UC-NRLF 
 
 SB 27fl 
 
 S 
 
 MKNT A!, v Nl " -MTTJ 
 
 jjrtiljmctxul <] 
 
 PERTAINING TO THE WORK OF HANDLING WHOLE 
 AND FRACTIONAL HUM^fiS AND A GREAT 
 VARIETY OF IMPORTANT PRACTICAL 
 'OBLEMS 
 
 BEING A COMPLETE IHTEOE7CTOHY vVOEZ 
 
 S ( .' ! K 
 
 i 377. 
 
IN MEMORIAM 
 FLOR1AN CAJORI 
 
SOULE'S 
 
 EMBRACING MENTAL AND WRITTEN 
 
 jirttijmettcal (J.xcrdses *ni Camples* 
 
 I'KKTAININ<; TO Til K WORK OF HANDLING NUM HERS AND THK 
 
 APPLICATION TUKREOF TO SIMPLE PRACTICAL QUESTIONS 
 
 INVOLVING THE PRINCIPLES OF 
 
 Addition, Subtraction, Multiplication and Division 
 
 OF WHOLE AND FRACTIONAL NUMBERS. 
 
 Also a great varii'ty ami ;i large number of important practical problems, such an 
 
 occur in the ('oiiiitiiigio.mi, Factory, Workshop, on the Plantation, 
 
 and throughout the various departments of business life. 
 
 /)rsi</nctf ns a Siiftjflctticnt to 
 
 Soule's Philosophic, Commercial and Exchange Calculator 
 
 AND, AS AN 
 
 S C I E N C E OF N U M B K R 8 . 
 
 By G^EO. SOTJLE, 
 
 ! 
 
 Practical and Consulting Accountant, Commercial Lawyer, President of 
 
 Sony's Commercial College arnf Literary Institute, author of "Con- 
 
 tractions in Numbers" and the " Analytic ai;d Philosophic, 
 
 ( 'cwnicrcial and Exchange Calculator." 
 
 NEW OKLK.L\S .- 
 
 Autlior. 
 
 1877. 
 
ENTERED ACCORDING TO ACT OF CONGRESS, IN THE YEAR 1877, 
 BY O E O . S O TJ L sT, 
 
 IN THE OFFICE OF THE LIBRARIAN OF CONGRESS, AT WASHINGTON. 
 
 FELINE, PRINTERS, ~| 
 avier St.,' N. 0. ) 
 
 CLARK & HOFELINE, PRINTERS, 
 112 Grav 
 

 PREF AC E. 
 
 |HE design of this work is twofold : 1. As a supplement 
 to the author's large and advanced treatise, which does 
 not contain sufficient primary work to meet the wants of 
 young pupils. 2. As an introductory treatise to the science of 
 numbers. It is especially designed to supply the requirements 
 of primary and intermediate classes, and, at the same time, it 
 presents much practical work of rare import to the advanced 
 student. Bills and invoices of various forms for many depart- 
 ments of business constitute a special feature of the work. It is 
 believed to possess superior merit on the following points : 1 . 
 In the arrangement and character of the mental exercises and 
 the logical methods of mental training. 2. In the extent, variety* 
 practical and scientific character oflhe problems. 3. In the 
 elucidation of subjects. 4. In the philosophic solution of prob- 
 lems, by which system all the, reasoning organs of the mind 
 are expanded and the learner capacitated, not only to produce 
 the results of problems, but to observe fine distinct! ms, reason 
 logically, and deduce correctly ; thus qualifying for a high plane 
 of usefulness in various vocations of life. 
 
 The philosophic system is believed to be the most valuable 
 improvement yet made to impart a thorough knowledge of the 
 principles of numbers and capacitate the learner to utilize the 
 same in the practical affair*' of business life, and a* it is the only 
 natural system, it is destined at no distant day, to be adopted by 
 all reasoning minds and efficient instructors. But, notwithstand- 
 ing its superiority and the fact that its advocates include manv 
 
IV PREFACE. 
 
 of the most profound mathematical minds, yet, like every other 
 improvement or discovery in education, commerce, art, or science^ 
 it has opponents and is regarded with indifference by those who 
 are satisfied with the non-progressive and non-reasoning methods 
 of past ages. 
 
 With an experience extending through a period of twenty 
 years with nearly four thousand students, the author has tested 
 the advantages of the philosophic system, and from a full knowl- 
 edge of its superior merits, he conscientiously assures his co-labor- 
 ers in the mathematical field of education that a more thorough 
 knowledge of the science of numbers can be imparted, and in 
 far less time, by this system than by the usual methods and sys- 
 tems of work. 
 
 The science of numbers is the front door to the grand Temple 
 of mathematics, in which are displayed some of the most beauti- 
 ful principles of logic and profound syllogistic, analogical and 
 axiomatical truths to l>e found in the vast fields of thought. 
 And all who aspire to pre-eminence in brain power all who 
 hope to ascend to the highest planes of mathematical knowledge, 
 must devote themselves earnestly to this subject. 
 
 In the selection of the material and the elements for it prob- 
 lems, this work does not present the toys and play things of the 
 nursery, nor does it confine itself to the articles bought and sold 
 on 'change. Instead of gyrating in the non-practical and non- 
 progressive paths described by its hundreds of predecessors, it 
 has diverged into new channels and derived the facts and ele- 
 ments of many of its problems from geography, history and 
 chronology ; from educational and commercial statistics ; from 
 Natural Philosophy, Astronomy, Geology and Chemistry ; from 
 Anatomy, Physiology and Hygiene; and from many other 
 
PREFACE. V 
 
 departments of scientific knowledge. Through this means, the 
 work is rendered far rno^e interesting, and as it brings into use 
 different organs of the mind from those which consider the com- 
 putation of numbers only, it thereby imtaes the mind of the 
 learner with much valuable information without the cost of 
 additional study or the expenditure of additional time. 
 
 The work has been prepared during such intervals of time 
 as the author could command from his professional duties as 
 teacher of Business Sciences, and Consulting and Practical 
 Accountant ; and, nowithstanding great care has been bestowed 
 upon it, it is not improbable that some typographical or other 
 errors may have escaped notice. Should any such be found, the 
 author will esteem it a favor to be informed of them, in order 
 that they may be expunged in future editions. 
 
 The author avails himself of this occasion to extend his 
 thanks to his associate instructors and advanced students for 
 their kindly aid in proof reading, and especially does he express 
 his gratitude to his assistant instructor, Mr B. I). Rowlee, for 
 services cheerfully rendered in proof reading and the re-working 
 of problems. 
 
 Soliciting for the work a thorough examination and a just 
 measure of its merits, with the earnest hope that it may prove 
 acceptable, and be of service in unfolding the principles of the 
 beautiful science of numbers, and aid in advancing the interests 
 of the rising generation, it is now submitted to the public. 
 
 THE AUTHOR. 
 NEW ORLEANS, Jan. 4, 1877. 
 
TOPICAL INDEX. 
 
 Pago. 
 
 DEFINITIONS 1 to 5 
 
 FRENCH SYSTEM OF NUMERATION 6 
 
 ENGLISH SYSTEM OF NUMERATION 7 
 
 THE ROMAN SYSTEM OF NOTATION 8 
 
 TABLE OF ROMAN CHARACTERS 8 
 
 EXERCISES IN NOTATION AND NUMERATION 9 
 
 ADDITION (Definitions and Introduction.) 10 
 
 Addition Tables 11 to 12 
 
 Addition and Subtraction Tables 13 to 14 
 
 Signs and Abbreviations 15 
 
 Mental Exercises in Addition 10 to 17 
 
 Examples in Addition IT to 23 
 
 Proof of Addition 18 
 
 Dollar and Cent Signs 23 
 
 Addition of Dollars and Cents 23 
 
 Miscellaneous Examples in Addition 24 to 31 
 
 STBTRACTION Definitions and Introduction 31 to 32 
 
 Oral Exercises in Subtraction 32 
 
 Examples in Subtraction 32 to 39 
 
 To subtract Dollars and Cents 35 
 
 MULTIPLICATION Definitions and Introduction... 40 
 
 Multiplication Table 41 
 
 Oral Exercises in Multiplication 42 to 43 
 
 To Multiply when the Multiplier consists of 
 
 only one figure 43 
 
 Examples in Multiplication 43 to 53 
 
 To Multiply when the Multiplier consists of 
 
 more than one figure 45 
 
 To Multiply when either the Multiplicand or 
 Multiplier or both have naughts on the 
 
 right 47 
 
 To Multiply by the Factors of a number 48 
 To Multiply when the Multiplicand or Mul- 
 tiplier contains dollars and cents 
 
 Miscellaneous Problems in Multiplication... 49 to 53 
 
 DIVISION Introduction and Definition 53 
 
 Principles of Division 53 to 54 
 
 Proof of Division .. 54 
 
TOPICAL INDEX. Vll 
 
 DIVISION Oral Exercises in 54 to 55 
 
 Fractional Numbers and Examples 55 to 57 
 
 Written Exercises 57 
 
 To Divde when the Divisor does not exceed 
 
 12 57 
 
 Short Division , 57 to 60 
 
 To Divide when the Divisor exceeds 12 59 
 
 Examples in Short Division 58 to 59 
 
 Long Division 60 
 
 To divide when there are naughts on the 
 
 right of the Divisor 61 
 
 Examples in Long Division 60 to 70 
 
 To Divide by the Factors of a number 63 
 
 Problems involving the English Money of 
 
 Account 69 to 70 
 
 Miscellaneous Problems involving the Prin- 
 ciples of Addition, Subtraction, Multipli- 
 cation and Division 70 to 72 
 
 CANCELLATION Fully explained 72 to 75 
 
 PROPERTIES and Definitions of Numbers 75 to 77 
 
 Divisibility of Numbers 77 to 78 
 
 FRACTIONS and Definitions of Fractions 78 
 
 Classification of Fractions 79 to 80 
 
 General Principles of Fractions 80 
 
 REDUCTION OF FRACTIONS 80 
 
 Oral Exercises in Fractions 80 to 86 
 
 Greatest Common Divisor and Examples... 86 to 87 
 
 Least Common Multiple and Examples 88 
 
 To Reduce Fractions to their lowest terms.. 89 
 To Reduce Whole or Mixed numbers to 
 
 Improper Fractions 90 
 
 To Reduce Improper Fractions to Whole or 
 
 Mixed numbers 90 
 
 To Reduce Compound Fractions to Simple 
 
 Fractions 91 
 
 To reduce fractions of different denomina- 
 tors to equivalent fractions of a common 
 denominator or of the least common de- 
 nominator 92 to 94 
 
 DENOMINATE FRACTIONS 94 
 
 To reduce a denominate fraction from a 
 
 greater unit to a less 94 
 
 To reduce a denominate fraction from a less 
 
 unit to a greater 94 
 
 ADDITION OF FRACTIONS 94 to 98 
 
 SUBTRACTION OF FRACTIONS... 99 to 102 
 
Vlll TOPICAL INDEX. 
 
 MULTIPLICATION OF FRACTIONS 103 to 111 
 
 To multiply Abstract Fractional numbers... 108 
 Miscellaneous examples in Multiplication of 
 
 Fractions 108 to 111 
 
 DIVISION OF FRACTIONS 112 to 121 
 
 Division of Abstract numbers 116 
 
 Miscellaneous examples in Division of Frac- 
 tions 117 to 121 
 
 Miscellaneous examples involving the prin- 
 ciples of Addition, Subtraction, Multipli- 
 cation and Division of Fractions 121 to 127 
 
 DECIMAL FRACTIONS Definition* and Intro- 
 duction 127 to 144 
 
 Exercises 131 
 
 Principles 133 
 
 Reduction of Decimals 133 
 
 To reduce Decimal Fractions to a common 
 
 denominator 133 
 
 To reduce a decimal to a common fraction.. 133 
 To reduce common fractions to equivalent 
 
 decimals 134 
 
 ADDITION OF DECIMALS l.:<; 
 
 SUBTRACTION OF DECIMALS 137 
 
 MULTIPLICATION OF DECIMALS 138 
 
 To multiply a decimal or mixed number by 
 
 10, 100, 1000, etc 140 
 
 DIVISION OF DECIMALS 140 
 
 To divide Decimal Fractions by 10, 100, 
 
 1000, etc 143 
 
 MISCELLANEOUS PRACTICAL PROBLEMS 
 
 ^uch as occur in the counting-room, factory, 
 
 workshop, on the plantation, and in the 
 
 various departments of business life 144 to 149 
 
 BILLS AND INVOICES 149 to 165 
 
 WEIGHTS AND MEASURES 165 
 
 Definitions and comparison of different units. 165 to 167 
 
 Tables of 167 to 173 
 
DEFINITIONS. 
 
 1. Definition is the meaning or import of a word 
 ex pressed by other words. 
 
 is classified knowledge. 
 
 >. Quantity i s nn y thin^ that can he increased or 
 diminished. 
 
 4. A I llit is ti single tiling of whatsoever denomination 
 or nature. 
 
 5. A Number is a unit or a collection of units. 
 
 <>. All Abstract Xmilber is one in which the kind of 
 unit or (juantity is not designated, thus: three, four, five, 
 
 etc. 
 
 7. A Denominate or Concrete Number is one in 
 
 which the kind of unit is designated. Thus: two pounds, 
 five vards. nine dollars, etc. 
 
 K. A Compound Number is a denominate numher 
 expressed in two or more denominations, thus: ,"> years. 4 
 months and S days: '1 milrs, ."> furlongs and 10 rods: 2 
 yards, '2 feet and 5 inches. 
 
 !>. An Arithmetical Complement of a Number 
 
 i.sthe dillerenc<- hetwern the numher and a unit of the next 
 higher nrder, thus: .'I is the arithmetical Complement of 7 ; 
 -(> is the arithmetical complement of 71 : l!> is the arith- 
 metical complement of J)Sl. 
 
 1 (> . A Problem is a <|uestion ju-oposed or given for 
 solution. 
 
 11. Philosophy the knowledge of phenomena as 
 explained hy and resolved into causes and reasons, powers 
 
 aud hi\vs. 
 
4 Arithmetical Exercises and Examples. 
 
 12. ^ Arithmetic is tin- Science of Numbers: or to 
 define it more extemledly, it is that branch of .Mathematics 
 which treats of the properties and relations of numbers 
 when expressed by the aid of figures, either singly or com- 
 bined. These principles and relations of numbers combined 
 with the facts relating to problems, aie applied, by the rea- 
 soning powers of man to the solution of all numerical prob- 
 lems of business affairs and practical life. 
 
 13. Figures; in Arithmetic _ ///////> x are characters used 
 to represent numbers. The ten Arable figures which we 
 use, are 
 
 Nan-lit or Cipher One T\v.> Tim-.. F"iir Fi\i- Six Seren Ki^lit Nino 
 
 1 2 3 4 5 (I 7 8 !) 
 
 By properly combining these ten figures all possible num- 
 bers may be represented. 
 
 The 1, 2, 3, 4, 5, 6, 7, 8, and H are sometimes called dibits. 
 They are also called the significant figures because each sig- 
 nifies a number when alone. 
 
 The is so called because by itself it docs not signify or 
 represent any number. It expresses number only when 
 used in connection with other figures. 
 
 14. Yalue of Figures Figures have two values, a 
 afin}rfc and a local value: thus when we write 2, independ- 
 ent of other figures, it has only a simple value, representing 
 two units ; but when we write it to the left of another fig- 
 ure or figures, thus, 23 or 241, it has a local value as well 
 as a simple value. This local value depends on the scale or 
 system of numbers employed and its location in the scale. 
 
 
 
 15. Order of Figures The successive places occu- 
 pied by figures are often called orders. Thus a figure in 
 the first place is called a figure of \\\Q first order, or of the 
 order of units ; a figure in the second place is a figure of the 
 second order, or of the order of tens ; in the third place, 
 of the third order, or of the order of hundreds ; and so on, 
 each figure next to the left belonging to a distinct order, 
 the unit of which is tenfold the size or value of a unit of 
 
Definitions. 
 
 the order of the figure on its right ; and this increase in 
 value from right to left Ly ten constitutes the Decimal Scale 
 or fystem of numbers. 
 
 16. Notation is a method of writing numbers. There 
 are two systems, the Arabic and Roman. 
 
 By the Arabic Notation numbers are expressed or writ- 
 ten by Jipures. 
 
 By the Roman Notation numbers are expressed or writ- 
 ten in letters. 
 
 pi 
 
 17. Numeration is the method of reading written 
 numbers. 
 
 There are two systems of numerating or reading numbers, 
 the French and the English. 
 
 The French system is the one in general use in the 
 United States and the Continent of Europe. 
 
 The Enf/1Ixh system is that generally used in England 
 and the English Provinces. 
 
Arithmetical Examples and Exercises. 
 
 FRENCH S VST KM < K M MKKATIoN. 
 
 18. The Fivnch system separates liirure^ into Croups or 
 periods of three figuivs each. and -ives a different name to 
 each period, thus : 
 
 ' 2, 
 = 7 f 
 |J|J 
 
 " ' ft v 
 
 i - 
 
 - 
 
 " ' ft I 4^ 
 
 -= , " 
 
 ? , r ic Bundreda of 
 
 |ft|J cc Tens of Octillions. 
 ( * Octillions 
 
 | "ii Hundreds of ! 
 f-f !'] ~ Tens of 
 " 2, I J-* Se|lilli)iis. 
 I hmdivd- of 
 Tt-us of SrxtilliniH. 
 
 / >rXtiHi<MH. 
 
 ^ Hundreds of ( t )uintilli<ins. 
 ^-i-i ~' Tens of Quintillionfl, 
 
 i ( ;- < v )uintil lions. 
 3:^5 / ~- Hundreds <.f (Quadrillions. 
 Ti-n.- of Quadrillions. 
 Quadrillions. 
 
 Hundn-d-- 
 
 Tens of ' 
 
 Trillions. 
 
 Hundreds of Uillions. 
 
 Tens of Billions. 
 
 Billions. 
 
 5 - | tc Hundreds of Millions. 
 |^|-j o Tens of Millions. 
 7 - ( & Millions. 
 . & ^ ( t_i Hundreds of Thousands. 
 Ifg-j ?o Tens of Thousands. 
 ! ' ft I j^- Thousands. 
 
 Hundreds. 
 
 Tens. 
 
 Units, 
 
 The periods aliove Octillions, in regular oulf-r, are !u>nilliun>, Decillions 
 I'ndecillions, Duodecillions, Tredecillions, Quatuordecillioiis, Qnindecilliona 
 
 SexdeciUions, Se|>t^ndecillions, Octod'.'cilli'.!is >'tvfuidecil lions Vii:;intillious, 
 &c. 
 
English System of Numeration. 
 
 ENGLISH SYSTEM OF NUMERATION. 
 
 19. The English system of numeration separates the 
 figures into groups or periods of six figures each, and desig- 
 nates each period by a distinct name, thus : 
 
 ic Hundreds of Thousands of Quadrillions. 
 
 GO Tens of Thousands of Quadrillions. 
 
 4-* Thousands of Quadrillions. 
 
 ^i Hundreds of Quadrillions. 
 
 cs Tens of Quadrillions. 
 
 ^ Quadrillions. 
 
 .: Hundreds of Thousands of Trillion-. 
 
 i~ Tens of Thousands of Trillions. 
 
 -j. Thousands of Trillions. 
 
 Hundreds of Trillions. 
 - 1 Ten- nf Trillions. 
 
 j> Trillions. 
 
 - Hundreds of Thousands of Billion-. 
 
 o Tens of Thousands of Millions. 
 
 4- Thousands of Billions. 
 
 - I lundreds of Billions. 
 
 -: Tens of Billions. 
 
 t~ I i II ion-. 
 
 4- Hundreds of Thousands of Millions. 
 
 / Tt-ns of Thousands of Millions. 
 
 -i Thousand^ of Millions, 
 
 tc 1 lundreds of Millions. 
 
 ~- Tens of Millions. 
 
 :; Millions. 
 
 Hundreds of Thousands. 
 co Tens of Thousands. 
 
 r> Thousands. 
 
 ^. Hundreds. 
 
 C5 Tens. 
 
 /. I'n its. 
 
 By examining and comparing the two systems, it will be 
 observed that they are the same to the ninth figure or the 
 hundreds of millions, but at that figure a variation is made. 
 Hence, if we wish to know the value of numbers higher 
 than hundreds of millions, when we hear them spoken or 
 see them in print, we must know whether they are ex- 
 pressed according to the French or the English system of 
 numeration, 
 
Arithuwtical K.\ari^cs and Examples. 
 
 TIIK KO.MAN SYSTFM OF NOTATION. 
 
 *JO. hi tin- Roman system of notation the letter 1 repre- 
 sents '///.; Vjfive; X. ////; L. //////; (', nut- Jntmlnd; 1>, 
 ///v hundred &ud M. <//// ///////> ///J. The intermediate and 
 'lini: nnml.-is are express. -d aeeordinL: to (lie Ibllow- 
 iiiLT principles : 
 
 First. Kvery time a letter i.- repeated, its value is re- 
 peat- d : thu> II ivpivseiits fteo; XX rejr->ents ///v uti/. 
 
 Srcond. Wlieiia Irt ter o{' / xs/ / value is jilaeed before one 
 of i/rt-utt r value, the lesser i> taken iroin the greater : ifplaeed 
 afii-r the greater, it is to lie added to it. Thus, IV repre- 
 sents f<'i\ while V I represent.- >/./; XL represents Jrtij^ 
 I A represent.^ >/.////. 
 
 Third. A line or Kir . plaeed over a letter, increase> 
 it> value a thou>and /inn*. Thus X re]rt'-ent> t> n 
 sitnd ; L re]>resents ////// 
 
 OF KO.MAN CHARACTERS. 
 
 I 
 
 II 
 
 111 
 
 IV 
 
 V 
 
 VI 
 
 VI I 
 
 II 11 
 IX 
 
 X 
 
 XI 
 
 XII 
 
 XIII 
 
 XIV 
 
 XV 
 
 XVI 
 
 XVII 
 
 XVIII 
 
 XIX 
 
 XX 
 
 XXI 
 
 XXII 
 
 XXIII 
 
 XXIV 
 
 one. 
 
 XXV 
 
 twenty-live. 
 
 two. 
 
 XXVI 
 
 twenty-six. 
 
 three. 
 
 XXVII 
 
 lweuty->even. 
 
 four. 
 
 XXVIIJ 
 
 twenty-ei.uht. 
 
 five. 
 
 X X 1 X 
 
 twenty-nine. 
 
 ->ix. 
 
 X X X 
 
 thirty. 
 
 seven. 
 
 XL 
 
 forty. 
 
 eight. 
 
 L 
 
 fifty. 
 
 nine. 
 
 LX 
 
 sixty. 
 
 ten. 
 
 LXX 
 
 seventy. 
 
 eleven. 
 
 LXXX 
 
 eighty. 
 
 twelve. 
 
 xc 
 
 ninety. 
 
 thirteen, 
 
 ( 
 
 one hundred. 
 
 fourteen. 
 
 cc 
 
 two hundred. 
 
 fifteen. 
 
 ( < 
 
 three hundred. 
 
 sixteen. 
 
 ( ( ( ( 
 
 four hundred. 
 
 seventeen. 
 
 I) 
 
 live hundred. 
 
 eighteen. 
 
 DC 
 
 six hundred. 
 
 nineteen. 
 
 DCC 
 
 seven hundred. 
 
 twenty. 
 
 IK.'v (' 
 
 ei<;lil hundred. 
 
 twenty-one. 
 
 pcccc 
 
 nine hundred. 
 
 twenty-two. 
 twenty-three. 
 
 M 
 MM 
 
 one thousand. 
 
 two thousand. 
 
 twenty-four 
 
 MixrrLXXVI 
 
 1876, 
 
Exercises in Notation and Numeration. 9 
 
 EXERCISES IN NOTATION AND NUMERATION. 
 
 21. In Writing Numbers begin at the left hand 
 with the highest order and write each period in regular 
 order, separating them by com mas. 
 
 Write in figures the following numbers and nuni'Titc 
 them according to the French .system of numeration. 
 
 1. One thousand, six hundred and ninety- four. 
 
 2. Eighteen hundred and seventy-seven. 
 .'{. Twenty-four hundred and six. 
 
 4. Three hundred forty-one thousand and twenty two. 
 
 5. Sixty-five million, one hundred thirty two thousand. 
 three hundred and eighty -seven. 
 
 6. Twelve billion, sixteen million, forty-throe thousand. 
 one hundred and eleven. 
 
 7. Nine hundred thousand, three hundred and iifiy. 
 
 8. Six million, one hundred and sixty-nine thousand, 
 four hundred and thirty-seven. 
 
 1). Seventy-six million, four hundred thousand, one 
 hundred. 
 
 10. Twenty-two billion, one hundred three million, 
 five hundred seventy-six thousand, one hundred and two. 
 
 11. One hundred two trillion, one hundred twenty live 
 million, four hundred and three. 
 
 12. Eight trillion, seven billion and seventy-six. 
 
 22. Write in figures the following numbers and nume- 
 rate them according to the English system of numeration: 
 
 1. Four hundred twenty-three thousand, five hundred 
 and fourteen. 
 
 2. Six hundred nineteen thousand, one hundred lil'ly- 
 two million, t\venfy-one thousand and fortv seven. 
 
 !>. Fifty-three billion, two hundred twelve tlmus-nd. 
 twenty-six million, seventy-live thousand three hundred 
 and eighty-four. 
 
 2:J. Write in the Koman System <;f Notation the follow- 
 ing numbers : 
 
 l< ->, 12. 14,37,49, s: 1 ,. His, ;,i!>. i:,ij. 11701. ssivrr,. 
 13140363. 
 
1<> Arithmetical Exercises and Examples. 
 
 A D hi T I o\. 
 
 -I. Addition -"/'"'/'"*''"// is the prm-ess if uniting 
 two or mure numbers oi' the same name or kind, so us t<> 
 
 make one equivalent IIUlllh'T. 
 
 -"). The number obtained }>\ tliis process is called the 
 
 Sum "i Amount. 
 
 L'O. The SiiCIl Of Addition is a perpendicular CFO8S, 
 
 . called plus: it means more- ; thus 7 -\ !' is read. 7 plus 
 
 !>, and means that 7 ami !> ar to he added. When used 
 
 after a niuiiher. thus. 5 --)- -. which is read ."> plus, it means 
 
 ."> and a small exo 
 
 '11. The SilTll Ol' Kqimlity is ^t is read equals, 
 or eijual to. and denotes that the numhers hetween which 
 it is placed are njual to each other ; thus 7 | !> ^= 1C 
 means that 7 and !> added art 1 njual to Hi. The expression 
 is read. 7 ]>lus !> equals 1 <i. 
 
 A Numerical Equation is an equality lu-tu-een 
 
 t \vo numerical expressions, which tliou^h ditleriiiL:; in form 
 from each other are equivalent. Macli expression is called a 
 term of the equation. Thus 5 + 8 = 13 is a numerical 
 equation in which the 5 ; S is called the first memhei 1 of 
 the equation and 13 the. see-md meinher. and hoth are called 
 the terms of the equation. 
 
 L >( J. rrhn-ijtli' <>f' Addition. Xumhers of the same kind, 
 order or character only, can he added. Thus we cannot add 
 - apples and 3 oranges, nor 5 pounds of sui^ar and (I boxes 
 oi' peaches ; nor (I units and f> hundreds; nor 2 and }. etc. 
 \Ve can only add apples to apples, oranges to oranges, suirar 
 to suu'ar, peaches to peaches, units to units, hundreds to 
 hundreds, halves to halves, fourths to fourths, etc. We 
 can enlist together things of different kinds, apples, peaches, 
 fiances, etc.. hut by coll-ectitii: them together we do not 
 increase the number or sum of either aud hcuce there is 
 no addition. 
 
Addition Tables. 1 1 
 
 ADDITION TABLE. 
 
 oU. No. I. 
 
 Kan. In h-iiniiiii:- thr.-M- tal.l.-s ami hamllin.ir all numliiTs. all int<Tiiirliat" 
 \\..n.l- ;l | M i thoughts that occur l,,-t\v.-cii th.. iitmil"-r- to I,,- .-..mliim-il ami the 
 
 iv-nlt of tin- d-iiv.l cuml, inatioii >lh,ul.| I mitt.'.l. Thus in.-t.-a.l ..f savini: 
 
 or thinking, that ~1 ami - ar.- 1. :' an-1 5 ura >. etc., M$ Ot tliinU 1 : v : etc. 
 
 /.\rj>/<(itn(i<jH In this t;ible 
 
 1 ., we show 20 different combioations 
 
 1 
 
 ol' (he ( J signitieant ii-iuvs to pro- 
 
 2 duce results iroin 1 to ( J. It 
 2 -^ may be said that three l v make 
 
 9 '-i 
 
 o, three 2' s make (J, etc., and that 
 
 5 they are regular combinations; but 
 
 4.3 
 
 we see by the table that two l' s 
 
 1.2.:; 
 
 ~ } . I.;; are 2, and that two IT are 1, ete. 
 
 \ -2 ;>> Hence, though the table does not 
 
 6.5.4 7 
 
 contain all the possible combma- 
 
 1 4> O 1 
 
 ._'""J' 8 tions, it does contain all that are 
 
 i .0.0.4 
 
 essential and of value in this con- 
 
 1.2.3.4 g 
 
 8.7.6.5 nection. 
 
Arithmetical \rn i; .srs un.l Examples. 
 ADDITION TAB LK. 
 
 ]_>;; j ;, 
 
 :i x - j; - } Explanation. In this table we 
 
 >lin\v ihi- LT) (litlt'ivnt combinations 
 L'.:5. I. :> 
 
 :>. S.T.I; of the :> significant ii^mvs th<> 
 
 .>', " ( . sum of which equals f> // or more. 
 
 1'2 
 ' s -< ( - '!> attain rapidity in adding, it 
 
 I ;,.i; is absolutely necessary that, the 
 
 MS" 
 
 '_^__ learner should be so familiar with 
 
 ">."'. 7 , . these eombinatiuns that he can 
 
 9.8.7 
 
 instantly see the result without 
 
 15 adding, i. e. he must know the 
 
 J.a 
 
 result by the combination, just as 
 
 r+ o 
 
 y'g l^ he knows the value of 4, or 5, by 
 
 the combination of lines forming 
 
 8 -j *r 
 
 9 the figure, or as he knows the pro- 
 
 nunciation of a word without 
 
 18 
 
 spelling it, 
 
 The rapid increasing and decreasing operations in 
 the science of numbers depend upon the capacity of the 
 calculator to instantly apprehend and accurately apply, the 
 result of two or more figures, no matter how they may 
 be combined. And the object of these tables is to aid in 
 acquiring the desired capacity* 
 
Addition and Subtraction Tables* 
 
 13 
 
 K w 
 
 H ^ 
 
 . 
 
 EC H 
 
 rH <M CO 
 
 rH ^1 CO 
 
 L- t- t-. t t t- 
 
 rH -M CO -t 10 CO 
 
 00 GO 00 00 GO GO 00 
 
 rH 1 CO ^ O CD 
 
 S. 
 
 
 
 M 
 
 H 
 
 c ^ o 
 
 * 1 S 
 
 O 03 C 
 
 "** 5 
 
 "S 
 
 C &^ 
 
 S o- 
 
 o s .. 
 
 o c 
 
 10 .2 
 
 .a *- ^ 
 
 .2 
 
14 Arithmetical Exercises and Examples. 
 
 ADDITION AND StT,TI!A< TK>N TAI'.LKS. 
 33. TA15LK IV. 
 
 1 A 
 
 i 7=100 
 
 26 A 
 
 ?=100 
 
 :.i , 
 
 loo 
 
 76 & 
 
 ?=lfO 
 
 1 
 
 100 
 
 27 
 
 100 
 
 52 
 
 loo 77 
 
 loo 
 
 3 
 
 loo 
 
 28 
 
 100 
 
 53 
 
 loo 78 
 
 100 
 
 4 
 
 100 
 
 29 
 
 100 
 
 54 
 
 loo 
 
 100 
 
 5 
 
 100 
 
 30 
 
 100 
 
 
 loo 
 
 
 10 
 
 6 
 
 100 
 
 31 
 
 100 
 
 56 
 
 loo 
 
 si 
 
 100 
 
 7 
 
 100 
 
 32 
 
 100 
 
 57 
 
 100 
 
 
 100 
 
 8 
 
 100 
 
 33 
 
 100 
 
 58 
 
 100 
 
 83 
 
 100 
 
 !> 
 
 100 
 
 34 
 
 loo 
 
 r>9 
 
 loo 
 
 
 100 
 
 10 
 
 100 
 
 
 100 
 
 
 100 
 
 
 100 
 
 11 
 
 100 
 
 36 
 
 100 
 
 til 
 
 100 
 
 
 100 
 
 12 
 
 100 
 
 37 
 
 loo 
 
 62 
 
 100 
 
 87 
 
 100 
 
 13 
 
 100 
 
 38 
 
 100 
 
 63 
 
 100 
 
 88 
 
 100 
 
 14 
 
 100 
 
 39 
 
 100 
 
 64 
 
 100 
 
 89 
 
 100 
 
 15 
 
 100 
 
 40 
 
 100 
 
 65 
 
 100 
 
 90 
 
 100 
 
 16 
 
 100 
 
 41 
 
 100 
 
 66 
 
 100 
 
 91 
 
 100 
 
 17 
 
 100 
 
 42 
 
 10* 
 
 67 
 
 100 
 
 92 
 
 loo 
 
 18 
 
 100 
 
 43 
 
 100 
 
 68 
 
 100 
 
 93 
 
 100 
 
 19 
 
 100 
 
 44 
 
 100 
 
 69 
 
 100 
 
 94 
 
 100 
 
 20 
 
 100 
 
 45 
 
 100 
 
 70 
 
 100 
 
 95 
 
 100 
 
 21 
 
 100 
 
 46 
 
 100 
 
 71 
 
 100 
 
 96 
 
 100 
 
 22 
 
 100 
 
 47 
 
 100 
 
 72 
 
 100 
 
 97 
 
 100 
 
 23 
 
 100 
 
 48 
 
 100 
 
 73 
 
 100 
 
 <J8 
 
 100 
 
 24 
 
 100 
 
 49 
 
 100 
 
 74 
 
 100 
 
 99 
 
 100 
 
 25 
 
 100 
 
 50 
 
 100 
 
 75 
 
 100 
 
 
 
 We present this table to aid the learner in instantly seeing 
 the difference between 100 and any number from 1 to 99. It 
 is of special value in addition and subtraction, and all who 
 expect to become rapid Calculators, must be proficient in this 
 character of work. 
 
Signs and Abbreviations. 15 
 
 SIGNS AND ABBREVIATIONS. 
 
 84. The following are the principal signs and abbrevia- 
 tions in general use among merchants and business men : 
 
 (a At. To. Company. 
 
 ''/,. Account. ( 1 r. Credit or creditor. 
 
 I 1 Oue and one-quarter. J>r. Debit or Debitor. 
 l a One and one-half. Gal. Gallons. 
 
 ] <H One and three-quarters. Ps. Pieces. 
 Per. Yd. Yards. 
 
 Pound (weight). Fr't. Freight. 
 
 $ Dollar or dolhns. ller'd. Keceived. 
 
 / Cent, or cents. Pay't. Payment 
 
 % Per cent, or per centum. lust. This month. 
 A int. Amount. Prox. The next month. 
 
 Blil. Barrel. VI. The last month. 
 
 Doz. Dnzrn. U Pound Sterling. 
 
 B. L. Bill .f Lading. 0. K. All Right. 
 
 Blk. Black. Fr. Franc, French coin. 
 
 Shipt. Shipment. Fwd. Forward. 
 
 Sunds. Sundries. Bal. Balance. 
 
 Dft. Draft, Cons't. Consignment. 
 
 Com. Commissioo. hhds, hogsheads. 
 
 Do. The same. Mdse. Merchandise. 
 
 / Shillings, thus 2 / 6 two shillings and sixpence. 
 iMk. Marks, the German monetary unit. 
 I Check mark, correct, approved. 
 
 Cifrao, used to separate the milreis from the rcis in 
 Brazil money. 
 
 17 doz. ,-*,. S/\. T 7 T =17 doz., 4 of which are at 810 
 per doz., G (<y $12 and 7 @ $15. 
 
 8 doz. | @ 5 / f (a> I, 2 doz. No. 4 (oj 5 shillings per 
 doz., and (5 doz. No. 5 (fV 4 shillings sixpence per dozen. 
 
16 Arithmetical Exercises and Examples. 
 
 35. Name the unit result of the following numbers : 
 !) 898989887 9 5 6 
 9866442298797 
 
 3 
 
 8 
 
 7 
 
 6 
 
 8 5 
 
 5 
 
 6 
 
 9 
 
 4 
 
 7 
 
 5 
 
 9 
 
 3 
 
 3 
 
 4 
 
 4 7 
 
 5 
 
 (i 
 
 1 
 
 7 
 
 7 
 
 8 
 
 3 
 
 4 
 
 5 
 
 1 
 
 1 
 
 9 
 
 8 
 
 g 
 
 3 
 
 6 
 
 5 
 
 5 
 
 2 
 
 2 
 
 3 
 
 3 
 
 2 
 
 1 
 
 5 
 
 7 
 
 4 
 
 7 
 
 7 
 
 9 
 
 8 
 
 7 
 
 8 
 
 2 
 
 4 
 
 6 
 
 8 
 
 9 
 
 3 
 
 7 
 
 1 
 
 8 
 
 4 
 
 6 
 
 5 
 
 4 
 
 9 
 
 7 
 
 9 
 
 7 
 
 4 
 
 8 
 
 2 
 
 7 
 
 7 
 
 8 
 
 5 
 
 6 
 
 8 
 
 8 
 
 5 
 
 5 
 
 8 
 
 9 
 
 9 
 
 8 
 
 9 
 
 8 
 
 8 
 
 5 
 
 6 
 
 9 
 
 For further explanation of Addition, the importance of 
 it, and the most rapid processes of adding see Soule's Con- 
 tractions in Numbers. 
 
 EXERCISES. 
 
 1. Write all the combinations of two figures that make 
 10, 11, 12, 13, 14, 15, 16, 17 and 18. 
 
 2. Commence with 1 and orally add thereto 2, and con- 
 tinue to add 2 to the successively occurring sums until you 
 produce 21. Thus 3, 5, 7, 9, 11, 13, etc. 
 
 3. Commence with 1 and in like manner add 3 until 
 you produce 31. Thus 4, 7, 10, 13, etc. 
 
 4. Commence with 1 and in like manner add 4 until 
 you produce 41. 
 
 5. Commence with 1 and in like manner add 5 until 
 you produce 51. 
 
 6. Commence with 1 and in like manner add 6 until 
 you produce 61. 
 
Examples in Addition. 17 
 
 7. Commence with 1 and in like manner add 7 until 
 you produce 71. 
 
 8. Commence with 1 and in like manner add 8 until 
 you produce 81. 
 
 9. Commence with 1 and in like manner add 9 until 
 you produce 91. 
 
 10. Oially add by 2'' until you produce 20. 
 
 11. " 3'* " " 30. 
 
 12. " " -1 v " " 40. 
 
 13. " " ") " " 50. 
 
 14. " " 6' 8 " " GO. 
 
 15. " 7' 8 " " 70. 
 
 16. " " 8' 8 " " 80. 
 
 17. t{ " 9" " " 90. 
 
 18. " " 10'" " k ' 100. 
 
 19. Commence at 1 and orally add by 3 and 5 altern- 
 ately until you produce 100. 
 
 20. Commence at 1 and orally add by 4 and 7 altern- 
 ately until you produce 100. 
 
 EXAMPLES IN ADDITION. 
 
 36. Add the following numbers : 6376, 564, 309, 485 
 and 5092. 
 
 OPERATION. 
 
 w j Explanation. In all addition pro- 
 
 || blems we firgt write the numbers so 
 
 |lc| that units of the same order will stand 
 
 in the same column, i. e., units in the 
 6o7o units or first column; tens in the tens 
 
 r f>(H or second column; hundreds in the 
 
 ;;OJI hundreds or third column and so on 
 
 through the numbers. We then begin 
 at the units or first column and add 
 the columns separately. In adding the 
 first column, we commence with the 2 
 Swn 12,826 and 5, and name only the successive 
 
 1^ results thus, 7, 16, 20, 26; which is 2 
 
 tens and 6 units; the 6 we write in the first place or column of 
 units and place the 2 tens which is to be carried to the column of 
 tens directly below the 6 in a small figure. Then adding the 2 
 
18 Arithmetical Exercises and Examples. 
 
 tens to the tens column, we say, 11, 10, !!:>, .".2; which is:* hundred 
 and 2 tens: the 2 tens we write in the column of tens and place 
 the 3 hundreds, which is to be carried to the hundreds column 
 directly under it. Then adding the 3 hundred to the hundreds 
 column, we say, 7, 10, 15, 18, which is 1 thnumtml and S hun- 
 dred; the 8 hundred we write in th hundreds column and the 
 carrying figure, 1 thousand, directly under. Then adding ihn 
 1 thousand, to the fourth or thousands' column, we say <;, \'2, 
 which is I (en thousand and 2 thnuxtinrl, and this being the last 
 column to add we write the figures in their respective columns 
 and produce 12826 as the siim of all the numbers. 
 
 When adding, set the result in pencil figures, being careful 
 to place the carrying figure or figures directly beneath the unit 
 figure of each column added as shown in the preceding problem. 
 
 PROOF OF ADDITION. 
 
 The best proof of the correctness of addition is to be 
 proficient in your work, and then re-add the columns in 
 the reverse direction. 
 
 What is the sums of the following groups of numbers ? 
 
