yC-NRLF. 
 
 7M7 
 
 
 RUDIMENTS 
 
 A? T 
 
 ONT 
 
 NHMEROUS EXERCISES 
 
 SLATE AND BTACKBOARD, 
 
 FOR BI1-- 
 
 BY OAMES B. iilOjViSON, A.M., 
 
 AUTHOR OF >*' , EXERCISKS IN ARITn!ft<CTICAIi ANfALYS 
 
 PRACTICAL ARITTIMKTIC ; UIGHKR ARITHMETIC; EDITOR OF DAV's 
 SCHOOL AL^KBK/t, '.KG^NDRE's GEOMSTRV> ETC, 
 
 oo 
 
 CM 
 O 
 
 ,W YOKE: 
 
 N , c \ T EY, 48 & 50 WALKER ST. 
 
 AGO : S. C. G RIGOS & CO., 89 & 41 LAK T 
 
 :vs & co. ST. LOUIS : KEITH & WOODS. 
 
 ,. x . -23 & CO. DETROIT I EAYMOIO & SKLLiiOK. 
 -.i.riMS. BTJFFALO: PH1NNKY A CO. 
 .vaUSfr: T. S. Q[JAOKENBU8H. 
 
. 
 
 LIBRARY 
 
 OF THE 
 
 UNIVERSITY OF CALIFOF 
 
 GIFT OF 
 
 Received .... 
 
 Accession No. 6 J ^ ^ 3~ - Class No. 
 
an& ffifjomsan's Series. 
 
 RUDIMENTS 
 
 OF 
 
 AEITHMETIC; 
 
 CONTAINING 
 
 NUMEROUS EXERCISES 
 
 SI-ATE AND S&ACKBOARD, 
 
 
 FOB BEGINNERS 
 
 BY JAMES IB. THOMSON, A.M., 
 JSFf P* fllf Tjfl B ^x^^ 1 ^^ 
 
 AUTHOR OK MENTAL ARiTHMETicr; EXERCISES IN ARIT&MKTICAL ANALYST** 
 
 PRACTICAL ARITHMETIC; HtaH^fa ARITHMETIC; E|<ITOR OP DA\ s 
 
 SCHOOL ALGEBRA; LEGENT>RK y SGEOMETR.V, ETC. 
 
 NEW TOEK: 
 
 IVISON & PHINNEY, 48 & 50 WALKER ST. 
 CHICAGO: S. C. GEIGGS & CO., 39 & 41 LAKE ST. 
 
 CINCINNATI : MOORE, WIL8TACH, KEYS & CO. ST. LOUIS : KEITH & WOODS. 
 
 PHILADELPHIA; SOWER, BARNES & oo. BUFFALO; PHINNEY fc CO. 
 NEWBUEQ: T. s. QTTAOKENBUSH. t 
 
 1859. 
 
o 
 
 Entered according to Act of Congress, in the year 1833, Ifjr 
 
 JAMES B. THOMSON, 
 fa the Clerk's Office for the Southern District of New Yoft 
 
 TKREOTTPXD BY THOMAS B. 81CITK, 
 216 W1LLIAH 8TRKKT, N. V. 
 
PREFACE. 
 
 EDUCATION, in its comprehensive sense, is the business of 
 life. The exercises of the school-room lay the foundation ; 
 the superstructure is the work of after years. If these exer- 
 cises are rightly conducted, the pupil oh tains the rudiments 
 of science, and what is more important, he learns how to 
 study , how to think and reason, and is thus enabled to appro- 
 priate the means of knowledge to his future advancement. 
 Any system of instruction, therefore, which does not embrace 
 these objects, which treats a child as a mere passive recipient, 
 is palpably defective. It is destitute of some of the most 
 essential means of mental development, and is calculated to 
 produce pigmies, instead of giant intellects. 
 
 The question is often asked, " What is the best method of 
 proceeding with pupils commencing the study of Arithmetic, 
 or entering upon a new rule ?" 
 
 The old method. Some teachers allow every pupil to cipher 
 u on his own hook;" to go as fast, or as slow as he pleases, 
 without reciting a single example or rule, or stopping to in- 
 quire the " why and the wherefore" of a single operation. 
 This mode of teaching is a relic of by-gone days, and is prima 
 fade evidence, that those who practice it, are behind the 
 spirit of the times. 
 
 Another method. Others who admit the necessity of teach- 
 ing arithmetic in classes, send their pupils to their seats, and 
 tell them to u study the rule." After idling away an hour 
 or more, up goes one little hand after another with the de- 
 spairing question : " Please to show me how to do this sum, 
 sir ?" The teacher replies, " Study the rule ; that will tell 
 you." At length, to silence their increasing importunity, he 
 takes the slate, solves the question, and, without a word of 
 
5V PREFACE, 
 
 explanation, returns it to its owner. He thus goes through 
 the class. When the hour of recitation comes, the class is 
 not prepared with the lesson. They are sent to their seats 
 to make another trial, which results in no "better success. 
 And what is the consequence? They are discouraged and 
 disgusted with the study. 
 
 A more excellent way. Other teachers pursue a more ex- 
 cellent way, especially for young pupils. It is this : The 
 teacher reads over with the class the preliminary explanations, 
 and after satisfying himself that they understand the mean- 
 ing of the terms, he calls upon one to read and analyze the 
 first example, and set it down upon the blackboard, while 
 the rest write it upon their slates. The one at the bourd 
 then performs the operation audibly, and those with their 
 slates follow step by step. 
 
 Another is now called to the board and requested to set 
 down the second example, while the rest write the same 
 upon their slates, and solve it in a similar manner. He then 
 directs them to take the third example, and lets them try 
 their own skill, giving each such aid as he may require. In 
 this way they soon get hold of the principle, and if now sent 
 to their seats, will master the lesson with positive delight. 
 
 As to assistance, no specific directions can he given which 
 will meet every case. The best rule is, to afford the 
 learner just that kind and amount, which will secure the 
 greatest degree of exertion on his part. Less than this dis- 
 courages; more, enervates. 
 
 In conclusion, we would add, that this elementary work 
 was undertaken at the particular request of several eminent 
 practical teachers, and is designed to fill a niche in primary 
 schools. It presents, in a cheap form, a series of progressive 
 exercises in the simple and compound rules, which are 
 adapted to the capacities of beginners, and are calculated to 
 form habits of study, awaken the attention, and strengthen 
 the intellect. 
 
 J. B. THOMSON. 
 
 KBW YORK, January, 1858. 
 
CONTENTS 
 
 SECTION I. 
 
 ARITHMETIC defined, ....--7 
 
 Notation, 7 
 
 Roman Notation, ......--7 
 
 Arabic Notation. ...... -9 
 
 Numeration, -12 
 
 SECTION II. 
 
 ADDITION defined, 16 
 
 When the sum of a column does not exceed 9, - - -18 
 When the sura of a column exceeds 9,- - - - -19 
 General Rule for Addition, - - - - - - -20 
 
 SECTION III. 
 
 SUBTRACTION defined, 27 
 
 When a figure in the lower No. is smaller than that above it, - 28 
 When a figure in the lower No. is larger than that above it, 29 
 Borrowing 10, .... 30 
 
 General Rule for Subtraction, - - - .. -81 
 
 SECTION IV. 
 
 MULTIPLICATION defined, 36 
 
 When the multiplier contains but one figure, 89 
 
 When the multiplier contains more than one figure, - - 41 
 General Rule for Multiplication, .... 43 
 
 To multiply by 10, 100, 1000, <fcc., 45 
 
 When there are ciphers on the right of the multiplier, - 46 
 When there are ciphers on the right of the multiplicand, - 47 
 When there are ciphers on the right of both, 48 
 
 SECTION V. 
 DIVISION defined, ....... . . 49 
 
 Short Division, - 52 
 
VI CONTENTS. 
 
 Rule for Short Di nsion, 54 
 
 Long Division, 56 
 
 Difference between Short and Long Division, - - - 67 
 
 Rule for Long Division, 58 
 
 To dhide by 10, 100, 1000, <fec., - 61 
 
 When there are ciphers on the right of the divisor, - - 62 
 
 SECTION VI. 
 FRACTIONS, - 68 
 
 To find what part one given number is of another, - - 66 
 
 A part of a number being given, to find the whole, - - 66 
 
 To multiply a whole number by a fraction, - - - 67 
 
 To multiply a whole by a mixed number, - - - - 69 
 
 To divide a whole number by a fraction, 70 
 
 To divide a whole by a mixed number, - - - -71 
 
 SECTION VII. 
 
 TABLES in Compound Numbers, - - - 74 
 
 Paper and Books, - - .. . . . -85 
 Tables of aliquot parts, ------- 87 
 
 SECTION VIII. 
 
 ADDITION of Federal Money, 90 
 
 Subtraction of Federal Money, 92 
 
 Multiplication of Federal Money, 93 
 
 Division of Federal Money, 94 
 
 SECTION IX. 
 
 REDUCTION, 96 
 
 Rule for Reduction Descending, 97 
 
 Rule for Reduction Ascending, 100 
 
 Compound Addition, 106 
 
 Compound Subtraction, 108 
 
 Compound Multiplication, 110 
 
 Compound Division, - - - - - - - -11.1 
 
 Miscellaneous Exercises, - * - - - 113 
 Answers to Examples, - - - 119 
 
ARITHMETIC. 
 
 SECTION I. 
 
 ART, ! ARITHMETIC is the science of numbers. 
 
 Any single thing, as a peach, a rose, a book, is called a 
 unit, or one ; if another single thing is put with it, the 
 collection is called two ; if another still, it is called three ; 
 if another, four ; if another, five, &c. 
 
 The terms, one, tioo, three, four, <kc., are tke names of 
 numbers. Hence, 
 
 2. NUMBER signifies a unit, or a collection of units. 
 
 Numbers are expressed by words, by letters, and bj 
 figures. 
 
 3* NOTATION is the art of expressing numbers by letters 
 or figures. There are two methods of notation in use, the 
 Roman and the Arabic. 
 
 I. ROMAN NOTATION. 
 
 4r The Roman Notation is the method of expressing 
 numbers by letters ; and is so called because it was invented 
 by the ancient Romans. It employs seven capital letters, 
 viz : I, V, X, L, C, D, M. 
 
 When standing alone, the letter I, denotes one ; V, fiv-e ; 
 X, ten ; L, fifty ; C, one hundred ; D, five hundred ; M, 
 one thousand. 
 
 QUEST. 1. What is Arithmetic ? What is a single thing called? If an- 
 other is put with it, what is the collection called? If another, what ? What 
 are the terras one, two, three, &c. ? 2. What then is number ? How are 
 numbers expressed ? 3. What is Notation ? How many methods of notation 
 are in use? 4. What is the Roman notation? Why so called? How many 
 etters does it employ? What does the letter I, denote? V? X? L? C? D? M? 
 
NOTATION. !SECT. i. 
 
 5. To express the intervening numbers from to one a 
 thousand, or any number larger than a thousand, we re* 
 sort to repetitions and various combinations of these let- 
 ters, as may be seen from the following 
 
 TABLE. 
 
 I denotes 
 
 one. 
 
 XXXI 
 
 denotes thirty-one. 
 
 II " 
 
 two. 
 
 XL 
 
 " forty. 
 
 111 
 
 three. 
 
 XLI 
 
 " forty-one. 
 
 IV 
 
 four. 
 
 L 
 
 " fifty. 
 
 V " 
 
 five. 
 
 LI 
 
 " fifty-one. 
 
 VI 
 
 six. 
 
 LX 
 
 " sixty. 
 
 VII 
 
 seven. 
 
 LXI 
 
 " sixty-one. 
 
 VIII 
 
 eight. 
 
 LXX 
 
 " seventy. 
 
 IX " 
 
 nine. 
 
 LXXX 
 
 " eighty. 
 
 X " 
 
 ten. 
 
 xc 
 
 " ninety. 
 
 XI 
 
 eleven. 
 
 XCI 
 
 " ninety-one. 
 
 XII " 
 
 twelve. 
 
 C 
 
 " one hundred. 
 
 XIII 
 
 thirteen. 
 
 CI 
 
 " one hund. and one 
 
 XIV " 
 
 fourteen. 
 
 CIV 
 
 " one hund. and fout 
 
 XV 
 
 fifteen. 
 
 ex 
 
 " one hund. and ten. 
 
 XVI 
 
 sixteen. 
 
 cc 
 
 " two hundred. 
 
 XVII 
 
 seventeen. 
 
 ccv 
 
 " two hund. and five. 
 
 XVIII " 
 
 eighteen. 
 
 ccc 
 
 " three hundred. 
 
 XIX 
 
 nineteen. 
 
 cccc 
 
 " four hundred. 
 
 XX 
 
 twenty. 
 
 D 
 
 " five hundred. 
 
 XXI 
 
 twenty-one. 
 
 DC 
 
 " six hundred. 
 
 XXII 
 
 twenty-two. 
 
 DCC 
 
 " seven hundred. 
 
 XXIII 
 
 twenty-three. 
 
 DCCC 
 
 " eight hundred. 
 
 XXIV " 
 
 twenty- four. 
 
 DCCCC 
 
 " nine hundred. 
 
 XXV " 
 
 twenty-five. 
 
 M 
 
 " one thousand. 
 
 XXVI " 
 
 twenty-six. 
 
 MC 
 
 " one thousand and 
 
 XXVII " 
 
 twenty-seven. 
 
 
 one hundred. 
 
 XXVIII " 
 
 twenty-eight. 
 
 MM 
 
 " two thousand. 
 
 XXIX " 
 
 twenty-nine. 
 
 MDCCCL " one thousand eight 
 
 XXX 
 
 thirty. 
 
 
 hundred and fifty. 
 
 QUKST. 5. What do the letters IV, denote? VI? VIII? IX? XI? XI V 
 XVI? XVTII? XIX? XXIV? XL? LXXX? XC? CIV? Express 
 seven by Id icrs on the slate or black-board. How express eleven ? Thirteen 
 Tweuty-flvof Nineteen? Forty-lour? Eighty-seven? Ninety-nine? 
 
ARTS. 5 7.] NOTATION. 9 
 
 OBS. 1. Every time a letter is repeated, its value is repeated. 
 Thus, the letter I, standing alone, denotes one ; II, two ones or two, 
 <fcc. So X denotes ten ; XX, twenty, <fcc. 
 
 2. When two letters of different value are joined together, if the 
 less is placed before the greater, the value of the greater is dimin- 
 ished as many units as the less denotes ; if placed after the greater, 
 the value of the greater is increased as many units as the less de- 
 notes. Thus, V denotes five ; but IV denotes only four ; and VI, 
 six. So X denotes ten ; IX, nine ; XI, eleven. 
 
 Note. The questions on the observations may be omitted, by 
 beginners, till review, if deemed advisable by the teacher. 
 
 II. AKABIC NOTATION. 
 
 6. The Arabic Notation is the method of expressing 
 numbers by figures ; and is so called because it is supposed 
 to have been invented by the Arabs. It employs the fol- 
 lowing ten characters or figures, viz : 
 
 1234567890 
 
 one, two, three, four, five, six, seven, eight, nine, naught. 
 
 OBS. 1. The first nine are called significant figures, because each 
 one always expresses a value, or denotes some number. They are 
 also called digits, from the Latin word digitus, signifying a finger, 
 because the ancients used to count on their fingers. 
 
 2. The last one is called naught, because when standing alone, 
 it expresses nothing, or the absence of number. It is also called 
 cipher or zero. 
 
 7 All numbers larger than 9, are expressed by different 
 combinations of these ten figures. For example, to express 
 ten, we use the 1 and 0, thus 10 ; to express eleven, we 
 use two Is, thus 11 ; to express twelve, we use the 1 and 
 2, thus 12, <fec. 
 
 QuKST.Oi*. What is the effect of repeating a letter ? If a letter of lost 
 value is placed before another of greater value, what is the effect? If placed 
 after, what ? 6. What is the Arabic notation ? Why so called ? How many 
 figures does it employ? What are their names? Obs. What are the first nino 
 called ? Why ? What else are they sometimes called? What is the last one 
 called? Why? 7. How are numbers larger than nine expressed ? Express 
 ten by figures. Eleven. Twelve. Fifteen. 
 
10 NOTATION. [SECT, i 
 
 Hie method of expressing numbers by figures from 
 
 one to a thousand, may be seen from the following 
 
 TABLE. 
 
 1, one. 
 
 36, thirty-six. 
 
 71, seventy-one. 
 
 2, two. 
 
 37, thirty-seven. 
 
 72, seventy-two. 
 
 3, three. 
 
 38, thirty-eight. 
 
 73, seventy-three. 
 
 4, four. 
 
 39, thirty-nine. 
 
 74, seventy- four. 
 
 5, five. 
 
 40, forty. 
 
 75, seventy-five. 
 
 6, six. 
 
 41, forty-one. 
 
 76, seventy-six. 
 
 7, seven. 
 
 42, forty-two. 
 
 77, seventy-seven. 
 
 8, eight. 
 
 43, forty-three. 
 
 78, seventy-eight. 
 
 9, nine. 
 
 44, forty- four. 
 
 79, seventy-nine. 
 
 10, ten. 
 
 45, forty-five. 
 
 80, eighty. 
 
 11, eleven. 
 
 46, forty-six. 
 
 81, eighty-one. 
 
 12, twelve. 
 
 47, forty-seven. 
 
 82, eighty-two. 
 
 13, thirteen. 
 
 48, forty-eight. 
 
 83, eighty-three. 
 
 14, fourteen. 
 
 49, forty-nine. 
 
 84, eighty- four. 
 
 15, fifteen. 
 
 50, fifty. 
 
 85, eighty-five. 
 
 16, sixteen. 
 
 51, fifty-one. 
 
 86, eighty-six. 
 
 17, seventeen. 
 
 52, fifty-two. 
 
 87, eighty-seven 
 
 18, eighteen. 
 
 53, fifty-three. 
 
 88, eighty-eight. 
 
 19, nineteen. 
 
 54, fifty-four. 
 
 89, eighty-nine. 
 
 20, twenty. 
 
 55, fifty-five. 
 
 90, ninety. 
 
 21, twenty-one. 
 
 56, fifty-six. 
 
 91, ninety-one. 
 
 22, twen'y-two. 
 
 57, fifty-seven. 
 
 92, ninety-two. 
 
 23, twenty-three. 
 
 58, fifty-eight. 
 
 93, ninety-three. 
 
 24, twenty- four. 
 
 59, fifty-nine. 
 
 94, ninety-four. 
 
 25, twenty-five. 
 
 60, sixty. 
 
 95, ninety-five. / 
 
 26, twenty-six. 
 
 61, sixty-one. 
 
 96, ninety-six. 
 
 27, twenty-seven. 
 
 62, sixty-two. 
 
 97, ninety-seven 
 
 28, twenty-eight. 
 
 63, sixty-three. 
 
 98, ninety-eight. 
 
 29, twenty-nine. 
 
 64, sixty- four. 
 
 99, ninety-nine. 
 
 30, thirty. 
 
 65, sixty-five. 
 
 100, one hundred. 
 
 31, thirty-one. 
 
 66, sixty-six. 
 
 200, two hundred. 
 
 32, thirty-two. 
 
 67, sixty-seven. 
 
 300, three hundred 
 
 33, thirty-three. 
 
 68, sixty-eight. 
 
 400, four hundred. 
 
 34, thirty-four. 
 
 69, sixty-nine. 
 
 900, nine hundred. 
 
 35. thirty-five. 
 
 70, seventy. 
 
 1000, one thousand. 
 
 QUEST. How express fifteen? Twenty-five? Forty-seven? Thirty-six 
 Seventy-three One hundred and one ? One hundred and ten ? One hundrev 
 and twdnty T Two hundred and fifteen ? 
 
ARTS. 8 11.] NOTATION. 11 
 
 8. It will be peiceived from the foregoing table, that 
 the same figures, standing in different places, have differ- 
 ent values. 
 
 When they stand alone or in the right hand place, they 
 express units or ones, which are called units of the first 
 order. 
 
 When they stand in the second place, they express tens, 
 which are called units of the second order. 
 
 When they stand in the third place, they express hun- 
 dreds, which are called units of the third order. 
 
 When they stand in the fourth place, they express 
 thousands, which are called units of the fourth order, <fec. 
 
 For example, the figures 2, 3, 4, and 5, when arranged 
 thus, 2345, denote 2 thousands, 3 hundreds, 4 tens, and 5 
 units ; when arranged thus, 5432, they denote 5 thousands, 
 4 hundreds, 3 tens, and 2 units. 
 
 9 Ten units make one ten, ten tens make one hundred, 
 and ten hundreds make one thousand, &c. ; that is, ten of 
 any lower order, are equal to one in the next higher order 
 Hence, universally, 
 
 O. Numbers increase from right to left in a tenfold 
 ratio ; that is, each removal of a figure one place towards 
 the left, increases its value ten times. 
 
 11. The different values which the same figures have, 
 are called simple and local values. 
 
 The simple value of a figure is the value which it ex- 
 presses when it stands alone, or in the right hand place. 
 
 QUEST. 8. Do the same figures always have the same value ? When stand- 
 ing alone or in the right hand place, what do they express? What do they 
 express when standing in the second place? In the third place? In the 
 fourth ? 9. How many units make one ten ? How many tens make a hun- 
 dred ? How many hundreds make a thousand ? Generally, how many of any 
 lower order are required to make one of the next higher order ? 10. What is 
 the general law by which numbers increase ? What is the effect upon the value 
 of a figure to remove it one place towards the left? 11. What are the differ* 
 ent values of the same figure called ? What is the simple value of a figure ? 
 What the local value ? 
 
12 NUMERATION. [SECT. 1. 
 
 The simple value of a figure, therefore, is the numbel 
 which its name denotes. 
 
 The local value of a figure is the increased value which 
 : t expresses by having other figures placed on its right. 
 Hence, the local value of a figure depends on its locality, 
 or the place which it occupies in relation to other num- 
 bers with which it is connected. (Art. 8.) 
 
 OBS. This system of notation is also called the decimal system, 
 because numbers increase in a tenfold ratio. The term decimal if 
 derived from the Latin word decem, which signifies ten. 
 
 NUMERATION. 
 
 12* The art of reading numbers when expressed by 
 figures, is called Numeration. 
 
 NUMERATION TABLE. 
 
 123 861 518 924 263 
 
 Period V. Period IV. Period III. Period II. Period I. 
 Trillions. Billions. Millions. Thousands. Units. 
 
 13. The different orders of numbers are divided into 
 periods of three figures each, 'beginning at the right hand. 
 
 QUEST. Upon what does the local value of a figure depend ? Obs. What ia 
 this system of notation sometimes called ? Why ? 12. What is Numeration ? 
 Repeat the numeration table, beginning at the right hand. What is the first 
 place on the right called? The second place? The third? Fourth? Fifth 
 Sixth? Seventh? Eighth? Ninth? Tenth, &c.? 13. How are the orders of 
 numbers divided ? 
 
ARTS. 12 14.] NUMERATION. 13 
 
 The first, or right hand period is occupied by units, tens, 
 hundreds, and is called units' period ; the second is oc- 
 cupied by thousands, tens of thousands, hundreds of 
 thousands, and is called thousands' period, &c. 
 
 The figures in the table are read thus : One hundred 
 and twenty-three trillions, eight hundred and sixty-one 
 billions, five hundred and eighteen millions, nine hundred 
 and twenty-four thousand, two hundred and sixty-three. 
 
 1 4 To read numbers which are expressed by figures. 
 
 Point them off into periods of three figures each ; then, 
 beginning at the left hand, read the figures of each period 
 as though it stood alone, and to the last figure of each, add 
 the name of the period. 
 
 OBS. 1. The learner must be careful, in pointing 0$~ figures, always 
 to begin at the right hand ; and in reading them, to be^in at the 
 left hand. 
 
 2. Since the figures in the first or right hand period alw lys de- 
 note units, the name of the period is not pronounced. Hevxce, in 
 reading figures, when no period is mentioned, it is always u,ider- 
 stood to be the right hand, or units' period. 
 
 EXERCISES IN NUMERATION. 
 
 Note. At first the pupil should be required to apply to each fig- 
 ure the name of the place which it occupies. Thus, beginning at 
 the right hand, he should say, " Units, tens, hundreds," &c., and 
 point at the same time to the figure standing in the place which he 
 mentions. It will be a profitable exercise for young scholars to 
 write the examples upon their slates or paper, then point them off 
 into periods, and read them. 
 
 QUEST. What is the first period called ? By what is it occupied ? What is 
 the second period called? By what occupied? What is the third period 
 called ? By what occupied ? Wnat is the fourth called ? By what occupied f 
 What is the fifth called? By what occupied? 14. How do you read nurabere 
 expressed by figures ? Obs. Where begin to point them off? Where to read 
 them ? Do you pronounce the name of the right hand periol ? When no 
 period is named, what is understood ? 
 
14 
 
 NUMERATION. 
 
 Read the following numbers : 
 
 [SECT. I 
 
 Ex. 1. 
 
 97 
 
 16. 
 
 12642 
 
 Si. 7620 
 
 2. 
 
 110 
 
 17. 
 
 20871 
 
 32. 8040 
 
 3. 
 
 256 
 
 18. 
 
 17046 
 
 33. 9638 
 
 4. 
 
 307 
 
 19. 
 
 43201 
 
 34. 11000 
 
 5. 
 
 510 
 
 20. 
 
 80600 
 
 35. 12100 
 
 6. 
 
 465 
 
 21. 
 
 4203 
 
 36. 1402C 
 
 7. 
 
 1248 
 
 22. 
 
 65026 
 
 37. 10001 
 
 8. 
 
 2381 
 
 23. 
 
 78007 
 
 38. 5 020 
 
 9. 
 
 4026 
 
 24. 
 
 90210 
 
 39. 18022 
 
 10. 
 
 6420 
 
 25. 
 
 5025 
 
 40. 30401 
 
 11. 
 
 8600 
 
 26. 
 
 69008 
 
 41. 2506 
 
 12. 
 
 7040 
 
 27. 
 
 100 000 
 
 42. 402 321 
 
 13. 
 
 8000 
 
 28. 
 
 125 236 
 
 43. 65 007 
 
 14. 
 
 9007 
 
 29. 
 
 6005 
 
 44. 750 026 
 
 15. 
 
 10000 
 
 30. 
 
 462 400 
 
 45. 804 420 
 
 40. 
 
 2325672 
 
 50. 7289405287 
 
 47. 
 
 4502360 
 
 51. 185205370000 
 
 48. 
 
 62840285 
 
 52. 6423691450896 
 
 49. 
 
 425026951 
 
 53. 75894128247625 
 
 EXERCISES IN NOTATION. 
 
 15 To express numbers by figures. 
 
 Begin at the left hand of the highest period, and write 
 the figures of each period as though it stood alone. 
 
 If any intervening order, or period is omitted in the 
 given number, write ciphers in its place. 
 
 Write the following numbers in figures upon the slate 
 or black-board. 
 
 1. Sixteen, seventeen, eighteen, nineteen, twenty. 
 
 2. Twenty-three, twenty-five, thirty, thirty-three. 
 
 3. Forty-nine, fifty-one, sixty, seventy-four. 
 
 4. Eighty-six, ninety-three, ninety- seven, a hundred. 
 
 QUEST. 15. How are numbers expressed by figures ? If any intervening 
 order is omitted in the example, how is jts place supplied? 
 
ART. 15.J NUMERATION. 19 
 
 5. One hundred and ten. 
 
 6. Two hundred and thirty-five. 
 
 7. Three hundred and sixty. 
 
 8. Two hundred and seven. 
 
 9. Four hundred and eighty-one. 
 
 10. Six hundred and ninety-seven. 
 
 11. One thousand, two hundred and sixty-three. 
 
 12. Four thousand, seven hundred and ninety-nine. 
 
 13. Sixty-five thousand and three hundred. 
 
 14. One hundred and twelve thousand, six hundred 
 and seventy-three. 
 
 15. Three hundred and forty thousand, four hundred 
 and eighty-five. 
 
 16. Two millions, five hundred and sixty thousand. 
 
 17. Eight millions, two hundred and five thousand, 
 three hundred and forty-five. 
 
 18. Ten millions, five hundred thousand, six hundred 
 and ninety-five. 
 
 19. Seventeen millions, six hundred and forty-five 
 thousand, two hundred and six. 
 
 20. Forty-one millions, six hundred and twenty thou- 
 sand, one hundred and twenty-six. 
 
 21. Twenty- two millions, six hundred thousand, one 
 hundred and forty-seven. 
 
 22. Three hundred and sixty millions, nine hundred 
 and fifty thousand, two hundred and seventy. 
 
 23. Five billions, six hundred and twenty-one millions, 
 seven hundred and forty-seven thousand, nine hundred 
 and fifty-four. 
 
 24. Thirty-seven trillions, four hundred and sixty-three 
 billions, two hundred and ninety-four thousand, fire hun- 
 dred and seventy-two. 
 
 
1$ ADDITION. [SECT. IL 
 
 SECTION II. 
 
 ADDITION. 
 
 ART. 16. Ex. 1. Henry paid 4 shillings for a pair of 
 gloves, 7 shillings for a cap, and 2 shillings for a knife : 
 how many shillings did he pay for all ? 
 
 Solution. 4 shillings and 7 shillings are 11 shillings, 
 and 2 shillings are 13 shillings. He therefore paid 13 
 shillings for all. 
 
 OBS. The preceding operation consists in finding a single num- 
 ber which is equal to the several given numbers united together, 
 and is called Addition. Hence, 
 
 17 ADDITION is the process of uniting two or more 
 numbers in one sum. 
 
 The answer, or number obtained by addition, is called 
 the sum or amount. 
 
 OBS. When the numbers to be added are all of the same kind, or 
 denomination, the operation is called Simple Addition. 
 
 18 Sign of Addition (+). The sign cf addition is 
 a perpendicular cross ( + ), called plus, and shows that 
 the numbers between which it is placed, are to be added 
 together. Thus, the expression 6 + 8, signifies that 6 is 
 to be added to 8. It is read, " 6 plus 8," or <: 6 added to 8." 
 
 Note. The term plus, is a Latin word, orig'nally signifying 
 '* rcore." In Arithmetic, it means " added to." 
 
 QUEST. 17. What is addition? What is the answer called ? Obs. When 
 the numbers to bo added are all o? the same denomination, what is the ope- 
 ration called? 18. What is the si^n of addition? Who* does it show ? JVWe. 
 What is the moaning of the word plus ? 
 
ARTS. 16 19.] ADDITION. 17 
 
 19. Sign of Equality ( ). The sign of equality is 
 two horizontal lines ( ), and shows that the numbers be- 
 tween which it is placed, are equal to each other. Thus, 
 the expression 4 + 3 = 7, denotes that 4 added to 3 are 
 equal to 7. It is read, " 4 plus 3 equal 7," or " the sum 
 cf 4 plus 3 is equal to 7." 18 + 5 = 7 
 
 ADDITION TABLE. 
 
 2 and 
 
 3 and 
 
 4 and 
 
 5 and 
 
 1 are 3 
 
 1 are 4 
 
 1 are 5 
 
 1 are 6 
 
 2 " 4 
 
 2 " 5 
 
 2 " 6 
 
 2 " 7 
 
 3 " 5 
 
 3 " 6 
 
 3 " 7 
 
 3 " 8 
 
 4 " 6 
 
 4 " 7 
 
 4 " 8 
 
 4 " 9 
 
 5 " 7 
 
 5 " 3 
 
 5 " 9, 
 
 5 " 10 
 
 6 " 8 
 
 6 " 9 
 
 6 " 10 
 
 6 " 11 
 
 7 " 9 
 
 7 " 10 
 
 7 " 11 
 
 7 " 12 
 
 8 " 10 
 
 8 " 11 
 
 8 " 12 
 
 8 " 13 
 
 9 " 11 
 
 9 " 12 
 
 9 " 13 
 
 9 " 14 
 
 10 " 12 
 
 10 " 13 
 
 10 " 14 
 
 10 " 15 
 
 6 and 
 
 7 and 
 
 8 and 
 
 9 and 
 
 1 are 7 
 
 1 are 8 
 
 1 are 9 
 
 1 are 10 
 
 2 " 8 
 
 2 " 9 
 
 2 " 10 
 
 2 " 11 
 
 3 " 9 
 
 3 " 10 
 
 3 " 11 
 
 3 " 12 
 
 4 " 10 
 
 4 " 11 
 
 4 " 12 
 
 4 " 13 
 
 5 " 11 
 
 5 " 12 
 
 5 " 13 
 
 5 " 14 
 
 6 " 12 
 
 6 " 13 
 
 6 " 14 
 
 6 " 15 
 
 7 " 13 
 
 7 " 14 
 
 7 " 15 
 
 7 " 16 
 
 8 " 14 
 
 8 " 15 
 
 8 " 16 
 
 8 " 17 
 
 9 " 15 
 
 9 " 16 
 
 9 " 17 
 
 9 " 18 
 
 10 " 16 
 
 10 " 17 
 
 10 " 18 
 
 10 " 19 
 
 Note. It is an interesting and profitable exercise for young pupils 
 to recite tables in concert. But it will not do to depend upon this 
 method alone. It is indispensable for every scholar who desires to 
 De accurate either in arithmetic or business, to have the c ommon 
 
 QUKST. 19. What is the sign of equality? What does it show? 
 2 
 
18 ADDITION. [SECT. II. 
 
 ' tab es distinctly and indelibly fixed in his mind. Hence, 
 after a taole has been repeated by the class in concert, or individ- 
 ually, the teacher should ask many promiscuous questions, to prevent 
 its being recited mechanically, from a knowledge of the regular in- 
 crease of numbers. 
 
