THE 
 
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 f COLUMBIA]^ CALCULATOR; 
 
 A PRACTICAL AND COxNCISE SYSTEM 
 
 DECIMAL ARITHMETIC. 
 
 VHE USE OF SCHOOLS IN THE UNITED STATES, 
 
 BY ALMOIN TiO" NOR, 
 
 r -'lUhA. -ouNTANT's .^^sts-^'ant;' -math-^:^''- 
 
 T A r; r K : ' 7' T C 
 
 "Such beiug ino nature of Federal money, its operations can in no way be so ; 
 well understood as in obtaining a good knowledge of Decimals, and applying ! 
 Ll>eir I'everul rules to th\! various cases of money matters." — Prof. Dewey. 
 
 Fourr^ '< E[K ;mo^- ri; ^'* 
 
 /ND CORRECTED. 
 
 PHILADELPHIA: 
 K AY & TRO UTM A N, 
 
 I^ANIELS & SMITl , AND W. A. 1>EARV & CO. PITTSBURGH : ELLIOTT 
 & ExVGLlSH. Ne\/ YORK: J. S. REDFIELDj HUNTING JON^ 'AVAGE,. 
 AND GATF/i & STEDMAN. ALBANY: E. H. PEASfe. 6^ ^0. '^iUFFALO .* 
 O. Kr. STEELE. CINCINNATI : C. D. TRC^MAN. RICHMOND, VA.". J. 
 W. RANDOLPH ^ CO, ; ''■ : HALL & DICKSON. 
 
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( 
 
 THE 
 
 COLUMBIAI CALCULATOR 
 
 BEING 
 
 A PRACTICAL AND CONCISE STSTEIM 
 
 f 
 
 DECIMAL ARITHMETIC. 
 
 APAPTED TO 
 
 THE USE OF SCHOOLS IN THE UNITED STATES. 
 
 BY ALMON TICKNOR, 
 
 AUTHOR OF "THE accountant's ASSISTANT," " MATHEMATICAL TABLES," ETC. 
 
 " Such being the nature of Federal money, its operations can in no way be so 
 well understood as in obtaining a good knowledge of Decimals, and applying 
 their several rules to the various cases of money matters." — Prof. Dewey. 
 
 FOURTH EDITION. REVISED AND CORHECTED. 
 
 PHILADELPHIA; 
 KAY & T R O U T M A N , 
 
 DANIELS&SMITH, ANDW.A.LEARY AND CO. PITTSBURGH: ELLIOTT 
 <fc KNGLISH. NEW YORK! J. S. REDFIELD, HUNTINCDON i^J SAVAGE. 
 AND GATES & STEDMAN. ALBANY; E. H. PEASE & CO. BUFFALO: 
 O.G.STEELE. CINCINNATI: E. D. TRUMAN. RICHMOND, VA. : J. 
 W. RANDOLPH & CO. SYRACUSE: HALL & DICKSON. 
 
RECOMMENDATIONS. 
 
 From W. McCartney, Esq,, Professor of Mathematics, Laf ay » 
 ette College. 
 
 ,, rp, E ASTON, January 7, 1845. 
 
 Mr. Ticknor : — 
 
 Dear Sir : I have looked over some of the proof-sheets of 
 your treatise on Arithmetic, and am pleased to observe that you 
 have introduced many practical examples in illustration of the 
 rules. Your book is well adapted to those who desire a prac- 
 tical work on the subject, and is full in details and illustrations 
 for those who are commencing the study of this science. Prac- 
 tical books are the kind adapted to the business transactions of 
 the acre. Very truly, yours, &c. 
 
 w. McCartney. 
 
 From D. P. Yeomans, Professor of Chemistry and Principal of 
 the Model School in Lafayette College. 
 
 Easton, February 1, 1845. 
 I have examined the work entitled the " Columbian Calcu- 
 lator," by Mr. Ticknor, and deem it well adapted as a work on 
 practical Arithmetic for use in common schools. The numerous 
 examples employed to illustrate principles, will render it, in the 
 hands of competent instructors, peculiarly valuable, to both the 
 student and the man of business. D. P. YEOMANS. 
 
 From N. Olmstead, Teacher of a Public School in Easton, Pa, 
 
 Mr. Ticknor :- ^*^^°^' February, 1845. 
 
 Dear Sir : From a pretty thorough examination of your sys^ 
 tem of Arithmetic, I can say without hesitation, that in my 
 opinion it is decidedly superior, for the use of common schools, 
 la any now in use. The currency of our country, in every sys 
 tem of Arithmetic, should be of paramount importance ; and ir 
 this respect, I think your system may challenge competition. 
 Yours, &c. NICHOLAS OLMSTEAD. 
 
 Entered, according to Act of Congress, in the year 1846. by Almon Ticknor, 
 m the Clerk's Office of the District Court of the United States for the Eastern 
 District of Pennsylvania. 
 
 STEREOTYPED BY REDFIELti & SAVAGE, NEW YORK. 
 
REMARKS. 
 
 As the first edition of this work was favorably received by 
 the public, notwithstanding its deficiency and imperfections, it 
 is considered a sufiicient inducement to enlarge and improve 
 the present volume, and present it to those for whom it is in- 
 tended, in a more perfect and acceptable form, by the introduc- 
 tion of a variety of rules, examples, questions, and reviews, 
 arranged and explained in a manner that the author feels assured 
 will meet the general approbation of the competent and unprej- 
 udiced teacher. The approbation of a majority of popular 
 teachers and other persons of the first respectability, in relation 
 to the arrangement and selection of suitable and instructive 
 questions, is satisfactory evidence that the utility and merits of 
 the work have been duly appreciated, when it is considered that 
 the country is already abundantly supplied with Arithmetics, 
 which have been adopted, and claim precedence and superiority 
 over other treatises of this description. The author is now 
 more fully confirmed in his belief than ever, that this system is 
 the only true and correct method of communicating direct and 
 correct instruction in the science of numbers ; and the time is 
 not far distant when this opinion will be unanimous throughout 
 the country, that a systematic arrangement of the rules, with a 
 thorough knowledge, both theoretically and practically, of their 
 operations, together with a variety of well-selected, practical, 
 rational questions for solution, is of the first importance, and 
 absolutely necessary, in order to acquire an exact and expe- 
 ditious method of calculation. 
 
 As this edition embraces about 2,200 examples, or questions 
 for solution, the belief is entertained that this number will be as 
 many as are required or will prove beneficial for the exercise 
 of the pupil, at the same time having in view the expense of the 
 work, which it is desirable should be placed within the means 
 of every one, but more particularly those who are the least able 
 to procure works of this kind, who are the most in need, and 
 who will eventually receive the most benefit, and they are the 
 persons who must, and will, finally preserve and perpetuate the 
 institutions and liberties of our country. The greatest care and 
 diligence have been exerted to introduce such questions as 
 would be within the range of possibility, and likely to occur in 
 ihe course of business, and in adapting the language and phrase 
 
REMARKS. 
 
 ology to the occasion, and capable of being correctly understood 
 by those for whom it is intended. 
 
 Comparative Arithmetic^ Analogy ('* resemblance between 
 things with regard to circumstances or effects,") the inductive 
 'principle^ persuasive or mejital Arithmetic — these terms are 
 nearly synonymous, and may with propriety be so considered. 
 The idea that mathematics can be taught to any extent by 
 analogy f induction, or degrees of comparison, without rules 
 and actual calculation, merely by the effort or operation of 
 the mind, without the use of figures, is unquestionably 
 absurd and inconsistent in the extreme ; and, strange it may 
 appear, some authors and teachers advocate the principle, 
 and call it precocity or march of intellect ; the system is neither 
 founded in common sense, nor on philosophical or mathematical 
 principles, for it is self-evident that every effect must be pro- 
 duced by some corresponding or sufficient cause : hence arises 
 the necessity of rules or data, on which to found a system of 
 calculations, which must operate with so much certainty that 
 they shall self-evidently prove their own theory and correctness, 
 or which carry their own evidence with them — the eternal prin- 
 ciple of truth ; this can only be done by actual calculation — by 
 induction never. Can the surveyor calculate the different 
 angles of his survey by analogy, and without the use of decimal 
 tables, and the application of rules, prepared for that express 
 purpose ? Can the navigator take solar and lunar observations, 
 and calculate the places, southing, and setting of the stars, on the 
 inductive principle ? Can the astronomer from analogy (induc- 
 tion), and without figures, trace the abstruse paths or parabolic 
 orbits of the comets, and calculate the powers of attraction and 
 repulsion, the distances and motions of the planetary system ? 
 Does not every mathematician know, that the sciences mentioned 
 above require the most intense study, and the application of 
 certain unerring rules ? 
 
 A variety of questions and examples have been given in 
 square and cubical measure, mensuration, etc., which will be 
 found highly useful and entertaining to the pupil, and in num- 
 bers sufficient for the usual occurrences of practical business life. 
 
 A free use of the black-board in school is earnestly recom- 
 mended, to make the ready calculator. 
 j! To those who have used or recommended the preceding 
 ( edition of this work, the author would embrace the present op- 
 portunity of tendering his sincere acknowledgments, for the 
 kind and obliging manner uniformly manifested in his behalf. 
 
 A Key to this work is now published. 
 
A SHORT INTRODUCTION TO AMERICAN CURRENCY 
 
 When this country was subject to the crown of England, we 
 were governed by their laws, and adopted their method of reck- 
 oning money, in pounds, shillings, and pence, which differed in 
 value in the several colonies, as they were then called, but Lre 
 now called states ; neither was it of the same value of the 
 currency of England. The pound currency of England was 
 once equal to a pound avoirdupois, which was many times its 
 present value. Under the colonial government, the several 
 states issued bills of credit to supply the want of specie, and 
 to answer a medium of trade ; but as these bills were not re- 
 ceived by the British merchants in payment for goods at their 
 par value, holders of the bills had to pay more. Thus, in New 
 York they had to pay in the bills of the state at the rate of 8 
 shillings for 4 shillings and 6 pence sterling, and so in propor- 
 tion to the depreciation of the bills of the other states. Taking 
 4 shillings 6 pence sterling as the value of a dollar, the currency 
 of the New England states was 6 shillings to the dollar : or, 6 
 shillings New England currency was worth 4 shillings and 6 
 pence English currency ; New York and North Carolina, 8 
 shillings to a dollar, or equal to 4 shillings and 6 pence Eng- 
 lish ; Virginia, Kentucky, Tennessee, and Ohio, 6 shillings ; 
 New Jersey, Pennsylvania, Delaware, and Maryland, 7 shillings 
 and 6 pence ; South Carolina and Georgia, 4 shillings and 8 
 pence ; (Canada and Nova Scotia, 5 shillings). This state of 
 the currency was attended with much inconvenience, and is at 
 the present day, but is fast going into disuse, much to the relief 
 of the community. After the war of the revolution we became a 
 separate and independent nation, framed our own laws, coined 
 money, fixed the value of currency, &c. Our coins are of gold, 
 silver, and copper, and their weight and value so proportioned 
 as to increase from the lowest to the highest by tens, or in a 
 tenfold proportion ; that is, ten of every lower denomination, or 
 less value, make one of the next higher, and consequently one 
 of every higher makes ten of the next lower. Thus we say, 
 10 mills make, or are equal to 1 cent in value; 10 cents are 
 equal to 1 dime; 10 dimes are equal to 1 dollar; and 10 dol- 
 lars are, equal to 1 eagle. There are also the half cent, half 
 dime, quarter and half dollar, quarter and half eagle ; but in. 
 
 1* 
 
A SHORT IXTIIODUCTION TO AMHRICAX CUI^REXCY. 
 
 reckoning money, ir is customary to use dollars, cents, and 
 mills, omitting the other denominations. In writing numbers in 
 dollars, cents, and mills, we leave a small space between them, 
 or separate them with a comma (,) : — 
 
 V' 25 '"5''" five dollars twenty-five cents and five mills ; 
 dollars 5, 25, 5 ; or dollars 5, 2, 5, 5 — five dollars two dimes 
 five cents and five mills. 
 
 The rule for adding several sums together in dollars, cents 
 and mills, is this : — add all the mills, and carry one for every 
 ten, and add this one to the next figure in the place of cents ; 
 but if it is less than ten, set it down directly under the column 
 of mills, then add the cents in the first column, and carry one 
 for every ten to the next column of cents, which add, and carry 
 one for every ten, as before, to the place of dollars ; add the dol- 
 lars and set down the full amount ; for even tens write a cipher 
 (0). If A owes you dollars 5, 25, 5, and B dollars 8, 82, 9, 
 how much do both owe you? A 5, 25, 5 
 
 B 8, 82, 9 
 
 14, 08, 4 
 
 In this example, we say 9 and 5 are 14, set down 4, and 
 carry 1 to the next figure 2, which will make 3, and 5 are 8 ; 
 set down the 8, and nothing to carry, because it is less than 10 ; 
 then 8 and 2 are 10, set down a cipher, because it is even (10), 
 and carry 1 to the next figure 8, will make 9, and 5 are 14 ; set 
 down all the last column, so that the two sums owing by A and 
 B, when added together, will make dollars 14, 08, 4 — fourteen 
 dollars, eight cents, and four mills. 
 
 Note. — No scholar should be permitted the use of an arith- 
 metic until he has been exercised in the first four rules, both 
 mentally and on the hoard ; every school should have several 
 boards, one for each class, and not let a day pass without their 
 use, by having the teacher give examples first, then the pupils, 
 beginning with the first, and continue through the class, without 
 the book ; prove their questions, and give their reasons : this 
 will exercise the mind, and bring forward all the faculties ; in 
 this way, there will be more knowledge acquired in one month 
 than there can be in ten in conning over a book, which is a 
 blank to them, and but little better to their teachers. It M^ill 
 attract attention and excite emulation, more than any other 
 course, which is the mainspring in the acquisition of knowledge 
 
 ' — STRICT ATTENTION. 
 
TABLE OF CONTENTS. 
 
 V PAOB. 
 
 Simple Numbers, from - - - - - -9 to 48 
 
 Decimal Fractions 43 to 59 
 
 Tables of Weights and Measures ----- 60 
 
 Simple Reduction 66 
 
 Practical Questions 70 
 
 Compound Numbers 72 to 87 
 
 Reduction of Decimals --*-.-- 88 
 
 Proportion ---------- 101 
 
 Vulgar Fractions 114 to 135 
 
 Practice 136 
 
 Simple Interest -- 140 
 
 Coins, Currency, &c. - - - 159 
 
 Discount ---------- 162 
 
 Equation 166 
 
 Barter 168 
 
 Profit and Loss 172 
 
 Partnership 177 
 
 Taxing 179 
 
 Percentage, do. on Lands - - - - - - -183 
 
 Gross, Tare, &c. --.- 185 
 
 United States Duties 187 
 
 Position 188 
 
 Involution --*----•-- 194 
 
 Evolution 195 
 
 Square Root .-- 195 
 
 Cube Root 202 
 
 Allegation 208 
 
 Progression ------- - 212 
 
 Permutation 219 
 
 Combination --------- 220 
 
 Compound Interest -------- 221 
 
 Annuities ----- 225 
 
 Duodecimals ---- 23C 
 
 Appendix ---,------ 235 
 
 Artificers' Work 235 to 240 
 
 Mensuration ---- 240 
 
 Miscellaneous Matter in Mensuration ----- 249 
 
 Promiscuous Questions ------- 258 
 
 Black-board 363 
 
EXPLANATION OF CHARACTERS. 
 
 Signs. Significations. 
 
 = Equal ; as, 10 mills = 1 cent, 10 cents = 1 dimCj 10 dimes = 1 
 dollar, 10 dollars = 1 eagle; and when placed between two 
 numbers it denotes that they are equal to each other. 
 
 + More, or addition; as, 6+2 = 8, 3-f-2 = 5; when the num- 
 bers are small, we can say 6 and 2 are 8, 3 and 2 are 5. 
 This sign is sometimes placed at the right of the quotient, or 
 answer, denoting that there is a small remainder. It is also 
 called plus, meaning more. 
 
 — Less (minus), or subtraction ; when placed between two num- 
 bers, it denotes that the one on the right is to be taken from 
 the one on the left; thus, 5 — 3 = 2 denotes that 3 is to be 
 taken from 5, will =2, or 2 over, which is the remainder ; 5 
 is the minuend, and 3 the subtrahend ; when the numbers are 
 small, you can say 3 from 5 leaves 2. It is sometimes used 
 in division, signifying a remainder, as 2 in 5, 2 — 1 remain- 
 der ; 2 in 5 twice and 1 over. 
 
 X Into, or multiplied by, sign of multiplication ; thus, 4x4=16, 
 
 5 X 5=125, denotes when placed between two numbers that 
 they are to be multiplied together ; 6 X 6 = 36, that is, 6 times 
 
 6 are 36. X is sometimes used. 
 
 •r Division, or divided by; as, 25-^-5 = 5, or 8)64(8, or ^8^ = 8, 
 meaning that 25 divided by 5, the answer will be 5, : 64 
 divided by 8 = 8 answer : again, ^-^ denotes that 64 is divided, 
 8 : and 8)^3^ signifies that 64 is divided by 8. E. eagle, D. 
 or $, dollar, d. dime, c. or cts. cents, m. mills. 
 
 f Period, or decimal point, separatrix, used to distinguish deci- 
 mals from whole numbers, as those on the right are decimals, 
 and on the left (if any) are whole numbers ; thus, .25 ; 46 
 .24 : it is used to separate D. c. m. ; the comma (,) is also 
 used for this purpose. 
 
 : :: : Proportion ; as, 2 : 4 :: 8 : 16, that is, as 2 is to 4, so is 8 
 to 16 ; or the same proportion that 4 is to 2, so is 16 to 8. 
 
 12 — 3 + 5=4, a vinculum; the line over the 3 and 5 connects, 
 all the numbers over which it is drawn as simple numbers, 
 as 3 and 5 are 8, from 12=4. 
 
 -/ or 2/ Square Root; as, ^64 = 8: denotes that the square 
 root of 64 is 8. 
 
 y Cube Root; as, ^64=4 : ^ Biquadrate Root, as ^64=2. 
 Dollars, dimes, and mills increase, and are calculated the 
 
 same as whole numbers. 
 
THE COLUMBIAN CALCULATOR. 
 
 ARITHMETIC. 
 
 Arithmetic is a part of the science of Mathematics, and ia 
 the art of computing by numbers, by the operation of six rules, 
 namely, Notation, Numeration, Addition, Subtraction, Multipli- 
 cation, and Division ; two of which may be considered primary 
 rules, namely, Addition and Subtraction, and the other four sec- 
 ondary, as they naturally arise from the operation of the former. 
 The knowledge of this science is so universally necessary, that 
 scarcely anything in life, and nothing in trade, can bo done 
 without it. 
 
 NOTATION. 
 
 Notation teaches to express words, or numbers, by ten Arabic 
 characters, or digits, namely, 1, one ; 2, two ; 3, three ; 4, four ; 
 5, five ; 6, six ; 7, seven ; 8, eight ; 9, nine ; 0, cipher ; by the 
 use of which, all numbers are expressed, and increase in value 
 from right to left, in a tenfold proportion ; thus, 1824, the figure 
 (4) in the place of units, denotes only its simple value, 4 ; that 
 in the second place, or place of tens (2), is ten times its simple 
 value, 24 ; that in the third place, or place of hundreds (8), one 
 hundred times its simple value, 824 ; that in the fourth place, or 
 place of thousands (1), one thousand times its simple value, or 
 1824, one thousand, eight hundred and twenty-four. 
 
 A cipher when alone is of no value, but when placed to the 
 right of a figure, increases the value of that figure in a tenfold 
 proportion ; thus, 5 alone is only five ; but annex a cipher to the 
 tight, thus, 50, and it increases the value tenfold, or fifty. 
 
10 
 
 NOTATION. 
 
 EXAMPLES. 
 
 Write the following numbers in figures : twenty-five ; seven* 
 ty-six ; ninety-one ; eighty-four ; sixty-five , nineteen ; eleven ; 
 one hundred ; one hundred and sixty-seven ; two hundred 
 four hundred ; one thousand ; two thousand, nine hundred and 
 ninety-nine. 
 
 In the Roman method of notation by letters, I represents 
 one ; V, five ; X, ten ; L, fifty ; C, one hundred ; D, five hun- 
 dred ; M, one thousand, &c. 
 
 As often as any letter is repeated, so many times its value is 
 repeated, unless it be a letter representing a less number, placed 
 before one representing a greater^ then the less number is taken 
 from the greater ; thus, IV represents four, IX, nine, &;c., as 
 will be seen in the following table, which the pupil should com- 
 mit to memory. 
 
 ROMAN METHOD, OR NOTATION BY LETTERS. 
 
 One 
 
 I Ninety 
 
 XC 
 
 Two 
 
 II One hundred 
 
 c 
 
 Three 
 
 III Two hundred 
 
 cc 
 
 Four 
 
 IIII, or IV Three hundred 
 
 COG 
 
 Five 
 
 V Four hundred 
 
 CCCC 
 
 Six 
 
 VI Five hundred 
 
 D, or io» 
 
 Seven 
 
 VII Six hundred 
 
 DC 
 
 Eight 
 
 VIII Seven hundred 
 
 DCC 
 
 Nine 
 
 Villi, or IX Eight hundred 
 
 DCCC 
 
 Ten 
 
 X Nine hundred 
 
 DCCCC 
 
 Twenty 
 
 XX One thousand 
 
 M, or Clot 
 
 Thirty 
 
 XXX Five thousand 
 
 100. or ^X 
 
 Forty 
 
 XL Ten thousand 
 
 CCIqo, or X 
 
 Fifty 
 
 L Fifty thousand 
 
 1000 
 
 Sixty 
 
 LX Hundred thousand 
 
 CCCIqoo, or C 
 
 Seventy 
 
 LXX One million 
 
 M 
 
 Eighty 
 
 LXXX Two millions 
 
 mm: 
 
 MDCCCXLIV = A. D. 1844 
 
 • l3 is used instead of D, to represent five hundred, and for every ad- 
 ditional (3 annexed at the right, the number is increased ten times. 
 
 t CI J is used to represent one thousand, and for every C and q put at 
 each end, the number is increased ten times. 
 
 X A line over any number increases Its value one thousand times. 
 
NUMERATION. 
 
 11 
 
 NUMERATION. 
 
 By Numeration we are taught to read any number of figures, 
 and ascertain their relative value, when taken in connexion with 
 each other, which is determined by the situation in which they 
 are placed, and more correctly and perfectly exemplified by the 
 following tables : — 
 
 
 # 
 
 
 1 
 
 TABLE 
 
 I. 
 
 
 CO 
 
 
 
 
 
 
 
 S 
 
 
 
 
 
 
 
 o 
 
 
 
 CO 
 
 
 
 
 
 
 
 
 
 CQ 
 
 
 
 ^ 
 
 . 
 
 
 o 
 
 '^ 
 
 
 
 *a 
 
 CO 
 
 g 
 
 
 rg 
 
 q 
 
 rt 
 
 CQ 
 
 . 
 
 
 o 
 
 
 
 CO 
 
 t3 
 
 CO 
 
 r^ 
 
 
 CO 
 
 T3 
 
 p 
 
 fl 
 
 cd 
 
 t3 
 
 <o 
 
 r3 
 
 ^ 
 
 ? 
 
 o 
 
 2 
 
 •5 
 
 s 
 
 o 
 
 V4 
 
 -5 
 
 1 
 
 
 M 
 
 fl 
 
 s 
 
 3 
 
 1 
 
 fl 
 
 3 
 
 i 
 
 rC3 
 
 9 
 
 8 
 
 7 
 
 6 
 
 5 
 
 TABLE 
 
 4 
 11 
 
 3 
 
 1 one. 
 12 twelve, 
 123 1 hundred 23. 
 1234 1 thousand 2 hundred and 34. 
 12345 12 thousand 3 hundred and 45. 
 123456 123 thousand 4 hundred and 56, 
 1234567 1 million 234 thousand 5 hundred and 67. 
 12345678 12 millions 345 thousand 6 hundred and 78. 
 123456789 123 millions 456 thousand 7 hundred and 89 
 
 To enumerate figures where the numbers are large, it will be 
 found convenient to divide, or separate them, into periods of 
 three figures, the first being hundreds, the second, hundreds of 
 thousands, &c. Then begin at the right hand, or place of units, 
 and read toward the left, as in table 1st; thus 1 is in the place 
 of units, 2 is in the place of tens, and 3 in the place ef hun- 
 dreds, which three figures, taken in connexion, would express 
 (321) three hundred and twenty one. 
 
 In table 2d, the numbers may be enumerated in the same 
 way, and then read from left to right, in the order in which they 
 stand. 
 
12 NUMERATION. 
 
 TABLE III. 
 
 1 units, 
 12 tens. 
 123 hundreds. 
 1234 thousands. 
 12345 tens of thousands. 
 123456 hundreds ;/ thousands. 
 1234567 millions. 
 12345678 tens of millions. 
 123456789 hundreds of millions. 
 1234567891 thousands of millions. 
 12345678912 tens of thousands of millions. 
 1*23456789123 hundreds of thousands of millions. 
 1234567891234 billions. 
 12345678912345 tens of billions. 
 123456789123456 hundreds of billions. 
 1234567891234567 thousands of billions. 
 12345678912345678 tens of thousands of billions. 
 } 23456789123456789 hundreds of thousands of billions. 
 
 Quintillions. Quadrillions. Trillions. 
 555,555 555,555 555,555 
 
 Nonillions. Octillions. Septillions. Sectillions. 
 555,555 555,555 555,555 555,555 
 
 The third table is given to express the higher powers, and 
 «an be read and applied in the same manner as the two prece- 
 ding tables. When even hundreds, thousands, &c., are to be 
 written, the vacant places of units, tens, hundreds, &;c., are to 
 be supplied by ciphers ; thus, to write one thousand (1000), 
 place three ciphers to the right of the 1, and so in all cases of 
 a similar nature. 
 
 EXAMPLieg. 
 
 Write down in words, the following numbers : 10, 15, 20, 25, 
 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 150, 
 476, 891, 999, 1001. Let the following numbers be expressed 
 in figures : twenty, thirty, forty-five, seventy-six, eighty-three, 
 ninety-one, one hundred and one, one hundred and nine, one 
 hundred and fifteen, one hundred and thirty, one hundred and 
 ninety-four, two hundred and sixty, three hundred and forty-five, 
 five hundred and eighty-five, six hundred and ninety-one, on© 
 hundred and fifty thousand. 
 
NUMERATION. 13 
 
 The preceding is the English method of Numeration ; the 
 following is the French. The teacher can use either at his 
 pleasure : 
 
 r " > 
 
 Quadrillions. Trillions. Billions. Millions. Thousands. Units. 
 321 321 321 321 321 321 
 
 What is Arithmetic? By what means are operations i 
 Arithmetic performed ? — Name "them. What is Notation : 
 What is Numeration ? How must figures be enumerated ? In 
 what manner should figures be read ? Why do you enumerate 
 from right to left? In what proportion do they increase in 
 value ? Recite the Numeration Table. Write down seventeen 
 millions : — seventeen hundred thousand. — Eleven billions 
 and nine hundred thousand. 
 
 SIMPLE ADDITION. 
 
 Addition is the first primary rule in Arithmetic, the use of 
 which is to ascertain the amount or sum total of two or more 
 numbers, when put or added together ; as, 5 + 5 ==10: that is 
 5 and 5 make 10. 
 
 RULE. 
 
 Set the given numbers under each other, with units under 
 units, tens under tens, hundreds under hundreds, &:c. Then 
 draw a line under the lowest number, and begin at the right 
 hand column, or place of units, and add (upward) all the column 
 together ; set down the sum when less than ten ; if ten or more, 
 set down the right hand figure, and add the left hand figure to 
 the next column ; and thus proceed to the last column, and set 
 down the wholes amount of it. 
 
 PROOF. 
 
 Perform the operation a second time, agreeably to the Rule ; 
 but in this case, begin at the top ; or reserve one of the given 
 numbers, find the sum of the rest, and thereto add the number 
 reserved. 
 
 Note. — The reason why you carry one for every ten, is this • 
 in the place of units, it requires ten to make one in the place 
 of tens ; and in the place of tens, it requires ten to make one 
 in the place of hundreds ; therefore, you always carry one from 
 one denomination to another, as it requires of that denomination 
 to make one in the next. 
 
 2 
 
SIMPLE ADDITION. 
 
 ADDITION TABLE. 
 
 2 and 2 
 
 are 4 
 
 3 and 9 
 
 are 12 
 
 5 and 9 are 14 
 
 8 and 8 are 16 
 
 2 3 
 
 5 
 
 3 
 
 10 
 
 13 
 
 5 
 
 10 
 
 15 
 
 8 
 
 9 
 
 17 
 
 2 4 
 
 6 
 
 3 
 
 11 
 
 14 
 
 5 
 
 11 
 
 16 
 
 8 
 
 10 
 
 18 
 
 2 5 
 
 7 
 
 3 
 
 12 
 
 15 
 
 5 
 
 12 
 
 17 
 
 8 
 
 11 
 
 19 
 
 2 6 
 
 8 
 
 4 
 
 4 
 
 8 
 
 6 
 
 6 
 
 12 
 
 8 
 
 12 
 
 20 
 
 2 7 
 
 9 
 
 4 
 
 5 
 
 9 
 
 6 
 
 7 
 
 13 
 
 9 
 
 9 
 
 18 
 
 2 8 
 
 10 
 
 4 
 
 6 
 
 10 
 
 6 
 
 8 
 
 14 
 
 9 
 
 10 
 
 19 
 
 2 9 
 
 11 
 
 4 
 
 7 
 
 11 
 
 6 
 
 9 
 
 15 
 
 9 
 
 11 
 
 20 
 
 2 10 
 
 12 
 
 4 
 
 8 
 
 12 
 
 6 
 
 10 
 
 16 
 
 9 
 
 12 
 
 21 
 
 2 11 
 
 13 
 
 4 
 
 9 
 
 13 
 
 6 
 
 11 
 
 17 
 
 10 
 
 10 
 
 20 
 
 2 12 
 
 14 
 
 4 
 
 10 
 
 14 
 
 6 
 
 32 
 
 18 
 
 10 
 
 11 
 
 21 
 
 3 3 
 
 6 
 
 4 
 
 11 
 
 15 
 
 7 
 
 7 
 
 14 
 
 10 
 
 12 
 
 22 
 
 3 4 
 
 7 
 
 4 
 
 12 
 
 16 
 
 7 
 
 8 
 
 15 
 
 11 
 
 11 
 
 22 
 
 3 5 
 
 8 
 
 5 
 
 5 
 
 10 
 
 7 
 
 9 
 
 16 
 
 n 
 
 12 
 
 23 
 
 3 6 
 
 9 
 
 5 
 
 6 
 
 11 
 
 7 
 
 10 
 
 17 
 
 12 
 
 12 
 
 24 
 
 3 7 
 
 10 
 
 5 
 
 7 
 
 12 
 
 7 
 
 11 
 
 18 
 
 13 
 
 13 
 
 26 
 
 3 8 
 
 11 
 
 5 
 
 8 
 
 13 
 
 7 
 
 12 
 
 19 
 
 14 
 
 14 
 
 28 
 
 To read t 
 
 he t£ 
 
 ible, 
 
 say 2 . 
 
 md 2 
 
 are 4 
 
 3 and 3 
 
 are 6, 
 
 &c. 
 
 All of the tables, and rules generally, should be committed to 
 memory, before the pupil attempts the solution of the questions. 
 
 QUESTIONS. 
 
 2463 Explanation. — Begin by saying, 1 and 3 are 
 
 4, and 2 are 6, and 3 are 9 ; set it down. Then, 
 
 4532 4 and 7 are 11, and 3 are 14, and 6 are 20 ; set 
 
 0773 down 0, and carry 2 to 8, which will make 10, 
 
 9841 and 7 are 17, and 5 are 22, and 4 are 26 ; set 
 
 down the 6, and carry 2 to the next figure, 9, 
 
 17609 amount, which will make 11, and is nothing, but 4 are 
 
 15, and 2 are 17; always set down the whole 
 
 15146 amount of the last column. Then, to obtain the 
 
 proof, draw a line under the numbers at the top, 
 
 17609 proof. and add the remaining numbers as before, omit- 
 ting the numbers at the top, and you find this 
 
 sum to be 15146 ; this last sum is now added to the numbers 
 at the top, which agrees with the first addition ; hence it is sup- 
 posed to be correct. Note. — A question may be proven accord- 
 ing to rule, and be incorrect, because the same error may be 
 committed in the proof that was in the first calculation. 
 
 1. Henry has 7 apples, and George will give liim 11 ; how 
 many will he then have ? 
 
SIMPLE ADDITION. 
 
 15 
 
 2. Thomas has 17 cents, John 19, William 14; how many 
 in all? 
 
 3. Samuel has 35 cents, he had lost 6 ; how many had he at 
 first ? . 
 
 4. How many are 17, 18, 19, 20, and 21 ? How many are 
 52,76, 83, 82, 2, 1, and 17? 
 
 5. William paid for a knife 31 cents, pocket-book 25 cents, 
 %late 18 cents, book 42 cents, paper 7 cents ; what did they 
 ail cost ? 
 
 6. A farmer has 19 cows, 157 sheep, and 84 calves; how 
 rjany in all ? 
 
 8 
 
 9 
 
 10 
 
 11 
 
 12 
 
 
 to 
 
 J3 
 
 3 § 
 
 
 CO w 
 
 hun'ds, 
 
 tens. 
 
 units. 
 
 Ij 
 
 «3 n 
 II 
 
 
 4 
 
 6 
 
 3 7 
 
 
 6 5 
 
 8 6 4 
 
 6 4 
 
 3 2 
 
 
 3 
 
 3 
 
 8 4 
 
 
 8 7 
 
 3 2 
 
 1 5 
 
 8 6 
 
 
 5 
 
 2 
 
 6 5 
 
 
 4 3 
 
 1 7 6 
 
 5 4 
 
 9 6 
 
 
 6 
 
 4 
 
 8 4 
 
 
 2 9 
 
 5 8 3 
 
 7 8 
 
 6 7 
 
 
 7 
 
 6 
 
 7 
 
 
 8 5 
 
 4 7 6 
 
 4 5 
 
 7 6 
 
 
 4 
 
 8 
 
 6 5 
 
 
 7 6 
 
 5 3 
 
 6 8 
 
 9 6 
 
 
 
 
 9 
 
 8 
 
 
 4 3 
 
 9 6 8 
 
 4 2 
 
 5 3 
 
 
 2 
 
 7 
 
 6 1 
 
 
 2 1 
 
 4 7 9 
 
 2 1 
 
 3 8 
 
 
 1 
 
 8 
 
 4 8 
 
 
 6 
 
 6 4 
 
 6 5 
 
 6 4 
 
 
 7 
 
 6 
 
 6 5 
 
 
 5 4 
 
 2 4 
 
 2 1 
 
 3 
 
 
 13 
 
 
 14 
 
 
 15 
 
 16 
 
 
 17 
 
 
 5314 
 
 
 4673 
 
 
 46785 
 
 24605 
 
 71456 
 
 
 2302 
 
 
 2145 
 
 
 43123 
 
 03123 
 
 04523 
 
 
 1435 
 
 
 3210 
 
 
 65320 
 
 64321 
 
 13072 
 
 
 2134 
 
 
 4532 
 
 
 13042 
 
 45671 
 
 34562 
 
 
 1435 
 
 
 3245 
 
 2 
 
 56231 
 
 31230 
 
 53103 
 
 
 12620 
 
 17805 
 
 24501 
 
 168950 
 
 176716 
 
 18 
 
 19 
 
 
 20 
 
 21 
 
 22 
 
 23 
 
 24 
 
 25 
 
 ^is. 
 
 dolls 
 
 sec'ds. minutes, hours. 
 
 days. 
 
 months, years. 
 
 78 
 
 80 
 
 
 464 
 
 431 
 
 841 
 
 302 
 
 197 
 
 6784 
 
 64 
 
 19 
 
 
 876 
 
 782 
 
 674 
 
 407 
 
 642 
 
 5378 
 
 41 
 
 16 
 
 
 221 
 
 641 
 
 531 
 
 681 
 
 396 
 
 9605 
 
 82 
 
 42 
 
 
 642 
 
 421 
 
 486 
 
 764 
 
 421 
 
 7432 
 
 99 
 
 78 
 
 
 381 
 
 764 
 
 798 
 
 842 
 
 789 
 
 1867 
 
 71 
 
 97 
 
 
 496 
 
 491 
 
 642 
 
 596 
 
 462 
 
 5978 
 
16 
 
 SIMPLE ADDITION. 
 
 26. What is the amount of 427, 632, 781, 1001 4765 
 32014? Answer, 39620 
 
 27. Add 672, 1021, 846, 27, 4762, 7820, and 48 together. 
 
 Ans. 15196 
 
 28. Add the following numbers, 5876, 7890, 6874, 9658, 
 1234. Ans. 31532. 
 
 29. W., in the collection of money, received of B. 45 dolls., 
 of C. 74 dolls., of D. 96 dolls., of E. 121 dolls. ; how much 
 did he receive in all ? Ans. 336 dollars. 
 
 30. B. has four fields ; in the first he has 85 sheep, in the 
 second 97, in the third 142, in the fourth 234 ; how many has 
 he in all ? 
 
 31. C. purchased at a store, 76 pounds of 
 coflfee, 27 pounds of sugar, 36 pounds of 
 cheese, 40 pounds of salt, 9 pounds of tea, 4 
 pounds of raisins, and 2 pounds of spice ; how 
 
 Ans. 558. 
 
 37 
 
 many pounds in all ? 
 
 32 
 4763 
 2184 
 5763 
 7298 
 6042 
 5769 
 6586 
 
 33 
 
 68978 
 47697 
 58321 
 79642 
 63426 
 48967 
 48302 
 
 34 
 
 21345 
 
 7896 
 
 214 
 
 30 
 
 2 
 
 67 
 
 478 
 
 35 
 76321 
 49687 
 24768 
 21324 
 13214 
 13214 
 10204 
 
 Ans. 194. 
 
 36 
 64057 
 32165 
 48732 
 17694 
 21879 
 36582 
 14768 
 
 rJ» "^ b J» • Oi 
 
 O J'^ O O rJ^ 1:5 
 O O O O O S 
 
 6 
 3 
 6 
 
 7 
 2 
 
 4 
 
 2 
 
 2 
 
 38405 415333 30032 208732 235877 3 19 6 110 
 
 D. 
 
 42 
 187 
 
 96 
 176 
 340 
 180 
 652 
 
 96 
 
 38 
 
 dimes. 
 7 
 6 
 7 
 3 
 6 
 7 
 3 
 7 
 
 m. 
 
 4 
 3 
 
 1 
 2 
 7 
 2 
 1 
 4 
 
 39 
 D. c. m. 
 
 1879 25 5 
 
 470 21 
 6741 57 
 4202 87 
 
 118 50 
 
 642 09 
 19 20 
 
 142 78 
 
 40 
 D. c. 
 
 1921 47 
 1482 78 
 1720 50 
 1471 97 
 1682 31 
 1476 91 
 2978 77 
 9642 31 
 
 41 
 Miles. 
 46890 
 97642 
 76960 
 43219 
 84298 
 64967 
 36899 
 47947 
 
 42 
 Seconds. 
 896747 
 342651 
 986742 
 489764 
 651876 
 981974 
 532763 
 876490 
 
 43. Required the amount of the following sums; five thousand 
 seven hundred and eighty ; seventy-nine thousand four hundred 
 and forty-two ; eight hundred and eighty-nine thousand ; one mil* 
 
SIMPLE ADDITION. 
 
 17 
 
 aon ; five hundred ; twenty-one ; nine ; seven hundred ; twenty- 
 one thousand; fourteen ; one hundred and nine ? Ans, 1996575. 
 
 44 
 
 45 
 
 46 
 
 47 
 
 48 
 
 486543 
 
 9874658 
 
 8479643 
 
 78964532 
 
 4798764 
 
 213456 
 
 6750234 
 
 5896745 
 
 47809214 
 
 2156845 
 
 789407 
 
 8456796 
 
 8421354 
 
 87642305 
 
 2156845 
 
 801508 
 
 5834978 
 
 6743214 
 
 78967408 
 
 9764058 
 
 764314 
 
 7640857 
 
 5679434 
 
 97814234 
 
 7428405 
 
 827641 
 
 6978574 
 
 5307642 
 
 70974584 
 
 6489675 
 
 583670 
 
 8328976 
 
 2131478 
 
 64897984 
 32487023 
 32487023 
 
 4287345 
 1472021 
 1472021 
 
 4466539 
 
 53865073 ^ 
 
 42659510 
 
 49 
 
 50 
 
 51 
 
 52 
 
 53 
 
 67809782 
 
 31829176 
 
 879678 
 
 5789630 
 
 79857690 
 
 31918976 
 
 15896502 
 
 527659 
 
 2139567 
 
 87235278 
 
 52031607 
 
 321 
 
 768659 
 
 8976387 
 
 76897652 
 
 65897031 
 
 78652 
 
 318796 
 
 6705896 
 
 13783658 
 
 21317215 
 
 65897 
 
 528781 
 
 3125789 
 
 21356785 
 
 78975873 
 
 551789 
 
 709152 
 
 2157682 
 
 10239653 
 
 54. A man commenced a journey, and travelled 9 days ; the 
 first day he travelled 34 miles, and after that he gained 1 mile 
 every day ; how many miles did he travel in the 9 days ? 
 
 Ans, 342. 
 
 55. A drover purchased the following drove, namely: 102 
 oxen, for which he paid 5764 dolls.; 176 cows for 3784 dolls.; 
 50 calves for 250 dolls.; 420 sheep for 960 dolls.; of how 
 many did his drove consist, and how much did he pay ? 
 
 Ans. the drove 748, paid 10758 D. 
 
 56. A miller purchased of a farmer 10 bags of wheat, which 
 were to be paid for by weight, which was as follows : No. 1, 
 165 pounds ; No. 2, 168 ; No. 3, 171 ; No. 4, 175 ; No. 5, 178 ; 
 No. 6, 180; No. 7, 182; No. 8,185; No. 9, 189 ; No. 10, 190; 
 how many pounds of grain did he buy? Ans. 1783 pounds. 
 
 57. The plantation of a gentleman will produce in one sea- 
 eon, 1500 bushels of wheat, worth 1875D. ; 1350 bushels of 
 rye, worth 1283D. ; 2134 bushels of barley, worth 1696D. , 
 450 bushels of corn worth 275D. ; and 876 bushels of oats 
 worth 421 D. ; required the number of bushels of grain, and the 
 value? Ans. 6310 bushels, value 5550D. 
 
 58. What number of dollars are in 6 bags, each bag contain 
 \\g 37542D. Ans. 2^5252. 
 
19 SIMPLE ADDITION. 
 
 99. There is a valuable farm, one quarter of which is worth 
 15674D. ; how much is the whole farm worth ? Ans, 62696D. 
 
 60 61 62 63 
 
 D. c. m. D. d. c. m. 389678 D. 
 
 52 
 
 76 
 
 2 
 
 178 
 
 7 
 
 6 
 
 3 
 
 58321 
 
 586 
 
 15 
 
 78 
 
 3 
 
 6 
 
 9 
 
 8 
 
 7 
 
 76521 
 
 7635 
 
 17 
 
 19 
 
 7 
 
 189 
 
 6 
 
 5 
 
 8 
 
 0586 
 
 63528 
 
 378 
 
 17 
 
 6 
 
 25 
 
 7 
 
 3 
 
 9 
 
 216 
 
 875697 
 
 1837 
 
 65 
 
 7 
 
 6 
 
 3 
 
 2 
 
 1 
 
 790 
 
 5563216 
 
 1316 
 
 00 
 
 9 
 
 268 
 
 7 
 
 9 
 
 8 
 
 210 
 
 11610780 
 
 64. Add 4764608, 407, 76, 46876541, 76084763, together. > 
 
 Ans. 127726395. 
 
 65. Add 48, 96, 423, 8765, 6420, 4876, 904078, together. 
 
 Ans. 924706. 
 
 66. Add 1, 7, 6, 941, 784, 54204, 56476096, together. 
 
 Ans. 56532039. 
 
 67. Add 746796, 48, 50701, 30, 40, 17, 17645, together. 
 
 Ans. 815277 
 
 68. Add 9, 11, 642, 4786, 3000104, 68588479, 879643978," 
 together. 
 
 REVIEW. 
 
 Which is the first primary rule in Arithmetic ? How many 
 primary rules are there in Arithmetic ? What are they called ? 
 How many secondary rules are there 1 Name them. Why 
 are they called secondary ? What is Addition 1 What is the 
 use of Addition? What is the rule for writing the numbers? 
 After the numbers are all written, how do you proceed ? When 
 you have found the amount of all the figures in the left or last 
 column, what will you do ? How do you prove the correctness 
 of the work ? When you have proved your sum according to 
 rule, do you know it to be correct ? Why do you carry 1 foi 
 every 10 ? Recite the Addition table. 
 
 69. Ten partners have each 87452D. 20c. in trade ; what if 
 3ie amount of their capital ? 
 
 70. Add 4444444, 5555555, 6666666, 7777777, 888888a 
 together. 
 
SIMPLE SUBTRACTION. 19 
 
 SIMPLE SUBTRACTION. 
 
 This is the second primary rule in Arithmetic, and is the 
 very reverse of Addition. It teaches to take a less number 
 from a greater of the same name, and to show the difference, or 
 remainder, as 10 — 5, 5 remain; that is, 5 subtracted from 10, 
 it will leave 5, which is the difference. 
 
 1. Write down the greatest number first, then write the less 
 number directly under it, observing to place units under units ; 
 tens under tens, (fee. ; draw a line underneath. 
 
 2. Begin with the units, or right-hand figure, and subtract 
 that figure from the figure over it, and set down the difference. 
 
 3. When the figure in the lower number is more than the one 
 above it, you must subtract from 10, and the difference between 
 that figure and 10 must be added to the figure in the upper num- 
 ber, then set down that figure. 
 
 4. When you subtract from 10, carry 1, and add it to the next 
 left-hand figure ; proceed in this manner with all the figures, 
 and the number thus obtained will be the difference between 
 the two given numbers. 
 
 RULE IT. 
 
 After stating the sum as above directed, then if either of the 
 lower figures be greater than the upper one, conceive 10 to be 
 added to the upper figure ; then take the lower from it, and set 
 down the remainder. When 10 is thus added to the upper 
 figure, there must be 1 added to the next lower figure. 
 
 PROOF. 
 
 Add the remainder, or difference, to the less number, and 
 their sum will be equal to the greater number. 
 
 [It is of the greatest importance that the pupil should be 
 thoroughly exercised in the primary rules previously to entering 
 on others ; a few examples are given in money for exercise. 
 As the calculations can be the same as whole numbers, that sub- 
 ject being decimal, it necessarily requires a knowledge of those 
 rules to a correct understanding of our currency, which will 
 follow Long Division.] 
 
30 
 
 SIMPLE SUBTRACTION. 
 
 SUBTRACTION TABLE. 
 
 1- 
 
 -1=0 
 
 3- 
 
 -3=0 
 
 5- 
 
 -5=0 
 
 7- 
 
 -7=0 
 
 9- 
 
 -9=0 
 
 2 
 
 1 
 
 4 
 
 1 
 
 6 
 
 1 
 
 8 
 
 1 
 
 10 
 
 1 
 
 3 
 
 2 
 
 5 
 
 2 
 
 7 
 
 2 
 
 9 
 
 2 
 
 11 
 
 2 
 
 4 
 
 3 
 
 6 
 
 3 
 
 8 
 
 3 
 
 10 
 
 3 
 
 12 
 
 3 
 
 5 
 
 4 
 
 7 
 
 4 
 
 9 
 
 4 
 
 11 
 
 4 
 
 13 
 
 4 
 
 6 
 
 5 
 
 8 
 
 5 
 
 10 
 
 5 
 
 12 
 
 5 
 
 14 
 
 5 
 
 7 
 
 6 
 
 9 
 
 6 
 
 11 
 
 6 
 
 13 
 
 6 
 
 15 
 
 6 
 
 8 
 
 7 
 
 10 
 
 7 
 
 12 
 
 7 
 
 14 
 
 7 
 
 16 
 
 7 
 
 9 
 
 8 
 
 11 
 
 8 
 
 13 
 
 8 
 
 15 
 
 8 
 
 17 
 
 8 
 
 10 
 
 9 
 
 12 
 
 9 
 
 14 
 
 9 
 
 16 
 
 9 
 
 18 
 
 9 
 
 11 
 
 10 
 
 13 
 
 10 
 
 15 
 
 10 
 
 17 
 
 10 
 
 19 
 
 10 
 
 12 
 
 11 
 
 14 
 
 1] 
 
 16 
 
 11 
 
 18 
 
 11 
 
 20 
 
 11 
 
 13 
 
 12 
 
 15 
 
 12 
 
 17 
 
 12 
 
 19 
 
 12 
 
 21 
 
 12 
 
 14 
 
 13 
 
 16 
 
 13 
 
 18 
 
 13 
 
 20 
 
 13 
 
 22 
 
 13 
 
 15 
 
 14 
 
 17 
 
 14 
 
 19 
 
 14 
 
 21 
 
 14 
 
 23 
 
 14 
 
 16 
 
 15 
 
 18 
 
 15 
 
 20 
 
 15 
 
 22 
 
 15 
 
 24 
 
 15 
 
 17 
 
 16 
 
 19 
 
 16 
 
 21 
 
 16 
 
 23 
 
 16 
 
 25 
 
 16 
 
 18 
 
 17 
 
 20 
 
 17 
 
 22 
 
 17 
 
 24 
 
 17 
 
 26 
 
 17 
 
 19 
 
 18 
 
 21 
 
 18 
 
 23 
 
 18 
 
 25 
 
 18 
 
 27 
 
 18 
 
 20 
 
 19 
 
 22 
 
 19 
 
 24 
 
 19 
 
 26 
 
 19 
 
 28 
 
 19 
 
 2- 
 
 -2=0 
 
 4- 
 
 -4=0 
 
 6- 
 
 -6=0 
 
 8- 
 
 -8=0 
 
 10-10=0 
 
 3 
 
 1 
 
 5 
 
 1 
 
 7 
 
 1 
 
 9 
 
 1 
 
 11 
 
 1 
 
 4 
 
 2 
 
 6 
 
 2 
 
 8 
 
 2 
 
 10 
 
 2 
 
 12 
 
 2 
 
 5 
 
 3 
 
 7 
 
 3 
 
 9 
 
 3 
 
 11 
 
 3 
 
 13 
 
 3 
 
 6 
 
 4 
 
 8 
 
 4 
 
 10 
 
 4 
 
 12 
 
 4 
 
 14 
 
 4 
 
 7 
 
 5 
 
 9 
 
 5 
 
 11 
 
 5 
 
 13 
 
 5 
 
 15 
 
 5 
 
 8 
 
 6 
 
 10 
 
 6 
 
 12 
 
 6 
 
 14 
 
 6 
 
 16 
 
 6 
 
 9 
 
 7 
 
 11 
 
 7 
 
 13 
 
 7 
 
 15 
 
 7 
 
 17 
 
 7 
 
 10 
 
 8 
 
 12 
 
 8 
 
 14 
 
 8 
 
 16 
 
 8 
 
 18 
 
 8 
 
 11 
 
 9 
 
 13 
 
 9 
 
 15 
 
 9 
 
 17 
 
 9 
 
 19 
 
 9 
 
 12 
 
 10 
 
 14 
 
 10 
 
 16 
 
 10 
 
 18 
 
 10 
 
 20 
 
 10 
 
 13 
 
 11 
 
 15 
 
 11 
 
 17 
 
 11 
 
 19 
 
 11 
 
 21 
 
 11 
 
 14 
 
 12 
 
 16 
 
 12 
 
 18 
 
 12 
 
 20 
 
 12 
 
 22 
 
 12 
 
 15 
 
 13 
 
 17 
 
 13 
 
 19 
 
 13 
 
 21 
 
 13 
 
 23 
 
 13 
 
 16 
 
 14 
 
 18 
 
 14 
 
 20 
 
 14 
 
 22 
 
 14 
 
 24 
 
 14 
 
 17 
 
 15 
 
 19 
 
 15 
 
 21 
 
 15 
 
 23 
 
 15 
 
 25 
 
 15 
 
 18 
 
 16 
 
 20 
 
 16 
 
 22 
 
 16 
 
 24 
 
 16 
 
 26 
 
 16 
 
 19 
 
 17 
 
 21 
 
 17 
 
 23 
 
 17 
 
 25 
 
 17 
 
 27 
 
 17 
 
 20 
 
 18 
 
 22 
 
 18 
 
 24 
 
 18 
 
 26 
 
 18 
 
 28 
 
 18 
 
 To read the table, say 1 from 10 and 9 remain; 4 from 7 
 and 3 remain. 
 
SIMPLE SUBTRACTIOxN. 
 
 21 
 
 QUE'STIONS. 
 
 From 47084 Explanation. — Begin and say, 2 from 4 
 
 Take 23192"! and 2 will remain, which set down; then, 
 
 < + you can not take 9 from 8, because 9 is more 
 
 Rem. 23892 J than 8; therefore you must say 9 from 10 
 
 and 1 will remain, which added to 8 will 
 
 Proof 47084 make 9, which set down; now, you have 
 
 borrowed 1, which you must pay by adding 
 it to the next figure at the left, which will make 2 ; then, you 
 can not take 2 from 0, but you must say 2 from 10 and 8 will 
 remain ; you now have one to carry to 3, which will make 4 ; 
 then say 4 from 7 and 3 remain ; now, you have none to carry, 
 because you did not borrow ; say 2 from 4 and 2 will remain : 
 then add the two lower numbers together for proof. 
 
 1 . William had 32 marbles, in playing he lost 8 ; how many 
 had he then ? 
 
 2. Joseph had 47 cents, he paid 18 at the store ; how many 
 remain ? 
 
 3. Out of 48 eggs, John sold 29 ; how many had he remain- 
 ing? 
 
 4. If a man should lose 25 dollars out of 75, how many 
 remain ? 
 
 From 24532 
 Take 12321 
 
 6 
 145632 
 034411 
 
 Diff. 12211 111221 
 
 7 
 478965 
 367843 
 
 111122 
 
 Proof 24532 145632 478965 
 
 8 
 8764321 
 6543210 
 
 2221111 
 
 8764321 
 
 9 
 
 7865435 
 7653204 
 
 212231 
 
 7865435 
 
 10. 1567053-— 1234730 rem. 332323 
 
 11. 1579085—1381964 
 
 12. 1100000— 1000 
 
 13. 1000000— 999999 
 
 14. 1111111— 999999 
 
 15. 1794321— 147808 
 
 16. 1963004—1837999 
 
 17. 2478432—2189760 
 
 26 
 8476535 
 7563258 
 
 27 
 964235 
 217943 
 
 197121 
 
 1099000 
 
 1 
 
 111112 
 1646513 
 
 125005 
 
 288672 
 
 28 
 1047658 
 146873 
 
 18. 8796784- 
 
 19. 9431476- 
 
 20. 1058760- 
 
 21. 2764583- 
 
 22. 3540506- 
 
 23. 4116181- 
 
 24. 7168097- 
 
 25. 9708769- 
 
 29 
 8643587 
 1234509 
 
 -7960583 
 -8942316 
 
 - 165843 
 
 - 181419 
 
 - 335060 
 
 - 201418 < 
 
 - 671497 
 
 - 701304 
 
 30 
 
 1047653 
 
 926782 
 
 913277 746292 
 
 900785 7409078 
 
 120871 
 
3t 
 
 SIMPLE SUBTRACTION. 
 
 31 
 
 4965832 
 1234679 
 
 33 
 
 18176085 
 465937 
 
 33 
 
 9684597 
 4107965 
 
 34 
 
 846587 
 100701 
 
 35 
 
 7800650 
 7799990 
 
 36. A had 2720 bushels of wheat, he disposed of 1987 bush- 
 els ; how many bushels will remain? Ans. 733. 
 
 37. A man in building a house that will require 47621 bricks 
 has received 21234 ; how many more will be required ? 
 
 Ans. 26387. 
 
 38. Subtract 4 from a million, and give the proof. 
 
 39. How much is seven thousand seven hundred and fifty-one 
 greater than two thousand six hundred and seventy-eight 1 
 
 40. From one million, take ninety-nine thousand. 
 
 41 
 4693698076 
 3729489635 
 
 42 
 58602973028764 
 40098806709875 
 
 43 
 329865782198 
 300982098709 
 
 44 
 4965786943 
 1476879678 
 
 45 
 9789804023784 
 2098479234678 
 
 46 
 74802136458 
 68497683597 
 
 47 
 98673982011 
 89658098701 
 
 48 
 6310618763980 
 1329019876543 
 
 49 
 982119860753289 
 765432098765420 
 
 50 
 
 5896478967 
 4678569748 
 
 51 
 211898784267 
 147989476984 
 
 52 
 4801213245630 
 1302469785487 
 
 53 
 539864298670 
 432986702100 
 
 106877596570 
 
 54 
 9865432701198 
 3762987502290 
 
 6102445198908 
 
 55 
 
 86543298765 
 29176807983 
 
 57366490783 
 
 56 
 4027058426 
 2134169874 
 
 57 
 980476589794 
 807547867982 
 
 58 
 48976480111 
 37766554223 
 
SIMPLE SUBTRACTION. 33 
 
 59. A has 4760D., he will pay 1786D. ; B has 6781 D., he 
 will pay 3789D. ; C. has 7420D., he will pay .4949 ; how much 
 did they all have ai first, and how much did they all pay ? 
 
 Ans,, they had 18961D., paid 10524D. 
 
 60. A gentleman has 3 farms, the first contains 421 acres ; 
 second, 687A. ; third, 582 A. ; the first is worth 18968D. ; 2d, 
 31486D. ; 3d, 25600D. ; how many acres has he in all ; and 
 how much are the three farms worth ; and if he should sell 742 
 acres, how many would remain ? 
 
 Ans. He has in all 1690A. ; worth 76054D. ; will remain 
 948 acres, after selling 742 acres. 
 
 61. What is the diflference between 87904032 and 23040978 ? 
 
 Ans, 64863054. 
 
 62. If you give 468D. 45cts. for 159 bushels of wheat, and 
 789D. 55cts. for 321 sheep; and you dispose of 87 bushels of 
 wheat for 220D., and 180 sheep for 350D. ; how many bush- 
 els of wheat will you have remaining ? and how many sheep ? 
 how much money will you receive for your wheat and sheep 1 
 
 Ans, 72 bushels wheat 5 141 sheep ; receive 570D, 
 
 ADDITION AND SUBTRACTION. 
 
 63. A guardian paid his ward at one time 721 D., at another, 
 984D., and again, 840D. ; how much would be left out of the 
 estate, which was but 2545D. ? 
 
 64. A gentleman held a bond of 1984D. ; he received at one 
 time 520D., at another time 640D. ; how much is due ? 
 
 Ans, 824D. 
 
 65. If I add 430, 621, 7840, 21, 76, 1, 97, 17, 490, and then 
 subtract that amount from 964082, what will remain ? 
 
 Ans. 954489. 
 
 66. A loaned B. at one time 17D., at another time 84D., and 
 at another time 500D. ; and B. paid 427D. ; how much is due ? 
 
 Ans. 174D. 
 
 67. A merchant purchased 674 pipes of wine for 87640D. ; 
 he sold 485 pipes for 75481D.; how many pipes has he re- 
 maining, and how much did they cost him ? 
 
 Ans. 189 pipes ; cost 12159D. 
 
 REVIEW. 
 
 Name the second primary rule in arithmetic ? How do you 
 write down numbers in this rule ? What next ? When the 
 figure below is more than the one directly over it, how do you 
 proceed ? What will you do when you borrow^ or subtract from 
 
-24 SIMPLE MULTIPLICATION. 
 
 ten ? How do you prove subtraction ? For what purpose is 
 subtraction used ? Repeat the rule ? 
 
 68. Three young men, A, B, and C, each inherited a large 
 estate; A's estate is worth 154760D. ; B's, 221784D.; C's, 
 341791D. : A has contracted debts to the amount of 47865D. ; B, 
 118464D.; C, 187642D.; required the amount of all their 
 debts, and what they will all be worth after payment? 
 
 Ans. Debts, 353971 D. ; have remaining, 364364D. 
 
 69. From 897640804235678 take 234796857912. 
 
 70. 4 + 9 + 62 + 49 + 78423 + 64027=142574; then~9876 
 £=132698. 
 
 SIMPLE MULTIPLICATION. 
 
 By multiplication we can perform a number of additions, in 
 a shorter and more easy method, or it is a number repeated a 
 given number of times ; as 5 multiplied by 5 is 25 ; so if five 
 5's be added together it would be the same. There are three 
 parts in multiplication which require particular attention. 
 
 1. The Multiplicand, the number given to be multiplied. 
 
 2. The Multiplier, the (less) number by which you multiply. 
 
 3. The Product, which is the result, or sum produced by the 
 operation of multiplying, and is just as many times larger than 
 the multiplicand, as there are units in the multiplier. The mul- 
 tiplicand and multiplier, together, are sometimes called y^c^or^. 
 
 EXAMPLES, FOR MENTAL EXERCISE. 
 
 1. What will 2 pounds of coffee cost, at 14 cents per pound? 
 
 2. There were 5 boys who gave a poor man 3 cents apiece ; 
 how many cents did the man receive? 
 
 3. How many dollars must I pay for 12 yards of cloth, that 
 IS worth 4 dollars a yard ? 
 
 4. How much will a Jerseyman get for 25 melons, if he sells 
 them at 5 cents a piece ? 
 
 5. A teacher had 8 classes in his school, and 7 scholars in 
 each class ; how many scholars had he in all ? 
 
 6. There are 24 hours in one day ; how many hours in 3 
 days ? 
 
 7. If 1 pound of honey is worth 12 cents, how many cents 
 are 9 pounds worth 1 
 
 8. 14 and what number make 20 ? 
 
 9. If 1 bushel of apples cost 21 cents, how much will 5 bush- 
 els cost ? 
 
SIMPLE MULTIPLICATION. 
 
 25 
 
 MULTIPLICATION TABLE. 
 
 Twice 
 
 5 times 
 
 8 times 
 
 11 times 
 
 1 make 2 
 
 1 make 5 
 
 1 make 8 
 
 1 make 11 
 
 2 4 
 
 2 10 
 
 2 16 
 
 2 22 
 
 3 6 
 
 3 15 
 
 3 24 
 
 3 33 
 
 4 8 
 
 4 20 
 
 4 32 
 
 4 44 
 
 :> 10 
 
 5 25 
 
 5 40 
 
 5 55 
 
 6 12 
 
 6 30 
 
 6 48 
 
 6 66 
 
 7 14 
 
 7 35 
 
 7 56 
 
 7 77 
 
 8 16 
 
 8 40 
 
 8 64 
 
 8 88 
 
 9 18 
 
 9 45 
 
 9 72 
 
 9 99 
 
 10 20 
 
 10 50 
 
 10 80 
 
 10 110 
 
 U 22 
 
 11 55 
 
 11 88 
 
 11 121 
 
 12 24 
 
 12 60 
 
 12 96 
 
 12 132 
 
 3 times 
 
 6 times 
 
 9 times 
 
 12 times 
 
 1 make 3 
 
 1 make 6 
 
 1 make 9 
 
 1 make 12 
 
 2 6 
 
 2 12 
 
 2 18 
 
 2 24 
 
 3 9 
 
 3 18 
 
 3 27 
 
 3 36 
 
 4 12 
 
 4 24 
 
 4 36 
 
 4 48 
 
 5 15 
 
 5 30 
 
 5 45 
 
 5 60 
 
 6 18 
 
 6 36 
 
 6 54 
 
 6 72 
 
 7 21 
 
 7 42 
 
 7 63 
 
 7 84 
 
 8 24 
 
 8 48 
 
 8 72 
 
 8 96 
 
 9 27 
 
 9 54 
 
 9 81 
 
 9 108 
 
 10 30 
 
 10 60 
 
 10 90 
 
 10 120 
 
 11 33 
 
 11 66 
 
 11 99 
 
 11 132 
 
 12 36 
 
 12 72 
 
 12 108 
 
 12 . 144 
 
 4 times 
 
 7 times 
 
 10 times 
 
 Twice 
 
 1 make 4 
 
 1 make 7 
 
 1 make 10 
 
 13 make 26 
 
 2 8 
 
 2 14 
 
 2 20 
 
 14 28 
 
 3 12 
 
 3 21 
 
 3 30 
 
 15 30 
 
 4 16 
 
 4 28 
 
 4 40 
 
 16 32 
 
 6 20 
 
 5 35 
 
 5 50 
 
 17 34 
 
 6 24 
 
 6 42 
 
 6 60 
 
 18 36 
 
 7 28 
 
 7 49 
 
 7 70 
 
 19 38 
 
 8 32 
 
 8 56 
 
 8 80 
 
 20 40 
 
 9 36 
 
 9 63 
 
 9 90 
 
 21 42 
 
 10 40 
 
 10 70 
 
 10 100 
 
 22 44 
 
 11 44 
 
 U 77 
 
 U 110 
 
 23 46 
 
 1? 48 
 
 12 84 
 
 12 120 
 
 24 48 
 
^ SIMPLE iMULTIPLIGJLTlON. 
 
 When the multiplier does not exceed 12. 
 
 1. Set down the multiplicand, and at the right hana, undei 
 the figure or figures, of the multiplicand set the multiplier. 
 
 2. Then begin with the imits, and multiply all the figures in 
 the multiplicand in succession, and set down their several prod- 
 ucts, observing to carry one for every ten to the product of the 
 next figure, and set down the whole of the last product. Proof, 
 multiply the multiplier by the multiplicand ; or by division. 
 
 64032 multiplicand. 
 5 X multiplier. 
 
 QUESTIONS. 
 
 Explanation. — Begin and say 5 times 
 2 are 10, and set down : now there 
 
 is 1 to carry ; then say 5 times 3 are ^ 
 
 320160 product, or ans. 15, and 1 is 16, set down 6 ; now there 
 is 1 to carry ; 5 times is 0, but 1 is 
 is 1, this set down ; now you have none to carry, because it is 
 less than 10 ; then 5 times 4 are 20, set down ; now you have 
 2 to carry ; then 5 times 6 are 30, and 2 are 32 ; set down the 
 whole number. 
 
 1. 
 
 12343 
 2X 
 
 2. 
 345624 
 2 
 
 3. 
 
 4532145 
 3 
 
 4. 
 465832 
 4 
 
 5. 
 1035678 
 4 
 
 24686 
 
 691248 
 
 13596435 
 
 1863328 
 
 4142712 
 
 6. 
 4789650 
 5 
 
 7. 
 9650489 
 5 
 
 8. 
 694763 
 6 
 
 9. 
 141817614 
 6 
 
 10. 
 95478654 
 7 
 
 11. 
 143567 
 8 
 
 12. 
 91456 
 9 
 
 13. 
 234567 
 10 
 
 14. 
 132454 
 11 
 
 15. 
 134021 
 12 
 
 1148536 
 
 823104 
 
 2345670 
 
 1456994 
 
 1608252 
 
 16. What cost 11 pounds of cheese at 11 cents per pound? 
 
 17. What would you give for 17 quarts of corn at 5 cents per 
 quart ? 
 
 18. What cost 24 quarts of salt, at 9 cents per quart? 
 
 19. When candles are worth 1^ cents per pound, what will 
 11 pounds cost? 
 
SIMPLE MULTIPLICATION. 
 
 27 
 
 20. What will 34 pounds of nails come tx) at 7 
 pound? 
 
 21. How much will 47 quarts of chestnuts cost at 9 cents 
 per quart ? 
 
 22. What cost 11 dozen of eggs, at 9 cents per dozen ? 
 
 23. Multiply 120 by 2 ; by 3. 
 
 Ans. i20x 2=240 
 
 24. Multiply 1211 by 5 ; by 6. 
 
 25. Multiply 1211 by 7; by 8. 
 Multiply 65321 by 9 ; by 6. 
 Multiply 65321 by 8 ; by 10. 
 
 26. 
 27. 
 28. 
 29. 
 30. 
 31. 
 
 cents per 
 
 120x3=360+240=600. 
 
 Ans. 13321. 
 
 Ans. 18165. 
 
 ^71^. 979815. 
 
 Ans. 1175778. 
 
 Multiply 123456 by 11 ; by 4. 
 Multiply 123456 by 3 ; by'5. 
 Multiply 345612 by 3; by 8. 
 Multiply 345612 by 12 ; by 7. 
 
 Ans. 1851840. 
 
 Ans. 987648. 
 
 Ans. 3801732. 
 
 Ans. 6566628. 
 
 32. Multiply 12345006789 by 3 ; by 4. Ans. 86415047523. 
 
 33. Multiply 12345006789 by 5 ; by 6. Ans. 135795074679. 
 Multiply 9784076987 by 8 ; by 12. 
 
 34. 
 35. 
 36. 
 37. 
 38. 
 39. 
 40. 
 41. 
 42 
 43. 
 44. 
 
 4218X 
 
 7321 
 
 87692 
 
 95698 
 
 10691 
 
 31078 
 
 109019 
 
 900078 
 
 278976 
 
 12569769 
 
 2 = 
 
 3 
 
 4 
 
 5 
 
 6 
 
 7 
 
 8 
 
 9 
 11 
 12 
 
 Ans. 8436. 
 
 21963. 
 
 350768. 
 
 478490. 
 
 64146. 
 
 217546. 
 
 872152. 
 
 8100702. 
 
 3068736. 
 
 150837228. 
 
 When the multiplier exceeds 12, and consists of two or more 
 
 figures. 
 
 RULE. 
 
 1 . Set down the multiplicand, and under it the multiplier, so 
 that units may stand under units, tens under tens, hundreds 
 ander hundreds, &c., and draw a line under them. 
 
 2. Multiply each figure in the multiplicand by each figure in 
 the multiplier separately, beginning at the right, or place of 
 units, placing the result directly under the multiplier, observing 
 to carry 1 for every 10, &c. ; then multiply by the next figure 
 in the place of ten, placing the^r^^ figure of every line directly 
 under its respective multiplier. 
 
 3. After multiplying by all the figures, add these products to- 
 gether, and their amount will be the product or answer required. 
 
2S 
 
 SIMPLE MULTIPLICATIOX. 
 
 4. When ciphers occur at the right hand of either the mul- 
 tiplier or multiplicand, or both, omit them in the operation and 
 annex them to the product. [Annex : to subjoin at the end- 
 right hand.) 
 
 QUESTIONS AND EXAMPLES. 
 
 45. 
 
 46. 
 
 47. 
 
 48 
 
 49. 50. 51. 
 
 14 
 
 13 
 
 28 
 
 25 
 
 35 30 42 
 
 13x 
 
 14 
 
 25 
 
 28 
 
 30 35 22 
 
 42 
 
 52 
 
 140 
 
 200 
 
 1050 150 84 
 
 14 
 
 13 
 
 182 
 
 56 
 700 
 
 50 
 700 
 
 10 X 90 84 
 
 182 
 
 10500 1050 924 
 
 52. 
 
 53. 
 
 
 54. 
 
 55. 56. 
 
 7649 
 
 145365402 
 
 23567894 
 
 356421345 3456784 
 
 22 
 
 
 14 
 
 15 
 
 23 24 
 
 15298 581461608 117839470 1069264035 13827136 
 15298 145365402 23567894 712842690 6913568 
 
 16827 
 
 8 2035115628 
 
 353518410 
 
 8197690935 82962816 
 
 57. 
 
 5234 X 
 
 
 145. 
 
 = Ans. 758930 
 
 58. 
 
 129186 
 
 
 98 
 
 12660228 
 
 59. 
 
 23430 
 
 
 230 
 
 5388900 
 
 60. 
 
 5101 
 
 
 300400 
 
 1532340400 
 
 61. 
 
 674200 
 
 62. 
 7853214650 
 462 
 
 
 2104 
 
 1418516800 
 
 63. 
 8653214578 
 304* 
 
 
 64. 
 
 65. 
 
 89700544315x4032 
 
 
 
 6489784X2001 
 
 66. 365420145321x3215: 
 
 -~ 
 
 Ans. 1174825767207015 
 
 67. 10145034X2031 = 
 
 
 
 Ans. 20604564054 
 
 68. Multiply 62123 by 
 
 13. 
 
 
 Ans. 807599 
 
 • IVIien there are ciphers between the significant figures of the multi- 
 vlier, we may omit the ciphers, multiplying by the significant figures 
 wtiy, placing the first figure of each product directly under the figure by 
 which we multiply. 
 
SIMPLE MULTIPLISATION. 
 
 29 
 
 69. Multiply 35432 by 14 ; by 15. Ans. 1027528 
 
 70. Multiply 65217 by 16 ; by 17. 2152161 
 
 71. Multiply 207812 by 19 ; by 21. 8312480 
 
 72. 207812 X25, X35= 12468720 
 
 73. 32100421 65, 85 4815063150 
 
 74. 32100421 27, 33 1926025260 
 
 75. 79814 29, 89 
 
 76. 1978987 4809 9516948483 
 
 77. 9807094 5047 49496403418 
 
 When the multiplier is exactly equal to any two figures in the 
 multiplication table. 
 
 RULE. 
 
 Multiply first by one of those figures, and that product by the 
 other, and the last product will be the answer. 
 
 78. 79. 80. 81. 
 
 521x16, 4X4 = 16, or thus, 521 4350x25 473213x35 
 4 16 5 7 
 
 2084 
 
 
 
 3126 21750 
 
 3312491 
 
 4 
 
 
 
 521 5 
 
 5 
 
 8336 
 
 8336 108750 
 
 16562455 
 
 82. 
 
 87698 X 
 
 
 72 = 
 
 Ans, 6314256 
 
 83. 
 
 75687 
 
 
 56 
 
 4238472 
 
 84. 
 
 34075 
 
 
 36 
 
 1226700 
 
 85. 
 
 47696 
 
 
 144 
 
 6868224 
 
 86. 
 
 23453 
 
 
 81 
 
 1899693 
 
 87. 
 
 45346 
 
 
 49 
 
 2221954 
 
 88. 
 
 74032 
 
 
 64 
 
 4738048 
 
 89. 
 
 85303 
 
 
 100 
 
 8530300 
 
 90. 
 
 63421 
 
 
 110 
 
 6976310 
 
 91. 
 
 79049 
 
 
 81 
 
 
 To multiply by 10, 
 
 100, 
 
 1000, &;c., annex 
 
 to the multiplicand 
 
 all the 
 
 ciphers in the 
 
 multiplier, and it will be the product re- 
 
 quired. 
 
 
 
 
 
 92. 
 
 421 X 
 
 
 10= 
 
 Ans. 4210 
 
 93. 
 
 4732 
 
 
 100 
 
 473200 
 
 94. 
 
 8619 
 
 
 1000 
 
 8619000 
 
 95. 
 
 18780 
 
 
 10000 
 
 187800000 
 
 96. John was employed by a gentleman 67 days, for which 
 he received 18 cents a day; how many cents did his wages 
 amount to? Ans. 1206. 
 
 3* 
 
30 SIMPLE MULTIPLICATION. 
 
 97. A man has 4 horses, each horse has 4 shoes, and each 
 shoe has 8 nails ; how many nails are there in all their shoes ? 
 
 Arts, 128. 
 
 98. If 32 quarts make 1 bushel, how many quarts are there 
 in ] 5 bushels ? Ans. 480. 
 
 99. If 112 pounds make one hundred weight (gross), how 
 many pounds would there be in 17 hundred weight? 
 
 Ans. 1904. 
 
 100. If 60 minutes make an hour, how many minutes are 
 there in 1124 hours? Ans. 67440. 
 
 101. If a man shoidd travel 247 days, at the rate of 38 miles 
 a day, how many miles would he travel in that space of time ? 
 
 Ans. 9386, 
 
 102. If a company of 98 men should receive 46D. each, as 
 their wages, how much would they all receive ? Ans. 4508D. 
 
 103. What is the product of 8505x64 ? Ans. 544320. 
 
 104. What sum of money must be divided among 500 men, 
 so that each man may receive 20D. ? Ans. lOOOOD. 
 
 105. If it requires 5768 shingles for the roof of a house, how 
 many would it require for 16 houses ? Ans. 92288. 
 
 106. If it takes 3242 bricks to build a house, how many will 
 it take for 240 houses ? Ans. 778080. 
 
 107. There are 4 houses, each house has 22 windows, and 
 each window has 16 panes of glass ; how many panes of glass 
 in the 4 houses? Ans. 1408. 
 
 108. In a certain town there are 572 houses, in each house 
 6 rooms, and in each room 7 persons ; required the number of 
 persons. Ans. 24024. 
 
 109. What is the product of 47865378X 13402 ? 
 
 Ans. 641491795956. 
 
 ADDITION AND MULTIPLICATION. 
 
 110. A drover purchased 84 oxen for 54D. each, and 147 
 cows for 27D. each ; how many cattle had he, and how much 
 did they cost? Ans. 231 cattle, cost 8505D. 
 
 111. Multiply 46 by 11 ; 71x18 ; 36x42 ; add their sev- 
 eral products together, and tell their amount. Ans. 3296. 
 
 112. A merchant purchased 7 pieces of carpeting, each piece 
 containing 27 yards; and 4 pieces, each piece containing 18 
 yards ; how many yards in all ? Ans. 261. 
 
 SUBTRACTION AND MULTIPLICATION. 
 
 113. Multiply 617 by 45, from that product subtract 6784. 
 
 Ans. 20981. 
 
SIMPLE MULTIPLICATION. 31 
 
 114. A man purchased 9 bags of salt, each bag weighed 231 
 pounds, and the sacks without the salt weighed 46 pounds ; 
 required the weight of salt without the sacks. Ans. 2033. 
 
 115. Multiply 47 by 16 ; 24 by 32 ; from their sum subtra.ct 
 542. Ans. 978. 
 
 116. If there be 16 ounces in one pound, how many ounces 
 in 3 pounds ? in 5 pounds ? in 8 pounds ? in 1 1 pounds ? in 
 129 pounds ? 
 
 117. If 12 inches make 1 foot, how many inches in 13 feet? 
 in 29? in 1700? 
 
 118. If 1 yard of riband cost 42 cents, how many cents will 
 17 yards cost? 24 yards ? 66 yards ? 114 yards ? 750 yards ? 
 
 119. If 24 hours make 1 day, how many hours in 4 days ? 
 in 1 week ? 
 
 120. If 60 minutes make an hour, how many in 6 hours ? in 
 96 hours ? in 789 hours ? 
 
 121. At 42 cents a bushel, what will 315 bushels of corn 
 cost? Ati^. 13230. 
 
 122. If 63 gallons make a hogshead, how many gallons in 
 24 hogsheads ? in 38 ? in 1974 ? 
 
 123. If you can earn QQ cents in a day, how much in 97 
 days ? in 159 ? in 428 ? 
 
 124. If the velocity of a car on a railroad is 18 miles an 
 hour, required the number of miles in 24 hours ; 76 hours ; 84 
 hours ; 497 hours. 
 
 REVIEW. 
 
 What is multiplication ? What is the number to be multiplied 
 called ? What is the number to multiply by called ? What 
 number is written down first ? How must the multiplier be set 
 under the multiplicand ? What is the result or answer called ? 
 How many times larger is the product than the multiplicand ? 
 Where do you begin to multiply ? Why do you begin to multi- 
 ply at the right hand figure ? How do you place the first product 
 figure of each line ? What rule does multiplication shorten ? 
 How many numbers are given in multiplication ? How many 
 numbers are required ? By what name are multiplier and mul- 
 tiplicand together called ? Repeat the rule when the multiplier 
 does not exceed 12. When the multiplier exceeds 12, what 
 is the rule ? How do you prove multiplication ? Repeat the 
 multiplication table. 
 
 125. If 250 men can do a piece of work in 218 days, how 
 many days will it require for 1 man to do the same ? 
 
 Ans. 54500< 
 
32 SHORT DIVISION. 
 
 126. What will 100 bushels of corn cost at 95 cents a bush- 
 el ? Ans. 9500 cts. 
 
 127. What will 1000 bushels of apples cost at 50 cents a 
 bushel ? Ans. 50000 cts. 
 
 128. Multiply 62123 by 13. Arts, 807599 
 
 129. 4078945X16= ^n^. 65263120. 
 
 130. Multiply 9876543210 by 970845. 
 
 SHORT DIVISION. 
 
 Division teaches to divide a larger number, by a less, into 
 equal parts, and is a short method of performing a number of 
 subtractions ; thus, if we wished to divide 25 D. between 5 men, 
 instead of subtracting 5 from 25 five times, we would divide 25 
 by 5, which would give 5D. for the answer, being the share of 
 each man, because 5 is contained in 25 just 5 times, and 5 X 5 =25. 
 
 There are four parts in Division that require attention, name- 
 ly : Dividend, Divisor, Quotient, and Remainder. 
 