 (3) W (5) (6) 
 
 780 89 777 9040 
 
 1261 706 888 1288 
 
 537 73 999 9907 
 
 309 4009 666 6543 
 
 6987 8888 645 2018 
 
Examples in Addition. 19 
 
 Add the following groups of numbers : 
 
 818 
 
 (8) 
 
 412 
 
 582 
 
 (10) 
 
 328 
 
 (ii) 
 809 
 
 (12) 
 
 981 
 
 390 
 
 297 
 
 578 
 
 346 
 
 523 
 
 350 
 
 970 
 
 318 
 
 757 
 
 386 
 
 605 
 
 269 
 
 270 
 
 824 
 
 420 
 
 672 
 
 848 
 
 789 
 
 752 
 
 932 
 
 731 
 
 793 
 
 945 
 
 696 
 
 843 
 
 373 
 
 542 
 
 64 
 
 397 
 
 136 
 
 805 
 
 570 
 
 853 
 
 905 
 
 684 
 
 169 
 
 129 
 
 876 
 
 684 
 
 448 
 
 976 
 
 295 
 
 768 
 
 444 
 
 743 
 
 404 
 
 666 
 
 468 
 
 9U4 
 
 102 
 
 915 
 
 151 
 
 217 
 
 687 
 
 972 
 
 814 
 
 080 
 
 148 
 
 879 
 
 825 
 
 114 
 
 331 
 
 637 
 
 263 
 
 516 
 
 951 
 
 340 
 
 554 
 
 917 
 
 295 
 
 259 
 
 784 
 
 545 
 
 101 
 
 650 
 
 101 
 
 890 
 
 122 
 
 022 
 
 m 
 
 411 
 
 401 
 
 864 
 
 440 
 
 749 
 
 490 
 
 237 
 
 874 
 
 565 
 
 450 
 
 717 
 
 876 
 
 349 
 
 898 
 
 150 
 
 414 
 
 222 
 
 902 
 
 489 
 
 769 
 
 514 
 
 654 
 
 234 
 
 390 
 
 698 
 
 243 
 
 446 
 
 789 
 
 100 
 
 484 
 
 228 
 
 174 
 
 576 
 
 458 
 
 305 
 
 235 
 
 433 
 
 952 
 
 489 
 
 747 
 
 272 
 
 380 
 
 949 
 
 683 
 
 394 
 
 030 
 
 729 
 
 .624 
 
 087 
 
 574 
 
 407 
 
 241 
 
 955 
 
 897 
 
 702 
 
 956 
 
 812 
 
 477 
 
 177 
 
 477 
 
 849 
 
 658 
 
 798 
 
 081 
 
20 Arithmetical Examples and Exercises. 
 
 Add the following groups of numbers: 
 
 (13) 
 
 864 
 
 ,H) 
 
 B77 
 
 595 
 
 (16) 
 
 849 
 
 (17) 
 
 539 
 
 257 
 
 363 
 
 305 
 
 249 
 
 
 : ) >77 
 
 476 
 
 629 
 
 420 
 
 n;.; 
 
 027 
 
 702 
 
 426 
 
 145 
 
 982 
 
 830 
 
 651 
 
 235 
 
 684 
 
 174 
 
 217 
 
 221 
 
 543 
 
 
 492 
 
 144 
 
 326 
 
 232 
 
 502 
 
 950 
 
 343 
 
 176 
 
 111 
 
 151 
 
 113 
 
 446 
 
 002 
 
 767 
 
 871 
 
 :;^7 
 
 438 
 
 834 
 
 182 
 
 644 
 
 512 
 
 516 
 
 455 
 
 540 
 
 955 
 
 747 
 
 814 
 
 247 
 
 328 
 
 919 
 
 
 156 
 
 376 
 
 331 
 
 633 
 
 358 
 
 989 
 
 106 
 
 468 
 
 281 
 
 (524 
 
 149 
 
 855 
 
 872 
 
 189 
 
 S2S 
 
 581 
 
 268 
 
 954 
 
 694 
 
 177 
 
 986 
 
 491 
 
 002 
 
 126 
 
 788 
 
 885 
 
 817 
 
 888 
 
 693 
 
 136 
 
 866 
 
 264 
 
 918 
 
 992 
 
 682 
 
 564 
 
 044 
 
 294 
 
 IN!) 
 
 202 
 
 355 
 
 163 
 
 922 
 
 896 
 
 259 
 
 548 
 
 223 
 
 764 
 
 116 
 
 597 
 
 
 365 
 
 521 
 
 921 
 
 911 
 
 814 
 
 329 
 
 208 
 
 530 
 
 515 
 
 866 
 
 277 
 
 678 
 
 662 
 
 874 
 
 735 
 
 179 
 
 476 
 
 040 
 
 704 
 
 528 
 
 393 
 
 129 
 
 716 
 
 821 
 
 387 
 
 584 
 
 550 
 
 659 
 
 778 
 
 802 
 584 
 
 457 
 
 587 
 
 848 
 255 
 
 025 
 202 
 
 888 
 932 
 
Examples in Addition. 21 
 
 Add the following groups of numbers: 
 
 19 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 883 
 
 792 
 
 743 
 
 153 
 
 919 
 
 547 
 
 356 
 
 414 
 
 560 
 
 214 
 
 620 
 
 380 
 
 595 
 
 454 
 
 871 
 
 248 
 
 922 
 
 616 
 
 638 
 
 366 
 
 349 
 
 636 
 
 369 
 
 874 
 
 679 
 
 464 
 
 955 
 
 549 
 
 158 
 
 682 
 
 594 
 
 933 
 
 936 
 
 f>!>4 
 
 862 
 
 232 
 
 953 
 
 686 
 
 746 
 
 783 
 
 874 
 
 713 
 
 178 
 
 641 
 
 793 
 
 225 
 
 935 
 
 499 
 
 215 
 
 939 
 
 798 
 
 61!) 
 
 951 
 
 874 
 
 119 
 
 201 
 
 324 
 
 232 
 
 959 
 
 779 
 
 753 
 
 S71 
 
 687 
 
 478 
 
 865 
 
 622 
 
 311 
 
 438 
 
 843 
 
 484 
 
 724 
 
 718 
 
 1 S2 
 
 218 
 
 421 
 
 252 
 
 645 
 
 180 
 
 686 
 
 869 
 
 586 
 
 648 
 
 148 
 
 477 
 
 896 
 
 189 
 
 518 
 
 551 
 
 227 
 
 396 
 
 996 
 
 595 
 
 959 
 
 995 
 
 193 
 
 495 
 
 293 
 
 521 
 
 152 
 
 475 
 
 947 
 
 568 
 
 2(52 
 
 727 
 
 572 
 
 D77 
 
 797 
 
 130 
 
 515 
 
 259 
 
 425 
 
 362 
 
 736 
 
 111 
 
 833 
 
 585 
 
 458 
 
 485 
 
 328 
 
 682 
 
 745 
 
 177 
 
 217 
 
 631 
 
 803 
 
 685 
 
 125 
 
 413 
 
 841 
 
 194 
 
 729 
 
 996 
 
 245 
 
 825 
 
 972 
 
 698 
 
 169 
 
 492 
 
 968 
 
 868 
 
 489 
 
 258 
 
 845 
 
 194 
 
 540 
 
 799 
 
 386 
 
 277 
 
 477 
 
 864 
 
Arithmetical Exercises and Example*. 
 
 -11 28 
 
 15656:; 13331H2 
 
 IMJ4SJH' -.VJ7 8368G9 7391 573 
 
 f>7sr,7* 78754d 2. M24<> 35175(11) 
 
 5775!>4 !U 14:12 7H5183 8598674 
 
 GGSU7S wwi< :;i5!>^7 
 
 6<;<M;:>7 :7s:;iji ;.viG78 
 
 5398 678789 I5(;.i:^ 
 
 (;c,47:); :H;H;T:{ :ur>7l8 
 
 795568 895437 7;:>391 
 
 (IDDC.sn 569128 \\r.\\X\ 7893344 
 
 i;s!i7si; i;7snsi> 137987 
 
 688968 ><;'J771 r)i;;7-: 
 
 i):;r,7 668339 :>4 1321 
 
 7788JHJ 'jrHiL'.-U 891389 
 
 431348 4235564 
 
 L>1>. Add 6, s. <>, 7, 6, 8, r>, 4, 9 5 4, 8, 7, 6, 9, 14, 19, 
 IS, ITT, HS. 47, 59, 65, 74, 83, 9i'. Ans. 632. 
 
 30. Add 528,791, 14389,888,91361,587,301,7004, 
 52800. 71 <H5. 42S81. Am. 218,636. 
 
 31. Add 476010, 51873, 98,- 48932, 3581427, 67843, 
 21050, 3672. Ans. 4,250,905. 
 
 82. Add 63, 94, 85, 74, 63, 52, 41 , 3<, 48, 57, 66, 75, 
 84, 93, 27, 18, 60, 80, 19, 88, 99, 77, 66, 55, 44, 33, 22, 
 11, 98, 97, !>6, Sii, 76, 65, 54, 43. Ans. 2248. 
 
 33. Add seven million four thousand and ninety-six, 
 and three hundred eighty-seven thousand five hundred and 
 sixty- tvo. Ans. 7391658. 
 
 34. Find the sum of 4888765, 92238, 1600084, 
 8888888, 99999999999, 4100000808707 and 222222333- 
 333444444. Ans. 222226533349723125. 
 
 35. Find the sum of 999999999, 88888888, 7777777, 
 666666, 55555, 4444, 333, 22, 1, and sixty-three millions. 
 
 Ans. 1160393685. 
 
Examples in Addition. 23 
 
 80. Add 781), 67!). <J87, 140018, 11)1070, 871230432, 
 4!)70G, 40000, 80000000 and eleven hundred and eleven. 
 
 Ans. 951,654,71)2. 
 
 37. Add five hundred thousand nine hundred thirty- 
 nine, and eleven thousand eleven hundred and eleven. 
 
 Ans. 513050. 
 
 37. Dollar and (-cut si^ns. The dollar sign is $, 
 and the cent sign is c. When the dollar sign is placed 
 before numbers they are read as dollars. Thus $45 is read 
 45 dollars. When the cent sign is placed after numbers 
 they are read as cents. Thus 14^ is read 14 cents. When 
 dollars and cents are written together the cents are separated 
 from the dollars by a point (.) and the sign of cents is 
 omitted. Thus $10. 45 is read 1(5 dollars and 45 cents. 
 
 Since there are 100 cents in 1 dollar, cents always occupy 
 two places and only two in connection with dollars. When 
 the number of cents is less than 10 a mnif/lit must be used 
 to iill the tens column or the first place at the right of the 
 point. Thus 8 dollais and 5 cents are written $8.05. 
 
 When cents only are written they are expressed as fol- 
 lows : 25 cents, or 25/ or $.2."). 
 
 When writing numbers representing dollars and cents for 
 the purpose of addition, they must be set so that dollars 
 will be under dollars and cents under cents in the regular 
 order of units, tens, hundreds, etc., and the points (.) that 
 separate dollars and cents must be in a vertical line. 
 
 The dollar sign ($) and the point (.) should never be 
 omitted when writing dollars and cents. 
 
 38. Add $14.50 $34. Ki *75. $ .88 $180.40 
 
 8. D.OS !.45 11. 48.08 
 
 4.25 14.83 <7.0(i 5.13 91.16 
 
 12.15 8. .35 7.02 7.05 
 
 S21.03 $326.G!) 
 
24 Arithmetical Exercises and Examples. 
 
 Add $821. 8521.16 8 1U5 $431, $194.15 
 
 040. SO 88.2.") 80. 124. 8.05 
 
 9.13 19.30 17. 381. 73.75 
 
 75.20 8. .65 569. 0.13 
 
 100. 1C) 4.07 6.10 S27. .95 
 
 $1146.18 $635.78 SI 13.20 82332. 8283.03 
 
 48. Add $8.12, $1), $.50, $3.40, $37.05, $.75 and 
 $12.12. Ans. 870.D4. 
 
 4!). Add 8 i:5. 10, $17. $5, 48C, 75/, $11, $24.14, $3. 
 
 Ans. $104,17. 
 
 50. Add $108, $97.16, 881.12, $.75, $8, $6.40, 25/, 
 $18. Ans. $322.68. 
 
 51. Add $580.10, $671.23, 87!) 1.1)8, $88, 45/, 5/, 
 $3.10. Ans. $2,137.91. 
 
 52. Add 999.99. $sss.S8, 8777.77, $<><>;.ii<;. 8555.55, 
 84-14.44, $333.33, $222.22, $111,11 and I/. 
 
 Ans. 8-4. 999. 96. 
 
 53. Add $1)87.65. $876.54. 8765.4:;, 8(554 32, 8543.- 
 21, $123.45, $234.56, $345.67, $456.78, $567.89, $678.90 
 and $789. Any. $7,123.40. 
 
 54. Middle-miss bought a hat for $2, a coat for $9.50, 
 a pair of shoes for $2.75, a pair of pants for $4, a vest for 
 $1.75, and had $41.05 left. How much money had he at 
 first? Ans. $61.05. 
 
 55. Miss Smith paid for a broom 35/, for soap $1.60, 
 for starch 75/, for matches 5^, for salt 15/, for sugar 
 $1.50, for rice $2, for butter 80/, Graham flour $1.25 and 
 for a hygienic cook book $1. What was the sum paid for 
 all? Ans. $9.45. 
 
 56. Prophet paid for a reader $1.35, for an arithmetic 
 $1.50, for a history $2,' for a set of drawing instruments 
 $3.70, for paper $.60, for pens $.15, for ink $.05, for a 
 pair of Indian clubs $3.50, and for the boy's own book $1. 
 What did all cost ? Ans. $13.85. 
 
 57. Conrad paid $1.75 for Chesterfield's letters ; $1.80 
 for Cutter's Anatomy, Physiology and Hygiene ; $1.75 
 
^Examples in Addition. 25 
 
 for Comb's Constitution of Man ; $1-25 for How to Read 
 Character by Wells; $1.50 for .Nordhoff's Politics for 
 Young Americans; $1.75 for Physical Perfection by Jac- 
 ques ; $4 for Plutarch's Lives ; $8 for Shakspeare's Works ; 
 $2 for the Literary Header ; $6 for Carey's Social Science ; 
 $5 for Parson's Laws of Business; $5 for Soule's Philoso- 
 phic Work on Commercial and Exchange Calculations, and 
 $1 for Cushing's Manual. How much did he pay for all? 
 
 Ans~ $40.80. 
 
 58. If you should travel by rail 160 miles, by steamer 
 214, and walk 8, how far would you travel? 
 
 Ans. 382. 
 
 59. A planter raises 9842 pounds of sugar, 2351 
 pounds of cotton, 1827 pounds of rice, 3840 bushels of 
 corn, 325 bushels of sweet potatoes and 194 bushels of 
 beans. How many pounds and how many bushels does he 
 raise in all? Ans. 14020 pounds, 4359 bushels. 
 
 (JO. Conrad loaned to Purccll 8:> : to ( Iresham $3.50 ; 
 to Ilanna 75/ ; to Mitchell 85;'; to Sweeney 5/ ; to 
 IJothick $1 ; to Keen 25/J to Abbott 75/ ; to Prophet 
 50/. What sum did he loan to all? Ans. 16.65. 
 
 til. Keen has Si I3.o:> ; Courct $91 ; McCoard $18.30; 
 Bush90/; Nevers 25 c ; Fischer *:>.<>.'> : Heck $9 ; Meyers 
 s<; ; Levy $7 : Brown $7 ; Kk-e $45 ; Shotwell 27 ; Wise 
 $15.80 ; Moffett $5.50 ; Limlsey 888.70. How much have 
 all? Ans. $460.55. 
 
 62. A merchant bought four adjacent lots of ground 
 for $6850. He built a house thereon which cost $11875. 
 Paid for fences $912 ; for flagging $1819.55 ; for furniture 
 $3481.12. How much did the whole cost? 
 
 Ans. $24,937.67. 
 
 63. If you pay $175 for a horse, $450 for a carriage, 
 $75 for a set of harness, s:;s for a saddle and bridle and 
 $6.50 for a whip. What will the whole cost? 
 
 Ans. $744.50. 
 
 64. A planter has 54 cows, 321 sheep, 174 mules, 23 
 horses, 42 oxen, 43 calves, 7 colts. How much live stock 
 has he altogether ? Ans. 664. 
 
 Q 
 
- Arithmetical Exercises and Examples. 
 
 !.">. A merchant bought at one time 250 barrels Flour 
 for 81 ")()() ; at another .'> if) banvls for 8- H5 ; and at another 
 21)0 barrels for 1625. How many barrels did he buy and 
 what was the total cost 7 Ans. 795 Bbls., 
 
 $5540 Cost. 
 
 66. The weight of ten bales of cotton is as follows : 481, 
 
 503, :-ws, 4r>2, 470. IT ( J, mi, :;:T. n;:i, 511 pounds, what 
 
 is the total weight? Ans. 4565. 
 
 67. Bought at one time 43 yards of calico and 32 yards 
 of silk ; at another 104 yards of calico and 24 yards of 
 silk, and at another 96 yards of calico and 48 yards of silk. 
 How many yards of each kind did I buy? 
 
 Ans. Calico 2415, Silk 104. 
 
 68. Paid $425 for a lot of BUgar, s 1 20 i;, r rice and 75 
 for potatoes. Sold the sugar at a profit of $ H and the rice 
 and potatoes at cost. What did I get for the whole? 
 
 Ans. 661. 
 
 69. From New Orleans to the Iligolets is 31 miles ; 
 hence to Montgomery, 18; hence to Bay St. Louis, 3; 
 hence to Pass Christian, 6 ; hence to Mississippi City, 13 ; 
 hence to Biloxi, 9 ; hence to Ocean Springs, 4 ; hence to 
 East Pascagoula, 16; hence to St. Elmo, 21; hence to 
 Mobile, 20. How many miles to Mobile? Ans. 141. 
 
 70. From New Orleans to Kenner is 10 miles ; hence 
 to Manchac, 27 ; hence to Ponchatoula, 11 ; hence to Ham- 
 mond, 4; hence to Amite, 16 ; hence to Tangipahoa, 10 ; 
 hence to Osyka, 10 ; hence to Magnolia, 10 ; hence to Me 
 Comb City, 7; hence to Summit, 3 ; hence to Bogue Chitto, 
 10 ; hence to Brookhaveu, 10 ; hence to Beauregard, 11 ; 
 hence to Crystal Springs, 19 ; hence to Terry, 9 ; hence to 
 Jackson, 15; hence to Madison, 13; hence to Canton, 11. 
 How many miles is it to Canton ? Ans. 206. 
 
 71. A young man paid $125 for a year's tuition at col- 
 lege, $22.50 for books, lost $40, and has $378.35 on hand. 
 How much had he at first? Ans. $565.85. 
 
 72. A boy gave Jane 6 oranges. Kate 4, John 3, he 
 ate 2, and had 5 remaining. How many had he at first ? 
 
 Ans. 20. 
 
Examples in Addition. '11 
 
 73. Louisiana contains 41 255 square miles ; Mississippi, 
 47156; Texas, 237504; Arkansas, 52198; Tennessee, 
 45600; Kentucky, 37680; Alabama, 50722; Georgia, 
 52009; South Carolina. 293S5 : North CJardifta, 50704; 
 Missouri, 07380; Virginia, r, 1:552: Maryland. 11124: 
 Florida, 592C.8 ; California. 1KS982. How many square 
 miles in the fifteen states? Ans. 1,032,319. 
 
 74. The population of London is 3311000; Paris. 
 1852000; Si. Petersburg, Iili7000 : Pun Janeiro. 2750(10: 
 Constantinople. 400000 ; Vienna. S3 1-000 ; Berlin, S25000 : 
 Lisbon, 224000; IVkin. ir,!SOOO; Tokio or Jeddo, 790- 
 000; Bomhav. 1117000; Madrid. 332000; (ilasgow, 
 489000; Dublin, 31 1000; Amsterdam. 27SOOO ; Brussels, 
 176000; Stockholm. 139000; Copenhagen, 181000; Cairo, 
 (Egypt) 351000; Tunis. 125000. What is the popula- 
 tion of all? Ans. 13,858,000. 
 
 75. The lenizth of the Mississippi Hiver is 1200 miles : 
 of the Nile, 4000: Ama/on. 3750 : Yenisei. 3 400 ; ()l,i. 
 3000: Yang-tse-Kian- 3320; tfiger, 3000 ; Lena, 2700 : 
 Amoor, 2<i50; Yol-a. 2000; (Jan-vs. liJOO; Brahmapoo- 
 tra, 2300 . | j;l Plata. 2300; Macken/.ie. 2:;oo ; St. Law- 
 rence, 2000; Saskatchewan. 1900; OriiHH-o. 1550; Colum- 
 bia, 1020; Colorado. (iOO ; Yukon. HJOO; Kcd lliver. 
 1500. What is the combined len-th of all ? 
 
 Ans. 50.090. 
 
 7<>. Lake Sii])erior is 100 miles in Iciiiith . Lake Michi- 
 gan, 320 ; Lake Ihmm, 210; Lake El ie, 240 ; Lake On- 
 tario. 180; Lake Baikal, 375: L-ike Pontehartrain, 40. 
 What is the comhined length oi' all ? Ans. 1795. 
 
 77. There are in the world 3910(10000 Christians: 
 500000000 Buddhists: 1 150000m) Brahmins; 100000000 
 Confucians; 15000000 Slmdoan : 199000000 Mohamme- 
 dans ; 7000000 Isndites. 1 low many combined ? 
 
 Ans. 1.3(iO,000.000. 
 
 78. Mount Kvei-est of the Himalaya chain in Asia the 
 highest point on the globe, is 29002 feet high; Mt. St. 
 E.ias, the highest mountain in North America, is 17900 
 feet; -Mt. Illampu, the highest mountain in South America, 
 
28 Arithmetical Exercises and Examples. 
 
 is 24812 feet; Mt. Blanc, the highest mountain in Europe, 
 is 15780 feet; Mt. Kilima Xj.iro, the highest mountain in 
 Africa, is 200G5 feet; Mt. Koseiusko. the highest mountain 
 in Australia, is 717U feet. What is (lie nunhined heiirht 
 of all? Ans. 114.795 feet. 
 
 70. l>v the census of 1 STO. the population of New York 
 was 942992; Philadelphia, 674022; Brooklyn. :J!M>niM) ; 
 St. Louis, 310864 ; Chicago, 2!.77 ; Baltimore, 2U7354; 
 Boston, 2:>or>2i; ; Cincinnati. 2Hi2:!'J: New Orleans'. 
 19141S: San Francisco, 140473 ; Buffalo, 117714 ; Wash- 
 ington, 100199; Newark, 105059; Louisville, 100753; 
 Mobile, 32034; Galveston, 13818; Memphis, 40226. 
 What is the population of all combined ? 
 
 . ADS. 4211571)7. 
 
 80. On Monday 85482 persons entered the gate -s at the 
 Centennial Exhibition, Philadelphia; on Tuesday. 10S121 : 
 on Wednesday, 98792; on Thursday. HI 9515; on Friday, 
 103819, and on Saturday, 174587. How many entered in 
 the six days? Ans. (>(;:. ()5 4. 
 
 81. The standing army of the United States is liL'OOO , 
 of Great Britain and Ireland, 192000 ; of France, 454000 ; 
 of the German Empire, 402000; of Russia. 7<J<iOOO ; of 
 Spain, 284000; of Switzerland, 201000; of Italy, 205000 ; 
 of Brazil, 25000 ; of Mexico, 21000; of Turkish Empire, 
 93000; of Sweden, 150000; of Holland, 62000; of Por- 
 tugal, 33000; of Belgium, 40000. How many men in 
 all? Ans. 2,960,000. 
 
 82. Homer was born 733 years before the Christian 
 Era. How many years from the birth of Homer to the 
 year 1876? Ans. 2609. 
 
 83. During the fiscal year ending Sept. 1st 1876, the 
 receipts of cotton at various points were as follows : 
 
 New Orleans, 1401563 bales; Galveston, 465529 ; Mobile, 
 371298; Savannah, 521437; Charleston, 389698; Wil- 
 mington, 78267; Norfolk, 469997; Baltimore, 18821; 
 New York, 219609 ; Boston, 75065 ; Philadelphia, 58632 ; 
 Various, 57976. How many bales were received during 
 the year? Ans, 4127892, 
 
Examples in Addition. 29 
 
 84. From Aug. 31st 1875 to Sept. 1st 1876, the pro- 
 duction of Sugar in Louisiana was as follows : 
 
 Parish of Livingston, 4 hogsheads; St. Tammany, 16; 
 East Feliciana, 37 ; Lafayette, 187 ; West Feliciana, 339 ; 
 Vermillion, 609 ; Avoyelles, 1582 ; St. Landry, 1768 ; St. 
 Martin, 1884; Orleans, 1041; St. Bernard, 2097; East 
 Baton Rouge, 2544 ; Rapides, 2453 ; Pointe Coupee, 2762 ; 
 Iberia, 3632; Jefferson, 3671 ; West Baton Rouge, 4155 ; 
 St. Charles, 5808; St. John, 8335; Plaquemines, 9068; 
 Iberville, 9814; Lafourche, 11302; Terrebonne, 10888; 
 St. James, 13437; Ascension, H267 ; St. Mary, 14318 ; 
 Assumption, 14712. How many hogsheads were produced 
 during the year ? Ans. 140730. 
 
 85. From July 1st 1875 to July 1st 1876, the monthly 
 receipts of coffee in New Orleans, were as follows : 
 
 July, 9635 bags ; August, 25987 ; September, 24851 ; 
 October, 8832 ; November, 34452 ; December, 4800 ; Jan- 
 uary, 32219; February, 16042, March, 4000; April, 
 10512 ; May, 9000 ; June, 15 120. How many bags were 
 received during the year? Ans. 195450. 
 
 86. From New Orleans to Carrolton is 7 miles; hence 
 to Donaldsonville, 71 ; hence to Pln^m-mines. 32 ; hence to 
 Baton Rouge, 20; hence to Port Hudson, 23; hence to 
 Bayou Sara, 12 ; hence to mouth Red River, 40 ; hence to 
 Natchez, 72 ; hence to Rodney, 45 ;' hence to Grand Gulf, 
 18 ; hence to Vicksburg, 61 ; hence to the Louisiana Line, 
 97 ; hence to Helena, 230 ; hence to Columbus, 329 ; hence to 
 Cairo, 20 ; hence to (/ape Ginwdeau, 50 ; hence to St. Louis, 
 151. How many miles to St. Louis by river? 
 
 Ans. 127S miles. 
 
 87. From New Orleans to tin; mouth of Red River is 
 210 miles ; hence to Black River, 40 ; hence to Alexandria, 
 110; hence to Grand Ecore, 120; hence to Grand Bayou, 
 95 ; hence to New Hope, 60 ; hence to Waterloo, 30 ; hence 
 to Shreveport, 35. How many miles to Shreveport by the 
 river? Ans. 700 miles. 
 
 88. From New Orleans to Algiers Depot is 1 mile . 
 hence to Gretna, 3 ; hence to Jefferson, 9 ; hence to St'. 
 
 v* 
 
30 Arithmetical Exercises and Examples. 
 
 Charles, 0; hence to Bouttc, 6; hence to Huyou des 
 Alemedes, 8; hence to Kaeeland, S ; henee to Kwin 
 hence to Lafourche, ; henue to Tenvbonne, -> : hemv t 
 Chucahoula, ft ; hence to Tigerville. 5 ; IUMICC to L'0;i 8 -. 
 4; hence to Bayou Beuf, 3; hence to Ramos, 3 : benee to 
 Morgan City, 4; hence to Galveston, 240. How many 
 miles to Ualveston ? A us. 321 miles. 
 
 89. 24 peaches were eaten, 5 being spoiled, were thrown 
 away, and 32 remained in the basket. How many were 
 there at first ? A us. (51. 
 
 90. A man was 2H years of age when lie was married. 
 TIow old will he be when he has been married 1 I years ? 
 
 AIIS. 1<> years. 
 
 !H. A young man graduated from college when he was 
 22 years of age. lie married l> years afterward.-. 2 year- 
 after that he was presented with a son. What will he his 
 age when the son is 21 years old? A us. ,">! year<. 
 
 92. A lady paid (f.50 for a dress, $8 for a shawl, $1 
 for a bonnet and 83.75 for a pair of shoes. What was tho 
 total cost? Ans. 822.2."). 
 
 93. A boy sold his pony for $45, and lost 15 by the 
 sale. What did the pony cost him? Ans. $60. 
 
 94. A merchant paid for a lot of goods $580, he sold 
 them and gained $190. How much did he receive for 
 them? Ans. 8770. 
 
 95. Henry is 16 years old, James is 3 years older, and 
 William is 2 years older than James. How old are James 
 and William? Ans. James 1 ( J, William 21. 
 
 96. The internal framework of the human body con- 
 sists of bones, which united by strong ligaments constitute the 
 skeleton. In the skull are 8 bones; in the face 14 ; in each 
 ear 3 ; in the tongue 1 ; in the trunk and spinal column 
 and pelvis 55 ; in each shoulder 2 ; in each arm 3 ; in 
 each wrist 8 ; in the palm of each hand 5 ; in each thumb 
 
 2 ; in each finger 3 ; in each leg 4 ; in each ank'e 7 ; in 
 each foot 5 ; in each great toe 2 ; in each of the other toes 
 
 3 ; and there are_ 32 teeth. How inaiiy bones in the whole 
 body? Ans. 240, 
 
Examples in Addition. 31 
 
 97. How many pupils in a school in which there are 6 
 grades, the first containing 63, the second 58, the third 27, 
 the fourth 41), the fifth 35 and the sixth 24? Ans. 256. 
 
 98. Bothick has $420; Conrad has $130 more than 
 Bothick, and Prophet has as much as Bothick and Conrad 
 together. What sum have all three ? Ans. $1940. 
 
 99. Keen, Soule and Abbott form a copartnership, 
 Keen invests 83400, Soule $4000, and Abbott $500 more 
 than both Keen and Soule. What is the capital of the 
 linn? Ans. $15,300. 
 
 100. A father gave his son seven thousand eight hun- 
 dred dollars ; his daughter nineteen hundred and fifty dol- 
 lars ; and his wife three thousand five hundred more than 
 he gave to both, the son and daughter. What sum did he 
 give away ?- Ans. $23,000. 
 
 SUBTRACTION, (Decreasing.) 
 
 39. Subtraction is the process or operation of finding 
 the difference between two numbers of the same kind. 
 
 40. The result obtained by subtraction is called the 
 
 Difference or.Remainder. 
 
 41. The greater number is called the Minuend, which 
 means a number to be decreased. 
 
 4'J. The lesser number is called the Subtrahend, winch 
 moans the number to be subtracted. 
 
 4:>. The sijjn of subtraction is a horizontal line, . 
 It is read minus and means less. 
 
 When this sign is placed between two numbers it nu,in> 
 that the number nfte.r it, is to be subtracted trom the nuin- 
 I'cr Injure it. Thus 8 3 is read 8 minus 3. 
 
 For Subtraction Tables and contracted method^ of sub^ 
 fraction see Soule's Contractions in Numbers. 
 
32 Arithmetical Examples and Exercises. 
 
 \ I. The sign, ( ), parenthesis, or , vinculum, indi- 
 cate that the numbers included within the parenthesis, or 
 below the vinculum, are to be considered as one, or together. 
 Thus (9+3) 57, or with the vinculum thus 9+3 5 
 
 7. 
 
 45. ORAL EXERCISES. 
 
 1. Commence at 50 and orally count to by continually 
 subtracting 1, thus : 49, 48, 47, 46, 45, etc. 
 
 2. Commence at 50 and orally count to by continually 
 subtracting 2, thus : 48, 46, 44, 42, etc. 
 
 3. Commence at 50 and orally count to by successively 
 subtracting 3, thus 47, 44, 41, 38, etc. 
 
 4. In like manner commence at 50 and subtract respect- 
 ively 4, 5, 6, 7, 8, 9 and 10 until you produce or a num- 
 ber less than the subtracted number, thus 46, 41, 35, 28, 
 etc. 
 
 5. Commence at 50 and subtract alternately 2 and 5 
 until you produce 1, thus, 48, 43, 41, 36, etc. 
 
 6. Commence at 50 and subtract alternately 8 and 3 
 until you produce 6, thus 42, 39, 31, etc. 
 
 46. To subtract one number from another irhrti any 
 figure of the subtrahend is less than the corretpondiitg figure 
 
 of the minuend. 
 
 1. From 897 subtract (>41. 
 
 OPERATION. 
 
 897 641 Explanation. First set the numbers 
 
 p } ^| or gQ-7 with the less under or over the greater, 
 
 so that units of the same order \\ill 
 
 ~ , stand in the same column. Then com- 
 
 2o6 Lob mence with the units figure and subtract 
 
 each order separately ; thus, 1 from 7 leaves 6 ; 4 from 9 leaves 
 
 5 ; 6 from 8 leaves 2. By this work we obtain the difference or 
 
 remainder, 256. 
 
 Subtract the following : 
 
 Gi) (:*) W (5) (6) 
 
 843 384 978 425 9876 
 
 521 762 655 679 3456 
 
Subtraction Decreasing. 33 
 
 47. 1o subtract one number from another when any 
 figure of the subtrahend is greater than the corresponding 
 figure of the minuend. 
 
 1. From 4173 subtract 2345. 
 
 FIRST OPERATION. 
 
 Minuend 4173 Subtrahend 2346 
 
 Subtrahend 2346 Minuend 4173 
 
 Difference 1827 Difference 1827 
 
 Explanation. Having written the numbers as in the preceding 
 problem, with the lesser number either above or below the 
 greater, we observe that 6 units cannot be taken from 3 units ; 
 we therefore mentally add 10 to the 3 units making 13 units, 
 and then say 6 from 13 leaves 7; then as we added 10 to the 
 minnend we now mentally add its equivalent, 1 ten, to the tens 
 figure of the subtrahend, and say 5 from 7 leaves 2 ; we next 
 observe that 3 hundreds cannot be taken from 1 hundred, we 
 therefore mentally add 10 hundreds to the 1 hundred making 11 
 hundreds, and then say !' from 11 leaves 8 ; then having added 
 10 hundreds to the hundreds figure of the minuend we now 
 mentally add I thousand, the equivalent of the 10 hundreds, to 
 the thousands figure of the subtrahend and say 3 from 4 leaves 
 1. This completes the operation and gives 1827 as the difference 
 of the two numbers. 
 
 The addition of 10 to the units and 10 hundreds to the hun- 
 dreds of the minuend, and its equivalent 1 ten and 1 thousand 
 to the tens and thousands columns of the subtrahend, is done 
 upon the principle that the difference between two numbers is 
 the same as the difference between the same two numbers 
 f^unllti incrcasi'd. 
 
 In all problems of subtraction the operation of adding 10 to 
 the minuend and its equivalent, 1, of the next higher order to 
 the subtrahend is repeated as often as the subtrahend figure is 
 greater than its corresponding minuend figure. 
 
 To Prove subtraction add the difference or remainder to 
 the subtrahend and if the sum is equal to the -minuend the 
 work may be considered correct, 
 
Arithmetical Exercises and Examples. 
 
 SECOND OPERATION. 
 11 To 
 
 -117:5 
 
 
 1Q97 
 
 1<s -' 
 
 Explanation. We will here perform 
 the operation by addition which is 
 a simpler and better method than 
 the preceding, and consists simply 
 in a(J(lin K to tlie subtrahend such a 
 number as will make it equal to the 
 
 minuend. Thus commencing with the unit figure of the sub- 
 trahend or smaller number, we say 6 and ? make 13; and set 
 the 7 in the units place of the difference; then carrying 1 we 
 say 5 and 2 make 7, and set the 2 in the tens column of the 
 difference ; then we say 3 and 8 make 11, and write the 8 in the 
 third column or hundreds place of the difference; then carry- 
 ing 1 we say 3 and 1 make 4, and write the 1 in the fourth place 
 of the difference. This completes the operation. 
 
 2. From 7:215 subtract 122*. 
 
 FIRST OPERATION. 
 
 fj\^> j - Explanation* Here we say 8 from 15 
 
 , .".,' leaves 7 ; 3 from 4 leaves 1 ; 2 from 2 
 
 leaves 0; 1 from 3 leaves 2 ; from 7 
 leaves 7. 
 
 72017 
 
 SECOND OPERATION. 
 
 73245 
 
 122S 
 
 72017 
 
 Explanation. Here we say 8 and 7 
 make 15; 3 and 1 make 4; 2 and 
 make 2 ; 1 and 2 make 3 ; and 7 
 make 7. 
 
 3. From 56802 subtract 50531. 
 
 FIRST OPERATION. 
 
 56802 
 50531 
 
 Explanation. Here we say 1 from 2 
 1 ; 3 from 10, 7 ; G from 8, 2 ; from 
 6, 6 ; 5 from 5, 0, which being the last 
 figure on the left has no value, and 
 hence is not set. 
 
 Explanation. Here we say 1 and 1 
 2 ; 3 and 710 ; and 2 =8 ; and 
 6=6 ; 5 and 0=5. The naught is not 
 set for the reason given in the first 
 solution. 
 
 EXAMPLES. 
 
 Write the following groups of numbers as they are here 
 
 6271 
 
 SECOND OPERATION. 
 
 56802 
 50531 
 
 6271 
 
Subtraction. 31> 
 
 written and subtract the lesser from the greater of each 
 group : 
 
 467 1807 3842 607 3001 6879 
 
 342 4251 1291 8013 1009 9640 
 
 Subtract the following numbers : 
 
 7. From 5307 take 309. Ans. 4998. 
 
 8. From 1090 take 1009. Ans. 81. 
 
 9. From 7608 take 3705. Ans. 3903. 
 
 10. From 184240 take 39460. Ans. 144780. 
 
 11. From 41074089 take 1875429. 
 
 Ans. 39198G60. 
 
 12. From 9876543210 take 123456781)0. 
 
 Ans. 8641975320. 
 
 48. To Subtract Dollars and Cents. 
 
 1. What is the difference between $483 and $51.65. 
 
 Ans. $431.35. 
 
 OPERATION. Explanation. In all problems of this 
 
 $483.00 kind we first set the numbers in the 
 
 .- i\.r same manner as when adding dollars 
 
 and cents, with dollars under dollars 
 
 and cents under cents, so that units of 
 
 $431.35 the same order will stand in the same 
 
 column and the points in a vertical line. 
 
 When there are no cents in the minuend, we fill the 
 place of cents with naughts. 
 
 The operation of subtraction is performed with dollars 
 and cents, the same as with other numbers. 
 
 What is the difference between the numbers in each of 
 the following groups ? 
 
 $16.25 $8~00 $.75 $4l!o4 $10.50 $L93 
 9.38 3.75 .59 6.61 4.78 .47 
 
 $6.87 $4.25 
 
*3ft Arithmetical Exercises and Example*. 
 
 $681.85 $127.05 $24S.oo silUl $8527.09 
 90.38 lo:>.r>n 181.15 !>.s<> 
 
 12 1:; 14 IT, 
 
 875.00 1971.50 lti-10.10 5184.62 
 
 43.50 >.oo 1270.UO 52!). 7S 
 
 10. Paid for rice 85500, and fur sugar 80875.40. 1 1 >w 
 much more was paid for sugar than rice ? 
 
 Ans. 1375.10 
 
 17. Bought a lot of flour for $2225, and sold the same 
 for 8'2SOO, what was the gain ? ADS. $575. 
 
 IS. It is 700 niilos to Shivveport and 320 to (Jalveston. 
 How much farther is it to Shivvepurt than to (Jalveston? 
 
 Ans. 3SO miles. 
 
 10. The ant has fifty eyes, the dra.-on fly 12000 ; how 
 many more has the dragon fly than the ant ? 
 
 Ans. 11 950 eyes. 
 
 20. The total coinage of gold and silver at the different 
 mints of the U. 8. during the fiscal year ending June 30th 
 1875, was 843,854,708. Of thls-aniount $33,553,965 was 
 g< Id, what was the amount of silver coined. 
 
 Ans. $10.300,743. 
 
 21. A student had 40 problems to work, he worked 
 17, how many has he yet to work ? Ans. 23. 
 
 22. Man has 26 bones in each foot, and 1!> in each 
 hand, how many more has he in the foot than in the hand ? 
 
 Ans. 7. 
 
 23. Sound travels through the air at the rate of 1118 
 feet per second, and a bullet fired from a rifle travels 1750 
 feet per second ; how much faster does the ball travel than 
 sound ? Ans. 632 feet per second. 
 
 24. Physiologists have determined with the aid of the 
 microscope, that the lungs of man contain not less than 
 600,000,000 air cells ; they have also determined that a sin- 
 gle drop of human blood contains more than 4jOOOjOOO,000 
 
Subtraction. 37 
 
 of corpuscles ; how many more corpuscles in one drop of 
 blood than air cells in the lungs ? Ans. 3,400,000,000. 
 
 25. Geologists have demonstrated that the formation of 
 the stalactites and stalagmites in the Mammoth Cave of 
 Kentucky, required not less than 75000 years of time ; and 
 that the wearing away of the rock of Niagara Falls by 
 friction, from Queenstown where they first were after the 
 glacial epoch, to their present location, 7 miles above, 
 required at least 40,000 years ; how much longer did it 
 require to form the stalactites and stalagmites, than for the 
 Falls of Niagara to recede to their present location? 
 
 Ans. 35000 years. 
 
 How many years from the date of each of the following 
 events to the present year ? 
 
 20. Quills first used for writing IJ30 A. I>. 
 
 27. Figures used by the Arabs, borrowed from the 
 Indians, 813 A. D. 
 
 2H. High towers first erected on churches, 1000 A. D. 
 
 2!). Glass Windows first used in Kn-land, 11 SO A. D. 
 
 30. Chimneys built in Kngland, 1230 A. D. 
 
 31. Spectacles invented by Spinu. 1299 A. D. 
 32^_\Voolen cloths first made in England, 1331 A. D. 
 
 33. Muskets used in .Knglaml. 1421, A. D. 
 
 34. Printing invented, 14 K) A. D. 
 
 35. Almanacs first published in Hilda, 1400 A. D. 
 30. Tobacco discovered in St. Domingo, 1490 A. D. 
 37. Spinning- wheel invented at Brunswick, 1530 A. D. 
 33. Needles first made in Kniiland by an Kast Indian, 
 
 1545 A. D. 
 
 39. Decimal Arithmetic invented at Bruges, 1602 A. D. 
 
 40. Circulation of the blood discovered by Harvey, 
 1619 A. D. 
 
 41. Newspapers first published, 1030 A. D. 
 
 42. Coffee brought to England, 1041 A. D. 
 
 43. Steam engines invented by the Marquis of Wor- 
 cester, 1049 A. D. 
 
 44. Cotton first planted in the United States. 1709 A. 
 D. 
 
38' Arithmetical Exercises and Examples. 
 
 45. Cotton first spun in America. 1787 A. D. 
 
 4G. Steam first used to propel boats by Fulton, in Amer- 
 ica, 1807 A. D. 
 
 47. First Locomotive was made at Liverpool, 1820 A. 
 D. 
 
 H. Electro-Magnetic Telegraphy invented by Morse, 
 of America, IS.'!:* A. D. 
 
 4 ( J. America was discovered in 141)2 A. 1). 
 
 50. The electric telegraphy was first used in the Tinted 
 States in 1844 A. D. 
 
 51. (Jem-rul (Jeorge. Washington was burn in 1732 and 
 died in 171W; (Jeneral 11. H. Lee was born in 1807 and 
 died in 1870. How much older was Ueiieral Washington 
 thaa General Lee, when he died? Ans. 4 years. 
 
 52. What is the difference between 23222 and 11 thou- 
 sand 11 hundred and 1 1 ? Ans. 11111. 
 
 53. What is the difference between dozen dozen and 
 half a dozen dozen ? Ans. 792. 
 
 54. What number must be ndded to G8741 to make a 
 million? Ans. 931251). 
 
 55. Philadelphia has 153151 buildings; New Orleans 
 35600. How many more has Philadelphia than New 
 Orleans? Ans. 117551. 
 
 5G. James, who is 23, is 7 years older than Henry; 
 how old is Henry? Ans. lb' years. 
 
 57. William has 8500 which is $150 more than I, and 
 I have $75 more than Lewis, how much has Lewis, and 
 how much have I ? Ans. Lewis has $275. 
 
 I have $350. 
 
 58. There are* two parties who owe me $8000, one of 
 them owes $4250. The other wishes to pay me $1700 on 
 account ; how much will he then owe ? 
 
 Ans. $2050. 
 
 59. A speculator bought a lot of apples for $215., and 
 sold them for such a price, that if he had got $22.50 more 
 he would have gained as much as they cost him. How 
 much did he sell them for ? Ans. $407.50. 
 
 60. From New Orleans to Vicksburg is 401 miles, and 
 
Subtraction. 39 
 
 to Natchez 277 miles ; how far is it from Natchez to Vicks- 
 burg? Ans. 124 miles 
 
 61. What is the difference between one million, seven- 
 teen thousand and seven, and one thousand sixteen hundred 
 and sixteen ? Ans. 1,014,391. 
 
 62. The sum of two numbers is 1463, one of the num- 
 bers is 628, what is the other ? Ans. 835. 
 
 63. The velocity of our earth on its yearly voyage 
 through space, around the sun, is 99733 feet per second ; 
 the velocity of a 12 pound cannon ball fired from a gun 
 with an average charge of powder is 1734 feet per second, 
 how many feet farther does the earth travel, in each second, 
 than a cannon ball ? Ans. 97999 feet, 
 
 or 18 miles and 2959 feet. 
 