 EXAMPLES. 
 
 !3O When the sum of a column does not exceed 9. 
 
 Ex. 1. George gave 37 cents for his Arithmetic, and 
 42 cents for his Reader : how many cents did he give for 
 both? 
 
 Directions. Write the numbers Operation. 
 under each other, so that units ^ & 
 may stand under units, tens under jf g 
 tens, and draw a line beneath them. 3 7 price of Arith. 
 Then, beginning at the right hand 4 2 " of Read. 
 or units, add each column sepa- - 
 rately in the following manner : 7 9 Ans. 
 2 units and 7 units are 9 units. Write the 9 in units 
 place under the column added. 4 tens and 3 tens are 
 Y tens. Write the 7 in tens' place. The amount is 79 
 cents. 
 
 Write the following examples upon the slate or black- 
 board, and find the sum of each in a similar manner : 
 
 (2.) (3.) (4.) (5.) 
 
 26 231 623 5734 
 
 42 358 145 4253 
 
 (6.) (7.) (8.) (9.) 
 
 425 3021 5120 3521 
 
 132 1604 2403 1043 
 
 321 2142 1375 4215 
 
 10. What is the sum of 4321 and 2475 ? 
 
 11. What is the sum of 32562 and 56214? 
 
 12. What is the sum of 521063 and 465725 ? 
 
ARTS. 20 22. J ADDITION. 19 
 
 21. When the sum of a column exceeds 9. 
 
 13. A merchant sold a quantity of flour for 458 dollars, 
 a quantity of tea for 887 dollars, and sugar for 689 dol- 
 lars : how much did he receive for all ? 
 
 Having written the numbers as Operation. 
 
 Defore, we proceed thus: 9 units 458 price of flour, 
 and 7 units are 16 units, and 8 887 " of tea. 
 are 24 units, or we may simply 689 " of sugar, 
 say 9 and 7 are 16, and 8 are 24. 2034 dollars. Ans. 
 Now 24 is equal to 2 tens and 
 
 4 units. We therefore set the 4 units or right hand figure 
 in units' place, because they are units ; and reserving the 
 2 tens or left hand figure in the mind, add it to the column 
 of tens because it is tens. Thus, 2 (which was reserved) 
 and 8 are 10, and 8 are 18, and 5 are 23. Set the 3 or 
 right hand figure under the column added, and reserving 
 the 2 or left hand figure in the mind, add it to the column 
 of hundreds, because it is hundreds. Thus, 2 (which was 
 reserved) and 6 are 8, and 8 are 16, and 4 are 20. Set 
 the or right hand figure under the column added ; and 
 since there is no other column to be added, write the 2 
 in thousands' place, because it is thousands. 
 
 N. B. The pupil must remember, in all cases, to set down the 
 whole sum of the last or l$ft hand column. 
 
 22. The process of reserving the tens or left hand fig- 
 ure, when the sum of a column exceeds 9, and adding it 
 mentally to the next column, is called carrying tens. 
 
 Find the sum of each of the following examples in a 
 similar manner : 
 
 (14.) (15.) (16.) (17.) 
 
 856 364 6502 8245 
 
 764 488 497 4678 
 
 1620 Ans. 602 8301 362 
 
20 ADDITION. [SECT. IL 
 
 23. From the preceding illustrations and principles 
 we derive the following 
 
 GENERAL RULE FOR ADDITION". 
 
 I. Write the numbers to be added under each other, so 
 that units may stand under units, tens under tens, &c. 
 
 II. Beginning at the right hand, add each column sepa- 
 rately, and if the sum of a column does not exceed 9, write 
 it under the column added. But if the sum of a column 
 exceeds 9, write the units' figure under the column and 
 carry the tens to the next column. 
 
 III. Proceed in this manner through all the orders, and 
 finally set down the whole sum of the last or left hand 
 column. 
 
 24. PROOF. Beginning at the top, add each column 
 downward, and if the second result is the same as the 
 first, the work is supposed to be right t 
 
 EXAMPLES FOR PRACTICE. 
 
 (1.) (2.) (3.) (4.) 
 
 Pounds. Feet. Dollars Yards. 
 
 25 113 342 4608 
 
 46 84 720 635 
 
 _84 2_16 898 43 
 
 (5.) (6.) (7.) (8.) 
 
 684 336 6387 8261 
 
 948 859 593 387 
 
 569 698 3045 13 
 
 203 872 15 7^ 
 
 9. What is the sum of 46 inches and 38 inches? 
 
 QUEST. 23. How do you write numbers for addition? When tho mim of a 
 column does not exceed 9, how proceed ? When it exceeds 9, how proceed ? 
 22. What is meant by carrying the tens ? What do you do with the sum of 
 the last column ? 24. How is addition proved ? 
 
< 23, 24.] ADDITION. 21 
 
 10. What is the sum of 51 feet and 63 feet ? 
 
 11. What is the sum of 75 dollars and 93 dollars? 
 
 % 
 
 12. Add together 45 rods, 63 rods, and 84 rods. 
 
 13. Add together 125 pounds, 231 pounds, 426 pounds. 
 
 14. Add together 267 yards, 488 yards, and 6 25 yards. 
 
 15. Henry traveled 256 miles by steamboat and 320 
 miles by Railroad : how many miles did he travel ? 
 
 16. George met two droves of sheep ; one contained 
 461, and the other 375 : how many sheep were there in 
 both droves ? 
 
 17. If I pay 230 dollars for a horse, and 385 dollars for 
 a chaise, how much shall I pay for both ? 
 
 18. A farmer paid 85 dollars for a yoke of oxen, 27 
 dollars for a cow, and 69 dollars for a horse : how much 
 did he pay for all ? 
 
 19. Find the sum of 425, 346, and 681. 
 
 20. Find the sum of 135, 342, and 778. 
 
 21. Find the sum of 460, 845, and 576. 
 
 22. Find the sum of 2345, 4088, and 260. 
 
 23. Find the sum of 8990, 5632, and 5863. 
 
 24. Find the sum of 2842, 6361, and 523. 
 
 25. Find the sum of 602, 173, 586, and 408. 
 
 26. Find the sum of 424, 375, 626, and 75. 
 
 27. Find the sum of 24367, 61545, and 20372. 
 
 28. Find the sum of 43200, 72134, and 56324. 
 
 29. A young man paid 5 dollars for a hat ; 6 dollars 
 for a pair of boots, 27 dollars for a suit of clothes, and 19 
 dollars for a cloak : how much did he pay for all ? 
 
 30. A man paid 14 dollars for wood, 16 dollars for a 
 stove, and 28 dollars for coal : how many dollars did he 
 pay for all ? 
 
 31. A farmer bought a plough for 13 dollars, a cart 
 for 46 dollars, and a wagon for 61 dollars : what was the 
 price of all ? 
 
22 ADDITION. [SECT. II 
 
 32. What is the sum of 261+31+256 + 17 ? 
 
 33. What is the sum of 163+478+82 + 19 ? 
 
 34. What is the sum of 428 + 632 + 76+394 ? 
 
 35. W3iat is the sum of 320 + 856 + 100+503? 
 
 36. What is the sum of 641+108 + 138+710 ? 
 
 37. What is the sum of 700 + 66 + 970+21 ? 
 
 38. What is the sum of 304+971+608+496 ? 
 
 39. What is the sum of 848 + 683+420+668 ? 
 
 40. What is the sum of 868+45 + 17+25+27+38? 
 
 41. What is the sum of 641 + 85+580+42+7+63 ? 
 
 42. What is the sum of 29+281+7+43+785+46? 
 
 43. A farmer sold 25 bushels of apples to one man, IT 
 bushels to another, 45 bushels to another, and 63 bushels 
 to another : how many bushels did he sell ? 
 
 44. A merchant bought one piece of cloth containing 
 25 yards, another 28 yards, another 34 yards, and an- 
 other 46 yards : how many yards did he buy ? 
 
 45. A man bought 3 farms ; one contained 120 acres, 
 another 246 acres, and the other 365 acres : how many 
 acres did they all contain ? 
 
 46. A traveler met four droves of cattle ; the first con- 
 tained 260, the second 175, the third 342, and the fourth 
 420 : hOw many cattle did the four droves contain ? 
 
 47. A carpenter built one house for 2365 dollars, an- 
 other for 1648 dollars, another for 3281 dollars, and an- 
 other for 5260 dollars : how much did he receive for all 9 
 
 48. Find the sum of six hundred and fifty- four, eighty- 
 nine, four hundred and sixty-three, and seventy- six. 
 
 49. Find the sum of two thousand and forty-seven, 
 three hundred and forty-five, thirty-six, and one hundred. 
 
 50. In January there are 31 days, February 28, March 
 81, April 30, May 31, June 30, July 31, August 31, .Sep- 
 tember 30, October 31, November 30, and December 31 : 
 how many days are there in a year ? 
 
ART. 24.a.] ADDITION. 23 
 
 24ra. Accuracy and rapidity in adding can be ac- 
 quired only by practice. The following exercises are de- 
 signed to secure this important object. 
 
 OBS. 1. In solving the following examples, it is recommended 
 to the pupil simply to pronounce the result, as he adds each suc- 
 cessive figure. Thus, in Ex. 1, instead of saying 2 and 2 are 4, 
 and 2 are 6, &/c., proceed in the following manner : " two, four, six ; 
 eight, ten, twelve, fourteen, sixteen, eighteen, twenty." Set down 
 naught and carry two. " Two, (to carry) three, six, nine," &c. 
 
 2. When two or three figures taken together make 10, as 8 and 2, 
 7 and 3, or 2, 3, and 5, k accelerates the process to add their sum 
 at once. Thus, in Ex. 4, the pupil should say: " ten (1+9), six- 
 teen (6), twenty-six (5+5), thirty-six (2+8)," &c. 
 
 (1.) (2.) (3.) (4.) 
 
 32 654 987 463 
 
 32 654 987 647 
 
 32 654 987 455 
 
 32 654 987 258 
 
 32 654 987 572 
 
 32 654 987 595 
 
 32 654 987 615 
 
 32 654 987 346 
 
 32 654 987 729 
 
 12 114 117 181 
 
 (5.) (6.) (7.) (8.) 
 
 614 2140 8675 9244 
 
 452 8963 2433 1432 
 
 528 1232 6182 7234 
 
 539 7855 2921 2523 
 
 420 2123 2209 8440 
 
 385 3333 4863 4346 
 
 355 7674 6558 6704 
 
 134 4521 5434 1852 
 
 976 6589 5276 9258 
 
 468 2637 8789 8106 
 
24 
 
 ADDITION. 
 
 [SECT. IL 
 
 (9.) 
 
 (10.) 
 
 . (11.) 
 
 (12.) 
 
 4360 
 
 9201 
 
 42671 
 
 62125 
 
 7046 
 
 7283 
 
 68439 
 
 31684 
 
 5724 
 
 4627 
 
 32074 
 
 22435 
 
 5385 
 
 6436 
 
 47616 
 
 16725 
 
 8275 
 
 9874 
 
 30045 
 
 94381 
 
 9342 
 
 8400 
 
 26765 
 
 25036 
 
 6768 
 
 6645 
 
 10850 
 
 85474 
 
 5020 
 
 4365 
 
 25232 
 
 10325 
 
 9384 
 
 8640 
 
 43679 
 
 42312 
 
 (13.) 
 
 (14.) 
 
 (15.) 
 
 (16.) 
 
 2720 
 
 5764 
 
 27856 
 
 47639 
 
 4382 
 
 5346 
 
 32534 
 
 23421 
 
 2640 
 
 3042 
 
 20631 
 
 34323 
 
 3047 
 
 5268 
 
 34327 
 
 71036 
 
 2163 
 
 3161 
 
 53102 
 
 62342 
 
 6741 
 
 2560 
 
 92763 
 
 57654 
 
 1360 
 
 7304 
 
 51834 
 
 32103 
 
 7056 
 
 2723 
 
 23452 
 
 53728 
 
 8554 
 
 8459 
 
 62327 
 
 61342 
 
 4275 
 
 6715 
 
 50632 
 
 23201 
 
 (17.) 
 
 (18.) 
 
 (19.) 
 
 (20.) 
 
 4521 
 
 6845 
 
 75360 
 
 89537 
 
 3432 
 
 3151 
 
 27838 
 
 23264 
 
 4327 
 
 2327 
 
 42627 
 
 41728 
 
 6238 
 
 4235 
 
 34872 
 
 74263 
 
 5494 
 
 2835 
 
 63538 
 
 21031 
 
 3217 
 
 5473 
 
 54321 
 
 53426 
 
 2382 
 
 9864 
 
 63054 
 
 91342 
 
 4723 
 
 3103 
 
 29872 
 
 23465 
 
 3604 
 
 7382 
 
 63541 
 
 38754 
 
 2352 
 
 5461 
 
 53279 
 
 94642 
 
ADDITION. 
 
 (21.) 
 
 (22.) 
 
 (23.) 
 
 (24.) 
 
 8564 
 
 56,430 
 
 84,703 
 
 341,725 
 
 4736 
 
 31,932 
 
 19,384 
 
 227,265 
 
 3405 
 
 29,754 
 
 21,705 
 
 311,265 
 
 037 
 
 46,536 
 
 43,641 
 
 200,378 
 
 6571 
 
 86,075 
 
 27,469 
 
 421,850 
 
 7439 
 
 30,235 
 
 52,267 
 
 370,432 
 
 4525 
 
 41,623 
 
 61,383 
 
 174,370 
 
 3137 
 
 45,810 
 
 75,604 
 
 831,031 
 
 2743 
 
 56,239 
 
 43,876 
 
 580,456 
 
 (25.) 
 
 (26.) 
 
 (27.) 
 
 (28.) 
 
 7243 
 
 31,625 
 
 68,901 
 
 460,732 
 
 2034 
 
 51,482 
 
 50,345 
 
 804,045 
 
 3710 
 
 49,061 
 
 75,005 
 
 346,325 
 
 5634 
 
 80,604 
 
 29,450 
 
 450,673 
 
 1730 
 
 24,540 
 
 80,063 
 
 859,721 
 
 5613 
 
 67,239 
 
 91,700 
 
 236,548 
 
 3005 
 
 24,307 
 
 43,621 
 
 632,462 
 
 7206 
 
 58,392 
 
 47,834 
 
 753,324 
 
 4354 
 
 70,300 
 
 83,276 
 
 970,300 
 
 7821 
 
 56,749 
 
 25,327 
 
 267,436 
 
 (29.) 
 
 (30.) 
 
 (31.) 
 
 (32.) 
 
 6458 
 
 75,340 
 
 64,268 
 
 346,768 
 
 2435 
 
 6,731 
 
 405 
 
 21,380 
 
 4678 
 
 748 
 
 1,708 
 
 4,075 
 
 4962 
 
 68,451 
 
 43,671 
 
 126,849 
 
 5143 
 
 396 
 
 72,049 
 
 257 
 
 8437 
 
 7,503 
 
 492 
 
 L305 
 
 7643 
 
 46,075 
 
 1,760 
 
 24,350 
 
 6850 
 
 1,290 
 
 25,357 
 
 439,871 
 
 7063 
 
 25,738 
 
 1,434 
 
 40,306 
 
 8324 
 
 46,803 
 
 84,162 
 
 601,734 
 
26 
 
 ADDITION. 
 
 [SECT. II, 
 
 (S3.) 
 
 (34.) 
 
 (35.) 
 
 (36.) 
 
 423,674 
 
 632,153 
 
 317,232 
 
 412,783 
 
 307,316 
 
 420,432 
 
 203,671 
 
 631,432 
 
 730,248 
 
 323,680 
 
 334,263 
 
 572,316 
 
 506,213 
 
 507,325 
 
 210,600 
 
 231,254 
 
 110,897 
 
 383,734 
 
 356,237 
 
 673,323 
 
 206,341 
 
 634,156 
 
 264,871 
 
 217,067 
 
 324,563 
 
 450,071 
 
 531,634 
 
 306,421 
 
 185,174 
 
 803,463 
 
 342,106 
 
 764,315 
 
 364,230 
 
 160,705 
 
 768,342 
 
 207,254 
 
 150,176 
 
 300,430 
 
 407,821 
 
 843,552 
 
 843,204 
 
 461,007 
 
 311,289 
 
 321,634 
 
 370,679 
 
 297,313 
 
 564,735 
 
 502,543 
 
 445,168 
 
 813,792 
 
 470,334 
 
 617,405 
 
 370,432 
 
 200,406 
 
 436,216 
 
 506,032 
 
 5,338,315^4? 
 
 is. 6,388,667 
 
 Ans. 621,353 
 
 762,573 
 
 (37.) 
 
 (38.) 
 
 (39.) 
 
 (40.) 
 
 674,326 
 
 783,457 
 
 863,725 
 
 958,439 
 
 453,403 
 
 675,306 
 
 755,387 
 
 843,670 
 
 561,734 
 
 858,642 
 
 964,845 
 
 784,561 
 
 789,867 
 
 246,468 
 
 836,450 
 
 976,435 
 
 645,275 
 
 587,649 
 
 645,265 
 
 833,406 
 
 576,182 
 
 523,731 
 
 783,842 
 
 797,624 
 
 934,922 
 
 445,372 
 
 532,653 
 
 845,358 
 
 423,641 
 
 832,148 
 
 647,412 
 
 978,262 
 
 561,232 
 
 465,363 
 
 481,735 
 
 784,643 
 
 143,671 
 
 642,742 
 
 824,364 
 
 865,343 
 
 238,406 
 
 830,423 
 
 537,572 
 
 976,736 
 
 453,762 
 
 256,372 
 
 463,489 
 
 853,974 
 
 984,651 
 
 662,456 
 
 827,343 
 
 467,852 
 
 845,359 
 
 572,834 
 
 642,536 
 
 948,685 
 
 967,423 
 
 864,213 
 
 725,342 
 
 896,872 
 
A.RTS. 25 27.] SUBTRACTION. 2T 
 
 SECTION III. 
 
 SUBTRACTION. 
 
 ART. 25. Ex. 1. Charles having 15 cents, gave 6 cents 
 for an orange : how many cents did he have left ? 
 
 Solution. 6 cents taken from 15 cents leave 9 cents. 
 Therefore he had 9 cents left. 
 
 OBS. The preceding operation consists in taking a less number 
 from a greater, and is called Subtraction. Hence, 
 
 26. SUBTRACTION is the process of finding the differ- 
 ence between two numbers. 
 
 The answer, or number obtained by subtraction, is called 
 the difference or remainder. 
 
 OBS. 1. The number to be subtracted is often called the subtra- 
 hend, and the number from which it is subtracted, the minuend. 
 These terms, however, are calculated to embarrass, rather than 
 assist the learner, and are properly falling into disuse. 
 
 2. When the given numbers are all of the same kind, or denomi- 
 nation, the operation is called Simple Subtraction. 
 
 27. Sign of Subtraction ( ). The sign of subtrac- 
 tion is a horizontal line (), called minus, and shows 
 that the number after it is to be subtracted from the one 
 before it. Thus the expression 7 3, signifies that 3 is to be 
 subtracted from 7 ; and is read, " 7 minus 3," or " 7 less 3." 
 Bead the following: 187 = 20 9. 23 10=16 3 
 35 8 = 31 4. 
 
 Note. The term minus is a Latin word signifying less. 
 
 QUEST. C 26. What is subtraction? What is the answer called? Obs. 
 What is the number to be subtracted sometimes called ? That from which it 
 Is subtracted ? When tho given numbers are of the same denomination, what 
 is the operation called ? 27. What is the sign of subtraction ? What does it 
 ihow 1 Note. What is t<he moaning of the term minus? 
 
SUBTRACTION. 
 
 [SECT, ill 
 
 SUBTRACTION TABLE. 
 
 2 from 
 
 3 from 
 
 4 from 
 
 5 from 
 
 2 leaves 
 
 3 leaves 
 
 4 leaves 
 
 5 leaves 
 
 3 " 1 
 
 4 " 1 
 
 5 " 1 
 
 6 " 1 
 
 4 " 2 
 
 5 " 2 
 
 6 " 2 
 
 7 " 2 
 
 5 " 3 
 
 6 " 3 
 
 7 " 3 
 
 8 " 3 
 
 6 " 4 
 
 7 " 4 
 
 8 4 
 
 9 " 4 
 
 7 " 5 
 
 8 " 5 
 
 9 " 5 
 
 10 " 5 
 
 8 6 
 
 9 " 6 
 
 10 " 6 
 
 11 " 6 
 
 9 " 7 
 
 10 " 7 
 
 11 " 7 
 
 12 " 7 
 
 10 " 8 
 
 11 " 8 
 
 12 " 8 
 
 13 " 8 
 
 11 " 9 
 
 12 " 9 
 
 13 " 9 
 
 14 9 
 
 12 " 10 
 
 13 " 10 
 
 14 " 10 
 
 15 " 10 
 
 6 from 
 
 7 from 
 
 8 from 
 
 9 from 
 
 6 leaves 
 
 7 leaves 
 
 8 leaves 
 
 9 leaves 
 
 7 " 1 
 
 8 " 1 
 
 9 " 1 
 
 10 " 1 
 
 8 " 2 
 
 9 " 2 
 
 10 " 2 
 
 11 " 2 
 
 9 " 3 
 
 10 " 3 
 
 11 " 3 
 
 12 " 3 
 
 10 " 4 
 
 11 " 4 
 
 12 " 4 
 
 13 " 4 
 
 11 " 5 
 
 12 " 5 
 
 13 " 5 
 
 14 " 5 
 
 12 " 6 
 
 13 " 6 
 
 14 " 6 
 
 15 " 6 
 
 13 " 7 
 
 14 " 7 
 
 15 " 7 
 
 16 " 7 
 
 14 " 8 
 
 15 " 8 
 
 16 " 8 
 
 17 ' 8 
 
 15 " 9 
 
 16 " 9 
 
 17 " 9 
 
 18 " 9 . 
 
 16 " 10 
 
 17 " 10 
 
 18 " 10 
 
 19 " 10 1 
 
 OBS. This Table is the reverse of Addition Table. Hence, if the 
 pupil has thoroughly learned that, it will cost him but little time or 
 trouble to learn this. (See observations under Addition Table.) 
 
 EXAMPLES. 
 
 28. When each figure in the lower number is smaller 
 than the figure above it. 
 
 1. A farmer raised 257 bushels of apples, and 123 
 bushels of pears : how many more apples did he rais 
 than pears ? 
 
ARTS. 28, 29.] SUBTRACTION. 29 
 
 Directions. Write the less Operation. 
 
 number under the greater, so 
 that units may stand under units, *g ^ jg 
 
 tens mder tos, &c., and draw a Jj _ 
 
 line beneath them. Beginning 257 apples, 
 
 with the units or right hand fig- 123 pears, 
 
 ure, subtract each figure in the JRem. 134 bush. 
 ower numDor from the figure 
 
 above it, in the following manner : 3 units from 7 units 
 leave 4 units. Write the 4 in units' place under the 
 figure subtracted. 2 tens from 5 tens leave 3 tens ; set 
 3 in tens' place. 1 hundred from 2 hundred leaves 1 hun- 
 dred ; write the 1 hundred in hundreds' place. 
 
 Solve the following examples in a similar manner : 
 
 (2.) (3.) (4.) (5.) 
 
 From 45 68 276 698 
 
 Take JU 123 453 
 
 (6.) (7.) (8.) (9.) 
 
 From 54 delis. 76 pounds. 257 yds. 325 shil. 
 Take Cd dolls. 64 pounds. 142 yds. 103 shil. 
 
 10. Samuel having 436 marbles, lost 214 of them: 
 how many had he left ? 
 
 29 When a figure in the lower number is larger than 
 the figure above it. 
 
 11. A man bought 63 bushels of wheat, and after- 
 wards sold 37: how many bushels had he left? 
 
 It is bv> jus tb'.tt we cannot take 7 Ifirst Method. 
 units fi < t 3 UIP',S, for 7 is larger than 63 
 
 3 ; we tl erefove add 10 to the 3 units, 37 
 
 and it w U raake 13 units ; then 7 from Rem. 26 bu. 
 13 leave i 6 ; write the 6 in units' place 
 under the figure subtracted. To compensate for the 10 
 
30 SUBTRACTION. (SECT [11 
 
 units we added to the upper figure, we add 1 ten to the 
 
 3 tens or next figure in the lower number, and it makes 
 
 4 tens ; and 4 tens from 6 tens leave 2 tens : write the 2 
 in tens' place Ans. 26 bushels. 
 
 We may also illustrate the process of borrowing in tfec 
 following manner : 
 
 63 is composed of 6 tens and 3 Second Method. 
 units. Taking 1 ten from 6 tens, 63 = 50 + 13* 
 and adding it to the 3 units, we 37=30+7 
 have63 = 50+13. Separating the JRem. = 2Q + 6, or 26 
 lower number into tens and units, 
 
 we have 37 = 30 + 7. Now, substracting as before, 7 
 from 13 leaves 6. Then as we took 1 ten from the 6 tens, 
 we have but 5 tens left ; and 3 tens from 5 tens leave 2 
 tens. The remainder is 26, the same as before. 
 
 3O The process of taking one from a higher order in 
 the upper number, and adding it to the figure from which 
 the subtraction is to be made, is called borrowing ten, and 
 is the reverse of carrying ten. (Art. 22.) 
 
 OBS. When we borrow ten we must always remember to pay it 
 This may be done, as we have just seen, either by adding 1 to tte 
 next figure in the lower number, or by considering the nextjigur* 
 in the upper number 1 less than it is . 
 
 12. From 240 subtract 134, and prove the operation. 
 
 Since 4 cannot be taken from 0, we Operation. 
 
 borrow 10; then 4 from 10 leaves 6. 1 240 
 
 added to 3 (to compensate for the 10 we 134 
 borrowed) makes 4, and 4 from 4 leaves 0. 106 Ans. 
 1 from 2 leaves 1. 
 
 PROOF. We add the remainder Proof. 
 
 to the smaller number, and since the 134 less No. 
 
 sum is equal to the larger number, 106 remainder, 
 
 the work is right. 240 greater No. 
 
ARTS. 30 32. j SUBTRACTION. SI 
 
 Solve the following examples, and prove the operation. 
 
 (13.) (14.) (15.) (16.) 
 
 From 375 5273 6474 8650 
 
 Take_238 2657 3204 5447 
 
 17. From 8461875, take 3096208. 
 
 31 From the preceding illustrations and principles 
 we derive the following 
 
 GENERAL RULE FOR SUBTRACTION. 
 
 I. Write the less number under the greater, so that units 
 may stand under units, tens under tens, &c. 
 
 II. Beginning at the right hand, subtract each figure in 
 the lower number from the figure above it, and set the re- 
 mainder under the figure subtracted. 
 
 III. When a figure in the lower number is larger than 
 that above it, add 10 to the upper figure ; then subtract as 
 before, and add 1 to the next figure in the lower number. 
 
 32* PROOF. Add the remainder to the smaller num- 
 ber ; and if the sum is equal to the larger number, the work 
 is right. 
 
 OBS. This method of proof depends upon the obvious principle, 
 that if the difference between two numbers be added to the less, the 
 sum must be equal to the greater. 
 
 EXAMPLES FOR. PRACTICE. 
 
 (1.) 
 
 From 325 
 
 (2.) 
 431 
 
 (3.) 
 562 
 
 (4.) 
 600 
 
 Take 108 
 
 249 
 
 320 
 
 231 
 
 (5.) 
 From 2230 
 
 (6.) 
 
 3042 
 
 (*) 
 
 6500 
 
 (8.) 
 8435 
 
 Take 1201 
 
 2034 
 
 3211 
 
 5001 
 
 
 QUEST. 31. How do you write numbers for subtraction ? Where do yon 
 begin to subtract ? When a figure in the lower number is larger than the one 
 ibovo it, how do you proceed ? 32, How is subtraction pro\*xl ? 
 
SUBTRACTION. [SEOT. Ill 
 
 (11.) 
 
 From 45100 826340 1000000 
 
 Take 10000 513683 999999 
 
 12. From 132 dollars subtract 109 dollars. 
 
 13. From 142 bushels subtract 85 bushels. 
 
 14. From 375 pounds subtract 100 pounds. 
 
 15. From 698 yards subtract 85 yards. 
 
 16. From 485 rods subtract 175 rods. 
 
 17. Take 230 gallons from 460 gallons. 
 
 18. Take 168 hogsheads from 671 hogsheads. 
 
 19. Take 192 bushels from 268 bushels. 
 
 20. From 275 dollars take 148 dollars. 
 
 21. From 468 pounds take 219 pounds. 
 
 22. From 3246 rods take 2164 rods. 
 
 23. From 45216 take 32200. 
 
 24. From 871410 take 560642. 
 
 25. From 926500 take 462126. 
 
 26. From 6284678 take 1040640. 
 
 27. 468423. 37. 1726513167. 
 
 28. 675367. 38. 2148020372. 
 
 29. 800560. 39. 3067126140. 
 
 30. 701643. 40. 4572331203, 
 
 31. 963421. 41. 8164757025. 
 
 32. 32631242. 42. 265328140300. 
 
 33. 41652340. 43. 170643106340. 
 
 34. 56003000. 44. 465746241680. 
 
 35. 72464161. 45. 694270 590S95. 
 
 36. 86707364. 46. 920486500000. 
 
 47. A man having 235 sheep, lost 163 of them: ho* 
 many had he left ? 
 
 48. A farmer having 500 bushels of wheat, sold 278 
 bushels : how much wheat had he left ? 
 
 49. A man paid 625 dollars for a carriage and 430 
 
ART. 32.] SUBTRACTION. 33 
 
 dollars for a span of horses : how much more did he pay 
 for his carriage than for his horses ? 
 
 50. A man gave 1263 dollars for a lot, and 2385 dol- 
 lars for building a house : how much more did his house 
 cost than his lot ? 
 
 51. If a person has 3290 dollars in real estate, and 
 owes 1631 dollars, how much is he worth? 
 
 52. A man gave his son 8263 dollars, and his daughter 
 5240 dollars : how much more did he give his son tha 
 his daughter? 
 
 53. A man bought a farm for 9467 dollars, and sold 
 it for 11230 dollars : how much did he make by his bar- 
 gain ? 
 
 54. If a man's income is 10000 dollars a year, and his 
 expenses 6253 dollars, how much will he lay up ? 
 
 55. The captain of a ship having a cargo of goods 
 worth 29230 dollars, threw overboard in a storm 13216 
 dollars' worth : what was the value of the goods left ? 
 
 56. A merchant bought a quantity of goods for 12645 
 dollars, and afterwards sold them for 13960 dollars: 
 how much did he gain by his bargain ? 
 
 57. A man paid 23645 dollars fora ship and after- 
 wards sold it for 18260 dollars : how much did he lose 
 by his bargain ? 
 
 58. The salary of the President of the United States is 
 25000 dollars a year ; now if his expenses are 19265 dol- 
 lars, how much will he lay up ? 
 
 59. A general before commencing a battle, had 35260 
 soldiers in his army ; after the battle he had only 21316: 
 how many soldiers did he lose ? 
 
 60. The distance of the sun from the earth is 95000000 
 miles ; the distance of the moon from the earth is 240000 
 miles : how much farther from the earth is the sun than 
 the moon ? 
 
 2 
 
34 SUBTRACTION. [SECT. Ill 
 
 EXAMPLES INVOLVING ADDITION AND SUBTRACTION. 
 
 61. Henry bought 63 oranges of one grocer, and 26 
 of another; he afterwards sold 72: how many oranges 
 did he have left ? 
 
 62. Charles had 47 marbles, and his father gave him 
 36 more; he afterwards lost 50: how many marbles did 
 he then have ? 
 
 63. A farmer having 158 sheep, lost 30 of them by 
 sickness and sold 52 : how many sheep did he have left? 
 
 64. Sarah's father gave her 60 cents, and her mother 
 gave her 54 cents ; if she spends 62 cents for a pair of 
 gloves, how many cents will she have left ? 
 
