"s 28 times 42 days, 21 men will re- 
 
 28 X 42 
 quire the -^\ part oi' 28 times 42 days, that is, — — — 
 
 = 50 days. Ana, 
 
 Example 5. — A can do a piece of work in 7 days which 
 B can do in 5 days ; in what time can they do it if thej 
 work together ? 
 
 OPERATION. 
 
 t 
 
 If A can do tlie whole work in 7 days, he can do \ of the 
 work in 1 duy ; and if B can do the whole work in o 
 days, he can do J^ of it in 1 day. 
 
 Then, since A does | and B ^ in one day, they will toj^ether 
 do I + ^, which is ^| of the work, in 1 day ; and to 
 do the whole work, they wi I require as many times 
 ^} of a day as ^| is contained times in 1. 
 
 Then the time re(|uired will be 1 ^ ^| = 1 x ^| = f| =: 
 2\^ days. Ans. 
 
 Example 6. — A, B, and C can together do a piece of 
 work in 30 days, A alone can do it in '75 ('ays, and B work, 
 ing alone can do it in 80 days ; in what time would 
 
 working alone do it V 
 
 operation. 
 
 A, B, and C can together do it in 30 days, therefore in 1 
 day they oan do 3^ of the work. 
 
 A working alone requires 75 days, therefore in 1 day hc 
 can (lo 7^< of the work ; B working alone requires 
 80 days, therefore in I day he can do „^o ^^ ^^^<3 work 
 Hence A and B working together will do -^K, -\- -J-q — 
 tIoT» of *he work in 1 day, but A, B, and C do 3^ 
 in 1 day. 
 
 Therefore C must do the difference between 
 
 that is, h — rHii = 
 
 >> 3 
 
 lioo 
 
 -3.. 
 
 3(1 
 
 ri^ 
 
 - liQiS — 49,11 
 
28 tiUii 
 
 men will ri»- 
 . 28 X 42 
 
 7 days which 
 do it if thej 
 
 do \ of the 
 )le work in o 
 
 will tof^ethev 
 
 day ; and to 
 
 many times 
 
 35. _ a.Q. — 
 ) J — ii — 
 
 o a piece of 
 and B work, 
 ae would 
 
 lierofore in 1 
 
 in ] day he 
 
 one requires 
 ^f the work 
 
 .1 
 
 1 r, 
 
 .1 . 
 so — 
 
 d C do h 
 /ij and yl-W 
 
 ANALYSIS. ,K^ 
 
 ^uldroqune 1 -<- ^i^ = i ^ ino ^ ^^o ^ jgg^ ^j^^,^^ 
 
 Po^?*'"'''n '^Tu ^'°^'^' "'•''^^ together 7 lbs. of sugar at 
 cent, per lb., 4 lbs. at 12 cents, and (5 lbs. at 10 cents- 
 what should he charge per lb. for Ihe mixture? ' 
 
 OPERATION. 
 
 7 lbs. at 9 cents will come to 63 cents. 
 4 lbs. at 12 " " 48 " 
 
 6 lbs. at 10 " «» 60 
 
 
 Therefore the mixture contains 11 lbs. of sugar and is worth 
 171 cents; but If 17 lbs. be worth fll cent^ T lb 
 should be worth ^ of I'fl, that is, 171 -M7 4 loX 
 cents. Ans, ^^ 
 
 -f n^.^r''':^ ®2~^ a. certain school ^ of the scholars are 
 which 1r?f'«t^'i """,5'"^' ^ ^' ge^g^'^Phy, and the rest, 
 Sool? how many scholars are there in the 
 
 OPERATION. 
 
 The sum of i f , and ^ = -fi^^,-, therefore the number at 
 play^ust be the whole minus /o«v, that is, |^f — 
 
 But the number at play is 38, and hence 38 is -k%- of the 
 whole school. 
 
 rf f -'^ f"o^''T*6 ^s A- of 88, which is 2. 
 " 2 IS yig Igf, i. e. the whole school, will be 2 x 105 = 
 210. Ans. 
 
 Example 9.— Two persons start at the same time in op. 
 posite dn-ections to walk from Toronto to HnmiUoH ' 
 cunce of 88 miles. A travels from Toronto at the ra 
 nules per hour, and B from Hamilton at the rate of - 
 perhoui; when and where will A and 13 meet ? 
 
 5 -- .>!,3- 
 
 j§ail"v..i«M*., 
 
174 
 
 llfli 
 
 ANALYSIS. 
 
 OPERATION. 
 
 It i3 evident they ,npproach each other at the rate of 6 + 4 
 =: 9 niilos ])cr hour, and hence the time will be 38 -i- 
 9 — 4:'^ hours. Next, A. travels 4| hours at the mte 
 of 5 miles per hour, therefore they will meet 4^ x 
 6 = 21^ miles from Toronto, or 4f x 4 = I65 miles 
 from Hamilton. 
 
 Exercise 80. 
 TTow many times 3 arc y\ of 77 1 
 How man J' times 5 ?ro f of 49? ^ 
 
 ITow many times 7 .ire ^^ of 130 ? 
 How many titnoe 9 are g of 70? 
 IIow many times 12 are § of 54? 
 72 is t"; of how many timea 5? 
 121 is H of how many timeiJllO? 
 48 is I of how many times 7 ? 
 78 is ^\ of how many times 11? 
 I of 26 is J of what number ? 
 f of 42 is TT of what number? 
 I of 81 is ^% of r.'hat number? 
 T*T of 9r "s I of what number? 
 
 If 8 cows srive 44 lbs. of butter per week, how intich tnay bf 
 ex]iectei' from 11 cows? 
 
 If $27 pay for 9 bariels of apples, what will 23 barrels of apples 
 cost ? 
 
 If 13 davs' work cost $7-80, for how many days workwil) 
 $19-80 pay ? 
 
 A can do a piece of work in 9 days Avhich A and B workinj? 
 together can do in 4 days ; in what time could B alone do it ? 
 
 A can do a piece of work in 10 days which 1) could do in 7 
 days, and C in 12 dayt* ; if they all three work together at it, 
 in what time ca, tlicy finish it? 
 
 A, B, and C can toeother do a piece of work in 18 days. A 
 alone c;in do it in 35 days and C alone in 42 days. In what 
 time could B working alone finish it ? 
 
 SO. ^ person boniE:ht a cow «nd paidflO 5 cash, and this was 3 of 
 
 a f^f !iH »<■ f I, ;„« . U»... .«..,»!, .i:.j i :.._ r — .i-_ « ' 
 
 1. 
 
 2. 
 
 8. 
 
 4. 
 
 b. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 10. 
 11. 
 12. 
 13. 
 14. 
 
 15. 
 
 16. 
 
 17. 
 
 18. 
 
 19. 
 
 27. 
 
e of 6 + 4 
 11 be 38 -^ 
 it the mte 
 leet 4^ X 
 16§ miles 
 
 21 
 
 INVOLUTIOK • 
 
 r» the mixt„re vvorih „„ Jillon i ° "' '" *^'"' 1 ^'■''='! 
 
 inect? '''^ """' > wbon and whuro will they 
 
 before he ovJrtakeg B? ' "^ "^^^y milcg will A travel 
 
 28. What number is it that the i o,,^ i j , 
 
 104 ; '^^^^ ^^® * «'-d i and i and ^^ of which make 
 
 29. mnt nnmbor is that i of which exceeds A of it bv 2 ? 
 oO. A certain iiumbpr in difiri.,/j i ^ , „ ^^ " ^^ 
 
 iB^ «"blracSd ; the rV,ttndor^ i«"1»f'"""!,^ '?^ ^^« ^"'^tlont 5 
 added to the auotfJt ^nl^.l '^ *^^" divided bv 11, 7 i" 
 ^f I of I of ^tS product t^u"^ '^^Itiplied b/S ; ,iow | 
 
 20. 
 
 27. 
 
 ^f I of I of "this product 'i'I^^""' '^^^P"'^ V^ ; "iiow j 
 number f proauct w 11^; what was tie origin J 
 
 ich tnay t)f 
 
 Is of apples 
 
 work wi I) 
 
 B workinj? 
 ilone do it i 
 
 uld do in 7 
 ether at it, 
 
 5 dayp. A 
 . In what 
 
 ff Avas } of 
 eow I 
 
 SECTION X. 
 
 INVOLUTION. 
 

 i 
 
 
 1 
 
 1 
 
 ^ I 
 
 176 
 
 mvor.uTioN. 
 
 2. Tho number wliicli, beinjjj iiiullipHo(l oju'C or oflohs'^p 
 
 by itself, produces the power, is calle:! the root of iluit 
 
 power. 
 
 Tluirt 5 if* t1io root of 25, sinco 5x5 = 21') ; 3 is tho rnot of 81, 
 6li;ce 8 X 8 X 3 X 3 .— 81. 
 
 3. The powers of a number are enUed the ./7>'sV, semvd 
 
 Ihiniy fourth^ JifilK &e., ucoonlinjj^ as the root is taken o/<(>, 
 
 iw^ J, th)'ic(\ four timcs^fivc thncs, &c., as faetor. 
 
 Tlius si is called the fourtli vo\\pr of 3, lu-ctiUKo 3 In trikon 4 
 tluu'8 a« fac.uu', ill order to protliui! !S1. 
 
 4. The second power of a nmnbor is also called it.i 
 sqnaiv^ b«»cnuse a square suii'ace, the length of one of 
 whose sides is expr«>ss(Ml by a <:;iven nunibci", will have its 
 area expressed by the second power of that nvnnber. 
 
 G. The third power of a number is also called its r?/ft«, 
 because if the length of one side of a cube be expressed l>y 
 a given nund)er, the solid contents of the cube will be ex' 
 pressed by the third power of that number. 
 
 9 
 
 81 
 729 
 
 TABLE OF SQUARES AND CUBES. 
 
 Roots... 1 2 3 4 5 () 7 8 
 Squares. 1 4 9 IG 25 Sf. 40 64 
 Cubes... 1 8 27 (54 125 216 843 512 
 
 6. The ivdci' or cxponcni of a powtM- is a small ri'!;urc 
 
 written to the rip:ht, hulicntinj; how ol'fcn the root hay 
 
 to be taken as factor in order to i)roduce the given power. 
 
 Tlius, 2=2 r: 2 = Fiivt povor of 2. 
 
 12' =: 2 X 2 = 4 =r Socoi d yowvY of 2. 
 
 23 zr "^ X 2 X 2 = 8 =r 'I'liird iH)AV(M-()f 2. 
 
 2< =;! X 2 X 2 X 2 = \^^ — I'\)!irt.h power of 2. 
 
 2» :z ■■' H 2 > 2 X 2 X 2 =: 32 rr Fifih power of 2. 
 
 7, The prof! :«! '>f fmding u power of a given number by 
 mnitlply^ug it into itself is called involution. 
 
 8. Tu mvcbre a number to any required pow'v •— 
 
 RULE. 
 
 Take. tJie rfiven number an facfor ok qffru as' ivdh'aUil 
 hji the inik,ic of Ui.c retjuiriid ^owcr a:id Jind ' ' jjroduci oj 
 ihcso J actors. 
 
•e or oflohs^r 
 ruoi of ituit 
 
 he root of 81, 
 
 /rrsf^ aecnrul 
 4 tukcu om i, 
 )r. 
 3 Irt tnkcn 4 
 
 jjo called ita 
 
 of ono of 
 
 ivill liave its 
 nher. 
 
 1 0(1 its athe^ 
 jxprossod l»y 
 I will bo ex- 
 
 8 9 
 
 I 64 81 
 ! 512 729 
 
 small fi'!;nrc 
 he, root hay 
 von power. 
 
 :)f 2. 
 i-.)f2. 
 of 2. 
 r of 2. 
 of 2. 
 
 1 niimbci' bv 
 
 W;'V— 
 
 r/.v ivdlrnuil 
 jfjt'oduci qJ 
 
 EXTHACTION OF SQUARE ROOT j-^. 
 
 .-n^^T^T^" ^♦»w/r«/rflC//on/»,tr<? multiply hoth nnmcrntoro and 
 t» tmproj,er/i avtfuns and (hen proceed an alwvn. 
 Example 1. — What is tlio Ctli power of 4? 
 
 OPERATION. 
 
 4» = 4x4x4x4^4 = 1024. An$ 
 Example 2.— What is the 8^' power of 2ii ? 
 
 OPERATION. 
 
 11 X 11 X 11 
 
 
 0x0x0 
 
 EXKRC'ISK S). 
 
 Find Uio value of— 
 I, Tho square of ^1. 
 £ Tiic cube of 23. 
 
 The square of 279. 
 
 Tho cube of 81. 
 
 T]\f fourth power of G. 
 
 'Vhiifi/th power of [i. 
 
 The sixth power of 4. 
 
 Tho seventh power of 3. 
 9. Tho cig-A^/i /jojf er of 2, 
 ('^ Tho m«^/iy)otcer of {'I 
 
 3 
 
 4 
 6. 
 
 6. 
 7. 
 
 8. 
 
 11. Tho third power on. 
 
 12. Thc'/«Mr/A /;oicf;- of H, 
 
 13. Tlio A'/(r//i power of 9. 
 
 14. TlH\fi/th power of I 
 10. TJie sq;uire of I'i'.'f). 
 10. TIio square of 4t;37 
 
 17. Thoc?/Af'of4i. 
 
 18. The cuhe of 29. 
 
 19. T]\Q fourth poirrr of 21, 
 
 20. The tenth power of 3. 
 
 iLXTRACTIOr OP Ti'lE SQVAUK KOOT. 
 
 9. To extract the squr.re root of a number : — 
 
 KULE. 
 
 T. Point off the pivcn number into periods of tv'ofgure9 
 tdfth, beriinmnrj at the. decimal point. 
 
 II. Find the higheat sejuare contained in the left-hana 
 fenod, and place its root to the right of the numOer, in the 
 
 12 
 
 sk^-' 
 

 178 
 
 TRACTION OF SQUARE ROOT. 
 
 1 !; 
 
 \i 
 
 i 
 
 i -, 
 
 ■I 
 
 'i}> n 
 
 m 
 
 h t-r. 
 
 III. Subtract the square of the digit put in the root from 
 
 the left-hand period, and to the remainder bring down t/iA 
 next period^ to the right, for a neio dividend. 
 
 IV. Double the part of the root alrcadg found for a 
 
 •TRIAL DIVISOR. 
 
 V. Fimi how many times the trial divisor is contained 
 in the dividend, exclusive of the right-hand digit, and plac^ 
 the figure thus obtained both in the root and also to the right 
 of the trial divisor. 
 
 VI. Mulliplg the divisor thus completed by the digit Uu\ 
 put in the root ; subtract the product from the dividend^ 
 and to the. remainder bring down the next period for a nen 
 dividend. 
 
 VII. Again, double the part of the root alreadi/ found 
 for a 7iew tuial divisor; proceed as in V. and VL, and 
 continue the process until all the periods are brought down. 
 
 Note.— To extract the square root of a fraction, extract tlio 
 square root of the numerator and of the denominator eeparatolv. 
 If tliey be complete squares ; if not, reduce tlie fraction to its 
 equivalent decimal and extract tlie square root by the ru'e. To 
 extract the pquare root of a mixed number, reduce the fractional 
 part to u decimal attached to the whole number aad then extract 
 the square root. 
 
 Example 1. — What is the square root of 576 ? 
 
 OI'ERATION. 
 
 576(24 
 4 
 
 14)176 
 176 
 
 Here we place a point between the 7 and 
 the 5 and thus divide the number into twc 
 periods. Then the highest square in 6, the 
 lirst period, is 4, the square root of which, 2, 
 we place in the root. Next we subtract tht 
 4 from 5 and bring down the next period, IQ 
 which giver; us I /6 for the next dividend. Then we double 
 t!ie 2 in the root for a trial divisor and a.^k how often this 
 4 will go into 17 (the dividend exclusive o7 the right-hand 
 iignre) ; obviously 4 times ; next we place this 4 both in 
 the root and in the divisor, multiply the complete divisor 
 thus formed by the 4 and subtract. 
 
EXTRACTION OF SQUARE ROOT. j^^g 
 
 Example 2. — What is the square root of 322G-j\ ? 
 
 •OPErwATION. 
 
 322(>-iV =:: 3220 235294 
 
 8226 23529-l(5G-7999, &c. 
 25 
 
 106) '?26 
 636 
 
 112Y) 90-23 
 
 '78-89 
 
 113-49) 11-3452 
 10-2141 
 
 113-589) 1-131194 
 1 • 022301 
 
 113-5989) 
 
 10889300 
 10223901 
 
 085399, &c. 
 
 Here, after reducing ,^7- to a decimal aud iiiinexing it 
 •■^o the whole number, 3220, we mark olf both ways fiom 
 the decimal point into periods of two figures each. Then 
 the highest square in 32, the left-lumd period, is 25, the 
 square root of which is 5, and we accordingly put 5 i:i the 
 root; next we subtract the 25 from the 32 anil biirig down 
 the next period, 26, which gives us 720 for a new dividend. 
 Then we take twice 5 — K), for a trial divisor ; iind how 
 often 10 will go into 72, aj.;;arcnt!y 7 times; bat when we 
 ivy 7 we find that it is too great, and therefore we try 6, 
 which we put both in the root and in the divisor. Next we 
 multiply the 100 by and subtract the product, 636, from 
 720, and to the remainder bring down the next period, 
 which gives us 90-23 for a new dividend. Next we take 
 twice 56 =112, for a tiial divisor, and find that it will go 
 into 90*2 sevr.i times, and we accordingly place 7 both in 
 the divisor and in the root, multiply, subtract, and brinjj; 
 
1 1 
 
 ,\ 
 
 liii 
 
 180 
 
 E.VfKACTIOx\ OF CUBE RQOT. 
 Exercise 82. 
 