 64. What number is that to which if 17821 be added 
 the sum will be 37907 ? Ans. 20086. 
 
 65. At an election the defeated candidate received 
 23742 votes; had he received 5112 votes more he would 
 have been elected by 1000 majority; how many votes did 
 the elected candidate receive ? Ans. 278.") I. 
 
 66. A father divided his plantation consisting of 4500 
 acres between his five sons Albert, Edward, William. 
 Frank and Robert. To Albert he ^ave SOO acres; to 
 Edward he gave 150 acres more than he gave Albert; to 
 William he gave 100 acres less than he gave Edward; to 
 Frank he gave as much as he gave I'M ward, and the re- 
 mainder he gave to Robert. How many acres did Robert 
 receive? Ans. 950 acres. 
 
 I 
 
40 Arithmetical Exercises and Examples. 
 
 MULTIPLIC ATIO>-( 
 
 -W. Multiplication is the processor operation of in- 
 creasing one of two numbers as many times as there are 
 units in the other. Or, differently explained, it is a short 
 method of performing addition. 
 
 The number to be multiplied is called the multiplicand. 
 
 The number which shows how many times the multipli- 
 cand is to be increased or repeated is called the innlfi/t/irr. 
 
 The result obtained by the operation of mttltiplyiog is 
 called the Product. 
 
 The multiplicand and multiplier are railed furffirs. The 
 meaning of the word factor is //< ///,/_/ or pm<lnc< r. 
 
 50. The Sigll Of Multiplication is an oblique cross, 
 X- It is read multiplied !>>/ or fiims. Thus 8 >< >, is read 
 8 multiplied by 3, or 3 times 8. 
 
 51. Principles of Multiplication. In all cases of 
 multiplication the multiplier must -be regarded as an abstract 
 number. Two denominate numbers cannot be multiplied 
 together as denominate numbers. 
 
 In all multiplication operations the product is the x<nn<' in 
 name or kind as the multiplicand. 
 
 For extended work on contracted methods of multiplying 
 see Soule's Contractions in Numbers. 
 
Multiplication Table. 
 
 41 
 
 112181 4| 5| 6| 7| 8| 9| 10 
 
 2| 4 1 6| 8| 10] 12| 14j T6| 18| 20 
 
 3| 6} 9|12| 15| 18| 21| 24 1 27| 30 
 
 4]_8J12|16|_20J_24J 2s :',2| 3G| 40 
 
 5J10J15J20 2-- | 30 1 36| 40| 45|_50 
 
 C. 12J18J24J 30| 3G| 42j 48| 54; (id 
 
 7 II 21J28 35] 42| 49] 56| 03 1 70 
 
 sin|24|32| Hi 48 .'.(. ; ''64|~72|80 
 
 "9|18|2"7|3l5| 40| 54j fir!; 72; M|~90 
 
 10[20|30J40| H(!| G0| 7d| 8d| 9o;inn 
 
 II 22|33 II 55j 66 77 8_8 99J110 
 
 1224|:-{r.|4S| Gdj 72| 84| 96|108|120 
 
 13|2j6|39j52| G5| 
 
 14|28|42|56| 7(l| 
 
 15)30 |.Vi.iij 7.">j 
 
 16J32 \>- M 80 '.MM 1 1- 12s 144J160 
 
 17J3J 5J 6J 85 
 
 78| 91J104J1I7130 
 
 84| !)8J112]l2G|140 
 '.MI I(i:.jl20|13."i 
 
 r,-{|l70 
 
 18J36J5472 ' ..... 8(126J144J162|180 
 L9|38J57|76 96 114J133J152 I71|190 
 20]40|6Tl|80 KHI l^n I 1(1,1(11) ISII^IKI 
 
 Erplanfilion WP rrcomnipnd tliif* taMp as being far superior to the one pre- 
 Bftitod in the School and College Text Nooks of the country, and urge all who 
 apire to proficiency in computing numlci> to learn it. In learning tint table, 
 or in the use of it. we caution the calculator apainet the use of all intermediate 
 words, whether he speak.* or thinks them; thus, instead of saying or thinking, 
 ( J times 3 are 27; 17 thm* r, are lu^, AT., say or think, 9, 3, 27; 17, 6, 102, <tc. 
 
 Tn reading we do not stop to spell orally or mentally the words that compose 
 the sentences; from the combination of the letters we see what the words are 
 without looking specially at each individual letter, and to read or operate with 
 rapidity in the combination of numbers, we muHt omit all superfluous talk or 
 Ih ought.. 
 
 P* 
 
42 Arithmetical Exercises and Examples. 
 52. ORAL EXERCISES. 
 
 1. At 8 cents a pound what will 9 pounds of rice cost ? 
 
 Solution. In all problems of this kind we reason thus : Since 
 1 pound costs 8/, 9 pounds will cost 9 times us much, which is 
 
 w, 
 
 2. What will 7 yards cost at 12/ per yard ? 
 
 3. What will 4 books cost at 20/ each ? 
 
 4. At 13/ per dozen what will 6 dozen cost ? 
 
 5. If 1 box (' -i |3 what will 23 boxes cost? 
 
 6. Flour is worth $7 per barrel, what are 25 barrels 
 worth ? 
 
 7. Bought 14 pounds of sugar at 8/ per pound, what 
 did it cost ? 
 
 8. At *i> per curd what will 34 cords of wood oos( ? 
 
 9. Paid $4 per barrel for potatoes and bought 47 bar- 
 rels, what did they cost? 
 
 10. If you receive 2 per day for labor and work 17 
 days how much money will you receive ? 
 
 Solution. Since 1 day's labor is worth $2, 17 days' labor is 
 worth 17 times as much, which is $3i. Or thus, since I receive 
 $2 for 1 day's work, for 17 days' work I will receive 17 limes as 
 much, which is $34. 
 
 11. Multiply from times 8 to 15 times 8 and reverse. 
 
 12. Multiply from times 9 to 16 times 9 and reverse. 
 
 13. Multiply from times 11 to 16 times 11 and reverse. 
 
 14. 12 inches make a foot, how many inches in 16 feet ? 
 
 15. 4 quarts mako a gallon. How many quarts in a 
 barrel that holds 42 gallons ? 
 
 16. What will 6 dozen shirts cost @ $18 per dozen ? 
 
 17. If you buy 15 boxes of peaches (a} $2 per box, 
 what will they cost ? 
 
 18. Multiply from times 12 to 19 times 12 and 
 reverse. 
 
 19. How many are 9 times 12 plus 8? 
 
 20. How many are 12 times 7 minus 6 ? 
 
 21. If you buy 7 pencils at 5 cents each and hand to 
 
Multiplication. 43 
 
 the seller 50^, how much change ought you to receive ? 
 
 22. A merchant bought 23 barrels of apples at $4 per 
 barrel and paid $65 on account. How much does he still 
 owe ? 
 
 53. To multi.ply whwi the multiplier consists of only <: 
 figure. 
 
 1. What is the product of 947 multiplied by 6 ? 
 
 OPERATION. Krplanation. In all problems of this 
 
 = r : ^ kind we place the multiplier under the 
 
 E.7 7 units' figure of the multiplicand and 
 
 i. then commencing with the units figure 
 
 we say, 6 times 7 are 42, which is 4 
 
 Multiplicand 947 /<//* and 2 unite; the 2 unit* we place 
 
 Multiplier (') i" the units place of the product and 
 
 retain in the mind the 4 tens to add 
 
 Prnrinff ^<N> to the column of tens ; we next sa . v 
 
 6 times 4 are 24 plus the 4 tens re- 
 tained in the mind, are 28, which is 2 hundreds and 8 *<*, the 8 
 /<;/.v-we write in the tens column ot the product, and retain in 
 the mind the 2 hundreds to add to the column of hundreds. 
 We then say (> times t> are 54. plus 2 hundreds are ,"G. which is 
 5 thousand and '> lnnnln<lt, wliich we write respectively in the 
 thousands and hundreds columns of the product. This com- 
 pletes the operation and gives a product of .> 
 
 In practice, instead of saying U times 7 are 42, G times 4 are 
 24, etc., we should only name the result of the combination, 
 thus 42, 24, etc. In handling fif/ur?x NV slum/if /////w//,s j>ron<nin<-r 
 t':< results of the combinations trithout )unni><</ the flour's tlxtt Jitaht: 
 the rryii1t,jm<t <ts ice pronounce words without xpclliny or naminj the 
 Utters that make the word*. 
 
 ">!. ToProvo the operations of multiplication, repeat the 
 
 work or multiply the multiplier by the multiplicand. If 
 the result is the same as the first, the work is probably 
 correct. 
 
 EXAMPLES. 
 
 Perform the following multiplications: 
 j -i 
 
 Multiplicand 5t:j '.)>:; 27(i!) 7<iS!)5 
 
 3Lultiplicr 785 9 
 
 Product :;S()1 7864 13845 692055 
 
44 Arithmetical Examples and Exercises. 
 
 s7<; t 
 
 5 
 
 2987 
 
 8 
 
 ;<; 
 
 7 
 
 <J 
 
 10 
 
 46532 
 14 
 
 11 
 58(17 I 
 15 
 
 ia 
 9861 
 
 17 
 
 18 
 
 si i:,:i 
 19 
 
 14. What will 4 pianos cost at $425 each ? 
 
 Ans. 81700. 
 
 15. At $65 each what will wagons cost ? 
 
 Ans. |585, 
 
 16. What will 7 lots of ground cost at $187") each ? 
 
 Ans. $13125. 
 
 17. At $6 per barrel what will be the cost of 245 bar- 
 rels of flour? 
 
 OPERATION. /.'.r/'litnution. The analysis for this 
 
 Multiplier 'Mf> problem is, since 1 barrel of Hour 
 
 ',, i " cost $6, 245 barrels will cost 245 
 
 MulUplicand I, times as much. The $6 is the real 
 
 multiplicand, but in the operation 
 #1470 Ans. we used it as the multiplier. This 
 wt> do for convenience in performing the operation, in all pro- 
 blems where the multiplicand is less than the multiplier. The 
 result is the same whichever factor we use as a multiplier. 
 
 18. What will 42 dozen hats cost at $9 per dnzen ? 
 
 Ans. 
 
 TJ. At $7 a piece what will 48 chairs cost ? 
 
 A us. 
 
Multiplication. 45 
 
 55. To multiply when the multiplier consists of more 
 than one figure. 
 
 1. What is the product of 397 multiplied by 653 ? 
 
 . 
 
 S*| = * s 
 
 ss .= = ? '3 
 
 Multiplicand ' 397 
 Multiplier 653 
 
 1st Partial product by 3 units 11 $\=3 times the multiplicand. 
 2d Partial product by 5 tens 1985 =50 times " 
 
 3d " " 6 h'ds- 2382 =600 " " 
 
 Total product 259,241= 653 " 
 
 Ksplanation. In all problems of this kind we first write the 
 multiplier under the multiplicand so that units of the same 
 order will stand in the same column, and then multiply by one 
 figure at a time. We first multiply by the units figure, then the 
 /<;/.*, hurn/rrt/A and so on in regular order through the multiplier 
 and add the several partial products together and thus obtain 
 the required product. 
 
 In this problem we first multiply by .?, the units figure, in the 
 same manner as explained in the first problem where there was 
 but one figure in the multiplier, and obtain 1191 as the first 
 partial product. This we write below the multiplier so that 
 units of the same order will stand in the same column. 
 
 Next we multiply by the 5 tens ; we say 5 times 7 are 35, which 
 is 3 hundreds and 5 tens ; we write the 5 tens in the tens column 
 directly below the multiplying figure and reierve in the mind 
 the 3 hundreds to add to the hundreds column. We then say 
 5 times 9 are 45 -f- 3 hundreds which were reserved are 48 hun- 
 dreds which is 4 thousands and 8 hundreds; we write the 8 
 hundreds in the column of hundreds and reserve the 4 thous- 
 ands to add to the thousands column. We then say 5 times 3 
 are 15 plus 4 thousands, reserved, are 19 thousands, which is 1 
 ten thousand and 9 thousands, which we write in tlmir respect- 
 ive columns. 
 
 We then in like manner multiply by the 6 hundreds in the 
 multiplier, being careful to write the first figure obtained (2) 
 in the hundreds column, directly under the G of the multiplier, 
 and the other figures in their respective columns, thousands, ten 
 
K > Arithmetical Exercises and Examples. 
 
 th<mxan<h and hnndr1 thu*andx. We then add the partial pro- 
 ducts together and obtain 259241 as the whole product of 397 
 multiplied by 653. 
 
 In practice remember to name or think only the results of the 
 numerical combinations when adding or multiplying. 
 
 EXAMPLES. 
 
 2. Multiply 3426 by 457. 
 
 OPERATION. 
 
 11. 
 
 Multiplicand 
 Multiplier 
 
 3426 
 457 
 
 1. Partial prod, by 7 units 23982= 7 times the multiplicand 
 
 2. Partial prod, by 5 Uns 17130 =3 times the multiplicand 
 
 3. Partial prod, by 4 h'ds 13704 = 4 times the multiplicand 
 Whole product 1,565,682 457 times the multiplicand 
 
 Multiply 647 by 58. 
 
 OPERATION. 
 647 
 
 58 
 
 5176 
 3235 
 
 37,526 Ans. 
 
 5. Multiply 28433 
 by 4172 
 
 4. Multiply 21794 by 2365 
 
 OPERATION. 
 
 21794 
 2365 
 
 108970 
 130764 
 65382 
 
 43588 
 
 51,542,810 Ans. 
 
 6. Multiply 989769 
 by 248193 
 
 Multiply the following numbers. 
 
 7. 483 by 569 
 
 8. 924 by 237 
 
 9. 1683 by 328 
 
 10. 581 by 76 
 
 11. 1847 by 84 
 
 12. 2346 by 127 
 
 13. 671 by 508 
 
 14. 8765 by 2043 
 
Multiplication. 47 
 
 Operation of the 13th 
 problem. 
 
 671 Explanation. In all problems where 
 
 508 there are naughts in the multiplier 
 
 we multiply by the significant figures 
 
 5368 only, for the reason that the product 
 
 3355 of_any number by is 0. 
 
 340868 
 
 56. To multiply when either the multiplicand or mul- 
 tiplier or both have naughts on the rijht. 
 
 1. Multiply 463 by 200. 
 
 OPERATION. Explanation. In all problems of this 
 
 ^gjj kind we write the significant figures so 
 
 900 *k ftt units of the same order may stand 
 
 in the same column and write the 
 naughts on the right of the significant 
 92600 figures. We then multiply the signifi- 
 
 cant figures and annex to the product as many naughts as there 
 are in the multiplier or multiplicand, or both. The basis or 
 reason of this is that the removal of a figure or number one 
 place to the left increases its value ten fold, and the annexing 
 of a naught removes the significant figures one place to the left, 
 thereby increasing them ten fold, and hence annexing a naught 
 is in effect multiplying by 10 ; aid for the same reason annexing 
 two naughts is multiplying by 100, the annexing of 3 naughts is 
 multiplying by 1100, etc., for other powers of 10. In this prob- 
 lem we first use the multiples 2 hundred as 2 units, hence the first 
 partial product, 926, was 100 times too small, we then by an- 
 nexing the two naughts multiply it by 100 and obtained 92600 
 as the correct product. 
 
 Multiply 3400 by 26. 
 
 OPERATION. 
 3400 
 
 26 
 
 204 
 68 
 
 88400 Ans. 
 
 3. Multiply 940 by 4700 
 
 OPERATION. 
 940 
 
 4700 
 
 658 
 376 
 
 4418000 Ana, 
 
I" ArUhmetieal "Exercises and Examples. 
 
 Multiply 5020 by 420. Multiply S2000 by 4S,'{. 
 
 OPERATION. S20UO 483 
 
 5020 4s:; s2ooo 
 
 420 
 
 1004 
 
 200S 
 
 2108400 
 
 210 966 
 
 [356 3864 
 
 39606000 
 
 39606000 
 
 4. Multiply 842 by GOO. 
 
 r>. Multiply 1208 by 1020. 
 I). Multiply DIMM) by 707. 
 
 7. Multiply 2:;:)0o by i2o:;o. 
 
 5. Multiply 1000 by IJ20S. 
 
 <). Multiply SI 00!) by <)0200. 
 
 10. Multiply 45U7H by 57SO. 
 
 11. Multiply !)>7000 by 4!>. 
 
 ,">7. T<* multiply ly the /'c/r/o/-.s < t f u 
 
 NOTK. FaM'tors of a nuinbiT are such numbers as will 
 wht3ii multiplie I together proluoe the numhor. 'J'htis <; 
 and (J are the factors of o<> ; 7 and 8 are the factors of 5f>, 
 or it is a number that will exactly divide a number. 
 
 1. Multiply 2435 by 42. 
 
 OPERATION. 
 
 >4;-f, Explanation. In all problems of this 
 1 kind we separate the multiplier into two 
 or more factors and multiply the multi- 
 plicand by one of the factors and the 
 17045 resulting product by another factor and 
 (J so on until we have used all the factors. 
 The last product will be the correct pro- 
 duct. 
 
 102270 
 2. Multiply 781 by 63. 
 :*. Multiply 3140 by 36. 
 4. Multiply 588 by 81. 
 
 5. Multiply 480 by 361. 
 
 6. Multiply 1756 by 125 
 
 7. Multiply 3281 by 128 
 
 58. To multiply when the Multiplicand or multiplier con- 
 tains dollars and cents. 
 
Multiplication. 49 
 
 1. Multiply $342.15 by 6. 
 
 OPERATION. Explanation. In all problems of 
 
 $342 15 *^ s kind we multiply in the regu- 
 
 lar manner and then prefix the 
 dollar sign $ and place the p<^it 
 (.) two places from the right. Wir 
 
 Product $2052.1)0 answer is then in dollars and 
 
 cents. 
 
 EXAMPLES. 
 
 2. What will 1082 pounds of sugar cost at 9/ per 
 pound ? Ans. $151.38. 
 
 3. A merchant's monthly expenses are $1342.75. What 
 are they for 12 months?. ' Ans. $1 Gil 3.00 
 
 4. It costs a family $2.30 a day for marketing, what 
 will be the expense for 30 days ? Ans. 861UM). 
 
 5. What will 37 boxes oranges cost at 83.75 per box? 
 
 Ans. $138.75. 
 
 (>. At 16 cents per pound what is the value of 23780 
 pounds Cotton ? Ans. $3804.80. 
 
 7. If it costs $17500 to construct one mile of railroad 
 what would be the cost to build 3(J4 miles .' 
 
 Ans. sr.:;70ooo. 
 
 8! What will S75 tons of railroad iron cost, at $55 per 
 ton ? Ans. 4S125. 
 
 !>. Multiply one million and twenty-six by nineteen 
 thousand seven hundred and ten. Ans. 11)710512460. 
 
 10. One cubic foot contains 1728 cubic inches. How 
 many cuijic inches in 324 cubic feet ? Ans. 559872. 
 
 11. One square foot contains 144 square inches. How 
 many square inches in !>5 square feet? Ans. 13680. 
 
 12. One gallon contains 231 cubic inches. How many 
 cubic inches in a cistern that holds 3500 gallons ? 
 
 Ans. 808500. 
 
 13. One bushel contains 2150.42 cubic inches. How 
 many cubic inches in 20 bushels ? Ans. 43008.40. 
 
 14. One mile contains 5280 feet. How many feet in 
 25 miles? Ans. 132000. 
 
 
 
r>n Arithmetical Exercises and Examples. 
 
 15. One year contains :>r>5 days. How many days in 
 21 years? Ans. 7665. 
 
 1(>. The human heart beats 4200 times an hour. How 
 many times does it boat in 10 years, there being 24 hours in 
 one day and :*i;5 day.- in each \var? Ans. .'Ji7i^OUUO. 
 17. Sound travels 11 IS foot per second. How far will 
 it travel in 10 minutes, there boini; 60 seconds in a minute. 
 
 Ans. 670800 feet. 
 
 1 S. Light (ravels 1 H2500 miles per second. How many 
 miles will it travel in 1 day, there being 24 hours in a day, 
 60 minutes in an hour, and GO seconds in a minute. 
 ^ Ans. 16,632,000,000. 
 
 It). A railroad train runs 25 miles an hour. How 
 far will it uu in :J days, allowing 3 hours for lost time in 
 stoppa.L-. Ans. 1725. 
 
 20. 1.1' a person respire 20 times in a minute, how many 
 times will he breathe in a day? Ans. 28,800. 
 
 21. If a person inhales 1 gallon of air at each respira- 
 tion, and respires 20 times per minute, how many gallons 
 will he inhale in 24 hours. Ans. 28800. 
 
 22. At 17 per ounce what is the worth of 9 pounds 
 of uold, there being 12 ounces in a pound Troy or Mint 
 weight? Ans.. $1836. 
 
 23. How many pounds of coffee in 180 bags if each 
 bag contains 162 pounds? Ans. 29160. 
 
 24. How many pounds of cotton in 87 bales, if each 
 bale weighs 475 pounds ? Ans. 41325. 
 
 25. What will 27893 pounds of tobacco cost at 56 
 cents per pound ? Ans. $15,620.08. 
 
 26. What will 1870 acres of land cost at $18 per acre ? 
 
 Ans. $33,660. 
 
 27. The Senate and House of Representatives of the 
 State of Louisiana consists of 138 members who receive $8 
 per day. The regular session continues 60 days. What 
 is the yearly expense for the salaries of the State's law 
 makers? Ans. $66240. 
 
 27. A contractor has 865 men employed at $1.50 per 
 day. What are the weekly wages of all for 6 days' labor? 
 
 Ans. $7785. 
 
Multiplication. 51 
 
 28. What will it cost to build 37428 cubic yards of 
 levee at 45 cents per cubic yard ? Ans. $16842.60. 
 
 29. A steamboat arrives with 3840 bales of cotton ; 
 1320 sacks cotton seed and 580 barrels molasses. Her 
 freight charges are $2 per bale for cotton, 25/ per sack far 
 cotton seed and 50/ per barrel for molasses. What is the 
 amount of her freight bills? Ans. $8300. 
 
 30. A drayman charges 75 cents a load, and he has 
 hauled 63 loads. How much is due him ? 
 
 Ans. 847.25. 
 
 31. What will it cost to slate the roof of a house contain- 
 ing 52 squares at $13.25 per square ? Ans. $689. 
 
 32. The walks around a dwelling contain 129 square 
 yards. What will it cost to flag them with German flans at 
 $3.10 per square yard? Ans. $399.90. 
 
 33. What will it cost to pave a street containing 20000 
 square yards, with stone at $4.75 per square yard ? 
 
 Ans. $95000. 
 
 34. Bought 2180 barrels of coal at 4S/ per barrel. 
 What was the cost? Ans. 104ti.4o. 
 
 35. Multiply 5 billions and 16 ly 5 millions and 1 
 thousand. Ans. 25,005,000,080,016,000. 
 
 36. A Hogshead of sugar contains 10S5 pounds; how 
 many pounds in 107 hogsheads of equal weight ? 
 
 37. A planter produced 68 bales of cotton, if the 
 average weight of the bales was 460 pounds, and the cotton 
 sold for 13 cents per pound, bow much money would it 
 bring? Ans. $4,066.40 
 
 38. What will 3 cases containing 2 dozen pairs each of 
 shoes cost (a) $2.90 per pair ? Ans. 2tS.sn. 
 
 39. If it costs $1.50 a day to support one person, what 
 will it cost to support a family of 13 for one year or .'>(>.") 
 days? Ans. $7117.50. 
 
 40. There are 35600 dwellings in New Orleans, allow- 
 ing 7 persons to each dwelling, what would be the popula- 
 tion of the city ? Ans. 249200. 
 
 41. A merchant sold thrw dor.cn dozen ladies' hose at 
 
52 Arithmetical Exercises and Examples. 
 
 one quarter of a <1<>\< n </<>:, , t cents a pair. How much 
 did he receive fur them? A us. 155.52 
 
 42. The pressure of tli o atmosphere is 15 puunds on 
 every square inch nf surface. The exterior surface of a 
 m;in of average size is aliout 2501} square inches. How 
 many pounds weight does he sustain ? 
 
 Ans. .'J7500 pounds. 
 
 43. How many dollars aiv .'>7-") *lo o-,,l,l pieces worth? 
 
 Ans. $-J750. 
 
 44. What is the value uf 21 -HI dimes? Ans. 82 14. (JO 
 -IT). AYhat is the valucof 1010 ,,uar. dol. Ans. $252.50 
 4(>. What is the value of 72S nirkels ? Ans. s.'ili.lO 
 -17. What is the value of 1612 hall 1 , dol. Ana. ssou.oo 
 
 48. During the fiscal year ending Sept. 1, ls7i>, there 
 
 was received .'501 Si hn^slu-ads nf Tuhaeeu. 11' each hlid. 
 contained 12 pounds of poison, how many pounds of poison 
 were there in the whole? Ans. .".U2172. 
 
 49. The circumfer'-nce of the earth is nearly 25000 
 miles, the distance to the sun is .'WOO times as many miles. 
 How far is it to the sun ? Ans. 95000060. 
 
 50. -1S75 is the thirteenth part of a number. What is 
 the number? Ans. (l!5!7r>. 
 
 51. The sun is loSl.^oo nines as large aft the earth; 
 the earth is 4f> times as larire as the moon. How many 
 times is the sun larger than the moon ? Ans. 62302500. 
 
 ___ 52. Lio-ht travels l!)2.")00 miles a second and it requires 
 100000 years to travel to us from some of the fixed stars 
 that are seen with the telescope. Allowing !>(>.') days, 5 
 hours, 48 minutes and 41) seconds to a year and remembering 
 that there are 24 hours in a day, (JO minutes in an hour 
 and 60 seconds in a minute, how far distant arc such stars? 
 Ans. 607470883250000000 miles. 
 
 53. A man's receipts are $1800 a year and his disburse- 
 ments are 1125 a year. How much arc his net receipts 
 in 3 years? Ans. 2025. 
 
 54. It is estimated by Astronomers that 7500000 vis- 
 ible meteors fall upon the earth daily ; it is also estimated 
 that the average weight of each is 100 grains. From thcs<- 
 
Division Decreasing. 53 
 
 figures and allowing 365 days to the year, what is the annual 
 growth of the earth in weight by the accession of the visible 
 meteoric matter ? Ans. 273750000000 grains. 
 
 DIVISION. (Decreasing.) 
 
 59. Division is the process of finding how many times 
 one number is equal to another. Or it is the process of find- 
 ing one of the factors of a given product when the other 
 factor is known. 
 
 60. The Dividend is the number to be divided or it is 
 the number to be measured. 
 
 61. The Divisor is the number by which we divide or 
 it is the number used as a unit of measure. 
 
 62. The Quotient is the result of the division, and 
 shows how many times the dividend is equal to the divisor. 
 
 63. The Remainder is the number left after dividing 
 dividends, which are not multiples of the divisor, or which 
 are not an exact number of times equal to the divisor. It 
 
 * must always be less than the divisor. 
 
 64. The Sign Of Division is a horizontal line with a 
 point above and below, thus -f-. It is read divided l>y ; and 
 it indicates that the number before it, is to be divided by the 
 number after it ; thus 25 -j- 5, is read 25 divided by 5. 
 
 The horizontal line, and the vertical or curved line when 
 placed between two numbers also indicates division. Thus, 
 - 3 j 6 , 4|36 or 4)36, are all read 36 divided by 4. 
 
 65. PRINCIPLES OF DIVISION. 
 
 1. When the divisor and dividend are both denominate 
 or both abstract numbers, the quotient will be an abstract 
 number. 
 
 2. When the divisor is an abstract number and the 
 dividend a denominate number, the quotient will be a 
 denominate number. 
 
 3. When there is a remainder it is a part of the divi- 
 dend and, is therefore the same in name or kind. 
 
 B* 
 
54 Arithmetical Exercises and Examples. 
 
 4. Multiplying the dividend or dividing the divisor 
 i))nltij)/irs the quotient. 
 
 5. Dividing the dividend or multiplying the divisor 
 divides the quotient. 
 
 (5. Multiplying or dividing both the divisor and divi- 
 dend by the same number does not change the quotient. 
 
 66. Prnnf of Division. Multiply the quotient by the 
 divisor and if there is no remainder the product should he 
 equal to the dividend ; when there is a remainder add it to 
 the product, and it' the work is correct the sum will equal 
 the dividend. 
 
 I >i vision operations may be performed by the process 
 of addition or subtraction. Hut as tlies processes are too 
 lengthy for practical purposes, we will not give them place 
 here. 
 
 For contracted methods in division, see Sonle's Con- 
 tractions in Numbers. 
 
 1. 
 
 3. 
 4. 
 5. 
 6. 
 
 7, 
 8. 
 9. 
 
 10. 
 
 11. 
 
 12. 
 
 13. 
 
 14. 
 
 15. 
 
 16. 
 
 17, 
 
 67. ORAL EXERCISES. 
 How many times is equal to 1 ? or 1 
 " " 1 u 0?or 1 
 
 Ans. An infinite number of times. 
 
 How many times is 1 equal to 1 ? or 1 
 
 2 " 1 ? or 2 
 
 " 3 " 1 ? or 3 
 
 " " 4 " 2 ? or 4- 
 
 " " 8 " 2 ? or 8- 
 
 " " !l " :*? or 9 
 
 " " 12 u 4? or 12 
 
 " 20 " 5? or 20- 
 
 " 24 " 6 ? or 24 
 
 -35 " 7 ? or 35 
 
 " " 56 " 8 ? or 56- 
 
 " 63 9 ? or 63 
 
 72 
 80 
 88 
 96 
 
 9 ? or 72 
 10 ? or SO- 
 11 ? or 88- 
 12 ? or 96 
 
 = ? 
 
Division Fractional Numbers. 55 
 
 19. 3g 6 -=? 4)42=? 9)45=? 77-=- 7? 
 -%*=? 6)48=? 5)55=? 8412? 
 
 20. How many times is 24 equal to 3, to 4, to 6, to 8, 
 to 12, to 24? 
 
 21. How many times is 36 equal to 3, to 4, to 6, to 9, 
 to 12, to 36? 
 
 22. How many times is 42 equal to 2, to 6, to 7, to 42 ? 
 
 23. " " 64 " 2, to 4, to 8, to 64 ? 
 
 24. " " 72 " 2, to 8, to 9, to 72 ? 
 
 68. FRACTIONAL NUMBERS. 
 
 When we divide a unit or a number of units of any 
 kind into equal parts, these parts are sometimes called frac- 
 tions. The name of the equal parts varies according to 
 the number of parts into which the thing or number was 
 divided. 
 
 When the unit or number is divided into 2 equal parts 
 1 of the parts is called om--h<i1j\ and is written thus J. If 
 divided into 4 equal parts 1 of the parts is called one-fourth, 
 and is written thus \ ; 3 of the parts are called three-fourths 
 and are written thus j. 
 
 In like manner we obtain fifth x, .s/./-/Ax. * ?/ ,/fhs, eighths, 
 twelfths^ sixteenths, twenty-firsts^ etc. 
 
 In writing fractional numbers in figures we place the 
 number which shows the name of the parts below a hori- 
 zontal line as a divisor, and the number which shows lum: 
 many parts are taken or used, above the line as a dividend. 
 
 The following examples will fully elucidate this work : 
 
 Two-thirds are written |. 
 Three-fourths are written , : 
 Five-eighths are written 
 Seven-ninths are written /. 
 
 Seven-twelfths are written 
 Nine-tenths are written ,'',, 
 Fifteen-sixteenths written 
 Eleven-eightieths written 
 
 How do you find ], .1, 1 , ^, etc. of any number? 
 How do you finu . A, etc. of any number? 
 
 What is \ uf 4? 
 " " J of <J? 
 " 1 of 8? 
 
 What is 4- of 15? 
 " " i of 18? 
 " " \ of 28? 
 
 What is 5 of !)? 
 of 12? 
 of 40? 
 
 7 
 
5(> Arithmetical Exercises and Examples. 
 
 EXAMPLES. 
 
 1 . If 3 hats cost 8l> what will 1 hat cost ? 
 
 . Ans. 2. 
 
 Analytic solution. Since 3 hats cost $0, 1 hat will cost J part 
 of SJ, which i 
 
 L > . If 8 yards cost 56 cents what will 1 yard cost ? 
 
 :>. Paid *:>< for (> barn-Is of flour, what did 1 bam-1 
 cost ? 
 
 4. gallons of molasses cost 4.50, what did 1 gallon 
 cost? AHS. 50/. 
 
 5. Bought 12 shirts for $30, how much did 1 cost? 
 
 Ans. $2.50. 
 
 (>. Paid SI. 00 for 8 pounds of sugar, what was the 
 price per pound ? Ans. &.12]. 
 
 7. 7 dozen oranges cost $2.10, what was the price per 
 dozen? Ans. 8. .'Jo. 
 
 8. Bought 20 peaches for 60^, how much did 1 peach 
 cost? Ans. $.03. 
 
 9. At $2 a yard how many yards can you buy for $'1 1 ? 
 
 Ans. 12 yards. 
 
 Analytic solution. Since $2 buy 1 yard, $1 will buy .] of a 
 yard, and $24 will buy 24 times J a yard, which is, y or 12 
 yards. 
 
 Or thus. Since $2 buy 1 yard, for $24 we can buy as many 
 yards as $24 is equal to $2, which is 12 times. 
 
 10. At 9 cents per pound how many pounds can be 
 bought for 45 cents ? Ans. 5 pounds. 
 
 Analytic solution. Since 9 cents buy 1 pound, 1 cent will buy 
 ^ of a pound, and 45 cents will buy 45 times ^ of a pound, 
 which are -\ 5 - or 5 pounds. 
 
 11. Flour is worth 8 per barrel, how many barrels can 
 be purchased for $56 ? Ans. 7 barrels. 
 
 Analytic solution. Since $8 buy 1 barrel, $1 will buy J of a 
 barrel, and $56 will buy 56 times J of a barrel, which are -% 6 or 
 7 barrels. 
 
 12. For $,95 how many papers can you buy at 5 cents 
 a paper? Ans. 19 papers. 
 
Short Division. 57 
 
 13. If 25 cents buy 1 yard how many yards will 75 
 cents buy ? Ans. 3 yards. 
 
 14. At $3 a piece how many chairs can be bought for 
 $36? Ans. 12 chairs. 
 
 15. If the printer charges 1.50 to set 1 page of this 
 book, how many pages can be set for $75 ? 
 
 Ans. 50 pages. 
 
 WRITTEN EXERCISES. 
 69. To divide irJten the divisor docs not exceed 12. 
 
 1. Divide 3048 by 5. 
 
 OPERATION. Kffhniation. In all problems of 
 
 Divisor 5) 3048 dividend this kind we write the numbers as 
 
 shown in the operation, and then 
 
 begin on the left of the dividend 
 Quotient 729 and 3rem. to (livide> We bc?in on the left 
 
 in order to carry the remainder, it' any, of the higher order of 
 units to the next lower order. In this problem we first take 
 the 3 (thousands,) and as it is not equal to 5, we therefore 
 unite it with the <> hundreds, making 36 hundreds, which by 
 trial multiplication and subtraction mentally performed, we 
 find is equal to 5, 7 (hundreds) times and 1 remainder; the 7 
 we write in the hundreds column of the quotient line, directly 
 under the > the last figure used of the dividend ; then to the 
 1 remainder we mentally annex the -1 tens, making 14 tens, as 
 the second partial dividend, and which by mental multiplication 
 and subtraction, we find is equal to 5, 2 (tens) times and 4 
 remainder; the 2 we write in the tens column of the quotient 
 line, and to the 4 we mentally annex the units figure of the 
 dividend, making 48 units as the third and last partial dividend ; 
 this we find, by mental multiplication and subtraction to be 
 equal to i">, !> times and 3 remainder. 
 
 The remainder is usually expressed fractionally by writing 
 it over the divisor, thus 2, this expresses the part of a unit of 
 times that the remainder is equal to the divisor. 
 
 SHORT DIVISION". 
 
 Operations in division according to the foregoing method 
 are called short divixittn, because tlu i multiplication and 
 subtraction work in finding the remainder of the partial 
 dividends were mentally performed. 
 
 2. II ow many times is 840 e<jual to ? 
 
58 Arithmetical Exercises and Example*. 
 
 OPERATION. 
 Divisor 0)846 Dividend 
 
 Quotient 141 
 
 tinn. In the* pre- 
 cxMling problem we gave a 
 full and explicit explanation 
 of each step of the operation. 
 
 In practice much of the explanation therein given is omitted, 
 and the work performed thus : Commencing with the left hand 
 figure we say 8 is equal to 0, 1 time and 2 remainder; 24, is 
 equal to 6, 4 times ; G is equal to 6, 1 time. 
 Work the following indicated divisions. 
 
 7)847 8)12327 9)1 n< ll)2:;-i; 
 
 121 
 8)1471 
 
 1540J 
 
 4)1 1 
 
 120| 
 9)81018 
 
 n 
 2344 
 
 7 1 93020 
 
 21345--8 
 
 12)10824 
 
 9 76451 
 
 Divide the following numbers : 
 
 15. 9872 by 4 
 
 16. 1483 " 7 
 
 17. 1691 " 9 
 
 18. 41070 " 8 
 
 23. What is i of $528? 1 
 
 24. u aref of 1005? \ 
 
 Operation for the 24th 
 
 problem. 
 5)$1005 
 
 3 
 
 $603 Ans. 
 
 19. 10286 by 6 
 
 20. 48710 " 7 
 
 21. 1001)8 " 9 
 
 22. 199999 " 8 
 W hut are 1 of $448? 
 
 " "I of 6444? 
 
 Operation for the 26th 
 problem. 
 
 8)$6444 
 
 25. 
 
 26. 
 
 805.50= 
 
 7 
 
 $5638.50 Ans. 
 
 27. How many apples can be bought for $ 2.25 at 5 
 cents a piece ? Ans. 45 apples. 
 
Short Division. 59 
 
 28. At 15 cents a pound, how many pounds can you 
 buy -for S3. 15? Ans. 21 pounds. 
 
 29. Paid $90 for 10 volumes of Chambers' Cyclopedia, 
 what was the price of one volume ? Ans. $i). 
 
 30. If 8 men are to receive $5791 in equal parts, what 
 wijl be each man's share ? Ans. $723-J. 
 
 31. The dividend is 63, and the quotient is 9, what is 
 the divisor ? Ans. 7. 
 
 32. The quotient is 15, the divisor 3, and the remain- 
 der 2, what is the dividend ? Ans. 47. 
 
 33. The quotient is 3G, and the divisor 6, what is the 
 dividend? Ans. 216. 
 
 34. The dividend is 72, and the divisor is 4, what is 
 the quotient? Ans. 18. 
 
 35. "/l^ow many pounds of cotton at 11 cents a pound 
 will be required to pay for 33 pounds of sugar @ 8 cents 
 a pound ? Ans. 24. 
 
 70. To divide when the divisor exceeds 1 .. 
 1. Divide 7387 by 36. 
 
 OPERATION. Kjcjtlnnation. We first write the 
 
 Divisor, Dividend, Quotient numbers as shown in the opera- 
 
 ->r\ '7'-*Q-' />nr 7 tion and commence to divide as 
 
 fttU ^Vo explained in the first written ex- 
 
 '- ample. But as the divisor is too 
 
 large to be conveniently used men- 
 
 1 S7 tally, we therefore write tho oper- 
 
 jgQ ation of multiplying the divisor 
 
 by the quotient figures, and sub- 
 
 ^ . , ttacting the successive products 
 
 7 remainder f rom t ne several partial dividends. 
 In performing the division we first see that 7, (thousands) 
 are not equal to 36, and hence there will be no thousands in the 
 quotient. We then annex to the 7 thousands the 3 hundreds, 
 making 73 hundreds as the first partial dividend; this is equal 
 to 30, 2 times, and a remainder; we write the 2 in the hundreds 
 column of the quotient, multiply the divisor by it, write the 
 product under and subtract the same from the 73 hundreds of 
 the dividend. This work gives us 1 hundred remainder, to 
 which we annex the 8 tens, making 18 tens as the second partial 
 dividend ; this partial dividend not being equal to 36, we write 
 
00 Arithmetical Exercises and Examples. 
 
 (no tens) in the tens column of the quotient, ami annex to 
 the 18 tons the 7 units, making 187 units as the third and last 
 partial dividend. This is equal to 36, 5 times and a remainder, 
 \ve write the 5 in the quotient, and multiply and subtract as we 
 did with the first obtained figure of the quotient, and thus pro- 
 duce 7 remainder, which we write over the divisor as explained 
 in short division. 
 
 LONG DIVISION. 
 
 Operations in division, according to the above method, 
 are called long tl iris fun, for the reason that the multiplica- 
 tion and subtraction work in finding the remainders of the 
 partial dividends is written. 
 
 2. How many times is 66804 equal to 5'1 ? 
 
 OPERATION 
 
 Divisor Dividend 
 
 53) i;r,sn4 0-< ; 
 53 
 
 138 
 106 
 
 Proof. 
 
 1260 Quotient, 
 
 53 Divisor. 
 
 3780 
 6300 
 
 24 Remainder. 
 
 320 
 318 
 
 24 
 
 3. What is the quotient 
 of 107941 --396? 
 
 OPERATION. 
 
 396)107941(272 Quotient. 
 792 
 
 2874 
 2772 
 
 1021 
 792 
 
 229 Remainder. 
 
 66804 Dividend. 
 
 4. Divide 7167901 by 
 11 207. 
 
 OPERATION. 
 
 11267)7167901(636 Quoti't. 
 67602 
 
 40770 
 33801 
 
 69691 
 67602 
 
 2089 Remainder. 
 
Long Division. 61 
 
 5. Divide 784 by 82. 
 
 STATEMENT. 
 
 82)784(9|f Ans. 
 
 6. Divide 91070 by 8761. 
 
 STATEMENT. 
 
 8761)91070(10-fff Ans. 
 
 7. Divide 2461 by 74. 
 
 8. Divide 4809 by 91. 
 
 9. Divide 13872 by 263. 
 
 10. Divide 54123 by 1423. 
 
 11. Divide 628100 by 156. 
 
 12. Divide 10000 by 304. 
 
 13. Divide 37021 by 2002. 
 
 4 Divide 8888888 by 332211. 
 Divide $6805 equally between 5 men, and what 
 will be the share of each ? Ans. 1361. 
 
 16. What is the sir.iy-fniirth part of $44800? 
 
 Ans. $700. 
 
 17. 145 men picked 1305000 pounds of cotton, sup- 
 posing they all picked an equal quantity, how much did 
 one man pick ? Ans. 9000 pounds. 
 
 18. A father gave his 7 sons a Christmas present of 
 1353.50 to l>c shared equally, what was each one's share? 
 
 Aus. $50.50. 
 
 71. To divide w/trn th<*i-< r> nnnylitu on the right of 
 tin 1 divisor. 
 
 1. Divide 2843 by 200. Ans. 14^fr 
 
 OPERATION. Explanation. Since by our scale 
 
 'M))2843( of n m bers tnc y increase from 
 
 I ' ~* ( __ right to left in a tenfold ratio, and 
 
 ~~ decrease from left to right in a 
 
 14 and 4.5 Kern, corresponding manner, it is clear 
 
 t.h:it the removal of any order of figures from left to right dimin- 
 
 ishes its value ten times for each place of removal. And as 
 
 previously shown, that the annexing of naughts multiplies num- 
 
 bers, by removing them to places of higher value, so in like 
 
 manner cutting figures off from the right of a number removes 
 
 the remaining orders to the right, and hence decreases them 
 
 tenfold for every figure cut off. Hence to cut off one figure is 
 
 dividing by 10 ; to cut off two figures divides by 100 ; to cut off 
 
 three figures divides by 1000 and so on. 
 