 65. A merchant purchased a piece of silk containing 
 78 yards; he then sold 18 yards to one lady, and 17 to 
 another : how many yards had he left ? 
 
 66. If a man has property in his possession worth 
 215 dollars, and owes 39 dollars to one person, and 
 54 dollars to another, how much money will he have left, 
 when he pays his debts ? 
 
 67. If a man's income is 185 dollars per month, and 
 he pays 35 dollars for house rent, and 63 dollars for pro- 
 visions per month, how many dollars will he have left for 
 other expenses ? 
 
 68. George having 74 pears, gave away 43 of them ; 
 if he should buy 35 more, how many would he then 
 have? 
 
 69. If you add 115 to 78, and from the sum take 134, 
 what will the remainder be ? 
 
 70. If you subtract 93 from 147, and add 110 to the 
 remainder, what will the sum be ? 
 
 71. A merchant purchased 125 pounds of butter of 
 one dairy-man, and 187 pounds of another ; he afterwards 
 sold 163 pounds: how many pounds did he have left? 
 
ART. 32.] SUBTRACTION. 35 
 
 72. A miller bought 200 bushels of wheat of one 
 farmer, and 153 bushels of another; he afterwards sold 
 180 bushels : how many bushels did he have left? 
 
 73. A man traveled 538 miles in 3 days ; the first day 
 he traveled 149 miles, the second day, 126 miles : how 
 far did he travel the third day ? 
 
 74. A grocer bought a cask of oil containing 256 gal- 
 lons ; after selling 93 gallons, he perceived the cask was 
 leaky, and on measuring what was left, found he had 38 
 gallons : how many gallons had leaked out ? 
 
 75. A manufacturer bought 248 pounds of wool of one 
 customer, and 361 pounds of another ; he then worked 
 up 430 pounds : how many pounds had he left ? 
 
 76. A man paid 375 dollars for a span of horses, and 
 450 dollars for a carriage ; he afterwards sold his horses 
 and carriage for 1000 dollars; how much did he make 
 by his bargain ? 
 
 77. A grocer bought 285 pounds of lard of one farmer, 
 and 327 pounds of another; he afterwards sold 110 
 pounds to one customer, and 163 pounds to another : how 
 much lard did he have left ? 
 
 78. A flour dealer having 500 barrels of flour on hand, 
 sold 263 barrels to one customer and 65 barrels to an- 
 other : how many barrels had he left ? 
 
 79. Harriet wished to read a book through which con- 
 tained 726 pages, in three weeks ; the first week she read 
 165 pages, and the second week she read 264 pages : 
 how many pages were left for her to read the third week ? 
 
 80. A man bought a house for 1200 dollars, and hav- 
 ing laid out 210 dollars for repairs, sold it for 1300 dol- 
 lars : how much did he lose by the bargain ? 
 
 81. A young man having 2000 dollars, spent 765 the 
 first year and 843 the second year : how much had he 
 left?' 
 
36 MULTIPLICATION. [SECT. IV, 
 
 SECTION IV. 
 
 MULTIPLICATION. 
 
 ART. 33. Ex. 1. What will three lemons cost, at 2 
 cents apiece ? 
 
 Analysis. Since 1 lemon costs 2 cents, 3 lemons will 
 cost 3 times 2 cents ; and 3 times 2 cents are 6 cents. 
 Therefore, 3 lemons, at 2 cents apiece, will cost 6 cents. 
 
 OBS. The preceding operation is a short method of finding how 
 much 2 cents will amount to, when repeated or taken 3 times, and 
 is called Multiplication. Thus, 2 cents -J- 2 cents -f- 2 cents are 
 6 cents. Hence, 
 
 34r MUTIPLICATION is the process of finding the amount 
 of a number repeated or added to itself, a given number of 
 times. 
 
 The number to be repeated or multiplied, is called the 
 multiplicand. 
 
 The number by which we multiply, is called the mul- 
 tiplier, and shows how many times the multiplicand is to 
 be repeated or taken. 
 
 The answer, or number produced by multiplication, is 
 called the product. 
 
 Thus, when we say 5 times 7 are 35, 7 is the multipli- 
 cand, 5 the multiplier, and 35 the product. 
 
 OBS. When the multiplicand denotes things of one kind, or de- 
 nomination only, the operation is called Simple Multiplication. 
 
 QUEST. 34. What is multiplication 1 What, is the numbr r to be repeated 
 or multiplied called 1 What the number by which we multijyly ? What does 
 the multiplier show ? What is the answer called ? When wo say 5 times 7 
 are 35, which is the multiplicand? Which the multiplier? Which the 
 product^ Obs. When the multiplicand denotes things of one denomination 
 only, what is the operation called 1 
 
ARTS. 36 39.] MULTIPLICATION. 39 
 
 38 The product cf any two numbers will be the same, 
 whichever factor is taken for the multiplier. Thus, 
 
 If a garden contains 3 rows of trees as 
 represented by the number of horizontal * * * * * 
 
 f JL : ^ ' 1 -L * * * * * 
 
 rows of stars m the margin, and each row . 
 
 has 5 trees as represented by the number of 
 
 stars in a row, it is evident, that the whole 
 
 number of trees in the garden is equal either to the number 
 
 of stars in a horizontal row, taken three times, or to the 
 
 number of stars in a perpendicular row taken five times; 
 
 that is, equal to 5 X 3, or 3X5. 
 
 EXAMPLES. 
 
 39. When the multiplier contains but ONE figure. 
 
 Ex. 1. What will 3 horses cost, at 123 dollars apiece? 
 
 Analysis. Since 1 horse costs 123 dollars, 3 horses 
 will cost 3 times 123 dollars. 
 
 Directions. Write the multi- Operation, 
 
 plicr under the multiplicand; 123 multiplicand, 
 
 then, beginning at the right 3 multiplier. 
 
 hand, multiply each figure of the 
 
 u . ,. \\ ., ' 1f . r Dolls. 369 product. 
 multiplicand by the multiplier. 
 
 Thus, 3 times 3 units are 9 units, or we may simply say 
 3 times 3 a^e 9 ; set the 9 in units' place under the figure 
 multiplied. 3 times 2 are 6 ; set the 6 in tens' place. 
 3 times 1 are 3 ; set the 3 in hundreds' place. 
 
 Note. The pupil should be required to analyze every example, 
 and to give the reasoning in full ; otherwise the operation is liable 
 to become mer* guess-icork, and a habit is formed, which is alike 
 destructive to mental discipline and all substantial improvement. 
 
 Solve the following examples in a similar manner : 
 (2.) (3.) (4.) (5.) 
 
 Multiplicand 34 312 2021 1110 
 
 Multiplier 2345 
 
40 MULTIPLICATION. [SECT. IV 
 
 (6.) (7.) (8.) (9.) 
 
 Multiplicand, 4022 6102 7110 8101 
 
 Multiplier, _J3 _4 5 7 
 
 10. What will 6 cows cost at 23 dollars apiece. 
 
 Suggestion. In this example the product of the differ- 
 ent figures of the multiplicand into the multiplier, exceeds 
 9 ; we must therefore write the unit*' figure under the 
 figure multiplied, and carry the tens to the next product 
 on the left, as in addition. Thus, begin- 
 ning at the right hand as before, 6 times Operation. 
 3 units are 18 units, or we may simply 23 dolls, 
 
 say 6 times 3 are 18. Now it requires 6 
 
 two figures to express 18; we there- Ans. 138 dollars* 
 fore set the 8 under the figure multi- 
 plied, and reserving the 1, carry it to the product of the 
 next figure, as in addition. (Art. 23.) Next, 6 times 2 
 are 12, and 1 (to carry) makes 13. Since there are no 
 more figures to be multiplied, we set down the 13 in full. 
 The product is 138 dollars. Hence, 
 
 4O When the multiplier contains but one figure. 
 
 Write the multiplier under the multiplicand, units un- 
 der units, and draw a line beneath them. 
 
 Begin with the units, and multiply each figure of the 
 multiplicand by the multiplier, setting down the result and 
 carrying as in addition. (Art. 23.) 
 
 Multiply the following numbers together. 
 
 11. 78X4. 18. 524X6. 
 
 12. 96X5. 19. 360X7. 
 
 13. 83X3. 20. 475X4. 
 
 14. 120X7. 21. 792X5. 
 
 15. 138X6. 22. 820X.8. 
 
 16. 163X5. 23. 804x7. 
 
 17. 281X8. 24. 968X9. 
 
ARTS. 40, 41.] MULTIPLICATION. 41 
 
 25. What will 11 5 barrels of flour cost, at 6 dollars 
 per barrel ? 
 
 26. A man bought 460 pair of boots, at 5 dollars a pair : 
 ,10 w much did he pay for the whole ? 
 
 27. What cost 196 acres of land, at 7 dollars per acre? 
 
 28. What cost 310 ploughs, at 8 dollars apiece? 
 
 29. What cost 691 hats, at 7 dollars apiece? 
 
 30. What cost 865 heifers, at 9 dollars per head? 
 
 31. What cost 968 cheeses, at 8 dollars apiece? 
 
 32. What cost 1260 sheep, at 7 dollars per head? 
 
 33. What cost 9 farms, at 2365 dollars apiece? 
 
 4: 1 When the multiplier contains more than ONE figure. 
 
 34. A man sold 23 sleighs, at 54 dollars apiece : how 
 much did he receive for them all ? 
 
 Suggestion. Eeasoning as before, if 1 sleigh costs 
 54 dollars, 23 sleighs will cost 23 times as much. 
 
 Directions. As it is not Operation. 
 
 convenient to multiply by 23 54 Multiplicand, 
 
 at once, we first multiply by 23 Multiplier, 
 
 the 3 units, then by the 2 162 cost of 3 s. 
 
 tens, and add-the two results 108 " " 20 s. 
 
 together. Thus, 3 times 4 Dolls. 1242 " "23 s. 
 are 12, set the 2 under the 
 
 figure 3, by which we are multiplying, and carry the 1 
 as above. 3 times 5 are 15, and 1 (to carry) makes 16. 
 Next, we multiply by the 2 tens thus : 20 times 4 units 
 are 80 units or 8 tens ; or we may simply say 2 times 4 
 are 8. Set the 8 under the figure 2 by which we are 
 multiplying, that is, in tens' place, because it is tens. 
 2 times 5 are 10. Finally, adding these two products 
 together as they stand, units to units, tens to tens, &c., 
 we have 1242 dollars, which is the whole product re- 
 quired. 
 
42 
 
 MULTIPLICATION. 
 
 [SECT. IV 
 
 Note. When She multiplier contains more than one figure, the 
 several products of the multiplicand into the separate figures of the 
 multiplier, are called partial products. 
 
 35. Multiply 45 by 36, and prove the operation. 
 
 Operation. 
 
 Beginning at the right hand, we 
 proceed thus : 6 times 5 are 30 ; 
 set the under the figure by which 
 we ar multiplying ; 6 times 4 are 
 24 and 3 (to carry) are 27, &c. 
 
 45 Multiplicand 
 36 Multiplier. 
 "270 
 135 
 
 PROOF. We multiply the mul- 
 tiplier by the multiplicand, and 
 since the result thus obtained is 
 the same as the product above, 
 the work is rio-ht. 
 
 1620 Prod. 
 
 Proof. 
 36 
 45 
 180 
 144 
 
 1620 Prod. 
 
 36. What is the product of 234 multiplied by 165 ? 
 
 Operation. 
 
 Suggestion. Proceed in the same man- 234 
 
 ner as when the multiplier contains but 165 
 
 two figures, remembering to place the 1170 
 right luand figure of each partial product 1404 
 directly under the figure by which you 234 
 multiply. 38610 Ant 
 
 37. What is the product of 326 multiplied by 205 ? 
 
 Suggestion. Since multiplying by a Operation. 
 
 cipher produces nothing, in the operation 326 
 
 we omit the in the multiplier. Thus, 205 
 
 having multiplied by the 5 units, we next ] 630 
 
 multiply by the 2 hundreds, and place the 652 
 first figure of this partial product under 66830 Ans, 
 the figure by which we are multiplying. 
 
ARTS. 42, 43.] MULTIPLICATION. 43 
 
 42 From the preceding illustrations and principles 
 we derive the following 
 
 GENERAL RULE FOR MULTIPLICATION 
 
 I. Write the multiplier under the multiplicand, units 
 under units, tens under tens, &c. 
 
 II. When the multiplier contains but ONE figure, begin 
 with the units, and multiply each figure of the multipli- 
 cand by the multiplier, setting down the result and carry- 
 ing as in addition. (Art. 23.) 
 
 III. If the multiplier contains MORE than on,e figure, 
 multiply each figure of the multiplicand by each figure 
 of the multiplier separately, and write the first figure of 
 each partial product under the figure by which you are 
 multiplying. 
 
 Finally, add the several partial products together, and 
 the sum will be the whole product, or answer required. 
 
 43 PROOF. Multiply the multiplier by the multipli- 
 cand, and if the second result is the same as the first, the 
 work is right. 
 
 OBS. 1. It is immaterial as to the result which of the factors is 
 taken for the multiplier. (Art. 88.) But it is more convenient and 
 therefore customary to place the larger number for the multipli- 
 cand and the smaller for the multiplier. Thus, it is easier to mul- 
 tiply 254672381 by 7, than it is to multiply 7 by 254672381, but 
 the product will be the same. 
 
 2. Multiplication may also be proved by division, and by casting 
 out the nines; but neither of these methods can be explained here 
 without anticipating principles belonging to division, with which 
 the learner is supposed as yet to be unacquainted. 
 
 QUEST. 42. How do you write numbers for multiplication? When the 
 multiplier contains but one figure, how do you proceed ? When the multi- 
 plier contains more than one figure, how proceed? 41. JVbt. What \& meant 
 by partial products? What is to be done with the partial products? 43, 
 How is multiplication proved ? 
 
44 MULTIPLICATION. [^ECT. IV 
 
 EXAMPLES FOR PRACTICE. 
 
 1. Multiply 63 by 4. 10. Multiply 46 by 10. 
 
 2. Multiply 78 by 5. 11. Multiply 52 by 11. 
 
 3. Multiply 81 by 7. 12. Multiply 68 by 12. 
 
 4. Multiply 97 by 6. 13. Multiply 84 by 13. 
 
 5. Multiply 120 by 7. 14. Multiply 78 by 15. 
 
 6. Multiply 231 by 5. 15. Multiply 95 by 23. 
 
 7. Multiply 446 by 8. 16. Multiply 129 by 35. 
 
 8. Multiply 307 by 9. 17. Multiply 293 by 42. 
 
 9. Multiply 560 by 7. 18. Multiply 461 by 55. 
 
 19. If 1 barrel of flour costs 9 dollars, how much 
 will 38 barrels cost ? 
 
 20. If 1 apple-tree bears 14 bushels of apples, how 
 many bushels will 24 trees bear ? 
 
 21. In 1 foot there are 12 inches : how many inches 
 are there in 28 feet? 
 
 22. In 1 pound there are 20 shillings : how many shil- 
 lings are there in 31 pounds ? 
 
 23. What will 17 cows cost, at 23 dollars apiece ? 
 
 24. What will 25 tons of hay cost, at 1 9 dollars per ton ? 
 
 25. What will 37 sleighs cost, at 43 dollars apiece ? 
 
 26. What will a drove of 150 sheep come to, at 13 
 shillings per head ? 
 
 27. What cost 105 acres of land, at 15 dollars per acre? 
 
 28. How much will 135 yards of cloth come to, at 18 
 shillings per yard ? 
 
 29. In 1 pound there are 16 ounces : how many ounces 
 are there in 246 pounds ? 
 
 30. A drover sold 283 oxen, at 38 dollars per head: 
 how much did he receive for them ? 
 
 31. If you walk 22 miles per day, how far will you 
 walk in 305 days ? 
 
 32. In one day there are 24 hours : how many hours 
 are there in 365 days ? 
 
ARTS. 44, 45.] MULTIPLICATION. 45 
 
 33. In 1 year there are 52 weeks : how many weeks 
 are there in 175 years? 
 
 34. In 1 hour there are 60 minutes : how many min- 
 utes are there in 396 hours? 
 
 35. In 1 hogshead there are 63 gallons : how many 
 gallons are there in 450 hogsheads ? 
 
 36. What will 475 horses cost, at 73 dollars apiece? 
 
 37. In 1 square foot there are 144 square inches : how 
 many square inches are there in 235 feet ? 
 
 38. How far will a ship sail in 158 days, if she sails 
 165 miles per day ? 
 
 44. It is a fundamental principle of notation, that 
 each removal of a figure one place towards the left, in- 
 creases its value ten times; (Art. 10;) consequently an- 
 nexing a cipher to a number, increases its value ten times, 
 or multiplies it by 10; annexing two ciphers, increases 
 its value a hundred times, or multiplies it by 100 ; an- 
 nexing three ciphers, increases it a thousand times, or mul- 
 tiplies it by 1000, <fec. ; for each cipher annexed, removes 
 each figure in the number one place towards the left. 
 Thus, 12 with a cipher annexed becomes 120, and i? 
 the same as 12X10; 12 with two ciphers annexed, be- 
 comes 1200, and is the same as 12X100; 12 with three 
 ciphers annexed, becomes 12000, and is the same as 
 12X1000, &c. Hence, 
 
 45. To multiply by 10, 100, 1000, &c. 
 
 Annex as many ciphers to the multiplicand as there are 
 ciphers in the multiplier, and the number thus formed will 
 t>e the product required. 
 
 Note. To annex means to place after, or at the right hand. 
 
 QUEST. 44. What effect does it have to remove a figure />ne place towards 
 the left hand ? Two piaces ? 45. How do you proceed when the multiplier 
 Is 10, 100, 1000, &c ? Note. What is the meaning of the term annex? 
 
46 MULTIPLICATION. [SECT. IT 
 
 40. What will 10 dresses cost, at 18 dollars apiece? 
 
 Solution. If 1 dress costs 18 dollars, 10 dresses will 
 cost 10 times 18 dollars. But annexing a cipher to a 
 number multiplies it by 10. We therefore annex a cipher 
 to the multiplicand, (18 dollars,) and it becomes 180 dol- 
 ars. The answer therefore is 180 dollars. 
 
 Multiply the following numbers in a similar manner : 
 
 41. 26X10. 46. 469X10000. 
 
 42. 37X100. 47. 523X100000. 
 
 43. 51X1000. 48. 681X1000000. 
 
 44. 226X1000. 49. 85612X10000. 
 
 45. 341X1000. 50. 960305X100000. 
 51. What will 20 wagons cott, at 67 dollars apiece? 
 
 Suggestion. Since multiplying by Operation, 
 ciphers produces ciphers, we omit mul- 67 
 
 tiplying by the 0, and placing the sig- 20 
 
 nificant figure 2 under the right hand Ans. 1340 dollars, 
 figure of the multiplicand, multiply 
 by it in the usual way, and annex a cipher to the product. 
 The answer is 1340 dollars. Hence, 
 
 46. When there are ciphers on the right hand of the 
 multiplier. 
 
 Multiply the multiplicand by the significant figures of 
 the multiplier, and to this product annex as many ciphers, 
 as are found on the right hand of the multiplier. 
 
 (52.) (53.) (54.) (55.) 
 85 97 123 234 
 200 3000 40000 50000 
 
 (56.) 
 261 
 
 (57.) 
 329 
 
 (58.) 
 462 
 
 (59.) 
 571 
 
 130 
 
 2400 
 
 3501/0 
 
 460000 
 
 
 Q.UET. 46. When there are ciphers on the right of the multiplier, how da 
 you proceed ? 
 
ARTS. 46, 47.] MULTIPLICATION. 47 
 
 60. In one hour there are 60 minutes : how many min- 
 utes are there in 125 hours? 
 
 61. What will 300 barrels of flour cost af 8 dollars per 
 barrel ? 
 
 62. What cost 400 yds. of cloth, at 17 shills. per yd. ? 
 
 63. If the expenses of 1 man are 135 dollars per month, 
 how much will be the expenses of 200 men ? 
 
 64. If 1500 men can build a fort in 235 days, how long 
 will it take one man to build it ? 
 
 4:7. When there are ciphers on the right of the mul- 
 tiplicand. 
 
 Multiply the significant figures of the multiplicand by 
 tJie multiplier, and to tlie product annex as many ciphers, 
 as are found on the right of the multiplicand. 
 
 65. What will 43 building lots cost, at 3500 dollars a 
 lot? 
 
 Having placed the multiplier under Operation. 
 the significant figures of the multipli- 3500 
 
 cand, multiply by it as usual, and to 43 
 
 the product thus produced, annex two 105 
 
 ciphers, because there are two ciphers 140 
 
 on the right of the multiplicand. Ans. 150500 dolls 
 
 (66.) (67.) (68.) (69.) 
 
 1300 2400 21000 25000 
 
 15 17 24 32 
 
 70. What is the product of 132000 multiplied by 25 ? 
 
 71. What is the product of 430000 multiplied by 34 ? 
 
 72. What is the product of 1520000 multiplied by 43 ? 
 
 73. What is the product of 2010000 multiplied by 52 ? 
 
 74. What is the product of 3004000 multiplied by 61 ? 
 
 Q.CBST. 47. When there are ciphers on the right of the multiplinawl) how 
 do you proceed ? 
 
48 
 
 MULTIPLICATION. 
 
 [SECT. 
 
 48. When the multiplier and multiplicand both have 
 ciphers on the right. 
 
 Multiply the significant figures of the multiplicand by 
 the significant figures of the multiplier, and to this pro- 
 duct annex as many ciphers, as are found on the right of 
 both factors. 
 
 75. Multiply 16000 by 3200. 
 
 Having placed the significant figures 
 of the multiplier under those of the mul- 
 tiplicand, we multiply by them as usual, 
 and to the product thus obtained, annex 
 five ciphers, because there are five ci- 
 phers on the right of both factors. 
 
 Solve the following examples : 
 76. 2100X200. 
 78. 12000X210. 
 80. 38000X19000. 
 82. 2800000X26000. 
 84. 1000 milesX!40. 
 86. 120 dollars X 4200. 
 88. 867 poundsX424. 
 00. 6726 rodsX627. 
 92. 25268 penceX4005. 
 
 Operation. 
 
 16000 
 3200 
 
 32 
 
 48 
 
 Ans. 51200000 
 
 94. 376245X3164. 
 
 96. 600400X7034. 
 
 98. 432467X30005. 
 100. 680539X80406. 
 102. Multiply seventy -three 
 
 77. 3400X130. 
 
 79. 25000X2600. 
 
 81. 500000X42000. 
 
 83. 140 yards XI 6000. 
 
 85. 20 dollars X 35000, 
 
 87. 75000 dolls. X 365. 
 
 89. 6830 feetX562. 
 
 91. 7207 galls. X 807. 
 
 93. 36074 tons X 4060. 
 
 95. 703268X5346. 
 
 97. 864325X6728. 
 
 99. 4567832X27324. 
 101. 7563057X62043. 
 thousand and seven by 
 
 twenty thousand and seven hundred. 
 
 103. Multiply six hundred thousand, two hundred and 
 three by seventy thousand and seventeen. 
 
 QUEST. 48. When there are ciphers on tho right of both tl e inultipliel and 
 multiplicand, how proceed ? 
 
ABTS. 48 50.] DIVISIDN. 49 
 
 SECTIt N V. 
 DIVISION. 
 
 ART. 49o Ex. 1. How many lead pencils, at 2 cents 
 apiece, can I buy for 1 cents ? 
 
 Solution. Since 2 cents will buy 1 pencil, 10 cents 
 will buy as many pencils, as 2 cents are contained times in 
 10 cents ; and 2 cents are contained in 10 cents, 5 times. 
 I can therefore buy 5 pencils. 
 
 2. A father bought 12 pears, which he divided equally 
 among his 3 children : how many pears did each re- 
 ceive ? 
 
 Solution. Reasoning in a similar manner as above, it 
 is plain that each child will receive 1 pear, as often as 3 
 is contained in 12 ; that is, each must receive as many 
 pears, as 3 is contained times in 12. Now 3 is contained in 
 12, 4 times. Each child therefore received 4 pears. 
 
 OBS. The object of the first example is to find how many times 
 one given number is contained in another. The object of the second 
 is to divide a given number into several equal parts, and to ascertain 
 the value of these parts. The operation by which they are solved 
 is precisely the same, and is called Division. Hence, 
 
 5O. DIVISION is the process of finding how many times 
 one given number is contained in another. 
 
 The number to be divided, is called the dividend. 
 
 The number by which we divide, is called the divisor. 
 
 The ansiuer, or number obtained by division, is called 
 the quotient, and shows how many times the divisor is 
 contained in the dividend. 
 
 QUEST. 50. What is division ? What is the number to be divided, called ? 
 The number by which we divide ? What is the answer called? What does 
 .ne quotient show ? 
 
 4 
 
50 DIVISIO.V. [SECT. V 
 
 Note. The term quotient is derived from the Latin word qttotiea 
 which signifies how )ften, or how many times. 
 
 51. The number which is sometimes left after division, 
 in called the remainder. Thus, when we say 4 is con- 
 tamed in 21, 5 times and 1 over, 4 is the divisor, 21 the 
 dividend, 5 the quotient, and 1 the remainder. 
 
 OBS. 1. The remainder is always less than the divisor; for if it 
 were equal to, or greater than the divisor, the divisor could be con- 
 tained once more in the dividend. 
 
 2. The remainder is also of the same denomination as the divi- 
 dend; for it is a part of it. 
 
 52. Sign of Division (-r). The sign of Division is 
 a horizontal line between two dots (-7-), and shows that 
 the number before it, is to be divided by the number 
 after it. Thus, the expression 246, signifies that 24 is 
 to be divided by 6. 
 
 Division is also expressed by placing the divisor under 
 the dividend with a short line between them. Thus the 
 expression A 7 *, shows that 35 is to be divided by 7, and is 
 equivalent to 35-7-7. 
 
 53* It will be perceived that division is similar in prin- 
 ciple to subtraction, and may be performed by it. For 
 instance, to find how many times 3 is contained in 12, 
 subtract 3 (the divisor) continually from 12 (the dividend) 
 until the latter is exhausted ; then counting these repeated 
 subtractions, we shall have the true quotient. Thus, 3 
 from 12 leaves 9 ; 3 from 9 leaves 6 ; 3 from 6 leaves 3 ; 
 3 from 8 leaves 0. Now, by counting, we find that 3 has 
 
 QUEST. 51. What is the number called which is sometimes left after divi- 
 sion? When we say 4 is in 21, 5 times and 1 over, what is the 4 cabled? The 
 21 ? The 5 ? The 1 V Obs. Is the remainder greater or less than the divisor? 
 Why? Of what denomination is it? Why? 52. What is the sign of divi- 
 aion ? What does it show ? In what other way is division expressed ? 
 
ARTS. 51 53.] DIVISION. 
 
 51 
 
 been taken from 12, 4 times; consequently 3 is contained 
 t~ ?2, 4 times. Hence, 
 
 Division is sometimes defined to "be a short way of per- 
 forming repeated subtractions of the same number. 
 
 OBS. 1. It will also be observed that division is the reverse of 
 multiplication. Multiplication is the repeated addition of the same 
 number ; division is the repeated subtraction of the same number. 
 The product of the one answers to the dividend of the other : but 
 the latter is always given, while the former is required. 
 
 2. When the dividend denotes things of one kind, or denominar 
 tion only, the operation is called Simple Division. 
 
 DIVISION TABLE. 
 
 1 is in 
 
 2 is in 
 
 3 is in 
 
 4 is in 
 
 5 is in 
 
 1, once. 
 
 2, once. 
 
 3, once. 
 
 4, once. 
 
 5, once. 
 
 2, 2 
 
 4, 2 
 
 6, 2 
 
 8, 2 
 
 10, 2 
 
 3, 3 
 
 6, 3 
 
 9, 3 
 
 12, 3 
 
 15, 3 
 
 4, 4 
 
 8, 4 
 
 12, 4 
 
 16, 4 
 
 20, 4 
 
 5, 5 
 
 10, 5 
 
 15, 5 
 
 20, 5 
 
 25, 5 
 
 6, 6 
 
 12, 6 
 
 18, 6 
 
 24, 6 
 
 30, 6 
 
 7, 7 
 
 14, 7 
 
 21, 7 
 
 28, 7 
 
 35, 7 
 
 8, 8 
 
 16, 8 
 
 24, 8 
 
 32, 8 
 
 40, 8 
 
 9, 9 
 
 18, 9 
 
 27, 9 
 
 36, 9 
 
 45, 9 
 
 10, 10 
 
 20, 10 
 
 30, 10 
 
 40, 10 
 
 50, 10 
 
 6 is in 
 
 7 is in 
 
 8 is in 
 
 9 is in 
 
 10 is in 
 
 6, once. 
 
 7, once. 
 
 8, once. 
 
 9, once. 
 
 10, once. 
 
 12, 2 
 
 14, .2 
 
 16, 2 
 
 18, 2 
 
 20, 2 
 
 18, 3 
 
 21, 3 
 
 24, 3 
 
 27, 3 
 
 30, 3 
 
 24, 4 
 
 28, 4 
 
 32, 4 
 
 36, 4 
 
 40, 4 
 
 30, 5 
 
 35, 5 
 
 40, 5 
 
 45, 5 
 
 50, 5 
 
 36, 6 
 
 42, 6 
 
 48, 6 
 
 54, 6 
 
 60, 6 
 
 42, 7 
 
 49, 7 
 
 56, 7 
 
 63, 7 
 
 70, 7 
 
 48, 8 
 
 56, 8 
 
 64, 8 
 
 72, 8 
 
 80, 8 
 
 54, 9 
 
 63, 9 
 
 72, 9 
 
 81, 9 
 
 90, 9 
 
 60, 10 
 
 70, 10 
 
 80, 10 
 
 90, 10 
 
 100, 10 
 
 QUEST. -Obs. When the dividend denotes things cf g>ne denomination only, 
 Irhat iR the operation called ? 
 
52 DIVISION. [SECT. V 
 
 SHORT DIVISION. 
 
 ART. 54. Ex. 1. How many yards of cloth, at 2 dol 
 lars per yard, can I buy for 246 dollars ? 
 
 Analysis. Since 2 dollars will buy 1 yard, 246 dol- 
 lars will buy as many yards, as 2 dollars are contained 
 times in 246 dollars. 
 
 Directions. Write the divisor on Operation. 
 the left of the dividend with a curve w *>'- v ^ n<L 
 line between them; then, beginning ' 
 
 at the left hand, proceed thus: 2 is uot ' l 
 contained in 2, once. As the 2 in the dividend denotes 
 hundreds, the 1 must be a hundred ; we therefore write 
 it in hundreds' place under the figure divided. 2 is con- 
 tained in 4, 2 times ; and since the 4 denotes tens, the 2 
 must also be tens, and must be written in tens' place. 2 is 
 in 6, 3 times. The 6 is units ; hence the 3 must be units, 
 and we write it in units' place. The answer is 123 yards. 
 
 Solve the following examples in a similar manner : 
 
 2. Divide 42 by 2. 6. Divide 684 by 2. 
 
 3. Divide 69 by 3. Y. Divide 4488 by 4. 
 
 4. Divide 488 by 4. 8. Divide 3963 by 3. 
 
 5. Divide 555 by 5. 9. Divide 6666 by 6. 
 
 55 When the divisor is not contained in the first 
 figure of the dividend, we must find how many times it 
 is contained hi the first two figures. 
 
 10. At 2 dollars a bushel, how much wheat can be 
 bought for 124 dollars? 
 
 Since the divisor 2, is not contained in Operation. 
 the first figure of the dividend, we find 2)124 
 how many times it is contained in the first Ans. 62 bu, 
 two figures. Thus 2 is in 12, 6 times ; set 
 the 6 under the 2. Next, 2 is in 4, 2 times. The an- 
 Ewer is 62 bushels. 
 
ARTS 54 57.] DIVISION. 53 
 
 11. Divide 142 by 2. 13. Divide 1648 by 4. 
 
 12. Divide 129 by 3. 14. Divide 2877 by 7. 
 
 56 After dividing any figure of the dividend, if there 
 is a remainder, prefix it mentally to the next figure of the 
 dividend, and then divide this number as before. 
 
 Note. To prefix means to place before, or at the left hand. 
 
 15. A man bought 42 peaches, which he divided 
 equally among his 3 children : how many did he give to 
 each? 
 
 When we divide 4 by 3, there is 1 re- Operation, 
 mainder. This we prefix mentally to the 3)42 
 next figure of the dividend. We then say, 14 Ans. 
 
 3 is in 12, 4 times. 
 
 16. Divide 56 by 4. 18. Divide 456 by 6. 
 
 17. Divide 125 by 5. 19. Divide 3648 by 8. 
 
 57. Having obtained the first quotient figure, if the 
 divisor is not contained in any figure of the dividend, place 
 a cipher in the quotient, and prefix this figure to the next 
 one of the dividend, as if it were a remainder. 
 