 1 . Dividend. The number given to be divided, which is al- 
 ways more than the divisor. 
 
 2. Divisor. The number given by which the dividend is to 
 be divided. 
 
 3. Quotient. That which gives the number of times the divisor 
 is contained in the dividend, which is also the answer. 
 
 4. Remainder. That which remains after all the figures have 
 been brought down from the dividend, and the last subtraction 
 performed (if any remain) will be of the same denomination 
 with the dividend, and always less than the divisor. 
 
 When the divisor does not exceed 12. 
 
 RULE. 
 
 1 . Write down the dividend, and draw a curve line at the left 
 hand side, and a straight line under the dividend, and place the 
 divisor at the left hand of it ; thus : Divisor 9)81 dividend. 
 
 9 quotient. 
 
 2. Consider how many times the divisor is contained in the 
 first figure or figures of the dividend, and set down the result, 
 observing whether there be any remainder, and, if any, carry it 
 to the left of the next figure, and consider it placed there as so 
 many tens, into which divide as before, &c. ; but if no remainder, 
 see how many times the divisor is contained in the next figure. 
 
 Proof. — Multiply the quotient by the divisor, and add in the 
 remainder, if any, and the product will agree with the dividend; 
 if the operation has been correctly performed. 
 

 
 
 SHORT DIVISIOI^. 
 
 
 
 
 33 
 
 DIVISION TABLE 
 
 
 
 
 
 2 in 2 10 rem. 
 
 4 
 
 in 5 
 
 1 
 
 1 
 
 
 6 
 
 in 12 
 
 2 
 
 2 3 11 
 
 4 
 
 6 
 
 1 
 
 2 
 
 
 7 
 
 7 
 
 1 
 
 2 4 2 
 
 4 
 
 7 
 
 1 
 
 3 
 
 
 7 
 
 8 
 
 1 1 
 
 Z 5 2 1 
 
 4 
 
 8 
 
 2 
 
 
 
 
 7 
 
 9 
 
 1 2 
 
 2 6 3 
 
 4 
 
 9 
 
 2 
 
 1 
 
 
 7 
 
 10 
 
 1 3 
 
 2 7 3 1 
 
 4 
 
 10 
 
 2 
 
 2 
 
 
 7 
 
 11 
 
 1 4 
 
 2 8 4 
 
 4 
 
 11 
 
 2 
 
 3 
 
 
 7 
 
 12 
 
 1 5 
 
 2 9 4 1 
 
 4 
 
 12 
 
 3 
 
 
 
 
 8 
 
 8 
 
 1 
 
 2 10 5 
 
 5 
 
 5 
 
 1 
 
 
 
 
 8 
 
 9 
 
 1 ] 
 
 2 11 5 1 
 
 5 
 
 6 
 
 1 
 
 1 
 
 
 8 
 
 10 
 
 1 2 
 
 2 12 6 
 
 5 
 
 7 
 
 1 
 
 2 
 
 
 8 
 
 11 
 
 1 3 
 
 3 3 10 
 
 5 
 
 8 
 
 1 
 
 3 
 
 
 8 
 
 12 
 
 1 4 
 
 3 4 11 
 
 5 
 
 9 
 
 1 
 
 4 
 
 
 9 
 
 9 
 
 1 
 
 3 5 12 
 
 5 
 
 10 
 
 2 
 
 
 
 
 9 
 
 10 
 
 1 1 
 
 3 6 2 
 
 5 
 
 11 
 
 2 
 
 1 
 
 
 9 
 
 11 
 
 1 2 
 
 3 7 2 1 
 
 5 
 
 12 
 
 2 
 
 2 
 
 
 9 
 
 12 
 
 1 3 
 
 3 8 2 2 
 
 6 
 
 6 
 
 1 
 
 
 
 
 10 
 
 10 
 
 1 
 
 3 9 3 * 
 
 6 
 
 7 
 
 1 
 
 1 
 
 
 10 
 
 11 
 
 1 1 
 
 3 10 3 1 
 
 6 
 
 8 
 
 1 
 
 2 
 
 
 10 
 
 12 
 
 1 2 
 
 3 11 3 2 
 
 6 
 
 9 
 
 1 
 
 3 
 
 
 11 
 
 11 
 
 1 
 
 3 12 4 
 
 6 
 
 10 
 
 1 
 
 4 
 
 
 11 
 
 12 
 
 1 1 
 
 4 4 10 
 
 6 
 
 11 
 
 1 
 
 5 
 
 
 12 
 
 12 
 
 1 
 
 To read the table, 
 
 say 2 is in 3 
 
 once 
 
 and 1 
 
 over; 
 
 3 in 5 
 
 once and 1 over, or re 
 
 mainder. 
 
 
 
 
 
 
 
 2 in 2 1 
 
 4 in 4 
 
 I 
 
 6 in 6 
 
 1 
 
 8 in 
 
 8 
 
 1 
 
 10 in 
 
 10 1 
 
 2 4 2 
 
 4 8 J 
 
 Z 
 
 6 
 
 12 
 
 2 
 
 8 1 
 
 6 
 
 2 
 
 10 
 
 20 2 
 
 2 6 3 
 
 4 12 , 
 
 3 
 
 6 
 
 18 
 
 3 
 
 8 2 
 
 4 
 
 3 
 
 10 
 
 30 3 
 
 2 8 4 
 
 4 16 ^ 
 
 i 
 
 6 
 
 24 
 
 4 
 
 8 3 
 
 2 
 
 4 
 
 10 
 
 40 4 
 
 2 10 5 
 
 4 20 , 
 
 5 
 
 6 
 
 30 
 
 5 
 
 8 4 
 
 
 
 5 
 
 10 
 
 50 5 
 
 2 12 6 
 
 4 24 
 
 5 
 
 6 
 
 36 
 
 6 
 
 8 4 
 
 8 
 
 6 
 
 10 
 
 60 6 
 
 2 14 7 
 
 4 28 ' 
 
 7 
 
 6 
 
 42 
 
 7 
 
 8 5 
 
 6 
 
 7 
 
 10 
 
 70 7 
 
 2 16 8 
 
 4 32 
 
 3 
 
 6 
 
 48 
 
 8 
 
 8 6 
 
 4 
 
 8 
 
 10 
 
 80 8 
 
 2 18 9 
 
 4 36 
 
 9 
 
 6 
 
 54 
 
 9 
 
 8 7 
 
 2 
 
 9 
 
 10 
 
 90 9 
 
 3 3 1 
 
 5 5 
 
 1 
 
 7 
 
 7 
 
 1 
 
 9 
 
 9 
 
 1 
 
 11 
 
 11 1 
 
 3 6 2 
 
 5 10 
 
 2 
 
 7 
 
 14 
 
 2 
 
 9 1 
 
 8 
 
 2 
 
 11 
 
 22 2 
 
 3 9 3 
 
 5 15 
 
 3 
 
 7 
 
 21 
 
 3 
 
 9 2 
 
 7s 
 
 3 
 
 11 
 
 33 3 
 
 3 12 4 
 
 5 20 
 
 4 
 
 7 
 
 28 
 
 4 
 
 9 3 
 
 6 
 
 4 
 
 11 
 
 44 4 
 
 3 15 5 
 
 5 25 
 
 5 
 
 7 
 
 35 
 
 5 
 
 9 4 
 
 5 
 
 5 
 
 11 
 
 55 5 
 
 3 18 6 
 
 5 30 
 
 6 
 
 7 
 
 42 
 
 6 
 
 9 5 
 
 4 
 
 6 
 
 11 
 
 66 6 
 
 3 21 7 
 
 5 35 
 
 7 
 
 7 
 
 49 
 
 7 
 
 9 6 
 
 3 
 
 7 
 
 11 
 
 77 7 
 
 3 24 8 
 
 5 40 
 
 8 
 
 7 
 
 56 
 
 8 
 
 9 7 
 
 2 
 
 8 
 
 11 
 
 88 8 
 
 3 27 9 
 
 5 45 
 
 9 
 
 7 
 
 63 
 
 9 
 
 9 8 
 
 I 
 
 9 
 
 11 
 
 99 9 
 
34 SHORT DIVISION. 
 
 28-r 7, or 2^^ =:how many ? 49-t- 7, or Y=^ow"^^^y^ 
 
 42-7- 6, or ^2_]2ow many ? 32-4- 4, or ^2_}jq^ niany ? 
 
 54-f- 9, or ^^^=howmany? 99-f-ll,or ^|^=how many ? 
 
 32^ 8, or 3^2_-how many? 84-f-12, or f|-=:how many ? 
 
 33^11, or ff =how many? 108-f-12, or i^^^nzhow many? 
 
 Note. — It is well for scholars to read the tables in class, the 
 same as reading lessons, by which means they will learn them 
 in a few hours. The most of the Arithmetic may be read occa- 
 sionally. 
 
 How many oranges could you buy for 24 cents at 6 cents an 
 orange ? 
 
 If you ride 4 miles an hour, how many hours would it take 
 you to ride 36 miles ? 
 
 With 72 cents, how many knives could you buy at 12 cents 
 a-piece ? 
 
 How many times 9 in 81 ? how many times 11 in 80, and 
 how many over ? 
 
 How many times 12 in 84 ? how many times 12 in 136, and 
 how many over ? 
 
 QUESTIONS AND EXAMPLES. 
 
 Divisor 3)976432 dividend. Explanation. — Begin and 
 
 say 3 is in 9 3 times, which 
 
 Quotient 325477 — 1 over, or rem. set down ; then 3 is in 7 
 
 X3 + l twice and 1 over; this 1 is 
 
 now supposed to be placed 
 Proof 976432 at the left of the next figure 
 
 6, which would be 16 ; then 
 say 3 is in 16 5 times and 1 over ; then 3 in 14 4 times and 2 
 over ; now this 2 is supposed to be placed at the left of the 
 next figure 3, which will make it 23 ; then 3 is in 23 7 times 
 and 2 over ; this 2 placed at the left of the next figure 2 is 22 ; 
 then 3 is in 22 7 times and 1 over, a remainder which must be 
 placed at the right. Then multiply the quotient by the divisor, 
 and add in the 1, for the proof. 
 
 JSlcte. — If the divisor is not contained in the next figure of the 
 dividend, write a cipher in the quotient, and join this figure in 
 the dividend, to the figure next to it, as so many tens, 
 
 1. 2. 3. 4. 5. 6. 
 
 2 )684 2)868 3 )963 4)840 5 )86421 6 )784023 
 
 342 ^434 321 210 17284—1 130670—3 
 
SHORT DIV SIGN. 
 
 36 
 
 7. 
 7)94065832 
 
 13437976 
 
 11. 
 11)2456321 
 
 8. 9. 
 
 8)4123045 9)6304532 
 
 10. 
 10)8465324 
 
 15. 
 4)9653450 
 
 515380—5 700503—5 
 
 12. 13. 
 
 12)5840235 2)5678432 
 
 223301—10 
 
 486686—3 
 
 16. 
 5)8432105 
 
 17. 
 6)123465789 
 
 846532—4 
 
 14. 
 3)6745076 
 
 18. 
 7)40345658 
 
 19. 20. 21. 22. 
 
 8)45673250 9)7847321 10)1213141516 11)84213567 
 
 23. 
 12)84213567 
 
 24. 
 12)64789057 
 
 25. 
 2)34789 
 
 26. 
 3)421562 
 
 27. 240653-^4= 60163—1 
 
 28. 321450-f-5== 64290—0 
 
 29. 845354-7-6 = 140892—2 
 
 30. 798543-^7=114077—4 
 
 31. 643257-8= 80407—1 
 
 32. 583460-f- 9=64828—8 
 
 33. 146148-10=14614-8 
 
 34. 367854-r- 11 = 33443— 3 
 
 35. 845678-^12=70473—2 
 
 36. 989789-^-100 
 
 37. A farmer sold 540 bushels of grain to 9 men ; how much 
 did each man receive for his share ? Ans. 60 bush. 
 
 38. Divide 320D. equally among 5 persons. Ans, 64D. 
 
 39. How many times is 9 contained in 6597? Ans. 733. 
 
 40. If a man spend 424D. in 8 months, how much is that 
 per month? Ans. ^^D. 
 
 41. If 9 persons sell property to the amount of 20763D., 
 how much will each man receive for his share ? Ans. 2307D. 
 
 42. Divide 378567 by 2 ; by 3 : 2)378567 3)378567 
 
 Ans. 315472 — 1 remainder. 
 
 43. Divide 278934 by 2, by 3. 
 
 44. 256788 by 3, by 4. 
 
 45. 256788 by 5, by 6. 
 
 46. 65342167by4, by 5. 
 
 47. 65342167by6, by 7. 
 
 48. 523467898 by 4, by 6. 
 
 49. 523467898 by 7, by 8. 
 
 50. 2653286 by 7, by 8. 
 
 189283—1 + 126189 
 
 Ans. 232445. 
 
 149793. 
 
 94155— S 
 
 29403974—5. 
 
 20224956—3. 
 
 218111623—6. 
 
 140214615—4 
 
 710700—12 
 
36 SHORT DIVISION. 
 
 What is Division ? How many parts are there in Division f 
 Which is the Dividend ? Which is the Divisor ? Which is the 
 Quotient ? Which is the Remainder ? When the Divisor does 
 not exceed 12, what is the rule ? What is this called ? How 
 do you prove Division ? Having the Dividend and Quotient 
 given, how is the Divisor found ? If you have the Divisor and 
 Quotient given, how can you find the Dividend ? Recite the 
 Table. 
 
 Note to Teachers. — It is expected that in every case the pu- 
 pil will commit to memory the tables and rules, before he shall 
 attempt the solution of the questions. It is evidently erroneous 
 for a scholar to attempt to work by, or follow, a given math- 
 ematical rule, unless he has learned it and understands its im- 
 port, merely from a given example, which I regret to say is too 
 frequently the case ; consequently, the scholar never has a cor- 
 rect knowledge of the science of numbers. The questions, or 
 interrogatories, given in the Review, can be used by the teacher 
 at the beginning of the rule, or as they are given, or, which 
 would be still better, at the beginning and close of the solution 
 of the questions in the rule. 
 
 LONG DIVISION. 
 
 Long Division is generally used when the divisor is more 
 than 12. 
 
 RULE. 
 
 m 
 
 1. Write down the dividend, and draw curved lines at the 
 right and left sides of the dividend, thus : )90( , and place the 
 divisor at the left hand, as in Short Division. 
 
 2. See how often the divisor is contained in the least number 
 of figures into which it can be divided, and set that number at 
 the right hand of the dividend, which is the multiplier. 
 
 3. Multiply the divisor by this figure in the quotient, and 
 place the result under the figures in the left of the dividend, 
 into which you are dividing. 
 
 4. Then subtract the result from the number directly over it, 
 and set down the remainder, which must always be less than 
 the divisor. 
 
 5. Bring down the next figure of the dividend, and place it 
 at the right of the remainder 5 if this number is less than the 
 divisor, place a cipher in the quotient, and bring down another 
 
LONG DIVISION. 
 
 37 
 
 {\fruYe ; which forms another dividend, into which divide as be- 
 fore, and so continue until all the figures are biought down and 
 divided. 
 
 ]\^ote. — If there be ciphers at the right of the dividend and 
 divisor, you can omit an equal number of each, by placing a 
 period at the left, thus : 1.00)10.00(10 answer. 
 
 EXAMPLES AND 
 
 Dividend. 
 Divisor 7)54306(7758 quotient. 
 49 
 
 53 
 49 
 
 40 
 35 
 
 QUESTIONS. 
 
 Explanation. — You can not 
 say 7 in 5, because 5 is less 
 than 7 ; but say 7 in 54 7 times, 
 7 times 7 being 49, which set 
 down under 54, and subtract, 
 and you have 5 for a remain- 
 der ; then bring down the next 
 figure, which is 3, and place it 
 to the right of the 5, which will 
 make 53 — this a new dividend , 
 then say 7 is in 53 7 times 
 and 4 remain ; then bring down 
 the next figure, which is ; 
 then say 7 is in 40, 5 times 7 
 is 35 ; set this down under 40, 
 then subtract 35 from 40, and 5 remain ; then bring down the 
 next figure 6 at the right of the 5, which will make 56 ; then 7 
 is in 56, 8 times 7 is 56, and remainder. 
 
 1. 2. 3. 4. 
 
 2)468(234 3)9636(3212 4)8456(2114 4)7324(1831 
 4 2X 9 8 4 
 
 56 
 56 
 
 remainder. 
 
 6 468 
 6 
 
 proof 
 
 6 
 6 
 
 
 4 
 4 
 
 33 
 32 
 
 8 
 8 
 
 
 3 
 3 
 
 
 5 
 
 4 
 
 12 
 12 
 
 
 
 6 
 6 
 
 
 16 
 16 
 
 4 
 4 
 
 5. 
 9)87643(9738 
 
 1] 
 
 6. 
 
 1)967884(87989 
 
 7. 
 12)138450(11537 
 
 8. 
 14)87631(6259 
 
 
 9. 
 15)161702(10780 
 
 10. 
 24)47653(1986 
 
LONG DIVISION. 
 
 11. 
 
 36)87642(2434 
 
 14. 
 42)778614( 
 
 17. 
 
 72)840658( 
 
 20. 
 125)2486542( 
 
 23. 
 165)976405( 
 
 26. 
 5.00)10000.00( 
 
 12. 
 42)765431(18224 
 
 15. 
 55)460245( 
 
 18. 
 89)442356( 
 
 21. 
 139)5006789( 
 
 24. 
 185)372405( 
 27. 
 672)89746705( 
 
 13. 
 44)653145(14844 
 
 16. 
 64)128240( 
 
 19. 
 114)6743214( 
 
 22. 
 
 147)864221( 
 
 25. 
 200)5(>06030( 
 28. 
 842)1 0647896685( 
 
 Divisor. Dividend. Quotient. 
 
 29. 75 40231 536 
 
 30. 422 253622 601 
 
 31. 342 13699840 40058 
 
 32. 3467 4586841 1323 
 
 33. 1234 46447786 37640 
 
 34. 1478 8769826000402 5933576454 
 
 35. 1600 15463420 9664 
 
 36. 27000 99607765 3689 
 
 37. How many times is 176 contained in 146524 ? 
 
 Ans. 832, 92 rem. 
 
 38. How many times is 250 contained in 925500 ? Ans. 3702. 
 
 39. Divide 5814D. among 9 persons. Ans, 646. 
 
 40. What is the quotient of 78656 divided by 45 1 
 
 Ans. 1747, 41 rem. 
 
 41. The crew of a ship consisting of 150 men, are entitled to 
 24750D. prize money ; what is the share of each man 1 
 
 Ans. 165D. 
 
 42. What number must be multiplied by 150 to produce 
 24750? Ans, 165. 
 
 43. A garrison of 2441 men have received 9764 pounds of 
 flour ; how much is the share of each man ? Ans. 4 pounds. 
 
 44. A gentleman bequeathed his estate, which was estimated 
 at 247655D., to his 4 sons and only daughter ; how much did 
 each receive ? Ans. 49531 D 
 
 45. What is the quotient of 678976500 divided by 6789 ? 
 
 46. What is the quotient of 10000000 divided by 99 ? 
 
 47. Divide 500D. equally among 10 men; 40 men; 100 
 men; 150 men. 
 
LONG DIVI3I0X. 39 
 
 48. Divide 6780D. among 15 men. Ans. 452 each 
 
 49. Divide 975 pounds of flour among 25 persons. 
 
 Ans. 39 pounds each. 
 
 50. Divide 100786 pounds by 43. Ans. 2343, 37 rem. 
 
 51. Divide 1600 bushels of wheat among 40 men. 
 
 Ans. 40 bush, each 
 
 52. There are 24 hours in a day ; how many days are there 
 in 7248 hours? ^n^. 302. 
 
 53. There are 60 minutes in an hour ; how many hours in 
 97680 minutes ? Ans, 1628 hours. 
 
 54. There are 365 days in a year ; how many years in 3285 
 days ? Ans. 9 years. 
 
 55. There are 63 gallons in a hogshead ; how many hogsheads 
 in 8796 gallons ? Ans. 139, 39 rem. 
 
 56. At 12 cents per pound, how many pounds can you have for 
 1728 cents ? Ans. 144 pounds. 
 
 57. If the dividend is 4200, and divisor 48, required the 
 quotient ? Ans, 87, 24 rem. 
 
 58. If the divisor is 25, and the dividend 5025, what is the 
 quotient? -4^^.201. 
 
 59. If you sell 84 bushels of wheat for 150 cents a bushel, and 
 take your pay in cloth at 4D. a yard, how many yards will you 
 have? Ans. 31^. 
 
 60. Divide 256976 by 41. 6267, 29 rem. 
 
 61. Divide 997816 by 59. 16912, 8 rem. 
 
 62. Divide 6283459 by 29. 216671. 
 
 63. Divide 37895429 by 112. 338352,5 rem. 
 
 64. Divide 29070 by 15 ; by 18 (+ quot.) 3553. 
 
 65. Divide 29070 by 19; by 17. 3240 
 
 66. Divide 10368 by 27; by 36. 672. 
 
 67. Divide 10368 by 54; by 18. 768. 
 
 68. Divide 2688 by 1 12 ; by 224. 36. 
 
 69. Divide 101442075 by 4025. 25203. 
 
 70. Divide 978098745 by 78978. 
 
 APPLICATION OF THE PRECEDING RULES. 
 
 1. If you add 476, 361, 842, together, and divide their amount 
 by 20, what number will result ? Ans. 83, 1 9 rem. 
 
 2. A. has 2 farms, one of 732 acres, the other 241 acres ; if he 
 should divide the land equally among his 7 sons, how many acres 
 would each receive ? Ans. 139. 
 
40 APPLICATION OF THE PRECEDING RULES. 
 
 3. A gentleman has 2 sons and 1 daughter; he is worth 
 697833D. ; he will give his daughter 23261 ID., and his sons 
 may share the remainder between them ; was the estate equally- 
 divided ? 
 
 4. If you multiply 1750 by 17, and divide it by 16, what will 
 be the result? Ans. 1859, 6 rem. 
 
 5. Divide 420D. among three men and a boy. 
 
 6. If you purchase 120 pounds of flour for 7D. and sell it for 
 6 cents a pound, will you gain or lose, and how much ? 
 
 7. Make out a bill of the following articles, namely: 18 
 pounds of nails at 14 cents per pound ; 47 pounds of sugar at 
 1 1 cents per pound ; 5 gallons of molasses at 44 cents per gal- 
 lon ; 4 pounds of tea at 75 cents per pound ; 9 yards of calico at 
 22 cents per yard ; and 2 brooms at 25 cents each. 
 
 Ans. 15D. 37c. 
 
 8. A gentleman has 4 notes of hand : the first is for 474D. 
 20c. ; the second for 760D. 42c. ; the third for 285D. 68c. ; the 
 fourth for 369D. 31c. ; when received, he must pay a debt of 
 1562D. 96c. ; how much will remain ? Ans. 326D. 65c, 
 
 9. "What is the amount of 40 bushels of corn at 84 cents a 
 bushel ; 25 bushels of rye at 97 cents a bushel ; and 30 bushels 
 of wheat at ID. 25c. a bushel? Ans. 95D. 35c. 
 
 10. What is the quotient of 55 multiplied by 22 and -7-15 ? 
 
 Ans. 80, 10 rem 
 
 11. Purchased 1 set of chairs for 4D. 25c. ; table 7D. 50c. ; 
 bureau for 27D. ; a watch for 60D. 50c. ; writing-desk for 18D. ; 
 a clock for 40D. ; required the amount. Ans. 157D. 25c. 
 
 12. A gentleman of Philadelphia lately purchased 4 tracts 
 of unimproved land in the western country ; in the first tract 
 there are 2780 acres ; in the second, 4027 acres ; third, 3012 
 acres ; fourth, 8760 acres, all of which is worth 5D. per acre ; 
 he will divide all the lands among his 3 sons ; how many acres 
 did he purchase, and how much land will each son receive, and 
 how much is it all worth ? 
 
 Ans. He purchased 18579 acres ; each son will receive 6193 
 acres ; worth 92895D. 
 
 13. A farmer has 3764 acres of land; he will sell A. 642 
 acres, B. 224 acres, C. 180 acres, and D. 354 acres ; the re- 
 mainder is to be equally divided among his 4 children ; how 
 much land wiU each receive, and how much will each share be 
 worth at 65D. an acre ? Ans. 591 A. ; worth 384 15D. 
 
 14. If you sell 44 bushels of oats for 35c. per bushel, 25 
 bushels of corn for 76c. per bushel, and 6 cords of wood for 
 4D. per cord, how much would you receive ? Ans. 58D. 40c. 
 
; APPLICATION OF THE PRECEDING RULES. 41 
 
 15. Required the amount of the following articles, namely : 
 1 set of knives and forks at 3D. ; 1 Cashmere shawl at 750.^ 
 47 yards Irish sheeting at 90c. per yard ; 7 yards of broadcloth 
 at 6D. per yard ; one Russian hat at 5D. ; 11 yards of silk at 
 2D. per yard ? Ans, 189D. 30c. 
 
 16. A gentleman deposited in bank, at one time 4638D., at 
 another 21 6D., at another 8329D., at another 1212D. ; required 
 the amount in bank. Ans, 14395D. 
 
 17. A ship in sailing to a distant part of the world : from one 
 port to another was 6243 miles, to another port 4123 miles, to 
 another 9401 miles, and thence home 130 miles; required the 
 number of miles she sailed. Ans. 19897. 
 
 18. Supposing you gain 34568D., then 1245D., again 2467D., 
 and then lose 2365D. ; again, you gain 41210D., and then lose 
 39300D. ; how much will you have left? 
 
 19. A man gains 367D.,then loses 423D. ; a second time he 
 gains 875D., and loses 912D. ; he then gains 1012D. ; how 
 much has he gained in all ? Ans. 91 9D. 
 
 20. A farmer agrees to furnish a merchant 40 bushels of rye 
 at 60 cents a bushel, and take his pay in coffee at 16 cents a 
 pound ; how much coffee will he receive ? Ans. 150 pounds. 
 
 21. A farmer sold 3 cows at 25D. each, and 1 pair of oxen at 
 65D., he agrees to take in payment 60 sheep ; how much per 
 head do his sheep cost him? Ans. 2D. 33c. .3-f- 
 
 22. The exports of the United States from October, 1841, to 
 October, 1842, were as follows : of products of the sea, 2823010 
 D. ; of the forest, 5518262D. ; of agriculture, 73688113; of 
 manufactures, 1094061 ID. The total value of the imports for 
 the same period was 100162087D. How much did the total 
 value of the imports exceed that of the exports ? 
 
 23. If M. R. is worth fourteen millions, two hundred and 
 fifty thousand D., how many men could he make comfortably 
 rich by giving them 25 thousand D. each ? Ans. 570. 
 
 24. A celebrated personage of England has a salary of seven 
 hundred and fifty thousand dollars annually ; how much is that 
 daily, and how many teachers would it pay at a salary of 500D. 
 a year? -4w5. daily income 2054D. 79c. 4+ 1500 teachers. 
 
 25. What cost 6400 yards of riband at 25c. per yard ? 
 
 Ans. 25rr:iD. 4)6400=1600D. 
 
 26. What cost 3600 yds. at 12^c. a yard? 12i=-iD. 8)3600. 
 
 27. What cost 2400 pounds of cheese at 6^0 . a pound ? 
 
 Ans.6l=j\D. 16)2400. 
 
 28. What cost 600 bushels of potatoes at -J of a dollar a 
 bushel? Ans. 3)600=200D. 
 
 4* 
 
43 APPLICATION OF THE PRECEDING RULES. 
 
 29. What cost 50 bushels of wheat at ID. 25c. a bushel? 
 
 Ans, 62D. 50c 
 
 30. Supposing 1800 apple-trees to be planted in 72 rows, 
 how many trees are there in each row ? Ans. 25. 
 
 31. A merchant bought 8200 barrels of flour; he then sold 
 3756 barrels ; he again bought 5000 barrels, after which he sold 
 4879 barrels ; how many has he on hand? Ans. 4565. 
 
 32. A man sets out on a journey, intending to travel 2450 
 miles ; how far must he go every day to perform the journey in 
 50 days ? Ans. 49 miles. 
 
 33. The quotient of an operation in division is 1763, the 
 dividend 8435955 ; required the divisor. Ans, 4785. 
 
 34* What number is that which, being multiplied by 7969, 
 the product will be 1864746 ? Afis. 234. 
 
 REVIEW. 
 
 What is Long Division ? What is the rule ? At which hand 
 of the dividend must the divisor be written ? Why do you be- 
 gin at the left hand of the dividend to divide ? Ans. Because 
 numbers decrease, &c. Under this rule, where must the quo- 
 tient be written ? How many figures of the dividend must first 
 be taken ? How can you find the one half of any number ? one 
 fourth ? one sixth ? one eighth ? &c. How many rules have 
 you now been through ? Name them ? Why are they called 
 the principal or fundamental rules of the science ? Ans. Be- 
 cause they are the foundation of all the other rules, and by their 
 use and operation all calculations in arithmetic are performed. 
 Perform the following examples on the slate, as the signs indi- 
 cate : 878344-284+65 + 32 + 100=88315 Ans. 876345723 
 —267001345 = 609344378 Ans. 692784578 X 27839421 rzr 
 19286721529249338 Ans. 202884150-^4025 = 50406 Ans. 
 2600—600=2000 + 1828 = 3828 Ans. 9788x97^29x17 + 
 79—400x92. 
 
 35. From 95,000,000 take 18,999,999; from 777,777-- 
 688,888. 
 
 36. If a miller should purchase 578 bushels of wheat for 
 482D., and sell 482 bushels for 375D., how many bushels will 
 he have left, and what sum will he have paid for the quantity 
 remaining? Ans. 96 bushels; paid 107D. 
 
 37. From eleven millions take eleven hundred and eleven. 
 
 Ans. 10998889, 
 
 38. What is the difference between 84 and 76 ? 90 and 22 ! 
 400 and 380 ? 7868 and 9897 ? 
 
DECIMAL FRACTIONS. 43 
 
 Methuselah died aged 969 years, and Adam aged 930 ; what 
 is the difference of their ages ? How many years were required 
 to have extended the life of Methuselah to 1000 ? How many 
 for Adam 1 How many years are their united ages ? How 
 many years since the death of Adam? How long from the 
 death of Adam to the deluge ? How many years from the del- 
 uge to the present time ? In what year was America discov- 
 ered by Columbus ? How many years since ? How many 
 years since the declaration of independence 1 How are you 
 pleased with the science of Arithmetic ? 
 
 DECIMAL FRACTIONS, 
 
 Adapted to the Currency of the United States. 
 
 We now come to treat of Decimals in connexion with the 
 currency of this country, by which it is believed that calcula- 
 tions generally, at least where money is concerned, will be much 
 facilitated, and bring to view a better and easier method of com- 
 putation than has heretofore been in general use. That fractions 
 must of necessity occur in almost every transaction, none will 
 presume to deny, and all will admit that no part of the science 
 of Arithmetic is more difficult of comprehension to the juvenile 
 mind than fractions, especially those termed vulgar ; but in this 
 treatise, this part of Arithmetic is introduced in a pleasing and 
 agreeable manner, particularly calculated to attract the attention 
 of youth, and overcome every obstacle and difficulty ; and the 
 author entertains the belief that he has in part succeeded. 
 
 Decimal is derived from the Latin word decern (signifying 
 ten, because they increase and decrease in a tenfold proportion, 
 like whole numbcBs), and is a part of a whole number, or unit, 
 which is distinguished by a period, decimal point, or separatrix, 
 placed at the left of the figure or figures, thus: .5=:y^Q-, or one 
 half; .25 =y2_5^= twenty-five hundredths = one quarter ; 6 .8, six 
 and eight tenths; .15 =y^^5__ fifteen hundredths; .144=:j^qYo» 
 one hundred and forty-four thousandths, &c. ; the number above 
 the line, ^^"^j is called the numerator, and the number below the 
 line, Yooo? ^^ called the denominator, and must consist of one 
 place more than the numerator, and shows the number of j:|^rta 
 into which a unit or anything is divided, thus, ^Vo ' ^^ 7^^ ^^^ 
 25 parts in 100 parts, you own one fourth, because 25x4 = 100== 
 4"§§, the numerator and denominator being alike, are equal to 1 
 The word fraction implies broken parts of a unit or whole number 
 
44 DECIMAL FRACTrONS. 
 
 In counting decimals from left to right, they decrease in a tenfold 
 proportion, with a view to give the decimal expression ; but to 
 enumerate them from right to left, they increase in a tenfold pro- 
 portion, the same as whole numbers. Ciphers annexed to a deci- 
 mal do not increase its value, as in whole numbers ; their value is 
 determined by their distance from the units' place, or decimal 
 point; thus .1, or .10, are equal, being read one tenth and ten 
 hundredths, which is the same ; but when prefixed, that is, placed 
 at the left of a figure, it decreases in a tenfold proportion, thus, 
 .1 .01 ; in this position their value is different ; figure 1 and 
 having changed places, it has decreased the value of 1 in a 
 tenfold proportion, it being only one tenth part as much as figure 
 .1 ; .5, .05, .005=-3-^o, j^, j^^, .75=1=/^%, &c. 
 
 A mixed number is that which consists of a whole number 
 and decimal, as 4.4, 21.42, 100.75, &;c. 
 
 There are various kinds of decimals, such as circulating 
 decimals ; but the limits of this work will not permit a full in- 
 vestigation ; neither does the occasion require it, suffice to say ; 
 in circulating decimals, if one figure only repeats, it is called a 
 single repetend, as .1111, .6666, &;c. 
 
 A compound repetend, thus : .0101, .379379, &;c. There 
 are a variety of examples under this head, but in most cases 
 three or four places of decimals will be sufficient, unless great 
 accuracy is required. "We will now introduce decimals as ap- 
 plicable to the currency of this country, which in its nature and 
 operations is purely decimal. The laws of our country require 
 that all transactions in money, both as relates to government and 
 individuals, shall be performed in dollars and cents, or, as it is 
 termed, "federal currency." This currency increases and 
 decreases like decimals, in a tenfold proportion, thus : — 
 
 10 mills equal 1 cent, is - 10 mills, m. mill. 
 
 10 cents " 1 dime - - 100 " d. dime. 
 
 10 dimes '' 1 dollar - - 1000 " D. dollar. 
 
 10 dollars" 1 eagle - -10000 " E. eagle. 
 It is customary with accountants to use only two of the above 
 denominations, namely, Dollars and Cents, the cents being 
 hundredths of a dollar ; and the fractional part of a cent is ex- 
 pressed thus : i, one quarter; -i, one half; |, three quarters of 
 a cent ; but it would be more correct to give the exact number 
 of mills. There is but one difficulty in learning decimals — that 
 is, where to place the decimal point or separatrix, which will 
 depend entirely upon the nature of the question, for which there 
 is a sufficient number of rules and examples ; and in every other 
 respect, their operation is the same as whole numbers. 
 
DECIMAL FRACTIONS. 45 
 
 AMERICAN COINS. 
 
 The Mill is not a coin, but the tenth part of a cent. 
 
 The Half-Cent is a copper coin, 200 being equal to one dol- 
 lar. 
 
 The Cent is a copper coin, TOO being equal to one dollar. 
 
 The Half-Dime is the smallest silver coin, being equal in 
 value to 5 cents, or 20 to one dollar. 
 
 The Dime is a silver coin, being equal to 10 cents, or 10 to 
 one dollar. 
 
 The Quarter-Dollar is a silver coin, =25 cents, or 4 to a dollar. 
 
 The Half-Dollar is a silver coin, =50 cents, or 2 to a dollar. 
 
 The Dollar is the largest silver coin, equal to 100 cents, -^^ 
 of an Eagle. 
 
 The Quarter-Eagle is the smallest gold coin, =2D. 50cts. 
 
 The Half-Eagle is a gold coin, =5D. 
 
 The Eagle is the largest gold coin. =10D. 
 
 The above coins, in their composition, are not of pure gold 
 and silver, but composed in part of alloj/, a table of which will 
 be given in its proper place. 
 
 To write sums in dollars and cents, the dollars are placed at 
 the left, and the cents at the right hand, separated by a period. 
 
 D. c. D. c. D. c. 
 
 Thus, 24 .25. or 20 .09: or 18 .00: (see the following 
 table.) 
 
 
 
 
 
 
 
 :=: ^ r^ ^ S 
 
 
 
 
 
 
 ;©;=! o ^ V- 
 
 
 
 
 
 
 •no o c 
 
 
 
 
 
 
 s of 
 ofd 
 tens 
 
 tens 
 ens i 
 
 
 TABLE I. 
 
 
 
 
 
 
 
 
 
 A 
 
 
 
 
 
 ee 
 
 S-gJs Ss 
 
 
 
 
 
 B 
 
 o c bD:=i s c 
 
 
 
 
 
 o 
 
 ^ P rt o .5 O 
 EhJUhqqo 
 
 D. 
 
 c. 
 
 parts. 
 
 c. 
 
 
 1 .23 
 
 1 
 
 and .23 
 
 
 or 123 
 
 
 k 1 3 .12 
 
 13 
 
 .12 
 
 
 1312 
 
 
 r 4 2 .09 
 
 42 
 
 .09 
 
 
 4209 
 
 
 6 7 2 .18 
 
 672 
 
 .18 
 
 
 67218 
 
 
 5 6 9 .04|- 
 
 569 
 
 .04 
 
 and f- 
 
 56904 
 
 .75 
 
 7 15 .42^ 
 
 7015 
 
 .42 
 
 t 
 
 701542 
 
 .50 
 
 2 14 6 .81A 
 
 2146 
 
 .81 
 
 214681 
 
 .25 
 
 To reduce oents to dollars, divide by 100, or separate two 
 figures from the right, and those on the left will be dollars ; to 
 
Too 
 Tooo 
 
 46 DECIMAL FRACTIONS. 
 
 change fourths of a cent to cents, divide by 4 ; to change halves 
 of a cent to cents, divide by 2 ; to change thirds of a cent to 
 cents, divide by 3 ; to change mills to dollars, divide by 1000, 
 or separate three figures from the right of their number, thus : 
 1.000=1D. 
 
 TABLE II. 
 
 . o g « -5 
 
 TO » o ^'^ ^ to 
 
 c« P! «> rt c 03 . ^ 
 
 2 g S S 2 ^ ^ ^^^ 'S ^ 
 
 y5_^ r= .5 « 5 tenths. 
 
 -J^i =.0 7 '* 7 hund's. 
 
 = .0 4 5 "45 thous's. 
 
 illio = .14 2 5 " 1425 tenth's 
 
 6J^ = 6 .7 « 6 & 7 tenths. 
 
 4iWo75(r(5 = 4 .0 7 " 4 & 7 mill's. 
 
 22j2^% = 22 .2 4 " 22 & 24 hun'ths. 
 
 3/00-0^0-000 = 3 .0 4 " 3 & 4 10 mill's 
 
 444 4 4 4 .0 " 444 4 h. & 44 
 
 TABLE III. 
 
 Whole numbers. Decimals. 
 
 • nd CO *J '-' S 
 
 "s 3 ^ § . ^- ^ <=^ g.'^ 1 .0 
 
 0«4-ic!<^'^Sl5 5 ^ ?^ 3 C =^ "S S 
 
 owj=!bJ»gflg.-i^ rSoo^onH^oo 
 
 oo-?oo_2pgj3 00000 -^^ 000 
 
 987654321 .123456789 
 
 QUESTIONS AND ANSWERS. 
 
 Quest. When the folio w^ing fractions occur, such as -f^, -f^^^ 
 yJI^, how is a unit supposed to be divided ? Ans. Into 10 equal 
 parts, called tenths ; and each tenth into 10 other equal parts, 
 called hundredths; and each hundredth into 10 more equal 
 
ADDITION OF DECIMALS. 47 
 
 parts, called thousandths, &c. Quest. How will you write them 
 down so as to give the decimal expression ? Ans. By taking 
 away the denominator, and placing a period, or decimal point, 
 before the numerator, thus : .5, .25, .75, .8, .2, &c. Quest, 
 What is the use of a decimal point in fractions ? Ans. It shows 
 the place of the units, and separates the fraction from the figures 
 at the left (if any) which are whole numbers, and may be called 
 the separatrix. Quest. Write down y^^ (7 hundredths) decimally. 
 Ans. .07, also yoVo'=-^^^ ^^^ T5W = •004= eight thousandths 
 and four thousandths. Quest. How do decimals decrease in 
 value from left to right? Quest. How do you determine the 
 value of a decimal ? Ans. By its distance from the decimal point. 
 ^ For a more full and perfect explanation of fractions, see Vul- 
 gar Fractions, which, if the teacher please, can follow decimals. 
 
 ADDITION OF DECIMALS. 
 
 RULE. 
 
 1 . Write down the numbers under each other, observing to 
 place tenths under tenths, hundredths under hundreths, &c 
 Be particular that the decimal points stand directly under each 
 other in a perpendicular line, both in the given number and in 
 the sum. 2. Then perform the operation the same as in Sim- 
 ple Addition. Proo/— as Simple Addition. 
 
 EXAMPLES AND QUESTIONS. 
 
 I. 2. 3. 4. 5. 6. 
 
 41 .653 46 .23456 12 .3456 48 .9108 21 .037 49 .607 
 
 36 .05 24 .90400 7 .891 1 .8191 15 .122 50 .421 
 
 24 .009 17 .00411 2 .34 3 .1030 12 .042 18 .1610 
 
 1 .6 3 .01111 5 .6 .7012 10 .120 71 .65843 
 
 103 .312 91 .15378 28 .1766 54 .5341 58 .321 
 
 7. Add 1?8 .34565+7 .891+2 .34 + 14 .0011 together. 
 
 ^71^.36 57775. 
 
 8. Add .7509+ 0074+.69+.8408+.6109 together. 
 
 Ans. 2 .9000. 
 
 9. Add .7569 + . 25+, 654+.199 together. Ans. 1 .8599 
 
48 SUBTRACTION OF DECIMALS. 
 
 10. Add 71 .467+27 .94+16 .084+98 .009+86 .5 to- 
 gether. Ans. 300 .000 
 
 11. Add to 9 .999999 the millionth part of a unit. 
 
 Ans. 10 000000 
 
 12. 13. 14. 15. 
 
 D. c. m. D. d. c. m. D. d. c. m. E. D. d. c. ra.dec.of m 
 
 14 .25 .6 12 .2 .6 .4 147 .6 .3 .4 121 .7 .6 .5 .3 .25 
 13 .14 .4 10 .8 .7 .3 121 .7 .5 .4 324 .6 .5 .4 .5 .75 
 13 .12 .3 15 .7 .6 .4 134 .3 .4 .6 242 .5 .6 .0 ,4 .25 
 
 15 .10 .2 16 .4 .3 .2 147 .6 .5 .4 221 .3 .4 .5 .G .50 
 
 55 .62 .5 55 .3 .3 .3 551 .3 .8 .8 910 .3 .2 .5 .9 .75 
 
 16. Add 278D. 9d. 7c. 4m. .21 + 87D. 6d. 9c. 5ni. .75 + 
 396D. 4d. 7c. 6m. .25 + 464D. 6d. 3c. 5m. .35, together. 
 
 Ans. 1227D. 7d. 8c. Im. .56. 
 
 17. A gentleman has the following charges entered in his 
 ledger: A. 420D. 27c. 5m.; B. 671D. 87c. 6m.; C. 742D. 
 33c. 4m. ; D. 621D. 25c. 7m.; E. 520D. 47c. 6m. ; required 
 the amount. Ans. 2976D. 21c. 8m. 
 
 18. A farmer sold his wheat for 1724D. 87c. 5m. ; rye, 
 1296D. 18c. .75 ; corn, 964D. 25c. 8m. ; oats, 250D. 37c. 5m. ; 
 required the amount of the value of his produce. 
 
 Ans. 4235D. 69c. .55. 
 
 19. A drover paid for his drove as follows: oxen, 1800D. 
 75c.; cows, 878D. 50c.; sheep, 502D. 31c. ; he also purchased 
 2 horses for 75D. 50c. each ; required the cost of his drove. 
 
 Ans. 3332D. 56c. 
 
 20. Add 420E. 8D. 27c. 6m. .75 + 371E. 8D. 29c. 4m. .25 
 +781E. 5D. 17c. .8m. 37+416E. 7D. 14c. 5m. .50, together. 
 
 Ans. 1990E. 8D. 89c. 4m. .87 
 
 SUBTRACTION OF DECIMALS. 
 
 RULE. 
 
 1. Write down the numbers the same as in Simple Subtrac- 
 tion, observing that the decimal points stand directly under each 
 other. 2. Then subtract in the same way as in whole numbers 
 and place the decimal point in the remainder under those above 
 it. Proof— 0.8 Simple Subtraction. 
 
SUBTRACTION OF DECIMALS. 
 
 41» 
 
 From 125 .64000 
 Take 95 .58756 
 
 145 .00 14 
 96 .84 5 
 
 .674 
 .91 
 
 2 .764 
 2 .371 
 
 761 .8109 
 18 .9113 
 
 Rem. 
 
 30 .05244 
 
 48 .16 8 
 
 .764 
 
 : .393 
 
 742 .899« 
 
 6, 
 271 .0 
 215 .7 
 
 7, 
 464 .000 
 376 .784 
 
 8. 
 
 719 .10009 
 
 7 .121 
 
 9. 
 270 .200 
 75 .4075 
 
 10. 
 480 .OOOCi 
 245 .0075 
 
 
 11. 12. 
 107 .0000 236 .OOC 
 .0007 .54S 
 
 ) 
 
 1 
 
 13. 
 
 1000000 
 .1 
 
 
 D. 
 
 % 431 
 
 271 
 
 14. 
 c. m. 
 .76 .8 
 .25 .6 
 
 15. 
 D. c. m. 
 671 .37 .4 
 581 .65 .3 
 
 16. 
 E. D. d 
 
 331 .7 .8 
 224 .8 .5 
 
 . c. m. 
 .5 .4 
 .6 .2 
 
 160 
 
 .51 .2 
 
 89 .72 .1 
 
 106 .9 .2 
 
 ; .9 .2 
 
 17. 
 E. D. d. c, m. dec. 
 336 .4 .0 .2 .4 .25 
 237 .8 .1 .3 .2 .10 
 
 18. 
 
 496 .0784 
 379 .6809 
 
 19. 
 
 700 .94870 
 199 .48397 
 
 20. 
 
 42 .84 
 19 .96432 
 
 21. 22. 23. 
 
 658 .00000 66666 .555556 78888 . 
 
 1 .77777 57777 ,666666 69999 . 
 
 7777 1 J 
 8888 J 
 
 24. 
 000000009 
 300000011 
 
 25. From 1078D. 7d. 8c., take 984D. 4d. 9c. 
 
 Ans. 94D. 2d. 9c. 
 
 26. From 99D. 99c. 9m., take 9d. 9m. Ans. 99D. 09c. 
 
 27. From 19E., take ID. 99c. .5. Ans. 188D. 00c. 5m. 
 
 28. A merchant deposited in bank 900E., he drew out at one 
 time 2434D. 50c. 5 at another 3224D. 18c. .75 ; how much has 
 he in bank ? 
 
 29. A. sold wheat to B. to the amount of 7840. 50c. ; corn, 
 347D. 75c. ; B. paid in part for the wheat, 525D. 25c. ; for 
 the corn, 235D. 12c. 5m. ; how much remains due from B. to 
 A.? Ans.37]I).S7c.5m, 
 
 30. A merchant in the city sent his clerk into the country to 
 pollect debts ; of A. be received 475D, 65c. ; of B. 8740. 87o 
 
M) MULTIPLICATION OF DECIMALS. 
 
 5m. ; of C. 721D. 35c. 5m. ; on his return he lost 565D. ^H/^ 
 5m. ; how much will he be able to pay his employer? "^ j 
 
 Ans, 1506D. 50c. 5m. 
 
 31. A. paid at the store, for flax 12D. 25c., tallow 13D. 
 27c. 5m., butter 14D. 85c., groceries 18D. 25c. 4m., for which 
 he gave the merchant a bank-note of lOOD. ; how much change 
 must A. receive back? Ans. 41D. 37c. Im. 
 
 32. What is the difference between 900d. and 900c. ? 
 
 Ans. SID. 
 
 33. A man, on leaving home, left in his desk lOOOOD.; during 
 his absence, his family had occasion to use a part of it, but could 
 not tell how much ; on counting it, he found that he had 7321D. 
 47c. 6Tn. ; how much was taken? Ans. 2678D. 52c. 4m. 
 
 34. From 799999 .8888888, take 654321 .9999999. 
 
 MULTIPLICATION OF DECIMALS. 
 
 RULE. 
 
 1. Write down the multiplicand, and under it the multiplier, 
 in the same manner as Simple MuUiplication, and multiply 
 without regard to decimal points. 
 
 2. When you have finished multiplying, begin at the right- 
 hand figure of the product, and count off as many figures toward 
 the left as there are decimal figures in the multiplicand and 
 multiplier, and there place the decimal point. If the number 
 of figures in the product be less than the decimal figures in the 
 multiplicand and multiplier, prefix a suflficient number of ciphers 
 at the left of the product, to equal those above the line, then 
 place the decimal point at the left of the ciphers. 
 
 EXAMPLES AND QUESTIONS. 
 
 1. 
 
 D.3.024 
 D.2.023X 
 
 2. 
 
 Feet 25.238 
 12.17 
 
 3. 
 
 .007853 
 .035 
 
 9072 
 6048 
 6048 
 
 176666 
 25238 
 50476 
 25238 
 
 39265 
 23559 
 
 000274855 
 
 D.6 117552 
 
 
 F.307,14646 
 
 
MULTIPLICATIOJf 0F DKCIMALg. 51 
 
 4. 5. 6. 
 
 .004 Yards 6.21 D.7.02 
 
 .004 D.2.25 5.27 
 
 R .000016 3105 4914 
 
 Multiply 4 by 4=16, then 1242 1404 
 
 ring down the 4 ciphers and 1242 3510 
 
 make the point ; or thus : . , , 4 
 
 X . , , 4 (four thousandths)^ D13.9725 D36.99.5.4 
 
 16; prefix 4 ciphers, .000016. 
 
 Explanation. — In table 2, the pupil will notice that ID. is the 
 unit in our currency, and that all of the less denominations are 
 so many decimals, or fractional parts of the dollar, decreasing 
 in a tenfold proportion from the decimal point, as dimes, cents, 
 mills, &c. ; therefore, in the (1) example, the multiplicand is 
 D3.02 cents and 4 mills, or y^^^ of a dollar. After you have 
 multipled byall the figures in the multiplier, that is, D2.02 cts. 
 and 3 mills, or yffo" ^^ ^ dollar, you count off as many decimal 
 figures in the product as you have in ±e multiplicand and mul- 
 tiplier, namely 6, and there make the decimal point, and the figure 
 or figures at the left of the point will be dollars=6D. lie. 7m. 
 and live hundred and fifty*two thousandths of a mill, which is a 
 little more than half a mill, &;c. In the (2) example 25.238 feet 
 are given to be multiplied by 12.17 feet ; multiply and count off 
 as before, and you have five decimal places, and three integers, 
 or whole numbers, which is 307 feet ; .14646 of a foot, equal to 
 about 1|- inches. In the (3) example, after multiplying, we find 
 the product to consist only of 6 figures ; therefore we must prefix 
 
 3 ciphers to make the number equal to the multiplicand and 
 multipler. In the (4) example, the product is even, or equal to 
 the multiplicand and multiplier. In the (5) example, we multi- 
 ply 6.21 yards by D2.25, this will give the value of the cloth at 
 that price, there are 4 given decimal figures, therefore count off 
 
 4 in the product, and we have Dl 3.9725 = 13 dollars 97 cents 
 and one quarter of a cent. In the (6) example we multiply dollars 
 by dollars, or dollars and fractions, and the product may be point- 
 ed off thus : D.36.9954, or 36D. 99c. 5.4m. ; observe that the 
 cents must always have two places, thus : (75) unless you divide 
 it into dimes and cents, then but one place, as, 7d. 5c., which 
 is the same =r 75 cents. (As a fraction, or decimal, is always 
 less than one, or unity, it follows that unity can not be produced 
 by the multiplication of decimals, for .9x.9 = .81; or .999 X 
 .999 = .998001, (fee; but whole numbers maybe produced by 
 the addition of several fractions, thus : .5 + .5-|-.54-.5=2.0 ; or 
 ^+Tiy+l^TJ+T^=T^=2 ' ^^^ ^ fraction may be divided by a 
 
63 MULTIPLICATION OF DECIMALS. 
 
 whole number, whether it be a decimal or vulgar ; but the quo- 
 tient must decrease from the decimal point in proportion to 
 the value of the divisor, thus : .624672-^482, v/hole number, 
 = .001296 Ans, 
 
 7. What cost 27.5 pounds of butter at 12.5 cents per pound? 
 27.5 pounds 
 
 .125 X Explanation. — First multiply in the usual way 
 
 ' by the multiplier, which is y\fo\ of a dollar:^ 
 
 1375 12^ cents, or -i of a dollar, and the product is 
 
 550 34375, count off 4 figures for decimals of a dol- 
 
 275 lar, and there place the period, and you have one 
 
 figure (3) at the left, Avhich is 3D. 43c. .75. 
 
 D3.4375 Ans. And 12.5 pounds at 27.5 cents per pound would 
 amount to the same. 
 
 8. What cost 30.5 pounds of nails at 9c. 5m. per pound? 
 
 Ans, 2D. 89c. .75. 
 
 9. What cost 42.25 pounds of leather at 20c. per pound? 
 
 Ans. 8D. 45c. 
 
 10. What cost 21.5 pounds of flax at 4c. 5m. per pound? 
 
 Ans. 96c. .75. 
 
 11. What cost 32.5 yards of sheeting at 14c. 5m. per yard ? 
 
 Ans. AD. 71c. .25. 
 
 12. What cost 14.75 pounds of sugar at 12c. 5m. per pound I 
 
 Ans. ID. 84c. 3m. .75. 
 
 13. What cost 18.25 yards of calico at 13c. 5m. per yard ? 
 
 Ans. 2D. 46c. 3m. .75. 
 14 What cost 16 pounds of coffee at 13c. .75 per pound ? 
 
 Ans. 2D. 20c. 
 
 15. What cost 14 pounds of indigo at 2D. 25c. per pound ? 
 
 Ans. 31D. 5Cc. 
 
 16. What cost 4.25 yards of broadcloth at 3D. 75c. per yard ? 
 
 Ans. 15D. 93c. .75. 
 
 17. What cost 6.5 yards of broadcloth at 4D. 75c. per yard ? 
 
 Ans. 30D. 87c. .5. 
 J.S. What cost 3.5 bushels of wheat at ID. 50c. per bushel? 
 
 Ans. 5D. 25c. 
 
 19. What cost 18.22 bushels of oats at 37.5c. per bushel? 
 
 Ans. 6D. 83c. .25. 
 
 20. What cost 12.5 bushels of flaxseed at 87.5c. per bushel? 
 
 Ans. lOD. 93c. .75. 
 
 21. What cost 4.25 bushels of cloverseed at 4D. 25c. per 
 bushel? Ans. 18D. 06c. .25. 
 
 22. What cost 742.5 pounds of pork at 8.5c. per pound ? 
 
 Ans. 63D. lie. .25. 
 
FDLTIPLICATION OF DECIMALS. 
 
 53 
 
 23. What cost 434 pounds of ham at 12.5c. per pound? 
 
 Ans. 54D. 25c. 
 
 24. What cost 75.5 pounds of lard at 7.5c. per pound? 
 
 Ans. 5D. 66c. .25. 
 
 25. What cost 630.5 pounds of cheese at 6.25c. per pound? 
 
 Ans. 39D. 40c. .625. 
 
 26. What cost 500 pounds at 3m. per pound ? Ans. ID. 50c. 
 
 We have remarked, that prefixing ciphers to a decimal dimin- 
 ishes its value ten times for every cipher prefixed. Thus, take 
 i^"*^, .05=y%%, by prefixing one cipher; -005=^^^^ by prefixing 
 two ciphers ; again, the value of a decimal figure depends on its 
 place f or distance from the decimal pom^, whether there be ciphers 
 prefixed or not, thus : .005 x .005. 
 
 . 5 
 .000025 
 
 Here the multiplicand and the multiplier are the same, 
 that is, (five thousandths) ; the ciphers are omitted in the mul- 
 tiplier, but their places are counted in the product. Multiply 
 2.034 by .014 = .028476. In this example the multiplicand is 
 a w^hole number and decimal, vi^hile the multiplier is a small 
 decimal (fourteen thousandths) ; in this case the product is less 
 than the multiplicand, it having diminished in consequence of 
 the small value of the multiplier, it being many times less than 
 unity ; it follows that the product must diminish in the same 
 proportion, and is one kind of subtraction. When a decimal 
 number is to be multiplied by 10, 100, 1000, &c., the multipli- 
 cation may be made by removing the decimal point as many 
 places to the right hand as there are ciphers in the multiplier, 
 and if there be not so many figures on the right of the decimal 
 point, supply the deficiency by annexing ciphers, thus : — 
 
 
 
 r 1^ 1 
 
 100 
 
 r 
 
 67.9 
 
 
 
 . 
 
 679. 
 
 
 6.79 X< 
 
 1000 y = 
 
 6790. 
 
 
 
 10000 1 
 
 
 67900. 
 
 
 
 100000 J 
 
 679000. 
 
 *>.7, 
 
 31.00467X10.03962. 
 
 Ans. 
 
 311.2751050254. 
 
 M. 
 
 596.04X0.00004. 
 
 Afta. 
 
 
 29. 
 
 Multiply 341.45D. by .007. 
 
 Ans. 
 
 
 30. 
 
 D.26.000: 
 
 ^75 X .00007. 
 
 Ans. 
 
 .00182002625. 
 
 31. Multiply three hundred, and twenty-seven hundredths 
 by 31. Ans. 9308.37. 
 
 5* 
 
54 DIVISION" OF DECIMALS. 
 
 DIVISION OF DECIMALS. 
 
 The operations in division of decimals are the same as whole 
 numbers, with the exception of the decimal points. We have 
 shown in multiplication of decimals, that one decimal multiplied 
 by another, the product will contain as many places of decimals 
 as there were in both factors ; then it follows, that if this product 
 be divided by one of the factors, the quotient will be the other 
 factor ; therefore, in division, the dividend must contain as many 
 places as the divisor and quotient together. The quotient will 
 contain as many places as the dividend^ less those of the divisor. 
 
 RULE. 
 
 1. Write down the dividend and divisor, then divide in the 
 same way as in Simple Division, without regard to the decimal 
 points. 2. Whe.n the division is finished, count oiF as many 
 ilgures from the right of the quotient, as the decimal figures in 
 the dividend exceed those in the divisor, and there place the 
 decimal point. 3. If the decimal figures in the quotient should 
 be less than is required above, prefix as many ciphers to the left 
 of the quotient as are required, and there place the decimal point. 
 4. When the divisor is larger than the dividend, then annex 
 ciphers until it exceeds the divisor ; then place the decimal point 
 accordingly ; when there is a remainder, annex ciphers and 
 continue the operation. (See examples, &c.) 
 1. .1812)4.18000(23.0 Ans. 1812 In this example we 
 3624 23.0 X have a whole num- 
 bej. and decimal in 
 
 5560 54360 the dividend, and a 
 
 5436 3624 decimal for a divi- 
 
 rem, \- 1240 sor ; divide as in 
 
 Rem. 1240 Long Division, and 
 
 Proof, 4.1 8000 you have three quo- 
 
 tient figures, a part of which must be a whole number ; therefore, 
 by the rule, you will observe you have 5 places of decimals in 
 the dividend and 4 in the divisor, difference 1 ; therefore count 
 ofif from the right one place, and there make the point. 
 2. .957)7.25406(7.58 In this example we have 5 decimal 
 6699 places in the dividend and 3 in the 
 
 divisor, difference 2 ; therefore, 
 
 5550 count off* 2 places from the right 
 
 4785 in the quotient, and there make the 
 
 point. 
 
 7656 
 7656 
 
I 
 
 DIVISION OF DECIMALS. 55 
 D. d. 
 
 3. D. 304 81)186513.239(611.9 In this example the denom 
 
 182886 inations are dollars, deci 
 
 mals of the Doll, unit , 
 
 36272 we find 3 decimal piaces 
 
 30481 in the dividend and 2 in 
 
 the divisor, difference 1 ; 
 
 57913 therefore count one place 
 
 30481 from the right in the 
 
 quotient, and there make 
 
 274329 the point, and you have 
 
 274329 611D. 9=:/^D. = 9d.= 
 
 90c. 
 
 4. Divide 600D. equally among 25 men. Ans. 24D. 
 
 5. Two men received D235.25 ; what is the share of each? 
 
 Ans. 117D. 62c. .5. 
 
 6. A. bought 1584 yards of sheeting for 250D. ; how much 
 was it per yard ? Ans. 15c. 7m. 
 
 7. A man was paid for his labor of 26 days, 16D. 75c. ; how 
 much did he earn daily ? Ans. 64c. 4m. 
 
 8. Seven men have the sum of 2764D. 25c. ; how much did 
 each receive? Ans. 394D. 89c. 3m. 
 
 9. A farmer purchased 75 sheep for 255D. ; how much did 
 he pay per head? Ans. 3D. 40c. 
 
 10. B. gave 150D. for 60 bushels of wheat; how much was 
 it per bushel ? Ans. 2D. 50c. 
 
 11. If I paid 400D. for 64 yards of broadcloth, how much 
 was it per yard ? Ans. 6D. 25c. 
 
 12. How many times are 12.5c. contained in 125D. ? 
 
 Ans. 1000. 
 
 13. How many half dimes in 75D. ? Ans. 1500. 
 
 14. Paid 20D. for 130 pounds of sugar ; how much per pound ? 
 
 / Ans. 15c. 3m. 
 
 15. Paid 25D. for 400 pounds of flax ; how much was it per 
 pound? Ans. 6c. .25 
 
 16. Gave 75D. for 200 bushels of oats ; what is the cost per 
 bushel ? Ans. 37Jc. 
 
 17. A farmer raised 480 bushels of grain from 30 acres; 
 how many bushels per acre ? Ans. 16. 
 
 18. C. paid 750D. for 25 i.:ows ; what cost 1 cow ? Ans. SOD. 
 
 19. If a man spend 250D. in a year, how much is that daily ? 
 
 Ans, 68.5c. (nearly). 
 
 20. If wool is worth 37.5c. per pound, how much can you 
 buy for 500D. ? Ans. 1333 pounds, .3=,^^+ 
 
56 DIVISION OF DECIMALS. 
 
 When there are more decimal places in the divisor than in 
 the dividend, annex as many ciphers to the dividend as are 
 necessary to make its decimal places equal to those of the divi' 
 sor ; all the figures of the quotient will then be whole numbers, 
 but if there be a remainder and the operation continued, then 
 the quotient figure will be decimal. 
 
 Example 1.— Divide 8794.8 by 6.98 : 
 6.98)8794.80(1260 Ans. 
 
 698 In this example there is one decimal 
 
 place in the dividend, and two in the 
 
 1814 divisor; therefore you will annex a ci- 
 
 1396 pher to the dividend, and this will make 
 
 the decimal places equal, and the quo 
 
 4188 tient a whole number. 
 
 4188 
 
 Example 2. — Divide 4.64 by 2.22 : 
 2.22)4.64(2.09 
 
 4 44 In this example you annex a ciphei 
 
 to the remainder ; the next quotient ^g 
 
 2000 ure will be a decimal. 
 
 1998 
 
 Rem. 2 
 When circulating decimals occur, it will be sufficient to con- 
 tinue the division to three or four places, unless great accuracy 
 is required, then it may be continued farther ; but it is indef- 
 inite, and will never terminate. 
 
 Examples 9)10.00(1.11+ .06).20(3.33+ 6)10.00(1.66 + 
 When a decimal number is to be divided by 10, 100, 1000, 
 <fec., the division is made by removing the decimal point as 
 many places to the left as there are ciphers in the divisor ; and 
 if there be not so many figures on the left of the decimal point, 
 the deficiency must be supplied by prefixing ciphers. 
 
 10 1 r4.274 101 bo r ■].2f765.4 
 
 7654 I J 
 
 U 
 
 fe . 1 100 I I .4274 100 , ;S ] ^c^^A I S J 76.54 
 
 101 
 
 bO 
 
 100 
 1000 
 
 > 
 
 10000 J 
 
 n3 
 
 ;2"^'{ 1000 f - ] .04274 1000 f T 1 f "■§ 1 7.654 
 
 [ 10000 J { .004274 lOOOOJ ^ { J §.[-7654 
 
 21. Divide 33.66431 by 1.01. Ans, 33.331. 
 
 22. Divide .01001 by .01. 
 
 23. Divide 2194.02194 by .100001. Ans. 21940. 
 
 24. Divide .1 by .0001. Ans, 1000 
 
 25. Divide 37.4 by 4.5. Ans, 8.311 1-f 
 
APPLICATION OF DECIMALS. ' 57 
 
 26. Dmde 94.0369 by 81.032. Ans, 1.160-f 
 
 27. If 2D. 25c. will board one man a week, how man)'* weeks 
 can he be boarded for lOOlD. 25c.? Ans. 445. 
 
 28. If 3.355 bushels of corn will fill one barrel, how many 
 barrels will 352.275 bushels fill? Ans, 105. 
 
 ,29. 1561.275~-24.3r=6425 ^n;?. 30. .264^2=. 132 An;?. 
 
 APPLICATION OF DECIMALS. 
 
 1. Add 7482D, 97c. 6m. .50; 4642D. 34c. 9m. .50 ; 3765D. 
 37c. 5m. .25 ; 2320D. 63c. 7m. .37, together. 
 
 Ans. 18211D. 33c. 8m. .62. 
 
 2. What cost 7.25 yards of broadcloth at 6D. 50c. per yard ? 
 
 Ans, 47D. 12c. .5. 
 
 3. What cost 7.25 bushels of wheat at ID. 37c. .5, per 
 bushel ? Ans. 9P, 96c. 8m. .75. 
 
 4. What cost 9.75 bushels of wheat at ID. 62c. .5, per bush- 
 el? Ans. 15D. 84c. 3m. .75. 
 
 5. What cost 15.35 bushels of com, at 87c. .5, per bushel ? 
 
 Ans. 13D. 43c. Im. .25. 
 
 6. A man purchased 1500 pounds of cheese, for which he 
 paid 6c. .25 per pourwl ; it cost him 7D. 50c. to convey it to 
 market ; he then disposed of it for 8c. .5 per pound ; did he 
 gain or lose, and how much ^ Ans, gained 26D. 25c. 
 
 7. If you buy 2000 bushels of wheat for 2500D. and sell it 
 for ID. 45c. per bushel, how much will you gain in all ? and 
 how much on a bushel ? 
 
 Ans. gain in all, 400D., which is 20c. per bushel, 
 
 8. James Wilson, Dr, 
 
 To 4.5 pounds of candles, at 12c. .5 per pound. 
 To 9 pounds of soap, at 8c. .25 per pound. 
 To 12.25 yards of calico, at 16c. .75 per yard. 
 To 11.5 pounds of butter, at 12c. .5 per pound. 
 Required the amount of the above bill. A71S. 4D. 79c. 4. 
 
 9. John Thompson, Dr. 
 
 To 6.5 bushels of oats, at 37.5c. per bushel. 
 To 8.75 pounds of feathers, at 43c, per pound. 
 To 12.5 pounds of coffee, at 16c. per pound. 
 To 1 side of upper leather, at 3D. 25c. 
 To 1 set of knives and forks, at 2D. 25c. 
 There has been paid 4D. 75c. ; required the balance due. 
 
 Ans. 8D. 95c 
 
68 RETIEW OF DECIMALS. 
 
 10. A. had an order on a merchant for 87D. 50c. to be paid 
 in goods ; after purchasing the following articles, how will the 
 balance stand, viz. : 5 sets of cups and saucers at 75c. per set; 
 38 yards of sheeting at 14.5c. ; 8 gallons of wine at ID. 25c. 
 per gallon ; 4.25 yards of broadcloth at 5D. 25c. per yard ; 4.5 
 pounds of tea at ID. 25c. per pound ; 9.5 bushels salt at 62.5c« 
 per bushel; 18 pounds of nails at 12.5c. per pound ; 4 barrels 
 of flour at 6D. 25c. per barrel 1 
 
 Ans. balance due on order 7D. 11.5. 
 
 11. Divide 764D. 76c. among 4 men and a boy. 
 
 12. Multiply 500D. by 500 cents. 
 
 13. What will 225 bushels of wheat cost ot ID. 12.5c. per 
 bushel? Ans. 253D. 12.5c. 
 
 14. What will 47.25 acres of land cost at 54D. per acre ? 
 
 Ans 2551D. 50c. 
 
 15. If 1 gallon of wine is w^orth ID. 87.5c., wbat are 20 , 
 gallons worth ? Ans. 37D. 50c. 
 
 16. If the income of an estate is 3650D. 40c. per annum, 
 how much for one day ? Ans. lOD. 00c. Im. 
 
 17. How much would it be for 1 week ? for 1 month ? &;c. 
 
 18. If 383 yards of cloth cost 50D.36.5c. how much for one 
 yard 1 Ans. OD. 13.15c. 
 
 19. If you can earn 600D. in a year, how much is it a week ? 
 
 Ans. 1 ID. 54c 
 
 20. Divide 10489 by .9846=10653.0+ ; 8.47-f-.7=12.1. 
 
 In the last two examples, where the dividend is a whole num- 
 ber and the divisor is a decimal, the quotient must consist of 
 one figure more than the divisor, which will be a whole number, 
 and the next figure will be a decimal. 
 
 REVIEW OF DECIMALS, NO. I. 
 
 What do you understand by decimal ? What is a fraction ? 
 When ciphers are annexed to a decimal, do they increase the 
 value of the decimal ? What is the value when prefixed ? Are 
 there more than one kind of decimals ? What is a circulating 
 decimal *? Is the currency of the United States decimal ? Why ? 
 What do the laws of the country require in relation to accounts, 
 book-keeping, &c. ? When the unit 1 is divided into 10 equal 
 parts, what is each part called ? When in money of the United 
 States, how are they called "? Is it necessary to set down the 
 denominator of a decimal? If you should, would it make any 
 difference in the value ? What is the place next to the decimal 
 point called ? The second ? The third ? How do you nu- 
 merate decimals ? Does the value of a figure depend upon the 
 
b 
 
 REVIEW OF DECIMALS. 69 
 
 distance from the decimal point, or the place which it occupies ? 
 How does the value change from the left toward the right ? 
 What part of a dollar is a dime ? A cent ? A mill ? Does the 
 annexing of ciphers to a decimal, alter its value ? When a ci- 
 pher is prefixed to a number? When prefixed, does it increase 
 the numerator or denominator ? Will it produce any effect on 
 the value of a fraction ? What will .8 become by prefixing a 
 cipher 1 How do you write down numbers in addition ? How 
 are you to place the decimal points ? How will you then pro- 
 ceed ? Where will you place the decimal point in the sum ^ 
 Repeat the rule ? Give an example in addition on your slate. 
 What is the rule for Subtraction of Decimals ? How do you 
 write down the numbers 1 How will you then proceed ? Where 
 will you place the decimal point in the remainder ? What is 
 the rule for Multiplication of Decimals ? After multiplying, 
 how many decimal places will you count off in the product ? 
 When there are not so many in the product, what will you do ? 
 How do you multiply a decimal by 10, 1000, &c. 1 What more 
 can you say of Multiplication of Decimals ? Give a few ex- 
 amples on your slate. What is the rule for Division of Deci- 
 mals ? How does the number of decimal places in the dividend 
 compare with those in the divisor and quotient ? How do you 
 determine the number of decimal places in the quotient ? Sup- 
 pose there are two places in the divisor, and three in the divi- 
 dend, how many would you count off in the quotient ? How do 
 you divide a decimal by 10, 100, &c. 1 When there are more 
 decimal places in the divisor than is in the dividend, what will 
 you do ? In this case what will the figures in the quotient he ? 
 How can you continue division after you have brought down all 
 the figures in the dividend ? Give a few examples of your 
 knowledge of decimals on the slate, with an explanation of the 
 same. Divide .2142 by 3.2 = .066+. Divide 2.00385 by 931 
 = .0021523. Can there be a decimal figure in the quotient, 
 unless you have a decimal in the dividend, or by annexing a 
 cipher, which is the same ? When you have brought down a 
 decimal figure to the remainder, what will the next quotient fig- 
 ure be ? When the price of one article is given, and you wish 
 to know the value of 100, 1000, &c., how will you place the 
 decimal point without midtiplying? 
 
 Ans. Thus, 1 yard, 37.5c. ; 10 yards, D.3.75 ; 1000 yards, 
 D.375.00, (fee. 
 
 The denominator to a decimal fraction, although not express- 
 ed, is always understood. 
 
60 TABLES OF WEIGHTS AND MEASURES. 
 
 TABLES OF WEIGHTS AND MEASURES. 
 
 Let the pupil commit all the tables to memory. 
 
 AVOIRDUPOIS WEIGHT. 
 
 By this weight are weighed things of a coarse or drossy na- 
 ture, and all metals, except silver and gold. 
 
 Denominations. Marked, 
 
 16 drams (dr.) equal, or make 1 ounce, oz. 
 
 16 ounces " 1 pound, lb. 
 
 28 pounds " 1 qr., (gross weight.) 
 
 4 quarters, or 112 lb. " 1 hund.cwt.(grosscwt.) 
 
 100 pounds (in general use) " 1 hund. C. (lb. wt.) 
 
 20 hund. wt. (2240 lb.) " 1 ton T. 
 
 TROY WEIGHT. 
 
 Gold, silver, jewels, and liquors, are weighed by this weight. 
 
 Denominations, Marked. 
 
 24 grains (gr.) make 1 pennyweight, pwt. or dwt. 
 
 20 pennyweights " 1 ounce, oz. 
 
 12 ounces " 1 pound, lb. 
 
 Note. — 1 lb. troy =5760 grains, and 1 lb. avoirdupois — 7000 
 grains, 175 oz. troy =: 192 avoirdupois oz., 175 lb. troy=144 
 avoirdupois lb. 
 
 APOTHECARIES' WEIGHT. 
 
 By this weight apothecaries mix their medicines, but buy and 
 sell by avoirdupois weight. 
 
 Denominations. 
 
 20 grains (gr.) make - 1 scruple. 
 
 3 scruples " 1 dram. 
 
 8 drams " 1 ounce. 
 
 12 ounces " 1 pound. 
 
 Note. — The lb. and oz. apothecaries' weight, and the lb. and 
 z. troy, are the same, only differently divided and subdivided. 
 
 CLOTH MEASURE. 
 
 By this measure cloth, tape, ribands, calico, &;c., are measured. 
 Denominations. Marked, 
 
 2.25 (2^) inches, (in.) make 1 nail. na, 
 
 4 nails, or 9 in. "1 quarter of a yard> qr 
 
TABLES? OF WEIGHTS AND MEASURES. 
 
 ai 
 
 n 
 
 Denaminations, 
 
 
 
 Marked. 
 
 10 nails, or 22^ in. 
 
 make 
 
 1 ell Hamburgh, 
 
 E. H. 
 
 3 quarters, or 27 in. 
 
 (( 
 
 1 ell Flemish, 
 
 E. F. 
 
 27.2 inches 
 
 *• 
 
 1 ell Scotch, 
 
 E. S. 
 
 33 inches 
 
 <( 
 
 1 var Spanish, 
 
 V. S. 
 
 4 quarters, or 36 in. 
 
 u 
 
 1 yard. 
 
 yd. 
 
 5 quarters, or 45 in. 
 
 C( 
 
 1 ell English, 
 
 E. E. 
 
 ^ 6 quarters, or 54 in. 
 
 {< 
 
 1 ell French, 
 
 E. F. 
 
 The yard is used in measuring all kinds of piece goods in the 
 nited States. 
 
 LONG MEASURE. 
 This measure is applied to distance, or length. 
 Denominations. 
 
 Marked. 
 
 make 1 inch, 
 
 in. 
 
 (( 
 
 1 foot. 
 
 ft. 
 
 ti 
 
 1 yard. 
 
 yd. 
 
 u 
 
 1 rod,po.. 
 
 or perch, R. P. 
 
 (( 
 
 1 furlong. 
 
 fur. 
 
 yds 
 
 1 mile, 
 
 M. 
 
 3 barleycorns (be.) 
 12 inches 
 
 3 feet, or 36 in. 
 
 H (5.5) yards, or 16i ft. 
 40 poles, or 220 yards. 
 
 69^ (69.5) statute, or ) ^ ^^^ ^ ^ 
 
 60 geographic J ' ^ ' ^ 
 
 360 degrees " 1 circle, cir. 
 
 A circle is the circumference of the earth. 
 6 points =1 line, 12 lines 1 inch, applied to the measure of 
 pendulums. 
 
 Note. — A hand is 4 inches, used in measuring the height of 
 horses ; a fathom is 6 feet, used in measuring the depth of wa- 
 ter (fm.) ; a league is 3 miles, used in measuring distances at 
 sea (L.). 
 
 LAND OR SQUARE MEASURE. 
 
 This measure is applied to land, and has respect to length 
 and breadth, but not to depth. 
 Denominations. Marked. 
 
 144 square inches, (in.) 
 9 square feet (1296 in.) 
 
 30.25 sq. yds. or 272.25 sq. ft. 
 40sq. P. (1210sq. yds.) 
 
 4 roods, or 160 P. 
 640 acres or 1 section of land, 
 
 1 foot 
 square 
 
 rnUe 
 
 .1 1 1 11 
 
 9 ft. or sq. yd. 
 
 make 1 square foot, ft. 
 " 1 sq. yard, sq. yd. 
 
 " 1 sq. perch, sq. P. 
 1 rood, R. 
 
 1 acre, A. 
 
 1 square mile, M. 
 
6a 
 
 TABLES OF WEIGHTS AND MBASXTRES. 
 
 CHAIN. 
 
 Denominations. 
 
 Marked 
 
 7.92 inches - - equal 1 link, li. 
 
 25 links - . « 1 rod,pole,orpeTch, P 
 
 4 rods, or 6Q ft. Gunter*s ch., or 100 li. 1 chain, ch. 
 
 1 square chain - ** 16 square poles. 
 
 80 chains, or 320 rods - "1 mile, m. 
 
 10 square chains - - "1 acre, A. 
 
 The American or English mile is 5280 feet. 
 
 French mile - - 5328 " 
 
 Italian mile - - 5566 " 
 
 German mile - - 26400 " 
 
 Dutch, Spanish, or Polish mile 21120 " 
 
 Scotch mile - - 7920 " 
 
 Indian mile - - 15840 " 
 
 Irish mile - - 6720 " 
 
 LIQUID MEASURE. 
 
 This measure is used for beer, wine, cider, &c. The gallon 
 contains 231 cubic inches, or .13368 feet. 
 
 Denominations, Marked, 
 
 4 gills (gi.) 
 
 - 
 
 equal 1 pint. 
 
 pt. 
 
 2 pints 
 
 - 
 
 " 1 quart. 
 
 qt 
 
 4 quarts 
 
 - 
 
 " 1 gallon. 
 
 gal. 
 
 16 gallons 
 
 . 
 
 " 1 half-barrel, 
 
 hf. bar 
 
 31^ (31.5) gallons- 
 
 - 
 
 " 1 barrel. 
 
 bar 
 
 42 gallons 
 
 • 
 
 " 1 tierce, 
 
 tier 
 
 63 gallons 
 
 - 
 
 " 1 hogshead, 
 
 hhd 
 
 84 gallons 
 
 - 
 
 " 1 puncheon, 
 
 punch 
 
 2 hogsheads 
 
 - 
 
 " 1 pipe or butt, 
 
 p. or b. 
 
 2 pipes or butts - 
 
 - 
 
 " 1 tun, 
 
 T. 
 
 
 DRY MEASURE. 
 
 
 This measure is used for all dry goods, or such as are sold 
 by the bushel. 
 
 Denominations, Marked 
 
 2 pints (pt.) - - equal 1 quart, qt. 
 
 8 quarts - - " 1 peck, pk. 
 
 4 pecks = 32 qts.=i64 pts. - " 1 bushel, bush. 
 
 i96 pounds flour, 200 pounds pork, '* 1 barrel, bar. 
 
 A bushel is 18 J inches in diameter, and 8 inches in depth. 
 
TABLES OF WEiOHTS AND MEASURES. 63 
 
 MOTION OR CIRCLE MEASURE. 
 
 This measure is used by astronomers, navigators, &c. 
 
 Denominations. Marked, 
 
 60 seconds ('') equal 1 minute, ' 
 
 60 minutes - - ** 1 degree, ° 
 
 30 degrees - - - "1 sign, sig. 
 