 Find tho square root of ; 
 
 1. 1296. 
 
 2. C9C9. 
 
 3. 15870. 
 
 4. 63361. 
 
 5. 1428S4. 
 C. 99S001. 
 
 7. -244006. 
 
 8. -305641. 
 0. 75(3-25. 
 
 lU. 11397 -4840. 
 
 11. 0S120-47891. 
 
 12. 6712914-23. 
 
 13. 918767. 
 
 14. 429|. 
 15, 
 10. tV 
 
 17. 428f. 
 
 18. 629*. 
 
 19. 1127893}. 
 
 20. 213798 1237 
 
 Tn> (tu iutf) 
 e 
 II XT' 
 
 
 EXTRACTION OF CUBE ROOT. 
 
 10. To extract the cube root of a number is to find a 
 number which, taken three times as factor, will produce the 
 given number : — 
 
 RULE. 
 
 I. Point of the number into periods of three fmireh 
 each, hegimnnj at the decimal point. 
 
 II. Find the highest cube contanicd in the left-hand 
 period and place its root to the right of the number, in the 
 place occupied by the quotient in division. 
 
 III. SuUract the cube of the digit put in the root from 
 t/ie left-hand period, and to the ronaindcr bring doun th' 
 next period to the right for a new dividend. 
 
 .' ^^/."'{''i^'-^'?/ '^^''^ ^^'"^^^''' ''/^^'^ P^^'^ ^f '^*« ^oot already 
 Jonnd bg oOO/o?- a trial divisor. 
 
 ^ y. nnd how mmiy times the trial divisor is contained 
 in the dividend and put the figure thus obtained in th 
 root. 
 
 VI. Complete the trial divisor by adding to it ; 
 
 1st. Tlie part of the root previously found x t\ 
 
 last digit put in the root x 30 ; and 
 2d. The square of ihe last digit put tn he root. 
 
EXTRACTION OP CUDE KOOT. ,01 
 
 J 81 
 
 hrntio place nc:H in the rootash.vT'/',.,?' 
 
 period, are hrouguLT '' ^'''''' ^"'^^'^ «^^ '/''' 
 
 dec&aJ t.:;;;^^^,.!:!? ^^^ -^ ? ^---tion, voOnco U to a 
 
 cube root of each. To ox ra ot t L ?.ni /""^ ''•'"■ e^'ract Iho 
 redu^o tbo fractional inir{^ to , ec.x^^v V)f?vl'\r''^';'' ',"^"''^^'-' 
 ber, a:.d ti.cn extract [he cube roora.dir' 'tS abc/v^' '"^'''^'' ""^"' 
 
 EXAMPLE—What is the cubo n>rt of 429172^^5200'?? 
 
 OPER/.TIOaV, 
 
 let trial divisor — 73 x 300 = 
 iHt increment =7x6x30 = 
 
 2i 
 
 52 - 
 
 tsi complete divisor 
 
 42v\-6326o-(75>3. ^„,. 
 11;00 8617J'vTlst dividend. 
 
 lUOO 
 
 25 ' 
 15775 78875 = ptTl,cto''comp. 
 
 42= 10 
 
 2J complete divisor 
 
 - 169G51G r;78G064 = pro'''nct nf 
 coin p. div. by 4 
 
 = 170022669 511868007 =: product of 
 
 coin;), d.v. by 3. 
 
 8J coinplotp. d' .f'ior 
 
 Iowa 
 
 thua obtuu. 80172 fur a i;ew dividend'. 
 
182 
 
 MISCELLANEOUS PROBLEMS. 
 
 i' 
 
 I I 
 t 
 
 j 
 
 ;|i I 
 
 Next we take 7, the part of the root alrondy found, pqiiaro it ,f4 
 multiply tlio 49 tliuri obtained by 300. This ijivos the fiivt vri:ii 
 divisor, 14700, which we find will go into the dividend 8017:2 (mak- 
 ing due allowance for the increaBe of the divieor) 5 times. 
 
 Next wc conipleto the divinor by adding to it 
 
 iHt, 7x5x30=1050, and id, 5'' = 25, which gives us 1.5775 for a 
 coiui^lete divieor. 'I'hirt we niu!ti])]y l)y 5, the diirit la.-t i)nt i'l llni 
 rout, subtract the product 78875 from the 1st div'dend, and to tht. 
 remainder, 7297, bring down the next period, 932, «kc., &c. 
 
 Exercise 83. 
 
 Extract the cube root of : 
 1. 32708. 
 fS. 658503. 
 S. 13S24. 
 
 4. 250047. 
 
 5. 970299. 
 C. 1953125. 
 
 7. 15813251. 
 
 8. 48228544. 
 
 9. 245314376. 
 10. C86' 128968. 
 
 11. 091028-973. 
 
 12. -915498011. 
 13 
 14 
 
 S 1-JG S4S «« 
 II s 
 
 Tai Tc* 
 
 15. -9, -1, 1. 
 
 16. 427986-7143. 
 
 17. 816?. 
 
 18. 917167tV 
 
 19. 8111471rV 
 
 20. 27J. 
 
 ; It. 
 
 Exercise 84. 
 Miscellaneous Prohlems. 
 
 1. 'Divido $7994 70 equally amo c: 29 per-ons. 
 
 2. The iliU'ei'ince of tv.o numbers i.> 127 and the greater is 24& 
 
 what irt the sn.allcr ? 
 
 B. Reduce £291 O.j. 4 J 1. to dollars and cents and divide Ihe iesu\ 
 
 by 9. 
 
 4. De<iiict 29 ner cc;:t. from $429 80 and divide the remainder bj 
 
 $10 20. 
 
 6. Find the A'a'ue of 2J + -1.\ + ' of ■' of 4^ + ^ - 5^. 
 
 6. Wliat irf the simple interest of $94o-70for 11-2 years at 9^ per 
 
 cent, per annum ? 
 
 • • • • • • 
 
 7. Kedtice 7, '42, 2357 and 376 to their oq^uivalcut vii'gar frac- 
 
 tione. 
 
MISCELLANEOUS PUOBLLMS. 
 
 183 
 
 8. Bought 729 barrels of flonr for $2916 ; for what mut^t I Bell it 
 
 per baiTel in order to jcain 28 jicr ccrtt. ? 
 
 9. ; of S of ^ of 63 Is I of how nm'.'.y t.mce 8 ? 
 10. Extract the cube root of SGI 72-1913. 
 
 ^1. In 1858 tlicrc was exported from Canada $370951 worth of 
 dried and smoked fish, J27C104 wortli of pickled fisli, $19592 
 worth of fie,-h lisli, :ii,d |o893G Avorth of tish oil ; wl.al was 
 llie totnl value of the lisli and iioii cil exported from Canada 
 in 1558? 
 
 ?2. What is the value of 27 Ibe. 4 oz. 6 dwt. 17 gre. multiplied by 
 C29i? 
 
 >3. Reduce Jf, i||, f 5«|2, and if«»J to their lowest terms. 
 
 l>, What is tlie value of -714625 of a mile ? 
 
 lb. Divide 90-478 by -002693. 
 
 13. If li of a vessel cost $6294-iV, what will /f of the vetoed co^-t ? 
 
 17. Find the price of 914 ]h». 7 oz. 5 dr3. Avoir, at $11-49 per lb. 
 
 18. What ip> the bank d'scount, and wliat the true discount on a 
 
 note for $1100 due months hence, dit?coui. ting at 7 percent, f 
 
 19. A, B, ai;d C can do a piece of work in 10 days, A woiking 
 
 alo: c can do it in '28 days, and C worki; ir aloi e can do it iu 
 32 da\ 8 ; in wiiat time can I> working uloi.e do it ? 
 
 20. What is the square root of 149, -'f ? 
 
 21. From the upper end of Lnke Superior to the mouth of the St. 
 
 Lawrence is about 2000 miles ; what time wottld a vesKcl rc- 
 quin- to make this voyage with an average speed of C| miles 
 per hour? 
 
 22. What is the difference between i;219 83. Hid. and g of 4| of | 
 
 of ^ of 24i times $976-53? 
 8| 
 
 23. Divide 978 acres 2 r. 1 per. 7 yds. by 8 a. 3 r. 27 per. 2 yds. 
 
 24. Express 27, 393, 4700, 78904 and 9136718 in Roman Numerals. 
 
 25. What is the ratio compounded of 19 .-4, 11:5, 13 OJ, and 
 
 33:17? 
 
 26. Find the Q. C. M. of 27051 and 16013. 
 
 i |s< 
 
 li 
 
184 
 
 !»]] 
 
 
 I 
 
 Mil 
 
 
 MISCELLANEOUS PHOnLEMS 
 
 
 3- Find ,ho 1. c. ,„. Of 1.,, 18, 20, 24. 60, 72, 80, 88, 90, „„u :oo: 
 
 3f) R'clucc 9146714 inches to acrc8 
 
 SS. > in,l iiio value of -14072 of 17 bushels 1 pk. 1 gul. 
 30. JJxlract the cube root of 7149-"-. 
 
 charirin:. pi/.o by vh o^ t r mlT''"*'''-' ^S'^^^ also a dig 
 NowifthJcsternbeinmvau^ in 30 minutes, 
 
 what thue will it be liUoU ^ ^^ ^^^ ^'""'^ l"t>os be opened, in 
 
 '' ""'S^^. ''' '''''' -^^ ^-° the quotient trn. to thre« deo^ 
 
 ''• ""2^;;;:^: tytl^^^ '^^^^°^ ^^ - 2 r. 7 per. ... i« a. 3 r 
 43. Reduce 278 yd.. 3 ar. 1 na. 2 In. into inches. 
 
 1 ^ 
 ' 3 
 
MiSCELLANEOns PKOiiLEMS. 
 
 loo 
 
 «inip:o quantity. ^^* -^'* I "i w lo a 
 
 !». W|,-.. i, .h„ CO,, of 2, „. , , ^ ,„„ ,, ,^,.,„ . 
 »o. »> ii.ii ifi till' worl}) of *-U(»fi u'.v I. • ^, 
 
 'Imaufu of 00 nii'08) ""■""" '° ■Ijo la.ls of Kiagani, a 
 
 ♦9. So!,l my farm foi-JTSM, whiel, wq« in „ . 
 
 co«t luc ; whal ,U I givofor U J "^ °"'- "'"'''' ""•" " 
 
 '" ^l^mtJi:';:::'^^^^"'^ 7yea„.S„,„„th,,20 cay, 
 
 M. Eyre., 2 lu,. 3 oz. 4 dwf:;:7;rac,i„„ of n ,D,. 7 oz. 9 dwt. 
 
 82. D:vi,loi:493I33,4j,I,byi;8n8.M 
 
 63. Find the ,. 0. ,„. Of 5, 8, n, u. U, 20, 22, na, 016, and 42 
 
 64. My MROnl soils f„r „„. 415 l,.irr,.U nf (! . « 
 
 allow per UusI.el for ih c'oaU J ' '''"^' """•■'' ''""S ha 
 
 67. Three persoMs roit .1 pasture for 4!onn a , . 
 
 4nio.,ti,s,K J BhU, f,romo"?^;„ ^^ P" « '" 207 shoep for 
 niontlis;\v jatroiM on nfii,« :''f' "'"^ ^' 43 cows for 41 
 1 cow to' be iiulvaSto 5 a;'^:;;'/''""^^ ^^^^'^ 1^^^^'- ^^l°^vi'^ 
 
 68, If ft grocfr mxes 23 lbs of tc-i -if sn 
 
 30 lbs. ut 40, .iKl 42 b'* at eo n,"^ 1?^ "'".^' ""''"^ ^^ >^^- «t 75, 
 
 {9. Di.-.tnbiito^noOamon'- A r? r on.n% 
 
 as A, B, a..d togetlier. lo^'ether, and D as muo] 
 
 18 
 
 uoh 
 
 day ni„.<t la mc:, u.,rk in o- J^i to ^^i ' 'i'"'''-"'-Vi.v hours 
 S leei Wide, ui.d icet deep , in 12 d'a^ s ? ''''"''^ *^ ^"'^^ ^°»if» 
 
 ■et 
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IMAGE EVALUATION 
 TEST TARGET (MT-S) 
 
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 1.8 
 
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 23 WEST MAIN STREET 
 
 WEBSTER, NY. 14580 
 
 (716) 872-4503 
 

 fA 
 
186 
 
 MENTAL ARITHMETIC. 
 
 MENTAL ARITHMETIC. 
 
 i -N 
 
 ! 
 
 SUGGESTIONS. 
 
 Tho following oxoroises will be fourd to he too difficult, Jn mont 
 infiiunci'Pj fur l-ginmMs, Hoforu ei,'a-i-h g ll,cni, the pupil is eup, 
 po.-cn ;,j ]i;ive been tliormiiiiiiy uvUWi in i rolik'iiis of it.i t-nsier dl" 
 Bc.'iption, as, foroxaini.lc, t!i()8i t,'ivcii in tliu boiiy of the book at 
 the commeiiceim'nt <.f eacii of t}ie tiinpU' rales, under tlie head of 
 Mcnta, Exorc KOH. Wherever the teacher finds the ex-^rcisc toO 
 nuicJi advanced tor liife clar^s, iie must brealt down tlie difficulty by 
 propos iig •unuion.^ i anii r quej>l:ous, of his ow;i co;.Btruction, hi 
 vo.V'ni,' the same ) r nc p'(>. • 
 
 I '^'l'" 'mm '•i''""'^l»<'''l'i be grneral'y conducted -with the text-bo- ca 
 closnU. i lie tea-her should readout the pmljlein slowly and dis- 
 tinctly, and hut onve, -mA the class should be required to solve it 
 m^ntalh/ and in perfect sHenre, and w.thout givin,,' any sign or siff 
 11*1 when they are ready to answer. After a r^pace of time sufficient 
 in- the hOiiiUon of the problem l.r.s eiai^scd, t^ie teacher gives a 
 signal upon wir.ch those ^v]lo have completed the process raise thv 
 hand. One of the.^e is then required to give* llic ref^ult. 'J'he 
 teacher ascertains how many a'/ree with it,''}ind calN upon some 
 one of t.icm to solve and analyse it for the -^las.-. Then another 
 problem IS proceeded with in tho same manner, OccasionaUv, es- 
 pecially i!i review le>-^8ons, the clasd may be '.llowed to recite' with 
 open books. 
 
 The pupil should be inquired to adhere *o the form of analvsia 
 given, unless the teacher can devise a bettei for him. In the same 
 8cnot)l a uni/hrin pliruseology in the solutious should invariably be 
 adopted. ^ 
 
 In order to secure the attention of the ' iitire clasp, no intima* 
 tion byword or glance should be g ven .'^ n to the member of tho 
 c.aso to be called \\\m)\\ for an sniswer or solution ; ho tliat every 
 one considering himself liable to be sclect'^d for that purpo.-^e, shall 
 concentrate his mind upon the question. 
 
 Finally, the instructor of youth should Always remember tJ.at 
 Mental Arithmetic, as a branch of schooi iftudy, is i.o iesigned to 
 be merely a nieans of daiizling and bcwi deriig theadalt publ.c by 
 the rapidihj w\{\\ which the pupils ari trained to solve certain 
 c.as-eso! problems, but rather us a mjiital trainii g of iho very 
 hi- !ie..t character. Its object maybe said to be threefold, viz. : 
 Jnthellrst place, to enable the pupil to solve mentally and with 
 taci.ity the ma.iorty of the problems that arise in tlie bu-iness of 
 cvi ryday life, and Avliich otherwise hfj could work only by tedious 
 projesees on his slate ; in the second pace, to fmniliarize him with 
 tne processes employed in written nri'.limetic, to render that part 
 ol the study of numbers clear to liis c^miirehension, and to make 
 lurn in a manner independent of m-M-e book-rules; and in the 
 third place dvuVprwiari/i/, to cultivate his powers of analysis and 
 '•s ability to concentrate his attention on a given eubject.~in a 
 x\>>rd, to develop and invigorate the most important of his intellec- 
 tual tacultJes. 
 
 i 
 
'm^*m^M*miii^-' 
 
 Mental arithmetic. 
 Exercise I. 
 
 187 
 
 lilt, J n mont 
 ipil is eup. 
 
 I cnsier du' 
 he book, at 
 lie head of 
 x'irciso too 
 ifficulty by 
 ruction, iu- 
 
 e text- bo- rs 
 ly and dis- 
 to solve it 
 eign or sig 
 esu&icieilt 
 fier gives a 
 58 raiee thv 
 ■f'lilt. 'J'iie 
 upon eonio 
 en another 
 ionaUy, es- 
 recite'with 
 
 of analysis 
 
 II t !ie banie 
 .'ariably be 
 
 no intinia- 
 ibt r of tho 
 Hint every 
 poae, Bhaii 
 
 mber tliat 
 'esigned to 
 : pubLc by 
 vo certain 
 f the very 
 sfold, viz. : 
 / and Avith 
 u^iiiess of 
 by tedious 
 » fiinj with 
 • that part 
 d to make 
 md in the 
 aiysis and 
 ject. — in a 
 is intellec- 
 
 2. How many are 12 and 11? 11 and 17? 13 and 14? 17 and 19? IS 
 aim l3 f 
 
 5. How many are 23 and 71 ? 38 and 47 ? 63 and £9 ? 29 and 81 ? 4' 
 4. How many are 123 and 47? 276 and 93? 489 and 29? 714 and 8C^ 
 
 ^' ^Zd^m ""'' ^^^ ""''^ ^^^* ^^^ """^ ''^^^ ^'^ ""^ ^^^^ ^'^9 
 
 6. How many arc 1478 and 976 ? 2913 and 579 ? 287 and 9163 ? 
 
 7. How many arc 4916 and 7189 ? 9612 apd 3407 ? 9161 and 7863 7 
 
 8. How many ore 19 and 18 and 27 ? 
 
 9. How many arc 28 and 143 and 729? 
 30. How many are 493 and 7816 ? 
 
 H. How many arc 9167 and 2G47 ? 
 
 12. Hr V many are 7 + 19 + 23 + 47 + 98 + 127 + 246 ? 
 
 EXKRCISE II. 
 