 20. If hats are 2 dollars apiece, how many can be 
 bought for 216 dollars ? 
 
 As the divisor is not contained in 1, Operation. 
 the second figure of the dividend, we 2)216 
 
 put a in the quotient, and prefix the Ans. 108 hats. 
 
 1 to the 6 as directed above. Now 2 
 is in 16, 8 times. 
 
 21. Divide 2545 by 5. 23. Divide 6402 by 6. 
 
 22. Divide 3604 by 4. 24. Divide 4024 by 8. 
 25. A man divided 17 loaves of bread equally between 
 
 2 poor persons : how many did he give to each ? 
 Suggestion. Reasoning as before, he gave each as 
 
 many loaves as 2 is contained times in 17 : 
 
54 DIVISION. [S2CT. V 
 
 Thus, 2 is contained in 17, 8 Opemtion-. 
 times and 1 over; that is, after 2)17 
 
 giving them 8 loaves apiece, there Quot. 8-1 remainder, 
 is one loaf left which is not divid- Ans. 8-J- loaves. 
 ed. Now 2 is not contained in 1 ; 
 
 hence the division must be represented by writing the 2 
 under the 1, thus , (Art. 52,) which jnust be annexed to 
 the 8. The true quotient, is 8-J. He therefore gave eight 
 and a half loaves to each. Hence, 
 
 58 When there is a remainder after dividing the last 
 figure of the dividend, it should always be written over the 
 divisor and annexed to the quotient. 
 
 Note. To annex means to place after, or at the riglit hand. 
 
 59* When the process of dividing is carried on in the 
 mind, and the quotient only is set down, the operation it 
 called SHORT DIVISION. 
 
 6O From the preceding illustrations and principles, we 
 derive the following 
 
 RULE FOR SHORT DIVISION. 
 
 I. Write the divisor on the left of the dividend, with a 
 curve line between them. 
 
 Beginning at the left hand, divide each figure of the 
 dividend by the divisor, and place each quotient figure 
 under the figure divided. 
 
 II. When there is a remainder after dividing any fig- 
 ure, prefix it to the next figure of the dividend and divide 
 this number as before. If the divisor is not contained in 
 
 QUEST. 59. What is Short Division ? GO. How do you write numbers 
 for short division? Where begin to divide ? Where place each quotient fig- 
 ure? When there is a remainder after dividing a figure of the dividend, 
 what must be done with it 1 If the divisor is not contained in a fl ore of th*> 
 dividend, how proceed? When there is a remainder, after dividing the luat 
 fl#me of the dividend, what must be done with it ? 
 
ARTS. 58 61.] DIVISION. 55 
 
 any figure of the dividend, place a cipher in the quotient, 
 and prefix this figure to the next one of the dividend, as if 
 it were a remainder. (Arts. 56, 57.) 
 
 III. When there is a ramainder after dividing the last 
 figure y write it over the divisor and annex it to the quotient, 
 
 61 PROOF. Multiply the divisor by the quotient, to 
 the product add the remainder, and if the sum is equal to 
 the dividend, the work is right. 
 
 OBS. Division may also be proved by subtracting the remainder, 
 if any, from the dividend, then dividing the result by the quotient. 
 
 EXAMPLES FOR PRACTICE. 
 
 1. Divide 426 by 3. 10. Divide 3640 by 5. 
 
 2. Divide 506 by 5. 11. Divide 6210 by 4. 
 
 3. Divide 304 by 4. 12. Divide 7031 by 7. 
 
 4. Divide 450 by 6. 13. Divide 2403 by 6. 
 
 5. Divide 720 by 7. 14. Divide 8131 by 9. 
 
 6. Divide 510 by 9. 15. Divide 7384 by 8. 
 
 7. Divide 604 by 5. 16. Divide 8560 by 7. 
 
 8. Divide 760 by 8. 17. Divide 7000 by 8. 
 
 9. Divide 813 by 7. 18. Divide 9100 by 9. 
 
 19. How many pair of shoes, at 2 dollars a pair, cau 
 you buy for 126 dollars ? 
 
 20. How many hats, at 4 dollars apiece, can be bought 
 for 168 dollars? 
 
 21. A man bought 144 marbles which he divided equally 
 among his 6 children : how many did each receive ? 
 
 22. A man distributed 360 cents to a company of poor 
 children, giving 8 cents to each : how many children were 
 there in the company ? 
 
 23. How many yards of silk, at 6 shillings per yard, 
 can I buy for 450 shillings ? 
 
 QUEST. 61. How is division proved? Obs* What other wav of proving 
 division is mentioned? 
 
56 DIVISION. [SECT. V 
 
 24. A man having 600 dollars, wished to lay it out 
 in flour, at 7 dollars a barrel: how many whole barrels 
 could he buy, and how many dollars would he have left ? 
 
 25. If you read 9 pages each day, how long will it 
 take you to read a book through which has 828 pages? 
 
 26. If I pay 8 dollars a yard for broadcloth, how many 
 yards can I buy for 1265 dollars? 
 
 27. If a stage coach goes at the rate of 8 miles per 
 hour, how long will it be in going 1560 miles ? 
 
 28. If a ship sails 9 miles an hour, how long will it 
 be in sailing to Liverpool, a distance of 3000 miles ? 
 
 LONG DIVISION. 
 
 ART. 62. Ex. 1. A man having 156 dollars laid it 
 
 out in sheep at 2 dollars apiece : how many did he buy ? 
 
 Analysis. Reasoning as before, since 2 dollars will 
 
 buy 1 sheep, 156 dollars will buy as many shee$ as 2 
 
 dollars are contained times in 156 dollars. 
 
 Directions. Having written the di- Operation. 
 visor on the left of the dividend as in Di - Divid - ^ uot - 
 short division, proceed in the follow- 34 . 
 
 ing manner : 
 
 First. Find how many times the -^ 
 
 divisor (2) is contained in (15) the 
 first two figures of the dividend, and place the quotient 
 figure (7) on the right of the dividend with a curve line 
 between them. Second. Multiply the divisor by the 
 quotient figure, (2 times 7 are 14,) and write the product 
 (14) under the figures divided. Third. Subtract the 
 product from the figures divided. (The remainder is 1.) 
 Fourth. Bringing down the next figure of the dividend, 
 and placing it on the right of the remainder we have 16. 
 Now 2 is contained in 16, 8 times; place the 8 on the 
 right hand of the last quotient figure, and multiplying 
 
ARTS. 62, G 3.] DIVISION. 57 
 
 the divisor by it, (8 times 2 are 16,) set the product undei 
 the figures divided, and subtract as before. Therefore 156 
 dollars will buy 78 sheep, at 2 dollars apiece. 
 
 63. When the result of each step in the operation is 
 set down, the process of dividing is called LONG DIVISION. 
 
 It is the same in principle as Short Division. The 
 only difference between them is, that in Long Division 
 the result of each step in the operation is written down, 
 while in Short Division we carry on the whole process 
 in the mind, simply writing down the quotient. 
 
 Note. To prevent mistakes, it is advisable to put a dot under 
 each figure of the dividend, when it is brought down. 
 
 Solve the following examples by Long Division : 
 
 2. Divide 195 by 3. Ans. 65. 
 
 3. Divide 256 by 2. 6, Divide 2665 by 5. 
 
 4. Divide 1456 by 4. V. Divide 4392 by 6. 
 
 5. Divide 5477 by 3. 8. Divide 6517 by Y. 
 
 OBS. When the divisor is not contained in the first two figures of 
 the dividend, find how many times it is contained in the first threc t 
 or the fewest figures which will contain it, and proceed as before. 
 
 9. How many times is 13 contained in 10519? 
 
 Thus, 13 is contained in 105, Operation. 
 
 8 times; set the 8 in the quo- 13)l0519(809- t 2 sr Ans. 
 tient then multiplying and sub- 104 
 
 tracting, the remainder is 1. 119 
 
 Bringing down the next figure 117 
 
 we have 11 to be divided by 13. 2 rem. 
 
 But 13 is not contained in 11 ; 
 
 therefore we put a cipher in the quotient, and bring down 
 
 the next figure. (Art. 57.) Then 13 is sontained in 119, 
 
 CUTEST. 63. What is long division ? Wiiat is the diflerence between long 
 ed short division ? 
 
58 DIVISION. [SECT "V. 
 
 9 times. Set the 9 in the quotient, multiply and sub- 
 tract, and the remainder is 2. Write the 2 over the di- 
 visor, and annex it to tho quotient. (Art. 58.) 
 
 O4. After the first quotient figure is obtained, for 
 each figure of the dividend which is brought down, either 
 a significant figure or a cipher must be put in the quotient. 
 
 Solve the following examples in a similar manner : 
 
 10. Divide 15425 by 11. Ans. 1402-ft-. 
 
 11. Divide 31237 by 15. Ans. 2082-ft. 
 
 65. From the preceding illustrations and principles 
 we derive the following 
 
 RULE FOR LONG DIVISION. 
 
 I. Beginning on the left of the dividend, find liow many 
 times the divisor is contained in the fewest figures that will 
 contain it, and place the quotient figure on the right of 
 the dividend with a curve line between them. 
 
 II. Multiply the divisor by this figure and subtract 
 the product from the figures divided ; to the right of 
 the remainder bring down the next figure of the dividend. 
 and divide this number as before. Proceed in this man- 
 ner till all the figures of the dividend are divided, 
 
 III. When there is a remainder after dividing the last 
 figure, write it over the divisor, and annex it to the quo- 
 tient, as in short division. 
 
 OBS. 1. Long Division is proved in the same manner as Short 
 Division. 
 
 2. When the divisor contains but one figure, the operation by 
 Short Division is the most expeditious, and si ould therefore be 
 practiced; but when the divisor contains two or *r* \re figures, it will 
 generally be the most convenient to divide by Long Division. 
 
 QUEST. 65. How do you divide in long division? Where place the quo- 
 tient ? Aftei obtaining the first quotient figure, how proceed ? When there is 
 a remainder after dividing the L'ist figure of the dividend what must be done 
 with it? Ols. How is long division proved? When should short division 
 be used ? Wheii long division ? 
 
ARTS. 64, 65.] DIVISION. 59 
 
 EXAMPLES FOR PRACTICE. 
 
 1. Divide 369 by 8. 10. Divide 675 by 25. 
 
 2. Divide 435 by 9. 11. Divide 742 by 31. 
 
 3. Divide 564 by 7. 12. Divide 798 by 37. 
 
 4. Divide 403 by 10. 13. Divide 834 by 42. 
 
 5. Divide 641 by 11. 14. Divide 960 by 48. 
 
 6. Divide 576 by 12. 15. Divide 1142 by 53. 
 
 7. Divide 274 by 13. 16. Divide 2187 by 67. 
 
 8. Divide 449 by 14. 17. Divide 3400 by 75. 
 
 9. Divide 617 by 15. 18. Divide 4826 by 84. 
 
 19. How many caps, at 7 shillings apiece, can I buy 
 for 168 shillings? 
 
 20. How many pair of boots, at 5 dollars a pair, can 
 be bought for 175 dollars ? 
 
 <}1. A man laid out 252 dollars in beef, at 9 dollars a 
 barrel : how many barrels did he buy ? 
 
 22. In 12 pence there is 1 shilling : how many shillings 
 are there in 198 pence? 
 
 23. In 20 shillings there is 1 pound : how many pounds 
 are there in 2 1 5 shillings ? 
 
 24. In 16 ounces there is 1 pound: how many pounds 
 are there in 268 ounces ? 
 
 25. How many trunks, at 15 shillings apiece, can be 
 bought for 255 shillings ? 
 
 26. If 27 pounds of flour will last a family a week, 
 how long will 810 pounds last them? 
 
 27. How many yards of broadcloth, at 23 shillings per 
 yard, can be bought for 756 shillings? 
 
 28. If it takes 18 yards of silk to make a dress, how 
 many dresses can be made from 1350 yards? 
 
 29. How many sheep, at 19 shillings per head, can be 
 bought for 1539 shillings? 
 
 30. A farmer having 1840 dollars, laid it out in land, 
 at 25 dollars per acre : how many acres did he buy? 
 
60 DIVISION. [SECT. V 
 
 31. A man wishes to invest 2562 dollars in Railroad 
 stock : how many shares can he buy, at 42 dollars per 
 share ? 
 
 32. In 1 year there are 52 weeks: how many years 
 are there in 1640 weeks ? 
 
 33. In one hogshead there are 63 gallons: how many 
 hogsheads are there in 3065 gallons ? 
 
 34. If a man can earn 75 dollars in a month, he wmany 
 months will it take him to earn 3280 dollars ? 
 
 35. If a young man's expenses are 83 dollars a month, 
 how long will 4265 dollars support him? 
 
 36. A man bought a drove of 95 horses for 4750 dol- 
 
 o 
 
 lars : how much did he give apiece ? 
 
 37. If a man should spend 16 dollars a month, how 
 long will it take him to spend 172 dollars? 
 
 38. A garrison having 2790 pounds of meat, wished to 
 have it last them 3 1 days : how many pounds could they 
 eat per day ? 
 
 39. How many times is 54 contained in 3241, and how 
 many over ? 
 
 40. How many times is 68 contained in 7230, and how 
 many over ? 
 
 41. How many times is 39 contained in 1042, and how 
 many over? 
 
 42. How many times is 47 contained in 2002, and how 
 many over? 
 
 43. What is the quotient of 1704 divided by 56 ? 
 
 44. What is the quotient of 2040 divided by 60 ? 
 
 45. What is the quotient of 2600 divided by 49 ? 
 
 46. What is the quotient of 2847 divided by 81 ? 
 
 47. Divide 1926 by 75. 51. Divide 9423 by 105. 
 
 48. Divide 2230 by 85. 52. Divide 13263 by 112, 
 
 49. Divide 6243 by 96. 53. Divide 26850 by 123, 
 
 50. Divide 8461 by 99. 54. Divide 48451 by 224. 
 
ARTS. 66, 67.] DIVISION. 6! 
 
 6G. It has been shown that annexing a cipher to a 
 number, increases its value ten times, or multiplies it by 
 10, (Art. 44.) Reversing this process, that is, removing 
 a cipher from the right hand of a number, will evidently 
 diminish its value ten times, or divide it by 10; for, each 
 Sgure in the number is thus restored to its original place, 
 and consequently to its original value. Thus, annexing a 
 cipher to 12, it becomes 120, which is the same as 12 X 10. 
 On the other hand, removing the cipher from 120, it be- 
 comes 12, which is the same as 12010. 
 
 In the same manner it may be shown, that removing 
 two ciphers from the right of a number, divides it by 100; 
 removing three, divides it by 1000; removing four, di- 
 vides It by 10000, &c. Hence, 
 
 67. To divide by 10, 100, 1000, &c. 
 
 Cut off as many figures from the right hand of the divi- 
 dend as there are ciphers in the divisor. The remaining 
 figures of the dividend will be the quotient, and those cut 
 off the remainder. 
 
 55. Divide 2456 by 100. 
 
 Since there are 2 ciphers on Operation. 
 
 the right of the divisor, we cut 1JOO)24|56 
 off 2 figures on the right of the Quot. 24 and 56 rera 
 dividend. The quotient is 24 
 and 56 remainder, or 24- 1 V<r- 
 
 50. Divide 1325 by 10. Ans. 132 and 5 rem. 
 
 57. Divide 4620 by 100. 
 
 58. Divide 5633 by 1000. 
 
 59. Divide 8465 by 1000. 
 
 60. Divide 26244 by 1000. 
 
 61. Divide 136056 by 10000. 
 
 QuEST.~6<5. What is the effect of annexing a cipher to a number? What i* 
 the effect of removing a cipher from the right of a nr"aber ? 67, How proceed 
 wben the divisor is 10, 100, 1000, &e.? 
 
DIVISION* 
 
 [SECT. V- 
 
 62. Divide 2443667 by 100000. 
 
 63. Divide 23454631 by 1000000. 
 
 68 When there are ciphers on the right hand of tho 
 divisor. 
 
 Cut off the ciphers from the divisor ; also cut of as 
 many figures from the right of the dividend. Then divide 
 the remaining figures of the dividend by the remaining fig- 
 ures of the divisor, and the result will be the quotient. 
 
 Finally, annex the figures cut off from the dividend to 
 the remainder, and the number thus formed will be the true 
 remainder. 
 
 64. At 200 dollars apiece, how many carriages can be 
 bought for 4765 dollars ? 
 
 Having cut off the two ciphers on 
 the right of the divisor, and two fig- 
 ures on the right of the dividend, we 
 divide the 47 by 2 in the usual way. 
 
 65. Divide 2658 by 20. 
 
 Ans. 132 and 18 rem 
 
 Operation. 
 
 2|OQ)47|65 
 
 Ans. 23 165 rera. 
 
 66. 3642 by 30. 
 68. 76235 by 1400. 
 70. 93600 by 2000. 
 72. 23148 by 1200. 
 74. 50382 by 1800. 
 76. 894000 by 2500. 
 78. 7450000 by 420000. 
 80. 348676 235. 
 82. 762005 401. 
 84. 6075071623. 
 86. 4367238-7-2367. 
 88. 8230732-7-3478. 
 90. 93670S58-f-67213. 
 
 67. 6493 by 200 
 69. 82634 by 1600. 
 71. 14245 by 3000. 
 73. 42061 by 1500. 
 75. 88317 by 2100. 
 77. 9203010 by 3100. 
 79. 9000000 by 300000. 
 81. 467342 341. 
 83. 506725 603. 
 85. 736241 2764. 
 87. 6203451-7-3827. 
 89. 823o762-f-42316. 
 91. 98765421-7-84327. 
 
 QUEST. 68. When there are ciphers on the right of the divisor, how pro 
 ceed ? \Vhat is to be done with figures cut off fro in the dividend ? 
 
ARTS. 68 73.] FRACTIONS. 63 
 
 SECTION VI. 
 
 FRACTIONS. 
 
 7 1 When a number or thing, as an apple or a pear, 
 is divided into two equal parts, one of these parts is called 
 one half. If divided into three equal parts, one of the 
 parts is called one third ; if divided into four equal parts, 
 one of the parts is called one fourth, or one quarter / if 
 into ten, tenths ; if into a hundred, hundredths, &Q. 
 
 When a number or thing- is divided into equal parts, as 
 halves, thirds, fourths, fifths, &c., these parts are called 
 Fractions. Hence, 
 
 72. A FRACTION denotes a part or parts of a number 
 or thing. 
 
 An Integer is a whole number. 
 
 Note. The term fraction, is derived from the Latin fractio, 
 which signifies the act of breaking, a broken part or piece. Hence, 
 fractions are sometimes called broken numbers. 
 
 73. Fractions are commonly expressed by two num- 
 bers, one placed over the other, with a line between them. 
 Thus, one half is written, -J-; one third, -J- ; one fourth, J; 
 three fourths, f- ; two fifths, f ; nine tenths, -f^-, &c. 
 
 The number below the line is called the denominator, 
 and shows into how many parts the number or thing is 
 divided. 
 
 QUEST. 71. What is meant by one half? What is meant by one third 7 
 What is meant by a fourth ? What are fourths sometimes called ? What is 
 meant by fifths ? By sixths ? Eighths ? How many sevenths mako a whole 
 one ? How many tenths ? What is meant by twentieths ? By hundrodths? 
 72. What is a Fraction? What is an Integer? 73. How are fractions com- 
 monly expressed 1 What is the number below the line callud ? What does it 
 show ? 
 
64 FRACTIONS. [SECT. VL 
 
 The number above the line is called the numerator, and 
 shows how many parts are expressed by the fraction. 
 Thus in the fraction -, the denominator .3, shows that the 
 number is divided into three equal parts ; the numerator 2, 
 shows that two of those parts are expressed by the fraction. 
 
 The numerator and denominator taken together, are 
 called the terms of the fraction. 
 
 7 4. A proper fraction is a fraction whose numerator is 
 less than its denominator ; as, -|-, f , -f-. 
 
 An improper fraction is one whose numerator is equal 
 to, or greater than its denominator, as f, f-. 
 
 A simple fraction is a fraction which has but one nu- 
 merator and one denominator, and may be proper, or im- 
 proper ; as, f, 
 
 A compound fraction is a fraction of a fraction ; as, J- of 
 i of I. 
 
 A complex fraction is one which has a fraction in its 
 
 2i 4 24- -3- 
 
 numerator, or denominator, or in both ; as , -, f, ~. 
 
 5 5 8f f 
 
 A mixed number is a whole number and a fraction writ- 
 ten together ; as, 4f , 25i. 
 
 7G The vato of a fraction is the quotient of the nu- 
 merator divided by the denominator. Thus, the value of 
 f is two ; of % is one ; of -J is one third ; &c. 
 
 Read the following fractions, and name the kind of each : 
 
 1. f; f; f ; f; if; if; Y; W; W- 
 
 2. -foff; f offof-V; * of of of 75. 
 
 QJ. 07 AI 3. 
 
 3. 2i; 14f ; 86; 
 
 QUEST. What is the number above the line called? What does it show? 
 What are the denominator and numerator, taken together, culled ? 74. What 
 is a proper fraction ? An improper traction V A simple fraction? A com- 
 pound fraction ? A complex fraction ? A mixed number ? 70. What is II* 
 value of a fraction? 
 
ARTS. 74 78.] FRACTIONS. x 65 
 
 To find a fractional part of a given number. 
 
 Ex. 1. If a loaf of bread costs 4 cents, what will half a 
 loaf cost ? 
 
 Analysis. If a whole loaf costs 4 cents, 1 half a loaf 
 will cost 1 half of 4 cents ; and 1 half of 4 cents is 2 cents 
 Half a loaf of bread will therefore cost 2 cents. 
 
 2. If a pound of sugar costs 12 cents, what will 1 third 
 of a pound cost ? 
 
 Analysis. Reasoning as before, if a whole pound costs 
 12 cents, 1 third of a pound will cost 1 third of 12 cents; 
 and 1 third of 12 cents is 4 cents. Ans. 4 cents. 
 
 77. From these examples the learner will perceive that 
 
 A half of a number is equal to as many units, as 2 is 
 contained times in that number. 
 
 A third of a number is equal to as many units as 3 is 
 contained times in that number. 
 
 A fourth of a numbei is equal to as many units, as 4 is 
 contained times in that number, &c. Hence, 
 
 78 To find a HALF of a number, divide it by 2. 
 To find a THIRD of a number, divide it by 3. 
 To find a FOURTH of a number, divide it by 4, &c. 
 Note. For mental exercises in Fractions, see Mental Arith- 
 metic, Section VII. 
 
 3. What is half of 257 ? 
 
 Dividing 257 by 2, the quotient is 128 Operation. 
 
 and 1 over. Placing the 1 over the 2 2)257 
 and annexing it to the quotient, we ha\e 128i -Ans* 
 128-J-, which is the answer required. 
 
 4. What is a third of 21 ? 33 ? 48 ? 78 ! 151 ! 
 
 5. What is a fourth of 45 ? 68 ? 72 ? 81 ? 130 I 
 
 6. What is a fifth of 7o ? 95 ? 135 ? 163 ? 
 
 7. What is an eighth of 73 ? 98 ? 104 ? 128 ? 
 
 QCKBT. 78. How do you find a half of a number ? A third ? A fonit t 
 3 
 
66 FRACTIONS. [SECT. VI. 
 
 8. What is a seventh of 88 ? 133 ? 175 ? 250 ? 
 
 9. What is a ninth of 126 ? 163 ? 270 ? 316 ? 
 
 7 9 To find what part one given number is of another. 
 
 Make the number called the part, the numerator, and 
 the other given number the denominator. The fraction 
 thus formed will be the answer required. 
 
 1. What part of 3 is 2 ? Ans. f. 
 
 2. What part of 4 is 1? Is 2 ? Is 3 ? Is5? 
 
 3. What part of 7 is 2 ? Is 4 ? Is 5 ? Is 6 ? 
 
 4. What part of 9 is 1 ? Is 2 ? Is 4 ? Is 5? 
 
 5. 5 is what part of 11 ? Of 12 ? Of 13 ? 
 
 6. 8 is what part of 17 ? Of 19 ? Of 45 ? 
 
 7. 15 is what part of 38 ? Of 57 ? Of 85 ? 
 
 8O A part of a number being given to find the whole. 
 
 Multiply the given part by the number of parts into 
 which the whole is divided, and the product will be the 
 answer required. 
 
 1. 27 is 1 ninth of what number? 
 
 Suggestion. Since 27 is 1 ninth, 9 ninths, or the whole, 
 must be 9 times 27; and 27x9 243. Ans. 
 
 The given part is 27, and the Operation. 
 
 number of parts into which the 27 = 1 ninth, 
 
 whole is divided, is 9 ninths; 9= no. parts, 
 
 we therefore multiply 27 by 9. Ans. 2 43= the whole. 
 
 2. 18 is 1 fifth of what number? 
 
 3. 23 is 1 fourth of what number ? 
 
 4. 34 is 1 seventh of what number ? 
 
 5. 45 is 1 fifteenth of what number ? 
 
 6. 58 is 1 twelfth of what number? 
 
 7. 63 is 1 sixteenth of what number ? 
 
 QUEST. /9. How do you find what part one number is of another 1 
 80 When a part of a number is given, how do you find the whole ? 
 
ARTS. 79 83.] FRACTIONS. 67 
 
 Multiplying a whole number by a fraction. 
 
 81. We have seen that multiplying by a whole num- 
 ber is taking the multiplicand as many times as there are 
 units in the multiplier. (Art. 36.) On the other hand, 
 
 If the multiplier is only a part of a unit, it is plain we 
 must take only a part of the multiplicand. Hence, 
 
 82 Multiplying by a fraction is talcing a certain 
 PORTION of the multiplicand as many times as there are 
 like portions of a unit in the multiplier. That is, 
 
 Multiplying by -J-, is taking 1 half of the multiplicand 
 once. Thus, 6Xi 3. 
 
 Multiplying by --, is taking 1 third of the multiplicand 
 once. Thus, 6xi=2. 
 
 Multiplying by f, is taking 1 third of the multiplicand 
 twice. Thus, 6X1=4. 
 
 Obs. If the multiplier is a unit or 1, the product is equal to the 
 multiplicand ; if the multiplier is greater than a unit, the product 
 is greater than the multiplicand ; (Art. 36 j) and if the multiplier is 
 less than a unit, the product is less than the multiplicand. Hence, 
 
 83. To multiply a whole number by a fraction. 
 Divide the given number by tJie denominator, and mul- 
 tiply the quotient by the numerator. 
 
 Obs. 1. The result will be the same if vre first multiply the given 
 number by the numerator, then divide this product by the denomi- 
 nator. 
 
 2. When the numerator is 1, it is unnecessary to multiply by it; 
 for, multiplying by 1 does not alter the value of a number. (Art. 
 82. Obs.) 
 
 QUEST. 81. What is meant by multiplying by a whole number? 82. By a 
 fraction? What is meant by multiplying by 1? By 4? By ? By |? 
 Obs. If the multiplier is a unit or 1, what is the product equal to ? When the 
 multiplier is greater than 1, how is the product compared with the multipli- 
 cand ? When less, how ? 83. How do you multfply a whole number by a frac- 
 tion ? Obs. What other method is mentioned ? When flie muneiator ie 1, is 
 tt aece&sas y to multiply by it ? Why not ? 
 
68 FRACTIONS. [SECT. VI, 
 
 Ex. 1. If a ton of coal costs 9 dollars, what -will a 
 ton cost ? 
 
 Suggestion. Since a whole ton costs Ojperc&i&ri. 
 9 dollars, 1 half a ton will cost 1 half of 2)9 
 
 9 dollars. Now 2 is contained in 9, 4 Ans. 4 dolls, 
 times and 1 over. Place the 1 over the 
 divisor 2, and annex it to the quotient. (Art. 58.) 
 
 2. What will f of a yard of cloth cost, at 36 shilling* 
 per yard ? 
 
 Suggestion. First find what 1 third First Operation. 
 
 of a yard will cost, then 2 thirds. 3)36 
 
 That is, divide the given number by 12 
 
 the denominator 3, then multiply the 2 
 
 quotient by the numerator 2. Ans. 24 shil. 
 
 Or, we may first multiply the given Second Operation 
 
 number by the numerator, then di- 36 
 
 vide the product by the denomina- 2 
 
 tor. The answer is the same as be- 3)72 
 
 fore. Ans. 24 shil. 
 
 3. If an acre of land costs 30 dollars, what will of 
 an acre cost ? 
 
 4. What will -J- of a barrel of flour cost, at 40 shillings 
 per barrel ? 
 
 5. What will of a hogshead of molasses cost, at 37 
 dollars per hogshead ? 
 
 6. What will -f- of a barrel of apples cost, at 28 shil- 
 fings per barrel ? 
 
 7.. Multiply 48 by f . 12. Multiply 56 by f. 
 
 8. Multiply 35 by f. 13. Multiply 72 by f. 
 
 9. Multiply 54 by i. 14. Multiply 120 by f. 
 
 10. Multiply 49 by f. 15. Multiply 168 by f. 
 
 11. Multiply 64 by -f. 16, Multiply 243 by \. 
 
ABT 84.] FRACTIONS. 69 
 
 Multiplying a whole number by a mixed number. 
 
 17. What will 5 yards of cloth cost, at 18 shillings 
 per yard ? 
 
 Suggestion. Since 1 yard costs 2)18 cost of 1 yd. 
 18 shillings, 5- yards will cost 5- Si- 
 
 times as much. We first multiply 90 cost of 5 yds. 
 
 18 shillings by 5, then by , and 9 "of -J- yd. 
 
 add the products together. Hence, Ans. 99s. " of 5| yds. 
 84. To multiply a whole number by a mixed number 
 Multiply first by the whole number, then by the fraction, 
 and add the products together. (Art. 83.) 
 
 18. Multiply 26 by 2. Ans. 65. 
 
 19. Multiply 30 by 2-J-. 25. Multiply 75 by 2-f. 
 
 20. Multiply 36 by 3-J-. 26. Multiply 63 by 4-f. 
 
 21. Multiply 45 by 4. 27. Multiply 100 by 5f. 
 
 22. Multiply 42 by 5K 28. Multiply 165 by 7f. 
 
 23. Multiply 36 by 3*- 29. Multiply 180 by 8f. 
 
 24. Multiply 56 by H. 30. Multiply 192 by 9$. 
 
 31. Multiply 41 rods by 5-J-. Ans. 225 rods. 
 
 32. Multiply 68 rods by 16-J-. 
 
 33. What cost 21- acres of land, at 35 dollars pei 
 acre ? 
 
 34. What cost 34-J- hundred weight of indigo, at 47 
 dollars per hundred ? 
 
 35. What cost 63f tons of iron, at 96 dollars per ton? 
 
 Dividing a whole number by a fraction. 
 
 Ex. 1. How many apples at -J- a cent apiece, can you 
 buy for 5 cents ? 
 
 Analysis. If % a cent will buy 1 apple, 5 cents will 
 buy as many apples, as -J- a cent is contained times in 5 
 cents ; that is, as many as there are halves in 5 whole ones, 
 
 QUEST. 84. How do you multiply a whole number by a mixed number ? 
 
70 FRACTIONS. [SECT. VI 
 
 Now in 1 cent there are 2 halves, therefore in 5 cents 
 there are 5 times 2, which are 10 halves ; and 1 half is 
 containad in 10 halves, 10 times. Ans. 10 apples. 
 
 2. How many plums, at -f of a cent apiece, can you 
 buy for 8 cents ? 
 
 Analysis. Reasoning as before, you can buy as many 
 plums as | of a cent are contained times in 8 cents. Now 
 in 1 cent there are 3 thirds, therefore in 8 cents there are 
 8 times 3, which are 24 thirds, and 2 thirds are contained 
 in 24 thirds, 12 times. Ans. 12 plums. Hence, 
 
 8 5 To divide a whole number by a fraction. 
 
 Multiply the given number by the denominator, and 
 divide the result by the numerator. 
 
 OBS. When the numerator is 1, it is unnecessary to divide bj 
 it ; for it is plain that dividing by 1 does not alter the value of e 
 number. 
 
 3. Divide 17 by i. 
 
 We multiply the 17 by the denominator Operation. 
 2; and since dividing by 1 does not alter 17 
 the value of a number, we do not divide 2 
 
 by it. 34 Ans 
 
 4. Divide 19 by -f. 
 
 Operation. 
 
 Multiply the 19 by 3, and divide the 19 
 product by 2. Place the remainder 1 3 
 
 over the divisor, and annex it to the 28. 2)57 
 
 " 
 
 6. Divide 25 by -J-. 10. Divide 89 by f. 
 
 6. Divide 38 by i. 11. Divide 123 by 
 
 7. Divide 47 by \. 12. Divide 156 by 
 
 8. Divide 63 by - 2 V 13. Divide 190 by - 
 
 9. Divide 72 by f. 14. Divide 256 by - 
 
 QUEST. 85. II ow do you divide a whole number by a fraction ? (Ift*. 
 the numerator is 1, is it necessary to divide by it? Why not? 
 