 12 sitjns, 360 degrees - "1 revolution of a ) 
 
 ° 1 * • 1 c rev. 
 
 planet or circle ) 
 
 Sound moves at the rate of 1142 feet per second, equal to 
 
 12 miles, 3 furlongs, 32 poles, 4 yards, per minute. Light 
 passes from the sun to the earth, 95 millions of miles, in 8.2 
 minutes, or 11585365 miles per minute. 
 
 SOLID OR CUBIC MEASURE. 
 
 By this measure is ascertained the solid content of all things 
 in which length, breadth, and thickness, are considered ; such 
 as stone, timber, wood, bark, grain, coal, &c, A cube is a solid 
 of six eq lal sides. 
 
 Denominations. Marked. 
 
 1728 cubic inches (c. in.) equal 1 cubic foot, c. ft. 
 
 27 cubic feet, or 46656 in. " 1 cubic yard, c. y. 
 
 40 feet of round timber, or j 
 
 50 feet of hewn timber, 
 2150.4252 c. in., or 1.24446 c. ft. " 1 bushel, of strick 
 
 measure, c.bu. 
 2553.6299 c.in.,or 1.47779+ eft. " 1 bush, of heaped 
 
 measure. 
 128 cubical feet, or 221 184 c. in.,] 
 
 or 8 feet long, 4 feet wide, and ^ ** 1 cord of wood, C. 
 
 4 ft. high: 8x4=32 X 4=128, j 
 A cord foot is one foot in length of the pile which makes a 
 
 cord, and is equal to 16 solid feet, or -I- of a cord. 
 ] 6^ solid feet= 1 solid perch. 
 
 TIME. 
 Denominations. 
 60 seconds, sec. 
 60 minutes, - - - 
 24 hours, - - - 
 
 7 days, - - 
 
 4 weeks . - - 
 
 13 months, 1 day, 6 hours (lunar), or 
 12 mo. of 30.4375 days each, or 
 365.25, 365J days, or 52 weeks, 
 
 1 ton, T. 
 
 
 Marked 
 
 equal 1 minute, 
 
 m. 
 
 " 1 hour, 
 
 hr. 
 
 " 1 day. 
 
 da. D. 
 
 " 1 week, 
 
 w. 
 
 " 1 mo. lunar, 
 
 mo. 
 
 >" 1 Julian year. 
 
64 
 
 TABLE OF WEIGHTS AND MEASURES 
 
 365 days, 5 hours, 48 minutes, 57 seconds, or 365.2423+ 
 = 1 solar year, 100 years =1 century. Every 4th or leap-year 
 has 366 days. 
 
 The year is also divided into 12 calendar months, as follows : 
 
 Istmonth, January, has 31 days. 
 
 7th month, 
 
 July, has 31 days 
 
 2d " February, 
 
 28 " 
 
 8th " 
 
 August, 31 ** 
 
 3d " March, 
 
 31 " 
 
 9th « 
 
 Sept., 30 " 
 
 4th " April, 
 
 30 " 
 
 10th " 
 
 Oct., 31 " 
 
 5th " May, 
 
 31 " 
 
 11th " 
 
 Nov., 30 " 
 
 6th " June, 
 
 30 " 
 
 12th *' 
 
 Dec, 31 " 
 
 The diurnal motion of the earth on its axis is 17^ miles per 
 minute at the equator=1035 miles per hour. The earth moves 
 in its orbit 68249 miles per hour= 1637965.6 per day. 
 " Thirty days hath September, 
 April, June, and November ; 
 February twenty-eight alone ; 
 All the rest have thirty-one." 
 N. B. — In bissextile, or leap-year, February has 29 days. 
 Note. — To know when it is leap-year, divide the years over 
 even centuries by 4 ; if no remainder, it is leap-year ; but if any 
 remain, it is so many years after leap-year. (See the note at 
 the close of the tables.) 
 
 Denominations. 
 24 sheets, or 23.6 (sh.) 
 20 quires 
 2 reams 
 
 PAPER. 
 
 make 
 
 1 quire, 
 1 ream, 
 1 bundle, 
 
 Marked. 
 qr. 
 re. 
 bun. 
 
 PARTICULARS, 
 
 Denominations. 
 
 12 single things 
 
 12 dozen 
 
 12 gross (144 dozen) - 
 
 Marked. 
 make 1 dozen, doz. 
 
 ** 1 gross, gro. 
 
 1 great gross, g. gro. 
 
 20 single things (112 lbs.=:l quintal fish) 1 score, sco. 
 
 5 scores (2 are 1 pair, or couple) " 1 hundred, hund. 
 
 BOOKS. 
 
 Denominations. Marked 
 
 36mo. is when 1 sheet makes 36 leaves, or 72 pages, pp. 
 
 24mo. 1 
 
 it 
 
 24 
 
 
 48 " 
 
 18mo. 1 
 
 u 
 
 18 
 
 
 36 " 
 
 12mo. 1 
 
 « 
 
 12 
 
 
 24 " 
 
 Octavo, or 8vo. 1 
 
 i( 
 
 8 
 
 
 16 " 
 
 Quarto, or 4to. 1 
 
 tt 
 
 4 
 
 
 . 8 " 
 
 Folio, or fol. 1 
 
 ti 
 
 2 
 
 (( 
 
 4 " 
 
TABLES OF WEIGHTS AND MEASURES. 65 
 
 Note. Extracts : The civil solar year of 365 days being 
 Bhort of the true year by 5h. 48m. 48sec., occasioned the be- 
 ginning of the year to run forward through the seasons nearly 1 
 day in 4 years. On this account Julius CcBsar ordained that 
 one day should be added to February every 4th year, by causing 
 the 24th day to be reckoned twice ; and because this 24th day 
 was the sixth (sextilis) before the Kalends of March, there were 
 in this year two of these sextiles^ which gave the name of Bis^ 
 sextile to this year, which being thus corrected, was thence 
 called the Julian year. 
 
 Pope Gregory the 13th made a reformation of the calendar. 
 The Julian calendar, or old style, had, before that time, been in 
 general use all over Europe. The year, according to the Julian 
 calendar, consists of 365 days and 6 hours ; which 6 hours being 
 one fourth part of a day, the common years consisted of 365 days, 
 and every fourth year, one day was added to the month of Feb- 
 ruary, which made each of those years 366 days, which are usu- 
 ally called leap-years. This computation, though near the truth, 
 is more than the solar year by eleven minutes which, in one hun- 
 dred and thirty-one years will amount to a whole day. By this 
 calculation, the vernal equinox was anticipated ten days from 
 the time of the general council of Nice, held in the year 325 of 
 ,lhe Christian era to the time of Pope Gregory, who therefore 
 caused ten days to be taken out of the month of October iu 1582, 
 to make the equinox fall on the 21st of March, as it did at the 
 time ot that council. And to prevent the like variation for the 
 future, he ordered that three days should be abated in every four 
 hundred years, by reducing the leap-year at the close of each 
 century, for three successive centuries, to common years, and 
 retaining the leap-year at the close of each fourth century only. 
 This was at that time esteemed as exactly conformable to the 
 true solar year ; but Dr. Halley makes the solar year to be 365 
 days, five hours, forty-eight minutes, fifty-four seconds, forty-one 
 thirds, twenty-seven fourths, and thirty-one fifths ; according to 
 which, in 400 years, the Julian year of 365 days 6 hours, will 
 exceed the solar year by three days, one hour, and fifty-five 
 minutes, which is near two hours, so that in 50 centuries it will 
 amount to a day, that is, the Ih. 55m. in 400 years, would in 50 
 centuries=23h. 57m. 30sec. ; +3 daysr=3d. 23h. 57m. 30sec. ; 
 which would be 2>Jm. less than 4 days, and would be the true 
 difference in 5,000 years. Though the Gregorian calendar, or 
 new style, had long been used throughout the greatest part of 
 Europe, it did not take place in Great Britain and America till 
 ^lie first of January, 1752 ; and in September following, the 
 
66 SIMPLE REDUCTION. 
 
 eleven days were adjusted, by calling the third day of that month 
 ihe fourteenth, and continuing the rest in their order. 
 
 A just and equal measure of the year is called the periodical 
 year, as being the time of the earth's period about the sun, in 
 departing from any fixed point in the heavens, and returning to 
 the same again. The Zodiac is a great circle of the sphere, 
 containing the twelve signs, through which the sun passes. 
 
 SIMPLE REDUCTION. 
 
 There are two kinds of Reduction, termed simple and com- 
 pound, by which we are taught how to change a sum or quantity 
 of one denomination to another, whether it be greater or less, still 
 retaining the same value ; when the sum consists of only one de- 
 nomination, and that is to be changed, or reduced to another, it is 
 called Simple Reduction. The operations are all performed either 
 by multiplication or division ; when by multiplication, it is called 
 reduction descending; when by division, reduction ascending; 
 for instance, to reduce pounds (avoir.) to drams, multiply by 16, 
 which will reduce it to ounces, then by 16 again, and it will be 
 in drams ; then take those drams and divide by 16, and it will be 
 ounces ; then divide again by 16, and it will be pounds, because 
 16 ounces make 1 pound, and 16 drams 1 ounce, &c. By the 
 above process, sums in reduction will reciprocally prove each 
 other ; and it would be well for the teacher to require it of the pupil. 
 
 RULE. 
 
 1 . Multiply the sum or quantity by that number of the next 
 lower denomination which it requires to make one of its own. 
 
 2. If there be one or more denominations between the de- 
 nomination of the given sum or quantity, and that to which it is 
 to be reduced, first reduce it to the next lower than its own, and 
 then to the next lower, &c. 
 
 3. When low denominations are to be brought to higher de- 
 nominations, as for example, drams to pounds, cents to dollars, 
 inches to miles, &;c., divide by as many of the lower as make 
 one of the higher, and set down what remains (if any) at the 
 right ; so proceed till you have brought it into that denomination 
 which your question requires. See example. 
 
I 
 
 SIMPLE REDUCTION. 6T 
 
 Example. Bring 2cwt. to drams. 
 
 2 cwt. Explanation. First multiply tho 2cwt. 
 
 4xqr. = l cwt. by 4, and this will reduce it to quar- 
 - ters, because 4 quarters = 1 cwt , 
 
 8 qr.=2 cwt. then multiply the quarters by 28, and 
 28xlb.=l qr. this will reduce it to pounds, because 
 
 28 lb.=:l qr. ; then multiply by 16 
 
 224 lb. = 8 qr. which will reduce it to ounces 
 
 16xoz.=l lb. because 16 ounces = 1 lb.; then 
 
 S" multiply the ounces by 16, which 
 
 ft 1344 will reduce it to drams: because 16 
 
 5^ 224 drams = 1 ounce, &c. Then to 
 
 Q prove it, take the sum of the drams, 
 
 3584 oz.=224 lb. and divide by the same denomina- 
 16xdrams=l oz. tions by which you multiplied, and 
 
 this will bring back the 2 cwt. 
 
 16)57344 drams =3584 oz. 
 
 ^JO 
 
 16)3584 ounces. 
 
 .1^ 28)224 pounds. 
 
 Si ~~ 
 
 V3 4)8 quarters. 
 
 2 cwt. proof. 
 Bring 1 year to seconds. Ans» 31536000 
 
 Thus ; 1 year =365 days. 
 4- 
 
 1460 
 6 
 
 4x6=24 hours =1 day. 
 
 8760 hours. 
 
 60 minutes=l hour. 
 
 525600 minutes. 
 
 60 seconds=l minute. 
 
 6,0)3153600,0 seconds. Ans, 
 6,0)52560,0 minutes. 
 
 6)8760 hours. 
 
 4)1460(365 days=l year, proof. 
 
i 
 
 68 SIMPLE REDUCTION. 
 
 Avoirdupois Weight. 
 
 1. In 120 pounds, how many ounces ^ Ans. 1920 
 
 2. Bring 1 cwt. to ounces. 1792 
 
 3. Bring 10 cwt. to ounces. 17920. 
 
 4. Bring 1284 ounces to pounds. 80 lbs. 4 oz. 
 
 5. Bring 1642 quarters to cwts. 
 
 6. Bring 184 drams to ounces. 
 
 7. Bring 1674 drams to pounds 
 
 8. Bring 1 ton to drams. 
 
 Troy Weight, 
 
 9. Reduce 15 lbs. to ounces. Ans, 180. 
 
 10. Bring 20 lbs. to grains. 115200. 
 
 11. Bring 35 ounces to grains. 16800. 
 
 12. Bring 1476 ounces to pounds. 123. 
 
 13. Bring 34 lbs. to ounces. 408 
 
 14. Bring 97 lbs. to grains. 
 
 15. Bring 1796 grains to ounces. 
 
 16. Bring 1100 ounces to pounds. 
 
 Apothecaries^ Weight, 
 
 17. Bring 72 ounces to drams. Ans, 576. 
 
 18. Bring 10 pounds to grains. 57600. 
 
 19. Bring 17 pounds to ounces. 204. 
 
 20. Bring 57 ounces to scruples. 1368. 
 
 21. Bring 27 drams to grains. 1620. 
 
 22. Bring 480 scruples to pounds. 
 
 Long Measure, 
 
 23. Bring 40 yards to feet. Ans, 120. 
 
 24. Bring 74 poles to feet. 1221. 
 
 25. Bring 120 furlongs to poles. 4800. 
 
 26. Bring 27 yards to inches. 972. 
 
 27. Bring 1 mile to yards. 
 
 Cloth Measure. 
 
 28. Bring 5 yards to quarters. Ans. 20 
 
 29. Bring 10 yards to nails. 160. 
 
 30. Bring 72 quarters to nails. 288. 
 
 31. Bring 144 quarters to nails. 
 
 Land or Square Measure, 
 
 32. Bring 18 roods to poles. Ans. 720 
 
SIMPLE REDUCTION. 
 
 69 
 
 33. Bring 45 acres to poles. 
 
 34. Bring 12 acres to poles. 
 
 35. Bring 27 roods to poles. 
 
 Liquid Measure, 
 
 36. Bring 32 gallons to pints. 
 
 37. Bring 4 hogshecfQs to quarts. 
 
 38. Bring 1 hogshead to gills. 
 
 39. Bring 1 barrel to pints. 
 
 40. Bring 504 pints to barrels. 
 
 Dry Measure, 
 
 41. Bring 14 bushels to pecks. 
 
 42. Bring 1664 quarts to bushels. 
 
 43. Bring 1152 pints to bushels. 
 
 44. Bring 84 bushels to quarts. 
 
 45. Bring 70 bushels to pints. 
 
 Time, 
 
 46. Bring 7 weeks to hours. 
 
 47. Bring 2 days to minutes. 
 
 48. Bring 11 days to seconds. 
 
 49. Bring 32 weeks to hours. 
 
 50. Bring 47 years to days. 
 
 51. Bring 259200 seconds to days. 
 
 Promiscuous Questions, 
 
 52. Bring 15 years to weeks. 
 
 53. Bring 1 mile to inches. 
 
 54. Bring 97 leagues to poles. 
 
 55. Bring 88 degrees to miles. 
 
 56. Bring 4730 yards to inches. 
 
 57. Bring 400 pints to gallons. 
 
 58. Bring 6 hogsheads to quarts. 
 
 59. Bring 1800 minutes to hours. 
 
 60. Bring 50 yards to nails. 
 
 61. Bring 131bs. avoirdupois to drams. 
 
 62. Bring 6784 drams to pounds. 
 
 63. Bring 1 ton to drams. 
 
 64. Bring 15 years to seconds. 
 
 65. Bring 1664 pints to bushels. 
 
 66. Bring 90 furlongs to yards. 
 
 67. Bring 47 acres to poles. 
 68 Bring 48 poles to inches. 
 
 Ans 7200 
 1920. 
 1080. 
 
 Ans. 256. 
 
 1008. 
 
 2016. 
 
 252. 
 
 2. 
 
 Ans. 56 
 
 52. 
 
 18. 
 
 2688. 
 
 Ans, 1176. 
 
 2880. 
 
 950400. 
 
 5376. 
 
 17155. 
 
 Ans. 780. 
 
 63360. 
 
 93120. 
 
 6116. 
 
 170280. 
 
 50. 
 
 1512. 
 
 30. 
 
 800. 
 
 3328. 
 
 26.5. 
 
 573440. 
 
 473040000. 
 
 26. 
 
 19800. 
 
 7520 
 
 9504 
 
70 
 
 PRACTICAL EXAMPLES. 
 
 69. Bring 4 signs to seconds. Ans, 432000 
 
 70. Bring 7 hogsheads to gallons. 
 
 71. Bring 27 barrels to pints. 
 
 72. Bring 8 miles to yards. 
 
 73. Bring 60 poles to inches. 
 
 74. Bring 1478 pints to gallons. 
 
 75. Bring 7644 inches to yards. • 
 
 76. Bring 82800 seconds to hours. 
 
 77. Bring 2764 ounces to cwt. 
 
 78. Bring 87960 hours to weeks. 
 
 79. Bring 116424 cubic inches to tons. 
 
 80. Bring 4014489600 square inches to square acres.^y . 
 
 SHORT PRACTICAL EXAMPLES, 
 
 APPLICABLE TO REDUCTION AND DECIMALS. 
 
 3. What cost 15lbs. 
 
 I. What cost 4^1bs. sugar 
 at 12c. per pound? 
 12 
 4X 
 
 48 
 ^Ib. =6 
 
 2 
 
 per lb. ? 
 
 54c. Ans, 
 What cost 5|lbs. at 14c. 
 
 14 
 5X 
 
 70 
 
 731c 
 
 per pound 1 
 
 at 16^c 
 
 16i 
 15x 
 
 2.40 
 
 71-rr^of 15 
 
 D2.47^ Ans, 
 4. What cost 22jlbs. at lie 
 per pound ? 
 11 
 22 X 
 
 2.42 
 
 8| = I of 11 
 
 5. 
 
 6. 
 
 7. 
 
 8. 
 
 9. 
 10. 
 11. 
 12. 
 
 . Ans. D2.50i Ans. 
 
 5 yards at D2.25 per yard ? Ans. D 11.25. 
 
 What cost 
 
 What cost 6| yards at D2.50 per yard ? 
 
 What cost 
 What cost 
 What cost 
 What cost 
 What cost 
 What cost 
 
 13. What cost 
 
 D16.25. 
 51- lbs. at 12^c. per pound? 68|c. 
 
 25| lbs. at 17c. per pound? D4.37f 
 
 7 galls. 3 qts. at 62^c. per gall. ? D4.84.375. 
 2 qts. 1 pt. at 17jc. per quart? 43|c. 
 
 2i cwt. at D4.50 per cwt. ? Dl 1 .25. 
 
 5 cwt. 1 qr. 14 ]bs. at D5 per cwt. ? D26.87.5 
 6| cwt. 7 lbs. at D4.50 per cwt. ? D30.65 
 
1>RACTICAL EXAMPLES. 71 
 
 14. What cost 97 lbs. at 75c. per lb. ? Ans, D72.75 
 
 15. What cost 49 lbs. at 12lc. per lb. ? D6.121. 
 
 , 16. What cost 42^ lbs. at ST^c. per lb. ? Dl5.93^*. 
 
 17. What cost 6.87 lbs. indigo at D2.25 per lb. ? Dl5.45f. 
 
 18. What cost 1 3. bushels at D 1.25 per bushel? D2.18|. 
 
 19. What cost 3 pecks at Dl. 50 per bushel? Dl.l2^ 
 
 20. What cost J- bushel at Dl.37i per bushel ? .45.8. 
 
 21. What cost 1 peck clover-seed at D6.25 per bush."? Dl.56|-. 
 
 22. What cost 2 galls. 1 pt. at D1.12^ per gall. ? D2. 39.06. 
 
 23. What cost 1 ^ tons of hay at D 17.50 per ton ? D26.25. 
 
 24. What will 1 cwt. 1 qr. cost at 6c. per lb. ? D8.40. 
 ^ 25. Wh^t will 2 cwt. 17 lbs. cost at 6.5c. per lb. ? D15.66.5. 
 : 26. What will 1 bush. 3 pecks cost at 7c. per qt. ? D3.92. 
 [27. What will 1 cwt. 12 lbs. cost at 2^c. per oz. ?, D49.60. 
 [28. What will 1 cwt. cost at S^c. per dram ? D1003.52. 
 ^ 29. If ^ of a yard cost ^ of a D., how much will 17 yds. cost ? 
 
 _ 30. If ^D. will pay for | of a yard of riband, how many yards 
 can you buy for D4 ? for D12 ? for D20 ? for D26 ? 
 
 REVIEW. 
 
 What things are weighed by Avoirdupois Weight? What 
 are the denominations ? Repeat the table. How do they 
 weigh in the principal cities? Ans. By the 100 lbs. What 
 is the use of Troy Weight? Repeat the table. When is 
 Apothecaries' Weight used, and for what purpose ? When is 
 Long Measure used ? What are the denominations ? Repeat 
 the tables, &c. For what is Land Measure used ? What is a 
 square ? Ans, A figure bounded by four equal sides, or lines, 
 and whose angles are all equal, or right angles. What is a 
 square number? Ans. Any number multiplied into itself, as 
 4x4=^16, &c. How can you find the number of small squares 
 contained in a large square ? What chain is used in surveying 
 land ? How long is it ? How is land generally estimated ? 
 When is Solid or Cubic Measure used ? What are the denom- 
 inations ? How many inches in a cubic foot ? How many feet 
 in a cord of wood ? in a quarter of a cord ? How many cubic 
 inches in a gallon ? What are the dimensions of a bushel ? 
 Time. Repeat the table. Which of the months have 30 days, 
 and which 31 days? For what is Circular Measure used? 
 How many things make a dozen ? How many make a score . 
 How many sheets make a quire of paper ? How many quires 
 3fiake a ream ? How many sheets are there in 17 reams ? How 
 many kinds of Reduction are there ? What are they called ? 
 For what purpose is Reduction used ? What is Simple Reduc- 
 
72 ADDITION OF DIFFERENT DENOMINATIONS. 
 
 tion? "When high denominations are to be brought to lower 
 denominations, how do you proceed? What is this kind of 
 Reduction sometimes called? When low denominations are to 
 to be brought higher, how will you proceed ? What is this 
 called ? What is the rule ? How can you prove Reduction? 
 
 ADDITION OF DIFFERENT DENOMINATIONS. 
 
 When it is required to add together several numbers of dif- 
 ferent denominations, such as pounds, ounces, &;c., for the pur- 
 pose of finding the sum, or amount total, it has generally been 
 designated as Compound Addition; or, by some. Denominate 
 Numbers^ which signifies *' to name," and Denomination, which 
 signifies " a name given to a thing, title, &c.," either of which 
 may be correct, as it is unimportant what the title is, provided 
 the subject is understood, although we have given the last-men- 
 tioned the preference. The Cwt. avoirdupois (112lbs.) has 
 been in use many years, but is not so much used of late, as the 
 merchants in the principal cities have substituted the lOOlbs. 
 net weight for the 112lbs. avoirdupois. This change will be 
 attended with many'advantages, it being by far the most agree- 
 able and expeditious method of computation, as it could easily 
 be reduced to a system of decimal calculations, and the division 
 of the lOOlbs. into lOths ; and this system should extend to 
 LENGTH, and our weight, measure, &:c., would correspond with 
 our currency, which may, and probably will, some day not far 
 distant, be the case. 
 
 1. Write down the numbers so that each denomination may 
 stand directly under each other, leaving a small space between 
 them. 
 
 2. Add the right-hand denomination, the same as in Simple 
 Addition. 
 
 3. Then divide that amount by as many as it requires of that 
 denomination to make one in the next. 
 
 4. Set down the remainder under that denomination, and 
 carry the quotient to the next denomination, and add it in ; if 
 there be no remainder, set down a cipher. 
 
 5. Continue in this way through to the last denomination, 
 
ADDITION OF DIFFERENT DENOMINATIONS. 
 
 7a 
 
 which add, the same as in simple addition, and set down the 
 whole amount. Proof : the same as in whole numbers. 
 
 EXAMPLES. 
 
 10 20 
 
 T. cwt. 
 
 2 3 
 
 4 1 
 
 2 4 
 
 4. 28. 16. 16. 
 qrs. lbs. oz. dr. 
 
 2 11 6 8 
 
 3 16 11 12 
 18 10 14 
 
 8 9 2 18 13 2 
 
 Explanation. This example be* 
 longs to avoirdupois weight, but the 
 same general rule will hold good in 
 all cases of compound numbers. The 
 first denomination at the right hand 
 is drams, which add, the same as in 
 simple addition, and the amount is 
 34 ; this is divided by 1 6, because 1 6 
 drams make 1 oz., and we have 2 ounces in the quotient, and 
 2 drs. over ; set down the 2 drs. and carry the 2 oz. to the de* 
 nomination of that name, which add in, and it will make 29 oz. ; 
 divide the 29 oz. by 16, to get the pounds, and it will give 1 lb. 
 and 13 oz. over ; set down the 13 oz., and carry the 1 lb. to the 
 next denomination ; then add that denomination, and it will give 
 46 lbs. ; divide the lbs. by 28, to bring them to quarters, and 
 you have 1 qr. and 18 lbs. ; set down the 18 lbs., and carry the 
 1 qr. to the denomination of that name, which add in, and you 
 have 6 qrs., which divide by 4, because 4 qrs. make 1 cwt., anc^^ 
 you have 1 cwt. and 2 qrs. ; set down the 2 qrs., and carry tho 
 1 cwt. to the next denomination, and add it in ; you will then 
 have 9 cwt., there being none to carry now, because it is less 
 than a ton ; then add and set down all of the next denomina- 
 tion, and you have the sum total. 
 
 1. 
 
 lbs. 
 478 
 321 
 214 
 321 
 
 oz. 
 
 8 
 4 
 8 
 4 
 
 decimally. 
 
 lbs. 
 238 
 342 
 137 
 320 
 
 oz. 
 6 
 10 
 4 
 
 
 1335 
 
 8=1335.5 at 4c, 
 4X 
 
 1038 4 = 1038.25 at 5c. 
 5X 
 
 D.53.42.0 
 
 D.51.91.25 
 
 AVOIRDUPOIS WEIGHT. 
 
 T.cwt.qr.lb, 4. cwt.qr.lb. oz. 
 
 3 19 3 27 
 
 2 12 1 12 
 
 10 5 00 
 
 3 2 10 
 
 T.17 00 3 21 
 
 6 2 U 
 
 3 1 14 
 
 4 16 
 
 5 10 
 3 3 9 
 
 TROY WEIGHT. 
 
 5. lb. OZ. dwt. 6. lb. oz. dwt. gr. 
 
 93 11 18 
 
 6 1 
 
 14 4 12 
 
 72 11 3 
 
 lbl87 3 14 
 
 27 9 
 
 20 
 24 
 18 
 16 
 
 16 
 4 
 11 
 11 
 18 
 
 18 
 11 
 19 
 16 
 10 
 
74 
 
 ADDITION OF DIFFERENT DEXOMINATIO.^ . 
 
 APOTHECARIES WEIGHT. 
 
 7. Ib.oz.dr.scr. 8. lb.oz.dr.scr.gr. 
 
 6 3 12 21 6 4 1 13 
 
 19 9 5 1 27 4 3 2 14 
 
 182 7 3 2 34 2 2 1 15 
 
 57 6 1 20 7 2 2 10 
 
 40 000 7630 11 
 
 10 3 4 1 18 
 
 306 2 3 2 
 
 LONG MEASURE. 
 
 9. yd. ft. in. 10. L. m. fur. po. 
 
 2 2 9 8 2 7 16 
 
 110 20 1 6 17 
 
 10 1 3 24 2 4 19 
 
 12 4 50 2 11 
 
 18 1 1 12 
 
 26 2 4 
 
 CLOTH MEASURE. 
 
 11. yd. qr. na. 12. EE. qr. na. 
 
 46 2 1 86 4 2 
 
 79 2 3 44 3 3 
 
 25 2 2 21 2 1 
 
 14 2 1 18 1 2 
 
 20 4 1 
 
 166 ^1 3 
 
 LAND OR SQUARE MEASURE. 
 
 13. yd. ft. in. 14. A. R, po. 
 
 8 2 12 150 3 39 
 
 10 1 95 265 2 11 
 
 12 1 115 284 1 12 
 
 20 46 326 20 
 
 171 3 18 
 
 50 5 124 
 
 LIQUID MEASURE. 
 
 15. gall. qt. pt. 16. hhd. gall, qt 
 
 24 2 1 2 11 3 
 
 14 1 4 16 2 
 
 6 3 10 10 1 
 
 8 2 1 11 9 
 
 12 11 2 
 
 54 1 
 
 17. DRV MEASURE. 18. 
 
 bush. pk. qt. bush. pk. qt. pt. 
 
 10 3 2 36 1 7 1 
 
 117 1 3 48 2 6 
 
 215 2 4 71 2 4 1 
 
 450 3 16 3 
 
 18 3 5 1 
 
 794 2 1 . 
 
 19. CUBIC MEASURE. 20. 
 
 T. ft. C. ft. ft. in. 
 
 41 43 3 122 13 1446 
 
 12 43 4 114 16 1726 
 
 49 6 7 83 3 866 
 
 4 27 10 127 17 284 
 
 108 19 
 
 21. TIME. 22. 
 da. h. m. sec. w. da. h. m. 
 4 20 56 54 2 1 10 40 
 3 19 25 22 14 11 27 
 2 8 3 4 6 14 16 
 6 6 3 16 18 
 7 2 10 19 
 
 11 6 22 19 
 
 MOTION OR CIRCLE MEASURE. 
 
 23. 24. 
 
 sig. ° ' '' sig. "^ ' " 
 
 1 5 7 32 2 7 32 29 
 
 1 7 26 12 4 21 18 
 
 4 8 26 11 1 5 20 16 
 
 1 4 32 17 3 6 10 19 
 
 3 6 47 4 3 11 17 
 
 2 2 16 20 
 
 11 1 32 59 • 
 
 25. 
 deg. m. fur. po. 
 
 94 65 6 38 
 
 37 47 3 28 
 
 50 34 1 12 
 152 14 4 
 
 26. 
 Y. m. da. 
 28 10 26 
 34 11 17 
 63 9 12 
 21 3 7 
 
ADDITION OF DIFFEREx\T DENOMINATIONS. 75 
 
 27. A purchased 4- hogsheads of sugar, which weighed as 
 follows : No. 1, 5 cwt. 3 qrs. 21 lbs. ; No. 2, 4 cwt. 2 qrs. 27 
 lbs. ; No. 3, 7 cwt. qrs. 1 8 lbs. ; No. 4, 6 cwt. 2 qrs. 13 lbs. ; 
 what is the weight of them all ? Ans. 24 cwt. 1 qr. 23 lbs. 
 
 28. A man bought 5 pieces of sheeting: 1st piece, 35 yds. 
 1 qr. 2 na. ; 2d, 40 yds. 2 na. ; 3d, 46 yds. 3 qrs. 3 na. ; 4th, 
 50 yds. 1 qr. 3 na. ; 5th, 54 yds. 3 qrs. 3 na. ; required the 
 number of yards in the 5 pieces. Ans. 227 yds. 3 qrs. 1 na. 
 
 29. A farmer disposed of 4 bags of grain, which measured 
 as follows : 1st, 3 bush. 2 pks. 6 qts. ; 2d, 2 bush. 2 pks. 7 qts. ; 
 3d, 3 bush. 1 pk. 4 qts. ; 4th, 3 bush. 1 pk. 3 qts. ; how much 
 grain did he sell? Ans. 13 bush. 4 qts. 
 
 30. A merchant had 4 hogsheads of wine, on measuring 
 which, they were found to fall short of the quantity purchased : 
 the 1st contained 59 galls. 3 qts. 1 pt. ; 2d, 58 galls. 2 qts. 1 pt. ; 
 3d, 60 galls. 1 qt. 1 pt. ; 4th, 62 galls. 3 qts. 1 pt. ; how many 
 gallons did he have ? A?is. 241 galls. 3 qts. 
 
 31. C. has five fields : the 1st contains 25 A. 1 R. 30 po.; 2d, 
 28 A. 3 R. 18 po. ; 3d, 30 A. 1 R. 27 po. ; 4th, 34 A. 20 po. ; 
 5th, 35 A. 2 R. 24 po. ; how many acres in the five fields 1 
 
 Ans. 154 A. 1 R.'39 po. 
 
 32. Add 172 years, 1 week, 4 hours, 52 sec. ; 34 m. 18 sec. ; 
 15 y. 4 mo. 5da. 3 h. 27 m. ; 1 w. 3 da. 21 h. 35 m. 18. sec. 
 together. Ans. 187 y. 4 mo. 3 w. 2 da. 5 h. 37. m. 28 sec. 
 
 33. Add 19 T. 2 hhds. 19 galls. ; 45 T. 1 qt. 1 pt. ; 3 hhds. 
 17 galls. 2 qts.; 21 galls. 1 pt. together. 
 
 Ans. 65 T. 1 hhd. 58 galls. qt. pt. 
 
 34. A tailor bought 4 pieces of broadcloth : the 1st piece 
 contained 25 yds. 2 qrs. 2 na. ; 2d, 21 yds. qrs. 2 na. ; 3d, 
 26 yds. 1 qr. ; 4th, 27 yds. ; for the 1st piece he paid D85.50 ; 
 2d, D73.40; 3d, D90.10; 4th, D91.00; how many yards did 
 he have ? how much did it all cost ? and how much per yard 1 
 
 A?is. 100 yds. ; cost in all, D340 ; price per yard, D3.40. 
 
 35. Add 27.25 yds. ; 16.75 yds. ; 18.5 yds. together, and 
 give the value at D4.25 per yd. 
 
 36. Add 106.6 pounds; 218.25 pounds; 374.18 pounds; 
 97.75 pounds together, and give the value at 7.5c. per pound. 
 
 37. Add 56.16 acres ; 97.75 acres ; 84.62 acres ; 45.21 acres 
 together, and givethe value, alD50. 50 per acre. Ans. D14328.87. 
 
 REVIEW. 
 
 What is the use of Addition of Different Denominations ? 
 What is a denominate number ? How do you set down the 
 numbers in this rule ? Where do you begin to add ? How do 
 
76 SUBTRACTION OF DIFFERKNT DENOMINATIONS. 
 
 you proceed after placing tlie denominations under each other ' 
 By what do you divide the amount of a column, or denomina- 
 tion ? If, after dividing, there is any remainder, what will you 
 do with it ? and how do you proceed if there be no remainder ? 
 What will you do with the quotient ? Proof. 
 
 38. Add together 20 yrs. 3G3 da. 20h. 50m. 30 sec. ; 20 
 yrs. 40 da. 10 h. 30 m. 20 sec; 12 yrs. 110 da. 13 h. 16 sec; 
 13 yrs. 8 da. 10 h. 20 m. 14 sec; 7 yrs. 20 da. 8 h. 10 m. 12 
 sec A?is. 73 yrs. 178 da. 14h. 51 m. 32 sec 
 
 SUBTRACTION OF DIFFERENT DENOMINATIONS. 
 
 This rule is used when numbers of different denominations 
 are given to be subtracted, or a smaller number from a greater 
 of a like denomination, and show their difference or remainder. 
 
 1. Write down the larger number, and directly under it the 
 less number, so that the same denominations shall stand under 
 each other. 
 
 2. Begin at tlie right hand, and subtract the lower from the 
 upper number, if that be the largest, and set down the remainder. 
 
 3. But if the lower number is more than the one above it, 
 then subtract from as many as it takes of that denomination to 
 make one in the next ; take the difference, and add it to the 
 upper number, set it down ; then carry one, and add it to the 
 next lower denomination, and continue through the sum in this 
 manner, and in the last denomination subtract the same, as in 
 simple subtraction. Proof the same as in simple subtraction ; 
 observing to carry as above directed. 
 
 EXAMPLES AND QUESTIONS. 
 
 10. 
 
 100. 
 
 12. 
 
 4. 
 
 7. 
 
 24. 
 
 60. 
 
 60. 
 
 Cent. 
 
 yrs. 
 
 mo. 
 
 w. 
 
 da. 
 
 h. 
 
 m. 
 
 sec 
 
 7 
 
 97 
 
 7 
 
 3 
 
 2 
 
 3 
 
 20 
 
 30 
 
 5 
 
 98 
 
 9 
 
 
 
 3 
 
 5 
 
 50 
 
 40 
 
 98 10 2 5 21 29 50 
 
 97 7 3 2 3 20 30 proof 
 
SUBTRACTION OF DIFFERENT DENOMINATIONS. 
 
 77 
 
 Explanation. — Place the numbers over their respective de- 
 nominations, as in Compound Addition ; then begin at the right 
 hand with 40 ; 40 from 60 will leave 20, which added to 30 
 will make 50, which set down, and carry 1 to 50 = 51 from 60, 
 9 will remain, add this to 20=29, which set down ; now carry 
 1 to 5 = 6 from 24 will leave 18, and 3 are 21 ; then 1 to carry 
 to 3 is 4, 4 from 7 will leave 3, and 2 are 5 ; then 1 to carry to 
 
 is 1, 1 from 3 and 2 remain ; 9 from 12 and 3 remain, which 
 added to 7 are 10 ; then 1 to carry to 98 is 99 ; 99 from 100, 
 
 1 remains, which added to 97 makes 98 ; then 1 to carry to 5 is 
 6, 6 from 7 and 1 remains, whiah set down. For the proof, 
 add and divide as in the example. 
 
 1. AVOIRDUPOIS WEIGHT. 2. 
 
 T. cwt.qrs.lb. T.cwt.qr.lb.oz. 
 52 12 3 15 24 18 3 12 9 
 24 10 26 20 19 2 18 2 
 
 28 2 2 17 
 
 3. TROY WEIGHT. 4. 
 
 lb.oz.dwt.gr. lb.oz.dwt.gr. 
 
 45 9 
 15 6 
 
 4 3 
 18 17 
 
 21 7 14 1 
 16 9 18 12 
 
 30 2 5 10 
 
 5. apothecaries' weight. 6. 
 lb. oz. dr. scr. lb. oz. dr. scr. 
 49 8 6 2 84 7 5 2 
 21 3 5 1 75 8 6 1 
 
 9. CLOTH MEASURE. 10. 
 
 yds. qr. na. yds. qr. na. 
 
 784 2 1 64 2 3 
 
 651 3 2 18 3 
 
 11. LAND MEASURE. 12. 
 
 A. R. po. A. R. po. 
 
 540 2 20 764 2 13 
 
 332 25 684 3 17 
 
 7* 
 
 13. LIQUID MEASUDE. 14. 
 
 T. hd.gal.qt.pt. T.hd.gal.qt pt. 
 2 2 40 1 1 27 1 18 2 1 
 1 1 14 3 1 18 2 20 3 1 
 
 1 1 
 
 25 
 
 2 
 
 
 
 
 
 
 15. 
 
 
 DRY 
 
 MEASURE. 
 
 16. 
 
 bush 
 
 pk. 
 
 qt. pt. 
 
 bush 
 
 pk. qt.pt. 
 
 220 
 
 2 
 
 
 
 1 
 
 643 
 
 2 
 
 6 
 
 214 
 
 1 
 
 1 
 
 1 
 
 324 
 
 1 
 
 7 1 
 
 6 7 
 
 17. CUBIC MEASURE. 18. 
 
 T. fh. C. ft. T. ft. in. 
 
 116 24 72 114 45 18 140 
 
 109 39 41 120 16 14 145 
 
 28 
 7. 
 L. 
 56 
 10 
 
 5 
 
 m. 
 
 1 
 
 
 1 1 
 
 LONG MEASURE. 8. 
 
 fur. po. fur po. yd.ft.in. 
 19 29 6 3 2 7 
 7 20 18 7 4 2 8 
 
 6 25 
 19. TIME. 20, 
 
 Y. m. w. d. h. mi. Y. m. w. d. h. 
 27 3 2 4 18 40 28 9 2 4 16 
 13 2 1 5 19 20 24 10 3 5 18 
 
 46 
 
 
 
 39 
 
 14 1 5 23 20 
 
 MOTION OR CIRCLE MEASURE. 
 
 21.sig. ** ' ''22.sig.° ' " 
 
 10 2 3 20 24 7 42 27 
 
 4 8 20 30 18 9 14 28 
 
 5 23 42 50 
 
 23. 
 
 Y.mo. w. d. ho. mi. sec. dec. 
 
 6 3 1 3 40 20 .5 
 
 1 2 6 2 57 36 .2f 
 
78 SUBTRACTION OF DII-FKRENT DENOMINATIONS. 
 
 24. W. has a cask of sugar weighing 7 cwt. 2 qrs. 18 lbs. 12 
 oz. ; if he shoiilcl sell 3 cwt. 2 qrs. 20 lbs., how much would he 
 have remaining? Ans. 3 cwt. 3 qrs. 26 lbs. 12 oz. 
 
 25. C. had 35 yds. 2 qrs. 1 na. of sheeting ; he sold two 
 pieces, one of 7 yards. 1 qr. 2 na., the other 1 1 yds. 2 qrs. 1 na. ; 
 how many yards remain of the piece ? Ans. 1 6 yds. 2 qrs. 2 na. 
 
 26. A farmer has 3 farms; the first contains 321 A. 2 R. 10 
 no., the second 231 A. 18 po., the third 180 A. R. 14 po. ; 
 if he should sell 274 A. 1 R. 19 po., how many acres would ho 
 have remaining ? Ans. 458 A. 1 R. 23 po. 
 
 27. W. purchased three hogsheads of wine ; if he should 
 sell 127 gallons, 3 quarts, 1 pint, how many gallons would re- 
 main ? Ans. 61 gallons, 1 pint. 
 
 28. Subtract 500 bushels, 1 peck, 4 quarts, from 740 bush- 
 els, 3 quarts. Ans. 239 bushels, 2 pecks, 7 quarts. 
 
 29. If it be 415 miles, 3 furlongs, 20 poles, to Pittsburgh, and 
 375 miles, 1 furlong, 30 poles, to Baltimore, how much farther 
 is it to Pittsburgh than to Baltimore ? Ans. 40 m. 1 fur. 30 po. 
 
 30. From 76 years take 27 years, 9 months, 3 weeks, 2 days. 
 
 Ans. 48 years, 2 months, weeks, 5 days. 
 
 To find the difference between two given dates. 
 
 Write down the larger number, and under it the less number. 
 \{ the number of days in the less number be more than in the 
 greater, subtract from as many as there be days in the month 
 mentioned in the less number, and add in the days in the greater 
 number, which set down, and carry one to the month, then sub 
 tract in the usual way. 
 
 31. W. was born the 18th of the 4th month, 1775 ; how old 
 was he on the 25th of the 3d month, 1824 1 
 Year, month, day. 
 
 Thus: 1824 3 25 
 
 1775 4 18 
 
 48 11 7 Ans. 
 
 32. A person was born March 17, 1796 ; required his agu 
 August 24, 1824. Ans. 28 years, 5 months, 7 days. 
 
 33. A. had 500 bushels of wheat; *^he sold to B. 150.75, to 
 C. 62.25, to D. 18.5, for Dl.75 per bushel; how many bushels 
 had he remaining, and how much money did he receive ? 
 
 34. A merchant had 3 hogsheads of sugar, which weighed 
 2764.5 pounds ; he sold 1421.25 pounds for DUO; how much 
 
MULTIPLICATION OF DIFFERENT DENOMINATIONS. 79 
 
 did he receive per pound ? and how much would the balance he 
 has remaining come to, at the same price ? 
 
 35. Take 252.76 acres from 431.18 acres, and calculate the 
 value of the remainder at D67.27 per acre ? 
 
 36. There are two men, the oldest is 81 years, 6 months, 3 
 weeks, 1 day, 21 hours, 16 seconds ; the youngest 29 years, 10 
 months, 2 weeks, 4 days, 16 hours, 34 minutes, 45 seconds; 
 what is the difference of their ages ? 
 
 37. What is the difference of time between 31 years, 10 
 months, 2 weeks, 4 days, 7 hours, 24 minutes, 49 seconds, and 
 10 years, 10 months, 2 weeks, 2 days, 7 hours, 59 minutes, 14 
 seconds? Ans. 21 years, 1 day, 23 hours, 25 min. 35 sec. 
 
 38. A merchant bought 375 tons, 15 cwt. 3 quarters, 19 lbs. 
 7 oz. 12 dr. of sugar, and sold 205 tons, 17 cwt. 1 quarter, 27 
 lbs. 9 oz. 15 dr. ; how much had he remaining 1 
 
 Ans. 169 tons, 18 cwt. 1 qr. 19 lbs. 13 oz. 13 dr. 
 
 39. Bought a piece of cloth containing 145 yards, 3 quarters, 
 and sold 95 yards, 2 quarters, 3 nails, how much remains ? 
 
 Ans. 50 yards, 1 nail. 
 
 40. From 174 hhds. 10 galls. 1 qt. 1 pt. take 86 hhds. 17 
 galls. 2 qts. 1 pt. Ans. 87 hhds. 55 galls. 3 qts 
 
 REVIEW. 
 
 What do you understand by Subtraction of Different Denom- 
 inations, and when is it used? How do you write down 
 numbers in this rule ? What do you place over the several de- 
 nominations ? Where do you begin to subtract ? When the 
 number to be subtracted is less than the one above it, what will 
 you do ? When it is greater, what will you do ? What is the 
 rule ? How do you prove questions in this rule ? How do you 
 set down two given calendar dates, in order to find the difference 
 between them ? After having set down the two given dates, 
 where do you begin to subtract ? How do you proceed when 
 the number of days in the less date is greater than the number 
 of days in the greater date ? What is the rule ? 
 
 MULTIPLICATION OF DIFFERENT DENOMINA- 
 TIONS. 
 
 The object of this rule is to perform a number of additions 
 of different denominations, or when numbers of that description 
 are to be multiplied. 
 
80 MULTIPLICATION OF DIFFERENT DENOMINATIONS. 
 
 When the multiplier does not exceed twelve. 
 
 Write down the multiplicand, and write the quantity of the 
 several denominations over each, as directed in subtraction. 
 Then write the multiplier under the lowest denomination, at the 
 right hand ; then multiply that denomination, and divide it by 
 as many as it takes of that denomination to make one in the 
 next, and set down the remainder, and carry the quotient to the 
 product of the next denomination, &c. Proof as in simple 
 multiplication. 
 
 EXAMPLES. 
 
 10. 20. 4. 28. 16. 16; 
 
 T. cwt. qr. lb. oz. dr. 
 
 4 9 2 17 12 13 
 
 7 
 
 31 7 2 12 9 11 
 
 Explanation. First 7x13 = 91 
 — 16 = 5 and 11 over; now 12x7 
 = 84 and 5 are 89-f-16=5 and 9 
 over, set down the 9 ; then 17x7 
 = 119 and 5 are 124^28=4 and 
 12 over, set down the 12 ; 2x7 = 
 14 and 4 are 18-^4 are 4 cwt. and 
 
 2 qrs. over, set down the 2 qrs. ; now 9 X 7 = 63 and the 4 cwt. 
 
 make 67-:- 20 = 3 tons and 7 cwt. over, set down the 7 cwt. ; 
 
 now 4x7=28 and the 3 tons are 31 tons. 
 
 1. 2. 
 
 lb. oz. dwt. gr. L. m. fur. po. 
 
 17 5 12 6 15 2 7 30 
 
 3 6 
 
 52 4 16 18 95 2 6 20 
 
 bush. pk. qt. 
 
 14 3 2 
 
 6 
 
 A. R. po. 
 47 3 15 
 
 2 
 
 95 2 30 
 
 5. 6. 
 
 yr. da. h. m. sec. lb. oz. pwt. gr. 
 4 5 20 32 10 4 1 15 22 
 
 7 11 
 
 88 
 
 33 5 23 45 10 45 7 15 2 
 
 7. 
 
 8. 
 
 yds. ft. in. 
 
 yr. mo. w. da. 
 
 4 2 7 
 
 7 8 3 6 
 
 6 
 
 9 
 
 9. 10. 
 
 yr. da. h. m. sec. L.m.fur.po.yd. 
 8 125 18 47 50 47 1 1 9 4 
 4 5 
 
 11. 12. 
 
 lb.oz.dr.scr.gr. lb.oz.dr.scr.gr 
 
 19 10 6 1 15 26 11 5 2 16 
 
 8 11 
 
 13. 14. 
 
 deg. m. fur. po. yd. ft. in. 
 
 4 62 7 34 184 2 11 
 
 7 9 
 
MULTIPLICATION OF DIFFERENT DENOMINATIONS. 
 
 8. 
 
 15. ^ IG. 
 
 T. cvvt. qr. lb. T. cwt. qr. lb. oz. 
 
 18 14 2 18 27 4 1 10 7 
 9 8 
 
 17. 18. 
 
 mo. w. da. h. m. sec. yr. m.da. 
 8 3 6 215122 15 11 22.5 
 5 11 
 
 19. Multipl)' 24 bushels 1 peck, by 5, 
 
 20. Multiply 18 yds. 1 qr. 2 na. by 7. 
 
 Ans. 121 bush. 1 pk. 
 128 yds. 2 qrs. 2 na. 
 
 When the multiplier exceeds twelve^ and is the product of any two 
 figures in the multiplication table. 
 
 RULE. 
 
 Multiply the sum by one of the figures, and that product by 
 the other, and the last product will be the answer. 
 
 21. Multiply 20 yds. 2 qrs. 1 na. by 42. 6x7=42. 
 
 A?is. 863 yds. 2 qrs. 2 na. 
 
 22. Multiply 3 A. 2 R. 10 po. by 49. 174 A. 2 R. 10 po. 
 
 23. Multiply 47 A. 3 R. 20 po. by 54. 2585 A. 1 R. po. 
 
 24. Multiply 48 M. 7 fur. 25 po. by 88. 4307 M. 7 fur. po. 
 
 25. Multiply 56 lb. 9 oz. 6 dr. by 84.(apoth.) 47721b. 3 oz. dr. 
 
 26. What will 176.75 acres come to, at 1)47.62 per acre ? 
 
 27. What will 67.47 yards of cloth cost, at D3.75 per yard ? 
 
 28. What will 47.5 bushels of grain cost, at Dl.25 per bush. ? 
 
 29. In 6 parcels of wood, each containing 5 cords, 96 feet, 
 how many cords 1 Ans. 34^, or 34,5. 
 
 When the rnultiplier exceeds twelve^ and is not a composite 
 number, that isy the product of two figures, 
 
 RULE. 
 
 Multiply the simple numbers by each of the denominations 
 separately, and reduce each product to the highest denomination 
 named. Then add the several products together, and their sum 
 will be the answer sought. 
 
 30. Multiply 7 cwt. 3 qrs. 22 lbs. by 51 . Ans. 405 cwt.l qr. 2 lb. 
 
 31. Multiply 12 lb. 5 oz. 8 dwt. by 39. 485 lb. 6 oz. 12 dwt. 
 
 32. Multiply 4 M. 6 fur. 21 po. by 87. 418 M. 7 fur. 27 po. 
 
 33. Multiply25M.3fur.l8po.byl265. 32170 M. 4 fur. 10 po. 
 
 34. Multiply 48 ft. 4 in. 2 be. by 2587. 125182 ft. in. 2 be. 
 
 35. Multiply 4 hhd. 37 gall. 2 qt. by 4250. 1 9529 hhd. 48 gall. 
 
 36. If 1 cord of wood cost D5.62^, what will 12 cords cost ? 
 
 Ans. 67.47, 
 
82 DIVISION OF DIFFERENT DENOMINATIONS. 
 
 37. What costs a box of sugar, weighing 106 pounds, a 
 151 c. per pound ? Ans. Dl6.16f 
 
 38. At D9.10 per day, how much a year ? D3321.50. 
 
 What is Multiplication of Different Denominations ? When 
 IS it used 1 Under what part of the number to be multiplied do 
 you place the multiplier ? How will you then proceed ? By 
 Avhat do you di^ade the product ? If there is a remainder, what 
 is to be done ? How do you carry ? If there be no remainder 
 after the division, what will you then do? What will you do 
 with the quotient ? Proof? Repeat the rule. Repeat the rule 
 when the multiplier is the exact product of any two figures in 
 the multiplication table. When the multiplier is not a compos- 
 ite number, and more than twelve, how will you perform the 
 operation ? Repeat this rule. 
 
 DIVISION OF DIFFERENT DENOMINATIONS. 
 
 By this rule we can divide a number containing different de- 
 nominations. 
 
 If the divisor does not exceed twelve, proceed as in short 
 division of whole numbers ; only observe this difference, that in 
 this rule there are several denominations ; therefore divide the 
 left hand denomination, and place the quotient under it ; reduce 
 the remainder, if any, to the next less denomination, and add in 
 the given number of that denomination ; then proceed as before. 
 
 2. When the divisor exceeds twelve, it will be more conve- 
 nient to work by long division, observing the above rule. Proof; 
 by multiplication. 
 
 EXAMPLES. 
 
 Divide 10 yards, 3 quarters, 2 nails, among 5 persons. 
 
 4. 4. Explanation. First say, 5 in 10 
 
 yds.qrs.na. twice and over ; then 5 in 3 qrs. 
 
 5)10 3 2 times, 3x4 = 12+2=14; then 5 in 
 
 1 4, 2 times and 4 remainder ; and the 
 
 2 2-f 4 sum is done. 
 
DIVISION OF DIFFERENT DENOMINATIONS. 
 
 83 
 
 3 If Ans, 
 
 4. 28. 16. 16. 
 
 cwt. qr. lb. oz. dr. cwt. qr. lb. oz. drams. 
 
 40)50 2 14 9 11(1 1 1 12 
 
 40 Explanation. — The first denomination into 
 
 — which you divide is cwt. ; here you have 
 10 1 cwt. and 10 cwt. over, or remainder, mul- 
 
 4 X tiply this remainder by 4 qr. and add in the 
 
 — 2 qr. in the dividend, and you have 42 qr. ; 
 40)42(1 qr. divide as before, and you have 1 qr. and 2 
 
 40 qr. over, multiply this remainder by 28 lb. 
 
 — and add in the 14 lb., and you have 70 lb. ; 
 2 divide as before, and you have 1 lb. and 30 
 
 28 X lb. remaining, multiply the 30 lb. by 16 oz. 
 
 — and add in 9 oz., and you have 489 oz. ; di- 
 40)70(1 lb. vide by 40 and you have 12 oz. and 9 re- 
 
 40 maining, multiply the 9 oz. by 16 dr. and 
 
 — add in the 11 dr., and you have 155 dr. ; 
 30 divide, and you have 3 dr. and 35 remain- 
 16 X der: these collectively are the answer. 
 
 40)489(12 oz. 
 40 
 
 89 
 80 
 
 9 
 16x 
 
 40)155(3 dr. 
 120 
 
 EXAMPLES. 
 
 1. cwt. qr. lb. 2. yd. *qr. na. 
 5)7 2 .15 7)25 2 1 
 
 1 2 3 
 
 3. M. fur. po. 
 6)45 7 18 
 
 3 2 2+3 
 
 4. A. R. po. 
 9)35 2 16 
 
 7 5 9 + 4 
 
 3 3 32+8 
 
 rem. 35 
 
 7. 
 L.M.fur. 
 12)62 2 18 
 
 9. 
 cwt. qr. lb. 
 5)45 3 27 
 
 11. 
 yds. qr. na. 
 7)44 1 2 
 
 5. yd. qr. na. 6. A. R. po. 
 10)87 2 3 7)59 2 37 
 
 Y. da. h. m. sec 
 
 6)47 96 18 47 55 
 
 10. 
 
 cwt. qr. lb. 
 
 9)10 15 
 
 12. 
 
 yds. qr. na. 
 
 11)56 3 3 
 
 22 
 
 I 13. 
 
 M. fur. po. 
 12)105 5 
 15, 
 hhd. gall, qt 
 63)44 28 2 
 17. 
 D. c. 
 78)196 75 
 
 14. 
 M. fur. po. 
 6)45 7 18 
 
 16. 
 
 hhd. gall. qt. 
 
 120)150 47 3 
 
 18. 
 
 D. c. m. 
 
 97)496 87 5 
 
84 DIVISION OF DIFFERENT DENOMINATIONS. 
 
 QUESTIONS. 
 
 19. Divide 482 bushels, 2 pecks, 4 quarts, among 3 persons. 
 
 Ans. 160 bushels, 3 pecks, 4 quarts, each. 
 
 20. A gentleman bequeathed to his 5 sons a tract of land con- 
 taining 842 acres, 2 roods, 20 poles, to be divided equally among 
 them; required the share of each. Ans. 168A. 2R. 4po. 
 
 21. Divide 17L. IM. 4fur. 21po. by 21. Ans, 2M. 4fur. Ipo. 
 
 22. Divide 4.5 gallons of wine equally among 144 persons. 
 
 Ans. 1 gill each. 
 
 23. From a piece of cloth containing 64 yards, 2 nails, a tai- 
 lor wishes to make 9 coats, which will take one third of the 
 whole piece ; how many yards did each coat contain ? 
 
 Ans. 2 yards, 1 quarter, 2 nails. 
 
 24. If 90 hogsheads weigh 56 tons, 13 cwt. 3 quarters, 10 lb., 
 what is the weight of one hogshead ? Ans. 12 cwt. 2 qr. 11 lbs. 
 
 25. If the earth revolve on its axis 15 degrees in one hour, 
 how far does it revolve in one minute ? Ans. 15'. 
 
 26. If 59 casks contain 44 hhds. 53 gallons, 2 quarts, 1 pint, 
 what are the contents of each cask ? 
 
 27. When 175 galls. 2 quarts of beer are drunk in 52 weeks, 
 how much is consumed in 1 week? Ans. 3 galls. 1 qt. 1 pt. 
 
 28. Divide 168 bushels, 1 peck, 6 quarts of wheat among 35 
 men. 
 
 29. What will be the share of one man, if 810 tons, 11 cwt 
 20 lb. 10 oz. 11 dr. be divided equally among 346 men? 
 
 Ans. 2 tons, 6 cwt. 3 qrs. 11 lb. 8 oz. 13 dr 
 
 30. Suppose a man has 246 miles, 6 furlongs, 36 poles, to 
 travel in 12 days, how far will that be in a day ? 
 
 Ans. 20 miles, 4 fur. 23 po. 
 
 31. If a steamboat should go 224 miles a day, how long 
 would it take her to go to China, it being 12000 miles ? 
 
 Ans. 53 days, 13 hours, 42 min. 51 sec. 
 
 32. If one man can lift 201 pounds, 12 ounces, how much 
 can a boy lift, if a man can lift 8 times as much as a boy ? 
 
 Ans. 25 lbs. 3 oz. 8 dms. 
 
 33. If 15 loads of hay contain 35 tons, 4 cwt., what is the 
 eight of each load ? 
 
 34. Divide 371 bushels, 1 peck of wheat equally among 270 
 men. Ans. 1 bushel, 1 peck, 4 quarts. 
 
 35 Divide 19 cwt. 3 qrs. 27 lb. 12 oz. of sugar among 48 
 families. Ans. I qr. 18 lb. 10 oz.+28. 
 
 36. Divide 4 cwt. 2 qrs. 7 lbs. 12 oz. of rice among 14 per- 
 sons. Ans. 1 qr. 8 lbs, 8 oz +13 
 
APPLICATIOX OF THE COMPOUND RULES. 85 
 
 37. Divide 19 bushels, 2 pecks, 7 quarts, 1 pint, 2 gills. 
 among fi2 persons. 
 
 38. Divide 1749 acres, 2 roods, 30 poles, equally among 9 
 men and 7 women. 
 
 REVIEW. 
 
 When is Division of Different Denominations used ? Where 
 will you place the divisor? If, when you divide the highest 
 denomination by the divisor, a remainder occurs, how do you 
 proceed ? When the divisor exceeds twelve, but is a composite 
 numbers, how can the operation be performed 1 Ans. First 
 divide the given sum by one of the numbers, and that quotient 
 by the other, and the last quotient will be the answer. When 
 the divisor exceeds twelve, and is not a composite number, how 
 will you proceed? Repeat the rule. How can you prove 
 questions in this rule ? 
 
 APPLICATION OF THE COMPOUND RULES. 
 
 1. A man travelled in one day 27 miles, 3 furlongs ; another 
 day, 30 miles, 2 furlongs, 25 poles ; required the distance. 
 
 Ans. 57 miles, 5 furlongs, 25 poles. 
 
 2. A man has three farms, the first contains 150 acres, 2 
 roods, 25 poles; the second, 200 acres, 1 rood, 15 poles; the 
 third, 100 acres, 1 rood, 10 poles ; how many acres in all^ 
 
 Ans. 451 acres, 1 rood, 10 poles. 
 
 3. Add 2 cwt. 3 quarters, 27 pounds; 1 cwt. 2 quarters, 16 
 pounds; 3 cwt. 1 quarter, 25 pounds; 5 cwt. 2 quarters, 12 
 pounds; 2 cwt. 2 quarters, 14 pounds; 5 cwt. 1 quarter, 15 
 pounds. Ans. 21 cwt. 2 qrs. 25 lbs. 
 
 4. Bought 3 gallons, 2 quarts of wine ; 5 gallons, 3 quarts of 
 vinegar ; 4 gallons, 1 qt. of molasses ; how many gallons in all ? 
 
 5. A tailor purchased a piece of broadcloth containing 40 
 yards, of which he sold 36 yards, 1 quarter, 2 nails ; how much 
 will he have remaining ? Ans. 3 yards, 2 quarters, 2 nails. 
 
 6. If a cask of wine contains 54 gallons, 2 quarts, 1 piht, 2 
 gills, and there should be 10 gallons, 3 quarts, 1 pint, 1 gill sold, 
 how much would remain ? ^4;?^. 43 galls. 3 qts. pts. 1 gilL 
 
 7. From 40 bush, take 23 bush. 2 pks. Ans. 16 bush. 2 pks. 
 
 8. From 1 square yard take 3.5 square feet. 
 
 9. From 1 lb. troy take 15 grains. Ans. lloz. 19dwt. 9gr. 
 
 10. From 1 gall, take 1 gill. Ans. 3 qts. 1 pt, 3 gills. 
 
 8 
 
86 APPLICATION OF THE COMPOUND RULES. 
 
 11 A merchant bought 5 chests of tea, each weighin/^ ^cwt. 
 2 qrs. 9 lbs. ; what was the weight of the whole ? 
 
 Ans. 17 cwt. 3 qrs. 17 lbs. 
 12. From 1 acre take 5^ poles. Ans. 3 R. 34^ po. 
 
 .13. A person purchased 6 hhds. of sugar, each weighing 8 
 cwt. 1 qr. 18 lbs. ; required the weight of the whole. 
 
 Ans. 50 cwt. 1 qr. 24 lbs. 
 
 14. A merchant purchased 8 casks of oil, each containing 41 
 galls. 3 qts. 1 pt. ; required the number of gallons. Ans. 335 
 
 15. In 28 pieces of calico, each containing 35 yards, 2 quar- 
 ters, 1 nail, how many yards in all ? Ans. 995 yds. 3 qrs. 
 
 16. A farmer has 6 bins of wheat, containing 53 bushels, 3 
 pecks, 5 quarts, 1 pint each ; how much in all 1 
 
 Ans. 323 bushels, 2 pecks, 1 quart. 
 
 17. Add 3 quarts, 3 pints, 1 gill ; 5 quarts, 4 pints, 3 gills , 
 6 quarts, 1 gallon, 1 pint, together. 
 
 18. From 1 ton of round timber take 50 cubic inches. 
 
 19. From 1 year take 12 hours. Ans. llmo. 3w. 6da. 12h. 
 
 20. From 12 miles, 15 rods, take 3 furlongs. 
 
 Ans. 11 m. 5 fur. 15 rods. 
 
 PROBLEM : 
 
 Resulting from a comparison of the compound rules, by the 
 operation of which many questions may be solved in a short 
 and elegant manner. 
 
 1. Having the sum of two numbers, and one of them given 
 to find the other. 
 
 Rule. — Subtract the given number from the given sum, and 
 the remainder will be the number required. 
 
 Let 154 be the sum of two numbers, one of which is 84, the 
 other is required. 154 sum— -84 given number =70 the other. 
 
 2. Having the greater of two numbers, and the difference 
 between that and the less given, to find the less. 
 
 Rule. Subtract one from the other. 
 
 If the greater number f)e 672, and the difl^erence between 
 that and the less 389, required the other number, 672 given— 
 389 diflrerence=283 less. 
 
 3. Having the least of two numbers given, and the diff*erence 
 between that and the greater, to find the greater. 
 
 Rule. — Add them together. 
 
 The less number is 142, and the difference 167; required 
 .he greater number. 309 greater number. 
 
APPLICATION OF THE COMPOUND RULES. 87 
 
 4. Having the product of two numbers, and one of them 
 given, to find the other. 
 
 Rule. Divide the product by the given number, and the quo- 
 tient v^rill be the number required. 
 
 If the product of two numbers be 196, and one of them 4, re- 
 quired the other. Thus: 196-4-4=49. Ans. 
 
 5. Having the dividend and quotient, to find the divisor. 
 
 Rule. Divide the dividend by the quotient. This will prove 
 division. 
 
 Let the dividend be 144, and the quotient 16, required the 
 divisor? 144—16=9. Ans. 
 
 6. Having the divisor and quotient given, to find the dividend. 
 
 Rule. Multiply them together. 
 
 Let the divisor be 6, and the quotient 72, required the divi- 
 dend. 72X6=432. Ans. 
 
 EXAMPLES. 
 
 1. Suppose a man born in the year 1743, when will he be 
 77 years of age ? Ans. 1820. 
 
 2. What number is that, which if it be added to 19418 will 
 make 21802? Ans. 2384. 
 
 3. What number must you multiply by 9, that the product 
 may be 675 ? Ans. 75. 
 
 4. What is the difference between thrice five and thirty, and 
 thrice thirty-five ? ^ Ans. 60. 
 
 5. If a man spend 192D. in a year, how much is that per cal- 
 endar month ? 
 
 6. A. borrowed at different times the following sums, namely : 
 of B. D625 ; of C. D721.50 ; of D. D842 ; and he is indebted 
 to others as much as he has borrowed, abating D125.50 ; he is 
 now prepared to retire from business ; required the amount of 
 his debts. Ans. D4251.50 
 
 7. General Burgoyne and his army were captured at Saratoga, 
 N. Y., by General Gates, October 17, 1777,%nd Earl Cornwallis 
 and his army surrendered to General Washington at York, Va., 
 October 19, 1781 ; required the space of time between. 
 
 Ans. 4 years, 2 days 
 
 8. Bought 5 loads of wood : the first containing 1 cord, 32 
 feet ; the second, 1 cord, 64 feet ; the third, 112 feet ; the fourth, 
 1 cord, 28 feet ; the fifth, 1 cord, 20 feet ; how much in all ? 
 
 Ans. 6 cords 
 
88 REDUCTION OF DECIMALS. 
 
 9 How many bottles holding 1 pint, 2 gills each, are required 
 for bottling 4 barrels of cider ? 
 
 10. 9 A. 7 R. 50po.+ 17 A. IIR. 70po.; then-12A. 5R. 
 45 po. ; then X by 62 ; then -^-by 49. 
 
 REDUCTION OF DECIMALS. 
 
 The two following cases of Reduction of Decimals are the 
 very reverse of each other, and by reversing the rule, one will 
 prove the other. The use of the rule is to change a denomina- 
 tion, or several denominations, from their given expression into 
 a decimal quantity having the same value, for the purposes of 
 multiplication, division, &c., which will then become the same 
 as whole numbers. Thus, if you wish to find the decimal ex- 
 pression of 2 quarters, 14 pounds, avoirdupois, first reduce it to 
 pounds, 2 x28=:56+ 14=70 pounds ; now annex ciphers to the 
 70 pounds, and divide by 112 lbs.=z=l cwt. of which you wish 
 to make it a decimal, and you have .625; jiwo—TT2~i^^^ 
 JDy^Y^^f. Again, you wish to reverse the rule, and give the 
 .625 the expression in the terms of the integer, or to reduce it 
 to its proper value. Thus .625 cwt. multiply by the next less 
 denomination, which is 4 qrs., and count off the three places for 
 decimals, according to the rule in' multiplication of decimals, 
 and the quantity at the left of the decimal point is of the same 
 name with the multiplier, that is, 2 qrs. ; then multiply the re- 
 maining decimal, .500, by the next less denomination, namely, 
 28 lbs., and count off as before, and you have 2 quarters, 14 
 pounds = j%2_5__^7^o__ 5 c^i-^ ^c. The correctness of the op- 
 eration of the rule is evident from the nature of decimals, and 
 the explanations already given in the preceding pages. 
 
 In the third case : to reduce a vulgar fraction to its equiva- 
 lent decimal, by annexing one, two, three, or more ciphers to 
 the numerator, the value of the fraction is increased ten, a hun- 
 dred, or more times. After dividing, the quotient will, of course, 
 be ten, a hundred, or more times, too -much ; the quotient must, 
 therefore, be divided by ten, a hundred, (fcc, to give the true 
 quotient or fraction. In the first example, ^ is i<^^ = i^*, 
 which divided by 1000 is ^^^j^z=:.l25, and this is the rule. 
 
REDUCTION OF DECIMALS. 89 
 
 To reduce numbers of different denominations, as of money, 
 weight, measure, <Sfc., to their equivalent decimal value. 
 
 RULE. 
 
 1 . Begin at the left hand and multiply by as many as it takes 
 of the next lower denomination to make one of the higher, add- 
 ing in the denominations respectively, as you multiply, until 
 they are reduced to the lowest denomination in the question, 
 and this is the dividend. 
 
 2. Then take one of that denomination of which you wish to 
 make it a decimal, and reduce it to the same denomination with 
 the one above-mentioned, and this last number is the divisor. 
 Divide as in whole numbers, and the quotient is the answer 
 Put the point before it, for it is always a decimal. 
 
 EXAMPLES. 
 
 1. Reduce 3 roods, 20 poles, to the decimal of an acre. 
 
 lA. Explanation. — 
 
 4R. X R. po. .875 A. You will first mul- 
 
 77 3 20 4R.=1A. tiply the 3 roods 
 
 ^^ 40 X by 40, because 40 
 
 R. 3.500 poles make 1 rood, 
 
 po. 1 60) 140.000(-Yyjy\ Ans. 40po.=lR. and add in the 20 
 
 1280 po., this will make 
 
 po. 20.000 proof. 140 poles ; then 
 
 1200 annex ciphers, and 
 
 1120 the dividend will 
 
 be formed ; now 
 
 800 * you wish to make 
 
 800 it the decimal of an 
 
 an acre, then take 
 
 1 acre and reduce it to the same denomination as the dividend, 
 that is, poles, 160 = 1 acre, and this is the divisor ; then to prove 
 it, multiply the quotient by the next lower denomination, observ- 
 ing to count off for the decimal point, and the denominations at 
 the left of the point make the answer, &c. 
 
 2. Reduce 20 poles to the decimal of an acre. Ans. .125. 
 Reduce 2 R. 4 po. to the decimal of an acre. Ans. .525. 
 Reduce 3 quarters, 2 nails, to the decimal of a yard. 
 
 4 qrs. 3 qrs. 2 na. 
 
 4 na. X 4 
 
 nails, 16 divisor. 16)14.000(.875 Am. 
 
 5. Reduce 1 gall, to the decimal of a hhd. Ans. .015873 
 
 8* 
 
90 REDUCTION OF DECIMALS. 
 
 6. Reduce 7 oz. 19 dwt. to the decimal of a pound troy. 
 
 240)159.0000(.6625. Ans 
 
 7. Reduce 3 quarters, 21 pounds, avoirdupois, to the decimal 
 of a cwt. Ans. .9375, 
 
 8. Reduce 2 feet, 6 inches, to the decimal of a yard ? 
 
 Ans, .83333+ 
 
 9. Reduce 5 furlongs, 16 poles to the decimal of a mile. 
 
 Ans. .675 
 
 10. Reduce 4.5 calendar months to the decimal of a year. 
 
 Ans. .375. 
 
 11. Reduce 40 minutes to the decimal of an hour. .666 -j- 
 
 12. Reduce 20 minutes to the decimal of an hour. .333-f- 
 
 13. Reduce 1 day, 6 hours, to the decimal of a week. .1785. 
 
 14. Required the cost of 1 yard, 2 quarters, 2 nails of cloth, 
 at D5.00 per yard. 
 
 2 qrs. 2 na. yd. 1.625 
 
 4 5 
 
 16)10.000(.625. D8.12.5 Ans. 
 
 15. What is the value of 3 yards, 1 quarter, 3 nails of cloth, 
 al D4.50 per yard? 
 
 1 qr. 3 na. yd, 3.4375 
 
 4 4.50D. 
 
 16)7.0000(.4375. D15.46.87.5 Ans. 
 
 16. Required the value of 1 acre, 2 roods, 18 poles, at D20 
 per acre ? 
 
 2 roods 18 poles. acres 1.6125 
 
 40 20D. 
 
 poles 160)98.0000(.6125 D32.25.00 Ans. 
 
 17. What cost 2 acres, 1 rood, 30 poles, at D25 per acre ? 
 
 Ans. D60.93.75 
 
 18. What cost 2 pecks, 7 quarts of wheat, at Dl.50 per bush 
 
 Ans. Dl. 07.8.125. 
 
 19. What cost 7 bushels, 2 pecks, 7 quarts of corn at 52 cts. 
 per bushel? D4.01.375. 
 
 20. What is the cost of 3 bushels, 3 pecks, 4 quarts of rye, 
 at 75 cents per bushel? Ans. D2.90.6.25. 
 
 21. What is the cost of 2 quarters, 15 pounds of coffee, at 
 Dll per cwt. ? Ans. D6.97.312. 
 
 22. Reduce 375678 feet to miles and decimals. 71.151-f 
 Note. — This last question is an exception to the general rule, 
 
 ©r at least questions of this kind are not of frequent occurrence 
 
REDUCTION OF DECIMALS. 91 
 
 although the rule will apply in this as well as in other cases ; 
 the number of feet given are equal to a number of miles ; there- 
 fore, take the feet for the dividend, reduce 1 mile to feet=5280 
 for a divisor, then divide according to the rules of Division of 
 Decimals, and you have 71.151-1- miles. This example will 
 apply and be sufficient in cases of this kind. 
 
 To reduce a Decimal to its proper value. 
 
 RULE IT. 
 
 1. Multiply the decimal by the number of parts in the next 
 less denomination, and cut off so many places for a remainder, 
 counting from the right, as there are decimal places in the given 
 decimal, and there make the decimal point. 
 
 2. Multiply the remainder, that is, the decimal, by the next 
 less denomination, and cut off a remainder as before ; continue 
 in this way through all the parts of the integer, and the several 
 denominations standing on the left hand of the decimal point 
 make the answer. 
 
 23. Required the value of .9075 of an acre. 
 
 Explanation. — Begin and multiply 
 by the next lower denomination that, 
 the one mentioned in the question 
 which is roods, 4 of which make ap 
 acre. After multiplying, count off a* 
 many places or figures as there are 
 in the given question for decimals, and 
 there place the decimal point, and the 
 figure or figures at the left of the 
 decimal point, will be of the same 
 name as the multiplier, namely, roods. 
 Then take the next less denomination, 
 which is poles, 40 of which make 1 
 rood ; multiply all the figures on the 
 
 right of the decimal point, then count 
 
 1800000 off from right to left as many figures 
 
 1800000 as there were decimals in the number 
 
 450000 multiplied ; there place the decimal 
 
 In. 64.800000 point, and the figures at the left of the 
 
 point will be poles. Thus far we have 
 
 3 roods, 25 poles, and .2000 ; now 30.25 yards make one pole 
 
 in square measure ; therefore, multiply by that number those 
 
 figures at the right of the decimal point, which are .2000, and 
 
 count off in the product 4 figures for those in the multiplicand, 
 
 and 2 for the multiplier, 6 figures, and there place the decimal 
 
 
 .9075 
 X4R. 
 
 Roods 
 
 3.6300 
 40 po. 
 
 Poles, 25.2000 
 
 30.25 yd. 
 
 Yards, 6.050000 
 
 9 feet 
 
 Feet, 
 
 0.450000 
 
 144 in. 
 
92 REDUCTION OF DECIMALS. 
 
 point, and you have 6 yards ; then multiply by 9, because 9 
 feet make 1 yard, and count off as before, which gives feet^ 
 yet the remaining decimal of a foot must be reduced to inches, 
 144 being equal to 1 foot, and you have 64 inches. Thus we 
 find that .9075 of an acre is equal to 3 roods, 25 poles, 6 yards, 
 feet, 64 inches. 
 
 24. What is the value of .046875 of a pound avoirdupois ? 
 Thus, .046875x16=750000x16—12.000000 drams, Ans. 
 
 25. What is the value of .617 of a cwt. ? 
 
 Ans. 2qr. 131b. 1 oz. 10 dr. 
 
 26. What is the value of .569 of a year of 365 days ? 
 
 Ans. 207 da. 16 ho. 26 min. 24 sec. 
 
 27. Find the proper quantity of .9071 of an acre. 
 
 28. Find the value of .76442 of a pound troy. 
 
 Ans. 9 oz. 3 dwt. 11 gr. 
 
 29. What is the value of .875 of a yard ? Ans. 3 qr. 2 na. 
 
 30. Find the proper quantity of .089 of a mile 1 
 
 Ans. 28 po. 2 yds. 1 ft. 11 in.+ 
 
 31. Find the proper quantity of .897 of a degree. 
 
 To reduce a Vulgar Fraction to its equivalent decimal, 
 
 RULE III. 
 
 1 . Annex ciphers to the numerator, and then divide it by the 
 denominator. 
 
 2. There must be as many places pointed off in the quotient 
 as you annex ciphers to the given numerator ; if there should 
 not be so many places of figures in the quotient, the deficiency 
 must be supplied by prefixing ciphers at the left hand of the 
 quotient ; if the numerator is larger than the denominator, there 
 must be a whole number, or whole number and decimal. 
 
 32. Reduce i to a decimal having the same value 1 8)1.000 
 
 Ans. .125 
 
 33. What decimal is equal to J? 2)1.0 
 
 ^ — 
 
 Ans. .5 
 
 34. Reduce ^i to a decimal. Ans, .6875. 
 
 35. Reduce i to a decimal. .2. 
 
 36. Reduce ^j to a decimal. .037037 + . 
 
 37. Reduce ^ to a decimal. .33333 + . 
 
 38. Reduce -^^ to a decimal. .85. 
 
 39. Reduce 3% to a decimal. .09375. 
 
 40. Reduce jyVs ^^ ^ decimal. .008 
 
 41. Reduce /^ to a decimal. .1923076 
 
REDUCTION OF DECIMALS. 93 
 
 42. Reduce ^^ to a decimal. Ans, .153846 
 
 43. Reduce }| to a decimal. -25=:^ 
 
 44. Reduce ^^ to a decimal. .125 = 1 
 
 45. Reduce ^-| to a decimal. .1875. 
 
 46. Reduce /g- to a decimal. .083333 + 
 
 47. Reduce ^§ to a decimal. 
 
 48. Reduce %* to a whole number. 8.00 
 
 49. Reduce ||- to a whole number. 
 
 50. Reduce y/ to a whole number. 
 
 ' 51. Reduce ^U® to a whole number. 
 
 REVIEW OF DECIMALS. NO. 11. 
 
 . When you wish to reduce several denominations to the deci- 
 mal of a larger denomination, still retaining the same value, how 
 do you proceed? (See rule, (fee.) In question 1st the quotient is 
 .875 ; what do you understand by this ? Ans. Eight hundred 
 seventy-five thousandths of an acre, which is equal to 140 rods • 
 that is, 140 rods bear the same proportion to 160 rods, or one 
 acre, as. 875 does to .1000, or y^^^^, or 875 to 1000, &c. Sup- 
 pose the number at the left of the decimal point in the dividend 
 had been more than 160 (the divisor) would the quotient have 
 been a decimal ? Ans. In that case it must have been a whole 
 number and decimal, but after a decimal figure is brought down 
 for a new dividend that quotient figure will be a decimal, as there 
 can be no decimal in the quotient until there is one in the divi- 
 dend. How do you reduce a decimal to its proper value 1 (See 
 the rule.) Can you explain the first example ? Wherein does 
 this rule differ from the one immediately preceding it ? Ans. 
 It is the very reverse of the first, and they reciprocally prove 
 each other. How do you reduce a vulgar fraction to it*? equiva- 
 lent decimal? What is the rule ? What do you understand by 
 numerator? Ans. It is the number above the line, thus ^, 
 which serves as the common measure to others. What is the 
 meaning of denominator ? Ans. It is the number below the 
 line, thus ■§-, and is the number of parts a unit, or anything, is 
 divided into, and gives the name or denomination of the parts. 
 What i^ understood by the expression of |-D. ? ^^^qD. ? Ans. 
 Seven eighths of a dollar, or 87.5 cents, and the other is twenty- 
 five hundredths, which is 25 cents. If you owned If of the cap- 
 ital in trade, what part would be yours ? Ans. Two thirds (§•); 
 that is, the stock would be divided into 24 shares, and I should 
 have sixteen of them, and 16 is two thirds of 24. Suppose the 
 stock divided into 32 shares, what part would then be yours ? 
 Ans. One half, because \^ is equal to one half. If divided into 
 
94 APPLICATION OF THE RULES OF DECIMALS. 
 
 64 shares how much 1 Ans. One fourth (J) ; for as the de 
 nominator increases, the vahie of the numerator decreases in the 
 same proportion ; and the same of the denominator, for ^|-, in 
 stead of being one half as above, it would be just 4 times as 
 much=D2.00, or my share would be 4xl6=:64, which is 
 more than the given capital. What is this last (^|-) fraction 
 called? Ans. An improper fraction. Why? Ans. Because 
 the numerator is larger than the denominator, consequently the 
 fraction expresses one whole, or more than one whole number, 
 as f =1, V^ = 2, &c. Any number may be expressed or writ-, 
 ten fractionwise, wheiher it be a whole number, decimal, or vul- 
 gar fraction, retaining its value, thus: J§tH-50=-3|Q ; fl-T-S 
 =/^rr.06974+ ; f§~8=f=1.25; 746-^f =1989i, &c. 
 