 1. From 17 take 8 and how many remain ? From 81 lake 43 and 
 how many remain ? 
 
 2.' From l-^S take 48 and hov many remain ? From 217 take 109 
 and how many remain ? 
 
 a How many are 43-27 ? 93-42? 6'^-43? 128-89 J 
 
 4. How many arc 768-400 ? i'9:;-150? 671-428? 678-434? 
 
 5. How many are 47-29 ? 78? -43? 675-71? 891-476? 
 
 6. How nany are 893-473 ? 9.S1 -671 ? 49G-r39? 781-407? 
 7 How many are 471-89 ? 90- -?3 ? 471-426 ? 711-189 ? 
 
 8. IIow many are 8146-23? 7167 -93? 9146-217? 
 
 9. How n\any are 8371-986 ? 6242-555 ? 9167-S147 ? 
 IP How many are 9187-8674? 9321-296? 817-4901 
 V How many arc 784-27-98-423-11 ? 
 
 "^ Ho-c wiany are 9807-2143-^7S~3iS-276-43| 
 
w 
 
 188 
 
 jl 
 
 If 
 
 1,1 
 
 MENTAL AlilTriMETIO. 
 
 KxicnciSK Iir. 
 
 1. Wow nmnynro 0+8+17 + 43-11 >J4»72 + lS + 0-l7 + 2f 
 
 2. »l'»wiu.uiyaivli-.a + s-2 + (i;j-.47 + joy_„7, 
 
 a Ilaw nv.ny u.-o li7 + 4n-l..-,UH7-8:i + 14 + ,y.,„3_,„o, 
 4. liow nmiiy lu-o 8(,9 + 47(»~3J7-217 + 4f)i ? 
 
 ^ IIo.v uuxuy ,uv Ul«7-.m-097-27-.i4-9 + 8 + Dl + 7fl + 12ft 'iSJ 
 6. How .u:uiy aro 1070-310 + 278-0+ 78-03 + 217 + 411 1 
 
 EXKHCISE IV. 
 Comjult to inomory tht» following :— 
 
 \ EXTKNDKD MULTIPLICATION TAnLR. 
 
 13 times 14 times 
 2 nru 20 2 — l'8 
 8 - 31) a - 4' 
 
 •] - U J - 4>, ;{ _ f,, 
 
 
 6 — Hf) .) _ 
 
 - 7.S ti - 
 
 7 - 1 : 7 - OS 
 
 8 - 104 8 - 11-2 
 
 0-117 9 - Ijti 
 
 Volo^ 7» ^- 8(! 5- 8a 
 
 S4 - 9(1 ,) - 9,1 ., _ ■,„:; 
 
 7 - lOA 7 - 11-j 7 _ -jir, 
 
 'J - KJf. 9 - 144 : 9 - 153 
 
 18 times ;» times 
 
 : - «6 ) ^. as 
 
 « - 54 a - f,7 
 
 70 
 i - 9(' 5 - 05 
 
 '» - \m tt - 114 
 
 7 - 12^ 7 - 1:53 
 S - 144 8 - u,> 
 9 - lOS I 9 - 171 
 
 3. How mnny nro 16 tlinos 43? 20 limcR 'Kit aa .. 
 
 tunes 19? ^ — umcB 37? 44 times 26 « 19 
 
 4. How mtuiy are 83 timr-a -roj nn <i 
 
 tiin.8419? ^^ ^'^^ ^^ ^'"^«-^ ^'a? 34 timofl 97f lo'S 
 
 6. How many aro 47 tlmpa a«j "r n ^^. 
 
 "A 
 
 6. II..W n,any are 717 tin.cs 25? 101 tunc. 102? 227 times 40." 
 
 : ' ;" r T '"'"'' "' ^° >^ «1 ^ n X S3 ? 4207 . 8 ? 71.4 . . 
 
 8. ^V hat 18 the product of 9137 x 8 ? 21074 x 11 ? 704 ^ 05 j 
 
 9. What is the product of 217 S 17 ? 4079 x 9 ( 2703 x 3^? 
 10. What iB the product oi mi x f S888 x 77 ? 8967 x 64 ? 
 

 180 
 
 I- 100 1 
 
 (5 + 129 :i8J 
 ? 
 
 on 
 
 y*tiino8 
 
 'M 
 
 ) -. 
 
 3« 
 
 64 
 
 a --. 
 
 67 
 
 72 
 
 4 - 
 
 70 
 
 W 
 
 5 - 
 
 05 
 
 )^ 
 
 tt - 
 
 111 
 
 2(i 
 
 7 - 
 
 1:5a 
 
 U 
 
 S ~ 
 
 liV2 
 
 !,' 
 
 y - 
 
 171 
 
 ICH 
 
 19? 
 
 14 
 
 0? 
 
 tiinos 
 
 nos 26 « 
 
 19 
 
 Ofl 
 
 97 f 
 
 102 
 
 loa 
 
 13 
 
 17 
 
 a.s4fi?'" 
 
 . 
 
 i4> 
 
 .Q 
 
 
 MIC.VTAL AIMTIIAIETIO. 
 K.VKRCi.sj.; V. 
 
 '•'?;;,!;.,}"'»' »"n.< ,.,..(•«, .■.„f„„ »„,„„ 
 
 '. W;;;m» , or .00, ,„f,.„, .„„,,„,, ^^^^^^^^ ^^^ 
 ». " ".I... .„„„„■„„. „f,„„^„3 , ^,_,„ '; • 
 
 KXKHCISK Vr 
 
 I.8.1mo.„nml{ofo,u.cl.ou'ma.,ytlmc«lo.? 
 
 ANAI.VSJS. 
 8 UliU'rt niv 7'> nii/l r .<• n 1 - 
 
 . a...3™,,„r„.,„„„„,„„, ,^^ ' 
 
 ■>. m,.,.„,,, J „,,,„„ ,,„^^,___,,,^, ^____^^^^. 
 
 "• f> times 6 aid » of "r» .... v 
 
 '^- » time, i Ina l If U "" """^' "'"" * "^ ^« M of 15 , 
 una I of 44 aro Low many tim«. | of 2i , ^ ^^ ^M 
 
 a. 
 
wm 
 
 190 
 
 MENTAL ARITHMETIC. 
 
 EXERCMSE VII. 
 . 1. What is ^ of that number of whicli 12 ia j\ ? 
 
 ANALYSIS. 
 
 If 12 1b W of a certain iiuinb«r, j\vf\]] bo Ihv J of 12, which 
 iH2. 
 
 If 2 IB j\ of :i certain number, 11 times i, which is 22, wU' 
 be tliat number. 
 
 Then ^ of 22 is equal to } of 22 x 3=3^ x 3=9^. 
 
 Therefore 9J- is ? of tliat number of v/hich 12 Is xV • 
 
 2. What is -^V of that number of which 21 is f? 
 
 8. "What is g of that number of wliich 81 is g ? 
 
 !4. What is g of that number of whiclj 36 is ^\ ? 
 
 5. What is ^ of tliat number of which 18 is ^ f 
 
 6. What is } of that number oC which 51 is |J ? 
 
 7. What is I of that number of wliich 77 is |} ? 
 
 8. What is 42 times that number of which 80 is xVt 
 
 Exercise VIII. 
 
 1. 25 is f of how many times 9 ? 
 
 ANALYSIS. 
 
 If 25 is f of a certain number, ^ will be J of 25, whicli Is & 
 
 If } of a number is 5, the number must be 6 x 7, which la «L 
 Then 35 -*- 9 = 3g. 
 
 Therefore 25 is f of 35 times 0. 
 
 2. 84 is ^ of how many times 10 ? 7 ? 9 ? 11 ? 
 8. 63 is T«V of how many times 7? 8? 5? i^| 
 4. 21 is { of how many times 11 ? 6 ? 5 ? 
 
 6. 96 is I of how many times 5 ? 7 ? 13) 
 6. 121 is {\ of how many times 12 ? 10 1 
 1 106 is I of how many tImcB 6 ! ni 
 
MilNTjVL ARITHMJJTIO. 
 
 191 
 
 I of 12, which 
 Ich is 22, wU" 
 
 EXKRCISE IX. 
 t A 0^44 is ,»j- of how many ninths of 54? 
 
 ANALYSIS. 
 
 Then 30 i3 ,T, of how many 6'«. 
 
 ^^l«V>"-'^.ul if ?!''■/:"" ";•""'"''' T^ ''''" 1'*' ^ of 38, which 
 
 Then 51^ divided l>y equals 8f. 
 Therefore ,«j of 44 is ^^ of 8^ times J of 64. 
 
 2. I of 64 is J of how many times J of 16 ? 
 
 3. I of 4S is 5 of liow many times | of 91? 
 
 4. /r of 77 is | of liow many times ,«. of 88? 
 ^ of 91 is T"r of how many tinuri J of 30 ? 
 I of 104 U I of how many times | of 50 ? 
 i of 63 is ^ of liow many times ^% of 150? 
 
 8. rV of 121 is JJ of how many times f cf 21 1 
 
 I '■ '''' 
 
 , whlcli Is 6. 
 which is 8& 
 
 Exercise X. 
 
 I' w^! "^.'n ^ ?' "^''^ ^^ ^^ «^'"^P *^ ^4-20 each f 
 2. What will be the cost of 11 ho.«es at $79-80 each? 
 
 la!l:;rf ''"' """^^^^^ '« ^«3 ' "^<' g^^-ter Is 284. what Is the 
 *• '^t.Sr'P^^''*' ^*'' '^^ P^^'^"^^^ »840, what is the multlpli. 
 
 6. What Is the difference between $278-80 and $127-63? 
 u. What is the ninth part of $29S7 80? 
 
 7. What is the product of 783 x 72 ? 
 
 8. How many are | of 639 ? 
 
 9. 8 times 5 and } of 20 are how many times 7? 8 ? 9? 
 
 11 What is i or I of tlmt number of which 84 IB Vt 
 
192 
 
 MENTAL ARITHMETIC. 
 
 12. What is J of » of that numher of which 27 Ib ^ f 
 
 13. 42 Is I of hi)W many timcH 6 ? 7? 11? 
 
 14. ,»j of 65 i:i I of how many times J of IS? 
 
 15. J of 4 times G^ ih ? of liow many times g of 3 of 2 t!mc8l5f 
 10, j»r of U timcrt 9» is J of bow many tiiues '{ of J of 6 iliuea 8j f 
 
 Exercise XL 
 
 1, If 3 of a barrel of apples cost $1-80, what is that per barrel ? 
 
 ANALYSIS. 
 
 If I coBt 11-80, i will cost i of $180, which \s $0-90. 
 
 If ^ cost $000, tho whole barrol will cost [^ times $0M 
 
 wliich is $2'70. 
 Therefore iCf of a Itarrel of apples cost fl'SO, the whola 
 
 barrel will cost i52-70. 
 
 2. If A o^ '*■ 1^^- *^f ^^'^^ ^^""^^ ^"^ cents, what will 1 lb. oo^t ? 
 
 3.- If g of a day's work cost 87 cents, to what will 4 days' work 
 amount ? 
 
 4. If i of 6 lbs. of cotfce co.'^t ? of $2, what will Vr of 5 lbs. cost ? 
 
 5. How much will } of a barrel of Hour cost, if ? cost $1-00? 
 6 IIow much will a basket of peaches cost, if J cost $270? 
 
 7. IIow much will 4 stone of meal come to if 'i of a stoiie cost 23 
 
 cents ? 
 
 8. How luuch will 6 cor 'h of wood amount to if ^V of 2 cords cost 
 
 $2-20 ? 
 
 Exercise XII. 
 
 Note.— TAe teacher must ihoronghhj explain how fractions ar* 
 added, subtracted, reduced, iradtiplied, and divided. 
 
 1. What is the quotient of 7J + 6* ? 
 
 SOLUTION. 
 
 7J = -V- and 6f = ■?^. Then 7J + 6^ = -V -»■ ¥ = ?- * 
 
 5*,= Jx5=:l?=l,V 
 
 2. Wliat is the value of ^ + ,\ - i ? 
 
 SOLUTION. 
 
 \ 7z II and 1= If . Then 5 + ,*« - * = ij + iV - W 
 = U-iS = ii = f ^'^' 
 
MKNTAL ARITHMETIC. 
 
 times 15 f 
 liiuea 8j ? 
 
 per barrel f 
 
 I \b $0-90. 
 
 La timed $0'OIV^ 
 
 1-SO, the whole 
 
 PO?t? 
 
 II 4 days' work 
 
 >f 5lbs. cost? 
 ost $1-00? 
 i8t$270? 
 a Btoiie cost 23 
 
 of 2 cords coat 
 
 w fractions ar* 
 
 :-y ^.^f = V-x 
 
 193 
 
 ** "i7o?J?? "^ * °^*' * °^* '*'*' * **^^ ^'^'^ <>^26? ♦ of f of 
 4. WjJ^aUs Uie value of | + |f | + ^K j + ^, gj + gj? 16| + 8,»,| 
 6. What Jg^the va\ue of ,V - ^ ? 7* - 2J ? BJ - 2,V ? 11 J - 7| f 27 A 
 
 ^* OJa'i'o how many fourths? 2Mre how many sevenths? S^V are 
 how many elevcnthfl ? "luj^"*" 
 
 '* ^^wh^tnumbSt?""""^*" 2|l8/,of what numlor? 5? is J oi 
 
 8. 13 Is 6 times what number ? 11 Is 4 times what number » 17 is 1 ^ 
 
 times what number? ^« •» .< 
 
 9. 68 is tV of what number? 29 is { of what number » 16 ig ♦ r' 
 
 what number? ' 
 
 10. What is the product of i x -» x J x {^ x | « ? 6J x ^ J ? 9» 
 
 11. What is the quotient of 8J -«- 1 of ^ of J of 2^ ' 
 
 12. What is the value of t x A -♦- - » 
 
 Exercise XIII. 
 .^ cost $37, what should 9 sheep cost! 
 
 1. " ANALYSIS. 
 
 Since 11 Bheep cost $37, 1 fiheep should cost ^V of $37, 
 whicli is$3X ; and if 1 slieep cost $3^, 9bhec"p should 
 cost 9 times $3^, which is $30fV. 
 
 Therefore if 11 sheep cost $37, 9 slicep should coet $30/^. 
 
 2. If 8 cords of wood cost 27 dollars, what will 17 cords co^t,? 
 
 3. If 3 barrels of flour cost 22 dollars, what will 11 barrels cost ? 
 
 4. If 7 davs' work amount to 17 dollars, to what will 3 days' work 
 
 amount ? 
 6. If 9 acre's of land cost 57 dollars, what will 13 acres coct? 
 & If 11 men do a piece of work In 40 days, in how many days 
 
 can 7 men do it ? 
 1. If 8f tons of hay be bought for $105, what wQuld be tho cost of 
 
 0^ tons I 
 
 J=U + «V-"i5 
 
 13 
 

 2 94 
 
 MKXTAL Ar.ITHMKTIC. 
 
 9. -At Jft7 for 11 buB;i"l3 of barloy, what -svouUl bo the cortt of 51 
 
 nilr'lli'iW ? 
 
 At Ibt. often for J5, how nmny 11 g. can be l.nd for $23? 
 10. If .)8 pay for 7 dayn' work, lor how many da>8 will $iO j ay ? 
 
 Fr -ciHE XI7. 
 
 1. If 3 hor?o^ con^nn? mhi»1;c:h of oats ',:\ 2 wcoke, how mail" 
 burtSie.rt would 5 ' ;ri ( o , ,un>c in 3 wtn 1:«? ' ' 
 
 iNA LYSIS. 
 
 If 3 horaos consume 8? bimlicl;^, one horrto will co:iBi!mo 
 
 I of 8?, whicli is 2.if busliols. If 1 horso con.-nmo 
 
 2JI bushels In 2 weeks, in on-j week he will (oii«iinio 
 
 i of 2if, which is 1^?. If a horcc to .suine 1J| bush« 
 
 els in 1 week, in 3 weeks ho Avill conpume 3 tinii.«» 
 
 2. If a " i' ^^'°^ '^^A bushels; and if one horse cciiKutn j 
 
 in tlonAvouUi' ^ ^'"''''' '''"' cou>nine 6 times 4,\, wl.ieh 
 men ? *' '' "'^''. Therefore, &c. 
 
 8. If 5 men can accnmT.i;.!, ''""'' ^"""' ^^"^ 1^' "i^" 4? d 3-^, 
 time can 3"r,'en7a^j oVrho'JlL^''^' ^"'^' "' '"' ""'"'" "^ 
 4. If 7 men in 4 (Iivum,,,,.!,: ,- , 
 
 in order .obnil,w.;S,!;;-l-^;^rn. 
 
 /f $60 pay 7 men for 9 d-ivs' work h "" '*'''**''T 
 
 11 men for 11 days' w-ork ? ' °^ "^""^ ^'^Ha^s tv^V/"'* 
 
 6. If 24 men can mow 6n irMv.. «^ 
 
 cuii 14 men mow fu 7 days / ^'''''^" ^ ^^^'^^' ^'o^^' ™any acre, 
 
 Exercise XV. 
 