ARTS. 85, 8G.] FRACTIONS. Yl 
 
 Dividing a whole number by a mixed number. 
 
 Ex. 1. How many lemons, at 5-J- cents apiece, can you 
 buy for 22 cents? 
 
 Analysis. Since 5j cents will buy 1 lemon, 22 cents 
 will buy as many lemons, as 5-J- cents are contained times 
 in 22 cents. Now in 5|- cents there are 11 halves, and in 
 22 cents there are 44 halves ; but 11 halves are contained 
 in 44 halves, 4 times. Ans. 4 lemons. 
 
 Suggestion. We change the divi- Operation,. 
 sor to halves by multiplying the whole 51^22 
 number by the denominator 2, and 2 2 
 adding the numerator, we have 11 ji ^44 
 halves ; then reducing the dividend to ^ ns ^ j emong 
 halves by multiplying it by 2, we have 
 44 halves. Now 11 is contained in 44, 4 times. Hence, 
 
 86. To divide a whole number by a mixed number. 
 
 Multiply the whole number in the divisor by the denomi- 
 nator, and to the product add the numerator. Then mul- 
 tiply the dividend by the same denominator, and divide as 
 in ivhole numbers. 
 
 Note. For further illustrations of the principles of fractions see 
 Practical Arithmetic, Section VI. It is incompatible with the 
 design of the present work to treat of fractions more exten 
 sively than is necessary to enable the pupil to understand the 
 operations in Reduction. 
 
 EXAMPLES. 
 
 2. How many times is 4f- contained in 15 ? 
 
 Suggestion. Multiplying the 4 and 15 Q perat i ont 
 by 3, reduces them to thirds. Now it is 
 
 . 
 
 plain we can divide thirds by thirds as 33 
 
 well as we can divide one whole number 
 
 by another ; for the divisor is of the same _. "^ ~Q~T 
 name or denomination as the dividend. 
 
 QUEST. 86. How "lo you divide a whole number by a mixed number 7 
 
72 FRACTIONS. [SECT. VI 
 
 3. Divide 18 by 1J. 7. Divide 46 by 7f. 
 
 4. Divide 20 by 3-J. 8. Divide 60 by 5i. 
 
 5. Divide 25 by 5f. 9. Divide 75 by 8f . 
 
 6. Divide 37 by 6-J-. 10. Divide 100 by lOf. 
 
 EXAMPLES FOB PRACTICE. 
 
 1. How many apples can you buy for 4 cents, if you 
 pay a cent apiece ? 
 
 2. How many peaches can you buy for 6 cents, if you 
 pay -J of a cent apiece ? 
 
 3. How many yards of tape can Sarah buy for 8 cents, 
 if she pays -J- of a cent a yard ? 
 
 4. How many yards of ribbon can Harriet buy for 9 
 shillings, at -f of a shilling per yard ? 
 
 5. How many pounds of tea, at f of a dollar a pound, 
 can be bought for 6 dollars ? 
 
 6. How many yards of calico, at -J of a dollar per yaid, 
 can you buy for 3 dollars ? 
 
 7. At of a penny apiece, how many marbles can 
 George buy for 14 pence ? 
 
 8. At -f of a dollar a bushel, how many bushels of corn 
 can a man buy for 6 dollars ? 
 
 9. At -f of a dollar a yard, how many yards of silk can 
 a lady buy for 1 5 dollars ? 
 
 10. At -fc of a dollar apiece, how many lambs can a 
 drover buy for 27 dollars? 
 
 11. In 1 rod there are 5 yards : how many rods are 
 there in 88 yards ? 
 
 12. In 1 rod there are !<> feet: how many rods are 
 therein 132 feet? 
 
 13. How many yards of cloth, at 5f dollars per yard, 
 can be bought for 100 dollars? 
 
 14. How many cows, at 12 dollars apiece, can be 
 bought for 125 dollars ? 
 
ART. 86. J FRACTIONS. 73 
 
 15. How many acres of land, at 20-f- dollars per acre, 
 can a man buy for 540 dollars ? 
 
 16. A grocer bought a quantity of flour for 239 dol- 
 1 irs, which was 8- dollars per barrel : how many barrels 
 did he buy ? 
 
 17. A merchant bought a quantity of broadcloth, at 
 7f dollars per yard, and paid 372 dollars for it: how 
 many yards did he buy ? 
 
 18. A man hired a horse and chaise to take a ride, and 
 paid 275 cents for the use of it, which was 12-J- cents per 
 mile : how many miles did he ride ? 
 
 19. If a man hires a horse and carriage to go 1 Si- 
 miles, and pays 315 cents for it, how many cents does he 
 pay per mile ? 
 
 20. A young man hired himself out for 16f dollars per 
 month, and at the end of his time he received 201 dollars : 
 how many months did he work ? 
 
 21. A farmer having 261 dollars, wished to lay it out 
 in young cattle which were worth 10-f dollars per head : 
 how many could he buy ? 
 
 22. A man having 100 acres of land, wishes to find 
 how many building lots it will make, allowing -fa of an 
 acre to a lot : how many lots will it make ? 
 
 23. How many barrels of beef, at 9-J- dollars per barrel, 
 can be bought for 156 dollars ? 
 
 24. How many hours will it take a man to travel 250 
 miles, if he goes 12-J- miles per hour ? 
 
 25. In 1 barrel there are 31 gallons: how many bar- 
 rels are there in 315 gallons ? 
 
 26. A farmer paid 843 dollars for some colts, which 
 was 35^ dollars apiece : how many did he buy? 
 
 27. A wagon maker sold a lot of wagons for 1452 dol- 
 lars, which was 45f dollars apiece : how many did he 
 sell? 
 
74 COMPOUND [SECT. VIL 
 
 
 
 . SECTION VII. 
 
 COMPOUND NUMBERS. 
 
 ART. 87 SIMPLE Numbers are those which express 
 units of the same kind or denomination ; as, one, two, 
 three ; 4 pears, 5 feet, &c. 
 
 COMPOUND Numbers are those which express units 
 of different kinds or denominations ; as the divisions of 
 money, weight, and measure. Thus, 6 shillings and 7 
 pence ; 3 feet and 7 inches, &c., are compound numbers. 
 
 Note. Compound Numbers are sometimes called Denominate 
 Numbers. 
 
 FEDERAL MONEY. 
 
 88. Federal Money is the currency of the United 
 States. Its denominations are, Eagles, dollars, dimes, 
 cents, and mills. 
 
 10 mills (m.) make 1 cent, marked ct. 
 
 10 cents " 1 dime, " d. 
 
 10 dimes " 1 dollar, " doll or $. 
 
 10 dollars " 1 eagle, " E. 
 
 89 The national coins of the United States are of 
 three kinds, viz : gold, silver, and copper. 
 
 1. The gold coins are the eagle, half eagle, and quarter 
 eagle, the double eagle* and gold dollar.* 
 
 2. The silver coins are the dollar, half dollar, quarter 
 dollar, the dime, half dime, and three-cent-piece. 
 
 QUEST. 87. What are simple numbers ? What are compound numbers'? 
 88. What is Federal Money ? Recite the Table. 89. Of how many kinds are 
 the coins of the United States ? What are the gold coins ? What are the 
 silver coins ? 
 
 * Added by Act of Congress, Feb. 20th, 1849. 
 
ARTS. 87 91.] NUMBERS. 75 
 
 3. The copper coins are the cent and half cent, 
 Mills are not coined. 
 
 Obs. Federal money was established by Congress, August 8th, 
 1786. Previous to this, English or Sterling money was the princi- 
 pal currency of the country. 
 
 STERLING MONEY. 
 
 90, English or Sterling Money is the national cur- 
 rency of Great Britain. 
 
 4 farthings (qr. or far.) make 1 penny, marked d. 
 12 pence " 1 shilling, " s. 
 
 20 shillings " 1 pound or sovereign, . 
 
 21 shillings " 1 guinea. 
 
 OBS. The Pound Sterling is represented by a gold coin, called 
 a Sovereign. Its legal value, according to Act of Congress, 1842, is 
 $4.84; its intrinsic value, according to assays at the U. S. mint, is 
 $4.861. The legal value of an English shilling is 24-1 cents. 
 
 TROY WEIGHT. 
 
 91. Troy Weight is used in weighing gold, silver, 
 jewels, liquors, &c., and is generally adopted in philo- 
 sophical experiments. 
 
 24 grains (gr.) make 1 pennyweight, marked pwt. 
 20 pennyweights " 1 ounce, " oz. 
 
 12 ounces " 1 pound, " Ib. 
 
 Note. Most children have very erroneous or indistinct ideas of 
 the weights and measures in common use. It is, therefore, strongly 
 recommended for teachers to illustrate them practically, by referring 
 to some visible object of equal magnitude, or by 'exhibiting the ounce, 
 the pound ; the linear inch, foot, yard, and rod ; also a square and 
 cubic inch, foot, &c. 
 
 QUEST. What are the copper coins ? Obs. When and by whom was Federal 
 Money established ? 90. What is Sterling Money ? Repeat the Table. Obs. By 
 what h the Pound Sterling represented ? What is its legal value in dollars and 
 tents ? What is the value of an English shilling ? 91. in what is Troy Weight 
 deed ? Recite the Ttible, 
 
*o COMPOUND [SECT. VII, 
 
 AVOIRDUPOIS WEIGHT. 
 
 92. Avoirdupois Weiylit is used in weighing groceries 
 and all coarse articles ; as sugar, tea, coffee, butter, cheese, 
 flour, hay, &c., and all metals except gold and silver. 
 
 16 drams (dr.) make 1 ounce, marked oz. 
 
 16 ounces " 1 pound, " Ib. 
 
 25 pounds " 1 quarter, " qr. 
 
 4 quarters, or 100 Ibs. " 1 hundred weight, ctvt. 
 20 hund., or 2000 Ibs. " 1 ton, marked T. 
 OBS. 1. Gross weight is the weight of goods with the boxes, or bags 
 which contain them, allows 112 Ibs. for a hundred weight. 
 Net weight is the weight of the goods only. 
 2. Formerly it was the custom to allow 112 pounds fora hundred 
 weight, and 28 pounds for a quarter : but this practice has become 
 nearly or quite obsolete. The laws of most of the states, as well as 
 general usage, call 100 Ibs. a hundred weight, and 25 Ibs. a quarter. 
 In estimating duties, and weighing a few coarse articles, as iron, 
 dye-woods, and coal at the mines, 112 Ibs. are still allowed for a 
 hundred weight. Coal, however, is sold in cities, at 100 Ibs. for a 
 hundred weight. 
 
 APOTHECARIES' WEIGHT. 
 
 93. Apothecaries' Weight is used by apothecaries and 
 physicians in mixing medicines. 
 
 20 grains (yr.) make 1 scruple, marked sc. or S. 
 3 scruples " 1 dram, " dr. or 3. 
 8 drams " 1 ounce, " oz. or g. 
 
 12 ounces " 1 pound, " Ib. 
 
 OBS. 1. The pound and ounce in this weight are the same as the 
 Troy pound and ounce; the subdivisions of the ounce are different. 
 2. Drugs and medicines are bought and sold by avoirdupois 
 weight. 
 
 QUEST. 92. In what is Avoirdupois Weight used ? Recite the Table. Obs 
 What is gross weight? What is net weight? How many pounds were for- 
 merly allowed for a quarter ? How many for a hundred weight ? 93. In what 
 is Apothecaries Weight used? Repeat the Table. Obs. To what are the Apo- 
 thecaries' porn<l and o*ce equal? How are drugs and medicines bought 
 and sold ? 
 
ARTS. 92 95.] NUMBERS. 77 
 
 LONG MEASURE. 
 
 O4r Long Measure is used in measuring length or 
 distances only, without regard to breadth or depth- 
 
 12 inches (in.) make 1 foot, marked ft. 
 
 3 feet " 1 yard, " yd. 
 
 5i yards, or 16 feet " 1 rod, perch, or pole, r. orp 
 
 40 rods " 1 furlong, marked fur. 
 
 8 furlongs, or 320 rods " 1 mile, " m. 
 
 3 miles " 1 league, " L 
 60 geographical miles, or ) 
 
 691 statute .miles \" l d ^ ree - " **'"* 
 
 360 deg. make a great circle, or the circum. of the eart li, 
 
 OBS. 1. 4 inches make a hand; 9 inches, 1 span; 18 inches, 1 
 cubit ; 6 feet, 1 fathom ; 4 rods, 1 chain ; 26 links, 1 rod. 
 
 2. Long measure is frequently called linear or lineal measure. 
 "Formerly the inch was divided into 3 barleycorns ; but the barley- 
 corn, as a measure, has become obsolete. The inch is commonly 
 divided either into eighths, or tenths ; sometimes it is divided into 
 twelfths, which are called lines. 
 
 CLOTH MEASURE. 
 
 95 Cloth Measure is used in measuring cloth, lace, and 
 all kinds of goods, which are bought or sold by the yard. 
 
 2 J inches (in.) make 1 nail, marked na. 
 
 4 nails, or 9 in. " 1 quarter of a yard, " qr. 
 
 4 quarters " 1 yard, " yd. 
 
 3 quarters " 1 Flemish ell, " Fl. e. 
 
 5 quarters " 1 English ell, " E. e. 
 
 6 quarters " 1 French ell, " F. e. 
 
 QUEST. 94. In what is Long Measure used 1 Repeat the Table. Draw a 
 line an inch long upon your slate or black-board. Draw one two inches long. 
 Draw another a foot long. Draw one a yard long. How long is your teacher 7 ! 
 desk 1 How long is the school-room ? How wide ? Obs. What is Long Meas- 
 ure frequently called 7 How is the inch commonly divided at the present 
 dav ? 95. In what is Cloth Measure used 1 Repeat the Table. 
 
COMPOUND 
 
 [SECT. VIL 
 
 OBS. Cloth mear ire is a species of long measure. The yard is 
 the eame in both. Cloths, laces, <fec., are bought and sold by the 
 linear yard, without regard to their width. 
 
 SQUARE MEASURE. 
 
 96* Square Measure is used in measuring surfaces, 
 or things whose length and breadth are considered with- 
 out regard to height or depth ; as land, flooring, plaster- 
 ing, &c. 
 144 square in. (sq. in.) make 1 square foot, marked sq.ft. 
 
 1 square yard, 
 
 sq. yd. 
 
 1 sq. rod, perch, (( 
 
 
 or pole, 
 
 sq. 
 
 1 rood, " 
 
 R. 
 
 1 acre, 
 
 A. 
 
 1 square mile, " 
 
 M. 
 
 9 square feet 
 
 30-J- square yards, or ) (( ( 
 2 7 2i square feet ) ( 
 40 square rods 
 
 4 roods, or 160 sq.rds. " 
 640 acres " 
 
 OBS. 1. A square is a figure, which has four equal sides, and all 
 its angles right angles, as seen in the adjoining diagram. Hence, 
 
 2. A Square Inch is a square, whose sides 9 sq.ft. I sq. yd. 
 are each a linear inch in length. 
 
 A Square Foot is a square, wiiose sides 
 are each a linear foot in length. 
 
 A Square Yard is a square, whose sides 
 are each a linear yard or three linear feet in 
 length, and contains 9 square feet, as re- 
 presented in the adjacent figure. 
 
 3. In measuring land, surveyors use a 
 chain which is 4 rods long, and is divided 
 
 into 100 links. Hence, 25 links make 1 rod, and 7-j^j- inched 
 make 1 link. 
 
 This chain is commonly called Counter's Chain, fiom the name 
 of its inventor. 
 
 4. Square Measure is sometimes called Land Measure, because 
 ?t is used in measuring land. 
 
 QUEST. Obs. Of what is Cloth Measure a species? 96. In what is Squaro 
 Measure used 1 Repeat the Table. Obs. What is a square ? What is a square 
 inch? What is a square foot? A square jard? Can you draw a 
 inch ? Can you draw a square foot ? A square yard ? 
 
ARTS. 96, 97.] 
 
 NUMBERS. 
 
 79 
 
 1 cubic yard, " 
 
 cu. yd. 
 
 1 ton, 
 
 T. 
 
 1 ton of shipping, " 
 1 cord foot, or a M 
 foot of wood, 
 
 T. 
 c.ft. 
 
 1 cord, 
 
 0. 
 
 CUBIC MEASURE. 
 
 97. Cubic Measure is used in measuring solid bodies, 
 or things which have length, breadth, and thickness; 
 such as timber, stone, boxes of goods, the capacity of 
 rooms, &c. 
 1 72 8 cubic inches (cu. in.) make 1 cubic foot, marked cu.ft. 
 
 27 cubic feet 
 
 40 feet of round, or 
 
 50 ft. of hewn timber, 
 
 42 cubic feet 
 
 16 cubic feet 
 
 8 cord feet, or '. 
 128 cubic feet 
 
 OBS. 1. A pile of wood 8 feet long, 4 feet wide, and 4 feet high, 
 contains 1 cord. For 8 into 4 into 4=128. 27 cu. ft.=l cu. yd. 
 
 2. A Cube is a solid body bounded by 
 six equal squares. It is often called a hex- 
 acdron. Hence, 
 
 A _Cubic Inch is a cube, each of whose 
 sides is a square inch, as represented by 
 the adjoining figure. 
 
 A Cubic Foot is a cube, each of whose 
 sides is a square foot. 
 
 3. The Cubic Ton is chiefly used for estimating the cartage and 
 transportation of timber. By a ton of round timber is meant, such 
 a quantity of timber in its rough or natural state, as when hewn, 
 will make 40 cubic feet, and is supposed to be equal in weight to 
 50 feet of hewn timber. 
 
 4. The cubic ton or load, is by no means an accurate or uniform 
 standard of estimating weight ; for, different kinds of timber, are of 
 very different degrees of density. But it is perhaps sufficiently ac- 
 curate for the purposes to which it is applied. 
 
 QUEST. 97. In what is Cubic Measure used? Recite the Table. How 
 long, wide, and high, must a pile of wood be to make a cord ? What is a 
 cube ? What is a cubic inch ? What is a cubic foot ? Can you draw a cubio 
 Inch on your slate ? 
 
80 COMPOUND [SECT. VIL 
 
 WINE MEASURE. 
 
 98. Wine Measure is used in measuring wine, alco- 
 hol, molasses, oil, and all other liquids except beer, ale, 
 and milk. 
 
 4 gills (gi.) make 1 pint, marked pt. 
 
 2 pints " 1 quart, " qt. 
 
 4 quarts " 1 gallon, " gal. 
 
 3 1 gallons " 1 barrel, " bar.orbbl. 
 
 42 gallons " 1 tierce, " tier. 
 
 63 gallons, or 2 bbls. " 1 hogshead, " hkd. 
 
 2 hogsheads " 1 pipe or butt, " pi. 
 
 2 pipes " 1 tun, " tun, 
 
 OBS The wine gallon contains 231 cubic inches. 
 BEER MEASURE. 
 
 99. Beer Measure is used in measuring beer, ale, and 
 milk. 
 
 2 pints (pt.) make 1 quart, marked qt. 
 
 4 quarts " 1 gallon, " gal. 
 
 36 gallons " 1 barrel, " bar. or bbl. 
 
 54 gals, or 1^ bbls. " 1 hogshead, " hhd. 
 
 OBS. The beer gallon contains 282 cubic inches. In many place* 
 
 milk is measured by wine measure. 
 
 DRY MEASURE. 
 
 1 GO. Dry Measure is used in measuring grain, fruit 
 salt, &c. 
 
 2 pints (pts.) make 1 quart, marked qt. 
 
 8 quarts " 1 peck, " pJc. 
 
 4 pecks, or 32 qts. " 1 bushel, " bu. 
 
 8 bushels " 1 quarter, " qr. 
 
 32 bushels " 1 chaldron, " cA. 
 
 e.-t-ln England, 36 bushels of coal make a chaldron. 
 
 QUEST. 98. In what is Wine Measure used? Recite the Table. Obs. Ho* 
 many cubic inches in a wine gallon? 99. In what is Beer Measure uced 
 Repeat the Table. Obs. How many cubic inches in a boer gallon ? 
 
ARTS. 98 102.] NUMBERS. 81 
 
 TIME. 
 
 1O1 Time is naturally divided into days and years ; 
 the former are caused by the revolution of the Earth on its 
 axis, the latter by its revolution round the Sun. 
 60 seconds (sec.) make 1 minute, marked min. 
 
 60 minutes " 1 hour, " hr. 
 
 24 hours " 1 day, " d. 
 
 7 days " 1 week, " wk. 
 
 4 weeks " 1 lunar month, " mo. 
 
 12 calendar months, or > .. . ., 
 
 ' x > " 1 civil year, " yr. 
 365 clays, 6 hrs., (nearly,) $ 
 
 13 lunar mo., or 52 weeks, " 1 year, " yr. 
 100 years " 1 century, " cen. 
 
 OBS. 1. Time is measured by clocks, watches, chronometers, dials, 
 hour-glasses, &c. 
 
 2. A civil year is a legal or common year ; a period of time es- 
 tablished by government for civil or common purposes. 
 
 3. A solar year is the time in which the earth revolves round 
 the sun, and contains 365 days, 5 hours, 48 min., and 48 sec. 
 
 4. A leap year, sometimes called bissextile, contains 866 days, 
 and occurs once in four years. 
 
 It is caused by the excess of 6 hours, which the civil year con- 
 tains above 365 days, and is so called because it leaps or rims over 
 one day more than a common year. The odd day is added to Feb- 
 ruary, because it is the shortest month. Every leap year, there- 
 fore, February has 29 days. 
 
 1Q2. The names of the days are derived from the 
 names of certain Saxon deities, or objects of worship. Thus, 
 
 Sunday is named from the sun, because this day was dedicated 
 to its worship. 
 
 Monday is named from the moon, to which it was dedicated. 
 
 QUEST. 100. In what is Dry Measure used ? Recite the table. 101. How 
 is Time naturally divided ? How are the former caused ? How the latter t 
 Repeat the Table. Obs. How is Time measured 1 What is a civil year ? A 
 solar year? A leap year? How is Leap Year caused ? To which month is 
 the odd day added 1 From what are the namua of the days derived 1 
 
 6 
 
82 COMPOUND [SECT. VII, 
 
 Tuesday is derived from Tuisco, the Saxon god of war. 
 Wednesday is derived from Woden, a deity of northern Europe. 
 TJiursday is from Thor, the Danish god of thunder, storma, <fec. 
 Friday is from Frig a, the Saxon goddess of beauty. 
 Saturday is from the planet Saturn, to which it was dedicated 
 
 1O3. The following are the names of the 12 calendar 
 months, with the number of days in each : 
 
 January, 
 
 February, 
 
 March, 
 
 April, 
 
 May, 
 
 June, 
 
 July, 
 
 August, 
 
 September, 
 
 October, 
 
 November, 
 
 December, 
 
 (Jan.) the first month, has 31 days. 
 
 (Feb.) " second " " 28 " 
 
 (Mar.) " third " "31 " 
 
 (Apr.) "fourth " " 30 " 
 
 (May) " fifth " "31 " 
 
 (June) " sixth " " 30 " 
 
 (July) " seventh " " 31 " 
 
 (Aug.) " eighth " "31 " 
 
 (Sept.) " ninth " " 30 " 
 
 (Oct.) " tenth " "31 " 
 
 (Nov.) " eleventh " " 30 " 
 
 (Dec.) " twelfth " "31 ' 
 
 OBS. 1. The number of days in each month may be easily re- 
 membered from the following lines : 
 
 " Thirty days hath September, 
 April, June, and November ; 
 February twenty-eight alone, 
 All the rest have thirty-one ; 
 Except in Leap Year, then is the time, 
 When February has twenty-nine." 
 
 2. The names of the calendar months were borrowed from the 
 Romans, and most of them had a fanciful origin. Thus, 
 
 January was named after Janus, a Roman deity, who WAS sup- 
 posed to preside over the year, and the commencement of all 
 undertakings. 
 
 February was derived fromfebrno, a Latin word which signifies 
 to purify by sacrifice, and was so called because this month was 
 devoted to the purification of the people. 
 
 QUEST. 103. What is the origin of the narr.es of the month*? 
 
ARTS. 103 105.] NUMBERS. 88 
 
 March was named after Mars, the Roman god of war ; and was 
 originally the first month of the Roman year. 
 
 April, from the Latin aperio, to open, was so called from the 
 opening of buds, blossoms, cfec., at this season. 
 
 May was named after the goddess Maia, the mother of Mercury, 
 to whom the ancients used to offer sacrifices on ike first day of 
 this month. 
 
 June was named after the goddess Juno, the wife of Jupiter. 
 
 July was so called in honor of Julius Ccesar, who was born in 
 this month. 
 
 August was so called in honor of Augustus Ccesar, a Roman 
 Emperor, who entered upon his first consulate in this month. 
 
 September, from the Latin numeral septem, seven, was so called, 
 because it was originally the seventh month of the Roman year. 
 It is the ninth month in our year. 
 
 October, from the Latin octo, eight, was so called because it was 
 the eighth month of the Roman year. 
 
 November, from the Latin novem, nine, was so called because it 
 was the ninth month of the Roman year. 
 
 December, from the Latin decem, ten, was so called because it 
 was the tenth month of the Roman year. 
 
 104. The year is also divided mtofour seasons of 
 three months each, viz: Spring, Summer, Autumn or 
 Fall, and Winter. 
 
 Spring comprises March, April, and May ; Summer, 
 June, July, and August ; Autumn or Fall, September, 
 October, and November ; Winter, December, Jan. and Feb. 
 
 CIRCULAR MEASyRE. 
 
 105. Circular Measure is applied to the divisions of 
 the circle, and is used in reckoning latitude and longitude, 
 and the motion of the heavenly bodies. 
 
 60 seconds (") make 1 minute, marked ' 
 60 minutes " 1 degree, 
 
 30 degrees " 1 sign, " s. 
 
 12 signs, or 360 " 1 circle, " c. 
 
 QUKST. 104. Name the seasons. 105. To what is Circular Measure applied 1 
 
84 
 
 COMPOUND 
 
 [SECT. VII. 
 
 OBS. J. Circular Measure is 
 c ten tailed Angular Measure, 
 %nd is chiefly used by astrono- 
 mers, navigators, and surveyors. 
 
 2. The circumference of every 
 circle is divided, or supposed to 
 be divided, into 360 equal parts, 
 called degrees, as in the sub- 
 icined figure. 
 
 3. Since a degree is yfcr part 
 of the circumference of a circle, 
 it is obvious that its length 
 must depend on the size of the circle. 
 
 270o 
 
 MISCELLANEOUS TABLE. 
 
 1O6. The following denominations not included in 
 the preceding Tables, are frequently used. 
 
 12 units 
 
 12 dozen, or 144 
 
 12 gross, or 1728 
 
 20 units 
 
 56 pounds 
 100 pounds 
 
 30 gallons 
 
 200 Ibs. of shad or salmon 
 196 pounds 
 200 pounds 
 
 14 pounds of iron, or lead 
 21 stone 
 8 pigs 
 
 OBS. Formerly 112 pounds were allowed for a quintal. 
 
 QUEST. Obs. What is Circular Measure sometimes called ? By whom ia i| 
 chiefly used ? Into what is the circumference of every circle divided ? On 
 what does the length of a degree depend ? 10G. How many units make a 
 dozen ? How many dozen a gross 1 A great gross ? How many units maka 
 a score ? Pounds a flrjdn ? 
 
 make 1 dozen, (doz.) 
 " I gross. 
 " 1 great gross. 
 " 1 score. 
 " 1 firkin of butter. 
 " 1 quintal of fish. 
 " 1 bar. of fish in Mass. 
 
 1 bar. in N. Y. and Ct 
 
 1 bar. of flour. 
 
 1 bar. of pork. 
 
 1 stone. 
 
 1 P^ 
 1 fother. 
 
ARTS. 106 108.] NUMBERS. 85 
 
 PAPER AND BOOKS. 
 
 1OT. The terms, folio, quarto, octavo, &c., applied to 
 books, denote the number of leaves into which a sheet 
 0f paper is folded. 
 
 24 sheets of paper make 1 quire. 
 
 20 quires " 1 ream. 
 
 2 reams " 1 bundle. 
 
 5 bundles " 1 bale. 
 
 A sheet folded in two leaves, is called & folio. 
 A sheet folded in four leaves, is called a quarto, or 4te. 
 A sheet folded in eight leaves, is called an octavo, or 8vo. 
 A sheet folded in twelve leaves, is called a duodecimo. 
 A sheet folded in sixteen leaves, is called a 16 wo. 
 A sheet folded in eighteen leaves, is called an 18 wo. 
 A sheet folded in thirty-two leaves, is called a 39mo. 
 A sheet folded in thirty-six leaves, is called a 3 6 wo. 
 A sheet folded in forty-eight leaves, is called a 4 8 wo. 
 
 1O8 Previous to the adoption of Federal money in 
 1786, accounts hi the United States were kept in pounds, 
 shillings, pence, and farthings. 
 
 In New England currency, Virginia, Ken- i 
 
 tucky, Tennessee, Indiana, Illinois, Mis- >6 shil. make $1. 
 
 Bouri, and Mississippi, j 
 
 In New York currency, North Carolina, ) 
 
 8 shil makc & 1 - 
 
 , ) 
 ] 
 
 Ohio, and Michigan, 
 In Pennsylvania currency, New Jersey, ) _, 
 
 Delaware, and Maryland, \ 7s ' 6d ' make $1 
 
 In Georgia currency, and South Carolina, 4s. 8d. make $1. 
 In Canada currency, and Nova Scotia, 5 shil. make $1. 
 
 QUEST. 107. When a sheet of paper is folded in two leaves, what is it 
 called ? When in four leaves, what ? When in eight ? In twelve ? In 
 sixteen 1 In eighteen 1 In thirty-six ? 108. Previous to the adoption of Fed- 
 eral Money, in what were accounts kept in the U. S. ? How many shillings 
 make a dollar in N. E. c\irrency 1 In N. V. currency ^ !\n Penn. currency 1 
 In Georgia currency 1 In Canada currency 7 
 
86 COMPOUND [SECT. VIL 
 
 OBS. At the time Federal money was adopted, the colonial cur* 
 rency or bills of credit issued by the colonies, had more or less de- 
 preciated in value : that is, a colonial pound was worth less than a 
 pound Sterling; a colonial shilling, than a shilling Sterling, &e. 
 This depreciation being greater in some colonies than in others, 
 gave rise to the different values of the State currencies. 
 
 ALIQUOT PARTS OF $1 IN FEDERAL MONEY. 
 
 50 cents = 
 
 33i cents = 
 
 25 cents = 
 
 20 cents = 
 
 16| cents = 
 
 12 J cents ~ 
 
 10 cents =* 
 
 8| cents = 
 
 6| cents = 
 
 5 cents = 
 
 PARTS OF $1 IN NEW YORK CURRENCY. 
 
 4 shillings == 
 
 2 shil. 8 pence = 
 2 shillings = 
 
 1 shil. 4 pence = 
 1 shilling = 
 
 6 pence = 
 
 OBS. 1. In New York currency, it will be seen, (Art. 108,^ that 
 A six-pence, written 6d. = 6^ cents, 
 
 A shilling, " Is. = 12 " 
 
 One (shil.) and 6 pence, 1/6. = 18J " 
 Two shillings, " 2s. = 25 " 
 
 PARTS OF $1 IN NEW ENGLAND CURRENCY. 
 
 3 shillings = $- 1 shilling = $f 
 
 2 shillings = $ -J- 9 pence = $fc 
 
 I shil. and 6d. = $i 6 pence = $fV 
 
 OBS. 2. In New England currency, it will be seen, that 
 A four-pence-half-penny, written 4d. = 6 cents. 
 A six-pence, " Gd. = 8 " 
 
 A nine-pence, " 9d. = 12^ {C 
 
 A shilling, " Is. = 16 f 
 
 One (shil.) and six-pence, " 1/6. = 25 " 
 Two shillings, " 2s. = 33 J " 
 
 QITEST. What are the aliquot parts of $1 in Federal Money 7 In New York 
 currency 7 In Now England currency 7 What are the aliquot parts of a pou 
 Sterling 7 Of a shilling 7 
 
ART. 108.] 
 
 NUMBERS. 
 
 87 
 
 ALIQUOT PARTS OF STERLING MONEY. 
 
 Parts of 1. 
 
 Parte o/ Is. 
 
 10 shil. = i 
 
 6 pence = -J- shil. 
 
 6s. 8d. = i 
 
 4 pence = - shil. 
 