 Note. This part of arithmetic is fully explained in its proper 
 place. (See Vulgar Fractions.) 
 
 APPLICATION OF THE RULES OF DECIMALS, &c, 
 
 1. Add 163-7- days, 183-2^%, 143-7^% together. 
 
 2. A. purchased 4 hogsheads of molasses : the first contained 
 62y\^ gallons ; second, 72^q^-^-^q gallons ; third, 50j2_ gallons ; 
 fourth, 55-^-^Q gallons ; how many gallons in all ? 
 
 Ans, 240.6157. 
 
 3. What is the amount of one and five tenths ; forty-five and 
 three hundred and forty-nine thousandths ; and sixteen hun- 
 dredths ? Ans, 47.009. 
 
 4. What is the difference between 4.75 minutes and 6.25 
 minutes? Ans. 1.5 mi. 
 
 5. From .65 cwt. take .125 cwt. Ans. .525. 
 
 6. From 769.7 hogsheads take 596.92. Ans. 172.78. 
 
 7. What will 9.7 lb. of sugar cost at D0.07 per lb. ? 
 
 Ans. DO.67.9. 
 
 8. At D0.90 per bushel, what will 7.25 bushels cost? 
 
 Ans. D6.52.5. 
 
 9. What is the quotient of 1561.275-^-24.3 ? Ans. 64.25 
 
 10. What is the quotient of .264-^2? Ans.. 132. 
 
 11. What decimal is equal to f ? Ans. .25 
 
 12. What decimal is equal to -^q ? .175. 
 
APPLICATION OF THE RULES OF DECIMALS. 95 
 
 13. What decimal is equal to ^ ? Ans, .3333, circulating rept. 
 
 14. What is the value of 2 yds. 3 na. French calico, at 70 c. 
 per yd '' Ans. Dl.53.125. 
 
 15. What cost 2 cwt. 17 lb. of chocolate, at 15D. per cwt. ? 
 
 2.151786 X15 = D32.27.6790. Ans. 
 
 16. What cost 1.25 tons of hay, at llD. per ton ? 
 
 Ans. 13D. 75c. 
 
 17. If 31.25 yds. cost 75D. how much is it per yard ? 
 
 Ans. D2.40. 
 
 18. If 35 bushels of wheat cost D97.50, what did 1 bushel 
 cost ? Ans. D2.78.5 + 
 
 19. What is the cost of 1 gall, and 1 pt. of wine at D1.75 
 per gall.? Ans. D 1.96.875. 
 
 20. If I pay 14D. for 112 lb. of sugar, and sell it for 12.5c. 
 per pound, do I gain or lose, and how much ? 
 
 21. If I pay 7D. for 80 pounds of flax, »nd sell it for 8c. per 
 pound, do I gain or lose, and how much ? Ans. Loss 60c. 
 
 22. When rye is selling at Dl.25 per bushel, how many 
 bushels can I have for D62.50. Ans. 50 bushels. 
 
 23. A gentleman bequeathed his estate of one hundred thou- 
 sand dollars to his son and daughter ; the son was to have 15 
 thousand, 15 hundred and 15 dollars more than the daughter; 
 required the share of the daughter. Ans. 41742.50. 
 
 24. If the sum of 33485D. was expended in the purchase of 
 United States land, at Dl.25 per acre, how many acres would 
 it purchase 1 Ans. 26788 acres. 
 
 25. Then if you should sell the 26788 acres for D2.5 per 
 acre, how much would you gain ? Ans. 33485D. 
 
 26. If you should buy a chest of tea containing 135 lb. for 
 150D., and sell the same for Dl.30 per lb., what would be the 
 difference ? Ans. Gain D25.5. 
 
 27. If a pipe of wine contains 126 galls., how many pinta 
 are in the pipe, and what is it worth at 14.5c. per pt. ? 
 
 Ans. 1008 pints, worth D146.16. 
 
 28. K farmer purchased 120 sheep, for which he paid D1.75 
 per head, and 47 young cattle at D 14.50 per head ; he sold the 
 sheep for D2.14 per head, and the cattle for 14D. per head ; did 
 he gain or lose, and how much ^ Ans. Gained D23.30. 
 
 29. A certain field will produce 127 bushels of grain; how 
 much will a field produce, just seven and a half times as large ? 
 
 Ans. 952.5 b. 
 
 30. A. and B. start from the same place ; A. goes east 17 days, 
 at the rate of 39 miles per day ; B. goes west at the rate of 43 
 miles per day for 14 days ; how many miles will each travel, 
 
96 APPLICATION OF THE RULES OF DECIMALS. 
 
 and how many distant from each other ? A71S. A. 663 m., B. 602 
 m., distance, 1265 m. 
 
 31. If 57 yards of cloth cost 197D., what cost 1 yard? 
 
 32. If 18 yds. 1 qr. of cloth cost D91.16, what is the cost 
 of 1 yd. ? 
 
 33. If you have 7D. and pay away .07, how much will re 
 main ? 
 
 34. What is the value of 23 grains of silver, at 14D. per lb 
 troy ? 
 
 35. If you earn ID. and Im. per day, how much is that h 
 one year 1 
 
 36. John Armstrongs Dr. 
 
 To 1 pound of tea at D 1.56c. 
 
 To 10 pounds loaf sugar at 19c. 
 
 To 3 gallons of wine at Dl.17. 
 
 To 9 pounds of steel at 10c. 
 
 To 6i yards of sheeting at 54c. 
 
 To 3 yards of cotton at 18c. 
 
 To 2 pounds of butter at 31c. 
 
 To 1 pound of indigo at Dl.46. 
 
 To goods per order, D3.72. 
 
 To 4 bushels of oats at 50c. D19.72. 
 
 37. Levi Woodbury^ Cr. 
 
 By 5.5 bushels rye at 50c, - D2.75 
 
 By 3i cords of wood at 1)4.50 - 15.75 
 
 By 3.25 bushels of wheat at Dl.75 - 5.68.7.5 
 
 By cash - 2.75 
 
 By 25 pounds of cheese at 7c. - 1.75 
 
 By 20.5 pounds of butter at 18c, - 3.69 
 
 By li cwt. of iron at D6.58 - 8.22.5 
 
 By 12 pounds of lard at 12ic. - 1.50 
 
 By 7^ pounds of raisins at 9c. - 67.5 
 
 By 6 quarts of salt at 5c. - 30 
 
 38. How many quarts in 7 hogsheads ? in 18 ? in 27 ? in 34 ? 
 
 39. How much is a hogshead of molasses worth, at 9c. per 
 quart? at 14? 
 
 40. A cask of sugar weighs 5 cwt., what is it worth at 12^c. 
 per pound ? Ans. D70.00. 
 
 41. If .25 of a yard is worth D0.25, what cost 17.5 yards ? 
 what cost 27.25 yards ? 
 
 42. How many cubic inches in 5 feet? in 11 feet ? in 40 ft. ? 
 
 43. How many inches in 35 poles ? in 65 poles ? in 75 poles ? 
 
Kj-:dUCTIO.S of DIFFERENT DENOMINATIONS. 0'^ 
 
 44. How many secpnds in 5 hours ? in 2 days ? in 1 week ? 
 
 45. How many hours in 5 years? in 12 years? in 170 
 days ? 
 
 46. How many rods in 7 acres, and how much would it be 
 worth at 75 cents per rod ? Arts, 840 D. 
 
 47. In 6795 square feet, how many square yards ? 
 
 Ans. 755. 
 
 48. In 78479 mills how many dollars ? how many eagles ? 
 
 49. What cost 60 galls. 1 pt. of wine, at .75D. per gallon ? 
 
 Ans. D45.09.34- 
 
 50. What will 26 yards 2 quarters of cloth cost at D4.75 per 
 yard? Ans. D125.87.5. 
 
 51. Required the value of 25.874 yards; 39.42 yards ; 67 
 yards, 2 nails ; 47 yards, 2 quarters, at D3.074 per yard. 
 
 52. What cost 91.27 gallons; 218.087 gallons; 8 gallons, 
 2 quarts ; 91 gallons, 3 quarts, at Dl.25 per gallon ? 
 
 53. 47963.0521 +897.009 + 81.81 +9.0009 + .629108-M.08. 
 
 54. What number is that, which being added to 9.0146 will 
 make 20.7092 ? 
 
 REDUCTION OF DIFFERENT DENOMINATIONS. 
 
 In Reduction of Different Denon^inations we have several 
 denominations given to be reduced to a lower denomination, 
 still retaining the same value ; it is called compound reduction. 
 The remarks and explanations in simple reduction will apply ou 
 the present occasion. 
 
 1 . Multiply the highest denomination by as many of the next 
 hess as it requires to make one of that, and add in the second 
 
 denomination, if any ; then multiply by the next less denomina- 
 tion, and add in, if any ; continue in this way through the sum. 
 
 2. When small denominations are to be reduced to larger de- 
 nominations, reverse this operation, and begin by dividing in the 
 same manner as you are directed above to multiply ; by this 
 reversion of the operatior. the proof is obtained. See the ev 
 amples. 
 
 9 
 
98 
 
 REDUCTION OF DIFFERENT DENOMINATION^. 
 
 EXAMPLES AND QUESTIONS. 
 
 4 28 16 16 
 
 1. Reduce 5 cwt. 2 qrs. 12 lb. 11 oz. 10 drs. to drams. 
 4x 
 — 16)1609^1^ drams. 
 
 20 
 
 2+ 16)10059+10 dr. 
 
 22 qrs 
 28 X 
 
 176 
 
 28)628+11 oz. 
 4)22 + 12 lb. 
 
 44 
 
 12 + 
 
 628 lb. 
 
 cwt. 5+2 qr. proof 
 5 cwt. 2 qrs. 12 lb. 11 oz. 10 dr. 
 
 16x 
 
 Explanation. Multiply the 5 cwt. 
 
 3768 
 628 
 
 10048 
 11 + 
 
 10059 oz. 
 16x 
 
 60354 
 10059 
 10 + 
 
 by 4 quarters, and add in the 2 quarters 
 and you have 22 quarters ; then multi- 
 ply 22 quarters by 28 pounds, and add 
 in the 12 pounds, and you have 628 
 pounds ; then multiply 628 pounds by 
 16 ounces and add in the 11 ounces, 
 and you have 10059 oz. ; then multi- 
 ply those ounces by 16 drams, and add 
 in 10 drams, and you have 160954, 
 which is the answer ; then for the 
 proof you can divide by short or long 
 division, as you like, &c. 
 
 160954 drams. 
 *Z In 126230400 seconds how many years of 365 days ? 
 6,0)12623040,0 
 
 6,0)210384,0 
 
 4X6=24 h. 
 
 4)35064 
 
 I 6)8766 
 
 365)1461(4 years. 
 1460 
 
 Ans, 4 years, 1 day. 
 
 } 4ay. 
 

 REDUCTION OF DIFFERENT DENOMINATIONS. % 
 
 3. Reduce 3124446 drams to tons 
 { 4)3124446 
 
 4X4=16 drams < 
 
 4x4 = 16 oz. 
 
 4X7=28 lb. 
 
 ( 4)781111 ... 2 
 
 ( 4)195277 ... 3x4+2 = 14 dr. 
 
 ( 4)48819 ... 1 
 
 4)12204 . . . 3x4+1 = 13 oz. 
 
 7)3051 
 
 qrs. 4)435 ... 6x4=24 lbs 
 
 ewt. 2,0)10,8 ... 3 qrs. 
 
 5 T. 8 cwt. 
 Ans. 5 T. 8 cwt. 3 qrs. 24 lbs. 13 oz. 14 drs 
 
 4 Reduce 2 cwt. 3 qrs. 17 lbs. to pounds. Ans. 325 lbs. 
 
 5 Bring 16 bushels 1 peck to pecks. 65 pks. 
 
 6 Reduce 2 hhds. 10 gallons to quarts. 544 qts. 
 
 7. Reduce 18 years, 6 months, to months. 222 mo. 
 
 8. Reduce 15 yards, 2 feet, to inches. 564 in. 
 
 9. Reduce 3 leagues, 2 miles, 7 furlongs, to furlongs 95 fur. 
 
 10. Reduce 11 acres, 2 roods, 19 poles, to poles. 1859 po. 
 
 11. In 12 cords of wood how many solid feet and inches ? 
 
 Ans. 1536 ft.; 2654208 in. 
 
 12. In 21 cords of wood how many solid feet ? 2688 ft, 
 
 13. In 24796800 seconds, how many weeks ? 41 w. 
 
 14. In 20692 square rods, how many acres ? 129 A. 1 R. 12 po. 
 
 15. In 15840 yards how many miles and leagues ? 9 m. 3 1. 
 
 16. In 51 miles how many furlongs and poles ? 
 
 Ans. 408 fur.; 16320 po 
 J7. In 3783 nails how many yards ? 236 yds. 1 qr. 3 na. 
 
 18. Bring 892245 ounces into tons. 
 
 Ans. 24 T. 17 cwt. 3 qr. 17 lb. 5 oz. 
 
 19. In 12 hhds. of sugar, each 11 cwt. 25 lb., how many 
 ounces ? Ans. 241344 oz. 
 
 20. How many revolutions will the wheel of a steamboat, 27 
 feet in circumference make, in navigating the Hudson from 
 New York to Albany, supposing the distance 160 miles, with- 
 out making any allowance for current or tide ? Ans. 31288. 
 
100 REDUCTION' OF DIFFERENT DENOMINATIONS. 
 
 21. How many inches between the earth and the sun, the 
 distance being 95,000,000 miles ? Ans. 6019200000000. 
 
 22. If a cannon ball move at the rate of 1 mile in 7 seconds, 
 how long wonld it be in passing from the earth to the sun 1 
 (Multiplyby 7,divideby60,&;c.),l;i?i5. 21yr.31da. 18h.13m.20s. 
 
 23. Admitting your age to be 16 years, 1 week, 17 days, 20 
 minutes, 45 seconds, how many seconds old are you ? (52w.= ly.) 
 
 Ans. 50526S445S. 
 
 24. Reduce 184320 grains to pounds. Ans. 32 lbs. 
 
 25. In J 6 tons how many hundred w^eight ? quarters ? pounds ? 
 ounces 1 and drams ? 
 
 A;^^. 320cwt. ; 1280qrs.; 320001bs.; 51200007..; 8192000drs. 
 
 26. In 1332005 grains (apoth.) how many pounds ? 
 
 Ans, 231 lb. 3 oz. 5 gr. 
 
 27. In 9173 nails how many yards ? Ans. 573 yds. 1 qr. 1 na. 
 
 28. In 3 ells French, how many inches ? A7is. 162. 
 
 29. in 34594560 inches how many miles ? Ans. 546 ra. 
 
 30. What is the circumference of the earth in yards ? 
 
 Ans. 44035200 yds. 
 
 31. Reduce a solar year to seconds. Ans. 31556937 sec. 
 
 32. In 622080 solid inches how many tons of round timber? 
 
 Ans. 9 T. 
 
 33. In 5529600 cubical inches how many cords. Ans. 25. 
 
 34. How many minutes and seconds in one complete revolu- 
 tion of a planet ? Ans. 21600 minutes ; and 1296000 sec. 
 
 35. In 89763 square yards how many acres ? 
 
 A?is. 18 A. 2 R. 7 po. 101 ft. 36 in. 
 
 36. In 5054 pints how many bushels ? A?is. 78 b. 3 pk. 7 qts. 
 
 37. How many quarters of rice, at 6c. per pound, may be 
 bought for D3.36 ? Ans. 2 qrs. 
 
 38. How many tuns of wine, at 6 cents a gill, may be bought 
 for D483.84. Ans. 1 tun. 
 
 39. What is the value of a silver cup weighing 10 oz. 5 pw^t. 
 18 gr. at 5 mills per grain 1 Ans. 24D. 69c. 
 
 40. At 9 cents an hour what will 2 yrs. 6 mo. 3 w. 6 da. 12 
 h. labor be? Ans. D 1873.80. 
 
 41. At 320 cents a yard, what will 64 nails of cloth cost? 
 
 Ans. D 12.80. 
 
 42. In 86400 grains how many scruples ? drams ? and ounces ? 
 
 ^71^. 4320 sc. ; 1440 dr. ; 180 oz 
 
 43. In 8903304 barleycorns how many leagues ? 
 
 Ans. 15 1. 1 m. 6 fur. 28 po. 4 yds. 
 
 44. In 20 cords of wood how many solid inches ? 
 
 Ans. 4423680 in. 
 
PROPORTION ; OR, SINGLE RULK OF THREE. 101 
 
 io. In 20 tuns of wine how many gills J Ans. 161280 g 
 
 46. In 11378955037 seconds, how many years? 
 
 Ans. 360 yrs. 300 da. 20 ho. 50 m. 37 sec. 
 
 47. How many barley-corns will reach round the globe, it 
 oemg 360 degrees ? -Ani-. 4755801600. 
 
 ' 48. In running 300 miles, how many times will a wheel 9 ft 
 2 in. in circumference turn round? Ans. 172800. 
 
 t 49. How many seconds in 1828 years, allowing 365-1- days 
 to the year? Ans. 57687292800. 
 
 50. What is the cost of a tun of wine at 3 cents per gill ? 
 
 II Ans. D241.92. 
 
 f REVIEW. 
 
 When is Compound Reduction used ? When several de- 
 nominations are given to be reduced, how will you proceed ? 
 What is the rule ? How will you prove questions in this rule 1 
 
 PROPORTION; OR, SINGLE RULE OF THREE. 
 
 The first idea that naturally presents itself to the mind in the 
 consideration of Proportion, is Ratio, or the connexion existing 
 between Ratio and Proportion ; for ratio is a Latin word, and 
 means proportion, although it can only exist between quantities 
 of the same kind. The numbers given are called terms, and 
 may be considered as a divisor and dividend ; and the ratio, the 
 required 'term, as a quotient in a division sum. The terms are 
 written thus, 5 : 15, ratio 3 ; 3 x5 = 15 ; the ratio of 84 : 336 
 = 4, because 84 is contained in 336, 4 times ; so may any two 
 numbers of the same kind form a ratio. What is the ratio of 
 9 to 18 ? Ans. 2. What is the ratio of 6 to 24 ? What is the 
 ratio of 11 to 13 ? Ans. 1^\. What is the ratio of 100 to 30 ? 
 Ans. ^^. 2 : 4 :: 8 : 16 ; here the ratio is 2, because 2 is in 
 4=:2, and 8 is in 16 = 2. 2:10:: 12 : 60 ; here the ratio is 5, 
 because 2 is in 10=5 ; 12 in 60=5. 16 : 15 :: 48 : 45 ; here 
 the ratio is reversed, and will stand thus, -|f , first and second 
 terms ; ||— i|., the third and fourth terms. 
 
 Proportion is a comparative relation of one thing, or number, 
 to another ratio. Proportion, when complete, consists of four 
 numbers or terms, and is a combination of two equal ratios ; 
 and the product of the first and fourth terms (which are the 
 extremes) is equal to the product of the second and third terms 
 
102 proportion; or, stnglk rule of three. 
 
 (middle term or means.) Therefore, it is evident that any one 
 of the four terms is rSadily ohtained, when we have the other 
 three given. 
 
 2 : (4 :: 8) : 16 Thus: if 4 lbs. of tea cost 8 dollars, 
 
 4x 2x what will 12 lbs. cost? 
 
 — — lbs. lbs. I). Here the price of 
 
 32 32 4 : 12 :: 8 the tea is 2 dollars 
 
 8 per pound ; conse 
 
 — quently, 4 pounds will 
 4)96 cost 8 dollars, 12 
 
 — pounds will cost 24 
 D24 cost, dollars, &c. 
 
 D. lbs. D. 
 Again: If 24 : 12 :: 8 : 4 lbs. Ans. 
 
 8 Hence we find, to obtain either the 
 
 — first or fourth term, divide the product 
 24)96(4lb. of the second and third (middle) 
 terms, by the given extreme, and the 
 quotient will be the other extreme, or term sought. And to 
 obtain either of the middle terms, divide the product of the 
 extremes by the given middle terms, and the quotient will be 
 the middle term required. The fu-st and second terms of a 
 proportion always express quantities of the same kind, and so 
 likewise do the third and fourth terms. Thus : — 
 lbs. lbs. D. D. 
 2 : 12 :: 6 : 36 
 That is, if 2 pounds cost 6 dollars, what will 12 pounds cost? 
 Ans. 36 dollars ; it is 3 dollars per pound, and 3 X 12 r=: 36 dol- 
 lars. This last example is the modern method of stating Pro- 
 portion, and now considered the most correct. The old method 
 was thus: If 21b. : cost 6D. :: what cost 12lb. : 36D. ; 12 X 
 6=72-^2=36 as above. 
 
 The following short rules can often be used to advantage : — 
 
 1. Divide the second term by the first, multiply the quotient 
 by the third, and the product will be the answer. 
 
 lbs. D. lbs. D. 
 
 5 : 10 :: .15 : 30 
 
 2. Divid-e the third term by the first, multiply the quotient by 
 the second, and the product will be the answer. 
 
 lbs. D. lbs. D. 
 
 5 : 10 :: 15 : 30 
 
 3. Divide- the first term by the second, and the third term by 
 that quotient, and the last quotient will be the answer. 
 
 D. lbs. D. lbs. 
 
 10 • 5 :: 30 : 15 
 
proportion; or, single rule of three. 103 
 
 4. Divide the first term by the third, yid the second term by 
 that quotient, and the last quotient will be the answer. 
 
 lbs. 
 
 D. lbs. 
 
 D. 
 
 15 
 
 : 30 :: 5 : 
 
 GENERAL RULES. 
 
 10 
 
 Rule for stating. — 1. Place that term in the third place, 
 which is of the same name or kind, with that in which the 
 niiswer is required ; and if the answer requires to be greater 
 than the third term, set the greater of the two remaining num- 
 bers for the second term, and the other number for the first 
 term. 
 
 2. But if the answer requires to be less than the third term, 
 set the less of the two remaining numbers in the second place, 
 and the other in the first place. 
 
 Rule for working. — 1. Reduce the third term to the lowest 
 denomination mentioned in it. 
 
 2. Reduce the first and second terms to the lowest denom- 
 ination mentioned in either of them. 
 
 3. Multiply the second and third terms together, and divide 
 by the first, and the quotient will be the answer, in the samo 
 name to which the third was reduced ; then bring this denom- 
 ination into the answer required. 
 
 When 1, or unity, is one of the terms, a formal statement is 
 not required ; it can be done by multiplication or division 
 
 1. Multiply the first and fourth terms together (extremes). 
 Multiply the second and third terms together (means). 
 
 If the four numbers are proportional, these products will be 
 equal. 
 
 2. Divide the larger of the first and second terms by the less ; 
 divide the larger of the third and fourth terms by the less, and 
 the two ratios will be equal. 
 
 3. Invert the question. 
 
 The terms may be distinguished generally in the following 
 manner : the first term is preceded with an if, or supposition, 
 and the second by a demand, " what will," " what cost," <fcc. 
 
104 
 
 PROPORTION ; OR, SINGLE RULE OF THREE. 
 
 QUESTIONS. 
 
 1. If 4 lbs. sugar cost 64 cts, 
 
 what will 8 lbs. cost? 
 
 lbs. lbs. cts. 
 
 4 : 8 :: 64 
 
 8 
 
 4. If 5 lbs. of cheese cost 45 
 cents, what will 18 lbs. cost? 
 lbs. lbs. cts. 
 5 : 18 :: 45 
 18 
 
 4)512 
 
 
 360 
 45 
 
 D1.28 Ans. 
 2. If 2 dollars will purchase 
 
 
 
 
 5)810 
 
 8 pounds of butter, how many- 
 
 
 
 pounds will 50 cents buy ? 
 
 
 Dl.62 Ans. 
 
 c. c. lb. 
 
 5. If 8 lbs. 
 
 of pork cost 60 
 
 2.00 : 50 :: 8 
 
 cents, what will 24 lbs. cost? 
 
 8 
 
 lb. lb. cts. 
 
 or, lbs. cts. lbs. 
 
 
 8 : 24 :: 60 
 
 8 : 60 :: 24 
 
 200)400 
 
 60 
 
 24 
 
 Ans, 2 lbs. 
 
 8)1440 
 
 8)1440 
 
 3. If 1 bushel of rye cost 75 
 
 
 
 cts., what will 25 bushels come 
 
 D1.80 Ans. 
 
 m. SO Ans. 
 
 to at that price ? 
 
 6. If 1 bushel of wheat cost 
 
 75 cents. 
 
 Dl.12.5, what will 70 bu. cost ? 
 
 25 X bushel. 
 
 bush. bush. D. 
 
 
 1 : 70 
 
 :: 1.12.5 
 
 375 
 
 D. 1.12.5 
 
 150 
 
 X 
 
 70 bush. 
 
 D18.75 Ans. 
 
 D78.75.0 Ans. 
 
 7. If 4 yards of cloth cost D9.50, what will 18 yards cost? 
 
 Ans. D42.75. 
 
 8. If 1 bushel of clover-seed cost D3.25, what will 7 bushels 
 cost ? Ans. D22.75. 
 
 9. If 6 bushels of turnips cost D1.25, what will 30 bushels 
 cost? ♦ Ans. D6.25. 
 
 10. When eggs are 18 cents a dozen, what will 47 dozen 
 cost? Ans. D8.46. 
 
 11. When eggs are 12^ cents per dozen, how many can you 
 have for 11.50? Ans. 92 dozen. 
 
 12. Purchased a barrel of mackerel for Dll.50, on counting 
 shem there were 210 fish ; what was the cost of 1 ? 
 
 Ans. 5 cents, 4 mills. -^ 
 
PROPORTION, OH SINGLE RULE OF THREE. 105 
 
 13. Purchased a box containing 70 hats, for which I paid 
 D280, what was the cost of 1 liat ? Ans. D4.00. 
 
 14. If 1 pound of sugar cost 11 cents, what will 1200 pounds 
 cost? Ans. D132.00. 
 
 15. If 5 bushels of salt cost D3.12.5, what will 50 bushels 
 cost? Ans. D31.25. 
 
 16. If 4 men can perform a piece of work in 20 days, how 
 many days will it require for 3 men to do the same ? 
 
 Ans. 26.666=1 . 
 
 17. If 7 bushels of corn cost 13D., what will 40 bushels cost? 
 
 Ans, D74.28.5. 
 
 18. When hay is 68 cents per cwt. what is it per ton ? 
 
 Ans. D13.60. 
 
 19. When sugar is 11.5 cents per pound how much is it per 
 cwt.? Ans. D12.88. 
 
 20. WHiat costs a hogshead of molasses, when it is selling at 
 52 cents per gallon ? Ans. D32.76. 
 
 21. If a barrel of wine cost 50 dollars, how much is that per 
 quart? Ans. 39.7 cents nearly. 
 
 22. If a bushel of cloverseed cost 6 dollars, what is that per 
 pint ? Ans. 9.37 cents. 
 
 23. If 30 men can perform a piece of work in 11 days, how 
 many men will it require to perform a piece of work 4 times as 
 large in 12 days ? Ans. 110 men. 
 
 24. How many men must be employed to finish a piece of 
 work in 15 days which 5 men can do in 24 days ? Ans. 8 men. 
 
 25. If 6 muli can harvest a piece of grain in 12 days, in 
 what time will 24 men do it ? Ans. 3 days. 
 
 26. How long must a board be, that is 9 inches wide, to make 
 12 square feet? 9 in. : 12 :: 12 ft. Ans. 16 feet long. 
 
 27. if a pound of sugar cost 9.5 cents, what will be the price 
 of a hogshead weighing 5 cwt. 2 qrs. 17 lbs. ? Ans. D60.13.5. 
 
 28. If a person's daily expenses be D2.25, and his annual 
 income 1250D., what sum will he save at the end of the year? 
 
 Ans. 428.75. 
 
 29. How much tea can you purchase with 500D., when it is 
 selling at Dl.25 per pound? Ans. 400 pounds. 
 
 30. If a man is paying D2.12.5 per week for board, how* 
 much is it for 1 year? (52 weeks=l year.) Ans. DllO.50. 
 
 31. If 1 cwt. of sugar cost D11.50, what will 15 cwt. 2 qrs. 
 18 lbs. cost ? Ans. D180.09.8. 
 
 32. If the annual income of a person is 800D., how much 
 may he use daily, and have 300 D. remaining at the expiration 
 of the year? Ans, D1.37 nearly. 
 
106 PROPORTION, OR SINGLE RULE OF THREE. 
 
 33. If 4.25 yards of cloth cost D16.25, what will 16.5 yaiis 
 cost? Ans. D63.08.8. 
 
 34. If 1.25 yards of sheeting cost 12.5 cents, what will 50.5 
 yards cost ? Ans. D5.05. 
 
 35. If 1.5 bushels of wheat cost D2.25, what will 20.5 bush- 
 els cost ? Ans. D30.75 
 
 36. If 20.5 bushels of wheat cost D30.75, how much is it 
 per bushel ? Ans. Dl.50. 
 
 37. How much will 20.5 bushels of wheat cost, at Dl.50 per 
 bushel ? Ans. D30.75. 
 
 38. A farmer sold 6 bags of wheat at 1.12 dollars per bushel 
 of 60 pounds ; the bags on being weighed, contained the follow- 
 ing number of pounds of grain, namely: No. 1, 186 pounds; 
 No. 2, 182 pounds ; No. 3, 176 pounds; No. 4, 181 pounds ; 
 No. 5, 190 pounds; No. 6, 185 pounds ; it is required to know 
 how many bushels he had, and how much money he received. 
 
 lbs. lbs. lbs. D. lbs. 
 
 186 or 60 : 1100 :: 1.12 then 60)1100(18 bushels. 
 
 182 1.I2X 60 
 
 176 D. 
 
 181 60)123200(20.53.3J Ans. 500 
 
 190 480 
 
 185 
 
 bush. 20 
 
 60)1100(18.333=^ 4xpecks. 
 
 D. 60)80(1 peck. 
 
 1)1.12 60 
 
 + 18i- bushels. IS-J bushels. — 
 
 60 X pounds. 20 
 
 896 8 X quarts. 
 
 112 1080 
 
 20=1 of 60 60)160(2 quarts. 
 
 20.16 120 
 
 .371 price i bu. 1100 lbs. proof. 
 
 40 
 
 D20.531 2 X pints. 
 
 2d Ans. 18 bush. 1 pk. 2 qt. 1 pt. 1 J g. — 
 
 60)80(1 pint. 
 39. What is the cost of 1890 pounds 60 
 
 of wheat, at 1.15 dollars per bushel of 60 — 
 
 pounds? 1890 lbs. -^60=31.5 bushels 20 
 
 XD1.15 = D36.22.5. Ans, 4 X gills. 
 
 Or, 60 : 1890 :: 1.15 Ans. D36.22.5. — 
 
 60)80(lf§ = |=^ 
 
PROPORTION, OR SINGLE RULE OF THREE. 107 
 
 40. When corn is selling at 62.5 cents a bushel, how much 
 can you have for 125 dollars ? Ans. 200 bush. 
 
 41. Paid 175 dollars for 120 bushels of rye, what did one 
 bushel cost? Ans. D1.45.8 + 
 
 42. Bought a barrel of flour for 11.50 dollars, what is the 
 cost of 1 pound? Ans, 5. Sets. + 
 
 43. What will 16.5 bushels of oats cost, at 37.5 cents per 
 bushel? Ans. D6.18.75. 
 
 44. What will 2 loads of corn come to, at 75c. per bushel, 
 one load has 35.5 bushels, the other 37 bushels ? Ans. D54.37.5. 
 
 45. If 4 cords of wood cost 15.50 dollars, what will 12.5 
 cords cost? Ans. D48.43.7. 
 
 46. If 7.5 acres will produce 87.25 bushels of grain, what 
 quantity will 18.75 acres produce ? Ans, bush. 218.125=4 qts. 
 
 47. When hay is selling at 18 dollars per ton, how much is 
 It per cwt. ? Ans. 90 cents. 
 
 48. When wood is selling at 7D. per cord, how much is that 
 per foot ? Ans. 5.4 cents -f- 
 
 49. What is the cost of a stove weighing 2 cwt. 1 qr. 18 lbs., 
 at 2.50 dollars per cwt. ? Ans. D6.02.6. 
 
 50. What is the cost of 1800 chestnut rails, at 3.75 dollars 
 per thousand ? Ans. D6.75. 
 
 51. If 1 cwt. of sugar cost 13 dollars, 50 cents, what will 17 
 cwt. 3 qrs. 14 lb. cost ? Ans. D241.31.2. 
 
 52. If 1 tun of wine cost 300 dollars, what cost 1 gallon ? 
 
 Ans. 1 dollar 19 cents. 
 
 53. If a hogshead of molasses cost 50 dollars, how much 
 must it be sold for per gallon, to gain ten dollars on the cost ? 
 
 Ans. 95.2 cents. 
 
 54. Purchased a pipe of wine for 150 dollars: 15 gallons 
 leaked out ; how much must the remainder be sold for per gal- 
 lon, to get the first cost? Ans. D 1.35.1 
 
 55. Bought a cask of wine at 87.5 cents per gallon, and paid 
 130D., how many galls, did it contain ? Ans. 148 galls. 2 qts. 2 gi. 
 
 ^Q, A. is indebted to B. 965 dollars, but A. compounds with 
 him, and agrees to pay him 75 cents on the dollar ; how much 
 will B. receive ? Ans. 723 dollars, 75 cents. 
 
 57. If a man spend 9 cents a day, what will it amount to in 
 a year ? Ans. 32 dollars, 85 cents. 
 
 58. How much wheat, at 1 dollar, 15 cents, per bushel, can 
 you purchase for 125 dollars ? Ans. 108 bush. 2 pks. 6 qts. 
 
 59. In what time will 48 men perform a piece of work which 
 12 men can do in 24 days ? Ans. 6 days, 
 
 60» Sound moves at the rate of 1142 feet per second : if the 
 
108 PROPORTION, OR SINGLE RULE OF THRKE. 
 
 report of a piece of ordnance be heard 1 minute, 3 seconds after 
 the flash was observed, the distance is required ? Arts. 13m. 5fur. • 
 .■ 61. Of what length must a board be, that is 4.5 inches in 
 "wi3th,,to make a square foot ? Ans. 32 inches. 
 
 62. Ml dollar and 25 cents per week, how many weeks can 
 you board for 150 dollars ? Ans. 120 weeks. 
 
 63. If 25 horses will consume 85 bushels of grain in 4 weeks, 
 how many bushels will 8 horses require in the same time ? 
 
 Ans. 27 bushels and 6 quarts. 
 
 64. The president of the United States receives a salary of 
 25000 dollars annually, what sum does he receive daily ? 
 
 Ans. D68.49.3-I- 
 
 65. When butter is 12.5 cents per pound, how much can you 
 have for 137 dollars ? Ans. 1096 pounds'. 
 
 66. If 1 bushel of corn cost 78 cents, what is the cost of 
 64.56 bushels ? .472^. D50.35.6. 
 
 67. If 5 pounds of venison cost 45 cents, what must you give 
 for 165 pounds ? Ans. 14 dollars, 85 cents. 
 
 68. When leather is selling at 22.5 cents per pound, what 
 will 750 pounds cost? Ans, 168 dollars, 75 cents. 
 
 69. If .5 yards of cloth cost 37.5 cents, what will 1.25 yards 
 cost? Ans. 93.75 cents. 
 
 70. If .75 pounds of indigo cost 87.5 cents, what will 1.5 
 pounds cost ? Ans. 1 dollar and 75 cents. 
 
 71. If 1 acre of land cost D18.25, what will 5 acres, 2 roods, 
 2 poles cost ? Ans. DlOO.60.312. 
 
 72. If 1.5 yards of silk cost D2.50, what will 1 quarter, 2 
 nails cost? Ans. 62.5 cents. 
 
 73. What cost 6 cwt. of sugar, at 10 cents per pound? 
 
 Ans. D67.20. 
 
 74. If 100 dollars in 12 months bring 6 dollars interest, what 
 sum will bring the same in 8 months ? Ans. 150 dollars. 
 
 75. How many yards of carpeting that is .75 yards wide, are 
 sufficient to cover a floor that is 18 feet wide and 60 feet long ? 
 
 Ans. 160 yards. 
 
 76. If 8 acres of land cost D182.50, what will 12.7 acres 
 ost? Ans. D289.71.9. 
 
 77. If 9.5 acres of land cost 240 dollars, what is the value 
 of 7.25 acres? Ans. 183.15.7. 
 
 78. If 15 yards of cloth cost D39.45, how many yards can 
 you buy for 21 dollars. Ans. 7 yards, 3 qUoLrters, 3 nail* 
 
 79. If 95 bushels of corn cost D68.25, what will 320 cost : 
 
 Ans. D229.89.4, 
 
 80. If 40 dollars will pay for 14^ yards of cloth, how many 
 ean you buy for 75 dollars ? Ans. 26 yards, 2 quarters, 3 nail^ 
 
FKOroilTK)X ; OR, SINGLE RULE OF THREE. 100 
 
 81. If a car will run 552 miles on a railroad in 24 hours, 
 w f:ir will it run in 13 hours ? Arts. 299 miles. 
 
 82. II" 15 pounds of coflee cost Dl.80, what will 18 pounds 
 cost? Ans. D2.16. 
 
 83. How much land at D2.50 per acre should be given in 
 exchange for 360 acres at D3.75 per acre 1 Ans. 540 acres. 
 
 84. How many feet of window-glass in a box containing 144 
 panes,7by9? 144x7x9=9072; 12 X 12=144)9072(63 ^n^. 
 
 85. How many cubical inches in a cubical mile ? 
 
 Thus, 1760 yds. x3x 12 = 63360 inches in 1 mile (L. M.) 
 63360x63360x63360=254358061056000 cubical inches = l 
 cubical mile. Ans. 
 
 86. How many cubical miles in the globe ? 
 
 Thus, as 355 : 113 :: 360x69.5 : 7964 diameter of the earth. 
 Then, 360 x 69.5 x 7964 X 1327.33=264482820122 cubical 
 miles in the globe. Ans. 7964-^-6=1327.33+ —^ diam. 
 
 87. How many cubical inches in the globe ? 
 
 Ans. 67273337308854741368832000 inches. 
 
 88. If you purchase 7 yards of cloth for 42 dollars, how 
 many yards can you have for 600 dollars ? and if you sell the 
 same at the rate of 40 dollars for 6^ yards, will you gain or lose, 
 and how much? Ans. gain D15.38.4-f 
 
 89. A merchant bought 49 tuns of wine for 910 dollars, the 
 freight cost 90 dollars, duties 40 dollars, cellar 31 dollars 67 
 cents, other charges 50 dollars, and he would gain 185 dollars 
 by the bargain ; what must I give him for 23 tuns ? 
 
 Ans. D613.33. 
 
 90. The earth, being 360 degrees in circumference, turns 
 round on its axis in 24 hours ; how far does it turn in 1 minute 
 in the 43d parallel of latitude, the degree of longitude in this 
 latitude being about 51 statute miles? Ans. 12| miles. 
 
 91. A. and B. depart from the same place, and travel the 
 same road, but A. goes 5 days beforelB. at the rate of 20 miles 
 per day; B. follows at the rate of 25 miles per day; in what 
 time and distance will he overtake A . ? 
 
 Ans. 20 days ; distance 500 miles. 
 
 92. If f of a yard of cloth cost D8.25, what will | yard cost ? 
 
 (1=2.25) Ans. 24.75. 
 
 93. If 300 barrels of flour cost 5070 dollars, what will 200 
 cost? What is f of 5070 ? 
 
 94. If I of a barrel cost -^^ of a dollar, what will f cost? 
 What is HXtt • ^^^- 4if=42.9c.-f- 
 
 95. What will 7 acres 2 roods 38 poles of land cost at 
 D64.50 per acre ? Ans. D499.06.875. 
 
 10 
 
110 proportion; or, single rule of three. 
 
 96. What is the cost of 328 yards of calico at 13J cents per 
 yard ? Ans. D43.733333^^. 
 
 97. What is the sum of 60 dollars and nine hundredths, 12 
 dollars and three tenths, 18 dollars and three thousandths, and 
 44 dollars and three hundredths ? Ans. Dl 34.423. 
 
 98. What is the cost of 19 gallons, 2 quarts, 1 pint, 1 gill of 
 wine, at 3 cents per gill ? 
 
 99. When a merchant compounds with his creditors for 40 
 cents on the dollar, how much is A.'s part, to whom he owes 
 2500 dollars? how much is B.'s part, to whom he owes 1600 
 dollars ? Ans. A. 1000 dollars ; B. 640 dollars. 
 
 100. If 30 bushels of rye may be bought for 120 bushels of 
 potatoes, how many bushels of rye may be bought for 600 bush- 
 els of potatoes? Ans. 150, ratio 5, 
 
 review. 
 
 What do you understand by ratio ? What does the ratio of 
 two numbers express ? What is it equal to ? What is the ratio 
 of 1 to 5 ? of 2 to 8 ? of 6 to 36 ? of 10 to 100 ? of 12 to 2 ? &c. 
 What are we able to ascertain by Proportion, or Single Rule of 
 Three ? Why is it called Proportion ? Which of the terms 
 will you place in the third place ? After having set down the 
 third term, what will you do next? If the answer ought to 
 be greater than the third term, how will you do ? If the answer 
 ought to be less than the third term, how will you proceed ? 
 After stating the question, what will you do with the third term ? 
 What will you do with the first and second terms ? Which of 
 the two terms will you multiply together for a dividend ? Which 
 of the terms will be the divisor ? After division, in what de- 
 nomination will your answer be ? What must be done to com- 
 plete the operation ? What can you say of the remainder, if 
 any ? How can you prove Proportion ? How do you know 
 when questions belong to Proportion? When 1, or unity, is 
 one of the terms, how can the question be solved ? What is the 
 rule for stating ? What is the rule for working ? Proof. How 
 many rules have you been through ? Name them. How many 
 questions have you solved in those rules? Ans, 1125. What 
 is your opinion of Proportion ? 
 
 Note. — All the rules in Proportion may be deduced from the 
 algebraical expression, a : b :: c : d, meaning that a divided by 
 b, gives the same quotient as c divided by d. A and D are 
 called the extremes, B and C the means; the product of the 
 means, divided by one extreme, will give the other extreme ; 
 and the product of the extremes, divided by one of the means, 
 will give the other mean ; and this furnishes the various rules 
 of Proportion. 
 
proportion; or, double rule of three. Ill 
 
 COMPOUND PROPORTION; OR, DOUBLE RULE OF 
 THREE. 
 
 Compound Proportion, or, as it is usually called, Double 
 Rule of Three, is different from Simple Proportion, in always 
 having an odd number of terms given, as five, seven ; it teaches 
 to solve such questions as require two or more statements by 
 Simple Proportion,, and hance its name. The remarks and 
 explanations in Simple Proportion are equally applicable in the 
 present case. All questions in Compound Proportion may be 
 solved by the following rules : — 
 
 1. Place that term in the third place, which is of the same 
 name with that in which the answer is required. 
 
 2. Then with this third term, and each pair of similar terms, 
 finish the statement as in the preceding directions in the Single 
 Rule of Three, and set them in the first or second places, as 
 directed for stating in that rule. 
 
 3. Reduce the third term to the lowest denomination men- 
 tioned in it, and the other term to similar denominations. 
 
 4. Multiply the first two terms together for a divisor, then 
 multiply the two terms in the second place, and that product by 
 the third term for a dividend. 
 
 5. Divide, and the quotient is the answer in the same de- 
 nomination with the third. Proof, invert the terms. 
 
 Note. — There are several methods by which the rules of pro- 
 portion may be cancelled or contracted ; for sufficient reasons, 
 those rules will be omitted here. 
 
 EXAMPLES. 
 
 1 . If 4 men in 12 days can harvest 48 acres of grain, how 
 many acres can 8 men harvest in 16 days ? 
 Thus: if men, 4 : 8 > . .g Explanation. — In the 
 
 days, 12 16 J " * first place, we are told 
 
 that 4 men in 1 2 days can 
 
 divisor — 48 128 harvest 48 acres of grain, 
 
 48 12x4=48 days; this is 
 
 just 1 acre per day for 
 
 48)6144(128 acres Ans, each man, because they 
 or, 16x8 = 128 acres. labor 48 days, and har- 
 
 vest 48 acres ; consequently, if 8 men labor 16 days, they will 
 harvest 128 acres, because 16 x 8 = 128 acres. The product of 
 4 and 12 bears the same proportion to 48 acres as the product 
 
113 proportion; or, double RULli OF THREE. 
 
 of 8 and 16 does to the answer, 128, which in this case are 
 the same ; then, if we multiply the product of 8 and 16 by 48 
 acres, the 5th term, and divide by the product of 4 and 12, tho 
 result or proportion will still be the same, &;c. 
 
 2. If lOOD. in 12 months will gain 7D. interest, how much 
 will 750D. gain in 48 months ? 
 
 lOOD. : 750D. :: > 7D. or 750D. 
 12 mo. 48 mo. > • 7x per cent. 
 
 1200 36000 5250 
 
 7D. 4x years. 
 
 12,00)2520,00(210 Ans. D210.00 Ans. 
 If lOOD. in 12 months will gain 7D. then it follows that in 
 the same proportion 750D. will gain 210D. in 48 months. 
 
 3. I lOOD. in 365 days will gain 6D., how much will GOOD 
 gain in 95 days ? 
 
 DlOO : 900D. :: ) 6 
 365 X da. 95da. 
 
 36500 8550 
 6X 
 
 D. c. 
 
 365)513000(14.05.4+ Ans. 
 
 4. If 400D. will gain in 7 months 14D., what is the rate per 
 cent, per annum ? Ans. 6D. 
 
 5. If lOOD. will gain 6D. in a year, in what time will 400D. 
 gaiti 14D.? Ans. 7 months. 
 
 6. If 16 men can moAv 112 acres of grass in 7 days, how 
 many acres can 24 men mow in 19 days ? Ans. 456. 
 
 7. If 2 men can construct 12 rods of wall in 6 days, how 
 many rods can 8 men build in 24 days ? Ans. 192. 
 
 8. If 5 men can make 300 pairs of shoes in 40 days, how 
 many men will it take to make 900 pairs in 60 days ? Ans. 10. 
 
 9. If 4D. will hire 8 men 3 days, how many days must 20 
 men work for 40D. ? Ans. 12. 
 
 10. If 4 men receive 12D. for 3 days' work, how many men 
 will it require to earn 48D. in 16 days ? Ans. 3. 
 
 11. If 100 bushels of oats be sufficient for 18 horses 20 days, 
 how many bushels will 60 horses require in 36 days ? 
 
 Ans. 600 bushels, 
 
 12. If the transportation of 8 cwt. cost D12.80 for 128 miles, 
 what sum must be paid for the carriage of 4 cwt. 32 miles ? 
 
 Ans. D1.60 
 
PROPORTION, OR DOUBLE RULE OF THREE. 113 
 
 13. How many dollars will it require to gain 6D. in 1 year 
 if wuth 560D. 1 gain 56D. in 1 year and 8 months ? Ans. lOOD. 
 
 14. If 6 tons of hay be sufficient for 8 horses 7 months, how 
 much will serve 20 horses 1 year and 5 months ? 
 
 Ans. 36 T. 8 cwt. 2 qrs. 8 lbs. 
 
 15. If 10 horses in 18 days consume the grass of 2 acres, how 
 many acres will 20 horses require in 27 days ? Ans. 6 acres. 
 
 16. If 20 bushels of wheat are sufficient for a family of 8 
 persons 5 months, how much will be sufficient for 4 persons 12 
 months ? Ans. 24 bushels. 
 
 17. If 8 men can build a wall 20 feet long, 6 feet high, 4 feet 
 thick, in 12 days, in what time will 24 men build one 200 feet 
 long, 8 feet high, and 6 feet thick 1 
 
 84-12=20x6 divisor. 200x6x8 dividend. 
 
 18. If a family of 8 persons use 360D. in nine months, how- 
 much will serve a family of 18 persons, 12 months ? 
 
 Ans. 1080U. 
 
 19. If lOOD. in 52 weeks gain 6D. interest, how much will 
 200D. gain in 26 weeks 1 Ans. 6D. 
 
 20. If 350D. in six months gain DlO.50 interest, what will be 
 the interest of 400D. for 4 years? Ans. 96D. 
 
 21. If a family of 6 persons expend 300D. in 8 months, how 
 much will serve a family of 15 persons for the same time ? 
 Thus: per. per. D. D. 
 
 6 : 15 :: 300 : 750 Ans. solved by a single statement. 
 
 22. If 750D. will support a family of 15 persons for 8 months, 
 how much will serve them 20 months ? 
 
 mo. mo. D. D. or 8 mo. : 20 :: > 750 : 1875. 
 8 : 20 :: 750 : 1875 -4;i^. 15 per. 15 5 Ans. 
 
 The number of persons (15) maybe omitted in statements of 
 this kind. 
 
 REVIEW. 
 
 What is Compound Proportion ? When you are about to 
 make a statement in this rule, which of the terms is first to be 
 , set down ? In what place ? What is then to be considered ? 
 What is to be done with the two terms which stand in the first 
 place ? What is to be done with the two terms which stand in 
 the second place ? By what do you multiply the product of the 
 two terms standing in the second place, and by what do you 
 divide that product for the answer, when they consist of differ- 
 ent denominations ? Repeat the rule. 
 
 10* 
 
114 VULGAR FRACTIONS. 
 
 VULGAR FRACTIONS. 
 
 Remarks. — A fraction is either vulgar or decimal, and as the 
 word implies, is a broken number, or parts ; a proper fraction is 
 less than unity. Those parts may be expressed by figures, as 
 well as whole things ; a whole is called an integer, but a part, 
 or some parts of a thing are denoted by figures as one third, one 
 fourth, seven tenths, &;c., of a thing; the expression of those parts 
 of figures are called fractions. The term fraction is derived from 
 a Latin word which signifies to break, as an integer or unity is 
 supposed to be broken or divided into a certain number of equal 
 parts, one or more of which parts are denoted by the fraction ; 
 thus, one fourth (1) denotes one of the four equal parts, &c., into 
 which a thing is broken, or integer divided. We may also view 
 it as a part of a certain number of units ; -J may either be con- 
 sidered as two thirds of one or one third of two, for one third 
 of two is the same quantity as two thirds of one, consequently 
 if the numerator of a fraction be viewed as an integer, and divi- 
 ded into as many equal parts as the denominator indicates, the 
 fraction may be regarded as expressing one of those parts ; thus, 
 if 4 be divided into 5 equal parts, the fraction } would express 
 one of them, and the fraction |- would express four of them, 
 40-^5 = 8x4=32 + 8=40; 8x5 = 40, &c. Fractions natu- 
 rally arise from the operation of division, when the divisor is 
 not contained a certain number of times, exactly, in the dividend. 
 For the remainder, after the division is performed, is a part of the 
 dividend which has not been divided, the divisor being the num- 
 ber of parts into which the integer is divided, and the remainder 
 showing the number of those parts expressed by the fraction. 
 Thus 4 is contained in 9 twice and one fourth times, and hence the 
 quotient can not be fully expressed in such cases, except by a 
 whole number and a fraction. As before observed, fractions are 
 of two kinds, vulgar and decimal (the latter have already been 
 explained) which may be reduced, or changed from one to the 
 other, still retaining the same value ; thus, f , which means three 
 parts or three quarters of anything, would, if expressed deci- 
 mally, be .75, or -^^q, which is the same, namely, three quarters. 
 
 Vulgar fractions are expressions for any assignable parts of 
 a unit or whole number, and are represented by two numbers 
 placed one above another, with a line drawn between them, thus : 
 |, |, &c. The number above the line is called the numerator, 
 and that below the line the denominator. The denominator 
 shows how many parts the integer is divided into, and the nu- 
 merator shows how many of those parts are meant by the frac* 
 
VULGAR FRACTIONS. 115 
 
 tion. If an apple or anything be divided into 4 equal parts, one 
 part is called ^ ; if divided into two equal parts, each part is 
 called ^f two of which =1 ; if divided into three equal parts, 
 one part is ^, two of those parts |, two thirds and one third =1, 
 when divided into eight equal parts, one part is ^, which is the 
 half of ^, two eighths =i, four eighths =^, six eighths =^, 
 seven eighths and one eighth =1 ; -jgg- of a dollar is 3 cents : 
 Jfj^Q is 70 cents ; ^^ is 6-J- cents ; | is 12^ cents ; -J- is 25 cents ; 
 Y is l^- dollars ; \^ is 2 dollars, &c. 
 
 DEFINITIONS. 
 
 Fractions are either proper, improper, single, compound, or mixed, 
 
 1. A Single or Simple Fraction is a fraction expressed in a 
 simple form, as ^, |-, y^g, &c. 
 
 2. A Compound Fraction is a fraction expressed in a com- 
 pound form, being a fraction of a fraction, as -^ of J, ^ of ^^ of 
 ^^, which are read thus : one half of three fourths, two sevenths 
 of five elevenths of nineteen twentieths. 
 
 3. A Proper Fraction is a fraction whose numerator is less 
 than its denominator, as |-, |, &c. Less than unity, because 
 the numerator is less than the denominator. 
 
 4. An Improper Fraction is a fraction whose numerator ex- 
 ceeds its denominator, as -I, |-, ^y^, &c. More than unity, be- 
 cause the denominator is less than the numerator. 
 
 5. A Mixed Number is composed of a whole number and a 
 fraction, as 7|-, 35^3, (fee, that is, seven and three fifths, &c. 
 
 6. A Complex Fraction is one that has a fraction for its 
 numerator, or for its denominator, or for both its terms, as — 
 z ^ el 
 
 2.' |.' _i ^ Sic. An equal or even fraction is f ==1, f =1, &c. 
 
 7. A fraction is said to be in its lowest or least term when il 
 is expressed by the least number possible. 
 
 8. The common measure of two or more numbers is that num- 
 ber which will divide each of them without a remainder ; thus, 5 
 is the common measure of 10, 20, and 30 ; and the greatest num- 
 ber which will d(»this is called the greatest common measure. 
 
 9. A number which can be measured by two or more num- 
 bers is called their common multiple ; and if it be the least num- 
 ber which can be so measured, it is called the least common 
 multiple, thus, 40, 60, 80, 100, are multiples of 4 and 5; but 
 their least common multiple is 20. 
 
 Note. — The product of two or more numbers is a common 
 multiple of those numbers, thus, 3x4x5=60; and 60, or 
 3x4x5, is evidently divisible without a remainder by each of 
 those numbers, and the same must be true in every similar case. 
 
116 VULGAR FRACTIONS. 
 
 10. A prime number is one which can be measured only by 
 itself or a unit, as 3, 7, 23, &c. 
 
 1 1 . A perfect number is equal to the sum of all its aliquot 
 parts. An aliquot part of a number is contained a certain num- 
 ber of times exactly in the number. 
 
 The following perfect numbers are all which are at presen 
 
 known : — 
 
 6 8589869056 
 
 28 137438691328 
 
 496 2305843008139952128 
 
 812 2417851639228158837784576 
 
 33550336 9903520314282971830448816128 
 Before proceeding farther, it will be proper to introduce the 
 
 two following problems, which will be found very useful and 
 
 important in the reduction and solution of questions involving 
 
 vulgar fractions : — 
 
 PROBLEM I. 
 
 To find the greatest common measure of two or more numbers. 
 
 RULE. 
 
 1. If there be two numbers only, divide the greater by the 
 less, and this divisor by the remainder, and so on, always divi- 
 ding the last divisor by the remainder, till nothing remains, then 
 will the last divisor be the greatest common measure required. 
 
 2. When there are more than two numbers, find the greatest 
 common measure of them as before ; then of that common meas- 
 ure and one of the other numbers, and so on, through all the 
 numbers, to the last ; then will the greatest common measure, 
 last found, be the answer. 
 
 3. If 1 happens to be the common measure, the given num- 
 bers are primes to each other, and found to be incommeasurable, 
 or in their lowest terms. 
 
 EXAMPLES. 
 
 1. What is the greatest common measure of 1836, 3996, and 
 1044? 
 
 1836)3996(2. So 108 is the greatest com. meas. of 3996, 1836 
 3672 Hence 108)1044(9 
 
 972 
 
 324)1836(5 
 
 1620 72)108(1 
 
 — — 72 
 
 216)324(1 — 
 
 216 (last greatest com. meas. 36)72(2 
 
 72 
 
 Common meas. 108)216(2 — 
 
 216 (Therefore 36 is the ans. required. 
 
VULGAR FRACTIONS. 117 
 
 2. Find the greatest common divisor of the two number^ ^« 
 *nd 81. 63)81(1 
 
 63 
 
 18)63(3 
 54 
 
 Greatest common divisor, 9)18(2 
 
 18 
 
 3. What is the greatest common measure of 1224 and 1080 ? 
 
 Ans. 72. 
 
 4. What is the greatest common measure of 1440, 672, and 
 3472? Ans. 16. 
 
 5. What is the greatest common divisor of 492, 744, and 1044 ? 
 
 Ans. 12. 
 
 6. Find the greatest common divisor of 24, 48, and 96. 
 
 Ans. 24. 
 
 PROBLEM II. 
 
 To find the common multiple of two or more numbers, 
 
 DEFINITION. 
 
 Multiple is a number which contains another several times, 
 as 9 is the multiple of 3, containing it 3 times ; 16 of 4, &c. 
 
 RULE I. 
 
 Divide by any number that vi^ill divide two or more of the 
 given numbers without a remainder, and set down the quotients, 
 together with the undivided numbers, in a line beneath. 
 
 2. Divide the second line as before, and so on, till there are 
 no two numbers that can be divided ; then the continued product 
 of the divisors and quotients will give the multiple required. 
 
 EXAMPLES. 
 
 1. What is the least common multiple of 6, 10, 16, and 20 ? 
 
 Explanation. — I survey my given 
 
 numbers, and find that 5 will divide two* 
 
 of them, namely, 10 and 20, which I 
 
 divide by 5, bringing into a line with the 
 
 quotients, the numbers which 5 will not 
 
 measure : Again, I view the numbers 
 
 3 1 *4 1 in the second line and find two will 
 
 ***** measure them all, and get 3, 1, 8, 2, in 
 
 5y 2 X2 x3x4=240 the third line, and find that 2 will meas- 
 
 Ans.) ure 8 and 2, and ia the fourth line get 
 
 *^)Q 
 
 10 16 
 
 20 
 
 *2)6 
 
 2 16 
 
 4 
 
 ♦2)3 
 
 1 8 
 
 2 
 
118 VULGAR FRACTIONS. 
 
 3, 1, 4, 1, all prime; I then multiply the prime numbers and 
 the divisors continually into each other for the number sought, 
 and find it to be 240, answer. 
 
 2. What is the common multiple of 6, 3, and 4 1 
 Operation. 3)6 . 3 . 4 
 
 2)2 . 1 . 4 
 
 1.1.2, 3X2X2 = 12 Ans, 
 
 3. Find the least common multiple of 3, 12, and 8. Ans. 24. 
 
 4. Find the least common multiple of 2, 7, 14, and 49. 
 
 Ans. 98. 
 
 5. What is the least common multiple of 6 and 8 ? Ans. 24. 
 
 6. What is the least number that 3, 5, 8, and 10, will meas- 
 ure ? Ans. 120. 
 
 7. What is the least number which can be divided by the 9 
 digits, separately, without a remainder? Ans. 2520. 
 
 RULE II. 
 
 Divide the number by any prime number, which will divide 
 it without a remainder ; then divide the quotient in the same 
 way, and so continue until a quotient is obtained which is a 
 prime. Then will the successive divisors, together with the 
 last quotient, form the prime factors required. Select all the 
 different factors which occur, observing, that when the same 
 factor has different powers, to take the highest power. The 
 continued product of the factors thus selected will give the least 
 common multiple. 
 
 8. What is the least common multiple of 12, 16, and 24 ? 
 These number, resolved into their prime factors, give — 
 
 12=22x3 
 
 16=2* 
 
 24=2^X3 
 Therefore, 2x2 = 4x2 = 8x2=16x3=48. Ans. 
 Note. — This rule is considered the most correct. 
 ' Note, — Any number ending with an even number or cipher is 
 divisible by 2 ; any number ending with 5 or is divisible by 5. 
 If the right-hand place of any number be 0, the whole is divisible 
 by 10. If the two right-hand figures of any number be divisible 
 by 4, the whole is divisible by 4. If the three right-hand figures 
 of any number be divisible by 8, the whole is divisible by 8. 
 
 INTRODUCTORY QUESTIONS. 
 
 If you pay 50 cents for ^ of a yard, what will 1 yard cost ? 
 
REDUCTION or VULGAR FRACTIONS. 119 
 
 If 1 yard cost ID., what will 1 part cost? If 1^ yards cost 3 
 dimes, what is 1 yard worth ? If 2| yards cost 4D., what is \ 
 of a yard worth ? If 1 yard cost 2^D., what is the cost of |- of 
 a yard ? If 1 bushel of wheat cost l^D., what did you pay for 
 1 peck ? If 2j bushels cost S^D., what cost 1 bushel ? How 
 many are 6 thirds? How many are 15 thirds? How many 
 are 24 eighths ? How much is f , i/, ^^, J-|- of a dollar ? At 
 y^^ of a dollar per yard, what will 4 yards cost ? What cost | 
 of a yard, at Dl'^ per yard ? How much is 3 times f ? 
 
 REDUCTION OF VULGAR FRACTIONS. 
 
 The reduction of fractions is bringing them from one form 
 into another, in order to prepare them for the operations of ad- 
 dition, subtraction, &c. 
 
 To abbreviate, or reduce fractions to their lowest terms. 
 
 Divide the terms of the given fractions by the least number . 
 which will divide them without a remainder, and the quotients 
 again in the same manner, and so on, till it appears that there is 
 no number greater than 1 which will divide them, and the frac- 
 tion will be in its lowest terms. Or, divide both the terms of 
 the fraction by their greatest common measure, and the quotients 
 will be the terms of the fraction required. 
 
 Reduce |-|| to its lowest terms. 
 (4) (3) 
 
 Thus: 8)f||=:f§=y«^=|. A;i^. Or, 288)480(1 
 
 288 
 
 192)288(1 
 192 
 
 Common measure, 96 
 Then 96)f||=|. Ans., as before. 
 96 is the greatest common measure, and f the answer. 
 
 1. Reduce -3-2_5o to its lowest terms. ^)2|, ^)^§§, 5)|, ^)2|=r-J. ' 
 
 2. Reduce i|- to its lowest terms. Ans. j^j, 
 
 3. Reduce If to its lowest terms. }, 
 
 4. Reduce j2_4_ to its lowest terms. | 
 
 5. Reduce ^^ to its lowest terms. ^. 
 
120 REDUCTION OF VULGAR FRACTIONS. 
 
 6. Reduce -^^jr to its lowest terms. Ans, | 
 
 7. Reduce jYA ^^ ^^^ lowest terms. |- 
 
 8. Reduce jjW to its lowest terms. ^^. 
 
 9. Reduce ||- to its lowest terms. |. 
 10. Reduce g-^-|- to its lowest terms. ^, 
 
 To reduce an improper fraction to a whole or mixed number. 
 
 RULE II. 
 
 Divide the numerator by the denominator, and the quotient 
 will be the whole number. If there be any remainder, set it 
 over the given denominator, for the numerator of the fraction. 
 
 11. Reduce ^-^ to its proper terms. 25-^6— Ans. 4^, 
 
 12. Reduce '^^-^ to its proper terms. 7^^^. 
 
 13. Reduce %^ to its proper terms. 8. 
 
 14. Reduce ^/ to its equivalent, or whole number. 9. 
 
 15. Reduce '^j-^^ to its equivalent, or mixed number. 127j*^. 
 
 To reduce a 7nixed number to its equivalent improper fraction, 
 
 RULE III. 
 
 Multiply the whole number by the denominator of the frac- 
 tion, and add the numerator of the fraction to the product, under 
 which subjoin the denominator, and it will form the fraction 
 required. 
 
 16. Reduce 19i|- to an improper fraction. 
 
 19xl8z:r342 + 12i=354. ^ Ans.^^^. 
 
 17. Reduce lOOJf to an improper fraction. ^M^- 
 
 18. Reduce 514^^^ to an improper fraction. ^xf^* 
 
 19. Reduce 127Y^y to an improper fraction. ^ff^* 
 
 To reduce a compound fraction to a simple one. 
 
 RULE IV. 
 
 Multiply all the numerators together for a new numerator, and 
 all the denominators for a new denominator. 
 
 20. Reduce t of |- of -J to simple fractions. 
 
 1X2X3 = 6 : 2X3X4=24 : ^\=1 
 
 21. Reduce |^ of f of f to a simple fraction. -g^. 
 
 22. Reduce i of | of f to a simple fraction. -jj. 
 
 23. Reduce f of | of ^| to a simple fraction. f^. 
 
 To reduce fractions of different denominations to equivalent fraC' 
 tions having a common denominator. 
 
 RULE V. 
 
 Multiply each numerator into all the denominators except 
 
REDUCTION OF VULGAR FRACTIONS. 121 
 
 its own, for a new numerator, and all the denominators into each 
 other, continually, for a common denominator. 
 
 24. Reduce ^, f , and |, to equivalent fractions having common 
 denominators. Thus, Ix5x8=:40, new numerator for ^ : 
 2x4x8=64 for | : 5x4x5 = 100, num. for f : 4x5x8 = 
 160, common denominator. Thus the equivalent fractions are 
 
 Reduce ^, f , and f, to a common denominator. 
 
 Thus, 7x5x7=245 : 4x9x7=252 : 3x5x9=135 : 
 9X5X7=315. Ans,§immi. 
 
 The least common denominator may be found by dividing 
 both the terms of a fraction by any numbers that will make the 
 denominators alike, for a common denominator. Thus jq^^q and 
 i^250=f , i ; -8% and ^\^^=^\ and /^. 
 
 To reduce a fraction of one denomination to an equivalent frac- 
 tion of a higher denomination, retaining the same value. 
 
 RULE VI. 
 
 Multiply the given denominations by the parts in the several 
 denominations between it and that denomination to which it is 
 to be reduced, for a new denominator, which is to be placed 
 under the given numerator, 
 
 25. Reduce I- of a cent to the fraction of a dollar. 
 
 i_ 
 -5* 
 
 7Xt^o=to-o=^^^-tK 
 
 26. Reduce -g- of a pwt. to the fraction of a lb. iroy. tfjo 
 
 27. Reduce \\ of a mill to the fraction of a dollar. y^Voo- 
 
 28. Reduce f of a mill to the fraction of an eagle. -nr^TTo- 
 
 29. Reduce | of a lb. avoir., to the fraction of a cwt. ji^. 
 
 To reduce a fraction of one denomination to an equivalent fraction 
 of a lower denomination, retaining the same value, 
 
 RULE VII. 
 
 Multiply the given numerator by the parts in the denomina- 
 tions between it and that denomination you would reduce it to, 
 for a new numerator, which place over the given denominator. 
 
 This rule is the very reverse of rule 6 ; they will prove each 
 other. 
 
 30. Reduce ^ts ^^ ^ dollar to the fraction of a cent. 
 
 ^1^ of y of V> ; then ^i^ X \o = \o = m=Ans. j. 
 
 31. Reduce ^ of a lb. avoir., to the fraction of an oz. y. 
 
 32. Reduce j^ of a cwt. to the fraction of a lb. avoir. ^. 
 
 33. Reduce J5V00 ^^ ^ dollar to the fraction of a mill. J^' 
 
 34. Reduce ytth ^f ^ ^^* ^^^7' ^ ^^^ fraction of a pwt, | pwt, 
 
122 REDUCTION OF VUl^OAR FRACTIONS. 
 
 To find the value of a fraction in tjie known parts of the integer 
 as of coins, weights, measures, <5^c, 
 
 RULE VIII. 
 
 Multiply the numerator by the parts of the next less denom- 
 ination, and divide the product by the denominator ; and if any- 
 thing remains, multiply by the next less denomination, and di- 
 vide by denominator as before, and so continue, as far as is neces- 
 sary, and the quotients placed after one another will be the answer. 
 
 35. Reduce -f of a dollar to its proper quantity. 
 
 Thus, 4 multiplied by 100—5=^^^ 80 cents. 
 
 36. Reduce If of a day to its proper quantity. 6 hours. 
 
 37. Reduce -^-^ of an eagle to its proper quantity. D 1.87. 5. 
 
 38. What is the value of | of a lb. troy? 7 oz. 4 pwt. 
 
 39. What is the value of f of a yard ? 2 qrs. 2|- na. 
 
 40. What is the value of f of a mile 1 6 fur. 26 po. 1 1 ft. 
 
 41. What is the value of j^- of a dollar ? 43c. 7im. 
 
 42. What is tho value off of an acre ? 3 R. 17-1- po. 
 
 43. What is the value of y^ of a day ? 16 h. 36 m. 55/3 s. 
 
 To reduee any given quantity to the fraction of any greater de- 
 nomination, of the same kind, 
 
 RULE IX. 
 
 Reduce the given sum to the lowest denomination mentioned 
 for a numerator, and the denomination of which to make it a 
 fraction, to the same for a denominator. 
 
 44. Reduce 6dimes,2 cents, and 5 mills, to the fraction of a D. 
 
 6—2—5 X 10 X 10 = 625— Ans. 62J cents 
 
 45. Reduce D 1.87. 5 to the fraction of an eagle. -^^ 
 
 46. Reduce c43.7|m. to the fraction of a D. -j^^ 
 
 47. Reduce 7 oz. 4 pwt. to the fraction of a lb. troy 
 
 48. Reduce 6 fur. 26 po. 11 ft. to the fraction of a mile. -I 
 
 49. Reduce 3 R. 17^ po.' to the fraction of an acre. |. 
 
 To reduce a whole number to an equivc^lent fraction, having a 
 given denominator, 
 
 RULE X. 
 
 Multiply the whole number by the given denominator ; place the 
 product over said denominator, and it forms the fraction required 
 
 50. Reduce 6 to a fraction whose denominator shall be 8. 
 Thus, 6x8=48 and ^^ Ans, Proof: 4.8=48-^8=6 
 
 51. Reduce 15 to a fraction whose denominator shall be 12 
 
 A71S, \y^, 
 
 52. Reduce 100 to a fraction whose denominator shall be 70. 
 
ADDITION OP VULGAR FRACTIONS. 123 
 
 To reduce a given fraction to another equivalent one^ having a 
 given numerator, 
 
 RULE XI. 
 
 The numerator of the given fraction is the first term ; the 
 numerator of the proposed fraction, the second term ; the de- 
 nominator of the given fraction is the third term, to the denom- 
 inator required. 
 
 53. Reduce f to a fraction of the same value, the numerator of 
 which shall be 15. Thus, as 3 : 16 :: 4 : 20 denom. Ans. \^. 
 
 54. Reduce | to a fraction of the same value having its nu- 
 merator 42. Ans. \^, 
 
 55. Reduce f to an equivalent fraction, the numerator of 
 which shall be 27. Ans. ^-^3. 
 
 To reduce a given fraction to another equivalent one, having a 
 given denominator. 
 RULE xir. 
 As the denominator of the given fraction is to the denom- 
 inator of the intended fraction, so is the numerator of the given 
 fraction to the numerator required. 
 
 56. Reduce |; to an equivalent fraction, having its denomina- 
 tor 24. As 8 : 24 :: 7 : 21 numerator. Ans. |i 
 
 57. Reduce |- to a fraction of the same value, the denomina- 
 tor of which shall be 45. , Ans. |-|-. 
 
 58. Reduce ^j to an equivalent fraction, having its denom- 
 inator 68. Ans. 1|-- 
 
 59. Reduce | to a fraction of the same value, the denomina- 
 tor of which shall be 46. Ans. 34^. 
 
 46 
 
 60. Reduce ^j to an equivalent fraction, having its denom- 
 inator 20. Ans. 12 8 ^ 
 
 5f . 20^^ 
 
 61. Reduce the mixed fraction zrr to a simple fraction, thus, 
 5|=2-^ and 7^=^|, then it becomes a complex fraction, 
 A., then 23x11=253 and 85x4=340. ^ns.^^^. 
 
 " # 
 
 ADDITION OF VULGAR FRACTIONS. 
 
 RULE. 
 
 1. If it is necessary, reduce the given fractions to simple 
 fractions ; that is, compound fractions to single ones ; mixed 
 numbers to improper fractions ; fractions of different integers to 
 
124 SUBTRACTION OF VULGAR FRACTIONS. 
 
 those of the same ; and all of them to a common denominator , 
 then the sum of the numerators, written over the common de- 
 nominator, will be the sum of the fractions required. 
 
 2. After the fractions are prepared, multiply each numerator 
 into all the denominators but its own, and take their sum for a 
 new numerator ; multiply all the denominators for a new de- 
 nominator. 
 
 61. What is the sum of ^^ of 4|, f of -^, and Q-J ? 
 
 Ans. i^-^^ = 12^^. 
 
 First, 4|= V, and 9-| = ^ , and /^ of ^ = 2^5_9 ^nd | of ^=|. 
 
 After reducing them to improper fractions, they will stand thus : 
 
 2_5_9 3 ^^A si Thpn 
 8 ' 8' **"^ 4 • -*^-iAt;ii, 
 
 259 X 8x4==z8288 Explanation. — After the 
 
 3 X 80 X 4= 960 , fractions are prepared or re- 
 
 37 X 8 X 80r::23680 duced, according to the pre- 
 
 ceding rules, you multiply 
 
 32928 each numerator into all the 
 
 ^=:i|^9 = 12|-§ denominators but its own, 
 
 80 X 8 X --^==2560 and take their sum for a new 
 
 numerator; then multiply all 
 
 the denominators for a new denominator. Sic. 
 
 62. Add together ^^, |, and A. Thus, 2x9x4=72; 
 8xl3x4=i416; 1x9x13 = 117; 13x9x4=468; ^j^n, 
 
 72 + 117+416=605 ) 
 
 \ —113 7 Anv 
 
 468 468 468 468 ) 
 
 63. Add f and y^^ together. 
 5)5 10 10-> 5x2=4) 4 + 5 = 9 
 
 :) 4+5=9 
 
 >num. Ans. 
 
 ) 10 10 10 
 
 1 2 com. denom. 10-f- 10x5=5. 
 
 64. Add ^-Q, ^^, and |, together. . Ans. 2^^-^. 
 
 65. Add f and 17^ together. 18^. 
 
 66. Add iD. |c. ^^^c. and | mills together. 20c. 9.m. 
 
 67. Add f , 9i and f of i together. 9^gi. 
 
 68. Add f of a dollar to -| of a dollar. Dl.12.5. 
 
 69. Add 4 of a ton to ^^ of a cwt. 
 
 Ans. 12 cwt. 1 qr. 8 lbs. 12 oz. 12f dr. 
 
 SUBTRACTION OF VULGAR FRACTIONS. 
 
 RULE. 
 
 1. Prepare the fractions, when necessary, as in Addition, and 
 ake the difference of the numerators, under which write the 
 
MULTIPLIOATION OF VULOAR FRACTIONS. 125 
 
 comraoa denominator, which will give the difference of th« 
 fraction required. 
 
 2. Multiply each numerator by the other denominator, and 
 subtract the less (product) from the greater, for the numerator 
 sought ; multiply one denominator by the other for a common 
 denominator. 
 
 70. From }| take f 15x7=105; 4x16=64; 16x7= 
 
 n2;if|-A%=TV2 ^^^- 
 
 71. From 5 take y\. 1 X 14=14 common denominator. 
 
 14— 1X5 = 70 > 7^ 8 _62_->l 6 A„r, 
 
 14^14x8= 8 (T^T^-Tl-^T^ •^^^• 
 
 72. From 96i take 14|.' c-rn /} v^y-v.,^ Ans. 81y^ 
 75. From 8^ take 5| i^ ' •' 
 
 74. From f take f . yf. r) /* 
 
 7fi From 8-1 take .51. '' ^ ' 
 
 Tod! 
 
 76. From 5i take f . 4 
 
 77. From 12 take f. 11 
 
 78. From | take |. | 
 
 MULTIPLICATION OF VULGAR FRACTIONS. 
 
 RULE. 
 
 Reduce compound fractions to simple ones, and mixed num- 
 bers to improper fractions ; if an integer be given, reduce it to 
 an improper fraction, by putting a unit for its denominator ; then 
 multiply the numerators together for a new numerator, apd thtp 
 denominators together for a new denominator. 
 
 Note, — In some cases, when the nmnerators and denomina- 
 tors are equal, they may be omitted 
 
 79. Multiply 
 f Ans. 
 
 6| b| 8. 
 
 Thus, 6|=V; 
 
 t^n 
 
 Vx|=='p-- 
 
 80. 
 
 Multiply 
 
 |bf|. 
 
 
 
 
 
 81. 
 
 Multiply 
 
 0^7 by 
 
 1- 
 
 
 
 Ans. 1| 
 
 82. 
 
 Multiply 
 
 H by 9^. 
 
 
 
 
 Ans. 69|- 
 
 83. 
 
 Multiply 
 
 \ by 5|. 
 
 5*=¥ 
 
 ; |xy 
 
 =H= 
 
 =44^ Ans. 
 
 84. 
 
 Multiply 
 
 Sibyf 
 
 
 
 
 Ans. I. 
 
126 MULTIPLICATION OF VULGAR FRACTIONS. 
 
 To multiply a whole number by a fraction, or a fraction by a 
 whole number. 
 
 Multiply the whole number by the numerator of the fraction, 
 and divide the product by the denominator ; if the numerator be 
 1, divide by the denominator only ; if there is a whole number 
 multiply by it, and add the products. 
 
 85. 86. 87. 88. 
 
 Multiply 15 or 15 35 68 42 
 
 by 3-1- 3i 5f 
 
 7^ 45 11§ 
 
 45 7^ 175 
 
 m 
 
 9f 
 
 12)748 
 
 4)126 
 
 62^ 
 . 476 
 
 31* 
 
 378 
 
 52^A. 52^ 186fA 
 
 538^A. 409^A. 
 89. At 30 cts. per bushel, what will l^j-^ bushels cost^ 
 Thus: 19tV 15)30 
 
 30 — 
 
 5.70 
 
 2 
 
 D5.72 Ans. 
 
 90. How many are 50 roods, multiplied by 5| ? 
 50x5^=250; 50-^-2=25 + 250=275 Ans. 
 
 91. What will 22 j\ of cloth cost at 11 dollars per yard? 
 245. What will 22^? 246. What will 23^^^ 260. W^hat 
 willS,-^? 42. What will 31^? 351. What will 99y\ ? 1095. 
 An;?. 2239 dollars. Thus, 22^^ 11)11 
 
 11 — 
 
 1X3=3 
 
 245 + 3 
 
 92. If a man's salary be 1200 dollars a year, what will 2^-^ 
 years come to?<|2420. What will 3/^? 3630. What will 
 5^2.o_? 6040. What will 8|§f? 10701. What will 12||^? 
 15598. Ans. 38398 dollars. 
 
 93. If lib. of butter cost ^^ of a dollar, what will 205lb8. cost? 
 
 3 X 205=615 ) 
 
 Thus, — \ =30i|D. Ans. 
 
 20 20 S 
 
DIVISION OF VULGAR FRACTIONS. 127 
 
 94. What cost 400 yards of calico, at | of a dollar per yard ? 
 3x400=1200) ^ 
 
 > =D150. Ans, /s2, . 
 
 How much IS 13 times fi^r ^ ^*^ Ans. 3|||. 
 
 How much is 513 times y\? ^{\^ =326/y. 
 
 Multiply 3^ by 367. 1192f 
 
 Multiply 6| by 211. 1450|. 
 
 99. Multiply 3/^ by 42. 1291. 
 
 100. How much is 314 multiplied by f ? 235^. 
 
 101. How much is 513 multiplied by y^^.? 3263-^. 
 
 102. Multiply 42 by \^. 11. 
 
 ;-> 
 
 
 DIVISION OF VULGAR FRACTIONS. 
 
 Prepare the fractions as before ; then invert the divisor, and 
 proceed exactly as in multiplication. The product will be the 
 quotient required. 
 