 AJIALTSIS, 
 
MENTAL AIMTHMKTIC. 
 
 105 
 
 5 cost of 21 
 
 $23 ? 
 M'J 1 ay ? 
 
 Iiow many 
 
 11 co:iBi!mo 
 
 J CoilHUlllO 
 11 (Ulleillino 
 
 il\l bush' 
 le 3 tinii.',' 
 c cciiKutn J 
 4j\, wliicJi 
 
 If a», lie » tlio tinmbcr \U'Af will be 30 fmrs •, >vMch i« 
 42';. 1 l:c'icr(.'<- ,f 56 in U.c si.ju of 1 i, k and |, tho 
 
 1 II 111 cr ilM',1 l^4u',^ 
 
 % Ihiving con ttMl i„y ! nok., I f.,mi<l t'lrit |, J, and i addoil to- 
 
 L,«'tin.T ;iii.uii,,UM t.i &'JJ ; !:ov.- m:;iiy h i.. l? 
 
 3 \fu;r paving away » oC my ino.,oy mim il,»;ii Jnif tl e reiuaiiulcr. 
 1 l.ad JJsu leiiiaiuu g : luiw mucli li.id 1 ul llivt i 
 
 if to j of, D H nut- you udd 14 yearn, tlio hvm. w.ll be 1 .K tinu a 
 
 li. ai.e ; Low oU iH lu' '/ 
 
 If fi-oii. j'j of C's .-go you «iil)tiMct fiyewrs, tho rornulnder will 
 bo ;;,', ol ii,H ago : how old 's lir i 
 
 ;rto J of the v.nsl of my hoii.-e you ndd f200, the 8um will bo 4 
 tho cost of uiy house ; wl.al Avas ilie cmt of the houHe i 
 
 7. if from my iiiiv you Bubtiaci i }, an J i of my aijo, the remain- 
 der Will be U y» Juv ; ho-v (,;d am I / 
 
 S. •'. boy belli,',' asked h h a^e, replied th.-it it was 2 years more 
 than f of his moih.ei'n :ijre, and tl at 12 years before that t mo 
 hirt mothe:- was 2 jears mere thiui half as o|<l us hw father 
 wiiO Was tht.u TJ. years of age ; hjw old was t!ie hoy ? 
 
 ■n 4J d j-^, 
 number ol 
 
 », in what 
 
 ExERCisn XVI. 
 
 f loan do a piocc> of work in 4} (hyy^ whlc'i H can do in 51 
 days ; in what time eould tliey do .t working togothe;-? 
 
 ' a pertain 
 ars T^^r/"'l« 
 
 many acrea 
 
 ANALYSIS. 
 
 ed together 
 
 iimhor wi!I 
 
 If A can do thn w]:ole woik i i 4 J days, in 1 dav he 
 would do 2 of 't ; and if P, c m do ine whole work In 5& 
 days, in 1 day he woul<l do 5'^- of it. 
 
 Then since A does = and B .J^ of V.,c work in 1 day 
 worki;ig togetiier, II ey would do I + ,/,-, which is f'^f. 
 
 Then, if they -lo ^j; in 1 day, t!;ev will re«iuire as many 
 times 1 day as J»5 is co. tamed times in 1, wli ch is M\ 
 and this isiqual 1o2tVo tliiye. '\ •"•efon, «fc-'. 
 
 ? A can do a piece of Mcrk in 7 days which B can do in 6 days, 
 in what time will tiiey do it working togeiher ? 
 
 5 A can d'g a certain pirdcn in 4 days which B can do in G and 
 C m 8 days; Iv what time will tliey finish it work.iiir to- 
 gether? ^ 
 
 '\ A can do a piece of work in 30 days which B can do in 25, C in 
 20, and I) in 15 (lays ;, in what time will. they finish it work- 
 ing together » 
 
 *. A and I? can cradle a field of wheat in 10 days, which A nlono 
 could ao Ip 17 daj's ; in wimt time could li aiono do it f 
 
 w 
 
 i v.\ 
 
w 
 
 leo 
 
 MENTAL AUITIIMKTU;. 
 
 0. A, n, nnd C cmh finish a certain iunount of work In 20 (1uy» 
 Willi li A could <io iihtn- III 40 Uiijij u.iil C ill tiOciuyn • in whul 
 tlmu cuiild U \vurU:u|{ alono Un.tth lif 
 
 EXKHCMHK XVir. 
 
 1. Whtii \n the value of 7800 UuhIioU of oatg at 87* centa nel 
 luahol ? * 
 
 o«nt to 
 Hnt 
 
 ANALYHI8. 
 
 37* ccnln Ih I of H ; 7K(HJ biinhrlB ft 41 wonld nmoiii 
 i?7hyu. Tlun, hU.w u7* Ci-jitH Irt il oi' Jl, 7890 '.>unlit'.„ ... 
 y7| wiil tniioiiiit to t ori?7800 ; | or$7890 Ih H tinu'« A of 
 $7S!m, I of $7800 Is $087, ami U times 1087 1» «20(51 
 'riicrofoii, iio. 
 
 2. What in llic valno .,f 798 buBhda of wheat at $160 per buahclf 
 (8. What Irt tho value of WO /bo. of tea at 76 coiitB lui lb. ? 
 4. What irt tho va"uo of .HIS yds. of cotton at JiOc.iits jier yd.? 
 6. What irt tlie value of '^012 doz. ei,'>,'H at 10§ oeiits ler doz, ? 
 
 6. Wliat Irt Mio value of 6780 ydH, of linen at 87* cenlH yvryi], ? 
 
 7. What is tiie ♦aluo of 7667 buBii. of oats at 50 ceniri j'er bush. I 
 
 8. What is tho value of 719 dayn' woik at S3J ceiilB per day ? 
 
 9. What lb the value of 6796 yds. of drupgc t at G'i* cents per yd. I 
 
 10. What Is tho vidue of 478 ft. of chetnut lumber at 6 cents per 
 
 foot? * 
 
 11. What Irt the value of 7864 lbs. of buttn- at IC* cents ])cr lb. ? 
 
 12. What Is tlio value of 1160 bushels of '.urnipa at 40 cents pel 
 
 bushel t 
 
 ExERc; K XVIII. 
 
 1. How much is 6| per cent, of $94e? 
 
 ANALYSIS. 
 
 H 
 H per "'Cnt. is -•^, which is ^V, a"d iV of $049 is |79xV 
 
 Therefore, &c. ^ 
 
 2. How much is 20 per cent, of |555-60? 
 
 3. How much ifl 6r>| per cent, of $540 33? 
 
 4. How much is 10 per cent, of $89 ? 
 
 6. How much is I'J* per cent, of $978? 
 6- How much is 60 per cent, of $42960? 
 
 7. How much is 6* per cent, of |727'20l 
 
 8. How muoh is 26 p«r csnt of |698'i0f 
 
liENTAL ARlTFlMiniO. 
 
 EXKRCISE XIX. 
 1. V/httt I. tho premium ,.f ln«„ra,.o« on |704 at 8 po. cent.f 
 
 ANALYSIS, 
 Sporoont. «H8tlme«or.oiM)rcont 
 
 ^';""«I'^-'-<^n..,or|7()4i. 17-64 X 8. which i«^.r 
 
 I W ' ! '" "-"koruK.. „n mm at 12 por ooi.t. V 
 4. What H the I.n.keraKo „n $r,980 at 2 p.-r cunt, f 
 
 9 mil ' '° ''"'"'"!*"'"" "" «^"« '^^ 1« I'or oonf. » 
 
 9. What is the prciniuni of iMHunu.eo on iSdo n,\ J 
 
 ^0. W) . ■ the premium of lnHwrancoou«U79U at a per cent.f 
 
 lor 
 
 ExKRCfSE XX. 
 ANALYSIS. 
 
 *• ""fnKf?'- '"'■''^•"'"•' ^"'"" '^■"«''" of H- principal !, .^e 
 '■ ^'ISi;"',';- f-^iJ-'-vhat f,.,,c.l„:, I, ,1,0 l„.er..tof ,„„ 
 *■ '"h,K.»r»"- '"■■''''•■'■•-'vh.t («,c,lo„„f the „H„elp„, u .l>e 
 ' '"prKi;"!'/" >"»■■■"■•'-'■"• <>"■""■» l» the In.,™, Of , ho 
 " ""iH^Ep;;',","- '■"'•^ J"""-' -l'»l f.„o,lo„ 1. ,h„ ,.,e,.„,, „f ,„„ 
 
 '• "" or'" ."prLlp.tr" ' ""•■'"" ^••■"'" '■••»«!<'■ i» ■!>« l"torc,t 
 '■ ''■' ptllSpai'i'- '■'"■ ° ^■^"™ *"■" ^'«='""' i» "■» i"to«Bt of th« 
 
lOi 
 
 ME.NTAL AUITIIMKTIO. 
 
 ■ 
 
 EXKRCISK XXI. 
 
 ] TTliivt is tlio Interest of $74300 for 8 yo:ir» 4 nionthu at pot 
 » c nt. f 
 
 ANALYSIS. 
 
 8 ,'rn. 4 m. = 8J yonr:* uimI J-^ x ^f.fj =: ^'J',, — J, nml lipiioi' tl'e 
 iiiloii'Mt ia cqunl iv) | of tho principal: i of $748(30 l^ 
 $o7 I -oO. 
 
 Tliooibi-c $a74;iO iH tho Intercrst of $74800 for 8 yrw. 4 
 inontlirt at i)i!r ci'iit. 
 
 2. \'>'l at iH the interest of |407-84 for 12 yours months at 8 po» 
 (■("It. ? 
 
 ". What irt the iiittMVHt of ^01070 for 5 years at 6 por cent. ?" 
 
 4. What irt tho li.tcrost of ,tr)4;V'20 for 3 yeara at 10 per cei.t. ? 
 
 6. What is tho ii tercet of $!M3 for 4 years at 12^ por cent. ? 
 
 fe. What is tlic interest of $T«9 for 3 yrs. 4 ni. at 3 por cei:t. ? 
 
 7 What is tlio interest of |47"J;} for 7 years at 2? per cent. ? 
 
 C. Wiiat is tho intyrer<t of $47'f-9 for 8 yearej 4 niojjlhs at 3 per 
 eent. i 
 
 9. What iM tho int(!rest of $S0n80 for 9^\ years at 11 per cent. ? 
 10. What is the ina-rest of $1027-40 for 4 years at 2^ per cent. ? 
 
 ■)0(> 
 
 Exercise XXII. 
 
 1. What is tho interest of ?560 for 6 years at 7 per cent. ? 
 
 ANALVSIi^. 
 
 7 per cent, is jj,^, per unit ai (1 -^-^tt x f» = \if„ = ^\,\ heire thf. 
 interest 's ./..oTli epri' e,i| a!,tliat i-t7 tinieCi/,, ; rj'rtO*'.fo"" 
 i.-t $28. Therefore 'he iiitore.-t i,s $28 x 7, wlilch i.s .fiyti 
 
 2. What i.A tlie intere>'t of $SiO for 8 years at 10 per cent. ? 
 '.^. What IS the interest of Slloa lor 7 y^'ars at 20 ]ier cent.i 
 d. What is tlie intere. t of ;?7G!t I'or v\ years at 4 jier cei t. ? 
 .». What is the inlerc st of :?')40 lor 8 yeara at 8 per cent- 1 
 C. Wl:it is tho interest of $500 for 7 years at 7 per oe; t. > 
 
 7. V.'!iat is tho interest of $1000 for 4 years at 0} per cut, f 
 
 8. What is the interest of .'♦•SOO lor years at a per eei.t ? 
 
 9. Wliut is tlie inlerest of !r.72U tor tt years at 4 wr eeni. I 
 10. Wluit is tliC Intercfct of sJSSO '*»r 2 years ul i,} p: r cent, i 
 
19» 
 
 MKNTAL ATUTIIMETIO. 
 
 EXEUCISK XXI II. 
 
 1. Vn\ai Ih till! Interest ol|l()8 for 1 ycaj- 2 m. la 6 niT cent, f 
 
 ANALYSIS.* 
 
 Ttic ItitcrPHt (if a for 14 moMlliH Ht vor cent. Ih 7 cct.t*. 
 
 Tlnrd'o (>tlu' inici-(Kt on ,tlu8\vlll he 10« Iiik-H 7 cd t-oi 
 
 I tuiii'rt lOH 1.1'iilrtor 7 Jlii.iv-, iJJ-ub le.li. v.liich ;« :J7f)(i 
 
 2. WJi;U i.s tl!-. i:itorfHt of $700 for 17 rno-thK M « p«,r cciif. f 
 
 y. Wl.!it Ih tho li.ton-htcf ibm for 2 ycMn 4 uiouU.h -.a (i p. r 
 
 I. WliJit in tho IJUunBt of ,4703 forij yearn 4 mouiUn at (I jkt 
 f). What irt (he hitcri'St of $4'J0 for 6 \«)aiH 2 inuii'.hs at G wr 
 
 CVllt. ? ' 
 
 6. What is the lilprcwt of $S10 for nioiKlirt at G per cci.l.? 
 V. Wluit l« the iiiUrcHt oC^ia for 7 uioiiths ut 3 percent.? 
 S. What Is (he Intcnwt of $809 for 11 nioutha ul 12 per coiit. f 
 o What Irt tlic Interest of f*.i70 for R niontliH at 18 per cent, ? 
 10. WJjai is tlio interest of $893 for 4 niontlirt iit 8 per cent, f 
 
 EXKRCISK XXIV. 
 
 ^' ^I'n'l .P'*^"«'P'^1 Will la yeiuv^ at G^ pc-r ceut, nmouut to 
 $7^U ' 
 
 ANALYSIS. 
 
 At ?^ percent, for hIk yearn trie intefopt In I of tho 
 p.'i. cipal, and the ntuoiinl, wliieii U equal to t'lie priia- 
 eipa) added to tlie iiiteie.sl, jh equal to J + ' - ? of iho 
 l)rineipal, bos 
 
 If V of (he i)Hnoip!il U $7-2n, « of tho princ ptd i,* i of 
 $.20, Av . el, is $]2i> • an.l if ,^120 is i. tlic whole priuci- 
 pal IK $J2(J X 5, wliiuh is $G00. 
 
 There foro, <S:c. 
 
 2. What principal win In 8 years 4 m. at 6 per cent, amount to 
 7200 I 
 
 8. W hat principal will in 9J years at 9 per cent, amount to $700 f 
 4. SVhat principal will In 4 year.4 ut 5 per cent, amount to $408? 
 6. What nnncirml will In 3?^ ves 
 
 lAua-OS? 
 
 yeai-fi ut ? per cent, arni/unt to 
 
 tieo Rule, jiago 161 
 
1 ? 
 
 200 
 
 MENTAL ARITHMETIC. 
 
 iii;iii 
 
 ': 
 
 7. What ,.ri.,cipal will in 8 years at 5 per cent, amount to il735 f 
 
 '■ ^^mS'"'^'^ ''^" '" ''y^^^^ ^ '"• ^' 8 per cent, amount to 
 10. What principal will ii, 1 year u 33J per cent, amount to |963-24 » 
 
 1. What pr.nc.pH wlUm 7 years at 10 per cent, amount to $561 » 
 12. WhatpruicipalwiUinSyearaatS percent.amountto$678->0? 
 
 Exercise XXV. 
 
 SOLUTION. 
 
 ® -i25.^vhfd! ^r "°^ '"'^ ""' ^''' ""^ ^'*" ^-'^^ ^'* 
 
 If I gain $4 on |25, on $1 i 'lall ~iin ,\ of $4 which n 
 
 ,*,.ofadoliar;a,,d if I , m $,*>" $'l on $lo5^V ehail 
 
 ffrl nSili."'"'' f^*^' ^^hich.slie!* ThlreforemygainVi 
 |16 on $100, or 16 per cent. ^ ^ 
 
 ^ ^^gutV^'Jelft."/ ^' ''"^' "^'^ ""^ «°^^ '^^ IS 5 ^h-t was my 
 
 ^' ^""verleutT '''' ^^' ^""^ ^'^^ ^' ^-^^50; what was mygai, 
 '• ^Tnf,!a?;r?/ir1l?t.'?"^' ''''^'''' ^'''' per cord ; what wa» 
 
 ^" ^"v^lolS^i"' ^'' ^°^ ^"^'^ ^^ ^^ *'^^ P^^ t°« ; ^hat was mylos* 
 ^' ^ pcS- cciuT"'' ^'^ ^^^ ""^ ^^^'^ ^* ^^'^ *1« ; ^l^^t was my losa 
 
 JO. Bought lumber 
 
 vas my gain per cent. I 
 
 sold it at III -25 per 1000 ft wi-* 
 
amount to 
 
 . to $735 ? 
 imouiit to 
 
 amount to 
 
 to 1963-24 » 
 It to $561 ? 
 to 1678-20 )f 
 
 I mygai- 
 
 II wns |29 
 
 which ts 
 30 I Bhall 
 lygaiu id 
 
 t was my 
 ■s ; what 
 
 my gair 
 vhat wa» 
 
 bushel ; 
 JmyloBS 
 I my losa 
 my gall 
 
 ANSWERS TO THE EXERCISEa gOl 
 
 ANSWERS TO THE EXERCISES. 
 
 Exercise 4. 
 
 • 333 ; 10000 ; 90000 ; GOOO ; 5977- 
 27027; 40444. '°»"^ 
 
 -^; ^8; 33; 44; 99; 478 ; 330 ; 4 
 
 /. 777; 296; 843; MO; looi ; 
 x4U0. 
 
 ^' ^^iL^^^ ' ^^2^ ■» 2030 ; 2858 ; 
 301. 
 