 5 shil. = i 
 
 3 pence = } shiL 
 
 4 shil. = i 
 
 2 pence = -J- shil. 
 
 3s. 4d. ~ | 
 
 1 pence = i shil. 
 
 2s. 6d. = i 
 
 1 penny = iV shil. 
 
 2 shil. = 1 1 (r 
 
 1 far. = i penny. 
 
 Is. 8d. -iV 
 
 2 far. = -J- penny. 
 
 1 shil. = -2 1 o 
 
 3 far. = f penny. 
 
 ALIQUOT PARTS OR A TON. 
 
 10 hund. lbs.=i ton. 
 5 hund. lbs.=-J- ton. 
 4 hund. lbs.=-- ton. 
 
 2 hund. 2 qrs.=i ton. 
 2 hund. Ibs. =-fo ton. 
 1 hund. Ibs. ^ ton. 
 
 ALIQUOT PARTS OF A POUND AVOIRDUPOIS. 
 
 8 ounces =% pound. 
 4 ounces =-J- pound. 
 
 2 ounces =i pound. 
 1 ounce =iV pound. 
 
 ALIQUOT PARTS OF TIME. 
 
 Parts of 1 year. 
 
 Parts of 1 month. 
 
 6 months = year. 
 
 15 days=i- month. 
 
 4 months =% year. 
 
 10 day s==^- month. 
 
 3 months =-J- year. 
 
 6 days= month. 
 
 2 months =-J- year. 
 
 5 days=-J- month. 
 
 1J- month =-J- year. 
 
 3 days=-iV month. 
 
 1-J- month =| year. 
 
 2 days^iV month. 
 
 1 month ==-i 1 2 year. 
 
 1 day =-sV month. 
 
 QUEST. How many shillings in half a pound Ster. ? In a fourth ? A fifth ? 
 A tenth ? A twentieth V How many pence in half a shilling ? In a third ? A 
 fourth? A sixth 7 A twelfth? How many hundreds in half a ton? In a 
 fourth ? A fiftn ? A tenth ? How many ounces in half a pound ? In a fourth ? 
 An eighth? A sixteenth? How many months in half a year? la a third? 
 A fourth? A sixth? A twelfth 7 
 
88 FEDERAL MONEY. [SECT. VIIL 
 
 SECTION VIIL 
 
 FEDERAL MONEY. 
 
 1 1 0. Accounts in the United States are kept in dol- 
 lars, cents, and mills. Eagles are expressed in dollars, and 
 dimes in cents. Thus, instead of 4 eagles, we say, 40 dol- 
 lars ; instead of 5 dimes, we say, 50 cents, &c. 
 
 Ill* Dollars are separated from cents by placing a 
 point or separatrix ( . ) between them. Hence, 
 
 112. To read Federal Money. 
 
 Call all the figures on the left of the point, dollars ; the 
 first two figures on the right of the point, are cents ; the 
 third figure denotes mills ; the other places on the right are 
 parts of a mill. Thus, $5.2523 is read, 5 dollars, 25 
 cents, 2 mills, and 3 tenths of a mill. 
 
 OBS. 1. Since two places are assigned to cents, when no cents 
 are mentioned in the given number, two ciphers must be placed 
 before the mills. Thus, 5 dollars and 7 mills are written $ 5.007. 
 
 2. If the given number of cents is less than ten, a cipher must 
 always be written before them. Thus, 8 cents are written .08, <fcc. 
 
 1. Read the following expressions: $83.635 ; $75.50. 
 $126.607; $268.05; $382.005; $2160. 
 
 2. Write the following sums : Sixty dollars and fifty 
 cents. Seventy-five dollars, eight cents, and three mills. 
 Forty-eight dollars and seven mills. Nine cents. Six 
 cents and four mills. 
 
 QUEST. 88. What is Federal Money? What are its denominations? Re- 
 cite the Table. 110. How are accounts kept iu the United States? How are 
 Eagles expressed ? Dimes? 111. How are dollars distinguished from cents 
 and mills ? 112. How do you read Federal Money ? Obs. How many places 
 are aseigned to cents ? When the number of cents is less than ten, what must 
 be done ? When no cents are mentioned, what do you do ? 
 
ARTS. 110 113.] FEDERAL MONEY. 89 
 
 REDUCTION OF FEDERAL MONEY. 
 
 CASE I. 
 Ex. 1, How many cents are there in 65 dollars? 
 
 Suggestion. Since in 1 dollar there are Operation. 
 100 cents, in 65 dollars there are 65 times as 65 
 many. Now, to multiply by 10, 100, &c.,we 
 
 annex as many ciphers to the multiplicand, 6500 cents. 
 as there are ciphers in the multiplier. (Art. 45.) Hence, 
 
 113. To reduce dollars to cents, annex TWO ciphers. 
 To reduce dollars tc mills, annex THREE ciphers. 
 To reduce cents to mills, annex ONE cipher. 
 
 OBS. To reduce dollars and cents to cents., erase the sign of dollars 
 and the separatrix. Tfefes, $25.36 reduced te cents, become 2536 
 cents. 
 
 2. Reduce $4 to cents. Ans. 400 cents. 
 
 3.* Reduce $15 to cents. 7. Reduce $96 to mills. 
 
 4. Reduce $27 to cents. 8. Reduce $12.23 to cents. 
 
 5. Reduce $85 to cents. 9'. Reduce $86.86 to cents. 
 
 6. Reduce $93 to cents. 10. Reduce $9.437 to mills. 
 
 CASE II. 
 1. In 2345 cents, how many dollars ? 
 
 Suggestion. Since 100 cents make 1 dol- Operation. 
 lar, 2345 cents, will make as many dollars 1|00)23|45 
 as 100 is contained times in 2345. Now to Ans. $23.45 
 divide by 10, 100, &c., we cut off as many 
 figures from the right of the dividend as there are ciphera 
 in the divisor. (Art. 67.) Hence, 
 
 QUEST. 113. Ho\v are dollars reduced to cents? Dollars to mills ? Centa 
 to mills ? Obs. Dollars and cents to cents ? 
 
90 FEDERAL MONET. [SECT. VIIL 
 
 114* To reduce cents to dollars. 
 
 Point off TWO figures on the right ; the figures remain* 
 ing on the left express dollars ; the two pointed off, cents. 
 
 1 1 5. To reduce mills to dollars. 
 
 Point off THREE figures on the right ; the remaining 
 figures express dollars ; the first two on the right of the 
 point, cents ; the third one, mills. 
 
 116* To reduce mills to cents. 
 
 Point off ONE figure on the right, and the remaining 
 figures express cents ; the one pointed off, mills. 
 
 2. Reduce 236 cts. to dolls. 3. Reduce 21 63 cts. to dolls. 
 4. Reduce 865 mills to dolls. 5. Reduce 906 mills to cts. 
 6. Reduce 2652 cts. to dolls. 7. Redfce 3068 cts. to dolls. 
 
 ADDITION OF FEDERAL MONEY. 
 Ex. 1. What is the sum of $8.125, $12.67, $3.098, $11 ? 
 
 Suggestion. Write the dollars under 
 
 dollars, cents under cents, mills tinder 1267 
 
 mills, and proceed as in Simple Addition. 3 098 
 
 From the right of the amount point off 11.00 
 
 three figures for cents and mills. Ans. $34.893 
 
 117* Hence, we derive the following general 
 
 RULE FOR ADDING FEDERAL MONEY. 
 
 Write dollars under dollars, cents under cents, mills 
 under mills, and add each column, as in simple numbers. 
 
 From the right of the amount, point off as many figures 
 for cents and mills, as there are places of cents and mills 
 in either of the given numbers. 
 
 QUEST.- 114. How are cents reduced to dollars? 115. Mills to dollars? 117 
 How do you add Federal Money? How point off the amount? 
 
ARTS. 114 11*7.] FEDERAL MONEY. 91 
 
 OBS. If either of the given numbers have no cents expressed, 
 upply their place by ciphers. 
 
 (2.) 
 
 $375.037 
 
 (3.) 
 
 $4869.45 
 
 $760.275 
 
 (5.) 
 
 $4607.375 
 
 60.20 
 
 344.00 
 
 897.008 
 
 . 896.084 
 
 843.462 
 
 6048.07 
 
 965.054 
 
 95.873 
 
 (6.) 
 
 $782.206 
 
 $609.352 
 
 (8.) 
 $2903.76 
 
 (9.) 
 
 $4668.253 
 
 84.60 
 
 830.206 
 
 453.06 
 
 430.064 
 
 379.007 
 
 408.07 
 
 25.89 
 
 307.60 
 
 498.015 
 
 631.107 
 
 6842.07 
 
 7452.349 
 
 10. What is the sum of $63.072, $843.625, and $71:60 ! 
 
 11. Add $873.035, $386.23, $608.938, $169.176. 
 
 12. Add 463 dolls. 7 cts. ; 248 dolls. 15 cts. ; 169 dolls. 
 9 cts. 7 mills. 
 
 13. Add 89 dolls. 8 cts.; 97 dolls. 10 cts. 3 mills; 40 
 dolls. 6 cts. ; 75 dolls. 
 
 14. Add 365 dolls. 20 cts. 2 mills; 68 dolls. 6 cts. 5 
 mills ; 7 cts. 3 mills ; 286 dolls. ; 80 dolls. 6 mills ; 30 dolls. 
 15 cts. 
 
 15. A man bought a cow for $16.375, a calf for $4.875, 
 and a ton of hay for $13.50 : how much did he pay foi 
 the whole ? 
 
 16. A lady paid $23 for a cloak, $7.625 for a hat, $25.75 
 for a muff, and $18 for a tippet : how much did she pay 
 for all ? 
 
 17. A farmer sold a cow for $16.80, a calf for $4.08, a 
 horse for $78, and a yoke of oxen for $63.18 : how much 
 did he receive for all ? 
 
 QUEST. Obs. When any of the given numbers have no cents expressed, 
 how ia their place supplied? 
 
92 FEDERAL MONEY. [SECT. VIIL 
 
 SUBTRACTION OF FEDERAL MONEY. 
 
 Ex. 1. What is the difference between $845.634, and 
 $86.087 ? 
 
 Suggestion. Write the less number Operation. 
 under the greater, dollars under dollars, $845.634 
 
 &c., then subtract, and point off the an- 86.087 
 
 swer as in addition of Federal Money. Ans. $759.547. 
 
 118* Hence, we derive the following general 
 
 EULE FOR SUBTRACTING FEDERAL MONEY. 
 Write the less number under the greater, with dollars 
 under dollars, cents under cents, and mills under mills ; 
 then subtract, and point off the answer as in addition 
 of Federal Money. 
 
 (2.) 
 From $856.453 
 Take $387.602 
 
 (3.) 
 
 $960.78 
 $463.05 
 
 (4.) 
 
 $605.607 
 78.36 
 
 (5.) 
 $6243.760 
 327.053 
 
 (6.) 
 From 965.005 
 Take 87.85 
 
 840.000 
 378.457 
 
 (8.) 
 483.853 
 
 48.75 
 
 (9.) 
 4265.76 
 2803.98 
 
 10. From $86256.63 take $4275.875 ? 
 
 11. From $100250, take $32578.867 1 
 
 12. From 1 dollar, subtract 11 cents. 
 
 13. From 3 dolls. 6 cts. 7 mills, take 75 cents. 
 
 14. From 110 dolls. 8 mills, take 60 dolls, and 8 cents. 
 
 15. From 607 dolls. 7 cents, take 250 dolls, and 3 cts. 
 
 16. A lad bought a cap for $2.875, and paid a five- 
 dollar-bill : how much change ought he to receive back ? 
 
 17. Henry has $7.68, and William has $9.625 : how 
 much more has the latter than the former ? 
 
 18. From $865275.60, take $340076.875. 
 
 QUEST. 106. How do you subtract FederaJ Money 1 How point off tht 
 nnswer 1 
 
ARTS. 118, 119.] FEDERAL MONEY. 93 
 
 MULTIPLICATION OF FEDERAL MONEY. 
 
 Ex. 1. What will 3 caps cost, at $1.625 apiece ? 
 
 Suggestion. Since 1 cap costs $1.625, Operation. 
 8 caps will cost 3 times as much. We *^ ^25 
 
 therefore multiply the price of 1 cap by 3, 3 
 
 the number of caps, and point off three Ans. $4.875. 
 places for cents and mills. Hence, 
 
 111). When the multiplier is a whole number, we have 
 the following 
 
 RULE FOR MULTIPLYING FEDERAL MONEY. 
 
 Multiply as in simple numbers, and from the right of 
 the product, point off as many figures for cents and mills, 
 as there are places of cents and mills in the multiplicand. 
 
 OBS. 1. In Multiplication of Federal Money, as well as in simple 
 numbers, the multiplier must always be considered an abstract 
 number. 
 
 2. In business operations, when the mills are 5 or over, it is 
 customary to call them a cent ; when under 5, they are disregarded. 
 
 (2.) (3.) (4.) (5.) 
 
 Multiply $633.75 $805.625 $879.075 $9071.26 
 By 8 9 24 37 
 
 (6.) 
 
 (7.) 
 
 (8.) 
 
 (9.) 
 
 Multiply $4063.36 
 
 $5327.007 
 
 $6286.69 
 
 $8265.68 
 
 By 63 
 
 86 
 
 123 
 
 264 
 
 10. What cost 8 melons, at 17 cents apiece ? 
 
 11. What cost 12 lambs, at 87 cents apiece ? 
 
 12. What cost 8 hats, at $3.875 apiece 1 
 
 13. At $8.75 a yard, what will 9 yards of silk come to! 
 
 14. At $1.125 apiece, what will 11 turkeys cost? 
 
 QUEST. 119. How do you multiply Federal Money ? How point off the 
 product? 065. What must the multiplier always be considered ? When 
 the mills are 5, or over, what is it customary to call them ? When lees than 
 5 what may be done with them ? 
 
94 FEDERAL MONEY. [SECT. VIIL 
 
 15. At $2.63 apiece, what will 15 chairs come tof 
 
 16. What costs 25 Arithmetics, at 37-^ cents apiece! 
 
 17. "What cost 38 Readers, at 62|- cents apiece? 
 
 18. What cost 46 over-coats, at $25.68 apiece ? 
 
 19. What cost 69 oxen, at $48.50 a head ? 
 
 20. At $23 per acre, what cost 65 acres of land ? 
 
 21. At $75.68 apiece, what will 56 horses come to ? 
 
 22. At 7-J cents a mile, what will it cost to ride 100 
 miles ? 
 
 23. A farmer sold 84 bushels of apples, at 87-J- cents per 
 bushel : what did they come to 1 
 
 24. If I pay $5.3 7|- per week for board, how much will 
 it cost to board 52 weeks ? 
 
 DIVISION OF FEDERAL MONEY. 
 Ex. 1. If you paid $18.876 for 3 barrels of flour, how 
 much was that a barrel ? 
 
 Suggestion. Since 3 barrels cost $18.- 
 876, 1 barrel will cost 1 third as much, 
 We therefore divide as in simple division, 
 and point on three places for cents and 
 mills, because there are three in the dividend. Hence, 
 
 1 2O. When the divisor is a whole number, we have 
 the following 
 
 RULE FOR DIVIDING FEDERAL MONEY. 
 Divide as in simple numbers, and from the right of the 
 quotient, point off as many figures for cents and mills, as 
 there are places of cents and mills in the dividend. 
 
 OBS. "When the dividend contains no cents and mills, if there 
 is a remainder annex three ciphers to it ; then divide a& before, 
 and point off three figures in the quotient. 
 
 QUKST. 320. How do you divide Federal Money ? How point off the 
 quotient ? Obs> When the dividend contains no cents and mills, how f roceed ? 
 
AAT. 120.] FEDERAL MONEY 95 
 
 Note. For a more complete development of multiplication and 
 division of Federal Money, the learner is referred to the author's 
 Practical and Higher Arithmetics. 
 
 When the multiplier or divisor contain decimals, or cents and 
 mills, to understand the operation fully, requires a thorough 
 knowledge of Decimal Fractions, a subject which the limits of this 
 work will not allow us to introduce, 
 
 (2.) (3.) (4.) 
 
 6) $856.272. 8) $9567.648. 9) $7254,108. 
 
 5. Divide $868.36 by 27. 6. Divide $3674.65 by 38. 
 
 7. Divide $486745 by 49. 8. Divide $634.075 by ofi. 
 
 9. Divide $6634.25 by 60. 10. Divide $5340.73 by 78 
 11. Divide $7643.85 by 83. 12. Divide $4389.75 by 89. 
 13. Divide $836847 by 94. 14. Divide $94321.62 by 97. 
 
 15. A man paid $2563.84 for 63 sofas : what was that 
 apiece ? 
 
 16. A miller sold 86 barrels of flour for $526.50 : how 
 much was that per barrel ? 
 
 17. If a man pays $475.56 for 65 barrels of pork, what 
 is that per barrel ? 
 
 18. A man paid $1875.68 for 93 stoves: how much 
 was that apiece ? 
 
 19. If $2682.56 are equally divided among 100 men, 
 how much will each receive ? 
 
 20. A cabinet-maker sold 116 tables for $968.75 : how 
 much did he get apiece ? 
 
 21. A farmer sold 168 sheep for $465 : how much did 
 he receive apiece for them ? 
 
 22. A miller bought 216 bushels of wheat for $375.50 : 
 how much did he pay per bushel ? 
 
 23. If $2368.875 were equally divided among 348 per- 
 sons, how much would each person receive ? 
 
98 REDUCTION. [SECT. IX. 
 
 SECTION IX 
 
 REDUCTION. 
 
 ART. 121* REDUCTION is the process of changing 
 Compound Numbers from one denomination into another 
 without altering their value. 
 
 REDUCING HIGHER DENOMINATIONS TO LOWER. 
 
 122. Ex. 1. Reduce 2, to farthings. 
 
 Suggestion. First reduce the Operation. 
 
 given pounds (2) to shillings, by 2 
 
 multiplying them by 20, because 20s. in l. 
 
 20s. make l. Next reduce the 40 shillings, 
 
 shillings (40) to pence, by multi- I2d. in Is. 
 
 plying them by 12, because 12d. 480 pence. 
 
 make Is. Reduce the pence (480) _1 far - in ld - 
 
 to farthings, by multiplying them ^ ns - 192 farthings. 
 by 4, because 4 far. make Id. 
 
 2. Reduce l, 2s. 4d. and 3 far. to farthings. 
 
 Suggestion. In this example Operation. 
 
 there are shillings, pence, and far- * d. far. 
 
 things. Hence, when the pounds * 2 . 4 3 
 
 are reduced to shillings, the given 
 shillings (2) must be added men- itd^iJT 
 
 tally to the product. When the 
 
 ' x , , , 268 pence, 
 
 shillings are reduced to pence, the 4 f - ' id 
 
 given pence (4) must be added; ^ ^^ ^ ' 
 
 and when the pence are reduced to 
 
 farthings, the given farthings (3) must be added. 
 
 Q,UEST. 121. What is reduction? 122. Ex. 1. How reduce pounds to shil- 
 lings? Why multiply by 20 ? How are shillings reduced to pence? Why ? 
 How pence to farthings 1 Why ? 
 
ARTS. 121 124.] REDUCTION. 0Y 
 
 OBS. lu these examples it is required to reduce higher denomi 
 nations to lower, as pounds to shillings, shillings to pence, <fcc. 
 
 123* The process of reducing higher denominationa 
 to lower, is usually called Reduction Descending. 
 
 It consists in successive multiplications, and may with 
 propriety be called Reduction by Multiplication. 
 
 124* From the preceding illustrations we derive the 
 following 
 
 RULE FOB REDUCTION DESCENDING. 
 
 Multiply the highest denomination given by the num- 
 ber required of the next lower denomination to make ONE 
 of this higher, and to the product, add the given number of 
 this lower denomination. 
 
 Proceed in this manner with each successive denomina- 
 tion, till you come to the one required. 
 
 EXAMPLES. 
 
 3. Reduce 4 miles, 2 fur., 8 rods and 4 feet to feet. 
 
 Operation. 
 
 Suggestion. Having reduced the m.fur. r. ft. 
 
 miles and furlongs to rods, we have 
 1368 rods. We then multiply by . 
 
 10-J-, because 16^- feet make 1 rod. ,Q 
 
 (Art. 94.) Now 16 J is a mixed 2 )1368 rods, 
 number; we therefore first multi- jgi 
 
 ply by the whole number (16), 8212 
 
 then by the fraction (-), and add 1368 
 
 the products together. (Art. 84.) 684 
 
 Ans. 22576 feet. 
 
 QUEST. 123. What is reducing compound numbers to lower denominations 
 usually called? Which of the fundamental rules is employed in reduction 
 descending? 124. What is the rule for Reduction Defending ? 
 
98 REDUCTION. | SECT. IX 
 
 4. In 5, 16s. 7d., how many farthings ? Ans. 5596 far 
 
 5. In 18 how many pence? 
 
 6. In 23, 9s., how many shillings ? 
 
 7. In 17s. 2d. 3 far., how many farthings? 
 
 8. Reduce 5 Ibs. 6 oz. Troy weight, to grains. 
 
 Ans. 31680 grs. 
 
 9. Reduce 13 Ibs. Troy, to ounces. 
 
 ' 10. Reduce 4 Ibs. 3 oz. Troy, to penny weights. 
 
 11. Reduce 15 oz. 6 pwts. 4 grs. Troy, to grain*. 
 
 12. In 2 cwt. 3 qrs. 7 Ibs. 4 .oz. 3 drams, avoirdupow 
 weight, how many drams? Ans. 72259 drams. 
 
 13. In 13 Ibs. 4 oz. avoirdupois, how many ounces? 
 
 14. In 2 qrs. 17 Ibs. avoirdupois, how many pounds f 
 
 15. In 6 Ibs. 12 oz. avoirdupois, how many drams? 
 
 16. In 12 cwt. 1 qr. 6 Ibs. avoir., how many ounces * 
 
 17. In 16 miles, how many rods? 
 
 18. In 28 rods and 2 feet, how many inches ? 
 
 19. In 19 fur. 4 rods and 2 yds., how many feet ? 
 
 20. In 25 leagues and 2 m., how many rods ? 
 
 21. Reduce 14 yards cloth measure to quarters. 
 
 22. Reduce 21 yards 2 quarters to nails. 
 
 23. Reduce 17 yards 3 quarters 2 nails, to nails. 
 
 24. How many quarts in 23 gallons, wine measure ? 
 
 25. How many gills in 30 gallons and 2 quarts? 
 
 26. How many gills in 63 gallons ? 
 
 27. How many quarts in 41 hogshead* ? 
 
 28. How many pecks in 45 bushels ? 
 
 29. How many pints in 3 pecks and 2 quarts ? 
 
 30. How many quarts in 52 bu. and 2 peck* ? 
 
 31. How many hours in 15 weeks? 
 
 32. How many minutes in 25 days ? 
 83. How many seconds in 265 hours ? 
 
 34. How many minutes in 52 weeks ? 
 
 35. How many seconds in 68 days ? 
 
ARTS. 125, 126.] REDUCTION. 99 
 
 REDUCING LOWER DENOMINATIONS TO HIGHER. 
 
 125. Ex. 1. Reduce 1920 farthings to pounds. 
 
 Suggestion. First reduce the given far- Operation. 
 things (1920) to pence, the next higher 4)1920 far, 
 denomination, by dividing them by 4, be- 12)480d. 
 cause 4 far. make Id. Next reduce the 20)40s. 
 pence (480) to shillings, by dividing them 2 Ans. 
 
 by 12, because 12d. make Is. Finally, re- 
 duce the shillings (40) to pounds, by dividing them by 20, 
 because 20s. make l. The answer is 2. That is, 1920 
 far. are equal to 2. 
 
 2. In 1075 farthings, how many pounds? 
 
 Suggestion. In dividing the Operation. 
 
 given farthings by 4, there is a 4)1075 far. 
 remainder of 3 far., which should 12)268d. 3 far. over, 
 be placed on the right. In di- 20)22s. 4d. over, 
 
 viding the pence (268) by 12, 1, 2s. over, 
 
 there is a remainder of 4d., which Ans. l, 2s. 4d. 3 far. 
 should also be placed on the 
 
 right. In dividing the shillings (22) by 20, the quotient 
 is l and 2s. over. The last quotient with the several 
 remainders is the answer. That is, 1075 far. are equal to 
 1, 2s. 4d. 3 far. 
 
 OBS. In the last two examples, it is required to reduce lower de- 
 nominations to higher, as farthings to pence, pence to shillings, &c. 
 The operation is exactly the reverse of that in Reduction Descending. 
 
 126* The process of reducing lower denominations to 
 higher is usually called Reduction Ascending. 
 
 It consists in successive divisions, and may with propri- 
 ety be called Reduction by Division. 
 
 QUFST. 125. Ex. 1. How are farihings reduced to pence ? Why divide by 4 1 
 How reduce pence to shillings 1 WLy? How shillings to pounds? Why? 
 120. What is reducing compound numbers to higher denominations usually 
 called ' Which ol the fundamental rules ia employed in Reduction Ascending ? 
 
100 REDUCTION. [SECT. IX. 
 
 127. From the preceding illustrations we derive the 
 following 
 
 RULE FOR REDUCTION ASCENDING. 
 
 Divide the given denomination ~by that number which it 
 takes of this denomination to ma fee ONE of the next higher. 
 Proceed in this manner with each successive denomination) 
 till you come to the one required. The last quotient, with 
 tlie several remainders, will be the answer sought. 
 
 128* PROOF. Reverse the operation; that is, reduce 
 back tlie answer to the original denominations, and if the 
 result correspond with the numbers given, the work is right. 
 
 OBS. Each remainder is of the satne denomination as the divi- 
 dend from which it arose. (Art. 51, Obs. 2.) 
 
 EXAMPLES. 
 
 3. In 429 feet, how many rods ? Operation, 
 Suggestion. We first reduce the feet 3 )429 feet. 
 
 to yards, then reduce the yards to rods 5i-)143 yds. 
 
 by dividing them by 5-J-. (Art. 86.) 2 2 
 
 Or, we may divide the given feet by 11 )286 
 
 16, the number of feet in a rod, and the Ans. 26 rods, 
 quotient will be the answer. 
 
 Proof. 
 
 We first reduce the rods back to yards, 26 rods. 
 
 (Art. 84,) then reduce the yards to feet. 5^ 
 
 The result is 429 feet, which is the same 130 
 
 as the given number of feet. 13 
 
 Or, we may multiply the 26 by 16i, 143 yds. 
 
 and the product will be 429. 3 
 
 429 feet. 
 
 4. Reduce 256 pence to pounds. Ans. l, Is. 4d. 
 
 5. Reduce 324 pence to shillings. 
 
 QUEST. 127, What is the rule for reduction ascending ? 328, Hovr is re- 
 duction proved 1 Obs. Of what denomination is each remainder 7 
 
ARTS. 127 129.] REDUCTION. 101 
 
 6. Reduce 960 farthings to shillings. 
 
 7. Reduce 1250 farthings to pounds. 
 
 8. In 265 ounces Troy weight, how many pounds ? 
 
 9. In 728 pwts., how many pounds Troy? % 
 
 10. In 54'8 grains, how many ounces Troy? 
 
 11. In 638 oz. avoirdupois weight, how many pounds? 
 
 12. In 736 Ibs. avoirdupois, how many quarters? 
 
 13. In 1675 oz. avoirdupois, how many hundred weight ? 
 
 14. In 1000 drams avoirdupois, how many pounds? 
 
 15. In 4000 Ibs. avoirdupois, how many *,ons? 
 
 16. How many yards in 865 inches ? 
 
 17. How many rods in 1000 feet? 
 
 18. How many miles in 2560 rods ? 
 
 19. How many miles in 3261 yards ? 
 
 20. How many leagues in 2365 rods ? 
 
 EXAMPLES IN REDUCTION ASCENDING AND DESCENDING. 
 
 129* In solving the following examples, the pupil 
 tnust first consider whether the question requires higher 
 denominations to be reduced to lower, or lower denomina- 
 tions to higher. Having settled this point, he "ill find no 
 difficulty in applying the proper rule. 
 
 FEDERAL MONEY. (ART. 88.) 
 
 1. In 3 dollars and 16 cents, how many cents ? 
 
 2. In 81 cents and 2 mills, how many mills? 
 
 3. In 245 cents, how many dollars? 
 
 4. In 321 mills, how many dimes? 
 
 5. In 95 eagles, how many cents ? 
 
 6. In 160 dollars, how many cents ? 
 
 7. In 317 dollars, how many dimes? 
 
 8. In 4561 mills, how many dollars? 
 
 9. In 8250 cents, how many eagles ? 
 
 10. In 61 dolls., 12 cts., *md 3 mills, how many milk? 
 
02 REDLCTION. [SECT. IX. 
 
 STERL .NG MONEY. (ART. 90.) 
 11. Keduce 17, 16s. to shillings. 
 12: Reduce 19s. 6d. 2 far. to farthings. 
 
 13. ^Reduce 1200 pence to pounds. 
 
 14. Reduce 3626 farthings to shillings. 
 
 15. Reduce 19 to farthings. 
 
 16. Reduce 2880 farthings to shillings. 
 
 17. Reduce 21, 3s. 6d. to pence. 
 
 18. Reduce 3721 farthings to pounds. 
 
 TROY WEIGHT. (ART. 91.) 
 .19. In 7 Ibs., how many ounces ? 
 
 20. In 9 Ibs. 2 oz., how many pennyweights ? 
 
 21. In 165 oz., how many pounds? 
 
 22. In 840 grains, how many ounces ? 
 
 23. In 3 Ibs. 5 oz. 2 pwts. 7 grs., how many grains? 
 
 24. In 6860 grains, how many pounds? 
 
 AVOIRDUPOIS WEIGHT. (ART. 92.) 
 
 25. In 200 oz., how many pounds ? 
 
 26. In 261 Ibs., how many ounces? 
 
 27. In 3 tons, 2 cwt., how many pounds? 
 
 28. In 1 cwt. 2 qrs., how many ounces ? 
 
 29. In 1000 oz., how many pounds? 
 
 30. In 4256 Ibs., how many tons ? 
 
 APOTHECARIES' WEIGHT. (ART. W.) 
 
 31. Reduce 45 pounds to ounces. 
 
 32. Reduce 71 oz. to scruples. 
 
 33. Reduce 93 Ibs. 2 oz. to grains. 
 
 34. Reduce 165 oz. to pounds. 
 
 35. Reduce 962 drams to pounds. 
 
 LONG MEASURE. (ART. 94 ) 
 
 36. In 636 inches, how many yards ? 
 87. In 763 feet, how many rods? 
 
ART. 129 ] REDUCTION'. 103 
 
 38. In 4 miles, how many feet ? 
 
 39. In 18 rods 2 feet, how many inches ? 
 
 40. In 1760 yards, how many miles? 
 
 41. In 3 leagues, 2 miles, how many inches? 
 
 CLOTH MEASURE. (ART. 95.) 
 
 42. How many yards in 19 quarters? 
 
 43. How many quarters in 21 yards and 3 quaiters? 
 
 44. How many nails in 35 yards and 2 quarters? 
 
 45. How many Flemish ells in 50 yards ? 
 
 46. How many English ells in 50 yards ? 
 
 47. How many French ells in 50 yards ? 
 
 SaUARE MEASURE. (ART. 96.) 
 
 48. In 65 sq. yards and 7 feet, how many feet ? 
 
 49. In 39 sq. rods and 15 yds., how many yards? 
 
 50. In 27 acres, how many square feet? 
 
 51. In 345 sq. rods, how many acres ? 
 
 52. In 461 square yards, how many rods? 
 
 53. In 876 sq. inches, how many sq. feet ? 
 
 CUBIC MEASURE. (ART. 97.) 
 
 54. In 48 cubic yards, how many feet ? 
 
 55. In 54 cubic feet, how many inches ? 
 
 56. In 26 cords, how many cubic feet ? 
 
 57. In 4230 cubic inches, how many feet? 
 
 58. In 3264 cubic feet, how many cords ? 
 
 WINE MEASURE. (ART. 98.) 
 
 59. Reduce 94 gallons 2 qts. to pints. 
 
 60. Reduce 68 gallons 3 qts. to gills. 
 
 61. Reduce 10 hhds. 15 gallons to quarts. 
 
 62. Reduce 764 gills to gallons. 
 
 63. Reduce 948 quarts to hogsheads. 
 
 64. Reduce 896 gills to gallons. 
 
J V4 REDUCTION. [SECT. IX. 
 
 BEER MEASURE. (ART. 99.) 
 
 65. How many quarts in 1 1 hogsheads of beer ? 
 
 66. How many pints in 110 gallons 2 qts. of beer ? 
 
 67. How many hogsheads in 256 gallons of beer? 
 
 68. How many barrels in 320 pints of beer? 
 
 69. How many pints in 46 hhds. 10 gallons ? 
 
 70. How many hhds. in 2592 quarts ? 
 
 DRY MEASURE. (ART. 100.) 
 