 103. Divide 5^ by 7f . Thus, ^^^, and 7f =3_i 5 then ^ 
 multiplied by ^ =if|=li- ^'^^' 
 
 104. Divide 84 by i|f. 84 multiplied by 320=26880-t- 
 168=160. Ans. 
 
 In the last example I multiplied the dividend by the denomi- 
 nator of the dividing fraction, and then divided the product by 
 the numerator. 
 
 105. Divide 100 by 2f. Ans, 36 /y. 
 
 106. Divide \\ by |. 3)i7^8^^l2|. ^^^. 
 
 107. Divide 4 by|. Ans. 4^ 
 ' 108. Divide 4| by f of 4. 2 " 
 
 109. Divide 99 by 108. 
 
 110. Divide 5i by 7f. 
 
 111. Divide! by 9. 
 
 112. Divide 92 by 4f * 20|. 
 
 rule II. 
 
 Divide the numerator by the whole number, writing the de- 
 nominator under the quotients ; or, multiply the denominator by 
 th© whole number, writing the result under the numerator. 
 
 ^1- 
 
 I 
 
128 APPLICATION OF VULGAR FRACTIONS. 
 
 113. If 8 yards cost jg-D., how much per yard ? 
 
 i^g-:-8z=:yL.D.=-A7i5. 6\ cenis 
 
 114. Divide I by 8. =z-^\=z 
 
 115. Divide 5-^ by 6. =/o = 
 
 116. Divide 6-| dollars among 5 men. 
 
 ^4 — 4 • ^ — 20" 
 
 117. Divide 16f by5. 
 
 118. Divide Sf by 6. 
 
 119. Divide 114^ by 280. 
 
 120. Divide 620 by 8^-2- . 
 
 8y2_=:ff : 620xll=6820-^90=75Jg= 
 
 121. Divide 92 by 41. 
 
 122. Divide 86 by 15|-. 
 
 123. How many square rods in 1210 square yards ? 
 30i sq. rods=i|i : 1210 X4==4840-^121 
 
 124. Divide f by j\. f x V^|f = 
 
 125. Divide If by^6_, 
 
 126. How many times are -^^ contained in f ^ ? 
 
 127. Divide 9J by J of |. '37. 
 
 % 
 
 Ans. l^V 
 
 Ans. 75J. 
 20f 
 
 ? 
 
 =Ans. 40. 
 
 2. 
 
 .» 
 
 APPLICATION OF VULGAR FRACTIONS. 
 
 128. What part of 1 month is 19 days ? Ans, if. 
 
 129. What part of 1 month is 25 days, 13 hours ? fif . 
 
 130. What is the value of f of a cwt. ? 1 = 3 qrs. 
 
 131. What is the value of fif of a hhd. ? 49 galls. 1 29/3 qt. 
 
 132. What is the value of -^^ of a day ? 16 h. 36 m. 55^^^ sec. 
 
 133. Reduce y||y ^^ ^ minute to the fraction of a day. 1:^. 
 
 134. Reduce f of a pint to the fraction of a hhd. 2"sVo=-6^o* 
 
 135. What part of a yard is 3 qrs. 3 na. ■? i|. 
 
 136. Reduce | and | to a common denominator. ^^, |-|. 
 
 137. Reduce ^ of f , and |, to a common denominator. ^^, |^. 
 
 138. How much are 5^ times 5^ ? 30^. 
 
 139. How much are 16^ times 16^ ? 272i. 
 
 140. Multiply ^\ of ^\ by f tV4\- 
 
 141. Divide 8f by 6. fl = lH' 
 
 142 How much is 60 times j|q ? f =2i' 
 
 143 Redace J of | of | to a single fraction. ^. 
 144. Reduce f of | of 5 to a single fraction 21 . 
 
SIMPLE PROPORTION IN VULGAR FRACTIONS. 
 
 12d 
 
 145. Reduce ^ off of | to a single fraction. Ans, ^j, 
 
 146. Add 74, f off, and 7, together. 
 
 First, 7f =Y, Y of |=i|- and 7=^ ; then the fractions are 
 3_9^l|^ and |; therefore, 39 x 56x1 ==2184 ; 15x5x1 = 75; 
 
 7 X 5X56 Xr::^ 1960: + =:4219-r-5x 56x1 ==289; = 152VV^^^- 
 Or thus: 2184x75x1960 
 
 280 =:152fo^^^. 
 
 147. From 5 take ^. Ans. 4^^» 
 
 148. Multiply 7] by 8-| ; 6| by 6f . 
 
 Y multiplied by 2_7_7^2_9^45_9_ . ^^ 45.5625. Ans. 
 
 149. From |J take f. 40x6=240; 5x41=205; 41x6 
 z=:246 ; ^^Q 246^^24"6' Ans, 
 
 1150. From fl take j\, Ans. j^^, 
 
 151. From I- take f. i-|. 
 
 152. From ll| take f . 1/^. 
 
 153. From } of j^^ take ^ of ^^. .^Vo- 
 
 154. Add together -j?, |, and i. li||. 
 
 155. Add together 14f and 15f . 30^. 
 
 SIMPLE PROPORTION IN VULGAR FRACTIONS. 
 
 RULE. 
 
 1. Reduce the given numbers, if necessary ; mixed numbers 
 to improper fractions ; like terms to the same denominations, 
 and each to their lowest terms. 
 
 2. Then state the question as in whole numbers ; invert the 
 first term, and multiply all the numerators together for a new 
 numerator, and all the denominators for a new denominator, and 
 divide the upper term by the lower for the answer. Or you can 
 reduce the given fractions to decimals, and work by the general 
 rule of proportion. 
 
 1. If |- of a yard of cloth cost D2f , what will 5^^ yards cost ? 
 
 Thn« • 'S 5 65. r)95._2_l . t}.p„ 8 y 65 y 21 = 10920 6_5 
 
 inUS. Oj.^_j2, -L'^8^— 8 > ^"^" T'^T^'^ ¥= ^72 — 4» 
 
 65-i-4 = Dl6.25; | : ff :: V ; Ans. D161. 
 
 2. How much silk that is f of a yard wide, will line 6| yards 
 of cloth, which is 1^ yards wide? 
 
4» 
 
 130 SIMPLE PROPORTION IN VULGAR FRACTIONS. 
 
 1. Common way: — 
 
 1|- yds. : 6f yds. :: 3 qrs. ; then 5 qrs. : 27 qrs. :: 3 qrs. 
 X4 X4 X5 
 
 2. By vulgar fractions : — f) 
 
 First, l\=l 6f = V» and 3 qrs.rz:| ; then, as f : 2/ .. 3 
 ^nd^X'-i Xi=ix\' X^=%\^ = '-^ = Ui yds. Ans. 
 
 3. By decimals : — 
 
 11=1.25, 6f = 6.75, andf =.75; then as 1.25 : 6.75 :: .75, 
 (inverse.) Ans. 11.25 yards. 
 
 3. How much in length, that is 13|- poles in breadth, will 
 make a square acre?. Ans. llj^^i-. 
 
 4. If a suit of clothes can be made of 4| yards of cloth, 1| 
 yards wide, how many yards of cloth |- of a yard wide will it 
 require for the same person ? Ans.»6 yds. 1 qr. 3f na. 
 
 5. If I of a yard cost |-D., what will 401 yards cost ? 
 
 Ans. D59.06J. 
 
 6. A merchant sold 5^ pieces of cloth, each containing 12§ 
 yards, at 12f cents per yard, what did the whole amount to ? 
 
 Ans. D8.82f . 
 
 7. If f of a yard of cloth cost |- of a D., what will 2J yards 
 cost? Ans. D4|l. 
 
 8. If 14i yards cost 19^D., how much will 19|- yards cost? 
 
 9. If D9 will buy 76-1- lbs. of sugar, how much will D17| buy ? 
 
 10. If D155 will buy 3^ acres, how many acres will D1576 
 buy? 
 
 11. If ^ of f of a gallon cost f D., what will 5^ gallons cost ? 
 
 Ans. D91. 
 
 12. If f oz. of gold be worth D1.50, what is the cost of 1 oz. ? 
 
 Ans. Dl.80. 
 
 COMPOUND PROPORTION IN VULGAR FRACTIONS. 
 
 RULE. 
 
 Prepare the terms, if necessary, then state them agreeably to 
 the directions given for whole numbers ; invert the dividing terms 
 and multiply the upper figures continually for the numerator, 
 and those below for the denominator of the fractional answer. 
 
COMPOUND PROPORTION IN VULGAR FRACTIONS. 131 
 
 IS. If 3|- yards of cloth, | of a yard in width, cost D7|, what 
 fvill 9f yards cost, | of a vard in width ? 
 
 Thus, D7f = V> 9j-=:f yd, 3^=1 yd. ; then if 7 . 3_9 7 ^ 
 1= ^1 divisor. \^ X I X V = Hir -- ¥ = ¥3t¥ = V¥ =D24. 
 78. 1| f yd. : I yd. :: D^, or invert the terms, |xf X\^ X 
 iX V = V9V-V¥= . ^^^- D24.78.1-1. 
 
 14. If a footman travel 294 miles in Of days of 12|^ hours 
 long, how many days of 10-| hours will it require him to travel 
 762 niiies ? Ans. 2\^ days 
 
 15. If D50 in 5 months gain 02^-^4, what time will D13J- 
 require to gain Dly^^? Ans. 9 months. 
 
 16. A. and B. are on opposite sides of a circular field, 268 
 poles about ; they begin to go round it, both the same way and 
 at the same time ; A. goes 22 rods in 2 minutes, and B. 34 rods 
 in 3 minutes ; how many times do they go round the field before 
 the swifter overtakes the slower? Ans. A. travels 16^ times 
 
 [round=4422 po. B. 17 times=4556 po. 
 
 17. If A. can do a piece of work alone in 7 days, and B. in 
 12 days ; let them both go about it together, in what time will 
 they finish it? Ans. 4^-^ days. 
 
 18. When 12 persons use 1|- pounds of tea per month, how 
 much should a family of 8 persons provide for a year ? 
 
 Ans. 9 lbs. 
 
 19. If 5 persons drink 7|- gallons of beer in a week, what 
 quantity will serve 8 persons 22^ weeks ? Ans. 280| galls. 
 
 REVIEW OF VULGAR FRACTIONS. 
 
 What is a fraction ? What is a vulgar fraction ? Can frac- 
 tions be expressed in various forms and retain the same value ? 
 What is common measure ? How will you find the greatest 
 common measure of two numbers only ? How will you find 
 the greatest common measure when there are more than two 
 numbers ? How will you find the common multiple of two or 
 more numbers? Explain the operation. What is the least 
 common multiple of two or more numbers ? What is the rule 
 for the least common multiple? When is a fraction in its lowest 
 terms ? How will you reduce a fraction to its lowest terms ? 
 What are the lowest terms of nine twelfths ? What is an 
 improper fraction ? How will you reduce an improper fraction 
 to a whole or mixed number ? What is a mixed number ? 
 How-will you reduce a mixed number to its equivalent improper 
 fraction ? What is a compound fraction ? How will you re- 
 duce a compound fraction to a simple one ? How will you re - 
 duce ^ fraction of one denomination to an equivalent fraction of 
 
132 MISCELLANEOUS MATTER IN VULGAR FRACTIONS. 
 
 a higher denomination ? How will you reduce a fraction of 
 one denomination to an equivalent fraction of a lower denomina 
 tion ? How will you find the value of a fraction in the known 
 parts of the integer ? How will you reduce a given quantity to 
 the fraction of any greater denomination of the same kind? 
 How will you reduce a whole number to an equivalent fraction, 
 having a given denominator ? How will you reduce a given 
 fraction to another equivalent one, having a given numerator ? 
 How will you reduce a given fraction to an equivalent one, hav- 
 ing a given denominator ? What is the rule for performing ad- 
 dition of vulgar fractions ? What will you do when a mixed 
 number is given ? What is to be observed when different de- 
 nominations are given ? when all the denominations are alike ? 
 Add I of a yard to f of an inch. Ans. f|-| yards, or 14j\- 
 inches. Add |- of a cwt. 8-| lbs. and 3j^q oz. together. Ans. 2 
 qrs. 17 lbs. 12f^ oz. How do you perform subtraction of vul- 
 gar fractions ? when the numerator is the greater ? What is to 
 be done when the fractions are of different denominations ? 
 From I of a league take j^q of a mile. Ans. 1 m. 2 fur. 16 po. 
 Repeat the rule for performing multiplication of vulgar fractions. 
 How will you prepare the given terms 1 Required the product 
 of 4^, f of i, and 18|. Ans. 9y|^. What is the rule for per- 
 forming division of vulgar fractions ? Divide 14| oi ^ by 3 J 
 of 6., Ans. ^^Q. How will you prepare the fractions for 
 stating in simple proportion ? How will you then proceed ? 
 What is the rule? How will you prepare the fractions for 
 stating in compound proportion ? Repeat the rule. The ques- 
 tions may be continued by the teacher. 
 
 MISCELLANEOUS MATTER IN 
 VULGAR FRACTIONS. 
 
 1. If 7 lbs. of coffee cost |i of a dollar, what is that per lb.? 
 
 1^ -f-7=^5~ of a dollar. Ans. 
 
 2. If 6 bushels of wheat cost 4| dollars, what is it per bush- 
 1? 41=11=1-1=81^ cents Ans. 
 
 3. How many times is 3 contained in 462^ ? Ans. 154i. 
 
 4. If a man spend f of a dollar per day, how much will he 
 spend in 8 days ? I ^^ 8 = ^ = 5 dollars. Ans. 
 
 A fraction can be multiplied by multiplying the nwnerator, 
 without changing the denominator. 
 
I 
 
 MISCELLANEOUS MATTER IN VULGAR FRACTIONS. 133 
 
 5. At g^Tj of a dollar for 1 pound, what will 12 pounds cost? 
 
 g5_ X 12 — 60 _ 6. —.3 Q^ a dollar. Ans, 
 
 6. What will 9J§ tons of hay come to at 17 dollars per ton? 
 
 Ans. 164^L dollars. 
 
 At f of a dollar per yard, what cost 7f yards ? 
 
 Ans. 6^i dollars. 
 Reduce \, |, and |-, to fractions having the least common 
 denominator, and add them together. Thus, H+A"+l4~4i 
 r=li|- amount. 
 
 9. What is the amount of ^ of | of a yard, ^, and \ of 2 yards ? 
 Reduce the compound fraction to a simple fraction, thus : — 
 
 ^ of 1=1, and i of 2=1 ; then, |+|+|- =i^=l^=^ yds. 
 
 10. -Reduce -^^ of a hogshead to the fraction of a gallon. 
 
 11. Reduce -^-^ of a bushel to the fraction of a quart. 
 2. Reduce ^^g^ of a day to the fraction of a minute. 
 
 (33. Reduce -^-^-^ of a cwt. to the fraction of a pound. 
 14. Reduce y^Vo ^^ ^ hogshead to the fraction of a pint. 
 * 5. Reduce -^-^-^ of a furlong to the fraction of a yard. 
 
 6. Reduce -^^ of a gallon to the fraction of a hogshead. 
 
 7. Reduce ^^ of a quart to the fraction of a bushel. 
 
 18. Reduce ^l^- of a minute to the fraction of a day. 
 
 19. Reduce ^^ of a pound to the fraction of a cwt. 
 
 20. Reduce y/g^ of a pint to the fraction of a hhd. 
 
 21. At 6| dollars per barrel for flour, what will y^g of a bar- 
 rel cost ? Ans. 2\^\ dollars 
 
 22. At D2i per yard, what cost 6|- yards? Ans. D14|-|. 
 
 23. At D4f per yard, how many yards of cloth may be 
 bought for D37 ? Ans. 8^^ yards. 
 
 24. How many times is ^^ contained in 6 ? Ans. f of 1 . 
 
 25. If 4| pounds of butter serve a family 1 week, how many 
 weeks will 36| pounds serve them ? Ans. 8y| ^ weeks. 
 
 26. How many times is f contained in 4|? Ans. llf 
 
 27. Multiply f of 2 by ^ of 4=3. Ans. Multiply 1000000 
 by|=555555|. Ans. 
 
 28. A man bought 27 gallons, 3 quarts, 1 pint, of molasses ; 
 what part is that of a hogshead ? 
 
 29. From 4 days 7| hours, take 1 day 9y^g- hours. 
 
 Ans. 2 days, 22 hours, 20 minutes. 
 
 30. If 6-^ yards of cloth cost 3 dollars, what cost 91 yards ? 
 
 Ans. D4.269, 
 
 31. If 4 oz. of silver cost i^ of a dollar, what cost 1 oz. ? 
 
 Ans. D1.283, 
 ,^^2. If T^ of a ship cost 251 dollars, what is 3^ of her worth ? 
 H Ans. D53.785. 
 
 m 
 
134 MISCELLANEOUS MATTER IN VULGAR FRACTIONS. 
 
 33. Add together |E. |D. and licts. Ans. 7D. 53c. 2fm 
 
 34. Take 3lcts. from l of 2^ dollars. Ans. 43lcts. 
 
 Note. — The reason of the rules, both for Addition and Sub- ' 
 traction of Fractions, is manifest. If the given fractions have 
 the same denominator and are of the same denomination, the 
 sum of the numerators written over the given denominator, will 
 be the sum of the fractions. A fraction is subtracted from a 
 whole number, by taking the numerator of the fraction from its 
 denominator, and placing the remainder over the denominator, 
 then taking one from the whole number, thus : — 
 
 From 123 12.2 12 
 
 Take 7| 7| f 
 
 H 4| llf 
 
 Note. — To multiply by a fraction, whether the multiplicand 
 be a whole number or a fraction, as before observed, you can 
 divide by the denominator of the multiplying fraction, and mul- 
 tiply the quotient by the numerator, thus: 20 multiplied by ^, 
 the product is 15. To divide by a fraction, whether the div- 
 idend be a whole number or a fraction, we multiply by the de- 
 nominator of the dividing fraction, and divide the product by the 
 numerator, thus : 20 divided by |- is 26|-, and 12 divided by | 
 is 16, &c. You will observe in multiplication, the multiplier 
 being less than unity, or 1, will require the product to be less 
 than the multiplicand ; and in division, the divisor being less 
 than unity or 1 , it will be contained a greater number of times, 
 and consequently will require the quotient to be greater than the 
 dividend, to which it will be equal when the divisor is 1, and 
 less when the divisor is more than 1 . 
 
 By Vulgar and Decimal Fractions, 
 
 35. If 2-| bushels sow an acre, how many acres will 22 
 bushels sow ? 
 
 Thus, 2| X 4+3 = y improper fraction ; 22 x 4~ 1 1 =8 acres 
 Ans. Or, 22-^2.75 = 8 Ans. 
 
 36. At 4 1 dollars per yard, how many yards of cloth may be 
 bought for 37 dolls. ? Thus, 4f x5 + 2 = ^ ; 37x5-^22 = 8^ 
 yards Ans. D37~4.40 = 8 yards, 1 quarter, 1 ndl\-\-Ans. 
 
 37. If 14| yards cost 75 dollars, how much per yard ? 
 Thus, 14f x8 + 3 = i-i^; 75 x8 = 600-^115=5jy.^rz:^V Ans 
 Or, D75~- 14.375 yards = D5.2i.7 Ans. 
 
 38. How many times is | contained in 746 1 
 
 Thus, 746 X 8-^-3== 1989i; 746~375 = 1989^lf =--i Ans. 
 
MISCELLANEOUS MATTER IN YULOAR FRACTIONS. 135 
 
 39. Divide ^ by ^=1 ; divide f by i = 3 ; divide I by -J =2. 
 
 40. If f of a yard cost 5 dollars, how much per yard ? 
 
 41. If f of a bushel cost 3 dollars, how much for 3 bushels ? 
 how much for 4- a bushel ? 
 
 A9 Aflfl 2 4 i_7 . r.AA 2 3_ 4 5 _14 . ^AA 1,2 9 18 
 
 4^. Add -g, -g, g — 3 , aaa -^-^^ y^, j^, jg— jg , aaa -g , ^, j^j 
 ¥t^ together. 
 
 43. There are 3 pieces of cloth, one containing 7-| yards, 
 another 13^ yards, and the other 15|- yards, how many yards 
 in all 1 First reduce the fractional parts to their least common 
 denominator, thus ; — 
 
 Qv9vft 4R 48 I 32156 1 3_6 — 0_8_ — 1 
 
 ^X-iXO_40 6'4 1^64 ' 6'4— 64 — -^dJ— 8* 
 
 1x4x8=32 Then, 7|-f -^ "^ -^ 
 
 7x4x2z=56 13f| /TO 
 
 4x2x8^64 
 
 13H 
 15M 
 
 37^=1 Ans. 371 
 
 Adding together all the 64ths, namely, 48, 32, and 56, you 
 have 136, that is "^-^-f —2-^-^—\ -, write down the fractions g^^ 
 under the other fractions, and carry the two integers to the 
 other integer, and you have 37|-. Ans. 
 
 Or, thus: 7.75 + 13.5 + 15.875 = 37.125=:|. The last method 
 is the most convenient. 
 
 44. From | of an ounce take |- of a pwt. Ans. 11 pwt. 3 grs. 
 
 45. From 4 days, 7^ hours, take 1 day 9j^ hours. 
 
 Ans. 2 days, 22 hours, 20 min. 
 
 46. A man purchased -^^ of 7 cwt. of sugar, how ijiuch did 
 he purchase ? , >^/t. 2,7^^ ^ ) i , 1 . J f- /• 7 -^ , 
 
 ATI. 14 hours, 48 minutes, 54|^ seconds, is what part of a 
 day ? 
 
 48. Add Jib. troy to ^-^ of anoz. Ans. 6 oz. 11 pwt. 16 grs 
 
 49. At 3|- dollars per cwt., what will 6| lbs. cost ? 
 
 50. If |- of a yard cost -§• of a dollar, what will -^^ of a yard 
 cost? 
 
 51. Add together f D., j-\ cents, \ mills. 
 
 52. Add f of a yard, f of a foot, and ^| of a mile, together. 
 
 53. Add 1 of a week, f of a day, \ of an hour, and \ of a 
 minute, together. 
 
 54. From 5 weeks, 2| days, 9f hours, take 1\ days, 141^ 
 hours. 
 
 55. Multiply f of f , by f of i of 1 If. Ans. /^. 
 
 56. What is the continual product of 7, J, f of f , and 3^ ? 
 
 Ans. 2|J 
 
136 PRACTICE. 
 
 57. In a certain school -^^ of the pupils study Greek, ^^ 
 Latin, |- arithmetic, ^ read and write, and 20 attend to other 
 studies ; required the number of pupils. Ans. 100. 
 
 53. In 342f gallons, how many i of a gallon ? 
 
 Ans. ^Y^ of a gallon, or 1371 quarts. 
 
 Note. — Any whole number may be made an improper frac- 
 tion by drawing a line under, and putting a unit or 1 for a de- 
 nominator, as 5 may be expressed thus, |^, and 10, ^j, &c. 
 
 Reduce .275 to a vulgar fraction. 
 
 Thus: j2_7_5__i.25 com. m.=ii. Ans. 
 
 SECOND REVIEW OF VULGAR FRACTION. 
 
 What do you mean by common divisor? by the greatest 
 common divisor ? How can you multiply a fraction by a whole 
 number ? How will you multiply a mixed number ? When 
 the multiplier is less than a unit, what is the product compared 
 with the multiplicand ? How do you multiply a whole number 
 by a fraction ? How can you multiply one fraction by another, 
 and can you do it in more than one way ? How do you multi- 
 ply a mixed number by a mixed number ? How can you di- 
 vide a fraction by a whole number ? What is understood by 
 common denominator 1 the least common denominator ? Do 
 cases frequently occur in vulgar fractions, where the terms may 
 be reduced, and solved by decimals ? What can you say of the 
 importance of a correct knowledge of all kinds of fractions ? 
 
 PRACTICE. 
 
 RACTiCE Is a contraction of simple proportion, when the first 
 " lerm happens to be a unit, or one, and has its name from its daily 
 }j use among merchants and tradesmen ; being an easy and con- 
 7 cise method of working many questions which occur in trade and 
 business ; also, many questions in simple interest may be read- 
 ily and easily solved by aliquot parts, after the pupil has made 
 himself thoroughly acquainted with the tables, which he should 
 commit to memory. One number is an aliquot, or even part of 
 another when it forms an exact part of it, as 25c. is an aliquot 
 partof aD=i; 75c. =f; 10c. = ^1^; 5c. = 2V> ^^* ^^ ^^'^^ ^^ 
 aliquot part of a cwt.=:^; 28 lbs.=^; 14lbs.=:^; 7 lbs.=:yi^, ^g. 
 
PRACTICE. 
 
 137 
 
 Many questions that occur in practice may be easily solved by 
 decimals ; as it is more immediately connected with sterling 
 money, and when pounds, shillings, and pence, occur, no method 
 of calculation is better than the various rules of practice. 
 
 1 . When the price is in dollars and cents, multiply the given 
 quantity by the dollars, and take aliquot parts for the cents, and 
 add the products together, and you will have the answer in 
 D., c, &c. 
 
 2. But when the quantities are of various denominations, 
 write down the given price of one of the highest denomination, 
 and multiply it by the whole of the highest denomination ; then 
 take aliquot parts of the next lowest denomination continually, 
 then add the products together for the answer. 
 
 Proof, by Simple Proportion. 
 
 A TABLE OF ALIQUOT, OR EVEN PARTS. 
 
 Time. 
 
 6 mo. is 
 
 4 mo. is 
 3 mo. is 
 
 2 mo. is 
 
 1 mo. is 
 equal to 
 15 days 
 10 days 
 7^ days 
 6 days 
 
 5 days 
 
 3 days 
 
 2 days 
 
 i a 
 h a 
 i a 
 i a 
 tV a 
 -i-of 
 ^of 
 lof 
 I of 
 I of 
 iof 
 
 1 
 
 of 
 
 year 
 year 
 year 
 year 
 year 
 
 3 mo. 
 
 a mo. 
 
 a mo. 
 
 a mo. 
 
 a mo. 
 
 a mo. 
 
 a mo. 
 
 a mo. 
 
 Cubic measure. 
 
 96 ft. is 
 64 ft. is 
 32 ft. is 
 16 ft. is 
 8 ft. is 
 4 ft. 
 2 ft. 
 
 Dolls. 
 
 50c. 
 
 33ic. 
 
 25c. 
 
 20c. 
 
 12.5c. 
 
 10 
 
 H 
 
 5 
 
 4 
 2 
 1 
 
 i of a D, 
 i of a D. 
 :! of a D. 
 ^ of a D. 
 1 of a D. 
 of aD. 
 
 1 
 
 10 
 
 yV of a D. 
 
 /q of a D. 
 
 ^0 of a D. 
 is ^ of a D. 
 is -J^T of a D. 
 
 Avoir. Weight. 
 
 84 lb. is 
 56 lb. is 
 28 lb. is 
 14 lb. is 
 7 1b. is- 
 
 is 
 
 50 
 
 T^of 
 
 D. 
 
 f of a cord 
 
 ^ of a cord 
 
 i of a cord 
 
 i of a cord 
 
 is j^g of a cord 
 
 is ^^ of a cord 
 
 is -^-g of a cord 
 
 Land Measure. 
 
 80po.or2R.ianA 
 
 40po.orlR.^an A. 
 
 32 po, 
 
 20 po 
 
 16po 
 
 8po. 
 
 4 po. 
 
 2po. 
 
 12* 
 
 ^an A 
 J- an A. 
 
 yi^anA. 
 
 2V an A. 
 
 4V an A. 
 
 ^anA. 
 
 f of a cwt. 
 
 J of a cwt. 
 
 ^ of a cwt. 
 
 -|- of a cwt. 
 Jg of a cwt. 
 3.5 lb. is ^^2 of a cwt. 
 If lb. is gif of a cwt. 
 
 Cloth Measure. 
 
 27 in. or 3 qrs. |- yd 
 18 in. or 2 qrs. -i yd 
 9 in. or 1 qr. ^ yd 
 
 iVyd 
 
 4.5 in. or2na. 
 2.25 orlna. 
 Uin. 
 
 32 
 
 yd 
 
 Parts of a i cwt. 
 
 14 lb. is J of a qr 
 7 lb. is ^ of a qr 
 4 lb. is ^ of SL qr 
 2 lb. is -^^ of a qr 
 1 lb. is ^ of a qr. 
 
138 PRACTICE. 
 
 EXAMPLES. 
 
 1 . What is the value of 5 cwt. 1 qr. 14 lbs., at D2.50 per cwt. ? 
 Thus : J D2.50 First multiply D2.50 by 5 cwt., 
 
 5 X and this will give the value of 5 
 
 cwt. ; then A of D2.50 is 62ic., 
 
 12.50=5 cwt. the price or value of the 1 qr. ; 
 
 I 62 J i cwt. then the ^ of 62 J is 31^ the value 
 
 =31-J- 14 lbs. of 14 lbs., and their products give 
 
 the whole value, &c. 
 
 Dl3.43f 
 
 2. What cost 2 cwt. 3 qrs. 7 lbs. 8 oz. of iron, at 5 dollars 
 per cwt. ? 
 
 J)5.00D. Or, 1792 oz. : 5048 oz. :: 5.00D. 
 2 cwt. 5.00 
 
 D, c. m. 
 
 10.00 =2 cwt. 1792)2524000(14.08.4+ Arts proof. 
 
 3.75 =:$ cwt. ^ ^ * ^ >^. 
 
 .31^=7 
 .02 
 
 s.^f=^>7y ^^^^^""^^^Hi 
 
 D14.08i Arts. 
 
 3. What is the value of 498 yards of tape, at 6 mills per 
 yard ? Ans. D2.98.8. 
 
 4. What cost 1724 yards of flannel, at 37^ cents per yard ? 
 
 Ans, D646.50. 
 
 5. What cost 190 pounds of cotton, at 20 cents per pound ? 
 
 Ans. D38.00. 
 
 6. What cost 16 cwt. 2 qrs. of sugar, at D5.18 per cwt. ? 
 
 Ans. D85.47. 
 
 7. What cost 560 yards of sheeting, at c.20.5 per yard ? 
 
 Aw5.Dll4.80. 
 
 8. What cost 270.5 yards, at 15 cents per yard ? 
 
 Ans. D40.57.5. 
 
 9. What cost 45 gallons, at 18| cents per gallon ? 
 
 Ans. D8.43.75. 
 
 10. What is the cost of 5^.5 pounds of butter, at c.12.5 per 
 pound ? Ans. D7.06.25. 
 
 11. What is the cost of 15 A. 1 R. at D 10.25 per acre ? 
 
 Ans. D156.31.25 
 
 12. What cost 5.30 cords of wood, at D2.50 per cord ? 
 
 Ans. D13.25. 
 
 13. What cost 6 yards, 3 quarters of cloth, at D1.50 per yard.? 
 
 Ans. 010.13^, 
 
PRACTICE. -^^^^^ I%^X 
 
 14. What cost 30.5 bushels of oats, at 37| cts. per bushel ? 
 
 Ans. Dll.43.75. 
 
 15. What cost 8 bushels, 3 pecks of wheat, at Dl.12 per 
 bushel? Ans. D9.80. 
 
 16. What cost 18.25 bushels of rye, at 93.75 cts. per bushel ? 
 
 Ans. D17.11. 
 
 17. What cost 25 bushels of com, at 56.25 cts. per bushel ? 
 
 Ans. D14.06.25. 
 
 18. What cost 2.75 bushels of cloverseed, at D4.25 per 
 bushel? Ans. Dl 1.68.75 
 
 19. What cost 12 cwt., 2 qrs., 14 lbs. of rice, at D4.75 per 
 cwt. ? Ans. D59.96.8f, 
 
 20. What is the value of 130 cwt, 1 qr., at Dl5 per cwt. ? 
 
 Ans. D1953.75. 
 
 21. What is the value of 25 cwt. 1 qr. 9 lbs., at Dl.75 per 
 cwt.? Ans. D44.32.84- 
 
 22. What is the value of 6 lbs. 5 oz. 10 pwt. 5 gr., at D4.16 
 per lb. ? Ans. D26.86.9. 
 
 23. What is the value of 428 gallons, 3 quarts, at Dl.40 per 
 gallon? Ans. D600.25 
 
 24. What is the value of 5 hogsheads, 31^ gallons, at D47.1Sj 
 per hogshead ? Ans. D259.16 
 
 25. What is the value of 35 A. 2 R. 18 po., at D54.35 pei 
 acre? Ans. D1935.53.9 
 
 26. What is the value of 750 A. 1 R. 4 po., at D12.25 pei 
 acre? Ans. D9190.86.8+ 
 
 27. What cost 927 yards, at 53j cents per yard? 
 
 Ans. D494.40 
 
 28. What cost 2691 lbs., at 16f cents per lb. ? D44.91.7, 
 
 29. What cost 300^ lbs., at 171- cents per lb. ? D52.58.7^. 
 
 30. What cost 265 lbs., at 12^ cents per lb. ? (-^8)z= 
 
 Ans. D33.12.5. 
 
 31. What cost 1000 yards, at 12.5 cents per yard ? D125 
 
 32. If 1000 yards cost D1625, what cost 1 yard ? Dl.62.5 
 
 33. What cost 862^ feet of boards, at at D12.00 per M. ? 
 
 Ans. DIO.34.6 
 
 34. What cost 35^ feet, at 35 cents per foot ? D12.32 
 
 35. What cost 842 yards, at 66f cents per yard ? D561.33i. 
 
 36. What cost 4 A. 7 R. 20 po., at D25 per acre ? D146.87^. 
 
 37. What cost 164 A. 2 R. 15 po., at D75 per acre ? 
 
 Ans. D12344.53.125. 
 
 38. What cost 94 yards, 3 qrs. 2 na., at D5.50 per yard ? 
 
 Ans. D521.81.25. 
 
 39. What cost 30 bushels, 1 peck, 7 quarts of oats, at 50 
 cents per bushel ? Ans. D15.23.4 
 
140 SIMPLE INTEREST. 
 
 40. What cost 7 cwt. 2 qrs. 9 lbs. 12 oz., at D2.50 per cwt. ? 
 
 REVIEW. 
 
 What is Practice ? What is an aliquot part ? Repeat the 
 table of aliquot parts ? In what cases will practice apply ? 
 Repeat the rule. 
 
 SIMPLE INTEREST. 
 
 Interest is an allowance, or premium, paid by the borrower 
 to the lender, or the debtor to the creditor, on money, notes, bonds, 
 mortgages, &;c., which is generally established by law at a cer- 
 tain rate, per centum, per annum ; which means, so many cents 
 for the use of one dollar, or so many dollars for the use of one 
 hundred dollars, according to the laws of the state, or the terms 
 agreed upon between the parties, which, in most cases in this 
 country is 6 per cent., or 6 dollars for the use of 100 dollars, for 
 one year, and in the same proportion for a larger or less sum, 
 and for a longer or shorter period. In England the rate is five 
 per cent. In the New England states it is six, and in the state 
 of New York it is seven per cent. In all transactions with the 
 general government, interest is computed at six per cent. The 
 courts of the United States allow interest according to the 
 practice of the state where the suit was commenced. The 
 banks throughout the Union compute interest at six per cent., 
 allowing the year to consist of 360 days, or 30 days to the 
 month. When interest is regulated by a statute of the state or 
 country, at a certain rate per cent., to extort, or receive inten- 
 tionally, more interest than is allowed by the statute, is called 
 usury, and is a punishable offence. This, however, is not intend- 
 ed to apply where the method of calculation is erroneous, but 
 where it is evident fraud, or extortion is actually intended. 
 Interest is of two kinds, namely, simple and compound. 
 Simple interest is that which is allowed on the principal only. 
 Commission, brokerage, insurance, &c., anything rated at so 
 much per cent., may be calculated by the rules of interest. In 
 computing interest, there are four parts which should be partic- 
 ularly attended to, namely : 
 
 1 . Principal is the sum for which interest is to be computed. 
 
 2. Ratio, or rate per cent, per annum, is the interest of one 
 nundred dollars for one year. 
 
SIMPLE INTEREST. I^' 
 
 3. Time is the number of years, months, days, &c., for which 
 interest is to be computed. 
 
 4. Amount is the principal and interest added together. 
 
 To compute interest for days, 
 
 RULE I. 
 
 Multiply the principal by the number of days ; that product 
 by the rate per cent., and divide by 365, and the quotient will 
 be the answer in dollars, cents, and mills. 
 
 QUESTIONS. 
 
 1. Required the interest of 550 dollars for 92 days, at 6 and 
 7 per cent. 1 
 
 First, thus : 550 dollars. Second, 550 dollars. 
 
 92 days. 92 days. 
 
 50600 
 
 50600 
 
 6 ratio. 
 
 7 ratio, 
 
 D. 
 
 D. 
 
 365)303600)8.31. 7+ J.n5. 365)354200(9.70.4+ Ans. 
 
 Or, 100 dollars 550 dollars ) a .- oj 
 
 365 days. •' 92 days. ''' I ^ ^^^^^- ^^^ 
 
 > 7 ratio. 4th. 
 
 100 dollars. , 550 dollars. 
 365 days. * 92 days. 
 2. Required the interest of 700 dollars for 365 days, at 6 per 
 cent, per annum. 
 
 365 days. 700 dollars. 
 
 700 dollars. 6 ratio. 
 
 255500 D42.00 proof. 
 
 6 
 D. 
 
 365)1533000(42.00 Ans. 
 
 3. Required the interest of D150 for 63 days, at 6 per cent 
 
 Ans. D1.55.3 + 
 
 4. What is the interest of D250 for 60 days, at 4.5 per cent. ? 
 
 Ans. Dl.84.9-j- 
 
 5. What is the interest of D 104.25 for 90 days, at 5 per cent. 
 
 Ans. Dl.28.5 
 
 6. What is the interest of D85 for 30 days, at 7 per cent. ? 
 
 Ans. c,48.9 
 
142 SIMPLE INTEREST. 
 
 7. What is the interest of D40 for 50 days, at 9 per cent. ? 
 
 Ans, 0.49.3 
 
 8. Required the interest on a bond for D45, for 150 days, at 
 5 per cent. ^ Ans. c.92.4. 
 
 9. Required the interest of D300 for 180 days, at 7 per cent. 
 
 Ans. DlO.35.6. 
 
 10. Required the interest of D34.2.76 for 176 days, at 7i per 
 cent. Ans. D 11. 98.2 + 
 
 11. What is the interest of D989.94 for 489 days, at 5^ per 
 cent. ? Ans. D72.94.3 + 
 
 12. What is the amount of D289.64 for 579 days„at 16i per 
 cent. ? Ans. D365.45. 
 
 Interest for months^ weeks, <SfC, 
 
 1. At 6 per cent., each dollar will draw 1 cent a month, there- 
 fore multiply by half the number of months ; the product will be 
 in cents, which is the answer. 
 
 2. For the interest at any other rate per cent., take aliquot 
 parts of the interest at 6 per cent., and add it or subtract it as 
 the case requires. 
 
 3. Or you can state the questioil in compound proportion. 
 13. Required the interest of DlOO for 10 months, at 6 and 7 
 
 per cent. 
 
 100 D. 
 
 5 half no. mo. 
 
 |-)5.00 at 6 per cent. 
 +83.3 
 
 D5.83.3 at 7 per cent. 
 14. What is the interest of D94.50 for 19 months, at 9 per ct. 
 
 Thus : 100 D. : 94.50 J). :: ) ^ ^ ,. 
 TO in ^9 ratio. 
 
 12 mo. 19 ) 
 
 Or, 94.50 D. 
 
 9.5 half no. mo. 
 
 for 3^)8.97.75=6 per cent. 
 4.48.87 
 
 D13.46.625 Ans. at 9 per cent. 
 
SIMPLE INTEREST. 14S 
 
 15. What is the interest of D649.76 for 74 months at 6 per^ 
 cent " X37. Ans. D240.41.1. 
 
 16. What is the interest of D331.42 for 150 weeks at 7 per 
 cent.? DlOO 331.42 
 
 W. 52 • 150 
 
 I 7 : D66.92.1 Ans. 
 
 17. What is the interest of D284.10 for 64 weeks at 3^ per 
 tent.? Ans. D12.23.8 
 
 18. What is the interest of D80 for 9 months at 6 per cent. 
 
 Ans. D3.60. 
 
 19. What is the interest of D94 for 13 months at 6 per cent. ? 
 
 Ans. D6.11. 
 
 20. What is the interest of D75 for 3 months at 6 per cent. ? 
 
 Ans. Dl.12.5. 
 
 21. What is the interest of D98 for 4 months at 6 per ^ent. ? 
 
 Ans. Dl.96. 
 
 22. What is the interest of D400 for 9 months at 6 per cent. ? 
 
 Ans. D18. 
 
 23. What is the interest of D700 for 10 months at 6 per cent. ? 
 
 Ans. D35. 
 
 24. What is the interest of D80 for 10 months at 9 per cent. ? 
 
 Ans. D6. 
 
 25. What is the interest of D300 for 15 months at 6 per cent. ? 
 
 Ans. D22.50. 
 
 26. What is the interest of DlOO for 11 months at 5 per cent. ? 
 
 Ans D4.58.4. 
 100 
 5.5 
 
 i)5.50D. at 6 per cent. 
 —91.6 
 
 4.58.4D. Ans. 
 Interest for years. 
 
 RULE III. 
 
 Multiply the principal by the rate per cent., and separate the 
 two right-hand figures for cents, and those at the left of the point 
 will be dollars, and this will be the answer for 1 year ; if more 
 than one year is required, multiply this interest by the number 
 of years, and the product will be the answer. If there are frac- 
 tional parts of a year, you can work by aliquot parts, &c. 
 
144 SIMPLE INTEREST. 
 
 27. What is the interest of D500 for 1 year, for 3 years, foi 
 3^ years, and for 4J years, at 9 per cent. ? 
 
 Ans. D45 for 1 year, D135 for 3 years, D 157.50 for 3^ years, 
 D] 91.25 for 4^ years. 
 
 500 500 
 
 9 per cent. 9 
 
 ^)45.00=^1 year. ^)45.00c=l year. 
 
 3X 4x 
 
 D135.00 = 3 years. D180.00=4 years. 
 
 22.50=-| year. 1 1 .25=^ year. 
 
 D157.50=r3iyrs. Ans. 0191.25=^4^ years Ans. 
 
 28. What is the interest of D 150.25 for 1 year at 6 per cent. 
 
 Ans. D9.01.5. 
 
 29. What is the interest of D67for 7 years at 6 per cent, per 
 annum? Ans. D28.14. 
 
 30. What is the interest of D75 for 10 years at 6 per cent. ? 
 
 Ans. D45. 
 
 31. What is the interest of D89 for 11 years at 6 per cent. 1 
 
 Ans. D58.74. 
 
 32. What is the interest of D700 for 12 years at 6 per cent. ? 
 
 Ans. D504. 
 
 33. What is the interest of 2000 for 9 years at 6 per cent. ? 
 
 Ans. D1080. 
 
 34. What is the interest of 300 for 14 years at 6 per cent. ? 
 
 Ans. D252. 
 
 35. What is the interest of D250.25 for 3 years at 7 per ct.? 
 
 Ans. D52.55.25. 
 
 36. What is the interest of D442 for 4 years at 8 per cent. ? 
 
 Ans. D141.44. 
 
 37. What is the interest of D222 for 6 years at 5 per cent. ? 
 
 Ans. D66.60. 
 
 38. What is the interest of D250 for 9f years at 5j per ct. ? 
 
 Ans. D134.06J 
 
 Interest for years, months, weeks, days, <SfC, 
 
 RULE IV. 
 
 Find the interest of the given sum for 1 year ; then as 1 year 
 ie to the given time, so is the interest of the given sum for 1 
 year to the interest required ; or, take part of the yearly interest 
 for the aliquot parts of a year that are in the given time, and 
 add the interest of the days (if any), which find by rule 1st. 
 When fractions occur in the rate per cent., work decimally. 
 
SIMPLE INTEREST. 
 
 145 
 
 39. What is the interest of D95.25 for 4 years, 1 month, 5 
 days, at 7 per cent. "? 
 
 Thus: D95.25 
 
 7 rate per cent. 
 
 ^)D6.66.75— 1 year, 
 4X 
 
 26.67.00—4 years. 
 I) .55.56—1 month, 
 9.25 — 5 days. 
 
 da. 
 365 
 
 D27.31.81 Ans, 
 
 da. D. 
 
 Or 365 : 1495 ;; 6.66.75 : 27.31 -f 
 
 40. What is the interest of D350 for 3 years and 10 months, 
 at 6 per cent, per annum ? 
 
 D350 
 
 6 rate per cent. 
 
 J &^)21.00— 1 year. 
 3X 
 
 63.00-— 3 years. 
 = 10.50- 
 21-^3)= 7.00 
 
 21-^2)3=10.50-^6 months, } 
 4 months. > 
 
 :10 
 
 D80.50 Ans, 
 DlOO : D350 
 
 l^ n^onths, 46 mo 
 
 ■* > 6 rate per cent, 
 no. ) ^ 
 
 13.00 
 
 16100 
 6x 
 
 12)96600 
 
 D80.50 Ans. D. 
 
 Or, 12 months : 46 months : D21 interest. 80.50. 
 
 41. Required the interest of D200 for 15 months at 6| per 
 cent. Ans. 200xl5mo. 6^ X rate per ct.~12mo. — D16.25. 
 
 42. What is the interest of D500 for 2 years and 9 months, 
 ftt 6 per cent. ? Aris. 82.50. 
 
 13 
 
146 SIMPLE INTEREST. 
 
 Thus, 500 
 
 6 XT. As 12 mo. : 33 mo. :: 30 
 
 ID 
 
 1 
 
 Ist/ 
 znt. : 82.51 < 
 
 D30.00 interest. 
 
 2d. 3d. 
 100 500 > ^ 5 500 
 12 • 33 •• $^ > 16ix 
 
 4th. 
 500 
 6x 
 
 D82.50 
 
 1)30.00—1 year. 
 2x 
 
 
 60.00 — 2 years 
 22.50 — 9 mo. 
 
 D82.50 Ans, 
 
 43. What is the interest of D250 for 1 year and 9 months 
 at 6 per cent. ? Ans. D26.25. 
 
 44. Required the interest of D7500 for 4 months at 7 per ct. 
 
 Ans. D175. 
 
 45. What is the interest of D967 for 2 years and 4 months 
 at 6 per cent. ? Ans. D135.38. 
 
 46. What is the interest of D1260 for 8 months at 7 per ct.? 
 
 Ans. D58.80. 
 
 47. What is the interest of D450 for 6 months and 20 days 
 at 5.5 per cent. ? Ans. D13.75. 
 
 48. Required the amount of D240 for 61 days at 4.75 per 
 cent. Ans. D241.90. 
 
 49. What is the interest of Dl for 15 months at 6 per cent. ? 
 
 Ans. DO.07.5. 
 
 50. Required the amount of D672.45 for 7 years, 4 months, 
 25 days, at 7\ per cent. 
 
 51. Required the interest of D379.50 for 27 years, 9 months, 
 5 days, at 6.7 per cent. 
 
 Bank interest. 
 
 Remarks. — The method of computing interest in the banks is 
 at the rate of 30 days to the month, or 360 days to the year ; this 
 will make the interest too much ; the difference may be discov- 
 ered by comparing the calculations below with rule 1st ; but this 
 is not all ; it is usual to take the interest in advance, which is 
 sometimes improperly called discount ; this error is greater than 
 the first, for instead of 6, it is little less than 7 per cent. Thus, 
 if A. should borrow DlOO of the bank, or give his note for that 
 
SIMPLE INTEREST. 147 
 
 sum, and the bank should deduct the interest for one year, which 
 would be six dollars, and keep that interest for one year ; the 
 interest of which would be 36 cents. It would stand thus : If 
 the bank will gain D6.36 on D94 loaned to A , how much will 
 DlOO gain? As D94 : D6.36 :: DlOO : D6.76 57 Ans. I 
 believe the above is generally practised in all banks throughout 
 the United States, with the addition of 3 days (commonly called 
 days of grace) which are allowed the payer on all notes, after 
 the time expires for which they were drawn. 
 
 RULE V. 
 
 Multiply the dollars by the number of days, and divide the 
 product by 6, and the quotient will be the answer in mills ; 
 which point off into dollars, &c. Or, work by rule 1st, and 
 divide by 360, the answer will be the same. 
 
 52. What is the interest of D1542 for 90 days, at 6 per cent. ? 
 
 Thus: 1542 D. Or, 1542 D. 
 
 90 days. 90 days. 
 
 6)138780 138780 
 
 6x 
 
 D23.13 Ans. D. 
 
 360)832680(23.13 
 
 53. What is the interest of D70 for 90 days, at 6 per cent. T 
 
 Ans. Dl.05. 
 
 54. What is the interest of D98 for 90 days, at 6 per cent. ? 
 
 Ans. D1.47. 
 
 55. What is the interest of D400 for 90 days, at 6 per cent. ? 
 
 Ans. D6.00. 
 
 56. What is the interest of D950 for 90 days, at 6 per cent. ? 
 
 Ans. D14.25. 
 
 57. What is the interest of D499 for 57 days, at 6 per cent. ? 
 
 Ans. D4.74. 
 
 58. What is the interest of D900 for 59 days, at 6 per cent. ? 
 
 Ans. D8.85. 
 
 59. What is the interest of D3000 for 60 days, at 6 per cent. ? 
 
 Ans. D30. 
 
 60. What is the interest of D600 for 103 days, at 6 per cent. ? 
 
 Ans. D10.30 
 
 61 . What is the interest of D3000 for 104 days, at 6 per cent. ^ 
 
 Ans. D52.00 
 
148 SIMPLE INTEREST. 
 
 Interest on notes, cj-c, where partial payments have been ?nade. 
 
 As it respects the computation of interest on notes, or bonds, 
 where partial payments or endorsements have been made, there 
 are various ways : in some states the rule is established by law, 
 in others it is not regulated by statute, but the courts of equity 
 adopt such rules as they think proper. 
 
 The limits of this work will not permit a full investigation ; 
 several rules and examples will be given, and the principle left to 
 the action of those concerned, as no rule given by any author 
 will be strictly adhered to. The following appears to be a fair 
 and equitable method of computing interest, when endorsements 
 are made ; it is always presumed that the note is due when pay- 
 ments are made, and that the holder of the note is entitled to the 
 interest accruing thereon, whether the note calls for periodical 
 payments or not ; the intention is to give the interest to the per- 
 son who has a just demand, and is entitled to receive the prin- 
 cipal, as well as interest. Cases, hovvev^er, may occur where a 
 departure from this rule may be necessary, as when there are 
 endorsements made before the note becomes due ; in this case 
 the person making the endorsement is entitled to compute inter- 
 est on it to the time the note becomes due, but no longer, for 
 then the holder of the note may demand payment in full. 
 
 62. On demand, I promise to pay A. B., or ®rder, the sum of 
 five hundred dollars, with interest, for value received. D500.00. 
 
 January 1, 1843. C. P. 
 
 Endorsements : April 1, 1843, D200 ; Oct. 1, 1843, D200. 
 
 1st. D500 princ. 
 
 6 per cent., 
 
 D30.00 int. 1 year. 
 Dl2. — int. on endors. 
 
 D18 bal. int. 
 DlOO+Jan 1, 1844. 
 
 D118 amount due. 
 
 ' In this case C. P. has the use of the ' 
 interest of the endorsement. 
 D9— 9mo. = 40.5 c. . . 
 3 3" 4.0 ^ ^' 
 
 200 end. 3 months. 
 1.5 
 
 D3.00 
 
 interest. 
 
 D200- 
 4.5 
 
 -9 mo. 
 
 D9.00 
 3.00 
 
 
 D12.00 interest on en- 
 dorsement. 
 
SIMPLE INTEREST. 14& 
 
 2d. D500 princ. April 1, 3 months. 
 
 1.5 
 
 D7.50 interest for 3 months paid 
 
 500 princ. 
 — 200 endorsement. 
 
 D300 April 1, balance, princ. 
 3 for 6 months, 1st October. 
 
 D9.00 interest paid. 
 
 
 interest paid. 
 
 3.00 princ. D7.50 
 
 — 2.00 endorsement. 9.00 
 
 1.50 
 
 DlOO princ. balance for~ 
 
 
 1.5 mo. 3 mo. to 
 
 18.00 
 
 Dl.50 int. p. Jany. 1. 
 
 > 
 
 DlOO— 1844. 
 
 
 The 2d example is correct. A. B. had a right to receive the 
 interest on D9 9mo. : D3 3mo.=44.5, this is the difference in 
 the 1st and 2d examples. 
 
 63. A. received of B. his note for DlOO, dated January 1, 
 1820, payable in 1 year, w^ith 6 per cent, interest ; and July 1, 
 A. received of B. fifty dollars, which A. credited on the note, 
 what was the balance due from B. to A. on January 1st, 1821, 
 one year ? 
 
 Dr. B. in account with A. Cr. 
 
 1821. 1820. 
 
 January 1, to his note of DlOO Julv 1, by cash, D50.00 
 
 1821. 
 To 1 year's interest, 6 Jan. l,byint.of D50, 
 
 paid as above, 1.50 
 
 D106 By balance, 54.50 
 
 D106.00 
 In this case D54.50 is the equitable balance. If the above 
 note had been on demand, or due at the time of payment (July 
 1, 1820) the balance on the 1st of January, 1821, would have 
 been D54.59, difference 9 cents, which would have arisen on 
 the D3 for 6 months ; this is the same as the 1st and 2d examples. 
 
 13* 
 
1^0 SIMPLE I>rrEREST. 
 
 " We have no special acts of assembly in this sraie. [Penn.] 
 Chief Justice McKean, in 1785, fixed the following rule for 
 calculating interest [see Dallas, 1st, p. 124] : That the inter- 
 est of the money paid in before the time be deducted from the 
 interest of the whole sum due at the time appointed by the 
 instrument for making the payment.'* As per examples above. 
 
 64. Due A. B. or order, the sum of one thousand dollars on 
 demand, with interest, for value received. 
 
 January 1, 1842. DlOOO.OO. C. D. 
 
 Endorsements : July 1, 1842, D300 ; January 1, 1843, D300 ; 
 July J, 1843, D200 ; required the balance due January 1, 1844, 
 interest 6 per cent. 
 
 DIOOO January 1, 1842. 
 
 3 ) interest July 1, 1842. 
 
 I 6 months. 
 
 Paid D30.00 
 
 DIOOO 
 endors.— 300 
 
 700 princ. July 1, 1842. 
 3 
 
 paid, D21.00 interest July 1, 1843, , 
 
 6 months. ; 
 
 princ. 700 
 endors.— 300 
 
 400 July 1, 1843, 
 3 6 months. 
 
 paid, D12.00 interest. 
 
 400 
 endors.— 200 
 
 200 princ. January 1, 1844, 
 3 6 months. 
 
 interest, 6.00 
 200 
 
 D206.00 due on the note. 
 
SIMPLE INTEREST. 
 
 151 
 
 The following is the rule established in the state of New 
 York (see Johnson's Chancery Reports, vol. 1, page 17). The 
 same rule is also adopted in Massachusetts and in most of the 
 other states. 
 
 RULE I. 
 
 Compute the interest on the principal to the time of the first 
 payment, and if the payment exceed this interest, add the inter- 
 est to the principal and from the sum subtract the payment ; the 
 remainder forms a new principal. 
 
 2. But if the payment is less than the interest, take no notice 
 of it until other payments are made, which in all shall exceed 
 the interest computed to the time of the last payments ; then 
 add the interest so computed to the principal, and from the sum 
 subtract the sum of the payments ; the remainder will form a 
 new principal on which interest is to be computed as before. 
 
 Note, — The above rule does not differ materially from the 
 preceding examples, and is the same in its general application ; 
 the object in computing interest on notes, should be to keep 
 down the interest, that it may not form a part of the principal 
 carrying interest, and in the second place to pay the interest to 
 the one entitled to receive it. 
 
 (When partial payments are made at short periods, the fol- 
 lowing rule will apply.) 
 
 RULE. 
 
 Subtract the several payments from the original sum in their 
 order, placing their dates in the margin. 
 
 65. Suppose a bill of D359 was due January, 1839, and D75 
 was paid February 3, D50 March 5, D80 April 9, D145 June 7 ; 
 what interest is due at 5, 6, and 7 per cent. ? 
 Dates. Bill. 
 
 January 1 , 
 Feb. 3, paid 
 
 Balance, 
 March 5, paid, 
 
 Balance, 
 April 9, paid, 
 
 Balance, 
 June 7, paid, 
 
 D. 350 
 —75 
 
 275 
 —50 
 
 225 
 —80 
 
 145 
 145 
 
 33 
 
 jrruuut 
 
 11550 
 
 30 
 
 8250 
 
 35 
 
 7875 
 
 59 
 
 8555 
 
 
 36230 
 
 7300)36230(4.963D. 
 at 5 per ct. 
 
 6083)36230(5.95.5 
 at 6 per cent. 
 
 5214)36230(6.94.8 
 at 7 per ct. 
 Ans, 
 
 (See next page to make a divisor, &c.) 
 (See Note at the conclusion of Simple Interest.) 
 
152 SIMPLE INTEREST. 
 
 The amount, time, and rate per cent., given to find the prii cipdl. 
 
 RULE VI. 
 
 1. Find the amount of DlOO for the time at the given rate 
 per cent. 
 
 2. Then as the amount of DlOO for the time required, at the 
 given rate per cent., is to the amount given, so is DlOO to the 
 principal required. 
 
 66. What principal at interest 6 years at 6 per cent, will 
 amount to D500 ? thus D6x6 years=:36D interest for 6 years, 
 + 100r=:136D. amount of DlOO ; then D136 : D500 :: DlOO : 
 
 Ans. D367.65. 
 
 67. What sum being put to interest 6 years at 6 per cent, 
 will amount to D44.88 1 Ans. D33. 
 
 68. What sum being put to interest 8 years at 6 per cent, 
 will amount to D74 ? Ans. D50. 
 
 69. What sum being put to interest 10 years at 6 per cent, 
 will amount to D 160? Ans. DlOO. 
 
 70. What sum being put to interest 11 years at 6 per cent, 
 will amount to D830 ? Ans. D500. 
 
 The principal, amount, and time, given to find the rate per cent. 
 
 RULE VII. 
 
 Find the interest for the whole given time, by subtracting the 
 principal from the amount ; then as the principal is to DlOO, so 
 is the interest of the principal for the given time to the interest 
 of DlOO, for the same time ; divide the interest last found by 
 the time, and the quotient will be the rate per cent. 
 
 71. At what rate per ct. will D500 amount to D605 in 3 years? 
 
 D500 
 3x 
 
 605 amount then 1500 : 105 :: 100 
 — 500 principal 100 
 
 1500 
 
 105 interest, 1500)10500 
 
 7 per cent. Ans, 
 
 72. At what rate per cent, will D750 amount to D930 in 4 
 years ? Ans. 6 per cent. 
 
 73. At what rate per cent, will D200 amount to D255 in 5 
 years ? Ans. 5.5 per cent. 
 
I SIMPLE INTEREST. 153^ 
 
 I To find the time when the principal amount and rate per cent, are 
 
 RULE VIII. 
 
 Divide the whole interest, by the interest of the principal for 
 one year, and the quotient will be the time required. 
 
 74. In what time will D400 amount to D568, at 6 per cent. 1 
 
 D. y. D. 
 Thus: D400 568 then 24 : 1 :: 168 : 7 years ^n^. 
 6 —400 
 
 D24.00 168 
 75. In what time will D600 amount to D750 at 5 per cent. ? 
 
 Ans. 5 years. 
 
 A short method to find the interest for one or more months, at any 
 rate per cent, 
 
 RULE IX. 
 
 Multiply the principal by the rate per cent., which will give 
 the interest for 1 year ; divide this by 12, and the quotient will 
 be the interest for 1 month ; multiply this last interest by the 
 number of months required, &:c. 
 
 Example, — D250 princ. 5 per ct. D120 at 7 per ct. 
 5X 7X 
 
 12)12.50 for 1 year. 12)8.40 for 1 year. 
 
 D 1.04.2 for 1 month. 0.70 for 1 month. 
 
 7x 8X 
 
 7.29.4 for 7 months. D5.60 for 8 months. 
 
 then Dl.04.2 inter, for 1 month. 
 
 4x D75 .70c. 
 
 7 per ct. 11 x mo. 
 
 D4.16.8for4mo. 
 
 2x 12)5.25.= 1 yr. D7.70forllmo. 
 
 D8.33.6 for 8 mo. 0.43.7J for 1 mo. 
 
154 SIMPLE INTEREST. 
 
 To make a divisor for any rate per cent. 
 
 RULE X. 
 
 Multiply 365 days by DlOO, and divide by the rate per cent 
 • Thus, 365x100-^6=6083 divisor at 6 percent. ; 365x100 
 -f-5=:7300 divisor ; 365 X 100-^-7=5214 divisor at 7 per cent. ; 
 and so for any other rate per cent. 
 
 76. Required the interest of D400 for 26 days at 5, 6, and 7 
 per cent. 
 
 Thus, 400 x26=:10400-f-6083=Dl.70.9f at 6 per cent. Ans. 
 10400^7300nrDl.42.44. at 5 per cent. Ans. 10400—5214 
 =rDl.99.4f at 7 per cent. Ans. Or thus, 100 rate 400 
 
 365 : 6 :: 26 • 
 Dl.70.9|. Or 400x26x7-^365 = Dl.99.4f at 7 per cent.; 
 Again, as 365 da. : 5 per ct. :: 7300 : 100 per ct., and as 12 
 mo. 5 per ct. :: 240 mo. : 100 per ct. 
 
 Hence it is evident, that if the rate be 5, any principal will 
 give 100 per cent. ; that is, it will double in 7300 days, or 240 
 months ; and at 6 per cent, any sum will double in 6083 days, 
 or 200 months ; and at 7 per ct. in 5214|^ days, or 171|- months ; 
 from which the following rule has its origin, as exemplified 
 above : — 
 
 RULE XI. 
 
 Multiply the principal by the days, and for 6 per cent, divide 
 by 6083 ; for 5 per cent, by 7300, &c., and the quotient is the 
 answer. These being the number of days, in which any sum 
 will double at those respective rates. For months, multiply the 
 principal by the months, and divide by 200 for 6 per cent., or 
 240 for 5 per cent., &c., and the quotient is the answer. 
 
 77. Required the interest of D 150 at 5 and 6 per cent, for 30 
 months. 
 
 Ansl D150 X 30-r240=rDl8.75 at 5 per cent. ; D150 X 30 - 
 200 = D22.50 at 6 per cent. 
 
 Hence, when: interest is to be calculated on cash accounts, or 
 accounts current, where partial payments are made, or partial 
 debts contracted, multiply the several balances into the days 
 they are a* interest, which should be done at every transaction, 
 and the sum of tho&e products divided by 6083 and 7300 will 
 give the interest at 6 and 5 per cent., and for any other rate find 
 a divisor as above directed (see quest. 6b). 
 
SIMPLE INTEREST. 155 
 
 APPLICATION. 
 
 78. What is the interest of D22 for 25 days, at 6 per cent. ? 
 (Rule 1.) Ans. c.9. 
 
 79. What is the interest of D58 for 29 days, at 6 per cent. ? 
 
 Ans, c.27.6. 
 
 80. What is the interest of D400 for 26 days, at 6 per cent. 1 
 
 Ans. D1.71. 
 8 1 What is the interest of D90 for 28 days, at 6 per cent. ? 
 
 Ans. c.41.4. 
 
 82. What is the interest of D50 for 9 months, at 6 per cent. 1 
 
 Ans. D2.25. 
 
 83. What is the interest of D57 for 14 months, at 6 per cent. ? 
 
 Ans. D3.99. 
 
 84. What is the interest of D700 for 1 1 months, at 6 per cent. 1 
 
 Ans. D38.50. 
 
 85. What is the interest of D900 for 1 year and 1 month, at 
 6 per cent. ? Ans. D58.50. 
 
 86. What is the interest of D200 for 1 year and 3 months, at 
 6 per cent. ? Ans. D15.00. 
 
 87. What is the interest of D500 for 1 year and 4 months, at 
 6 per cent. ? Ans. D40.00. 
 
 88. What is the interest of D31 for 10 years and 6 months, 
 at 6 per cent. ? Ans. D19.53. 
 
 89. What is the interest of D500 for 14 years, at 6 per cent. ? 
 
 Ans. D420. 
 
 90. What is the bank interest of D89 for 89 days, at 6 per 
 cent.? Ans. Bl. 32, 
 
 91. What is the interest of D764 for 420 days, at 5.5 per 
 cent.? (Rule 1.) 
 
 92. What is the interestof D642.25for 900days,at7.5 per ct.? 
 
 [n the solution of the following questions, the year will con- 
 sist of 365 days, and the months of 30^ days. (Rule 1.) 
 
 93. What is the interest of D1062.80 for 2 months, at 9 per 
 cent.? Ans. 1062.80x61 x9^365 = Dl5.98.5. 
 
 94. What is the interest of D1S896.25 for 4 months, at 4 per 
 cent.? ^71^. D 172. 42.1 + 
 
 95. What is the amount of D97.21 for 1 year and 1 month, 
 at 7 per cent. ? Ans D104. 58. 3. 
 
 96. What is the interest of D3 19.25 for 2 years and 7 months, 
 at 7 per cent. ? Ans. D57.76.6. 
 
 97. What is the amount of D205.10 for 1 year, 4 months, 
 and 2 days, at 7 per cent. ? Ans. D224.33.4 
 
156 SIMPLE INTEREST. 
 
 98. What is the interest of D4008.50 for 1 year and 9 months 
 at 7i per cent. ? Ans. D526.73.3 
 
 99. What is the amount of D107.70 for 7 months and 5 days 
 at 7 per cent. ? Ans. D112.21.3. 
 
 100. What is the interest of D1121.42 for 11 months, at 5.5 
 per cent.? Ans. 56.69-f 
 
 101. What is the amount of D1428.50 for 1 year, 5 months, 
 and 5 days, at 5 per cent. ? Ans. D 1438.72.4. 
 
 102. What is the interest of D 1892. 50 for 1 year, at 20 per 
 cent. ? Ans. Dl892.50x20^D378.50. 
 
 103. What is the amount of D 1050. 25 for 1 year, at 4 per 
 cent.? Ans. D1092.26.0. 
 
 104. What is the interest of D742.18 for 120 days, at 6 per 
 cent.? A71S. D14:64. 
 
 105. What is the amount of D 19.60 for 1 year and 10 months, 
 U 41 per cent. ? Ans. D21.21.9. 
 
 Note 1. — For the sake of greater exactness in calculating 
 interest, should any require it, the divisor, or number of days 
 in a year may be 365i, but the difference will be unimportant ; 
 in this case the hour the note was dated and the hour of pay- 
 ment, should be considered. 
 
 Required the interest of D2500 for the first 6 months of the 
 year 1841, allowing the year to consist of 360, 365, and 365-J- 
 days, at 6 per cent, per annum. (Rule 1.) 
 
 January, 31+ February, 28 + March, 31+April, 30 + May, 
 31+June, 30z=181 days; 2500x181=442500x6=2715000 
 dividend- 360, &c. 1st Ans. D75.41.6 ; 2d, D74.38.3 ; 3d, 
 D74.33.2. It will be seen that by omitting the 5 days in the 
 year, the difference is Dl.03.3 too much. 
 
 Note 2. — The erroneous method of computing interest on 
 endorsements, and then deducting those amounts from the face 
 of the note, except in particular cases, may be seen in the fol- 
 lowing statement : Suppose I borrow DlOO at 6 per cent., for 
 10 years, and pay D6 at the end of each year, what will be due 
 at the end of 10 years ? The amount of DlOO is D160. But 
 the first endorsement of D6 has borne interest for 9 years, the 
 second for 8 years, the third for 7 years, and so on ; so that six 
 dollars have, in fact, been drawing interest forty-five years, and 
 thus produced D 16.20 of interest. This added to the nine en- 
 dorsements of D6 each, gives D70.20 ; that is, while I have 
 paid only the annual interest of D6, the principal has actually 
 been reduced D16.20; by paying D6 annually for 25 years, 
 
INSURANCE, COMMISSION, AND BROKERAGE. 15 
 
 and computing interest on the several endorsements by this 
 method, the whole principal would be paid, and the lender 
 would be indebted to the borrower the sum of D2.00, while in 
 fact, the lender had received no part of the sum lent. The 
 dillerence of the rules depends on the principle assumed in 
 respect to the time when the interest becomes due, and which 
 party has a right to receive it. 
 
 REVIEW. 
 
 What is simple interest ? How many parts are there which 
 require attention in interest ? Name them. How do you com- 
 pute interest for days ? What is the interest of D1799.89 for 
 1700 days, at 9| per cent. ? How can you calculate interest 
 for months at 6 per cent. "? How can you calculate at any 
 other rate per cent. ? How do you compute interest for years.? 
 When interest is required for years, months, days, &c., how 
 will you proceed ? What is the interest of 1)979.99 for 9 
 years, 9 months, 9 weeks, and 9 days, at 9.9 per cent. ? What 
 is bank interest ? What is the rule ? What can you say of 
 the correctness of the method of calculating bank interest ? When 
 the amount and rate per cent, are given to find the principal, by 
 what rule will you work ? What is the rule when the principal, 
 amount, and time, are given, to find the rate per cent. ? What is 
 the rule to find the time, when the principal, amount, and rate 
 per cent., are given ? What can you say in relation to the compu- 
 tation of interest on notes, &c., where partial payments are made l 
 
 INSURANCE, COMMISSION, AND BROKERAGE. 
 
 Insurance is an agreement by which an individual or a com- 
 pany agrees to exempt the owners of certain property from loss 
 or hazard. The written agreement is called the jwlicy. The 
 premium is the amount paid by him who owns the property to 
 those who insure it, as a compensation for their risk. It is gen- 
 erally so much per cent, on the value of property insured. Com- 
 mission is an allowance made to a factor, or a commission mer- 
 chant, for buying and selling. Brokerage is an allowance made 
 to dealers in money or stocks. The calculations may all be made 
 according to the rule of simple interest for one year (See rule 3. 
 
 14 
 
158 INSURANCE, COMMISSION, AND BROKERAGE. 
 
 QUESTIONS. 
 
 1. Required the insurance on D60, at 2^ per cent. ? 
 
 Arts. 60X2.5=D1.50. 
 
 2. Required the commission on D50 at 2^- per cent. 
 
 Ans. D1.25. 
 
 3. Sold goods to the amount of D200 at .75 per cent, commis- 
 sion ; how much was received ? Ans. Dl.50. 
 
 4. What is the commission on D89 at 3 per cent. ? D2.67. 
 
 5. What is the insurance on D300 at 4.5 per cent. ? D] 3.50. 
 
 6. What is the insurance on D700 at 5.5 per cent. ? D38.50. 
 
 7. What is the insurance on a ship and cargo valued at 
 D20000 at 61 per cent. ? A7is. D1300. 
 
 8. A house valued at D 10000 which is insured at 1.5 per 
 cent., required the insurance. Ans. D150.00. 
 
 9. B. collected a county tax of D 10000, for which he re- 
 ceived a commission of 3^ per cent., what sum did he receive ? 
 
 Ans. D350.00. 
 
 10. A. sold to B. goods to the amount of .D99, and gained 
 5.5 per cent, on the sale ; what was his gain? Ans. D5.44.5. 
 
 11. What sum must you pay for the use of D85 at 6^ per 
 cent ? Ans. D5.52.5. 
 
 12. If I borrow D160, and agree to pay 1.5 per cent, for the 
 use of it, how much must I pay ? Ans. D2.40. 
 
 13. Sixty-nine shares of bank stock, of which the par value 
 is D125, is at a discount of 8 per cent. ; what is its value ? 
 
 Ans. D7935. 
 
 14. What would be the insurance on a ship valued at D47520, 
 at i per cent. ? also at ^ per cent. ? Ans. D237.60 ; D158.40. 
 
 15. What would be the insurance on a house valued at 
 D 14000, at 1-1^ per cent. ? at | per cent.? at J per cent. ? at i per 
 cent. ? at 1 per cent. ? Ans. D210 ; D105 ; D70 ; D46.66 ; D35. 
 
 16. What is the insurance on a store and goods valued at 
 D15000, at l-J- per cent. ? at 1| per cent. ? at 2^ per cent. ? at 
 3| per cent. ? at 4^ per cent. ? at 5^ per cent. ? 
 
 17. If a policy be taken out for D781.25 to cover D625, re- 
 quired the premium per cent. 
 
 Thus, D781.25 : 625 :: 100 : 80 : then 100— 80= Ans. 20 per cU 
 
 18. It is required to cover D759, premium 8 per cent. ; for 
 what sum must the policy be taken ? 
 
 Ans. 1)100-8 = 92 : 100 :: 759 : 825. 
 
 REVIEW. 
 
 What is insurance ? commission ? brokerage ? and policy ? 
 
COINS AND CURRENCY. 
 
 159 
 
 COINS, CURRENCY, &c. 
 
 A table showing the weight and value of the several coins of the 
 United States. 
 
 Gold 
 
 COINS. 
 
 C Eagle, 
 ] Half-Eagle, 
 f Quarter-Eagle, 
 f Dollar, 
 I Half-Dollar, 
 Silver -^ Quarter-Dollar, 
 Dime, 
 [ Half-Dime, 
 Cent, 
 Half-Cent, 
 
 Copper 
 
 Pure. 
 
 Standard. 
 
 Value. 
 
 dwt. gr. 
 
 dwt. 
 
 gr. 
 
 D. c. m 
 
 10 7| 
 5 3| 
 
 11 
 
 6 
 
 10 00 
 
 5 
 
 15 
 
 5 00 
 
 2 13J 
 
 2 
 
 19i 
 
 2 50 
 
 15 111 
 
 17 
 
 8 
 
 1 00 
 
 7 19^f 
 
 8 
 
 16 
 
 50 
 
 3 20i| 
 
 4 
 
 8 
 
 25 
 
 1 13i 
 
 1 
 
 17f 
 
 10 
 
 18^»^ 
 
 
 
 20A 
 
 5 
 
 7 00 
 
 
 
 
 
 1 
 
 3 12 
 
 
 
 
 
 5 
 
 All Gold of equal fineness to be valued at 89c. per dwt., and 
 all Silver Coins of the same fineness at Dl.ll per ounce. The 
 standard for Gold is 1 1 parts of fine and 1 part of alloy. The 
 standard Silver is 1485 parts of pure silver to 179 parts of 
 alloy, which is to be wholly of Copper. 
 
 A TABLE OF GOLD COINS. 
 
 Names of Coins. 
 
 BRAZIL. 
 
 Johannes, half in pro- 
 portion - - - - 
 
 Dobraon - - - - 
 
 Dobra 
 
 Moidore, half in propor- 
 tion 
 
 Crusado - - - - 
 
 ENGLAND. 
 
 Guinea, half in propor- 
 tion 
 
 Sovereign, half in pro- 
 portion - - - - 
 
 Seven-shilling piece - 
 
 FRANCE. 
 
 Double Louis, coined 
 before 1786 - ■ 
 
 Weight, 
 dwt. gr. 
 
 Grains 
 
 of pure 
 
 gold. 
 
 Standard 
 before 1st 
 Aug.1834 
 D. c. m. 
 
 18 
 
 
 
 16 
 
 34 
 
 12 
 
 759 
 
 30 66 6 
 
 18 
 
 6 
 
 401 5 
 
 16 22 2 
 
 6 
 
 22 
 
 152 2 
 
 6 14 9 
 
 
 14 
 
 15 8 
 
 59 8 
 
 5 
 
 H 
 
 118 7 
 
 4 79 6 
 
 5 
 
 H 
 
 113 1 
 
 4 57 
 
 1 
 
 19 
 
 39 9 
 
 1 60 
 
 10 
 
 11 
 
 224 9 
 
 9 68 7 
 
 Standard 
 after 1st 
 Aug.1834 
 D. c. m. 
 17 6 8 
 32 71 4 
 17 30 5 
 
 6 56 
 63 
 
 4 87 5 
 1 70 6 
 
 9 69 4 
 
160 
 
 COINS AND CURRENCY. 
 
 TABLE OF GOLD COINS— Continued. 
 
 Names of Coins. 
 
 Louis, coined bef. 1786 
 Double Louis, coined 
 
 since 1786 - - 
 Louis, do. - - - 
 Double Napoleon, or 40 
 
 francs - - - 
 Napoleon, or 20 francs 
 
 COLOMBIA, S. AMERICA. 
 
 Doubloon - - - 
 
 MEXICO. 
 
 Doubloons, shares in 
 proportion - - 
 
 PORTUGAL. 
 
 Dobraon - - - 
 Dobra - - - - 
 Johannes - - - 
 Moidore - - - 
 Piece of 16 testoons, or 
 
 1600 rees - - 
 Old Crusado, 400 rees 
 New Crusado, 480 rees 
 Milree, coined 1776 - 
 
 SPAIN. 
 
 Quadruple pistole, or 
 
 doubloon,1772, shares 
 
 in proportion - - 
 Doubloon, 1801 - - 
 Pistole, 1802 - - - 
 Coronilla, gold dollar, 
 
 or Vintem, 1801 - 
 
 UNITED STATES. 
 
 Eaffle, coined before Ju- 
 ly 31, 1834 - - - 
 
 Eagle, coined since Ju- 
 ly 31, 1834 - - - 
 
 The above tables are calculated and arranged to the actual 
 use at the United States mint, for estimating the value of gold 
 
 Note. — In England, where accounts are kept in pounds, shil 
 lings, pence, and farthings, the pound is always estimated at 20 
 
 Weight. 
 5 5^ 
 
 Grains 
 of pure 
 
 gold. 
 112 4 
 
 Standard 
 
 before 1st 
 
 Aug.1834 
 
 4 54 1 
 
 9 20 
 
 4 22 
 
 212 
 106 
 
 6 
 3 
 
 8 59 
 4 29 5 
 
 8 7 
 4 ^ 
 
 179 
 89 
 
 7 
 
 7 23 2 
 3 62 4 
 
 17 8^ 
 
 360 
 
 5 
 
 14 56 
 
 17 8i 
 
 360 
 
 5 
 
 14 56 
 
 34 12 
 
 18 6 
 
 18 
 
 6 22 
 
 759 
 401 
 
 152 
 
 5 
 
 2 
 
 30 QQ 6 
 
 16 22 2 
 
 16 00 
 
 6 14 9 
 
 2 6 
 15 15 
 
 19| 
 
 49 
 13 
 14 
 18 
 
 3 
 6 
 8 
 1 
 
 1 99 2 
 54 9 
 59 8 
 73 2 
 
 17 81 
 17 9 
 4 81 
 
 437 
 
 360 
 
 80 
 
 2 
 5 
 
 1 
 
 15 03 
 
 14 56 
 
 3 64 
 
 1 3 
 
 22 
 
 8 
 
 92 1 
 
 11 6 
 
 247 
 
 5 
 
 10 00 
 
 10 18 
 
 232 
 
 
 
 
CO/NS AND CURRENCY 
 
 IM 
 
 Btiillings, the shilling at 12 pence, and a penny at 4 farthings. 
 The pound sterling is not a coin, although there are bank notes 
 of that denomination. The pound sterling in this country is 
 now estimated at D4. 87. 5, which wbuld make the shilling=24-J 
 cents nearly. 
 
 REMARKS. 
 
 The value of the Eagle coined prior to the 31st of July, 1834, 
 is DlO. 66. 8-f-, which contains 247.5 grains of pure gold-|-22.5 
 grains alloy =270 grains, standard weight. By computation, it 
 will be found that every 23.2 grains of pure gold are equal in 
 value to Dl, and 25.8 grains of United States standard are also 
 equal to Dl. 
 
 To find the value of any Gold Coin whatsoever, 
 
 RULE. 
 
 Reduce its weight of pure gold to grains (troy), and divide by 
 23.2 for the value in dollars, cents, and mills. 
 
 1. Required the value of lib. of pure gold. 
 
 Thus, llb.Xl2x20x24-f-23.2 = D248.27.5ff Ans. 
 
 2. Required the value of lib. standard gold of the U. States. 
 
 Thus, llb.Xl2x20x24^25.8=D223.25.5|f Ans, 
 A Table of Foreign Coins, &c. — Showing their value in 
 the United States as established by acts of Congress — not 
 given in the preceding tables : — 
 
 I Silver. 
 
 [ English or French Crown - , - 
 ] dollar of Spain, Sweden, and Denmark 
 Dollar of Mexico and S. A. S. 
 Five-franc Piece - - - - 
 
 Franc 
 
 Pistareen 
 
 Pound of Ireland - - - - 
 Pagoda of India - - - - 
 Tale of China - - - - 
 
 Millrea of Portugal - - - - 
 Ruble of Russia - - - - 
 Rupee of Bengal - - - - 
 Guilder of the United Netherlands - 
 Mark-Banco of Hamburg 
 Livr6 of Francois . - - - 
 Gold Ducat of Russia - - - 
 
 D 
 
 c. 
 
 m. 
 