 5. 1899 ; 2222 ; 4505 ; 9604 ; 8888 
 6-^^<^II;^XLVII; xcii 
 LXXX; XX; IXXVll' 
 ^i;x CXl'; DCVI. ' 
 
 S. MMCCXXXIII; MMMCCXXXII; MMMCCCXXXIII- 
 MVCCCXXI ; MCCXXXIV : VDCLXXVIII • 
 VMMMDCCLXV. ' 
 
 9. MXCMXCIX; XXVDCLXXI ; DCCCXCAICCCXLVII • 
 _CMXMMCCGXLII ; X VMDOCXIII. 
 10. CXCMCMXIX ; XXMXCXXXIV ; XXMMMCDLXXVI • 
 CMXMMCCCXLV ; MDCLSvMMMCMXLII • 
 MMMCDLVMDCCXIII. ' 
 
 1. 
 
 879. 
 
 2. 
 
 8785. 
 
 3. 
 
 9536, 
 
 4. 
 
 6424. 
 
 1. 
 
 189930. 
 
 2. 
 
 19168. 
 
 8. 
 
 2062S62 
 
 4. 
 
 203883. 
 
 6. 
 
 176258. 
 
 6. 
 
 63665. 
 
 1. 
 
 181220. 
 
 8. 
 
 113092. 
 
 
 1111 rni 
 
 10. 
 
 372639a 
 
 Exercise 5. 
 
 
 5. 38904. 
 
 9. 5647. 
 
 6. 997688. 
 
 10. 95878. 
 
 7. 796. 
 
 11. 89745. 
 
 8. 25734. 
 
 12. 3499. 
 
 Exercise 6. 
 
 
 11. 670614. 
 
 22. 345482. 
 
 12. 1108958. 
 
 23. 746456. 
 
 13. 1563804. 
 
 24. 1069843784. 
 
 14. 28140244. 
 
 25. 864. 
 
 15. 32287760. 
 
 26. 320 bushels. 
 
 16. 5586789. 
 
 27. $29431920. 
 
 17. 98536. 
 
 28. 20213174 bushcla. 
 
 18. 172686. 
 
 29. 4035 busheld. 
 
 iy. ii347. 
 
 30. 8638prs. ofeboee. 
 
 20. 26132. 
 
 31. In the year 1877. 
 
 21. 46429. 
 
 
202 
 
 ANSWEflS TO THE EXERCISES. 
 
 J. 51144002. 
 2. o41()6o0. 
 S. CU0002. 
 
 1. 064072. 
 
 2. 802909. 
 a. 10817. 
 
 4. 172723. 
 
 5. 7253073. 
 
 6. 1719034. 
 
 7. 730139. 
 p. 80S4031. 
 0. 70850871. 
 
 10. 4013S39. 
 IJ. 950017022. 
 
 12. 5S1259iG3. 
 
 13. G1374. 
 
 1. 142982742. 
 il. 275450841. 
 3. 304^38876. 
 i. 335719585. 
 5. 5512592526. 
 G. 0154158486. 
 
 7. 98SG3888. 
 
 8. 8S844S9482. 
 
 9. 10714913450. 
 10. 86476406301. 
 
 1. S4553616L 
 I. 7 7003 130. 
 C. 19287909. 
 
 4. 2587424. 
 
 5. 5;>07538. 
 C. 9909537. 
 
 Exercise 7. 
 
 
 4. 1050. 
 
 8 611803. 
 
 
 9. 204104053. 
 
 6. 109113. 
 
 10. 400340. 
 
 7. 702512. 
 
 
 EXKKCISE 8. 
 
 
 14. 701353. 
 
 27. 2C993838. 
 
 15. 4CG(j82. 
 
 28. or.5000. 
 
 1(3. 64021479. 
 
 81. 08024. 
 
 17. oo89;;86. 
 
 32. 71152. 
 
 18. 0342920. 
 
 33. 2644. 
 
 19. 384. 
 
 34. 2585. 
 
 20. 18294. 
 
 37. 744985, 
 
 21. 1260000. 
 
 :.S. 56867. 
 
 22. 111. 
 
 l;9. 796098. 
 
 23. $1072. 
 
 41. 717258. 
 
 24. $662. 
 
 42. 28518. 
 
 25. $3826. 
 
 43. (k;29G7. 
 
 ' 26. 922. 
 
 i 
 
 Exercise 9. 
 
 • 
 
 11. 51497984052. 
 
 16. 09871590: 
 
 12. 629399344. 
 
 13. 2375628 ; 1683752 ; 
 3167504 ; 9502512. 
 
 14. 5730338929; 
 6548968776 ; 
 7367578623 ; 
 9004818317. 
 
 4Mt]i»113; 
 55M»7272 ; 
 41'J2-.^y54. 
 
 17. 8570028. 
 
 18. 17221284. 
 
 19. 012784. 
 
 20. 44000 7S056. 
 
 15. 18537 ; 30895 ; 
 43253 ; 56611 ; 
 74148. 
 
 21. 4998 ; 78rv4 •, 
 4284 ; 3570 \ 
 8568. 
 
 Exercise 10. 
 
 
 7. ia3235G16. 
 
 14. 439455324. 
 
 8. 9026040. 
 
 15. $7203. 
 
 9. 8058870. 
 
 16. $3024. 
 
 10. 57054404. 
 
 17. $78960. 
 
 11. 99249624. 
 
 18. 969024. 
 
 12. 101037948. 
 
 19. 37873. 
 
 13. 43529GS8. 
 
 
 \ 
 
 
 
 1. 
 
 353. 
 
 
 
 3.^ 
 
 I 
 
 4. 
 
 
 5. . 
 
 
 6. . 
 
 58. 
 
 7. ' 
 
 
 8. 1 
 
 ' 
 
 u. ( 
 
ANSWEIIS TO THE EXEKCiaES. 
 
 1. 165404178. 
 
 2. V2532804. 
 
 3. '572161070. 
 
 4. 7099S0(J912. 
 
 5. 52588916400. 
 
 6. 578577 15S82. 
 
 7. 772820915328. 
 
 8. 2G079731 80020. 
 U. C4311091G24800. 
 
 coa 
 
 ExERCrSK 11. 
 
 10. 1210703092GSC3. 
 
 11. 82GG0927GSO2G8. 
 
 12. 89432071615010653 
 
 13. G5G6oOCl2. 
 
 14. 749188440. 
 
 15. G5f)17S570fi30. 
 10. 2&G4C2jG5o9y9. 
 
 17. 13n4lG15'220. 
 
 18. 110l7SG4o80. 
 
 , 19. ISOS.'OR??©. 
 j 241. 4;.:'l 1 u.^1 t'.'s. 
 
 i21. 
 I 22. 
 ! 23. 
 
 .?230U2. 
 695LG7. 
 115200 1 crcljc'8. 
 
 M. 501 29G. 
 
 26. 1081. 
 
 
 EXERCISK 12. 
 
 1 
 
 1. 357349Ci. 
 
 11. 7G21305G-,\. 
 
 21, 7407407 ,V 
 
 22. 13717421. 
 
 2. 2971522^. 
 
 12. 1-345G790122. 
 
 3. 228524^. 
 
 13. 457oHf;|. 
 
 23. 834 79685 r\- 
 
 24. 28571428'. 
 
 .4. 26009439?. 
 
 14. 23821 Og. 
 
 5. 104606405.^-. 
 
 15. 30KGi,2J. 
 
 25. 1672747511. 
 2G. $126. 
 
 6. 15592451". 
 
 10. 1400000. 
 
 7. 30602;;82.' 
 
 17. 13500G7,t. 
 
 27. G7,»i bnsl:e!s. 
 
 8. 101G3522. 
 
 18. 1285947. 
 
 -8. 161. 
 
 9. 2223:J344jV,. 
 
 19. 1257G25SJ. 
 
 
 10. 2S3755923/r. 
 
 20. 9999999,V 
 
 
 Exercise 13. 
 
 
 1. 44685,'ij. 
 
 8. 81828384/>„-. 
 
 15. 16404«°. Inisl-els. 
 
 2. 3737-,V 
 
 9. 104342025-1. 
 
 16. GU77,^i bu^Lels. 
 
 3 113416^1. 
 
 10. 1462940351*>. 
 
 17. $36:». 
 
 4 n21G4y^V 
 
 11. S33G800rfj;. 
 
 18. 60 lb--. 
 
 6. 39S07i'.^ 
 
 12. 1111174,^5. . 
 
 19. 1 28909.^ n. 
 
 6 205761/,V. 
 
 13. 58CGG23.f. 
 
 20. V^hbli. 
 
 ■». 51O04O7jVt. 
 
 14. 1.1915" bushels. ' 
 
 ■ %i> 
 
 Exercise 14. 
 
 2. 7313,';^. 
 
 5. 310536^5. 
 
 6. 3G3006i^. 
 
 7. 
 8. 
 9. 
 
 S35G,W;. 
 
 I 
 
,ll 
 
 it 
 
 204 ^ 
 
 10. 1002^6'vys. 
 
 11. 828xV^VV. 
 
 19 C'Sfi? 
 
 13 r>7fi «««■« 
 
 14. 91874163. 
 
 15. 8050. 
 
 1. $287927H. 
 
 2. $2686-531. 
 8. $6a817xV. 
 
 .Rfi 
 
 TO THB 
 
 _ 1 
 
 16. 
 
 £258. 
 
 
 22. 836i|arres. 
 
 17. 
 
 32S;|f. 
 
 
 23. 6722i5?f. 
 
 18. 
 
 3S60Sf. 
 
 
 24. l983283-3Mg. 
 
 19. 
 
 205x2iV. 
 
 
 25. 8865f3J|J. 
 
 20. 
 
 847ii?^. 
 
 
 26. $23 J g. 
 
 21. 
 
 42-aYf days. 
 
 
 
 Exercise 
 
 4. $71 -521. 
 
 5. $2623 94t\. 
 C. $3110-25. 
 
 7. $446 &8^. 
 
 15. 
 
 8. $2269-75t", 
 
 9. $2-29 C9,^ 
 10. $281999^ 
 
 I 
 
 ExERCrSE 16. 
 
 1. £179178.; £229 5^.6(3.-, 
 i;i7 8s. 6d.; £104 9r^ 9d. 
 
 2. £42 15s. C|d.; i;47 10s. 5?fl.; 
 je418 1l3.1|d.- £194 86. lOJd. 
 
 a £11? ?€, Jifd.; £9 <fk ^,d,^ 
 £17 108. 41 d.; ,C4» .9h. l}a. ' 
 
 4. £178 lis. 11-^ ; Xo <Jt. ;./)J(i 5 
 £4 0s. 92d /xWo U^ tOlcL 
 
 1. $12110-12. 
 
 2. 6981GG-53. 
 
 3. $752-80^. 
 
 4. 15732 49. 
 
 5. $1778-37. 
 
 6. $972-GGf 
 
 7. $529-57. 
 
 8. $32218-08. 
 
 Exercise 17. 
 
 
 
 9. $409096-14. 
 
 1' 
 
 ^61-68tV 
 
 10. $8903-07. 
 
 1" 
 
 fS6-87i. 
 
 11. $91-78Af. 
 
 1^. 
 
 ^29665. 
 
 12. $12G-13r\<t, i 
 
 13. 24,fJ,. 
 
 14- 62^-iff J. 
 15. $452-14. 
 
 % 
 
 $123-801^, 
 
 21 
 
 $22';28969. 
 
 •^2. 
 
 $6349558. 
 
 23. 
 
 $9-84jVsVVd"^ 
 
 16. $33-955^. 
 
 
 
 1. 6016 ctib. ft. 
 
 2. 103952 oz. 
 
 3. 93922 far. 
 
 5. 291454023 drB. 
 
 6. 2472 qts. 
 
 Exercise \ 
 
 7. 13608552 lino 
 
 8. 190G9". 
 
 9. 5738 pts. 
 
 1f\ Knrt S_ 
 J-". UXJO IIU 
 
 11. 10661367 sec. 
 
 12. 299624 mills. 
 
 13. l02G35fr8. 
 
 14 1742935 :!uV^ h 
 
 15 745011(1-9. 
 
 10 54575 tifi, ft. 1 jSlp. 
 jl7 o728iJri. 
 1 18. 2259 cub. A.. 
 
 
 I. 
 2. 
 
••/d 
 
 09- 2853 v^ta. 
 to. 20721 dor. 
 
 ?2. £90984 iu. 
 
 ANSWERS TO THE EXEUCISES. 
 
 23. 595834J sq. yda. 
 
 24. 853437840 aec. 
 
 25. 3878 hre. 
 2G. 1899734 far. 
 
 20A 
 
 27. 1112 pt8. 
 28 62248 gr^. 
 
 29. 36136690704 eq. la. 
 
 30. lG4695far. 
 
 I 
 
 2 
 S. 
 4. 
 5. 
 6. 
 7. 
 8. 
 9. 
 
 10. 
 11. 
 12. 
 
 ly. 
 
 14. 
 
 x5, 
 
 Exercise 
 
 lm;lolfur.3per.3yds.7in. 
 ■idaysshrs. 22min. 
 14 Ibd. 1 oz. 5 drs. 7 grg. 
 
 41c.2c.f(.l0cu.ft.642cu.in. 
 31 cli. 5 bush. 2pk. 1 pt. 
 33 cub. yds. 23 cub. ft 
 1I38F. e. Sqrs. 
 12 wka. 5 days. 
 
 19. 
 
 1 r. 3 per. 22 ydn. 4 ft. 101 in. 
 
 l.atuaslOgal. 3qt8. 
 
 6 yrs. 158 days 14 hra -u. 
 
 jei 78642 11.-. 11^1, 
 298E.$7 1diniu4ots.0mill8 
 
 16. 
 
 17. 
 
 18. 
 
 19. 
 
 20. 
 
 21. 
 
 22. 
 
 
 23. 
 124. 
 '25. 
 
 26. 
 
 27. 
 
 28. 
 29. 
 
 30. 
 
 136 c. 90 c. ft. 
 
 je3832 lis. 5d. 
 
 4 Iba. 1 oz 19 dwt. 
 
 221buflh. Ipk. Igai. 
 45S33 F. e. 
 
 8 tons 11 cwt. 1 qr. 24 Ibg. 
 3 rnilea 1 fur. 36 per. 5 yds. 
 
 308° 38' 31". 
 14 cub. yds. 8 ft. 781 in. 
 1:^70 lbs. 3 drs. 2 scr. 7 grs. 
 117 Ibe. 10 oz. 5 dwt. 6 grs. 
 
 "o Z[' ^ '^'■'- ^^ ^^'- ^^ «^ 
 9 a. 3 r. 8 per. 6 yds. 1ft, 75 in. 
 ft 6?;/ '"^'^2 'ur. 7per.l 
 9930 fa. 2 ft. 2 in 
 
 1. 
 o 
 
 i 
 
 4. 
 
 6. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 10. 
 H. 
 
 jei29;) 0^. 2fd. 
 
 \^?,";.''.<;f«^"'--18per.2yds. 
 
 2S1 11)8. 5 oz. 6 drs. 
 
 212 cwt. 2 qrH. 12 lbs. 15 oz. 
 
 100 yd^. 2 ft. 1 in. 
 
 179 bu. Ipk. IgaJ. 
 
 104 c. 4 c. ft. 
 
 433 a. 2 r. 
 
 28 yds. 2 qrs. 2 na, 2 in. 
 
 154 lbs. 3 oz. 6 dwt. 18 grs. 
 
 Exercise 20. 
 
 4.9 
 
 14. 
 
 15. 
 10. 
 
 17. 
 18. 
 19. 
 20. 
 21. 
 22. 
 
 23. 
 
 i2. 226 sq. per. 7 yds. 6 ft. 
 13. 100 wks. 4 days 3 brs. 
 
 36 in. 
 
 je260 14s. 9id. 
 1G9 qrs. 2 Ib.s. 2 oz. 
 178 Eng. ells 4 qr.-i. 
 51 plis. gul. 3 qts. 
 440 oz. 10 dwt. 14 grs 
 i:478 58. l^a. 
 
 321r.28pcr. 9yds.2ft 36in. 
 223 yrs. 27 wks. 3 days. 
 
 287 lbs. oz. 6 drs. 2 fcr 8 
 grs. 
 
 2?l ^- 1 '■• 11 per. 20 yds. 1 fL 
 
 24. 73 lea. 2 miles 7 fur 
 
 ill 
 I' I 
 
 2yd3. 2 ft. lOin. 6Iinei, 
 
 27 per. 
 
200 
 
 ANSWKRS TO THE EXKUCISKS. 
 
 1. 03 iinyA 4 hr(t. 2 inln. 
 
 2. 73iM!le«0 \uy. 10 puj-. 
 
 8. 1 qr. 15 \ha. 11 02. 
 4. 481 « il. 2 qu. 1 pt. 
 
 6. ;i78 a. 1 r. liO per, 
 0. jCKH-Jd. 1J<1. 
 
 7. 78 lb-. Ooz. 7ihNt. 
 !{. 78 yd-'. 3 qra. 'I n:i. 
 
 9. 175 hra 54 inlii. 55 hpc. 
 jlo. 177 c, 5o. ft. 1*1 cub. ft. 
 11. 27 oz. 6 (liN. 2 «cv. 
 U. 4 fur. 39 per. 4 yd«. 1 ft. in 
 
 ExERClSK 21. 
 
 la. 7'l fWl. Oqi«. 21 Ibw. 
 
 14. (12 luifili. y pkH. 1 t;:»l. 
 
 15. (j'J yt'Uis 47 wcekB 4 dnyi. 
 Iti. :.9 r. G8 pir. 24 yclB. 2 n.36in 
 
 17. £57 «iH. CJl. 
 