 71. In 156 pecks, how many bushels ? 
 
 72. In 238 quarts, how many bushels ? 
 
 73. In 360 pints, how many pecks ? 
 
 74. In 58 bushels, 3 pecks, how many pecks ? 
 
 75. In 95 pecks, 2 quarts, how many quarts ? 
 
 76. In 373 quarts, how many bushels ? 
 
 77. In 100 bushels, 2 pecks, how many pints? 
 
 TIME. (ART. 101.) 
 
 78. How many minutes in 16 hours ? 
 
 79. How many seconds in 1 day? 
 
 80. How many minutes in 365 days ? 
 
 81. How many days in 96 hours ? 
 
 82. How many days in 3656 minutes ? 
 
 83. How many seconds in 1 week ? 
 
 84. How many years in 460 weeks? 
 
 CIRCULAR MEASURE. (ART. 105.) 
 
 85. Reduce 23 degrees, 30 minutes to minutes. 
 
 86. Reduce 41 degrees to seconds. 
 
 87. Reduce 840 minutes to degrees. 
 
 88. Reduce 964 minutes to signs. 
 
 89. Reduce 2 signs to seconds. 
 
 90. Reduce 5 signs, 2 degrees to minutes. 
 
 91. Reduce 960 seconds to degrees. 
 
 92. Reduce 1800 minutes to signs. 
 
ART. 117.] REDUCTION. 105 
 
 93. In 45 guineas, how many farthings \ 
 
 94. In 60 guineas, how many pounds ? 
 
 95. In 62564 pence, how many guineas ? 
 
 96. In 84, how many guineas ? 
 
 97. How many grains Troy, in 46 Ibs. 7 oz. 5 pwts. ? 
 
 98. How many pounds Troy, in 825630 grains ? 
 
 99. Reduce 62 Ibs. 10 pwts. to grains. 
 
 100. In 16 tons, 11 cwt. 9 Ibs., avoir., how many pounds ? 
 
 101. Reduce 782568 ounces to tons. 
 
 102. In 18 rods, 2 yds. 3 ft. 10 in., how many inches ^ 
 
 103. How many feet in 3 leagues, 2 miles, 12 rods ? 
 
 104. In 2738 inches, how many rods ? 
 
 105. In 2 tons, 3 cwt. 2 qrs. 15 Ibs., how many ounces ! 
 
 106. Reduce 53 Ibs. 11 pwts. 10 grs. Troy, to grains. 
 
 107. How many English ells in 45 yards ? 
 
 108. How many yards in 45 English ells ? 
 
 109. How many Flemish ells in 54 yards 1 
 
 110. How many French ells in 60 yards ? 
 
 111. In 13 m. 2 fur. 6 ft. 7 in., how many inches ? 
 
 112. In 84256 feet, how many leagues ? 
 
 113. In 135 bu. 3 pks. 2 qts. 1 pt. how many pints ? 
 
 114. In 84650 pints, how many quarters ? 
 
 115. How many gills in 48 hhds. 18 gal. wine measure ? 
 
 116. How many pipes in 98200 quarts? 
 
 117. How many seconds in 15 solar years ? 
 
 118. How many weeks in 8029200 seconds? 
 
 119. How many square feet in 82 acres, 36 rods, 8 yds. ! 
 
 120. How many cords of wood in 68600 cubic inches ? 
 
 121. How many inches in 10 cords and 6 cubic feet ? 
 
 122. In 246 tons of round timber, how many inches I 
 
 123. In 65200 square yards, how many acres? 
 
 124. In 8 signs, 43 deg. 18 sec., how many seconds I 
 
 125. In 75260 minutes, how many signs ? 
 
106 COMPOUND ADDITION. [SECT. VII. 
 
 COMPOUND ADDITION. 
 
 ART. 129. Compound Addition is the process of 
 Uniting two or more compound numbers in one sum. 
 
 Ex. 1. What is the sum of 2, 3s. 4d. 1 far.; 1, 6s. 
 9d. 3 far. ; 7, 9s. 7d. 2 far. 
 
 Suggestion First write the Operation. 
 
 numbers under each other, pounds St $ f art 
 
 under pounds, shillings under shil- 2 " 4 " 4 " 1 
 
 lings, &c. Then, beginning with 1 " 6 " 9 " 3 
 
 H It Q ff H ft o 
 
 the lowest denomination, we find * tf ' * 
 
 the sum is 6 farthings, which is Ans. 11 '0 '9 '2 
 equal to 1 penny and 2 far. over. Write the 2 far. under 
 the column of farthings, and carry the Id. to the column of 
 pence. The sum of the pence is 21, which is equal to Is. 
 and 9d. Place the 9d. under the column of pence, and 
 carry the Is. to the column of shillings. The sum of the 
 shillings is 20, which is equal to l and nothing over. 
 Write a cipher under the column of shillings, and carry the 
 l to the column of pounds The sum of the pounds is 
 11, which is set down in full. 
 
 13O* Hence, we derive the following general 
 
 RULE FOB COMPOUND ADDITION. 
 
 I. Write the numbers so that the same denominations 
 shall stand under each other. 
 
 II. Beginning at the right hand, add each column sepa- 
 rately, and divide its sum by the number required to make 
 ONE of the next higher denomination. Setting the remain- 
 der under tke column added, carry the quotient to the next 
 column, and thus proceed as in Simple Addition. (Art. 23.) 
 
 PROOF. The. proof is the same as in Single Addition. 
 
 QUEST. 121). What ia Compound Addition 1 130. How do you write com- 
 pound numbers for addition 1 Where do you begin to add, and how pro* 
 coed 1 How is Compound Addition proved 1 
 
ARTS. 129, 130.] COMPOUND ADDITION. 
 
 (2-) 
 
 (3.) 
 
 W . 
 
 
 
 s. 
 
 d. 
 
 far. 
 
 Ib. 
 
 oz. 
 
 pwt. gr. 
 
 m. 
 
 r. 
 
 /* 
 
 ^fl 
 
 1 
 
 3 
 
 6 
 
 2 
 
 2 
 
 5 
 
 7 
 
 4 
 
 7 
 
 15 
 
 20 
 
 8 
 
 3 
 
 
 
 8 
 
 3 
 
 2 
 
 
 
 5 
 
 19 
 
 6 
 
 4 
 
 8 
 
 7 
 
 9 
 
 18 
 
 9 
 
 1 
 
 6 
 
 8 
 
 
 
 3 
 
 9 
 
 6 
 
 4 
 
 4 
 
 14 3 2^W6\11 1 13 2^/15.22 27 
 
 (5.) 
 5. d. /ar. 
 10 17 1 
 
 (6.) 
 Ib. os. pwt. gr. 
 17 10 13 5 
 
 (*) 
 r. yd. ft. in. 
 
 4426 
 
 19 6 5 2 
 
 8928 
 
 6602 
 
 7820 
 
 10 4 11 3 
 
 6814 
 
 8263 21 11 16 6 2 3 2 S 
 
 (8.) (9.) (10.) 
 
 cwt. qr. Ib. oz. wk. d. hr. mm. yd. qr. na. in. 
 
 5345 13 4 19 30 6312 
 
 6298 15 13 16 3231 
 
 8172 73 5 10 7024 
 
 6096 12 14 25 5112 
 
 11. Add 4 tons, 5 cwt. 3 qrs. 2 Ibs. 10 oz. 4 drs. ; 6 
 tons, 4 cwt. 17 Ibs. 15 oz. 9 drs. ; 3 tons, 2 cwt. 1 qr. 15 Ibs. 
 
 12. Add 4 hhds. 10 gals. 3 qts. 1 pt. ; 15 hhds. 19 gals. 
 
 2 qts. ; 8 hhds. 7 gals. 2 qts. 1 pt. wine measure. 
 
 13. Add 1 pipe, 1 hhd. 8 gals. 2 qts. 1 pt. 2 gills ; 1 
 pipe, 6 gals. 1 qt. ; 3 pipes, 1 hhd. 3 gals. 3 qts. 1 pint. 
 
 14. A man sold the following quantities of wheat : 5 bu. 
 
 3 pks. 2 qts. ; 10 bu. 1 pk. 4 qts. ; 21 bu. 2 pks. 5 qts. : 
 how much did he sell in all ? 
 
 15. A merchant bought 3 pieces of silk, one of which 
 contained 21 yds. 2 qrs. 3 nails ; another 19 yds. 3 qrs. 1 
 nail ; and the other 26 yds. 1 qr. and 2 nails : how many 
 yards did they all contain ? 
 
108 COMPOUND SUBTRACTION. [SECT. VIIL 
 
 COMPOUND SUBTRACTION. 
 
 AIIT. 131. Compound Subtraction is the process of 
 finding the difference between two compound numbers. 
 
 Ex. 1. From 11, 8s. 5d. 3 far., subtract 5, 10s. 2d. 1 
 farthing. 
 
 Suggestion. Write the less number Operation. 
 under the greater, pounds under pounds, s . d.far. 
 shillings under shillings, &c. Then, be- 11 8 5 3 
 
 ginning with the lowest denomination, 5 10 2 1 
 
 proceed thus : 1 far. from 3 far. leaves 2 5 18 3 2 
 far. Set the remainder 2 under the farthings. Next, 2d. 
 from 5d. leave 3d. Write the 3 under the pence. Since 
 10 shillings cannot be taken from 8 shillings ; we borrow 
 as many shillings as it takes to make one of the next 
 higher denomination, which is pounds ; and 1, or 20s., 
 added to the 8s. make 28 shillings. Now 10s. from 28s. 
 leave 18s., which we write under the shillings. Finally, 
 carrying 1 to the next number in the lower line, we have 
 6 ; and 6 from 11 leave 5, which we write under 
 the pounds. The answer is o, 18s. 3d. 2 far. 
 
 132, Hence, we derive the following general 
 
 RULE FOR COMPOUND SUBTRACTION. 
 
 I. Write the less number under the greater, so that the 
 same denominations may stand under each other. 
 
 II. Beginning at the right hand, subtract each lower 
 number from the number above it, and set the remainder 
 under the number subtracted. 
 
 III. When a number in the lower line is larger than 
 that above it, add as many units to the upper number as it 
 
 UUKST. 131. What is Compound Subtraction ? 132. How do you write 
 compound numbers for subtraction 1 Where begin to subtract, and how 
 proceed? When a number in the lower line is laiger than that above it, 
 what is to be done 1 * 
 
ARTS. 131, 132.] COMPOUND SUBTRACTION. 109 
 
 takes to make ONE c/ the next higher denomination ; then 
 subtract as before, and adding I to the next number in the 
 lower line, proceed as in Simple Subtraction. 
 
 PROOF. The proof is the same as in Sim. Subtraction. 
 
 (2.) (3.) 
 
 From 13, 7s. 8d. 3 far. 19 Ibs. 3 oz. 7 pwts. 12 grs. 
 Take 6, 5s. lid. 1 far. 15 Ibs. 8 oz. 3 pwts. 4 grs. 
 
 (4.) (5.) 
 
 From 12 T. 7 cwt. 1 qr. 3 Ibs. 15 m. 3 fur. 10 r. 8 ft. 4 in. 
 Take 7 T. 9 cwt. 3 qrs. 4 Ibs. 9 m. 6 fur. 3 r. 4 ft. 7 in. 
 
 6. From 24 yds. 2 qrs. 3 nails, take 16 yds. 3 qrs. 2 
 nails. 
 
 7. A lady having 18, 4s. 7d. in her purse, paid 8, 7s. 
 3d. for a dress : how much had she left ? 
 
 8. If from a hogshead of molasses you draw out 19 gals. 
 
 3 qts. 1 pi, how much will there be left in the hogshead ? 
 
 9. A person bought 8 tons, 3 cwt. 19 Ibs. of coal, and 
 having burned 3 tons, 6 cwt. 45 Ibs. sold the rest: how- 
 much did he sell ? 
 
 10. From 17 years, 7 mos. 16 days, take 15 years, and 
 
 4 months. 
 
 11. From 39 yrs. 3 mos. 7 days, 4 min., take 23 yrs. 5 
 mos. 3 days, 16 hrs. 
 
 12. From 43 A. 2 roods, 15 rods, take 39 acres and 11 
 rods. 
 
 13. From 38 leagues, 2 miles, 5 fur. 17 rods, take 29 
 leagues, 2 miles, 7 fur. 13 rods. 
 
 14. From 125 bushels, 3 pecks, 4 quarts, 2 pints, take 
 108 bushels, 2 pecks, 7 quarts. 
 
 15. From 85 guineas, 13 shillings, 4 pence, 2 far. take 
 39 guineas, 15 shillings, 8 pence. 
 
 QUEST, -How is Compound Subtradfcon proved 7 
 
110 COMPOUND MULTIPLICATION. [SjECT. VIIL 
 
 COMPOUND MULTIPLICATION. 
 
 ART. 133. Compound Multiplication is the process 
 of finding the amount of a compound number repeated or 
 added to itself, a given number of times. 
 
 Ex. 1. What will 3 barrels of flour cost, at 1, 7s. 5d. 2 
 far. per barrel ? 
 
 Suggestion. Write the multiplier un- ~ . 
 der the lowest denomination of the multi- ~ , 1 
 
 plicand, and proceed thus : 3 times 2 far. i " V " 5* 2 * 
 are 6 far. which are equal to Id. and 2 3 
 
 far. over. Write the remainder 2 far. 4 2 4 2 
 under the denomination multiplied, and 
 carry the Id. to the next product. 3 times 5d. are 15d., 
 and 1 to carry makes 16d., equal to Is. and 4d. over. 
 Write the 4d. under the pence, and carry the Is. to the 
 next product. 3 times 7s. are 21s. and 1 to carry makes 
 22s., equal to l, and 2s. Write the 2 under the shillings 
 and carry the l to the next product. Finally, 3 times 
 1 are 3, and 1 to carry makes 4. Write the 4 under 
 the pounds. The answer is 4, 2s. 4d. 2 far. 
 
 134;* Hence, we derive the following general 
 
 RULE FOR COMPOUND MULTIPLICATION". 
 Beginning at the right hand, multiply each denomina- 
 tion of the multiplicand by the multiplier separately, and 
 divide its product by the number required to make ONE of 
 the next higher denomination, setting down the remainder 
 and carrying the quotient as in Compound Addition. 
 
 2. Multiply 4, 6s. 2d. 3 far. by 15. 
 
 3. Multiply 19 Ibs. 8 oz. 9 pwts. 7 grs. by 12. 
 
 4. If a man walks 3 miles, 3 fur. 18 rods in 1 hour, how 
 far will he walk in 10 hours ? 
 
 QUEST. 133. What is Compound Multiplication ? 134. What is the role 
 for Compound Multiplication ? 
 
ARTS. 133, 134.] COMPOUND DIVISION. Ill 
 
 5. Multiply 7 leagues, 1 m. 31 rods, 12 ft. 3 in. by 9. 
 
 6. Multiply 18 tons, 3 cwt. 10 Ibs. 7 oz. 3 drs. by 11. 
 
 7. A man has 7 pastures, each containing 6 acres, 25 
 rods, 5 1 square feet : how much do they all contain ? 
 
 8. A man bought 9 loads of wood, each containing 1 
 cord and 21 cu. ft. : how much did they all contain ? 
 
 9. Multiply 17 yds. 3 qrs. 2 nails by 35. 
 
 10. Multiply 53 days, 19 min. 7 sec. by 41. 
 
 11. Multiply 36 years, 3 weeks, 5 days, 12 hours, by 63. 
 
 12. Multiply 65 hhds. 23 gals. 3 qts. 1 pt. by 72. 
 
 COMPOUND DIVISION. 
 
 135* Compound Division is the process of dividing 
 compound numbers. 
 
 Ex. 1. A father divided 10, 5s. 8d. 2 far. equally 
 among his 3 sons : how much did each receive ? 
 
 Suggestion. Write the divisor ~ 
 
 A , , ., ~ ,. ,. ., , Operation. 
 
 on the left of the dividend, and x , - 
 
 , . , TV . . ' , s. d. far. 
 
 proceed as in Snort Division. Thus, 3^10 " 5 " 8 2 
 
 3 is contained in 10, 3 times and A Q // Q ///>// Ql ' 
 
 n . ^f, . . , ^LnS. O O O O-ar 
 
 1 over. We write the 3 under 
 
 the pounds, because it denotes pounds ; then reducing the 
 .remainder l to shillings and adding the given shillings 5, 
 we have 25s. Again, 3 is in 25s. 8 times and Is. over. We 
 set the 8 under the shillings, because it denotes shillings ; 
 then reducing the remainder Is. to pence and adding the 
 given pence 8, we have 20d. Now 3 is in 20d. 6 times 
 and 2d. over. We set the 6d. under the pence, because it 
 denotespence. Finally, reducing the rem. 2d. to farthings 
 and adding the given far. 2, we have 10 far. ; and 3 is in 
 10, 3 times and 1 far. over. Write the 3 under the far 
 
 QUEST. 135. What is Compound Division 2 
 
112 COMPOUND DIVISION. [SECT. VIIL 
 
 136. Hence, we derive the following general 
 
 KULE FOR COMPOUND DIVISION. 
 
 1. Beginning at the left hand, divide each denomination 
 of the dividend by the divisor, and write the quotient fig- 
 ures under the figures divided. 
 
 II. If there is a remainder, reduce it to the next lower 
 denomination, and adding it to the figures of the correspond- 
 ing denomination of the dividend, divide this number as 
 before. Thus proceed through all the denominations, and 
 the several quotients will be the answer required. 
 
 OBS. 1. Each quotient figure is of the same denomination as that 
 part of the dividend from whitfia it arose. 
 
 2. When the divisor exceeds 12, and is a composite number, we 
 may divide first by one factor and that quotient by the other. 
 
 2. Divide 14 Ibs. 5 oz. 6 pwts. 9 grs. by 3. 
 
 3. Divide 5, 17s. 8cl. 1 far. by 4. 
 
 4. Divide 25 Ibs. 3 ounces, 8 pwts. 7 grs. by 5. 
 
 5. Divide 15 T. 15 cwt. 3 qrs. 10 Ibs. by 6, 
 
 6. Divide 23 yards, 2 qrs. 1 nail, by 7. 
 
 7. Divide 35 leagues, 1 rn. 3 fur. 17 rods by 8. 
 
 8. Divide 45 hhds. 18 gals. 39 qts. 1 pint by 9. 
 
 9. A farmer had 34 bu. 3 pks. 1 qt. of wheat in 9 bags . 
 how much was in each bag? 
 
 10. If you pay 25, 17s. 8^d. for 5 cows, how much 
 will that be apiece ? 
 
 11. Divide 38 tons, 5 cwt. 2 qrs. 15 Ibs. by 17. 
 
 12. Divide 41 hhds. 13 gals. 2 qt. wine measure by 23 
 
 13. Divide 54 acres, 2 roods, 25 rods, by 34. 
 
 14. Divide 29 cords, 19 cu. feet, 18 cu. inches by 41. 
 
 15. Divide 78 years, 17 weeks, 24 days, by 63. 
 
 QUEST. 136. What is the rule for Compound Division ? Obs. Of what de- 
 nomination is each quotient figure ? 
 
MISCELLANEOUS EXERCISES. 118 
 
 MISCELLANEOUS EXERCISES. 
 
 1. From the sum of 463279 + '734658, take 926380. 
 
 2. To the difference of 856273 and 4671 9, add 420376. 
 
 3. To 476208 add 5207568 4808345. 
 
 4 Multiply the sum of 863576 + 435076 by 287. 
 
 5. Multiply the difference of 870358 640879 by 365. 
 
 6. Divide the sum of 439409 + 87646 by 219. 
 
 7. Divide the difference of 607840 23084 by 367 
 
 8. Divide the product of 865060X406 by 1428. 
 
 9. Divide the quotient of 55296+144 by 89. 
 
 10. What is thesum of 4845 + 76 + 1009 + 463+407 ? 
 
 11. What is the sum of 836X46, and 784x76? 
 
 12. What is the sum of 1728+72, and 2828-+ 96? 
 
 13. What is the sum of 85263 45017, and 68086? 
 
 14. What is the difference between 38076 + 16325, and 
 20268 + 45675? 
 
 15. What is the difference between 40719 + 6289, and 
 31670 18273. 
 
 16. What is the difference between 378X 96, and 9419 I 
 
 17. What is the difference between 7560-7-504, and 
 7560X504? 
 
 18. Froml45X87, take 12702+87. 
 
 19. Multiply 83X19 by 75X23. 
 
 20. How many times can 34 be subtracted from 578 f 
 
 21. How many times can 1512 be taken from 7569 ? 
 
 22. How many times can 63 X 24 be taken from 27640 1 
 
 23. How many times is 68 + 31 contained in 45600? 
 
 24. Divide 832 + 1429 by 45 + 84. 
 
 25. Divide 467 + 2480 by 346 187. 
 
 26. Divide 6824016226 by 10405 6200. 
 27 Divide 320X160 by 2125 960. 
 
 28. Divide 826340 36585 by 126X84. 
 
 29. From 62345 + 19008, take 2134X38. 
 
 30. From 2631X216, take 57636. 
 
1 14 MISCELLANEOUS EXERCISES. 
 
 33. A young man having 50 dollars, bought a coat frf 
 15 dollars, a pair of pants for 8 dollars, a vest for 5 dol- 
 lars, and a hat for 3 dollars : how much money did he 
 have left ? 
 
 34. A farmer sold a cow for 18 dollars, a calf r or 4 
 dollars, and a lot of sheep for 35 dollars: how much 
 more did he receive for his sheep than for his cow and 
 calf? 
 
 35. A man having 90 dollars in his pocket, paid 27 
 dollars for 9 cords of wood, 35 dollars for 7 tons of coal, 
 and 1 1 dollars for carting both home : how much money 
 had he left ? 
 
 36. A young lady having received a birthday present 
 of 100 dollars, spent 17 dollars for a silk dress, 26 dol- 
 lars for a crape shawl, and 8 dollars for a bonnet : how 
 many dollars did she have left ? 
 
 37. A dairy- woman sold 23 pounds of butter to one 
 customer, 34 pounds to another, and had 29 pounds left: 
 how many pounds had she in all ? 
 
 38. A lad bought a pair of boots for 16 shillings, a 
 pair of skates for 10 shillings, a cap for 17 shillings, and 
 had 20 shillings left : how many shillings had he at first ? 
 
 39. A grocer having 500 pounds of lard, sold 3 kegs 
 of it ; the first keg contained 43 pounds, the second 45 
 pounds, and the third 56 pounds : how many pounds did 
 he have left ? 
 
 40. A man bought a horse for 95 dollars, a harness 
 for 34 dollars, and a wagon for 68 dollars, and sold them 
 all for 225 dollars : how much did he make by his bar- 
 gain? 
 
 41. A person being 1000 miles from home, on his re- 
 turn, traveled 150 miles the first day, 240 miles the sec- 
 ond day, and 310 miles the tfiird day: how far from 
 home was he then ? 
 
MISCELLANEOUS EXERCISES. 115 
 
 \ 42. George bought a pony for 78 dollars and paid 3 
 dollars for shoeing him ; he then sold him for 100 dol- 
 lar : how much did he make by his bargain ? 
 
 43. A man bought a carriage for 273 dollars, and paid 
 27 dollars for repairing it ; he then sold it for 318 dol- 
 lars : how much did he make by his bargain ? 
 
 44. A man bought a lot for 275 dollars, and paid a 
 carpenter 850 dollars for building a house upon it : he 
 then sold the house and lot for 1200 dollars : how much 
 did he make by the operation ? 
 
 45. A farmer having 150 sheep, lost 17 and sold 65 ; 
 he afterwards bought 38 : how many sheep had he then ? 
 
 46. A man bought 27 cows, at 31 dollars per head: 
 how many dollars did they all cost him ? 
 
 47. A miller sold 251 barrels of flour, at 8 dollars ^ 
 barrel : how much did it come to ? 
 
 48. A merchant sold 218 yards of cloth, at 8 dollars 
 per yard : how much did it come to ? 
 
 49. A merchant sold 18 yards of broadcloth, at 4 dol- 
 lars a yard, and 21 yards of cassimere, at 2 dollars a yard : 
 how much did he receive for both ? 
 
 50. A farmer sold 12 calves, at 5 dollars apiece, and 
 35 sheep, at 3 dollars apiece : how much did he receive 
 for both ? 
 
 51. A grocer sold to one person 25 firkins of butter, 
 at 7 dollars a firkin, and 13 to another, at 8 dollars a fir- 
 kin : how much did both lots of butter come to ? 
 
 52. A shoe dealer sold 100 pair of coarse boots to one 
 customer, at 4 dollars a pair, and 156 pair of fine boots 
 to another, at 5 dollars a pair: what did both lots of 
 boots come to ? 
 
 53. A miller bought 165 bushels of corn, at 5 shillings 
 a bushel, and 286 bushels of wheat, at 9 shilliigs a 
 bushel : how much did he pay for both ? 
 
. 16 MISCELLANEOUS EXERCISES. 
 
 54. A man bought 45 clocks, at 3 dollars apiece, and 
 sold them, at 5 dollars apiece : how much did he make by 
 his bargain? 
 
 55. A bookseller bought 87 books, at 7 shillings apiece, 
 and afterwards sold them, at 6 shillings apiece : how much 
 did he lose by the operation? 
 
 56. How many yards of calico, at 18 cents a yard, can 
 be bought for 240 cents ? 
 
 57. A little girl having 326 cents, laid it out in ribbon, 
 at 25 cents a yard : how many yards did she buy ? 
 
 58. If a man has 500 dollars, how many acres of land 
 can he buy, at 15 dollars per acre? 
 
 59. How many cows, at 27 dollars apiece, can be bought 
 for 540 dollars ? 
 
 60. How many barrels of sugar, at 23 dollars per bar- 
 rel, can a grocer buy for 575 dollars? 
 
 61. Henry sold his skates for 87 cents, and agreed to 
 take his pay in oranges, at 3 cents apiece : how many 
 oranges did he receive ? 
 
 62. William sold 80 lemons, at 4 cents apiece, and took 
 his pay in chestnuts, at 5 cents a quart : how many chest- 
 nuts did he get for his lemons ? 
 
 63. A milkman sold 110 quarts of milk, at 6 cents a 
 quart, and agreed to take his pay in maple sugar, at 11 
 cents a pound : how many pounds did he receive ? 
 
 64. A farmer bought 25 yards of cloth, which was 
 worth 6 dollars per yard, and paid for it in wood, at 2 
 dollars per cord : how many cords did it take ? 
 
 65. A pedlar bought 4 pieces of silk, at 24 dollars 
 apiece : how much did he pay for the whole ? 
 
 66. A farmer sold 8-j- bushels of wheat, at 96 cents 
 per bushel : how much did he receive for his wheat ? 
 
 67. A man sold a lot of land containing 15f acres, at 
 16 dollars per acre : how much did he receive for it ? 
 
MISCELLANEOUS EXERCISES. 117 
 
 \ 68. If a man can walk 45 miles in a day, how far caa 
 ha wak in 2 Of days? 
 
 69. What cost 75 yds. of tape, at f of a cent per yd. ? 
 
 70. What will 100 pair of childrens' gloves come to, 
 at -ft of a dollar a pair ? 
 
 71. What will 160 boys' caps cost, at f of a dollar 
 apiece ? 
 
 72. What will 210 pair of shoes cost, at -f- of a dollar 
 a pair ? 
 
 73. How many childrens' dresses can be made from a 
 piece of lawn which contains 54 yards, if it takes 4 yards 
 for a dress ? 
 
 74. A farmer wishes to pack 100 dozen of eggs in 
 boxes, and to have each box contain 6-J- dozen : how many 
 boxes will he need ? 
 
 75. A lad having 275 cents, wishes to know how many 
 miles he can ride in the Railroad cars, at 2 cents per mile : 
 how many miles can he ride ? 
 
 76. How many apples, at a cent apiece, can Horatio 
 buy for 75 cents ? 
 
 77. If Joseph has to pay f of a cent apiece for marbles, 
 how many can he buy for 84 cents ? 
 
 78. At f of a dollar apiece, how many parasols can a 
 .shopkeeper buy for 168 dollars? 
 
 79. If I am charged -f- of a dollar apiece for fans, how 
 many can I buy for 265 dollars ? 
 
 80. How many yards of silk, which is worth ^ of a 
 dollar a yard, can I buy for 227 dollars ? 
 
 81. How many pair of slippers, at -J of a dollar a pair, 
 can be bought for 448 dollars ? 
 
 82. In 45, 13s. 6d., how many pence ? 
 
 83. In 63, 7s. 8d. 2 far., how many farthings? 
 
 84. How many yards of satin can I buy for 75, 10s., 
 If I have to pay 5 shillings per yard ? 
 
118 MISCELLANEOUS EXERCISES. 
 
 85. How many six-pences are there in 100 ? 
 
 86. A grocer sold 10 hogsheads of molasses, at 3 shit 
 lings per gallon : how many shillings did it come to ? 
 
 87. A milkman sold 125 gallons of milk, at 4 cents pel 
 quart : how much did he receive for it ? 
 
 88. A man made 30 barrels of cider which he wished to 
 put into pint bottles : how many bottles would it require ? 
 
 89. How much would 85 bushels of apples cost, at 12 
 cents a peck ? 
 
 90. What will 97 pounds of snuff cost, at 8 cents per 
 ounce ? 
 
 91. What will 5 tons of maple sugar come to, at 11 
 cents a pound ? 
 
 92. A farmer sold 34 tons of hay, at 65 cents per hun- 
 dred : how much did he receive for it ? 
 
 93. A blacksmith bought 53 tons of iron for 3 dollars 
 per hundred : how much did he pay for it ? 
 
 94. A young man returned from California with 50 
 pounds of gold dust, which he sold for 16 dollars per 
 ounce Troy : how much did he receive for it ? 
 
 95. A man bought 36 acres of land for 3 dollars per 
 square rod : how much did his land cost him ? 
 
 96. John Jacob As tor sold five building lots in the city 
 of New York, containing 560 square rods, for 13 dollars 
 per square foot : how much did he receive for them ? 
 
 97. A laboring man engaged to work 5 years for 16 
 dollars per month : what was the amount of his wages ? 
 
 98. What will 17 cords of wood cost, at 6 cents per 
 cubic foot ? 
 
 99. If it takes 35 men 18 months to build a fort, how 
 many years would it take 1 man to build it ? 
 
 ] 00. If it takes 1 man 360 days to build a house, how 
 many weeks would it take 15 men to build it, allowing 6 
 working days to a week ? 
 
ANSWERS TO EXAMPLES. 
 
 119 
 
 ANSWERS TO EXAMPLES. 
 ADDITION. 
 
 Sx. Aus. 
 
 Ex. Ana. 
 
 Ex. Ans. 
 
 ART. 20. 
 
 4. 5286 yards. 
 
 28. 171658. 
 
 1. Given. 
 
 5. 2404. 
 
 29. 57 dollars. 
 
 2. 68. 
 
 6. 2765. 
 
 30. 58 dollars. 
 
 3. 589. 
 
 7. 10040. 
 
 31. 120 dollars. 
 
 4. 768. 
 
 8. 8668. 
 
 32. 565. 
 
 5. 9987. 
 
 9. 84 inches. 
 
 33. 742. 
 
 6. 878. 
 
 10. 114 feet 
 
 34. 1530. 
 
 V. 6767. 
 
 11. 168 dollars. 
 
 35. 1779. 
 
 8. 8898. 
 
 12. 192 rods. 
 
 36. 1597. 
 
 9. 8779. 
 
 13. 782 pounds. 
 
 37. 1757. 
 
 10. 6796. 
 
 14. 1380 yards. 
 
 38. 2379. 
 
 11. 88776. 
 
 15. 576 miles. 
 
 39. 2619. 
 
 12. 986788. 
 
 16. 836 sheep. 
 
 40. 1020. 
 
 
 17. 615 dollars. 
 
 41. 1418. 
 
 ART. 22. 
 
 18. 181 dollars. 
 
 42. 1191. 
 
 13, 14. Given. 
 
 19. 1452. 
 
 43. 150 bushels. 
 
 15. 1454. 
 
 20. 1255. 
 
 44. 133 yards. 
 
 16. 15300. 
 
 21. 1881. 
 
 45. 731 acres. 
 
 17. 13285. 
 
 22. 6693. 
 
 46. 1197 cattle. 
 
 
 23. 20485. 
 
 47. 12554 dollars. 
 
 ART. 24. 
 
 24. 9726. 
 
 48. 1282. 
 
 1. 155 pounds. 
 
 25. 1769. 
 
 49. 2528. 
 
 2. 413 feet. 
 
 26. 1500. 
 