 1 
 
 10 
 
 
 
 1 
 
 00 
 
 
 
 1 
 
 00 
 
 
 
 
 
 93 
 
 6 
 
 
 
 18 
 
 8 
 
 
 
 20 
 
 
 
 4 
 
 10 
 
 
 
 1 
 
 94 
 
 
 
 1 
 
 48 
 
 
 
 1 
 
 28 
 
 
 
 
 
 66 
 
 
 
 
 
 55 
 
 5 
 
 
 
 39 
 
 
 
 
 
 35 
 
 5 
 
 
 
 18 
 
 5 
 
 2 
 
 00 
 
 
 
 3. In 50 pounds sterling, how many dollars ? 
 
 Ans. 4.87.5x50=243.750 
 14* 
 
162 COINS AND CURRENCY. 
 
 4 In 75 tales of China, how much United States money ? 
 
 Ans. Dlll.OO 
 
 5. In 25 sovereigns, how many dollars? Ans, D121.87.5 
 
 6. How much United States money in 500 francs ? 
 
 Ans. D94.00 
 
 DISCOUNT. 
 
 Discount is an allowance made for the payment of any sum 
 of money before it becomes due, and is the difference between 
 that sum, due some time hence, and its present worth. 
 
 The present worth of any sum or debt not yet due, is so much 
 as would, if put to interest, produce a sum equal to the discount ; 
 or the interest of the present worth and interest of the discount 
 for the given time and rate per cent, shall be equal to the inter- 
 est of the given sum, or debt, for the same time and rate per 
 cent. Thus the present worth and discount of D 100 for 1 year 
 at 6 per cent. ; the present worth is D94.34, which subtract from 
 DlOO, gives D5. 66 discount, and the interest of D94.34 is D5.66, 
 so that neither party is wronged, provided they are both agreed ; 
 but in no case should the interest be allowed on the given sum 
 as discount, because the interest would be D6.00, which is 34 
 cents too much. Again, ijf I give my note for 1)106, payable 
 one year hence, th-e present value of the note will be less than 
 D106 by the interest on its present value for one year ; that is, 
 its present value will be DlOO. The amount named in a note 
 is called the face of the note; thus D106 is the face of the 
 above note ; the discount is the difference between the face of 
 a note and its present value — that is, D6 is the discount on the 
 above note. 
 
 EXTRACT. 
 
 That an allowance ought to be made for paying money before 
 it becomes due, which is supposed to bear no interest till after 
 it is due, is very just and reasonable ; but if I pay it before it is 
 due, I give that benefit to another. Therefore, we have only to 
 inquire what discount ought to be allowed. Many suppose that 
 by not paying till it becomes due they may employ it at interest ; 
 therefore, by paying it before due, they shall lose that interest. 
 
DISCOUNT. 163 
 
 and for that reason all such interest ought to be discounted ; but 
 the supposition is false, for they can not be said to lose that in- 
 terest till the time arrives when the debt becomes due ; in other 
 words, they can not lose what they do not possess, whereas we 
 are to consider what would be lost at present by paying the debt 
 before it becomes due ; this can in point of equity be no other 
 .,han such a sum, which being put out at interest till the debt 
 shiil become due, would amount to the interest of the debt for 
 the same time, as before observed. 
 
 The truth of the rule for working is evident from the nature 
 of simple interest ; for since the debt, or face of the note, may 
 be considered as the amount of some principal (called here pres- 
 ent worth), at a certain rate per cent., and for the given time, that 
 amount must be in the same proportion, either to its principal or 
 interest, as the amount of any other sum, at the same rate, and 
 for the same time, to its principal or interest. In what is termed 
 hank discount, the interest is taken for, or called discount. The 
 word is misapplied ; the banks loan money and receive interest, 
 not discount. The difference will stand D5. 66 to D6.76i on 
 DlOO for 1 year, at 6 per cent. (See Bank Interest.) 
 
 I. Find the interest of DlOO for the time, and at the rate per 
 cent, mentioned in the question ; then add this interest to the 
 DlOO, and this is the first term. 2d. The given sum in the 
 question is the 2d term, and DlOO is the 3d term. 3d. Multi- 
 ply the 2d and 3d terms together, and divide by the first, and 
 the quotient will be the present worth. 4th. When the discount 
 is required, subtract the present worth thus found, from the 
 given sum, and the remainder is the discount. 
 
 QUESTIONS. 
 
 1 . Required the present worth and discount of D500, due I 
 year hence, at 6 per cent. ? 
 Thus: DlOO 
 
 6 rate per ct., then 106 : 500 :: 100 
 
 100 
 
 interest — 6.00 Ans, 
 
 100 106)500(471.98.8 pr't worth, 
 
 D106 
 
 500 given sum. 
 Or, 106)500(471.69.8 present worth. 
 
 D28.30.2 discount. Ans. 
 
164 
 
 DISCOUNT. 
 
 RULE II. 
 
 Assume any principal at pleasure, and find the amount for the 
 time and rate per cent. Then, as the amount found is to the 
 amount or debt given, so is the principal assumed to the required 
 principal, or present worth. 
 
 2. Suppose a debt of D810, were to be paid three months 
 hence, allowing 5 per cent., what is its worth in cash 1 
 
 Assume D80 
 5 
 
 then, 81 
 
 3 months i)4.00 
 
 1 
 + 80 
 
 810 
 80 
 
 80 
 
 81)64800(800 Ans. 
 
 D81 
 
 Or, 
 
 DlOO then 101.25 : 810 :: 100 
 5 100 
 
 101.25)81000(800 An^. 
 
 i)5.00 
 
 1.25 
 100 
 
 D101.25 
 
 3. Purchased goods to the amount of D750, on a credit of 
 9 months, at 5 per cent., but wishing to make immediate pay- 
 ment, it is required to know what sum in ready money would 
 discharge the debt. Ans. D722.89.1 + 
 
 4. What is the discount on D280, due in 6 months, at 7 per 
 cent. ? Ans. D9.46.9-f 
 
 5. What is the present worth of D840, due in 1 year, 6 
 months, at 6 per cent. ? Ans. D770.64.2. 
 
 6. What is the present worth of D954, due in 3 years, at 4.5 
 per cent. ? Ans. D840.52.8. 
 
 7. If you purchased goods to the amount of D796.49, on a 
 credit of 4 months, at 3 per cent., what sum in ready money 
 would discharge the debt ? Ans. D788.60.3. 
 
 8. What difference is therebetween the interest of D1200 at 
 5 per cent, per annum, for 12 years, and the discount of the 
 same sum at the same rate and time ? Ans. D270. 
 
DISCOUNT. 165 
 
 9. What sum in ready money must be received for a bill of 
 900D. due 73 days hence, discount at 6 per cent, per annum ? 
 
 Ans. D889.32.8. 
 
 10. What sum will discharge a debt of D615.75, due in 7 
 months at 4J per cent, per annum ? Ans. D600. 
 
 11. B. has D2000 due him from A., of which D500 are pay- 
 able in 6 months, D800 in 1 year, and the remainder at the ex- 
 piration of 3 years at 6 per cent. ; but if A. should make present 
 payment, how much would he have to pay ? 
 
 Ans. D1833.37.4+ 
 Note. — When payments are to be made at different times, find 
 the present value of the several sums separately, and their sum 
 will be the present value of the note or debt. 
 
 12. What sum will discharge a debt of DlOOO, whereof 
 D600 is payable in one year, and the remainder in 6 months, at 
 4 per cent. ? Ans. D969.08. 
 
 13. What is the present worth of D600 due in 5 years at 7 
 per cent. ? Ans. D444. 44.44- 
 
 14. What is the difference between the interest and discount 
 of DlOOO for 1 year at 6 per cent. ? Ans. D3.39.6-f- 
 
 15. What is the present value of a note for D3500, on which 
 D300 are to be paid in 6 months, D900 in 1 year, D1300 in 18 
 months, and the remainder at the expiration of 2 years, the rate 
 of interest being at 6 per cent, per annum ? 
 
 Ans. D3225.83-f 
 
 16. What is the present value of D2880, one half payable in 
 3 months, one third in 6 months, and the remainder in 9 months, 
 at 6 per cent, per annum ? Ans. D2810.08-|- 
 
 17. Bought goods to the amount of D1854 for which I gavo 
 my note for 8 months at 6 per cent. ; but being desirous of tak- 
 ing it up at the expiration of 2 months, what sum does justice 
 require me to pay ? Ans. D1800 
 
 18. What is the present worth of D515 due 6 months hence ? 
 1)500— due 1 year hence? D485.84.9 — due 15 months hence ? 
 D479.06.9— due 20 months hence? D468.18.1— due 4 years 
 hence ? D415.32.2~at 6 per cent. ? Ans. D2348.42.1-f 
 
 19. What is the present worth of D1350, due 5 years 10 
 months hence, at 6 per cent. ? Ans. DlOOO. 
 
 20. What is the discount of D460, due 2 years 6 month-s 
 hence at 6 per cent. 1 Ans. D60. 
 
 REVIEW. 
 
 What is discount ? What is present worth ? How will you 
 first proceed to find the present worth ? After having found the 
 
166 EQUATION. 
 
 interest of DlOO at the given time and rate per cent, what is 
 next to be done ? After having added the interest so found to 
 DlOO, by what rule do you work to find the discount ? Repeat 
 the rule. Is it correct to take the interest for the discount 1 
 What is the difference ? What is the face of a note ? When 
 payments are to be made at different times, how do find the 
 present value ? 
 
 21. What is the difference between the discount of D227.66 
 for 2 years 3 months and 20 days, and the interest of the same 
 sum for the same time, at 6 per cent. ? Ans. D3.81.2.^yfc. 
 
 22. What is the present worth of D1500 for 90 days at 7 per ^ 
 cent, per annum ? ^n^. D1474.54+ 9 
 
 RULE III. • /■ [\}i^ 
 
 1. Divide the given sum or debt by the amount of Dl for the 
 given time. 2. The quotient will represent the present worth, 
 which taken from the debt will le-ave the discount. Thus to find 
 the present worth of D 133.20 payable 1 year and 10 months 
 hence, and discount. The amount of Dl for 1 year 10 months, 
 is Dl.ll ; then D133.20-f-Dl.ll=Dl20 present worth; and 
 D133.20— D120=D13.20 discount. 
 
 EQUATION. 
 
 Equation is a rule used to find the mean or equated time of 
 several payments which are due at different times, so that no 
 loss shall be sustained by either party. 
 
 RULE I. 
 
 Multiply each payment by its time, and divide the sum of the 
 several products by the whole debt, and the quotient will be the 
 equated time for the payment of the whole. 
 
 QUESTIONS. 
 
 1. A. owes B. D380 to be paid as follows, namely : DlOO in 
 6 months ; D 120 in 7 months ; and D160 in 10 months ; what 
 is the equated time for the payment of the whole debt ? 
 Thus: 100x6 = 600 
 120x7 = 840 
 160x10=1600 
 
 380 )3040(8 mo. Ans, 
 
 2. I have D200 due me, of which DlOO is to be paid in 6 
 months, and DlOO in 12 months, but it is agreed to make one 
 TMiyment ; required the time. Ans, 9 months. 
 
EQUATION. 117 
 
 3. A gentleman has due him D600 to be paid as follows, D400 
 in 10 months, and D200 in 6 months ; what is the equated time 1 
 
 Ans, 8| mpnths. 
 
 4. A. has due him a certain sum of money to be paid, ^ in 2 
 months, 5 in 3 months, and the remainder in 6 months ; what is 
 the equated time "? Ans. 4^ months. 
 
 5. What is the equated time for paying D2000, of which D500 
 is due in 3 months, D360 in 5 months, D600 in 8 months, and 
 the balance in 9 months ? Ans. 6^ months (nearly). 
 
 6. What is the equated time for paying D380 : whereof DlOO 
 is payable in 180 days, D120 in 210 days, D160 in 300 days 1 
 
 Thus, 100x180=18000 : 120x210^:25200 : 160x300 = 
 48000=91200 dividend -^3 80 =240 days, An5. 
 
 See by rule 1, at what time the first man mentioned, ought to 
 pay in his whole money ; then, as his money is to his time, so is 
 the other's money to his time ; inversely, which, when found, 
 must be added to, or subtracted from, the time at which the sec- 
 ond ought to have paid in his money, as the case may require, 
 and the sum, or remainder, will be the true time of the second 
 payment. 
 
 7. P. is indebted to Q. D150, to be paid, D50 at 4 months, 
 and DlOO at 8 months ; Q. owes P. D250, to be paid at 10 
 i^onths ; it is agreed between them that P. make present pay- 
 ment of his whole debt, and that Q. shall pay his so much soon- 
 er, as to balance that favor ; I demand the time at which Q. 
 must pay the D250. 
 
 Thus, 50x4=200; 100x8 = 800 = 1000^ 150=6f months, 
 P.'s equated time ; then, D150 : 6f months :: D250 : 4 months ; 
 then 10 months— 4=6 months, time of Q.'s payment. Ans. 
 
 Note. — Notwithstanding the general use of the rules of equa- 
 tion, they are manifestly incorrect. It is argued by those who 
 defend the principle, that what is gained by keeping some of the 
 debts after they are due, is lost by paying others before they are 
 due ; but this can not be the case, for though by keeping a debt 
 after it is due, there is gained the interest of it for that time, yet 
 by paying a debt before it is due, the payer does not lose the in- 
 terest for that time, but the discount only, which is less than the 
 interest ; consequently, the rule can not be strictly correct ; al- 
 though in most questions which occur in business the error is so 
 trifling that it will always be made use of as the most eligible 
 method. The same system of erroneous calculation in regard 
 
166 BARTER. 
 
 to interest and discount is exhibited in equation, by a misappli- 
 cation of terms, and a false principle. 
 
 Note 2. — Suppose a sum of money be due immediately, and 
 another at the expiration of a certain given time forward, and it 
 is proposed to find a time, so that neither party shall sustain loss ; 
 now, it is plain that the equated time must fall between the two 
 payments, and that what is gained by keeping the first debt after 
 '' is due, should be equal to what is lost by paying the second debt 
 before it is due ; but the gain arising from the keeping of a sum 
 of money after it is due, is evidently equal to the interest of the 
 debt for that time ; and tbe loss which is sustained by the paying ' 
 of a sum of money before it is due, is evidently the discount of 
 the debt for that time ; therefore it is obvious that the debtor must 
 retain the sum immediately due, or the first payment, till its inter- 
 est shall be equal to the discount of the second sum for the time 
 it is paid before due; because in that case the gain and loss will 
 be equal, and consequently neither party can be a loser. 
 
 REVIEW. 
 
 For what purpose is equation used ? What is the first rule ? 
 The second 1 What can you say in relation to the correctness 
 of the principle of calculating equations of time of payment ? 
 
 BARTER. 
 
 This is a rule by which merchants and others exchange one 
 commodity for another, and by which they know how to make 
 the exchange, or proportion the quantities without loss to either 
 party ; the rules in Barter, Profit and Loss, and Partnership, 
 are only applications of the rules of proportion which have been 
 explained and are easily understood. 
 
 RULE I. 
 
 Find the value of what you propose to exchange at the price 
 at which you wish to exchange it, by any rule most convenient. 
 Then, as the price of one of the articles which you receive, is to 
 the whole quantity, so is the whole value of what you give in 
 exchange, to the answer required. 
 
 QUESTIONS. 
 
 1. A. has 200 pounds of tea at D1.25 per pound, which he 
 will let B. have in exchange for sugar at lie. per pound ; how 
 much su^ar must A. receive for his tea ? 
 
 I 
 
D1.25 per lb. tea. 
 200 lb. tea. 
 
 c.ll)250.00(2272lb. II0Z.+ A71S. 
 Or, as lie. : 2001b. :: Dl.25 : 22721b. lloz. 
 
 2. A. and B. barter, A. sold B. 4.5 yards of broadcloth at 
 D5.25 per yard, 150 lb. of pork at 8.5 per lb., and one barrel of 
 mackerel at D10.25 ; B. let A. have 2.5 cords of M^ood at D3.75 
 per cord, 10 bushels of wheat at Dl.25 per bushel, and 7 bush 
 els of rye at 80c. per bushel ; how does the account stand ? 
 
 Ans. B. must pay to A. D19.15. 
 
 3. D. sold a horse for D75, one half was paid in cash down 
 and the remainder in oats at 37.5c. per bushel ; how many 
 bushels of oats did he receive ? Ans. 100. 
 
 4. How much corn at 87.5c. per bushel, is equal to 464 bush- 
 els of wheat at Dl.25 per bushel ? 
 
 Ans. 662 bushels, 3 pecks, 3 quarts. 
 
 5. How much tallow at 8.5c. will you receive in barter, for 
 12cwt. 2qrs. 81b. of sugar at 15c. per lb. ? Ans. 24841b, lloz. 
 
 6. C. has 2 pieces of broadcloth ; 1 piece contains 30 yards 
 at D4.50 per yard ; the other 25 yards at D3.75 per yard, which 
 he will barter with P. for 20001b. of pork at 10.5c. per lb. and 
 the balance in flax at 12^c. per lb. ; how much flax will C. re- 
 ceive ? Ans. 1501b. 
 
 7. W. has 3 hogsheads of wine at Dl.12 per gallon, for tho 
 value of one half he will take wheat, at Dl.lO per bushel, and 
 for the remainder he will take 250 yards of domestic cloth ; how 
 much wheat will he receive, and how much will the cloth cost 
 him per yard ? 
 
 Ans. 96 bushels, 6 quarts, 1 pint, wheat ; and cost of cloth, 
 c.42.34- per yard. 
 
 8. How much butter at 14c. per pound must be given for 85 
 pounds of ham at 16c. per pound, and 8 pounds of tea at Dl.25 
 per pound ? Ans. 1681b. 9oz. 
 
 9. Two farmers bartered ; A. had 120 bushels of wheat at 
 Dl.50 per bushel, for which B. gave him 100 bushels of barley 
 worth 65c. per bushel, and the balance in oats at 40c. per bush- 
 el ; what quantity of oats did A. receive ? Ans. 287.5 bush. 
 
 10. A merchant sold 14.5 yards of broadcloth at D4.25 per 
 yard, and received in payment 95lb. of wool at 31.25c. per lb., 
 15doz. of eggs at 16c. per dozen, 4.25 bushels of wheat at Dl .12 
 per bushel, a quarter of beef weighing 1641b. at 6-^c. per lb., 471b. 
 of tallow at 12^c, per lb., 8 bushels of rye at 93^0. per bushel ; 
 
 15 
 
 h 
 
170 BARTER. 
 
 and 78lc. in cash ; how much more must he (A ) receive loi 
 his cloth ? Ans. 0. 
 
 11. A. and B. bartered ; A. had 8.25 cwt. of sugar at 12 cts. 
 per lb., for which B. gave him 18 cwt. of flour ; how much was 
 the flour per lb. ? Ans. 5.5c. 
 
 12. B. has 5 pieces of muslin, each piece containing 95 yards 
 at 23c. per yard, for which C. is to give him 32 sheep at D2.50 
 each, and the remainder in rye flour at Dl.50 per cwt. ; how 
 many cwt. of flour must C. receive ? Ans. 19cwt. 2qr. 
 
 13. A. purchased a flock of sheep of B. consisting of 75 in 
 number, at D1.75 each ; he paid B. D87.50 in cash, 1.5 tons of 
 hay at D7.50 per ton, and 9 bushels of corn at 62ic. per bush- 
 el ; required the balance due. A7is. 1)26.87.5. 
 
 Having the ready money and bartering price of one article 
 given and the ready money price and quantity of another article 
 given, to find the bartering price and the quantity of the other. 
 
 As the ready money price of the one : is to the ready money 
 price of the other : : so is the bartering price of the first : to 
 the bartering price of the second. 
 
 14. A. and B. barter ; A. has 145 gallons of wine at Dl.20 
 per gallon ready money, but in barter will have Dl.35 per gal- 
 lon ; B. has Hnen at 58c. per yard, ready money; how must B. 
 sell his linen per yard, in proportion to A.'s bartering price, and 
 ho.v many yards are equal in value to A.'s win© ? 
 
 Ans. 65.25c. linen per yard, and 300 yards. 
 
 15. A. has wheat worth Dl.13 per bushel ready money, but 
 in barter he will have D1.33 per bushel ; B. has butter worth 20c. 
 per lb. ready money; how must B. rate the butter to be equal 
 to the wheat ? A7is. 23.5c. 
 
 16. P. has indigo worth Dl.OO per pound ready money, but in 
 barter he will have Dl.13 per pound ; C. has sugar worth 10c. 
 per pound ready money ; how must C. rate his sugar that it may 
 be equal Avith the indigo? Ans. 11.3c. 
 
 17. A. has coffee worth 20c. per lb. in ready money, but in 
 barter he will have 25c. per lb. ; 13 has broadcloth worth D2 per 
 yard ready money ; at what price ought the broadcloth to be ra- 
 ted in barter 1 Ans. D2.50. 
 
 18. A. has 320 doz. of candles at Dl.20 per dozen, for which 
 B. agrees to pay him D160 in cash, and the rest in cotton at 20c. 
 per lb I how much cotton must B. give A. 1 
 
 Ans. n20lb. 
 
BARTER. ^ l1 
 
 19. A person purchased 120 tons of iron at D 10.25 per toil 
 and paid as follows, namely : In cash D500 ; 27 bushels of sali 
 at 65c. per bushel ; 1501b. of leather at 25c. per lb. ; 8 bushelsX 
 of clover-seed at D4.75 per bushel ; 15 bushels of flax-seed at' 
 D 1.10 per bushel ; 75 gallons of currant wine at Dl.25 per 
 gallon ; 250 bushels of oats at 37.5 per bushel ; and he is to 
 pay the balance in honey at 75c. per gallon ; required the bal- 
 ance due, and the quantity of honey to be paid. 
 
 Arts. Due D432.95 ; honey 577 gallons, 1 quart. 
 
 20. H. has 75 sheep at Dl.75 each, for which VV. is to give 
 him D109 in cash, and the rest in corn at 62.5c. per bushel; 
 how much corn must W. give H. ? 
 
 Ans. 35 bushels, 2 pecks, 3 quarts. 
 
 21. P. and Q. barter; P. has Irish linen at 60c. per yard, but 
 in barter he will have 64c. per yard ; Q. delivers him broad- 
 cloth at D6 per yard, worth only 1)5.50 per yard ; which has the 
 advantage in the bargain, and how much linen does P. give Q. 
 for 148 yards of broadcloth ? 
 
 As 60 : 64 :: 5.50 : 5.86|- ; therefore, Q. by selling at D6 
 has the advantage ; then 6 : 148 yards :: 64 : 1578| yds. Ans. 
 
 22. A. has linen cloth at 30c. per yard ready money, in bar- 
 ter 36c., B. has 3610 yards of riband at 22c. per yard ready 
 money, and would have of A. D200 in ready money, and the 
 rest in linen cloth ; what rate does the riband bear in barter per 
 yard, and how much linen must A. give B. ? The rate of rib- 
 and is 26.4c. per yard, and B. must receive 1980|- yards of lin- 
 en and D200 in cash. 
 
 23. A. has 150 yards of linen, at 25c. per yard, which he 
 wishes to exchange with B. for muslin at 50c. per yard ; how 
 much muslin must A. receive 1 Ans. 75 yards. 
 
 24. A. had 200 barrels of flour at D 10.50 per barrel, for which 
 B. gave him D1090 in money, and the rest in molasses at 20c. 
 per gallon ; how many hogsheads of molasses did he receive ? 
 
 Ans. 80 hogsheads, }0 gallons. 
 
 25. A merchant, in bartering with a farmer for wood at Do per 
 cord, rated his molasses at D25 per hogshead, which was worth 
 no more than D20 ; what price ought the farmer to have a cord 
 for his wood to be equal to the merchant's bartering price ? 
 
 Ans. D6.25. 
 
 26. A farmer sold a grocer 20 bushels of rye at 75c. per bush- 
 1 ; 2001b. of cheese at 10c. per pound ; in exchange for which 
 
 he received 20 gallons of molasses, at 22c. per gallon, and the 
 balance in money ; how much money did he receive ? 
 
 Ans. D30.60 
 
172 PROFIT AND LOSS. 
 
 REVIEW. 
 
 What is barter ? What will you first do when you wish to 
 barter one commodity for another? What is the first rule? 
 After having found the value of the article you wish to exchange, 
 how will you find the answer ? Repeat the second rule, and 
 explain its operation. 
 
 PROFIT AND LOSS. 
 
 This rule is used to find the profit or loss sustained in the 
 purchase or sale of property of every description, and to ad- 
 vance or reduce the value of any article so as to gain or lose so 
 much per cent. 
 
 When property or an article is purchased at a certain price, and 
 is to be sold at any other price, either more or less, to ascertain the 
 profit or loss on the whole work by the following rule. 
 
 1. Find the whole amount of the cost. 2. Find the amount 
 it sold for. 3. Then if the sum it sold for be more than the 
 cost, subtract the sum it cost from the sum it sold for, and the 
 remainder will be the profit. 4. But if it sold for less than it 
 cost, subtract the sum it sold for from the sum it cost, and the 
 remainder will be the loss. 
 
 QUESTIONS. 
 
 1. A. purchased 501b. of cheese at 12.5c. per pound, and sold 
 it for 15c, per pound ; what is the profit on the whole ? 
 
 Thus : price per lb. 12.5 cents. 15 cents, price it sold for. 
 
 50 pounds. 50 pounds. 
 
 ^ D6.25.0 cost. D7.50 sold for. 
 
 *-6.25 
 
 Di.25 profit. 
 
• PROFIT AND LOSS. 17S 
 
 Or thus, 15c. 
 
 — 12.5c. 
 
 2.5 gain per pound. 
 50 pounds. 
 
 Dl.25 profit. 
 
 2. Purchased 12961b. of sugar at 9c. per pound, and sold it 
 at 11.5c. per pound ; what is the profit on the whole ? 
 
 Ans. D32.40. 
 
 3. When 250 bushels of wheat are purchased at Dl.25 per 
 bushel, and sold for D 1.3 1.25 per bushel; what is the profit on 
 the whole 1 
 
 Ans. D15.62.5. 
 
 4. If you purchase 180 bushels of rye at D0.93| per bushel, 
 and sell it for Dl.02 ; what is the profit on the whole ? 
 
 Ans. D15.75. 
 
 5. If 1500 pounds of liam cost 12.5 cents per pound, and is 
 sold for 18.75 ceiits per pound, what is the profit on the whole ?' 
 
 ■ Ans. D93.75. 
 
 6. If a merchant should purchase 1150 bushels of corn, at 
 87.5c. per bushel, and sell it at 85c. per bushel, what would his 
 loss be on the whole ? Ans. D28.75. 
 
 7. If you purchase 25 barrels of flour at D9 per barrel, and 
 sell it for 5c. per pound, will you gain or lose, and how much ? 
 
 Ans. D20, giin. 
 
 8. If a hogshead of sugar weighing 9c wt. 2qr. cost D120, and 
 be sold for 10c. per pound, what is gained or lost on the whole ? 
 
 Ans. D13.60, lost. 
 
 9. A merchant purchased 30 ydrds of broadcloth at 1)4.75 
 per yard, and sold it for D5.06.75 ; required the profit on the 
 whole. , Ans. D9.52.5. 
 
 10. A. paid D 130 for a pipe of wine, which he sold for Dl.02 
 per gallon ; did he gain or lose, and how much on the whole ? 
 
 Ans. ri.48, lost. 
 
 When property is purchased or soldy to know what is gavned or 
 lost per cent. 
 
 The price it cost is the first term ; the difference of what it 
 cost and what it sold for is the second term; and 100 is the 
 third term ; then work by simple proportion. 
 
 15* 
 
174 TROFIT AND LOSS-. 
 
 11. Bought 120 bushels of oats at 50c. per bushel, and sold 
 them for 55c. per bushel ; what was the gain on the whole, and 
 gain per cent. ? 
 
 Thus : 120 bushels. then 120 bushels. 
 
 50c. cost per bustiel. 55c. 
 
 D60.00 cost. D66.00 sold for. 
 
 — 60.00 
 
 D6.00 gained on the whole 
 
 Or: 120 bushels. then 55 
 
 5c. gain on 1 bushel. —50 
 
 D6.00 A71S. 50 : 5c. :: 100 li 
 
 5 :'• 
 
 60)500 ' 
 
 10 per cent. 
 
 12. Bought a yard of linen cloth for 37.5c. and sold it for 32c. , 
 what is the loss per cent. ? Ans. 14. 6c. per cent. 
 
 13. If I pay 25 cents per pound for butter, and purchase 550 
 pounds, then sell the same for 31 cents per pound, what do I 
 gain on the whole, and how much per cent. ? 
 
 Ans. D33, and 24 per cent. 
 
 14. A. paid D225 for 200 pounds of tea, and sold it for Dl.05 
 per pound ; did he gain or lose ? and how much, and what per 
 cent. 1 Ans. Lost D15, which is 6|- per cent. 
 
 15. Bought 150 pounds of pork for Dll.oO, and 1 wish to 
 gain D2.50 profit ; how much must I sell it i'or per pound 1 
 
 Ans. 9c. 3m. 
 
 16. Purchased 200 pounds of tallow for D18, and sold it for 
 D25; how much did I get per pound, and how much profit per 
 cent. ? e^/z.s'. 12.5c, per pound, and 89 per cent. 
 
 TTie prime cost and the profit or loss per cent, given, to find 
 what it sold for. 
 
 DlOO is the first term ; the prime cost is the second term ; 
 and DlOO, wnth the gain added, or loss subtracted, is the third 
 term ; and the quotient is the answer. 
 
 17. If a bushel of corn cost 50c., what must it be sold for io 
 ^in 25 per cent. ? Thus: 100 : 50 :: 125 : 62.5c. Ans, 
 
PROFIT AND LOSS. 175 
 
 18. If I purchase sheeting at 8c. per yard, at what price 
 must it be sold to gain 25 per cent. ? Ans. 10c. 
 
 19. At 25c. profit on a dollar, how much per cent. ? 
 
 Ans. 25 per cent. 
 
 20. If I purchase wine at Dl.05 per gallon, how must I sell 
 it to gain 30 per cent. ? Ans. Dl.36.5. 
 
 When the profit or loss per cent, is given, with the selling price oj 
 the article, to find the prime cost. 
 
 RULE IV. 
 
 DlOO with the gain per cent, added, or loss per cent, subtract- 
 ed, is *he first term ; DlOO is the second term ; the prime cost 
 is the thiil term. 
 
 21. If by selling wheat at D5 there is 25 per cent, gained, 
 what is the first cost ? 
 
 Thus: 100+25=125 : 100 :: 5 : D4.. Ans. 
 
 22. If in purchasing cloth at D30 there is 10 per cent, gain- 
 ed, what was the prime cost? Ans. D27.27.2. 
 
 23. If I sell a bushel of oats for 45c. and lose 10 per cent., 
 required the prime cost. Ans. 50c. 
 
 24. If I sell rye at 87.5c. and lose 12.5 per cent., how much 
 did it cost? Ans.Dl.QO. 
 
 The selling price and the gain or loss per cent, given, to find what 
 would be the profit or loss per cent, if sold at any other price. 
 
 RULE V. 
 
 The first price is the first term ; the second price is the sec- 
 ond term ; DlOO with the gain per cent, added or loss per cent, 
 subtracted, is the third term ; the quotient is the answer. 
 
 25. A person sold a yard of cloth for D2.23, and gained 10 
 per cent. ; if he had sold it for D2.75, what would have been 
 the gain per cent. ? 
 
 Thus: 2.23 : 2.75 :: 110 : D135.65— Dl00=35| per ct. Ans. 
 
 26. Bought a hogshead of molasses containing 119 gallons at 
 52c. per gallon, paid for transportation Dl.21, and by accident 
 9 gallons leaked out ; at what rate must I sell the remainder per 
 gallon to gain Dl3 on the first cost? Ans. 69.17c. 
 
 27. If I purchase 12cwt. 2qrs. of rice at D3.45 per cwt. and 
 sell it at 4c. per pound, what is my profit? Ans. D12.87.5. 
 
 28. Bought Icwt. of cotton for D34.86 and sell it for 41.5 
 per pound ; what do I gain or lose, and what per cent. ? 
 
 Ans. Gain D11.62, and 33.3 per cent. 
 
176 PROFIT AND LOSS. 
 
 29. If by selling cloth at D3.25 per yard, I lose at the rate of 
 20 per cent., what is the prime cost of said cloth per yard ? 
 
 Ans. D4.06.25 
 
 30. Bought a chest of tea weighing 4751b. for D420 and sole* 
 the same for D350 ; how much did I lose on the pound ? 
 
 Ans. 14.7c. 
 
 31. Bought cloth at Dl.25 per yard, which proving bad, 1 
 wished to sell it at a loss of 18 per cent. ; hpw much must I ask 
 per yard ? 
 
 32. Bought a cow for D30 cash, and sold her for D35 at a 
 credit of 8 months ; reckoning interest (on D35) at 6 per cent. 
 how much did I gain ? 
 
 33. A merchant buys 158 yards of calico for which he pays 
 20 cents per yard ; one half is so damaged that he is obliged to 
 sell it at a loss of 6 per cent., the remainder he sells at an advance 
 of 19 per cent. ; how much did he gain ? Ans. D2.054- 
 
 34. A tailor bought 60 yards of cloth at D4.50 per yard, and 
 38 yards at D2.50 per yard, and sold the whole quantity at 
 D4.25 per yard ; did he gain or lose, and what per cent. 1 
 
 Ans, Gained D51.50, and Dl4.ll gained per ct. (on DlOO). 
 
 35. A. purchased cloth at D2.50 per yard, but being damaged 
 he is willing to lose 17-| per cent, on it ; how must he sell it 
 per yard? \ Ans. 1)2.06.25. 
 
 36. If I buy a bale of cotton weighing 8cwt. 201b. at a cost of 
 D45.55, how must it be sold per cwt. to lose 8 per cent, by it ? 
 
 What do you understand by profit and loss ? In what cases 
 can you apply the several rules ? When an article is purchased 
 at a certain price, and is sold at any other price, how will you 
 ascertain the profit or loss ? Repeat rule 1. When property is 
 sold, to know what is gained or lost per cent., how will you pro- 
 ceed ? What is rule 2 ? What will you do when the prime cost 
 and the profit or loss per cent, are given, to find what it sold for ? 
 Repeat rule 3, and give an example in this rule. When the 
 profit or loss per cent, is given with the selling price of the arti- 
 cle, to find the prime cost, what is to be done ? What is rule 4 ? 
 When the selling price and the gain or loss per cent, are given, 
 to find what would be the profit or loss per cent, if sold at any 
 other price what is to be done ? Repeat rule 5. Now solve 
 the following question : — 
 
 37. I sold a watch for D50, and by so doing, lost D17 pei 
 cent., whereas in trading I ought to have cleared D20 per cent. ; 
 how much was it sold under its real v-alue ? Ans. D22.28.8. 
 
- PARTNERSHIP. 177 
 
 PARTNERSHIP. 
 
 Partnership, or as it is frequently termed, fellowship, is a 
 rule used by merchants, tradesmen, and others who deal or 
 transact business in company, or partnership, to ascertain the:r 
 equitable share of profit or loss when their proportion of stock 
 is unequal, or when the stock has been invested for a longer or 
 shorter time ; and also by this rule the estate of an insolvent 
 may be justly divided between his creditors, and the equitable 
 distribution of property of every description, the assessment of 
 taxes, &c., (fee. The money employed is the capital stock. 
 The gain or loss to be shared is called the dividend. The op- 
 eration of the rules is proportional. 
 
 Wheji time is not considered 
 rule I. 
 
 The amount of stock, or whole debt, is the first term ; either 
 the partners' shares or sum in trade is the second term (for 
 his share) ; the whole gain or loss is the third term ; and the 
 quotient is the amount of that partner's share of the profit or 
 loss ; and so continue the statement for each partner, respec- 
 tively. Proof: add the shares of the profit or loss of each of 
 the partners together, and if the amount agrees with the sum 
 mentioned in the question, it will be correct. 
 
 1 . Divide the whole gain or loss by the whole stock, and the 
 quotient will be the gain or loss per dollar. 
 
 2. Multiply the gain or loss per dollar by each particular 
 stock, and the product is the proportional gain or loss required. 
 
 QUESTIONS. 
 
 1. A. and B. formed a connexion in business ; A. advanced 
 D750, B. D250 ; after trading together, they find their profits 
 amount to D400 ; required a just division of their gain. 
 
 A. B. 
 
 A. 750 > 1000 : 750 :: 400 G. ( 1000 : 250 :: 400 ) 300 A. 
 
 B. 250 J 400 I 400 5 100 B. 
 
 1000 S 1000)300000(300 A. \ 1000)100000(100 B. \ 400 Pf. 
 
 2. A. and B. purchased goods to the amount of DlOOO; A. 
 paid D600, and B. D400 ; they gained DlOO ; required the 
 share of each. 
 
178 PARTNERSHIP. 
 
 Rule 2d. 600 A. ) A. 600 x 10 = 60 A. > share? 
 
 400 B. S B. 400 X 10 = 40 B. 5 Ans 
 
 gain D]00)1000(10c. per D. \ 100 proof. \ 
 
 3. A., B., C, and D., formed a partnership : A. put in D3200, 
 
 B. D2100. C. D2150, and D. D550 ; on examination they find 
 their gain to be D2500 ; what is the share of each ? 
 
 Ans, A. DIOOO; B. D656.25 ; C. D671.87.5 ; D. D171.87.5. 
 
 4. A., B., and C, freighted a ship with 108 tons of wine, of 
 which A. had 48 T., B, 36 T., and C 24 T. ; but in stress of 
 weather they were obliged to throw 45 T. overboard ; how 
 much must each sustain of the loss ? An^. A. 20, B. 15, C. 10 T. 
 
 5. Three farmers, A., B., and C, occupy a farm of 350 acre.s, 
 of which A. has 100 acres, B. 110 acres, C. 140 acres, for which 
 thev pay D750 ; what must each man pay in proportion to the 
 land he holds ? Ans. A. D214.28.5 ; B. D235.71.4 ; C. D300. 
 
 6. Four men traded with a stock of D800, by which they 
 gained D307 ; A.'s stock was DUO, B.'s D260, C.'s D300 ; 
 required D.'s stock, and what each man gained. 
 
 Ans. D's stock DlOO ; A. gained D53.72.5 ; B. D99.77.5 ; 
 
 C. D115.12.5; D. D38.37.5. 
 
 7. A., B., and C, traded in partnership ; A. put in D140, 
 B. D250, and C. put in 120 yards of cloth at cost price ; they 
 gained D230, of which C. took DlOO for his share of the gain ; 
 how did C. value his cloth per yard in common stock, and what 
 was A. and B.'s part of the gain ? 
 
 Ans. C's cloth D2.50 per yard; A. D46.66.6+ ; B. 
 D83.33.3+ ; C. share, D. 100. 
 
 8. Three merchants trading together lost goods to the amount 
 of D1920 ; suppose A.'s stock was D2880, B.'s D11520, C.'s 
 D4800 ; what share of the loss must each sustain ? 
 
 Ans. A. D288 ; B. D1152 ; C. D480. 
 
 9. A., B., C, and D., gain D200 in trade, of which as often 
 as A. has D6, B. must have DlO, C. Dl4, and D. D20 ; what 
 is the share of each ? Ans. A. D24 ; B. D40 ; C. D56 ; D. D80. 
 
 10. An insolvent estate of D633.60 is indebted to A. D3 12.75, 
 o B. D297.00, to C. D50.25, to D. D0.25, to E. D200, to F. 
 
 D 142.50, and to G. D21.25 ; what proportion will each credi- 
 tor receive ? 
 
 Ans. A. D193.51.41; B. Dl83.76.87; C. D31.09.23 ; D 
 DO. 15.41; E. D123.75 ; F. D88.17.18; G. D13.14.87 = 
 D633.59.97+ proof. 
 
». 
 
 PARTNERSHIP. 179 
 
 11. A., B., and C, put their money into joint stock ; A. put 
 in D40, B. and C. together D170 ; they gained D126, of which 
 B. took D42 ; what did A. and C. gain, and B. and C. put in 
 respectively ? 
 
 Ans. A.'s gain D24, B.'s stock D70, C.'s stock DlOO, and 
 C.'s gain D60. 
 
 12. A. and B. companied ; A. put in D45, and took f of the 
 gain ; what did B. put in 1 Ans. D30. 
 
 13. A. and B. venture equal stocks in trade, and clear D164; 
 hy agreement A. was to have 5 per cent, of the profits, because 
 he managed the concerns ; B. was to have but 2 per cent. ; what 
 was each one's gain ? how much did A receive for his trouble ? 
 
 Ans. A.'s gain was Dll7.142-f, and B.'s P46.857J, and A. 
 received D70.285^ for his trouble. 
 
 TAXING. 
 
 A tax is a sum required to be paid to the government for its 
 pport, or for other purposes, and is generally collected from 
 each individual in proportion to his property ; this, however, is 
 different in some states, where every white male citizen over the 
 age of 21 years is required to pay a certain tax, called a poll- 
 tax ; in this case each person is called a poll : hence the expres- 
 sions, *' polling votes," " attending the polls," &;c. In the assess- 
 ment of taxes, we must first make an inventory of all the prop- 
 erty, both real and personal, of the whole town or district to be 
 taxed, and also of each individual who is to be taxed ; and as 
 the number of polls are rated at so much each, the tax on all the 
 polls must be taken out from the whole tax, and the remainder 
 is to be assessed on the property. Then to find how much any 
 individual must be taxed for his property, we need only find 
 how much the remainder of the whole tax is on Dl, and multi- 
 ply his inventory by it. 
 
 14. If a town raise a tax of D1920, and all the property in 
 town be valued at D64000, what will that be on Dl, and what 
 will be A.'s tax whose property is valued at D1200 ? 
 
 Ans. DO. 03c. on a dollar; or r36.00 : for Dl920-^64000= 
 3c.xDl200=36.00 A's tax. 
 
 15. A certain town, valued a D64530, raises a tax of 
 D2259.90; there are 540 polls which are taxed D60 each; 
 what is the tax on a dollar, and what will be A.'s tax, whoso 
 real estate is valued at D1340, his personal property at D874, 
 and who pays for 2 polls ? 
 
 Thus, D540 X .60=:D324 amount of the poll-taxes ; D2259.90 
 --D324=D1 935.90, to be assessed on property ; then 64530 : 
 1935 :: 1 : .03, or 1935.90^64530 = .03, or 3c., tax Dl. 
 
 ^ 
 
180 
 
 PARTNERSHIP. 
 
 Tax on Dl is .03 
 
 2 .06 
 
 3 
 4 
 5 
 6 
 
 7 
 8 
 9 
 
 .09 
 .12 
 
 .15 
 .18 
 .21 
 .24 
 .27 
 
 TABLE. 
 Tax on DIO is .30 Tax on D 100 is D3 
 
 20 
 30 
 40 
 50 
 60 
 70 
 80 
 90 
 
 .60 
 .90 
 1.20 
 1.50 
 1.80 
 2.10 
 2.40 
 2.70 
 
 200 
 
 6 
 
 300 
 
 9 
 
 400 
 
 12 
 
 500 
 
 15 
 
 600 
 
 18 
 
 700 
 
 21 
 
 800 
 
 24 
 
 900 
 
 27 
 
 1000 
 
 30 
 
 Now, to find A.'s tax, his real estate being D1340 ; I find by 
 the table, that the tax on DlOOO is D30 
 The tax on 300 is 9 
 
 The tax on 40 is 1 .20 
 
 Tax on his real estate 
 In like manner, I find the tax on 
 his personal property to be 
 
 2 polls at .60 each are 
 
 D40.20 
 
 D26.22 
 
 1.20 
 
 Amount, D67.62 Ans, 
 
 When the stocks are considered with regard to time. 
 
 RULE III. 
 
 1. Multiply each man's stock by its time, and add the prod* 
 ucts together. 
 
 2. Then, as the sum of the whole stock, multiplied by the 
 time, is to the product of each individual share, multiplied by 
 its time, so is the whole gain or loss of each individual. 
 
 16. Three merchants traded together for 12 months ; A. put 
 in D120 for 4 months, B. put in D250 for 9 months, and C. put 
 in D300 for 12 months, they gained D175 ; what will each man 
 receive for the gain ? 
 
 A. 120 X 4mo.= 480 ) A. 6330 : 480 :: 175g. D13.27 Ans 
 B.250X 9 " =2250 \ B. 6330 : 2250 :: 175" 62.20.4 
 C. 300x12 " =3600lC. 6330 : 3600 :: 175" 99.52.6 
 
 D175,00.0 Pf, 
 
PARTNERSHIP. ~ 181 
 
 RULE IV. 
 
 1st. Multiply each man's stock by the time it was in trade, 
 for its product. 
 
 2d. Divide the gain or loss by the sum of the products for 
 their gain or loss per dollar. 
 
 3d. Multiply the gain or loss per dollar, by each product, for 
 each man's proportional gain or loss, and the product is the 
 answer. 
 
 17. A. and B. traded 12 months ; A. put in D300 for 12 
 months, B. put in D600 for 6 months, their gain is D288 ; re- 
 quired the share of each. 
 
 A. DSOOx 12=3600 > 72)288(.04c. per D, 
 
 B. 600 X 6 = 3600 
 
 7200 
 
 A. 3600 B. 3600 A. 144.00 
 
 4 4 B. 144.00 
 
 D144.00 D144.00 D288.00 Proof. 
 
 18. A. and B. formed a partnership for 18 months ; A. put 
 in D2750 for 15 months, and B. D3500 for 10 months ; at the 
 expiration of that time he took out D2000 ; at the close of their 
 business they find their gain to be D1500 ; what is each man's 
 share of the gain ? 
 
 Ans, A.'s D701.13.3+; B.'s D798.86.7+. 
 
 19. A. put in stock to the amount of D1800, B. advanced 4 
 mo. after ; what sum must he put in to receive equal profit with 
 A. at the end of the year ? A?is. D2700. 
 
 20. A. and B. formed a partnership; A. put in, the 1st of 
 January, DlOOO, but B. could not put in till the 1st of May; 
 what sum did he then put in, to have an equal share with A. at 
 the expiration of the year? Ans. D1500. 
 
 21. A., B., and C, entered into partnership for 12 months; 
 A. put in at first D873.60, and 4 months after he put in D96 
 more ; B. put in at first D979.20, and at the end of 7 months 
 he took out D206.40 ; C. put in at first D355.20, and 3 months 
 after he put in D206.40, and 5 months after that he put in 
 D240 more ; at the end of 12 months their gain is D3446.40; 
 what is each man's share of the gain ? 
 
 Ans. A. D1334.82.5, B. D1271,61.4, C. D839.96. 
 16 
 
182 
 
 PARTNERSHIP. 
 
 22. Two merchants entered into company for 18 months ; A. 
 at first put in D500, and at the end of 8 months he put in DlOO 
 more : B. at first put in D800, and at the end of 4 months he took 
 out D200 ; at the expiration of the time they find their gain to 
 be D700 ; what is each man's share of the gain ? 
 
 Ans. A. D324.07.4, B. D375.92.5+ 
 
 23. Three persons, A., B., and C, made a stock for 12 
 months ; A. put in at first D580, 3 months after he put in DlOO^ 
 more ; B. put in at first DIOOO, and after 9 months he put in 
 D200 ; C. put in at first D486, 3 months after he took out D300, 
 and 2 months after he put in D500 ; 3 months after this he took 
 out D400, and 1 month after he put in DlOOO ; at the end of 12 
 months their gain was D2108.44 ; what is each man's share "? 
 
 A. D580x3=l740 
 680x9=6020 + 
 
 7860A. 
 
 B. DIOOO X 9:^9000 1 
 
 200 + 
 
 C. D486x3 = 1458 
 -300 
 
 + 186X2X 372 
 500 
 
 686x3= 2058 
 —400 
 
 1200X3 = 3600 I +286x1= 286 
 
 j 1000 
 
 12600B. J 
 
 1286x3= 3858 
 
 7860 A. 
 
 12600 B. 
 
 8032 C. 
 
 28492 div. 
 
 ;>A.581.64.8+ 
 B.932.41.4 
 C.594.37.7 
 
 2108.44.0Pr. 
 
 8032C. 
 
 24. A gentleman left an estate at his death of D30000, to be 
 divided among his 5 children, in such a manner that their 
 shares should be to each other as their ages, which are 7, 10, 
 12, 15, 16 ; required the share of each. 
 
 Ans. D3500, 5000, 6000, 75000, 8000. 
 
 25. A. and B. entered into partnership for 16 months ; A. put 
 in D1200 at first, and 9 months after D200 more ; B. put in at 
 first D1500, at the end of 6 months he took out D500 ; their 
 gain was D772.20 ; what is the share of each ? 
 
 Ans. A.'s D401.70, B.'s D370.50. 
 Note. — As an evidence of the correctness of the rules of part- 
 nership, and that their loss or gain is in proportion to their 
 stocks in trade, let A.'s stock be 200, and B.'s DlOO, and their 
 OSS or gain D37.50, which is at the rate of 12j per cent. ; A. 
 will gain or lose D25,and B. D12. 50, because A.'s stock is just 
 
PERCENTAGE. 183 
 
 • 
 
 twice as much as B.'s, consequently his loss or gain must be 
 twice as much ; and the same principle will hold good in partner- 
 ship with or without time, that is, the gain or loss must be in 
 proportion to the stock, and time that the partnership continued. 
 If A. and B. enter into partnership for one year, and A. puts in 
 D200, and B. DlOO, but at the expiration of 6 months, A. finds 
 it convenient to withdraw his D200, and the partnership con- 
 tinues to the end of the year, and their gain is D37.50, it must 
 be divided equally. Again : If A. commence trade with a cap- 
 ital of D500, and at the expiration of 6 months he shall receive 
 B. into partnership with a capital of DlOOO, their loss or gain at 
 the end of 12 months will be equal ; if at 6 per cent. (D60) it 
 will be D30 each. 
 
 What is partnership ? When is it used ? When time is not 
 considered, what is the rule ? How will you prepare the ques- 
 tion for solution ? What can you say of the assessment of taxes ? 
 What is the first thing to be done ? How will you proceed to 
 find the amount of taxes to be paid by A., B., C, &c. ? When 
 time is considered in partnership, what is the rule ? Repeat the 
 2d and 4th rules. Method of proof. What is capital or stock ? 
 On what principles are the rules of partnership founded ? 
 
 26. E., F., and G., formed a partnership ; E. put in D400for 
 .75 of a year, F. D300 for .5 of a year, and G. D500 for .25 
 of a year, with which they gained D720 ; required the share 
 of each. 
 
 Ans. E. D375l| ; F. Dl87jf ; G. Dl56i|=D720 proof. 
 
 27. A. put in ^ for | of a year, B. f for ^ a year, and C. the 
 rest for 1 year ; their joint stock was 1, and their gain 1 ; what 
 is each man's share 1 Ans. A.'s is i| ; B.'s ^j ; C.'s ^^y=|| ~1 
 
 PERCENTAGE. 
 
 When we speak of per cent., it generally has reference to 
 interest or discount on money, and means the hundredth part of 
 the thing spoken of ; for we can say so much per cent, of a 
 bushel, yard, &c., as well as money ; when we say 10 per cent., 
 we mean DlO on the hundred, or the 10th part of 100 ; as B. 
 spends 10 per ct. of his DlOO, he would have but D90 remaining 
 
184 PERCENTAOE. 
 
 EXAMPLES. 
 
 1. What is the difference between D500 at 7.5 per cent., and 
 D500 at 8 per cent. ? 
 
 Thus: D500x7.5=D37.50; D500x8=D40.00; difference 
 2.50. Ans. 
 
 2. Two men had each D240 ; one of them spends 14 per 
 cent., and the other 18^ per cent. ; how much more did one 
 spend than the other ? 
 
 D240X 14 = D33.60 ; D240+18^ X D44.40 ; difference 
 DlO.80. Ans. 
 
 To find the rate per cent. 
 
 RULE I. 
 
 1. Bring the number to hundreds by annexing two ciphers, oi 
 removing the decimal point two places to the right. 
 
 2. Divide the numbers so formed by the sum on which the 
 percentage is estimated ; the quotient will express the per cent. 
 
 3. A merchant goes to New York with D1500 ; he first lays 
 out 20 per cent., after which he expends D660 ; what per cent, 
 was his last purchase of the money that remained after his first ? 
 Thus: D1500X20 per ct.=zD300: 1500 — 300=1200)66000(55 
 per cent. Ans. 
 
 4. If I pay D679.84 for 750 bushels of wheat, and sell the 
 same for 1)874.50 ; how much do I make per cent, on what I 
 paid, and on the sum received ? 
 
 5. If I contract a debt of D500 and make a payment of 
 D350, what per cent, of the debt do I pay ? 
 
 When the per cent, of loss or gain is given, and the amount re- 
 ceived, to find the principal cost. 
 
 6. I sell a quantity of goods for D170, by which I lose 15 
 per cent. ; what did they cost 1 
 
 Aw^.DlOO — 15 = 85)170x100(200 D. cost. 
 
 7. Sold goods to the amount of D225, and made 20 per 
 cent. ; what did they cost ? 
 
 To find the percentage on lands, or allowance for roads, Sfc. 
 
 It is customary in Pennsylvania, and probably in many other 
 states, to deduct 6 acres out of 106, for roads, &;c. ; the land 
 before the deduction is made may be termed the gross, and that 
 remaining after each deduction, the net or strict measure. 
 
GROSS, TARE, AND NET WEIGHT. 185 
 
 teauce the gross to perches, and divide by 1.06, and the quo- 
 tient will be the answer in perches, strict measure. 
 
 Multiply the net or strict measure by 1.06, and the product 
 will give the gross measure or quantity ; or work decimally. 
 
 EXAMPLES. 
 
 1 . How much net land or strict measure is there in a tract 
 :)f 901 A. 2 R. 26 po. gross? 
 
 Thus, 901 A. 2 R. 26 po. =901.6625-^ 1.06 = 850.625 A.= 
 850 A., 2 R., 20 po. Ans. 
 
 2/* How much land must I enclose to have 850 A. 2 R. 20 po., 
 net? Thus 850.625x1.06=901.6625 A.=901 A.2R.26po., 
 gross. Ans. 
 
 Note. — These two operations prove each other. 
 
 RULE IT. 
 
 ^^B Divide the content in perches by 169.6, which will give the 
 '^ret in all cases, where the given quantity is 106 A., and ratio 6. 
 
 Thus : 901 A. 2 R. 26 po. = 144266 po.-f- 169.6 = 850.625 A. 
 = 850 A. 2 R. 20 po. Ans. 
 
 Or: 850.625x169.6 = 144266 po.=901 A. 2 R. 26 po. as 
 above. 
 
 Again: 106 A. = 16960po.~169.6=100 A. ; or 16960-7-1.06 
 = 16000po. = 100 A. 
 
 Note. — The deduction to be made on every 106 A. is 9.6 
 perches ; this added to 160 po.= 169.6 po., hence the divisor. 
 
 GROSS, TARE, AND NET WEIGHT. 
 
 The following questions are usually denominated under the 
 rule or appellation of Tare and Tret, The use and application 
 of the several rules are for deducting certain allowances which 
 are made by merchants and tradesmen in selling their goods by 
 weight, for the purpose of making the proper deduction to ascer- 
 tain the net weight. In England these rules are in constant 
 use, but a better system is being introduced into this country, 
 that of taking 100 pounds as the true weight, in place of 112 lb. 
 gross ; then all that will be required will be to ascertain the 
 weight of the box, cask, bag, &c., containing the article, and 
 
 16* 
 
186 GROSS, TARE, AND NET WEIGHT, 
 
 the remainder will be the net or true weight. Tlie collections 
 at the customhouse, or United States duties, are connected with 
 gross weight. 
 
 Gross weight is the whole weight of the goods, together with 
 the weight of the cask, bag, &c., which contains them. 
 
 Tare is an allowance made to the buyer, or the deduction of 
 the weight of the cask, box, bag, (fee, containing the articles 
 sold. 
 
 Net weight is what remains after all the deductions are made 
 
 Note. — The following questions are only the application of 
 the rules of proportion and practice. 
 
 When the tare is so much on a given quantity » 
 
 RULE I. 
 
 Subtract the given tare from the given quantity, and the re 
 mainder will be the net weight. 
 
 1 . What is the net weight of a cask of sugar, weighing 7 cwt 
 1 qr. 16 lb. gross, tare 3 qrs. 18 lb. ? 
 
 7 cwt. 1 qr. 16 lb. 
 - 3 18 
 
 6 1 26 Ans. 
 
 2. What is the net weight of a cask of rice, weighing 5 cwt. 
 gross ; tare 2 qrs. 13 lb. ? Ans. 4 cwt. 1 qr. 15 lb. 
 
 3. Required the net weight of a hogshead of sugar, weighing 
 gross, 8 cwt. 1 qr. 22 lb. ; tare 3 qrs. 9 lb. ? 
 
 Ans. 7 cwt. 2 qrs. 13 lb. 
 
 4. What is the net weight of 175 cwt. 2 qrs. 20 lb. ; tare 
 6 cwt. 2 qrs. 25 lb. ? Ans. 168 cwt. 3 qrs. 23 lb. 
 
 5. What is the net weight of 4 casks of sugar, each weigh- 
 ing 5 cwt. 2 qrs. 12 lb. gross ; tare per cask, 2 qrs. 18 lb. 
 
 Ans. 19 cwt. 3 qrs. 4 lb. 
 
 When the tare is so much per cask, box, bag, <SfC. 
 
 RULE II. 
 
 Multiply the given tare per bag, box, &c., by the number of 
 bags, boxes, &c., and subtract the product from the gross weight, 
 and the remainder is the net weight. 
 
 6. A. sold 5 casks of rice, which weighed 1137 lbs. each, and 
 each cask weighed 75 lbs. ; required the net weight and value 
 of the tobacco at 14 cents per pound ? 
 
 Ans. Net weight, 5310 lb. ; value, D743.40. 
 
UNITED STATES DUTIES. 187 
 
 7. Received per ship Napoleon, from South Atneiica, 55 bags 
 of coffee, the gross weight of which is 2201bs. each, and the 
 weight of each sack is olbs. ; required the net weight and value 
 of the coffee at 16 cents per pound ? 
 
 A?is. net weight 11825lbs.; value D1892. 
 
 8. Received from Salina 72 bags of salt, the weight of each 
 bag is 2101b. gross, and each sack weighs 71bs. ; what is the 
 net weight, and what does it come to at 4 cts. per pound ? 
 
 Ans. weight 146161bs. ; value D584.64. 
 
 9. Bou^ it 1 10 hogsheads of sugar, gross 723lbs. each, weight 
 of each hogshead 62 lbs. ; what is the net weight, and what 
 will it come to at 12.5 cts. per pound ? 
 
 Ans. 727101bs.; amount D9088.75. 
 
 10. Bought 5 casks of rice, which weighed gross 18 cwt., 2 
 qrs., 12 lbs., tare per cask 45 lbs., for which I paid D5.50 per 
 cwt., and sold the same for D6.25 per cwt. ; required the net 
 weight, cost, amount it sold for, and profit ? 
 
 Ans. net weight 1859 lbs. ; cost D91.29 ; sold for D103.74 ; 
 profit Di2.45. 
 
 REVIEW. 
 
 What is gross weight ? What is tare ? What is net weight ? 
 Repeat rule Ist. Repeat rule 2d. 
 
 UNITED STATES DUTIES. 
 
 In all civilized countries, where merchandise or goods are 
 imported, importers are required to pay a certain amount of their 
 value, at a certain rate per pound, hundred, yard, or gallon ; this 
 is called duty, which is established and collected by the laws 
 of the country where the goods are landed ; for this purpose, 
 customhouses are erected in the seaport towns to collect the 
 custom or duties, tonnage of vessels, port duties, &c., which to- 
 gether are called the revenue. An ad-valorem duty is such a 
 per cent, on the actual cost of the goods in the country from 
 which they are imported ; thus an ad-valorem duty of 20 per 
 cent, on tea from China, is a duty of 20 per cent, on the cost of 
 the tea in China ; the duties are computed on the net weight. 
 
 EXAMPLES. 
 
 1. What is the duty on 1400 lbs. of coffee at 2 J- cts. per lb. "? 
 
 Ans. 1400x2^=D3.50 
 
188 SINGLE POSITION. 
 
 2. If the duty on molasses is 5 cents on a gallon when im 
 ported in an American vessel, and 10 per cent, more in a foreigii 
 vessel ; what is the duty on 3950 gallons in both vessels ? 
 
 Ans. D197.50 and D217.25 
 
 3. What is the duty on goods which cost in Calcutta 
 D2780.50, at 12^ per cent, ad valorem? Ans. D347.561 
 
 POSITION. 
 
 Position is a rule founded on the principles of proportion, 
 and by working with one or more supposed numbers, as real 
 numbers, we can discover the true number or answer to the 
 question. It is of two kinds, termed single and double. Single 
 Position is when only one supposed number is necessary for 
 the operation. Such questions as are usually given in arith- 
 metic for solution, by working with supposed (improperly called 
 false) numbers, are equations in algebra, by which they are 
 more conveniently and readily solved by those who are ac- 
 quainted with that science. (For an illustration of the rule, see 
 Double Position.) 
 
 SINGLE POSITION. 
 
 RULE. 
 
 Suppose a number, and work with it as though it was the 
 true number, according to the nature of the question ; then, as 
 the result of that operation : is to the given number : : so is the 
 number supposed : to the number required. Proof, add the 
 several results together. 
 
 QUESTION. 
 
 1 . A. owes a certain sum of money : |, g^, }, is D500 ; what 
 is the amount of the debt ? 
 
 Suppose debt D1200 ^=300 ^ then, 650 : 500 :: 1200 
 -1 = 150 > 1200 
 
 ^=200 ) D 
 
 650)600000(923.07.7 A 
 
 Sum of shares, D650 
 
DOUBLE POSITION. 18^ 
 
 2. A certain sum of money is given to 4 persons : to A. J, 
 B. \, C. -g-, and D. draws the rest, which is D28 ; required th^ 
 sum. (This question can be solved by Double Position.) 
 Suppose 60, iz:z20 A. ] Then, 15 : 28 :: 60 : 112. Ans. 
 45 i=15 B. I Proof, 112, J=37.33^ A. 
 — i=10C. ;> 112,1=28.00 B. 
 
 ffference, 15 — 
 Sum of parts, 45 
 
 112, ^ = 18.66f C. 
 D. took, 28.00 D. 
 
 D112.00 
 f 3. What sum is that of which ^, -J, and ^, make D94 ? 
 
 Ans. D120. 
 
 ^4. A person said that ^, ^, J, and i of the money in his pos- 
 
 kssion would make D57 ; how much had he ? Ans. D60. 
 
 I5. The ages of A., B., and C, amount to 133 years ; B. is J 
 
 cler than C, and A. is J older than B. ; required their separate 
 
 fires. Ans. A. 56 yrs., B. 42 yrs., C. 35 yrs. 
 
 6. What sum of money at 6 per cent, per annum, simple in« 
 
 terest, will amount to D500 in 10 years 1 Ans. D312.50. 
 
 !7B.'s age is 1.5 of A.'s, C.'s twice as much as B.'s, and their 
 
 united ages make ] 32 years ; required the age of each 1 
 
 Ans. A. 24, B. 36, C. 72 years. 
 
 8. WKM is the age of a person who says that if -f^ of the 
 years he has lived be multiplied by 7, and f of them be added 
 to the product, the sum would be 292 1 Ans. 60 years. 
 
 9. A bankrupt is indebted to A. D284.60, to B. D794.18, to 
 C. D651.44, and to D. D491.25, and his estate is worth 
 D1460.5; if the whole were divided, how much would each 
 creditor receive l^Z Yf^- i " ^C^ S- b @* 1 (^ ir^ l) ) ^ 
 
 DOUBLE POSITION 
 
 Teaches to solve such questions as require two supposed numbers 
 in the operation, 
 
 RULE I. 
 
 Suppose two numbers, and work with each agreeably to the 
 tenor of the question, observing the errors of the results ; multi- 
 ply the errors of each operation into the supposed number of the 
 othw ; then, if the errors be alike (that is, both too much, or both 
 
190 DOUBLE POSITION. 
 
 too small), take their difference for a divisor, and the difference 
 of the products for a dividend ; but if not alike, then take their 
 sum for a divisor, and the sum of the products for a dividend. 
 
 QUESTIONS. 
 
 1. A., B., and C, agree to pay a debt of D500, of which A. is 
 to pay a certain sum ; B. is to pay DlO more than A., and C. is to 
 pay as much as both A. and B. ; how much must each one pay ? 
 Suppose 1st, A. 90 Suppose 2d, A. 100 500 500 
 
 B. 100 B. 110 —380 —420 
 
 C. 190 C. 210 
 
 120 err. 80 error 
 
 Sum 380 420 —80 
 
 40 Diff. error. 
 Then, 120 error, xlOO A=. 12000 120 A. 
 
 80 error, X 90 A.= 7200 130B.}Ans. 
 
 250 C 
 
 :l 
 
 Difference of error, 40)4800(120 ^w^. 
 
 500 proof. 
 Note. — This rule is founded on the supposition that the first 
 error is to the second, as the difference between the true and 
 supposed number is to the difference between the true and sec- 
 ond supposed number. When that is not the case, the exact 
 answer to the question can not be found by this rule. 
 
 KULE II. 
 
 1. ''Take any*two convenient numbers and proceed with each 
 according to the conditions^of the question. 2. Place the result 
 or errors against their positions or supposed numbers, thus "^ ^ X ^ ^ 
 and if the error be too great, mark it with + ; if too small, with 
 — . 3. Multiply them crosswise ; that is, the first position by 
 the last error, and the last position by the first error. 4. If the 
 errors be alike, that is, both too small or both too great, divide 
 the difference of the products by the difference of the errors, 
 and the quotient will be the answer. 5. If the errors be unlike, 
 that is, one too small and the other too great, divide the sum of 
 the products by the sum of the errors, and the quotient will be 
 the ariswer. 
 
 Note, — When the errors are the same in quantity, and unlike 
 in quality, half the sum of the supposition is the number sought 
 Observe the following examples and explanation. 
 
DOUBLE POSITION. 
 
 19] 
 
 500 
 420 
 
 2 er. 80 
 
 EXAMPLE 2. (1st ques.) 
 
 Supposition 1. A. 90 Sup. 2. A. 100 given num. 500 
 B. 100 B. 110 am. of sup. 380 
 
 IK C. 190 C.210 
 
 V^m 1 er'r 120 
 
 H[ Am't of sup.— 380 Am. of sup. 420 
 
 l^lfer. 120") Istsup. 90 120 Ister. f then, 12000 
 
 2^" ^n2dsup.l00^-802der.^ "J^ 
 
 Dif.er.40j 120 x 90 x (^ d.er.40)4800(120D.A.pM 
 
 12000 7200 130 B. " 
 
 250 C. »' 
 
 500D. proof. 
 
 EXAMPLE III. 
 
 then the errors are unlike^ that is, one plus and the other minus ^ 
 
 1st, A. 100 2d, A. 140- 
 
 B. 110 B. 150 
 
 C. 210 C. 290 y 
 
 I minus 
 
 420 580 J 
 
 80 100x80= 8000 
 80 ■^140x80 = 11200"^ 
 
 500 
 •420 
 
 580 
 —500 
 
 80-1 er. SO + plus 
 (*see remarks.) 
 120 A. 
 130 B. 
 250 C. 
 
 160 
 
 160)19200(120 A. 
 
 500 proof, 
 
 2. A. is 20 years of age ; B.'s age is A.'s, and half of C.'s; 
 and C's is equal to them both ; their several ages are required. 
 
 Ans. A. 20, B. 60, C. 80. 
 
 3. Three persons pay the sum of DlOO ; B. paid DlO more 
 than A., and C. as much as A. and B. ; how much did each pay ? 
 
 Ans. A. 20, B. 30, C. 50. 
 
 4. What sum is that, which being increased by its \, its \, 
 and 18 more, will be doubled? Ans. 72. 
 
 5. A., B., and C, receive D324, but not agreeing in the divis- 
 ion of ir, each took as many as he could : A got a certain num- 
 ber ; B. as many as A. and Dl5 more; C. got a fifth part of 
 the sums added together ; how much did each get ? 
 
 Ans. A. D127.50, B. D142.50, C. D54. 
 
 6. Divide DlOO, so that^B. may have twice as much as A., 
 wanting D8 ; and C. three' times as much, wanting D15 ; what 
 is the share of each ? *Ans. A. D20.50, B. D33, C. D46.50. 
 
192 DOUBLE POSITfON. 
 
 7. A certain sum of money is given to four persons : to A 
 •J, B. i, and C. |, and D. draws the rest, which is D28 ; re 
 quired the sum. (See question 2, Single Position.) 
 
 Sup. 1, 
 
 H 22.50 i, 24.75 2.25 difference of error. 
 
 1 
 
 J).] 28 D"., 28 99x5.5 =544.5 
 
 
 First operation. 
 
 90 Sup. 2, 
 
 99 5.5 
 
 — 
 
 — 3.25 
 
 30 i, 
 
 33 
 
 22.50 I, 
 
 24.75 2.25 i 
 
 15 h 
 
 16.50 
 
 90x3.25=292.5— 
 
 95.50 102.25 
 
 —90.00 —99.00 2.25)252.0(112 Ans, 
 
 5.5 error. 3.25- error. 
 
 Second operation. 
 Sup. 1,90 Sup. 2, 60 13 error. 
 
 3' 
 
 30 
 
 20 
 
 T' 
 
 22.5 
 
 15 
 
 "B» 
 
 15 
 
 10 
 
 \ 
 
 28 
 
 28 
 
 
 95.5 
 
 73 
 
 
 90 
 
 —60 
 
 5.5 error. 
 
 7.5 difference of errors 
 
 90x13 =1170 
 60X5.5= 330— 
 
 7.5)840(112 ^n^'. 
 
 5.5 error. 13 error. 
 Ih the above case the suppositions are varied to illustrate the 
 rule, but in both statements the errors are alike, that is, both too 
 small, or minus. In the following example the errors are un- 
 like, that is, one is minus and the other pkis. 
 
 
 
 Third operation. 
 
 Sup. 1, 
 
 99 Sup 
 
 .2, 120 
 
 120 102.25 3.25 
 
 
 — 
 
 
 118 99 2 
 
 h 
 
 33 
 
 h ^0 
 
 
 i^ 
 
 24.75 
 
 h 30 
 
 2 er. 3.25 er. 5.25 sum. er. 
 
 h 
 
 16.50 
 
 h 20 
 
 
 D., 
 
 28 
 
 D., 28 
 
 99x2 =198 
 130x3.25=390^ 
 
 
 
 
 102.25 118 
 
 ' 5.25)588(112 Ans. 
 
BOUBLE POSITION. 193 
 
 REMARKS. 
 
 In the above examples, I find the errors to be alike --that is. 
 both too small; because the sum in either case is less than 
 D500 the given number ; therefore, after proceeding according 
 to rule, till I find the difference of the errors, and then make the 
 statement X for obtaining the answers, you will observe that the 
 statement or terms are muhiplied crosswise ; then take the dif 
 ference of the products and divide it by the difference of th 
 errors, which is 40, and yoti get the first or A.'s answer ; from 
 this you can easily get the others. Had the errors been unlike, 
 that is, one too great and the other too small, you would divide 
 the sum of the products by the sum of the errors, and the quo- 
 tient would be the answer. It is immaterial what numbers are 
 used for the suppositions, provided you observe the principles 
 of the rule and the examples above, for the result will always 
 be the same. As has been before observed, this rule has its 
 origin in proportion ; the reason for changing the errors or sup- 
 positions to obtain a different result or answer, is accounted for 
 on the same principles that the 1st and ^d terms in Proportion 
 are varied, or changed, to gain the required result ; in that rule 
 we change the first and second terms ; if the answer is to be 
 less, then the greater of the two terms is the divisor j if the 
 answer is to be more, then the less of the two terms is the 
 divisor, &c. 
 
 Algebraical deinonstration, — Let A and B be two numbers, 
 produced from a and h by similar operations ; it is required to 
 find the number from which N is produced by a like operation. 
 Put a:=number required, and let N — A=r, and N — Br=5. 
 Then, according to the supposition on which the rule is found- 
 ed, r : s :: x—a : x—b; whence, by multiplying means and 
 extremes, rx—rb=sx—sa; and by transposition, rx~sx=rb — 
 sa ; and by division, rb~as 
 
 x=-r-. — — r=the number sought ; and if r and 
 r — s 
 s be both negative, the theorem is the same ; and if r or ,y be 
 
 rb-^-sa 
 
 negative, x will be equal to —■ which is the rule. 
 
 r -\- s 
 
 ' REVIEW. 
 
 What is Position ? How many kinds are there ? What is 
 Single Position ? What is the rule ? What is Double Position T 
 What is first to be done when you commence an operation iu 
 
 17 
 
194 INVOLUTION' OR THE RAISING OF POWERS. 
 
 Single Position ? After having ascertained the result of the 
 operation, how will you proceed? How will you first begin an 
 operation in Double Position ? After having obtained the first 
 error, how will you proceed? When you have obtained the 
 second error, what is to be done ? What will yon do after you 
 have multiplied the second supposition by the first error, and 
 the first supposition by the second error? When you have 
 ascertained whether the errors are both of the same kind, how 
 do you proceed ? If they are not of the same kind, how will 
 you proceed ? What is the rule fir Double Position ? 
 
 8. A laborer was hired for 60 days upon this condition : that 
 for every day he wrought, he should receive 75 cents, and for 
 every day he was idle, he should forfeit 37i cents ; at the ex- 
 piration of the time he received Dl8 ; how many days did he 
 work, and how many 'W.% ha idle ? 
 
 ^ Ans. worked 36 days, was idle 24 days. 
 
 9 Two persons, A.^apd B., have the same income ; A. saves 
 } of his yearly ; but B?V^b^ spending D150 per annum more than 
 A., at the end of 8 years *finds himself D400 in debt:; what is 
 their income, and what does each spend per annum ? 
 
 Ans, income D400 ; A spends D300 • B. D450. 
 
 INVOLUTION, OR THE RAISING OF POWERS, 
 Teaches the method of finding the powers of numbers, 
 
 A POWER is the product arising from multiplying any given 
 number into itself continually a certain number of times ; thus, 
 2x2=4, is the second power or square of 2; 2x2x2=: 8, the 
 third power or cube of 2 ; 2x2x2x2 = 16, the fourth power 
 of 2, &LC. The mimber whicl;^ denotes a power is called its in- 
 dex. If two or more powers are multiplied together, their prod- 
 uct is that power whose index is the sum of the exponents of 
 the factor ; thus 2x2=4, square of 2 ; 4x4=16, fourth power 
 of 2; and 16x16=256, eighth power of 2, &c. The power 
 often denoted by a figure placed at the right and a little abovt) 
 -he number, which figure is called the index or exponent of that 
 power (thus, 2^, 3^), and is always one more than the number 
 of multiplications to produce the power, or is equal to the num- 
 
INYOLUTION, OK TWE RAISING OF POWERS. 195 
 
 ber of times the given number is used as a factor in producinig 
 the power. In producing the square of 2, there is only one mul- 
 tiplication, or two factors ; in producing the cube, there are 
 two multiplications or three factors, 3^ x 3x3=27, &c. This 
 subject will be more fully illustrated in progression. 
 
 Multiply the given number or first power continually by itself 
 till the number of multiplications be 1 less than the index of the 
 power to be found, and the last product will be the power re- 
 quired. Fractions are multiplied by taking the products of their 
 numerators, and of their denominators; they will be involved 
 by raising each of their terms to the power required, and if a 
 mixed number be proposed, either reduce it to an improper frac- 
 tion, or reduce the vulgar fraction to a decimal, and proceed by 
 the rule. 
 
 EXAMPLES. 
 
 1 . Required the third power or cube of 35. 
 
 Ans. 35 X 35 X 35=42875. 
 
 2. What is the 5th power of 7 ? Ans. 16807. 
 
 3. What is the 5th power of 9 ? Ans. 59049=9^. 
 
 4. What is the 5th power of f ? Ans, -^yts' 
 
 5. What is the 4th power of .045 ? 
 
 Ans. .000004100625. 
 
 Here we see that in raising a fraction to a higher power, we de^ 
 crease its value. 
 
 6. What is the third power of .263 ? Ans. .018191447. 
 
 7. What is the eighth power of i ? Ans. -g-^j^^. 
 
 8. What is the square of 60 ? Ans. 3600. 
 
 9. What is the square of i ? Ans. i. 
 
 10. What is the square of .01 1 Ans. .0001. 
 
 11. What is the cube of 2i ? Ans. ll||. 
 
 12. What is the ninth power of 747 ? 
 
 13. What is the seventh power of 298.75 ? 
 
 REVIEW. 
 
 What is a power ? How do you raise a number to any pow- 
 er ? What is the rule ? 
 
196 . SQUARE ROOT. 
 
 EVOLUTION, OR THE EXTRACTION OF ROOTS. 
 
 Evolution, or the extraction of roots, properly belongs to 
 mathematics, and without a knowledge of that science, it will 
 require strict attention and close application to arrive at any 
 degree of perfection in the use and principles of those rules. 
 The most correct and convenient method of extracting the roots 
 of the several powers, particularly those of the higher order, is 
 by logarithmic tables, as far preferable to any rules that can be 
 given in common arithmetic. 
 
 The root of a number, or power, is such a number as, being 
 multiplied into itself a certain number of times, will produce that 
 number or power, and is denominated the square, cube, biquad- 
 rate, &:c., or 2d, 3d, and 4th root, accordingly as it is, when 
 raised to the 2d, 3d, and 4th power, equal to that power. Thus 
 4 is the square root of 16, because 4x4 = 16 : and 4 is the cube 
 root of 64, because 4x4x4=64 : and 4 is the fourth root or 
 biquadrate of 256, because 4 x4 x4 X4=:256, &c. The roots 
 are proportional, but their proportion is different from simple or 
 compound proportion; the raising of powers increase in a 
 uniform ratio, but this will not always — indeed but seldom — 
 occur in the extraction of roots. Although there is no number 
 of which we can not find any power exactly, yet there are 
 many numbers of which precise or exact roots can never be 
 determined ; but by the use of decimals we can approximate 
 toward the root to any assigned degree of accuracy ; those 
 roots are called surds ; and those which are perfectly accurate, 
 rational roots ; surd roots sometimes have tbeir origin in circu- 
 lating decimals, or vulgar fractions. As few numbers are com- 
 plete powers, surds must very often occur in arithmetical opera- 
 tions, but the result can be obtained nearly by continuing the 
 extraction of the root. 
 
 SQUARE ROOT. 
 
 RULE I. 
 
 ] . Separate the given number into periods of two figures 
 each, beginning at th(3 right hand or place of units. 
 
liqUAilE ROOT. 107 
 
 2. Begin at the left hand, and find the quotient root in thai 
 . jriod, and place it on the right of the given sum in the quo- 
 wont, and its square under said period, which subtract from the 
 number above. 
 
 3. Then bring down the next period of 2 figures, and place 
 it on the right of the remainder, as in division, and this forms 
 a new dividend. 
 
 4. Now double this figure or root in the quotient, and place 
 it on the left of the new dividend for a divisor. 
 
 5. Then consider how often the divisor is contained in the 
 dividend, omitting the last figure, and place the result on the 
 right of the root in the quotient, and then place this figure on 
 the right of the number produced by doubling for a divisor, and 
 multiply as in division, until the periods are all brought down. 
 
 For Decimals. — When decimals occur in the given number, 
 it must be pointed both ways from the decimal point, and the 
 root must consist of as many figures, of whole numbers and 
 decimals respectively, as there are periods of integers or deci- 
 mals in the given number. When a decimal alone is given, 
 annex one cipher, if necessary, so that the number of decimal 
 places shall be equal ; and the number of decimal places in the 
 root will be equal to the number of periods in the given decimal 
 
 For Vulgar Fractions. — 1. Reduce mixed numbers to im 
 proper fractions, and compound fractions to simple ones, and 
 then reduce the fraction to its lowest terms. 
 
 2. Extract the square root of the numerator and denominator 
 separately, if they have exact roots ; but if they have not, reduce 
 the fraction to a decimal, and then extract the root, as above, &c. 
 
 Proof: square the root and add in the remainder. 
 
 EXAMPLES. 
 
 Illustration. — A square number can not have more places of 
 figures than double the places of the root, and at least but one 
 less. A square is a figure of four equal sides, each pair meet- 
 ing perpendicularly, or a figure whose length and breadth are 
 equal. As the area, or number of equal feet, inches, &c., in a 
 square, is equal to the products of the two sides, which are 
 equal, the second power is called the square. Let the follow^ 
 ing figure represent a board one foot square, and one inch in 
 thickness, which being sawn or cut into square or solid inches, 
 will make 144 inches, or 144 blocks one inch square ; and the 
 square root of 144 is 12 ; because 12 x 12 = 144, which in this 
 case will be inches = one side of the square. 
 