 18. 80 oz. 18(l\vt. Ggrs. 
 10. A'Hl IOh. fiid. 
 
 20. 62 cub. ydH. 12 ft. 1401 In. 
 
 21. 82 lihdH. bnr. 24 gal. 2 qt. 
 
 22. To dru. 1 tcv. 8 gre. 
 
 23. 6 Floni. 0. qi-ri. 1 nr^. 
 
 24. 47 eq. por. 10 yds. 7 A. SO In. 
 
 ExKRCISK 
 
 1. 41 days 10 Iws. 10 mln. 32 boo 
 
 2. 73 qi-B. 14 lbs. 12 oz. 
 
 a 134 bual). 1 i)k. 1 gul. 2 qtfl. 
 
 4. £2150 19s. Od. 
 
 6. 125 gal. Oqt. Ipt 
 
 6. 838 Iht.. 1 i)z. 10 dw-t. 
 
 7. 298 ml.csOfur. 10 per. 
 
 8. 1148 years 81 dayb 23 hra. 
 
 9. 608 cub. ft. 53 in. 
 
 10. 1343 c. 1 c. ft, 4 cub. ft. 
 
 11. 211 r. 9 por. 1 yd. ft. 108 in. 
 
 12. 233 cwt. 3 qvB. 1 lb. 
 
 13. 
 14. 
 16. 
 10. 
 
 17. 
 18. 
 10. 
 JO. 
 21. 
 22. 
 23. 
 24. 
 
 Exercise 
 
 12. 
 
 1. £1199 108. 4d. 
 
 2. 1350 IbB. oz. dwt. 
 
 8. 339 days 17 liotra 57 min. 
 4. 1321 Flem. ells qrs. 3 iia. 
 
 6. 3557 mdes 1 fur. 32 per. 
 C. 5515 gal. qt. 1 pt. 
 
 7. 3039 hours 20 mln. 24 sec. 
 
 8. 1820 a. 1 r. 30 per. 
 
 0. 11115 oz. 4 dra. 2 scr. 19 gri. 
 iO. £4897 Is. 4f 1. 
 il. 16582 sq. per. 22 yds. 4 ft. 721n. 
 
 13. 
 14. 
 15. 
 16. 
 17. 
 18. 
 19. 
 
 22. 
 
 £1150 lOs. 2|d. 
 783 per. 1 yd. ft. 1 fn. 
 67 sq. per. 23 ydn. ft. 
 1975 yds. 3 (irs. na. If In. 
 109 11b. 4oz. 7dri». Isor. 
 58 qrs. 8 ll)s. 12 oz. 
 567 rnilcri4 fur. 22 per. 
 705 wks. 1 day 1 hr. 68 mln. 
 123 plis. 1 gal. 2qtB. 1 pt. 
 1316 cwt. 1 qr, 7 lbs. 
 £1629 Id. 9fd. 
 
 070 a. 3 r. 24 per, 5 yds. 6 ft 
 
 108 in. 
 
 'J.I 
 
 929S cwt. qr. 8 lbs. 
 
 815 years 44 days 4 hours. 
 
 1423 cub. yds. ft. 1440 lu- 
 
 6806 bush. 2 pk, 1 qt. 
 
 £2780 6e. 3d. 
 
 44942 per. 2 yds. 1 ft. 
 
 102029 hours 5 min. 
 
 3105 a. 2 r. 6 per. 18 ydfi.4» 
 
 20. £516290 12s. 6d. 
 
 15. 6 
 
 !«. 1 
 
K 
 
 \ (InyB. 
 2 1l.C6>ti 
 
 4fil In. 
 
 al. 
 
 2qt. 
 
 no. 
 
 
 I ft 
 
 .sola. 
 
 Ifn. 
 5 ft. 
 
 II. U In. 
 1 Bor. 
 
 per. 
 
 '. 68 niln. 
 
 3. 1 pt. 
 
 8. 
 
 > yds. 6 fl 
 
 8. 
 
 hours. 
 1440 Ir.- 
 It. 
 
 ft. 
 
 AJS'.'JWKItH TO TIIK KJiEIiCISKS. 
 
 K\i:j5(;i8e 24, 
 
 207 
 
 
 :;. 241:] (la\n L"J IioiirK CI min. 
 4. .*:!'-2>ni(ls 1 n. f, in. 
 
 6. 12:<:niH. 4i'z. lOdwi. 
 
 fl. WIO !.. 2 r. yopcr. 20 yds. 7 
 n. 72 ill. 
 
 7. 10:m yiU. 2(;rp. 1 rii. 
 
 8. 1f-r>274 <.z. 1 fcr. I'j ki'O. 
 ft. X7J!3I)2 1(1 . 1(1^(1. 
 
 JO. 4018 rwf. ] qr. 22 Ibn. 
 
 14. 2:)14t5 :.. 1 ,.. (5 j,» ,.. IG y.K 
 - ft. 4j n. 
 
 15. iKnf) ytais 140 di\y» 11 hours 
 
 60 inin. G K?c, 
 10. o07c. 3('. ft. 2 (Mil', f!. 
 17. 2;:88;j hii-h. 2 ikn. 3 (itf. 
 IS. 00700 ll-H. 2 di-H. 2 ^cv. 5 ^m'S. 
 10. Xl.':(;344 10-^. 3.1. 
 
 II. 48S5,n.le«7fur.6i,or.lyd.2 20. SOiO Ibn. 8 oz. 8 chvt 13 
 ^'" ' grfl. * * 
 
 ExEnciSE 25. 
 
 4 
 5. 
 
 6. 
 7, 
 8. 
 9 
 
 10. 
 li 
 12, 
 
 1.1. 
 14. 
 
 £244 lis. 7d. 
 
 11 cMt. 2 (lis. 10 11)8. 10 o^. 
 
 2f^ tlrs. 
 24 days 10 hr. 4 min. 8} ncc. 
 70 blHli. 3 plvH. 1 qt. Oi ];t. 
 090 milcfl 6 fur. 8 per. 2 ft. 
 
 in. 
 C6 <q. per. 22 ydP. 7 ft. 24 in. 
 22 lbs, 3 o/. 7 dro. 1 (»cr. 
 .") I'a^r. 3 qtf. OJ pt. 
 £1147 1.-^. 10,\d. 
 79 c,v1. 2 qrH. 14 Ibf. 1 oz. 
 
 4 Tfl. 7.]Jd. 
 
 in. ner. 4 yds. 1 foot OJJ 
 IShrs. 10 nil,. 
 
 17. 
 
 18. 
 19. 
 
 20. 
 
 21. 
 22, 
 23. 
 24. 
 
 8 yrs. S8 days 21 hrtf, 2 min. 
 
 6 pcc. 
 10 oz. Idr. 1 err. S;;* pri*. 
 14 r. 14 jicr. 1 y.-ird 6 feet 
 
 «i in I,, 
 
 3vkK. 4diiyH21hrP. I8min. 
 
 4 "J.I fit'c. 
 
 £820 ISh. 5d. ^y.« far. 
 
 fie. 3c. ft. 2=5 cub. ft. 
 
 48''47'3,W''. 
 
 5 Frcncli ells 2 qrs. 3 
 
 na. 
 
 StoVit'". 
 
 1 
 
 0. 
 
 5 «0.. p«r. 4 yardH'V' 
 
 J«« i 'b. (> 
 
 35rV^ h 
 
 foi.f 
 
 "?. oik ars. 
 
 13 niilop 5 fur. 18 rr-r. 2 yds. 
 
 2 ft. O;,? In. 
 £11 178. f.fl. I*? far. 
 9 1b>«. 6oz. 10 dwt. li;,«|* grs 
 17 bush. 1 e::»!. i»p^ pt. 
 4 dtiys 12 liours 7 minutes 
 
 nn* (It* 
 
 3<'-ll cwt, 2 qrs. 2 lbs. 12 oz. 
 iMif drs. 
 
 20 
 27, 
 28. 
 29. 
 
208 
 
 ANSWKttB TO TllK KXKRCISKS. 
 ExKUCisE 26. 
 
 1. wmn- 
 
 10. ur>^?o. 
 
 11. 3".)g3;. 
 
 12. 46Vi8. 
 
 13. miMU 
 
 H. iDiiisra- 
 
 15. 24yn- 
 
 la. iBHU- 
 
 17. 287%. 
 
 18. 4HU. 
 I'i). 1067v'6¥i. 
 20. 87,Wif 
 
 
 EXKHCISK 27. 
 
 t 402:)J ft. I 2. no IbH. 9 07.. 6 «hvt. 6 gr«. 
 
 3. DCCXIV; MCXI; MMDCOIV. Xr MiXM.XXl, 
 
 DC'CCXMMMCDLXXl ; XXXMCJM XVMMCLXTX. 
 
 4. JEW 0^. 1A»». I ^- ^*'^ '^"'- 1 "'• ^^ ''''"'• "'■•^ '^''"' 
 
 A Oiu> tnllUmlwo blUIoim forty-st-vi'ii thoiisiuid ii;:(l six ; iiina 
 ■ huiuimi (uuulrilhous ckni'ii tnllioiiH o hu <livcl and ten 
 billioiH elovon lulllioim 0110 liuiulriil ;ui*l ton tlioUKUiu luid 
 M cloven • sixtooii billion^* seven hiiu.lird iuul fmrteei. nu llon« 
 nine huudrtMl imd rtixty-scveii Itiousuid nine liuudred iu;a 
 four; Huventyoi.o blUiouB throe huadred millluuB lour bun- 
 dved'thoiU'^and and two hundred. 
 
 17. Out' X8 6fl. and each of the 
 
 7. 2 nvllos 2 fur. 20 per. 3 yds. 
 
 i Soet. 
 
 8. 7889 tor,8 2 cwt. 24 lbs; 
 $157"82t!4. 
 
 9. 3 wks. 4 d:iyB 13 hrs. 10 min. 
 19,V, ^00. 
 
 V). $79205. 
 
 11. 12 Ib.^. 2oz. 5dwt, 
 
 i2. $14-11. 
 
 15. lOOOl ; 50500 ; 3333 ; 2650090 ; 
 
 89044 ; 450001709 ; 1094000 , 
 
 1004900702. 
 
 M. 1625 w,^. 
 
 ^d. 
 
 15. 6545233. ^ I -• 
 
 10. 7 a. 1 r. 31 per. 18 yar-^ ^f* 28. 11911 Ijush. 
 36 In. 
 
 others X4 3.<- 
 
 18. 1st $124, 2d $155, and 3d 
 ftnd4tb oaeh $23260. 
 
 19. A $10 19r\. 
 
 20. 5089 yda. 3 qre. 1 na. 
 
 21. 3 tont^ 11 cwt. 3 qr- 
 6 oz. CJ dra. 
 
 22. 664f)f fathr 
 
 23. 85203 
 
 24. ''^. 
 ;!!?• $412-50 and 1266-60. 
 
 '27. 9566172. 
 
 • • i-¥ 
 

 
 ANSWEEfl TO THE EXERCrSEB. 
 
 «o. 7ri. 
 
 «8. 217 yard« ? flsot. 
 
 **• ^/i%*-4^«"'l B '^n<> Conch 
 
 B. l«t r«ceiyM rai cub. ft., 2d 
 M6 ciib. ft., wjd sa 1183 
 oao. ft. 
 
 209 
 
 86. 60 a. 13 per. 10 ydi. 108 In, 
 
 87. 8 ml led. 4 fUr. 88 per. 6 y<h. 
 
 « In. 
 
 38. je78260 or $318000, 
 
 39. 418. 
 
 40. 114 1bM5dwt.;$M7M0. 
 
 41. 612|{ft barrel.. 
 
 t. 6 gr«. 
 
 fA'TX. 
 
 vt. 12 Ufi'fl. 
 
 I Hi.\ ; iiina 
 nd and ton 
 i)Urt:\iul luid 
 '(Ml niilUoim 
 \iiulred lud 
 iB lour huu- 
 
 each of the 
 
 1.%, and 3d 
 260. 
 
 1 na. •^*' 
 qr- 
 
 50. 
 
 C. 00. 
 
 i. 3S0. 
 J. 420. 
 4. 5040. 
 
 6. 720. 
 «. 3168. 
 
 7. 11088a 
 
 EXBRCISS 28. 
 
 8. 117. 
 ». 76. 
 
 10. 1127. 
 
 11. 201. 
 
 12. They have none. 
 
 13. 28. 
 U. 13. 
 
 MxiDrcisE 29. 
 
 8. 4200. 
 
 9. 2020. 
 
 10. 1008. 
 
 11. 6OJ0f 
 12 12464 
 la 144. 
 
 lU 25200 
 
 «. ^m m\ Hii, ^i^ 
 
 4. mi. l^^^ la^^u. ^1. 
 
 V. i 
 
 
 IS. 8. 
 10. 17. 
 
 17. 58. 
 
 18. They hav« 
 10. 97. 
 
 20. 4. 
 
 1 16. 1280. 
 1 16. 8466. 
 
 17. 1441440 
 
 18. 69300. 
 
 19. 160320 
 
 20. i9aiii 
 
 i I 
 
 i 
 
 u 
 
210 
 
 AKSWKRB TO TUB EXKROIBIS. 
 
 1. H*. 
 
 2. 
 8. 
 4. 
 
 l'.. 
 0. 
 
 
 ,1. n|. 
 
 2. 164. 
 H, 1282. 
 4. 61076H. 
 6. 37+i. 
 6. 28f 
 1 17Mi 
 
 1. tV. 
 
 2. tWt. 
 
 3. ee. 
 
 4. H. 
 
 Exercise 81. 
 
 8. *?««^. 
 
 10. J-^f». 
 
 11. ?i 
 
 12. aa,V^. 
 
 13. "W^. 
 
 14. MF. 
 
 15. flVuW'X 
 10. -^-f^U*. 
 
 17. »n^*. 
 
 18. -^JJ^. 
 
 10. ^HU'^** 
 20. av»V*» 
 
 Exercise 32. 
 
 
 8. ISOJIHi 
 
 16. 4|f. 
 
 9. 26iV 
 
 16. 8U?|J. 
 
 10. 167||. 
 
 17. 2(v70i5i. 
 
 11. 1048i|J. 
 
 18. V^^l, 
 
 12. 6H. 
 
 1*3. 1'09|H 
 
 13. 94^. 
 
 i.0. 2H3^, 
 
 14. 9t«^. 
 
 1 
 
 Exercise 33. 
 
 6. m"^' 
 
 '7. iW*. 
 8. H. 
 
 9. H. 
 
 10. Vo^ 
 
 11. y>,V 
 
 ^5 /,!i 
 
 ExEBCisx 84 
 
 1. ^r ii ^^» «^^ H. 
 
 2. IM, IV8, Ul m> and IM. 
 
 3. .^tf, iK, •i'^iV ^l^i *"^ t^V 
 
 4. iB, sa liii, m. m. and m 
 
 6. \n. m. t'At, Hi IM-, an<i tt». 
 *. ^VV, *, ^V, tV^, WVi a^^i V>^ 
 
 '(f^l'i- 
 
^nsweks to the KXEncrsm. 
 ^. il If!, iii\i\ iiji. 
 
 ;;»• Jjp: mi j^sa. and f.gft. 
 
 ^''' ^^'^i> mi mi mi mi VM ^vA rm. 
 
 211 
 
 
 3. a. 
 
 ExKRCisE 35. 
 5. 1. 
 
 1. 19^. 
 8. Ifg. 
 
 Exercise 86. 
 
 T). 2|. 
 
 18. AV 
 
 0. /y. 
 
 11. lUOlJf. 
 
 12. Of. 
 
 0. i^A. 
 
 10. iHf. 
 12. /AV 
 
 ^ i\ of a week. 
 ^- iff of a qr. 
 U- ii-ofFIem.eiI. 
 4. ii6-i4^ per. 
 
 
 yt%m> 
 
 Exercise 87. 
 
 5- VA of a yd. 
 
 0. i^i. 
 
 8. vWofa£. 
 
 Exercise 38. 
 8. -M'Wk 
 
 » " t u .•» ' 
 
 ] 9. U^ 
 
 ^- Ta^? of a week. 
 
 10. i^;|na. • 
 
 11. ii^sio, 
 
 12. rV\jTjofai: 
 
 1 11. %nh' 
 
 12. -.Wi>%. 
 
 •j Q _2j n 
 
 14. -/.VAV 
 
 t i 
 
iil2 
 
 ANSWERS TO tHB EXERCISJ^ 
 Exercise 39. 
 
 1. 1 duy. 
 
 2. 1 pk. 1 gal. 
 
 3. 1 bar. 10 gale. 2 qts. |S! P^. 
 
 4. oz. 
 
 0. 6 per. 7 yds. fl. 92f in. 
 
 6. 2 na. l^^jf in. 
 
 7. 6s. 
 
 8. 36 a. 2 r. 20 per. 
 
 9. 13 miles 2 fur. 
 
 10. 2 cwt. 2 qrs. 16 lb*. 10 -^ 
 
 10? drs. 
 11 2 oz. 3 drs- 2 scr. A6ff jf^ 
 12. Is. Offd. 
 
 ' iv .li 
 
 
 1. 2-m. 
 
 ■2. llflf. 
 
 8. S^V. 
 
 4. 13H^. 
 
 5. 243|t§. 
 
 6. ssm- 
 
 v. 283iff|. 
 
 1. 
 
 fsV 
 
 2. 
 
 a. 
 
 3. 
 
 ,m- 
 
 4. 
 
 US-ij^. 
 
 5. 
 
 1614V 
 
 1. ^^. 
 
 o R 
 
 — 7 f* 
 
 8. 2^. 
 
 4. stsa. 
 