 50. 365 days. 
 
 3. 1960 dollars. 
 
 27. 106284. 
 
 
 ART. 2 4. a. 
 
 10. 65471. 
 
 20. 551452. 
 
 30. 279,075. 
 
 1. 300. 
 
 11. 327371. 
 
 21. 46157. 
 
 31. 295,306. 
 
 2. 6000. 
 
 12. 390497. 
 
 22. 424634. 
 
 32. 1,606,895. 
 
 3. 9000. 
 
 13. 37938. 
 
 23. 430032. 
 
 35. 6,140,704. 
 
 4. 4861. 
 
 14. 50342. 
 
 24. 3458772. 
 
 36. 7,569,904. 
 
 5. 4871. 
 
 15. 449458. 
 
 25. 48350. 
 
 37. 9,253,854. 
 
 6. 47067. 
 
 16. 466789. 
 
 26. 514299. 
 
 38. 9,247,176. 
 
 7. 53340. 
 
 17. 40290. 
 
 27. 595522. 
 
 39. 10,531,960 
 
 8. 59139. 
 
 18. 50676. 
 
 28. 5781566. 
 
 40. 12,811,860. 
 
 9. 61304. 
 
 19. 508302. 
 
 29. 61993. 
 
 
120 
 
 ANSWERS. [PAGES 28 35. 
 
 SUBTRACTION. 
 
 Ex. A.ns. 
 
 Ex. Ans. 
 
 Ex. Ana. 
 
 ART. 28. 
 
 14. 275 pounds. 
 
 48. 222 bushels. 
 
 1. Given. 
 
 15. 613 yards. 
 
 49. 195 dollars. 
 
 2. 24. 
 
 16. 310 rods. 
 
 50. 1122 dollars. 
 
 3. 12. 
 
 17. 230 gallons. 
 
 51. 1659 dollars. 
 
 4. 153. 
 
 18. 503 hhds. 
 
 52. 3023 dollars. 
 
 5. 24S. 
 
 19. 76 bushels. 
 
 53. 1763 dollars. 
 
 6. 31 dollars. 
 
 20. 127 dollars. 
 
 54. 3747 dollars. 
 
 7. 12 pounds. 
 
 21. 249 pounds. 
 
 55. 16014 dollars. 
 
 8. 115 yards. 
 
 22. 1082 rods. 
 
 56. 1315 dollars. 
 
 9. 222 shillings. 
 
 23. 13016. 
 
 57. 5385 dollars. 
 
 10. 222 marbles. 
 
 24. 310768. 
 
 58. 5735 dollars. 
 
 
 25. 464374. 
 
 59. 13944 soldiers 
 
 ART. 3O. 
 
 26. 5244038. 
 
 60. 94760000 m. 
 
 11, 12. Given. 
 
 27. 45. 
 
 61. 17 oranges. 
 
 13. 137. 
 
 28. 308. 
 
 62. 33 marbles. 
 
 14. 2616. 
 
 29. 240. 
 
 63. 76 sheep. 
 
 15. 3270. 
 
 30. 58. 
 
 64. 52 cents. 
 
 16. 3203. 
 
 31. 542. 
 
 65. 43 yards. 
 
 17. 5365667. 
 
 32. 2021. 
 
 66. 122 dollars. 
 
 
 33. 1825. 
 
 67. 87 dollars. 
 
 ART. 32. 
 
 34. 2600. 
 
 68. 66 pears. 
 
 1. 217.* 
 
 35. 3085. 
 
 69. 59. 
 
 2. 182. 
 
 36. 1306. 
 
 70. 164. 
 
 3. 242. 
 
 37. 4098. 
 
 71. 149 pounds. 
 
 4. 369. 
 
 38. 1108. 
 
 72. 164 bushels. 
 
 5. 1029. 
 
 39. 4531. 
 
 73. 263 miles. 
 
 6. 1008. 
 
 40. 14520. 
 
 74. 125 gallons. 
 
 7. 3289. 
 
 41. 24622. 
 
 75. 179 pounds. 
 
 8. 3434. 
 
 42. 125028. 
 
 76. 175 dollars. 
 
 9. 35100. 
 
 43. 64303. 
 
 77. 339 pounds. 
 
 10. 312657. 
 
 44. 224066. 
 
 78. 172 barrels. 
 
 11. 1. 
 
 45. 103875. 
 
 79. 297 pages. 
 
 12. 23 dollars. 
 
 46. 420486. 
 
 80. 110 dollars. 
 
 13. 57 bushels. 
 
 47. 72 sheep. 
 
 81. 392 dollars. 
 
 * It is an excellent exercise for the pupil to prove all the examples. This is 
 one of the beet means to give him confidence in his own powers. 
 
PAGES 39 46 ] ANSWERS. 
 
 121 
 
 MULTIPLICATION. 
 
 Ex. Ans. 
 
 Ex. Ans. 
 
 Ex. Ana. 
 
 ART. 39. 
 
 ART. 41. 
 
 33. 9100 weeks. 
 
 1. Given, 
 
 34 37. Given. 
 
 34. 23760 min. 
 
 2. 68. 
 
 
 35. 28350 gallons. 
 
 3. 936. 
 
 ART. 43. 
 
 36. 34675 "dolls. 
 
 4. 8084. 
 
 1. 252. 
 
 37. 33840 sq. in. 
 
 5. 5550. 
 
 2. 390. 
 
 38. 26070 miles. 
 
 6. 12066. 
 
 3. 567. 
 
 
 7. 24408. 
 
 4. 582. 
 
 ART. 45. 
 
 8. 35550. 
 
 5. 840. 
 
 40. Given. 
 
 9. 56707. 
 
 6. 1155. 
 
 41. 260. 
 
 10. Given. 
 
 7. 3568. 
 
 42. 3700. 
 
 
 8. 2763. 
 
 43. 51000. 
 
 ART. 40. 
 
 9. 3920. 
 
 44. 226000. 
 
 11. 312. 
 
 10. 460. 
 
 45. 341000. 
 
 12. 480. 
 
 11. 572. 
 
 46. 46900*00. 
 
 13. 249. 
 
 12. 816. 
 
 47. 52300000. 
 
 14. 840. 
 
 13. 1092. 
 
 48. 681000000. 
 
 15. 828. 
 
 14. 1170. 
 
 49. 856120000. 
 
 16. 815. 
 
 15. 2185. 
 
 50. 96030500000 
 
 17. 2248. 
 
 16. 4515. 
 
 51. Given. 
 
 18. 3144. 
 
 17. 12306. 
 
 
 19. 2520. 
 
 18. 25355. 
 
 ART. 46. 
 
 20. 1900. 
 
 19. 342 dollars. 
 
 52. 17000. 
 
 21. 3960. 
 
 20. 336 bushels. 
 
 53. 291000. 
 
 22. 656C, 
 
 21. 336 inches. 
 
 54. 4920000. 
 
 23. 5628. 
 
 22. 620 pounds. 
 
 55. 11700000 
 
 24. 8712. 
 
 23. 391 dollars. 
 
 56. 33930. 
 
 25. 1050 dollars. 
 
 24. 475 dollars. 
 
 57. 789600. 
 
 26. 2300 dollars. 
 
 25. 1591 dollars. 
 
 58. 16170000. 
 
 27. 1372 dollars. 
 
 26 1950 shil. 
 
 59. 262660000. 
 
 28, 2720 dollars. 
 
 27. 1575 dollars. 
 
 60. 7oOO minutes. 
 
 29. 4837 dollars. 
 
 28. 2430 shil. 
 
 61. 2400 dollars. 
 
 30, 7785 dollars. 
 
 29. 3936 ounces. 
 
 62. 6800 shillings. 
 
 31. 7744 dollars. 
 
 30. 10754 dollars. 
 
 63 27000 dollars. 
 
 S2. 8820 dollars. 
 
 31. 6710 miles. 
 
 64. 352500 days. 
 
 &3. 2 1285 dollars. 
 
 32. 8760 hours. 
 
 
122 
 
 ANSWERS. 
 
 [PAGES 47 55. 
 
 MULTIPLICATION CONTINUED. ARTS. 47, 48. 
 
 Ex. Ana. 
 
 Ex. Ans. Ex. Ans. 
 
 65. Given. 
 
 78. 2520000. [ 91. 5816049 galls. 
 
 66. 19500. 
 
 79. 65000000. 92. 101198340 d. 
 
 67. 40800. 
 
 80. 722000000. 93. 146460440 T. 
 
 68. 504000. 
 
 81. 21000000000. 94. 1190439180. 
 
 69. 800000. 
 
 82. 72800000000. 95. 3759670728. 
 
 70. 3300000. 
 
 83. 2240000yds. 96. 4223213600. 
 
 71. 14620000. 
 
 84. 140000 miles. 97. 5815178600. 
 
 72. 65360000. 
 
 85. 700000 dolls. 98. 12976172335. 
 
 73. 104520000. 
 
 86. 504000 dolls. 99. 124811441568 
 
 74. 183244000. 
 
 87. 27375000 d. 100. 54719418834. 
 
 75. Given. 
 
 88. 367608 Ibs. 101. 469234745451 
 
 76. 420000. 
 
 89. 3838460ft. 102. 197118900. 
 
 77. 442000. 
 
 90. 4217202 r. 103. 420152303451, 
 
 SHORT DIVISION. 
 
 ART. 54. 
 
 17. 25. 
 
 9. 116*. 
 
 1. Given. 
 
 18. 76. 
 
 10. 728. 
 
 2. 21. 
 
 19. 456. 
 
 11. 1552f. 
 
 3. 23. 
 
 
 12. 1004f 
 
 4. 122. 
 
 ART. 57. 
 
 13. 400f. 
 
 5. 111. 
 
 20. Given. 
 
 14. 903*. 
 
 6. 342. 
 
 21. 509. 
 
 15. 923. 
 
 7. 1122. 
 
 22. 901. 
 
 16. 1222f. 
 
 8. 1321. 
 
 23. 1067. 
 
 17. 875. 
 
 9. 1111. 
 
 24. 503. 
 
 18. 1011-|. 
 
 
 25. Given. 
 
 19. 63 pair. 
 
 ART. 55. 
 
 
 20. 42 hats. 
 
 10. Given. 
 
 ART. 61. 
 
 21. 24 marbles. 
 
 11. 71. 
 
 1. 142. 
 
 22. 45 children. 
 
 12. 43. 
 
 2. 101-J-. 
 
 23. 75 yards. 
 
 13. 412. 
 
 3. 76. 
 
 24. 85 barrels, an C 
 
 14. 411. 
 
 4. 75. 
 
 5 dolls, over. 
 
 
 5. 102f 
 
 25. 92 days. 
 
 ART. 56. 
 
 6. 56|. 
 
 26. 158-J- yards. 
 
 15. Given. 
 
 7. 120f. 
 
 27. 195 hours. 
 
 16. 14. 
 
 8. 95. 
 
 28. 333| hours 
 
PAGES 56 62.] ANSWERS. 
 
 123 
 
 LONG DIVISION. 
 
 far 
 
 Ex. 
 
 Ex. 
 
 ART. 62. 
 
 1, 2. Given. 
 
 3. 128.* 
 
 4. 364. 
 
 5. 1825f. 
 
 6. 533. 
 
 7. 732. 
 
 8. 931. 
 911. Given. 
 
 ART. 65. 
 
 1. 46-i 
 
 2. 48-f. 
 
 3. 80f. 
 
 4. 40^. 
 
 5. 58-ft. 
 
 6. 48. 
 7. 
 
 8. 
 
 9. 41A- 
 
 10. 27. 
 
 11. 23f. 
 
 12. 21-fj, 
 
 13. 19ff. 
 
 14. 20. 
 15. 
 
 16, 
 
 17. 45f|. 
 
 18. 57ff. 
 
 19. 24 caps. 
 
 20. 35 pair. 
 
 21. 28 barrels. 
 
 82. 1900W. 
 
 83. 840ff. 
 
 QA g>7 \ 5 5 
 
 OT:. O t ^r~i ,;' .>T, 
 
 1 22. 
 23. 
 24. 
 25. 
 26. 
 27. 
 28. 
 29. 
 30. 
 31. 
 32. 
 33. 
 34. 
 35. 
 36. 
 37. 
 38. 
 39. 
 40. 
 
 41. 
 42. 
 
 43. 
 
 44. 
 45. 
 46. 
 
 47. 
 48. 
 49. 
 50. 
 
 85. 
 86. 
 
 87. 
 
 88. 
 
 16-ft- shillings 
 10^ pounds. 
 16|f pounds. 
 17 trunks. 
 30 weeks. 
 32f yards. 
 75 dresses. 
 81 sheep. 
 73-J-f- acres. 
 61 shares. 
 3 Iff years. 
 48ii hhds. 
 43ff- months. 
 5 Iff months. 
 50 dollars. 
 lOif months. 
 90 pounds. 
 60, and 1 over. 
 106, and 22 
 
 over. 
 26, and 28 
 
 over. 
 42, and 28 
 
 over. 
 30|f. 
 34. 
 53^. 
 
 25f. 
 
 51. 
 52. 
 
 53. 218-ftV 
 
 54. 216-iftfr. 
 
 ART. 67. 
 
 55. 56. Given. 
 
 57. 
 58. 
 59. 
 60. 
 61. 
 62. 
 63. 
 
 1620-fiif 
 
 ART. 68. 
 
 64, 65. Given, 
 
 66. 
 
 67. 
 
 68. 
 
 69. 
 
 70. 46<HHH!. 
 
 71. 
 
 72. 
 
 73. 
 
 74. 27rH. 
 
 75. 
 
 76. 
 
 77. 
 
 78. 
 
 79. 30. 
 
 80. 14834fJ-. 
 89. 
 
 90. 
 91. 
 
124 
 
 ANSWERS. 
 
 [PAGES 68 -92, 
 
 FRACTIONS. 
 
 Ex, Ans. 
 
 Ex. Ana. 
 
 Ex. Ans. 
 
 Ex. Ans. 
 
 ART. 83. 
 
 15. 147. 
 
 30. 1896. 
 
 11. 307. 
 
 3. 10 doll. 
 
 16. 135. 
 
 31. Given. 
 
 12. 273. 
 
 4. 10 shill. 
 
 19. 70. 
 
 32. 1122 r. 
 
 13. 304. 
 
 5. 7f doll. 
 
 20. 117. 
 
 33. 752| dol. 
 
 14. 329+. 
 
 6. 21 shill. 
 
 21. 189. 
 
 34. 1609fd. 
 
 ART. 86. 
 
 7. 36. 
 
 22. 217. 
 
 35. 6120 dol. 
 
 3. 12. 
 
 8. 28. 
 
 23. 112. 
 
 ART. 85. 
 
 4. 6. 
 
 9. 6. 
 
 24. 399. 
 
 5. 100. 
 
 5. 4-fr 
 
 10. 14. 
 
 25. 200. 
 
 6. 190. 
 
 6. 5ff. 
 
 11. 40. 
 
 26. 270. 
 
 7. 423. 
 
 V. 5. 
 
 12. 21. 
 
 27. 575. 
 
 8. 1260. 
 
 8. 11H. 
 
 13. 32. 
 
 28. 1287. 
 
 9. 108. 
 
 9. 9/ ff . 
 
 14. 100. 
 
 29. 1540. 
 
 10. 118f. 
 
 10. 9. 
 
 EXAMPLES FOR PRACTICE. 
 
 1. 8 a. 
 
 8. 9| bus. 
 
 15. 25 W a. 
 
 22. 277-f 1. 
 
 2. 8 p. 
 
 9. I7f yds. 
 
 16. 28 W bar. 
 
 23. 16 bar. 
 
 3. 24 yds. 
 
 10. 30 lambs. 
 
 17. 48 yds. 
 
 24. 20 hrs. 
 
 4. 13| yds. 
 
 11. 16 rods. 
 
 18. 22 miles. 
 
 25. 10 bar. 
 
 5. lOlbs. 
 
 12. 8 rods. 
 
 19. 20-H- cts. 
 
 26. 24 colts. 
 
 6. 24 yds. 
 
 13. 1 7-2^ yds. 
 
 20. 12 mo. 
 
 27. 32 wag. 
 
 7. 49 mar. 
 
 14. 10 cows. 
 
 21. 24 cattle. 
 
 
 ADDITION OF FEDERAL MONEY. ART. 117. 
 
 2.^1278.699. 
 
 6. $1743.828. 
 
 10. $978.297. 
 
 14.$829.49d 
 
 3.111261.52. 
 
 7.12478.735. 
 
 11. $2037.379. 
 
 15. $34.75. 
 
 4. $2622.337. 
 
 8.110224.78. 
 
 12. $880.317. 
 
 16. $74.375. 
 
 5.85599.332. 
 
 9. $12858.266, 
 
 13. $301.243. 
 
 17. $162.06. 
 
 SUBTRACTION OF FEDERAL MONEY. ART. 118. 
 
 2. $468.851. 
 
 6. $877.155. 1 
 
 0. $81980.755. 
 
 14. $49.928. 
 
 3. $497.73. 
 
 7. $461.543. ] 
 
 il. $67671. 133. 
 
 15. $357.04. 
 
 4. $527.247. 
 
 8. $435.103. 1 
 
 L2. $0.89. 
 
 16. $2.125. 
 
 5. 5916.707. 
 
 9. $1461.78.1 
 
 13. $2.317. 
 
 17. $1.945. 
 
PAGES 93 100.] ANSWERS. 125 
 
 MULTIPLICATION OF FEDERAL MONEY. ART. 119. 
 
 Ex. Ana, 
 
 Ex. Aria. 
 
 Ex. Ana. 
 
 Ex. Ans. 
 
 1. Given. 
 
 2. $5070. 
 3. $7250.625. 
 4, $21097. 80. 
 5. $335636.62 
 6. $255991.68 
 
 7. $458122.602 
 8. $773262.87. 
 9.$2182139.52 
 10. $1.36. 
 11. $10.44. 
 12. $31. 
 
 13. $78.75. 
 14.$12.375. 
 15. $39.45. 
 16. $9.375. 
 17. $23.75. 
 18.11181.28 
 
 19. $3346.50 
 20. $1495. 
 21. $4238.08 
 22. $7.50. 
 23. $73.50. 
 24. $279.50. 
 
 DIVISION OF FEDERAL MONEY. ART. 120. 
 
 1. Given. 
 
 7. $9933.57. 
 
 13. $8902.627. 
 
 19. $26.82 
 
 2. $142.712. 
 
 8. $11.322. 
 
 14. $972.38. 
 
 20. $8.35. 
 
 3. $1195.956. 
 
 9. $110.57. 
 
 15. $40.69. 
 
 21. $2.767. 
 
 4. $806.012. 
 
 10. $68.47. 
 
 16. $6.12. 
 
 22. $1.738. 
 
 5. $32.16. 
 
 11. $92.09. 
 
 17. $7.31. 
 
 23. $6.807. 
 
 6. $96.70. 
 
 12. $49.32. 
 
 18. $20.16. 
 
 
 REDUCTION DESCENDING. 
 
 ART. 124. 
 1-4. Given. 
 5. 4320d. 
 6. 469s. 
 
 11. 7348 gr. 
 13. 212 oz. 
 14. 67 Ibs. 
 15. 1728 dr. 
 
 20. 24640 r. 
 21. 56 qrs. 
 22. 344 na. 
 23. 286 na. 
 
 28. 180 pk. 
 29. 52 pts. 
 30. 1680 qts. 
 31. 2520 hrs. 
 
 7. 827 far. 
 8. Given. 
 9. 156 oz. 
 10. 1020pwt. 
 
 16. 19696 oz. 
 17. 5120 r. 
 18. 5568 in. 
 19. 12614 ft. 
 
 24. 92 qts. 
 25. 976 g. 
 26. 2016 g. 
 27. 10332 q. 
 
 32. 36000 m. 
 33. 954000s. 
 34. 524160 m. 
 35. 5875200s 
 
 REDUCTION ASCENDING. 
 
 ART. 12T. 
 14. Given. 
 5. 27 shillings. 
 6. 20 shillings. 
 7. l,6s.0d.2far. 
 8. 22 Ibs. 1 oz. 
 9. 3 Ibs. oz. 8 
 pwts. 
 
 10. 1 oz. 2 pwts. 
 20 grs. 
 11. 39 Ibs. 14 oz. 
 12. 29 qrs. 11 Ibs. 
 13. 1 cwt. 4 Ibs. 
 11 oz. 
 14. 3 Ibs. 14 oz. 8 
 drs. 
 
 15. 2 tons. 
 16. 24 >ds. 1 in. 
 17. 60 r. 10 ft. 
 18. 8 miles. 
 19. 1 m. 6 fur. 32 
 r. 5 yds. 
 20. 2 lea. 1 m. 3 
 fur. 5 r. 
 
126 
 
 ANSWERS. L PAaE * 101 1 
 
 REDUCTION ASJENDING AND DESCENDING. 
 
 Ex. Ans. 
 
 Ex. Ans 
 
 Ex. Ans. 
 
 ART. 129. 
 
 31. 540 ounces. 
 
 62. 23 gals. 3 qts. 
 
 1. 316 cents. 
 
 32. 1704 scruples. 
 
 Ipt. 
 
 2. 812 mills. 
 
 33. 536640 grs 
 
 63. 3hhds. 48 gls, 
 
 3. 2 dolls. 45 cts. 
 
 34. 13 Ibs. 9 oz. 
 
 64. 28 gals. 
 
 4. 3 dimes 2 cts. 
 
 35. 10 Ibs. oz. 
 
 65. 2376 qts. 
 
 1 mill. 
 
 2 drs. 
 
 66. 884 pints. 
 
 5. 95000 cents. 
 
 36. 17 yds. 2 ft. 
 
 67. 4hhds.40gls. 
 
 6. 16000 cents. 
 
 37. 46 rods 4 ft. 
 
 68. 1 bbl. 4 gals. 
 
 7. 3170 dimes. 
 
 38. 21120 feet. 
 
 69. 19952 pts. 
 
 8. 4 dolls. 56 cts. 
 
 39. 3588 inches. 
 
 70. 12 hhds. 
 
 1 mill. 
 
 40. 1 mile. 
 
 71. 39 bushels. 
 
 9. 8E.2dolls.50c. 
 
 41. 696960 in. 
 
 72. 7 bu. 1 pk 
 
 10. 61123 mills. 
 
 42. 4 yds. 3 qrs. 
 
 6 qts. 
 
 11. 356 shillings. 
 
 43. 87 qrs. 
 
 73. 22 pks. 4 qts. 
 
 12. 938 farthings. 
 
 44. 568 nails. 
 
 74. 235 pecks. 
 
 13. 5. 
 
 45. 66Fl.e.2qrs. 
 
 75. 762 quarts. 
 
 14. 75s. 6d. 2 far. 
 
 46. 40 E. e. 
 
 76. 11 bu. 2 pks 
 
 15. 18240 far. 
 
 47. 33 F. e. 2 qrs. 
 
 5 qts. 
 
 16. 60 shillings. 
 
 48. 592 sq. ft. 
 
 77. 6432 pints. 
 
 17. 5082 pence. 
 
 49. 1194f sq. yds. 
 
 78. 960 minutes. 
 
 18. 3, 17s 6d. 
 
 50. H76120sq.ft. 
 
 79. 86400 sec. 
 
 1 far. 
 
 51. 2 A. 25 sq.r. 
 
 80. 525600 min. 
 
 19. 84 ounces. 
 
 52. 15 sq. r. 7f 
 
 81. 4 days. 
 
 20. 2:00 pwts. 
 
 sq. yds. 
 
 82. 2 days 12 hr? 
 
 21. 13 Ibs. 9 oz. 
 
 53. 6 sq. ft. 12 
 
 56 min. 
 
 22. 1 oz. 15 pwts. 
 
 sq. in. 
 
 83. 604800 sec, 
 
 23. 19735 grains. 
 
 54. 1296 cu. ft. 
 
 84. 8 yrs. 11 mo. 
 
 24. 1 Ib. 2 oz. 5 
 
 55. 93312 cu. in. 
 
 85. 1410'. 
 
 pwts. 20 grs. 
 
 56. 3328 cu. ft. 
 
 86. 147600". 
 
 25. 12 Ibs. 8 oz. 
 
 57. 2 cu. ft. 774 
 
 87. 14. 
 
 26. 4176 ounces. 
 
 cu. in. 
 
 88. Os. 16 4-. 
 
 27. 6200 Ibs. 
 
 58. 25 cords, 64 
 
 89. 216000". 
 
 28. 2400 ounces. 
 
 cu. ft. 
 
 90. 9120'. 
 
 29. 62 Ibs. 8 oz. 
 
 59. 756 pts. 
 
 91, 16'. 
 
 30. 2 tons, 2 cwt. 60. 2200 gills. 
 
 92. 1 sign.. 
 
 2 qrs. 6 Ibs. 1 61. 2580 qts. 
 
 
PAGES 105 111.] ANSWERS. 127 
 
 REDUCTION ASCENDING AND DESCENDING. 
 
 Ex. Ans. 
 
 Ex. Ans. 
 
 Ex. Ans. 
 
 93. 45360 far. 
 94. 63. 
 95. 248G.5s.8d. 
 96. 80 G. 
 
 104. 13r.l3f. 8 i. 
 105. 69840 oz. 
 106. 205554 grs. 
 107. 36 E. ells. 
 
 116. 194 p. Ik 43 
 gals. 
 117. 473353920s 
 118. 13 wks. 1 d. 
 
 97. 268440 grsr 
 98. 143 1. 4 o. 1 
 
 108. 56 yds. 1 qr. 
 109. 72. Fl. ells. 
 
 22 hrs. 20 min. 
 119. 3581793s. ft. 
 
 p. 6g. 
 99. 357360 grs. 
 100. 33109 Ibs. 
 
 110. 40 F. ells. 
 111. 839599 in. 
 112. 5 1. 306 r. 7 f. 
 
 120. 39- ft. 1208 i. 
 121. 2222208 c.in. 
 122. 17003520 in. 
 
 101. 24 T. 9 cwt. 
 10 Ibs. 8 oz. 
 102. 3682 in. 
 103. 58278 ft. 
 
 113. 8693 pts. 
 114. 165 qrs.2bu. 
 2 pks. 5 qts. 
 115. 97344 gills. 
 
 123. 13 A. 75 r. 
 11^ yds. 
 124. 1018818 sec. 
 125. 418.24, 20'. 
 
 COMPOUND ADDITION. 
 
 5. 40, 14s.2d.2f. 
 
 9. 35 w. 4h. 21m. 
 
 13. 6 pi. 18 gals. 
 
 6. 59 1. 2 p. 22 g. 
 
 10. 23 yds. 3 na. 
 
 3 qts. 2 gi. 
 
 7. 22 r. 1 yd. 5 in. 
 
 11. 13T.12c.lqr. 
 
 14. 37 bu. 3 pks. 
 
 8. 26 cwt. 3 qrs. 5 
 
 10 1. 9 o. 13 d. 
 
 3 qts. 
 
 Ibs. 5 oz. 
 
 12. 27hhds. 38 g. 
 
 15. 67 y. 3 q. 2 na. 
 
 COMPOUND SUBTRACTION. 
 
 2. 7, Is. 9d. 2 far. 
 
 6. 7 yds. 3 qrs. 1 n. 
 
 12. 4 A. 2 roods, 4 
 
 3. 3 Ibs. 7 oz. 4 
 
 7. 9, 17s. 4d. 
 
 rods. 
 
 pwts. 8 grs. 
 
 8. 44 gals. 1 pt. 
 
 13. 8 lea. 2 mi. 
 
 4. 4 T. 17 cwt. 1 
 
 9. 4T. 16 c. 74 Ib. 
 
 fur. 4 r. 
 
 qr. 24 Ibs. 
 
 10. 2y. 3 mo. 16 d. 
 
 14. 17 bu. 5 q. 2 p. 
 
 5. 5 m. 5 fur. 7 r. 
 
 11. 15 y. 10 mo. 3 
 
 15. 45 G. 18s. 8d 
 
 3 ft. 9 in. 
 
 d. 8 h. 4 m. 
 
 2 far. 
 
 COMPOUND MULTIPLICATION. 
 
 1. Given. 
 
 5. 661. 285 r. 11 f. 
 
 9. 625 y. 2 q. 2 n 
 
 2. 64, 13s. 5d. 1 
 
 3 i. 
 
 10. 2173 d. 13 h. 3 
 
 farthing. 
 
 6. 199 T. 14 c. 14 
 
 m. 47 s. 
 
 3. 236 1.5 o. 11 p. 
 
 1. 15 o. 1 I. 
 
 11. 2272 y. 30 w 
 
 12 g. 
 
 7. 43A.16r.84f f. 
 
 3 d. 12 h. 
 
 4, 34 mi. 2 f. 20 r. 
 
 8. 10 cords, 61 c. f. 
 
 12. 4707 h. 18 g. 
 
128 
 
 ANSWERS. [PAGES 113 118. 
 COMPOUND DIVISION. 
 
 Ex. A us. 
 
 Ex. Ans. 
 
 Ex. Ans. 
 
 1. Given. 
 2. 4 1. 9 oz. 15 p. 
 11 g. 
 3, 1, 9s. 5d. f. 
 4 51. 13 p. 15f g. 
 5. 2 T. 12 c. 2 q. 
 14*1 
 
 6. 3 y. 1 q. If na. 
 7. 4 1. 1 in. 2 f. 
 I7i r. 
 8. 5h.2g.4q.0-fp. 
 9. 3 b. 3 p. 3f q. 
 10. 5, 3s. 6d. 2 f. 
 11.2 T.5 c. 3-H-l- 
 
 12. 1 hhd. 49 gals. 
 3|f qts. 
 13. 1 A. 2 rx>ds, 
 l**r. 
 14. 91 c. f. OH i- 
 15. 1 yr. 12 wks, 
 4 d. 
 
 MISCELLANEOUS EXERCISES. 
 
 1. 271557. 
 
 23. 460|f. 
 
 51. $279. 
 
 76. 150 ap. 
 
 2. 1229930. 
 
 24. 17-ftV 
 
 52. $1180. 
 
 77. 112 mar. 
 
 3. 875431. 
 
 25. 18-ftV 
 
 53. 3399 s. 
 
 78. 192 par. 
 
 4. 372713- 
 
 26. 12HH- 
 
 54. $90. 
 
 79. 371 fans. 
 
 124. 
 
 27. 43i+H- 
 
 55. 87 s. 
 
 80. 25 2| yds. 
 
 5. 837598- 
 
 28. 74V L ( /V i 8V 
 
 56. 13-f-g yds. 
 
 81. 512 pair. 
 
 35. 
 
 29. 261. 
 
 57. 13-/g- yds. 
 
 82. 10962d. 
 
 6. 24064H- 
 
 30. 568280. 
 
 58. 33-ft- a. 
 
 83. 60850 f. 
 
 7. 1593i. 
 
 33. $19. 
 
 59. 20 cows. 
 
 84. 302 yds. 
 
 8. 245948, 
 
 34. $13. 
 
 60. 25 bar. 
 
 85. 4000. 
 
 616 rem. 
 
 35. $17. 
 
 61. 29 or. 
 
 86. 1890 s. 
 
 9. 4-f* 
 
 36. $49. 
 
 62. 64 quarts. 
 
 87. $20. 
 
 10. 6800. 
 
 37. 86 Ibs. 
 
 63. 60 Ibs. 
 
 88. 7560 bot 
 
 11. 98040. 
 
 38. 63 s. 
 
 64. 75 c. 
 
 89. $40.80. 
 
 12. 53f. 
 
 39. 356 Ibs. 
 
 65. $108. 
 
 90. $124.16 
 
 13. 108332. 
 
 40. $28. 
 
 66, $7.92. 
 
 91. $1100. 
 
 14. 11542. 
 
 41. 300 m. 
 
 67. $252. 
 
 92. $442. 
 
 15. 33611. 
 
 42. $19. 
 
 68. 936 m. 
 
 93. $3180. 
 
 16. 26869. 
 
 43. $18. 
 
 69. 45 cents. 
 
 94. $9600. 
 
 17. 3810225. 
 
 44. $75. 
 
 70. $30. 
 
 95- $17280. 
 
 18. 12469. 
 
 45. 106 sk 
 
 71. $140. 
 
 96. $1981980. 
 
 19. 2720325. 
 
 46. $837. 
 
 72. $150. 
 
 97. $960. 
 
 20. 17. 
 
 47. $2008. 
 
 73. 12 dress's. 
 
 98. $130.56. 
 
 21. 5 and 9 r. 
 
 48. $1744. 
 
 74. 16 boxes. 
 
 99. 52 y. 6 m. 
 
 22. 18 and 
 
 49. 8114. 
 
 75. 110 miles. 
 
 1^0. 4 weeks, 
 
 23. 424 over. 
 
 50.:$165. 
 
 
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