 17* 
 
198 
 
 SQUARE ROOT. 
 
 Thus, 12 inches. . operation. 
 
 - raiSfflllBHBBHHHB Z 
 
 X mnmmmmmmmmjam ^ 
 
 \ ^BBBHHHHBHB ii 
 
 \ IBBriBBIBHBHHHB ^ 
 
 .BBBBBBBBBBBH "^ 
 
 \ BillBBJBiiaBBBBil g* 
 
 ^ HiJEBBeiBHBBBB o 
 
 aflflBBBBBBBBB ^ 
 
 I^LimSSSlBBBBflBB % 
 
 Figure 1. 
 1. Required the square root of 154682 ? 
 
 144 
 1 
 
 22)44(12=1 foot, or length 
 of one side of the square =: 12 
 inches. 12x12 = 144 inches 
 1 square fool. Proof 
 
 Thus, 154682(393 root. 
 3x3= 9 
 
 3932 
 393 
 
 Explanation. — 
 First seek the 
 
 3X2 = 69)646 
 621 
 
 1179 
 3537 
 
 square of the 
 first period (15), 
 3X3 = 9, which 
 
 
 1179 
 
 is 9, it can not 
 
 39x2=783)2582 
 2349 
 
 
 be 4, because 
 4x4 = 16 which 
 
 154449 
 
 
 233 rem. 
 
 is more than 15 ; 
 
 233 rem. 
 
 
 place the 3 in 
 the quotient and 
 
 154682 proof. 
 
 the 9 under the first period (15), which subtract and you have 6 ; 
 then bring down the next period (46), which place to the right 
 of the remainder (6) ; now double the quotient figure, 3x2 = 6, 
 and place it in the divisor, one place to the left ; now consider 
 how many times 6 in 64, which for trial call 9 ; write 9 in the 
 quotient, and to the right of 6 in the divisor, which makes the 
 divisor 69 ; then multiply 69 by 9, and place the result under 
 the dividend ; subtract as before, and bring down the next pe- 
 riod (82); now double the two quotient figures, 39 + 2 = 78, 
 which place in the divisor ; consider how many times 78 is 
 contained in 258, suppose 3 times ; place 3 in the quotient and 
 in the divisor, as before ; then multiply the divisor by the last 
 quotient figure, and write the result as above directed, and the re- 
 mainder is 233 ; which by annexing ciphers may be continued. 
 
 2. What is the square root of 74770609 ? Arts. 8647 
 
 3. What is the square root of 54990.25 ? 234.5, 
 
 4. What is the square root of .3271.4007? 57.19-h 
 
 5. What is the square root of 14876.2357? 121.968175. 
 
 6. What is the square root of 96385103 ? 9817 + 
 
SQUARE ROOT. 199 
 
 7. What is the square root of 4.372594 ? Ans. 2.091 + 
 
 8. What is the square root of .00103041 ? Ans, .0321. 
 
 9. What is the square root of f|§i? Ans. |. 
 
 10. What is the square root of IJf 1 A7is. .89802+ 
 
 1 1 . What is the square root of -} ? A7is. -|. 
 ^2. What is the square root of 6^ ? Ans. 2^. 
 
 13. What is the difference between ^81 and Sl^? 
 
 Ans. 6552. 
 
 14. What is the square root of |-|-^? Ans. |. 
 
 15. What is the square root of J-^-? Ans. .95744- 
 
 16. What is the square root of 912^^5-? Ans. 30-}. 
 
 17. What is the square root of 2-1? Ans. 1.5. 
 
 18. What is the square root of 9980.01 ? Ans. 99.9. 
 
 19. What is the square root of 2 ? Ans. 1.414213. 
 
 20. What is the square root of ^J{|^ ? Ans. ^^^ = .09756 + 
 
 21. What is the square root of .0000316969? Ans. .00563. 
 
 22. What is the square root of 964.5192360241 ? 
 
 Ans. 31.05671. 
 
 23. What is the square root of m| ? Ans. yV- 
 
 24. What is the square root of ^^-|^ ? 
 
 25. What is the square root of 6.9169 ? Ans. 2.63. 
 
 26. What is the square root of 106929 ? Ans. 327. 
 
 27. What is the square root of -if ? Ans. ^. 
 
 28. What is the square root of ^l Ans. .7745. 
 
 29. What is the square root of 20A ? Ans. 4-^. 
 
 APPLICATION. 
 
 To find a mean proportional between two numbers. 
 
 RULE. 
 
 Multiply the given numbers together, and extract the square 
 root of the product, which will be the mean proportional sought. 
 
 30. What is the mean proportional between 24 and 96 ? 
 
 Ans. ^96X24=48. 
 
 To find the side of a square equal in area to any given superficies 
 whatever. 
 
 RULE. 
 
 Find the area, and the square root is the side of the square 
 sought. 
 
 31. If the area of a circle be 184.125, what is the side of a 
 square equal in area thereto ? Ans. y^l84. 125 = 13.569+ 
 
200 
 
 SQUARE ROOT. 
 
 32. A general has an army of 5625 men ; how many must h« 
 place in ranlc and file to form them into a square ? 
 
 Ans. ^5625 — 75. 
 
 33. Suppose that Napoleon, at the battle of Marengo, com- 
 manded an army of 256036 men ; how many did he place in 
 rank and file to form them into a solid square ? Ans. 506. 
 
 Having the area of a circle to find the diameter. 
 
 Multiply the square root of the area by 1.12837, and the 
 product will be the diameter. Or multiply the area by 1 .2732 
 and take the square root of the product. 
 
 34. Required the diameter of a circle whose area is 82 feet 
 81 inches. Ans. 10 feet 3.13 inches. 
 
 35. Admit a leaden pipe | of an inch in diameter will fill a 
 cistern in 3 hours ; I demand the diameter of another pipe which 
 will fill the same cistern in 1 hour. 
 
 Thus f=.75 and .75 X .75^.5625 ; then 3 hours : 5625 :: 1 
 hour: 1.6875 and ^1.6875=1.3 inches nearly, ^n^. (inversely.) 
 
 36. If a circular pipe of 1.5 inches diameter fill a cistern in 
 5 hours, in what time would it be filled by one 3.5 inches di- 
 ameter? Ans. 55 minutes, 4.8 seconds. 
 
 37. If 784 trees be planted in a square orchard, how many 
 must be in a row ? Ans. 28. 
 
 The square of the longest side or hypotenuse of a right-angled 
 triangle, is equal to the sum of the squares of the other two sides ; 
 consequently, the difference of the square of the longest, and either 
 of the others, is the square of the remaining side. 
 
 38. The wall of a fort is 17 feet high, which is surrounded 
 by a ditch 20 feet in breadth ; required the length of a ladder to 
 reach from the outside of the ditch to the top of the wall. 
 
 ^n^. 17xl7=i289 : 20x20=400+ 289 = ^689=26.2-1- 
 
 ^^. 
 
 ditch 20 feet. 
 
SQUARE ROOT. 
 
 201 
 
 39. Two ships leave tlie same port, one sails due east 40 
 miles' and the other due north 50 miles ; required the distance 
 irom each other. 
 
 40 m. East. 
 
 40. There is a church 48 feet in height from the ground to 
 the eaves or roof, and the street is 64 feet in width ; what length 
 of cord would it require to reach from the opposite side of the 
 street to the eaves of the house ? Ans. 80 feet. 
 
 41. In a circle, whose area or superficial content is 4096 
 feet, I demand what will be the length of one side of the square . 
 containing the same number of feet. Ans. 64 feet. 
 
 42. A certain square garden measures 4 rods on each side ; 
 what will be the length of one side of a garden containing 4 
 times as many square rods ? Ans. 8 rods. 
 
 43. If one side of a square piece of land measure 5 rods, 
 what will be the side of one measuring 4 times as large ? 16 
 times as large ? 36 times as large ? Ans. 10, 20, 30. 
 
 44. In a load of 120 cherry-boards of one inch in thickness, 
 27 inches in width, and 11.5 feet in length, how many square 
 feet, and what is the value of the boards at D22 per M. ? 
 
 Ans. 3105 feet; value D68.31. 
 
 45. In a tract of land of 640 acres, required the length of one 
 side ; and suppose the tract to contain 16 times as much, what 
 would be the length of one side ? and suppose each rod valued 
 at D75, what would be the amount ? 
 
 Ans. 1st, 320 poles, length of one side ; 2d, 1280 po., or 16 
 times as much ; 3d, D 122880000 value. 
 
 46. One side of a square field is 400 rods in length ; required 
 the number of acres in the field. 
 
 Note. — For examples in square and cub. measure, see appendix. 
 
 REVIEW. 
 
 What is evolution ? What is the square root of a number ? 
 What is the cube root ? When you wish to extract the square 
 root of a number, what must first be done ? After you have 
 counted otT the given number into periods, how will you pro- 
 
202 
 
 SQUARE ROOT. 
 
 ceed ? Repeat the rule. When the given number is a whole 
 number, how many figures will there be in the root? When 
 you can not get the exact root, what can be done ? How can 
 you extract the root of a decimal fraction 1 What will you do 
 with a whole number and decimal 1 What is the rule ? How 
 will 5^ou extract the square root of a vulgar fraction ? Repeat 
 the rule ? Give an example on the slate. What is the differ- 
 ence between a square and the square root ? 
 
 47. Required the square root of 9876.4792 iy*2* ? (This is 
 given for a trial question.) 
 
 CUBE ROOT. 
 
 The cube of a number is the product of that number multiplied 
 into its square. To extract the cube root, is to find a number 
 which being multiplied into itself, and then into that product, 
 will produce the given number. 
 
 ILLUSTRATION. 
 
 The solid called a cube, has its length, breadth, and thick- 
 ness, all equal. As the number of solid feet, inches, &c., in a 
 cube, are found by multiplying the length, height, and breadth, 
 together — that is, by multiplying one side into itself twice, the 
 third power of a number is called the cube of that number ; thus 
 a cubical or solid foot has 6 equal sides of 1 foot square, con- 
 sequently it follows that a cubical foot contains 1728 solid or 
 cubical inches, because 12 X 12=144 X 12=1728 inches : that 
 is, a cubical foot or block of wood may be sawn into 1728 
 blocks, each of which will be a solid inch, which may be rep- 
 resented in the following manner : — 
 
 2 3 4 5 6 
 
 144 144 
 
 144X12=^ 
 
 1 solid foot. 
 
 
 u:jji:i:::: 'iJiiiiJlUS sHiJlKf^ rI!I:ku!I :::~::n:i:' i::::;::;::; 
 
 12 6 equal sides 
 
 144 
 
 144 
 
CUBK ROOT. 203 
 
 Figure 2 is a solid or cubical foot with 6 equal sides ; sup- 
 pose this block be sawn into 12 pieces at the distance of 1 inch, 
 it would make 12 pieces, as represented above, each 1 foot 
 square, and 1 inch in thickness, as in figure 1 (S. R.) ; now, 
 each of those 12 square pieces may be divided into 144 solid 
 blocks: the 12 blocks divided would be 12x12 = 144 inches 
 for each of the 12 pieces, as represented by tbe 12 squares, 
 which is equal to 1728 inches in a cubical or solid foot, and 12 
 is the cube root of 1728. 
 
 Thus, the root 12 is equal to 12 square feet of ] inch in thick- 
 ness (as 1, 2, 3, &c.), or 12 inches in length, breadth, and 
 thickness, which is equal to 1 solid foot. 
 
 Thus, 1x300=300 : 2x30=60 : 2x2=4 
 60 
 4+ 1728(12 root. 
 
 364 divisor, 
 
 364) 728 
 
 728 
 
 In the example above, I first point off the three figures from 
 the right, and place the period over the 7 ; this leaves 1, or unity, 
 for the first operation ; consequently the quotient figure must in 
 this case be 1, which I put in the quotient, and under 1 in the 
 dividend ; then bring down the next period (728) ; now seek for 
 the divisor ; first multiply 1 by 300 = 300 for a trial divisor, then 
 suppose 300 in 728, say 2, which put in the quotient; then 
 2 X 30=60 ; then 2x2 = 4 the square of 2 ; then add these sev- 
 eral products together and you have the true divisor (364) which 
 multiply by the last quotient figure (2) and the work is finished. 
 
 1 . Point the given number into periods of three figures each, 
 beginning at the right or units' place. 
 
 2. Find the greatest cube in the left-hand period, and sub- 
 tract it therefrom, and set do\vn the root in the quotient, and 
 then to this remainder bring down and annex the next period 
 for a dividend. 
 
 3. Square the quotient by multiplying it into itself, and mul- 
 tiply that product by 300 for a trial divisor ; see how often it i? 
 contained in the dividend, and set the result in the quotient. 
 
 4. Multiply the figure last put in the quotient by the other 
 figures in the quotient, and that product by 30. 
 
204 CUBE ROOT. 
 
 5. Then square this last figure in the quotient, and add it to 
 ihe product just mentioned, for the second part of the divisor; 
 all these products added, form the true divisor. 
 
 6. Now multiply this true divisor by the last figure put in the 
 quotient ; subtract the product from the dividend as in division 
 bring down the next period of three figures ; proceed as before. 
 
 For Decimals, — Annex ciphers to the decimal, if necessary, 
 so that it shall consist of 3, 6, 9, &c., places ; then put the first 
 point over the place of thousandths, the second over the place 
 of millionths, and so on over every third place to the right ; and 
 then extract the root as in whole numbers. Observe, 1. There 
 will be as many places in the root as there are periods in the 
 given number. 2. The same rule applies when the given num- 
 ber is composed of a whole number and a decimal. 3. If there 
 be a remainder in a whole number, after all the periods have 
 been brought down, you can annex periods of ciphers, consid- 
 ering them as decimals. 
 
 For Vulgar Fractions. — 1. Reduce compound fractions to 
 simple ones, mixed numbers to improper fractions, and then re- 
 duce the fraction to its lowest terms. 2. Then extract the cube 
 root of the numerator and denominator separately, if they have 
 exact roots ; but if either of them have not an exact root, reduce 
 the fraction to a decimal, and extract the root as above. 
 
 Proof. — Cube the root and add in the remainder. 
 
 EXAMPLES. 
 
 1. What is the cube root of 5382674 ? . . 
 
 5382674(175.2 root, 
 i \^Ans 
 
 
 1X1X300=300 
 1X7X 30=210 + 
 7x7 =49 
 
 17X17X300 = 
 
 17x 5X 30 = 
 
 5x 5 = 
 
 86700 ) 
 2550+ \ 
 25 ) = 
 
 =559)4382 
 3913 
 
 :89275)469674 
 446375 
 
 1752x300 = 9187500 
 175x2x30=10500 + 
 2X2 = 4 
 
 od Divisor, 9198004, 
 
 9198004)23299000 
 18396008 
 
 4902992 rem. 
 
 RULE II. 
 
 1. Point off the given number into periods of three figures 
 each, if whole numbers commencing at the right haiid ; but if 
 decimals, at the left. 
 
CUBE ROOT. 
 
 2Qd 
 
 2. Find the greatest cube in the left hand period, and place 
 ihe root to the right of the given number, and subtract the cube 
 of the root from the left hand period, and to the remainder bring 
 down the next period for a dividend. 
 
 3. Square the root, and multiply it by three for a defective 
 divisor. 
 
 4. Reserve mentally the units and tens of the dividend, and 
 try how often the defective divisor is contained in the remainder; 
 place this result to the root, and the square of it to the right of 
 the defective divisor, but if the square is less than ten, supply 
 the ten's place by a cipher. 
 
 5. Complete the divisor by adding thereto the product of the 
 last figure of the root, and the remaining figure or figures of the 
 root, and that again by 30 ; then divide and subtract as in long 
 division. 
 
 6. The defective divisors after the first may be easily found 
 by adding to the last divisor the number that completed it, with 
 twice the square of the last figure of the root. 
 
 7. The divisor is then completed according to the 4th and 
 5th paragraphs above. 
 
 2. V3796416 . . 
 
 Greatest cube of 3 is 1. 
 Square of 1 multiplied by 
 the square of 5 added, is 
 The 5 paragraph produces 
 
 3 and 
 
 325 
 
 = 150 
 
 3796416( 
 
 1 
 
 2796 
 2375 
 
 156 
 
 Complete divisor 
 The number that com- 
 
 =475 
 
 421416 
 
 
 pleted it. 
 
 150 
 
 421416 
 
 
 Twice the square of the 
 last figure of the root. 
 
 50 
 
 
 
 
 
 Square of 68, defective 
 divisor. 
 
 
 
 
 67536 
 
 
 
 Paragraph 5 produces, 
 
 2700 
 
 
 
 Complete divisor, 
 
 70236 
 
 
 3. What is the cube root 
 
 of 99252.847 
 
 ? 
 
 
 4. What is the cube root 
 
 of 259694072 
 
 ? 
 
 Ans. 638. 
 
 5. What is the cube root of 34328125 ? 
 
 
 325. 
 
 6. What is the cube root 
 
 of 12.977875 
 
 ? 
 
 2.35. 
 
 7. What is the cube root 
 
 of 171.46776406? 
 
 
 8 What is the cube root of .5 ? 
 
 5.555 + 
 
 17+87 rem 
 
 
 18 
 
 
 
206 CUBE ROOT. 
 
 9. What is the cube root of 32461759 ? Ans. 319, 
 
 10. What is the cube root of 27054036008 ? 3002 
 
 11. What is the cube root of .751089429? .0909. 
 
 12. What is the cube root of 3.408862625 1 1.505 
 
 13. What is the cube root of |ff ? ^, \, 
 
 14. What is the cube root of 31 J/g ? 3f "' 
 
 15. What is the cube root of f ? .822-f 
 
 16. What is the cube root of |fif 1 ' ^^ 
 
 17. What is the cube root of -j^q ? 13 + 
 
 18. What is the cube root of 84.604519? 4.39, 
 
 19. What is the cube root of ^-|| ? f 
 
 20. What is the cube root of 16194277? 253. 
 
 21. What is the cube root of 54854153 ? 379.958793 + 
 
 22. What is the cube root of i||M 
 
 23. What is the cube root of j\^^j 1 y\. 
 
 24. What is the cube root of 7 1 1.9129. 
 
 25. What is the cube root of 15625 ? 25 
 
 26. What is the cube root of 436036424287 ? 
 
 27. What is the cube root of 99 ? 4.62+ 
 
 APPLICATION. 
 
 To find two mean proportionals between any two given numbers 
 
 RULE. 
 
 Divide the greater by the less, and extract the cube root of 
 the quotient. 
 
 2. Multiply the root so found by the least of the given num- 
 bers, and the product will be the least. 
 
 3. Multiply this product by the same root, and it will give the 
 greatest. 
 
 28. What are the two mean proportionals between 6 and 
 750? 
 
 Thus: 750-4-6=125 and 3/125 = 5, then 5x6 = 30, least,* 
 and 30x5 = 150 greatest; 30 and 150. Ans, 
 
 Note. — The solid contents of similar figures are in proportion 
 to each other, as the cubes of their similar sides or diameters. 
 
 29. If a bullet 6 inches in diameter weigh 32 lb., what will 
 a bullet of the same metal weigh, whose diameter is 3 inches ? 
 
 As6x6x6=216; 3x3x3=27; as216: 32 lb. :: 27 : 4 lb. 
 
CUBE ROOT. 207 
 
 The side of a cube being given, to find the side of that cube which 
 shall be double , triple, <^c., in quantity to the given cube. 
 
 RULE. 
 
 Cube your given side, and multiply it by the given propcrtion 
 between the given and required cube, and the cube root of the 
 product will be the side sought. 
 
 30. If a cube of silver whose side is 4 inches, be worth D50, 
 I demand the side of a cube of the like silver, whose value 
 shall be 4 times as much ? 
 
 ^725. 4X4X4 = 64, and 64x4=256 : V256=6-349 + in.ches. 
 
 31. There is an oblong cellar, the content of which is 
 1953.125 cubic feet ; what is the side of a cubical cellar that 
 shall contain just as much? Ans. 12.5 feet. 
 
 32. What is the difTerence between a solid half foot and half 
 of a solid foot 1 Ans. 3 half feet. 
 
 33. In 221184 solid inches how many cords 1 Ans. 1 cord. 
 
 34. Admitting a room to be 11 feet high, 21 feet in length, 
 and 16 feet in width, what number of cubic feet of space in it ' 
 
 ^^_ Ans. 3696 cubic feet. 
 
 |^K35. The diameter of a bushel measure being 18^ inches, and 
 
 ^flre height 8 inches, what is the side of a cubic box which shall 
 
 contain that quantity ? Ans. 12.907+inches. 
 
 36. In a cubic foot, how many cubes of 6 inches, and how 
 many of 4, of 3, of 2, of 1, are contained therein ? 
 
 Ans. 8 of 6 inches*; 27 of 4 inches ; 64 of 3 inches ; 216 
 of 2 inches ; 1728 of 1 inch. 
 
 37. Suppose a cubical cellar to contain 1728 solid feet, what 
 will one of its cubic sides measure ? 
 
 38. In a square box that will contain 1000 marbles, how many 
 will it take to reach across the bottom of the box, in a straight 
 row ? Ans. 10. 
 
 39. What is the difference between the cube root of 27 and 
 the square root of 9 ? Ans. 
 
 40. If a globe of silver 3 inches in diameter be worth D160, 
 what is the value of one 6 inches in diameter ? 
 
 Ans. 3^ : 6^ :: D160 : D1280. 
 
 41. If the diameter of the planet Jupiter is 12 times as much 
 as the earth, how many globes of the earth would it take to 
 make one as large as Jupiter ? Ans. 1728. 
 
 42. There are two small globes ; one of them is one inch in 
 diameter, and the other two inches ; how many of the smaller 
 globes will make one of tbe larger ? Ans. 8. 
 
208 ALLIGATION MEDIAL. 
 
 RULE III. 
 
 1. Find the quotient root of the left hand period, which sub- 
 tract from the same, and then bring down the next period. 2. 
 Multiply the square of the quotient figure by 300 for a divisor ; 
 then find the next figure ; square this quotient figure ; multiply 
 that square by the other quotient figure, and then by 30 ; find 
 the cube of this last quotient figure ; add both these products to 
 the product of the divisor and the quotient figure, the sum of 
 which subtract from the dividend. 3. Then bring down the 
 next period, which will complete the next dividend ; square 
 your two quotient figures, and multiply by 300 for your next 
 divisor, and so continue till the operation is completed. 
 
 Note. — The roots of the 4, 6, 8, 9, and 12 powers may be 
 obtained in the following manner : — 
 
 For the 4th root, take the square root of the square root. 
 
 For the 6th, take the square root of the cube root. 
 
 For the 8th, take the square root of the 4th root. 
 
 For the 9th, take the cube root of the cube root. 
 
 For the 12th, take the cube root of the 4th root, &c. 
 
 What is a cube ? What is the cube root 7 What will you first 
 do to extract the cube root of a whole number ? Repeat the rule ? 
 How do you extract the cube root of a decimal fraction ? When 
 there is a decimal and whole number, how will you point them 
 off? Why do you point decimals from the left or decimal point 
 toward the right ? In extracting a root, if there be a remainder, 
 what may be done ? What is the rule for decimals ? How do 
 you extract the cube root of a vulgar fraction ? What is the 
 rule ? How do you extract the cube root of a mixed number ? 
 What is the difference between a cube and the cube root ? 
 
 43. What is the cube root of 36|f 1 Ans. 3.32 + 
 
 ALLIGATION MEDIAL. 
 
 Alligation is used when the quantities and prices of several 
 things are given, to find the mean price of the mixture composed 
 of those materials ; or the method of mixing two or more sim- 
 ples of different qualities, so that the composition may be of a 
 mean or middle quality. There are two kinds, Alligation Me- 
 dial and Alligation Alternate. 
 
ALLIGATION ALTERNATE. 209 
 
 Divide the entire cost of the whole mixture by the sum of the 
 simples ; the quotient will be the price of the given mixture. 
 
 1. If 19 bushels of wheat at 75c. per bushel, 40 bushels of 
 rye at 50c. per bushel, and 12 bushels of barley at 37.5c. per 
 bushel, be mixed together, what is a bushel of the mixture 
 worth? 19x75 =14.250. 
 
 40x50 =20.00 
 12X37^= 4.50 
 
 71 71)38.75(54c. 6m. Ans. 
 
 2. If 4 ounces of silver, worth 62.5c. per ounce, be melted 
 with 8 ounces at 50c. per ounce, what is an ounce of this mix- 
 ture worth? Ans. 54c. 
 
 3. A goldsmith melted together 8 ounces of gold, 22 carats 
 fine ; 1 lb. 8 oz. of 21 carats fine ; and 10 oz. of 1 8 carats fine ; 
 what is the quality of fineness of this composition ? 
 
 Ans. 20y^^ carats fine. 
 
 4. In buying tea, I pay 90c. per lb. for 12 lbs. and Dl.20 per 
 lb. for 16 lbs.; what is the mixture worth per lb. ? Ans. Dl .07.1 -|- 
 
 5. A wine merchant mixed 12 gallons of wine, at 75c. per 
 gallon, with 24 gallons at 90c. per gallon, and 16 gallons at 
 Dl.lO; what is 1 gallon worth? Ans. 92c. 6m. 
 
 6. On a certain day the mercury in the thermometer was ob- 
 served to average the following heights : from 6 in the morning 
 to 9, 64° ; from 9 to 12, 74^ ; from 12 to 3, 84^ ; and from 3 to 6, 
 70^ ; what was the mean temperature of the day ? Ans, 73^. 
 
 ALLIGATION ALTERNATE 
 
 Is the method of finding what quantity of each of the ingre- 
 dients whose rates are given, will compose a mixture of a given 
 rate ; so that it is the reverse of Alligation Medial, and may be 
 proved by it. 
 
 To find the proportion in vMch several simples of given prices 
 must be mixed together, that the compound may he worth a given 
 price. 
 
 W 1. Set down the prices of the simples under each other iu 
 the order of theii values, beginning with the lowest 
 
 1^* 
 
210 
 
 ALLIGATION* ALTERNATE. 
 
 2. Link the least price with the greatest, and the next least 
 with the next greatest, and so on, until the price of each simple 
 which is less than the price of the mixtures is linked with one 
 or more that is greater, and every one that is greater with one 
 or more that is less. 
 
 3. Write the difference between the price of the mixture and 
 that of each of the simples opposite that price with which the par- 
 ticular simple is linked ; then the difference standing opposite 
 any one price, or the sum of the differences when there is more 
 than one, will express the quantity to be taken at that price. 
 
 1. If you would mix wines worth 16c., 18c., and 22c. per 
 quart in such a way that the mixtures be worth 20c. quart, how 
 much must be taken of each sort 1 
 
 Thus: (^^1 2 at 16c. )2 qts. at 16c. ) . 
 
 given price 20c. < 18s 2 at 18c. V 2 " 18c. > § 
 
 ( 22'' -J 4 + 2 = 6 at 22c. ) 6 " 22c. ) ^ 
 
 2. How much corn at 42c., 60c., 67c., and 78c., per bushel, 
 must be mixed together that the compound may be worth 64c. 
 per bushel ? 
 
 Ans, 14 bushels at 42c., 3 bushels at 60c., 4 bushels at 67c. ^ 
 and 22 bushels at 78c. 
 
 3. What quantity of oats at 50c. per bushel, and 30c. per 
 bushel, must be mixed together, that the compound may be 
 worth 40c. per bushel ? Ans. an equal quantity of each sort. 
 
 4. How much rye at 50c. per bushel, barley at 37^c. per 
 bushel, and oats at 25c. per bushel, will make a mixture worth 
 31c. per bushel? 
 
 Ans. 6 bushels at 50c., 6^ at 37ic., aud 25^ at 25c. 
 JS^ote. — If all the given prices are greater or less than the mean 
 or given price, they must be linked to a cipher. A variety of 
 answers may be obtained, according to the method of linking. 
 
 RULE II. 
 
 1. Find the proportional quantities of the simples as in rule 
 1st. 2. Then say as the number opposite the simples whose 
 quantity is given, is to the given quantity, so is either propor- 
 tional quantity to the part of its simple to be taken. 
 
 5. What quantity of coffee at 20c. and at 16c. must be mixed 
 with 35 lbs. at 14c. per lb. to make a mixture worth 18c. per lb. ? 
 
 ( 14 V 2— : 35 :: 2 : 35lbs. at 16c. ) 
 
 18 } 16. 2— : 35 :: 6 : lOSlbs. at 20c. V 
 
 ^20''>' 4+2=6 ) 
 
 For 35x2-2= 35 at 16c. ) . 
 
 and 35x6-2 = 105 at 20c. J ^^* 
 
ALLLIGATION ALTERNATE. 311 
 
 6. A farmer would mix 14 bushels of wheat at D1.20 per 
 bushel, with rye at 72c., barley at 48c., and oats at 36c. ; how 
 much must be taken of each sort to make the mixture worth 
 64c. per bushel ? 
 
 Ans. 14 bushels of wheat, 8 bushels of rye, 4 bushels of bar- 
 ley, and 28 bushels of oats. 
 
 7. A person wishes to mix 10 bushels of wheat at 70c. per 
 bushel, with rye at 48c., corn at 36c., and barley at 30c. per 
 bushel, so that a bushel of this mixture may be worth 38c. ; 
 what quantity of each must be taken ? 
 
 Ans. 2J bushels of rye, 12J bushels of corn ; 40 bushels of 
 barley. 
 
 8. How much water must be mixed with 100 gallons of wine 
 worth 90c. per gallon, to reduce it to 75c. per gallon. 
 
 Ans. 20 gallons. 
 
 When the quantity of the compound is given as well as the price* 
 
 RULE III. 
 
 1. Find the proportional quantities as in rule 1st. 
 
 2. Then say, as the sum of the proportional quantities is to 
 the given quantity, so is each proportional quantity to the part 
 to be taken of each. 
 
 9. A grocer has currants at 4c. 6c., 9c., and lie, per pound, 
 and he wishes to make a mixture of 2401b., worth 8c. per 
 pound ; what quantity of each kind must be taken ? 
 
 '4—-. —3 " 
 
 8 
 
 I] 
 
 Lii- 
 
 ~2 Then 
 
 10 
 
 3 : 
 
 : 240 
 
 72lbs. 
 
 at 4c. 
 
 10 
 
 1 : 
 
 : 240 
 
 241bs. 
 
 at 6c. 
 
 10 
 
 2 ; 
 
 : 240 
 
 481bs. 
 
 at 9c. 
 
 10 
 
 4 : 
 
 : 240 
 
 961bs. 
 
 at lie. 
 
 ^Ans. 
 
 10 
 
 10. How much water, at per gallon, must be mixed with 
 wine at 80c. per gallon, so as to fill a vessel of 90 gallons, 
 which may be offered at 50c. per gallon ? 
 
 Ans. 56^ gallons of wine, 33| gallons of water. 
 
 11 . A goldsmith has several sorts of gold, namely : of 15, 17, 
 20, and 22 carats fine, and would melt together, of all these 
 sorts, so much as may make a mass of 40oz., 18 carats fine ; 
 how much of each sort is required ? 
 
 Ans. 16oz. of 15 carats fine, 8oz. of 17, 4oz. of 20, 12oz. of 22. 
 
 12. How much barley at 40c. per bushel, rye at 60c., and 
 wheat at 80c., must be mixed together that the compound may be 
 worth 62^0. per bush. ? Ans. 17^ bu. barley, 1 7^ rye, 25 wheat. 
 
212 ARITHMETICAL PROGRESSIOX. 
 
 13. A composition was made of 5 lbs. of tea at Dl^ per lb., 
 9 lbs. at D1.80 per lb., and 17 lb§. at Dl-^ per lb. ; what is a 
 pound of it worth? Ans. Dl.54.6.f 
 
 14. A grocer would mix different quantities of sugar, name- 
 ly : one at 20c., one at 23c., and one at 26c. ; what quantity ol 
 each sort must be taken to make a mixture worth 22c. per lb. ? 
 
 Ans. 5 lbs. at 20c., 2 at 23c., and 2 at 26c. 
 Demonstration, — By connecting the less rate with the greater, 
 and placing the difference between them and the mean rate al- 
 ternately, or one after the other, in turn ; the quantities are such, 
 that there is precisely as much gained by one quantity as is lost 
 by the other, and therefore the gain and loss, upon the whole, 
 are equal, and exactly the proposed rate. In like manner. Jet 
 the number of simples be what it may, and with how many 
 soever each one is linked, since it is always a less with a greater 
 than the mean price, there will be an equal balance of loss or 
 gain between every two, and consequently an equal balance on 
 the whole. The rule is founded on the principles of proportion. 
 
 REVIEW. 
 
 What is Alligation ? What is Alligation Medial ? How do 
 you find the price of the mixture ? What is the rule ? What 
 is Alligation Alternate ? How can you prove it 1 How do you 
 find the proportional parts when the price only is given ? Re- 
 peat rule 1st. What is the rule when a given quantity of one 
 of the simples is to be taken ? 
 
 What is the rule when the quantity of the compound as well 
 as the price is given ? What more can you say of Alligation ? 
 
 15. Bought a pipe of wine, containing 120 gallons, at Dl.30 
 a gallon ; how much water must be mixed with it to reduce the 
 first price to Dl^O a gallon ? Ans. 21 j\ gallons. 
 
 ARITHMETICAL PROGRESSION. 
 
 Any series of numbers more than two, increasing or decreas- 
 ing by an even ratio or common difference, is in Arithmetical 
 Progression. When the numbers are formed by continual ad- 
 dition of the ratio or common difference, they form an ascending 
 series ; but when formed by continual subtraction of the com- 
 mon difference, it is a descending series, 
 rp, ( 0, 2, 4, 6, 8, 10, &c., is an ascending arithmetical series. 
 
 I 1, 2, 4, 8, 16, 32, is an ascending geometrical series. 
 . T C 10, 8, 6, 4, 2, 0, is a descending arithmetical series. 
 
 I 32, 16, 8, 4, 2, 1, is a descending geometrical series. 
 
ARITHMETICAL PROGRESSION. 213 
 
 The numbers which form the series are called the terms of 
 the progression. 
 
 The first and last terms of a progression are called the ex- 
 tremes, and the other terms the means. Any three of the fol 
 lowing things being given, the other two may be easily found 
 1st, the first term ; 2d, the last term ; 3d, the number of terms ; 
 4th, the common difference ; 5th, the sum of all the terms. 
 
 The first term, the last term, and the number of terms being given t, 
 to find the common difference, 
 
 RULE I. 
 
 Divide the difference of the extremes by the number of terms, 
 less 1, and the quotient will be the common difference sought. 
 
 1 . The extremes are 3 and 39, and the number of terms is 
 19 ; what is the common dfference ? 
 Extremes. 
 39-.3=36~19-l = 18, or 18)36(2 Ans. 
 '2. A man had 10 sons whose several ages differed alike; 
 the youngest was 3 years old, and the eldest 48 ; what was the 
 common difference of their ages ? Ans. 5 years. 
 
 The first term, the last term, and the number of terms ^ being given, 
 to find the sum of all the terms. 
 
 RULE II. 
 
 Multiply the sum of the extremes by the number of terms, 
 and half the product will be the answer. 
 
 3. A lady purchased 19 yards of riband, for which she gave 
 Ic. for the first yard, 3c. for the second, and 5c. for the third 
 yard, increasing 2c. per yard ; required the cost ? 
 
 19 less 1 = 18 then 1 + 37=38 
 
 com. diflf. 2 X X 19 no. of terms. 
 
 36 2)722 
 1st term, +1 
 
 — half product, D3.61 Ans. 
 last term, 37 
 
 4. If 100 stones were laid two yards distant from each other, 
 in a right line, and a basket placed two yards from the first 
 stone ; what distance would a person travel, to gather them 
 singly into a basket? Ans. 11 m., 3 fur. 180 yds. 
 
 Note. — In this question, there being 1760 yards in a mile, 
 and the man returning with each stone to the basket, his travel 
 will be doubled. 
 
214 : ARITHMETICAL PROGRESSION. 
 
 Given the extremes and the common difference, to find the numhet 
 of terms. 
 
 RULE HI. 
 
 Divide the difference of the extremes by the common differ- 
 ence, and the quotient, increased by 1, will be the number of 
 terms required. 
 
 5. The extremes are 3 and 39, and the common difference 2 ; 
 what is the number of terms ? 
 
 39 — 3 = 36—2 com. diff. 18 quot. + l = 19. Ans, 
 
 6. A man going a journey travelled the first day 7 miles, the 
 last day 51 miles, and each day increased his journey by 4 miles ; 
 how many days did he travel, and how far? Ans. 12 days ; 348 m. 
 
 The extremes and common difference given, to find the sum of all 
 the series. 
 
 Multiply the sum of the extremes by their difference, increased 
 by the common difference, and the product, divided by twice 
 the common difference, will give the sum. 
 
 7. If the extremes are 3 and 39, and the common difference 
 2, what is the sum of the series? 39 + 3=42 sum of the ex- 
 tremes ; 39 — 3 = 36, difference of extremes ; 36 + 2 = 38, differ- 
 ence of extremes increased by the common divisor; 42x38 = 
 1596-i-4, twice the common difference =399. Ans. 
 
 The extremes and sum of the series given, to find the number of 
 
 terms. 
 
 Twice the sum of the series, divided by the sum of the ex- 
 tremes, will give the number of terms. 
 
 8. Let the extremes be 3 and 39, and the sum of the series 
 399 ; what is the number of terms ? Sum of the series = 399 
 X 2 = 798 ; then sum of the extremes =39 + 3=42 : 798-r-42 
 = 19. Ans. 
 
 The extremes and the sum of the series given, to find the common 
 difference, 
 
 RULE VI. 
 
 Divide the product of the sum antj the difference of the ex- 
 tremes by the difference of twice the sum of the series and the 
 sum of the extremes, and the quotient will be the common dif- 
 ference. 
 
ARITHMETICAL PROGRESSION. 215 
 
 9. Let the extremes be 3 and 39, and the sum 399 ; what is 
 whe common difference? Sum. — Extremes = 39-f 3=42 ; dif- 
 ference of extremes=39— 3=36 ; 42x36=1512 ; then 399 
 X2-42=756)1512(2 Ans. 
 
 The first term, the common difference, and the number of terms 
 given, to find the last term, 
 
 RULE VII. 
 
 The number of terms, less 1, muhiplied by the common dif- 
 ference, and the first term added to the product, will give the 
 last term. 
 
 10. A man bought 100 yards of cloth, giving 4c. for the first 
 yard, 7c. for the second, 10c. for the third, and so on, with a 
 common difference of 3c. ; what was the cost of the last yard ? 
 
 No. of terms, 100 — 1 = 99x3, com. difr.=297+4=the first 
 term = 3.0lD. Ans. 
 
 11. A man travelled 20 miles the first day, 24 the second, 
 and so on, increasing 4 miles every day ; how far did he travel 
 the 12th day ? Ans. 64 m. 
 
 Note. — The great variety of cases in arithmetical progres- 
 sion will not permit more space to be occupied in this work ; 
 such questions as usually occur may be solved by the above rules. 
 
 12. A man had 8 sons, whose ages differed alike ; the young- 
 est was 10 years old, and the eldest 45; what was the com- 
 mon difference of their ages ? Ans. 5 years. 
 
 13. A man is to travel from New York to a certain place in 
 19 days, and to go but six miles the first day, increasing every 
 day by an equal excess, so that the last day's journey may be 60 
 miles ; what is the common difference, and distance of the 
 journey ? Ans. com. diff. 3 miles ; distance 627 miles. 
 
 14. It is required to know how many times the hammer of a 
 clock would strike in a week, or 168 hours, provided it increases 
 1 at each hour ? Ans. 14196. 
 
 15. What will Dl at 6 per cent, amount to in 20 years, at 
 simple interest ? Ans. D2.20. 
 
 16. How many times does a regular clock strike in 12 hours ? 
 
 Ans. 78. 
 
 17. A man bought 100 oxen, and gave for the first ox Dl ; 
 for the second, D2 ; for the third, D3 ; and so on to the last ; 
 how much did they come to 1 Ans. 5050. 
 
 REVIEW. 
 
 What is Arithmetical Progression 1 Name the five things 
 that should be particularly attended to in this rule ? How do you 
 
216 GEOMETRICAL PROGRESSION. 
 
 form an arithmetical series ? What is the common difference ? 
 What is an ascending series ? What is a descending series ? 
 What are the several numbers called ? What are the first and 
 last terms called ? Ans. Extremes, and the intermediate terms 
 are called the means. How do you find the common difference, 
 when you know the two extremes and number of terms ? What 
 is lule 1st? 2d ? &c. 
 
 18. If a piece of land, 60 rods in length, be 20 rods wide at 
 one end, and at the other terminate in an angle or point, what 
 number of square rods does it contain ? Ans. 600. 
 
 GEOMETRICAL PROGRESSION. 
 
 Geometrical Progression is a series of numbers increasing 
 or decreasing by a common ratio ; thus, 2, 4, 8, 16, 32, increased 
 by the multiplier or ratio 2, and 32, 16, 8, 4, 2, decrease con- 
 tinually by the divisor 2, &;c. In this rule there are five denom- 
 inations, any three of which being given, the other two may be 
 found : 1st, the first term ; 2d, the last term ; 3d, the number of 
 terms ; 4th, the ratio ; 5th, the sum of the series. The ratio is; 
 the multiplier or divisor by which the series is founded. To 
 raise a power or series of numbers by the ratio, we place the 
 ratio at the left hand for the first power ; this (first power) mul- 
 tiplied by the ratio (its square) gives the second power, the sec* 
 ond by the ratio gives the third, and so on, until the power is 
 one number less than the first term. Let 2, 4, 8, 16, 32, &c.y 
 be the series whose ratio is 2. The second term is formed by 
 multiplying the first term by the ratio 2 ; the third term by mul- 
 tiplying the second by the ratio, and so on. The series may 
 therefore be written thus : 2, 2x2, 2x2x2, 2x2x2x2, 2x 
 2X2X2X2, &c., or thus: 2, 2x2^, 2x2^, 2x2^, 2x2^&c• 
 Any power after the first is evidently that power of the ratio 
 whose index is one less than the number of the term multiplied 
 by the first term ; thus the third term is 2 x22, the 4th term is 
 2 X 2^, and the 8th term would be 2 X 2^, &c. In an ascending 
 series, therefore, multiply the first term by that power of the 
 ratio whose index is one less than the number of the term sought, 
 as mentioned above, and the product is the term sought, la a 
 descending series, as 243, 81, 27, 9, 3, 1, whose ratio is 3, and 
 which is also lx3^ lx3S 1x33, Ix32,lx3\ 1, 
 
 1X3^ 
 the last term 1 = 
 
 3^ 
 
aEOMETRICAL PROGRESSION. 217 
 
 Hence we derive the following general rule, the correctness of 
 which is evident, whatever be the first term, or ratio. 
 
 RULE. 
 
 Raise the ratio to the power whose index is one less than the 
 number of terms given, which multiply by the first term ; that 
 product is the last term, or greater extreme. 
 
 2. Multiply the last term by the ratio, and from the produc 
 subtract the first term, and divide the remainder by the ratio 
 less one ; the quotient will be the sum of the series. 
 
 Or, raise the ratio to a power equal to the number of terms ; 
 subtract one from that power ; multiply the remainder by the 
 first term ; divide this product by the ratio, less one ; the quo- 
 tient will be the sum of a geometrical series. 
 
 1. The first term is 3, and the ratio 2 ; what is the 6th term? 
 2X2X2X2X2 = 2^=32; 32 x3, 1st term, z=:96. Ans, 
 
 Having given the ratio and the two extremes^ to find the surn of 
 the series. 
 
 Subtract the less extreme from the greater; divide the re- 
 mainder by one less than the ratio, and to the quotient add the 
 greater extreme ; the sum will be the sum of the series. 
 
 2. The first term is 3, and the ratio 2, and the last term 192 ; 
 what is the sum of the series ? 192 — 3=: 189, diff. of extremes, 
 
 2 — 1 = 1)189(189; then 189+ 192z=:381. Ans. 
 
 3. If the first term be 2, and the ratio 2, what is the 13th 
 term? 1,2,3, 4, 5x 5 x3==13, or, 2x21^ (less, 1) = 8192. 
 
 2, 4, 8, 16, 32x32x8=8192. Ans. 
 
 4. If the first term be 5, and the ratio 3, what is the 7th 
 term ? 
 
 0, 1, 2, 3,+2 +1 = 6 = indices to 6th t'm beyond 1st or 7th. 
 5, 15,45, 135,X45X 15=91125 dividend. 
 
 The number of terms multiplied is three, namely : 135x45 
 X 15, and 3 — 1=2, is the power to which the term 5 is to be 
 raised; but the second power of 5 is 5x5=25, therefore, 
 91125-^25 = 3645=7, term required. 
 
 Note. — The following examples will embrace the general op- 
 eration of the several rules of geometrical progression, sufficient 
 for common use. 
 
 5. A. purchased 24 yards of sheeting, for wnich he paid 2c. 
 for the first yard, 4c. for the second, 8c. for the third, <fec., 
 increasing in a duplicate proportion ; required the amount. 
 
 19 
 
218 
 
 GEOMETRICAL PROGRESSION. 
 
 Thus, 12 3 4 indices. 
 
 2 4 8 16 leading terms 
 16X 
 
 256= 8th term. 
 256 X 
 
 65536 = 16th term. 
 256 X 
 
 16777216=24th term. 
 
 2 X ratio, raultiplicL 
 
 33554432 
 
 2— ratio, subtracted. 
 
 2-^1 = 1)335544.30 Ans. in dollars and cents. 
 6. What debt can be discharged in a year, by paying 1 cen* 
 the first month, 10 cents the second, and so on, each mon^h ii' 
 a tenfold proportion 1 
 
 12 3 4 5 6 
 
 10 X 100 X 1000 X 10000 X 100000 X 1000000 
 
 1000000 
 
 1000000000000 
 
 1" 
 
 999999999999 
 IX 
 
 10-1 = 9)999999999999 
 
 Dllllllllll.il Ans. 
 
 7. What will one cent amount to, if you double it every year 
 for 21 years 1 Ans. D20971.51. 
 
 8. The first term of a series having 10 terms, is 4, and the 
 ratio 3, what is the last term ? Ans. 78732 
 
 9. What is the last term of a series having 18 terms, the 
 first of which is 3, and the ratio 3 ? Ans. 387420489. 
 
 10. A man was to travel to a certain place in 4 days, and to 
 travel at whatever rate he pleased ; the first day he went 2 miles, 
 the second 6 miles, and so on to the last, in a threefold ratio ; 
 how far did he travel the last day, and how far in all ? 
 
 Ans, last day, 54 miles ; in all, 80 ndlo^ 
 
PERMUTATION. 219 
 
 11. Sold 14 pairs of stockings, the first at 4 cents, the second 
 at 12 cents, and so on in a geometrical progression ; what did 
 the last pair bring him, and what did the whole bring him ? 
 
 Ans. last, D63772.92 ; whole, D95659.36. 
 
 12. If our ancestors who landed at Plymouth, A. D. 1620, 
 being 101 in number, had increased so as to double their num- 
 ber in every 20 years, how great would have been their popu- 
 lation at the end of 1840 ? Ans. 206747. 
 • 13. A sum of money is to be divided among 10 persons ; the 
 first to have DIO, the second, D30, and so on, m a threefold 
 proportion; what will the last have ? Ans D196830. 
 
 14. A man bought a horse, and by agreement was to give Ic. 
 for the first nail, 2c. for the second, 4c. for the third, &c. ; there 
 were 4 shoes, and 8 nails in each shoe ; what was the cost of 
 the horse ? Ans. D42949672.95. 
 
 What is Geometrical Progression ? How do you form a 
 geometrical progression ? What is the common ratio ? What 
 is the difl^erence between arithmetical and geometrical progres- 
 sion ? What is an ascending series ? What is a descending 
 series ? What are the several numbers called ? In every geo- 
 metrical progression how many things are to be considered ? 
 What are they ? What are the first and last terms called ? 
 What are the intermediate terms called ? 
 
 15. If the ratio be 4, the number of terms 6, and the greatest 
 term 3072 ; what is the sum of the series ? ^ns. 4095. 
 
 Divide the last term by the 5th power of the ratio, &c 
 
 PERMUTATION. 
 
 Permutation is the method of finding how many diflferent 
 ways any number of things may be changed. Thus take the 
 first three letters of the alphabet, a b c ; they will admit of six 
 changes, a b c, a c b, b a c, b c a, c b a, cab, and so on, 
 according to the given number of terms. 
 
 RULE. 
 
 Multiply all the terms of the natural series constantly from 1, 
 or unity, to the given number, inclusive ; the last product will 
 be the number of changes required. 
 
220 COMBINATION. 
 
 1. In how many different positions can five persons be placed 
 at a table? Thus: 1x2x3x4x5 = 120. Ans, 
 
 2. What time will it require for 8 persons to seat themselves 
 differently every day at dinner ? Ans. 110 years, 142^ days. 
 
 3. How many variations can be made of the English alpha- 
 bet, it consisting of 26 letters ? 
 
 ^;25. 403291461126605635584000000. 
 
 4. How many changes may be rung on 15 bells ; and in what 
 time may they be rung, allowing three seconds to every round ? 
 
 Ans. 1307674368000 changes; 3923023104000 seconds. 
 
 5. How many variations may there be in the position of the 
 nine digits ? Ans. 362880. 
 
 6. A man bought 25 cows, agreeing to pay for them 1 cent 
 for every different order in which they could all be placed ; how 
 much did the cows cost him ? 
 
 Ans. D155112100433309859840000. 
 
 7. Christ's church, in Boston, has 8 bells ; how many 
 changes may be rung upon them. Ans. 40320. 
 
 COxMBINATION. 
 
 Combination teaches how many different ways a less num- 
 ber of things may be combined out of a larger ; thus out of the 
 letters abed, are six different combinations of two, namely, 
 a b, a c, a d, d c, d b, b c. 
 
 Thus: 4x3 = 12; 1 x2=2)12( = 6. .4^^. 
 
 RULE. 
 
 Take a series, proceeding from and increasing by a unit, up 
 to the number to be combined ; then take a series of as many 
 places, decreasing by unity, from the number out of which the 
 combinations are to be made ; multiply the first continually for 
 a divisor, and the other for a dividend ; the quotient will be the 
 nswer. 
 
 1. How many combinations of five letters in ten? 
 
 Thus: 10x9x8x7x6 = 30240, dividend; 1x2x3x4x5 
 = 120, divisor. Ans. 252. 
 
 2. How many combinations of ten figures may be made out 
 of twenty? " Ans. 184756. 
 
 3. How many combinations may be made of seven letters out 
 of twelve. Ans. 792 
 
COMPOUND INTEREST 22i 
 
 4. How many combinations can be made of six letters out of 
 the 24 letters of the alphabet? Ans. 134596. 
 
 REVIEW. 
 
 What is Permutation? What is the rule? What is com- 
 bination ? What is the rule '' 
 
 5. How many changes may be rung with 4 bells out of 8 ? 
 
 Ans. 1680. 
 
 6. How many variations may be made of the letters in the 
 word Zaphnathpaaneah(15)? 2x6x1x2x120=2880 divi- 
 sor. Ans. 454053600. 
 
 7. How many different numbers can be made of the follow- 
 ing figures : 1223334444 ? Ans. 12600. 
 
 COMPOUND INTEREST, BY DECIMALS. 
 
 Compound Interest is that which arises from interest and 
 principal added together annually, as the interest becomes due, 
 by the continued multiplication of the new principal by the ratio 
 or rate per cent. ; thus, if I owe A. DlOO, payable on demand, 
 and neglect to pay either interest or principal for several years, 
 he would be justified in adding the interest to the principal an- 
 nually, and computing the interest on this amount: DlOO + 6 
 = 106, first year; then Dl06x6=6.36 interest second year, 
 which D6.004-6.36D. = 12.36 interest or Dl 12.36 amount for 
 the second year ; Dl 12.36 X 6=6.74 interest for the third year, 
 which D6. 00 + 6. 36 + 6.74 = 19. 10 interest for the third year, or 
 Dl 19.10.160 amount at compound interest for three years at 6 
 per cent. In many cases it is considered illegal to receive 
 compound interest, therefore it is seldom computed ; but when 
 a note, bond, or obligation, is given with a credit of several years, 
 on condition that the interest shall be paid annually, if the inter- 
 est is not paid until the obligation becomes due, it is no more 
 than just and right that compound interest should be paid, as 
 well as principal ; for the person having the use of the princi- 
 pal has likewise the use of the interest, which of right belongs 
 to another ; therefore interest should be computed accordingly. 
 
 The following table is computed by the continual multiplica- 
 tion of 1, or DlOO by the ratio as above; which may some- 
 times be found useful where the higher powers are required, or 
 »o test the accuracy of calculations. 
 
 19* 
 
2Q2 
 
 COMPOUND INTEREST. 
 
 A tkible showing the amount of Dl^ or DlOO, for 15 t^ears at 4, 
 5 J 6, and 7 per cent, at compound interest. 
 
 Years. 
 
 4 per cent. 
 
 5 per cent. 
 
 6 per cent. 
 
 7 per cent. 
 
 1 
 
 1.0400000 
 
 1.050000 
 
 1.060000 
 
 1.070000 
 
 3 
 
 1.0816000 
 
 1.102500 
 
 1.123600 
 
 1.144900 
 
 3 
 
 1.1248640 
 
 1.157625 
 
 1.191016 
 
 1.225043 
 
 4 
 
 1.1698585 
 
 1.215506 
 
 1.262477 
 
 1.310795 
 
 5 
 
 1.2166529 
 
 1.276281 
 
 1.338225 
 
 1.402552 
 
 6 
 
 1.2653190 
 
 1.340095 
 
 1.418519 
 
 1.500730 
 
 7 
 
 1.3159317 
 
 1.407100 
 
 1.503630 
 
 1.605781 
 
 8 
 
 1.3685690 
 
 1.477455 
 
 1.593848 
 
 1.718186 
 
 9 
 
 1.4233118 
 
 1.551328 
 
 1.689479 
 
 1.838459 
 
 10 
 
 1.4802842 
 
 1.628894 
 
 1.790848 
 
 1.967151 
 
 11 
 
 1.5394540 
 
 1.710339 
 
 1.898299 
 
 2.104852 
 
 12 
 
 1.6010322 
 
 1.795856 
 
 2.012197 
 
 2.252191 
 
 13 
 
 1.6650735 
 
 1.885649 
 
 2.132928 
 
 2.409845 
 
 14 
 
 1.7316764 
 
 1.979931 
 
 2.260904 
 
 2.578534 
 
 15 
 
 1.8009435 
 
 2.078928 
 
 2.396558 
 
 2.759032 
 
 TABLE II. 
 
 A table showing the amount of Dl or DlOO for one year, and 
 for quarters of a year, at compound interest, and simple inter' 
 est for one month. 
 
 rate per c. 
 
 ratio. 
 
 3quart'rs. 2quart'rs. 
 
 1 quarter. 
 
 S.int. Imo, 
 
 4 
 
 1.04 
 
 1.029852 
 
 1.019804 
 
 1.009853 
 
 .003333 
 
 4.5 
 
 1.045 
 
 1.033563 
 
 1.022252 
 
 1.011065 
 
 .003750 
 
 5 
 
 J. 05 
 
 1.037270 
 
 1.024675 
 
 1.012272 
 
 .004167 
 
 5.5 
 
 1.055 
 
 1.040973 
 
 1.027132 
 
 1.013475 
 
 .004583 
 
 6 
 
 1.06 
 
 1.044671 
 
 1.029536 
 
 1.014674 
 
 .005000 
 
 To compute interest by the use of the tables, find the amount 
 or tabular number, under the rate per cent., and opposite the num- 
 ber of years ; multiply ^is number by the principal, and you 
 have the amount, from which subtract the principal, and the re- 
 mainder will be the compound interest. If quarterly payments 
 ere required at compound the 4th root of the ratio • 
 
CX>MPOUND INTERES'^. 2S^8 
 
 for half-yearly, the square root ; and for three quarters, the prod- 
 uct of the quarterly and half yearly ; thus, the tabular number 
 at 6 per cent., will be for one quarter 1.014674 ; for two quar- 
 ters, 1.029536; for 3 quarters, 1.044671, at 6 per cent. Or 
 find the powers by the tables, which multiply together, and the 
 product will be the amount ; thus, for 4.25 years at 6 per cent, 
 per annum, 1.2624769 X 1.014674 =:1.2810024860306 amount 
 of DlOO, (fee, or D28.10 interest for 4A years. It may be found 
 very nearly by finding the compound interest for the years, and 
 then compute the simple interest on that amount for quarterly 
 or parts of a year. 
 
 Again, ^1.06 = 1.014674 quarterly amount: and ^1.06 = 
 1.029536 half-yearly amount: then, 1.014674x1.029536 = 
 1.044671: then, for 3^ years at 6 per cent., 1.191016X 
 ]. 0114674 = 1.2184929, amount for 3} yrs. at 6 per ct. per ann. 
 
 1. What is the compound interest of DlOO for 4 years, at 
 6 per cent, per annum ? By the table, 1.2624769 X 100 = 
 1.262476900-1.00 = D26.24,7,6900. Ans, 
 
 2. What is the amount of Dl50p for 12 years at 3.5 per ct. 
 per annum? Thus, tabular number, 1.5110686x1500 = 
 D2266.60.29. (The tabular number is not in the table.) 
 
 RULE II. 
 
 Multiply the principal by the ratio, which will give the simple 
 interest for 1 year ; add this interest to the principal and calcu- 
 late as before for the second year, and so continue up to the 
 number of years required, and the last product will be the 
 amount, from which subtract the principal and the remainder 
 will be the compound interest. 
 
 3. What is the compound interest of DlOO for 4 years at 6 per 
 cent, per annum ? Thus, DlOO principal, 
 
 DlOO x 6 ratio = 6 interest 1st year, 
 
 106 amount 1st year, 
 106 X 6= 6.36 interest 2d year, 
 
 112.36 amount 2d year. 
 112.36 X6=t 6.7416 interest 3d year. 
 
 119.1016 amount 3d year. 
 119.1016x6= 7.146096 interest 4th year. 
 
 126.24.7.696 amount 4th year. 
 Principal —100 
 
 D26.24,7.696. Ans. 
 
224 COMPOUND INTEREST. 
 
 4 What is the compound interest of D500 for 4 years ax 6 
 percent.? Ans. D131.23.8. 
 
 5. What is the compound interest of D700 for 7 years at 5 pet 
 cent. ? Ans. D284.97. 
 
 6. What is the compound interest of D850 for 10 years at 6 
 per cent. ? Ans. D672.22.08. 
 
 7. What is the compound interest of Dl 100 for 15 years at 5 
 per cent. ? Ans. D1186.82.08. 
 
 8. What is the compound interest of D4400 for 20 years at 6 
 per cent. ? Ans. D971 1 .39.6. 
 
 9. What is the compound interest of D20000 for 13 years at 6 
 per cent. ? Ans. D22658.56 
 
 10. What is the amount of D768 for 3 J years at 6 per cent 
 compound interest ? Ans. For 3 years, 1.1910160 X 1.014674 for 
 i=: 12084929687840 amount of lOOfor 3-i ; then x768D.princ. 
 = D928.12.2. Ans. 
 
 11. What is the difference between the simple interest of 
 D200 for 3 years, and the compound interest for the same time ? 
 
 Ans, D2.2033^. 
 
 When the amount, rate, and time, are given, to find the principal 
 
 RULE III. 
 
 Divide the amount by the amount of 1 or D 100 for the given 
 time, and the quotient will be the principal. 
 
 Or if you multiply the present value of 1 or DlOO for the 
 given number of years, at the given nate per cent, by the amount 
 the product will be the principal or present worth. 
 
 12. What principal must be put to interest 6 years, at 5^ pe ,' 
 cent, per annum, to amount to D689.4214033809453125 ? 
 
 Thus, 1.3788426)689.4214033809453125(500D. Ans. 
 
 Note. — We have a variety of cases in Compound Interes , 
 and Annuities, which, for want of room, are necessarily omitted, 
 a sufficient number will be given for all practical purposes. 
 
 REVIEW. 
 
 What is Compound Interest ? How can you compute inter- 
 est by the use of the tables ? What is rule 2 ? How can you 
 find the principal 1 In what time will any sum double at 6 per 
 cent, simple interest ? Ans. 16 years, 8 months. In what time 
 at compound interest? Ans. 11 years, 8 months, 22 days 
 When is it lawful to compute compound interest ? 
 
 13. What is the compound interest of the following sums 
 D1200 for 5 years, and D480 for 7 years, at 7 per cent. ? 
 
ANNUITIES. 32§ 
 
 ANNUITIES. 
 
 An Annuity is a sum of money payable at regular periods, 
 for a certain time, or for ever. Annuities sometimes depend on 
 some contingency, as the life or death of a person, and the an 
 nuity is then said to be contingent. Sometimes annuities are 
 not to commence till a certain number of years have elapsed, 
 and the annuities are then said to be in reversion. The annuity 
 is said to be in arrears when the debtor keeps it beyond the 
 time of payment. The present worth of an annuity is such a 
 sum, as being now put out at interest would exactly pay the an- 
 nuity, as it becomes due, and is the sum which must be given 
 for the annuity if it be paid at its commencement. The amount 
 is the sum of the annuities for the time it had been forborne, 
 with the interest due on each. 
 
 To Jind the amount of an annuity, at simple interest. 
 
 RULE I. 
 
 Multiply the natural series of numbers, 1, 2, 3, 4, &c., to the 
 number of years, less 1, by the interest of the annuity for one 
 year, and the product will be the interest which is due on the 
 annuity. Multiply the annuity by the time, and the sum of the 
 two products will be the amount. 
 
 1. What is the amount of an annuity of DlOO, for 4 years, 
 computing interest at 6 per cent. ? Thus : l-f-2+3 = 6, sum of 
 the natural series to the number of years, less 1. D6 interest 
 of annuity for one year ; then 6x6 = D36, the whole interest - 
 100x4=D400 + 36=436, amount. Ans. (See note at close 
 of annuities.) 
 
 2. If a pension of D20 be continued unpaid for 6 years, what 
 is its amount at 6 and 7 per cent. ? At 6 per cent., 1)138 ; at 7 
 per cent., D141. Ans. 
 
 REMARKS. 
 
 It is plain that upon the first year's annuity there will be duo 
 so many years' interest as the given number of years, less 1, 
 and gradually one year less upon each such succeeding year, to 
 that preceding the last, which has but one year's interest, and 
 the last bears none. There is, therefore, due in the whole as 
 many years' interest of the annuity as the sum of the series, 1, 
 2, 3, &;c., to the number of years, diminished one. It is evi- 
 dent then, that the whole interest due must equal this sum of the 
 natural series, multiplied by the interest for one year, and that 
 the amount will be all the anrmities, or the product of the an- 
 nuity and time added to the whole interest ; and this is the rule.- 
 
Z26 
 
 ANNUITIES. 
 
 The annuity J time, and ratio given, to find the amount at com" 
 pound interest. 
 
 RULE II. 
 
 1. Make 1 the first term of a geometrical progression, and 
 the amount of Dl, and tlie given rate per cent, the ratio. 
 
 2. Carry the series to as many terms as the number of years, 
 and find the sum. 
 
 3. Multiply the sum thus found by the given annuity, and the 
 product will be the amount sought. Or, multiply the amount of 
 Dl for one year, at the given rate per cent., into itself as many 
 times as there are years given; from the product subtract one, 
 then divide the remainder of the interest of Dl for 1 year, av 
 its given rate per cent., and multiply the quotient by the annuity 
 for the amount required. 
 
 
 TABLE 
 
 I. 1 
 
 TABLE II. 
 
 
 A table showing the amount of 
 
 A table showing the present 
 
 Dl or DlOO, from 1 to 15 
 
 worth of D I or BlOO, from 1 
 
 years, at 5 and 6 per cent. 
 
 to 15 years, at 5 and 6 percent. 
 
 Years. 
 
 5 per cent. 
 
 6 per cent. 
 
 5 per cent. 
 
 6 per cent. 
 
 Years 
 
 1 
 
 1. 
 
 1. 
 
 0.95238 
 
 0.94339 
 
 1 
 
 2 
 
 2.05 
 
 2.06 
 
 1.86941 
 
 1.83339 
 
 2 
 
 3 
 
 3.1525 
 
 3.1836 
 
 2.72325 
 
 2.67301 
 
 3 
 
 4 
 
 4.310125 
 
 4.374616 
 
 3.54595 
 
 3.46510 
 
 4 
 
 5 
 
 5.525631 
 
 5.637093 
 
 4.32988 
 
 4.21236 
 
 5 
 
 6 
 
 6.801913 
 
 6.975318 
 
 5.07569 
 
 4.91732 
 
 6 
 
 7 
 
 8.142008 
 
 8.393837 
 
 5.78637 
 
 5.58338 
 
 7 
 
 8 
 
 9.549109 
 
 9.897467 
 
 6.46321 
 
 6.20979 
 
 8 
 
 9 
 
 11.026564 
 
 11.491315 
 
 7.10782 
 
 6.80169 
 
 9 
 
 10 
 
 12.577892 
 
 13.180794 
 
 7.72173 
 
 7.36009 
 
 10 
 
 11 
 
 14.206787 
 
 14.971642 
 
 8.30640 
 
 7.88687 
 
 11 
 
 12 
 
 15.917126 
 
 16.869940 
 
 8.86325 
 
 8.38384 
 
 12 
 
 13 
 
 17.712983 
 
 18.882132 
 
 9.39357 
 
 8.85668 
 
 13 
 
 14 
 
 19.598632 
 
 21.015064 
 
 9.89864 
 
 9.29498 
 
 14 
 
 15 
 
 21.785635 
 
 23.275968 
 
 10.37965 
 
 9.71225 
 
 15 
 
 
 TABLE III. 
 
 ratio. 
 
 half yearly. 
 
 quarterly. 
 
 4 
 
 1.009902 
 
 1.014877 
 
 4.5 
 
 1.011126 
 
 1.016720 
 
 5 
 
 1.012348 
 
 1.018550 
 
 5.5 
 
 1.013567 
 
 1.020395 
 
 6 
 
 1.014781 
 
 1.022257 
 
 The construction of table 
 3 is from an algebraic 
 theorem, which may be in 
 words, thus : for half-year- 
 ly payments, take a unit 
 from the ratio, and from 
 the square root of the ratio ; 
 half the quotient of the firsr 
 
ANNUITIES. 22*^ 
 
 remainder, divided by the latter will be the tabular number ; for 
 quarterly payments, use the 4th root, and take one quarter of the 
 quotient. 
 
 Table 1st is calculated thus: take the first year's amount, 
 which is Dl, multiply it by 1.06-|- 1=2.06, second year's 
 amount, which also multiply by 1. 06+1 = 3. 1836, third year's 
 amount, &c., at 6 per cent., and in this manner calculate the 
 other tables. 
 
 RULE III. 
 
 Multiply the tabular number under the rate, and opposite to 
 the time, by the annuity, and the product will be the amount. 
 
 3. What will an annuity of D60 per annum amount to in 20 
 years, allowing 6 per cent., compound interest ? 
 
 Thus; 36.785592 X D60 ann.:rrD2207.13.5520. Ans, 
 
 4. What will an annuity of D60 per annum, payable yearly, 
 amount to, in 4 years, at 6 per cent. ? Thus : 1 + 1. 06 -f 1.062 
 + 1.063 = 4.374616, sum, xD60, ann. =D262.47.696. Ans. 
 (Rule 2.) 
 
 5. What will a pension of D75 per annum, payable yearly, 
 umcunt to in 9 years, at 5 per cent., compound interest ? 
 
 Ans. D826.99.2^V 
 
 The annuitr/, time, and rate given, to find the present worth, 
 
 RULE IV. 
 
 Divide the annuity by the amount of Dl for 1 year, and the 
 quotient will be the present worth of one year's annuity. 
 
 2. Divide the annuity by the square of the ratio, and the quo- 
 tient will be the present worth for 2 years. 
 
 3. In like manner, find the present worth of each year by it- 
 self, and the sum of all these will be the present value of the 
 annuity. 
 
 6. What ready money will purchase an annuity of D60, to 
 continue 4 years, at 6 per cent., compound interest ? 
 
 ratio = 1.06)60.00000(56.603=present worth 1 year, 
 
 ratio 2= 1.1236)60.00000(53.399 " 2 years, 
 
 ratio 3= 1.191016)60.00000(50.377 " 3 years, 
 
 ratio 4= 1.26247696)60.00000(47.525 « 4 years. 
 
 D207.904 Ans. 
 (See table, compound interest.) 
 
 OR, BY TABLE 2. 
 
 Multiply the number under the rate, and opposite the time, in 
 
228 PERPETUlriES AT COMPOUND INTEREST. 
 
 the table, by the annuity, the product will be the present worth 
 for yearly payments. 
 
 When the payments are to be made quarterly, or half-yearly, 
 the present worth, found as above, must be multiplied by the 
 proper number in table 3d. 
 Thus : question 6, tabular No. 3, 46510xD60 = 207.906. Ans. 
 
 7. What is the present worth of an annuity of D60 per an- 
 num, for 20 years, at 6 per cent. ? Ans, D688. 19.50. 
 
 8. What is the present worth of D75 per annum, for 7 years, 
 at 5 per cent.? Thus : 5.78637x75 = 0433.97.775. Ans. 
 
 Table 2d is thus made : Divide Dl by 1.06 = .94339 the pres- 
 ent worth of the first year, which, divided by 1.06 is equal to 
 .88999, which added to the first year's present worth is = 
 1.83339, the second year's present worth; then .88999 divided 
 by 1.06, and the quotient added to 1.83339 gives 2.67301 for 
 the 3d year's present worth, &c. 
 
 Annuities in reversion at compound interest. 
 
 RULE V. 
 
 Take two numbers under the given rate in table 2d, that 
 stand opposite the sum of the two given times, and the number 
 opposite the time when the annui.v is to commence, or time of 
 reversion, and multiply their difference by the annuity for the 
 present worth. 
 
 9. The reversion of a freehold estate of D60 per annum, for 
 4 years, to commence 2 years hence ; what is the present worth, 
 allowing 4 per cent, for present payment ? Thus : tab. no. 
 5.24214-1.88609=3.35605 xD60 = D201.36.300. Ans. 
 
 10. What is the present worth of a reversion of a lease for 
 D120 per annum, to continue 9 years, but not to commence till 
 the end of 4 years, at 4 per cent., to the purchaser ? 
 
 Thus: 9.98565 — 3.62989=6. 35576 X120=D762.69.1. Ans. 
 
 PERPETUITIES AT COMPOUND INTEREST. 
 
 Perpetuities are such annuities as continue for ever. 
 The annuity and rate given, to find the present worth. 
 
 RULE VI. 
 
 Multiply the amount of Dl for one year, at the given rate per 
 cent., involved to the time of reversion, by the ratio, for a divi- 
 sor, by which divide the yearly payments ; the quotient will be 
 the answer. 
 
ANNUITIES. 229 
 
 11. Suppose a freehold estate of D140 per annum, to com- 
 «nence three years hence, is to be sold ; what is it worth, allow 
 in^ the purchaser 7 per cent. ? Thus, 1 .07 X 1 .07 X 1 .07 X .07= 
 .08575301; then Dl40^08575301 = 1632.59.5D. Ans. 
 
 12. What is an estate of D260 per annum, to continue foi 
 ever, worth in present money, allowing 6 per cent, to the pur 
 chaser? D4333.33.3. Ans, 
 
 7\ find the present worth of a freehold estate, or an annuity to 
 continue for ever at compound interest, 
 
 RULE VII. 
 
 As the rate per cent, is to DlOO, so is the yearly rent to the 
 value required. 
 
 13. What is the worth of a freehold estate of D40 per an- 
 num, allowing 5 per cent, to the purchaser 1 
 
 Thus, D5 : 100 :: 40 : 800. Ans, 
 
 14. What is the amount of an annuity of D 180 for 9 years at 
 5 per cent. ?* Ans. D1984.781520. 
 
 15. What will an annuity of D200 amount to in 5 years, to 
 be paid by half-yearly payments at 6 per cent. ? Thus, 5.637093 
 X200=1127.4186x 1.014781 = 1144.08.2D. Ans. 
 
 16. Required the amount of an annuity of D 150 for 10 years 
 at 5 per cent. ? Ans. 1886.68.3. 
 
 17. If a salary of DlOO per annum, to be paid yearly, be for- 
 borne 5 yrs. at 6 per cent., what is the amount ? Ans. D563.70.9. 
 
 18. What sum of ready money will buy an annuity of D300, 
 to continue 10 years, at 6 per cent. 
 
 10 years=::7.36008x 300=2208.0240. Ans, 
 
 19. What salary, to continue 10 years, will D2208.024 buy? 
 This example is the reverse of the one above, consequently 
 D2208.024-^7.36008=300. Ans, 
 
 20. If the annual rent of a house, which is D150, remain in 
 arrears for 3 years at 6 per cent., what will be the amount dua 
 for that time ? Ans, D477.54. 
 
 Note. — From the nature of an annuity, as explained in the 
 proof of the rule in annuities at simple interest, there is due one 
 year's interest less than the number of years the annuity has 
 been continued. By the rule and examples in compound inter- 
 est, the amount of Dl at the given rate is equal to that power of 
 the amount for one year, which is indicated by the number of 
 years. This amount is obtained for one less than the number- 
 of years by forming the geometrical series as directed in the rule, 
 or beginning with unity ; thus, in question 1 (annuities) the series 
 
 * See preceding rules. 
 80 
 
230 DUODECIMALS. 
 
 is 1, 1.06, 1.062, 1.063, and the last term is the amount of Dl 
 for 1 less than 4, the number of years. The sum of this series 
 is the amount at compound interest, of an annuity of Dl for 4 
 years. The amount of any other annuity, for the same time and 
 rate, will be as much greater or less as the annuity is greater 
 or less than Dl ; that is, the amount of the annuity of Dl must 
 be multiplied by the annuity to obtain its amount, <fec. 
 
 REVIEW. 
 
 What is an annuity ? How will you find the amount of an 
 annuity at simple interest ? How will you find the amount of 
 an annuity at compound interest ? What is the rule ? What 
 will you do when the payments are half-yearly, or quarterly ? 
 How is table 1st calculated? How will you find the present 
 worth of an annuity ? What is the rule 1 How is table 2d cal- 
 culated? When quarterly payments are required^ what is to be 
 done ? How is table 3 constructed ? What can you say of 
 annuities in reversion ? What is the rule ? What name is 
 given to annuities which continue for ever ? What is the rule 
 for perpetuities ? What more can you say of annuities ? 
 
 DUODECIMALS, OR CROSS MULTIPLICATION. 
 
 Duodecimals, that is, numbering by 12, are fractions of a 
 foot, or of an inch, or parts of an inch, having 12 for their de- 
 nominator. It is a rule in general use with artificers in casting 
 up the contents of their work, and an excellent and useful rule 
 for that purpose. Considering a foot as the measuring unit, a 
 prime is the 12th part of a foot, a second the 12th part of a 
 prime, &c. It is to be observed, that in measures of length, 
 inches are primes, but in superficial measure they are seconds. 
 In both, a prime is jL. of a foot ; but -^^ ^^ ^ square foot is a 
 parallelogram, a foot long and an inch broad. The 12th part of 
 this is a square inch, which is yi-j of a square foot. This 
 method of multiplying crosswise is not confined to twelves, but 
 may be greatly extended, for any number, whether its inferior 
 denominations decrease from the integer in the same ratio or 
 not, may be multiplied crosswise. Thus, pounds multiplied by 
 pounds are pounds, pounds multiplied by shillings are shillings, 
 &c. ; shillings multiplied by shillings are twentieths of a shil- 
 ling ; shillings multiplied by pence are twentieths of a penny, 
 pence multiplied by pence are 240ths of a penny, <fec. The word 
 
DUODECIMALS. 
 
 231 
 
 duodecimal is derived from the Latin word duodecim, signifying 
 
 twelve. The denominations are — 
 
 12 fourths '''' make - - - 1 third "' 
 12 thirds " - - - 1 second ^' 
 
 12 seconds " - - . 1 inch I. 
 
 12 inches " - - - 1 foot ft. 
 
 ADDITION OF DUODECIMALS. 
 
 Add as in Compound Addition, carrying 1 for each 12 to th© 
 next denomination. 
 
 
 
 
 
 EXAMPLES. 
 
 
 
 
 
 
 (1) ft. 
 
 I. 
 
 // 
 
 /// 
 
 nil 
 
 (2) 
 
 ft. 
 
 I. 
 
 // 
 
 /// 
 
 III/ 
 
 14 
 
 4 
 
 3 
 
 5 
 
 6 
 
 
 28 
 
 4 
 
 3 
 
 7 
 
 10 
 
 85 
 
 7 
 
 8 
 
 6 
 
 6 
 
 
 71 
 
 7 
 
 8 
 
 4 
 
 2 
 
 56 
 
 10 
 
 5 
 
 7 
 
 9 
 
 
 67 
 
 11 
 
 3 
 
 7 
 
 5 
 
 43 
 
 1 
 
 6 
 
 4 
 
 3 
 
 
 32 
 
 
 
 8 
 
 4 
 
 7 
 
 87 
 
 11 
 
 10 
 
 8 
 
 5 
 
 
 46 
 
 3 
 
 8 
 
 11 
 
 10 
 
 48 
 
 5 
 
 2 
 
 10 
 
 11 
 
 
 67 
 
 10 
 
 1 
 
 4 
 
 11 
 
 ft. 336--5 — 1 — 7 — 4 
 
 3. Five floors in a certain building contain each 1295 feet, 9 
 inches, 8^^ ; how many feet in all ? Ans. 6479ft., Oin., 4'". 
 
 4. Several boards measure as follows, namely : 27ft., 3in. ; 
 25ft., llin. ; 23ft., lOin. ; 20ft., 9in. ; 20ft., 6in. ; and 18ft., 5 
 in. ; what number of feet in all? Ans. 136ft., 8in. 
 
 SUBTRACTION OF DUODECIMALS. 
 
 Work as in Compound Subtraction, borrowing 12, &c. 
 when necessary. 
 
 (5) ft. in. " "' "" (6) ft. - " '" "" 
 
 in 
 From 176 1 
 Take 97 10 
 
 6 
 
 7 
 
 10 
 11 
 
 Rem. 78 3 10 11 
 
 m. 
 3786 10 4 
 987 8 11 
 
 7 
 9 
 
t^S8 DtrODECIMALS. 
 
 7. From a board measuring 41ft. 7in., cut 19ft. lOin. ; wha 
 remains 1 Ans. 21ft. 9in. 
 
 MULTIPLICATION OF DUODECIMALS. 
 
 RULE I. 
 
 1. Under tlie multiplicand write the corresponding denomina- 
 tions of the multiplier. 
 
 2. Multiply each term of the multiplicand, beginning at the 
 lowest, by the highest denomination in the multiplier, and write 
 the result of each under its respective term, observing, in duo- 
 decimals, to carry a unit for every 12 from each lower denom- 
 ination to its next superior, and for other numbers accordingly. 
 
 3. In the same manner, multiply all the multiplicand by the 
 primes, or seconds, denomination in the multiplier, and set the 
 result of each term one place removed to the right of those in 
 the multiplicand. 
 
 4. Do the same with the seconds in the multiplier, setting 
 the result of each term two places to the right of those in the 
 multiplicand. 
 
 5. Proceed in like manner with all the rest of the denomina- 
 tions, and their sum will be the answer required. 
 
 Note. — If there are no feet in the multiplier, supply their 
 place with a cipher, and in like manner supply the place of any 
 other denomination between the highest and lowest. 
 
 Feet X by feet give feet. 
 Feet X by inches give inches. 
 Feet X by seconds give sec'ds. 
 
 Inches xby inches give seconds. 
 Inches X by seconds give thirds. 
 Seconds xby sec. give fourths. 
 
 Note. — Let it be remembered, that though the feet obtained 
 by the rule are square feet, the inches are not square inches, 
 but the twelfth part of a square foot, ^^, 
 
 8. Multiply 2-i feet by 2i feet. 
 
 Thus: 2— 6in. or, 2i or, 2.5 decimally. 
 
 2 6 2i 2.5 
 
 5 5-0 125 
 
 1 3 1—} 50 
 
 Ans. feet 6-3 ft. 6^ 6.25 
 
 So that 3 is not 3 inches, but 36 inches, or |^ or a square foot 
 t\V 12x3=tYt, &c. 
 
DUODECIMALS. $83 
 
 9. Multiply 10 feet 6 inches by 4 feet 6 inches ? 
 Thus : 10—6 in. or, 10-6 or, 10.5 
 4—6 in. 4-6 4.5 
 
 5 3 42 
 42 5 3 
 
 525 
 
 420 
 
 ft. 47—3 Ans. 47 3 
 
 As before, 47i feet.=^/y. 
 