62 
 
 Arithmetical Exercises and Examples. 
 
 Considering these principles, in all cases of this kind we cut 
 off the naughts from the right of the divisor and the same num- 
 ber of figures from the right of the dividend ; and then divide 
 the remaining figures of the dividend by the remaining figun-s 
 of the divisor. When there is a remainder annex the figures 
 nit, off, and we obtain the true remainder. 
 
 2. Divide S7JK51 by 1000. Ans. 87 T Vu- 
 
 OPERATION. 
 
 1|000) 87(931 
 
 Quotient 87 and 931 Remainder. 
 
 3. Divide 178 by 10. I 4. Divide 6581 by :;on. 
 
 OPERATION. OPERATION. 
 
 10;17S 3 00)65 H 
 
 Quotient 21 2S1 Rein. 
 Ans. LM 
 
 Quotient 17 8 Remainder 
 Ans. 17 /V 
 
 6. 
 
 Divide 714G8071 by'.341000. 
 
 OPERATION. 
 
 Ans. 
 
 682 
 
 3268 
 3069 
 
 6. 
 7. 
 8. 
 9. 
 10. 
 
 199 
 
 Divide 8897600 by 8100. 
 Divide 1000000 " 10000. 
 Divide 99999 by 9000. 
 Divide 33440 by 270. 
 Divide 140817 by 6800. 
 
 Ans. 1098-ff. 
 Ans. 100. 
 Ans. ll-fffo. 
 Ans. 123ffg, 
 Ans. 20f ffrf 
 
Division. 63 
 
 72. To divide by the Factors of a number. 
 
 1. Divide 936 by 24. Explanation. In ail problems 
 OPERATION. where the divisor is a compos- 
 
 4)936 ite number we may divide by 
 
 the factors and thus shorten the 
 
 operation In this example the 
 factors are 4 and 6, and we first 
 divide by 4 which gives a quo- 
 39 tient 6 times too large, for the 
 
 reason that 4 is but J of 24 the true divisor. We therefore divide 
 this quotient by 6 and obtain the true quotient. 
 
 2. Divide 588 by 28. The factors are 4 and 7. 
 
 ADS. 21. 
 
 3. Divide 6976 by 32. The factors are 4 and 8. 
 
 Ans. 218. 
 
 4. Divide 2583 by 63. The factors are 7 and 9. 
 
 Ans. 41. 
 
 5. Divide 10206 by 81. The factors are 9 and 9. 
 
 Ans. 126. 
 
 6. Divide 11984 by 56. The factors are 8 and 7. 
 
 Ans. 214. 
 
 7. Divide^ 1607 by 72, using the factors 3, 4 and 6, and 
 find the true 'remainder. 
 
 Ans. 22 quotient, and 23 remainder. 
 
 FIRST OPERATION. 
 3^) 1607 Explanation In this ex- 
 
 ' ample using as divisors 3, 
 
 . 4 and 6, the factors of 72. 
 
 4)535 2, 1st remainder, we obtain for remainders 2, 
 
 3 and 1. 
 
 6) 133 3, 2d remainder. The first remainder 2, is 
 
 clearly units of the given 
 
 9> i Qrl * v, ; 1 dividend, and hence a part 
 
 1, 3d remainder. of the tr ' ue remainder / 
 
 The second remainder, 3 being fourths of the second dividend. 
 535 which are reciprocal thirds of the given dividend, it is hence 
 | of the reciprocal of J of J 9, of the given dividend and 
 true remainder. 41 
 
 The third remainder, 1 being sixths of the third dividend 133, 
 which are reciprocal twelfths of the given dividend, it is hence 
 J of the reciprocal of J of ^ of ^ 12 of the given dividend 
 
G4 Arithmetical Exercises and Examples. 
 
 and true remainder. Therefore 2, the first remainder, plus 0, 
 the unit value of the second remainder, plus 12, the unit value 
 of the third remainder =_ 23, the true remainder. Or we may 
 obtain the true remainder without considering the reciprocal 
 
 relationship of the quotients ami divisors, tlni> : 
 
 First remainder.' _' 
 
 Plus 2d, remainder 3. X tnc preceding divisor 3, = ! 
 
 Plus 3d, remainder 1, > x ft U tlu ' preceding divisors, 4 and 3= 12 
 
 which added gives the true remainder 23 
 
 From the foregoing we see that the true remainder may be 
 obtained by adding to the first remainder the product of the 
 other remainders by all the divisors preceding the one which 
 produced it. 
 
 8. Divide 7851 by <i4. u>ini: tlu> factors S and S. 
 
 Ans. l'2'2 quoti. -nt. 4i> remainder. 
 
 OPERATION. Ks/'ftimrfinn. Here the 1st re- 
 
 8)7851 mainder is 3, to which we add the 
 
 ' _ product of the 2d remainder f>, 
 
 . multiplied by the preceding divi- 
 
 3, 1st; remainder. sor 8j equals 40, making 43, the 
 
 true remainder. 
 1225, 2d remainder. 
 
 9. Divide 17803 by 96, using the factor* 2, 3, 4 and 4. 
 
 Ans. IK, 
 
 OPERATION. Explanation. 
 
 2)17803 
 ' 
 
 __ 1st remainder 
 
 3)8901 1 2d remainder 3X3X 2 18 
 
 4^2967 3d remainder IX^X^X^^= 24 
 
 4)7413 
 
 True Remainder 43 
 
 ' 1851 
 
 10. Divide 27865 by the factors of 81. 
 
 Ans. 344 g\. 
 
 11. Divide 101041 by the factors of 84. 
 
 Ans. 1202|| 
 
Division. 65 
 
 12. Divide 899 by the factors of 108. Ans. 
 
 13. If $4691 are divided equally between 35 men, 
 what will each one receive ? Ans. $134^. 
 
 14. There are 32 quarts in one bushel, how many 
 bushels are there in 1536 quarts ? Aus. 48 bushels. 
 
 15. A hogshead of wine contains 63 gallons. How 
 many hogsheads in 2898 gallons ? Ans. 46 hogsheads. 
 
 16. One of the factors of 10800 is 225 ; what is the 
 other? Ans. 48. 
 
 17. What number multiplied by 137 will give 959137 
 fbrjhe_product ? Ans. 7001. 
 ""18. Multiplying 372 by an unknown number gives 
 44640 ; what is the number? Ans. 120. 
 
 19. What is the quotient of 9126 divided by 9 ? 
 
 Ans. 1014. 
 
 20. Divide four million, eight thousand and sixteen by 
 MMDCXLIV. Ans. ISlSfffl 
 
 21. What number is that to which, if sixteen be added 
 the sum multiplied by 8 and 13 substracted from the pro- 
 duct the remainder will be 33!) ? Ans. 2S. 
 
 22. Theie is a number from which if you subtract 55, 
 and divide the remainder by 12, your quotient will be 36. 
 What is that number? Ans. 487. 
 
 23. A merchant owes a debt of 1 < S 7.">, which he agreed 
 to pay by weekly installments ot 825. He has made 55 
 pavments, how many more payments has he to make. 
 
 Ans. 20. 
 
 24. A merchant bought 350 barrels of flour at $6 a 
 barrel, and sold it at $7.50 per barrel. The gain he gave 
 in equal parts to 4 worthy boys to aid them in obtaining an 
 education. What was the cost and selling price of the 
 flour, and how much money did each boy receive. 
 
 Ans. 82100 cost, 2625 selling price,$131.25 
 each boy received. 
 
 25. An acre contains 160 square rods ; how many acres 
 in a plantation containing 123200 square rods? 
 
 Ans. 770 acres^ 
 
66 Arithmetical Exercises and Examples. 
 
 2('>. A boy sold 50 oranges at 5^ each and thereby 
 gained $1.50. At what rate did he buy the oranges? 
 
 A i is. 2/ a piece. 
 27. How many times 13G will produce 1708? 
 
 Ans. i:^. 
 
 IN. Dividejhe product of 750 and 875 by their differ- 
 ence. Ans. 525o. 
 
 21). The diameter of the earth at the equator is 71)25 
 miles ; how long would it take a locomotive to travel that 
 distance at the rate >f 25 miles an hour? 
 
 Ans. 317 hours 13 days 5 hours. 
 
 30. The first Atlantic Telegraph Cable as originally 
 made cost $125^250. 10 miles of deep sea cable was made 
 at a cost of 81450 j H >r mile, and 25 miles of shore ends 
 was made at a. cost of $1250 per mile. The remainder 
 cost $485 per mile. How many miles of Cable were made ? 
 ^ Ans. 25i 55 mile,-. 
 
 31. A grocer wishes to put 3335 pounds of sugar in 3 
 kinds "of boxes, containing respectively 20, 50 and 75 
 pounds, using the same number of boxes of each kind or 
 size. How many boxes will he require ? 
 
 Ans. 23 of each size. 
 
 32. The Northern Pacific Railroad from Lake Superior 
 to Puget Sound, as located, is 2000 miles long. The esti- 
 mated co*t and equipment of the road, including interest 
 i- > ^5277000. What will be the average cost per mile ? 
 
 Ans. $42638.50. 
 
 33. It is estimated that, by reason of intemperance the 
 United States loses annually $08400000. How many 
 School Houses costing $5000 each, and how many Libraries 
 costing $3000 could be established with this amount of 
 money? Ans. 12300 of each. 
 
 34. ^Ten freedmen agreed to pick 20000 pounds of cot- 
 ton and receive for their labor I of the cotton picked. 
 After they had picked 7000 pounds 4 freedmen quit, 
 leaving the other 6 to finish the work. How much cotton 
 i> each entitled to when the work is finished ? 
 
 Ans. 140 pounds each for those who left, ami 
 573f each for those who remained. 
 
Divison. . 67 
 
 35. A merchant bought 800 gallons of molasses at 
 and sold J of it at 72^ a gallon. From the profit he 
 bought his children a set of Cutter's Anatomical and Phy- 
 siological charts, and had $8.20 left. What did the charts 
 cost? Ans. $19.80 
 
 36. The capacity of steam engines is measured by horse 
 power ; and 1 horse power is a force that will raise 33000 
 pounds 1 foot in 1 minute. How many horse power has a 
 steam engine that possesses a capacity of 1188000 pounds? 
 
 Ans. 36. 
 
 37. The average weight of man is 150 pounds. About 
 \ of this weight is blood. Allowing that the heart throws 
 out 2 ounces of blood at each pulsation, and that it beats 
 72 times a minute, and that 16 ounces make a pound, how 
 long will it take the heart to circulate all the blood in the 
 body ? Ans. 3|^ minutes. 
 
 38. Prof. Wilson, a physician and physiologist, has 
 counted in the skin of the palm of the hand 3528 perspi- 
 ratory pores to [the square inch ; but as there are less to 
 the square inch on some other parts of the body, he esti- 
 mates that 2800 is a fair average to allow to the square inch 
 for the whole surface of the body. The average size man 
 has 2 ")()() 5<iuare inches of body surface, which would give 
 7000000 perspiratory pores. Through these pores fully 2 
 pounds of perspiration, water, refuse matter and worn out 
 tissue pass every 24 hours. If a man weighs 150 pounds, 
 how long will it take for matter equal to the weight of the 
 body to pass through the perspiratory pores, if they are 
 kept open as they should be by daily bathing ? 
 
 Ans. 75 days. 
 
 :'/.). Our earth is about 95000000 miles from the sun, 
 Neptune, the most distant member of our Solar System, is 1 
 about 2850000000. How many times farther from the 
 Sun is Neptune than our earth ? Ans. 30. 
 
 40. The velocity of the earth on its yearly voyage 
 around the Sun is 99733 feet per second. The velocity of 
 a cannon, ball fired from a gun with an average charge of 
 
^ Arithmetical Exercises and Examples. 
 
 powder is 1750 feet a second. How many times fatter is 
 the velocity of the earth than a cannon ball? 
 
 Ans. :<;; 
 
 41. Gco. Peabody, of Ma- while living, to '21 
 
 schools and colleges, library associations, benevolent socie- 
 ties, state public schools, etc., not including many private 
 presents, $7S7.">000. Of this an:uunt $3300000 \\vrc -ivm 
 to the public schools of the South. What part of the 
 whole spc-iliid donation did he gives to the South ? 
 
 v... 8800000 ' 
 Ans - b ;;i.v 
 
 4'2. Stephen (Jinml, of Philadelphia, gave SI;O<MMHIO 
 for the founding and support of Girard College. Soule's 
 College in New Orleans is worth .'iO,000. How many 
 such colleges could be built with the amount of money 
 -iven by Mr. tlirard to establish one collc-_ 
 
 Ans. 200. 
 
 43. The air which surrounds our earth, and of which 
 we eaih inhale (500 gallons every hour, is composed of four 
 parts of Nitrogen and 1 pait of ( )xygen. Il<>w many gal- 
 lons of each are iherc in a room 2'2 feet long and 21 leet 
 wide and 10 feet high, which contains IM^UO gallons of 
 air? Ans. (JIU2 Oxygen. L'Tn'iS Nitrogen. 
 
 44. A room contains iM-^liO gallons of air, a man inhales 
 u'OO gallons per hour, how long will it take for 10 men to 
 inhale the air in the room. Ans. .") ( |;";|;;;- hours. 
 
 4,"). A room 1(5 feet long, 10 feet wide, and 8 feet high, 
 contains 12SO cubic feet of air. Every time a person 
 breathes he throws out from his lungs a sufficient quantity 
 of carbonic acid, or carbon di-oxide, (a most deadly gas.) 
 to pollute or render poisonous and unfit for breathing 3 
 cubic feet of air, and he breathes '20 times a minute. How 
 long- will it take for the air of a room of the above dimen- 
 sions to become poisonous if occupied by 5 persons, and DO 
 change of air is made by ventilation. 
 
 Ans. 4-3%-^ minutes. 
 
 46. A man produces by breathing at least 6 gallons of 
 carbonic acid gas every minute, a single burning gas jet, 10 
 gallons, an ordinary stove, 60 gallons, How many gallons 
 
Division. 69 
 
 of carbonic acid gas will an audience of 1000 people, 2 heated 
 stoves, and 50 burning gas jets produce in 3 hours, and 
 how many times would the quantity fill a room 100 feet long, 
 50 feet wide and 30 feet high ? 
 
 Ans. 1191600 gallons. l^WoVA time - 
 
 NOTE. Theie are 60 minutes in an hour, 231 cubic inches in 
 a gallon, and 1728 cubic inches in a cubic foot. 
 
 47. Astronomers estimate that 7500000 visible meteors 
 fall upon the earth daily, the average weight of which is 
 estimated to to 100 grains. Allowing for an equal quan- 
 tity of matter to be brought down by the invisible meteors 
 and the serolhes, how many pounds a year does our earth 
 increase in weight, there being 7000 grains in a pound, and 
 365 days in a year? Ans. 78214285f pounds. 
 
 73. PROBLEMS INVOLVING THE ENGLISH MONEY OF 
 ACCOUNT. 
 
 1. What will 13840 pounds of cotton cost at 8 pence 
 a pound. Ans. 461. 6s. 8d. 
 
 OPERATION. Estimation. In this problem the 
 
 13840 price is given in one of the subdivi- 
 
 o * sions of the English monetary unit, 
 
 and hence we must know what that 
 
 unit and its subdivisions are, before 
 
 12) 110720d. we can solve the problem. The En- 
 
 glish monetary unit is the Pound Ster- 
 
 20) 9226 8d liuff, which is divided into 20 Shillings ; 
 each shilling is divided into 12 Pen- 
 " nies, and each penny into 4 Farthings. 
 
 uS- With this knowledge of English money 
 we can work all problems of the above character. In this 
 example we first multiply the price of one pound by the num- 
 ber of pounds and thus produce the value of the whole in 
 pence. Then to reduce the pence to shillings, we divide them 
 by 12, and obtain 9*22(5 shillings and a remainder of 8, which 
 being a part of the dividend is therefore 8d. Then to reduce 
 the shillings to pounds we divide them by 20, and obtain 4GI 
 pounds and a remainder of 6, which being a part of the second 
 dividend is therefore 6s. N the English monetary system the 
 following abbreviations are used : . represents pounds, s. 
 represents shillings, d. represents pence, and f. represents far- 
 things. 
 
70 Arithmetical Exercises and Examples. 
 
 '1. What is the value of 3. What will 241 boxee 
 483 yards of cloth at 1(J shil- cheese cost at 3 per box ? 
 
 ling per yard V 
 
 Ans. JC3Sfi. 8s. 
 OPERATK 
 
 li 
 
 r72 shillings 
 
 Ans. .C723. 
 
 OPERATION. 
 241 
 
 3 
 
 723 pounds. 
 
 386 8s. 
 
 I. Sold 4S(' yards of calico at 5 pence a yard, what did 
 it amount to? Ans. .10 2s. (id. 
 
 T>. Bought 38495 pounds of good middling cotton at 7 
 pence a pound. .How much did it CM - 
 
 Ans. 1122 15a :>d. 
 
 U. What is the value of S,">0 ham-Is ol' flour at 34 
 shillings a harrel ? Ans. 14-1."). 
 
 7. How much will 1S12 tons of iron cost at 52, 4s. 
 per ton? Ans. !l4r>S(j 8s. 
 
 8. Bought 3> 121 pounds of cotton at 9 pence per pound 
 what did it cost? Ans. C1440 15s. 9d. 
 
 74. MISCELLANKors PROBLEMS INVOLVING THE I'HIN- 
 CIl'LKS OK ADDITION, SUBTRACTION, MULTIPLICA- 
 TION AM) DIVISION'. 
 
 1. The subtrahend is 210, and the remainder 184, what 
 is the minuend ? Ans. 400. 
 
 2. A irrocer paid 8350 for some tea and coffee ; for the 
 tea he paid $50 more than for the coffee, what did he pay 
 for each ? Ans. tea 8200, coffee $150. 
 
 3. John has 25 cents, and James has four times as 
 many lacking 10 cents, how many cents has James? 
 
 Ans. 90 cents. 
 
 4. A slate cost 15 cents; an arithmetic four times as 
 much as the slate, and a philosoj hy twice as much, lacking 
 25 cents, as the slate and arithmetic. What did they all 
 cost? Ans. $2.00. 
 
Division, 71 
 
 B. The sum of two numbers is 480, and their difference 
 is 80, what are the numbers ? Ans 200, 280. 
 
 6. A man purchased a horse and cow. For the horse 
 he paid $175, and for the cow $110 less than for the horse, 
 what did the cow cost ? Ans. 65. 
 
 7. The less of two numbers is 224 and their difference 
 100, what is the greater? Ans. 324. 
 
 8. The product of two numbers is G450, and one of 
 the numbers is 150, what is the other? Aus. 43. 
 
 9. A merchant bought 415 yards calico at 10 cts. per 
 yard and sold it for 13 cts. per yard. How much did he 
 gain? Ans. 812.45. 
 
 10. The dividend is 37500 and the quotient 75, what 
 is the divisor ? Ans. 500. 
 
 11. A boy sold 5 chickens at 25/ a piece ; 8 ducks at 
 50^ each ; received in payment 3 pigeons at 30^ each, and 
 the balance in money ; how much money did he receive ? 
 
 Ans. $4.35. 
 
 12. The divisor is 37; the quotient 21, and the re- 
 mainder 23, what is the dividend ? Ans. 800. 
 
 13. A news boy sold 20 papers at 5/ each, and with 
 the money bought oranges at 4c each, how many oranges 
 did he get? Ans. 25. 
 
 14. The first battle of the Revolution was fought April 
 19, 1775, how many years, months and days have passed 
 since then ? 
 
 15. H. Zuberbier has an orange orchard consisting of 
 480 trees, and each tree produces 5 barrels of oranges which 
 are worth in the market $4 a barrel ; what is the value of 
 his orange crop ? Ans. 81)600. 
 
 16. C. Quentell bought a barrel of sirop de batterie 
 containing 43 gallons at 95/ per gallon ; 4 gallons having 
 leaked out he sold the remainder at $1.U5 a gallon. How 
 much did he gain by the transactions? Ans. $.10 gain. 
 
 17. W. C. Martin bought ;354 barrels of flour for 
 $2478. He sold the same at $7.50 per barrel ; how mucn 
 did he gain? Ans. $177. 
 
 18. The Capital Stock of a Manufactory is $100000 
 
72 Arithmetical Exercises and Examples. 
 
 which is divided into 200 shares. What are 5 shares worth ? 
 
 Ans. j!f)00. 
 
 19. J. Muller sold to O. Braun 25 barrels of apples at 
 $4 per barrel and 124 barrels of potatoes at $3.25 per bar- 
 rel; he received in payment 1 hogshead of sugar containing 
 1143 pounds at 8/ and the remainder in money ; how much 
 money did he receive? Ans. $411.5(1. 
 
 20. A speculator bought 528 cords of wood at $(J.f>0 
 per cord. He re-corded the wood so that it measured 57!) 
 cords which he sold at $6.75 a cord ; how much did he 
 gain? Ans. $476.25. 
 
 CANCELLATION. 
 
 75. < ampliation is the proecss of shortening the 
 operations of division, or of the indicated result of multi- 
 plication and division operations combined, by rejecting 
 equal factors from both dividend and divisor or from both 
 increasing and decreasing numbers. 
 
 The operation is perfoimed by drawing a line across each 
 factor cancelled or cut out. 
 
 7G. The Principles of Cancellation, are, 1. lle- 
 
 jecting or Cancelling a factor from any number is in effect 
 dividing the number by that factor. 2. llejecting or can- 
 celling equal factors from both dividend and divisor, or from 
 both increasing and decreasing numbers in an indicated 
 result, does not change the quotient or result. 
 
 EXAMPLES. 
 
 Divide 7x3x4 by 7>(4 
 
 Operation by Cancellation. Explanation. In all prob- 
 y 7 lems where we have both niul- 
 
 ^ 3 triplication and division opera- 
 
 tions to perform, we use a 
 vertical or perpendicular line 
 
 ^ which we call the statement 
 
 3 Ans. line. This line is used to facil- 
 
 itate the work by separating the dividends and divisors, or the 
 increasing and decreasing numbers. The dividends or increas- 
 ing numbers are always placed upon the right hand side of the 
 
Cancellation. 7B 
 
 line and the divisors or decreasing numbers are always placed 
 ui)on the left hand side. 
 
 In this example having written the numbers that constitute 
 the dividend and divisor, respectively upon the right and left 
 hand side of the statement line, we cut out or cancel the equal 
 factors T's and 4's in the numbers constituting the dividend and 
 divisor and thus obtain 3 the answer to the problem. 
 
 To perform the work without the aid of Cancellation we 
 would be obliged to make the following figures: ly^'^='2l, which 
 X4:r=84 the dividend ; then 7X 4 2 8 the divisor ; then 
 28)84(3 Ans. 
 84' 
 
 2. Multiply 25, 48 and 88 together and divide the pro- 
 duct by the product of 10, 30 and 8. 
 
 Operation by Cancellation. AV y /,W/ >/<>//. In this exam- 
 ' lO'^o 5 P^ e we wr * te ^ ie numbers on 
 
 ~ " 
 
 .) . , <) the line as above directed and 
 
 then cancel the lo, and 2:. by 
 
 Kpp 11 f> ; then the 3G and 48 by 12 ; 
 
 then the 8 and 88 by 8 ; then 
 
 110 the 4 and 2 by 2. This is all 
 
 __ that can be cancelled and we 
 
 ., .., * then multiply together the 
 
 50-j Alls. 5 ^ ., . uul j l an j jjyide tne pr<J _ 
 
 duct by 3, and thus obtain the true result 
 
 Should the student experience any difficulty in this kind of 
 work, he should be orally drilled on the factors of numbers and 
 composite numbers*. 
 
 3. Divide the product of 32 X^ l>y 8 
 
 Operation by Cancellation. E, p hniti<,n. Having writ- 
 - 4 ten the numbers on the state- 
 
 -"> merit line, we first cancel the 
 
 3 ( ,) 
 | j^; 
 
 8 and 32 by 8 ; then the 9 and 
 3 by 3 ; then the 1G and 4 by 
 
 __ 
 
 -) _ i A '*' ^ W llav " 1 S no more num- 
 
 i -2 A ns - bers on the increasing side of 
 
 the line to cancel we multiply together the remaining numbers 
 on the decreasing side of the line and thus produce the correct 
 result ,\,. 
 
 In all cases where, after cancelling, no factor appears on either 
 
71 Arithmetical Exercises and Examples. 
 
 side of the statement line, the factor 1, is always understood, 
 as being there. Its non-appearance is in consequence of not 
 having written it when we cancelled a number by itself. 
 
 4. A merchant sold 25 boxes of candles containing 3G 
 pounds each at lt>c' per pound and received in payment 
 starch at (> cents per pound. How many boxes each con- 
 taining !><) pounds did he receive? Ans. 80 boxes. 
 
 Operation by Cancellation 
 
 tin; 
 p w > r, 
 
 80 boxes Ans. 
 Cancel and work the following line statements or results: 
 
 12 i:> 
 
 2<; 
 
 .;: 
 ISM 
 
 y . 
 
 ! rx; 
 
 784 
 
 in 4 
 
 51 
 
 124 
 
 17 
 
 10 
 
 20 
 70 
 
 76 
 91 
 140 
 25 
 
 II Ans. 70 Ans. f Ans. 1921 Ans. 
 
 5. Divide the product of 6, 7, 12 and 22 by the pro- 
 duct of 11, 3, 14 and 8. , Ans. 3. 
 
 (). What is the quotient of 28x66x7XT8-5-56Xl30 
 X42X13? Ans. J. 
 
 7. Multiply 21, 55 and 128 together and divide the 
 product by 14X25X64. Ans. 6f. 
 
 8. How many bushels of corn at 70^ each will pay for 
 140 gallons molasses at (55 cents a gallon ? 
 
 Ans. 130 bushels. 
 
 9. Bought 420 pounds of sugar at 6 cents a pound and 
 gave in payment 360 pounds of rice. What was the price 
 of the riee ? Ans. 7 cents. 
 
 10. Sold a drayman 64 bushels of oats at 75 cents a 
 bushel, for which he is to pay in drayage at 50 cents a load. 
 How many loads must he haul ? Ans. 96 loads. 
 
 11. How many pounds of butter at 35/ per pound will 
 
Properties of Numbers. 75 
 
 pay for 245 pounds of rice at 5 cents per pound ? . 
 
 Ans. 35 pounds. 
 
 12. Paid 65/ for 5 yards of calico, what will 27 yards 
 cost at the same rate ? 
 
 Analytic Solution by Can- Explanation. In all practical 
 cellation. problems of this kind we give a 
 
 |$j3 13 reason for each step of the opera- 
 
 tion, and make the whole state- 
 ment to indicate the final result 
 
 _ 
 
 ^ without performing any of the 
 
 $0.51 Ans. intermediate work. In this prob- 
 
 lem, we place the 6- r >? on the increasing side of the statement 
 
 line as our premise and reason thus: since 5 yards cost <J.V ? . 
 
 1 yard will cost i part of it, and 27 yards will cost 27 times as 
 
 much as 1 yard. 
 
 13. If 17 barrels of flour cost $110.50, what will 500 
 barrels cost at the same rate ? Ans. $3250. 
 
 14. If | of a dozen apples cost 40 cents, what will 
 13i- dozen cost? Ans. $7.20. 
 
 PROPERTIES OF NUMBERS. 
 
 DEFINITIONS. 
 
 77. All Integer is a whole number ; as 1, 5, 6, 18, etc. 
 Whole numbers are divided into two classes, Prime and 
 
 Composite. 
 
 78. A Prime number is one that can only be divided, 
 without a remainder, by itself and 1 ; as 1, 2, 3, 5, 7, 11. 
 13. 17, etc. 
 
 7!>. A Composite number is one that can be divided 
 without a remainder, by some other whole number than 
 itself and 1 ; as 4, 0, 12, 15, 24, etc. 
 
 All composite numbers are the product of two jjr more 
 other numbers. 
 
 Numbers are prime to each other when they have no 
 common factor that will divide each without a remainder; 
 as 0, 13, 20, etc. 
 
76 Arithmetical Exercises and Examples. 
 
 SO. An Kveil number is one that can be divided by 2. 
 without a remainder; as 4, 8, 12, 50, etc. 
 
 81. All 0(1(1 number is one that cannot be divided by 
 2, without a remainder, as 1,7, l'.. l.">. l.'J.'J. etc. 
 
 82. A Factor of a Number is a number that will 
 divide it without a remainder or by being taken an entire 
 number of times, will produce it ; as 4 is a factor of 1(>, and 
 5 a factor of 2.">. 
 
 .Kvery factor of a number is a divisor of it. 
 
 83. A Prime Factor of a number is a prime number 
 that will divide it without a remainder : thus, 1. 2. 3 and 
 5 are the prime factors of .'Jo. 
 
 84. A Composite Factor of a number is a composite 
 number that will divide it without a remainder ; thus, (j 
 and 8 are composite factors of IS 
 
 85. An Aliquot part of a number is such a part a> 
 will divide it without a remainder; thus, 1, 2, 3, 4, (\ and 
 8 are aliquot parts of 2 I. 
 
 86. TllO Reciprocal of a number is the quotient of 1 
 divided by the number ; thus the reciprocal of 8 is 1 8=|- ; 
 and the reciprocal of \ is 1 -=- ^ = 4. 
 
 87. The Power of a number is the product obtained 
 by multiplying the number by itself a certain number of 
 times ; thus, 36 is the second power of ; 125 is the third 
 power of 5. 
 
 88. The Multiple of a number is the product obtained 
 by multiplying the number by any other number any num- 
 ber of times. Or it is a number divisible by a given num- 
 ber without a remainder ; thus, 14 is a multiple of 7 and 
 2. and 54 is a multiple of 2, 3, !>, and 27. 
 
 89. A Common Multiple of two or more numbers is 
 a number divisible by each of them without a remainder ; 
 thus, 24 is a common multiple of 2, 3, 4, 6 and 12. 
 
 00, The Least Common Multiple of two or more 
 
Divisibility of Numbers. 77 
 
 numbers is the least number that is divisible by each of 
 them without a remainder; thus, 12 is the least common 
 multiple of 2, 3, 4, 6 and 12. 
 
 DIVISIBILITY OF NUMBERS. 
 
 91. A Divisor, or measure of a number, is any num- 
 ber that will divide it without a remainder ; thus, 4 is a 
 divisor or measure of 12, and 5 is a divisor or measure 
 of 20. 
 
 Cue number is said to be Divisible by another when 
 the remainder is 0. 
 
 92. A Common Divisor of two or more numbers is 
 a number that will divide each of them without a remain- 
 der; thus, 2 is a common divisor of 12, 18 and 24. 
 
 93. The Greatest Common Divisor of two or more 
 
 numbers is the greatest number that will divide each of 
 them without a remainder ; thus, 6 is the greatest common 
 divisor of 12, 18 and 24. 
 
 94. Every number ending with 0, 2, 4, 6 or 8, is divi- 
 sible by 2. 
 
 95. Every number is divisible by 4 when its units and 
 tens figures are divisible by 4 ; thus, 15(3, 264, 34312, 
 561308, are each divisible by 4. 
 
 96. Every number is divisible by 8 when the units, 
 tens and hundreds figures are divisible by 8 ; thus 3824, 
 12512, 190720 are each divisible by 8. 
 
 97. All numbers ending in or 5 are divisible by 5 ; 
 thus, 10, 15 and 35 are divisible by 5. 
 
 98. Every number, the sum of whose figures is divi- 
 sible by 3 or 9 without a remainder, is divisible by 3 or 9 ; 
 thus, 135, 3456, 12345912, etc., are each divisible by 3 
 and by 9. 
 
 99. Every even number, the sum of whose figures is 
 a* 
 
78 Arithmetical Exercises and Examples. 
 
 divisible by 3 without a remainder, is divisible by (\ ; thus 
 318, 12414, etc., are divisible by (5. 
 
 Fiucnoxs. 
 
 KM). A Fraction is one or more of the equal parts of 
 a unit of any kind, or of a collection of units taken tn^-eth'-r. 
 Or more briefly, a part of anything, or a numerical expres- 
 sion of a part of a unit. 
 
 101. A Fractional Tnit is one of the equal pans 
 
 into which any integral unit is divided. If the i,it- t irral 
 unit is divided into two equal pans, each is called n //////'. 
 it into three, each is called a thinl ; if into four. ea-h is 
 called a /'nurtli ; and so on according to the number f 
 parts into which the integral unit is divided. 
 
 102. Fractions are divided into two kinds. ( 1 <nnm<ni or 
 Ynlijiir. and ]>n-iin<il fr'nu-tinim. 
 
 Common Fractions are expressed by two numbers, one 
 
 written above the other, with a hori/.ontal line between 
 them. The number below the line is called the D> nnmin- 
 <ttur, and the number above the line is called the .Xmnrrator. 
 Thus 2 (one half,} J (three fourths,) f (Jive .\v>//ix, ) J 
 (,-rrn- r/V////x,) and } ^ (thirteen sevciitrrnthx) are fractions, 
 the denominators of which are, respectively, 2, 4, 6, 8, and 
 17. The Numerator and Denominator together, are called 
 the terms of the fraction. 
 
 The Denominator of a fraction shows the number of 
 equal parts into which the unit is divided. 
 
 Thus in the fraction | the 8 is the denominator and shows 
 that the unit is divided into 8 equal parts called eighths. 
 
 The Numerator of a fraction shows the number of equal 
 parts taken to form the fraction. 
 
 Thus in |-, the numerator is 5 and shows that 5 of the 
 8 equal parts are taken or expressed by the fraction. 
 
 All fractions aritie from division and are expressions of 
 unexecuted division in which the numerator is the dividend, 
 the denominator the divisor, and the fraction itself the 
 
Fractions. 79 
 
 Decimal Fractions are those in which the denominators 
 are not generally expressed, but are always 10, or a power 
 of ten; thus, .5, .75, .821, read respectively Jive tenths, 
 seventy-five hundredth*, and eight hundred and twenty-one 
 thousandths, are decimal fractions. To write these fractions 
 as common fractions, they would be written thus, y 5 ^, T 7 5 % 
 
 ' 
 
 The point (.) placed before the 5, 7 and 8, in the above 
 decimally expressed fractions, is called the decimal point, 
 and is used to abbreviate the work. 
 
 103. CLASSIFICATION OF FRACTIONS. 
 
 For convenience fractions are classed under the following 
 heads : Proper Fractions ; Improper Fractions ; Simple 
 Fractions; Mixed Numbers; Compound Fractions ; Com- 
 plex Fractions. 
 
 104. A Proper Fraction is one in which the nume- 
 rator is less than the denominator, as i, f , J. 
 
 105. An Improper Fraction, is one in which the 
 numerator is equal to or greater than the denominator ; as 
 
 ' and V'. 
 
 10l>. A Simple Fraction is one in which both terms 
 are whole numbers, and may be either a proper or hnpn^n r 
 fraction ; as f , f , fj- or 
 
 107. A Mixed Number, is a number composed of a 
 whole number and a fraction ; as 2], 5:] and -IfV 
 
 103. A Compound Fraction is a fractional part of 
 a fraction or mixed number ; as -J of | and 2 of r 7 ^ of 1'J:]. 
 
 109. A Complex Fraction is one that has one or 
 more of its terms fractional ; as : * 
 
 i 61 3 i 6} 
 
 - of r and - of of 
 I 5| II 8 
 
 UO, The Reciprocal of a Fractlqn is the result yf 
 
^ Arithmetical Exercises and Examples. 
 
 1 divided by the fraction. Thus the reciprocal of 
 
 -H=*=u. 
 
 111. The Value of a Fraction is the result of its 
 numerator divided by its denominator. Thus J--1. 
 
 2 . 
 
 112. <;KNKRAL PRINCIPLES OF FRACTIONS. 
 
 1. Multiplying the numerator, or di rid ing the denom- 
 inator, multiplies the fraction. 
 
 '1. Dividing the numerator, or multiplying the denmn- 
 inator, divides the fraction. 
 
 3. Multiplying or dlrldlny both numerator and denom- 
 inator by tlie sumo number does not change the value of 
 the fraction. 
 
 113 .REDUCTION OF FRACTIONS. 
 
 Reduction Of Fractions is the process of changing 
 theiryorw without altering their c<ilu< . 
 
 114. A fraction is reduced to Higher Terms when 
 the numerator and denominator are expressed in larger 
 numbers. Thus, }=| or f or T 8 F ; f or T \ or -J-J, etc. 
 
 115. A fraction is reduced to Lower Terms when 
 the numerator and denominator are expressed in smaller 
 numbers. Thus T V | or ; i^ f or j. 
 
 111). A fraction is reduced to its Lowest Terms when 
 its numerator and denominator cannot be divided by any 
 number greater than 1, or, when its numerator and denom- 
 inator are prime to each other. 
 
 117. Whole numbers maybe reduced to fractions 
 having any desired denominator. 
 
 118. ORAL EXERCISES. 
 
 1 unit, abstract, or denominate of any kind equals, how 
 many halves? thirds? fourths? fifths? sixths? sevenths? 
 eighths? ninths? 
 
 2 =how many halves? thirds? fourths? fifths? sixths? 
 sevenths? eighths? ninths? 
 
Reduction of Fractions. 
 
 81 
 
 3 how many halves? thirds? fourths? fifths? sixths? 
 sevenths? eighths? ninths? 
 
 4=how many halves ? thirds? fourths? fifths? sixths? 
 sevenths? eighths? ninths? 
 
 5 how many halves ? thirds? fourths? fifths? sixths? 
 sevenths? eighths? ninths? 
 
 6 how many halves ? thirds? fourths? fifths? sixths? 
 sevenths? eighths? ninths? 
 
 119. What kind of numerical work is the above called ? 
 
 i how many fourths? sixths? eighths? tenths? 
 twelfths? fourteenths? 
 
 = how many sixths ? ninths ? twelfths ? fifteenths ? 
 eighteenths ? twenty-firsts ? 
 
 J = how many eighths? twelfths? sixteenths? twen- 
 tieths ? twenty-fourths ? 
 
 = how many tenths? fifteenths? twentieths? twenty- 
 fifths? thirtieths? 
 
 J how many sixteenths ? twenty-fourths ? thirty- 
 seconds? fortieths? sixty-fourths? 
 
 y 1 ^ = how muuy thirty-seconds ? forty-eighths ? sixty- 
 fourths? eightieths? 
 
 What kind of numerical work is the above called ? 
 Answer the following numerical questions : 
 
 F' 
 
 120. What kind of numerical work is the above called? 
 
 How many halves = or make a unit? 
 
 " u thirds = 
 
 " " fourths 
 fifths = 
 
 " " sixths 
 " " sevenths 
 <' " eighths = 
 
Arithmetical Exercises and Examples. 
 
 Huw many ninths = or make a unit? 
 " " tenths = " " " 
 " " elevenths = " " " 
 " twelfths = " " " 
 
 121. 
 
 1 unit equals how 
 
 many eighths? 
 
 ^ U (( 
 
 twelfths? |' V 
 
 .') units equal 
 
 thirds? ! 
 
 ^ a a 
 
 fourths? X 
 
 - 
 
 halves? i 
 
 1J " 
 
 halves? | 
 
 1 '> 
 
 fourths? ; 
 
 1 :! " 
 
 thirds? 4 
 
 I! 
 
 sixths? ^ 
 
 1 ; | " 
 
 eiuliths? ] 
 
 ;;i u c< 
 
 fourths'/ \ 
 
 > : cc tc 
 
 JO / / f i 
 
 sixteenths? '/ 
 
 1.1 
 
 ? 
 
 *=? 
 
 r ; 
 
 1 B 
 
 122. What is the value of the following fractional ex- 
 pressions ? 
 
 if of of *=? 
 
 .1 of 
 
 I of J=? 
 .'> of J=? 
 i of i=? 
 ft of i=? 
 
 ff= 
 
 !i of J=? 
 
 
 123. What is the reciprocal of 1, of 2, of 3, of i, of :] 
 of 1^? 
 
 What is the value of f, of \ 2 , of *> of V of V ? 
 Analyse the fraction J. 
 
 Analysis: \ is a proper fraction, since the numerator is less 
 than the denominator; 4 is the denominator, and shows that 
 the unit is divided into 4 equal parts ; | is the fractional un.t, 
 since it is ONE of the four equal parts into which the unit is di- 
 vided ; H is the numerator and shows that three of these equal 
 parts are taken ; 3 and 4 are the terms of the fraction, and its 
 value is less than 1, or unity. 
 
Reduction of Fractions. 
 
 In like manner analyse the following fractions : 
 
 | 5 | j J 12^ ^5 ? | ? 4^ |4 
 
 124. Mentally add the following fractions : 
 
 f + * + 
 
 *+=? 
 f+l=? 
 
 i+t=? 
 
 125. Answer by mental work the following numerical 
 questions : 
 
 n 11 9 
 
 ,*=? 
 
 126. ORAL EXERCISES. 
 
 1. What will 2} yards cost at $J per yard ? 
 
 Ans. SI ,x. 
 
 Analytic solution. Since 1 yard cost SJ, J a yard will cost J as 
 much, which is $; ; and ] yards (2j yards reduced to halves) 
 will cost 5 times as much, which is $ 1 S " or $U- 
 
 2. AYluit will 2} pounds cost at 7-]/ per pound ? 
 
 Analytic solution. Since 1 pound cost - 1 ./' cents, J of a pound 
 will' cost } part which is y cents ; and Y will cost 11 tiiu^s ' s "' 
 cents, which is i|j- 5 - cents or 20 J- cents. 
 
 In like manner solve the following problems. 
 
 3. At $3 a yard what will 5J yards cost? 
 
 Ans. t2& 
 
 4. A dozen is worth $3o, what are 4^ dozen worth? 
 
 Ans. $15i-. 
 
 5. What is the value of 61 dozen apples at 1 per 
 dozen ? Ans. $'2\\. 
 
 (i. What cost 82 gross at $2J per gross? 
 
 Aiis. S2.-J 
 
 . 
 
 1. If ;| of a pound cost 12 cents, what is 1 pound 
 worth ? Ans. 16 cents. 
 
S4 Arithmetical Exercises and Examples. 
 
 Analytic solution. Since J of a pound cost 12 cents, J will 
 cost \ of 12 cents, which is 4 cents, and j or 1 pound will cost 
 4 times 4 cents, which is 16 cents. 
 
 2. f of a dozen cost $10, \Uiat is the value of 2j 
 dozen? Ans. $371. 
 
 Analytic solution. Since J of a dozen cost $10, J of a dozen 
 will cost J as much, which is $5, and jj or a whole dozen will 
 costs times as much, which is $15; and since 1 dozen cost 
 $1:"), j dozen will cost as much, which is $7J and } dozen will 
 cost 5 times as much which is $37J. 
 
 In like manner solve the following problems : 
 
 3. If | of a yard cost $2], what will 1 yard cost ? 
 
 Ans. $0. 
 
 4. If | of a yard cost $2, what will \ of a yard c 
 
 Ans. 8} 
 
 5. 4 of a number is 15, what is the number? 
 
 Ans. 20. 
 
 6. If s of a number is 8, what is 13 times the number ? 
 
 Ans. LM. 
 