 Exercise 40. 
 
 
 8, I88O7V 
 
 15. T38-,V 
 
 9. mmu- 
 
 16. 350|ffft 
 
 10. 51-,8(f5. 
 
 IV. mun 
 
 11. 5|m. 
 
 18. 81sV 
 
 12. I64V&-. 
 
 19. 5-5^%. 
 
 13. 64igV 
 
 20. 56ii|. 
 
 14. 50H. 
 
 
 Exercise -LI. 
 
 
 6. 886|H. 
 
 12. 5fmf 
 
 7. mh 
 
 13. smi 
 
 8. 195KB. 
 
 14. 85,V^. 
 
 9. 5|. 
 
 15. f,¥^^ 
 
 10. urn- 
 
 16. 158g»if4-. 
 
 11. 40TMiy- 
 
 
 Exercise 42. 
 
 
 6. eii 
 
 9. 2V12H^ 
 
 "lot 
 
 10. 190i-^. 
 
 \ 90tV- 
 
 11. 43tfg. 
 
 S tt 
 
 12. 2V0tH 
 
ANrfv^ERS TO THE EXLV.C18KS. 
 
 <*. i44. 
 
 
 18. if. 
 
 213 
 
 19- 211JH^. 
 
 1. 
 
 u. 
 
 8. 
 
 t>V<r. 
 
 8. 
 
 ^iiV- 
 
 €. 
 
 U^ 
 
 1^. 
 
 mi. 
 
 Exercise 43. 
 
 6. 40-,ii«V. 
 '7. 2,-T,-. 
 
 1 10. 52fg|. 
 
 12. 26i5j-. 
 I 15. 277ftf. 
 
 2. 17 bi.an. tf |,i-'p. 3 q(g_ qj pj^ 
 
 3. 5 Ibe. r-% M.. -iy dy.-i, 
 
 4. 1 1-. Siiv.r. l2>aB,8ft.97»in. 
 6 3 cwt. 24 Ib^. e o;-;. 2? diu 
 
 6. 2 a. 23jer. 25j(;s. 8:^J-} in. 
 
 7. i;3l;:}^. l(J|d. _>A f;^,.. 
 
 6 477 miles 7 fur. ::0) tr, 4 yds 
 
 lOilin. 
 fl. laibs. 7oz. l^Bj-drp. 
 to. 2Glbs. 9oz. 6drs. 6| .f. 
 
 Exercise 44.' 
 
 25 yds. 2qrs. Inn. ijin. 
 2c-wt. 10 lbs. l^oz. li-,vdi-g. 
 33bu6h.3pk8. 2qtH. l^j.is. 
 5wks. 5d:iys 20 hours 9 in in. 
 l2}-» pec. 
 
 lib. Qm. IS dwt. 22 grains. 
 3 a. 2 r. 5 j^er. 11 yds. 8 ft. 
 
 03 in. 
 i;i2 Gt*. 2j'',d. 
 2 cwt. 2 qre. S lbs. 15 ot 
 
 11. 
 12. 
 13. 
 14. 
 
 15. 
 10. 
 
 17. 
 
 ^8. 
 
 Gtn Urs. 
 
 T^XERCISE 45. 
 
 a«. 
 
ffifi : 
 
 Mil ' 
 
 214 
 
 yiNsw :r3 to tke exercises. 
 
 8. Scvcnty-ori" nnd c ithiy nino .luiidrodtlis of thousaiidtha ; ons 
 liuiulro'l ;ni<l s 5;ty-sev('ii, ml one liuiulrcd and nir.ety-tlireft 
 tlioii-aiHii ■ s ■. I'" c'ly-iiiif, :v\<] ('jrlil lliouBand six hundred 
 ftiid .-"cvoMiy-four tuiilimcf miHio.tlia. 
 
 4. Fivo iniM () h k'x hnndrol aril st*v( nty-foiir tlioiisand tlireo hun* 
 dsc'i! and PC'vciity-cijiht, ami iiiiif liundiod and tuurteoii tliou* 
 sand eever) liundrtd and clurliiy-six billioiiths ; scventyon^ 
 millio' s tliiTc hut d"('d tl onsaad I'dur liundrcd, nn<i fli* 
 hundred thou, and four jaindrcd ai.d ne\en trilliontht, 
 
 E 
 
 Iff 
 
 1. -09 ; -0070 , -000447. 
 
 2. -oooooToooir). 
 
 3. }00;)000502ii()ll. 
 
 4. -0057403. 
 6. -709. 
 
 1. 1742 41Sfi. 
 
 2. l:740;jl48. 
 S. I0185a)2!^493. 
 4. 910912::8, 
 
 ^EUCISK 40. 
 
 6. 49G710-0011004. 
 
 7. VG<^100G-00001.4700930. 
 8 1-71717. 
 9. 749-2000049. 
 
 10. 72-970704. 
 
 ExKRcrsR 47 
 
 5. 1027 -3^33. 
 
 G. 11478-9ir;G. 
 
 7. I'd -non:?. 
 
 S. 2oii84:04. 
 
 0. 43 9145714. 
 
 10. 88-88938. 
 
 11. 101 ■09300- 
 1?. 535-3531, 
 
 1. 705-753. 
 
 2. 92 14. 
 
 S. 44 73-190. 
 
 4. -07402713. 
 
 i, 72-922. . 
 
 2. ll-rs84. 
 
 3. 41S0-50n. 
 
 4. 1-U2-178 
 
 EXKRCISK 48. 
 5. 9100 07507104. 
 n. l')7'9>i55S70. 
 
 7. A-i{)^->. 
 
 8. •00104«098. 
 
 EXKRCISE 49. 
 5. 3-17*. 
 
 G. 1003. 
 
 7. 80-284. 
 
 8. 
 
 Jl«iv^>' 
 
 iViilK. 
 
 9. 077-080149. 
 10 10581-13006 
 
 11. -007:7028. 
 
 12. -0003444. 
 
 • 9. 137-28Ji. 
 10 G-341. 
 li. 8M3. 
 1-2. 2-33*. 
 
ANSWEKS TO THE EXEIICISES. 
 Exercise 60. 
 
 215 
 
 t 2857H ; -4. 
 
 2. -54 ; 1-8 ; -7. 
 
 e. -023076 ; -18 ; -95. 
 
 J. -69583 ; -57894736842105263i ; 
 
 •340938775510 + 
 •*»-S271GU193; 
 
 •4419551U3482G883 + 
 
 6. -063 ; 1550802139037431 
 
 7. -544809228039 + ; 
 31 -5802008965 + . 
 
 8. -865863 ; -61623. 
 
 9. -8600372419908+ ; 
 17-0280154849 + 
 
 10, l2-5294n7C47058823 : 
 -12499999SS6093+ 
 
 
 
 
 
 Exercise CL 
 
 
 1. 
 
 'h; n-, h 
 
 
 
 
 T fio-fa. .an ft, 
 
 •• yOdO ) 24 9 7 5* 
 
 2. 
 
 i/a ; Ul ; 
 
 -.¥r. 
 
 
 
 °- J 9 « t a ;iO' 
 
 3. 
 
 /i¥,Y.¥r ; 
 
 A. 
 
 
 
 ^' yuoooj "itioTT* 
 
 4. 
 
 -SLA. . lati 
 
 
 
 
 10. iV./^i^« ; 1 ; §f. 
 
 5. 
 
 2-111. LI. 
 
 2.1191 
 
 
 11. 273i; iVIU. 
 
 6. 
 
 6 60U 1 i)i)m)' 
 
 
 
 12. 4C7i 
 
 ^uVA; 16-.W«^ 
 
 
 Exercise />2. 
 
 1. 
 
 i5.a 
 
 -^9 0* 
 
 
 4. 
 
 elligt^. 
 
 '7. Wli 
 
 2. 
 
 qua 
 
 
 5. 
 
 uuh- 
 
 8. 46iWf^. 
 
 S, 
 
 "4 6U0' 
 
 
 G. 
 
 2^W3-. 
 
 9. -AWy. 
 10. iV6¥oV 
 
 
 Exercise 63. 
 
 
 I 
 
 •32738095. 
 
 
 5. 
 
 •25351 2'J9. 
 
 9. -0379585. 
 
 2. 
 
 ■6015625. 
 
 
 6. 
 
 -56018. 
 
 IQ, -2137093. 
 
 8. 
 
 •10449218. 
 
 
 7. 
 
 -847916 i. 
 
 11. -88200'25. 
 
 <- 
 
 •81918777. 
 
 , 
 
 8. 
 
 •12637. 
 
 
 12. 2-7001379. 
 
216 
 
 ANSWERS TO THE EXERCISES. 
 
 Exercise 54. 
 
 1. as. lid. -9136 far. 
 
 2. 6 days 3 min. 28224 sec. 
 
 3. 2 lbs. 1 oz. 6 dra. 6-72 grs. 
 
 4. 4 fur. 36 per. 3 yds. 6-048 in. 
 
 5. 3 r. 26 per. 17 yds. 5 ft. 
 124-31808 in. 
 
 L. 2 F. e. 3 na. 11431S in. 
 
 7. 9 hours 15 min. 52J sec. 
 
 8. 4 hhda. 1 bar. IS gals. 2 qts. 
 1-7208 pts. 
 
 9. 3 r. 13 per. IS yds. 5 ft. 73| vii. 
 
 10. £9 188. 3id. -44 far. 
 
 11. £21 138. lOid. -828 far. 
 
 12. 9 years 63 days 13 hrs. 68 
 min. 19-92 sec. 
 
 13. $0-343191. 
 
 14. 17 sq. yds. 6 ft. 129-492 in. 
 
 15. 2 tons 17 cwt. 1 qr. 2'S807; 
 
 lbs. 
 
 16. .£47 158. 4id. -049 fa*. 
 
 
 Exercise 55. 
 
 
 1. $2377883-3333 + 
 
 5. i;i628 4s. Hd. 
 
 8. $168729. 
 
 2. 13860. 
 
 6. 17l03vV,. 
 
 9. $117099. 
 
 8. $78-6425. 
 
 4. m, -^i§. n§, 
 
 7. A $2999-451 and 
 B and C each 
 $2249-62^. 
 
 10. $113-644. 
 
 11. DCCIV ; MCXI 
 
 MXDCCCLXXVI ; 
 
 XXMMMCDLXXI, 
 
 MXCXLMMC( 
 
 :CLXXI. 
 
 
 12. 70OO0C0O402O-0O00C 
 
 06200019. 
 
 
 13. 22816. 
 
 20. 4yd8. 2ft.2-566in. 
 
 27. 65902. 
 
 14. 12^. 
 
 15. -0902777+ 
 
 16. 274 
 
 17. 1, 1^, TTTTTI 
 
 97119 
 
 6»eoo' 
 
 18. 14791572 in. 
 
 19. 6fur.28pcr.4yf'8. 
 
 1 ft. 024 in. 
 
 21. Greatest j*j and 
 least ■^. 
 
 22. 2xVydB. 
 
 23. $25205i. 
 
 24. 602790 doz.; 
 $-119 and $107. 
 
 25. 110880. 
 
 26. H70/5«,. 
 
 28. 39 lbs. 6 oz. ISig 
 dra. 
 
 29. 118-6904002. 
 
 30. ^Ve. 
 
 31. $7-688167. 
 
 32. Sj'i. 
 
 
 Exercise 56. 
 
 
 1. 1| -^, T3, 24, m, t 
 
 )i. 
 
 S. 3r-5, -64i 
 
 I, 9, 9-6, -703, 6-6. 
 
 a.'*-; i|,4*64i,2« 
 
 f 
 
 4. 15, 5 5, 
 
 '^•758. 
 
 5 6, i5-857,»>'A31» 
 

 >. 
 
 ANSWERS TO THE EXERCISES. g^y ] 
 
 
 6. Greatest 47 : 79, least 18 : 33. 
 
 9. 2'7.-187, 
 
 ft. 73| til. 
 
 6. Greatest 11 : 3, least 164 : 55. 
 
 10. 12 : 1. 
 
 
 t Greatest 176 . 16 '4, least 8 . -89. 
 
 11. 12 : 5. 
 
 far. 
 
 8. 11 : 16. 
 
 
 12. 2048 : 7246. 
 
 ■ 
 
 13 hrs. M 
 
 
 1 
 
 
 
 Exercise 57. 
 
 
 1. 120. 
 
 8. 70. 1 ■•5. 30| days. 
 
 •492 in. 
 
 2.SA. 
 
 9. m 
 
 iO. 24^1 -weeks. 
 
 IT. 2'S807; 
 
 8. 4/r. 
 
 10. H- 
 
 17. JEIO 178. 2{f (!. 
 
 a*. ' 
 
 4. 161|. 
 
 11. 9900. 
 
 18. $36-627. 
 
 
 6. 161. 
 
 12. $143-50. 
 
 19. $20-3903. 
 
 
 " 6. 3J. 
 
 13. i;217 28. 7*d. 
 
 20. $186-30. 
 
 t 
 
 7. 12|. 
 
 14. $445-80. 
 
 
 ■ 
 
 Exercise 58. 
 
 
 *• 
 
 1. 40a. 2 r. 
 
 15. $2-3625. 
 
 29. 184 a. r. 26f5 p. 
 
 
 2. 16-50. 
 
 16. $311-48^ 
 
 30. £5 138, 7A<1. 
 
 DLXXI, 
 
 3. 116 men. 
 
 17. $0-7379. 
 
 81. 2a. Or. 31|§J5 per 
 
 
 4. 166|days. 
 
 18. £1584 Is. 10|i|d. 
 
 32. $2-648. 
 
 
 5. $4 0514. 
 
 19. $165-0375. 
 
 33. $8-2776. 1 
 
 6 oz. 15i3 
 
 6. $8936-26. 
 
 20. 254§. 
 
 34. 71fV llJS. 
 
 
 7. 1312-5518. 
 
 21. 1413-2421. 
 
 35. 9 dMy8. 
 
 1002. 
 
 8 $12511975. 
 
 22. $4-4789. 
 
 36. $104-5627. 
 
 
 9 $16787-4&. 
 
 23. 9| weeks. 
 
 87. $811-652. 
 
 »7. 
 
 iO $1264-8717. 
 
 24. 108 ft. 7| in. 
 
 38. JE7 178. e^^y^d. 
 
 
 21. £^lOB.^%d. 
 
 25. 1005 miles 6 fur. 
 
 39. $1-8214. 
 
 
 12. lOi months. 
 
 26. 683J yds. 
 
 40. 40i5. 
 
 
 J3 je291 128. OJid. 
 
 27. 53i. 
 
 
 
 14 $219-77 
 
 28. $50-46. 
 
 V 
 
 ,6«. 
 
 Exercise 69. 
 
 
 K/'aSI* 
 
 i oil AAVtAM 
 
 O QA » ^- 
 
 K rtrt K Jl 
 
 
 2. $712-72^. 
 
 4. 24U days, j 
 
 6. 138-65^1. ,* 1 
 
218 
 
 ANSWERS TO THE EXERCISES. 
 
 I 
 
 7. 252J ft. 
 
 «. 7 mi'ii. 
 
 9, 36d:iy8. 
 
 V). lG9/,cor'^8. 
 
 il. 12d:iyfl. 
 
 x2. 13-2 days. 
 
 13. 130 men. 
 
 15. 6277i bushels, 
 10. 7i2*-ai acrc'B. 
 
 17. $235-07 A'i. 
 
 18. 19237ilbfl. 
 
 19. 10000 Ibfl. 
 
 20. 93i|oz. 
 
 21. 1822i§*g|{ yds. 
 
 22. 182,V/o acres. 
 
 23. $5456-25, 
 
 24. 105JbuBhela. 
 
 
 Exercise 60. 
 
 
 
 1, $629-75. 
 
 11. £56622 7s. SJd- 
 
 21. 
 
 £24338 4s, 9»-d. 
 
 2. $2830-83. 
 
 12. £14693 15rf. 9iVs«»*. 
 
 '22. 
 
 £10771 4.^. 6d. 
 
 3. $690211-61. 
 
 13. $56435-16f. 
 
 ,23, 
 
 $30507-40. 
 
 4i £599 19s. 
 
 14. $21064-58tV 
 
 24. 
 
 £191 148. 7d. 
 
 5. £2906 148, 5id. 
 
 15, $42-90i|. 
 
 25. 
 
 $9353-03S||. 
 
 6. £90134 Is. lOJd. 
 
 16. $4289-33Ai, 
 
 26. 
 
 $70-91??. 
 
 7. $199175-92i. 
 
 17. $1677-75r'V 
 
 27, 
 
 $755-41/5. 
 
 8. $-2641 8 -40^^. 
 
 18. £46 15s, lOiJid- 
 
 28. 
 
 £274 10s. 7;|d. 
 
 9, $15371 -02f5. 
 
 19. $219-3li§. 
 
 29. 
 
 $20954-12JJj. 
 
 10, £487 16s, 4id, 
 
 20, £1540 88. 6^?gd. 1 
 
 30, 
 
 £1749 6s. 3i|H 
 
 
 Exercise 61. 
 
 
 1, -09 •, -045. 
 
 8, -002 ; 09375. 
 
 15. 
 
 100 ; -7. 
 
 2. -037 ; -2925. 
 
 9. -162; -0098, 
 
 16. 
 
 67i;3i. 
 
 8. -062 ; -082. 
 
 10, 1-472 ; -2612, 
 
 17. 
 
 12 ; 13. 
 
 4. 1-11 ; 1-47. 
 
 11. 7 ; 61. 
 
 18. 
 
 •95 J 121-7. 
 
 5. -0975 ; -6316. 
 
 12. 147 ; 
 
 19. 
 
 i;i3^. 
 