 10. Multiply 9 ft. 8', 6'' by 7 ft. 9', 3'\ 
 
 ft. ' " 
 
 9 8 6 
 
 7 9 3 
 
 47.25 
 
 67 — 11 6=product of the feet in the multiplier. 
 
 7 3 4 ^'" do. the primes. 
 
 2 5 1 ^''" do. by the seconds. 
 
 ft. 75 5 3—7-6 Ans, 
 
 11. Multiply 9 ft. 7 in. by 3 ft. 6 in. Ans, 33 ft. 6 in. 6'. 
 
 12. Multiply 3 ft. 11 in. by 9 ft. 5 in. 36 ft. 10 in. 7', 
 
 13. Multiply 8 ft. 6 in. 9^' by 7 ft. 3 in. 8'. 62 ft. 6 in. 7", 9"'. 
 
 14. In a load of wood, 8 feet 4 Thus : 8 ft. 4 in. length, 
 inches long, 2 feet 6 inches high, 2 6 X height, 
 and 3 feet 3 inches wide, how 
 
 many solid feet? 16 8 
 
 4 2+ 
 
 20 10 sum. 
 3 3 X width. 
 
 62 6 
 5 2 6'^--f- 
 
 Cub. p. od. 67 8 ^" , Ans. 
 
 15. How many solid feet are there in a stick of timber, 70 
 feet long, 15 inches thick, and 18 inches wide ? Thus : 70 ft. 
 Xl5 in. = 1050xl8 in. = 18900''~144 = 131, and 36'' rem., 
 fhen 36-^12 = 3 in. Ans. 131i ft. 
 
 16. How many cubic feet of wood in a load 6 ft. 7 in. long, 
 3 ft. 5 in. high, and 3 ft. 8 in. wide ? Ans. 82 ft. 5' 8'' 4''". 
 
 When the feet of the multiplier exceed twelve. 
 Rule. — Multiply by the feet of the multiplier as in compound 
 multiplication, and take parts for the inches &c. 
 
 20* 
 
284 DUODECIMALS. 
 
 17. Multiply 112 feet, 3 inches, 5'' by 42 feet, 4 in., 6'' ? 
 112 ft. 3 in. 5'' 
 
 6x7=42 
 
 673 
 
 8 
 
 6 
 7x 
 
 4715 
 4 in. 1 = 37 
 6'' i= 4 
 
 11 
 5 
 
 8 
 
 6 
 
 1 8''' 
 
 1 8 6"'^ 
 
 Ans. 4758 9 4 6''^' 
 
 18. What is the freight of a bale of goods containing 65 feet 
 9 inches, at D15 per ton of 40 feet? 
 
 20 ft. 1=15.00 Or, 65.75 
 
 5 ft. i= 7.50 15 X 
 
 6 in.yV= 1-87.5 
 
 2 in. 1= 18.7 40)986.25 
 
 9.3 
 
 D24.65.6 Ans. 
 
 D24.65.5 
 
 19. What will be the expense of plastering the wall of a room 
 8^ feet high, and each side 16 feet 3 inches long, at 50 cents 
 per square yard ? Ans. D30.69.4. 
 
 20. How many square feet in a board 17 feet 7 inches long, 
 1 foot 5 inches wide ? Ans. 24 ft. 10^^ 11''. 
 , 21. A load of wood is 4 feet 6 inches wide, 3 feet 10 inches 
 high, and 7 feet 8 inches long ; how many feet more than a cord 
 does it contain 1 Ans. 4^ feet. 
 
 22. What will the paving of a courtyard cost, which is 70 
 feet long, and 56 feet 4 inches wide, at 20c. per square foot ? 
 
 Ans. D788 66| 
 
 23. A man built a house consisting of 3 stories ; in the upper 
 story there were 10 windows, each containing 12 panes of 
 glass, each pane 14 by 12 inches ; the first and second stories 
 contained 14 windows, each 15 panes, and each pane 16 by 12; 
 how many square feet of glass were there in the whole house ? 
 
 Ans. 700 sq. ft 
 
 REVIEW. 
 
 What are Duodecimals ? By whom, and how can those rules 
 be applied ? What is the rule for addition ? subtraction ? mul- 
 tiplication ? &c 
 
235 
 
 APPENDIX, 
 
 Containing a variety of useful rules and examples in square and 
 cubical measure, mensuration, <Sfc. 
 
 ARTIFICERS' WORK. 
 
 Artificers estimate or compute the value of their work by 
 different measures ; but the best method of taking the dimen- 
 sions of all sorts of work is by feet, tenths, and hundredths ; in 
 other words, by decimals ; glazing, and masons' flat work, &c., 
 by the foot ; painting, plastering, paving, by the yard ; flooring, 
 partitioning, roofing, tiling, by the square of 100 feet; brick- 
 work, by the rod of 16-^ feet, whose square is 272.25. 
 
 Note. — In calculating the square feet it is usual to omit the 25, 
 but the more correct way is to use the perfect number (272.25). 
 
 The practice has formerly been to calculate the various prices 
 of mechanical work, according to the rules and regulations of the 
 several countries of Europe, particularly England and Germany, 
 but the method has, in some respects, been departed from, 
 which has rendered the calculations different in the several 
 states of the Union. The general system of computation will 
 DC given, from which any others may easily be derived. 
 
 BRICKLAYERS' WORK. 
 
 Bricklayers compute their work at the rate of one brick ana 
 a half thick, and if the wall be more or less than this standard, 
 it must be reduced to it as follows : — 
 
 Multiply the superficial content of the wall, in feet, by the 
 number of half bricks in thickness, and ^ of that product will 
 be the content required. 
 
 It is generally the practice in this country, at the present 
 time, to calculate bricks by the 1000. All windows, doors, &c., 
 are to be deducted out of the content of the walls in which 
 they are placed ; but this deduction is made only with regard to 
 materials ; for the value of their workmanship is added to the 
 bill at the rate agreed on. 
 
 1. How many square rods are there in a wall 52.5 feet long, 
 12.75 feet high, and 2.5 bricks thick ? Thus : 52.5 X 12.75-t- 
 272=2.4609x5 half bricks = 12.3045-^3=4.1015=4 rodS; 27 
 feet, 7 inches. Ans. 
 
236 masons' work. 
 
 2. How many square rods are there in a wall 62^ feet k)ng, 
 14 feet 8 inches high, and 2^ bricks thick 1 
 
 Ans. 5 rods, 167 feet, 9 inches, 4 pa. 
 
 3. How many bricks, 8 inches long, 4 inches wide, 2^ inches 
 thick, will build a wall in front of a garden, which is to be 240 
 feet long, 6 feet high, 1 foot 6 inches wide ? Ans. 51840 bricks. 
 
 4. How many bricks, 9 inches long, 5 inches wide, 2^ thick, 
 will it require to build a wall 90 feet long, 7 feet high, and 2 
 feet thick ; and what will it cost at 90 cents per square foot 1 
 
 MASONS' WOilK. 
 
 To masonry belong all sorts of stone-work ; and the measure 
 made use of is a solid perch, or a superficial, or a solid foot. 
 Solid measure is generally used for materials, and the superficial 
 for workmanship. 
 
 RULE. 
 
 In solid measure multiply the length, breadth, and thickness, 
 continually together, and in superficial measure the length and 
 breadth of every part of the projection must be taken. 
 
 1. Required the solid content of a wall, whose length is 48.5 
 feet, its height 10.75 feet, and thickness 2 feet. 
 
 48.5X10.75X2 = 1042.75 feet. Ans, 
 
 2. What is the solid content of a wall whose length is 60.75 
 feet, its height, 10.25 feet, and its thickness 2^ feet? 
 
 Ans, 1556.71875 feet. 
 
 3. What is a marble slab worth, whose length is 5 feet 7 
 inches, and breadth 1 foot 10 inches, at 80 cents per foot ? 
 Thus : 5 ft. 7 in.=5/2 ft.=:f| and 1 ft. 10 in.=lf ft. = U ft. 
 {\ X V = W ^content of the slab in feet, \y x 80c. = ^\^2^0c. 
 = 7 3_7 0c.^D8.18.8|. Ans. 
 
 4. How many solid perches of stone are contained in a cellar 
 wall, the length being 45.5 ft. on a side, and the breadth 24 ft. at 
 each end, 6.75 feet high, and 2 feet thick ? Ans. 1 13.72fperches. 
 
 5. Required the cost of making a stone wall under a build- 
 ing, whose length is 42 feet, breadth on the outside 26 feet, the 
 height of the wall being 6.5 feet, and 2 feet thick, at 40 cents 
 per solid perch. Ans. D40.33.6 + 
 
 6. In a block of marble 5 ft. long, 18 in. square, how many 
 cubical, and how many square feet? 18 in. X 18x60 in.= 
 19440-1728=111 cub. ft. ; 19440-r 144 = 135 sq. ft. Ans. 
 
SLATERS AND TILERs' WORK. 237 
 
 CARPENTERS AND JOINERS' WORK. 
 
 Carpenters and joiners' work is that of flooring, partitioning, 
 roofing, &c., and is measured by the square of 100 feet. 
 
 1. If a floor be 57.25 feet long, and 28.5 feet broad, how 
 many squares will it contain? Thus, 57.25x28.5 = 1631.625 
 square feet, =:16 squares, 31 feet, 7 inches, Q" . Ans. 
 
 2. A partition is 91.75 feet long, and 11.25 feet broad ; how 
 many squares does it contain? Ans, 10 squares, 32 feet. 
 
 3. A partition is 96.75 feet long, 11 .5 feet broad ; how many 
 squares will it contain? Ans. 11.12625 squares. 
 
 4. What is the expense of flooring a building 45.5 feet long 
 26.75 wide, 2 stories high, at Dl. 36 per square? Ans. D49.65.87. 
 
 5. If a floor be 60 feet long, 28.75 broad, how many squares 
 will it contain? Ans. 17.25 squares. 
 
 6. In a floor 46 by 24 feet, required the expense of flooring 
 at 15 cents per square foot, and cost of boards at D7.50 per M. 
 
 Ans. Cost of flooring, D16.56 ; cost of boards D8.28. 
 
 SLATERS AND TILERS' WORK. 
 
 In these works, the content of a roof is found by multiplying 
 the length of a side by the girth from eave to eave ; and in sla- 
 ting, allowance must be made for the double row at the bottom. 
 In taking the girth, the line is made to ply over the lowest row 
 of slates, and returned up the under side till it meet with the 
 wall or eaves-board ; but in tiling, the line is stretched down 
 only to the lowest part, without returning it up again. Double 
 measure is generally allowed for hips, valleys, gutters, &c., but 
 no deductions are made for chimneys. In all works of this 
 kind, the content is computed either in yards of 9 square feet, 
 or in squares of 100 feet, and the same allowance of hips and 
 valleys is to be made as in roofing. 
 
 1. The length of a slated roof is 45.75 feet, and its girth 
 34.25 feet; required the content. Thus: 45.75x34.25 = 
 1566.9375 square feet= 174.104 yards. Ans. 
 
 2. The length of a slated roof is 48.5 feet, and its girth 
 36.25 feet; required the content. Ans. 1758.125 square feet. 
 
 3. What will the tiling of a barn cost at D3.40 per square, 
 he length being 43 feet 10 inches, and the breadth 27 feet 5 
 
 inches on the flat, the eave-board projecting 16 inches on each 
 
238 PAINTERS AND GLAZERS' WORK. 
 
 side, allowing the roof to be a true pitch 1 Thus, 27 feet 5 
 
 inrliPQ — 975 329 ,,^^1 32J I /32_?_^9\ — 329il64.5 493.5 
 
 incnes_.i/y^_ j2^ , ana y2 ' V T2. ~'^i— T2 + T2 — T2 
 = true pitch. 16 x 2 = 32 inches, |^ feet, to be added for pro- 
 jection ; then, 4||-^ + ff=^^|^=girt of roof; again, 43 feet 
 10 inches=43f feetrrr2G3^ and ^ 25.5 ^ 2|3_i3 8|o 6.5 fg^^^ 
 138|^65 squares of roofing; hence 1382065 x D3.40~ 
 469|02.i cents =:D65.26.4m., cost of roofing. Ans. 
 
 PLASTERERS' WORK. 
 
 Plasterers' work is of two kinds, namely, plastering upon 
 laths, called ceiling ; and plastering walls, called rendering ; 
 and these different kinds must be measured separately, and their 
 content collected into one sum. Their work is measured by 
 the square foot, or yard of 9 square feet, and moulding by run- 
 ning measure. 
 
 1. If a ceiling be 59.75 feet long, and 24.5 feet broad, how 
 many yards does it contain? Thus: 59.75x24.5 = 1463.875 
 square feet= 162.6528 square yards. Ans. 
 
 2. If the partitions between rooms are 141.5 feet about, and 
 11.25 feet high, how many yards do they contain? 
 
 Ans. 176.87. 
 
 3. If a ceiling be 64.75 feet long, and 24.5 broad, how many 
 square yards does it contain? Ans. 176.263|. 
 
 PAINTERS AND GLAZERS' WORK. 
 
 Painters' work is measured in the same manner as that of 
 carpenters ; and in taking the dimensions, the line must be forced 
 into all the mouldings and corners. The work is estimated by 
 the yard, except sashes, which are calculated per light. Gla- 
 zers' work is calculated by the light. All work of this kind is 
 done by the square yard of 9 feet. 
 
 1. If a room be painted, whose height is 16.5 feet, and its 
 compass 97.75 feet, how many yards does it contain ? 
 
 Thus: 97.75X16.5 = 1612.875 square feet=179.208 square 
 yards. Ans, 
 
pavers' work. 239 
 
 2. The height of a room is 14 feet 10 inches, and the cir- 
 cumference 21 feet 8 inches ; how many square yards does it 
 contain? Thus: 14—10 
 21 8x 
 
 9—10-8 
 311 6 
 
 321 4 8 
 Then, 321.3888 feet =35.71 square yards +. Ans. 
 
 PAVERS' WORK. 
 
 Pavers' work is done by the square yard. The content is 
 found by multiplying the length by the breadth. 
 
 1. How many square yards in a rectangular (right-angled) 
 court-yard, the length being 27 feet 10 inches, and breadth 14 
 feet 9 inches ? Ans. 410 ft. 6 in. 6 pa.=:45 yds. 7 ft. 4 in. 8—8. 
 
 2. A rectangular court-yard is 64.75 feet long, and 45.5 in 
 breadth ; what is the expense of paving at 45 cents per square 
 yard? Ans. D 147.30. 
 
 The dimensions of the walls of a brick building being given, to 
 find how many bricks are required to build it* 
 
 From the whole circumference of the wall measured round 
 on the outside, subtract 4 times its thickness ; then multiply the 
 remainder by the height, and that product by the thickness of 
 the wall, will give the solid content of the whole wall, which 
 multiplied by the number of bricks contained in a solid foot, 
 will give the answer. 
 
 1. How many bricks 8 inches long, 4 inches wide, 2.5 inches 
 thick, will it require to build a house 44 feet long, 40 feet wide, 
 and 20 feet in height, the wall to be 1 foot in thickness ; thus, 
 8x4x2.5=80 solid inches brick , 1728-^80=21.6 bricks in a 
 solid foot; 44 + 40 + 44-|-40=168 feet length of wall; 168 — 
 4=164x20x21.6=1 70848.0 bricks. Ans. 
 
 2. In a room of 4 walls, 2 of them measure 12.5 feet in length, 
 and 7.5 in height, and the other two sides are 14.6 feet in length, 
 
240 MENSURATION. 
 
 by 7.5 feet in height ; required the number of square yards ; and 
 Avhat the plastering will amount to at I2c. per yard. Thus: 
 12.5x7.5x2 = 187.5 feet; 14.5 X 7.5 X 2=217.5 feet ; 217.5-f- 
 187.5=405.0~9=:45 yds. Xl2=D5.40. Ans. 
 
 3. How many square yards in a garden of 96 by 50 feet ? 
 
 Ans. 533^ yards. 
 
 To calculate the number of shingles for a roof 
 
 I 
 
 RULE. 
 
 1. Reduce the length and breadth of the space to be roofed to 
 inches separately. 
 
 2. Divide the breadth by the average width of the shingles, 
 and the quotient will be the number of shingles in one course. 
 
 3. Divide the length by the number of inches you intend lay- 
 ing the shingles or courses of the weather, and the quotient will 
 be the number of courses. 
 
 4. Multiply the number of courses by the number of shingles 
 in one course, and you will have the number of shingles re 
 quired 
 
 1. Required the number of shingles for a roof 16 feet in 
 width and 18 feet in length, the shingles to average 5 inches 
 each in width, and the course to run 8 inches to the weather. 
 
 Thus: 16x12 = 192-^5 = 38.4; 18x 12=216-~8:=27x 
 38.4 = 1036.8 shingles. Ans. 
 
 MENSURATION, &c. 
 To measure wood, (^c. 
 
 RULE. 
 
 Multiply the width by the height, that product by the length, 
 and divide by 128, and the quotient is the answer in cords. 
 
 1. Required the content, in cords, of a pile of wood 16 feet 
 in length 5.5 feet in width, and 4.5 feet in height. 4.5x5.5 X 
 16 = 396.00-^128 = 3 cords. Ans. 
 
 2. How many cubical feet in a piece of scantling 40 feet in 
 length, 1 .5 feet in width, and .5 feet in depth ? 1.5X .5 X 40= 
 30.00 feet. Ans. 
 
Mensuration. 241 
 
 3. In a tier of wood 25 feet in length, 18.5 feet in width, and 
 7.5 feet in height, how many cords ? Ans. 27 cords, 12 feet. 
 
 4. Required the cost of a load of wood 9 feet in length, 4.5 
 feet in height, and 4 feet 3 inches in width, at D5,50 per cord ? 
 
 Thus : 4.50 X 4.25 X 9= 172.125 ; then 128 : 5.50 ;: 172.125 : 
 D7.39.6. Ans. 
 
 5. How many solid feet of timber in a stick 8 feet long, 10 
 inches thick, and 6 inches in width 1 Ans. 3J 
 
 In 10 feet long, 12 inches thick, and 1 foot 3 inches wide ? 
 
 Ans. 121 
 
 6. In a pile of wood 10 feet wide, 3 feet 3 inches high, and 
 1 mile long, how many cord feet, and how many cords ? 10725 
 cord feet==1340|- cords. 
 
 7. How many cubical feet in a pile of rails 70 feet long, cut 
 12 J feet and 14^ feet high ; and how many cords of wood would 
 they make? ^/^/| £>viL^f 
 
 To find the area of a square having equal sides, i 
 
 Multiply the side of the square into itself, and the product 
 will be the area, or content. 
 
 1. In a garden 130 feet square, how many square yards ? 
 
 Ans. 1600. 
 
 To measure a rectangle parallelogram ^ or long square. 
 
 A rectangle is a four-sided figure like a square, in which the 
 sides are perpendicular to each other (right-angled), but the ad' 
 jacent sides are longer and parallel. 
 
 RULE. 
 
 Multiply the length by the breadth, and the product will be 
 the answer. 
 
 1. A garden is 76 feet in length, and 42 feet in width ; how 
 many square feet of ground are contained in it ? 
 
 Ans. 76X42=3192 feet. 
 
 2. What is the content of a field 40 rods square ? 
 
 Ans. 10 acres. 
 
 3. What is the content of a field 25 chains long by 20 chains 
 broad? Ans. 50 acres, 
 
 4. Required the content of a field 75 chains long and 75 
 chain* broad ? rr&rn^ - i C ti^ 
 
 21 
 
242 
 
 MENSURATlvJN. 
 
 Note. — In measuring boards, you can multiply the length ir> 
 feet by the breadth in inches, and divide by 12 ; the quotien 
 willgive the answer in square feet. 
 
 5. In a board 20 feet in length, 16 inches in width, how 
 many feet? ^;z5. 20x 16=320-M2 
 
 26f feet. 
 
 To measure a triangle, or to find the area. 
 
 A triangle is a figure bound- 
 ed by three straight lines ; 
 thus, B, A, C, is a triangle. 
 When a line like A, D, is 
 drawn, making the angle A 
 D B square to the angle A 
 D C, then A D is said 
 to be perpendicular to B C, 
 and A D is called the altitude 
 of the triangle. Each triangle, 
 B A D, or D A C, is called 
 a right-angled triangle. The 
 side B A, or the side A C, 
 opposite the right angle, is 
 called the hypotenuse. 
 
 Multiply the base of the given triangle into half its perpen* 
 dicular height, or half the base into the whole perpendicular, 
 and the product is the answer. 
 
 1. Required the area of a triangle whose base or longest side 
 is 32 inches, and the perpendicular height 14 inches. 
 
 Arts. '^2x1 (^ of 14) = 224 square inches. 
 
 2. In a triangular field the base is 40 chains, and the perpen- 
 dicular 15 chains ; how many acres ? 
 
 Ans. 15x40 = 600-^2=300—10 ch. =30 acres. 
 
 3. What is the area of a square piece of land, of which the 
 sides are 27 chains ? . Ans. 72 acres, 3 roods, 24 poles. 
 
 4. How many acres in a piece of land 560 rods long and 32 
 rods wide ? Ans. 112 aq|ps, 
 
 5. How many acres are contained in a road 40 miles long 
 and 4 rods wide ? Ans. 320. 
 
 6. What will a lot of land 1 mile square come to, at D20.75 
 per acre? Ans. D 1 3280. 
 
 7. How many yards of carpeting, that is 1^- yards wide, will 
 i;over a floor 21 feet, 3 inches long, and 13 feet, 6 inches wide ? 
 
 Am> ^5J yarcl^ 
 
MENSURATIO.V. 243 
 
 To measure a circle, area, circumference, d^c. 
 
 A circle is a portion of a plane 
 bounded by a curved line, every part 
 of which is equally distant from a 
 certain point within, called the cen- 
 tre. The curved line, A E B D, is 
 called the circumference ; the point 
 C the centre ; the line A B, pas- 
 sing through the centre, a diameter, 
 and C H the radius. The circum- 
 ference A E B D is 3.1416+times 
 greater than the diameter A B. 
 Hence, if the diameter is 1, the circumference will be 3. 1416 -f- 
 Also, if the diameter is known, the circumference is found by 
 multiplying 3.1416 by the diameter. Hence the following rule: — 
 
 RULE. 
 
 Multiply the diameter by 3.1416, and the product Avill be the 
 circumference ; or divide the circumference by 3.1416, and the 
 quotient will be the diameter. 
 
 Note 1. — As 7 is to 22, so is the diameter to the circumfer- 
 ence ; or, as 22 is to 7, so is the circumference to the diameter. 
 Or, more correctly: — 
 
 As 113 is to 355, so is the diameter to the circumference ; 
 or as 355 is to 113, so is the circumference to the diameter. 
 
 Note 2. — The problem of " squaring the circle," as it is 
 usually termed, has never been solved, nor can a square or any 
 other right-lined figure, be found, that shall be equal to a given 
 circle, as there rnust be a fraction in one case or the other ; it is 
 not in the power of numbers to bring them exactly alike ; the 
 numbers used above are sufficiently correct ; but the calculation 
 maybe extended to almost an indefinite number of decimals, with- 
 out apparently arriving any nearer the solution of the problem. 
 
 1. If the circumference of a circle be 354, what is the diam- 
 eter ? Thus: 354.000-^3.1416=112.681, diameter. Ans. 
 
 2. If the diameter of a circle be 17, what is the circumfer- 
 en<;e ? Thus : 3.1416 x 17 = 53.4072, circumference. Ans. 
 
 3. If the circumference of the earth be 25000 miles, what is 
 the diameter? Ans. 7958 miles (nearly). 
 
 4. The base of a cone is a circle ; what is its diameter, when 
 the circumference is 64 feet? Ans. 20.3718 '^ 
 
 5. What is the circumference of a wheel whose diameter is 
 5 feet, 2 inches ? Ans. 16.2316. 
 
 6. If the circumference of a carriage- wheel be 16 feet, 6 
 inches, what is the diameter ? Ans. 5 252Weet,'f 
 
2^4 MSNSURATtOIf. 
 
 7. The circumference of a circle is 16 chains ; what is tha 
 diameter? ^W5. 5.0929+chains, 
 
 The diameter given^ to find the area or content. 
 
 RULE. 
 
 Multiply the square of the diameter by the decimal .7854, 
 and the product will be the area. 
 
 1. How many square feet are contained in a circle, whose 
 diameter is 4 feet 3 inches? Thus: 4.25^ = 180625 X .7854 = 
 14.1862875 square feet. Ans. 
 
 2. What is the value of a circular garden, whose diameter is 
 6 rods, at the rate of 8 cents per square foot ? 
 
 Ans. D615.8L6.432. 
 The area af a circle given^ to find the diameter, 
 
 RULE. 
 
 Divide the area by .7854, and extract the square root of the 
 quotient. 
 
 1. The area of a circle is 5 acres, 3 roods, 26 perches, re- 
 quired the diameter. Thus : 946.000-^-.7854=34. 7 poles. Ans. 
 
 2. What is the length of a rope fastened to a stake in the 
 centre of a circular field, and the other end to the nose of a 
 horse, which will permit him to feed on 2 acres of land ? 
 
 Ans, 2.5231 chains. 
 
 The circumference given^ to find the area, 
 
 RULE. 
 
 Multiply the square of the circumference by the decimal 
 .07958, and the product will be the area. 
 
 If the circumference of a circle be 1, the diameter =1 — 
 3.14159=0.31831 ; and \ the product of this into the circum- 
 ference is .07958, the area. 
 
 1. If the circumference of a circle be 136 feet, what is the 
 area 1 Ans. 1472 feet. 
 
 2. The circumference of a circle being 37.7, required the 
 area. Thus: 37.72 ==1421.29=square,X. 07958 = 113.1062582 
 = area of the circle. 
 
 To find the surface of a sphere, globe ^ or hall. 
 
 RULE. 
 
 Square the diameter, and multiply it by the decimal 3.1416, 
 ~ and the product will be the answer. 
 
 1. What is the surface of a sphere whose diameter is 12 ? 
 
 Ans. 122-144x3.1416=452.3904 
 
 2. Required the number of square inches in the surface of a 
 sphere whose diameter is 2 feet, or 24 inches ? Ans. 1809.5616. 
 
MENSURATION. 245 
 
 To find the solidity of a sphere, ^^^^ 
 
 A sphere or globe is a round solid body. ^^to-.^^k 
 
 Multiply the surface by the diameter and divide the sphere. 
 product by 6 ; the quotient will be the solidity (multiply the 
 square of the diameter by 3.1416). 
 
 1. What is the solidity of a sphere whose diameter is 12 ? 
 Thus: 122=144x3.1416 = 452.3904x12 = 5428.6848-6= 
 
 904.7808, Arts. Or, multiply the cube of the diameter bv 
 .5236, thus: 12 X 12x12 X .5236 = 904.7808. 
 
 2. What is the solidity of the earth, its mean diameter being 
 7918.7 miles ? Ans, 259992792082.6374908. 
 
 Tofi^nd the solidity of a prism. 
 Definition. — A prism is a body with two equal ^^ ^ 
 or parallel ends, either square, triangular, or poly- '"'"^ 
 gonal, and three or more sides, which meet in Prism, 
 parallel lines, running from the several angles at one end to 
 those of the other. 
 
 RULE. 
 
 Multiply the area of the base by the altitude, and the product 
 will be the content. 
 
 1. What is the content of a prism, each side of the square 
 which forms the base being 15, and the altitude of the prism 20 
 feet? Ans. 152=225x20=4500 feet. 
 
 2. The side of a stick of timber is hewn 3 square, is 10 
 inches, and the length is 12 feet; required the content. 
 
 Ans. 10x4.33=1 perp. =43. 3 area at the end, X 12 feet 
 length, =519.6^144 = 3.6+ content. 
 
 3. Required the solidity of a triangular prism whose height 
 is 10 feet, and area of the base 350 feet, Ans. 3500 feet 
 
 To find the convex surface of a cylinder. 
 Definition. — A cylinder is a round body, whose 
 bases are circles, like a round column, or stick of ^ ________ 
 
 timber, of equal bigness at both ends. Cylinder. 
 
 RULE. 
 
 Multiply the circumference of its base by the altitude. 
 
 1. What is the convex surface of a cylinder, the diameter of 
 whose base is 20, and the altitude 50 feet ? 
 
 Ans. 3.1416x20x50 = 3141.6000- 
 
 2. Required the convex surface of a cylinder, the circumfer 
 •nr c^r whosft base is 6.509. and altitude 27. Ans, 175743 
 
246 MENSURATION. 
 
 To find the solidity of a cylinder. 
 
 Multiply the square of the diameter of the end by .7854, 
 which will give the area of the base ; then multiply the area of 
 the base by the length, and the product will be the content. 
 
 1. Required the solid content of a round stick of timber of 
 equal bisrness at both ends, whose diameter is 1.5 and length 20 
 feet. Thus: 1.5 X 1.5=2.25 X. 7854 X 20=35.3430. Ans. 
 
 Or, 18X 18 = 324 X .7854 X 20=5089.3920-r- 144 = 35.3430. 
 Ans. 
 
 2. What is the solidity of a cylinder, the diameter of whose 
 base is 12, and the altitude 30 ? Ans, 3392.928. 
 
 3. How many solid feet in a round stick of timber 16 feet 
 long, and the diameter at each end 15 inches? 
 
 Ans. 19.635 solid feet. 
 
 4. Required the solidity of a cylinder, the diameter of whose 
 base is 30 inches, and the height 50 inches ? 
 
 Ans. 20.4531 solid feet 
 
 To find the solidity of a cone. 
 Definition. — A cone is a round solid body of a true ta- 
 per from the base to a point, which is called the vertex. 
 
 RULE. 
 
 Cone. 
 
 Multiply the area of the base by the altitude, and divide 
 the product by 3 ; that is, square the diameter and multiply it 
 by .7854, which gives the area of the base ; then multiply by 
 the altitude and -f-by 3. Or, the square of the circumference 
 of the base X by .07958, and that product by ^ of the perpcn 
 dicular altitude, and the product will be the solidity. 
 
 1. Required the solidity of a cone, the diameter of who^e 
 base is 5 and the altitude 10. 
 
 Ans. 52=25 X. 7854= 19.635 XlO-r 3 = 65.45 
 
 2. What is the solidity of a cone whose altitude is 27 feet 
 and the diameter of the base 10 feet ? Ans. 706.86 
 
 3. Required the solidity of a cone, the diameter of whoso 
 base is 18 inches, and its altitude 15 feet. Ans. 8.83575 feet. 
 
 4. If the circumference of the base of a cone be 40 feet, and 
 the height 50 feet, what is the solidity? Ans. 2122.1333 feet 
 
 5. The top of a cistern is 5.5 feet, the bottom 4.75 feet, and 
 the height, or depth, 7.25 ; how many hogsheads will it con 
 tain? Ans. 17.7'=i-f- 
 
MENSURATION. 24T 
 
 To find the solidity of a pyramid. 
 Definition. — A pyramid is a solid, whose sides are all 
 triangles, meeting in a point at the vertex, and the base 
 any plane figure whatever. 
 
 Pyramid. 
 RULE. 
 
 Multiply the area of the base by the altitude, and divide the 
 product by 3. 
 
 1 . Required the solidity of a pyramid, of which the area of the 
 base is 95 and the altitude 15. Ans. 95 X 15=1425-^3=475. 
 
 2. What is the solidity of a pyramid, the area of whose base 
 is 403 and the altitude 30 ? Ans, 4030. 
 
 3. Required the solidity of a triangular pyramid, whose 
 height is 30, and each side of the base 3 ? 
 
 Thus: 32 X. 433013 = 3. 897117 area of the base, then 3.897117 
 X 33°n:38.97117=solidity required. 
 
 4. A pyramid with a square base, of which each side is 30, 
 has an altitude of 20 ; what is the solid content ? Ans. 6000. 
 
 To measure a parallelopipedon. 
 Definition. — A parallelopipedon is a solid of 
 three dimensions, length, breadth, and thickness ; 
 
 as a piece of timber exactly square, whose length paraiiei^ipedon. 
 is more than the breadth and thickness ; ^he ends are called 
 bases, which are equal. The solidity of a parallelopiped is 
 equal to the product of the base into the perpendicular altitude. 
 And a parallelopiped and a cylinder, which have equal bases 
 and altitudes, are equal to each other. 
 
 RULE. 
 
 Find the area of the base, then multiply that by the length, 
 and it will give the^solid content. 
 
 1. If the side of a stick of timber is 1.75 feet, and the length 
 9.5 feet, to find the content. Thus: 1.752=3.0625== area of 
 base x9.5 =29.09375 content. Ans. 
 
 2. A vessel 3.5 feet each side within, and 5 feet deep, to find 
 the content. Ans. 3.5x3.5=12.25x5 = 61.25 content. 
 
 3. A piece of timber is 1 foot 6 inches broad, 9 inches thick 
 9 feet 6 inches long ; required the content. 
 
 Ans. 1.5 X. 75=1. 125x9.5 = 10.6875 content. 
 
 4. What is the solidity of a cylinder whose height is 121 
 and diameter 45.2! Ans. 45.22x .7854 X 121 = 192442.6. 
 
 5. The Winchester bushel is a hollow cylinder 18-^ inches 
 in diameter, and 8 inches deep ; what is its capacity ? 
 
 Ans, 18.52X .7853982=268.8025 X 8=:2150.43 cubic inches 
 
248 
 
 MENSURATION. 
 
 6. There is a cistern under ground, in the form of a parallel 
 ogram, in length 16 feet, width 12 feet, depth 9 feet ; required 
 the number of hogsheads it will contain. Thus : 16x12x9= 
 1728 cubic feet of space, then 17282=2985984 cubical inches, 
 which divide by 231, the number of inches in a gallons 12926 
 gallons— 63=205 hogsheads, 11|J gallons. Ans. 
 
 7. How many hogsheads will a circular cistern contain 
 which is 7 feet in diameter, and 9 feet in depth ? 
 
 Ans, 41 hogsheads, 7 gallons, 3 quarts, 1 pint. 
 
 When the breadth and thickness of a piece of timber are given in 
 inches, to find how much in length will make a solid foot. 
 
 RULE. r^ 
 
 Divide 1728 by the product of the breadth and depth, aiid the 
 quotient will be the answer or length, making a solid foot. 
 
 In a piece of timber 1 1 inches broad and 8 inches thick, how 
 many inches in length will make a solid foot 1 
 
 Ans. 11x8 = 88)1728(19.6. 
 
 To find how many solid feet a round stick of timber of equal thick" 
 ness will contain, when hewn square, 
 
 RULE. 
 
 Multiply twice the square of its radius, in inches by the 
 length in feet, then divide the product by 144, and the quotient 
 will be the answer. 
 
 1. Admitting the radius of a round stick of timber to be 11 
 inches, and its length 20 feet, how many solid feet will it con- 
 tain when hewn'square ? Thus: llXllX2x20=:4840~144 
 = 33.6 solid feet of hewn timber. 
 
 To find how many feet of square-edged boards of a given thick' 
 ness, can tm sawn from a log of a given diameter. 
 
 Find the solid content of the log when made square by the 
 last rule ; then say, as the thickness of the board, including the 
 saw-calf, is to the solid foot :: so is 12 inches to the number of 
 feet of boards. 
 
 1. How many feet of square-edged boards 1.25 inches thick, 
 including the saw-calf, can be sawn from a log 20 feet long and 
 24 inches in diameter? Thus: 12x12x2=288x20 = 5760 
 -^144=40 feet solid content; then, 1.25 : 40 :: 12 : 384 fee^ 
 of boards, Ans. 
 
MENSURATION. 349 
 
 Miscellaneous Matter in Mensuration. 
 
 1. Purchased a box of window-glass containing 256 panes of 
 glass, 8 by 10 inches ; required the number of square feet, and 
 the cost at 5c. per foot. 
 
 ^715. 8X10 = 80 in. : 256x80=20480-^144 = 142.22 square 
 feet, D7.11.1. 
 
 2. Bought a box of window-glass containing 320 square feet 
 of glass 10X12 inches ; required the number of panes. 
 
 Ans. 10x12 = 120 : 320 X 144=46080-f- 120 = 384 panes. 
 
 3. In a pile of wood 7.5 feet long, Q\ feet high, and cut 4.75 
 feet, how many cords, and how much will it come to at D5 per 
 cord? Ans. 7.5x6.2.5x4.75=22265625^128 = 1.73950 
 cords xD5=8.69.75. 
 
 4\ Fjom a mahogany plank 26 inches broad, a yard and a 
 half is to be sawn off; wh^ distance from the end must the line 
 be struck? ^^:i IJ^^-f^ iw.fir -^^ -/^ ■■:: y'-V^/^n^. 6.23 feet. 
 
 5. A joist is 8^ inches 3eep, 3.5 broad; what will be the 
 dimensions of a scantling just as large again as the joist, that is 
 4j inches broad? Ans. 12.52 inches deep. 
 
 6. How many 3-inch cubes can be cut out of a 12-inch cube ? 
 
 Ans. 64. 
 
 7. A may-pole, whose top was broken off by a blast of wind, 
 struck the ground at 15 feet distance from the bottom of the 
 pole ; what was the height of the whole may -pole, supposing 
 the length of the broken piece to be 39 feet ? Ans. 75 feet. 
 
 8. The diameter of a circle is 25 rods ; required the length 
 of a stone wall that will enclose it. Ans. 78.54 rods. 
 
 9. How many men may stand on one acre of land, allowing 
 12 square feet to each man ? Ans. 3630. 
 
 10. How many solid yards of earth will be thrown out to 
 make a cellar 48 feet long, 27 wide, and 6 deep? Ans. 288. 
 
 11. There is an island 50 miles in circumference, and three 
 men start together to travel the same way about it; A. goes 7 
 miles per day, B. 8, and C. 9 ; when will they come together 
 again ? Thus, 50x7+50x8-f 50"x9"-f-7-|-8 + 9 = 
 50 days, time of meeting, and A. will travel 350 miles, B. 400, 
 C. 450. A71S. 
 
 12. A general disposing his army into a square battalion, 
 found he had 231 men remaining, but increasing each side with 
 one soldier, he wanted 44 to fill the square ; of how many men 
 did his army consist? Thus: 2314-44=275 and 
 275-1-7-2 = 137, then 137x137+231=19000. Ans. 
 
 Proof, 138x138=19044 : ^19044=138. 
 
250 3IENSURATI0N. 
 
 13. Suppose a lighthouse built on the top of a rock ; the dis* 
 tance between the place of observation and that part of the rock 
 level with the eye is 620 yards ; the distance from the top of 
 the rock to the place of observation is 846 yards ; and from the 
 top of the lighthouse, 900 yards ; the height of the lighthouse 
 is required. 
 
 Ans. y900x900-620x 620-^846x846-620x620 = 
 76.77 yards. 
 
 14. Suppose one of those meteors called fire-balls, to move 
 parallel to the earth's surface, and 50 miles from it, at the rate 
 of 20 miles per second : in what time would it move round the 
 earth ? 
 
 Thus : the earth's diameter is 7964 miles, then 7964 + 50 x 3 
 =r8064=the diameter of the circle described by the ball, 
 then 8064x3.1416=25333.8624 miles its circumference, and 
 25333.8624 ' '' ''' '''' '"" """ '""" 
 
 -^20=1266.69312sec'ds=21 6 41 35 13 55 12 ^n5. 
 
 1 5. The mean distances of the planets from the sun, in English 
 miles,are as follows: Mercury 36686617.5, Venus 68552135.83, 
 Earth 94772980 (or 95000000), Mars 144404783.33, Jupiter 
 492912533.33, Saturn 903957657.5 ; now, as a cannon-ball at 
 its first discharge flies a mile in 8 seconds, and sound 1142 feet 
 in a second, in what time, at the above rate, would a ball pass 
 from the earth to the sun, and sound move from the sun to 
 Saturn 1 
 
 Thus: 94772980 X8'' = 758183840=24 years, 15 days, 
 6 hours, 27 minutes, 20 seconds, for the passage of the ball. And 
 903957657.5 X 5280=4772896431600 feet, and 4772896431 600 
 -f-1142 = 132 years, 192 days, ^i^hours, 42 minutes, 21|ff 
 seconds, for the passing of sound from the sun to Saturn. Ans. 
 
 16. Light passes from the sun to the earth in 8.2 minutes; 
 in what time would it pass from the sun to the Georgium Sidus 
 (Herschel), it being 1803930416.66 English miles? 
 
 Ans. As 94772980 : 8.2 :: 1803930416.66 : 2 h. 36 m. 4" 50'^'. 
 
 17. If a cubic foot of iron were hammered or drawn into a 
 square bar, an inch about, that is, \ of an inch square, required 
 its length, supposing there is no waste of metal. 
 
 Thus, 12 X 12 X r2-f-.25 X .25 X 4 = 6912 inches = 576 ft. Ans. 
 
 18. How many square feet of boards in a load consisting of 
 150, 12^ fe^tln length, and 11 inches in width? 
 
 Thus: 12.5x11 = 137-^12 = 11.46 feet in one board; 11.46 
 X 150=1719 square feet (nearly), Ans. 
 
MBNSURATION'. 25a 
 
 19. The solid content of a square stone is found to be 136^ 
 feet, its length is 9^ feet ; what is the area of one end ? and if 
 the breadth be 3 feet 11 inches, what is the depth? 
 
 ^, 136.5X1728 on«on^or • a 2069.0526 
 
 Thus: == area 2069.0526 m. and 
 
 9.5x12 47 
 
 = 44.022 inches. Ans. 
 
 20. How many perches of stone are there in a wall 48.25 
 feet in length, 1.5 in width, and 12 feet in height? 
 
 Ans. 35.09+ perches, 
 
 21. Required the quantity of wood in a tier 64.5 feet in 
 length, 4 feet in width, and 4 feet in height. 
 
 Ans. 8.0625 = 83^1^ cords. 
 
 22. In a pile of stone 25 feet in length, 12j feet in width, 
 and 5i feet in height ; required the cubical feet, the number of 
 perches, and the value at 75c. per perch ? 
 
 Ans. 1718.75 cubical feet=69.444 perches, value D52.08.3-f- 
 
 23. How much wood in a pile 14^ feet in length, 6 J feet 
 high, and 3^ feet in width ; and value at D4i per cord ? 
 
 Ans. 2.47802 cords ; Dll.15.1090 value. 
 
 24. What is the breadth of a piece of cloth which is 36 
 yards long, and which contains 63 square yards ? 
 
 Ans. 63^36 = If yards. 
 
 25. How many yards of carpeting 1^ yards wide will cover 
 a iloor 30 feet long and 22-^ feet broad ? 
 
 Thus: 30x22.5=1:675 ft.-r9 = 75 yds. -f-ll=60 yds. Ans. 
 
 26. What is the side of a square piece of land containing 289 
 square rods ? ^^/^ ^j>. i i 
 
 27. If the diameter of the earth be 7930 miles, what is the 
 circumference? . ^71^.24912.8. 
 
 28. How many miles does the earth move in revolving round 
 the sun, supposing the orbit to be a circle whose diameter is 
 190y{nillions of miles ? Ans. 59690S«^ 
 
 29. What is the circumference of a circle whose diamefeVis 
 769843 rods? / f or) Ij 9>^//^/h>l V)^ 
 
 30. If the circumference of the sun be 2800000 miles, what j^ 
 is the diameter? Ans. 891267.^^* 
 
 31. What is the diameter of a tree which is 5^- feet round ? f^'^^ 
 
 32. If the circumference of the moon be 6850 miles, what is 
 her diameter? ^w^. 2180 miles. 
 
 33. If the whole extent of the orbit of Saturn be 5650 mill- 
 ons of miles, how far is he from the sun ?di?,jy / 7 9 ^,^^1^4 
 
 134. What is the area of a circle whose diameter is 623 feet : 
 Ans. 304836 square feet 
 I 
 
252 MENSURATION. 
 
 To measure stone in a well. 
 
 In taking the diameter of the wall, measure from centre to 
 centre, or to the clear; add the thickness of the wall; and 
 when the diameter is taken, or obtained, find the circumference, 
 which is the length of the wall. 
 
 As 7 : is to 22 :: so is the diameter (taken) : to the circumfer 
 ence required ; having found the length of the wall, and meas- 
 ured its thickness and depth, it is then prepared for calculation 
 as explained above. 
 
 35. A well walled with stone 1 foot 2 inches thick, its diam- 
 eter in the clear 2 feet 4 inches, and depth 50 feet ; required 
 the length of the wall, and the quantity of stone. 
 
 As 7 : 22 :: 3.5 : 11 feet circumference, or length; then 1 ft. 
 2 inches = l. 1666 + 6 the thickness, 11 length. 
 50 depth. 1.1666 + 6 
 
 11 
 
 12.8332 
 50 
 
 24.75)641.66(25.925+ 
 
 Is what the well measures in the clear (inside the walls) ; add 
 the thickness of the wall, and you have the diameter of the wall 
 or circle. Ans. 25.925+ perched. 
 
 36. Required the number of perches of stone in a well, whose 
 diameter is 4 feet 9 inches in the clear, the depth of the well 20 
 feet, and the thickness of the wall 15 inches. 
 
 Ans. 19.047 perches, or 19+ perches. 
 
 37. How many acres are there in a circular island whose 
 diameter is 124 rods ? Ans. 75 acres, 76 rods. 
 
 38. If the diameter of a circle be 113 and the circumference 
 355, what is the area ? Ans. 10029. 
 
 39. What is the diameter of a circle whose area is 380.1338 
 feet ? 
 
 Thus: 380.1336-^.7854=484 and ^484=22. Ans, 
 
 40. What is the area of a square inscribed in a circle whosQ 
 area is 159? Thus : .7854 : \ :: 159 : 101,22, Am 
 
MENSURATION. 253 
 
 41. What is the side of a square whose area is equal to that 
 of a circle 452 feet in diameter 1 
 
 Ans. V (452)?'x. 7854 =400.574 
 
 42. What is the diameter of a circle which is equal to a square 
 whose side is 36 feet ? Ans. ^/(36)2 ^0.7854 = 40.6217. 
 
 43. If the height of a square prism be 2^ feet, and each side 
 of the base lO^, what is the solidity? 
 
 Ans. the area of the base lOi x 10^= 106 J square feet ; and 
 the solid content =106jx2i=:240| cubic feet. 
 
 44. How many gallons ale are there in a cistern, which is 11 
 feet, 9 inches deep, and whose base is 4 feet, 2 inches square ? 
 
 The cistern contains 352500 cubic inches-f-282 = 1250 gal- 
 lons. Ans. 
 
 45. Suppose a cellar dug 16 feet in length, 16 in width, and 
 6 feet in depth ; required the quantity of stone to enclose a wall 
 1.5 feet thick, and 6 feet in height, in side of said cellar. 
 
 Ans. 21i, perches. (24f ft. = l perch.) 
 
 46. A room is 11 feet high, 21 feet long, and 16 feet wide ; 
 how many cubic feet of space in it ? Ans. 3696 cubic feet. 
 
 47. How many square feet in a stack of 15 boards, 12 feet, 
 8 inches in length, and 13 inches wide ? 
 
 Ans. 205^=205 ft. 10 in. 
 
 48. Suppose a ship sails from latitude 43°, north, between 
 north and east, till her departure from the meridian be 45 leagues, 
 and the sum of the distance and difference of latitude be 135 
 leagues ; I demand her distance sailed ? and latitude come to ? 
 
 Ans. 135 X 135-45 x45^gQ ^ ^^^ 60x3 = 180 
 
 135X2 
 miles = 3 degrees, the difference of latitude 135 — 60=75 
 leagues, the distance. Now, the vessel is sailing from the equa- 
 tor, and consequently the latitude is increasing ; therefore, to the 
 latitude sailed from 43° add the difference of latitude 3°, and 
 the sum is the latitude come to, that is, 46 degrees. 
 
 49. Four men, E, F, G, and H, purchased a grindstone of 60 
 inches in diameter ; how much of its diameter must each grind 
 off, to have an equal share of the stone, if one first grind his 
 share, and then another, till the stone is ground away, making 
 
 no allowance for the eye. ^^ ^: ^2^:6^'=>^/V(^.Of^Z,^^ai'^'^ >Vi^/ 
 
 ak JqucL>^ M- S./9^^^9 RULE.^ ^'^ ^ ' mi 
 
 Divide the square of the diameter by the number of men , 
 subtract the quotient from the square, and extract the square 
 root of the remainder, which is the length of the diameter after 
 
254 
 
 MENSURATION. 
 
 the first man has ground his share ; this work being repeated by 
 subtracting the same quotient from the remainder, for every man 
 to the last ; extract the square root of the remainder, and sub- 
 tract those roots from the diameter, one after another, and the 
 several remainders will be the answers. 
 
 Thus : 
 
 60 in. diam, 
 60 X 
 
 Again : 60 in. diam. of stone. 
 —■51.9615 
 
 4)3600 
 
 Quot. 900 
 
 from 3600 
 take 900 
 
 2/2700 > 51.9615- 
 jL 900 ( — 60 
 
 8.0385= 1st, or A.'s share. 
 
 51.9615 
 —42.4264 
 
 9.5351z=2d, or B.'s share. 
 
 42.4264 
 -30.0000 
 
 2/1800 > =42. 
 JL 900 J— 51. 
 
 J 900 > =30.1 
 ^ S —42. 
 
 4264— 12.4264 = 3d, C.'s share. 
 
 9615 and the remaining 
 
 30 in. is D.'s share. 
 
 = 30.000— for 8.0385+9.5351 + 12.4264 
 
 4264 +30. =60 in. the dia.of the st 
 
 Or, geoniettically, thus : on the radius A B describe a semi- 1 
 circle; al^o divide' A B into as many equal parts as there are 
 shares, and draw perpendiculars from the points of division to 
 the semicircle on A B ; then with the centre B, and radii B D^ 
 B C, &c., describe circles, and the question is solved. 
 
MENSURATION. 
 
 255 
 
 50. If a load of wood is 8 feet long and 3 feet wide, how 
 high must it be to contain 1 cord ? Ans. 51 ft. = 5 ft. 4 in. 
 
 51. Required the quantity of stone in a pile 16.5 feet in 
 length, 4.5^ feet in width, and 4 feet in height, computed at 16.5 
 cubic feet per perch. Ans. 18 perches. 
 
 52. If the floor of a square room contain 36 square yards, 
 how many feet does it measure on each side ? Ans. 18 feet. 
 
 53. What is the difference between 100 square feet and 100 
 feet square 1 Ans. 9900. 
 
 54. Suppose a ladder 60 feet long so planted as to reach a 
 window 37 feet from the ground, on one side of the street, and 
 without moving it at the foot, will reach a window 23 feet high, 
 on the other side ; required the breadth of the street 1 
 
 Ans. 102.64. 
 
 55. The Bunker Hill monument 
 in Charlestown, near Boston, is 
 said to be 225 feet high ; the as- 
 cent is gained by winding stairs on 
 the inside, and also by steam-pow- 
 er, acting on machinery for that 
 purpose ; suppose a person stand- 
 ing 121 feet from the base, and 
 the ground level, what length of 
 cord would be required to reach 
 from the ground to the top of the 
 monument, allowing the side, or 
 wall to be perpendicular (which, 
 however, is not the case). 
 
 Thus: 225x225 = 50625; 12X2 
 = 14641 + 50625 = 65266; then 
 y/65266 =255.47 feet length of 
 cord. Ans. 
 
 56. Required the length of a rope that will be sufficient for a 
 horse to graze just 2 (circular) acres ? 
 
 Thus : 2 A. (y) = 9680 yards = 98.3869 + root, then 98.3869 
 X1.12837 = lli:il6826353 yards, whole diameter which-~2 
 = 55.55 + yards. Ans. ' % 
 
 57. What is the side of a square piece of land containing 
 124J acres ? Ans. 141 rods 
 
256 MENSURATION. 
 
 58. The height of a tree growing in the centre of a circula? 
 island 44 feet in diameter, is 75 feet, and a line stretched from 
 the top of it over the hither edge of the water, is 256 feet ; what 
 is the breadth of the stream, provided the land on each side of 
 the water be level ? 
 
 Thus: 256x256 = 65536; and 75 X 75=5625 and ^59911 
 =244.76 + and 244.76 -f^= (22) =::222.76 feet. Ans. 
 
 59. What is the side of a square field which contains 58f 
 acres ? ^^ns q(.(^i- 
 
 60. What i« the width of a piece of land which is 280 rods 
 long, and which contains 77 acres ? 
 
 Ans. 77 X 160-12320—280=44 rods. 
 
 61. The Winchester bushel is a hollow cylinder, 18^ inches 
 in diameter, and 8 inches deep ; what is its capacity ? 
 
 Ans. 2150.42. 
 
 62. What is the solidity of a cylinder, whose height is 424, 
 and circumference 213? Ans. 1530791-}- 
 
 63. What is the convex surface of a right cylinder, which is 
 42 feet long, 15 inches in diameter ? . 
 
 Ans. 42x1.25x3.14169 = 164.933 square feet. 
 
 64. What is the solidity of a cylinder, whose height is 121, 
 and diameter 45.2? A71S. 45.22 x .7854 x 121 = 194156.6. 
 
 65. What is the square root of 37|f ? 
 
 Ans. 37ff x49-|-36=:if|9y=6f 
 
 66. In a circular cistern, whose diameter at bottom is 9 feet, 
 and top 7 feet, depth 12 feet, required the number of hogsheads 
 it will contain. Thus: 8x8=64 square, mean diameter, 
 X. 7854=50.2656 area base, or mean diameter, X 12, height = 
 603.1872, content in feet, x 1728 = 1042307.4816 cubic inches, 
 -^231=4512iff gallons-^63=71 hogsheads, 39 gallons, 2-f- 
 quarts. Ans. 
 
 2d. As7 : 8 :: 22 : 25.1428-4-2 = 12.5714x4, half diameter, 
 = 50.2856x12, length, =603.4272X1728 = 1042722.2016-^ 
 231=4513.9489 gallons, -^63=71 hhds., 40 galls. ^- Ans. 
 
 67. Boston is 6** 40' east longitude, from the city of Wash- 
 ington ; when it is 6 o'clock, P. M., at Washington, what is the 
 hour at Boston? Thus: 6x4=24' : 40x4=160''=2' 40'^ 
 -h24'=26^ 40''' past 6. Ans. 
 
 68. How many planks 15 feet long, 15 inches wide, will floor 
 n barn which is 60^ feet long, and 33^ wide ? Ans. 108/^. 
 
 69. If we are 95 millions of miles from the sun, and if the 
 earth revolves round it in 365J days ; how far are we carried 
 in 24 hours ? Ans. 1 million 634 thousand miles. 
 
^ MENSURATION. 257 
 
 V 70. If the diameter of Saturn's larger ring be 205000 and 
 ^i90000 miles, how many square miles on one side of the ring? 
 ^ Thus : 395000 x 15000 x .7854=4653495000 miles. Ans. 
 
 *' 71. What is the expense of paving a street 20 rods long and 
 ^2 rods wide, at 5 cents for a square foot? Ans. D544|-. 
 
 >$" 72. How many hills of corn may be planted on an acre, 3 ft. 
 '^distance? Thus: 1 acre ==43560 square feet^ (3x3 =9) 
 
 ^=4840 hills, Ans. 
 
 s^ 73. How many trees may be planted on 10 acres, at 6 feet 
 ^distance? Ans. 12100. 
 
 ^ 74. How many rails will it require to enclose a field of 16 
 "Nacres? (length of rails 13 feet, and 6 rails high.) v ^ ; 
 
 ^ 75. What is the side of a square whose area is equal to that 
 
 > of a circle 452 feet in diameter ? ^^ ^ ^ '^ i I 
 
 ;;; Ans. y^ (452)2 X. 7854=400.574. 
 
 ^ 76. What is the diameter of a circle which is equal to a 
 vj square whose side is 36 feet? Ans, ^(36)2-^0.7854 = 40.6217. 
 ':^ 77. If a carriage-wheel 4 feet in diameter revolve 300 times 
 ^ in going round a circular green, what is the area of the green ? 
 ":« Ans. 4154 square rods, or 25 acres, 3 qrs., 34 rods. 
 
 V 78. What is the solidity of a wall which is 22 feet long, 12 
 >? feet high, and 2 feet 6 inches thick ? Ans. 660 cubic feet. 
 ^ 79. If the height of a square prism be 1\ feet, and each side 
 o of the base 10^ feet, what is the solidity? 
 
 o Ans. 106|- square feet, 240^ cubic feet. 
 
 ^ 80. How many ale gallons are there in a cistern which is 11 
 feet 9 inches deep, and whose base is 4 feet 2 inches square ? 
 Ans. (-^282) 352500 cubic inches = 1250 gallons. 
 
 81. The highest point of the Andes is about 4 miles above 
 the level of the ocean. If a straight line from this touch the 
 surface of the water at the distance of 178J[ miles, what is the 
 diameter of the earth ? 
 
 Thus: 178.252^4m.=7939.2656 miles, Ans, 
 
 82. If the diameter of the earth be 7939.2656 miles, and 
 Mount Etna 2 miles high, how far can its summit be seen at 
 sea? Thus: 7939.2656x2 miles+22= 15882 ; 
 then y 15882.5312 = 126-1- miles. Ans. 
 
 83. How much water can be put into a cubical vessel 3 feet 
 deep, which has been previously filled with cannon-balls of the 
 same size, 2, 4, 6, or 9 inches in diameter, regularly arranged 
 
 in tiers, one directly above another? Ans. 96^ wine gallons .^^/c^^U 
 
 84. How many such globes as the earth are equal in bulk to 
 the sun, if the former is 7930 miles in diameter, and the latter . 
 890000? Ans. 1413678,^/*^^ 
 
 22* 
 
258 PROMISCUOUS QUESTIONS. 
 
 What is the rule to measure wood ? What is a square ? 
 How will you find the area of a square having equal sides ? 
 What is a rectangle ? How will you find the area of a rectan- 
 gle ? What is a triangle ? How will you find the area of a 
 triangle ? What is a circle ? What is the diameter of a circle ?' 
 W^hen the diameter is given, how will you find the circumfer- 
 ence ? When the circumference is given, how will you find 
 the diameter ? How can you find the area of a circle ? If the 
 area is given, how will you find the diameter ? how the circum- 
 ference ? What is a sphere? How will you find the surface, 
 of a sphere ? How will you find the solidity of a sphere ?' 
 What is a prism ? How will you find the solidity of a prism ? 
 What is a cylinder 1 How will you find the convex surface of 
 a cylinder? How will you find the solidity of a cyhnder? 
 What is a cone ? How will you find the solidity of a cone ? 
 What is a pyramid ? How will you find the solidity of a 
 pyramid ? What is a parallelopipedon ? How will you find 
 the solidity of a parallelopipedon ? What are the rules for 
 measuring timber, &c. ? 
 
 PROMISCUOUS QUESTIONS. 
 
 1. What is the interest of D650.75 for 150 days at 7 per 
 cent.? Ans. Dl8.72.r 
 
 2. What is the interest of D440 for 30 days at 6 per cent. ? j 
 
 Ans. D2.17.^ 
 
 3. Required the interest of D212 for 60 days at 5 per cent. 
 
 Ans. m. 74,2.^ 
 
 4. A. sells goods on commission at 2.5 per cent. ; how much 
 will he receive for selling D1800 worth? Ans. D4.50. 
 
 5. How much must a broker receive for discounting D2000 
 at 3.5 per cent. ? Ans. D70. 
 
 6. What is the commission on D2176.50 at 2.5 per cent. ? 
 
 ^7^5. D54.41.2. 
 
 7. What is the premium of insuring D1650 at 15.5 per ct. ? 
 
 Ans. D255.75. 
 
 8. How many bushels of wheat can be purchased with 
 D81.76 at D1.12 per bushel ? Ans. 73. 
 
PROMISCUOUS QUESTIONS. 269 
 
 9. How much will 27c wt. of iron cost at D4.56 per cwt. ? 
 
 Ans. D123.12. 
 
 10. Purchased a quantity of goods for D250, and 3 months 
 after sold them for D275 ; how much per cent, per annum was 
 gained ? Ans. 40 per cent. 
 
 11. In what time will a sum double itself at 6 per cent, sim- 
 ple interest? . Ans. 16 years, 8 months. 
 
 12. Required the amount of the following bill, namely: 150 
 bushels of rye at Dl ; 200 bushels of wheat at Dl.60 per bush- 
 el ; 50 bushels of com at 80c. per bushel ; 20 bushels of oats 
 at 60c. per bushel ; 40 bushels of salt at 40c. per bushel ; 10 
 bushels of potatoes at 20c. per busliel ; 3.5 bushels of flax-seed 
 at Dl.25 per bushel ; 5 bushels of cloverseed at D5 per bushel ; 
 15 bushels of barley at Dl.25 per bushel ; 4 bushels of turnips 
 at 30c. per bushel ; 8 bushels of onions at 25c. per bushel ; 2 
 bushels of beans at Dl.25 per bushel ; 9 bushels of apples at 
 75c. per bushel. Ans. D600.57.5. 
 
 13. What is the amount of the following bill, namely : 4 parrs 
 of boots at D5 per pair ; 6 pairs of small shoes at 75c. per pair ; 
 7 yards of muslin at 20c. per yard ; 150 lbs. of flax at 20c. per 
 pound ; 4c wt. of flour at D3 per cwt, ; 50 bushels of malt at Dl 
 per bushel ; 70 bushels of apples at 20c. per bushel ; 50 lbs. of 
 sugar at 12.5c. per pound ; 208 bushels of potatoes at 30c. per 
 bushel ; 171 bushels of corn at Dl per bushel ; 43 lbs. of beef 
 at 7c. per pound ; 324 yards of linen at 50c. per yard ? 
 
 Ans. D536.56. 
 
 14. What is the weight and value of 13 bags of wheat, which 
 weigh as follows, at D1.35 per bushel, at 601bs. per bushel? 
 
 No. 1. leOlbs. ; No. 2, 165lbs. ; No. 3, 172lbs. ; No. 4, 173 
 
 lbs. ; No. 5, ISOlbs. ; No. 6, 1691bs. ; No. 7, 195lbs. ; No. 8, 
 
 185lbs. ; No. 9, 184lbs . ; No. 10, 1631bs. ; No. 11, 1701bs. ; No. 
 
 No. 12, 1901bs. ; No. 13, 1671bs. (the weight of the sacks 261bs.) 
 
 fi;- g^^<l'}X ^^)^^h^'\^Q'^^ ^"^' 22731b., value D50.55.7. 
 
 15. How much will trie following bill amount to, namely: 
 120 bushels of apples at 20c. per bushel ; 110 bushels of rye at 
 Dl per bushel ; 50 bushels of salt at 40c. per bushel ; 70 
 bushels of sea-coal at 30c. per bushel ; 30 bushels of malt at 
 50c. per bushel ; 40 bushels of peas at 30c. per bushel ; 30 
 bushels of beans at 40c. per bushel ; 80 bushels of oats at 70c. 
 per bushel; 90 bushels of wheat at Dl.60 per bushel; 100 
 bushels of corn at 70c. per bushel ; 80 bushels of potatoes at 
 80c. per bushel ; and 40 bushels of turnips at 20c. per bushel ? 
 
 Ans, D516.00. 
 
260 PROMISCUOUS QUESTIONS. 
 
 16. What is the value of 1403 lbs. of wheat at Dl.25 pel 
 bushel of 60 lbs. ? Ans. D29.22.9. 
 
 17. If the earth be 360 degrees in circumference, and each 
 degree 69.5 miles, what time would it require for a person to 
 travel round the globe at the rate of 20 miles per day, and th© 
 year to consist of 365.25 days? Ans. 3 years, 155.25 days 
 
 18. A line 35 yards long will exactly reach from the top of a 
 fort standing on the brink of a river, to the opposite bank, known 
 to be 27 yards broad ; required the height of the wall. 
 
 Ans. 22 yards, 9.^ inches. ^^ 
 
 19. How mTk;ii will 27cwt. of sugar come to at 9c. per lb. ? 
 
 Ans. D272.16. 
 
 20. How mucn will 281 yards of tape cost at 9m. per yard? 
 
 Ans. D2.52.9. 
 
 21. What is the cost of 29.5 yards of flannel at 75c. per yd. ? 
 
 Ans. D22.12.5. 
 
 22. How much will 371 yds. of broadcloth come to at D5.79 
 per yard? ^n^. D2148.09.0. 
 
 23. If a man receive yearly D949, how much is it daily? 
 
 Ans. D2.60. 
 
 24. How much will 32625 feet of boards come to at D8.25 
 per M.? Ans 1)269.15.6+ 
 
 25. At D5.50 per M. what will 21186 feet of boards come 
 to? Ans. 1)116.52.3. 
 
 26. When boards are sold at D18 per M. how much is it per 
 foot? Ans. Ic. 8m. 
 
 27. The sale of certain goods amount to D1875 40c. ; what 
 sum is to be received for them, allowing 2.5 per cent, for com- 
 mission, and .25 for prompt payment of the net proceeds ? 
 
 Ans. D1823.82.65. 
 
 28. If 4.5 yards of broadcloth cost D25, what will 17.25 
 yards cost? Ans. D95.83.3 
 
 29. If 30 yards of cloth cost DUO, how much is it per yd. ? 
 
 AnsfD3.66.6, " 
 
 30. A farmer purchased goods at a store to the amount of 
 D180 ; he is to pay one half in wheat at Dl.25 per bushel, and 
 the other half in rye at 87.5cts. per bushel ; required the quan- 
 tity of each kind. 
 
 Ans. wheat 72 bushels, rye 102 bush., 3 pecks, 3+ quarts. 
 
 31. When wheat is selling at Dl.60 per bushel, what is the 
 price of 1 quart ? Ans. 5c. 
 
 32. Paid D140 for 2000 lbs. of cheese; how much did it 
 cost per lb. ? and if I sell it for 2c. per lb. more than the cost, 
 what is my profit? Ans. cost 7c. per lb. ; and profit D40. 
 
PROMISCUOUS QUESTIONS. 261 
 
 33. If I give D9.50 for 2 lb. of indigo, how much do I pay 
 per ounce, and how much do I gain or lose by selling it at 25c, 
 per ounce, avoirdupois ? 
 
 Ans. paid 29.6c., and lost Dl.50 on the whole. 
 
 34. If I lost Dl.50 on the above indigo, at what price per 
 oz. should I have sold it to gain Dl.50, and at what rate per 
 cent, profit would it be 1 
 
 Thus : D9. 50+3.00, gain on 2 lbs. = Dl2.50-i-32oz.=2 lb. 
 = 39+ should have sold for ; then 9.50 : 1.50 :: 100 : 15.7+ 
 gain per cent. Ans. 
 
 35. How much is the difference between the interest and 
 discount of D500 for 1.5 years, at 6 per cent. ? 
 
 Ans. D3.71.54 
 
 36. What is the interest of D480, for 60 days, at 5 per cent.? 
 
 Ans. D3.94.5+ 
 
 37. A. loaned B. DlOO, at 5 per cent., compound interest, 
 and to C, DlOO at 10 per cent., simple interest; payment is to 
 be made when the interest of the two sums shall be equal ; re- 
 quired the interest and time. 
 
 (By approximation.) Ans. interest D265.70 on each princi- 
 pal ; time, 26 years, 208 days. 
 
 38. What is the square root of 368863 ? 
 
 Ans, 607.34092 + 
 
 39. What is the cube root of .467764060? c^mi'^^C -f- 
 
 40. If 4.44 bushels of wheat cost D7.50, what will 17.22 
 bushels cost ? Ans. D29.08.7. 
 
 41. If 7.5 pounds of coffee cost 65c., what will 1 12 lbs. cost ? 
 
 Ans. D9.70.6. 
 
 42. What is the cost of 4 hogsheads of wine, at 9c. per pint! 
 
 ^n^. D181.44. 
 
 43. If a man travel 96 miles in 3 days, how many could he 
 travel in 3 weeks ? Ans. 672. 
 
 44. If 675 yards cost Dl2, 8d. 2c., 5m., how many yards 
 may be had for 38m. ? Ans. 2 yards. 
 
 45. If a yard of riband cost 4.5c., how much will it require 
 to purchase 345 yards ? Ans. D15.52.5. 
 
 46. If 19 yards of sheeting cost D25, 7d. 5c., what will 
 435.5 cost ? Ans. D590, 2d. Ic. 7m. 
 
 47. What is the value of .3375 of an acre ? 
 
 Ans, 1 R. 14 poles. 
 
 48. What is the value of .3 of a year ? 
 
 Ans. 109 days, 13 hours, 48 minutes. 
 
 49. What is the value of .6875 of a yard ? 
 
 Ans. 2 qrs. 3 nail* 
 
263 PROMISCUOUS QUESTIONS, 
 
 50. What is the value of .76 of a league ? 
 
 Ans, 2 m. 2 fur. 9 po. 3 yd. 3 inches. 
 
 51. A person purchased 10 cords of wood ; on measuring it, 
 he found it 3 inches too short, which should have been 4 feet ; 
 now much did he lose in 10 cords ? 
 
 Length, 4 ft. = 48 inches^3 = 16 or J^ per cord too short; 
 128-^-16 = 8 ft. in 1 cord; 10x8 = 80 feet. Ans. 
 
 Or, 96x48x3 = 13824-^1728 = 8x10 = 80 feet. Ans. 
 
 52. Required the quantity of wood in a parcel 28 feet long, 
 4 feet high, and cut 3 feet ? 
 
 ^71^. 28x5x3-f-128 = 3 cords, 1 qr. 4 feet. 
 
 53. If 5 men can reap 52.2 acres in 6 days, how many men 
 will it require to reap 417.6 acres in 12 days ? Ans. 20 men. 
 
 54. If a ceiling be 60 feet, 3 inches, by 25 feet, 6 inches, re- 
 quired the number of square yards. 
 
 Ans. 60.25x25.54-9 = 170 yards, 6 feet. 
 
 55. Add I of a yard to f of an inch. 
 
 A?is. Ill yards, or 14j\ inch. 
 
 56. Add J of a week, ^ of a day, and -^ of an hour, together. 
 
 Ans. 2 days, 14^ hours. 
 
 57. Add 1|- miles, -^q furlongs, and 30 rods, together. 
 
 Ans. 1 mile, 3 furlongs, 18 poles. 
 
 58. Add |- of a cwt., ^^ of a lb., 13 oz., and ^ of a cwt., 6 
 lb. together? Ans. 1 cwt. 1 qr. 27 lb. 13 oz. 
 
 59. How much is the difference between | of a rod and f of 
 an inch? Ans. 10 feet, 11^ inches. 
 
 60. How much is the difference between ^^ of a hogshead, 
 and j^g of a quart ? Ans. 16 galls. 2 qts. 1 pt. 3/^ gills. 
 
 61. From f oz. take |- pwt. Ans. ih pwt. 3 grs 
 
 62. Required the product of 6, X by f of 5. Ans. 20. 
 
 63. Required the product of 3f X 4l| 1 Ans, li^^. 
 
 64. What will 3| boxes of raisins cost, at D2^ per box 1 
 
 Ans. D8y7^. 
 
 65. If I of a yard of cloth cost D3, what is the price pev 
 yard ? Ans. DSf 
 
 66. If 4| gallons of molasses cost D2f , how much is it pev 
 quart ? -4n^. 15ii^ cents. 
 
 67. If 1| hogshead of wine cost D250J, how much is the 
 wine per quart ? Ans. 85.3c. per qt. 
 
 68. What number is that, to which if you add -^ of f of itself, 
 the sum will be 39 ? Ans. 30. 
 
 69. What number is that, which being multiplied by |, the 
 product will be 3^ ? Ans, 3||^ 
 
PROMISCUOUS QUESTIONS. 263 
 
 Black Board. 
 
 1. Reduce 3658 inches to yards. 
 
 2. Reduce 6490 grains to pounds. 
 
 3. Reduce 32044 scruples to pounds. 
 
 4. Reduce 108910592 drams to tons. 
 
 5. Reduce 59 m. 7 fur. 38 poles, to poles. 
 
 6. In 3278 nails, how many yards ? 
 
 7. In 19 A. 2 R. 37 po. how many square poles ? 
 
 8. In 175 square chains, how many square rods ? 
 
 9. In 55 cords of wood, how many solid feet ? 
 
 10. Reduce 3058560 cubic inches to tons of round timber. 
 
 11. Reduce 1 tun to gills. 
 
 12. Reduce 47 barrels, 16 gallons, 4 quarts, to pints. 
 
 13. In 17408 pints, how many bushels 1 
 
 14. In 5927040 minutes, how many weeks? 
 
 15. In 27894 seconds, how many degrees ? 
 
 16. In 7 lb. 11 oz. 3 pwt. 9 gr. of silver, how many grains? 
 
 17. Reduce 45681 grains to pounds. 
 
 18. In 12 tons, 15 cwt. 1 qr. 18 lb. 4 oz. 12 dr. how many drs.? 
 
 19. In 34 Ib^ oz. 6 pwt. 16 grs., troy, how many lbs., avoir. ? 
 
 20. In 9173841 nails, how many yards? 
 
 21. How many barleycorns in 360 degrees ? 
 
 22. In 4755801600 barleycorns, how many degrees ? 
 
 23. In 30539520 inches, how many miles ? 
 
 24. In 96 square miles, how many square feet? 
 
 25. In 1784217600 square feet, how many square miles ? 
 
 26. In 96 A. 2 R. 16 po., how many square feet and inches ? 
 
 27. In 622080 cubic inches, how many tons of round timber ? 
 
 28. In 1^ tuns of v;ine, how many gills? and cost at 6Jc. 
 per gill ? 
 
 29. Reduce 12528 pints to hogsheads. 
 
 30. In 87^ bushels of wheat how many pints ? and value at 
 3.5c. per pint? 
 
 31. Reduce 475047465 seconds to years. 
 
 32. In 1020300'', how many degrees ? 
 
 33. Reduce 9s. 13°, 25', to seconds. 
 
 34. How many steps of 30 inches are contained in 95 miles ? 
 and how long would it require a man to walk it, at the rv^ tf 
 8000 steps per hour ? 
 
 35. Reduce \ and yyg^ to decimals. 
 
 36. Reduce |H||-H-|f ^^ a decimal. 
 
 37. Reduce .21 pints to the decimal of a peck. 
 
 38. Reduce .7 drams to the decimal of a pound. 
 
264 PROMISCUOUS QUESTIONS. 
 
 39. Reduce 73 feet, 96 inches, to yards. 
 
 40. Reduce 17 h. 6 m. 43 sec. to the decimal of a day. 
 
 41. Reduce 3 cwt. qr. 12 lb. 7 oz. 12 drs. to the decimal 
 of a ton. 
 
 42. If 1 yd. cost D2.5, what will .8 of a yard cost ? 
 
 43. How many acres in 4 fields ; the 1st contains 12 acres, 
 2 roods, 38 poles ; 2d, 4 acres, 1 rood, 26 poles ; 3d, 85 acres, 
 rood, 19 poles ; 4th, 57 acres, 1 rood, 2 poles? 
 
 44. In 2 casks of wine, one of 89 gallons, 17 quarts, the 
 other 120 gallons, 9 quarts, how many barrels of 31-^ gallons, 
 and how many gallons remain ? 
 
 45. How many yards in the four following pieces of cloth : 
 1st, 49 yards, 2 nails ; 2d, 148 yds. 3 qr. 1 na. ; 3d, 9 yds. 7 
 qr. 8 na. ; 4th, 6 yds. 1 qr. na. ? 
 
 46. How much paper, f of a yard in width, will be required 
 to paper the 4 walls of a room, whereof 2 walls are 36 feet 
 long and 8 feet high, and 2 are 28 feet long and 8 feet in 
 height, deducting 15 square yards for windows ? 
 
 47. Multiply 9 cwt. 3 qrs. 27 lb. 12 oz. by 7. 
 
 48. How much water will be contained in 96 hogsheads, 
 each containing 62 gallons, 1 qt. 1 pt. 1 gill ? 
 
 49. If 7 A. 20 po. cost D94.78, what will 4 A. 27 po. cost ? 
 
 50. If 19 hhd. 4 galls, molasses cost D147.5, what will 37 
 hhds. 19 qts. cost ? 
 
 51. What is the value of (60-T-2-4x9 + 7)x(12+8-r4- 
 3X2)? 
 
 52. What is the value of (7-4) X (12X3-2X7)^-3X11? 
 
 53. What is the value of ^ j-f%f g" °^ Ts • 
 
 54. What is the value of (f+i of |.)^(A-2) ? 
 
 55. Multiply f , 31 5, and ^ of f , together. 
 
 56. Multiply f, f, 6f , 5|, and 12, together. 
 
 57. What is the value of f of f-rf of | ? 
 
 58. What is the value of 4f -^1 of 4 ? 
 
 59. In 13f of a cwt. how many ^^ of a cwt. ? 
 
 60. Divide 147_t. cubic yards, by 13^ K- 
 
 61. 2/ 90764876.920097. 
 
 62. Y 690897624jff8- 
 
 63. 2V768480^|H-,. 
 
 64. 2y 8409709485.70904. 
 
 65. 3/ .0496742189. 
 
 66. %/ 97890985708|76, 
 
 67. 3j/ 478096847.09607. 
 
 THE END. 
 
\ 
 
lOi 
 
RECOMMENDATIONS. 
 
 The following recomm<?ndation is from Thomas H. Burrowes, ] 
 late able and talented Superintendent of the Common Schools of t 
 — and who may very properly be termed' !?«e ** father of our 
 School System;'^ — 
 
 *'I feel pleasure in expressing the opinion that the Cotiunbian Calculator, 
 by Mr. Almou Ticknor, is a most valuable school-book. The adherence to 
 our own beautiful and simple decimal system of money, and the exclusion 
 of the British currency of pounds, shillings and pence, which Morris one of 
 its cliief differences iVom other Arithmetics, I consider a decided and valu- 
 able imi:rovement. It always appeared to me: useless, if not worse, to puz- 
 zle the beginner in A*rithmetic with questions in aiiy other money than our 
 own, at a time, too, when the unavoidable jjitricacies of the Science are 
 sufficing* ly numerous and difficult to task all his patience, and wKen thfe 
 teacher's chief object should be to excite and sustain his intere*^* in the 
 study. After he has become well versed in the principles oi Ai'-hmetic, 
 imd complete master of all calculations in our own coin; it is nc> rdy pro- 
 per to give him a knowledge of those of oihc: lands, but it v1*il be found 
 practicable lo do so in one-tenth of the time requisite for that v ;p'i: e at an 
 earlier period. In. many other respects, also, the Coluiijfcian «J.4k5tjlator is a 
 superior work, and I therefore cordially recommend it. 
 
 Ti.^MAsH. E-:.. r.Es." 
 
 Lancaster, Nov. 26, 1847. 
 
 3f| 
 
 (From the Carbon County Gazette.) 
 Messes. Editors : Among the latest systems of Arithmetic which 
 been presented to the American public, and which are destined JVcni 
 rior excellence to supplant earlier school-books in this department oi 
 cation, it has seemed to me, that a notice of the "Columbian Golciilr 
 in your columns would subserve the interests of our county cf- 
 schdols. As its name indicates, it discards the sterling calculations 
 in use in our colonial relations with Great Britain, and which constitu* 
 body of the Arithmetics in use, adopting in their stead computation* 
 exercises wholly in American money, and nearly exclusive of ft 
 weights and measures, except for the convenience of reference. 
 
 To this peculiar feature, so desirable, so proper, and so patriotic! 
 work sacrifices nothing essential, but in perusing it Mr. Ticknor'B I^t 
 must be regarded as furnishing an era, ironi which, if real utility be 
 aim, future writers or compilers will date one of the most signal reforii} 
 the method of calculation, and follow in this truly national v/akc. For ^ 
 use of our common schools, this Work commends itself before most O' 
 rivals, in the multiplicity of its examples, always in i^ro}: ortion to th^*^- 
 tical nature of the rules ; in the immense variety presented to the sM 
 and in the adapting of explanatory exercises, and cross-t Aaminatid# 
 fixed and annexed to the several rules; in the early introductioi:'^^ 
 tions, as important and necessary as units and integer-, aud, in a ic^ 
 accurate inductive system, should precede them as their component 
 in the good arrangement of the subject, and the strict test*, adopted 
 vent superfifciality ; and last, but not least, a brief but iidmiryble :ih^ 
 mensuration, with a view to aid the mechanic in hisealcuiav'ona: for \> 
 reasons we hope soon to see it upon the desks of our common schools, 
 the place of inferior books. I. H. Siewers, 
 
 Frincipal of the, High School^ Carbon County ^ Fa. 
 
 Another eminent teacher, of Connecticut, writes: "1 should judge tW ^_ 
 by its systematic arrangement, perspicuity of explanation, and above aU,\i^ jj*:, I 
 practical adaptation to the currency of this country, its claims are euperjor fc; 
 ^0 any other work of the kind extant, and eminently worthy of its nitmt. f 
 and author.^* ' ^