 7. If % of a dozen cost $8, what will of a dozen cost 
 at the same rate ? Ans. $9. 
 
 8. What part of 4 is 3 ? Ans. J. 
 
 Analytic solution. Here by the terms of the question we have 
 3 to divide or measure by 4, and by the exercise of our reason 
 we proceed thus : since 3 is equal to 1, 3 times, it is equal to 4 
 J of 3 times, which is J. Or thus. Since 1 is J of 4, 3 is 3 
 times J, which is J. 
 
 9. What part of 5 is 5 ? Ans. T ^. 
 
 Analytic solution. Since $ is equal to 1, ij of a time, it is equal 
 to 5 the i part of f of a time, which is f$. 
 
 10. What part of | is 7 ? Ans. 8}. 
 
 Analytic solution. Since 7 is equal to one 7 times, it is equal 
 to 1, 5 times 7 which is 35, and to 4 instead af i to \ part of 35, 
 which is 8J. 
 
 What part of f is f ? Ans. 1 1 f 
 
 Analytic solution. Since | is equal to 1, $ of a time, it is 
 equal to J 8 times f which is 4 g , and to f instead of J, to J part 
 of \, which is | J or IJf. 
 
Reduction of Fractions. 85 
 
 12. What part of $ is 4? Ans. ff 
 
 13. What part of 3 J is 2 H Ans. T 9 . 
 
 14. What part of 5 is J of 2 ? Ans. T V 
 
 15. What part of 4 is f of f ? Ans. ^%. 
 
 16. 9 is i of what number? Ans. 72. 
 
 Analytic solution. Since 9 is ^ of a number, | or the whole 
 number is 8 times 9 or 72. 
 
 17. 13 is \ of what number? Ans. 91. 
 
 18. 21 T \ is | of what number ? Ans. 106J. 
 
 19. ^ is -}- of what number? Ans. f. 
 
 20. 24 is of how many times 3 ? Ans. 10. 
 Analytic solution. Since 24 is i of the number, ^ is J part of 
 
 24 which is 6, and |- or the whole number is 5 times 6, which is 
 30 ; and as 30 is equal to 3, 10 times, therefore '?4 is $ of 10 
 times 3. 
 
 21. 32 is \ of how many times 8 ? Ans. 7. 
 
 22. 28 is T 7 ^ of how many times 12 ? Ans. 5. 
 
 23. \ of 48 is % of what number ? Ans. 54. 
 
 Analytic solution. Since 48 is the whole of a number, J of 
 the number is \ part of 48, which is 1 2, and f is 3 times 12, which 
 is 30 ; and since 36 is jj of an unknown number, J of it is J of 
 .'56, which is 18, and | or the whole number is 3 times 18, which 
 
 is 54. 
 
 24. | of 63 is T 4 r of what number? Ans. 154. 
 
 25. 8 of % of 64 is^of what number? Ans. 104. 
 
 26. i of ^ of 42 is J- of what number ? Ans. 7. 
 
 27. J of 32 is f of 4 times what number? Ans. 9. 
 
 Analytic solution. Since 32 is the whole of a number, \ of the 
 number is \ part of 32, which is 8, and J is 3 times 8, which is 
 24 ; and since 24 is of 4 times an unknown number, j of 4 
 times the number is J of 24, which is 12, and f or the whole of 
 I limes the number is 3 times 12, which is 36 ; and since 36 is 
 4 times the number, \ of 3, which is 9, is the required number. 
 
 2S. I uf 40 is f of 7 times what number? Ans. 6. 
 25). i df .")() is J ot 6 times what number? Ans. 12. 
 30. f| of { of t>6 is 3| of 3 times what number? 
 
 Aus. 3. 
 
86 Arithmetical Exercises and Examples. 
 
 31. What is the i and \ of a J, of of 15 ? 
 
 Ans. 5. 
 
 127. GREATEST COMMON DIVISOR. 
 
 For a definition of a divisor, a common divisor, and the 
 greatest common divisor, see page 77. 
 
 1. What is the greatest common divisor of 42, 56 and 
 210? 
 
 OPERATION. Rjcp!(in<itwn.ln all problems of 
 
 42. 56. 210 tn ' 8 kind we first divide by any 
 
 ' factor that will divide all the num- 
 
 91 9Q 1 n^ bers ; then we divide in like man- 
 
 6l. o. 1UD ner tQe successive quotients thus 
 
 obtained, until we obtain quotient? 
 3 4 15 that have no common factor; then 
 
 we multiply all the divisors to- 
 
 j * i ^ gether and in the product we have 
 
 the greatest common divisor. 
 When there is no number greater than 1 that will divide all 
 the numbers without a remainder, then 1 is the greatest com- 
 mon divisor. 
 
 When there are two large numbers the operation may be 
 more easily performed by first dividing the larger number by 
 the smaller, and if there is a remainder divide the preceding 
 divisor by it,. .and thus continue until there is no remainder. 
 When there are more than two numbers, proceed as with two, 
 and then with the greatest common divisor of the two and one 
 of the other numbers, and thus continue until through all the 
 numbers. The last divisor will be the greatest common divisor. 
 
 
Greatest Common Divisor. 
 
 87 
 
 2. What is the greatest 
 common divisor of 88 and 24 ? 
 Ans. 8. 
 
 OPERATION. 
 
 24)88(3 
 72 
 
 16)24(1 
 16 
 
 8)16(2 
 16 
 
 3. What is the greatest 
 common divisor of 195, 285, 
 and 315? Ans. 15. 
 
 OPERATION. 
 
 285)315(1 
 285 
 
 30)285(9 
 '270 
 
 15)30(2 
 30 
 
 15)195(13 
 15 
 
 45 
 45 
 
 What is the greatest common divisor of the following 
 numbers ? 
 
 4. Of 441 and 567 ? Ans. 63. 
 
 5. Of 90, 315 and 810? Ans. 45. 
 
 6. Of 654, 216, and 108? Ans. 6. 
 
 7. Walker has 25 and Caruthers 45 dimes, how shall 
 they arrange them in packages, so that each shall have the 
 same number in each package ? Ans. 5 in each package. 
 
 8. A planter has 697 bushels of corn and 204 bushels 
 of rough rice, which he wishes to put in the least number 
 of bins containing the same jnumber of bushels, without 
 mixing the two kinds. Kow many bushels must each bin 
 hold? Ans. 17 uushels. 
 
 9. A Commission Merchant has 2490 bushels of wheat, 
 1886 bushels of corn and 8438 bushels of oats, which he 
 wishes to ship in the least number <f sacks of equal size 
 that will exactly hold either kind of grain. How many 
 sacks will he require ? Ans. 6407. 
 
Arithmetical Exercises and Examples. 
 
 128. LEAST COMMON MULTIPLE. 
 
 For a definition of a Mufti l<\ <i Cnmmun Miilriplf., and 
 a Lettsf CotniiHtn Mn/fi^ir. see pajje 76. 
 
 1. What is the least ennimon multiple <: ."). 6, ^. 21. 2* 
 
 "I'KKATION. fi^flllfltfon. In all prob- 
 
 - 5. f>. S. 21 l emg O f t hj s kj n< i we fi r!;t 
 
 arrange the numbers on a 
 
 '2) 5 3 4 -1 14 horizontal line, and then di- 
 
 ___ vide by the */nnl/<'itt prime 
 
 ., , >; %> _ number that will divide two 
 
 1 f> - -' * or more without a remainder 
 
 and write the quotient and 
 
 7 i ."> 1 '2 7 7 undivided numbers in a line 
 
 ___ below ; this process of divid- 
 
 - j .;, i ing we continue until there 
 
 are no two numbers that can 
 be divided by the same num- 
 
 2X2X3X7X^X2=840 ADS. her without a remainder; 
 
 then we multiply the divi- 
 
 sors and the numbers in the last line together, and the product 
 is the least common multiple. 
 
 When there is any number that will divide any of the others 
 without a remainder it may be cancelled before commencing to 
 divide. 
 
 2. What is the least common multiple of 4, 9, 12, 15, 
 and 24 ? Ans. -JGO. 
 
 What is the least common multiple of the following: 
 
 3. Of 8, 4. Hand HO? Aus. 360. 
 
 4. Of 50. 27, 3, 45 and 63 ? Ans. 9450. 
 :>. Of 21, 36, 11 and 22? Ans. 2772. 
 (>. Of 800, 600, 10, 40 and 12? Ans. 2400. 
 
 7. Of 8, 18, 20 and 70 ? Ans- 2520. 
 
 8. A drayman has 2 drays and 2 floats ; on 1 dray he- 
 can haul 9 barrels of flour, and on the oth T 12 barrels: 
 on 1 float he can haul 18 barrels, and on the other 21 bar- 
 rels ; what is the least number of barrels that will make 
 full loads for either of th ; drays or floats. 
 
 Ans. 252. 
 
Reduction of Fractions. 89 
 
 9. A fruit dealer desires to invest an equal amount of 
 money in oranges, peaches and grapes, and to expend as 
 small a sum as possible ; the price of oranges is $2.40 per 
 box ; peaches $1.60, and grapes for a medium article, 90 
 ^., and for first quality, $1.20 ; of these two qualities the 
 fruit dealer took the cheaper. How much more money did 
 he invest than he would had he taken the grapes at $1.20 
 per box ? Ans. $28.80. 
 
 WRITTEN EXERCISES. 
 REDUCTION OF FRACTIONS. 
 129. To rc(ln<c fraction* to tlidr lowest terms. 
 Reduce |^ to its lowest terms. 
 
 FIRST OPERATION. Explanation. In all problems 
 
 2)|| = !~| ; 4)|-| = I Ans. of this kind we divide both the 
 
 numeratorand denominatorsuc- 
 
 SECOND OPEATION. cessively by each of the com- 
 
 Q\56 7 A mon ' ac tors they contain. Or 
 
 '64 ^~ i" Ans - as shown in the second opera- 
 
 tion we may produce the same result with less figures, by 
 dividing both terms of the fraction by their greatest common 
 divisor. 
 
 By this reduction we change the form of the fraction |-J, 
 but we do not alter or change its value, lor the fractional 
 unit of the resulting fraction (i) is 8 times as great while 
 the number taken is ^ as great. 
 
 When the t rms of the fraction have no common factor 
 greater than 1, the fraction is in its lowest terms and is 
 called an irr/nci/>/c fraction. 
 
 The object of reducing fractions to their lowest terms is to 
 enable us to more easily and readily understand their value. 
 
 Reduce the following fractions to their lowest terms : 
 - if- ii-ff-ff Ans. J,f, J.ffr. 
 
 a - H*. Hi itt- An - A, t, 
 
90 Arithmetical Exercises and Examples. 
 
 
 
 130. To reduce whole or inir,<l nnml>*>r$ to improper 
 fractions. 
 
 1. Reduce 5ii to an improper fraction or to tkml*. 
 
 OPERATION. Explanation. In all problems of this 
 
 5jj kind we reason thus: Since there are 
 
 3 thirds in every unit or whole num- 
 ber, in 5 units there are 5 times as 
 JT Ans. many, which is 15 -(- the $ make l s 7 -. 
 
 2. Reduce 9 to a fraction whose denominator is 6. 
 
 OPERATION. 
 
 3=z:- 5 /- Ans. 
 
 1U. Reduce the following numerical expressions to Im- 
 proper fractions. 
 
 3. 
 
 8J 
 
 Ans. 
 
 " 
 
 8 
 
 71} 
 
 Ans. 
 
 J-f 3 - 
 
 4. 
 
 161 
 
 Ans. 
 
 
 
 9. 
 
 68$ 
 
 Ans. 
 
 479 
 
 5. 
 
 17} 
 
 Ans. 
 
 
 
 10. 
 
 2183? 
 
 Ans. 
 
 8.J3_5 
 
 6. 
 
 32| 
 
 An-. J 
 
 -fi. 
 
 11. 
 
 ^i 
 
 Ans. 
 
 w- 
 
 7. 
 
 4354 
 
 Ans. & 
 
 
 12. 
 
 
 Ans. 1 
 
 
 13. 
 
 Reduce 
 
 14 to a fraction whose denominator is 9. 
 
 14. 
 
 u 
 
 37 " 
 
 u 
 
 u u 
 
 a 
 
 24. 
 
 15. 
 
 11 
 
 54} " 
 
 u 
 
 " u 
 
 u 
 
 10. 
 
 132. To redu e improper fractions to whole or mixed 
 numbers. 
 
 Reduce -^ to a mixed number. 
 
 OPERATION. Explanation. In all problems of this 
 
 17 ..^.i Ans kind we reason thus. Since there are 
 
 * 4 fourths in 1 unit or whole number, in 
 
 11 fourths there areas many units as 17 
 
 17-i-4 -4ff Ans. j s e q ua i to 4, which is 4 times with 1 
 
 remainder, or altogether 4} as the proper quotient or answer. 
 
Reduction of Fractions. 91 
 
 Reduce the following improper fractions to whole or 
 mixed numbers. 
 
 Ans. 4. 6. % 9 - Ans. 3f. 
 
 Ans. 5|. 7. f g Ans. 5 T V 
 
 Ans. 48. 8. ^ 8 T 3 - Ans. 9 
 Ans. 18}. 9. *WP- Ans. 
 
 133. To reduce Compound Fractions to Simph' Frac- 
 tions. 
 
 Reduce \ of f of J to a simple fraction. 
 
 OPERATION Explanation. In all problems of 
 
 i v 3 v _ 21 A < this kind we multiply together all 
 
 X-j-Xa -g-Q Ans. the nuniera tors for a new nume- 
 rator and all the denominator for a new denominator. 
 
 When a compound fraction contains whole or mixed numbers 
 they must first be reduced to improper fractions. 
 
 When there are common factors in both terms of a compound 
 fraction they should be cancelled before multiplying. By this 
 cancelling the common factors the work is shortened, and the 
 result unchanged for the reason that dividing both terms, of a 
 fraction by the same number does not alter its value. 
 
 2. Reduce of | of f to a simple fraction. 
 
 OPERATION. 
 
 $ ? S 5 5 
 -X-X-= Ans. 
 4 8 16 
 2 
 
 3. Reduce | of 71 of | of 4 of ^ to a simple frac- 
 tion. 
 
 OPERATION. 
 
 -X X-X-X =- Ans. 
 
 9 p . i . # a 
 
 9 3 
 
 4. Reduc'j the following compound iractions to simple 
 ones. 
 
Arithmetical Exercises and Examples. 
 
 5. of j| of f^. Ans. 
 
 6. i of of T ^-. Ans. 
 
 7. | of 3J of J. Ans. 1 
 
 8. i of Ans. 2^, 
 i. & *' '"'' Ans. 54 
 
 in. ii of ff of 17 J. Ans. 6. 
 
 11. iXiXfV- Ans. 
 
 12. 5X^XA- Ans. 
 
 KM. To red nee fraction* of different denominator* to 
 
 equivalent fraction* of a common denominator or of ?/!< 
 
 least cummnn denominator. 
 
 1 >"> Definitions and Principles, pertaining to this 
 
 kind of reduction of fra -lions. 
 
 KM. A Common Denominator is a denominator 
 
 common to two or more fractions. 
 
 l >T. The Least Common Denominator of two 
 
 or more fractions is the least denominator to which all the 
 fractions can be reduced. 
 
 i:K A Common Denominator of two or more 
 fractions is a common multiple of their denominators-, and 
 the Least Counmm /)< nominator of two or more fractions 
 is the least common multiple of their denominators, for the 
 reason that all higher terms of a fraction are multiples of 
 its corresponding lower or lowest terms. 
 
 1. Reduce J, J, and -J, to equivalent fractions having 
 a common denominator. 
 
 OPERATION. Explanation.-^^} 
 
 35 it -$1 problems of this kind 
 
 3X4X8= 96 common denominator we obtain the common 
 
 $ of ) =32 hence fjL equivalent of J denominator by mul- 
 
 J of -91 -_-^72 hence , equivalent off- "plying together the 
 
 ,, i i . * denominators of all 
 
 -I of ) =84 hence |f, equivalent of $ lhe fractions Then 
 
 to find the respective numerators we take such a part of the 
 
Reduction of Fractions. 93 
 
 common denominator as the respective fractic ns are parts of a 
 unit, as shown in the operation. 
 
 Reduce the following fractions to equivalent fractions, 
 having a common denominator. 
 
 2. "f |, and f Ans. r <&, tffr, and 
 
 3. A *, and tV An s. Ml, *, and 
 
 4. ,,, and ff Ans. ||f , and ff 
 
 5. J, A, A. H. and f 
 
 Ans. iHH. ?W&> MM!, ftti* and 
 
 6. T^, i, 5, and 34. 
 
 Ans. T %<V, T VA, fWV and 
 
 7. Reduce i, :1 and f to equivalent fractions having the 
 least common denominator. 
 
 OPERATION. 
 
 2 :-!. 4. 8 
 
 L ; 8, 2. 4 
 
 3 1 2 
 
 X2=-4 Least Common Denominator. 
 i of ^ =8 hence ^ is the equivalent of J. 
 J of > 24=18 hence |~f is the equivalent ol ,. 
 A of J -~21 hi'iKT r! j is the equivalent of i. 
 Explanation. In all problems of this kind we first find the 
 least Common Mn/t>ji/c of the denominators of all the fractions 
 as explained in article 128 page 88 which is the h.ast com- 
 mon denominator. Then, having the least common denomin- 
 ator to find the respective numerators we take such a part of 
 tht' least common denominator as the respective fractions are 
 parts of a unit, as shown in the operation. 
 
 Reduce the following fractions to equivalent fractions 
 
 having a least common denominator. 
 8. *, |and f 
 D. ^,. 1 and /',.. 
 
 10. ^. Uand 
 
 11. A-i;^ H"i* 
 
 12. :, . 
 
 18. Jf 
 
94 Arithmetical Exercises and Examples. 
 
 14. a.], :(, Ji and ;>. Ans. ff, ^ -^ and if 
 
 15. T 8 r r , 3, \ and ||. 
 
 A'ns. If, 5328, yWff and 1665. 
 
 DENOMINATE FRACTIONS. 
 
 I3t. A Denominate Fraction is one whose unit is 
 denominate. Thus J of a yard is a denominate fraction. 
 
 140. To reduce " </< u<>niin<it<> fr<i<-t>nn from a greater 
 unit tn n A.s-y. 
 
 1. Reduce j { a yard to inches. 
 
 OPERATION. I-.r^annUnn.lu all problems of 
 
 this kind we multiply the fraction 
 3 by the units of the scale to which 
 
 ; ;; it belongs until we reach the unit 
 
 ^ required. 
 
 In this example we multiply the 
 
 2< inches. Ans. ; ;. v;ir ,j hy > to re( i u ce it to feet, 
 then by 12 to reduce it to inches, the unit required. 
 
 2. In I of a ton hw many ounces? Ans. 2SIMK). 
 
 141. To reduce (t denominate fraction from a /r.sx unit 
 to a yrrutt r. 
 
 1. Reduce I of a pound to a fraction of a ton. 
 
 OPERATION. Kxpl<in,ition.l* all problems of 
 
 p this kind we divide the fraction by 
 
 25 
 
 _> 4 
 
 the units of the scale to which it be- 
 lon^s until we reach the unit requir- 
 ed. Tn this example we divide the f 
 pound by 25 to reduce it to quarters ; 
 then by 4 to reduce it to hundred- 
 
 TWfr ton ^ ns - weights ; and then by 20 to reduce it 
 to tons, the unit required. 
 
 2. Reduce of a penny to a fraction of a pound ster- 
 ling, Ans. jrfa. 
 
 ADDITION OF FRACTIONS. 
 
 142. Addition of fractions is the process of adding two 
 or more fractional numbers of the same kind, or of the 
 same denomination. 
 
Addition of Fractions. 
 
 95 
 
 (1.) Add Jf.f |f and J together. 
 OPERATION. 
 
 ;2 3 4- & 7 
 
 p p p 
 
 Whole numbers 
 1 
 1 
 1 
 1 
 
 Explanation. In 
 this example there 
 are no two frac- 
 tions alike, hence 
 they cannot be ad- 
 ded until we shall 
 
 Fractions ob- have reduced them to 
 tained in a ddino- fractions of the same kind. 
 To facilitate and simplify 
 g the operation, we here re- 
 
 ji fi p p JLJ duce and add but two 
 
 fractions at a time, and 
 
 we first select such two as 
 may be the most easily 
 reduced and added. Ac- 
 cordingly, by inspection, 
 we select the J and } as the fractions to first reduce and add; 
 and by the exercise of our reason we see that \ is equal to f 
 which, added to the J, make J, which, for the reason that | 
 make 1 is equal to 1 and J. We set the 1 in the column of 
 whole numbers, and the J in the column of fractions. We then 
 cancel the and J, and select the 3 and J as the next two frac- 
 tions to reduce and add. Again, we see, reasoning analytically, 
 that J is equal to and f are equal to J, which, added to the f, 
 make |, which is equal to 1 and , which, reduced, equals J. 
 The 1 we set in the column of whole numbers, and the J in the 
 column of fractions, and cancel the 3 and ;]. We next add the 
 \ and , and by our reason we see that J is equal to |, which, 
 added to the , make J, which we set in the column of fractions, 
 and then cancel the J arid . We then select the J and | as the 
 next two fractions to add, and reducing tha J to 8ths, we see 
 by our reason that is equal to f and } are equal to 3 times as 
 many, which is f, which added to the | make ^ 3 which is equal 
 to 1 and ; we set the 1 in the column of whole numbers and 
 the f in the column of fractions, and cancel the J and | We 
 then proceed to reduce and add the t\vo remaining fractions $ 
 and . I>y inspection, the exercise of our reasoning faculties, 
 and the use ot our knowledge of the principles of numbers as 
 contained in the preceding \vork, we see that the i and f are 
 not only not alike, but that we can neither reduce the 1 to 8ths nor 
 the to 5ths, and, therefore, before we can add them we must 
 reduce both the and to equivalent fractions of the same 
 kind, or whose denominators are alike. To do this, we first 
 observe that the denominators are not divisible by the same 
 number, greater than 1, and hence the product of ; -'^rn. 4fl 
 
96 Arithmetical Exercises and Examples. 
 
 is the least number that both of the fractions are reducible to, 
 or, in other words their product 40 is the least common deno- 
 minator of the two fractions. Having this, we next reduce the 
 $ and | to 40ths, and by our reason we see that -J- is equal to 
 4 8 and | are equal to 4 times as many, which is { ; then that 
 | is equal to fa and are equal to "> times as many, which is 
 
 !f r w ich added to the Jj- make J, which for the reason that 
 $ make a whole one, is equal to I and .},\. which we place in 
 their respective columns and cancel tin- i and J. The opera- 
 tion c 1 ' adding the fractions is now completed, and by adding 
 the whole numbers and annexing the remaining fractions, we 
 have as the correct result 
 
 Tht- foregoing problem illustrates the most rational, easy and 
 rapid system of adding fractions known, and as fractions are 
 so indispensable and of so frequent occurrence in practical life, 
 the principles involved in the system should be thoroughly 
 understood. 
 
 In practical work, we would very much shorten the opera- 
 tion by adding several fractions at once, and mentally perform- 
 ing the most, if not all of the reduction and addition work, 
 without stating the results. Thus, in the above problem, we 
 would add the J, J and J at once. We can instantly sec- that 
 their sum is y or 2J, and without naming or setting the 2J, we 
 add to it mentally the result of 3 and j, whi'.-h we mentally see 
 is | or 1J, making 3f, which are the only figures we set. Thus 
 all tl.e fractions, except and , are added at one mental 
 operation. Then we mentally add the sum of -J and f by the 
 same process of reasoning as given in the illustration of the 
 above example, and obtain the correct result 4JJ. 
 
 (2.) Add |, % and ^ together. 
 
 OPERATION. 
 
 Fractions obtained 
 by adding. 
 
 Wholn 
 numbers. 
 
 4 
 
 27 
 
 9 ;; 
 
 28 288 
 
 209 
 
 Explanation. By in- 
 spection and reason we 
 see that there are no 
 two fractions alike, and 
 that we cannot reduce 
 either of them to an 
 equivalent denomina- 
 tion of any other; there- 
 fore we select the small 
 est two, f and |, and 
 reduce them to equiva- 
 lent fractions of the 
 same kind, or of the 
 
 same denominator, which wo find, by multiplying the denom- 
 inators together, to be :*;. 
 
 497 
 ijf 
 
 Ans. 
 
Addition of Fractions. 
 
 97 
 
 We now find by reasoning as in the preceding example, that 
 J and J are equal respectively to f J, and ff , and collectively to 
 f|, which is equal to 1 and if. Then proceeding as in the 
 first case, we add the T 8 T and j|. 
 
 (3.) Add J of f of 2i, 1 f and ^ together. 
 
 OPERATION. 
 
 Statement >h>wing the reduction 
 of the fractions. 
 
 3 
 1 ^ 3 5 9 
 
 -x-x- - 
 
 % $ 4 8 6 23 
 
 Statement showing the result of the. 
 reduction and tl\e iitMt'tioii of 
 fractions. 
 
 Whole I'nu-lioMs ol.tjiim-d 
 
 numbers, by ml <liiii;. 
 
 t 9 ; - /: > 
 
 20 L' 
 
 8 
 
 Explanation. Here we 
 have compound fractions 
 and mixed numbers, and 
 before adding, we reduce 
 the mixed numbers to im- 
 proper fractions, and the 
 compound fractions to 
 simple ones. Then we add 
 the J and f, which are 
 equal to 1 and J ; then 
 the | and J, which are 
 equal to || ; then the f J 
 and ^ 3 -, which are equal 
 to 1 and ^I'jj. Then adding 
 the whole numbers, and 
 annexing the fractions, we 
 have 2JJi| as the correct 
 result. 
 
 EXAMPLES. 
 
 (4.) Add }, }, f, f I, and ^. 
 
 (5.) Add ij and Jf 
 
 (6.) Add J, f, I, and ^. 
 
 (7.) Add f , |J, || and |f 
 
 (8.) Add f , ,% H and fj- 
 
 ~(9~) f of iJ and 21 of 4 of ^ 
 
 (10.) Add 2J, 6|, 5| and 21. 
 
 Ans. 
 Ans. 2 
 Ans. 2 
 Ans. Sj 
 Ans. 
 
 jVns. f| 
 Ans. 17 
 
s. Arithmetical Exercises and Examples. 
 
 (1 i and .jV. 
 
 nd f. 
 
 ,/.(;, IS^and 2^. 
 
 i .-). 
 
 and in;. 
 
 and -VJ. 
 
 i :>." 
 ,327^ and25i, 
 
 lulu of in sacks 
 
 wreigl) ;:.[ . 1 i 1571, L52J, 1 Uf, 
 
 and Kil , ; pounds? Ans, L53SH ll.s. 
 
 Of .', and 2j Of -\ Of 1. Ans. H. 
 Add 3- of ^ of 'l and n nf A of . Ans. 2||. 
 IIo\v many vards in s l,lts of dmncstic. nieasur- 
 tfi follows: 101,39), i: ii!' , ! ;. and 
 
 yards? Ans. 328|f. 
 
 I'D. 14 feags of Coffee weigh as follows: Kii^ 7 ,,. i 
 
 . 164J, 1654, 164|, 165J, L62J, \M&. 164f, L65|, 
 Hi.") i. 1 (i 1 } /. and 1 ('.") -I pounds: how many pounds in all? 
 
 Ans. ^:-iOl. 
 
 V merchant nought ll."):)', pounds of rice for 
 8i)-l; STl'J pounds of su-ar for s^7^: ."isoji pounds ol' 
 roifee for -Sii");; 'Ji-[(}\ pounds of cheese for ST5-]; and 
 ]ounds : f uraham flour for 8 IS:. What was the 
 total number of pounds, and the total cost of all he pur- 
 chased? Ans. 3254 f pounds, $357-$- cost, 
 ,,,and :-]'J of ^ of 1J. 
 
 Ans. 
 . Add ; v, 14- and } of 3. Ans. 
 
Subtraction of Fractions. 99 
 
 SUBTRACTION OF FRACTIONS. 
 
 114. Subtraction of fractions is the operation in num- 
 bers of finding the difference between two fractional num- 
 bers that are of the same denomination ; it is hence the 
 converse of addition. 
 
 (1.) What is the difference between J and f? Ans. ^ 
 OPERATION Explanation Here we see 
 
 f = 15 
 
 Ans. 
 
 or f 4 that the denominations are 
 
 ^5 IQ not the same, and there- 
 
 fore, before we can sub- 
 tract, we must reduce the 
 -0- Ans fractions to a ccmmon de- 
 
 nominator. By inspection 
 
 and in accordance with the principles as explained in the first 
 problem of addition, we see that the least common denominator 
 is 20 ; then, that J are equal to ^ and that | are equal to if, 
 and that the difference is 2 ^. 
 
 (2.) From 28 1- take 71. Ans. 21 1. 
 
 OPERATION. Explanation In perfonn- 
 
 28$ ing: the operation of the 
 
 71 question before us, we first 
 
 observe that the fractions 
 
 which constitute a part of 
 
 21i Ans. the numbers to be sub- 
 
 stractv-d are not of the 
 
 Bftme kind or denomination, and hence, before we can peifrrm 
 the work, we must reduce them to equivalent fractions. \Ve 
 next observe that the J may be reduced to Hths, and by the exer- 
 cise of our reason we see that it is equal to |, which taken from 
 -| leaves J; this completes the work with the fractions, and we 
 have but to find the difference between the whole numbers as 
 in simple subtraction. 
 
 (3.) What is the difference between 371 and 12/j- ? 
 
 Ans. 24* f, 
 
 OPERATION. Emanation. By inspec" 
 
 72 ) QQ tion we here see that the 
 
 37| = 27 j fractions belonging to the 
 
 J9JL- 56 whole numbers are not 
 
 ^ of the same denomina 
 
 and that neither can be 
 
 ^- n3 ' reduced to an equivalent 
 
 fraction of the same term 
 
100 Arithmetical Exercises and Examples. 
 
 as the other, and therefore we must reduce both to equivalent 
 fractions having a common denominator, before we can sub- 
 tract ; and by the exercise of our reason we see that 72 is the 
 smallest number to which both can be reduced, which, for con- 
 venience, we set below the fractions, and by the same reason- 
 ing as given in the preceding examples we see that J are equal 
 
 and that | are equal to |J, which, for convenience, we 
 carry to the right of the respective tractions, and to economise 
 time we set only the numerators. We now observe that the 
 upper traction, belonging to the greater number, is less than the 
 lower fraction, In-longing to the lesser number. Therefore, 
 before \ve can subtract the fractions, we must add 1, reduced 
 to 72ds, to 2 7 ,. which We now take C' from i! 1 ;! and 
 
 have a remainder iie fractional part of our answer. We 
 
 now add 1 to the subtrahend, because we previously added 1 to 
 thi- minuend, making it !.', which we -ubtract from 37 and 
 
 t remainder ot 'J I. wi. 1 com- 
 
 plete the operation. 
 
 4. What is tin- di n ] \ and i! ? An- 
 
 .">. What i> tin 1 dit \> 
 
 (I. What is tlu- difiL-ivinv U-twren .")- and !U ? An 
 
 7. What is th<- diffrivn.v bi-t \\von 7 and o^.- ? Ans. 
 
 8. What is the diffi n --y and 14? An- 
 What is the diflxTriii-i- li-t\v. en tin 1 following niun' 
 !). : and ;*. Ans. 
 
 10. 1-j and f Ans. 
 
 11. 'j and , 
 1'2. ^ and 
 
 13. [ian.i 
 
 14. ; , 4 , r and , 
 
 15. i^|and ! 
 Hi. Hi and ^. 
 
 17. A of |- and i of 
 and ;17-. 
 
 J. .T) | and U^. 
 
 lid 3-vyl-. 
 
 iV 
 
 75 4 and 4.' Ans 
 
Subtraction of Fractions. 101 
 
 24. 31J- and 17|. Ans. 13|. 
 
 25. From (>l of T 4 +13 take J- of } of 15^-f'of . 
 
 Ans. 13f|. 
 
 2(5. From 8J+6f ^ take J of | of 3 of lf+2 1 ,. 
 
 Ans. 
 
 27. From t7}+f take 6} f. Ans. 12^. 
 
 88. <i. K. Shotwell had $38}; he gave S2i- for a pair 
 of Indian clubs. S.Vj for books, $1} for a drawing board. 
 and $| for ink and pencils. How much had he left ? 
 
 Ans. SJ- 
 
 20. \V. A. Weaver had 67 .1 and his friend gave him .] 
 more ; A. Denis had $16o and he spent $5i ; How much 
 more has A. Denis than W. A. Weaver? Ans. SL^. 
 
 '.](). E. Schwartz bought 2 bags of coffee each weighing 
 lO.'J-l pounds; he sold 27 I pounds, 50f pounds, 871 1'Ound.s 
 and 45 \ pounds; how* many pounds has he left? 
 
 Ans. L16| 
 
 31. S. Myers bought 7.V. gallons of molasses; he used 
 4]. lost by lak.iu i 2f, and so,d 22 , : gallons. I low miu-li 
 has he left? Ans. 4(1' gallons. 
 
 32. What is tlu ditli-n-nce lu-tween a dozen times >. plus 
 (J.j, and (J tiiiHis a dozen minus one dozen and a halt".' 
 
 Ans. 2 i 
 
 J. Famll bi 'light (\ chests of tea weighing 3s. 
 1 r,.:i!)', and 43 i pounds; he sold 120^ pounds ; , n d 
 used 64 pounds. How many pounds has he on hand ? 
 
 Ans. IL' 
 
 34. JY Burba owned the Steamer Lsubel. he sold ;^ ; what 
 is -j of his present interest? Ans. -f^. 
 
 3r>. From the .-urn oi i) 1 and S^ take the difference of 14-2^ 
 and Of Ans. 10^. 
 
 36. What number is that to wliich if \(\\ be added the 
 sum will be. 44^ ? Ans. 27:. 
 
102 Arithmetical Exercises and Examples. 
 
 37. C. E. McNeil bought of of a vessel and sold f 
 of 3 of his share. . How much of the whole vessel has he 
 left? Ans. i. 
 
 38. E. Brinkman bought a barrel of molasses containing 
 H I gallons ; lie sold !) \ gallons ; how many gallons remain 
 in the barrel? Ans. 31 J gallons. 
 
 'V.>. C. GT. De Russy bought two sacks of coffee weighing 
 respectively 1(11-1 and lii:>:, : pounds. He sold to J. Walter 
 ISO. I pounds; how many has he left? 
 
 Ans. 138J pounds. 
 
 40. A. Buchanan sold to J. Bruns J of of his planta- 
 tion, what part, has he left? Ans. ^. 
 
 41. What is the difference between > of J plus 5 and % 
 of J plus J ? Ans. T y 
 
 4 .]. -I. Srhonekas and M. Shlenker were each o owner 
 of a broom and brush factory. Schonekas sold J of his 
 interest to E. F. Meyer, and then ] of his remaining inter- 
 est to M. Shlenker who subsequently sold % of f of his 
 whole interest to F. Kranz. What is the present interest 
 of each owner ? Ans. J. J. Slionekas i ; E. F. Meyer } ; 
 M. Shlenker f| and F. Kranz f 
 
 43. J. J. Manson owned J of the Steamer Natchez. He 
 sold to G. Lindsey ] interest in the Steamer, and to W. S. 
 Keaghoy .{ of his remaining interest. What is the present 
 interest of each in the boat? Ans. Manson f; Lindsey i 
 
 and Keaghey J. 
 
Multiplication of Fr actions. 103 
 
 MULTIPLICATION OF FRACTIONS. 
 
 115. Multiplication of fractions is the.proqess of multi- 
 plying when one or both of the factors contain fractional 
 numbers. 
 
 In the multiplication of simple numbers we saw that the 
 result of multiplication operations was increasing, but in the 
 multiplication of fractions, when the multiplier is less than 
 a unit, the result is decreasing. This is evident from the 
 fact that multiplication is the process of repeating the mul- 
 tiplicand as many times as there are units in the multiplier, 
 and therefore, when the multiplier is less than a unit, the 
 multiplicand will be repeated only a } art of a time, or such 
 a part of itself as the multiplier is p:\rt of a unit. 
 
 To elucidate the principles of the subject and render 
 clear the reasoning we present our first questions in prac- 
 tical language ; and to aid still farther in comprehending 
 the work, we give the following practical definition of mul- 
 tiplication. 
 
 116. MULTIPLICATION is that operation in the prac- 
 tical business computation of numbers of finding the cost of 
 either a part of one, or of many pounds, yards, barrels, etc., 
 when we have the cost of one pound, yard, barrel, etc. On 
 the principle or fact embraced in this definition, we found OUT 
 reasoning for the solution of every question tint can pos- 
 sibly be presented in multiplication, either of simple num- 
 bers or of fractions. 
 
 Considering the foregoing, we see that in all multiplica- 
 tion questions of a practical nature, we must necessarily 
 reason from one, or unity , to a part of one or many. Thus, 
 if 1 pound cost 50 /, \ of a pound will cost 1th part of 
 it ; and if 1 yard costs $2, 3 yards will cost three times as 
 much, or 3 times $2. 
 
 In the solution of abstract questions we apply the same 
 system of reasoning without naming the factors, and thereby 
 avoid all of the arbitrary rules given in the Arithmetics of 
 the day 
 
v KM- Arithmetical Exercises and Examples. 
 
 (1.) What will 41| pounds of coflee cost at 21 \f per 
 pound ? Ans. $8.97| 
 
 OPERATION. 
 
 j Explanation. In this example by 
 
 inspection and the exercise of our 
 
 
 reason, we see that we have in the 
 
 167 solution both increasing and decreas- 
 
 - ing work to perform, and hence to 
 
 facilitate the operation of our work, 
 we use a perpendicular or statement 
 line, on the right hand side of -which 
 Ans. we pi ac e all increasing numbers, and 
 on the left hand side all decreasing 
 numbers. But, be it remembered, we never place a number on 
 either side without giving a reason therefor, and in commencing 
 the solution statement of any problem we always place at the 
 top right hand side the mimber representing the article or thing 
 to be increased or decreased, or that which the conditions of 
 the question require the answer to be in. 
 
 By further inspection and reasoning we see that 21 \f are to 
 be increased 41} times, and hence we will place the same at the 
 top and on the increasingr side of our statement line ; but, before 
 doing so, in order to facilitate the work, we first reduce the 21J 
 to half cents, which equals 4 2 3 cents, the denominator of which 
 we place on the decreasing side and the numerator on the in- 
 creasing side of the statement line. We then reason as follows: 
 since 1 pound cost % 3 cents, J of a pound will cost i part of it, 
 and as this conclusion is a decreasing one, we write the 4 on 
 the decreasing side ; then, since J costs the result of the state- 
 ment thus far made, 1 J 7 , 41f reduced, will cost 167 times as 
 much, which, because the conclusion is an increasing one, we 
 write on the increasing side, and thus complete the reason and 
 statement. It may be asked how we know that if 1 pound costs 
 4j3^ J of a pound will cost J part of it. and that l J 7 will cost 
 lt>7 times as much. We answer, by the exercise of our reasoning 
 faculties, our common sense, our judgment, which is the only 
 way that mortal man knows anything. 
 
 In working out the statement, there being no common factors 
 in the increasing and decreasing numbers that can be cancelled, 
 we have but to multiply the increasing numbers together, which 
 produce 7181, and the decreasing numbers together, which pro- 
 duce 8 ; then we divide the 7181 by 8 and obtain $8.97f as the 
 result of the reasoning and operation. 
 
 In all simple statements the result is always of the same 
 kind or character as represented by the number first placed 
 on the statement lint 
 
Multiplication of Fractions. 105 
 
 Multiply by f , or to express the problem in practical 
 language : 
 
 (2.) What will I of a yard cost at -i of a dollar per 
 yard? Ans. $}. 
 
 OPERATION. Explanation. As explained in the 
 
 above example, by inspection and rea- 
 
 son we see that the i of a dollar is the 
 number to be multiplied, and also the 
 number representing the nature of the 
 
 answer; hence we first place the same 
 , on our statement line, and then reason 
 
 2 AnS. ag follows : if, or since, 1 yard costs | 
 
 of a dollar, J of a yard will cost $ 
 part of it, and f- wih cost 5 times as much as J. The 8 and 5 
 are placed respectively on the decreasing and increasing sides 
 of the statement line, because the reasoning, when, they were 
 respectively used, was decreasing and increasing. 
 
 In working out the statement, we first cancel the 5's, and then 
 the H by the 4, and thus obiain J a dollar as the correct result. 
 
 The reasoning and operation of the foregoing problems 
 vi 1 solve every question that can be presented in multipli- 
 ation of fractions. 
 
 (3.) What will 58} pounds cost at in*/ per pound? 
 
 OPERATION. I'.jjHiDKitinn. The reasoning for the 
 
 tf solution of this problem is the same 
 
 25 as that given in the first example: 
 
 "lit 39 hence we will very much abridge our 
 
 explanation. We first reduce and 
 \ place on the line the IGj-V ; then having 
 
 $r.75 Ans. the cost of 1 pound, we. see by our 
 reason that J a pound will cost J as 
 much, and H 7 pounds will cost 117 times as much as J. 
 
 This completes the statement, and in working the same we 
 first cancel the 50 by 2, then the 117 by the 3. this cancels all 
 of the decreasing figures, and \ve have but to multiply the 25 
 by 39 and produce the answer, 9.75. 
 
106 
 
 Arithmetical Exercises and Examples. 
 
 (4.) What will 3| dozen cost at $3| per dozen ? 
 
 OPERATION. 
 
 17 
 
 51 
 
 Ans. 
 
 F,.cplanation For reasons above giv- 
 en, we reduce and place the $3J on 
 tin- line; then we see that since 1 
 di-y.-n costs $y, J of a dozen will 
 ; part of it, and ^ will cost 15 
 times as much. 
 
 (5.) What will 5J bushels cost at 15J/ per pint? 
 
 OPK'IATION. l-'<f>l(in<ition. By inspection and 
 
 reason, we see that tin- 15^ is the 
 number to he increased; hence we 
 reduce and place the same on the line 
 and proceed to reason as follows: if 1 
 pint COStfi . _ pints or a quart will 
 cost 2 times as much, and if 1 quart 
 costs the resuft of the statement now 
 made, H quarts or a peck will cost 8 
 times as much, and if a peck costs 
 the result of this statement, that 4 
 pecks or a bushel will cost 4 times as much, and if a bushel 
 costs the result of this statement, that J of a bushel will cost 
 J part of it, and ^ will cost 43 times as much. 
 
 53.32 Ans. 
 
 (6.) What will 50} pounds of tea cost at 
 ounce? 
 
 per 
 
 OPERATION. 
 
 f 21 
 
 ! J(6 
 
 (I20J 
 
 2 Ans. 
 
 Explanation. Here we reduce 
 and place the 10J^ on the line, and 
 reason thus : since 1 ounce costs 
 ^^, 16 ounces or 1 pound will cost 
 16 times as much, and since 1 
 pound costs the result of this 
 statement, of a pound will cost 
 - part of it. and w will cost 201 
 
 times as much. This completes the reasoning and statement, 
 which worked gives $84.42, answer. 
 
Multiplication of Fractions. 107 
 
 (7.) What will 24 pounds cost at 9J/ per pound? 
 
 OPERATION. 
 
 X Explanation. Reducing and plac- 
 ing the 9J^ on the line, we reason 
 
 f iy thus: if 1 pound costs y% 24 
 ^ pounds will cost 24 times as ranch. 
 12 This completes the statement, 
 which worked gives $2.28, the 
 
 $2.28 An*. answer ' 
 
 (8.) What will 14? dozen oost at 5 per dozen? 
 
 OPERATION. 
 