 6. '08 ; -005. 
 
 13. 87i ; 220. 
 
 14, 111; 1.0-7. 
 
 20. 
 
 ife;27^ 
 
 7, -00375 ; '02621), 
 
 
 
 1. $1644-516. 
 
 2. $1079-75. 
 
 3. $5-6875. 
 
 Exercise 62. 
 
 4. $524-72. 
 6. 17a.lr.22rer.ir 
 yds. 7 ft. 25iJ irv 
 
 6. 1 gal. 2 qt». if^ 
 
 pt8. 
 
 7, 3 lbs. 1 oz. 2 dwt 
 '22m era. 
 
 i 
 
»k^^As&.i 
 
 
 ANS\VEES TO THE EXERCISEB 
 
 Zli) 
 
 i 
 
 l4^-» i.t dny 27 » 1^. $1114-35; «12G-20;;; 
 
 $Jlo-l-Jl;§atiS-Ol; 
 $22i:8-7U. 
 
 mn, 
 
 iO. v^54-74i 
 
 11. ^5-6807. 
 
 12. ira57-6U95. 
 
 1. $3«rrl. 
 
 2. $"1-Oo2. 
 ^. $1C2, 
 
 *. $288-62S95. 
 t. ;i33&1f 
 6. ^OST-lrof 
 
 i. $21-710i£, 
 I $179 GSi, 
 J. $e84-80. 
 i, $32'77i. 
 
 '^ $37G-3328. 
 2. $8575-78. 
 C. $1219 21875. 
 i. $195G-845;42. 
 
 f. $107-14. 
 
 2.- i«l057 707. 
 
 8. $182'G5ti. 
 J. <iT'd.-d;yo 
 
 6. $101235. 
 4 $77 •4-1271. 
 
 oatfl ; lS-10 X 
 i-r>')( croyj*; 4w'-S0 
 a. fill low. 
 
 14. 45-4a.\vhoat; 40-80; 15. 2r>3,*,V kilu'ti 
 a. )^r;i>H ; 38-59 a. I ^Vlmll<l(•.! 
 
 peas; 4313 a. I taken pi isoi 2 -,i 
 
 
 ExEiiciSK G3. 
 
 1. $11.-95. 
 
 8. $344-4444. 
 
 9. $6.'i-G768. 
 
 10. $104-105625. 
 
 11. $131-Ili. 
 
 12. $374-10. 
 
 Exercise 64. 
 
 ) 5. ((7-551. 
 
 0. $140 -59f 
 
 7 $65-72f 
 
 8 $714-74. 
 
 Exercise 65. 
 
 5. ^5693-CO." 
 
 0. >lS82-3529. 
 
 7 ^5511-81102. 
 
 £ !*5G4S-80. 
 
 EXKHCISE G6. 
 
 r. $413 -8ef. 
 
 6. .5-le)]7 291». 
 
 0. $4102 744. 
 
 A $292 41'X 
 
 ,1. $r)6r9-4c.?6, 
 
 .2. $5533 bO. 
 
 13. $328 -TOJ. 
 
 14. $546. 
 
 15. $136-17. 
 
 16. $4807-C8|. 
 
 17. $186-25. 
 
 18. $38 47J. 
 
 9. $186-78}. 
 10. $153094375, 
 
 9. $683-4375, 
 10. $9797 3588. 
 
 13. $120-77S2- 
 
 14. $51-8889.':^ 
 
 15. $ll-44r-lF.5. 
 
 16. $159-721iti. 
 
220 
 
 ANSWERS TO THE EXERCISEa 
 
 1. $-04 ; f 'OSb ; 1-055. 
 
 2. $0-165. 
 8. $008. 
 4. $0-355. 
 
 Exercise 67. 
 
 6. $0-065. 
 0. $0-66 
 
 7. $0-265. 
 
 8. $0-377. 
 
 0. $0-1966. 
 
 10. ifU-2706. 
 
 11. $0-10910. 
 
 12. $015788. 
 
 1. tsss-ae. 
 
 2. $81-066. 
 8. $88 6908. 
 4. $41 -694281. 
 ^ 6. $12-96295f 
 6. $201-9515. 
 
 1. $225'043. 
 
 2. $209-9815. 
 8. $304-403. 
 4. $101-9151. 
 6. $36-8807. 
 
 EXERCISh 68. 
 
 7. $10-7918. 
 
 8. $21 -685801. 
 
 9. $115 85069i. 
 
 10. $202-73545f 
 
 11. $514-31488f 
 
 12. $579'91816|. 
 
 Exercise 69. 
 
 6. $11 0229. 
 
 7. Am't = $621-1484. 
 Inter. =$1211484. 
 
 8. Am't = $4'<7-0074. 
 Inter. =$77-0074. 
 
 13. $88-1616. 
 
 14. $6-35026|. 
 
 15. $12-03616. 
 
 16. $1356-7605. 
 
 9. Am't = $871-0362. 
 
 Inter. =$156-136-2. 
 
 10. Am' I =$906-7706k 
 
 iMi-i- «*li2'170ft 
 
 1. $12-7273. 
 
 2. $1-3301. 
 8. $7-2816. 
 4. $65-4188. 
 
 
 2. $11-20. 
 
 Exercise 70. 
 
 
 5. $20-40. 
 
 0. $90-8581 
 
 6. $16 3095. 
 
 10. $8-1879. 
 
 7. $70-1926. 
 
 11. $9-6416. 
 
 8. $75-8843. 
 
 12. $77-4863 
 
 Exercise 71. 
 
 3. $13 49,'^. 
 
 4. $7-626l76. 
 
 6. $11-5650. 
 
 e. $o-co5|. 
 
 
 m T<-i 
 
AN8V/S118 TO TIIK EXkRCISES. 
 
 221 
 
 EXK 
 
 •65. 
 06. 
 
 oia. 
 
 78i 
 
 616. 
 026|. 
 3016. 
 5-7606. 
 
 ^*gt^\r\ =$6'j'71J. 
 V» " u 1020. 
 
 = 2428J. 
 
 -.■*f)4'.iS2. 
 
 = J>31'8f}. 
 
 =$76-84ff. 
 
 = 61 40. 
 
 = 169«75A. 
 
 i A's 8liarc=$1408-60. 
 
 B'^ •« = 2622-55. 
 
 C'8 " = 3371-86. 
 
 6. $1381 57U; I1973-68A' 
 I4144-73H. 
 
 >. Firat 
 Second 
 Third 
 
 UCISR 
 6. 
 
 7. 
 8. 
 
 r 
 
 •t 9. 
 
 t 
 
 72. 
 
 A's 8harc=:$l()S0R8J2. 
 
 H'a •' = ir.io ii»j. 
 
 A's '« =$1837-08^,V». 
 
 n'rt '« = 2573-1 7|J» 5- 
 
 CM «' = 4588'83iVA- 
 
 A's losB =$6326. 
 
 B's •« = 2130. 
 
 Cs «« = 3105. 
 
 $1011 62^?, ; $161843tVV; 
 
 and $231'/ J. 
 
 f \0. A should fiav, ')358-17A'iV. 
 : and B $2131-8: ,VA. 
 
 ■^XERCISK 73. 
 
 = $8710362. 
 = $166-136-2. 
 = $900-77061 
 « ♦li2-17/)ft 
 
 )8r 
 
 16. 
 
 !68 
 
 9. 
 
 1. A'B 8hare^$2031-02|f J 
 B'8 " Xi 2i68-97jf; 
 
 2. A*s share:s^63d 96||. 
 B'8 " c: 8*»,'60?X. 
 C'b «' = 8tttr43^|. 
 
 ft A's ■harfl=$772-T2^ 
 B'B " = l22:-27rt-. 
 
 I. A's Bbaro=$44-38?3. 
 B'B " = 41-09^1. 
 C'8 " = 34-52^,. 
 
 «. A's Bhare=$753 42f |. 
 
 B'B share: 
 C'8 " : 
 D's '« : 
 
 6. A's share: 
 B'8 ♦• : 
 C'8 " : 
 
 '!. A's share: 
 STb " = 
 
 bi A's share: 
 Fs •' r 
 <;'8 " •: 
 
 :$342-46H. 
 : 513-69f J. 
 : 890-41/,. 
 :$198-60,fy. 
 : 38902Sf. 
 : 361 -DO* f. 
 :$2222-22i. 
 : 777-77f 
 :$720571f. 
 : 31V7-65A. 
 : 316->SK 
 
 Exercise 74. 
 
 1 *1R-R7i 
 — -r— --3. 
 
 2. $11-6T. 
 8. |27-8a 
 
 A AflOfi.lA 
 
 6. $88-05. 
 6. |28-»7A 
 
 8. $37-36^ 
 
222 
 
 1. $322C0. 
 
 2. 11041 CO." 
 8. $712-008. 
 
 ANSWERS TO THE J2XEUCIS.BS. 
 
 EXKRCISE 75. 
 4. $5iJ3*04. 
 C. $2413 5075. 
 6. $1218-56. 
 
 7. 15640-264. 
 S. $4068 6630. 
 
 1. 7 times. 
 
 2. 51 « 
 8. 16f * 
 
 f 8| 
 
 
 ExnuciSK 76. 
 
 
 1. OJ per cent. 
 
 3. 23j5?s per cent. 
 
 5. 9{ if: J percent, 
 
 ^' ISjVsVi per cent. 
 
 4. 5iY«V pe»' cent. 
 
 6. 151? per cent. 
 
 
 Exercise 77. 
 
 • 
 
 1. ^'im. 
 
 4. 10i5 cts. per lb. 
 
 7. $133-0275. 
 
 2. $251-1627. 
 
 5. $6636-36. 
 
 8. $9-1603 per 100(; 
 
 d. 16-6179. 
 
 6. $7-39805. 
 
 
 
 Exercise 78. 
 
 
 1. 66 cents. 
 
 4. $3-61tV 
 
 7. $3-00. 
 
 2. 364 lbs. 
 
 6. $8-755»,K 
 
 8. 202-304 quarta. 
 
 3. 57i», yards. 
 
 6. $0-072JJ«. 
 
 
 
 Exercise 79. 
 
 
 1. £178 148. 7jd. 
 
 8. $358 -37i. 
 
 14. $1434- 84229. 
 
 2. £365 198. 2id. 
 
 9. £204 78. 6id. 
 
 15. $2004 37255. 
 
 3. jei83 78. 2f d. 
 
 10. jei84 5s. 9d. 
 
 16. $4105-07115. 
 
 4. jei7 lis. 4/jd. 
 
 11. £209 lis. Hid. 
 
 17. $785-577;. 
 
 6. $1175-46|. 
 
 12. £18 8s. 9d. 
 
 18. £605 13b. 5d. 
 
 6. $785-59|. 
 
 la $144-77804. 
 
 19. £516 89. 9Jd, 
 
 7. $1059-09376. 
 
 
 20. $2292-2809. 
 
 M 
 
 Exercise 80. 
 6. 4 times. 
 
 6. 26? " 
 
 7. 13J 
 
 8. 6f 
 
 (( 
 
 t* 
 
 9. IStimei. 
 
 10. 5|. 
 
 11. 79J. 
 
 12. sia . 
 
ANSWERS TO THE EXEIlCISEa. 
 
 223 
 
 r cent, 
 
 ont. 
 
 rl00(? 
 
 arts. 
 
 29. 
 55. 
 15. 
 
 5d. 
 
 n^ 
 
 J. 
 
 1 ts. 
 
 li. iO^lba. 
 
 'i5. <)69. 
 
 10. 33. 
 
 i7. 7} days, 
 
 18. 8-?, days. 
 
 319. Todays. 
 
 ». 289. 
 
 a. 12167. 
 
 U. 77841. 
 
 4. 531441. 
 
 5. 1296. 
 
 6. 3125. 
 
 7. 4096. 
 
 1, 36. 
 
 2. 63. 
 
 3, 126 
 
 4. 231. 
 6. 378. 
 6. «99. 
 •?. -494. 
 
 1. 32. 
 
 2. 87. 
 
 3. 24. 
 
 4. 63. 
 
 5. 9<>, 
 
 6. 125. 
 
 7. 2&1. 
 
 20. $106. 
 
 
 48j'»j in., or 63f| 
 
 21. jfTSOO. 
 
 
 njiii'8 from whove 
 
 2-2. |l-31rV 
 
 
 tluy started. 
 
 23. $3-92i^ 
 
 
 27. 200 iiiilfB. 
 
 '24. 84. 
 
 
 '28. 109^0. 
 
 25. 25;^ feet. 
 
 
 29. 88. 
 
 26. In 15 J*, hours and 
 
 30. 040. 
 
 Exercise 
 
 81. 
 
 
 8. 2187. 
 
 
 14. s??5. 
 
 9. 250. 
 
 
 15. 1500625. 
 
 10. 196S3. 
 
 
 10. 2339G5J9. 
 
 11. 343. 
 
 
 17. 76t». 
 
 12. 14641. 
 
 
 18. 24389. 
 
 13. 531441. 
 
 
 19. 60|f. 
 
 20. 59049. 
 
 ExF'CISE 
 
 82. 
 
 
 8. -629. 
 
 
 14. 20-7304. 
 
 9. 27-5. 
 
 
 15. *, S, H, ^V 
 
 10. 106-759. 
 
 
 16. -7977, -727* 
 
 11. 313-248. 
 
 
 17. 20-698. 
 
 12. 2590-929. 
 
 
 18 25095. 
 
 13. 958-523. 
 
 
 19. 1062-024. 
 
 20. 402-383. 
 
 Exercise 83. 
 
 
 8. 384. 
 
 
 14. -971, -985. 
 
 9. 6-26. 
 
 
 15. 1, -464, -4807. 
 
 10. 8-83. 
 
 
 16. 75-36. 
 
 11. 99-7. 
 
 
 17. 9 34G. 
 
 12. -OTl. 
 
 
 18. 971 iSS, 
 
 13. i,/i.l,i> 
 
 
 19. 200-92. 
 30. 8 0276. 
 
224 
 
 ANSWERS TO THE EXERCISES 
 
 t 
 
 
 Exercise 84. 
 
 
 1. $275-67! {. 
 
 10. 44-169. 
 
 17. $10507-11,%. 
 
 2. 122. 
 
 11. $714883. 
 
 18. $39-1413 ; 
 
 3. |1177-27f 
 
 12. 17-223 !l)9. 11 oz. 
 
 $37-19807. 
 
 4. 29-91. 
 
 6. S,h- 
 
 6. $1004-0968. 
 
 2 dwt. 2U f^^- 
 
 13. ia,hunh- 
 
 14. 5 fur. 28 per. S 
 
 , 10. 304? days. 
 
 20. 12-2177. 
 
 21. 1 wk. 6 days Iv 
 hours 41 mil*. 
 
 '^' S' i«» tVi'bj SSI- 
 
 yds. 2 ft. 24f in. 
 
 32t*3 «ec. 
 
 8. I51-2. 
 
 15. 33597 4749. 
 
 22. $7910-98^. 
 
 9. 5 times 8. 
 
 16. $3471-38j»,V 
 
 23. 109-708. 
 
 24. XXVIl ; CCCX 
 
 CIII; MVDCC; LXl 
 DCCXVTII. 
 
 CVMMMCMIV; 
 
 MXCXXXVM 
 
 
 25. 2178 . 85. 
 
 29. $3843-64226; 
 
 31. 3i, H, and |} per 
 
 |26. They have none. 
 
 £613 5rt. OH. 
 
 cent. 
 
 27. 46 J 1 hours. 
 
 30. A't -$954-01488, 
 
 32. 79200. 
 
 28. 11021 rods. 
 
 Int. =1154-01488. 
 
 
 33. 90104000000700&'00 
 
 000009030017. 
 
 
 84 A'8 tfharo, 
 
 41. 78-075. 
 
 63. 18480. 
 
 =$844-17?5. 
 
 42. 8928 a. 3 r. 15 per. 
 
 64. $251-714|. 
 
 B'Bdo.=$719 11?i. 
 
 2 yd3. 2 ft. 3f in. 
 
 65. 49^ cents. 
 
 C'8do.=$90e-70|§. 
 
 43. 10039i inches. 
 
 56. ^53-125. 
 
 35. m} per cent. 
 
 
 67. A's share. 
 
 86. 1 a. 1 r, 33 per. 9 
 
 45. $1144-21J. 
 
 =$70'75»wf 
 
 yds. 3 n. 86 in. 
 
 46. $2604. 
 
 B's «' =$4b-57,»AV 
 
 37. ;e890 6s. 9Jd.-, 
 
 47. 1U9241II pace 
 
 C'8 " =$82-67;i{?. 
 
 $78-625. 
 
 48. $6801■72|^ 
 
 88. 61jV« cc-ntP. 
 
 38. 2 bu. 2 pkB. 1 qt. 
 ipt. 
 
 49. $392-2201. 
 60. 9^ days. 
 
 69. A ai'd B h;!vi' 
 $27f,_ aiMl D $65Q 
 
 89. 19-26. 
 
 61. tVtVj. 
 
 (h. P/v liours pes 
 
 40. 31H minutes. 
 
 62. 67|m. 5 
 
 TaS END, 
 
0711,%. 
 
 1411 ; 
 
 19807. 
 days. 
 177. 
 
 k. 6 days Iv 
 rs 41 mil*. 
 
 0-98^. 
 • 08. 
 
 CMIV; 
 
 f , and 1} pelf 
 0. 
 
 D. 
 
 ■714|. 
 cents. 
 .25. 
 
 share. 
 
 =$70'75J«f. 
 =|4b-57,»AV 
 =$82-67;^^. 
 j ccntp. 
 
 !m1 B h;!Vi' 
 
 ai!«lD$55Q 
 iiours pes