 $ Explanation. Placing the $5 on 
 
 5 the line, we reason thus : if 1 
 
 Q A A dozen co*ts $5, of a dozen will 
 
 cost J of it, and </ will cost 44 
 times as much. This completes 
 the statement, which worked out 
 gives $73, the answer. 
 
 $73i Ans. 
 
 (10.) What will 6 dozen and 7 chickens cost at $4.87} 
 per dozen? Ans. $32.09 
 
 OPERATION. Explanation. Reducing and 
 
 6 placing the $4.8 7 on the line 
 
 2 .975 we reason thus: Since 1 dozen 
 
 12 
 
 79 
 
 cost 9750 ] chicken will cosf 
 the 12th part and 7U chickens 79 
 times as much as I ; or thus, 
 $32.09f Ans. since 1 dozen costs 9 J 5 ^, 6^ 
 dozen will cost (J^ times as raur:i. 
 
 (11.) What will 42 pounds and 11 ounces of butter cost 
 at 22 J/ per pound ? Ans. $9.60f 
 
 OPERATION. Explanation. As usual we 
 
 here place the price of one 
 
 2 
 16 
 
 A* pound on the statement line and 
 
 * reason as follows Since 1 
 
 pound cost ^ 1 ounce will 
 cost the 16th part and 683 
 Ans. ounces (which is 42 pounds and 
 
 11 onnces) will cost uss times as much. 
 
108 Arithmetical Exercises and Examples. 
 
 117. To multiply Abstract Fractional numbers. 
 
 1. Multiply 8J by 3|. 
 
 OPERATION. Explanation. In this problem both 
 
 ;j ->5 factors are abstract numbers, hence we 
 
 4 T 5 cannot give the same analogical reason- 
 
 ing as we gave in the foregoing problems 
 where the factors were denominate num- 
 125 bers ; although were we to do so, the 
 
 result so far as the figures are concerned, 
 31} Ans would be correct. We therefore, reduce 
 and place the 8J, the number to be mul- 
 tiplied, on the statement line and reason as follows : Since 1 
 time - 2 ^ 5 - is equal to - 2 S 5 , J- time the same is J part of *j- ami Y 
 are 15 times as many. 
 
 MISri-LLANEOUS EXAMPLES IN Mr LTII'LIOATION 
 OF FRACTIONS. 
 
 1. What will 1(> yards cost at 14:j^ per yard? 
 
 Ans. 82.31). 
 '1. What will 23$ pounds cost at 35^ per pound ? 
 
 Ans. $s.:-ni. 
 
 3. What will J of a yard cost at $| per yard? 
 
 " Ans. ^-|. 
 
 4. What will J of a yard cost at $J per yard? 
 
 - Ans. $}. 
 .">. What w r ill 8^ pounds cost at 7JX per pound? 
 
 Ans. 63J/. 
 
 6. What will 10| pounds cost at 9J^ per pound ? 
 
 Ans. 99-j^/. 
 
 7. What will 19f pounds cost at 18|/ per pound? 
 
 Ans. $3.60|/- 
 
 8. What will 25if yards cost at 17 \ff per yard ? 
 
 Ans. $4.55^. 
 
 9. What will 11 f yards cost at 12J/ per yard? 
 
 Ans. $ 
 
 10. What will 21} yards cost at 16J/ per yard? 
 
 Ans. $3.58-J. 
 
 11. What will 14t pounds cost at 12}^ per pound? 
 
 'Ans. $1.74 A, 
 
Multiplication of Fractions. 109 
 
 12. What will 31 J pounds cost at 11J/ per pound? 
 
 Ans. S3.46JJ. 
 
 13. Multiply %\ by 12. Ans. 5 J. 
 
 14. ,. Multiply If by 13. Ans. 10^. 
 
 15. Multiply ^ by 19. Ans. 10J-J-. 
 
 16. Multiply 7 by f Ans. 2f 
 
 17. Multiply 13 by iV Ans - 9 iV 
 
 18. Multiply 105 by ^. Ans. 12. 
 
 19. Multiply 136 by T 3 ^. Ans. 44-ff 
 
 20. Multiply 12 by 31|. Ans. 382. 
 
 21. Multiply 25 by 3|. Ans. 85. 
 22 Multiply 19 by T 3 r . Ans. 5^. 
 
 23. Multiply \\ by if. Ans. }}. 
 
 24. Multiply || by f Ans. |. 
 
 25. ailtiply 11 j by If. Ans. 18f 
 Multiply 2| by 21 J. Ans. 50*. 
 
 27. Find the value of f of J of | of f f of 4. 
 
 Ans. A. 
 
 28. Multiply 7^ by f Ans. 6^-. 
 
 29. What is the product of T 9 7 , f , f 'and \ ? 
 
 Ans. ^. 
 
 30. What is the product of If, f , 2 and 5 J ? 
 
 Ans. 
 
 31. What is the product of T \ of 2J by of 7J ? 
 
 Ans. 1H- 
 
 32. What is the product of 12} multiplied by 5} times 
 6|? Ans' 464^. 
 
 33. At i of a dollar a pound, what will $ of a pound 
 of tea cost? Ans. T 9 T of a dollar. 
 
 34. What will 51 dozen buttons cost at -g% of a dollar 
 per dozen ? Ans. J of a dollar. 
 
 35. What will 4} yards cost at 4 J/ per yard ? 
 
 Ans. 
 
110 Arithmetical Exercises and Eaxmples. 
 
 36. What will 9 J yards cost at 9f f per yard ? 
 
 Ans. 9: 
 
 37. What will 12 yards cost at 12 jy per yard? 
 
 Ans. $1.6C_ 
 
 38. What will 12} pounds cost at 12}/ per pound? 
 
 Ans. $1.56}. 
 
 39. What will 6} pounds cost at 6}/ per pound ? 
 
 Ans. 42 T * 
 
 40. What will 8f pounds cost at 8J/ per pound? 
 
 Ans. 72|/. 
 
 41. What will 19| pounds cost at 19f/ per pound? 
 
 Ans. $3.80. 
 
 42. What will 9} pounds cost at ll}/ per pound ? 
 
 Ans. $1.14^. 
 
 43. . What will 15} pounds cost at 10}/ per pound? 
 
 Ans. $1.62}. 
 
 44. What will 40| pounds cost at 22f / per pound ? 
 
 Ans. 39.19^. 
 
 45. What will 2812} gallons cost at $4.50 per gallon ? 
 
 Ans. $12656.25. 
 
 46. What cost 471} gallons at $3f per gallon ? 
 
 Am $1592^-. 
 
 47. Sold 937852} pounds of cotton at 14|f / per pound, 
 what did it amount to ? Ans. $135695.53f|. 
 
 48. If a man earns $2} in 1 day how much will he earn 
 in lr>} days? Ans. $41}. 
 
 49. A Contractor pays $1 } per day for labor and he has 
 370 men employed for six days. How much money will 
 it take to pay them ? Ans. $2775. 
 
 50. E. J. Denis paid ^j- of a dollar for a book and for 
 paper # of the cost of the book. How much did he pay 
 for paper ? Ans. 60 cents. 
 
 51. Distillers of the essence of rose have determined by 
 experience that it requires 48000 pounds of rose leaves to 
 
Multiplication of Fractions. Ill 
 
 make or distill one pound of the ottar of rose. How many 
 pounds of rose leaves will it require to distill 50$ pounds 
 of the ottar of rose ? Aus. 2442000 pounds. 
 
 52. If a pound and a half costs a cent and a half what 
 will 25 pounds cost? Ans. 25J cents. 
 
 53. F. Querens Jr. owned i| of the Steamer Katie and 
 sold f of his share to G. M. Leahy, what part of the whole 
 Steamer did he sell ? Ans. f . 
 
 54. K. E. Terregrossa can work the problems in this 
 book in 4f months, how many months would it take him 
 to work f of them ? Ans. 3^- months. 
 
 55. E. Schwartz paid $ for 1 gallon of molasses, what 
 is J of a gallon worth at the same rate? Ans. $|. 
 
 56. What will 7? boxes of raisins cost at $2| per box ? 
 
 Ans. $161. 
 
 57. iOn one occasion at the New Orleans Opera 2 of the 
 ladies and gentlemen present were French ; 2 of the re- 
 mainder Ameiican ; J of the remainder German, and the 
 others were of different nationalities. What part were 
 Americans, what part Germans and what part were of differ- 
 ent nationalities ? Ans. J Americans, T X T Germans and 
 
 J of different nationalities. 
 
 68. C. Reynolds owned I of a plantation and sold % of 
 his share to I). C. Williams, who sold \ of what he pur- 
 chased to E. Szymanowski, who sold J of what he pur- 
 chased to N. Forcheimer. What is Forcheimer's share in 
 the plantation ? Ans. -^. 
 
 59. J. Byrnes owned of 2000 acres of land and sold 
 f of his share to E. H. Wells, who sold | of what he pur- 
 chased to H. Clark. How many acres have each? 
 
 Ans. J. Byrnes 400 ; E. H. Wells 450 ; and H. Clark 
 750 acres. 
 
112 Arithmetical Exercises and Examples. 
 
 DIVISION OF FRACTIONS. 
 
 126. Division of fractions is the process of dividing 
 when the divisor or dividend, or both, contain fractional 
 numbers. 
 
 In the division of simple numbers we saw that the result 
 of division operations was decreasing, but in the division 
 of fractions, when the divisor is less than a unit, the result 
 is increasing. This iact is plain, for the reason that the 
 operation of division is the process of finding how many 
 times the dividend is equal to the divisor, and, hence, when 
 the divisor is less than 1. the dividend will be equal to the 
 divisor as many times itself as the divisor is part of 1. 
 
 In practical operations we usually have the thive follow- 
 ing cases or questions in division of fractional numbers. 
 
 1st. To find the cost of mu* pound, yard, or trticle of 
 of any kind, when we have the cost of many pounds, 
 yards or articles of any kind given. 
 
 2d. VTo/mt/ the cost of one pound, yard or article of 
 any kind, when we have the cost of n jxn-f of a pound, 
 yard or article of any kind given. 
 
 3d. To find the number of pounds, yards or articles 
 of any kind that can be bought with a specified sum. when 
 we have the price of one, or apart of one pound, yard or 
 article of any kind given. 
 
 From these questions we see that division is the converse 
 of multiplication and that from the nature of the question, 
 we must reason from many to one or from apart of one to 
 one. Thus : 1st. if 5 pounds cost 50^, 1 pound will cost 
 the ith part of it ; in the 2d. case, if J of a yard cost 
 $2, J- of a yard will cost the J part of it, and -^ths, or a 
 whole yard, will cost 4 times as much ; and in die third 
 case, if jfjt buy 1 yard, or any other thing, \ft will buy 
 the -^-th part of it, and f , or a whole cent, will buy 2 
 times as much. 
 
 For the full reasoning, for this, ease, see the explanation 
 of the 2d problem. 
 
Division of Fractions. 113 
 
 (1.) Bought 7J pounds of sugar for 78}/. What 
 was the cost per pound ? 
 
 OPERATION. Explanation. By inspection and 
 
 ft the exercise of our reasoning facul- 
 
 315 21 t * es > we see th*^ as the 78}^ are 
 
 the cost of 7J poui.ds. it must be 
 divided by 7J in order to obtain 
 the cost of 1 pound. We therefore, 
 Ans. reduce and place the 78} on our 
 
 statement line. In all division operations, in order to facilitate 
 the work, we thus place the number to be divided. 
 
 We then reason from mauy to 1, as follows : since -V 5 - pounds, 
 which is 7J reduced to halves, cost -S^p/, one-half a pound will 
 cost the 15th part of it, and 2 halves, or a whole pound, will 
 cost 2 times as much. This completes the reasoning and state- 
 ment. The 15 and 2 are placed respectively on the decreasing' 
 and increasing sides of the line, for the reason that when they 
 were used the conclusion arrived at were respectively decreasing 
 and increasing. 
 
 {%) At 10 \fl per pound, how many pounds can* be 
 bought for $3.92f? 
 
 OPERATION. Explanation. By inspection 
 
 - lb and reason, we see that the ques- 
 
 1 tion requires puunds for the 
 
 A result or answer. Therefore, in 
 
 -- *o all of the reasoning we must 
 
 either increase or decrease 
 pounds. To aid in rendering 
 36i lb Ans. the solution easily understood, 
 we first place the 1 pound that cost lOff on the right of our 
 statement line, and reason as follows: since 4^. which is 10}p 
 reduced, buy 1 pound, of a cent will buy the 4Jd part, and 4 
 fourths or a whole cent will buy 4 times as much; then, since 
 1 cent will buy the result of the statement now made, J of a 
 cent will buy J part, and 2-1-^3-p will buy 3139 times as much. 
 This completes the reasoning and statement. 
 
 The placing of the 1 pound on the statement line may be 
 omitted and the reasoning given in the same manner as when 
 t is thus placed. 
 
114 Arithmetical Exercises and Examples. 
 
 At $| per yard, how many yards can we buy for 
 
 ? 
 
 OPERATION. 
 
 Y 
 
 1 
 
 34 
 
 Explanation. For reasons given 
 in preceding examples, we first 
 place 1 yard on our line, and then 
 reason as follows: since J of a dol- 
 lar buy 1 yard, \ will buy the 3d 
 part, and J or a whole dollar, 4 
 times as much: then, sin (ye 1 dollar 
 6|l-J- yard, Ans. will buy the result of our state- 
 
 ment. J of a dollar will buy the 8th part, and J, 7 times as 
 much. This completes the reasoning and statement. 
 
 V-l.) At $l-g- JUT puuml, ho\v many pounds can we buy 
 
 OI'KKATION. 
 ft 
 
 A 
 
 38 
 
 H)0 
 
 It) Ans. 
 
 Having placed 1 
 pound on our line, we reason thus: 
 >in-:e 7 dollars buy 1 pound, will 
 buy the 7th part, and |, or a whole 
 dollar, f> times as much, and 38 
 dollars will buy 38 times as much 
 as $1. This completes the reason- 
 ing and state.meut. 
 
 (5.) At So per dozen, now many dozen can we buy 
 for V i ' 
 
 Ol'K.r.ATlON. 
 
 Duz. 
 
 1 
 
 I 
 
 9Q1 , . 
 484 doz Ans. 
 
 Explanation. We place 1 dozen, 
 the equivalent of $3, on the line, 
 and reason thus ; if $3 buy 1 
 dozen, $1 will buy the 3rd part; 
 then, if $1 buys the result of the 
 statement now made, J of a dollar 
 
 will buy the 8th part, and $ 6 J S 
 will bliy 675 times as manv> 
 
Division of Fractions. 
 
 115 
 
 1 o 
 
 (6.) Bought 9f yards for $22i. What was the price 
 per yard ? 
 
 Explanation. For reasons given 
 in the first example of division, 
 we reduce and place the $22J on 
 4 our line, and then reason thus : 
 
 if 'f yards cost V dollars, of 
 
 a yard will cost the 75th part, 
 and | or a whole yard will cost 
 8 times as much. This completes 
 Ans. the reasoning and statement. In 
 working out the statement, we first cancel the 75 and 45 by 15 ; 
 then the 8 by the 2 ; then we multiply the 3 and 4 together and 
 divide the result by the 5. 
 
 (7.) Bought f of a pound for 30/. What was the 
 price per pound ? 
 
 OPERATION. Explanation. Placing the cost on 
 
 the line, we reason thus ; if f of a 
 %(h 1ft pound cost 30f, will cost the 3d 
 
 I rr part, and f or a whole pound will 
 
 cost 4 times as much ; which 
 worked out gives 40f or the cost 
 40/ Ans. of ! pound. 
 (8.) Bought 5 boxes of indse. for SSI J. What was the 
 price per box ? 
 
 OPERATION. 
 
 Ans. 
 
 65 
 
 Explanation 1 lie ^81| being the 
 number to be divided, we reduce 
 and place the same on the line ; 
 then reason thus : if 5 boxes cost 
 3 J 5 dollars, 1 box will cost the 5th 
 
 EXAMPLES. 
 
 (9.) Bought 18| pounds for 37}/. What was the 
 price per pound ? Ans. 2^. 
 
 10. At 6i/ per pound, how many pounds can be 
 bought for 96IX ? Ans. 15ff 
 
 11. Bought 250J dozen for $1251f What was the 
 price per dozen ? Ans. $4.99ff . 
 
116 Arithmetical Exercises and Examples. 
 
 11 
 
 12. At 79-j$r/ per pound, how many pounds can bo 
 bought for 7287}/ ? Ans. 91|-J2|. 
 
 13. At $i a piece, how many chickens can be bought 
 for $25J ? Ans. 51. 
 
 127. DIVISION OP ABSTRACT NUMBERS. 
 
 (1.) Divide 22} by 5}. 
 
 jo] Explanation. The real question 
 
 to be determined in this example 
 is, how many times is 22J equal to 
 5J, and as both the dividend and 
 
 22 divisor are abstract numbers, we 
 
 cannot, therefore, logically reason 
 ~ . as in the preceding problems, and 
 
 , - ^ ns - accordingly proceed as follows: 
 
 the 22J being the number to be divided, we first reduce and 
 place the same on our statement line; then by inspection and 
 the exercise of our reason, we see ihat 22J is equal to 1, 22f 
 times, or reduced that -^ are equal to 1, ^- times ; and if equal 
 to I, -M- times, it is equal to ^, twice as many times, and to ty 
 instead of J, the T J T part. This completes the reasoning and 
 statement of the problem, and the same character of reasoning 
 and statement will solve all division problems in abstract 
 numbers. 
 
 (2.) Divide T ^ by i. 
 
 OPERATION. Explanation. In this example 
 
 15 
 
 3 
 
 2 the dividend being less than the 
 
 4 
 
 divisor, the question is, what part 
 of a time is the -f 5 equal to J. 
 Placing the on the line, we fea- 
 Ans. son, as in the above example, thus: 
 
 o are equal icTl, ^ of a time ; and if equal to 1, -fa times, it 
 iV equal to J, 4 times as many times, and to } instead of J, the 
 3d part. 
 
Division of Fractions. 117 
 
 (3.) Divide 3 by } . 
 
 OPERATION. 
 
 ;; Explanation. We first place the 
 
 *; j 3 on the line, and reason thus : 3 
 
 is equal to 1, 3 times, and to J 
 instead of 1, 3 times as many 
 times ; and to -f instead of , the 
 J part. 
 Ans. 
 
 i 1.) Divide 14f by 9. 
 
 OPERATION. Explanation. Placing the num- 
 
 ber to be divided on the line, we 
 reason thus : ^- are equal to 1, 
 - 7 ^ 2 - times, and to 9 instead of 1, the 
 9th part. 
 Ans. 
 
 8 
 
 The solution of the 4 preceding problems elucidates the 
 only correct reasoning for dividing abstract fractional num- 
 bers. But for practical work we would not advise a change 
 from the reasoning given where the numbers are (^nomi- 
 nate. 
 
 MISCELLANEOUS EXAMPLES IX DIVISION OF FRAC- 
 TIONS. 
 
 1. Bought 4 yards for 143, what was the cost per 
 yard? Ans. $3f. 
 
 2. Sold 8J. pounds for $1.87, what was the price per 
 pound ? Ans. 22 cents. 
 
 3. Paid 37J cents for 6} yards of calico, what was the 
 price per yard ? Ans. 6 cents. 
 
 4. At $1| per gallon, how many gallons can be bought 
 for $148.1 ? Ans. 108 gallons. 
 
 5. Divide ^ by 2. Ans. f. 
 
 6. Divide ^ 9 T by 3. Ans. f 
 
 7. Divide |f by 5. Ans. ||. 
 
 8. Divide ^ by 5. Ans. -^-. 
 
 9. Divide 7i by 9. Ans. 
 10. Divide 2 by f Ans. 2J. 
 
118 Arithmetical Exercises and Examples. 
 
 11. Divide 3 by J T . Ans. 7. 
 
 12. Divide 5 by jf Ans. 5 A. 
 
 13. Divide 21 by ^-. Ans. 33. 
 
 14. Divide 105 by jf Ans. 119. 
 
 15. Divide fj- by ? 5 5 , Ans. 4. 
 
 16. Divide ft by . Ans. 12. 
 
 17. Divide 2i by jj. Ans. 3. 
 PT87J Divide ff by T 4 T . Ans. 
 
 TOT Divide -5A by 21. Ans. 
 
 20. Divide $ of |f by f Ans. 
 
 21. If one pound of tea cost of a dollar, how many 
 pounds can be bought for $25 ? Ans. 30 Ibs. 
 
 22. Six barrel* of flour were divided among some poor 
 families in such a manner that each received f of a barrel; 
 how many families were there? Ans. 9. 
 
 23. If a boy can earn T 7 T of a dollar in one day ; how 
 many days will it take him to earn 21 ? Ans. 33 days. 
 
 24. Henry walked 25 miles, which was | of the dis- 
 tance Robert walked ; how many miles did Kobert walk ? 
 
 Ans. 30 miles. 
 
 25. At the battle of Germantown the British lost about 
 600 men ; this was f of the number lost by the Ameri- 
 cans ; and the number lost by the Americans was f of 
 the number they received as re-enforcements just before 
 the battle. How many men did the Americans lose, and 
 how many receive as re-enforcements ? 
 
 Ans. 1000 men lost, 2500 re-enforcements. 
 
 26. A man had his store insured for $9000, which was 
 f of T 9 T of its value ; what was the store worth ? 
 
 Ans. $12375. 
 
 27. Sulphur will fuse at 232 Fahrenheit. This is 7} 
 times the temperature required to melt ice. At what 
 temperature will ice melt ? 4ns. 32. 
 
 28. A quantity of mercury weighed 32062J Ibs., which 
 is 13i times the weight of an equal bulk of water. What 
 would an equal bulk of water weigh ? Ans. 2375 ft>s. 
 
 29. A pound of water at 212 F. was mixed with a 
 
Division of Fractions. 119 
 
 pound of powdered ice at 32. The united temperature 
 of the two was 4 T 9 g times the temperature of the mixture 
 when the ice became melted. What was the temperature 
 of the two pounds after the ice became melted ? 
 
 Ans. 52. 
 
 30. lx ^VVhen the air was at the freezing point, a cannon 
 
 27(>KJ-S feet distant from New Orleans was discharged. 
 
 25 J 'seconds elapsed after the discharge before the sound 
 
 Breached New Orleans. How many feet per second did the 
 
 sound travel ? Ans. 1090 feet. 
 
 31. Divido 2S7] by 5. Ans. 57 
 
 Operation without the Kjplanution. We first divide 
 
 line statement. ^ 28T b 7 th P rocess of short 
 
 r .,,0^3 division and obtain a quotient of 
 
 57 and a remainder of 2 ; this 
 
 remainder we reduce to a fraction 
 
 57^-Q- Ans. whose denominator is the same as 
 
 that of the fraction to be divided, add it to this fraction and 
 
 then divide the sum by 5 and annex the result to the quotient 
 
 57. Thus 2=|-f J=*fS and ^-5-6=4$. 
 
 32. Divide 1471 -,*. by !. Ans. 163 T ^. 
 
 33. Divide 1044$ by 12\ Ans. 87 T V 
 
 34. E. T. Churchill divided 14 T 7 ^ dozen apples among 3 
 boys and 2 girls ; he gave each girl twice as many as each 
 boy. How many did each boy and each girl receive ? 
 
 Ans. 2 r ^ doz. each boy, 4i doz. each girl. 
 
 35. Divide 1 by \. Ans. 5. 
 
 36. Divide \ by 1. Ans. \. 
 
 8J A- DJ i 
 
 37. Divide of by of - Ans. f 
 
 6J H 4$ * 
 
 38. If 4J pounds of coffee cost 90 cents, what will 22 if 
 pounds cost? Ans. $4.55. 
 
 39. R. E. L. Fleming owns 1 of the capital stock of a 
 factory valued at $24000; he gives \ of i to educational 
 societies, and the remainder he divides equally between his 
 
120 Arithmetical Exercises and Examples. 
 
 four children. How much does he give to educational 
 societies and how much does each child receive ? 
 
 Ans. $1500 to educational societies. 
 $1875 each child receives. 
 
 40. C. Craft has 65} yards of cloth, 2 yards wide, how 
 many yards of lining if of a yard wide will be required to 
 line it? Ans. 196} yards. 
 
 41. Divide IS oranges between A. and B. so that A. 
 wtllhave i more than B. What number will each have ? 
 
 Ans. A. 10 ; B. 8. 
 
 42. Divide 18 oranges between A. and B. so that A. 
 will have } less than B. What number will each have ? 
 
 Ans. A. 7f ; B. lOf 
 
 43. A.. B. and C. are to receive $26 in proportion to 
 ,}, } and J. What will each receive ? 
 
 Ans. A. $12 ; B. $8 ; C. $ii. 
 
 44. A. and B. can do a piece of work in 10 days ; A. 
 alone can do it in 15 days. How many days will it take 
 B. to do it ? Ans. 30 days. 
 
 45. A. and B. can do a piece of work in 14 days. A. 
 can do f as much as B, How many days will it take each 
 to do it, work in IT alone ? Ans. 24} days for B. 
 
 32| days for A. 
 
 46. Three persons, A., B. and C., do a piece of work; 
 A. and B. together do -J- of it, and B. and 0. do ^ ot it. 
 What part of the work is done by B ? Ans. |~|-. 
 
 -IT. A planter remits his factor $500 to invest in rice 
 and coffee, in equal sums. He pays 9i^ per pound for 
 rice, and 23|^ per pound for coffee. How many pounds 
 of each did he purchase ? 
 
 Ans. 2702|- ft> rice. 1069^ ft> coffee. 
 
 48: W. Quintel has $100 : he gives f of it for five bar- 
 rels of flour, and J of the remainder tor 4 barrels of pota- 
 toes, and with the remainder he buys coffee at 20^ per 
 pound. How much coffee did he buy ? 
 
 Ans. 200 Ib coffee. 
 
 49. C. Wehrmann owned -^ of a stock of goods : he 
 
Division of Fractions. 121 
 
 sold -J- of his share for $5000, and } of the remainder for 
 $5000 . and then the balance of his interst for $15000. 
 What part did he sell the last time, and what would the whole 
 stock be worth at that rate ? 
 
 Ans. || sold last. 
 
 $27777-J value of stock. 
 
 50. What quantity, from which if you subtract f of 
 itself, the remainder will be 15? Ans. 24. 
 
 148. MISCELLANEOUS EXAMPLES, INVOLVING THE 
 PRINCIPLES OF ADDITION, SUBTRACTION, MULTI- 
 PLICATION AND DIVISION OF FRACTIONS. 
 
 1. Find the difference between ^ and , $ and ^, ? and 
 T 7 T , 3| and 2f , 4 and } of 3J ? Ans. to last, 3. 
 
 2. Find the sum of f of ^ and of / T . Ans. . 
 
 3. To the quotient of 2f divided by 5, add the quo- 
 tient of 3 J divided by ^- r . Ans. 7 . 
 
 4. A number was divided by , and gave a quotient of 
 20, what was the number? Ans. 15. 
 
 5. What number is that, which being multiplied by ^ r 
 gives as a product ^ ? Ans. f . 
 
 G. What number is that, from which, if you take f of 
 itself, the remainder will be 12? Ans. 30. 
 
 7. What number is that, to which, if you add -| of 
 itself, the sum will be 40 ? Ans. 25. 
 
 8. A. owns f of a store which is worth $25000. He 
 sells |- of his share ; what part does he still own, and 
 what is it worth ? Ans. owns y 1 ^, worth $2500. 
 
 9. $mith owns y 5 r of a cotton mill and sells T 3 g- of his 
 share to Jones for $33000 ; what is the mill worth at that 
 rate? Ans. $242000. 
 
 10. John has 5 cents, and James J of 8 cents ; what 
 part of James' money is John's? Ans. f. 
 
 11. One planter raised 500 bales of cotton, another 
 raised 250 ; what part of the first one's crop is the second ? 
 
 Ans. i. 
 
\'1'1 Arithmetical Exercises and Examples. 
 
 12. The sum of four fractions is 1;}. Three of the 
 fractious are ^, i and f ; what is the fourth? Aus. -f 1 . 
 
 13. What number is that, to which it ^ of f of 1^- be 
 added, the sum will be 1 j ? Ans. 1. 
 
 14. Two boys bought a bushel of oranges, one paying 
 2J dollars and the other 4 dollars : what part of it should 
 each have? Ans. first, ^f ; second, f*. 
 
 15. A farmer sold - of his mules on Monday ; on Tues- 
 day lie bought | as many as he sold, and then had 40 ; how 
 many mules had he at first? Ans. f>(>. 
 
 16. F. Gernon gave , i and k of his money to different 
 benevolent institutions, and had $1000 left, How much 
 had heat first? Ans. S^IMMM). 
 
 17. J. D. Bothick owning -fa of a rice mill, sold of 
 his share for $8800. What was the value of the mill? 
 
 Ans. SLM200. 
 
 18. A book-keeper worked !>1 \ days, and after paying ii 
 of f of his earnings for board and washing, had $1 
 inaiiiing. How many dollars did he receive in all, and how 
 many per day ? Ans. $730 in all, $3 per day. 
 
 19. A planter gave 50 bales of cott >n at $50^ per bale 
 for flour at $75 per barrel. How many barrels of flour did 
 he receive ? Ans. 334. 
 
 20. Prophet ^an do a piece of work in 6, and Fisher 
 can do the same in 8 days ; how many days will it take 
 both together to do the work ? Ans. 3^ days. 
 
 21. Myers, Levy and Hoffman can do a piece of work: 
 in 10 days ; JJyers and Levy can do it in 15 days ; in what 
 time can Hoffman do it, working alone ? Ans. 30 days. 
 
 22. A man died and left his wife $14400, which was J 
 of f | of his estate. At her death she left of her share 
 to her daughter. How many dollars did the daughter 
 receive, and what part was it of her father's estate ? 
 
 Ans. $12000, ff of her father's estate. 
 
 23. A mule and dray cost $240; the mule cost If times 
 as much as the dray. What did each cost ? 
 
 Ans. $150 mule, $90 dray. 
 
Miscellaneous Problems. 123 
 
 24. A man engaging in trade lost ^ of the money he 
 invested, he then gained $1000, when he had $3800 ; what 
 did he have at first, and what was his loss? 
 
 Ans. #4900 at first, $2100 loss. 
 
 25. Forcheimer lost f of his fish-line, and then added 
 25 J feet when it was just J of its original length. What 
 was its original length ? Ans. 204 feet. 
 
 26. How many bushels of apples at $f a bushel, will 
 pay for ^ of a barrel oranges at $6ir a barrel ? 
 
 Ans. 7J bushels. 
 
 27. Sweeney paid % of his year's wages for board, f of 
 the remainder for clothes, and had $80 left ; how many 
 dollars did he receive for labor ? Ans. $560. 
 
 28. Purcell, having a certain number of cents, gave one- 
 half of them and half a cent over to one beggar ; one-half 
 of what he had remaining and half a cent ov> r to a second 
 beggar ; and to a third, one-half of what he then had and 
 half a cent over, and had left 3 cents. How many cents 
 had he at first? Ans. 31 cents 
 
 29. Jol.n lives with his parents, but works for Mr. 
 Smith who pays him $210 per year. His parents board 
 him, but he has his clothes to buy. He spends ^ of his 
 wages for cigars, -| of the remainder for theater tickets, 
 of the remainder for wine, J of what he tfren has for nov- 
 els. How much has he remaining at the end of the year 
 to pay for his clothes ? Ans. $30. 
 *30 Joseph worked on the same conditions as John. He 
 gave ^ of his wages to the cause of charity, fa of the 
 remainder for useful books, i of the remainder to be taught 
 evenings, paid $100 for clothes, and deposited the balance 
 in the bank. How many dollars did he put in the bank? 
 
 Ans. $50. 
 
 31. W. T. Harris and C. E. Jones have $1899, Jones 
 has 3J times as much as Harris ; how much has each ? 
 
 Ans. Harris $422, and Jones $1477. 
 
 32. J. C. Beals can solve 25 problems in 50 minutes and 
 
124 Arithmetical Exercises and Examples. 
 
 H. H. Barlow can solve them in 30 minutes. In what 
 time can both solve them ? Ans. 18:]- minutes. 
 
 33. E. Meyer purchased 200 barrels of flour for $1450 
 and sold J of it at a profit of $-1 per barrel, and the re- 
 mainder at $7 1 1 Tr per barrel. How much did he gain ? 
 
 Ans. $67.50. 
 
 34. What is the numerical value of 
 
 Ans. 1 4 V 
 
 35. M. Ernst bought 3S41 -I pounds of cotton at 7 , : pence 
 per pound; what did it cost ? Ans. 1-4, 11. (i 1 , <1. 
 
 36. J. W. Anderson has 3 dozen oranges which he 
 wishes to divide between Miss Kate and Miss Lucy, so that 
 Miss Kate shall receive ] more than Miss Lucy. How 
 many will each receive? 
 
 Ans. Mi>s K. 2n ami Miss L. 16. 
 
 37. A tree 110 feet high, had f of it broken off in a 
 storm ; how much of it was left standing ? Ans. 44 feet. 
 
 38. What cost L'L' J pounds of coffee at 21 J/ per pound ? 
 
 Ans. $4.94J|. 
 
 39. If 18? yards cost $3.37 what will 3J yards cost? 
 
 Ans. 60 cents. 
 
 40. W. D. Maxwell has $600 of which he wishes to 
 give to A. J, B. J, C. ^ and D. i ; how much will each 
 receive? Ans. A. $200, B. $150, C. 120 
 
 and D. $100. 
 
 41. R. L. Paul has $600 which he wishes to give to A* 
 B., C. and D. in the proportion of J, }, ^ and i ; how 
 much will each receive? Ans. A. $210j|, B. $157}f 
 
 C. $126^ and D. $105^. 
 
 42. C. M. Huber and A. J. Hohensee bought on specu- 
 lation $800 worth of merchandise, of which Huber paid 
 $500 and Hohensee $300 ; they sold to W. A. Tomlinson 
 i of the whole for $400. How much of the $400 must 
 Huber and Hohensee receive respectively, in order to con- 
 stitute each J owner in the renriinder of the goods ? 
 
 Ans. Huber $350 and Hohensee $50. 
 
Miscellaneous Problems. 125 
 
 43. If a yard and a half cost a dollar and a half what 
 will twelve and a half yards cost? Ans. $12J. 
 
 44. If 3 is the third of 6 what will the fourth of 20 be? 
 
 Ans. 7J. 
 
 45. Greo. Meyer owned a quantity of rice, of which he 
 sold i for $99.60 ; what is f of the remainder worth at the 
 same rate? Ans. $16.60. 
 
 46. F. Miller paid $60 for f of an acre of land ; what 
 is the value of f of an acre ? A.ns. $50. 
 
 47. S. Benavides bought 937852J pounds of cotton at 
 1415^ p er pound ; what was the cost? 
 
 Ans. $135695.53ff. 
 
 48. J. Koch invested } of his money in sugar, J in rice, 
 | in coffee and deposited in bank $2645. How much money 
 had he at first ? Ans. $63480. 
 
 49. L. Meyer spends i of his time in study, i in labor, 
 i in rest and recreation, and the remainder in sleep. How 
 many of the 24 hours of a day does he sleep? 
 
 Ans. 7 hours. 
 
 50. A loafer spends 4 hours per day sauntering on street 
 corners, 3 hours smoking and drinking, i of the day in 
 sleep, i of the day in drunkenness, y 1 ^ in eating, T V in 
 quarreling and the remainder of the day in gaming. How 
 many hours does he spend in guming ? Ans. 3 hours. 
 
 51. An industrious young lady spends i of her time in 
 the performance of household affairs, i in reading good 
 books, y 1 ^ in physical exercise in the open air and sunlight, 
 i in the practice of music, singing and parlor amusements, 
 or social intercourse, 2 hours per day in eating, and the 
 remainder of the day in sleeping. How many hours per 
 day does she devote to each ? 
 
 Ans. 6 hours to household affairs; 4 hours to reading; 
 2 hours to exercise ; 3 hours to music, etc.; 2 hours 
 to eating, and 7 hours to sleep. 
 
 52. A fashionable young lady spends of her time in 
 dressing, painting and making her toilet, J in reading nov- 
 els and papers of senseless fiction, in making calls and 
 
126 Arithmetical Exercises and Examples. 
 
 gossiping, y 1 ^- in street promenading, ^ in criticising indus- 
 trious young men, and speculating upon the qualities and 
 fortune of an anticipated husband, J ? in making remarks 
 derogatory to the ohanicti-r of those who labor, while her 
 own mother is perhaps cooking or washing, T ^ in enter- 
 taining young men, and the remainder in eating and sleep- 
 ing. How many hours does she devote to useful service, 
 and how many to eating and sleeping ? 
 
 Ans. hours to useful service; 8 hours to eating and 
 sleeping. 
 
 53. A man willed } of his property to his wife, J of the 
 remainder to his daughter, and the remainder to his SOD; 
 the difference between his wife and daughter's share was 
 $8000. How much did he give his son ? 
 
 Ans. $4800. 
 
 54. R. W. Tyler owned a J interest in a factory ; he 
 sold to C. Modinger \ of his interest for $15000. What 
 interest does he still own, and how much is it worth at the 
 rate received for the part sold ? 
 
 Anfe. he still owns f, worth $15000. 
 
 55. J. Cassidy owned I of the Steamer R. E. Lee. He 
 sold to Gr. Buesing i interest in the Steamer for $20000 ; 
 and to J. C. Beals \ of his remaining interest at the same 
 rate. What did he receive for the last sale, and what is 
 his remaining interest in the boat ? 
 
 Ans. he received $30000 ; T 9 F remain- 
 ing interest. 
 
 56. N. Puech and A. Palacio bought on joint account 
 each J the New Orleans Cotton Factory. N. Puech sold J 
 of his interest to 11. Krone, and subsequently J of his 
 remaining interest to A. Palacio, who subsequently sold J 
 of I of his whole interest to R. Lynd for $7500. What 
 is the factory worth at the same rate, and what is each 
 owner's interest? 
 
 Ans. $32000 value of Factory ; Puech owns J ; Krone 
 i ; Palacio |f , and Lynd -J-f . 
 
 57. L. Kaiser bought f of f of 28 J barrels of apples, 
 
Miscellaneous Problems. 127 
 
 and sold to S. L. Crawford f of 9 barrels for $20}, which 
 was $1.50 more than the same cost. What was the cost 
 of the whole, and how many barrels has he unsold ? 
 
 Ans. $39^- cost ; 7*S barrels unsold. 
 
 58. W. D.Maxwell gives | of his annual income to aid 
 meritorious young men in obtaining an education; J of the 
 remainder for the publication and free distribution of books 
 treating of the awful injury to the human race by the use 
 of tobacco, tea. coffee and wine ; J of the second remainder 
 for various benevolent purposes. The balance $5490 he 
 retains for his own personal use ; how much does he give 
 for each object named ? 
 
 Ans. $8235 for meritorious young men ; $8235 for the 
 publication and distribution of books ; and $2745 
 for various benevolent purposes. 
 
 59. What is the smallest sura of money for which I 
 could purchase a number of bushels of oats, at $-f^ a 
 bushel; a number of bushels of corn, at $f a bushel ; a 
 number of bushels of rye, at $1 J a bushel ; or a number of 
 bushels of wheat, tit $2| a bushel ; and how many bushels 
 of each could I purchase for that sum ? 
 
 Ans. $22 '. ; 72 bushels of oats ; 3G bushels of corn ; 
 15 bushels of rye; 10 bushels of wheat. 
 
 60. There is an island 15 miles in circuit, around which 
 A. can travel in J of a day, B. in i of a day, and a horse 
 car in $ of a day. Supposing all to start together from 
 the same point to travel around it in the same direction, * 
 how long must they travel before coming together again at 
 the place of departure, and how many miles will each have 
 traveled ? 
 
 Ans. 10} days; A. 210 miles; B. 180 miles; Horse 
 Car 525 miles. 
 
 DECIMAL FRACTIONS. 
 
 150. A Decimal Fraction is one whose integral unit is 
 divided according to the decimal scale ; therefore thedenomi- 
 
I'-.s Arithmetical Exercises and Examples. 
 
 nator is some power of ten ; as 10, 100, 1000, etc. The 
 word decimal is derived from the Latin word decem, which 
 means ten. 
 
 151. The Decimal Point (.) is used to distinguish 
 decimals from whole numbers. When there are mixed 
 numbers, it also separates th> whole numbers from the 
 decimals. 
 
 The following are decimal fractions : y 8 ^, y 1 -^, T W^, 
 a "d T^inr?; tn y are here written as common fractions, 
 but generally the denominator of decimal fractions is omit- 
 ted and the value i,s indicated by the location of the deci- 
 mal point before the numerator. 
 
 To write these fractions according to the decimal nota- 
 tion, they would be written thus : 
 
 T 7 ^ decimally expressed is . ,~. 
 
 iW Decimally expressed is .If). 
 
 yW<r decimally expressed is .137. 
 
 T$Mo decimally express.-,! is .0123. 
 
 152. Notation Of Decimals. Whenever decimal 
 fractions are expressed decimally, the numerator must 
 have as many decimal places as there are naughts in the 
 denominator. Thus T V=.4 ; .yV^- 16 5 .^WW=- 1456 - 
 When the number o* 1 naughts in th" denominator is greater 
 than the number of figures in the numerator, naughts 
 must be prefixed to the numerator until the number of 
 places is equal to the naughts in t!:e demominator. Thus 
 Y*T=-04; TrtW=-7; TTrVVm^-00125, etc. 
 
 When the number of naughts in the denominator is less 
 than the figures in the numerator, the result or value of 
 the fraction will embrace a whole number and a fraction. 
 
 153. A Pure or Simple Decimal consists of a dec- 
 imal fraction, decimally expressed or written. Thus .5, 
 .42, .875 and .1256 are pure decimals, and are read 
 respectively 5 tenths ; 42 hundredths ; 875 thousandths and 
 1250 ten thousandths. 
 
Decimal Fraction*. 129 
 
 154. A Mixed Decimal consists of a whole number 
 and a decimal. Thus 24.5 and 41.25 are mixed decimals. 
 They are read respectively, 24 and <> tenths ; 41 and 25 
 hundredths. 
 
 155. A Complex Decimal consists of a decimal 
 with a common fraction annexed. Thus .15J and .005i 
 are complex decimals. They are read respectively, 15 J 
 hundredths ; 5i thousandths. 
 
 156. A Circulating Decimal is one in which a fig- 
 ure or set of figures constantly repeats itself. Thus J= 
 .3333+, |=.142857 + , H=.7333o+. The figure or 
 set of figures which is repeated is called a Repeteml. If 
 the repetend consists of only one figure, a dot is placed 
 over it ; if of a set of figures, a dot i placed over the first 
 and last figures, as J=.3, J=.6, j*f=. i 9, '<^=. 142857. 
 
 157. .A Pure Circulating Dedmal is one which 
 
 contains only the repetend ; as $.6, \. 142857, % 
 
 158. A Mixed Circulating Decimal is one which 
 
 contains other figures than the repet ;nd ; at J=. ie, |jjj- 
 .647. 
 
 There are still other kinds of circulating decimals, but 
 as they are of very little practical im )ortauc3, we will not 
 here consider them. 
 
 159.- Decimal fractions, like whole numbers, decrease 
 towards the right and increase towaids the left in a ten- 
 fold ratio, and hence the prefixing of laughts between the 
 decimal figures and the decimal poin,, or the removal of 
 the decimal point towards the left diminishes their value 
 ten- fold, or divides the decimal by tea for oach order or 
 place removed, and conversely the removal ol the decimal 
 point to the right, increases the value ten- fold or multi- 
 plies the decimal by ten for each place retuov -d. 
 
 Annexing naughts "to decimals does not change their 
 
130 Arithmetical Exercises and Examples. 
 
 value, because the significant figures are not thereby 
 removed nearer to nor farther from the decimal point. 
 
 Decimal orders are also called decimal y/A/m<?. each order 
 being counted as one place. Thus in .0043 there -are four 
 decimal plates, although the 3 is of the fifth decfiii't/ //>!<, 
 from unity, the base of the system. 
 
 The following table will illustrate more fully the relation 
 of whole numbers and decimals, with their incrcasim: and 
 decreasing orders to the left and right of the decimal 
 point. 
 
 TABLE. 
 
 WHOLE NUMBERS. DECIMALS. 
 
 cc 
 
 TO 
 
 3 
 
 C3 
 
 
 
 .2 
 
 14 
 
 02 
 ^^ -f 
 
 s I 
 
 if 
 
 Is .j 
 
 *- rr 
 
 d 
 
 s. . 
 
 en HJ aa 
 
 sl'sll 
 
 . GO'S 2 | 
 & l| IF,: I 5 ? 
 
 73 s o ^ ^ - "^ 
 
 ^SlJSflSj 
 
 "O 
 
 .C ^J g ns jj 
 
 g.g'Tj g^^^o^^ 
 
 ~ ^ 
 
 S p S o a S ^ 
 
 g(3|gt2^5 
 
 lllllwlll 
 
 98 
 
 7654321 
 
 .23456789 
 
 Orders of ascending scale. Orders of descending scale 
 
 This number is read 987 million 654 thousand 321, and 
 23 million 456 thousand 789 hundred-millionths. 
 
 In order to clearly understand decimals, we must bear 
 in mind that the unit one is the basis of all numbers, inte- 
 gral and fractional, abstract and denominate, and that all 
 mathematical operations have this fundamental principle 
 for their origin, and every number is but a multiple, either 
 ascending or descending of unity or one. 
 
 The names of the decimal orders are derived from the 
 names of the orders of whole numbers. Thus the names 
 of the orders in the ascending scale, are, after units, tr.ns, 
 
Decimal Fractions. 131 
 
 hundreds etc., and the orders in the decending scale, are, 
 after units, tenths, hundredth^ etc., the decimal orders 
 being the reciprocal of the orders of whole numbers equal- 
 ly distant with themselves from the u iits. 
 
 Numeration of Decimals. In reading decimal frac- 
 tions the entire decimal is regarded as reduced to units of 
 the lowest order expressed, and the name of this order is 
 given to the entire number of decimal units. Thus .25 is 
 read twenty -five hundredths. 
 
 Before reading a decimal, we must determine 1st. How 
 many units are expressed. To do this, we numerate and 
 read the significant figures of the decimal as in whole num- 
 bers. 2nd. We must determine the name of the lowest order 
 in the decimal. To do this, we numerate the number dec- 
 imally. Thus to read .001073, we c< mmence at the 3 and 
 numerate to the 1 thousand, and thus find that 1073 units 
 are expressed ; then we commence ao the decimal point 
 and numerate decimally to the 3 and thus find that mil- 
 lionths is the lowest order, we then re id 1073 millionths. 
 
 160. EXERCISES. 
 
 Read the following numbers : 
 
 1. 16.008 ; reads thus, sixteen units and e ght thous- 
 andths. 
 
 2. .94f ; reads thus, ninety-four and three-eights hund- 
 reths. 
 
 3. 5067.4005 ; reads thus, 5067 units and 4005 ten- 
 thousandths. 
 
 4. Write and read 197.8; 4.68907; .00073; 48.769- 
 146. 
 
 5. Write and read 2.491; 10.0101089167; 582.400- 
 410905. 
 
 6. Write and read 5841. 291f; 8000.0000000217; 
 9876541.1000001. 
 
132 Arithmetical Exercises and Examples. 
 
 161. Writing Decimals. In writing decimals we 
 write down the given number as if it were a whole nnmber ; 
 then, to facilitate the operation, we numerate from right to 
 left, beginning the numeration with tenth*, and continue 
 until we come to the required place or order, always writing 
 O's to fill the places not occupied by significant figures. 
 Thus, to write 25 ten thousandths we first write the 2f> ; 
 then we begin at the right and numerate thus, tenths, hun- 
 dredths, thousandths, ten thousandths; by this we find that 
 four places are required and as there are but two figures in 
 the number we prefix two O's and obtain the correct result 
 .0025. 
 
 1. Write 104 hundred thousandths. Ans. .00104. 
 
 Explanation. According to the above 
 diiections we write the 104 and then 
 commence on the right and numerate 
 thus; tenths, huudredths, thousandths, 
 ten thousandths, hundred thousandths. 
 
 This numeration shows that five places are required and as we 
 
 have but tkrie we therefore prefix two O's. 
 
 OPERATION. 
 .00104. 
 
 162. EXERCISES. 
 
 1. Write 10101 hundred billionths. 
 
 Ans. 
 
 .00000010101. 
 
 Write decimally, numerate and read the following : 
 
 2 
 
 314 millionths. 
 
 6. 1205 ten millionths. 
 
 s! 
 
 12 thousandths. 
 
 7. 897 hundred billionths. 
 
 4. 
 
 107 billionths. 
 
 8. 1 isextillionth. 
 
 5. 
 
 1 trillionth. 
 
 9. 21001 ten vigintillionths. 
 
 10. 
 
 _5* 
 
 14. 
 
 60409 
 
 17 87 
 
 
 1 
 
 100000000 
 
 10000 
 
 11. 
 
 TOO 
 
 
 748^ 
 
 If 
 
 
 
 15 
 
 
 18 7 
 
 12. 
 
 TOOO 
 
 1000000 
 
 100 
 
 
 1042J 
 
 1 fi 
 
 1 
 
 -. ^ 990099 
 
 i 3 
 
 
 10. 1 
 
 i ii ii m: !i >MI K i 
 
 *' i<vwv LI infWKWtAfi 
 
 J.O. 
 
 100000 
 
 
Reduction of Decimals. 133 
 
 163. PRINCIPLES. 
 
 From the foregoing work we recapitulate the following 
 principles : 
 
 1. Decimals are governed by the same laws of notation 
 as whole numbers, hence the value of any decimal figure 
 depends upon the place it occupies. 
 
 2. Each removal of the decimal point one place to the 
 right is equivalent to multiplying the decimal by 10. 
 
 3. Each removal of the decimal point one place to the 
 left is equivalent to dividing the decimal by 10. 
 
 4. Annexing or rejecting naughts at the right of any 
 decimal does not change its value. 
 
 REDUCTION OF DECIMALS. 
 
 164. To reduce Decimal Fractions to a common de- 
 nominator. 
 
 1. Reduce .7, .18, .2581 and .045 to a common denom- 
 inator. 
 
 OPERATION. 
 
 .7000 Explanation. To reduce decimals to 
 
 1800 E common denominator we have but to 
 
 ' annex a sufficient number of O's to give 
 
 each decimal the same number of places. 
 .0450 
 
 165. 70 reduce a decimal to a common fraction. 
 
 1. Reduce .25 to a common fraction. 
 
 OPERATION Explanation. In all problems of this 
 
 25 . ' kind we simply write the decimal as a 
 
 1 00 ==4 A- ns - common fraction and then reduce it to 
 its lowest terms. 
 
 2. Reduce .125 to a common fraction. 
 
 OPERATION. 
 
 TTnfV^* Ans - 
 
 3. Reduce .59| to a common fraction. 
 
 FIRST OPERATION. 
 
 59f J 
 
 --d^^^-if An, 
 
134 Arithmetical Exercises and Examples. 
 
 Explanation. To reduce complex decimals to simple fractions 
 we first write the decimal as a common fraction; then we reduce 
 both the numerator and denominator to the fractional unit of 
 the denominator contained in the numerator term of the frac- 
 tion, and thus obtain a complex fraction, which we reduce to a 
 simple fraction. 
 
 SECOND OPERATION. Ksplanation. Here, when redu- 
 
 59| cing the fraction to the fractional 
 
 47 5 19 A ns unit of the denominator contained 
 
 10n~~ ^ in thenumerator term of the frac- 
 
 tion, we shorten the work by 
 
 omitting the denominator (8) in both terms of the complex 
 fraction, and writing the result as a simple fraction. By this 
 process we save the operation of division, the result of which, 
 is the cancelling of the denominator in both terms of the com- 
 plex fraction. 
 
 Reduce the following decimals to common fractions : 
 
 4. 
 
 .8. 
 
 Ans. A. 
 
 13. 
 
 .88. Ans. 
 
 5. 
 
 .05 
 
 Ans. T f -. 
 
 14. 
 
 .909. Ans. 
 
 6. 
 
 .25. 
 
 Ans. 
 
 15. 
 
 .00025. Ans. 
 
 7. 
 
 .125. 
 
 Ans. 
 
 1G. 
 
 .4j. Ans. 
 
 8. 
 
 .675, 
 
 Ans. 
 
 17. 
 
 .055|. Ans. 
 
 9. 
 
 .105. 
 
 Ans. 
 
 18. 
 
 .008f Ans. 
 
 10. 
 
 .07. 
 
 Ans. T fo. 
 
 19. 
 
 .00054/s-. Ans. 
 
 11. 
 
 .005. 
 
 Ans. rfa. 
 
 20. 
 
 .999. Ans. 
 
 12. 
 
 .1045. 
 
 Ans. 
 
 21. 
 
 4007 r V Ans. 
 
 167. To reduce common fractions to equivalent deci- 
 mals. 
 
 1. Reduce f to a decimal. 
 
 OPERATION. Explanation. To reduce common frac- 
 
 8)3 000 tions to decimals, we annex naughts to 
 
 _ ' the numerator and divide by the denom- 
 
 inator, then point off as many places 
 .375 Ans. f or decimals as there were O's annexed. 
 When a remainder continues beyond four or six places, we dis- 
 continue dividing and write the sign -j- to the right of the last 
 figure obtained, which indicates that the quotient is not com- 
 plete. The annexing of O's to the numerator is equivalent to 
 multiplying it by 10 for each naught annexed, consequently 
 the quotient obtained is as many times 10 too great as there 
 were O's annexed ; and hence the reason for pointing off as 
 
Reduction of Decimals. 135 
 
 many places in the quotient as there were O's annexed to the 
 numerator. 
 
 2. Reduce -f- to an equivalent decimal. 
 OPERATION. 
 7)5.000000 
 
 .714285+ Ans. 
 3. Roduce yl-j- to an equivalent decimal. 
 
 FIRST OPERATION. SECOND OPERATION. 
 
 725)3.000000(4137+ 725)3. (.004137-f Ans. 
 
 2900 .004137+ Ans. 30 tenths. 
 
 1000 300 hundredths. 
 
 725 
 3000 thousandths. 
 
 2750 2900 
 
 2175 
 
 1000 ten-thousandths. 
 
 5750 725 
 
 2750 hundred thousandths, 
 2175 
 
 5750 millionths. 
 5075 
 
 675 Remainder. 
 
 Explanation. Here, in the first operation, we annex six O's 
 and obtain but 4 figures in the quotient. Therefore in order 
 to point off as many decimal places as we annexed O's, we pre- 
 fix two O's and thus obtain the correct result. The reason for 
 this will appear clear if we consider each step of the work as 
 performed in the second operation. We are to divide or meas- 
 ure 3 by 725, and we first see that 3 is not equal to 725 any 
 whole or unit number of times ; we therefore write the decimal 
 point in the quotient, annex a to the 3 units and thus 
 reduce it to 30 tenths, which we also see i? not equal to 725 
 any tenth times, and hence we write in the tenth's place of 
 the quotient; we then annex another and thereby reduce the 
 30 tenths to 300 hundredths, which we see is not equal to 725 
 any hundredths times, and hence we write in the hundredths 
 place of the quotient ; we then annex another and thereby 
 reduce the 300 hundredths to 3000 thousandths, which we see 
 
136 Arithmetical Exercises and Examples. 
 
 is equal to 725 4 times, with a remainder. We have now 
 obtained the first significant figure of the decimal, and we con- 
 tinue the division in the usual manner to the sixth decimal 
 place and annex the + sign to indicate that there is still a 
 remainder. 
 
 4. Reduce 6} to a decimal. 
 
 FIRST OPERATION. SECOND OPERATION. 
 
 6f =^ and V=)27.00 61=6 and f ; and 
 
 1=4)3.00 
 
 6.75 Ans. 
 
 .75+6=6.75 Ans. 
 
 Reduce the following fractions to equivalent decimals not 
 exceeding 6 places. 
 
 5. |f Ans. .71875 9. f Ans. .625 
 
 6. Ans. .336 
 
 7. Ans. .032 
 
 8. f of | Ans. .107142+ 
 
 10. T V Ans. .076923+ 
 
 11. .37 jV Ans. .370625 
 
 12. 47.18J Ans. 47.1875 
 
 Reduce f to a complex decimal of 3 places. 
 
 OPERATION. 
 
 3)2.000 
 
 .6661 Ans. 
 
 13. Reduce ^ to a complex decimal of 4 places. 
 
 Ans. .4285f 
 
 14. Reduce % to a complex decimal of 6 places. 
 
 Ans. .222222f 
 
 168. ADDITION OF DECIMALS. 
 
 Since decimals increase from right to left, and decrease 
 from left to right in a tenfold ratio as do simple whole num- 
 bers, they may be added, subtracted, multiplied and divided 
 ia the same manner. 
 
 1. Add .785, .93, 166.8, 72.5487 and 4.17. 
 
Subtraction of Decimals. 137 
 
 OPERATION. Explanation. In all problems of 
 
 ^oc this kind we write the numbers so 
 
 Q that units of the same order will 
 
 "** stand in the same column, and the 
 
 166.8 decimal poim be in a vertical line ; 
 
 72.5487 then we add as in simple whole 
 
 4 -[7 numbers. 
 
 When the addition is completed 
 
 . we point off in the sum, from the 
 ^45.Zoo7 Ans. right hand, as many places for dec- 
 imals as equal the greatest number of decimal places in any of 
 the numbers added. 
 
 Add the following numbers. 
 
 2. 3.25, 42.348, 748.4 and 29.32. Ans. 823.318. 
 
 3. .0049, 47.0426, 37.041 and 360.0039. 
 
 Ans. 444.0924. 
 
 4. 1121.6116, 61.87, 46.67, 165.13 and 676.167895. 
 
 Ans. 2071.449495. 
 
 5. .8, .09, 34.275, 562.0785 and 1.01. 
 
 Ans. 598.2535. 
 
 6. 81.61356, 6716.31, 413.1678956, 35.14671, 3.1671 
 and 314.6. . Ans. 7564.0052656. 
 
 7. l.Olf, 240.06J, 999.9, 80.6051 and .17. 
 
 Ans. 132^7576. 
 
 8. What is the sum of the following numbers : twenty- 
 five, and seven millionths ; one hundred forty-five, and six 
 hundred forty-three thousandths; one hundred seventy -five, 
 and eighty-nine hundredths ; seventeen, and three hundred 
 forty-eight hundred-thousandths. Ans. 363.536487. 
 
 9. A farmer has sold at one time 3 tons and 75 hun- 
 dredths of a ton of hay, at another time 11 tons and 7 
 tenths of a ton, and at a third time 16 tons and 125 thou- 
 sandths of a ton. How much has he sold in all? 
 
 Ans. 31.575. 
 
 169 SUBTRACTION OF DECIMALS. 
 
 1. From 345.3046 subtract 92.1435847. 
 
138 Arithmetical Exercises and Examples. 
 
 OPERATION. Explanation. In all problems of this 
 
 345.3046 kin , d we write the numbers so that 
 
 92 1435847 units of the same order will stand in 
 
 *'" the same column, and the decimal 
 
 o.o unniKQ A P intS bC ^ * Vertical Hne 5 theQ WC 
 
 ^Oo.lDlUlDo Ans. subtract as in simple whole numbers, 
 and point off in the difference, from the 
 
 right hand, as many places for decimals as equal the greatest 
 number of decimal places in either the minuend or subtra- 
 hend. 
 
 Wnen the decimal places in the subtrahend exceed those in 
 the minuend, naughts are understood to occupy the vacant 
 places, and may be filled in if it is desired. 
 
 EXAMPLES. 
 
 81.04089 121.25 532.8 *' 
 14.587 109.054:;- 9.00451681 
 
 66.45389 Ans. 12.19562 Ans. 523.79548319 Ans. 
 
 5. From 461.072 take 427.125. Ans. 33.947. 
 
 6. From 17.5 take 4.19. Ans. 13.31. 
 
 7. From 4000.0004 take 4.3. Ans. 3995 7004. 
 
 8. From three million take three inillionths. 
 
 Ans 2999999.999997. 
 9* From 11 take 1 and 9 thousand trillionths. 
 
 * Ans. 9.999999999991. 
 
 10. From 24000 subtract 2.078. Ans. 23997.922. 
 
 11. From 886.333 subtract 98.5427. Ans. 787.7903. 
 
 170. MULTIPLICATION OF DECIMALS. 
 
 1. Multiply 26.58 by 4.3. 
 
 OPERATION Explanation. In all problems of this 
 
 yp ro kind we multiply as in whole numbers, 
 
 and point off on the right of the product 
 
 as many places for decimals as there are 
 
 decimal places in both the multiplicand 
 
 7974 and multiplier. The reason for thus 
 
 1063:4 pointing off the 3 decimal places in this 
 
 problem is obvious from the fact that in 
 
 the multiplicand we have 2 decimal 
 
 114.294 Ans. places or hundredths, which we used as 
 
Multiplication of Decimals. 
 
 139 
 
 whole numbers and thereby produced a product 100 times too. 
 great; and in the multiplier we have 1 decimal place or tenths 
 which we also used as a whole number and thereby produced 
 a product 10 times too great ; and both together gives a pro- 
 duct 1000 times too great ; hence to obtain the correct product 
 we divide or point off 3 decimal places. 
 
 Explanation. In this operation we 
 reduce the factors to common frac- 
 tions and then multiplying them 
 together, we obtain a product of 
 HMf 1 which written decimally is 
 114.294 This process shows in 
 
 SECOND OPERATION. 
 
 100|2658 
 101 43 
 
 114294 
 
 1000 
 or decimally written 
 
 114.294 Ans. 
 2. Multiply 4.024 by .0056. 
 
 Ans. another way why we point off on 
 the right of the product as many 
 places for decimals as there are dec- 
 imal places in both factors. 
 
 OPERATION 
 4.024 
 
 .0056 
 
 24144 
 20120 
 
 .0225344 Ans. 
 
 Explanation. In all problems of this 
 kind where the number of figures in the 
 product is not equal to the number of 
 decimal places in the two factors, we 
 must prefix a sufficient number of O's to 
 supply the deficiency. In this example, 
 we prefix one 0. The reason of this 
 will appear evident by working the ex- 
 ample as a common fraction as shown 
 in the second operation of the first 
 problem. 
 
 EXAMPLES. 
 
 3. Multiply 27 by .9. 
 
 4. Multiply .38 by 8. 
 
 5. Multiply .75 by .42. 
 
 6. Multiply .OOr. by .0103. 
 
 7. Multiply 340.012 by 61.23. 
 S. Multiply .1234 by 1234. 
 
 Multiply 1500 l.y .00014. 
 
 Ans. 24.3. 
 Ans. 3.04. 
 \n<. .3150. 
 Ans. .0000618. 
 Ans 20818.93476. 
 Aus. 152.2756. 
 Ans. .21. 
 
 9. 
 
 10. What is the product of one thousand and twenty- 
 five multiplied by three hundred and twenty seven ten- 
 thousandths ? Ans. 33.5175. 
 
 11. What is the product of seventy-eight million two 
 hundred five thousand and two, multiplied by fifty-three 
 hundredths? Aus. 41448651.06. 
 
//, *' 
 *''>* 
 
 140 Arithmetical Exercises and Examples. 
 
 J2. Multiply one hundred and fifty-three thousandths 
 by one hundred and twenty-nine millionths. 
 
 Ans. .000019737. 
 
 13. Multiply 1 thousand by 1 thousandth. Ans. 1. 
 
 14. Multiply 2 million by 2 billionths Ans. .004. 
 
 15. What will 37.23 tons of hay cost at $20.75 per ton? 
 
 Ans. $772.52+. 
 
 16. What will 428.431 bushels cost at SI. 125 per 
 bushel? Ans. $481.98+. 
 
 171. To multiply a decimal or mixed number by 10, 
 100, 1000, etc. 
 
 1. Multiply 428.375 by 100. 
 
 OPERATION Explanation In all problems where 
 
 4-937 ^ *A *^ e mu ltiplier i 8 10, 100, etc., we sim- 
 
 4400*. Ans. ply remove the decimal point as many 
 
 places to the right as there are naughts in the multiplier, annex- 
 
 ing naughts if required. 
 
 2. Multiply 271.32 by 1000. Ans. 271320. 
 
 3. Multiply .756 by 100. Ans. 75.6. 
 
 4. Multiply .025 by 10. Ans. .25. 
 
 5. Multiply 61.052 by 10000. Ans. 610520. 
 
 172. DIVISION OF DECIMALS. 
 
 1. Divide 17.094 by 8.14. 
 
 FIRST OPERATION. Explanation. In all problems 
 
 8.14)17.094(2.1 Ans. of this kind > we divide as in 
 1 fi 2ft whole numbers, and then point 
 
 off as many places for decimals 
 from the right of the quotient as the 
 814 number of decimal places in the div- 
 
 814 idend exceeds those in the divisor, 
 
 observing to -supply any deficiency 
 by prefixing naughts. In this 
 
 problem the excess is one, and we therefore point off one deci- 
 mal place in the quotient. The reason for thus pointing is 
 obvious from the fact that in the dividend we had 3 decimals^ 
 or thousandths, and in the divisor we hadi cU&imatjor frcTTrtrs^ ' 
 and thousandths divided or decreased by t^atts* gives Iwwi- 
 dretifcks as a quotient. The reason will also appear plain if we 
 observe that the dividend is the product of the divisor and 
 
Division of Decimals. 141 
 
 quotient multiplied together, and hence we point off enough 
 decimal places in the quotient to make the number in the two 
 factors equal to the number in the product or dividend, accord- 
 ing to the principles shown in the first problem of multiplica- 
 tion of decimals 
 
 SECOND OPERATION. Explanation. In this opera- 
 
 1000 
 
 814 
 
 1701)4 tion we reduce the decimals 
 
 100 
 
 to common fractions and then 
 proceed as in the division of 
 mixed numbers. The reduc- 
 | ^y 1 ^- Arns. tion of the dividend and divi- 
 Decimally written 2.1 Ans. sor to common fractions and 
 then the mixed numbers to 
 
 improper fractious, is performed thus : the dividend 17.094= 
 17 rVo 4 <y = Vo% 9 o 4 -5 the' divisor 8.14==8^=fjf This method 
 also shows the reason for pointing off and may be used for all 
 problems in decimal fractions. 
 
 2. Divide 7898.50 by 2.4G83. 
 
 OPERATION. 
 
 2.4083)7898.5600(3200. Ans. 
 7401!) 
 
 49366 
 19366 
 
 oo 
 
 Explanation. Here we have an excess of decimals in the di- 
 visor, and in all cases of this kind we first make them equal by 
 annexing naughts to the dividend, and the quotient will be in 
 whole numbers. The reason for annexing the naughts will 
 appear more obvious by solving the problem in the form of a 
 common fraction. 
 
 3. Divide 7.07t51 by (JS7. 
 
 OPERATION Explanation. In this problem 
 
 087)7. 07(J1 ( 10.') there are 4 decimal plares in the 
 
 <>S7 .0103 Ans. dividend and none in the divisor, 
 
 hence nccording to the forego- 
 
 ;()/ I i"? instruction we must point 
 
 ~ oft 4 decimal places in the quo- 
 
 tient, and as there are but 3 fig- 
 ures in the quotient we prefix 1 
 
 naught. In all problems of this kind, O's are prefixed to supply 
 :niy ilvtieii'iiry of figures that may orenr. 
 
14:! Arithmetical Exercises and Examples. 
 
 4. Divide 47.789 by 39.27. 
 
 OPERATION. Explanation. 
 
 39.27)47.789(1.2168+ Ans. lem we have 
 3927 
 
 8519 
 7854 
 
 GG50 
 3927 
 
 Iu this prob- 
 , remainder, 
 
 after dividing the dividend, 
 of 6G5 ; to this and the 2 suc- 
 cessive remainders we annex 
 O's and continue the division 
 until we have produced 4 dec- 
 imal places. The annexing 
 of O's reduces the successive 
 remainders to the next lower 
 order of tenths and hence all 
 quotient figures produced by 
 annexing O's are decimals. 
 We therefore, point off from 
 the right of the quotient as 
 many places for decimals as 
 the number of decimals in the 
 dividend exceed those of the 
 divisor, plus the number of 
 
 O's annexed. This is done in all division problems where O's 
 are annexed, and a sufficient number of O's should be annexed 
 to produce 4 or 6 decimal places. When there is a remainder 
 after the last division the plus (-{-) sign should be annexed to 
 the answer to indicate that the quotient is incomplete. 
 
 7. Divide .112233 by 12. 
 
 27230 
 
 235G2 
 
 3GG80 
 31416 
 
 5. 
 
 Divide 1.12233 by 12. 
 
 OPERATION. 
 
 12)1.12233 
 
 8. 
 
 9352+r=.09352 + Ans. 
 Divide 11.2233 by 12. 
 
 OPERATION. 
 
 12)11.2233 
 
 9. 
 
 .9352+ Ans. 
 Divide .0004869 by 396. 
 
 OPERATION. 
 
 396)4869(12+ Ans. 
 
 396 =.0000012+. 
 
 909 
 792 
 
 117 
 
 OPERATION. 
 
 12). 112233 
 
 9352+=.009352+ Ans. 
 , Divide 112.233 by 12. 
 
 OPERATION. 
 
 12)112.233 
 
 10. 
 
 9.3627+ Ans. 
 Divide .0004869 by 3.96. 
 
 OPERATION. 
 
 3.96). 0004869(12+ Ans. 
 396 =.00012 -f- 
 
 909 
 792 
 
 117 
 
Division of Decimals. 
 
 143 
 
 11. Divide .0004869 by .0396. 
 
 FIRST OPERATION. 
 
 .0396).00048t>9(122-r Ans. 
 396 =.0122 + 
 
 909 
 792 
 
 1170 
 792 
 
 378 
 
 SECOND OPERATION. 
 
 396)4869(122-f-=.0122-f Ans. 
 396 
 
 909 
 
 792 
 
 1170 
 792 
 
 378 
 
 12. 
 13. 
 14. 
 15. 
 16. 
 17. 
 18. 
 19. 
 20. 
 
 Divide 67.8632 by 32.8. 
 Divide 983 by 6.6. 
 Divide 13192.2 by 10.47. 
 Divide 67.56785 by .035. 
 Divide .00125 by .5. 
 Divide 7.482 by .0006. 
 Divide 1 by 999. 
 Divide 84375 by 3.75. 
 Divide 1081 by 39.56 
 Divide 35.7 by 485. 
 
 Ans. 2.069. 
 Ans. 148.939+. 
 Ans. 1260. 
 Ans. 1930.51. 
 Ans. .0025. 
 Ans. 12470. 
 Ans. .001001+. 
 Ans. 22500. 
 Ans. 27.3255+. 
 Ans. .0736+. 
 
 21. 
 
 22. If rice costs $.0775 per pound, how many pounds 
 can be bought for $40.64875 ? Ans. 524:5 pounds. 
 
 23. Sold 14.75 acres of land for $191.75. What was 
 the price per acre? Ans. $13. 
 
 24. Divide four thousand three hundred twenty-two, 
 and four thousand five hundred seventy-three ten-thous- 
 andths by eight thousand and nine thousandths. 
 
 Ans. .5403+. 
 
 173. To divide Decimal Fractions by 10, 100, 1000, 
 etc. 
 
 1. Divide 48.76 by 10. 
 
 OPERATION. Kxpltnation. In all problems of this 
 
 4.876 Ans. kind we simply remove the decimal 
 
 point as many places to the left as there are O's in the divisor. 
 The reason for this was fully shown on page 133. When there 
 are not a sufficient number of figures in the dividend to alkw 
 this to be done naughts must be prefixed to supply the defi- 
 ciency. 
 
IN Arithmetical Exercises and Examples. 
 
 2. Divide 875.25 by 100. Ans. 8.7525. 
 
 3. Divide .52:n by 1000. Ans. .0005231. 
 
 4. Divide 72 by LOOOO. Aris. .0072. 
 
 5. Divide 9.85 by loo. Ans. .0985. 
 0. Divide .025 by 200. Ans. .000125. 
 7. Divide 412.99 by 10. Ans. 41.299. 
 
 174. MISCELLANEOUS PRACTICAL PROBLEMS. 
 
 Much a* occur /;/ fhr c<;itnfin(/ rut nn. fnctnry. irurkxhop, on 
 the plantation^ am/ in the nir/W.s departments of bu- 
 i life. 
 
 1. What is the cost of 1465 pounds of corn at 84 cents 
 per bushel, and how maov bushels are there? 
 
 Ans. S21.97} cost. 
 2t; Bush., 9 flbs. 
 
 2. Sold 51*94 pounds of hay at $23.75 per ton. How 
 many tons were there, and what was the value of it? 
 
 Ans. 2 tons, 1294 Ibs. 
 $62.86| value. 
 
 I. Pxui-ht 320-42 bushels of wheat at $1.95 per bushel. 
 What was the cost? Ans. $625. HOA. 
 
 4. Bought 11361 pounds of dried peaches at $5.80 
 per bushel. How many bushels were there and what did 
 they cost? Ans. 34 bushels. 14} Ibs. 
 
 $199.74ff'cost. 
 
 5. Bought 15 bushels and 31 pounds of corn at 78} 
 cents per bushel. What was the cost ? 
 
 Ans. $12.20if|. 
 
 6. Bought 3 coops of chickens containing 2 dozen and 
 7 chickens each, at $4.35 per dozen. What did they 
 cost? Ans. $33.71}. 
 
 7. What will 74 pounds and 11 ounces of butter cost 
 at 42} cents per pound? Ans. $31.74-^. 
 
 8. Bought 36 pounds and 7 ounces of tea at $1.12} 
 per pound. What did it cost ? Ans. $40.99^-. 
 
 9. Butter is worth 45 cents per pound. How much 
 can be bought for 20 cents ? Ans. 7-^ ounces. 
 
Miscellaneous Practical Problems. 145 
 
 10. What is the cost of 31845 feet of lumber at 22.25 
 per M. ? Ans. $708.55J. 
 
 11. What will 183 feet of lumber cost at $25.75 per 
 M. ? Ans. $4.71 T V 
 
 12. Bought 3 bales of hay weighing as follows: (1) 
 421 pounds (2) 394 pounds, (3) 487 pounds, at $22.50 
 per ton. What did it cost ? Ans. $14.64f. 
 
 13. Sold 3] dozen boxes Spencerian pens at $108 per 
 gross. What did they amount to? Aus. $29. 2.1. 
 
 14. How much coffee can I buy fur 5 cents when a 
 pound costs 28 cents? Ans. 2^ ounces. 
 
 15. What is the cost of 400 T. 2 cwt. 3 qrs. 20 Ibs. of 
 iron at $60 per ton of 2240 pounds ? 
 
 Ans. $24008.784. 
 
 1 G. A planter shipped 6 dozen dozen boxes of peaches 
 to market, but being delayed on the way -I a dozen dozen 
 boxes spoiled ; the remainder were sold it 70 cents per box. 
 What did they amount to? Ans. $554.40. 
 
 17. Bought 12 dozen and 5 huts at $11 per dozen. 
 What did they cost ? Ans. $136.58$. 
 
 18. Wh;it. is the amount due for the freight of 40000 
 % pounds of merchandise fur JMJ5 miles at fn' for 100 pounds 
 
 for 100 miles? Ans. $193. 
 
 11). What is the cost of 2381 J pounds of cotton at 
 17|| cents IHT pound ? Ans. $ t24.24f|. 
 
 20. What is the cost of a 14 carat gold chain that 
 weighs 4 o/. 7 pwt. 15 gr. at $1.20 per pwt. for pure gold, 
 allowing > <' per irrain on full weight for manufacturing and 
 the alloy? Ans. ^71.85*. 
 
 21. A hardware merchant received from Liverpool an 
 invoice of iron weighing 2 T. 2 cwt. 3 <jrs. 20 Ibs., long 
 tori weight; the invoice price was 12, 17s. Gd. per ton; 
 What did the whole cost in sterling money? 
 
 Ans. 27, 12s. 8fd. 
 
 22. Bought 4(192 pounds of barley at $.88 per bushel. 
 How many bushels were there and what was the cost? 
 
 Ans. 97 bush. M Ibs. 
 , Cosi S<;.02. 
 
146 Arithmetical Exercises and Examples. 
 
 23. Bought 2765 pounds of oats at 76/ per bushel. 
 What was the cost, and how many bushels were there ? 
 
 Ans. $65.661 cost. 
 86 bush. 13 Ibs. 
 
 24. What is the cost of 4878 pounds of wheat at $2.45 
 percental? Aus. $119.511. 
 
 25. What is the cost of 200 sacks of guano each weigh- 
 ing 162 pounds, at $52 i per ton ? Ans. $846.45. 
 
 26. What is the value of 5790 hoop-poles at $18 per M ? 
 
 Ans. 8104.22. 
 
 27. What is the value of 8750 shingles at $8.75 per M? 
 
 Ans. $76.f>r>i. 
 
 28. What* is the value of 11428 fence pickets at $9 
 per M? Ans. $102.852. 
 
 29. What is the value of 1364 pine apples at 
 per C.?' Aus. 
 
 30. What is the cost of 2417 cocoanuts at 
 C.? Ans. 
 
 31. What is the value of 78420 railroad ties at $75 
 per M.? Ans. $5881.50. 
 
 ',\'2. What is the freight on 540 bales cotton, weighing 
 243084 pounds, |d. per pound from New Orleans to Liv- 
 erpool? Ans. 633 Os. 7 3d. 
 
 33. What is the freight in United States currency on 
 25000 bushels corn from New Orleans to Liverpool, at 24s. 
 per imperial quarter of 480 pounds ; allowing 1 to be 
 equal to $4.87 ? Ans. $17045. 
 
 34. How % many square feet in a pavement 120 feet 4 
 inches long and 10 feet wide? Ans. 12U3J sq. feet. 
 
 35. How many square yards in a plat of ground 140 
 feet 3 inches long and 64 ieet 6 inches wide ? 
 
 Ans. 1005i sq. yards. 
 
 36. How many squares in the roof of a building 78 feet 
 6 inches long, and 48 feet 4 inches wide ? 
 
 Ans. 37.94J squares. 
 
 37. How many square yards of plastering in the walls 
 and ceiling of a room which is 40 feet 6 inches long, 24 
 feet 8 inches wide, and 14 feet high, deducting 3 square 
 
Miscellaneous Practical Problems. 147 
 
 yards for doors, windows and base-hoard, and what will it 
 cost at 35 cents per square yard ? 
 
 Ans. 279ff sq. yards. 
 $97.90ff cost. 
 
 38. How many sq. feel in 8 boards, each measuring 16 
 feet long and 17 inches wide, and what will they cost at 
 2i/ per foot? Ans. 181i feet. 
 
 $4.53J cost. 
 
 39. How many square feet in 13 pieces of plank, each 
 measuring -0 feet 6 inches Jong, 14 inches w^de and 3 
 inches thick, and what is the cost at $23 per M.? 
 
 Ans. 932| feet. 
 $21-453} cost. 
 
 40. How many square feet in a circle, the diameter of 
 which is 12 yards? Ans. 1017.8784 sq. feet. 
 
 41. How many shingles will it require to shingle a 
 building, the roof of which measures 44 feet 7 inches from 
 cave to cave, without allowances, by 50 feet 4 inches long, 
 allowing a shingle to cover a space 4 inches wide and 5 
 inches long ? Ans. 16157. 
 
 42. A. yard is 24 feet 3 inches long by 11 feet 5 inches 
 wide ; how many brick, 4 by 8 inches will it take to pave 
 it, no allowance to be made for the openings between the 
 bricks? Ans. 1245$ J, 
 
 43. How many square yards of paving in a sidewalk 64 
 feet long and 11 feet 8 inches wide? 
 
 Ans. 82|^ square yards. 
 
 44. How many flags, each 16 inches square, will it 
 require to flag a walk 22 yards 1 foot 4 inches long and 6 
 feet 8 inches wide ? Ans. 252"J. 
 
 45. How many yards of carpeting that is 27 inches 
 wide, will it take to cover the floor of a room that is 25 
 feet 6 inches long, and 22 feet 9 inches wide, making no 
 allowance for waste in matching or turning under? 
 
 Ans. 85j- yards. 
 
 46. A water pipe is 50 feet 9 inches long, and its diam- 
 eter is 30 inches ; what is its concave surface ? 
 
 Ans. 57397.032 inches. 
 
148 Arithmetical Exercises and Examples. 
 
 47. How many cubic feet in a box 5 feet long, 3 feet 
 wide and 4 feet deep ? Ans. 60 cubic feet. 
 
 48. What is the freight on a box 6 feet 4 inches long, 
 4 feet wide and 3 feet 9 inches deep at 25 cents per cubic 
 foot? Ans. $232. 
 
 49. What will be the freight on a box 9 feet 3 indies 
 Ion-, 4 feet 6 inches wide, 2 feet 10 inelh- de p. at 30 
 c -iits a cubic foot? Ar.s. $3r> : 
 
 50. How many bushels will a bin hold, that is 10 feet 
 long, 8 feet 6 inches wide, and 5 feet 2 inches '.cep? 
 
 Ans. :;:>LVJO-|- bushels. 
 
 51. How many cords of wood in two ranks, each 4-1 fret 
 long and 6 feet 3 inches high? Ans. 1 7 , :l ( . cords. 
 
 52. How many barrels will a (jiiadi ilatcral cistern hold, 
 whosejieight is 12 feet and width of side f> icet S inches? 
 
 Ans. HI-,*; 2 , 1 ; ban 
 
 53. How many cubic yards in a levee 80 rods loiiir, IO 
 feet wide at the base, 12-j feet at the top, and 5 feet 1 ineln s 
 average depth? Ans. IM.V 
 
 54. How many gallons will a box hold, that is f> fed 
 long. 2 feet 4 inches wide, and 3 feet deep ? 
 
 Ans. 201.81 + gallons. 
 
 55. How many cubic feet in a cylinder U fret long 3 ieet 
 4 inches in diameter? Ans 52.36 cubic feet. 
 
 50. IIow many gallons in a cylindrical cistern, 9 feet b* 
 inches high, and 7 feet 2 inches in diameter? 
 
 Ans. 28(Hi.<)S!Mi gals. 
 
 57. How many pints in a cylindrical vessel, whose height 
 is 14 inches and diameter 12-1 inches? 
 
 Ans. 59.5 pints. 
 
 58. How many bushels in a cylinder shaped box. whose 
 height is 10 feH, and diameter 10 feet? 
 
 Ans. 031.125 bu. 
 
 59. How many cubic feet in a frustum of a cone, whose 
 height is 6 feet, diameter of the greater end is 4 feet and of 
 the smaller end 3 feet? Ans. 58.1196 cubic feet. 
 
 60. How many gallons in a cistern which is in the form 
 of a frustum of a cone, whose height is 9 feet inches, 
 
Bills and Invoices. 149 
 
 lower base 7 feet 2 inches, and upper base 6 feet 8 inches ? 
 
 Ans. 2671.3392 gals. 
 
 61. A farmer has a heap of grain in a conical form, the 
 diameter of which is 14 feet 4 inches, and the depth 5 feet 
 3 inches ; how many bushels does it contain ? 
 
 Ans. 226.906. 
 
 62. A barrel i* 26 inches long, 17 inches in diameter at 
 the head, and 20 inches in diameter at the bung or center. 
 The staves have a medium curve. How many gallons will 
 it hold? Ans. 3 1. 244 -f- gallons. 
 
 For full in ormation and a thorough elucidation of all 
 questions pertaininir to the mensuration of surfaced and 
 solids, as contained in the foregoing miscellaneous problems, 
 and in the following bills, see Soule's Philosophic Commer- 
 cial and Exchange Calculator, pages 741 to 796. 
 
 174. BILLS AND INVOICES. 
 
 Bills in a general sense, embrace all written statements 
 of accounts and many legal instruments of writing ; but 
 in a more common and limited sense they are statements of 
 goods sold or delivered, services rendered or work done, with 
 the price or value, quality or grade of each article or item. 
 They should state the place and date of each sale, the names 
 of the buyer and seller, the extra charges or discount to be 
 allowed, and the terms of the sale. 
 
 When goods are bought to sell again, or when bills are 
 rendered to a jobber or retailer, or consigned to an agent, 
 the bill is then called an invoice. 
 
 It is the custom of accountants and merchants, when making 
 bills to commence the name of each article with a capital. 
 
 When a charge is made for the box, barrel, jar etc. contain- 
 i Q g goods, it is customary to write its price above and to the 
 right of it and add the same to the cost of the goods it con- 
 tains. 
 
 In making extensions, fractions of cents are not used in the 
 
150 Arithmetical Exercises and Examples. 
 
 product ; when they are \ or more they are counted cents 
 when they are less than J they are not counted. 
 
 In making the following bills, students should use pen and 
 ink and give earnest attention to the proper form and spacing, 
 to plain, neat and rapid penmanship of both words and figures, 
 and above all to the accuracy of extentions and additions. 
 
 When notes or bills of exchange are given in payment, the 
 student should draw the same and correctly mature them. 
 
 No. 1. 
 NEW ORLEANS, Jan. 2, 1877. 
 
 H. A, & R. C. Spencer, 
 
 TERMS Cash. 
 
 Hot. of A. L & E. Soule. 
 
 1876 
 
 
 
 
 
 
 Dec. 
 
 u; 
 
 2 bags Rio Coffee, 325 Ibs. @ 
 
 $ 23ic. 
 
 $ 76 
 
 38 
 
 
 
 50 c. 
 
 
 
 
 
 
 1 bbl. Sugar, 234J Ibs. 
 
 9 c. 
 
 21 
 
 61 
 
 
 
 i Chest Black Tea, 35 tbs. " 
 
 87 Jc. 
 
 30 
 
 63 
 
 
 
 1 bbl. Rice, 24^-1 6 =227" 
 
 8 c. 
 
 18 
 
 16 
 
 
 
 40 gal. N. 0. Molasses 
 
 75 c. 
 
 30 
 
 00 
 
 
 
 6 doz. Brooms 
 
 4.15 
 
 24 
 
 DO 
 
 
 
 3 bbls. XXX Family Flour" 
 
 8.12J 
 
 24 
 
 38 
 
 
 
 25 Ibs. Cream Crackers 
 
 16 c. 
 
 4 
 
 00 
 
 
 
 50 Ibs. Graham do " 
 
 15 c. 
 
 7 
 
 50 
 
 
 
 20 Ibs. W. Butter 
 
 30 c. 
 
 6 
 
 00 
 
 Rec'd pay't, 
 
 $243.56 
 
 A. L. & E. SOULE, 
 Per S. Richardson. 
 
Bilk and Invoiees. 
 
 No. 2. 
 
 151 
 
 Nfcw ORLEANS, Jariy 31, 1877. 
 
 S. S. Packard and E, G, Folsom, 
 
 Bot. of W. E. and Frank Soule. 
 
 TERMS Note at 30 days. 
 
 1877 
 
 
 
 
 
 
 Jan. 
 
 31 
 
 453 Ibs. Mocha Coffee, @ 
 
 $ 25 c. 
 
 
 
 
 
 241 Rio Coffee, 
 
 18|c. 
 
 
 
 
 
 31 6 1 C. Sugar, 
 
 12Jc. 
 
 
 
 
 
 72 Duryea's Starch, 
 
 6ic. 
 
 
 
 
 
 64 N. Y. C. Cheese, 
 
 17ic. 
 
 
 
 
 
 52 W. F. Cheese, 
 
 15 c. 
 
 
 
 
 
 180 B. Sugar, 
 
 7*c. 
 
 
 
 
 
 80 doz. C. Eggs, 
 
 37 ic. 
 
 
 
 
 
 42 gals. N. 0. Molasses, 
 
 62c. 
 
 
 
 
 
 320 Ibs. G. Butter, 
 
 35 c. 
 
 
 
 
 
 23 " Almonds, 
 
 27 c. 
 
 
 
 
 
 76 Y H. Tea, 
 
 74 c. 
 
 
 
 
 
 68 boxes Shrimp, 
 
 48 c. 
 
 
 
 
 
 84 boxes Lobsters, 
 
 34 c. 
 
 
 
 
 
 92 gals. N. 0. G. Syrup, 
 
 96 c. 
 
 
 
 
 
 114 B. Whiskey, 
 
 1.08 
 
 
 
 
 
 112 bags Salt, 
 
 93 c. 
 
 
 
 
 
 320 Bbls. Sweet Potatoes, 
 
 1.25 
 
 
 
 
 
 S2 kits, No. 1 Mackerel, 
 
 i50 
 
 
 
 
 
 63 Ibs. S. Crackers, 
 
 11 c. 
 
 
 
 
 
 24f " P. L. Soap, 
 
 8Jc. 
 
 
 
 
 
 18i " Codfish, 
 
 9Jc. 
 
 
 
 
 
 Drayage $31.25, boxes $2.50. 
 
 
 
 
 Rec'd pay't by note at 60 days, $1,492 09 
 W. H. & F. SOULE. 
 
 per J. J. Manson. 
 
 NOTE. All of the extensions of this bill should be made mentally 
 apid mental work see SoulS's Contractions In Numbers. 
 
 For 
 
Arithmetical Exercises and Examples. 
 
 No. 3. 
 NEW ORLEANS, Jariy 31, 1877. 
 
 Bot. of J. B. Cundiff. 
 
 J. M. Butchee, 
 
 TERM r.Mlit. 
 
 
 
 875 bbls. Nes. Potatoes @ 
 
 $ 4.25 
 
 
 
 
 
 H() l P. B. Potatoes, 
 
 3.87 J 
 
 
 
 
 
 325 l Perfect' n Flour, 
 
 8.50 
 
 
 
 
 
 i:*.24 St. L. XX " 
 
 6.62J 
 
 
 
 
 
 1 1 2 l F'ily Clear Pork ' 
 650 ' Prime Pork, 
 
 17.50 
 13.75 
 
 
 
 
 
 220 kegs Pig Feet, 
 1LM h.lfbblsF.M. Beef, l 
 
 7.50 
 11. 
 
 
 
 
 
 1ST 2 Ibs. Choice Ham, 
 289 " B. Bacon, 
 10(J Piir Tonnes, 
 
 14 c. 
 9Jc. 
 8 c. 
 
 
 
 Rec'd pay't, $31167 27 
 
 NOTE All the extensions of this bill shall be made mentally. 
 
 E. C. Spencer & Co., 
 
 TERMS Cash. 
 
 o. 4. 
 NEW YORK, Dec. 8, 1876. 
 
 Bot. of B. D. Rowlee & Co. 
 
 20 doz. Missionary Bibles, @ $15.25 
 
 108 " small New Testam't, u 2.50 
 
 65 " Prayer Books, >< 2.25 
 
 65 " Hymn Books, " 3. 
 
 3 Bible Dictionaries, " 4. 
 
 i doz. Webster's Dict'ry u 50. 
 
 Rec'd pay't, $ 953 25 
 
 B. D. ROWLEE & CO. 
 
 Per E. Conrad. 
 
Bills and Invoices. 
 
 153 
 
 Wm. Melchert & Co., 
 
 No. 5. 
 NEW ORLEANS, Jariy 31, 1877. 
 
 Bot. of L L Willi&ms & Go. 
 
 
 
 321 
 
 Ibs. 
 
 Tobacco Low Lugs, 
 
 (a). 6 c. 
 
 
 
 
 
 1140 
 
 u 
 
 " 
 
 Med. Lugs, 
 
 - 7Jc. 
 
 
 
 
 
 509 
 
 u 
 
 i 
 
 Low Leaf, 
 
 ' 9ic. 
 
 
 
 
 
 965 
 
 u 
 
 
 
 Med. Leaf, 
 
 1 11 Jc. 
 
 
 
 
 
 398 
 
 u 
 
 i 
 
 Good Leaf, 
 
 ' 13fc 
 
 
 
 
 
 2416 
 
 u 
 
 i 
 
 Fine Leaf, 
 
 < 15 c 
 
 
 
 
 
 713 
 
 U 
 
 t 
 
 Selections. 
 
 ' 16|c. 
 
 
 
 Bryant and Nelson, 
 
 TERMS Dft. 30 days. 
 
 Rec'd pay't, $ 
 
 L. L. WILLIAMS & CO. 
 No. 6. 
 NEW ORLEANS, Dec. 17, 187 1). 
 
 Bot. of Wm. Horn & Go. 
 
 1876 
 
 
 1420 Ibs. Su^ar, Common, @ 5c. 
 1927 ' " Good, l 71c. 
 2810 4 ^ Fair. ' 7ic. 
 902 < "'Prime, ' 8Jc. 
 813 ' " Choice, 9}c. 
 2741 ' Yellow Cent'al ' lOJc. 
 
 
 
 Rec'd pay't, $ 
 
 No. 7. 
 
 NEW ORLEANS, Dec. 23, 1876. 
 Montgomery & Lettellier, 
 TERMs-3 mos. Bot. of Sadler & Smith. 
 
 1 Gross Chewing Tobacco @ $13. 
 180 Ibs. Smoking do 1.40 
 
 6 M. Havana Cigars ' 70. 
 
 2 M. N. 0. Manufacture do : > 30. 
 
 Rec d pay ? t. 
 
154 Arithmetical Exercises and Examples. 
 
 No. 8. 
 
 NEW ORLEANS, Jariy 21, 1877. 
 Jones and Carpenter/ 
 
 Bot. of Stewart & Henderson. 
 
 TERMS Note 60 days. 
 
 1877 
 
 34 bbls. La Orange, large, @ $5.75 
 27 boxes Messina Lemons, " 6.00 
 03 cases Malaga Grapes, " 1.75 
 45 boxes California Pears, " 4.50 
 5 mats Dates, 593 Ibs., " 7ic. 
 
 Rec'd pay't, by note at 60 days. $ 
 
 STEWART & HENDERSON. 
 
 No. 9. 
 NEW ORLEANS, Nov. 17, 1876. 
 
 Geo. B. Brackett & Co., 
 
 Bot. of R. Spencer Soule, 
 
 TEKM8 Cash. 
 
 1427 bu.No.l Winter Wheat $1.55 
 
 856 " No. 2 Winter Wheat " 1.47 
 
 420 " 111. No. 1 White do " 1.41 
 
 3145 " W. Corn " .70 
 
 1040 a B. Oats " .55 
 
 Rec'd pay't, $ 
 
 R. SPENCER SOULE. 
 
 No. 10. 
 NEW ORLEANS, Feb. J., 1877. 
 
 F. L & W. P. Richardson, 
 
 Bot. of P. W. Sherwood & Co. 
 
 TERMS 1 mo . 
 
 ;0 box. Sperm Candles, 596 Ibs. @ .35J 
 24 do. Adam. Ex. do. 483 " " .28 
 15 do. Sil. Gloss St'rch 360 .lOfi 
 Rec'd pay't. : 
 
Bills and Invoices. 
 
 155 
 
 Eeald & Howe, 
 
 TERMS Due Bill 1 mo. 
 
 No. 11. 
 
 NEW ORLEANS, Jan. 19, 1877. 
 
 Bot. of Cole & Montague. 
 
 
 
 342 Ibs. La. Pecans @ 
 
 $ .13 
 
 
 
 
 
 289 " Taragona Almonds 
 175 " Naples Walnuts " 
 196 " Brazil Nuts " 
 
 .21 
 .17 
 .11 
 
 
 
 
 
 268 " Western Chestnuts 
 
 .18 
 
 
 
 
 
 160 Boxes Figs 
 585 Cocoanuts @ $45 per M. 
 61 Bunches Bananas 
 
 .20 
 1.75 
 
 
 
 
 
 14 do. Plantains " 
 
 .85 
 
 
 
 
 
 327 Pine Apples @ $80 per M. 
 
 
 
 
 Hibbard & Gray, 
 
 TERMS Cash. 
 
 Rcc'd pay't, $ 
 
 COLE & MONTAGUE. 
 
 No. 12. 
 NEW ORLEANS, Feb. 1, 1877. 
 
 Bot. of Odell & Faddis. 
 
 
 
 2714 Ibs. Black Moss @ 4 / 
 1829 " Gray do. 1J/ 
 913 " Wool < 24 / 
 74 " Live Geese Feathers " 65 / 
 1528 Packages Broom Corn " 6 ^ 
 752 Ibs. Baling Twine " 14 / 
 800 yds. Indian Bagging " 11 ft 
 
 
 
 Rec'd pay't, $ 
 
 ODELL & FADDIS, 
 
 Per C. P. Meads. 
 
156 Arithmetical Exercises and Examples. 
 
 No. 13. 
 
 NEW ORLEANS, Jan. 7, 1877. 
 Spaulding & Musselman, 
 
 Bot. of Warr & Bogardus. 
 
 TERMS 60 days. 
 
 
 
 78i yds. Black Silk, @ $2.90 
 
 
 
 
 
 148 Muslin, < 16 c. 
 
 
 
 
 
 62 Cassimeres, 1.75 
 
 
 
 
 
 -38} Blk. F. Broadcloth 5.25 
 
 
 
 
 
 45 " " Doeskin, 1.50 
 
 
 
 
 
 324 American Satinets, ' 95 c 
 
 
 
 
 
 3 cases, each 40 psr Amos- 
 
 
 
 
 
 keag Sheetings, 
 
 
 
 
 
 1123 1 1 
 
 1204 L 3423} yds. " IT-'.c. 
 
 1096 2 j J 
 
 
 
 
 
 142 yds. 6-4 Alpaca, " 32 c. 
 
 
 
 
 
 560 " Union Ginghams, " ll}c. 
 
 
 
 
 
 491 " Am. Fancy Prints, " 12 c. 
 
 
 
 
 
 107 " Manch'ter Delains, " 21}c. 
 
 
 
 . 
 
 
 10 doz. Handkerchiefs, " 2.15 
 
 
 
 
 
 Ladies Hose. * 
 
 
 
 
 
 2 ps. 61 j yds. Can. Flannel u 18 c. 
 
 
 
 
 
 Rec'd pay't, \ 
 
 \ 
 
 
Tasker & Felton, 
 
 TERMS Cash. 
 
 Bills and Invoices. 157 
 
 No. 14. 
 NEW ORLEANS, Feb. 8, 1877. 
 
 Sot. of Allen & Shields. 
 
 
 50 
 
 Ibs. Casing Nails, 
 
 @ 
 
 $ 7c. 
 
 
 
 
 i; 
 
 duz. Mortice Locks, 
 
 u 
 
 7.50 
 
 
 
 
 * 
 
 " Porcelain Knobs, 
 
 U 
 
 4.75 
 
 
 
 
 5(1 
 
 pr. Butts, 
 
 U 
 
 25c. 
 
 
 
 
 ' 
 
 Gross Screws, 
 
 u 
 
 75c. 
 
 
 
 
 I 8 
 
 bars, \\ X 2 Bar Iron, 
 
 
 
 
 
 
 
 254 Ibs., 
 
 u 
 
 5c. 
 
 
 
 
 2 
 
 Rowland No. 2 Spades, 
 
 u 
 
 1. 
 
 
 
 Lee and Ward, 
 
 TERMS Due end mo. 
 
 Rec'd pay't, $ 
 
 ALLEN & SHIELDS. 
 
 No. 15. 
 NEW ORLEANS, Mar. 9, 1877. 
 
 Bot. of Harris & DeRussy, 
 
 ?> reams Cap Paper, -@ $3.25 
 
 2 doz. Ebony Rulers, 3.50 
 4 6 qr. Med. Ledgers 24 qrs. 1.75 
 33" Demy Journals, 9 " 1.25 
 33 " Cash Books, 9 " 1.25 
 
 3 6 " " Sales Books 18" 1.15 
 
 4 gross Pen-holders, 2.10 
 
 1 doz. Black Ink, 4.50 
 } ream Blotting Paper, 2.5() 
 
 2 doz. Mucilage, 2.75 
 
 3 " Carmine, 2.15 
 
 doz. Bill-books, 4.25 
 
 Rec'd pay't, 
 
158 Arithmetical Exercises and Examples. 
 
 No. 16. 
 NKW ORLEANS, April 1, Jf<S77. 
 
 E. J. & R. Paul, 
 
 TERMS Due bill 30 days. 
 
 Bot. of Gresham & Harp. 
 
 
 
 J doz. Comb's Con. of Man, (a), 
 
 H2. 
 
 
 
 
 
 2 ." Dana's Geological Story 
 
 
 
 
 
 
 briefly told, 
 
 10. 
 
 
 
 
 
 6 Soule's Phi. Arithmetics 
 
 42. 
 
 
 
 
 
 6 " Con. in Numbers, 
 
 if 
 
 
 
 
 
 6 " Prim. Arithmetic 
 
 9. 
 
 
 
 
 
 } Webster's Acad. Diet., 
 
 18. 
 
 
 
 
 
 2 Swinton's Lan. Lessons, 
 
 3.25 
 
 
 
 
 
 li ' Steel's Nat. Philosophy, 
 
 12. 
 
 
 
 
 
 i " Spencer's Science of 
 
 
 
 
 
 
 Sociology, " 
 
 10.50 
 
 
 
 
 
 2 copies Wood's Byron, 
 
 3. 
 
 
 
 
 
 4 k ' Dick's Shakspeare, ' 
 
 4.50 
 
 
 
 
 
 dray age 75^, box 50/. 
 
 
 
 
 T. Janney, 
 
 Eec'd pay't. $ 
 
 GRBSHAM & HARP. 
 
 No. 17. 
 
 NEW ORLEANS, Jan. 1, 1876. 
 
 To A. Laborde. 
 
 Dr. 
 
 For furnishing, making, and laying|| 
 Bru sels carpeting 22 inches wide|| 
 in 2 rooms measuring as follows :'| 
 No. 1. 24 ft. 3 in. X 20 ft. 9 in. 
 No. 2. 18ft. 6 in. X 17 ft. 8 in. 
 * yards, @ $1.95 
 
 68 " 6-4 Che. Mat, " 90c. 
 Rec'd pay't. $ 
 
 * In this bill no allowance is made for waste in matching or otherwise. 
 
Bills and Invoices. 
 
 159 
 
 Warner & Cornell, 
 
 TERMS Note at 60 days. 
 
 No. 18. 
 NEW ORLEANS, Jan. 25, 1877. 
 
 Bot. of Goldsmith and Clark. 
 
 2143 'l.s. bush. Yellow 
 
 Corn. @ $ 61c. 
 
 12-J1 ll.s.- " Texas 
 
 Wheat. 1.70 
 
 852 ibs. " White 
 
 Oats. " 54c. 
 
 7-!l Ibs. u Barley" 83c. 
 
 1427 " Cwt. Bran, " 75c. 
 
 - Tons Timo 
 thy Hay, " 18.50 
 
 1701 Ibs. -Tons Clover 
 
 Hay, " 20. 
 
 Rec'd pay't, by note at 60 days. $ 
 
 GOLDSMITH AND CLARK. 
 
 No. 19. 
 
 NEW ORLEANS, Jan. 3, 1877. 
 Geo. F, Bartley & Co., 
 
 To Steamship Knickerbocker and Owners, Dr. 
 
 For Freight on cubic feet @ 25/ 
 
 The same bein^r contents of 8 boxes 
 measuring as follows : 
 Nos. 1, 2 & 3, 5 ft. 4 in. x 4 ft. 6 in. x 
 
 2 ft. 8 in. = 
 Nos. 4, 5 & (I, (> ft. 2 in. x 3 ft, in. x 
 
 2 it. 11 in. = 
 Nos. 7 & 8, 12 ft. 3 in. x 2 ft. 1 in. x 
 1 ft. 6 in. = 
 
 ' Rec'd pay't, 
 
160 Arithmetical Exercises and Examples. 
 0. Emmet & Co., 
 
 No. 20. 
 NEW ORLEANS, Jan. 4, 1877. 
 
 To B. Criswell. 
 
 Dr. 
 
 For rent of house No. 386 Dryades 
 St., from Oct. 7, 1876, to Jan. 1, 
 1877, 2ff months, @ $35 
 
 For services as collector from Sept. 19, 
 187*, to Jan. 4, 1877, both inclu- 
 sive, 3-J-| months, @ $75 
 
 Rec'd pay't, 
 No. 21. 
 
 The la. Levee Co., 
 
 NEW ORLEANS, Jan. 9, 1877. 
 To James Selleck & Co. 
 
 For Constructing cubic yards 
 
 Levee @ 45/ as per the follow- 
 ing measurements : 
 1st Section 890 T % ft. long, 70 ft. wide 
 at the base and 30 ft. at the top, 
 with an average depth of 8-^ ft. 
 2nd Section 165 ft. long, 60 and 25 
 it. respectively for the lower and 
 upper widths, and 6, 7?, 5i, 8, 9, 
 and 6* ft. in depth at different 
 points. 
 
 For Excavating - cubic yards 
 Earth @ 45/, the same being the 
 contents of a cellar measuring as 
 follows : 
 
 92 ft. long and 50 ft. wide at the top 
 and 86 ft. long and 44 ft. wide at 
 the bottom, average depth 8 ft. 4 in 
 
 Rec'd pay't, 
 
Bills and Invoices. 
 
 161 
 
 Geo, Soule, 
 
 No, 22. 
 NEW ORLEANS, Jan. 16, 1877. 
 
 To Clark & Eofeline. 
 
 Dr. 
 
 For composition Arithmetical Exercises 
 and examples 180 pp., 1600 ems a 
 page, @ 75c. per m. 
 
 For press work on 54 token, @ $ 50c. 
 
 For 4 Reams Paper, u 6.00 
 
 For Binding 500 sep., " 25c. 
 
 Rec'd pay't, $ 
 
 CLARK & HOFELINE. 
 
 No. 23. 
 
 NEW ORLEANS, Jan. 18, 1877. 
 
 Western Union Telegraph Co., 
 
 To Jacob Simon & Co., Dr. 
 
 For 
 
 cubic feet Timber 
 
 $24 per 100 ; the same being the 
 contents of 50 Telegraph polos 
 measuring as follows : 
 40 Poles are 70 feet long, 16x16 
 inches at the larger end and so 12- 
 *inain for a distance of 10 feet, at 
 which point they begin and taper 
 regularly to the smaller cud', whi(-h 
 is 6x6 inches. 
 
 10 Poles are 60 feet long, 16x12 
 inches at the larger end, 6x4 inches 
 at the smaller end, and ti jer reg-i 
 larly the whole length. 
 
 Rec'd pay't. 
 
162 Arithmetical Exercises and Examples. 
 
 No. 24. 
 
 NEW ORLEANS, Jariy 29, 1877. 
 
 S. Drey fuss, 
 
 To V. Keiffer. 
 
 Dr. 
 
 Jan. 
 
 1 
 
 To old balance, as per bill 
 
 rendered. 
 
 91 
 
 10 
 
 
 6 
 
 12 cords Ash Wood 
 
 @$7. 
 
 84 
 
 00 
 
 
 6 
 
 4 cords Oak Wood 
 
 " $6.50 
 
 26 
 
 00 
 
 
 14 
 
 50 bbls. Pittsburg Coal 
 
 " 60c. 
 
 30 
 
 00 
 
 
 
 
 
 $231 
 
 10 
 
 
 
 Cr. 
 
 
 
 
 
 8 
 
 By Cash 
 
 $50 
 
 
 
 
 21) 
 
 " 6 days Labor, at $4, 
 
 824 
 
 74 
 
 00 
 
 Balance due Jan. 2 ( J, 1877. $ 157 10 
 
 Settled by note at 60 days. 
 
 
 
 V. KEIFFER. 
 
 A. & S. H. Souk, 
 
 No. 25. 
 NEW ORLEANS, Jan. 12, 1877. 
 
 To Z, M. Pike & Co. 
 
 Dr. 
 
 To 4378 feet Com. Boards @ 
 
 $21. per m. 
 " 1760 feet Dressed Flooring " 
 
 $28.50 per m. 
 " 5125 Bricks " 
 
 $14.25 per m. 
 " 9250 Cypress Shingles 
 
 $6.50 per m. 
 " Cartage and Labor 
 
 Rec'd pay't, 
 
 14 
 
 25 
 
Levy & King, 
 
 Bills and Invoices. 163 
 
 No. 26. 
 NEW ORLEANS, Dec. 81, 1876. 
 
 To R. & C. Rice, Dr. 
 
 For -- sq. yds. North River 
 Flags @ $7.50 as per the follow- 
 ing measurements : 
 Nos. 1, 2 & 3, are each 4 ft. 3 in. by 
 
 3 ft. 6 in. = sq. ft. 
 
 Nos. 4, 5 & 6 are each 4 ft. 8 in. by 
 
 3 ft. 4 in. = sq. ft. 
 
 Nos. 7 & 8 are each 4 ft. in. by 3 ft. 
 
 in. = sq. ft. 
 
 Nos. 9, 10 & 11 are each 3 ft. 4 in. by 
 
 2 ft. 9 in. = sq. ft. 
 
 For 
 
 sq. yds. German Flags @ 
 $2.25, comprising 152 Flags, each 
 22x16 inches. 
 
 For --- sq. yds. Brick Pavement 
 @ $1.15, contained in a side- 
 walk measuring 124 ft. 4 in. long 
 by 11 ft. 9 in. wide. 
 For 124 ft. 4 in. Curbing ty $ 1.30 
 For - cu. yds. Granite u $16.00 
 contained in 23 blocks of stone 
 measuring as follows : 
 Nos. 1 to 7 inclusive, are each 26xlx 
 
 10 inches, = 
 Nos. 8 to 20 inclusive, are each 2ox 
 
 16x9},= 
 
 Nos. 21 to 23 inclusive, are each 
 42x35x21 inches, = 
 
 Rec'd pay't, $ 
 
164 Arithmetical Exercises and Examples. 
 
 No. 27. 
 
 NEW ORLEANS, Jan. 28, 1877. 
 
 Invoice of Sundries, purchased by J. Simmons & Co. and 
 shipped per Steamer La Belle, for acc't. and risk of James 
 Byrnes, Shreveport, La. 
 
 
 
 87 bbls. Molas's, 3498 gals. @ 
 
 60c. 
 
 
 
 
 
 20 hhds. Sugar, 23780 Ibs. " 
 
 9c. 
 
 
 
 
 
 10 bbls. Rice, 2150 Ibs. " 
 
 5c. 
 
 
 
 
 
 
 
 4346 
 
 50 
 
 
 
 
 
 Drayage 
 
 
 17 
 
 50 
 
 
 
 Insurance on $4800.40 " 
 
 l%- 
 
 35 
 
 00 
 
 
 
 Commission on $4364.00 " 
 
 21%. 
 
 109 
 
 10 
 
 4503 10 
 
 No. 28. 
 
 W. H. Carey. 
 
 NEW ORLEANS, Jan. 14, 1877. 
 To Geo. Jumonville, Dr. 
 
 For Slating a roof measuring 72 ft. 4 
 in. by 49 ft. 10 in. and contain- 
 ing squares (a/, $14.50 
 
 For 239 ft. Guttering .90 
 
 Rec'd pay't. 
 No. 29. 
 
 NEW ORLEANS, Feb. 1, 1877. 
 Mississippi Valley Transportation Co., 
 
 To Buck & Richardson, Dr. 
 
 For Servises rendered in cause No. 
 55472. "Steamer R. E. Lee and 
 owners vs. Miss. V. T. Co." 
 
 Rec'd pay't. 
 
Bills and Invoices. 165 
 
 No. 30. 
 
 NEW ORLEANS, Nov. 4, 
 New Orleans, 'St. Louis and Chicago R. R., 
 
 To W. L. & E. Sail, Dr. 
 
 For 150 Cisterns holding 
 
 @ 2J^ per gal. The inside 
 measurement of each cistern is as 
 follows: 11 ft. o in. perpend cular 
 height, lower base 9 ft. '1 in. and 
 upper base 8 ft. 5 in. 
 
 Rec'd pay't. 
 
 TABLES OF WEIGHTS AND MEASURES. 
 
 175. Weight is that property of bodies by virtue of 
 which they tend toward the center of the earth, and the 
 resistance required to overcome this centralizing pressure, 
 or gravitating tendency of bodies, is what is named weight; 
 Weight varies according to the quantity of matter a body 
 contains, and its distance from the centre of the earth. 
 
 176. A Measure is a standard unit established by 
 law or custom, by which quantity, such as extent, dimension, 
 capacity, amount or value, is i: easured or estimated. 
 
 There are seven kinds of measure : 
 
 1st. Length. 2d. Surface or Area. 3d. Solidity or 
 Capacity. 4th. Weight or Force of Gravity. 5th. Time. 
 6th. Angles. 7th. Money or Value. 
 
 COMPARISON OF STANDARD UNITS OF MEASURE. 
 
 1. The Yard 3 feet 36 inches. 
 
 2. The Meter 3.2808 feet 39.37 inches'. 
 
 3. The Vanir-2.7778 feet 33J inches! 
 
 4. The Troy and Apothecaries pound=12 oz.=:5760 grains. 
 
 5. The Avoirdupois pound=16 oz.= 7000 grains. 
 
 6. The Wine gallon= 231 cubic inches. 
 
 7. The Beer gallon (nearly obsolete) 282 cubic inches. 
 
 8. The Dry gallon 268.8 cubic inches. 
 
H5(i Arithmetical Exercises and Examples. 
 
 9. The Imperial gallon of England 277.274 cubic inches. 
 
 10. The Bushel=r:4 pks.^32 qts.=64 pts.=: 
 
 2150.42 cubic inches. 
 
 11. The Imperial Bushel of England 2218.192 cubic inches. 
 
 12. The Diamond grain is equal to .8 of a grain Troy. 
 
 13. The Gallon, wine measure, of distilled water weighs 
 8.3388 pounds Avoirdupois or 10.134 pounds Troy. 
 
 14. The Civil Dan rrmmriK <.-.< ami oiids at midnight, and the 
 Astronomical Day, at noon of the Civil Day. 
 
 15. The Kntur /></// is the interval of time between two suc- 
 cessive p:is-:ijri's of the sun across the meridian of any place, 
 and they are of unequal length on account of the unequal or- 
 bital motion of the earth and the obliquity of the ecliptic. 
 
 RATIOS. 
 
 When the diameter of a circle is 1, the circumference is 
 
 3.1416. 
 Whenthe area of a square is 1, the area of a circle, the diameter 
 
 of which is equal to one side of the square, is .7854. 
 
 When the solidity of a cube is 1, the solidity of a sphere, the 
 
 diameter of which is equal to one side of the cube, is .5236. 
 
 \\K1GHT OF COIN. 
 
 $10000 Gold=258000 gr.=44 Ibs. 9 oz. 10 pwt. gr. Troy. 
 
 $1000 Silver dollars old issue=412500 gr.=71 Ibs. 7 oz. 7 
 pwt. 12 gr. 
 
 $1000000 Gold weigh 53750 ounces Troy or 3685.71 -Avoir- 
 dupois pounds. 
 
 $1000000 Silver Trade dollars weigh 875000 ounces Troy or 
 60000 pounds Avoirdupois. 
 
 $1000000 Silver, half and quarter dollars, 20 cent pieces and 
 dimes, weigh 803750 ounces Troy or 55114.28 Avoirdupois 
 pounds'. 
 
 VALUE OF COIN. 
 
 Gold coin=about 86 cents per pennyweight. 
 Silver com=about $1.11 per ounce. 
 
 For more extended tables of weights and measures, a 
 condensed history of time measure and the units of measure 
 in use in the early ages of civilization, see Soule's Philoso- 
 phic, Commercial and Exchange Calculator, pages 121 to 
 152 and 497 to 519. 
 
Tables of Weights and Measures. 
 
 167 
 
 'I._UNITED STATES MONEY. 
 
 10 Mills (m.) 
 10 Cents 
 10 Dimes 
 10 Dollars 
 
 = 1 Cent, f 
 
 1 Dime, d. 
 
 = 1 Dollar, $ 
 
 --= 1 Eagle, E. 
 
 II. ENGLISH MONEY. 
 
 TABLE. 
 
 4 Farthings (far.) 1 Penny, d. 
 
 12 Pence = 1 Shilling, a. 
 
 20 Shillings = 1 Pound (Sovereign,) 
 
 1 = $4.8665 
 
 III. FRENCH MONEY. 
 
 TABLE. 
 
 10 Centimes = 
 
 10 Decimes or 100 Centimes 
 Fr. 1 
 
 1 Decime. 
 1 Franc. 
 19 3 Cents. 
 
 TIME MEASURE. 
 
 60 Seconds (a.) 
 60 Minutes 
 .24 Hours 
 7 Days 
 
 365 Days 
 
 366 Days 
 100 Years 
 
 Minute, m. 
 
 Hour, h. 
 
 Day, d. 
 
 Week, wk. 
 Common Year, yr. 
 
 Leap Year, yr. 
 
 Century, c. 
 
 The names and orders of the months, and the number of days 
 contained in each, are now as follows; 
 
 Names. 
 
 No. 
 
 No. days. ' Names. 
 
 No. 
 
 No. days. 
 
 January, 
 
 1st, 
 
 31 
 
 July, 
 
 7th, 
 
 31 
 
 February 
 
 2d, 
 
 28 
 
 August, 
 
 8th, 
 
 31 
 
 March, 
 
 3d, 
 
 31 
 
 September, 
 
 9th, 
 
 30 
 
 April; 
 
 4th, 
 
 30 
 
 October, 
 
 10th, 
 
 31 
 
 May, 
 
 5th, 
 
 31 
 
 November, 
 
 llth. 
 
 30 
 
 June, 
 
 6th, 
 
 30 
 
 December, 
 
 12th, 
 
 M 
 
ir>^ Arithmetical Exercises and Examples. 
 
 The number of days in each, may be readily remembered by 
 committing to memory the following lines : 
 
 " Thirty days hath September, 
 April, June, and Novemlu-r ; 
 And all the resth;i\< thirty-one, 
 Save F'l.ru;iry, \vhich alone 
 Huth twenty-eight; and this, in fine, 
 One year in four hath twenty -nine." 
 
 (LINE) LINEAR OR LONG MEASURE. 
 
 TABLE. 
 
 12 Inches (in.) = 1 Foot, ft. 
 
 3 Feet = 1 Yard, yd. 
 
 5J Yards or 16Jifeet = 1 Rod or Pole, rd. or po. 
 
 40 Rods = 1 Furlong, tur. 
 
 8 Furlongs = 1 Mile (Statute Mile) m. 
 
 3 Miles = 1 League, lea. 
 
 MARINERS' MEASURE. 
 
 TABLE. 
 
 6 Feet = 1 Fathom. 
 
 120 Fathoms = 1 Cable-length. 
 
 880 Fathoms or 7J Cable-lengths = 1 Mile. 
 
 A knot, or geographical mile, is -fa of a degree, and is equiva- 
 lent to 1.15257 statute miles. 
 
 The length of a degree at the Equator is nearly equal to 69J 
 statute miles. The length of an average degree on the meridian 
 is 69.042 statute miles. 
 
 MISCELLANEOUS UNITS OF LINEAR MEASURE. 
 
 of an Inch = A Line (American). 
 
 of an Inch = A Line (French). 
 
 4 Inches = A Hand. 
 
 3 Inches = A Palm. 
 
 9 Inches = A Span. 
 
 3 Feet = A Pace. 
 
 2J Feet (28 in.) = A Military Pace. 
 
Tables of Weights and Measures. 169 
 
 CLOTH MEASURE. 
 
 TABLE. 
 
 2J Inches (in.) = 1 Nail, na. 
 
 4 Nails (9 inches) = 1 Quarter. qr. 
 
 4 Quarters = 1 Yard, yd. 
 
 This table formerly contained : 
 
 The Flemish Ell, which equaled 3 quarters or 2*7 inches ; 
 The English Ell, which equaled 5 quarters or 45 inches ; 
 The French Ell, which equaled 6 quarters or 54 inches. 
 
 These units of measure are nearly out of use. 
 
 SURVEYORS' AND ENGINEERS' MEASURE. 
 
 TABLE. 
 
 7.92 Inches = 1 Link, li. 
 
 25 Links = 1 Rod or Pole, rd. or po. 
 
 4 Poles or ) . n , . , 
 
 66 Feet } = l Cham ' ch " 
 
 80 Chains = 1 Mile, m. 
 
 Engineers use another chain which consists of 100 links, each 
 1 foot long. 
 
 SQUARE OR SURFACE MEASURE. 
 
 TABLE. 
 
 144 Square Inches (sq. in.) = 1 Square Foot, sq. ft. 
 
 9 Square Feet = 1 Square Yard, sq. yd. 
 
 301 Square Yards } ' ^J^ ^ 
 
 40 Square Rods or Perches = 1 Rood, r. 
 
 4 Roods = 1 Acre, a. 
 
 640 Acres 1 = l Square Mile, sq. m. 
 
 / or Section, sec. 
 
 36 Square Miles (6 miles sq.)= 1 Township, T. 
 
 16 Perches = 1 Square Chain, sq. ch. 
 
 10 Square Chains = 1 Acre, 
 
170 Arithmetical Exercises and Examples. 
 
 CUBIC OR SOLID MEASURE. 
 
 TABLE- 
 
 1728 Cubic Inches 
 27 Cubic Feet 
 16 Cubic Feet 
 
 8 Cord Feet or 128 Cubic Feet 
 24J Cubic Feet, or 16J feet long, 
 
 1} high and I foot wide 
 40 Cubic Feet of round timber, 
 50 Cubic Feet of hewn timber 
 
 I = 
 
 1 Cubic Foot. 
 1 Cubic Yard. 
 1 Cord Foot 
 1 Cord of Wood. 
 
 1 Perch. 
 
 1 Ton or Load. 
 
 A cubic foot contains 7 4805 Wine gallons. 
 A cubic foot contains .2374 barrels " 
 A cubic foot contains .8082 bushels. 
 
 LIQUID MEASURE. 
 
 TABLE. 
 
 4 Gills (gi.) 
 
 2 Pints 
 
 4 Quarts 
 
 31J Gallons 
 
 2 Barrels or 63 gallons 
 
 2 Hogsheads 
 
 2 Pipes 
 
 2 Pints (pt.) 
 8 Quarts 
 4 Pecks 
 8 Bushels 
 36 Bushels 
 
 1 Pint, pt. 
 
 1 Quart, qt. 
 
 1 Gallon, gal. = 
 
 1 Barrel, bbl. 
 1 Hogshead, hhd. 
 
 1 Pipe, P. 
 
 1 Tun, T. 
 
 :231 cubic in. 
 
 DRY MEASURE. 
 
 TABLE. 
 
 = 1 Quart, 
 
 = 1 Peck, 
 
 = 1 Bushel, 
 
 = 1 Quarter, 
 
 1 Chaldron, 
 
 qt. 
 pk. 
 bu. 
 qr. 
 ch. 
 
 TROY OR MINT WEIGHT. 
 
 TABLE. 
 
 24 Grains (gr.) 
 20 Peiiuywcigtits 
 12 Ounces 
 
 = 1 Pennyweight, 
 ^= 1 Ounce, 
 *= 1 Pound, 
 
 dwt. or pwt. 
 oz. 
 
 m. 
 
Tables of Weights and Measures. 171 
 
 AVOIRDUPOIS WEIGHT. 
 
 TABLE. 
 
 Grains = 1 Dram. dr. 
 
 16 Drams = 1 Ounce, oz. 
 
 16 Ounces 1 Pound, ft). 
 
 25 Pounds = 1 Quarter, qr. 
 
 4 Quarters or 100 pounds = 1 Hundredweight, cwt. 
 
 20 Hundredweight or 2000 pounds= 1 Ton, t. 
 
 480 Pounds = 1 Imperial Quarter. 
 
 100 Pounds is also called 1 Cental, c. 
 
 The cwt. in England, and in some cases in the United States, 
 is 112 pounds, or 4 quarters of 28 pounds. The ton English is 
 2240 pounds. Thi,s is called the long ton, and 2000 pounds, the 
 short ton. 
 
 APOTHECARIES' WEIGHT. 
 
 TABLE. 
 
 20 Grains (gr.) = 1 Scruple, 
 
 3 Scruples = 1 Dram, 
 
 8 Drams = 1 Ounce, 
 
 12 Ounces = 1 Pound, 
 
 MEDICAL DIVISIONS OF THE GALLON. 
 
 TABLE. 
 
 60 Minims (m.) = 1 Fluidram, fz 
 
 8 Fluidrama = 1 Fluidounce, f!$ 
 
 16 Fluidounces = 1 Pint, 0. 
 
 8 Pints = 1 Gallon, Cong. 
 
 0. is an abbreviation of octans, the Latin for one-eighth ; 
 Gong, for congiarium, the Latin for gallon. 
 
 DIAMOND WEIGHT. 
 
 TABLE. 
 
 16 Parts = 1 Grain. 
 
 4 Grains = 1 Carat. 
 
 1 Carat 3 Grain* Troy, nearly. 
 
172 Arithmetical Exercises and Examples. 
 ASSAYERS' WEIGHT. 
 
 TABLE. 
 
 1 Carat = 10 Pwts. Troy. 
 
 1 Carat grain = 2 Pwts. 12 grains, or 60 grains Troy. 
 24 Carats == 1 Pound Troy. 
 
 The term carat is also used to express the fineness of gold,- 
 each carat meaning a twenty-fourth part. 
 
 SHOEMAKERS' MEASURE. 
 
 No. 1 small size is 4J inches, and every succeeding No. increases 
 ^^f an inch to 13. 
 
 No. 1 large size is 8JJ inches, and every succeeding No. in- 
 creases J of an inch to 15. 
 
 CIRCULAR MEASURE. 
 
 TABLE. 
 
 60 Seconds " = 1 Minute, 
 
 60 Minutes = 1 Degree, 
 
 30 Degrees == 1 Sign, s. 
 
 12 Signs or 360 = 1 Circle, c. 
 
 90 degrees make 1 quadrant or right angle. 
 60 " " 1 sextant or sixth of a circle. 
 
 180 u " 1 semi-circle or half-circle. 
 
 MISCELLANEOUS TABLES. 
 
 BOOKS AND PAPER. 
 
 SIZE OF PAPER. 
 Inches. 
 
 Demy 17 by 22 
 
 Medium 19 " 24 
 
 Double medium 24 " 38 
 
 Super-royal 21 " 27 
 
 Inches. 
 
 Letter 10 by 15 
 
 Folio post 16 " 21 
 
 Foolscap 14 " 17 
 
 Crown 15 ' 20 
 
 Double Elephant 26 " 40 
 
 Imperial 22 " 32 
 
 A sheet (medium) folded in 2 leaves is called folio. 
 
 " " 4 " u quarto or 4to. 
 
 " 8 " " octavo or 8vo. 
 
 " u u 12 " " duodecimo or 12 mo. 
 
Tables of Weights and Measures. 
 
 1 Ream. 
 
 1 Bale 
 
 A sheet (medium) folded in 16 leaves is called 16mo. 
 " 18 " " 18mo. 
 
 a u 24 " " 24rao. 
 
 < " 32 " 32mo. 
 
 1 Quire. 
 = 20 Quires 
 
 = 1 Bundle ; 5 Bundles = 
 : 1 dozen. 
 
 i 12 dozen = 1 gross. 
 r 1 great gross. 
 1 score. 
 
 - 1 firkin of butter. 
 
 = 1 quintal of dried fish. 
 
 - 1 barrel of flour. 
 
 = 1 barrel of flour in California. 
 1 barrel of beef, pork, or fish. 
 = 1 barrel of salt. 
 
 1 cask 'of raisins. 
 14 11). iron or lead = 1 stone. 
 }'2 barrels of wheat = 7 English quarters. 
 :u.l stone= 1 Pig; 8 pigs = 1 fother. 
 256 pounds of soap = 1 barrel. 
 25 pounds of powder = 1 keg. 
 18 Inches = 1 Cubit. 
 
 24 Sheets 
 
 480 Sheets 
 
 2 Reams 
 
 12 units 
 144 units 
 12 gross 
 20 units 
 56 It). 
 100 It). 
 106 11.. 
 200 tt). 
 200 Ib. 
 280 II). 
 loo Ib. 
 
 WEIGHT OK GRAIN AND PRODUCE PER BUSHEL, 
 
 AS USBD IN NEW ORLEANS WIIBN THERE IS NO AGREEMENT TO T11K 
 CONTRARY. 
 
 Wheat, bu 
 
 sh. 60 Ib. 
 
 Flaxseed, bu 
 
 sh. 56 Ib. 
 
 Corn, < 
 
 56 " 
 
 Hempseed, 
 
 44 " 
 
 Rye, 
 
 ">6 " 
 
 Buckwheat, 
 
 52 " 
 
 Oats, 
 
 32 " 
 
 Castor Beans, 
 
 46 - li 
 
 Barley, * 
 
 48 " 
 
 Dried Peaches, 
 
 33 " 
 
 Irish Potatoes, 4 
 
 60 " 
 
 Dried Apples, 
 
 24 " 
 
 Sweet Potatoes, ' 
 
 (10 " 
 
 Onions, 
 
 57 " 
 
 Beans, ' 
 
 62 " 
 
 Coarse Salt, 
 
 50 ' 
 
 Bran, ' 
 
 24 u 
 
 Fine Salt, 
 
 50 " 
 
 Clover seed, ' 
 
 60 < ; 
 
 Stone Coal, 
 
 80 " 
 
 Timothy seed. ' 
 
 45 " 
 
 Corn Meal, 
 
 44 li 
 
 Barley Malt. ' 
 
 34 " 
 
 Plastering Hair. 
 
 7 4< 
 
 Peas, split. ' 
 
 60 M 
 
 Blue Grass seed, 
 
 10 " 
 
 Small Hominy, ' 
 
 50 u 
 
 
 
VB - 1 7433 
 
 M306O21 
 
 QA / 
 
 503 
 
 THE UNIVERSITY OF CALIFORNIA LIBRARY 
 
 

 Soule's 
 
 . 
 
 College and Literary Institute, 
 
 
 
 In wl 
 
 /*/. 
 
 jj, />/ Institute, 
 
 For b 
 
 f >///><' TwJthif/ SV//oo/, 
 
 There are 
 
 arious 
 
 fields of educqfjpp ; its curriculum 
 
 THIRTY-TWO BRANCHES OF STUr 5T, 
 
 taught by the most improved, ogresr- 
 meth 
 
 Lectures are given on i.he various subjects studied, an., also on 
 -IOLOGY, HYGIENE, PURE: OL- 
 ETC. 
 
 
 our stn 
 
 
 GEO. SOUT-E, 
 
 I* *0pt